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ff38ad909bd7cb5d914d00fa28895712b993a632d4d085147dcdfae6c66cc46c	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.utility_function import (Set, With, Module, Scan, MapAnd, FalseQ, ZeroQ, NegativeQ, NonzeroQ, FreeQ, List, Log, PositiveQ, PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, ComplexNumberQ, RealNumericQ, PositiveOrZeroQ, NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, FracPart, RationalQ, ProductQ, SumQ, NonsumQ, First, Rest, SqrtNumberQ, LinearQ, Sqrt, ArcCosh, Coefficient, Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, IntegerPart, AppellF1, PolynomialQuotient, ArcTan, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcCsch, Sinh, Coth, LessEqual, Less, Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ, InverseFunctionQ, EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, ExpandLinearProduct, GCD, ContentFactor, NumericFactor, NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, ExpandExpression, Apart, SmartApart, MatchQ, PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, RationalFunctionFactors, NonrationalFunctionFactors, Reverse, RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, GeneralizedBinomialParts, GeneralizedTrinomialDegree, GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, MinimumMonomialExponent, MonomialExponent, LinearMatchQ, PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, AlgebraicFunctionFactors, NonalgebraicFunctionFactors, QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, CombineExponents, FactorInteger, FactorAbsurdNumber, SubstForInverseFunction, SubstForFractionalPower, SubstForFractionalPowerOfQuotientOfLinears, FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, NormalizeIntegrandAux, NormalizeIntegrandFactor, NormalizeIntegrandFactorBase, NormalizeTogether, NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, SmartDenominator, ActivateTrig, ExpandTrig, TrigExpand, SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, KnownTangentIntegrandQ, KnownCotangentIntegrandQ, KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, FunctionOfLog, PowerVariableExpn, PowerVariableDegree, PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, ConstantFactor, SameQ, ReplacePart, CommonFactors, MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, _SimpFixFactor, Tanh, DerivativeDivides, SimpFixFactor, _FixSimplify, FixSimplify, _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, PureFunctionOfCothQ, _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, IntegralFreeQ, Sum_doit, rubi_exp, rubi_log, PolynomialRemainder, CoprimeQ, Distribute, ProductLog, Floor, PolyGamma, process_trig, replace_pow_exp, ExponentList) # TODO - Add tests for: Int, NFreeQ, PureComplexNumberQ, EllipticPi, EllipticE, # EllipticF, ArcCot, ArcCoth, Tanh, Cosh, Sech, ArcSec, ArcSech, Subst, # SqrtNumberSumQ, Sin, Cos, Tan, Cot, Sec, Csc, Csch, TrigHyperbolicFreeQ, # InverseFunctionFreeQ, RealQ, from sympy.core.add import Add from sympy.core.expr import unchanged from sympy.core.numbers import (E, I, oo, pi, zoo) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.core.symbol import (symbols, Symbol, Wild) from sympy.functions.elementary.exponential import exp, log as sym_log from sympy.functions.elementary.hyperbolic import acosh, asinh, atanh, acsch, cosh, sinh, tanh, coth, sech, csch, acoth from sympy.functions.elementary.miscellaneous import Min, sqrt from sympy.functions.elementary.trigonometric import (cos, cot, csc, sec, sin, tan, atan, acsc, asin, acot, acos, asec, atan2) from sympy.functions.special.error_functions import (Chi, Ci, Ei, Shi, Si, expint, li) from sympy.functions.special.gamma_functions import (gamma, loggamma, polygamma) from sympy.functions.special.hyper import hyper from sympy.functions.special.zeta_functions import (polylog, zeta) from sympy.integrals.integrals import Integral from sympy.simplify.simplify import (nsimplify, simplify) A, B, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B a b c d e f g h y z m n p q u v w F', real=True, imaginary=False) x = Symbol('x') def test_ZeroQ(): e = b(np + n + 1) d = a assert ZeroQ(ae - bd(n(p + S(1)) + S(1))) assert ZeroQ(S(0)) assert not ZeroQ(S(10)) assert not ZeroQ(S(-2)) assert ZeroQ(0, 2-2) assert ZeroQ([S(2), (4), S(0), S(8)]) == [False, False, True, False] assert ZeroQ([S(2), S(4), S(8)]) == [False, False, False] def test_NonzeroQ(): assert NonzeroQ(S(1)) == True def test_FreeQ(): l = [ab, x, a + b] assert FreeQ(l, x) == False l = [ab, a + b] assert FreeQ(l, x) == True def test_List(): assert List(a, b, c) == [a, b, c] def test_Log(): assert Log(a) == rubi_log(a) def test_PositiveIntegerQ(): assert PositiveIntegerQ(S(1)) assert not PositiveIntegerQ(S(-3)) assert not PositiveIntegerQ(S(0)) def test_NegativeIntegerQ(): assert not NegativeIntegerQ(S(1)) assert NegativeIntegerQ(S(-3)) assert not NegativeIntegerQ(S(0)) def test_PositiveQ(): assert PositiveQ(S(1)) assert not PositiveQ(S(-3)) assert not PositiveQ(S(0)) assert not PositiveQ(zoo) assert not PositiveQ(I) assert PositiveQ(b/(b(bc/(-ad + bc)) - a(bd/(-ad + bc)))) def test_IntegerQ(): assert IntegerQ(S(1)) assert not IntegerQ(S(-1.9)) assert not IntegerQ(S(0.0)) assert IntegerQ(S(-1)) def test_IntegersQ(): assert IntegersQ(S(1), S(0)) assert not IntegersQ(S(-1.9), S(1)) assert not IntegersQ(S(0.0), S(0)) assert IntegersQ(S(-1), S(0), S(2)) def test_FracPart(): assert FracPart(S(10)) == 0 assert FracPart(S(10)+0.5) == 10.5 def test_IntPart(): assert IntPart(mn) == 0 assert IntPart(S(10)) == 10 assert IntPart(1 + m) == 1 def test_NegQ(): assert NegQ(-S(3)) assert not NegQ(S(0)) assert not NegQ(S(0)) def test_RationalQ(): assert RationalQ(S(5)/6) assert RationalQ(S(5)/6, S(4)/5) assert not RationalQ(Sqrt(1.6)) assert not RationalQ(Sqrt(1.6), S(5)/6) assert not RationalQ(rubi_log(2)) def test_ArcCosh(): assert ArcCosh(x) == acosh(x) def test_LinearQ(): assert not LinearQ(a, x) assert LinearQ(3x + y*2, x) assert not LinearQ(3x + y*2, y) assert not LinearQ(S(3), x) def test_Sqrt(): assert Sqrt(x) == sqrt(x) assert Sqrt(25) == 5 def test_Util_Coefficient(): from sympy.integrals.rubi.utility_function import Util_Coefficient assert unchanged(Util_Coefficient, a + bx + cx3, x, a) assert Util_Coefficient(a + bx + cx3, x, 4).doit() == 0 def test_Coefficient(): assert Coefficient(7 + 2x + 4x3, x, 1) == 2 assert Coefficient(a + bx + cx3, x, 0) == a assert Coefficient(a + bx + cx3, x, 4) == 0 assert Coefficient(bx + cx3, x, 3) == c assert Coefficient(x, x, -1) == 0 def test_Denominator(): assert Denominator(-S(1)/S(2) + I/3) == 6 assert Denominator((-a/b)3) == (b)(3) assert Denominator(S(3)/2) == 2 assert Denominator(x/y) == y assert Denominator(S(4)/5) == 5 def test_Hypergeometric2F1(): assert Hypergeometric2F1(1, 2, 3, x) == hyper((1, 2), (3,), x) def test_ArcTan(): assert ArcTan(x) == atan(x) assert ArcTan(x, y) == atan2(x, y) def test_Not(): a = 10 assert Not(a == 2) def test_FractionalPart(): assert FractionalPart(S(3.0)) == 0.0 def test_IntegerPart(): assert IntegerPart(3.6) == 3 assert IntegerPart(-3.6) == -4 def test_AppellF1(): assert AppellF1(1,0,0.5,1,0.5,0.25).evalf() == 1.154700538379251529018298 assert unchanged(AppellF1, a, b, c, d, e, f) def test_Simplify(): assert Simplify(sin(x)2 + cos(x)2) == 1 assert Simplify((x3 + x2 - x - 1)/(x2 + 2x + 1)) == x - 1 def test_ArcTanh(): assert ArcTanh(a) == atanh(a) def test_ArcSin(): assert ArcSin(a) == asin(a) def test_ArcSinh(): assert ArcSinh(a) == asinh(a) def test_ArcCos(): assert ArcCos(a) == acos(a) def test_ArcCsc(): assert ArcCsc(a) == acsc(a) def test_ArcCsch(): assert ArcCsch(a) == acsch(a) def test_Equal(): assert Equal(a, a) assert not Equal(a, b) def test_LessEqual(): assert LessEqual(1, 2, 3) assert LessEqual(1, 1) assert not LessEqual(3, 2, 1) def test_With(): assert With(Set(x, 3), x + y) == 3 + y assert With(List(Set(x, 3), Set(y, c)), x + y) == 3 + c def test_Module(): # Same as With assert Module(Set(x, 3), x + y) == 3 + y assert Module(List(Set(x, 3), Set(y, c)), x + y) == 3 + c def test_Less(): assert Less(1, 2, 3) assert not Less(1, 1, 3) def test_Greater(): assert Greater(3, 2, 1) assert not Greater(3, 2, 2) def test_GreaterEqual(): assert GreaterEqual(3, 2, 1) assert GreaterEqual(3, 2, 2) assert not GreaterEqual(2, 3) def test_Unequal(): assert Unequal(1, 2) assert not Unequal(1, 1) def test_FractionQ(): assert not FractionQ(S('3')) assert FractionQ(S('3')/S('2')) def test_Expand(): assert Expand((1 + x)10) == x10 + 10x9 + 45x*8 + 120x*7 + 210x*6 + 252x*5 + 210x*4 + 120x*3 + 45x*2 + 10x + 1 def test_Scan(): assert list(Scan(sin, [a, b])) == [sin(a), sin(b)] def test_MapAnd(): assert MapAnd(PositiveQ, [S(1), S(2), S(3), S(0)]) == False assert MapAnd(PositiveQ, [S(1), S(2), S(3)]) == True def test_FalseQ(): assert FalseQ(True) == False assert FalseQ(False) == True def test_ComplexNumberQ(): assert ComplexNumberQ(1 + I2, I) == True assert ComplexNumberQ(a + b, I) == False def test_Re(): assert Re(1 + I) == 1 def test_Im(): assert Im(1 + 2I) == 2 assert Im(aI) == a def test_PositiveOrZeroQ(): assert PositiveOrZeroQ(S(0)) == True assert PositiveOrZeroQ(S(1)) == True assert PositiveOrZeroQ(-S(1)) == False def test_RealNumericQ(): assert RealNumericQ(S(1)) == True assert RealNumericQ(-S(1)) == True def test_NegativeOrZeroQ(): assert NegativeOrZeroQ(S(0)) == True assert NegativeOrZeroQ(-S(1)) == True assert NegativeOrZeroQ(S(1)) == False def test_FractionOrNegativeQ(): assert FractionOrNegativeQ(S(1)/2) == True assert FractionOrNegativeQ(-S(1)) == True assert FractionOrNegativeQ(-S(1)/2) == True assert FractionOrNegativeQ(S(1)) == False def test_NegativeQ(): assert NegativeQ(-S(1)) == True assert NegativeQ(S(1)) == False assert NegativeQ(oo) == False def test_ProductQ(): assert ProductQ(ab) == True assert ProductQ(a + b) == False def test_SumQ(): assert SumQ(ab) == False assert SumQ(a + b) == True def test_NonsumQ(): assert NonsumQ(ab) == True assert NonsumQ(a + b) == False def test_SqrtNumberQ(): assert SqrtNumberQ(sqrt(2)) == True def test_IntLinearcQ(): assert IntLinearcQ(1, 2, 3, 4, 5, 6, x) == True assert IntLinearcQ(S(1)/100, S(2)/100, S(3)/100, S(4)/100, S(5)/100, S(6)/100, x) == False def test_IndependentQ(): assert IndependentQ(a + bx, x) == False assert IndependentQ(a + b, x) == True def test_PowerQ(): assert PowerQ(ab) == True assert PowerQ(a + b) == False def test_IntegerPowerQ(): assert IntegerPowerQ(a2) == True assert IntegerPowerQ(a0.5) == False def test_PositiveIntegerPowerQ(): assert PositiveIntegerPowerQ(a3) == True assert PositiveIntegerPowerQ(a(-2)) == False def test_FractionalPowerQ(): assert FractionalPowerQ(a(S(2)/S(3))) assert FractionalPowerQ(asqrt(2)) == False def test_AtomQ(): assert AtomQ(x) assert not AtomQ(x+1) assert not AtomQ([a, b]) def test_ExpQ(): assert ExpQ(E2) assert not ExpQ(2E) def test_LogQ(): assert LogQ(rubi_log(x)) assert not LogQ(sin(x) + rubi_log(x)) def test_Head(): assert Head(sin(x)) == sin assert Head(rubi_log(x3 + 3)) in (sym_log, rubi_log) def test_MemberQ(): assert MemberQ([a, b, c], b) assert MemberQ([sin, cos, sym_log, tan], Head(sin(x))) assert MemberQ([[sin, cos], [tan, cot]], [sin, cos]) assert not MemberQ([[sin, cos], [tan, cot]], [sin, tan]) def test_TrigQ(): assert TrigQ(sin(x)) assert TrigQ(tan(x2 + 2)) assert not TrigQ(sin(x) + tan(x)) def test_SinQ(): assert SinQ(sin(x)) assert not SinQ(tan(x)) def test_CosQ(): assert CosQ(cos(x)) assert not CosQ(csc(x)) def test_TanQ(): assert TanQ(tan(x)) assert not TanQ(cot(x)) def test_CotQ(): assert not CotQ(tan(x)) assert CotQ(cot(x)) def test_SecQ(): assert SecQ(sec(x)) assert not SecQ(csc(x)) def test_CscQ(): assert not CscQ(sec(x)) assert CscQ(csc(x)) def test_HyperbolicQ(): assert HyperbolicQ(sinh(x)) assert HyperbolicQ(cosh(x)) assert HyperbolicQ(tanh(x)) assert not HyperbolicQ(sinh(x) + cosh(x) + tanh(x)) def test_SinhQ(): assert SinhQ(sinh(x)) assert not SinhQ(cosh(x)) def test_CoshQ(): assert not CoshQ(sinh(x)) assert CoshQ(cosh(x)) def test_TanhQ(): assert TanhQ(tanh(x)) assert not TanhQ(coth(x)) def test_CothQ(): assert not CothQ(tanh(x)) assert CothQ(coth(x)) def test_SechQ(): assert SechQ(sech(x)) assert not SechQ(csch(x)) def test_CschQ(): assert not CschQ(sech(x)) assert CschQ(csch(x)) def test_InverseTrigQ(): assert InverseTrigQ(acot(x)) assert InverseTrigQ(asec(x)) assert not InverseTrigQ(acsc(x) + asec(x)) def test_SinCosQ(): assert SinCosQ(sin(x)) assert SinCosQ(cos(x)) assert SinCosQ(sec(x)) assert not SinCosQ(acsc(x)) def test_SinhCoshQ(): assert not SinhCoshQ(sin(x)) assert SinhCoshQ(cosh(x)) assert SinhCoshQ(sech(x)) assert SinhCoshQ(csch(x)) def test_LeafCount(): assert LeafCount(1 + a + x2) == 6 def test_Numerator(): assert Numerator((-a/b)3) == (-a)(3) assert Numerator(S(3)/2) == 3 assert Numerator(x/y) == x assert Numerator(-S(1)/S(2) + I/3) == -3 + 2I def test_Length(): assert Length(a + b) == 2 assert Length(sin(a)cos(a)) == 2 def test_ListQ(): assert ListQ([1, 2]) assert not ListQ(a) def test_InverseHyperbolicQ(): assert InverseHyperbolicQ(acosh(a)) def test_InverseFunctionQ(): assert InverseFunctionQ(rubi_log(a)) assert InverseFunctionQ(acos(a)) assert not InverseFunctionQ(a) assert InverseFunctionQ(acosh(a)) assert InverseFunctionQ(polylog(a, b)) def test_EqQ(): assert EqQ(a, a) assert not EqQ(a, b) def test_FactorSquareFree(): assert FactorSquareFree(x5 - x3 - x2 + 1) == (x3 + 2x*2 + 2x + 1)(x - 1)2 def test_FactorSquareFreeList(): assert FactorSquareFreeList(x5-x3-x2 + 1) == [[1, 1], [x3 + 2x*2 + 2x + 1, 1], [x - 1, 2]] assert FactorSquareFreeList(x*4 - 2x2 + 1) == [[1, 1], [x2 - 1, 2]] def test_PerfectPowerTest(): assert not PerfectPowerTest(sqrt(x), x) assert not PerfectPowerTest(x5-x3-x2 + 1, x) assert PerfectPowerTest(x4 - 2x2 + 1, x) == (x2 - 1)2 def test_SquareFreeFactorTest(): assert not SquareFreeFactorTest(sqrt(x), x) assert SquareFreeFactorTest(x5 - x3 - x2 + 1, x) == (x3 + 2x*2 + 2x + 1)(x - 1)2 def test_Rest(): assert Rest([2, 3, 5, 7]) == [3, 5, 7] assert Rest(a + b + c) == b + c assert Rest(abc) == bc assert Rest(1/b) == -1 def test_First(): assert First([2, 3, 5, 7]) == 2 assert First(y*S(2)) == y assert First(a + b + c) == a assert First(abc) == a def test_ComplexFreeQ(): assert ComplexFreeQ(a) assert not ComplexFreeQ(a + 2I) def test_FractionalPowerFreeQ(): assert not FractionalPowerFreeQ(x*(S(2)/3)) assert FractionalPowerFreeQ(x) def test_Exponent(): assert Min(ExponentList(x2 + x + 1 + 5, x)) == 0 assert ExponentList(x2 + x + 1 + 5, x) == [0, 1, 2] assert ExponentList(x2 + x + 1, x) == [0, 1, 2] assert ExponentList(x2 + 2x + 1, x) == [0, 1, 2] assert Exponent(x3 + x + 1, x) == 3 assert Exponent(x2 + 2x + 1, x) == 2 assert ExponentList(x3, x) == [3] assert Exponent(S(1), x) == 0 assert Exponent(x(-3), x) == 0 def test_Expon(): assert Expon(x*2+2x+1, x) == 2 def test_QuadraticQ(): assert not QuadraticQ([x*2+x+1, 5x2], x) assert QuadraticQ([x2+x+1, 5x2+3x+6], x) assert not QuadraticQ(x2+1+x3, x) assert QuadraticQ(x2+1+x, x) assert not QuadraticQ(x2, x) def test_BinomialQ(): assert BinomialQ(x9, x) assert not BinomialQ((1 + x)3, x) def test_BinomialParts(): assert BinomialParts(2 + x(9x), x) == [2, 9, 2] assert BinomialParts(x*9, x) == [0, 1, 9] assert BinomialParts(2x*3, x) == [0, 2, 3] assert BinomialParts(2 + x, x) == [2, 1, 1] def test_BinomialDegree(): assert BinomialDegree(b + 2cxn, x) == n assert BinomialDegree(2 + x(9x), x) == 2 assert BinomialDegree(x9, x) == 9 def test_PolynomialQ(): assert not PolynomialQ(x(-1 + x2), (1 + x)(S(1)/2)) assert not PolynomialQ((16x + 1)/((x + 5)2(x*2 + x + 1)), 2x) C = Symbol('C') assert not PolynomialQ(A + bx + cx2, x2) assert PolynomialQ(A + Bx + Cx*2) assert PolynomialQ(A + Bx*4 + Cx2, x2) assert PolynomialQ(x*3, x) assert not PolynomialQ(sqrt(x), x) def test_PolyQ(): assert PolyQ(-2ad3e2 + x6(ae*5 - bde4 + cd*2e3)\ + x4(-2ade*4 + 2bd2e*3 - 2cd3e2) + x2(2ad2e*3 - 2bd3e*2), x) assert not PolyQ(1/sqrt(a + bx*2 - cx4), x2) assert PolyQ(x, x, 1) assert PolyQ(x2, x, 2) assert not PolyQ(x3, x, 2) def test_EvenQ(): assert EvenQ(S(2)) assert not EvenQ(S(1)) def test_OddQ(): assert OddQ(S(1)) assert not OddQ(S(2)) def test_PerfectSquareQ(): assert PerfectSquareQ(S(4)) assert PerfectSquareQ(a*S(2)bS(4)) assert not PerfectSquareQ(S(1)/3) def test_NiceSqrtQ(): assert NiceSqrtQ(S(1)/3) assert not NiceSqrtQ(-S(1)) assert NiceSqrtQ(pi2) assert NiceSqrtQ(pi*2sin(4)4) assert not NiceSqrtQ(pi2sin(4)3) def test_Together(): assert Together(1/a + b/2) == (ab + 2)/(2a) def test_PosQ(): #assert not PosQ((be - cd)/(ce)) assert not PosQ(S(0)) assert PosQ(S(1)) assert PosQ(pi) assert PosQ(pi3) assert PosQ((-pi)4) assert PosQ(sin(1)*2pi*4) def test_NumericQ(): assert NumericQ(sin(cos(2))) def test_NumberQ(): assert NumberQ(pi) def test_CoefficientList(): assert CoefficientList(1 + ax, x) == [1, a] assert CoefficientList(1 + ax3, x) == [1, 0, 0, a] assert CoefficientList(sqrt(x), x) == [] def test_ReplaceAll(): assert ReplaceAll(x, {x: a}) == a assert ReplaceAll(ax, {x: a + b}) == a(a + b) assert ReplaceAll(ax, {a: b, x: a + b}) == b(a + b) def test_ExpandLinearProduct(): assert ExpandLinearProduct(rubi_log(x), x2, a, b, x) == a2rubi_log(x)/b*2 - 2a(a + bx)rubi_log(x)/b2 + (a + bx)*2rubi_log(x)/b*2 assert ExpandLinearProduct((a + bx)n, x3, a, b, x) == -a*3(a + bx)n/b3 + 3a*2(a + bx)(n + 1)/b3 - 3a(a + bx)(n + 2)/b3 + (a + bx)(n + 3)/b3 def test_PolynomialDivide(): assert PolynomialDivide((ac - bcx)*2, (a + bx)*2, x) == -4abc*2x/(a + bx)2 + c2 assert PolynomialDivide(x + x2, x, x) == x + 1 assert PolynomialDivide((1 + x)3, (1 + x)2, x) == x + 1 assert PolynomialDivide((a + bx)3, x3, x) == a(a2 + 3abx + 3b2x2)/x3 + b3 assert PolynomialDivide(x3(a + bx), S(1), x) == bx4 + ax3 assert PolynomialDivide(x6, (a + bx)2, x) == -a5(5a + 6bx)/(b6(a + bx)2) + 5a4/b6 - 4a3x/b*5 + 3a*2x2/b4 - 2ax3/b3 + x4/b2 def test_MatchQ(): a_ = Wild('a', exclude=[x]) b_ = Wild('b', exclude=[x]) c_ = Wild('c', exclude=[x]) assert MatchQ(ab + c, a_b_ + c_, a_, b_, c_) == (a, b, c) def test_PolynomialQuotientRemainder(): assert PolynomialQuotientRemainder(x2, x+a, x) == [-a + x, a2] def test_FreeFactors(): assert FreeFactors(a, x) == a assert FreeFactors(x + a, x) == 1 assert FreeFactors(abx, x) == ab def test_NonfreeFactors(): assert NonfreeFactors(a, x) == 1 assert NonfreeFactors(x + a, x) == x + a assert NonfreeFactors(abx, x) == x def test_FreeTerms(): assert FreeTerms(a, x) == a assert FreeTerms(xa, x) == 0 assert FreeTerms(ax + b, x) == b def test_NonfreeTerms(): assert NonfreeTerms(a, x) == 0 assert NonfreeTerms(ax, x) == ax assert NonfreeTerms(ax + b, x) == ax def test_RemoveContent(): assert RemoveContent(a + bx, x) == a + bx def test_ExpandAlgebraicFunction(): assert ExpandAlgebraicFunction((a + b)x, x) == ax + bx assert ExpandAlgebraicFunction((a + b)*2x, x)== a*2x + 2abx + b2x assert ExpandAlgebraicFunction((a + b)*2x2, x) == a2x2 + 2abx2 + b2x2 def test_CollectReciprocals(): assert CollectReciprocals(-1/(1 + 1x) - 1/(1 - 1x), x) == -2/(-x2 + 1) assert CollectReciprocals(1/(1 + 1x) - 1/(1 - 1x), x) == -2x/(-x*2 + 1) def test_ExpandCleanup(): assert ExpandCleanup(a + b, x) == a(1 + b/a) assert ExpandCleanup(b2/(a2(a + bx)2) + 1/(a2x2) + 2b2/(a3(a + bx)) - 2b/(a3x), x) == b2/(a2(a + bx)2) + 1/(a2x2) + 2b2/(a3(a + bx)) - 2b/(a3x) def test_AlgebraicFunctionQ(): assert not AlgebraicFunctionQ(1/(a + cx(2n)), x) assert AlgebraicFunctionQ(a, x) == True assert AlgebraicFunctionQ(ab, x) == True assert AlgebraicFunctionQ(x2, x) == True assert AlgebraicFunctionQ(x2a, x) == True assert AlgebraicFunctionQ(x*2 + a, x) == True assert AlgebraicFunctionQ(sin(x), x) == False assert AlgebraicFunctionQ([], x) == True assert AlgebraicFunctionQ([a, ab], x) == True assert AlgebraicFunctionQ([sin(x)], x) == False def test_MonomialQ(): assert not MonomialQ(2x7 + 6, x) assert MonomialQ(2x*7, x) assert not MonomialQ(2x*7 + 5x*3, x) assert not MonomialQ([2x*7 + 6, 2x*7], x) assert MonomialQ([2x*7, 5x*3], x) def test_MonomialSumQ(): assert MonomialSumQ(2x7 + 6, x) == True assert MonomialSumQ(x2 + x*3 + 5x, x) == True def test_MinimumMonomialExponent(): assert MinimumMonomialExponent(x*2 + 5x*2 + 3x5, x) == 2 assert MinimumMonomialExponent(x2 + 5x2 + 1, x) == 0 def test_MonomialExponent(): assert MonomialExponent(3x7, x) == 7 assert not MonomialExponent(3+x3, x) def test_LinearMatchQ(): assert LinearMatchQ(2 + 3x, x) assert LinearMatchQ(3x, x) assert not LinearMatchQ(3x2, x) def test_SimplerQ(): a1, b1 = symbols('a1 b1') assert SimplerQ(a1, b1) assert SimplerQ(2a, a + 2) assert SimplerQ(2, x) assert not SimplerQ(x*2, x) assert SimplerQ(2x, x + 2 + 6x3) def test_GeneralizedTrinomialParts(): assert not GeneralizedTrinomialParts((7 + 2x*6 + 3x12), x) assert GeneralizedTrinomialParts(x2 + x3 + x4, x) == [1, 1, 1, 3, 2] assert not GeneralizedTrinomialParts(2x + 3x + 4x, x) def test_TrinomialQ(): assert TrinomialQ((7 + 2x*6 + 3x12), x) assert not TrinomialQ(x2, x) def test_GeneralizedTrinomialDegree(): assert not GeneralizedTrinomialDegree((7 + 2x6 + 3x12), x) assert GeneralizedTrinomialDegree(x2 + x3 + x4, x) == 1 def test_GeneralizedBinomialParts(): assert GeneralizedBinomialParts(3x(3 + x*6), x) == [9, 3, 7, 1] assert GeneralizedBinomialParts((3x + x*7), x) == [3, 1, 7, 1] def test_GeneralizedBinomialDegree(): assert GeneralizedBinomialDegree(3x(3 + x6), x) == 6 assert GeneralizedBinomialDegree((3x + x*7), x) == 6 def test_PowerOfLinearQ(): assert PowerOfLinearQ((6x), x) assert not PowerOfLinearQ((3 + 6x3), x) assert PowerOfLinearQ((3 + 6x)*3, x) def test_LinearPairQ(): assert not LinearPairQ(6x*2 + 4, 3x*2 + 2, x) assert LinearPairQ(6x + 4, 3x + 2, x) assert not LinearPairQ(6x, 3x + 2, x) assert LinearPairQ(6x, 3x, x) def test_LeadTerm(): assert LeadTerm(abc) == abc assert LeadTerm(a + b + c) == a def test_RemainingTerms(): assert RemainingTerms(abc) == abc assert RemainingTerms(a + b + c) == b + c def test_LeadFactor(): assert LeadFactor(abc) == a assert LeadFactor(a + b + c) == a + b + c assert LeadFactor(bI) == I assert LeadFactor(cab) == ab assert LeadFactor(S(2)) == S(2) def test_RemainingFactors(): assert RemainingFactors(abc) == bc assert RemainingFactors(a + b + c) == 1 assert RemainingFactors(aI) == a def test_LeadBase(): assert LeadBase(ab) == a assert LeadBase(abc) == a def test_LeadDegree(): assert LeadDegree(ab) == b assert LeadDegree(abc) == b def test_Numer(): assert Numer(a/b) == a assert Numer(a(-2)) == 1 assert Numer(a(-2)a/b) == 1 def test_Denom(): assert Denom(a/b) == b assert Denom(a(-2)) == a2 assert Denom(a*(-2)a/b) == ab def test_Coeff(): assert Coeff(7 + 2x + 4x3, x, 1) == 2 assert Coeff(a + bx + cx3, x, 0) == a assert Coeff(a + bx + cx3, x, 4) == 0 assert Coeff(bx + cx3, x, 3) == c def test_MergeMonomials(): assert MergeMonomials(x2(1 + 1x)3(1 + 1x)n, x) == x2(x + 1)(n + 3) assert MergeMonomials(x2(1 + 1x)*2(1(1 + 1x)1)2, x) == x*2(x + 1)4 assert MergeMonomials(b2/a3, x) == b2/a3 def test_RationalFunctionQ(): assert RationalFunctionQ(a, x) assert RationalFunctionQ(x2, x) assert RationalFunctionQ(x3 + x4, x) assert RationalFunctionQ(x*3S(2), x) assert not RationalFunctionQ(x3 + x(0.5), x) assert not RationalFunctionQ(x*(S(2)/3)(a + bx)2, x) def test_Apart(): assert Apart(1/(x2(a + bx)2), x) == b2/(a2(a + bx)2) + 1/(a2x*2) + 2b2/(a3(a + bx)) - 2b/(a3x) assert Apart(x*(S(2)/3)(a + bx)2, x) == x(S(2)/3)(a + bx)2 def test_RationalFunctionFactors(): assert RationalFunctionFactors(a, x) == a assert RationalFunctionFactors(sqrt(x), x) == 1 assert RationalFunctionFactors(xx*3, x) == xx*3 assert RationalFunctionFactors(xsqrt(x), x) == 1 def test_NonrationalFunctionFactors(): assert NonrationalFunctionFactors(x, x) == 1 assert NonrationalFunctionFactors(sqrt(x), x) == sqrt(x) assert NonrationalFunctionFactors(sqrt(x)rubi_log(x), x) == sqrt(x)rubi_log(x) def test_Reverse(): assert Reverse([1, 2, 3]) == [3, 2, 1] assert Reverse(ab) == ba def test_RationalFunctionExponents(): assert RationalFunctionExponents(sqrt(x), x) == [0, 0] assert RationalFunctionExponents(a, x) == [0, 0] assert RationalFunctionExponents(x, x) == [1, 0] assert RationalFunctionExponents(x(-1), x)== [0, 1] assert RationalFunctionExponents(x(-1)a, x) == [0, 1] assert RationalFunctionExponents(x(-1) + a, x) == [1, 1] def test_PolynomialGCD(): assert PolynomialGCD(x2 - 1, x2 - 3x + 2) == x - 1 def test_PolyGCD(): assert PolyGCD(x2 - 1, x2 - 3x + 2, x) == x - 1 def test_AlgebraicFunctionFactors(): assert AlgebraicFunctionFactors(sin(x)x, x) == x assert AlgebraicFunctionFactors(sin(x), x) == 1 assert AlgebraicFunctionFactors(x, x) == x def test_NonalgebraicFunctionFactors(): assert NonalgebraicFunctionFactors(sin(x)x, x) == sin(x) assert NonalgebraicFunctionFactors(sin(x), x) == sin(x) assert NonalgebraicFunctionFactors(x, x) == 1 def test_QuotientOfLinearsP(): assert QuotientOfLinearsP((a + bx)/(x), x) assert QuotientOfLinearsP(xa, x) assert not QuotientOfLinearsP(x2a, x) assert not QuotientOfLinearsP(x*2 + a, x) assert QuotientOfLinearsP(x + a, x) assert QuotientOfLinearsP(x, x) assert QuotientOfLinearsP(1 + x, x) def test_QuotientOfLinearsParts(): assert QuotientOfLinearsParts((bx)/(c), x) == [0, b/c, 1, 0] assert QuotientOfLinearsParts((bx)/(c + x), x) == [0, b, c, 1] assert QuotientOfLinearsParts((bx)/(c + dx), x) == [0, b, c, d] assert QuotientOfLinearsParts((a + bx)/(c + dx), x) == [a, b, c, d] assert QuotientOfLinearsParts(x2 + a, x) == [a + x2, 0, 1, 0] assert QuotientOfLinearsParts(a/x, x) == [a, 0, 0, 1] assert QuotientOfLinearsParts(1/x, x) == [1, 0, 0, 1] assert QuotientOfLinearsParts(ax + 1, x) == [1, a, 1, 0] assert QuotientOfLinearsParts(x, x) == [0, 1, 1, 0] assert QuotientOfLinearsParts(a, x) == [a, 0, 1, 0] def test_QuotientOfLinearsQ(): assert not QuotientOfLinearsQ((a + x), x) assert QuotientOfLinearsQ((a + x)/(x), x) assert QuotientOfLinearsQ((a + bx)/(x), x) def test_Flatten(): assert Flatten([a, b, [c, [d, e]]]) == [a, b, c, d, e] def test_Sort(): assert Sort([b, a, c]) == [a, b, c] assert Sort([b, a, c], True) == [c, b, a] def test_AbsurdNumberQ(): assert AbsurdNumberQ(S(1)) assert not AbsurdNumberQ(ax) assert not AbsurdNumberQ(a(S(1)/2)) assert AbsurdNumberQ((S(1)/3)(S(1)/3)) def test_AbsurdNumberFactors(): assert AbsurdNumberFactors(S(1)) == S(1) assert AbsurdNumberFactors((S(1)/3)(S(1)/3)) == S(3)(S(2)/3)/S(3) assert AbsurdNumberFactors(a) == S(1) def test_NonabsurdNumberFactors(): assert NonabsurdNumberFactors(a) == a assert NonabsurdNumberFactors(S(1)) == S(1) assert NonabsurdNumberFactors(aS(2)) == a def test_NumericFactor(): assert NumericFactor(S(1)) == S(1) assert NumericFactor(1I) == S(1) assert NumericFactor(S(1) + I) == S(1) assert NumericFactor(a*(S(1)/3)) == S(1) assert NumericFactor(aS(3)) == S(3) assert NumericFactor(a + b) == S(1) def test_NonnumericFactors(): assert NonnumericFactors(S(3)) == S(1) assert NonnumericFactors(I) == I assert NonnumericFactors(S(3) + I) == S(3) + I assert NonnumericFactors((S(1)/3)(S(1)/3)) == S(1) assert NonnumericFactors(rubi_log(a)) == rubi_log(a) def test_Prepend(): assert Prepend([1, 2, 3], [4, 5]) == [4, 5, 1, 2, 3] def test_SumSimplerQ(): assert not SumSimplerQ(S(4 + x),S(3 + x3)) assert SumSimplerQ(S(4 + x), S(3 - x)) def test_SumSimplerAuxQ(): assert SumSimplerAuxQ(S(4 + x), S(3 - x)) assert not SumSimplerAuxQ(S(4), S(3)) def test_SimplerSqrtQ(): assert SimplerSqrtQ(S(2), S(16x3)) assert not SimplerSqrtQ(S(x2), S(16)) assert not SimplerSqrtQ(S(-4), S(16)) assert SimplerSqrtQ(S(4), S(16)) assert not SimplerSqrtQ(S(4), S(0)) def test_TrinomialParts(): assert TrinomialParts((1 + 5x3)2, x) == [1, 10, 25, 3] assert TrinomialParts(1 + 5x*3 + 2x*6, x) == [1, 5, 2, 3] assert TrinomialParts(((1 + 5x3)2) + 6, x) == [7, 10, 25, 3] assert not TrinomialParts(1 + 5x3 + 2x*5, x) def test_TrinomialDegree(): assert TrinomialDegree((7 + 2x6)2, x) == 6 assert TrinomialDegree(1 + 5x3 + 2x*6, x) == 3 assert not TrinomialDegree(1 + 5x*3 + 2x5, x) def test_CubicMatchQ(): assert not CubicMatchQ(S(3 + x6), x) assert CubicMatchQ(S(x3), x) assert not CubicMatchQ(S(3), x) assert CubicMatchQ(S(3 + x3), x) assert CubicMatchQ(S(3 + x*3 + 2x), x) def test_BinomialMatchQ(): assert BinomialMatchQ(x, x) assert BinomialMatchQ(2 + 3x5, x) assert BinomialMatchQ(3x*5, x) assert BinomialMatchQ(3x, x) assert not BinomialMatchQ(x + x2 + x3, x) def test_TrinomialMatchQ(): assert not TrinomialMatchQ((5 + 2x6)2, x) assert not TrinomialMatchQ((7 + 8x*6), x) assert TrinomialMatchQ((7 + 2x*6 + 3x*3), x) assert TrinomialMatchQ(bx*2 + cx4, x) def test_GeneralizedBinomialMatchQ(): assert not GeneralizedBinomialMatchQ((1 + x4), x) assert GeneralizedBinomialMatchQ((3x + x7), x) def test_QuadraticMatchQ(): assert not QuadraticMatchQ((a + bx)(c + dx), x) assert QuadraticMatchQ(x2 + x, x) assert QuadraticMatchQ(x2+1+x, x) assert QuadraticMatchQ(x2, x) def test_PowerOfLinearMatchQ(): assert PowerOfLinearMatchQ(x, x) assert not PowerOfLinearMatchQ(S(6)3, x) assert not PowerOfLinearMatchQ(S(6 + 3x2)3, x) assert PowerOfLinearMatchQ(S(6 + 3x)*3, x) def test_GeneralizedTrinomialMatchQ(): assert not GeneralizedTrinomialMatchQ(7 + 2x*6 + 3x*12, x) assert not GeneralizedTrinomialMatchQ(7 + 2x*6 + 3x*3, x) assert not GeneralizedTrinomialMatchQ(7 + 2x*6 + 3x5, x) assert GeneralizedTrinomialMatchQ(x2 + x3 + x4, x) def test_QuotientOfLinearsMatchQ(): assert QuotientOfLinearsMatchQ((1 + x)(3 + 4x*2)/(2 + 4x), x) assert not QuotientOfLinearsMatchQ(x(3 + 4x*2)/(2 + 4x*3), x) assert QuotientOfLinearsMatchQ(x(3 + 4x)/(2 + 4x), x) assert QuotientOfLinearsMatchQ(2(3 + 4x)/(2 + 4x), x) def test_PolynomialTermQ(): assert not PolynomialTermQ(S(3), x) assert PolynomialTermQ(3x*6, x) assert not PolynomialTermQ(3x*6+5x, x) def test_PolynomialTerms(): assert PolynomialTerms(x + 6x3 + rubi_log(x), x) == 6x*3 + x assert PolynomialTerms(x + 6x*3 + 6x, x) == 6x3 + 7x assert PolynomialTerms(x + 6x3 + 6, x) == 6x*3 + x def test_NonpolynomialTerms(): assert NonpolynomialTerms(x + 6x*3 + rubi_log(x), x) == rubi_log(x) assert NonpolynomialTerms(x + 6x*3 + 6x, x) == 0 assert NonpolynomialTerms(x + 6x3 + 6, x) == 6 def test_PseudoBinomialQ(): assert PseudoBinomialQ(3 + 5(x)*6, x) assert PseudoBinomialQ(3 + 5(2 + 5x)6, x) def test_PseudoBinomialParts(): assert PseudoBinomialParts(3 + 7(1 + x)6, x) == [3, 1, 7(S(1)/S(6)), 7*(S(1)/S(6)), 6] assert PseudoBinomialParts(3 + 7(1 + x)3, x) == [3, 1, 7(S(1)/S(3)), 7*(S(1)/S(3)), 3] assert not PseudoBinomialParts(3 + 7(1 + x)*2, x) assert PseudoBinomialParts(3 + 7(x)5, x) == [3, 1, 0, 7(S(1)/S(5)), 5] def test_PseudoBinomialPairQ(): assert not PseudoBinomialPairQ(3 + 5(x)6,3 + (x)6, x) assert not PseudoBinomialPairQ(3 + 5(1 + x)6,3 + (1 + x)6, x) def test_NormalizePseudoBinomial(): assert NormalizePseudoBinomial(3 + 5(1 + x)6, x) == 3+(5(S(1)/S(6))+5(S(1)/S(6))x)*S(6) assert NormalizePseudoBinomial(3 + 5(x)*6, x) == 3+5x*6 def test_CancelCommonFactors(): assert CancelCommonFactors(S(xyS(6))S(6), S(xyS(6))) == [46656x*6y*6, 6xy] assert CancelCommonFactors(S(y6)*S(6), S(xyS(6))) == [46656y*6, 6xy] assert CancelCommonFactors(S(6), S(3)) == [6, 3] def test_SimplerIntegrandQ(): assert SimplerIntegrandQ(S(5), 4x, x) assert not SimplerIntegrandQ(S(x + 5x3), S(x2 + 3x), x) assert SimplerIntegrandQ(S(x + 8), S(x*2 + 3x), x) def test_Drop(): assert Drop([1, 2, 3, 4, 5, 6], [2, 4]) == [1, 5, 6] assert Drop([1, 2, 3, 4, 5, 6], -3) == [1, 2, 3] assert Drop([1, 2, 3, 4, 5, 6], 2) == [3, 4, 5, 6] assert Drop(abc, 1) == bc def test_SubstForInverseFunction(): assert SubstForInverseFunction(x, a, b, x) == b assert SubstForInverseFunction(a, a, b, x) == a assert SubstForInverseFunction(xa, xa, b, x) == x assert SubstForInverseFunction(ax*a, a, b, x) == aba def test_SubstForFractionalPower(): assert SubstForFractionalPower(a, b, n, c, x) == a assert SubstForFractionalPower(x, b, n, c, x) == c assert SubstForFractionalPower(a(S(1)/2), a, n, b, x) == x(n/2) def test_CombineExponents(): assert True def test_FractionalPowerOfSquareQ(): assert not FractionalPowerOfSquareQ(x) assert not FractionalPowerOfSquareQ((a + b)(S(2)/S(3))) assert not FractionalPowerOfSquareQ((a + b)*(S(2)/S(3))c) assert FractionalPowerOfSquareQ(((a + bx)(S(2)))(S(1)/3)) == (a + bx)S(2) def test_FractionalPowerSubexpressionQ(): assert not FractionalPowerSubexpressionQ(x, a, x) assert FractionalPowerSubexpressionQ(x(S(2)/S(3)), a, x) assert not FractionalPowerSubexpressionQ(ba, a, x) def test_FactorNumericGcd(): assert FactorNumericGcd(5a*2e*4 + 2abde3 + 2acd*2e2 + b2d2e*2 - 6bcd*3e + 21c2d*4) ==\ 5a*2e*4 + 2abde3 + 2acd*2e2 + b2d2e*2 - 6bcd*3e + 21c2d4 assert FactorNumericGcd(x(S(2))) == x*S(2) assert FactorNumericGcd(rubi_log(x)) == rubi_log(x) assert FactorNumericGcd(rubi_log(x)x) == xrubi_log(x) assert FactorNumericGcd(rubi_log(x) + xS(2)) == rubi_log(x) + xS(2) def test_Apply(): assert Apply(List, [a, b, c]) == [a, b, c] def test_TrigSimplify(): assert TrigSimplify(asin(x)*2 + acos(x)*2 + v) == a + v assert TrigSimplify(asec(x)*2 - atan(x)*2 + v) == a + v assert TrigSimplify(acsc(x)*2 - acot(x)2 + v) == a + v assert TrigSimplify(S(1) - sin(x)2) == cos(x)2 assert TrigSimplify(1 + tan(x)2) == sec(x)2 assert TrigSimplify(1 + cot(x)2) == csc(x)2 assert TrigSimplify(-S(1) + sec(x)2) == tan(x)2 assert TrigSimplify(-1 + csc(x)2) == cot(x)2 def test_MergeFactors(): assert simplify(MergeFactors(b/(a - c)3 , 8c3(bx + c)(S(3)/2)/(3b*4) - 24c*2(bx + c)(S(5)/2)/(5b*4) + \ 24c(bx + c)*(S(7)/2)/(7b*4) - 8(bx + c)(S(9)/2)/(9b*4)) - (8c*3(bx + c)(S(3)/2)/(3b*3) - 24c*2(bx + c)(S(5)/2)/(5b*3) + \ 24c(bx + c)*(S(7)/2)/(7b*3) - 8(bx + c)(S(9)/2)/(9b3))/(a - c)3) == 0 assert MergeFactors(x, x) == x*2 assert MergeFactors(xy, x) == x*2y def test_FactorInteger(): assert FactorInteger(2434500) == [(2, 2), (3, 2), (5, 3), (541, 1)] def test_ContentFactor(): assert ContentFactor(ab + ac) == a(b + c) def test_Order(): assert Order(a, b) == 1 assert Order(b, a) == -1 assert Order(a, a) == 0 def test_FactorOrder(): assert FactorOrder(1, 1) == 0 assert FactorOrder(1, 2) == -1 assert FactorOrder(2, 1) == 1 assert FactorOrder(a, b) == 1 def test_Smallest(): assert Smallest([2, 1, 3, 4]) == 1 assert Smallest(1, 2) == 1 assert Smallest(-1, -2) == -2 def test_MostMainFactorPosition(): assert MostMainFactorPosition([S(1), S(2), S(3)]) == 1 assert MostMainFactorPosition([S(1), S(7), S(3), S(4), S(5)]) == 2 def test_OrderedQ(): assert OrderedQ([a, b]) assert not OrderedQ([b, a]) def test_MinimumDegree(): assert MinimumDegree(S(1), S(2)) == 1 assert MinimumDegree(S(1), sqrt(2)) == 1 assert MinimumDegree(sqrt(2), S(1)) == 1 assert MinimumDegree(sqrt(3), sqrt(2)) == sqrt(2) assert MinimumDegree(sqrt(2), sqrt(2)) == sqrt(2) def test_PositiveFactors(): assert PositiveFactors(S(0)) == 1 assert PositiveFactors(-S(1)) == S(1) assert PositiveFactors(sqrt(2)) == sqrt(2) assert PositiveFactors(-rubi_log(2)) == rubi_log(2) assert PositiveFactors(sqrt(2)S(-1)) == sqrt(2) def test_NonpositiveFactors(): assert NonpositiveFactors(S(0)) == 0 assert NonpositiveFactors(-S(1)) == -1 assert NonpositiveFactors(sqrt(2)) == 1 assert NonpositiveFactors(-rubi_log(2)) == -1 def test_Sign(): assert Sign(S(0)) == 0 assert Sign(S(1)) == 1 assert Sign(-S(1)) == -1 def test_PolynomialInQ(): v = rubi_log(x) assert PolynomialInQ(S(1), v, x) assert PolynomialInQ(v, v, x) assert PolynomialInQ(1 + v*2, v, x) assert PolynomialInQ(1 + av2, v, x) assert not PolynomialInQ(sqrt(v), v, x) def test_ExponentIn(): v = rubi_log(x) assert ExponentIn(S(1), rubi_log(x), x) == 0 assert ExponentIn(S(1) + v, rubi_log(x), x) == 1 assert ExponentIn(S(1) + v + v3, rubi_log(x), x) == 3 assert ExponentIn(S(2)sqrt(v)v3, rubi_log(x), x) == 3.5 def test_PolynomialInSubst(): v = rubi_log(x) assert PolynomialInSubst(S(1) + rubi_log(x)3, rubi_log(x), x) == 1 + x*3 assert PolynomialInSubst(S(1) + rubi_log(x), rubi_log(x), x) == x + 1 def test_Distrib(): assert Distrib(x, a) == xa assert Distrib(x, a + b) == ax + bx def test_DistributeDegree(): assert DistributeDegree(x, m) == xm assert DistributeDegree(xa, m) == x*(am) assert DistributeDegree(ab, m) == am bm def test_FunctionOfPower(): assert FunctionOfPower(a, x) == None assert FunctionOfPower(x, x) == 1 assert FunctionOfPower(x3, x) == 3 assert FunctionOfPower(x*3cos(x6), x) == 3 def test_DivideDegreesOfFactors(): assert DivideDegreesOfFactors(ab, S(3)) == a(b/3) assert DivideDegreesOfFactors(abc, S(3)) == a(b/3)c*(c/3) def test_MonomialFactor(): assert MonomialFactor(a, x) == [0, a] assert MonomialFactor(x, x) == [1, 1] assert MonomialFactor(x + y, x) == [0, x + y] assert MonomialFactor(rubi_log(x), x) == [0, rubi_log(x)] assert MonomialFactor(rubi_log(x)x, x) == [1, rubi_log(x)] def test_NormalizeIntegrand(): assert NormalizeIntegrand((x2 + 8), x) == x2 + 8 assert NormalizeIntegrand((x*2 + 3x)2, x) == x2(x + 3)2 assert NormalizeIntegrand(a2(a + bx)2, x) == a2(a + bx)2 assert NormalizeIntegrand(b2/(a2(a + bx)2), x) == b2/(a2(a + bx)2) def test_NormalizeIntegrandAux(): v = (6Aac - 2Ab*2 + Bab)/(ax*2) - (6Aa2c*2 - 10Aab*2c - 8Aabc*2x + 2Ab*4 + 2Ab3cx + 5Ba2bc + 4Ba2c*2x - Bab*3 - Bab2cx)/(a2(a + bx + cx*2)) + (-2Ab + Ba)(4ac - b2)/(a2x) assert NormalizeIntegrandAux(v, x) == (6Aac - 2Ab2 + Bab)/(ax*2) - (6Aa2c*2 - 10Aab*2c + 2Ab*4 + 5Ba2bc - Bab3 + x(-8Aabc*2 + 2Ab3c + 4Ba*2c*2 - Bab2c))/(a*2(a + bx + cx*2)) + (-2Ab + Ba)(4ac - b2)/(a2x) assert NormalizeIntegrandAux((x*2 + 3x)2, x) == x2(x + 3)2 assert NormalizeIntegrandAux((x2 + 8), x) == x2 + 8 def test_NormalizeIntegrandFactor(): assert NormalizeIntegrandFactor((3x + x3)2, x) == x*2(x2 + 3)2 assert NormalizeIntegrandFactor((x2 + 8), x) == x2 + 8 def test_NormalizeIntegrandFactorBase(): assert NormalizeIntegrandFactorBase((x2 + 8)3, x) == (x2 + 8)3 assert NormalizeIntegrandFactorBase((x2 + 8), x) == x2 + 8 assert NormalizeIntegrandFactorBase(a*2(a + bx)2, x) == a2(a + bx)2 def test_AbsorbMinusSign(): assert AbsorbMinusSign((x + 2)5(x + 3)5) == (-x - 3)5(x + 2)5 assert AbsorbMinusSign((x + 2)5(x + 3)2) == -(x + 2)5(x + 3)2 def test_NormalizeLeadTermSigns(): assert NormalizeLeadTermSigns((-x + 3)(x*2 + 3)) == (-x + 3)(x2 + 3) assert NormalizeLeadTermSigns(x + 3) == x + 3 def test_SignOfFactor(): assert SignOfFactor(S(-x + 3)) == [1, -x + 3] assert SignOfFactor(S(-x)) == [-1, x] def test_NormalizePowerOfLinear(): assert NormalizePowerOfLinear((x + 3)5, x) == (x + 3)5 assert NormalizePowerOfLinear(((x + 3)2) + 3, x) == x*2 + 6x + 12 def test_SimplifyIntegrand(): assert SimplifyIntegrand((x2 + 3)2, x) == (x2 + 3)2 assert SimplifyIntegrand(x2 + 3 + (x6) + 6, x) == x6 + x2 + 9 def test_SimplifyTerm(): assert SimplifyTerm(a2/b2, x) == a2/b2 assert SimplifyTerm(-6x/5 + (5x + 3)2/25 - S(9)/25, x) == x2 def test_togetherSimplify(): assert TogetherSimplify(-6x/5 + (5x + 3)2/25 - S(9)/25) == x2 def test_ExpandToSum(): qq = 6 Pqq = e*3 Pq = (d+ex2)3 aa = 2 nn = 2 cc = 1 pp = -S.Half bb = 3 assert nsimplify(ExpandToSum(Pq - Pqqxqq - Pqq(aax(-2nn + qq)(-2nn + qq + 1) + bbx(-nn + qq)(nn(pp - 1) + qq + 1))/(cc(2nnpp + qq + 1)), x) - \ (d3 + x4(3de2 - 2.4e3) + x2(3d*2e - 1.2e3))) == 0 assert ExpandToSum(x2 + 3x + 3, x3 + 3, x) == x3(x2 + 3x + 3) + 3x2 + 9x + 9 assert ExpandToSum(x3 + 6, x) == x3 + 6 assert ExpandToSum(S(x*2 + 3x + 3)3, x) == 3x*2 + 9x + 9 assert ExpandToSum((a + bx), x) == a + bx def test_UnifySum(): assert UnifySum((3 + x + 6x3 + sin(x)), x) == 6x*3 + x + sin(x) + 3 assert UnifySum((3 + x + 6x*3)3, x) == 18x3 + 3x + 9 def test_FunctionOfInverseLinear(): assert FunctionOfInverseLinear((x)/(a + bx), x) == [a, b] assert FunctionOfInverseLinear((c + dx)/(a + bx), x) == [a, b] assert not FunctionOfInverseLinear(1/(a + bx), x) def test_PureFunctionOfSinhQ(): v = rubi_log(x) f = sinh(v) assert PureFunctionOfSinhQ(f, v, x) assert not PureFunctionOfSinhQ(cosh(v), v, x) assert PureFunctionOfSinhQ(f2, v, x) def test_PureFunctionOfTanhQ(): v = rubi_log(x) f = tanh(v) assert PureFunctionOfTanhQ(f, v, x) assert not PureFunctionOfTanhQ(cosh(v), v, x) assert PureFunctionOfTanhQ(f2, v, x) def test_PureFunctionOfCoshQ(): v = rubi_log(x) f = cosh(v) assert PureFunctionOfCoshQ(f, v, x) assert not PureFunctionOfCoshQ(sinh(v), v, x) assert PureFunctionOfCoshQ(f*2, v, x) def test_IntegerQuotientQ(): u = S(2)sin(x) v = sin(x) assert IntegerQuotientQ(u, v) assert IntegerQuotientQ(u, u) assert not IntegerQuotientQ(S(1), S(2)) def test_OddQuotientQ(): u = S(3)sin(x) v = sin(x) assert OddQuotientQ(u, v) assert OddQuotientQ(u, u) assert not OddQuotientQ(S(1), S(2)) def test_EvenQuotientQ(): u = S(2)sin(x) v = sin(x) assert EvenQuotientQ(u, v) assert not EvenQuotientQ(u, u) assert not EvenQuotientQ(S(1), S(2)) def test_FunctionOfSinhQ(): v = rubi_log(x) assert FunctionOfSinhQ(cos(sinh(v)), v, x) assert FunctionOfSinhQ(sinh(v), v, x) assert FunctionOfSinhQ(sinh(v)cos(sinh(v)), v, x) def test_FunctionOfCoshQ(): v = rubi_log(x) assert FunctionOfCoshQ(cos(cosh(v)), v, x) assert FunctionOfCoshQ(cosh(v), v, x) assert FunctionOfCoshQ(cosh(v)cos(cosh(v)), v, x) def test_FunctionOfTanhQ(): v = rubi_log(x) t = Tanh(v) c = Coth(v) assert FunctionOfTanhQ(t, v, x) assert FunctionOfTanhQ(c, v, x) assert FunctionOfTanhQ(t + c, v, x) assert FunctionOfTanhQ(tc, v, x) assert not FunctionOfTanhQ(sin(x), v, x) def test_FunctionOfTanhWeight(): v = rubi_log(x) t = Tanh(v) c = Coth(v) assert FunctionOfTanhWeight(x, v, x) == 0 assert FunctionOfTanhWeight(sinh(v), v, x) == 0 assert FunctionOfTanhWeight(tanh(v), v, x) == 1 assert FunctionOfTanhWeight(coth(v), v, x) == -1 assert FunctionOfTanhWeight(t2, v, x) == 1 assert FunctionOfTanhWeight(sinh(v)2, v, x) == -1 assert FunctionOfTanhWeight(coth(v)sinh(v)2, v, x) == -2 def test_FunctionOfHyperbolicQ(): v = rubi_log(x) s = Sinh(v) t = Tanh(v) assert not FunctionOfHyperbolicQ(x, v, x) assert FunctionOfHyperbolicQ(s + t, v, x) assert FunctionOfHyperbolicQ(sinh(t), v, x) def test_SmartNumerator(): assert SmartNumerator(x(-2)) == 1 assert SmartNumerator(x*(2)a) == x*2a def test_SmartDenominator(): assert SmartDenominator(x(-2)) == x2 assert SmartDenominator(x*(-2)1/S(3)) == x*23 def test_SubstForAux(): v = rubi_log(x) assert SubstForAux(v, v, x) == x assert SubstForAux(v2, v, x) == x2 assert SubstForAux(x, v, x) == x assert SubstForAux(v2, v4, x) == sqrt(x) assert SubstForAux(v*2v, v, x) == x*3 def test_SubstForTrig(): v = rubi_log(x) s, c, t = sin(v), cos(v), tan(v) assert SubstForTrig(cos(a/2 + bx/2), x/sqrt(x2 + 1), 1/sqrt(x2 + 1), a/2 + bx/2, x) == 1/sqrt(x2 + 1) assert SubstForTrig(s, sin, cos, v, x) == sin assert SubstForTrig(t, sin(v), cos(v), v, x) == sin(rubi_log(x))/cos(rubi_log(x)) assert SubstForTrig(sin(2v), sin(x), cos(x), v, x) == 2sin(x)cos(x) assert SubstForTrig(st, sin(x), cos(x), v, x) == sin(x)2/cos(x) def test_SubstForHyperbolic(): v = rubi_log(x) s, c, t = sinh(v), cosh(v), tanh(v) assert SubstForHyperbolic(s, sinh(x), cosh(x), v, x) == sinh(x) assert SubstForHyperbolic(t, sinh(x), cosh(x), v, x) == sinh(x)/cosh(x) assert SubstForHyperbolic(sinh(2v), sinh(x), cosh(x), v, x) == 2sinh(x)cosh(x) assert SubstForHyperbolic(st, sinh(x), cosh(x), v, x) == sinh(x)2/cosh(x) def test_SubstForFractionalPowerOfLinear(): u = a + bx assert not SubstForFractionalPowerOfLinear(u, x) assert not SubstForFractionalPowerOfLinear(u(S(2)), x) assert SubstForFractionalPowerOfLinear(u(S(1)/2), x) == [x*2, 2, a + bx, 1/b] def test_InverseFunctionOfLinear(): u = a + bx assert InverseFunctionOfLinear(rubi_log(u)sin(x), x) == rubi_log(u) assert InverseFunctionOfLinear(rubi_log(u), x) == rubi_log(u) def test_InertTrigQ(): s = sin(x) c = cos(x) assert not InertTrigQ(sin(x), csc(x), cos(h)) assert InertTrigQ(sin(x), csc(x)) assert not InertTrigQ(s, c) assert InertTrigQ(c) def test_PowerOfInertTrigSumQ(): func = sin assert PowerOfInertTrigSumQ((1 + S(2)(S(3)func(x2))S(5))*3, func, x) assert PowerOfInertTrigSumQ((1 + 2(S(3)func(x2))3 + 4(S(5)func(x2))S(3))2, func, x) def test_PiecewiseLinearQ(): assert PiecewiseLinearQ(a + bx, x) assert not PiecewiseLinearQ(Log(csin(a)S(3)), x) assert not PiecewiseLinearQ(x3, x) assert PiecewiseLinearQ(atanh(tanh(a + bx)), x) assert PiecewiseLinearQ(tanh(atanh(a + bx)), x) assert not PiecewiseLinearQ(coth(atanh(a + bx)), x) def test_KnownTrigIntegrandQ(): func = sin(a + bx) assert KnownTrigIntegrandQ([sin], S(1), x) assert KnownTrigIntegrandQ([sin], (a + bfunc)*m, x) assert KnownTrigIntegrandQ([sin], (a + bfunc)*m(1 + 2func), x) assert KnownTrigIntegrandQ([sin], a + cfunc*2, x) assert KnownTrigIntegrandQ([sin], a + bfunc + cfunc2, x) assert KnownTrigIntegrandQ([sin], (a + bfunc)*m(c + dfunc2), x) assert KnownTrigIntegrandQ([sin], (a + bfunc)*m(c + dfunc + efunc*2), x) assert not KnownTrigIntegrandQ([cos], (a + bfunc)*m, x) def test_KnownSineIntegrandQ(): assert KnownSineIntegrandQ((a + bsin(a + bx))m, x) def test_KnownTangentIntegrandQ(): assert KnownTangentIntegrandQ((a + btan(a + bx))m, x) def test_KnownCotangentIntegrandQ(): assert KnownCotangentIntegrandQ((a + bcot(a + bx))m, x) def test_KnownSecantIntegrandQ(): assert KnownSecantIntegrandQ((a + bsec(a + bx))m, x) def test_TryPureTanSubst(): assert TryPureTanSubst(atan(c(a + btan(a + bx))), x) assert TryPureTanSubst(atanh(c(a + bcot(a + bx))), x) assert not TryPureTanSubst(tan(c(a + bcot(a + bx))), x) def test_TryPureTanhSubst(): assert not TryPureTanhSubst(rubi_log(x), x) assert TryPureTanhSubst(sin(x), x) assert not TryPureTanhSubst(atanh(atanh(x)), x) assert not TryPureTanhSubst((a + bx)*S(2), x) def test_TryTanhSubst(): assert not TryTanhSubst(rubi_log(x), x) assert not TryTanhSubst(a(b + c)*3, x) assert not TryTanhSubst(1/(a + bsinh(x)*S(3)), x) assert not TryTanhSubst(sinh(S(3)x)cosh(S(4)x), x) assert not TryTanhSubst(a(bsech(x)3)c, x) def test_GeneralizedBinomialQ(): assert GeneralizedBinomialQ(axq + bx*n, x) assert not GeneralizedBinomialQ(ax*q, x) def test_GeneralizedTrinomialQ(): assert not GeneralizedTrinomialQ(7 + 2x*6 + 3x*12, x) assert not GeneralizedTrinomialQ(ax*q + cx*(2n-q), x) def test_SubstForFractionalPowerOfQuotientOfLinears(): assert SubstForFractionalPowerOfQuotientOfLinears(((a + bx)/(c + dx))(S(3)/2), x) == [x4/(b - dx2)2, 2, (a + bx)/(c + dx), -ad + bc] def test_SubstForFractionalPowerQ(): assert SubstForFractionalPowerQ(x, sin(x), x) assert SubstForFractionalPowerQ(x2, sin(x), x) assert not SubstForFractionalPowerQ(x(S(3)/2), sin(x), x) assert SubstForFractionalPowerQ(sin(x)(S(3)/2), sin(x), x) def test_AbsurdNumberGCD(): assert AbsurdNumberGCD(S(4)) == 4 assert AbsurdNumberGCD(S(4), S(8), S(12)) == 4 assert AbsurdNumberGCD(S(2), S(3), S(12)) == 1 def test_TrigReduce(): assert TrigReduce(cos(x)2) == cos(2x)/2 + S.Half assert TrigReduce(cos(x)*2sin(x)) == sin(x)/4 + sin(3x)/4 assert TrigReduce(cos(x)2+sin(x)) == sin(x) + cos(2x)/2 + S.Half assert TrigReduce(cos(x)*2sin(x)*5) == 5sin(x)/64 + sin(3x)/64 - 3sin(5x)/64 + sin(7x)/64 assert TrigReduce(2sin(x)cos(x) + 2cos(x)2) == sin(2x) + cos(2x) + 1 assert TrigReduce(sinh(a + bx)*2) == cosh(2a + 2bx)/2 - S.Half assert TrigReduce(sinh(a + bx)cosh(a + bx)) == sinh(2a + 2bx)/2 def test_FunctionOfDensePolynomialsQ(): assert FunctionOfDensePolynomialsQ(x2 + 3, x) assert not FunctionOfDensePolynomialsQ(x2, x) assert not FunctionOfDensePolynomialsQ(x, x) assert FunctionOfDensePolynomialsQ(S(2), x) def test_PureFunctionOfSinQ(): v = rubi_log(x) f = sin(v) assert PureFunctionOfSinQ(f, v, x) assert not PureFunctionOfSinQ(cos(v), v, x) assert PureFunctionOfSinQ(f2, v, x) def test_PureFunctionOfTanQ(): v = rubi_log(x) f = tan(v) assert PureFunctionOfTanQ(f, v, x) assert not PureFunctionOfTanQ(cos(v), v, x) assert PureFunctionOfTanQ(f2, v, x) def test_PowerVariableSubst(): assert PowerVariableSubst((2x)3, 2, x) == 8x*(S(3)/2) assert PowerVariableSubst((2x)*3, 2, x) == 8x*(S(3)/2) assert PowerVariableSubst((2x), 2, x) == 2x assert PowerVariableSubst((2x)*3, 2, x) == 8x*(S(3)/2) assert PowerVariableSubst((2x)*7, 2, x) == 128x*(S(7)/2) assert PowerVariableSubst((6+2x)*7, 2, x) == (2x + 6)*7 assert PowerVariableSubst((2x)*7+3, 2, x) == 128x*(S(7)/2) + 3 def test_PowerVariableDegree(): assert PowerVariableDegree(S(2), 0, 2x, x) == [0, 2x] assert PowerVariableDegree((2x)*2, 0, 2x, x) == [2, 1] assert PowerVariableDegree(x*2, 0, 2x, x) == [2, 1] assert PowerVariableDegree(S(4), 0, 2x, x) == [0, 2x] def test_PowerVariableExpn(): assert not PowerVariableExpn((x)*3, 2, x) assert not PowerVariableExpn((2x)*3, 2, x) assert PowerVariableExpn((2x)*2, 4, x) == [4x3, 2, 1] def test_FunctionOfQ(): assert FunctionOfQ(x2, sqrt(-exp(2x2) + 1)exp(x2),x) assert not FunctionOfQ(S(x3), x2, x) assert FunctionOfQ(S(a), x2, x) assert FunctionOfQ(S(3x), x2, x) def test_ExpandTrigExpand(): assert ExpandTrigExpand(1, cos(x), x*2, 2, 2, x) == 4cos(x2)4 - 4cos(x2)2 + 1 assert ExpandTrigExpand(1, cos(x) + sin(x), x2, 2, 2, x) == 4sin(x2)2cos(x2)2 + 8sin(x*2)cos(x2)3 - 4sin(x2)cos(x*2) + 4cos(x2)4 - 4cos(x2)2 + 1 def test_TrigToExp(): from sympy.integrals.rubi.utility_function import rubi_exp as exp assert TrigToExp(sin(x)) == -I(exp(Ix) - exp(-Ix))/2 assert TrigToExp(cos(x)) == exp(Ix)/2 + exp(-Ix)/2 assert TrigToExp(cos(x)tan(x2)) == I(exp(Ix)/2 + exp(-Ix)/2)(-exp(Ix*2) + exp(-Ix*2))/(exp(Ix*2) + exp(-Ix2)) assert TrigToExp(cos(x) + sin(x)2) == -(exp(Ix) - exp(-Ix))*2/4 + exp(Ix)/2 + exp(-Ix)/2 assert Simplify(TrigToExp(cos(x)tan(x*S(2))sin(x)*S(2))-(-I(exp(Ix)/S(2) + exp(-Ix)/S(2))(exp(Ix) - exp(-Ix))S(2)(-exp(IxS(2)) + exp(-Ix*S(2)))/(S(4)(exp(IxS(2)) + exp(-Ix*S(2)))))) == 0 def test_ExpandTrigReduce(): assert ExpandTrigReduce(2cos(3 + x)*3, x) == 3cos(x + 3)/2 + cos(3x + 9)/2 assert ExpandTrigReduce(2sin(x)*3+cos(2 + x), x) == 3sin(x)/2 - sin(3x)/2 + cos(x + 2) assert ExpandTrigReduce(cos(x + 3)2, x) == cos(2x + 6)/2 + S.Half def test_NormalizeTrig(): assert NormalizeTrig(S(2sin(2 + x)), x) == 2sin(x + 2) assert NormalizeTrig(S(2sin(2 + x)3), x) == 2sin(x + 2)*3 assert NormalizeTrig(S(2sin((2 + x)2)3), x) == 2sin(x2 + 4x + 4)3 def test_FunctionOfTrigQ(): v = rubi_log(x) s = sin(v) t = tan(v) assert not FunctionOfTrigQ(x, v, x) assert FunctionOfTrigQ(s + t, v, x) assert FunctionOfTrigQ(sin(t), v, x) def test_RationalFunctionExpand(): assert RationalFunctionExpand(xS(5)(e + fx)*n/(a + bx*S(3)), x) == -ax*2(e + fx)n/(b(a + bx3)) +\ e2(e + fx)n/(bf*2) - 2e(e + fx)*(n + 1)/(bf*2) + (e + fx)*(n + 2)/(bf2) assert RationalFunctionExpand(xS(3)(S(2)x + 2)*S(2)/(2x*2 + 1), x) == 2x*3 + 4x*2 + x + (- x + 2)/(2x*2 + 1) - 2 assert RationalFunctionExpand((a + bx + cx4)rubi_log(x)*3, x) == arubi_log(x)*3 + bxrubi_log(x)3 + cx*4rubi_log(x)*3 assert RationalFunctionExpand(a + bx + cx4, x) == a + bx + cx4 def test_SameQ(): assert SameQ(1, 1, 1) assert not SameQ(1, 1, 2) def test_Map2(): assert Map2(Add, [a, b, c], [x, y, z]) == [a + x, b + y, c + z] def test_ConstantFactor(): assert ConstantFactor(a + ax3, x) == [a, x3 + 1] assert ConstantFactor(a, x) == [a, 1] assert ConstantFactor(x, x) == [1, x] assert ConstantFactor(xS(3), x) == [1, x3] assert ConstantFactor(x(S(3)/2), x) == [1, x(S(3)/2)] assert ConstantFactor(ax3, x) == [a, x3] assert ConstantFactor(a + x3, x) == [1, a + x3] def test_CommonFactors(): assert CommonFactors([a, a, a]) == [a, 1, 1, 1] assert CommonFactors([xS(2), x*S(3)S(2), sin(x)xS(2)]) == [2, x, x*3, xsin(x)] assert CommonFactors([x, x*S(3), sin(x)x]) == [1, x, x*3, xsin(x)] assert CommonFactors([S(2), S(4), S(6)]) == [2, 1, 2, 3] def test_FunctionOfLinear(): f = sin(a + bx) assert FunctionOfLinear(f, x) == [sin(x), a, b] assert FunctionOfLinear(a + bx, x) == [x, a, b] assert not FunctionOfLinear(a, x) def test_FunctionOfExponentialQ(): assert FunctionOfExponentialQ(exp(x + exp(x) + exp(exp(x))), x) assert FunctionOfExponentialQ(a*(a + bx), x) assert FunctionOfExponentialQ(a*(bx), x) assert not FunctionOfExponentialQ(a*sin(a + bx), x) def test_FunctionOfExponential(): assert FunctionOfExponential(a*(a + bx), x) def test_FunctionOfExponentialFunction(): assert FunctionOfExponentialFunction(a*(a + bx), x) == x assert FunctionOfExponentialFunction(S(2)a(a + bx), x) == 2x def test_FunctionOfTrig(): assert FunctionOfTrig(sin(x + 1), x + 1, x) == x + 1 assert FunctionOfTrig(sin(x), x) == x assert not FunctionOfTrig(cos(x2 + 1), x) assert FunctionOfTrig(sin(a+bx)*3, x) == a+bx def test_AlgebraicTrigFunctionQ(): assert AlgebraicTrigFunctionQ(sin(x + 3), x) assert AlgebraicTrigFunctionQ(x, x) assert AlgebraicTrigFunctionQ(x + 1, x) assert AlgebraicTrigFunctionQ(sinh(x + 1), x) assert AlgebraicTrigFunctionQ(sinh(x + 1)2, x) assert not AlgebraicTrigFunctionQ(sinh(x2 + 1)2, x) def test_FunctionOfHyperbolic(): assert FunctionOfTrig(sin(x + 1), x + 1, x) == x + 1 assert FunctionOfTrig(sin(x), x) == x assert not FunctionOfTrig(cos(x2 + 1), x) def test_FunctionOfExpnQ(): assert FunctionOfExpnQ(x, x, x) == 1 assert FunctionOfExpnQ(x2, x, x) == 2 assert FunctionOfExpnQ(x2.1, x, x) == 1 assert not FunctionOfExpnQ(x, x2, x) assert not FunctionOfExpnQ(x + 1, (x + 5)2, x) assert not FunctionOfExpnQ(x + 1, (x + 1)2, x) def test_PureFunctionOfCosQ(): v = rubi_log(x) f = cos(v) assert PureFunctionOfCosQ(f, v, x) assert not PureFunctionOfCosQ(sin(v), v, x) assert PureFunctionOfCosQ(f2, v, x) def test_PureFunctionOfCotQ(): v = rubi_log(x) f = cot(v) assert PureFunctionOfCotQ(f, v, x) assert not PureFunctionOfCotQ(sin(v), v, x) assert PureFunctionOfCotQ(f*2, v, x) def test_FunctionOfSinQ(): v = rubi_log(x) assert FunctionOfSinQ(cos(sin(v)), v, x) assert FunctionOfSinQ(sin(v), v, x) assert FunctionOfSinQ(sin(v)cos(sin(v)), v, x) def test_FunctionOfCosQ(): v = rubi_log(x) assert FunctionOfCosQ(cos(cos(v)), v, x) assert FunctionOfCosQ(cos(v), v, x) assert FunctionOfCosQ(cos(v)cos(cos(v)), v, x) def test_FunctionOfTanQ(): v = rubi_log(x) t = tan(v) c = cot(v) assert FunctionOfTanQ(t, v, x) assert FunctionOfTanQ(c, v, x) assert FunctionOfTanQ(t + c, v, x) assert FunctionOfTanQ(tc, v, x) assert not FunctionOfTanQ(sin(x), v, x) def test_FunctionOfTanWeight(): v = rubi_log(x) t = tan(v) c = cot(v) assert FunctionOfTanWeight(x, v, x) == 0 assert FunctionOfTanWeight(sin(v), v, x) == 0 assert FunctionOfTanWeight(tan(v), v, x) == 1 assert FunctionOfTanWeight(cot(v), v, x) == -1 assert FunctionOfTanWeight(t2, v, x) == 1 assert FunctionOfTanWeight(sin(v)2, v, x) == -1 assert FunctionOfTanWeight(cot(v)sin(v)2, v, x) == -2 def test_OddTrigPowerQ(): assert not OddTrigPowerQ(sin(x)3, 1, x) assert OddTrigPowerQ(sin(3),1,x) assert OddTrigPowerQ(sin(3x),x,x) assert OddTrigPowerQ(sin(3x)3,x,x) def test_FunctionOfLog(): assert not FunctionOfLog(x2(a + bx)3exp(-a - bx) ,False, False, x) assert FunctionOfLog(rubi_log(2x*8)2 + rubi_log(2x8) + 1, x) == [3x + 1, 2x8, 8] assert FunctionOfLog(rubi_log(2x)2,x) == [x2, 2x, 1] assert FunctionOfLog(rubi_log(3x3)2 + 1,x) == [x*2 + 1, 3x*3, 3] assert FunctionOfLog(rubi_log(2x*8)2,x) == [2x, 2x*8, 8] assert not FunctionOfLog(2sin(x)2,x) def test_EulerIntegrandQ(): assert EulerIntegrandQ((2x + 3((x + 1)3)(S(3)/2))(-3), x) assert not EulerIntegrandQ((2x + (2x2)2)3, x) assert not EulerIntegrandQ(3x*2 + 5x + 1, x) def test_Divides(): assert not Divides(x, ax2, x) assert Divides(x, ax, x) == a def test_EasyDQ(): assert EasyDQ(3x2, x) assert EasyDQ(3x3 - 6, x) assert EasyDQ(x3, x) assert EasyDQ(sin(xrubi_log(3)), x) def test_ProductOfLinearPowersQ(): assert ProductOfLinearPowersQ(S(1), x) assert ProductOfLinearPowersQ((x + 1)3, x) assert not ProductOfLinearPowersQ((x2 + 1)3, x) assert ProductOfLinearPowersQ(x + 1, x) def test_Rt(): b = symbols('b') assert Rt(-b2, 4) == (-b2)(S(1)/S(4)) assert Rt(x2, 2) == x assert Rt(S(2 + 3I), S(8)) == (2 + 3I)(S(1)/8) assert Rt(x2 + 4 + 4x, 2) == x + 2 assert Rt(S(8), S(3)) == 2 assert Rt(S(16807), S(5)) == 7 def test_NthRoot(): assert NthRoot(S(14580), S(3)) == 92*(S(2)/S(3))5(S(1)/S(3)) assert NthRoot(9, 2) == 3.0 assert NthRoot(81, 2) == 9.0 assert NthRoot(81, 4) == 3.0 def test_AtomBaseQ(): assert not AtomBaseQ(x2) assert AtomBaseQ(x3) assert AtomBaseQ(x) assert AtomBaseQ(S(2)3) assert not AtomBaseQ(sin(x)) def test_SumBaseQ(): assert not SumBaseQ((x + 1)2) assert SumBaseQ((x + 1)3) assert SumBaseQ(3x+3) assert not SumBaseQ(x) def test_NegSumBaseQ(): assert not NegSumBaseQ(-x + 1) assert NegSumBaseQ(x - 1) assert not NegSumBaseQ((x - 1)2) assert NegSumBaseQ((x - 1)3) def test_AllNegTermQ(): x = Symbol('x', negative=True) assert AllNegTermQ(x) assert not AllNegTermQ(x + 2) assert AllNegTermQ(x - 2) assert AllNegTermQ((x - 2)3) assert not AllNegTermQ((x - 2)2) def test_TrigSquareQ(): assert TrigSquareQ(sin(x)2) assert TrigSquareQ(cos(x)2) assert not TrigSquareQ(tan(x)2) def test_Inequality(): assert not Inequality(S('0'), Less, m, LessEqual, S('1')) assert Inequality(S('0'), Less, S('1')) assert Inequality(S('0'), Less, S('1'), LessEqual, S('5')) def test_SplitProduct(): assert SplitProduct(OddQ, S(3)x) == [3, x] assert not SplitProduct(OddQ, S(2)x) def test_SplitSum(): assert SplitSum(FracPart, sin(x)) == [sin(x), 0] assert SplitSum(FracPart, sin(x) + S(2)) == [sin(x), S(2)] def test_Complex(): assert Complex(a, b) == a + Ib def test_SimpFixFactor(): assert SimpFixFactor((ac + bc)*S(4), x) == (ac + bc)4 assert SimpFixFactor((aComplex(0, c) + bComplex(0, d))S(3), x) == -I(ac + bd)*3 assert SimpFixFactor((aComplex(0, d) + bComplex(0, e) + cComplex(0, f))*S(2), x) == -(ad + be + cf)*2 assert SimpFixFactor((a + bx*(-1/S(2))xS(3))S(3), x) == (a + bx(S(5)/2))3 assert SimpFixFactor((ac + bcS(2)xS(2))S(3), x) == c*3(a + bcx2)3 assert SimpFixFactor((acS(2) + bc*S(1)xS(2))S(3), x) == c*3(ac + bx2)3 assert SimpFixFactor(acos(x)2 + asin(x)*2 + v, x) == acos(x)*2 + asin(x)*2 + v def test_SimplifyAntiderivative(): assert SimplifyAntiderivative(acoth(coth(x)), x) == x assert SimplifyAntiderivative(ax, x) == ax assert SimplifyAntiderivative(atanh(cot(x)), x) == atanh(2sin(x)cos(x))/2 assert SimplifyAntiderivative(acos(x)*2 + asin(x)*2 + v, x) == acos(x)*2 + asin(x)*2 def test_FixSimplify(): assert FixSimplify(xComplex(0, a)(vComplex(0, b) + w)*S(3)) == ax(bv - Iw)3 def test_TrigSimplifyAux(): assert TrigSimplifyAux(acos(x)*2 + asin(x)2 + v) == a + v assert TrigSimplifyAux(x2) == x2 def test_SubstFor(): assert SubstFor(x2 + 1, tanh(x), x) == tanh(x) assert SubstFor(x2, sinh(x), x) == sinh(sqrt(x)) def test_FresnelS(): assert FresnelS(oo) == S.Half assert FresnelS(0) == 0 def test_FresnelC(): assert FresnelC(0) == 0 assert FresnelC(oo) == S.Half def test_Erfc(): assert Erfc(0) == 1 assert Erfc(oo) == 0 def test_Erfi(): assert Erfi(oo) is oo assert Erfi(0) == 0 def test_Gamma(): assert Gamma(u) == gamma(u) def test_ElementaryFunctionQ(): assert ElementaryFunctionQ(x + y) assert ElementaryFunctionQ(sin(x + y)) assert ElementaryFunctionQ(E(xa)) def test_Util_Part(): from sympy.integrals.rubi.utility_function import Util_Part assert Util_Part(1, a + b).doit() == a assert Util_Part(c, a + b).doit() == Util_Part(c, a + b) def test_Part(): assert Part([1, 2, 3], 1) == 1 assert Part(ab, 1) == a def test_PolyLog(): assert PolyLog(a, b) == polylog(a, b) def test_PureFunctionOfCothQ(): v = rubi_log(x) assert PureFunctionOfCothQ(coth(v), v, x) assert PureFunctionOfCothQ(a + coth(v), v, x) assert not PureFunctionOfCothQ(sin(v), v, x) def test_ExpandIntegrand(): assert ExpandIntegrand(sqrt(a + bxS(2) + cx*S(4)), (fx)*(S(3)/2)(d + exS(2)), x) == \ d(fx)(S(3)/2)sqrt(a + bx2 + cx*4) + e(fx)(S(7)/2)sqrt(a + bx2 + cx4)/f2 assert ExpandIntegrand((6Aac - 2Ab2 + Bab - 2cx(Ab - 2Ba))/(x2(a + bx + cx*2)), x) == \ (6Aac - 2Ab*2 + Bab)/(ax*2) + (-6Aa2c*2 + 10Aab*2c - 2Ab*4 - 5Ba2bc + Bab3 + x(8Aabc*2 - 2Ab3c - 4Ba*2c*2 + Bab2c))/(a*2(a + bx + cx*2)) + (-2Ab + Ba)(4ac - b2)/(a2x) assert ExpandIntegrand(x*2(e + fx)3F*(a + b(c + dx)1), x) == F(a + b(c + dx))e*2(e + fx)3/f2 - 2F*(a + b(c + dx))e(e + fx)4/f2 + F*(a + b(c + dx))(e + fx)5/f2 assert ExpandIntegrand((x)(a + bx)2f*(e(c + dx)n), x) == a2f*(e(c + dx)n)x + 2abf(e(c + dx)n)x2 + b2f(e(c + dx)n)x3 assert ExpandIntegrand(sin(x)3(a + b(1/sin(x)))2, x) == a2sin(x)3 + 2absin(x)2 + b2sin(x) assert ExpandIntegrand(x(a + bArcSin(c + dx))*n, x) == -c(a + basin(c + dx))*n/d + (a + basin(c + dx))n(c + dx)/d assert ExpandIntegrand((a + bx)*S(3)(A + Bx)/(c + dx), x) == B(a + bx)*3/d + b(a + bx)2(Ad - Bc)/d*2 + b(a + bx)(Ad - Bc)(ad - bc)/d3 + b(Ad - Bc)(ad - bc)2/d4 + (Ad - Bc)(ad - bc)3/(d4(c + dx)) assert ExpandIntegrand((x*2)(S(3)x)(S(1)/2), x) ==sqrt(3)x*(S(5)/2) assert ExpandIntegrand((x)(sin(x))*(S(1)/2), x) == xsqrt(sin(x)) assert ExpandIntegrand(x(e + fx)*2F*(b(c + dx)), x) == -F(b(c + dx))e(e + fx)2/f + F(b(c + dx))(e + fx)3/f assert ExpandIntegrand(xm(e + fx)*2F*(b(c + dx)n), x) == F(b(c + dx)n)e*2x*m + 2F*(b(c + dx)n)efxxm + F(b(c + dx)n)f*2x*2x*m assert simplify(ExpandIntegrand((S(1) - S(1)xS(2))(-S(3)), x) - (-S(3)/(8(x2 - 1)) + S(3)/(16(x + 1)*2) + S(1)/(S(8)(x + 1)*3) + S(3)/(S(16)(x - 1)*2) - S(1)/(S(8)(x - 1)3))) == 0 assert ExpandIntegrand(-S(1), 1/((-q - x)3(q - x)3), x) == 1/(8q*3(q + x)*3) - 1/(8q*3(-q + x)*3) - 3/(8q*4(-q2 + x2)) + 3/(16q4(q + x)*2) + 3/(16q*4(-q + x)*2) assert ExpandIntegrand((1 + 1x)*(3)/(2 + 1x), x) == x*2 + x + 1 - 1/(x + 2) assert ExpandIntegrand((c + dx*1 + ex2)/(1 - x3), x) == (c - (-1)*(S(1)/3)d + (-1)*(S(2)/3)e)/(-3(-1)(S(2)/3)x + 3) + (c + (-1)*(S(2)/3)d - (-1)*(S(1)/3)e)/(3(-1)(S(1)/3)x + 3) + (c + d + e)/(-3x + 3) assert ExpandIntegrand((c + dx*1 + ex*2 + fx3)/(1 - x4), x) == (c + Id - e - If)/(4Ix + 4) + (c - Id - e + If)/(-4Ix + 4) + (c - d + e - f)/(4x + 4) + (c + d + e + f)/(-4x + 4) assert ExpandIntegrand((d + e(f + gx))/(2 + 3x + 1x*2), x) == (-2d - 2ef + 4eg)/(2x + 4) + (2d + 2ef - 2eg)/(2x + 2) assert ExpandIntegrand(x/(ax*3 + bSqrt(c + dx6)), x) == ax4/(-b2c + x6(a2 - b2d)) + bxsqrt(c + dx6)/(b2c + x6(-a2 + b2d)) assert simplify(ExpandIntegrand(x1(1 - x4)(-2), x) - (x/(S(4)(x2 + 1)) + x/(S(4)(x2 + 1)2) - x/(S(4)(x2 - 1)) + x/(S(4)(x2 - 1)2))) == 0 assert simplify(ExpandIntegrand((-1 + xS(6))(-3), x) - (S(3)/(S(8)(x6 - 1)) - S(3)/(S(16)(xS(3) + S(1))S(2)) - S(1)/(S(8)(xS(3) + S(1))S(3)) - S(3)/(S(16)(xS(3) - S(1))S(2)) + S(1)/(S(8)(xS(3) - S(1))S(3)))) == 0 assert simplify(ExpandIntegrand(u1(a + bu2 + cu4)(-1), x)) == simplify(1/(2b(u + sqrt(-(a + cu4)/b))) - 1/(2b(-u + sqrt(-(a + cu*4)/b)))) assert simplify(ExpandIntegrand((1 + 1u + 1u2)(-2), x) - (S(1)/(S(2)(-u - 1)(-u2 - u - 1)) + S(1)/(S(4)(-u - 1)(u + sqrt(-u - 1))2) + S(1)/(S(4)(-u - 1)(u - sqrt(-u - 1))2))) == 0 assert ExpandIntegrand(x(a + bLog(c(d(e + fx)p)q))*n, x) == -e(a + brubi_log(c(d(e + fx)p)q))*n/f + (a + brubi_log(c(d(e + fx)p)q))n(e + fx)/f assert ExpandIntegrand(xf*(e(c + dx)S(1)), x) == f*(e(c + dx))x assert simplify(ExpandIntegrand((x)(a + bx)*mLog(c(d + exn)p), x) - (-a(a + bx)*mrubi_log(c(d + exn)p)/b + (a + bx)(m + S(1))rubi_log(c(d + exn)p)/b)) == 0 assert simplify(ExpandIntegrand(u(a + bFv)S(2)(c + dFv)S(-3), x) - (b*2u/(d*2(F*vd + c)) + 2bu(ad - bc)/(d2(F*vd + c)*2) + u(ad - bc)2/(d2(Fvd + c)*3))) == 0 assert ExpandIntegrand((S(1) + 1x)*S(2)f*(e(1 + S(1)x)n)/(g + hx), x) == f*(e(x + 1)*n)(x + 1)/h + f*(e(x + 1)*n)(-g + h)/h2 + f(e(x + 1)n)(g - h)2/(h2(g + hx)) assert ExpandIntegrand((ac - bcx)2/(a + bx)*2, x) == 4a*2c*2/(a + bx)*2 - 4ac2/(a + bx) + c2 assert simplify(ExpandIntegrand(x2(1 - 1x2)(-2), x) - (1/(S(2)(x2 - 1)) + 1/(S(4)(x + 1)*2) + 1/(S(4)(x - 1)2))) == 0 assert ExpandIntegrand((a + x)2, x) == a*2 + 2ax + x2 assert ExpandIntegrand((a + bx)S(2)/x3, x) == a2/x3 + 2ab/x2 + b2/x assert ExpandIntegrand(1/(x*2(a + bx)2), x) == b2/(a2(a + bx)2) + 1/(a2x*2) + 2b2/(a3(a + bx)) - 2b/(a3x) assert ExpandIntegrand((1 + x)3/x, x) == x2 + 3x + 3 + 1/x assert ExpandIntegrand((1 + 2(3 + 4x2))/(2 + 3x*2 + 1x*4), x) == 18/(2x*2 + 4) - 2/(2x*2 + 2) assert ExpandIntegrand((c + dx*2 + ex*3)/(1 - 1x*4), x) == (c - d - Ie)/(4Ix + 4) + (c - d + Ie)/(-4Ix + 4) + (c + d - e)/(4x + 4) + (c + d + e)/(-4x + 4) assert ExpandIntegrand((a + bx)*2/(c + dx), x) == b(a + bx)/d + b(ad - bc)/d2 + (ad - bc)2/(d2(c + dx)) assert ExpandIntegrand(x2(a + bLog(c(d(e + fx)p)q))n, x) == e2(a + brubi_log(c(d(e + fx)p)q))n/f2 - 2e(a + brubi_log(c(d(e + fx)p)q))n(e + fx)/f2 + (a + brubi_log(c(d(e + fx)p)q))n(e + fx)2/f2 assert ExpandIntegrand(x(1 + 2x)3rubi_log(2(1 + 1x2)1), x) == 8x4rubi_log(2x2 + 2) + 12x*3rubi_log(2x2 + 2) + 6x*2rubi_log(2x2 + 2) + xrubi_log(2x2 + 2) assert ExpandIntegrand((1 + 1x)*S(3)f*(e(1 + 1x)n)/(g + hx), x) == f*(e(x + 1)*n)(x + 1)2/h + f(e(x + 1)n)(-g + h)(x + 1)/h2 + f(e(x + 1)*n)(-g + h)2/h3 - f*(e(x + 1)*n)(g - h)3/(h3(g + hx)) def test_Dist(): assert Dist(x, a + b, x) == ax + bx assert Dist(x, Integral(a + b , x), x) == xIntegral(a + b, x) assert Dist(3x,(a+b), x) - Dist(2x, (a+b), x) == ax + bx assert Dist(3x,(a+b), x) + Dist(2x, (a+b), x) == 5ax + 5bx assert Dist(x, cIntegral((a + b), x), x) == cxIntegral(a + b, x) def test_IntegralFreeQ(): assert not IntegralFreeQ(Integral(a, x)) assert IntegralFreeQ(a + b) def test_OneQ(): from sympy.integrals.rubi.utility_function import OneQ assert OneQ(S(1)) assert not OneQ(S(2)) def test_DerivativeDivides(): assert not DerivativeDivides(x, x, x) assert not DerivativeDivides(a, x + y, b) assert DerivativeDivides(a + x, a, x) == a assert DerivativeDivides(a + b, x + y, b) == x + y def test_LogIntegral(): from sympy.integrals.rubi.utility_function import LogIntegral assert LogIntegral(a) == li(a) def test_SinIntegral(): from sympy.integrals.rubi.utility_function import SinIntegral assert SinIntegral(a) == Si(a) def test_CosIntegral(): from sympy.integrals.rubi.utility_function import CosIntegral assert CosIntegral(a) == Ci(a) def test_SinhIntegral(): from sympy.integrals.rubi.utility_function import SinhIntegral assert SinhIntegral(a) == Shi(a) def test_CoshIntegral(): from sympy.integrals.rubi.utility_function import CoshIntegral assert CoshIntegral(a) == Chi(a) def test_ExpIntegralEi(): from sympy.integrals.rubi.utility_function import ExpIntegralEi assert ExpIntegralEi(a) == Ei(a) def test_ExpIntegralE(): from sympy.integrals.rubi.utility_function import ExpIntegralE assert ExpIntegralE(a, z) == expint(a, z) def test_LogGamma(): from sympy.integrals.rubi.utility_function import LogGamma assert LogGamma(a) == loggamma(a) def test_Factorial(): from sympy.integrals.rubi.utility_function import Factorial assert Factorial(S(5)) == 120 def test_Zeta(): from sympy.integrals.rubi.utility_function import Zeta assert Zeta(a, z) == zeta(a, z) def test_HypergeometricPFQ(): from sympy.integrals.rubi.utility_function import HypergeometricPFQ assert HypergeometricPFQ([a, b], [c], z) == hyper([a, b], [c], z) def test_PolyGamma(): assert PolyGamma(S(2), S(3)) == polygamma(2, 3) def test_ProductLog(): from sympy.core.evalf import N assert N(ProductLog(S(5.0)), 5) == N(1.32672466524220, 5) assert N(ProductLog(S(2), S(3.5)), 5) == N(-1.14064876353898 + 10.8912237027092I, 5) def test_PolynomialQuotient(): assert PolynomialQuotient(rubi_log((-ad + bc)/(b(c + dx)))/(c + dx), a + bx, e) == rubi_log((-ad + bc)/(b(c + dx)))/((a + bx)(c + dx)) assert PolynomialQuotient(x*2, x + a, x) == -a + x def test_PolynomialRemainder(): assert PolynomialRemainder(rubi_log((-ad + bc)/(b(c + dx)))/(c + dx), a + bx, e) == 0 assert PolynomialRemainder(x2, x + a, x) == a2 def test_Floor(): assert Floor(S(7.5)) == 7 assert Floor(S(15.5), S(6)) == 12 def test_Factor(): from sympy.integrals.rubi.utility_function import Factor assert Factor(ab + ac) == a(b + c) def test_Rule(): from sympy.integrals.rubi.utility_function import Rule assert Rule(x, S(5)) == {x: 5} def test_Distribute(): assert Distribute((a + b)c + (a + b)d, Add) == c(a + b) + d(a + b) assert Distribute((a + b)(c + e), Add) == ac + ae + bc + be def test_CoprimeQ(): assert CoprimeQ(S(7), S(5)) assert not CoprimeQ(S(6), S(3)) def test_Discriminant(): from sympy.integrals.rubi.utility_function import Discriminant assert Discriminant(ax*2 + bx + c, x) == b*2 - 4ac assert unchanged(Discriminant, 1/x, x) def test_Sum_doit(): assert Sum_doit(2x + 2, [x, 0, 1.7]) == 6 def test_DeactivateTrig(): assert DeactivateTrig(sec(a + bx), x) == sec(a + bx) def test_Negative(): from sympy.integrals.rubi.utility_function import Negative assert Negative(S(-2)) assert not Negative(S(0)) def test_Quotient(): from sympy.integrals.rubi.utility_function import Quotient assert Quotient(17, 5) == 3 def test_process_trig(): assert process_trig(xcot(x)) == x/tan(x) assert process_trig(coth(x)csc(x)) == S(1)/(tanh(x)sin(x)) def test_replace_pow_exp(): assert replace_pow_exp(rubi_exp(S(5))) == exp(S(5)) def test_rubi_unevaluated_expr(): from sympy.integrals.rubi.utility_function import rubi_unevaluated_expr assert rubi_unevaluated_expr(a)rubi_unevaluated_expr(b) == rubi_unevaluated_expr(b)*rubi_unevaluated_expr(a) def test_rubi_exp(): # class name in utility_function is `exp`. To avoid confusion `rubi_exp` has been used here assert isinstance(rubi_exp(a), Pow) def test_rubi_log(): # class name in utility_function is `log`. To avoid confusion `rubi_log` has been used here assert rubi_log(rubi_exp(S(a))) == a
ca342afb327a38980dda89bd0517abd5524a0d058a87e07d221168252a0a0241	''' Tests for Rubi Algebraic 1.2 rules. Parsed from Maple syntax All tests: http://www.apmaths.uwo.ca/~arich/IntegrationProblems/MapleSyntaxFiles/MapleSyntaxFiles.html Note: Some tests are commented since they depend rules other than Algebraic1.2. ''' import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.functions import log, sqrt, exp, cos, sin, tan, sec, csc, cot from sympy.functions.elementary.hyperbolic import atanh as arctanh from sympy.functions.elementary.hyperbolic import asinh as arcsinh from sympy.functions.elementary.hyperbolic import acosh as arccosh from sympy.functions.elementary.trigonometric import atan as arctan from sympy.functions.elementary.trigonometric import asin as arcsin from sympy.functions.elementary.trigonometric import acos as arccos from sympy.integrals.rubi.utility_function import EllipticE, EllipticF, hypergeom, rubi_test from sympy.core.numbers import (I, pi as Pi) from sympy.core.singleton import S from sympy.core.symbol import symbols from sympy.functions.elementary.exponential import exp_polar from sympy.functions.special.hyper import hyper from sympy.simplify.simplify import simplify a, b, c, d, e, f, m, n, x, u = symbols('a b c d e f m n x u') def test_1(): test = [ [ - S(3)/S(2), x, S(1), - S(3)/S(2)x], [Pi, x, S(1), Pix], [a, x, S(1), ax], [xm, x, S(1), x(S(1) + m)/(S(1) + m)], [xS(100), x, S(1), S(1)/S(101)xS(101)], [x(S(5)/S(2)), x, S(1), S(2)/S(7)x(S(7)/S(2))], [x(S(5)/S(3)), x, S(1), S(3)/S(8)x(S(8)/S(3))], [S(1)/x(S(1)/S(3)), x, S(1), S(3)/S(2)x(S(2)/S(3))], [xS(3)(a + bx), x, S(2), S(1)/S(4)axS(4) + S(1)/S(5)bxS(5)], [(a + bx)S(2)/xS(2), x, S(2), - aS(2)/x + bS(2)x + S(2)ablog(x)], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True) else: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) def test_2(): test = [ [(a + bx)/x, x, S(2), bx + alog(x)], [xS(5)/(a + bx), x, S(2), a*S(4)x/b*S(5) - S(1)/S(2)a*S(3)xS(2)/bS(4) + S(1)/S(3)aS(2)xS(3)/bS(3) - S(1)/S(4)axS(4)/bS(2) + S(1)/S(5)xS(5)/b - aS(5)log(a + bx)/bS(6)], [S(1)/(a + bx)*S(2), x, S(1), ( - S(1))/(b(a + bx))], [S(1)/(x(a + bx)S(3)), x, S(2), S(1)/S(2)/(a(a + bx)S(2)) + S(1)/(aS(2)(a + bx)) + log(x)/aS(3) - log(a + bx)/a*S(3)], [S(1)/(S(2) + S(2)x), x, S(1), S(1)/S(2)log(S(1) + x)], [S(1)/(x(S(1) + bx)), x, S(3), log(x) - log(S(1) + bx)], [x*S(3)sqrt(a + bx), x, S(2), - S(2)/S(3)a*S(3)(a + bx)(S(3)/S(2))/bS(4) + S(6)/S(5)a*S(2)(a + bx)(S(5)/S(2))/bS(4) - S(6)/S(7)a(a + bx)(S(7)/S(2))/bS(4) + S(2)/S(9)(a + bx)(S(9)/S(2))/bS(4)], [(a + bx)(S(3)/S(2)), x, S(1), S(2)/S(5)(a + bx)(S(5)/S(2))/b], [xS(4)/sqrt(a + bx), x, S(2), - S(8)/S(3)aS(3)(a + bx)(S(3)/S(2))/bS(5) + S(12)/S(5)a*S(2)(a + bx)(S(5)/S(2))/bS(5) - S(8)/S(7)a(a + bx)(S(7)/S(2))/bS(5) + S(2)/S(9)(a + bx)(S(9)/S(2))/bS(5) + S(2)aS(4)sqrt(a + bx)/bS(5)], [S(1)/sqrt(a + bx), x, S(1), S(2)sqrt(a + bx)/b], [S(1)/(x(a + bx)*(S(3)/S(2))), x, S(3), - S(2)arctanh(sqrt(a + bx)/sqrt(a))/a(S(3)/S(2)) + S(2)/(asqrt(a + bx))], [S(1)/(xS(2)( - a + bx)(S(3)/S(2))), x, S(4), - S(3)barctan(sqrt( - a + bx)/sqrt(a))/a*(S(5)/S(2)) + ( - S(2))/(axsqrt( - a + bx)) - S(3)sqrt( - a + bx)/(a*S(2)x)], [x*S(3)(a + bx)(S(1)/S(3)), x, S(2), - S(3)/S(4)a*S(3)(a + bx)(S(4)/S(3))/bS(4) + S(9)/S(7)a*S(2)(a + bx)(S(7)/S(3))/bS(4) - S(9)/S(10)a(a + bx)(S(10)/S(3))/bS(4) + S(3)/S(13)(a + bx)(S(13)/S(3))/bS(4)], [x*S(2)(a + bx)(S(2)/S(3)), x, S(2), S(3)/S(5)a*S(2)(a + bx)(S(5)/S(3))/bS(3) - S(3)/S(4)a(a + bx)(S(8)/S(3))/bS(3) + S(3)/S(11)(a + bx)(S(11)/S(3))/bS(3)], [x*S(2)/(a + bx)*(S(1)/S(3)), x, S(2), S(3)/S(2)a*S(2)(a + bx)(S(2)/S(3))/bS(3) - S(6)/S(5)a(a + bx)(S(5)/S(3))/bS(3) + S(3)/S(8)(a + bx)(S(8)/S(3))/bS(3)], [x*S(3)/( - a + bx)*(S(1)/S(3)), x, S(2), S(3)/S(2)a*S(3)( - a + bx)(S(2)/S(3))/bS(4) + S(9)/S(5)a*S(2)( - a + bx)(S(5)/S(3))/bS(4) + S(9)/S(8)a( - a + bx)(S(8)/S(3))/bS(4) + S(3)/S(11)( - a + bx)(S(11)/S(3))/bS(4)], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True) else: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) def test_3(): test = [ [x*m(a + bx), x, S(2), ax*(S(1) + m)/(S(1) + m) + bx(S(2) + m)/(S(2) + m)], [x(S(5)/S(2))(a + bx), x, S(2), S(2)/S(7)ax*(S(7)/S(2)) + S(2)/S(9)bx(S(9)/S(2))], [x(S(5)/S(2))/(a + bx), x, S(5), - S(2)/S(3)ax(S(3)/S(2))/bS(2) + S(2)/S(5)x(S(5)/S(2))/b - S(2)a*(S(5)/S(2))arctan(sqrt(b)sqrt(x)/sqrt(a))/b(S(7)/S(2)) + S(2)a*S(2)sqrt(x)/bS(3)], [x(S(3)/S(2))/(a + bx), x, S(4), S(2)/S(3)x*(S(3)/S(2))/b + S(2)a*(S(3)/S(2))arctan(sqrt(b)sqrt(x)/sqrt(a))/b(S(5)/S(2)) - S(2)asqrt(x)/bS(2)], [x(S(5)/S(2))/( - a + bx), x, S(5), S(2)/S(3)ax(S(3)/S(2))/bS(2) + S(2)/S(5)x(S(5)/S(2))/b - S(2)a*(S(5)/S(2))arctanh(sqrt(b)sqrt(x)/sqrt(a))/b(S(7)/S(2)) + S(2)a*S(2)sqrt(x)/bS(3)], [x(S(5)/S(2))sqrt(a + bx), x, S(6), - S(5)/S(64)aS(4)arctanh(sqrt(b)sqrt(x)/sqrt(a + bx))/b*(S(7)/S(2)) - S(5)/S(96)a*S(2)x*(S(3)/S(2))sqrt(a + bx)/bS(2) + S(1)/S(24)ax(S(5)/S(2))sqrt(a + bx)/b + S(1)/S(4)x*(S(7)/S(2))sqrt(a + bx) + S(5)/S(64)a*S(3)sqrt(x)sqrt(a + bx)/bS(3)], [x(S(3)/S(2))sqrt(a + bx), x, S(5), S(1)/S(8)aS(3)arctanh(sqrt(b)sqrt(x)/sqrt(a + bx))/b*(S(5)/S(2)) + S(1)/S(12)ax(S(3)/S(2))sqrt(a + bx)/b + S(1)/S(3)x*(S(5)/S(2))sqrt(a + bx) - S(1)/S(8)a*S(2)sqrt(x)sqrt(a + bx)/bS(2)], [x(S(5)/S(2))/sqrt(a + bx), x, S(5), - S(5)/S(8)a*S(3)arctanh(sqrt(b)sqrt(x)/sqrt(a + bx))/b*(S(7)/S(2)) - S(5)/S(12)ax(S(3)/S(2))sqrt(a + bx)/bS(2) + S(1)/S(3)x*(S(5)/S(2))sqrt(a + bx)/b + S(5)/S(8)a*S(2)sqrt(x)sqrt(a + bx)/b*S(3)], [sqrt(x)/sqrt(a + bx), x, S(3), - aarctanh(sqrt(b)sqrt(x)/sqrt(a + bx))/b(S(3)/S(2)) + sqrt(x)sqrt(a + bx)/b], [x(S(2)/S(3))(a + bx), x, S(2), S(3)/S(5)ax(S(5)/S(3)) + S(3)/S(8)bx(S(8)/S(3))], [x(S(1)/S(3))(a + bx), x, S(2), S(3)/S(4)ax(S(4)/S(3)) + S(3)/S(7)bx(S(7)/S(3))], [x(S(5)/S(3))/(a + bx), x, S(6), - S(3)/S(2)ax(S(2)/S(3))/bS(2) + S(3)/S(5)x(S(5)/S(3))/b - S(3)/S(2)a*(S(5)/S(3))log(a(S(1)/S(3)) + b(S(1)/S(3))x(S(1)/S(3)))/b(S(8)/S(3)) + S(1)/S(2)a*(S(5)/S(3))log(a + bx)/b(S(8)/S(3)) - a(S(5)/S(3))arctan((a*(S(1)/S(3)) - S(2)b*(S(1)/S(3))x(S(1)/S(3)))/(a(S(1)/S(3))sqrt(S(3))))sqrt(S(3))/b(S(8)/S(3))], [x(S(4)/S(3))/(a + bx), x, S(6), - S(3)ax(S(1)/S(3))/bS(2) + S(3)/S(4)x*(S(4)/S(3))/b + S(3)/S(2)a*(S(4)/S(3))log(a(S(1)/S(3)) + b(S(1)/S(3))x(S(1)/S(3)))/b(S(7)/S(3)) - S(1)/S(2)a*(S(4)/S(3))log(a + bx)/b(S(7)/S(3)) - a(S(4)/S(3))arctan((a*(S(1)/S(3)) - S(2)b*(S(1)/S(3))x(S(1)/S(3)))/(a(S(1)/S(3))sqrt(S(3))))sqrt(S(3))/b(S(7)/S(3))], [(S(1) - x)(S(1)/S(4))/(S(1) + x), x, S(5), S(4)(S(1) - x)(S(1)/S(4)) - S(2)S(2)*(S(1)/S(4))arctan((S(1) - x)(S(1)/S(4))/S(2)(S(1)/S(4))) - S(2)S(2)(S(1)/S(4))arctanh((S(1) - x)(S(1)/S(4))/S(2)(S(1)/S(4)))], [x*m(a + bx)S(2), x, S(2), aS(2)x*(S(1) + m)/(S(1) + m) + S(2)abx(S(2) + m)/(S(2) + m) + bS(2)x(S(3) + m)/(S(3) + m)], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True) else: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) def test_4(): test = [ [xm/(a + bx)S(2), x, S(1), x(S(1) + m)hypergeom([S(2), S(1) + m], [S(2) + m], - bx/a)/(a*S(2)(S(1) + m))], [x*m/sqrt(S(2) + S(3)x), x, S(1), x*(S(1) + m)hypergeom([S(1)/S(2), S(1) + m], [S(2) + m], - S(3)/S(2)x)/((S(1) + m)sqrt(S(2)))], [x*m(a + bx)n, x, S(2), x(S(1) + m)(a + bx)nhypergeom([S(1) + m, - n], [S(2) + m], - bx/a)/((S(1) + m)(S(1) + bx/a)n)], [x( - S(1) + n)/(a + bx)n, x, S(2), xn(S(1) + bx/a)*nhypergeom([n, n], [S(1) + n], - bx/a)/(n(a + bx)n)], [(c + d(a + bx))(S(5)/S(2)), x, S(2), S(2)/S(7)(c + d(a + bx))*(S(7)/S(2))/(bd)], [(c + d(a + bx))*(S(3)/S(2)), x, S(2), S(2)/S(5)(c + d(a + bx))*(S(5)/S(2))/(bd)], [(a + bx)S(3)/(ad/b + dx)S(3), x, S(2), bS(3)x/d*S(3)], [(a + bx)(ac - bcx)*S(3), x, S(2), - S(1)/S(2)acS(3)(a - bx)S(4)/b + S(1)/S(5)c*S(3)(a - bx)S(5)/b], [(ac - bcx)*S(3)/(a + bx), x, S(2), - S(4)aS(2)c*S(3)x + acS(3)(a - bx)S(2)/b + S(1)/S(3)c*S(3)(a - bx)S(3)/b + S(8)a*S(3)c*S(3)log(a + bx)/b], [S(1)/((a + bx)*S(2)(ac - bcx)), x, S(3), ( - S(1)/S(2))/(abc(a + bx)) + S(1)/S(2)arctanh(bx/a)/(aS(2)bc)], [(S(1) + x)(S(1)/S(2))/(S(1) - x)(S(9)/S(2)), x, S(3), S(1)/S(7)(S(1) + x)(S(3)/S(2))/(S(1) - x)(S(7)/S(2)) + S(2)/S(35)(S(1) + x)(S(3)/S(2))/(S(1) - x)(S(5)/S(2)) + S(2)/S(105)(S(1) + x)(S(3)/S(2))/(S(1) - x)(S(3)/S(2))], [(S(1) + x)(S(5)/S(2))/(S(1) - x)(S(1)/S(2)), x, S(5), S(5)/S(2)arcsin(x) - S(5)/S(6)(S(1) + x)*(S(3)/S(2))sqrt(S(1) - x) - S(1)/S(3)(S(1) + x)(S(5)/S(2))sqrt(S(1) - x) - S(5)/S(2)sqrt(S(1) - x)sqrt(S(1) + x)], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True) else: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) def test_5(): test = [ [(S(1) + ax)(S(3)/S(2))/sqrt(S(1) - ax), x, S(4), S(3)/S(2)arcsin(ax)/a - S(1)/S(2)(S(1) + ax)*(S(3)/S(2))sqrt(S(1) - ax)/a - S(3)/S(2)sqrt(S(1) - ax)sqrt(S(1) + ax)/a], [(S(1) - x)(S(1)/S(2))/(S(1) + x)(S(1)/S(2)), x, S(3), arcsin(x) + sqrt(S(1) - x)sqrt(S(1) + x)], [S(1)/((S(1) - x)*(S(1)/S(2))(S(1) + x)*(S(3)/S(2))), x, S(1), - sqrt(S(1) - x)/sqrt(S(1) + x)], [(a + ax)*(S(5)/S(2))(c - cx)(S(5)/S(2)), x, S(5), S(5)/S(24)acx(a + ax)*(S(3)/S(2))(c - cx)(S(3)/S(2)) + S(1)/S(6)x(a + ax)*(S(5)/S(2))(c - cx)(S(5)/S(2)) + S(5)/S(8)a*(S(5)/S(2))c*(S(5)/S(2))arctan(sqrt(c)sqrt(a + ax)/(sqrt(a)sqrt(c - cx))) + S(5)/S(16)aS(2)c*S(2)xsqrt(a + ax)sqrt(c - cx)], [S(1)/((a + ax)(S(5)/S(2))(c - cx)(S(5)/S(2))), x, S(2), S(1)/S(3)x/(ac(a + ax)(S(3)/S(2))(c - cx)(S(3)/S(2))) + S(2)/S(3)x/(a*S(2)c*S(2)sqrt(a + ax)sqrt(c - cx))], [(S(3) - x)(S(1)/S(2))( - S(2) + x)*(S(1)/S(2)), x, S(5), - S(1)/S(8)arcsin(S(5) - S(2)x) - S(1)/S(2)(S(3) - x)*(S(3)/S(2))sqrt( - S(2) + x) + S(1)/S(4)sqrt(S(3) - x)sqrt( - S(2) + x)], [S(1)/(sqrt(a + bx)sqrt( - ad + bdx)), x, S(2), S(2)arctanh(sqrt(d)sqrt(a + bx)/sqrt( - ad + bdx))/(bsqrt(d))], [S(1)/((a - Iax)*(S(7)/S(4))(a + Iax)*(S(1)/S(4))), x, S(1), - S(2)/S(3)I(a + Iax)(S(3)/S(4))/(aS(2)(a - Iax)*(S(3)/S(4)))], [(a + bx)*S(2)(ac - bcx)n, x, S(2), - S(4)a*S(2)(ac - bcx)(S(1) + n)/(bc(S(1) + n)) + S(4)a(ac - bcx)*(S(2) + n)/(bc*S(2)(S(2) + n)) - (ac - bcx)(S(3) + n)/(bc*S(3)(S(3) + n))], [(a + bx)S(4)(c + dx), x, S(2), S(1)/S(5)(bc - ad)(a + bx)S(5)/bS(2) + S(1)/S(6)d(a + bx)S(6)/bS(2)], [(a + bx)(c + dx), x, S(2), acx + S(1)/S(2)(bc + ad)x*S(2) + S(1)/S(3)bdx*S(3)], [(a + bx)*S(5)/(c + dx), x, S(2), b(bc - ad)S(4)x/d*S(5) - S(1)/S(2)(bc - ad)*S(3)(a + bx)S(2)/dS(4) + S(1)/S(3)(bc - ad)*S(2)(a + bx)S(3)/dS(3) - S(1)/S(4)(bc - ad)(a + bx)S(4)/dS(2) + S(1)/S(5)(a + bx)*S(5)/d - (bc - ad)S(5)log(c + dx)/dS(6)], [(a + bx)/(c + dx)S(3), x, S(1), S(1)/S(2)(a + bx)S(2)/((bc - ad)(c + dx)S(2))], [(a + bx)*S(5)(c + dx)(S(1)/S(2)), x, S(2), - S(2)/S(3)(bc - ad)*S(5)(c + dx)(S(3)/S(2))/dS(6) + S(2)b(bc - ad)S(4)(c + dx)(S(5)/S(2))/dS(6) - S(20)/S(7)b*S(2)(bc - ad)*S(3)(c + dx)(S(7)/S(2))/dS(6) + S(20)/S(9)b*S(3)(bc - ad)*S(2)(c + dx)(S(9)/S(2))/dS(6) - S(10)/S(11)b*S(4)(bc - ad)(c + dx)(S(11)/S(2))/dS(6) + S(2)/S(13)bS(5)(c + dx)(S(13)/S(2))/dS(6)], [(c + dx)*(S(1)/S(2))/(a + bx)*S(2), x, S(3), - darctanh(sqrt(b)sqrt(c + dx)/sqrt(bc - ad))/(b*(S(3)/S(2))sqrt(bc - ad)) - sqrt(c + dx)/(b(a + bx))], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True) else: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) def test_6(): test = [ [(S(1) + ax)*(S(3)/S(2))/sqrt(S(1) - ax), x, S(4), S(3)/S(2)arcsin(ax)/a - S(1)/S(2)(S(1) + ax)*(S(3)/S(2))sqrt(S(1) - ax)/a - S(3)/S(2)sqrt(S(1) - ax)sqrt(S(1) + ax)/a], [(S(1) - x)(S(1)/S(2))/(S(1) + x)(S(1)/S(2)), x, S(3), arcsin(x) + sqrt(S(1) - x)sqrt(S(1) + x)], [S(1)/((S(1) - x)*(S(1)/S(2))(S(1) + x)*(S(3)/S(2))), x, S(1), - sqrt(S(1) - x)/sqrt(S(1) + x)], [(a + ax)*(S(5)/S(2))(c - cx)(S(5)/S(2)), x, S(5), S(5)/S(24)acx(a + ax)*(S(3)/S(2))(c - cx)(S(3)/S(2)) + S(1)/S(6)x(a + ax)*(S(5)/S(2))(c - cx)(S(5)/S(2)) + S(5)/S(8)a*(S(5)/S(2))c*(S(5)/S(2))arctan(sqrt(c)sqrt(a + ax)/(sqrt(a)sqrt(c - cx))) + S(5)/S(16)aS(2)c*S(2)xsqrt(a + ax)sqrt(c - cx)], [S(1)/((a + ax)(S(5)/S(2))(c - cx)(S(5)/S(2))), x, S(2), S(1)/S(3)x/(ac(a + ax)(S(3)/S(2))(c - cx)(S(3)/S(2))) + S(2)/S(3)x/(a*S(2)c*S(2)sqrt(a + ax)sqrt(c - cx))], [(S(3) - x)(S(1)/S(2))( - S(2) + x)*(S(1)/S(2)), x, S(5), - S(1)/S(8)arcsin(S(5) - S(2)x) - S(1)/S(2)(S(3) - x)*(S(3)/S(2))sqrt( - S(2) + x) + S(1)/S(4)sqrt(S(3) - x)sqrt( - S(2) + x)], [S(1)/(sqrt(a + bx)sqrt( - ad + bdx)), x, S(2), S(2)arctanh(sqrt(d)sqrt(a + bx)/sqrt( - ad + bdx))/(bsqrt(d))], [S(1)/((a - Iax)*(S(7)/S(4))(a + Iax)*(S(1)/S(4))), x, S(1), - S(2)/S(3)I(a + Iax)(S(3)/S(4))/(aS(2)(a - Iax)*(S(3)/S(4)))], [(a + bx)*S(2)(ac - bcx)n, x, S(2), - S(4)a*S(2)(ac - bcx)(S(1) + n)/(bc(S(1) + n)) + S(4)a(ac - bcx)*(S(2) + n)/(bc*S(2)(S(2) + n)) - (ac - bcx)(S(3) + n)/(bc*S(3)(S(3) + n))], [(a + bx)S(4)(c + dx), x, S(2), S(1)/S(5)(bc - ad)(a + bx)S(5)/bS(2) + S(1)/S(6)d(a + bx)S(6)/bS(2)], [(a + bx)(c + dx), x, S(2), acx + S(1)/S(2)(bc + ad)x*S(2) + S(1)/S(3)bdx*S(3)], [(a + bx)*S(5)/(c + dx), x, S(2), b(bc - ad)S(4)x/d*S(5) - S(1)/S(2)(bc - ad)*S(3)(a + bx)S(2)/dS(4) + S(1)/S(3)(bc - ad)*S(2)(a + bx)S(3)/dS(3) - S(1)/S(4)(bc - ad)(a + bx)S(4)/dS(2) + S(1)/S(5)(a + bx)*S(5)/d - (bc - ad)S(5)log(c + dx)/dS(6)], [(a + bx)/(c + dx)S(3), x, S(1), S(1)/S(2)(a + bx)S(2)/((bc - ad)(c + dx)S(2))], [(a + bx)*S(5)(c + dx)(S(1)/S(2)), x, S(2), - S(2)/S(3)(bc - ad)*S(5)(c + dx)(S(3)/S(2))/dS(6) + S(2)b(bc - ad)S(4)(c + dx)(S(5)/S(2))/dS(6) - S(20)/S(7)b*S(2)(bc - ad)*S(3)(c + dx)(S(7)/S(2))/dS(6) + S(20)/S(9)b*S(3)(bc - ad)*S(2)(c + dx)(S(9)/S(2))/dS(6) - S(10)/S(11)b*S(4)(bc - ad)(c + dx)(S(11)/S(2))/dS(6) + S(2)/S(13)bS(5)(c + dx)(S(13)/S(2))/dS(6)], [(c + dx)*(S(1)/S(2))/(a + bx)*S(2), x, S(3), - darctanh(sqrt(b)sqrt(c + dx)/sqrt(bc - ad))/(b*(S(3)/S(2))sqrt(bc - ad)) - sqrt(c + dx)/(b(a + bx))], [(a + bx)*S(4)/(c + dx)*(S(1)/S(2)), x, S(2), - S(8)/S(3)b(bc - ad)S(3)(c + dx)(S(3)/S(2))/dS(5) + S(12)/S(5)b*S(2)(bc - ad)*S(2)(c + dx)(S(5)/S(2))/dS(5) - S(8)/S(7)b*S(3)(bc - ad)(c + dx)(S(7)/S(2))/dS(5) + S(2)/S(9)bS(4)(c + dx)(S(9)/S(2))/dS(5) + S(2)(bc - ad)*S(4)sqrt(c + dx)/dS(5)], [(a + bx)*S(2)/(c + dx)*(S(1)/S(2)), x, S(2), - S(4)/S(3)b(bc - ad)(c + dx)(S(3)/S(2))/dS(3) + S(2)/S(5)b*S(2)(c + dx)(S(5)/S(2))/dS(3) + S(2)(bc - ad)*S(2)sqrt(c + dx)/dS(3)], [(S(1) - x)(S(1)/S(3))/(S(1) + x), x, S(5), S(3)(S(1) - x)*(S(1)/S(3)) + S(3)log(S(2)(S(1)/S(3)) - (S(1) - x)(S(1)/S(3)))/S(2)(S(2)/S(3)) - log(S(1) + x)/S(2)(S(2)/S(3)) - S(2)*(S(1)/S(3))arctan((S(1) + S(2)*(S(2)/S(3))(S(1) - x)*(S(1)/S(3)))/sqrt(S(3)))sqrt(S(3))], [(c + dx)(S(1)/S(2))/(a + bx)*(S(1)/S(2)), x, S(3), (bc - ad)arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(3)/S(2))sqrt(d)) + sqrt(a + bx)sqrt(c + dx)/b], [(a + bx)*(S(1)/S(2))(c + dx)(S(3)/S(2)), x, S(5), S(1)/S(3)(a + bx)(S(3)/S(2))(c + dx)(S(3)/S(2))/b - S(1)/S(8)(bc - ad)*S(3)arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(5)/S(2))d*(S(3)/S(2))) + S(1)/S(4)(bc - ad)(a + bx)*(S(3)/S(2))sqrt(c + dx)/bS(2) + S(1)/S(8)(bc - ad)*S(2)sqrt(a + bx)sqrt(c + dx)/(bS(2)d)], [(a + bx)(S(1)/S(2))/(c + dx)*(S(1)/S(2)), x, S(3), - (bc - ad)arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(d*(S(3)/S(2))sqrt(b)) + sqrt(a + bx)sqrt(c + dx)/d], [S(1)/((a + bx)*(S(1)/S(2))(c + dx)(S(5)/S(2))), x, S(2), S(2)/S(3)sqrt(a + bx)/((bc - ad)(c + dx)(S(3)/S(2))) + S(4)/S(3)bsqrt(a + bx)/((bc - ad)*S(2)sqrt(c + dx))], [(a + bx)*m(c + dx)(S(1) + S(2)n - S(2)(S(1) + n)), x, S(2), (a + bx)*(S(1) + m)hypergeom([S(1), S(1) + m], [S(2) + m], - d(a + bx)/(bc - ad))/((bc - ad)(S(1) + m))], [a + bx + cxS(2) + dx*S(3), x, S(1), ax + S(1)/S(2)bx*S(2) + S(1)/S(3)cxS(3) + S(1)/S(4)dxS(4)], [a + d/xS(3) + c/xS(2) + b/x, x, S(1), - S(1)/S(2)d/x*S(2) - c/x + ax + blog(x)], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True) else: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) def test_7(): test = [ #[(a + bx)*(S(3)/S(2))(c + dx)(S(1)/S(3)), x, S(5), S(12)/S(187)(bc - ad)(a + bx)*(S(3)/S(2))(c + dx)(S(1)/S(3))/(bd) + S(6)/S(17)(a + bx)*(S(5)/S(2))(c + dx)(S(1)/S(3))/b - S(108)/S(935)(bc - ad)*S(2)(c + dx)(S(1)/S(3))sqrt(a + bx)/(bd*S(2)) - S(108)/S(935)S(3)*(S(3)/S(4))(bc - ad)*S(3)((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3)))EllipticF(( - b*(S(1)/S(3))(c + dx)(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) + sqrt(S(3))))/( - b*(S(1)/S(3))(c + dx)(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) - sqrt(S(3)))), sqrt( - S(7) + S(4)sqrt(S(3))))sqrt(((bc - ad)(S(2)/S(3)) + b(S(1)/S(3))(bc - ad)(S(1)/S(3))(c + dx)(S(1)/S(3)) + b(S(2)/S(3))(c + dx)(S(2)/S(3)))/( - b(S(1)/S(3))(c + dx)(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) - sqrt(S(3))))*S(2))sqrt(S(2) - sqrt(S(3)))/(b*(S(4)/S(3))d*S(3)sqrt(a - bc/d + b(c + dx)/d)sqrt( - (bc - ad)*(S(1)/S(3))((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)(S(1)/S(3)))/( - b(S(1)/S(3))(c + dx)*(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) - sqrt(S(3))))*S(2)))], #[(a + bx)*(S(3)/S(2))/(c + dx)*(S(1)/S(3)), x, S(6), S(6)/S(13)(a + bx)(S(3)/S(2))(c + dx)(S(2)/S(3))/d - S(54)/S(91)(bc - ad)(c + dx)*(S(2)/S(3))sqrt(a + bx)/dS(2) - S(162)/S(91)(bc - ad)*S(2)sqrt(a - bc/d + b(c + dx)/d)/(b(S(2)/S(3))d*S(2)( - b*(S(1)/S(3))(c + dx)(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) - sqrt(S(3))))) - S(54)/S(91)S(3)(S(3)/S(4))(bc - ad)*(S(7)/S(3))((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3)))EllipticF(( - b*(S(1)/S(3))(c + dx)(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) + sqrt(S(3))))/( - b*(S(1)/S(3))(c + dx)(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) - sqrt(S(3)))), sqrt( - S(7) + S(4)sqrt(S(3))))sqrt(S(2))sqrt(((bc - ad)(S(2)/S(3)) + b(S(1)/S(3))(bc - ad)*(S(1)/S(3))(c + dx)(S(1)/S(3)) + b(S(2)/S(3))(c + dx)(S(2)/S(3)))/( - b(S(1)/S(3))(c + dx)(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) - sqrt(S(3))))S(2))/(b(S(2)/S(3))dS(3)sqrt(a - bc/d + b(c + dx)/d)sqrt( - (bc - ad)*(S(1)/S(3))((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)(S(1)/S(3)))/( - b(S(1)/S(3))(c + dx)*(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) - sqrt(S(3))))*S(2))) + S(81)/S(91)S(3)*(S(1)/S(4))(bc - ad)*(S(7)/S(3))((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3)))EllipticE(( - b*(S(1)/S(3))(c + dx)(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) + sqrt(S(3))))/( - b*(S(1)/S(3))(c + dx)(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) - sqrt(S(3)))), sqrt( - S(7) + S(4)sqrt(S(3))))sqrt(((bc - ad)(S(2)/S(3)) + b(S(1)/S(3))(bc - ad)(S(1)/S(3))(c + dx)(S(1)/S(3)) + b(S(2)/S(3))(c + dx)(S(2)/S(3)))/( - b(S(1)/S(3))(c + dx)(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) - sqrt(S(3))))*S(2))sqrt(S(2) + sqrt(S(3)))/(b*(S(2)/S(3))d*S(3)sqrt(a - bc/d + b(c + dx)/d)sqrt( - (bc - ad)*(S(1)/S(3))((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)(S(1)/S(3)))/( - b(S(1)/S(3))(c + dx)*(S(1)/S(3)) + (bc - ad)(S(1)/S(3))(S(1) - sqrt(S(3))))*S(2)))], [(a + bx)*(S(2)/S(3))(c + dx)(S(1)/S(3)), x, S(3), S(1)/S(6)(bc - ad)(a + bx)*(S(2)/S(3))(c + dx)(S(1)/S(3))/(bd) + S(1)/S(2)(a + bx)*(S(5)/S(3))(c + dx)(S(1)/S(3))/b + S(1)/S(18)(bc - ad)*S(2)log(c + dx)/(b(S(4)/S(3))d*(S(5)/S(3))) + S(1)/S(6)(bc - ad)*S(2)log( - S(1) + d*(S(1)/S(3))(a + bx)(S(1)/S(3))/(b(S(1)/S(3))(c + dx)(S(1)/S(3))))/(b(S(4)/S(3))d*(S(5)/S(3))) + S(1)/S(3)(bc - ad)*S(2)arctan(S(1)/sqrt(S(3)) + S(2)d(S(1)/S(3))(a + bx)(S(1)/S(3))/(b(S(1)/S(3))(c + dx)(S(1)/S(3))sqrt(S(3))))/(b*(S(4)/S(3))d*(S(5)/S(3))sqrt(S(3)))], [(a + bx)(S(4)/S(3))/(c + dx)*(S(1)/S(3)), x, S(3), - S(2)/S(3)(bc - ad)(a + bx)*(S(1)/S(3))(c + dx)(S(2)/S(3))/dS(2) + S(1)/S(2)(a + bx)(S(4)/S(3))(c + dx)(S(2)/S(3))/d - S(1)/S(9)(bc - ad)*S(2)log(a + bx)/(b(S(2)/S(3))d*(S(7)/S(3))) - S(1)/S(3)(bc - ad)*S(2)log( - S(1) + b*(S(1)/S(3))(c + dx)(S(1)/S(3))/(d(S(1)/S(3))(a + bx)(S(1)/S(3))))/(b(S(2)/S(3))d*(S(7)/S(3))) - S(2)/S(3)(bc - ad)*S(2)arctan(S(1)/sqrt(S(3)) + S(2)b(S(1)/S(3))(c + dx)(S(1)/S(3))/(d(S(1)/S(3))(a + bx)(S(1)/S(3))sqrt(S(3))))/(b*(S(2)/S(3))d*(S(7)/S(3))sqrt(S(3)))], #[(a + bx)(S(5)/S(2))/(c + dx)*(S(1)/S(4)), x, S(10), - S(40)/S(117)(bc - ad)(a + bx)*(S(3)/S(2))(c + dx)(S(3)/S(4))/dS(2) + S(4)/S(13)(a + bx)(S(5)/S(2))(c + dx)(S(3)/S(4))/d + S(16)/S(39)(bc - ad)*S(2)(c + dx)(S(3)/S(4))sqrt(a + bx)/dS(3) - S(32)/S(39)(bc - ad)*(S(15)/S(4))EllipticE(b*(S(1)/S(4))(c + dx)(S(1)/S(4))/(bc - ad)(S(1)/S(4)), I)sqrt(S(1) - b(c + dx)/(bc - ad))/(b*(S(3)/S(4))d*S(4)sqrt(a - bc/d + b(c + dx)/d)) + S(32)/S(39)(bc - ad)*(S(15)/S(4))EllipticF(b*(S(1)/S(4))(c + dx)(S(1)/S(4))/(bc - ad)(S(1)/S(4)), I)sqrt(S(1) - b(c + dx)/(bc - ad))/(b*(S(3)/S(4))d*S(4)sqrt(a - bc/d + b(c + dx)/d))], [(c + dx)*(S(5)/S(4))/(a + bx)*(S(25)/S(4)), x, S(4), - S(4)/S(21)(c + dx)(S(9)/S(4))/((bc - ad)(a + bx)(S(21)/S(4))) + S(16)/S(119)d(c + dx)*(S(9)/S(4))/((bc - ad)S(2)(a + bx)(S(17)/S(4))) - S(128)/S(1547)d*S(2)(c + dx)(S(9)/S(4))/((bc - ad)S(3)(a + bx)(S(13)/S(4))) + S(512)/S(13923)d*S(3)(c + dx)(S(9)/S(4))/((bc - ad)S(4)(a + bx)(S(9)/S(4)))], [(a + bx)*(S(5)/S(4))/(c + dx)*(S(1)/S(4)), x, S(6), - S(5)/S(8)(bc - ad)(a + bx)*(S(1)/S(4))(c + dx)(S(3)/S(4))/dS(2) + S(1)/S(2)(a + bx)(S(5)/S(4))(c + dx)(S(3)/S(4))/d + S(5)/S(16)(bc - ad)*S(2)arctan(d*(S(1)/S(4))(a + bx)(S(1)/S(4))/(b(S(1)/S(4))(c + dx)(S(1)/S(4))))/(b(S(3)/S(4))d*(S(9)/S(4))) + S(5)/S(16)(bc - ad)*S(2)arctanh(d*(S(1)/S(4))(a + bx)(S(1)/S(4))/(b(S(1)/S(4))(c + dx)(S(1)/S(4))))/(b(S(3)/S(4))d*(S(9)/S(4)))], [S(1)/((a + bx)*(S(3)/S(4))(c + dx)(S(1)/S(4))), x, S(4), S(2)arctan(d*(S(1)/S(4))(a + bx)(S(1)/S(4))/(b(S(1)/S(4))(c + dx)(S(1)/S(4))))/(b(S(3)/S(4))d*(S(1)/S(4))) + S(2)arctanh(d*(S(1)/S(4))(a + bx)(S(1)/S(4))/(b(S(1)/S(4))(c + dx)(S(1)/S(4))))/(b(S(3)/S(4))d*(S(1)/S(4)))], #[(a + bx)*(S(3)/S(2))/(c + dx)*(S(1)/S(5)), x, S(2), S(2)/S(5)(a + bx)(S(5)/S(2))(b(c + dx)/(bc - ad))*(S(1)/S(5))hypergeom([S(1)/S(5), S(5)/S(2)], [S(7)/S(2)], - d(a + bx)/(bc - ad))/(b(c + dx)*(S(1)/S(5)))], #[(a + bx)*(S(5)/S(2))/(c + dx)*(S(1)/S(6)), x, S(7), - S(9)/S(28)(bc - ad)(a + bx)*(S(3)/S(2))(c + dx)(S(5)/S(6))/dS(2) + S(3)/S(10)(a + bx)(S(5)/S(2))(c + dx)(S(5)/S(6))/d + S(81)/S(224)(bc - ad)*S(2)(c + dx)(S(5)/S(6))sqrt(a + bx)/dS(3) + S(243)/S(448)(bc - ad)*S(3)(c + dx)(S(1)/S(6))(S(1) + sqrt(S(3)))sqrt(a - bc/d + b(c + dx)/d)/(b*(S(2)/S(3))d*S(3)((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3))(S(1) + sqrt(S(3))))) + S(243)/S(448)S(3)(S(1)/S(4))(bc - ad)*(S(10)/S(3))(c + dx)(S(1)/S(6))((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3)))sqrt(cos(arccos(((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3))(S(1) - sqrt(S(3))))/((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3))(S(1) + sqrt(S(3))))))*S(2))/cos(arccos(((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)(S(1)/S(3))(S(1) - sqrt(S(3))))/((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3))(S(1) + sqrt(S(3))))))EllipticE(sin(arccos(((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)(S(1)/S(3))(S(1) - sqrt(S(3))))/((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3))(S(1) + sqrt(S(3)))))), sqrt(S(1)/S(4)(S(2) + sqrt(S(3)))))sqrt(((bc - ad)(S(2)/S(3)) + b(S(1)/S(3))(bc - ad)(S(1)/S(3))(c + dx)(S(1)/S(3)) + b(S(2)/S(3))(c + dx)(S(2)/S(3)))/((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)(S(1)/S(3))(S(1) + sqrt(S(3))))S(2))/(b(S(2)/S(3))dS(4)sqrt(a - bc/d + b(c + dx)/d)sqrt( - b*(S(1)/S(3))(c + dx)(S(1)/S(3))((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3)))/((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)(S(1)/S(3))(S(1) + sqrt(S(3))))*S(2))) + S(81)/S(896)S(3)*(S(3)/S(4))(bc - ad)*(S(10)/S(3))(c + dx)(S(1)/S(6))((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3)))sqrt(cos(arccos(((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3))(S(1) - sqrt(S(3))))/((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3))(S(1) + sqrt(S(3))))))*S(2))/cos(arccos(((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)(S(1)/S(3))(S(1) - sqrt(S(3))))/((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3))(S(1) + sqrt(S(3))))))EllipticF(sin(arccos(((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)(S(1)/S(3))(S(1) - sqrt(S(3))))/((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3))(S(1) + sqrt(S(3)))))), sqrt(S(1)/S(4)(S(2) + sqrt(S(3)))))(S(1) - sqrt(S(3)))sqrt(((bc - ad)(S(2)/S(3)) + b(S(1)/S(3))(bc - ad)*(S(1)/S(3))(c + dx)(S(1)/S(3)) + b(S(2)/S(3))(c + dx)(S(2)/S(3)))/((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)(S(1)/S(3))(S(1) + sqrt(S(3))))S(2))/(b(S(2)/S(3))dS(4)sqrt(a - bc/d + b(c + dx)/d)sqrt( - b*(S(1)/S(3))(c + dx)(S(1)/S(3))((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)*(S(1)/S(3)))/((bc - ad)(S(1)/S(3)) - b(S(1)/S(3))(c + dx)(S(1)/S(3))(S(1) + sqrt(S(3))))*S(2)))], #[(a + bx)*m(c + dx)n, x, S(2), - (a + bx)*(S(1) + m)(c + dx)(S(1) + n)hypergeom([S(1), S(2) + m + n], [S(2) + n], b(c + dx)/(bc - ad))/((bc - ad)(S(1) + n)), (a + bx)*(S(1) + m)(c + dx)nhypergeom([S(1) + m, - n], [S(2) + m], - d(a + bx)/(bc - ad))/(b(S(1) + m)(b(c + dx)/(bc - ad))*n)], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True) else: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) def test_numerical(): test = [ [(a + bx)*(S(1)/S(2))(c + dx)(S(1)/S(4)), x, S(5), S(4)/S(7)(a + bx)(S(3)/S(2))(c + dx)(S(1)/S(4))/b + S(4)/S(21)(bc - ad)(c + dx)*(S(1)/S(4))sqrt(a + bx)/(bd) - S(8)/S(21)(bc - ad)(S(9)/S(4))EllipticF(b*(S(1)/S(4))(c + dx)(S(1)/S(4))/(bc - ad)(S(1)/S(4)), I)sqrt(S(1) - b(c + dx)/(bc - ad))/(b*(S(5)/S(4))d*S(2)sqrt(a - bc/d + b(c + dx)/d))], [S(1)/((a + bx)(ad/b + dx)S(3)), x, S(2), - S(1)/S(3)bS(2)/(dS(3)(a + bx)**S(3))], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True, _diff=True, _numerical=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True, _numerical=True) else: assert rubi_test(r, i[1], i[3], expand=True, _diff=True, _numerical=True)
df7e476d37696af420c1cb6bdfefd513857bb760cb670efa2fa01c4c51215121	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.functions import log, sqrt, exp, cos, sin, tan, sec, csc, cot, sinh, sech, atan, asin, acos, atanh, asinh, acosh from sympy.functions.elementary.hyperbolic import acsch as arccsch from sympy.functions.elementary.trigonometric import acsc as arccsc from sympy.integrals.rubi.utility_function import (EllipticE, EllipticF, Int, ArcCsch, ArcCsc, Gamma, hypergeom, rubi_test, AppellF1, EllipticPi, Log, Sqrt, ArcTan, ArcTanh, ArcSin, ArcSinh, ArcCosh, ArcTanh, ArcCos, Hypergeometric2F1) from sympy.core.numbers import (I, pi) from sympy.core.singleton import S from sympy.core.symbol import symbols from sympy.functions.elementary.exponential import (exp, exp_polar) from sympy.functions.special.error_functions import (Ei, erf, erfi) from sympy.functions.special.gamma_functions import (gamma, uppergamma) from sympy.functions.special.hyper import hyper from sympy.functions.special.zeta_functions import polylog from sympy.integrals.integrals import Integral from sympy.simplify.simplify import simplify from sympy.testing.pytest import SKIP a, b, c, d, e, f, m, n, x, u , k, p, r, s, t= symbols('a b c d e f m n x u k p r s t') A, B, C, D, a, b, c, d, e, f, g, h, i, y, z, m, n, p, q, u, v, w, E, F, G, H = symbols('A B C D a b c d e f g h i y z m n p q u v w E F G H') def test_1(): assert rubi_test(rubi_integrate(F*(c(a + bx))(d + ex)m, x), x, F(c(a - bd/e))(-bc(d + ex)log(F)/e)*(-m)(d + ex)mGamma(m + S(1), -bc(d + ex)log(F)/e)/(bclog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(c(a + bx))(d + ex)S(4), x), x, F(c(a + bx))(d + ex)S(4)/(bclog(F)) - S(4)F*(c(a + bx))e(d + ex)S(3)/(bS(2)cS(2)log(F)*S(2)) + S(12)F*(c(a + bx))e*S(2)(d + ex)S(2)/(bS(3)c*S(3)log(F)*S(3)) - S(24)F*(c(a + bx))e*S(3)(d + ex)/(bS(4)c*S(4)log(F)*S(4)) + S(24)F*(c(a + bx))eS(4)/(bS(5)cS(5)log(F)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(d + ex)S(3), x), x, F(c(a + bx))(d + ex)*S(3)/(bclog(F)) - S(3)F*(c(a + bx))e(d + ex)S(2)/(bS(2)cS(2)log(F)*S(2)) + S(6)F*(c(a + bx))e*S(2)(d + ex)/(bS(3)c*S(3)log(F)*S(3)) - S(6)F*(c(a + bx))eS(3)/(bS(4)cS(4)log(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(d + ex)S(2), x), x, F(c(a + bx))(d + ex)*S(2)/(bclog(F)) - S(2)F*(c(a + bx))e(d + ex)/(b*S(2)c*S(2)log(F)*S(2)) + S(2)F*(c(a + bx))eS(2)/(bS(3)cS(3)log(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(d + ex), x), x, F*(c(a + bx))(d + ex)/(bclog(F)) - F(c(a + bx))e/(b*S(2)c*S(2)log(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx)), x), x, F*(c(a + bx))/(bclog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/(d + ex), x), x, F*(c(a - bd/e))Ei(bc(d + ex)log(F)/e)/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(c(a + bx))/(d + ex)S(2), x), x, -F(c(a + bx))/(e(d + ex)) + F*(c(a - bd/e))bclog(F)Ei(bc(d + ex)log(F)/e)/eS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/(d + ex)S(3), x), x, -F(c(a + bx))bclog(F)/(S(2)e*S(2)(d + ex)) - F(c(a + bx))/(S(2)e(d + ex)S(2)) + F(c(a - bd/e))bS(2)c*S(2)log(F)*S(2)Ei(bc(d + ex)log(F)/e)/(S(2)eS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/(d + ex)S(4), x), x, -F(c(a + bx))bS(2)c*S(2)log(F)*S(2)/(S(6)e*S(3)(d + ex)) - F(c(a + bx))bclog(F)/(S(6)eS(2)(d + ex)S(2)) - F(c(a + bx))/(S(3)e(d + ex)S(3)) + F(c(a - bd/e))bS(3)c*S(3)log(F)*S(3)Ei(bc(d + ex)log(F)/e)/(S(6)eS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/(d + ex)S(5), x), x, -F(c(a + bx))bS(3)c*S(3)log(F)*S(3)/(S(24)e*S(4)(d + ex)) - F(c(a + bx))b*S(2)c*S(2)log(F)*S(2)/(S(24)e*S(3)(d + ex)S(2)) - F(c(a + bx))bclog(F)/(S(12)eS(2)(d + ex)S(3)) - F(c(a + bx))/(S(4)e(d + ex)S(4)) + F(c(a - bd/e))bS(4)c*S(4)log(F)*S(4)Ei(bc(d + ex)log(F)/e)/(S(24)eS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(d*S(4) + S(4)d*S(3)ex + S(6)d*S(2)e*S(2)x*S(2) + S(4)deS(3)xS(3) + eS(4)xS(4)), x), x, F(c(a + bx))(d + ex)S(4)/(bclog(F)) - S(4)F*(c(a + bx))e(d + ex)S(3)/(bS(2)cS(2)log(F)*S(2)) + S(12)F*(c(a + bx))e*S(2)(d + ex)S(2)/(bS(3)c*S(3)log(F)*S(3)) - S(24)F*(c(a + bx))e*S(3)(d + ex)/(bS(4)c*S(4)log(F)*S(4)) + S(24)F*(c(a + bx))eS(4)/(bS(5)cS(5)log(F)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(dS(3) + S(3)d*S(2)ex + S(3)deS(2)xS(2) + eS(3)xS(3)), x), x, F(c(a + bx))(d + ex)S(3)/(bclog(F)) - S(3)F*(c(a + bx))e(d + ex)S(2)/(bS(2)cS(2)log(F)*S(2)) + S(6)F*(c(a + bx))e*S(2)(d + ex)/(bS(3)c*S(3)log(F)*S(3)) - S(6)F*(c(a + bx))eS(3)/(bS(4)cS(4)log(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(dS(2) + S(2)dex + e*S(2)xS(2)), x), x, F(c(a + bx))(d + ex)*S(2)/(bclog(F)) - S(2)F*(c(a + bx))e(d + ex)/(b*S(2)c*S(2)log(F)*S(2)) + S(2)F*(c(a + bx))eS(2)/(bS(3)cS(3)log(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/(d*S(2) + S(2)dex + e*S(2)xS(2)), x), x, -F(c(a + bx))/(e(d + ex)) + F*(c(a - bd/e))bclog(F)Ei(bc(d + ex)log(F)/e)/eS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/(dS(3) + S(3)d*S(2)ex + S(3)deS(2)xS(2) + eS(3)xS(3)), x), x, -F(c(a + bx))bclog(F)/(S(2)eS(2)(d + ex)) - F(c(a + bx))/(S(2)e(d + ex)S(2)) + F(c(a - bd/e))bS(2)c*S(2)log(F)*S(2)Ei(bc(d + ex)log(F)/e)/(S(2)eS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/(dS(4) + S(4)d*S(3)ex + S(6)d*S(2)e*S(2)x*S(2) + S(4)deS(3)xS(3) + eS(4)xS(4)), x), x, -F(c(a + bx))b*S(2)c*S(2)log(F)*S(2)/(S(6)e*S(3)(d + ex)) - F(c(a + bx))bclog(F)/(S(6)eS(2)(d + ex)S(2)) - F(c(a + bx))/(S(3)e(d + ex)S(3)) + F(c(a - bd/e))bS(3)c*S(3)log(F)*S(3)Ei(bc(d + ex)log(F)/e)/(S(6)eS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/(dS(5) + S(5)d*S(4)ex + S(10)d*S(3)e*S(2)x*S(2) + S(10)d*S(2)e*S(3)x*S(3) + S(5)deS(4)xS(4) + eS(5)xS(5)), x), x, -F(c(a + bx))b*S(3)c*S(3)log(F)*S(3)/(S(24)e*S(4)(d + ex)) - F(c(a + bx))b*S(2)c*S(2)log(F)*S(2)/(S(24)e*S(3)(d + ex)S(2)) - F(c(a + bx))bclog(F)/(S(12)eS(2)(d + ex)S(3)) - F(c(a + bx))/(S(4)e(d + ex)S(4)) + F(c(a - bd/e))bS(4)c*S(4)log(F)*S(4)Ei(bc(d + ex)log(F)/e)/(S(24)eS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))((d + ex)n)m, x), x, F(c(a - bd/e))(-bc(d + ex)log(F)/e)*(-mn)((d + ex)n)mGamma(mn + S(1), -bc(d + ex)log(F)/e)/(bclog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(c(a + bx))(d*S(4) + S(4)d*S(3)ex + S(6)d*S(2)e*S(2)x*S(2) + S(4)deS(3)xS(3) + eS(4)xS(4))m, x), x, F(c(a - bd/e))(-bc(d + ex)log(F)/e)*(-S(4)m)((d + ex)S(4))mGamma(S(4)m + S(1), -bc(d + ex)log(F)/e)/(bclog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(c(a + bx))(d*S(3) + S(3)d*S(2)ex + S(3)deS(2)xS(2) + eS(3)xS(3))m, x), x, F(c(a - bd/e))(-bc(d + ex)log(F)/e)*(-S(3)m)((d + ex)S(3))mGamma(S(3)m + S(1), -bc(d + ex)log(F)/e)/(bclog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(c(a + bx))(d*S(2) + S(2)dex + e*S(2)xS(2))m, x), x, F*(c(a - bd/e))(-bc(d + ex)log(F)/e)*(-S(2)m)((d + ex)S(2))mGamma(S(2)m + S(1), -bc(d + ex)log(F)/e)/(bclog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(c(a + bx))(d + ex)m, x), x, F(c(a - bd/e))(-bc(d + ex)log(F)/e)*(-m)(d + ex)mGamma(m + S(1), -bc(d + ex)log(F)/e)/(bclog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(c(a + bx))(d + ex)(-m), x), x, F(c(a - bd/e))(-bc(d + ex)log(F)/e)*m(d + ex)(-m)Gamma(-m + S(1), -bc(d + ex)log(F)/e)/(bclog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(c(a + bx))(d*S(2) + S(2)dex + e*S(2)xS(2))(-m), x), x, F*(c(a - bd/e))(-bc(d + ex)log(F)/e)*(S(2)m)((d + ex)S(2))(-m)Gamma(-S(2)m + S(1), -bc(d + ex)log(F)/e)/(bclog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(c(a + bx))(d*S(3) + S(3)d*S(2)ex + S(3)deS(2)xS(2) + eS(3)xS(3))(-m), x), x, F(c(a - bd/e))(-bc(d + ex)log(F)/e)*(S(3)m)((d + ex)S(3))(-m)Gamma(-S(3)m + S(1), -bc(d + ex)log(F)/e)/(bclog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(S(5)x + S(2)), x), x, F*(S(5)x + S(2))/(S(5)log(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + bx), x), x, F*(a + bx)/(blog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(10)(S(5)x + S(2)), x), x, S(2)*(S(5)x + S(2))S(5)(S(5)x + S(1))/log(S(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + bx)x(S(7)/2), x), x, S(105)sqrt(pi)Faerfi(sqrt(b)sqrt(x)sqrt(log(F)))/(S(16)b(S(9)/2)log(F)(S(9)/2)) + F(a + bx)x*(S(7)/2)/(blog(F)) - S(7)F(a + bx)x(S(5)/2)/(S(2)b*S(2)log(F)*S(2)) + S(35)F*(a + bx)x(S(3)/2)/(S(4)b*S(3)log(F)*S(3)) - S(105)F*(a + bx)sqrt(x)/(S(8)b*S(4)log(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + bx)x*(S(5)/2), x), x, -S(15)sqrt(pi)Faerfi(sqrt(b)sqrt(x)sqrt(log(F)))/(S(8)b(S(7)/2)log(F)(S(7)/2)) + F(a + bx)x*(S(5)/2)/(blog(F)) - S(5)F(a + bx)x(S(3)/2)/(S(2)b*S(2)log(F)*S(2)) + S(15)F*(a + bx)sqrt(x)/(S(4)b*S(3)log(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + bx)x*(S(3)/2), x), x, S(3)sqrt(pi)Faerfi(sqrt(b)sqrt(x)sqrt(log(F)))/(S(4)b(S(5)/2)log(F)(S(5)/2)) + F(a + bx)x*(S(3)/2)/(blog(F)) - S(3)F(a + bx)sqrt(x)/(S(2)b*S(2)log(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + bx)sqrt(x), x), x, -sqrt(pi)Faerfi(sqrt(b)sqrt(x)sqrt(log(F)))/(S(2)b(S(3)/2)log(F)(S(3)/2)) + F(a + bx)sqrt(x)/(blog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + bx)/sqrt(x), x), x, sqrt(pi)Faerfi(sqrt(b)sqrt(x)sqrt(log(F)))/(sqrt(b)sqrt(log(F))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + bx)/x*(S(3)/2), x), x, S(2)sqrt(pi)Fasqrt(b)sqrt(log(F))erfi(sqrt(b)sqrt(x)sqrt(log(F))) - S(2)F(a + bx)/sqrt(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + bx)/x*(S(5)/2), x), x, S(4)sqrt(pi)Fab*(S(3)/2)log(F)*(S(3)/2)erfi(sqrt(b)sqrt(x)sqrt(log(F)))/S(3) - S(4)F(a + bx)blog(F)/(S(3)sqrt(x)) - S(2)F*(a + bx)/(S(3)x(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + bx)/x*(S(7)/2), x), x, S(8)sqrt(pi)Fab*(S(5)/2)log(F)*(S(5)/2)erfi(sqrt(b)sqrt(x)sqrt(log(F)))/S(15) - S(8)F(a + bx)bS(2)log(F)*S(2)/(S(15)sqrt(x)) - S(4)F(a + bx)blog(F)/(S(15)x(S(3)/2)) - S(2)F*(a + bx)/(S(5)x(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + bx)/x*(S(9)/2), x), x, S(16)sqrt(pi)Fab*(S(7)/2)log(F)*(S(7)/2)erfi(sqrt(b)sqrt(x)sqrt(log(F)))/S(105) - S(16)F(a + bx)bS(3)log(F)*S(3)/(S(105)sqrt(x)) - S(8)F(a + bx)bS(2)log(F)*S(2)/(S(105)x*(S(3)/2)) - S(4)F*(a + bx)blog(F)/(S(35)x(S(5)/2)) - S(2)F*(a + bx)/(S(7)x(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(d + ex)(S(7)/2), x), x, F(c(a + bx))(d + ex)(S(7)/2)/(bclog(F)) - S(7)F*(c(a + bx))e(d + ex)*(S(5)/2)/(S(2)b*S(2)c*S(2)log(F)*S(2)) + S(35)F*(c(a + bx))e*S(2)(d + ex)(S(3)/2)/(S(4)b*S(3)c*S(3)log(F)*S(3)) - S(105)F*(c(a + bx))e*S(3)sqrt(d + ex)/(S(8)b*S(4)c*S(4)log(F)*S(4)) + S(105)sqrt(pi)F(c(a - bd/e))e*(S(7)/2)erfi(sqrt(b)sqrt(c)sqrt(d + ex)sqrt(log(F))/sqrt(e))/(S(16)b(S(9)/2)c*(S(9)/2)log(F)(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(d + ex)(S(5)/2), x), x, F(c(a + bx))(d + ex)*(S(5)/2)/(bclog(F)) - S(5)F*(c(a + bx))e(d + ex)*(S(3)/2)/(S(2)b*S(2)c*S(2)log(F)*S(2)) + S(15)F*(c(a + bx))e*S(2)sqrt(d + ex)/(S(4)b*S(3)c*S(3)log(F)*S(3)) - S(15)sqrt(pi)F(c(a - bd/e))e*(S(5)/2)erfi(sqrt(b)sqrt(c)sqrt(d + ex)sqrt(log(F))/sqrt(e))/(S(8)b(S(7)/2)c*(S(7)/2)log(F)(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(d + ex)(S(3)/2), x), x, F(c(a + bx))(d + ex)*(S(3)/2)/(bclog(F)) - S(3)F*(c(a + bx))esqrt(d + ex)/(S(2)bS(2)c*S(2)log(F)*S(2)) + S(3)sqrt(pi)F(c(a - bd/e))e*(S(3)/2)erfi(sqrt(b)sqrt(c)sqrt(d + ex)sqrt(log(F))/sqrt(e))/(S(4)b(S(5)/2)c*(S(5)/2)log(F)(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))sqrt(d + ex), x), x, F*(c(a + bx))sqrt(d + ex)/(bclog(F)) - sqrt(pi)F*(c(a - bd/e))sqrt(e)erfi(sqrt(b)sqrt(c)sqrt(d + ex)sqrt(log(F))/sqrt(e))/(S(2)b*(S(3)/2)c*(S(3)/2)log(F)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/sqrt(d + ex), x), x, sqrt(pi)F*(c(a - bd/e))erfi(sqrt(b)sqrt(c)sqrt(d + ex)sqrt(log(F))/sqrt(e))/(sqrt(b)sqrt(c)sqrt(e)sqrt(log(F))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/(d + ex)*(S(3)/2), x), x, -S(2)F*(c(a + bx))/(esqrt(d + ex)) + S(2)sqrt(pi)F(c(a - bd/e))sqrt(b)sqrt(c)sqrt(log(F))erfi(sqrt(b)sqrt(c)sqrt(d + ex)sqrt(log(F))/sqrt(e))/e(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/(d + ex)*(S(5)/2), x), x, -S(4)F*(c(a + bx))bclog(F)/(S(3)eS(2)sqrt(d + ex)) - S(2)F*(c(a + bx))/(S(3)e(d + ex)*(S(3)/2)) + S(4)sqrt(pi)F(c(a - bd/e))b*(S(3)/2)c*(S(3)/2)log(F)*(S(3)/2)erfi(sqrt(b)sqrt(c)sqrt(d + ex)sqrt(log(F))/sqrt(e))/(S(3)e(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/(d + ex)*(S(7)/2), x), x, -S(8)F*(c(a + bx))b*S(2)c*S(2)log(F)*S(2)/(S(15)e*S(3)sqrt(d + ex)) - S(4)F*(c(a + bx))bclog(F)/(S(15)eS(2)(d + ex)(S(3)/2)) - S(2)F*(c(a + bx))/(S(5)e(d + ex)*(S(5)/2)) + S(8)sqrt(pi)F(c(a - bd/e))b*(S(5)/2)c*(S(5)/2)log(F)*(S(5)/2)erfi(sqrt(b)sqrt(c)sqrt(d + ex)sqrt(log(F))/sqrt(e))/(S(15)e(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))/(d + ex)*(S(9)/2), x), x, -S(16)F*(c(a + bx))b*S(3)c*S(3)log(F)*S(3)/(S(105)e*S(4)sqrt(d + ex)) - S(8)F*(c(a + bx))b*S(2)c*S(2)log(F)*S(2)/(S(105)e*S(3)(d + ex)(S(3)/2)) - S(4)F*(c(a + bx))bclog(F)/(S(35)eS(2)(d + ex)(S(5)/2)) - S(2)F*(c(a + bx))/(S(7)e(d + ex)*(S(7)/2)) + S(16)sqrt(pi)F(c(a - bd/e))b*(S(7)/2)c*(S(7)/2)log(F)*(S(7)/2)erfi(sqrt(b)sqrt(c)sqrt(d + ex)sqrt(log(F))/sqrt(e))/(S(105)e(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(13)/2)exp(-bx), x), x, -x(S(13)/2)exp(-bx)/b - S(13)x*(S(11)/2)exp(-bx)/(S(2)b*S(2)) - S(143)x*(S(9)/2)exp(-bx)/(S(4)b*S(3)) - S(1287)x*(S(7)/2)exp(-bx)/(S(8)b*S(4)) - S(9009)x*(S(5)/2)exp(-bx)/(S(16)b*S(5)) - S(45045)x*(S(3)/2)exp(-bx)/(S(32)b*S(6)) - S(135135)sqrt(x)exp(-bx)/(S(64)bS(7)) + S(135135)sqrt(pi)erf(sqrt(b)sqrt(x))/(S(128)b(S(15)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(d + ex)(S(4)/3), x), x, -F(c(a - bd/e))e(d + ex)*(S(1)/3)Gamma(S(7)/3, -bc(d + ex)log(F)/e)/(b*S(2)c*S(2)(-bc(d + ex)log(F)/e)*(S(1)/3)log(F)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*(S(4)/3)(F*(c(a + bx)))n, x), x, -F(-cn(a + bx) + cn(a - bd/e))e(d + ex)*(S(1)/3)(F*(c(a + bx)))nGamma(S(7)/3, -bcn(d + ex)log(F)/e)/(bS(2)c*S(2)n*S(2)(-bcn(d + ex)log(F)/e)(S(1)/3)log(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(d + ex), x), x, F*(c(a + bx))(d + ex)/(bclog(F)) - F(c(a + bx))e/(b*S(2)c*S(2)log(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(d + ex + fxS(2)), x), x, F(c(a + bx))d/(bclog(F)) + F*(c(a + bx))ex/(bclog(F)) + F(c(a + bx))fxS(2)/(bclog(F)) - F(c(a + bx))e/(b*S(2)c*S(2)log(F)*S(2)) - S(2)F*(c(a + bx))fx/(bS(2)c*S(2)log(F)*S(2)) + S(2)F*(c(a + bx))f/(b*S(3)c*S(3)log(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(d + ex + fxS(2) + gxS(3)), x), x, F(c(a + bx))d/(bclog(F)) + F(c(a + bx))ex/(bclog(F)) + F(c(a + bx))fxS(2)/(bclog(F)) + F(c(a + bx))gxS(3)/(bclog(F)) - F(c(a + bx))e/(b*S(2)c*S(2)log(F)*S(2)) - S(2)F*(c(a + bx))fx/(bS(2)c*S(2)log(F)*S(2)) - S(3)F*(c(a + bx))gxS(2)/(bS(2)c*S(2)log(F)*S(2)) + S(2)F*(c(a + bx))f/(b*S(3)c*S(3)log(F)*S(3)) + S(6)F*(c(a + bx))gx/(bS(3)c*S(3)log(F)*S(3)) - S(6)F*(c(a + bx))g/(b*S(4)c*S(4)log(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(d + ex + fxS(2) + gx*S(3) + hxS(4)), x), x, F(c(a + bx))d/(bclog(F)) + F(c(a + bx))ex/(bclog(F)) + F(c(a + bx))fxS(2)/(bclog(F)) + F(c(a + bx))gxS(3)/(bclog(F)) + F(c(a + bx))hxS(4)/(bclog(F)) - F(c(a + bx))e/(b*S(2)c*S(2)log(F)*S(2)) - S(2)F*(c(a + bx))fx/(bS(2)c*S(2)log(F)*S(2)) - S(3)F*(c(a + bx))gxS(2)/(bS(2)c*S(2)log(F)*S(2)) - S(4)F*(c(a + bx))hxS(3)/(bS(2)c*S(2)log(F)*S(2)) + S(2)F*(c(a + bx))f/(b*S(3)c*S(3)log(F)*S(3)) + S(6)F*(c(a + bx))gx/(bS(3)c*S(3)log(F)*S(3)) + S(12)F*(c(a + bx))hxS(2)/(bS(3)c*S(3)log(F)*S(3)) - S(6)F*(c(a + bx))g/(b*S(4)c*S(4)log(F)*S(4)) - S(24)F*(c(a + bx))hx/(bS(4)c*S(4)log(F)*S(4)) + S(24)F*(c(a + bx))h/(b*S(5)c*S(5)log(F)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(a + bx)*S(3)exp(-a - bx), x), x, -aS(3)x*m(bx)(-m)Gamma(m + S(1), bx)exp(-a)/b - S(3)aS(2)x*m(bx)(-m)Gamma(m + S(2), bx)exp(-a)/b - S(3)ax*m(bx)(-m)Gamma(m + S(3), bx)exp(-a)/b - x*m(bx)(-m)Gamma(m + S(4), bx)exp(-a)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a + bx)S(3)exp(-a - bx), x), x, -aS(3)x*S(3)exp(-a - bx)/b - S(3)a*S(3)x*S(2)exp(-a - bx)/bS(2) - S(6)a*S(3)xexp(-a - bx)/b*S(3) - S(6)a*S(3)exp(-a - bx)/bS(4) - S(3)a*S(2)x*S(4)exp(-a - bx) - S(12)a*S(2)x*S(3)exp(-a - bx)/b - S(36)a*S(2)x*S(2)exp(-a - bx)/bS(2) - S(72)a*S(2)xexp(-a - bx)/b*S(3) - S(72)a*S(2)exp(-a - bx)/bS(4) - S(3)abx*S(5)exp(-a - bx) - S(15)axS(4)exp(-a - bx) - S(60)axS(3)exp(-a - bx)/b - S(180)axS(2)exp(-a - bx)/bS(2) - S(360)axexp(-a - bx)/bS(3) - S(360)aexp(-a - bx)/bS(4) - bS(2)xS(6)exp(-a - bx) - S(6)bxS(5)exp(-a - bx) - S(30)x*S(4)exp(-a - bx) - S(120)x*S(3)exp(-a - bx)/b - S(360)x*S(2)exp(-a - bx)/bS(2) - S(720)xexp(-a - bx)/b*S(3) - S(720)exp(-a - bx)/bS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx)S(3)exp(-a - bx), x), x, -aS(3)x*S(2)exp(-a - bx)/b - S(2)a*S(3)xexp(-a - bx)/b*S(2) - S(2)a*S(3)exp(-a - bx)/bS(3) - S(3)a*S(2)x*S(3)exp(-a - bx) - S(9)a*S(2)x*S(2)exp(-a - bx)/b - S(18)a*S(2)xexp(-a - bx)/b*S(2) - S(18)a*S(2)exp(-a - bx)/bS(3) - S(3)abx*S(4)exp(-a - bx) - S(12)axS(3)exp(-a - bx) - S(36)axS(2)exp(-a - bx)/b - S(72)axexp(-a - bx)/bS(2) - S(72)aexp(-a - bx)/bS(3) - bS(2)xS(5)exp(-a - bx) - S(5)bxS(4)exp(-a - bx) - S(20)x*S(3)exp(-a - bx) - S(60)x*S(2)exp(-a - bx)/b - S(120)xexp(-a - bx)/b*S(2) - S(120)exp(-a - bx)/bS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)S(3)exp(-a - bx), x), x, a(a + bx)S(3)exp(-a - bx)/bS(2) + S(3)a(a + bx)*S(2)exp(-a - bx)/bS(2) + S(6)a(a + bx)exp(-a - bx)/b*S(2) + S(6)aexp(-a - bx)/b*S(2) - (a + bx)*S(4)exp(-a - bx)/bS(2) - S(4)(a + bx)S(3)exp(-a - bx)/bS(2) - S(12)(a + bx)S(2)exp(-a - bx)/bS(2) - S(24)(a + bx)exp(-a - bx)/bS(2) - S(24)exp(-a - bx)/bS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(3)exp(-a - bx), x), x, -(a + bx)*S(3)exp(-a - bx)/b - S(3)(a + bx)S(2)exp(-a - bx)/b - S(6)(a + bx)exp(-a - bx)/b - S(6)exp(-a - bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(3)exp(-a - bx)/x, x), x, aS(3)exp(-a)Ei(-bx) - S(3)aS(2)exp(-a - bx) - S(3)abxexp(-a - bx) - S(3)aexp(-a - bx) - bS(2)x*S(2)exp(-a - bx) - S(2)bxexp(-a - bx) - S(2)exp(-a - bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(3)exp(-a - bx)/xS(2), x), x, -aS(3)bexp(-a)Ei(-bx) - aS(3)exp(-a - bx)/x + S(3)a*S(2)bexp(-a)Ei(-bx) - S(3)abexp(-a - bx) - bS(2)xexp(-a - bx) - bexp(-a - bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(3)exp(-a - bx)/xS(3), x), x, aS(3)b*S(2)exp(-a)Ei(-bx)/S(2) + a*S(3)bexp(-a - bx)/(S(2)x) - aS(3)exp(-a - bx)/(S(2)x*S(2)) - S(3)a*S(2)b*S(2)exp(-a)Ei(-bx) - S(3)aS(2)bexp(-a - bx)/x + S(3)ab*S(2)exp(-a)Ei(-bx) - b*S(2)exp(-a - bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(3)exp(-a - bx)/xS(4), x), x, -aS(3)b*S(3)exp(-a)Ei(-bx)/S(6) - a*S(3)b*S(2)exp(-a - bx)/(S(6)x) + a*S(3)bexp(-a - bx)/(S(6)xS(2)) - aS(3)exp(-a - bx)/(S(3)x*S(3)) + S(3)a*S(2)b*S(3)exp(-a)Ei(-bx)/S(2) + S(3)aS(2)b*S(2)exp(-a - bx)/(S(2)x) - S(3)aS(2)bexp(-a - bx)/(S(2)xS(2)) - S(3)abS(3)exp(-a)Ei(-bx) - S(3)ab*S(2)exp(-a - bx)/x + bS(3)exp(-a)Ei(-bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx))x*m(e + fx)S(2), x), x, F(a + bc)eS(2)x*m(-bdxlog(F))(-m)Gamma(m + S(1), -bdxlog(F))/(bdlog(F)) - S(2)F*(a + bc)efxm(-bdxlog(F))(-m)Gamma(m + S(2), -bdxlog(F))/(bS(2)d*S(2)log(F)S(2)) + F(a + bc)f*S(2)x*m(-bdxlog(F))(-m)Gamma(m + S(3), -bdxlog(F))/(bS(3)d*S(3)log(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx))xS(3)(e + fx)S(2), x), x, F(a + bc + bdx)eS(2)x*S(3)/(bdlog(F)) + S(2)F*(a + bc + bdx)efxS(4)/(bdlog(F)) + F(a + bc + bdx)fS(2)x*S(5)/(bdlog(F)) - S(3)F*(a + bc + bdx)eS(2)xS(2)/(bS(2)dS(2)log(F)*S(2)) - S(8)F*(a + bc + bdx)efxS(3)/(bS(2)d*S(2)log(F)*S(2)) - S(5)F*(a + bc + bdx)fS(2)xS(4)/(bS(2)dS(2)log(F)*S(2)) + S(6)F*(a + bc + bdx)eS(2)x/(b*S(3)d*S(3)log(F)*S(3)) + S(24)F*(a + bc + bdx)efxS(2)/(bS(3)d*S(3)log(F)*S(3)) + S(20)F*(a + bc + bdx)fS(2)xS(3)/(bS(3)dS(3)log(F)*S(3)) - S(6)F*(a + bc + bdx)eS(2)/(bS(4)d*S(4)log(F)*S(4)) - S(48)F*(a + bc + bdx)efx/(bS(4)d*S(4)log(F)*S(4)) - S(60)F*(a + bc + bdx)fS(2)xS(2)/(bS(4)dS(4)log(F)*S(4)) + S(48)F*(a + bc + bdx)ef/(b*S(5)d*S(5)log(F)*S(5)) + S(120)F*(a + bc + bdx)fS(2)x/(b*S(5)d*S(5)log(F)*S(5)) - S(120)F*(a + bc + bdx)fS(2)/(bS(6)d*S(6)log(F)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx))xS(2)(e + fx)S(2), x), x, F(a + bc + bdx)eS(2)x*S(2)/(bdlog(F)) + S(2)F*(a + bc + bdx)efxS(3)/(bdlog(F)) + F(a + bc + bdx)fS(2)x*S(4)/(bdlog(F)) - S(2)F*(a + bc + bdx)eS(2)x/(b*S(2)d*S(2)log(F)*S(2)) - S(6)F*(a + bc + bdx)efxS(2)/(bS(2)d*S(2)log(F)*S(2)) - S(4)F*(a + bc + bdx)fS(2)xS(3)/(bS(2)dS(2)log(F)*S(2)) + S(2)F*(a + bc + bdx)eS(2)/(bS(3)d*S(3)log(F)*S(3)) + S(12)F*(a + bc + bdx)efx/(bS(3)d*S(3)log(F)*S(3)) + S(12)F*(a + bc + bdx)fS(2)xS(2)/(bS(3)dS(3)log(F)*S(3)) - S(12)F*(a + bc + bdx)ef/(b*S(4)d*S(4)log(F)*S(4)) - S(24)F*(a + bc + bdx)fS(2)x/(b*S(4)d*S(4)log(F)*S(4)) + S(24)F*(a + bc + bdx)fS(2)/(bS(5)d*S(5)log(F)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx))x(e + fx)S(2), x), x, F(a + bc + bdx)eS(2)x/(bdlog(F)) + S(2)F(a + bc + bdx)efxS(2)/(bdlog(F)) + F(a + bc + bdx)fS(2)x*S(3)/(bdlog(F)) - F(a + bc + bdx)eS(2)/(bS(2)d*S(2)log(F)*S(2)) - S(4)F*(a + bc + bdx)efx/(bS(2)d*S(2)log(F)*S(2)) - S(3)F*(a + bc + bdx)fS(2)xS(2)/(bS(2)dS(2)log(F)*S(2)) + S(4)F*(a + bc + bdx)ef/(b*S(3)d*S(3)log(F)*S(3)) + S(6)F*(a + bc + bdx)fS(2)x/(b*S(3)d*S(3)log(F)*S(3)) - S(6)F*(a + bc + bdx)fS(2)/(bS(4)d*S(4)log(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx))(e + fx)S(2), x), x, F(a + bc + bdx)(e + fx)S(2)/(bdlog(F)) - S(2)F*(a + bc + bdx)f(e + fx)/(bS(2)d*S(2)log(F)*S(2)) + S(2)F*(a + bc + bdx)fS(2)/(bS(3)d*S(3)log(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx))(e + fx)S(2)/x, x), x, F(a + bc)e*S(2)Ei(bdxlog(F)) + S(2)F*(a + bc + bdx)ef/(bdlog(F)) + F*(a + bc + bdx)fS(2)x/(bdlog(F)) - F*(a + bc + bdx)fS(2)/(bS(2)d*S(2)log(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx))(e + fx)S(2)/xS(2), x), x, F*(a + bc)bdeS(2)log(F)Ei(bdxlog(F)) + S(2)F(a + bc)efEi(bdxlog(F)) - F*(a + bc + bdx)eS(2)/x + F(a + bc + bdx)fS(2)/(bdlog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx))(e + fx)S(2)/xS(3), x), x, F(a + bc)bS(2)d*S(2)e*S(2)log(F)*S(2)Ei(bdxlog(F))/S(2) + S(2)F*(a + bc)bdeflog(F)Ei(bdxlog(F)) + F(a + bc)fS(2)Ei(bdxlog(F)) - F(a + bc + bdx)bdeS(2)log(F)/(S(2)x) - F(a + bc + bdx)eS(2)/(S(2)x*S(2)) - S(2)F*(a + bc + bdx)ef/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx))(e + fx)S(2)/xS(4), x), x, F(a + bc)bS(3)d*S(3)e*S(2)log(F)*S(3)Ei(bdxlog(F))/S(6) + F(a + bc)bS(2)d*S(2)eflog(F)*S(2)Ei(bdxlog(F)) + F(a + bc)bdfS(2)log(F)Ei(bdxlog(F)) - F*(a + bc + bdx)bS(2)d*S(2)e*S(2)log(F)*S(2)/(S(6)x) - F*(a + bc + bdx)bdeS(2)log(F)/(S(6)xS(2)) - F(a + bc + bdx)bdeflog(F)/x - F(a + bc + bdx)eS(2)/(S(3)xS(3)) - F(a + bc + bdx)ef/xS(2) - F(a + bc + bdx)fS(2)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx))(e + fx)S(2)/xS(5), x), x, F(a + bc)bS(4)d*S(4)e*S(2)log(F)*S(4)Ei(bdxlog(F))/S(24) + F(a + bc)bS(3)d*S(3)eflog(F)*S(3)Ei(bdxlog(F))/S(3) + F(a + bc)bS(2)d*S(2)f*S(2)log(F)*S(2)Ei(bdxlog(F))/S(2) - F(a + bc + bdx)bS(3)d*S(3)e*S(2)log(F)*S(3)/(S(24)x) - F*(a + bc + bdx)bS(2)d*S(2)e*S(2)log(F)*S(2)/(S(24)xS(2)) - F(a + bc + bdx)b*S(2)d*S(2)eflog(F)*S(2)/(S(3)x) - F*(a + bc + bdx)bdeS(2)log(F)/(S(12)xS(3)) - F(a + bc + bdx)bdeflog(F)/(S(3)xS(2)) - F(a + bc + bdx)bdf*S(2)log(F)/(S(2)x) - F(a + bc + bdx)eS(2)/(S(4)x*S(4)) - S(2)F*(a + bc + bdx)ef/(S(3)xS(3)) - F(a + bc + bdx)fS(2)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(4)(c + dx)S(3)exp(-a - bx), x), x, -dS(3)(a + bx)S(7)exp(-a - bx)/bS(4) - S(7)d*S(3)(a + bx)S(6)exp(-a - bx)/bS(4) - S(42)d*S(3)(a + bx)S(5)exp(-a - bx)/bS(4) - S(210)d*S(3)(a + bx)S(4)exp(-a - bx)/bS(4) - S(840)d*S(3)(a + bx)S(3)exp(-a - bx)/bS(4) - S(2520)d*S(3)(a + bx)S(2)exp(-a - bx)/bS(4) - S(5040)d*S(3)(a + bx)exp(-a - bx)/bS(4) - S(5040)d*S(3)exp(-a - bx)/bS(4) - S(3)d*S(2)(a + bx)S(6)(-ad + bc)exp(-a - bx)/b*S(4) - S(18)d*S(2)(a + bx)S(5)(-ad + bc)exp(-a - bx)/b*S(4) - S(90)d*S(2)(a + bx)S(4)(-ad + bc)exp(-a - bx)/b*S(4) - S(360)d*S(2)(a + bx)S(3)(-ad + bc)exp(-a - bx)/b*S(4) - S(1080)d*S(2)(a + bx)S(2)(-ad + bc)exp(-a - bx)/b*S(4) - S(2160)d*S(2)(a + bx)(-ad + bc)exp(-a - bx)/b*S(4) - S(2160)d*S(2)(-ad + bc)exp(-a - bx)/b*S(4) - S(3)d(a + bx)*S(5)(-ad + bc)*S(2)exp(-a - bx)/bS(4) - S(15)d(a + bx)*S(4)(-ad + bc)*S(2)exp(-a - bx)/bS(4) - S(60)d(a + bx)*S(3)(-ad + bc)*S(2)exp(-a - bx)/bS(4) - S(180)d(a + bx)*S(2)(-ad + bc)*S(2)exp(-a - bx)/bS(4) - S(360)d(a + bx)(-ad + bc)S(2)exp(-a - bx)/bS(4) - S(360)d(-ad + bc)S(2)exp(-a - bx)/bS(4) - (a + bx)*S(4)(-ad + bc)*S(3)exp(-a - bx)/bS(4) - S(4)(a + bx)S(3)(-ad + bc)*S(3)exp(-a - bx)/bS(4) - S(12)(a + bx)S(2)(-ad + bc)*S(3)exp(-a - bx)/bS(4) - S(24)(a + bx)(-ad + bc)*S(3)exp(-a - bx)/bS(4) - S(24)(-ad + bc)*S(3)exp(-a - bx)/bS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(4)(c + dx)S(2)exp(-a - bx), x), x, -dS(2)(a + bx)S(6)exp(-a - bx)/bS(3) - S(6)d*S(2)(a + bx)S(5)exp(-a - bx)/bS(3) - S(30)d*S(2)(a + bx)S(4)exp(-a - bx)/bS(3) - S(120)d*S(2)(a + bx)S(3)exp(-a - bx)/bS(3) - S(360)d*S(2)(a + bx)S(2)exp(-a - bx)/bS(3) - S(720)d*S(2)(a + bx)exp(-a - bx)/bS(3) - S(720)d*S(2)exp(-a - bx)/bS(3) - S(2)d(a + bx)*S(5)(-ad + bc)exp(-a - bx)/b*S(3) - S(10)d(a + bx)*S(4)(-ad + bc)exp(-a - bx)/b*S(3) - S(40)d(a + bx)*S(3)(-ad + bc)exp(-a - bx)/b*S(3) - S(120)d(a + bx)*S(2)(-ad + bc)exp(-a - bx)/b*S(3) - S(240)d(a + bx)(-ad + bc)exp(-a - bx)/bS(3) - S(240)d(-ad + bc)exp(-a - bx)/bS(3) - (a + bx)*S(4)(-ad + bc)*S(2)exp(-a - bx)/bS(3) - S(4)(a + bx)S(3)(-ad + bc)*S(2)exp(-a - bx)/bS(3) - S(12)(a + bx)S(2)(-ad + bc)*S(2)exp(-a - bx)/bS(3) - S(24)(a + bx)(-ad + bc)*S(2)exp(-a - bx)/bS(3) - S(24)(-ad + bc)*S(2)exp(-a - bx)/bS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(4)(c + dx)exp(-a - bx), x), x, -d(a + bx)S(5)exp(-a - bx)/bS(2) - S(5)d(a + bx)*S(4)exp(-a - bx)/bS(2) - S(20)d(a + bx)*S(3)exp(-a - bx)/bS(2) - S(60)d(a + bx)*S(2)exp(-a - bx)/bS(2) - S(120)d(a + bx)exp(-a - bx)/b*S(2) - S(120)dexp(-a - bx)/b*S(2) - (a + bx)*S(4)(-ad + bc)exp(-a - bx)/b*S(2) - (a + bx)*S(3)(-S(4)ad + S(4)bc)exp(-a - bx)/b*S(2) - (a + bx)*S(2)(-S(12)ad + S(12)bc)exp(-a - bx)/b*S(2) - (a + bx)(-S(24)ad + S(24)bc)exp(-a - bx)/bS(2) - (-S(24)ad + S(24)bc)exp(-a - bx)/bS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(4)exp(-a - bx), x), x, -(a + bx)*S(4)exp(-a - bx)/b - S(4)(a + bx)S(3)exp(-a - bx)/b - S(12)(a + bx)S(2)exp(-a - bx)/b - S(24)(a + bx)exp(-a - bx)/b - S(24)exp(-a - bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(4)exp(-a - bx)/(c + dx), x), x, -(a + bx)S(3)exp(-a - bx)/d - S(3)(a + bx)S(2)exp(-a - bx)/d - S(6)(a + bx)exp(-a - bx)/d - S(6)exp(-a - bx)/d + (a + bx)*S(2)(-ad + bc)exp(-a - bx)/d*S(2) + (a + bx)(-S(2)ad + S(2)bc)exp(-a - bx)/dS(2) + (-S(2)ad + S(2)bc)exp(-a - bx)/dS(2) - (a + bx)(-ad + bc)S(2)exp(-a - bx)/dS(3) - (-ad + bc)S(2)exp(-a - bx)/dS(3) + (-ad + bc)S(3)exp(-a - bx)/dS(4) + (-ad + bc)S(4)exp(-a + bc/d)Ei(-b(c + dx)/d)/d*S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(4)exp(-a - bx)/(c + dx)S(2), x), x, -bS(3)(c + dx)*S(2)exp(-a - bx)/dS(4) - S(2)b*S(2)(c + dx)exp(-a - bx)/dS(3) + S(4)b*S(2)(c + dx)(-ad + bc)exp(-a - bx)/d*S(4) - S(2)bexp(-a - bx)/d*S(2) + S(4)b(-ad + bc)exp(-a - bx)/dS(3) - S(6)b(-ad + bc)S(2)exp(-a - bx)/dS(4) - S(4)b(-ad + bc)S(3)exp(-a + bc/d)Ei(-b(c + dx)/d)/d*S(5) - b(-ad + bc)*S(4)exp(-a + bc/d)Ei(-b(c + dx)/d)/d*S(6) - (-ad + bc)S(4)exp(-a - bx)/(dS(5)(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(4)exp(-a - bx)/(c + dx)S(3), x), x, -bS(3)xexp(-a - bx)/dS(3) - bS(2)exp(-a - bx)/dS(3) + bS(2)(-S(4)ad + S(3)bc)exp(-a - bx)/d*S(4) + S(6)b*S(2)(-ad + bc)*S(2)exp(-a + bc/d)Ei(-b(c + dx)/d)/d*S(5) + S(4)b*S(2)(-ad + bc)*S(3)exp(-a + bc/d)Ei(-b(c + dx)/d)/dS(6) + bS(2)(-ad + bc)S(4)exp(-a + bc/d)Ei(-b(c + dx)/d)/(S(2)dS(7)) + S(4)b(-ad + bc)S(3)exp(-a - bx)/(dS(5)(c + dx)) + b(-ad + bc)*S(4)exp(-a - bx)/(S(2)d*S(6)(c + dx)) - (-ad + bc)S(4)exp(-a - bx)/(S(2)d*S(5)(c + dx)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(4)exp(-a - bx)/(c + dx)S(4), x), x, -bS(3)exp(-a - bx)/d*S(4) - S(4)b*S(3)(-ad + bc)exp(-a + bc/d)Ei(-b(c + dx)/d)/dS(5) - S(6)b*S(3)(-ad + bc)*S(2)exp(-a + bc/d)Ei(-b(c + dx)/d)/d*S(6) - S(2)b*S(3)(-ad + bc)*S(3)exp(-a + bc/d)Ei(-b(c + dx)/d)/dS(7) - bS(3)(-ad + bc)S(4)exp(-a + bc/d)Ei(-b(c + dx)/d)/(S(6)dS(8)) - S(6)b*S(2)(-ad + bc)*S(2)exp(-a - bx)/(dS(5)(c + dx)) - S(2)b*S(2)(-ad + bc)*S(3)exp(-a - bx)/(dS(6)(c + dx)) - bS(2)(-ad + bc)*S(4)exp(-a - bx)/(S(6)d*S(7)(c + dx)) + S(2)b(-ad + bc)S(3)exp(-a - bx)/(dS(5)(c + dx)S(2)) + b(-ad + bc)*S(4)exp(-a - bx)/(S(6)d*S(6)(c + dx)S(2)) - (-ad + bc)S(4)exp(-a - bx)/(S(3)d*S(5)(c + dx)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(4)exp(-a - bx)/(c + dx)S(5), x), x, bS(4)exp(-a + bc/d)Ei(-b(c + dx)/d)/dS(5) + S(4)b*S(4)(-ad + bc)exp(-a + bc/d)Ei(-b(c + dx)/d)/dS(6) + S(3)b*S(4)(-ad + bc)*S(2)exp(-a + bc/d)Ei(-b(c + dx)/d)/d*S(7) + S(2)b*S(4)(-ad + bc)*S(3)exp(-a + bc/d)Ei(-b(c + dx)/d)/(S(3)dS(8)) + bS(4)(-ad + bc)*S(4)exp(-a + bc/d)Ei(-b(c + dx)/d)/(S(24)dS(9)) + S(4)b*S(3)(-ad + bc)exp(-a - bx)/(d*S(5)(c + dx)) + S(3)b*S(3)(-ad + bc)*S(2)exp(-a - bx)/(dS(6)(c + dx)) + S(2)b*S(3)(-ad + bc)*S(3)exp(-a - bx)/(S(3)d*S(7)(c + dx)) + bS(3)(-ad + bc)*S(4)exp(-a - bx)/(S(24)d*S(8)(c + dx)) - S(3)b*S(2)(-ad + bc)*S(2)exp(-a - bx)/(dS(5)(c + dx)S(2)) - S(2)b*S(2)(-ad + bc)*S(3)exp(-a - bx)/(S(3)d*S(6)(c + dx)S(2)) - bS(2)(-ad + bc)*S(4)exp(-a - bx)/(S(24)d*S(7)(c + dx)S(2)) + S(4)b(-ad + bc)S(3)exp(-a - bx)/(S(3)d*S(5)(c + dx)S(3)) + b(-ad + bc)*S(4)exp(-a - bx)/(S(12)d*S(6)(c + dx)S(3)) - (-ad + bc)S(4)exp(-a - bx)/(S(4)d*S(5)(c + dx)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))x*m(en + e(bcxlog(F) + m + S(1))log(dx) + e)log(dx)n, x), x, F(c(a + bx))ex(m + S(1))log(dx)(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))x*S(2)(en + e(bcxlog(F) + S(3))log(dx) + e)log(dx)n, x), x, F(c(a + bx))exS(3)log(dx)(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))x(en + e(bcxlog(F) + S(2))log(dx) + e)log(dx)n, x), x, F(c(a + bx))ex*S(2)log(dx)(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(en + e(bcxlog(F) + S(1))log(dx) + e)log(dx)n, x), x, F(c(a + bx))exlog(dx)(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(bcexlog(F)log(dx) + en + e)log(dx)n/x, x), x, F(c(a + bx))elog(dx)(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(en + e(bcxlog(F) + S(-1))log(dx) + e)log(dx)n/xS(2), x), x, F*(c(a + bx))elog(dx)(n + S(1))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c(a + bx))(en + e(bcxlog(F) + S(-2))log(dx) + e)log(dx)n/xS(3), x), x, F*(c(a + bx))elog(dx)(n + S(1))/xS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)sqrt(exp(a + bx)), x), x, S(2)x*S(4)sqrt(exp(a + bx))/b - S(16)x*S(3)sqrt(exp(a + bx))/bS(2) + S(96)x*S(2)sqrt(exp(a + bx))/bS(3) - S(384)xsqrt(exp(a + bx))/b*S(4) + S(768)sqrt(exp(a + bx))/bS(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(exp(a + bx)), x), x, S(2)x*S(3)sqrt(exp(a + bx))/b - S(12)x*S(2)sqrt(exp(a + bx))/bS(2) + S(48)xsqrt(exp(a + bx))/b*S(3) - S(96)sqrt(exp(a + bx))/bS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(exp(a + bx)), x), x, S(2)x*S(2)sqrt(exp(a + bx))/b - S(8)xsqrt(exp(a + bx))/b*S(2) + S(16)sqrt(exp(a + bx))/bS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(exp(a + bx)), x), x, S(2)xsqrt(exp(a + bx))/b - S(4)sqrt(exp(a + bx))/b*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(exp(a + bx)), x), x, S(2)sqrt(exp(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(exp(a + bx))/x, x), x, exp(-bx/S(2))sqrt(exp(a + bx))Ei(bx/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(exp(a + bx))/xS(2), x), x, bexp(-bx/S(2))sqrt(exp(a + bx))Ei(bx/S(2))/S(2) - sqrt(exp(a + bx))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(exp(a + bx))/xS(3), x), x, bS(2)exp(-bx/S(2))sqrt(exp(a + bx))Ei(bx/S(2))/S(8) - bsqrt(exp(a + bx))/(S(4)x) - sqrt(exp(a + bx))/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(exp(a + bx))/xS(4), x), x, bS(3)exp(-bx/S(2))sqrt(exp(a + bx))Ei(bx/S(2))/S(48) - b*S(2)sqrt(exp(a + bx))/(S(24)x) - bsqrt(exp(a + bx))/(S(12)xS(2)) - sqrt(exp(a + bx))/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) def test_2(): assert rubi_test(rubi_integrate(f(c + dx)xS(3)/(a + bf*(c + dx)), x), x, x*S(3)log(S(1) + bf(c + dx)/a)/(bdlog(f)) + S(3)xS(2)polylog(S(2), -bf(c + dx)/a)/(bdS(2)log(f)*S(2)) - S(6)xpolylog(S(3), -bf*(c + dx)/a)/(bdS(3)log(f)*S(3)) + S(6)polylog(S(4), -bf(c + dx)/a)/(bdS(4)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c + dx)x*S(2)/(a + bf*(c + dx)), x), x, x*S(2)log(S(1) + bf(c + dx)/a)/(bdlog(f)) + S(2)xpolylog(S(2), -bf(c + dx)/a)/(bdS(2)log(f)*S(2)) - S(2)polylog(S(3), -bf(c + dx)/a)/(bdS(3)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c + dx)x/(a + bf(c + dx)), x), x, xlog(S(1) + bf*(c + dx)/a)/(bdlog(f)) + polylog(S(2), -bf(c + dx)/a)/(bdS(2)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c + dx)/(a + bf*(c + dx)), x), x, log(a + bf(c + dx))/(bdlog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c + dx)/(x(a + bf*(c + dx))), x), x, Integral(f*(c + dx)/(x(a + bf*(c + dx))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c + dx)/(x*S(2)(a + bf(c + dx))), x), x, Integral(f*(c + dx)/(x*S(2)(a + bf(c + dx))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c + dx)xS(3)/(a + bf*(c + dx))S(2), x), x, -xS(3)/(bd(a + bf(c + dx))log(f)) - S(3)x*S(2)log(af(-c - dx)/b + S(1))/(abd*S(2)log(f)*S(2)) + S(6)xpolylog(S(2), -af*(-c - dx)/b)/(abd*S(3)log(f)*S(3)) + S(6)polylog(S(3), -af(-c - dx)/b)/(abd*S(4)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c + dx)x*S(2)/(a + bf*(c + dx))S(2), x), x, -xS(2)/(bd(a + bf(c + dx))log(f)) - S(2)xlog(af*(-c - dx)/b + S(1))/(abd*S(2)log(f)*S(2)) + S(2)polylog(S(2), -af(-c - dx)/b)/(abd*S(3)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c + dx)x/(a + bf(c + dx))*S(2), x), x, -x/(bd(a + bf*(c + dx))log(f)) + x/(abdlog(f)) - log(a + bf(c + dx))/(abd*S(2)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c + dx)/(a + bf*(c + dx))*S(2), x), x, -S(1)/(bd(a + bf*(c + dx))log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c + dx)/(x(a + bf*(c + dx))S(2)), x), x, -Integral(S(1)/(xS(2)(a + bf*(c + dx))), x)/(bdlog(f)) - S(1)/(bdx(a + bf*(c + dx))log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c + dx)/(x*S(2)(a + bf(c + dx))*S(2)), x), x, -S(2)Integral(S(1)/(x*S(3)(a + bf(c + dx))), x)/(bdlog(f)) - S(1)/(bdx*S(2)(a + bf(c + dx))log(f)), expand=True, _diff=True, _numerical=True) # recursion assert rubi_test(rubi_integrate(f(c + dx)xS(3)/(a + bf*(c + dx))S(3), x), x, -xS(3)/(S(2)bd(a + bf*(c + dx))*S(2)log(f)) + S(3)xS(2)/(S(2)abd*S(2)(a + bf(c + dx))log(f)S(2)) + xS(3)/(S(2)a*S(2)bdlog(f)) - S(3)xS(2)log(S(1) + bf(c + dx)/a)/(S(2)aS(2)bdS(2)log(f)*S(2)) + S(3)xlog(af*(-c - dx)/b + S(1))/(a*S(2)bdS(3)log(f)*S(3)) - S(3)xpolylog(S(2), -bf*(c + dx)/a)/(a*S(2)bdS(3)log(f)*S(3)) - S(3)polylog(S(2), -af(-c - dx)/b)/(a*S(2)bdS(4)log(f)*S(4)) + S(3)polylog(S(3), -bf(c + dx)/a)/(a*S(2)bdS(4)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c + dx)x*S(2)/(a + bf*(c + dx))S(3), x), x, -xS(2)/(S(2)bd(a + bf*(c + dx))*S(2)log(f)) + x/(abd*S(2)(a + bf(c + dx))log(f)S(2)) + xS(2)/(S(2)a*S(2)bdlog(f)) - xlog(S(1) + bf*(c + dx)/a)/(a*S(2)bdS(2)log(f)S(2)) - x/(aS(2)bd*S(2)log(f)*S(2)) + log(a + bf*(c + dx))/(a*S(2)bdS(3)log(f)*S(3)) - polylog(S(2), -bf*(c + dx)/a)/(a*S(2)bdS(3)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c + dx)x/(a + bf(c + dx))*S(3), x), x, -x/(S(2)bd(a + bf(c + dx))*S(2)log(f)) + S(1)/(S(2)abdS(2)(a + bf(c + dx))log(f)S(2)) + x/(S(2)a*S(2)bdlog(f)) - log(a + bf(c + dx))/(S(2)aS(2)bdS(2)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c + dx)/(a + bf*(c + dx))*S(3), x), x, -S(1)/(S(2)bd(a + bf(c + dx))*S(2)log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c + dx)/(x(a + bf*(c + dx))S(3)), x), x, -Integral(S(1)/(xS(2)(a + bf*(c + dx))*S(2)), x)/(S(2)bdlog(f)) - S(1)/(S(2)bdx(a + bf(c + dx))*S(2)log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c + dx)/(x*S(2)(a + bf(c + dx))S(3)), x), x, -Integral(S(1)/(xS(3)(a + bf*(c + dx))*S(2)), x)/(bdlog(f)) - S(1)/(S(2)bdx*S(2)(a + bf(c + dx))*S(2)log(f)), expand=True, _diff=True, _numerical=True) def test_3(): assert rubi_test(rubi_integrate(exp(x)/(S(6)exp(x) + S(4)), x), x, log(S(3)exp(x) + S(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(a + bexp(x)), x), x, log(a + bexp(x))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(dx)/(a + bexp(c + dx)), x), x, exp(-c)log(a + bexp(c + dx))/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(c + dx)/(a + bexp(c + dx)), x), x, log(a + bexp(c + dx))/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(x))*nexp(x), x), x, (a + bexp(x))(n + S(1))/(b(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(c + dx))*nexp(dx), x), x, (a + bexp(c + dx))(n + S(1))exp(-c)/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(c + dx))*nexp(c + dx), x), x, (a + bexp(c + dx))(n + S(1))/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Fx/(Fxb + a), x), x, log(F*xb + a)/(blog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(dx)/(F*(c + dx)b + a), x), x, F(-c)log(F*(c + dx)b + a)/(bdlog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(c + dx)/(F*(c + dx)b + a), x), x, log(F(c + dx)b + a)/(bdlog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Fx(F*xb + a)n, x), x, (Fxb + a)(n + S(1))/(b(n + S(1))log(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(dx)(F(c + dx)b + a)n, x), x, F(-c)(F*(c + dx)b + a)(n + S(1))/(bd(n + S(1))log(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(c + dx)(F(c + dx)b + a)n, x), x, (F(c + dx)b + a)(n + S(1))/(bd(n + S(1))log(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx*S(2))xm, x), x, -fax(m + S(1))(-bxS(2)log(f))*(-m/S(2) + S(-1)/2)Gamma(m/S(2) + S(1)/2, -bxS(2)log(f))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx*S(2))xS(11), x), x, -faGamma(S(6), -bx*S(2)log(f))/(S(2)bS(6)log(f)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))xS(9), x), x, faGamma(S(5), -bx*S(2)log(f))/(S(2)bS(5)log(f)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))xS(7), x), x, f(a + bxS(2))x*S(6)/(S(2)blog(f)) - S(3)f*(a + bx*S(2))x*S(4)/(S(2)b*S(2)log(f)*S(2)) + S(3)f*(a + bx*S(2))xS(2)/(bS(3)log(f)S(3)) - S(3)f*(a + bxS(2))/(bS(4)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx*S(2))xS(5), x), x, f(a + bxS(2))x*S(4)/(S(2)blog(f)) - f(a + bx*S(2))xS(2)/(bS(2)log(f)S(2)) + f(a + bxS(2))/(bS(3)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx*S(2))xS(3), x), x, f(a + bxS(2))x*S(2)/(S(2)blog(f)) - f(a + bx*S(2))/(S(2)b*S(2)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))x, x), x, f*(a + bx*S(2))/(S(2)blog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))/x, x), x, faEi(bx*S(2)log(f))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bxS(2))/xS(3), x), x, bfalog(f)Ei(bx*S(2)log(f))/S(2) - f*(a + bx*S(2))/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))/xS(5), x), x, bS(2)f*alog(f)*S(2)Ei(bxS(2)log(f))/S(4) - bf(a + bx*S(2))log(f)/(S(4)xS(2)) - f(a + bx*S(2))/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))/xS(7), x), x, bS(3)f*alog(f)*S(3)Ei(bxS(2)log(f))/S(12) - b*S(2)f*(a + bx*S(2))log(f)*S(2)/(S(12)x*S(2)) - bf*(a + bx*S(2))log(f)/(S(12)xS(4)) - f(a + bx*S(2))/(S(6)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))/xS(9), x), x, -bS(4)f*aGamma(S(-4), -bxS(2)log(f))log(f)S(4)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))/xS(11), x), x, b*S(5)f*aGamma(S(-5), -bxS(2)log(f))log(f)S(5)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx*S(2))xS(12), x), x, -faxS(13)Gamma(S(13)/2, -bxS(2)log(f))/(S(2)(-bx*S(2)log(f))(S(13)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))xS(10), x), x, -faxS(11)Gamma(S(11)/2, -bxS(2)log(f))/(S(2)(-bx*S(2)log(f))(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))xS(8), x), x, f(a + bxS(2))x*S(7)/(S(2)blog(f)) - S(7)f*(a + bx*S(2))x*S(5)/(S(4)b*S(2)log(f)*S(2)) + S(35)f*(a + bx*S(2))x*S(3)/(S(8)b*S(3)log(f)*S(3)) - S(105)f*(a + bx*S(2))x/(S(16)bS(4)log(f)*S(4)) + S(105)sqrt(pi)faerfi(sqrt(b)xsqrt(log(f)))/(S(32)b(S(9)/2)log(f)(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))xS(6), x), x, f(a + bxS(2))x*S(5)/(S(2)blog(f)) - S(5)f*(a + bx*S(2))x*S(3)/(S(4)b*S(2)log(f)*S(2)) + S(15)f*(a + bx*S(2))x/(S(8)bS(3)log(f)*S(3)) - S(15)sqrt(pi)faerfi(sqrt(b)xsqrt(log(f)))/(S(16)b(S(7)/2)log(f)(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))xS(4), x), x, f(a + bxS(2))x*S(3)/(S(2)blog(f)) - S(3)f*(a + bx*S(2))x/(S(4)bS(2)log(f)*S(2)) + S(3)sqrt(pi)faerfi(sqrt(b)xsqrt(log(f)))/(S(8)b(S(5)/2)log(f)(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))xS(2), x), x, f(a + bxS(2))x/(S(2)blog(f)) - sqrt(pi)faerfi(sqrt(b)xsqrt(log(f)))/(S(4)b(S(3)/2)log(f)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2)), x), x, sqrt(pi)f*aerfi(sqrt(b)xsqrt(log(f)))/(S(2)sqrt(b)sqrt(log(f))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bxS(2))/xS(2), x), x, sqrt(pi)sqrt(b)f*asqrt(log(f))erfi(sqrt(b)xsqrt(log(f))) - f(a + bxS(2))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))/xS(4), x), x, S(2)sqrt(pi)b(S(3)/2)f*alog(f)*(S(3)/2)erfi(sqrt(b)xsqrt(log(f)))/S(3) - S(2)bf*(a + bx*S(2))log(f)/(S(3)x) - f(a + bx*S(2))/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))/xS(6), x), x, S(4)sqrt(pi)b(S(5)/2)f*alog(f)*(S(5)/2)erfi(sqrt(b)xsqrt(log(f)))/S(15) - S(4)bS(2)f*(a + bx*S(2))log(f)*S(2)/(S(15)x) - S(2)bf*(a + bx*S(2))log(f)/(S(15)xS(3)) - f(a + bx*S(2))/(S(5)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))/xS(8), x), x, S(8)sqrt(pi)b(S(7)/2)f*alog(f)*(S(7)/2)erfi(sqrt(b)xsqrt(log(f)))/S(105) - S(8)bS(3)f*(a + bx*S(2))log(f)*S(3)/(S(105)x) - S(4)bS(2)f*(a + bx*S(2))log(f)*S(2)/(S(105)x*S(3)) - S(2)bf(a + bx*S(2))log(f)/(S(35)xS(5)) - f(a + bx*S(2))/(S(7)xS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))/xS(10), x), x, -fa(-bxS(2)log(f))*(S(9)/2)Gamma(S(-9)/2, -bxS(2)log(f))/(S(2)xS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(2))/xS(12), x), x, -f*a(-bxS(2)log(f))*(S(11)/2)Gamma(S(-11)/2, -bxS(2)log(f))/(S(2)xS(11)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx*S(3))xm, x), x, -fax(m + S(1))(-bxS(3)log(f))*(-m/S(3) + S(-1)/3)Gamma(m/S(3) + S(1)/3, -bxS(3)log(f))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx*S(3))xS(17), x), x, -faGamma(S(6), -bx*S(3)log(f))/(S(3)bS(6)log(f)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))xS(14), x), x, faGamma(S(5), -bx*S(3)log(f))/(S(3)bS(5)log(f)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))xS(11), x), x, f(a + bxS(3))x*S(9)/(S(3)blog(f)) - f(a + bx*S(3))xS(6)/(bS(2)log(f)S(2)) + S(2)f*(a + bx*S(3))xS(3)/(bS(3)log(f)S(3)) - S(2)f*(a + bxS(3))/(bS(4)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx*S(3))xS(8), x), x, f(a + bxS(3))x*S(6)/(S(3)blog(f)) - S(2)f*(a + bx*S(3))x*S(3)/(S(3)b*S(2)log(f)*S(2)) + S(2)f*(a + bx*S(3))/(S(3)b*S(3)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))xS(5), x), x, f(a + bxS(3))x*S(3)/(S(3)blog(f)) - f(a + bx*S(3))/(S(3)b*S(2)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))xS(2), x), x, f(a + bxS(3))/(S(3)blog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))/x, x), x, faEi(bx*S(3)log(f))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bxS(3))/xS(4), x), x, bfalog(f)Ei(bx*S(3)log(f))/S(3) - f*(a + bx*S(3))/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))/xS(7), x), x, bS(2)f*alog(f)*S(2)Ei(bxS(3)log(f))/S(6) - bf(a + bx*S(3))log(f)/(S(6)xS(3)) - f(a + bx*S(3))/(S(6)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))/xS(10), x), x, bS(3)f*alog(f)*S(3)Ei(bxS(3)log(f))/S(18) - b*S(2)f*(a + bx*S(3))log(f)*S(2)/(S(18)x*S(3)) - bf*(a + bx*S(3))log(f)/(S(18)xS(6)) - f(a + bx*S(3))/(S(9)xS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))/xS(13), x), x, -bS(4)f*aGamma(S(-4), -bxS(3)log(f))log(f)S(4)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))/xS(16), x), x, b*S(5)f*aGamma(S(-5), -bxS(3)log(f))log(f)S(5)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx*S(3))xS(4), x), x, -faxS(5)Gamma(S(5)/3, -bxS(3)log(f))/(S(3)(-bx*S(3)log(f))(S(5)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))xS(3), x), x, -faxS(4)Gamma(S(4)/3, -bxS(3)log(f))/(S(3)(-bx*S(3)log(f))(S(4)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))x, x), x, -f*ax*S(2)Gamma(S(2)/3, -bxS(3)log(f))/(S(3)(-bx*S(3)log(f))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3)), x), x, -faxGamma(S(1)/3, -bx*S(3)log(f))/(S(3)(-bx*S(3)log(f))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))/xS(2), x), x, -fa(-bxS(3)log(f))*(S(1)/3)Gamma(S(-1)/3, -bxS(3)log(f))/(S(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxS(3))/xS(3), x), x, -f*a(-bxS(3)log(f))*(S(2)/3)Gamma(S(-2)/3, -bxS(3)log(f))/(S(3)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)exp(S(4)xS(3)), x), x, exp(S(4)xS(3))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x)xm, x), x, fax*(m + S(1))(-blog(f)/x)(m + S(1))Gamma(-m + S(-1), -blog(f)/x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x)xS(4), x), x, -bS(5)faGamma(S(-5), -blog(f)/x)log(f)S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x)xS(3), x), x, bS(4)f*aGamma(S(-4), -blog(f)/x)log(f)S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x)xS(2), x), x, -bS(3)f*alog(f)*S(3)Ei(blog(f)/x)/S(6) + bS(2)f*(a + b/x)xlog(f)S(2)/S(6) + bf*(a + b/x)x*S(2)log(f)/S(6) + f*(a + b/x)xS(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x)x, x), x, -bS(2)f*alog(f)*S(2)Ei(blog(f)/x)/S(2) + bf*(a + b/x)xlog(f)/S(2) + f(a + b/x)xS(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x), x), x, -bfalog(f)Ei(blog(f)/x) + f*(a + b/x)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x)/x, x), x, -faEi(blog(f)/x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x)/xS(2), x), x, -f*(a + b/x)/(blog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x)/xS(3), x), x, -f*(a + b/x)/(bxlog(f)) + f(a + b/x)/(bS(2)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x)/xS(4), x), x, -f(a + b/x)/(bxS(2)log(f)) + S(2)f(a + b/x)/(bS(2)xlog(f)S(2)) - S(2)f(a + b/x)/(bS(3)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x)/xS(5), x), x, -f(a + b/x)/(bx*S(3)log(f)) + S(3)f(a + b/x)/(bS(2)x*S(2)log(f)*S(2)) - S(6)f(a + b/x)/(bS(3)xlog(f)*S(3)) + S(6)f(a + b/x)/(bS(4)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x)/xS(6), x), x, -faGamma(S(5), -blog(f)/x)/(bS(5)log(f)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x)/xS(7), x), x, faGamma(S(6), -blog(f)/x)/(b*S(6)log(f)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x*S(2))xm, x), x, fax(m + S(1))(-blog(f)/xS(2))(m/S(2) + S(1)/2)Gamma(-m/S(2) + S(-1)/2, -blog(f)/xS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))xS(9), x), x, -bS(5)faGamma(S(-5), -blog(f)/xS(2))log(f)S(5)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x*S(2))xS(7), x), x, bS(4)faGamma(S(-4), -blog(f)/xS(2))log(f)S(4)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x*S(2))xS(5), x), x, -bS(3)falog(f)*S(3)Ei(blog(f)/xS(2))/S(12) + bS(2)f(a + b/xS(2))xS(2)log(f)*S(2)/S(12) + bf(a + b/xS(2))xS(4)log(f)/S(12) + f(a + b/xS(2))xS(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))xS(3), x), x, -bS(2)falog(f)*S(2)Ei(blog(f)/xS(2))/S(4) + bf(a + b/xS(2))xS(2)log(f)/S(4) + f(a + b/xS(2))xS(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))x, x), x, -bfalog(f)Ei(blog(f)/xS(2))/S(2) + f(a + b/x*S(2))xS(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/x, x), x, -faEi(blog(f)/xS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(3), x), x, -f(a + b/xS(2))/(S(2)blog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(5), x), x, -f(a + b/x*S(2))/(S(2)bxS(2)log(f)) + f(a + b/xS(2))/(S(2)bS(2)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(7), x), x, -f(a + b/xS(2))/(S(2)bx*S(4)log(f)) + f(a + b/xS(2))/(b*S(2)x*S(2)log(f)S(2)) - f(a + b/xS(2))/(bS(3)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(9), x), x, -f(a + b/xS(2))/(S(2)bxS(6)log(f)) + S(3)f(a + b/xS(2))/(S(2)b*S(2)x*S(4)log(f)*S(2)) - S(3)f(a + b/xS(2))/(b*S(3)x*S(2)log(f)*S(3)) + S(3)f(a + b/xS(2))/(b*S(4)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(11), x), x, -f*aGamma(S(5), -blog(f)/xS(2))/(S(2)b*S(5)log(f)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(13), x), x, f*aGamma(S(6), -blog(f)/xS(2))/(S(2)b*S(6)log(f)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x*S(2))xS(10), x), x, faxS(11)(-blog(f)/xS(2))(S(11)/2)Gamma(S(-11)/2, -blog(f)/xS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))xS(8), x), x, faxS(9)(-blog(f)/xS(2))(S(9)/2)Gamma(S(-9)/2, -blog(f)/xS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))x*S(6), x), x, -S(8)sqrt(pi)b(S(7)/2)f*alog(f)*(S(7)/2)erfi(sqrt(b)sqrt(log(f))/x)/S(105) + S(8)b*S(3)f(a + b/xS(2))xlog(f)*S(3)/S(105) + S(4)b*S(2)f(a + b/xS(2))xS(3)log(f)*S(2)/S(105) + S(2)bf(a + b/xS(2))x*S(5)log(f)/S(35) + f(a + b/xS(2))xS(7)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))x*S(4), x), x, -S(4)sqrt(pi)b(S(5)/2)f*alog(f)*(S(5)/2)erfi(sqrt(b)sqrt(log(f))/x)/S(15) + S(4)b*S(2)f(a + b/xS(2))xlog(f)*S(2)/S(15) + S(2)bf(a + b/xS(2))x*S(3)log(f)/S(15) + f(a + b/xS(2))xS(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))x*S(2), x), x, -S(2)sqrt(pi)b(S(3)/2)f*alog(f)*(S(3)/2)erfi(sqrt(b)sqrt(log(f))/x)/S(3) + S(2)bf(a + b/xS(2))xlog(f)/S(3) + f(a + b/xS(2))xS(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x*S(2)), x), x, -sqrt(pi)sqrt(b)fasqrt(log(f))erfi(sqrt(b)sqrt(log(f))/x) + f(a + b/xS(2))x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(2), x), x, -sqrt(pi)f*aerfi(sqrt(b)sqrt(log(f))/x)/(S(2)sqrt(b)sqrt(log(f))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(4), x), x, -f(a + b/xS(2))/(S(2)bxlog(f)) + sqrt(pi)faerfi(sqrt(b)sqrt(log(f))/x)/(S(4)b*(S(3)/2)log(f)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(6), x), x, -f(a + b/xS(2))/(S(2)bx*S(3)log(f)) + S(3)f(a + b/xS(2))/(S(4)b*S(2)xlog(f)S(2)) - S(3)sqrt(pi)faerfi(sqrt(b)sqrt(log(f))/x)/(S(8)b*(S(5)/2)log(f)(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(8), x), x, -f(a + b/xS(2))/(S(2)bx*S(5)log(f)) + S(5)f(a + b/xS(2))/(S(4)b*S(2)x*S(3)log(f)*S(2)) - S(15)f(a + b/xS(2))/(S(8)bS(3)xlog(f)S(3)) + S(15)sqrt(pi)faerfi(sqrt(b)sqrt(log(f))/x)/(S(16)b*(S(7)/2)log(f)(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(10), x), x, -f(a + b/xS(2))/(S(2)bx*S(7)log(f)) + S(7)f(a + b/xS(2))/(S(4)b*S(2)x*S(5)log(f)*S(2)) - S(35)f(a + b/xS(2))/(S(8)bS(3)x*S(3)log(f)*S(3)) + S(105)f(a + b/xS(2))/(S(16)bS(4)xlog(f)S(4)) - S(105)sqrt(pi)faerfi(sqrt(b)sqrt(log(f))/x)/(S(32)b*(S(9)/2)log(f)(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(12), x), x, f*aGamma(S(11)/2, -blog(f)/xS(2))/(S(2)x*S(11)(-blog(f)/xS(2))(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(2))/xS(14), x), x, faGamma(S(13)/2, -blog(f)/xS(2))/(S(2)x*S(13)(-blog(f)/xS(2))(S(13)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))xm, x), x, fax(m + S(1))(-blog(f)/xS(3))(m/S(3) + S(1)/3)Gamma(-m/S(3) + S(-1)/3, -blog(f)/xS(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))xS(14), x), x, -bS(5)faGamma(S(-5), -blog(f)/xS(3))log(f)S(5)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x*S(3))xS(11), x), x, bS(4)faGamma(S(-4), -blog(f)/xS(3))log(f)S(4)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x*S(3))xS(8), x), x, -bS(3)falog(f)*S(3)Ei(blog(f)/xS(3))/S(18) + bS(2)f(a + b/xS(3))xS(3)log(f)*S(2)/S(18) + bf(a + b/xS(3))xS(6)log(f)/S(18) + f(a + b/xS(3))xS(9)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))xS(5), x), x, -bS(2)falog(f)*S(2)Ei(blog(f)/xS(3))/S(6) + bf(a + b/xS(3))xS(3)log(f)/S(6) + f(a + b/xS(3))xS(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))x*S(2), x), x, -bf*alog(f)Ei(blog(f)/xS(3))/S(3) + f(a + b/x*S(3))xS(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))/x, x), x, -faEi(blog(f)/xS(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))/xS(4), x), x, -f(a + b/xS(3))/(S(3)blog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))/xS(7), x), x, -f(a + b/x*S(3))/(S(3)bxS(3)log(f)) + f(a + b/xS(3))/(S(3)bS(2)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))/xS(10), x), x, -f(a + b/xS(3))/(S(3)bx*S(6)log(f)) + S(2)f(a + b/xS(3))/(S(3)b*S(2)x*S(3)log(f)*S(2)) - S(2)f(a + b/xS(3))/(S(3)bS(3)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))/xS(13), x), x, -f(a + b/xS(3))/(S(3)bx*S(9)log(f)) + f(a + b/xS(3))/(b*S(2)x*S(6)log(f)*S(2)) - S(2)f(a + b/xS(3))/(b*S(3)x*S(3)log(f)*S(3)) + S(2)f(a + b/xS(3))/(b*S(4)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))/xS(16), x), x, -f*aGamma(S(5), -blog(f)/xS(3))/(S(3)b*S(5)log(f)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))/xS(19), x), x, f*aGamma(S(6), -blog(f)/xS(3))/(S(3)b*S(6)log(f)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/x*S(3))xS(4), x), x, faxS(5)(-blog(f)/xS(3))(S(5)/3)Gamma(S(-5)/3, -blog(f)/xS(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))xS(3), x), x, faxS(4)(-blog(f)/xS(3))(S(4)/3)Gamma(S(-4)/3, -blog(f)/xS(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))x, x), x, f*ax*S(2)(-blog(f)/xS(3))(S(2)/3)Gamma(S(-2)/3, -blog(f)/xS(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3)), x), x, fax(-blog(f)/xS(3))(S(1)/3)Gamma(S(-1)/3, -blog(f)/xS(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))/xS(2), x), x, f*aGamma(S(1)/3, -blog(f)/xS(3))/(S(3)x(-blog(f)/xS(3))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))/xS(3), x), x, faGamma(S(2)/3, -blog(f)/x*S(3))/(S(3)x*S(2)(-blog(f)/xS(3))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b/xS(3))/xS(5), x), x, faGamma(S(4)/3, -blog(f)/xS(3))/(S(3)x*S(4)(-blog(f)/xS(3))(S(4)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx*n)xm, x), x, -fax(m + S(1))(-bxnlog(f))*(-(m + S(1))/n)Gamma((m + S(1))/n, -bxnlog(f))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx*n)xS(3), x), x, -faxS(4)(-bxnlog(f))*(-S(4)/n)Gamma(S(4)/n, -bxnlog(f))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx*n)xS(2), x), x, -faxS(3)(-bxnlog(f))*(-S(3)/n)Gamma(S(3)/n, -bxnlog(f))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx*n)x, x), x, -f*ax*S(2)(-bxnlog(f))*(-S(2)/n)Gamma(S(2)/n, -bxnlog(f))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bxn), x), x, -fax(-bxnlog(f))*(-S(1)/n)Gamma(S(1)/n, -bxnlog(f))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bxn)/x, x), x, faEi(bx*nlog(f))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bxn)/xS(2), x), x, -f*a(-bxnlog(f))*(S(1)/n)Gamma(-S(1)/n, -bxnlog(f))/(nx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxn)/xS(3), x), x, -f*a(-bxnlog(f))*(S(2)/n)Gamma(-S(2)/n, -bxnlog(f))/(nxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxn)/xS(4), x), x, -f*a(-bxnlog(f))*(S(3)/n)Gamma(-S(3)/n, -bxnlog(f))/(nxS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx*n)x*(S(3)n + S(-1)), x), x, f*(a + bx*n)x*(S(2)n)/(bnlog(f)) - S(2)f(a + bx*n)xn/(bS(2)nlog(f)*S(2)) + S(2)f*(a + bxn)/(bS(3)nlog(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxn)x*(S(2)n + S(-1)), x), x, f*(a + bx*n)x*n/(bnlog(f)) - f(a + bxn)/(bS(2)nlog(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxn)x(n + S(-1)), x), x, f(a + bxn)/(bnlog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxn)/x, x), x, faEi(bx*nlog(f))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx*n)x*(-n + S(-1)), x), x, bf*alog(f)Ei(bx*nlog(f))/n - f*(a + bx*n)x(-n)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxn)x*(-S(2)n + S(-1)), x), x, b*S(2)f*alog(f)*S(2)Ei(bxnlog(f))/(S(2)n) - bf*(a + bx*n)x*(-n)log(f)/(S(2)n) - f(a + bx*n)x*(-S(2)n)/(S(2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx*n)x*(S(5)n/S(2) + S(-1)), x), x, f*(a + bx*n)x*(S(3)n/S(2))/(bnlog(f)) - S(3)f(a + bx*n)x*(n/S(2))/(S(2)b*S(2)nlog(f)S(2)) + S(3)sqrt(pi)faerfi(sqrt(b)x(n/S(2))sqrt(log(f)))/(S(4)b(S(5)/2)nlog(f)(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx*n)x*(S(3)n/S(2) + S(-1)), x), x, f*(a + bx*n)x*(n/S(2))/(bnlog(f)) - sqrt(pi)f*aerfi(sqrt(b)x(n/S(2))sqrt(log(f)))/(S(2)b(S(3)/2)nlog(f)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx*n)x*(n/S(2) + S(-1)), x), x, sqrt(pi)f*aerfi(sqrt(b)x(n/S(2))sqrt(log(f)))/(sqrt(b)nsqrt(log(f))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx*n)x*(-n/S(2) + S(-1)), x), x, S(2)sqrt(pi)sqrt(b)f*asqrt(log(f))erfi(sqrt(b)x*(n/S(2))sqrt(log(f)))/n - S(2)f(a + bx*n)x(-n/S(2))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bxn)x*(-S(3)n/S(2) + S(-1)), x), x, S(4)sqrt(pi)b*(S(3)/2)f*alog(f)*(S(3)/2)erfi(sqrt(b)x(n/S(2))sqrt(log(f)))/(S(3)n) - S(4)bf(a + bx*n)x*(-n/S(2))log(f)/(S(3)n) - S(2)f*(a + bx*n)x*(-S(3)n/S(2))/(S(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp(-0.1x), x), x, -10.0xexp(-0.1x) - 100.0exp(-0.1x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c(a + bx)S(2))xm, x), x, Integral(f(a*S(2)c + S(2)abcx + b*S(2)cxS(2))xm, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)*S(2))x*S(3), x), x, -sqrt(pi)a*S(3)erfi(sqrt(c)(a + bx)sqrt(log(f)))/(S(2)b*S(4)sqrt(c)sqrt(log(f))) + S(3)a*S(2)f*(c(a + bx)S(2))/(S(2)b*S(4)clog(f)) - S(3)af(c(a + bx)S(2))(a + bx)/(S(2)b*S(4)clog(f)) + S(3)sqrt(pi)aerfi(sqrt(c)(a + bx)sqrt(log(f)))/(S(4)b*S(4)c*(S(3)/2)log(f)(S(3)/2)) + f(c(a + bx)*S(2))(a + bx)S(2)/(S(2)b*S(4)clog(f)) - f(c(a + bx)S(2))/(S(2)b*S(4)c*S(2)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)*S(2))x*S(2), x), x, sqrt(pi)a*S(2)erfi(sqrt(c)(a + bx)sqrt(log(f)))/(S(2)b*S(3)sqrt(c)sqrt(log(f))) - af*(c(a + bx)S(2))/(bS(3)clog(f)) + f(c(a + bx)S(2))(a + bx)/(S(2)b*S(3)clog(f)) - sqrt(pi)erfi(sqrt(c)(a + bx)sqrt(log(f)))/(S(4)b*S(3)c*(S(3)/2)log(f)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)*S(2))x, x), x, -sqrt(pi)aerfi(sqrt(c)(a + bx)sqrt(log(f)))/(S(2)b*S(2)sqrt(c)sqrt(log(f))) + f(c(a + bx)S(2))/(S(2)b*S(2)clog(f)), expand=True, _diff=True, _numerical=True) # long time in rubi_test(1940 is matched before 1909) assert rubi_test(rubi_integrate(f(c(a + bx)S(2)), x), x, sqrt(pi)erfi(sqrt(c)(a + bx)sqrt(log(f)))/(S(2)bsqrt(c)sqrt(log(f))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c(a + bx)S(2))/x, x), x, Integral(f(c(a + bx)S(2))/x, x), expand=True, _diff=True, _numerical=True) # long time in rubi_test(1940 is matched before 1909) assert rubi_test(rubi_integrate(f(c(a + bx)S(2))/xS(2), x), x, S(2)abclog(f)Integral(f*(c(a + bx)S(2))/x, x) + sqrt(pi)bsqrt(c)sqrt(log(f))erfi(sqrt(c)(a + bx)sqrt(log(f))) - f*(c(a + bx)S(2))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)S(2))/xS(3), x), x, S(2)a*S(2)b*S(2)c*S(2)log(f)*S(2)Integral(f*(c(a + bx)S(2))/x, x) + sqrt(pi)abS(2)c*(S(3)/2)log(f)*(S(3)/2)erfi(sqrt(c)(a + bx)sqrt(log(f))) - abcf*(c(a + bx)S(2))log(f)/x + b*S(2)clog(f)Integral(f*(c(a + bx)S(2))/x, x) - f(c(a + bx)S(2))/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)*S(3))xm, x), x, Integral(f(c(a + bx)*S(3))xm, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)*S(3))xS(2), x), x, -aS(2)(a + bx)Gamma(S(1)/3, -c(a + bx)S(3)log(f))/(S(3)bS(3)(-c(a + bx)*S(3)log(f))*(S(1)/3)) + S(2)a(a + bx)*S(2)Gamma(S(2)/3, -c(a + bx)*S(3)log(f))/(S(3)bS(3)(-c(a + bx)*S(3)log(f))(S(2)/3)) + f(c(a + bx)*S(3))/(S(3)b*S(3)clog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)S(3))x, x), x, a(a + bx)Gamma(S(1)/3, -c(a + bx)S(3)log(f))/(S(3)bS(2)(-c(a + bx)*S(3)log(f))*(S(1)/3)) - (a + bx)*S(2)Gamma(S(2)/3, -c(a + bx)*S(3)log(f))/(S(3)bS(2)(-c(a + bx)*S(3)log(f))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)*S(3)), x), x, (-a/S(3) - bx/S(3))Gamma(S(1)/3, -c(a + bx)S(3)log(f))/(b(-c(a + bx)S(3)log(f))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)S(3))/x, x), x, Integral(f(c(a + bx)S(3))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)S(3))/xS(2), x), x, S(3)aS(2)bclog(f)Integral(f(c(a + bx)S(3))/x, x) - abc(a + bx)Gamma(S(1)/3, -c(a + bx)*S(3)log(f))log(f)/(-c(a + bx)S(3)log(f))*(S(1)/3) - bc(a + bx)*S(2)Gamma(S(2)/3, -c(a + bx)*S(3)log(f))log(f)/(-c(a + bx)S(3)log(f))(S(2)/3) - f(c(a + bx)S(3))/x, expand=True, _diff=True, _numerical=True) # difference in simplify of sympy and mathematica assert rubi_test(rubi_integrate(f(c(a + bx)S(3))/xS(3), x), x, S(9)aS(4)b*S(2)c*S(2)log(f)*S(2)Integral(f*(c(a + bx)S(3))/x, x)/S(2) - S(3)a*S(3)b*S(2)c*S(2)(a + bx)Gamma(S(1)/3, -c(a + bx)*S(3)log(f))log(f)S(2)/(S(2)(-c(a + bx)*S(3)log(f))*(S(1)/3)) - S(3)a*S(2)b*S(2)c*S(2)(a + bx)S(2)Gamma(S(2)/3, -c(a + bx)*S(3)log(f))log(f)S(2)/(S(2)(-c(a + bx)*S(3)log(f))*(S(2)/3)) - S(3)a*S(2)bcf*(c(a + bx)S(3))log(f)/(S(2)x) + S(3)abS(2)clog(f)Integral(f*(c(a + bx)S(3))/x, x) - bS(2)c(a + bx)Gamma(S(1)/3, -c(a + bx)S(3)log(f))log(f)/(S(2)(-c(a + bx)*S(3)log(f))(S(1)/3)) - f(c(a + bx)*S(3))/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmexp(aS(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3)), x), x, Integral(xmexp(a*S(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)exp(a*S(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3)), x), x, -aS(4)(a + bx)Gamma(S(1)/3, -(a + bx)S(3))/(S(3)b*S(5)(-(a + bx)S(3))(S(1)/3)) + S(4)a*S(3)(a + bx)S(2)Gamma(S(2)/3, -(a + bx)S(3))/(S(3)b*S(5)(-(a + bx)S(3))(S(2)/3)) + S(2)a*S(2)exp((a + bx)S(3))/bS(5) + S(4)a(a + bx)*S(4)Gamma(S(4)/3, -(a + bx)S(3))/(S(3)b*S(5)(-(a + bx)S(3))(S(4)/3)) - (a + bx)*S(5)Gamma(S(5)/3, -(a + bx)S(3))/(S(3)b*S(5)(-(a + bx)S(3))(S(5)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)exp(a*S(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3)), x), x, aS(3)(a + bx)Gamma(S(1)/3, -(a + bx)S(3))/(S(3)b*S(4)(-(a + bx)S(3))(S(1)/3)) - aS(2)(a + bx)S(2)Gamma(S(2)/3, -(a + bx)S(3))/(bS(4)(-(a + bx)S(3))(S(2)/3)) - aexp((a + bx)S(3))/bS(4) - (a + bx)*S(4)Gamma(S(4)/3, -(a + bx)S(3))/(S(3)b*S(4)(-(a + bx)S(3))(S(4)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)exp(a*S(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3)), x), x, -aS(2)(a + bx)Gamma(S(1)/3, -(a + bx)S(3))/(S(3)b*S(3)(-(a + bx)S(3))(S(1)/3)) + S(2)a(a + bx)*S(2)Gamma(S(2)/3, -(a + bx)S(3))/(S(3)b*S(3)(-(a + bx)S(3))(S(2)/3)) + exp((a + bx)*S(3))/(S(3)b*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp(a*S(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3)), x), x, a(a + bx)Gamma(S(1)/3, -(a + bx)S(3))/(S(3)b*S(2)(-(a + bx)S(3))(S(1)/3)) - (a + bx)*S(2)Gamma(S(2)/3, -(a + bx)S(3))/(S(3)b*S(2)(-(a + bx)S(3))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(aS(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3)), x), x, (-a/S(3) - bx/S(3))Gamma(S(1)/3, -(a + bx)*S(3))/(b(-(a + bx)S(3))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(aS(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3))/x, x), x, Integral(exp(aS(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(sqrt(S(3)x + S(5))), x), x, S(2)sqrt(S(3)x + S(5))exp(sqrt(S(3)x + S(5)))/S(3) - S(2)exp(sqrt(S(3)x + S(5)))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c/(a + bx))xm, x), x, Integral(f(c/(a + bx))xm, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx))xS(4), x), x, -aS(4)clog(f)Ei(clog(f)/(a + bx))/bS(5) + aS(4)f(c/(a + bx))(a + bx)/b*S(5) + S(2)a*S(3)c*S(2)log(f)*S(2)Ei(clog(f)/(a + bx))/b*S(5) - S(2)a*S(3)cf(c/(a + bx))(a + bx)log(f)/bS(5) - S(2)a*S(3)f*(c/(a + bx))(a + bx)S(2)/bS(5) - a*S(2)c*S(3)log(f)*S(3)Ei(clog(f)/(a + bx))/bS(5) + aS(2)cS(2)f*(c/(a + bx))(a + bx)log(f)S(2)/bS(5) + aS(2)cf(c/(a + bx))(a + bx)*S(2)log(f)/b*S(5) + S(2)a*S(2)f*(c/(a + bx))(a + bx)S(3)/bS(5) - S(4)ac*S(4)Gamma(S(-4), -clog(f)/(a + bx))log(f)S(4)/bS(5) - cS(5)Gamma(S(-5), -clog(f)/(a + bx))log(f)S(5)/bS(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx))xS(3), x), x, aS(3)clog(f)Ei(clog(f)/(a + bx))/bS(4) - aS(3)f(c/(a + bx))(a + bx)/b*S(4) - S(3)a*S(2)c*S(2)log(f)*S(2)Ei(clog(f)/(a + bx))/(S(2)bS(4)) + S(3)a*S(2)cf(c/(a + bx))(a + bx)log(f)/(S(2)b*S(4)) + S(3)a*S(2)f*(c/(a + bx))(a + bx)*S(2)/(S(2)b*S(4)) + ac*S(3)log(f)*S(3)Ei(clog(f)/(a + bx))/(S(2)bS(4)) - ac*S(2)f*(c/(a + bx))(a + bx)log(f)S(2)/(S(2)b*S(4)) - acf(c/(a + bx))(a + bx)*S(2)log(f)/(S(2)bS(4)) - af*(c/(a + bx))(a + bx)S(3)/bS(4) + c*S(4)Gamma(S(-4), -clog(f)/(a + bx))log(f)S(4)/bS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx))xS(2), x), x, -aS(2)clog(f)Ei(clog(f)/(a + bx))/bS(3) + aS(2)f(c/(a + bx))(a + bx)/b*S(3) + ac*S(2)log(f)*S(2)Ei(clog(f)/(a + bx))/b*S(3) - acf(c/(a + bx))(a + bx)log(f)/bS(3) - af*(c/(a + bx))(a + bx)S(2)/bS(3) - c*S(3)log(f)*S(3)Ei(clog(f)/(a + bx))/(S(6)bS(3)) + cS(2)f*(c/(a + bx))(a + bx)log(f)S(2)/(S(6)b*S(3)) + cf*(c/(a + bx))(a + bx)*S(2)log(f)/(S(6)bS(3)) + f(c/(a + bx))(a + bx)*S(3)/(S(3)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx))x, x), x, aclog(f)Ei(clog(f)/(a + bx))/bS(2) - af*(c/(a + bx))(a + bx)/bS(2) - cS(2)log(f)S(2)Ei(clog(f)/(a + bx))/(S(2)bS(2)) + cf*(c/(a + bx))(a + bx)log(f)/(S(2)bS(2)) + f(c/(a + bx))(a + bx)S(2)/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)), x), x, -clog(f)Ei(clog(f)/(a + bx))/b + f(c/(a + bx))(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c/(a + bx))/x, x), x, f*(c/a)Ei(-bcxlog(f)/(a(a + bx))) - Ei(clog(f)/(a + bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx))/xS(2), x), x, -f(c/(a + bx))/x - bf*(c/(a + bx))/a - bcf*(c/a)log(f)Ei(-bcxlog(f)/(a(a + bx)))/aS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx))/xS(3), x), x, -f(c/(a + bx))/(S(2)xS(2)) + bS(2)f*(c/(a + bx))/(S(2)aS(2)) + bcf(c/(a + bx))log(f)/(S(2)a*S(2)x) + b*S(2)cf(c/a)log(f)Ei(-bcxlog(f)/(a(a + bx)))/aS(3) + bS(2)cf*(c/(a + bx))log(f)/(S(2)aS(3)) + bS(2)cS(2)f*(c/a)log(f)*S(2)Ei(-bcxlog(f)/(a(a + bx)))/(S(2)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(2))xm, x), x, Integral(f(c/(a + bx)S(2))xm, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(2))x*S(4), x), x, -sqrt(pi)a*S(4)sqrt(c)sqrt(log(f))erfi(sqrt(c)sqrt(log(f))/(a + bx))/bS(5) + aS(4)f(c/(a + bx)*S(2))(a + bx)/bS(5) + S(2)a*S(3)clog(f)Ei(clog(f)/(a + bx)S(2))/bS(5) - S(2)aS(3)f*(c/(a + bx)*S(2))(a + bx)S(2)/bS(5) - S(4)sqrt(pi)aS(2)c*(S(3)/2)log(f)*(S(3)/2)erfi(sqrt(c)sqrt(log(f))/(a + bx))/b*S(5) + S(4)a*S(2)cf(c/(a + bx)*S(2))(a + bx)log(f)/b*S(5) + S(2)a*S(2)f*(c/(a + bx)*S(2))(a + bx)S(3)/bS(5) + ac*S(2)log(f)*S(2)Ei(clog(f)/(a + bx)S(2))/bS(5) - acf*(c/(a + bx)*S(2))(a + bx)S(2)log(f)/b*S(5) - af*(c/(a + bx)*S(2))(a + bx)S(4)/bS(5) - S(4)sqrt(pi)c(S(5)/2)log(f)*(S(5)/2)erfi(sqrt(c)sqrt(log(f))/(a + bx))/(S(15)bS(5)) + S(4)c*S(2)f*(c/(a + bx)*S(2))(a + bx)log(f)*S(2)/(S(15)b*S(5)) + S(2)cf(c/(a + bx)*S(2))(a + bx)S(3)log(f)/(S(15)bS(5)) + f(c/(a + bx)*S(2))(a + bx)S(5)/(S(5)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(2))x*S(3), x), x, sqrt(pi)a*S(3)sqrt(c)sqrt(log(f))erfi(sqrt(c)sqrt(log(f))/(a + bx))/bS(4) - aS(3)f(c/(a + bx)*S(2))(a + bx)/bS(4) - S(3)a*S(2)clog(f)Ei(clog(f)/(a + bx)*S(2))/(S(2)b*S(4)) + S(3)a*S(2)f*(c/(a + bx)*S(2))(a + bx)S(2)/(S(2)b*S(4)) + S(2)sqrt(pi)ac*(S(3)/2)log(f)*(S(3)/2)erfi(sqrt(c)sqrt(log(f))/(a + bx))/b*S(4) - S(2)acf*(c/(a + bx)*S(2))(a + bx)log(f)/b*S(4) - af*(c/(a + bx)*S(2))(a + bx)S(3)/bS(4) - cS(2)log(f)*S(2)Ei(clog(f)/(a + bx)*S(2))/(S(4)b*S(4)) + cf*(c/(a + bx)*S(2))(a + bx)S(2)log(f)/(S(4)bS(4)) + f(c/(a + bx)*S(2))(a + bx)S(4)/(S(4)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(2))x*S(2), x), x, -sqrt(pi)a*S(2)sqrt(c)sqrt(log(f))erfi(sqrt(c)sqrt(log(f))/(a + bx))/bS(3) + aS(2)f(c/(a + bx)*S(2))(a + bx)/bS(3) + aclog(f)Ei(clog(f)/(a + bx)S(2))/bS(3) - af(c/(a + bx)*S(2))(a + bx)S(2)/bS(3) - S(2)sqrt(pi)c(S(3)/2)log(f)*(S(3)/2)erfi(sqrt(c)sqrt(log(f))/(a + bx))/(S(3)bS(3)) + S(2)cf(c/(a + bx)*S(2))(a + bx)log(f)/(S(3)bS(3)) + f(c/(a + bx)*S(2))(a + bx)S(3)/(S(3)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(2))x, x), x, sqrt(pi)asqrt(c)sqrt(log(f))erfi(sqrt(c)sqrt(log(f))/(a + bx))/b*S(2) - af*(c/(a + bx)*S(2))(a + bx)/bS(2) - clog(f)Ei(clog(f)/(a + bx)S(2))/(S(2)bS(2)) + f(c/(a + bx)S(2))(a + bx)S(2)/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(2)), x), x, -sqrt(pi)sqrt(c)sqrt(log(f))erfi(sqrt(c)sqrt(log(f))/(a + bx))/b + f*(c/(a + bx)*S(2))(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(2))/x, x), x, Integral(f(c/(a + bx)S(2))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(2))/xS(2), x), x, Integral(f*(c/(a + bx)S(2))/xS(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c/(a + bx)S(2))/xS(3), x), x, Integral(f*(c/(a + bx)S(2))/xS(3), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c/(a + bx)*S(3))xm, x), x, Integral(f(c/(a + bx)S(3))xm, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(3))xS(4), x), x, aS(4)(-clog(f)/(a + bx)S(3))(S(1)/3)(a + bx)Gamma(S(-1)/3, -clog(f)/(a + bx)*S(3))/(S(3)b*S(5)) - S(4)a*S(3)(-clog(f)/(a + bx)S(3))(S(2)/3)(a + bx)*S(2)Gamma(S(-2)/3, -clog(f)/(a + bx)*S(3))/(S(3)b*S(5)) - S(2)a*S(2)clog(f)Ei(clog(f)/(a + bx)S(3))/bS(5) + S(2)aS(2)f*(c/(a + bx)*S(3))(a + bx)S(3)/bS(5) - S(4)a(-clog(f)/(a + bx)S(3))(S(4)/3)(a + bx)S(4)Gamma(S(-4)/3, -clog(f)/(a + bx)*S(3))/(S(3)b*S(5)) + (-clog(f)/(a + bx)S(3))(S(5)/3)(a + bx)S(5)Gamma(S(-5)/3, -clog(f)/(a + bx)*S(3))/(S(3)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(3))xS(3), x), x, -aS(3)(-clog(f)/(a + bx)S(3))(S(1)/3)(a + bx)Gamma(S(-1)/3, -clog(f)/(a + bx)*S(3))/(S(3)bS(4)) + aS(2)(-clog(f)/(a + bx)S(3))(S(2)/3)(a + bx)S(2)Gamma(S(-2)/3, -clog(f)/(a + bx)S(3))/bS(4) + aclog(f)Ei(clog(f)/(a + bx)S(3))/bS(4) - af*(c/(a + bx)*S(3))(a + bx)S(3)/bS(4) + (-clog(f)/(a + bx)S(3))(S(4)/3)(a + bx)S(4)Gamma(S(-4)/3, -clog(f)/(a + bx)*S(3))/(S(3)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(3))xS(2), x), x, aS(2)(-clog(f)/(a + bx)S(3))(S(1)/3)(a + bx)Gamma(S(-1)/3, -clog(f)/(a + bx)*S(3))/(S(3)b*S(3)) - S(2)a(-clog(f)/(a + bx)S(3))(S(2)/3)(a + bx)S(2)Gamma(S(-2)/3, -clog(f)/(a + bx)*S(3))/(S(3)b*S(3)) - clog(f)Ei(clog(f)/(a + bx)S(3))/(S(3)bS(3)) + f(c/(a + bx)S(3))(a + bx)S(3)/(S(3)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(3))x, x), x, -a(-clog(f)/(a + bx)S(3))(S(1)/3)(a + bx)Gamma(S(-1)/3, -clog(f)/(a + bx)*S(3))/(S(3)b*S(2)) + (-clog(f)/(a + bx)S(3))(S(2)/3)(a + bx)S(2)Gamma(S(-2)/3, -clog(f)/(a + bx)*S(3))/(S(3)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(3)), x), x, (-clog(f)/(a + bx)S(3))(S(1)/3)(a/S(3) + bx/S(3))Gamma(S(-1)/3, -clog(f)/(a + bx)S(3))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(3))/x, x), x, Integral(f(c/(a + bx)S(3))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c/(a + bx)S(3))/xS(2), x), x, Integral(f(c/(a + bx)S(3))/xS(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c/(a + bx)S(3))/xS(3), x), x, Integral(f*(c/(a + bx)S(3))/xS(3), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c(a + bx)n)xm, x), x, Integral(f(c(a + bx)*n)xm, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)*n)xS(3), x), x, aS(3)(-c(a + bx)nlog(f))*(-S(1)/n)(a + bx)Gamma(S(1)/n, -c(a + bx)*nlog(f))/(b*S(4)n) - S(3)aS(2)(-c(a + bx)*nlog(f))*(-S(2)/n)(a + bx)S(2)Gamma(S(2)/n, -c(a + bx)*nlog(f))/(b*S(4)n) + S(3)a(-c(a + bx)*nlog(f))*(-S(3)/n)(a + bx)S(3)Gamma(S(3)/n, -c(a + bx)*nlog(f))/(b*S(4)n) - (-c(a + bx)*nlog(f))*(-S(4)/n)(a + bx)S(4)Gamma(S(4)/n, -c(a + bx)*nlog(f))/(b*S(4)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c(a + bx)n)xS(2), x), x, -aS(2)(-c(a + bx)nlog(f))*(-S(1)/n)(a + bx)Gamma(S(1)/n, -c(a + bx)*nlog(f))/(b*S(3)n) + S(2)a(-c(a + bx)*nlog(f))*(-S(2)/n)(a + bx)S(2)Gamma(S(2)/n, -c(a + bx)*nlog(f))/(b*S(3)n) - (-c(a + bx)*nlog(f))*(-S(3)/n)(a + bx)S(3)Gamma(S(3)/n, -c(a + bx)*nlog(f))/(b*S(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c(a + bx)n)x, x), x, a(-c(a + bx)nlog(f))*(-S(1)/n)(a + bx)Gamma(S(1)/n, -c(a + bx)*nlog(f))/(b*S(2)n) - (-c(a + bx)*nlog(f))*(-S(2)/n)(a + bx)S(2)Gamma(S(2)/n, -c(a + bx)*nlog(f))/(b*S(2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(c(a + bx)n), x), x, (-c(a + bx)nlog(f))*(-S(1)/n)(-a - bx)Gamma(S(1)/n, -c(a + bx)*nlog(f))/(bn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)n)/x, x), x, Integral(f(c(a + bx)n)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)n)/xS(2), x), x, Integral(f(c(a + bx)n)/xS(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(c(a + bx)n)/xS(3), x), x, Integral(f(c(a + bx)n)/xS(3), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))(c + dx)m, x), x, -Fa(-b(c + dx)*S(2)log(F))*(-m/S(2) + S(-1)/2)(c + dx)(m + S(1))Gamma(m/S(2) + S(1)/2, -b(c + dx)*S(2)log(F))/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))(c + dx)S(11), x), x, -FaGamma(S(6), -b(c + dx)*S(2)log(F))/(S(2)bS(6)dlog(F)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))(c + dx)S(9), x), x, FaGamma(S(5), -b(c + dx)*S(2)log(F))/(S(2)bS(5)dlog(F)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))(c + dx)S(7), x), x, F(a + b(c + dx)S(2))(c + dx)S(6)/(S(2)bdlog(F)) - S(3)F(a + b(c + dx)S(2))(c + dx)S(4)/(S(2)b*S(2)dlog(F)S(2)) + S(3)F*(a + b(c + dx)S(2))(c + dx)S(2)/(bS(3)dlog(F)S(3)) - S(3)F*(a + b(c + dx)S(2))/(bS(4)dlog(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))(c + dx)S(5), x), x, F(a + b(c + dx)S(2))(c + dx)S(4)/(S(2)bdlog(F)) - F*(a + b(c + dx)S(2))(c + dx)S(2)/(bS(2)dlog(F)S(2)) + F(a + b(c + dx)S(2))/(bS(3)dlog(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))(c + dx)S(3), x), x, F(a + b(c + dx)S(2))(c + dx)S(2)/(S(2)bdlog(F)) - F*(a + b(c + dx)S(2))/(S(2)b*S(2)dlog(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))(c + dx), x), x, F(a + b(c + dx)S(2))/(S(2)bdlog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)S(2))/(c + dx), x), x, F*aEi(b(c + dx)*S(2)log(F))/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))/(c + dx)S(3), x), x, Fablog(F)Ei(b(c + dx)S(2)log(F))/(S(2)d) - F(a + b(c + dx)S(2))/(S(2)d(c + dx)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(2))/(c + dx)S(5), x), x, FabS(2)log(F)*S(2)Ei(b(c + dx)*S(2)log(F))/(S(4)d) - F(a + b(c + dx)S(2))blog(F)/(S(4)d(c + dx)S(2)) - F(a + b(c + dx)*S(2))/(S(4)d(c + dx)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(2))/(c + dx)S(7), x), x, FabS(3)log(F)*S(3)Ei(b(c + dx)*S(2)log(F))/(S(12)d) - F(a + b(c + dx)S(2))b*S(2)log(F)*S(2)/(S(12)d(c + dx)S(2)) - F(a + b(c + dx)*S(2))blog(F)/(S(12)d(c + dx)S(4)) - F(a + b(c + dx)*S(2))/(S(6)d(c + dx)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(2))/(c + dx)S(9), x), x, -FabS(4)Gamma(S(-4), -b(c + dx)*S(2)log(F))log(F)S(4)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)S(2))/(c + dx)S(11), x), x, FabS(5)Gamma(S(-5), -b(c + dx)*S(2)log(F))log(F)S(5)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)S(2))(c + dx)S(12), x), x, -Fa(c + dx)S(13)Gamma(S(13)/2, -b(c + dx)*S(2)log(F))/(S(2)d(-b(c + dx)*S(2)log(F))(S(13)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(2))(c + dx)S(10), x), x, -Fa(c + dx)S(11)Gamma(S(11)/2, -b(c + dx)*S(2)log(F))/(S(2)d(-b(c + dx)*S(2)log(F))(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(2))(c + dx)S(8), x), x, S(105)sqrt(pi)Faerfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(32)b*(S(9)/2)dlog(F)(S(9)/2)) + F(a + b(c + dx)S(2))(c + dx)S(7)/(S(2)bdlog(F)) - S(7)F(a + b(c + dx)S(2))(c + dx)S(5)/(S(4)b*S(2)dlog(F)S(2)) + S(35)F*(a + b(c + dx)S(2))(c + dx)S(3)/(S(8)b*S(3)dlog(F)S(3)) - S(105)F*(a + b(c + dx)S(2))(c + dx)/(S(16)b*S(4)dlog(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))(c + dx)S(6), x), x, -S(15)sqrt(pi)Faerfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(16)b*(S(7)/2)dlog(F)(S(7)/2)) + F(a + b(c + dx)S(2))(c + dx)S(5)/(S(2)bdlog(F)) - S(5)F(a + b(c + dx)S(2))(c + dx)S(3)/(S(4)b*S(2)dlog(F)S(2)) + S(15)F*(a + b(c + dx)S(2))(c + dx)/(S(8)b*S(3)dlog(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))(c + dx)S(4), x), x, S(3)sqrt(pi)Faerfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(8)b*(S(5)/2)dlog(F)(S(5)/2)) + F(a + b(c + dx)S(2))(c + dx)S(3)/(S(2)bdlog(F)) - S(3)F(a + b(c + dx)S(2))(c + dx)/(S(4)b*S(2)dlog(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))(c + dx)S(2), x), x, -sqrt(pi)F*aerfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(4)b*(S(3)/2)dlog(F)(S(3)/2)) + F(a + b(c + dx)S(2))(c + dx)/(S(2)bdlog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)S(2)), x), x, sqrt(pi)F*aerfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(2)sqrt(b)dsqrt(log(F))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)S(2))/(c + dx)*S(2), x), x, sqrt(pi)F*asqrt(b)sqrt(log(F))erfi(sqrt(b)(c + dx)sqrt(log(F)))/d - F(a + b(c + dx)S(2))/(d(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))/(c + dx)*S(4), x), x, S(2)sqrt(pi)Fab*(S(3)/2)log(F)*(S(3)/2)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(3)d) - S(2)F(a + b(c + dx)S(2))blog(F)/(S(3)d(c + dx)) - F*(a + b(c + dx)S(2))/(S(3)d(c + dx)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(2))/(c + dx)*S(6), x), x, S(4)sqrt(pi)Fab*(S(5)/2)log(F)*(S(5)/2)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(15)d) - S(4)F(a + b(c + dx)S(2))b*S(2)log(F)*S(2)/(S(15)d(c + dx)) - S(2)F(a + b(c + dx)S(2))blog(F)/(S(15)d(c + dx)S(3)) - F(a + b(c + dx)*S(2))/(S(5)d(c + dx)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(2))/(c + dx)*S(8), x), x, S(8)sqrt(pi)Fab*(S(7)/2)log(F)*(S(7)/2)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(105)d) - S(8)F(a + b(c + dx)S(2))b*S(3)log(F)*S(3)/(S(105)d(c + dx)) - S(4)F(a + b(c + dx)S(2))b*S(2)log(F)*S(2)/(S(105)d(c + dx)*S(3)) - S(2)F*(a + b(c + dx)S(2))blog(F)/(S(35)d(c + dx)S(5)) - F(a + b(c + dx)*S(2))/(S(7)d(c + dx)S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(2))/(c + dx)S(10), x), x, -Fa(-b(c + dx)S(2)log(F))*(S(9)/2)Gamma(S(-9)/2, -b(c + dx)*S(2)log(F))/(S(2)d(c + dx)S(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))/(c + dx)S(12), x), x, -Fa(-b(c + dx)S(2)log(F))*(S(11)/2)Gamma(S(-11)/2, -b(c + dx)*S(2)log(F))/(S(2)d(c + dx)S(11)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(3))(c + dx)m, x), x, -Fa(-b(c + dx)*S(3)log(F))*(-m/S(3) + S(-1)/3)(c + dx)(m + S(1))Gamma(m/S(3) + S(1)/3, -b(c + dx)*S(3)log(F))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(3))(c + dx)S(17), x), x, -FaGamma(S(6), -b(c + dx)*S(3)log(F))/(S(3)bS(6)dlog(F)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(3))(c + dx)S(14), x), x, FaGamma(S(5), -b(c + dx)*S(3)log(F))/(S(3)bS(5)dlog(F)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(3))(c + dx)S(11), x), x, F(a + b(c + dx)S(3))(c + dx)S(9)/(S(3)bdlog(F)) - F*(a + b(c + dx)S(3))(c + dx)S(6)/(bS(2)dlog(F)S(2)) + S(2)F*(a + b(c + dx)S(3))(c + dx)S(3)/(bS(3)dlog(F)S(3)) - S(2)F*(a + b(c + dx)S(3))/(bS(4)dlog(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(3))(c + dx)S(8), x), x, F(a + b(c + dx)S(3))(c + dx)S(6)/(S(3)bdlog(F)) - S(2)F(a + b(c + dx)S(3))(c + dx)S(3)/(S(3)b*S(2)dlog(F)S(2)) + S(2)F*(a + b(c + dx)S(3))/(S(3)b*S(3)dlog(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(3))(c + dx)S(5), x), x, F(a + b(c + dx)S(3))(c + dx)S(3)/(S(3)bdlog(F)) - F*(a + b(c + dx)S(3))/(S(3)b*S(2)dlog(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(3))(c + dx)S(2), x), x, F(a + b(c + dx)S(3))/(S(3)bdlog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)S(3))/(c + dx), x), x, F*aEi(b(c + dx)*S(3)log(F))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(3))/(c + dx)S(4), x), x, Fablog(F)Ei(b(c + dx)S(3)log(F))/(S(3)d) - F(a + b(c + dx)S(3))/(S(3)d(c + dx)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(3))/(c + dx)S(7), x), x, FabS(2)log(F)*S(2)Ei(b(c + dx)*S(3)log(F))/(S(6)d) - F(a + b(c + dx)S(3))blog(F)/(S(6)d(c + dx)S(3)) - F(a + b(c + dx)*S(3))/(S(6)d(c + dx)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(3))/(c + dx)S(10), x), x, FabS(3)log(F)*S(3)Ei(b(c + dx)*S(3)log(F))/(S(18)d) - F(a + b(c + dx)S(3))b*S(2)log(F)*S(2)/(S(18)d(c + dx)S(3)) - F(a + b(c + dx)*S(3))blog(F)/(S(18)d(c + dx)S(6)) - F(a + b(c + dx)*S(3))/(S(9)d(c + dx)S(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(3))/(c + dx)S(13), x), x, -FabS(4)Gamma(S(-4), -b(c + dx)*S(3)log(F))log(F)S(4)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)S(3))/(c + dx)S(16), x), x, FabS(5)Gamma(S(-5), -b(c + dx)*S(3)log(F))log(F)S(5)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)S(3))(c + dx)S(3), x), x, -Fa(c + dx)S(4)Gamma(S(4)/3, -b(c + dx)*S(3)log(F))/(S(3)d(-b(c + dx)*S(3)log(F))(S(4)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(3))(c + dx), x), x, -Fa(c + dx)S(2)Gamma(S(2)/3, -b(c + dx)*S(3)log(F))/(S(3)d(-b(c + dx)*S(3)log(F))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(3)), x), x, -Fa(c + dx)Gamma(S(1)/3, -b(c + dx)S(3)log(F))/(S(3)d(-b(c + dx)*S(3)log(F))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(3))/(c + dx)S(2), x), x, -Fa(-b(c + dx)S(3)log(F))*(S(1)/3)Gamma(S(-1)/3, -b(c + dx)*S(3)log(F))/(S(3)d(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(3))/(c + dx)S(3), x), x, -Fa(-b(c + dx)S(3)log(F))*(S(2)/3)Gamma(S(-2)/3, -b(c + dx)*S(3)log(F))/(S(3)d(c + dx)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(3))/(c + dx)S(5), x), x, -Fa(-b(c + dx)S(3)log(F))*(S(4)/3)Gamma(S(-4)/3, -b(c + dx)*S(3)log(F))/(S(3)d(c + dx)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bsqrt(c + dx)), x), x, S(2)f*(a + bsqrt(c + dx))sqrt(c + dx)/(bdlog(f)) - S(2)f*(a + bsqrt(c + dx))/(bS(2)dlog(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + b(c + dx)(S(1)/3)), x), x, S(3)f*(a + b(c + dx)(S(1)/3))(c + dx)(S(2)/3)/(bdlog(f)) - S(6)f*(a + b(c + dx)(S(1)/3))(c + dx)(S(1)/3)/(bS(2)dlog(f)S(2)) + S(6)f*(a + b(c + dx)(S(1)/3))/(bS(3)dlog(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx))(c + dx)m, x), x, Fa(-blog(F)/(c + dx))(m + S(1))(c + dx)(m + S(1))Gamma(-m + S(-1), -blog(F)/(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx))(c + dx)S(4), x), x, -FabS(5)Gamma(S(-5), -blog(F)/(c + dx))log(F)S(5)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx))(c + dx)S(3), x), x, FabS(4)Gamma(S(-4), -blog(F)/(c + dx))log(F)S(4)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx))(c + dx)S(2), x), x, -FabS(3)log(F)*S(3)Ei(blog(F)/(c + dx))/(S(6)d) + F(a + b/(c + dx))bS(2)(c + dx)log(F)*S(2)/(S(6)d) + F*(a + b/(c + dx))b(c + dx)S(2)log(F)/(S(6)d) + F(a + b/(c + dx))(c + dx)*S(3)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx))(c + dx), x), x, -F*ab*S(2)log(F)*S(2)Ei(blog(F)/(c + dx))/(S(2)d) + F(a + b/(c + dx))b(c + dx)log(F)/(S(2)d) + F(a + b/(c + dx))(c + dx)*S(2)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)), x), x, -F*ablog(F)Ei(blog(F)/(c + dx))/d + F*(a + b/(c + dx))(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx))/(c + dx), x), x, -FaEi(blog(F)/(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx))/(c + dx)S(2), x), x, -F(a + b/(c + dx))/(bdlog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx))/(c + dx)S(3), x), x, -F(a + b/(c + dx))/(bd(c + dx)log(F)) + F*(a + b/(c + dx))/(b*S(2)dlog(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx))/(c + dx)S(4), x), x, -F(a + b/(c + dx))/(bd(c + dx)S(2)log(F)) + S(2)F(a + b/(c + dx))/(b*S(2)d(c + dx)log(F)S(2)) - S(2)F*(a + b/(c + dx))/(b*S(3)dlog(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx))/(c + dx)S(5), x), x, -F(a + b/(c + dx))/(bd(c + dx)S(3)log(F)) + S(3)F(a + b/(c + dx))/(b*S(2)d(c + dx)*S(2)log(F)*S(2)) - S(6)F*(a + b/(c + dx))/(b*S(3)d(c + dx)log(F)S(3)) + S(6)F*(a + b/(c + dx))/(b*S(4)dlog(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx))/(c + dx)S(6), x), x, -FaGamma(S(5), -blog(F)/(c + dx))/(b*S(5)dlog(F)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx))/(c + dx)S(7), x), x, FaGamma(S(6), -blog(F)/(c + dx))/(b*S(6)dlog(F)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(2))(c + dx)m, x), x, Fa(-blog(F)/(c + dx)S(2))(m/S(2) + S(1)/2)(c + dx)*(m + S(1))Gamma(-m/S(2) + S(-1)/2, -blog(F)/(c + dx)*S(2))/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))(c + dx)S(9), x), x, -Fab*S(5)Gamma(S(-5), -blog(F)/(c + dx)*S(2))log(F)*S(5)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))(c + dx)S(7), x), x, Fab*S(4)Gamma(S(-4), -blog(F)/(c + dx)*S(2))log(F)*S(4)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))(c + dx)S(5), x), x, -Fab*S(3)log(F)*S(3)Ei(blog(F)/(c + dx)*S(2))/(S(12)d) + F*(a + b/(c + dx)*S(2))b*S(2)(c + dx)S(2)log(F)*S(2)/(S(12)d) + F*(a + b/(c + dx)*S(2))b(c + dx)*S(4)log(F)/(S(12)d) + F(a + b/(c + dx)*S(2))(c + dx)S(6)/(S(6)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))(c + dx)S(3), x), x, -Fab*S(2)log(F)*S(2)Ei(blog(F)/(c + dx)*S(2))/(S(4)d) + F*(a + b/(c + dx)*S(2))b(c + dx)*S(2)log(F)/(S(4)d) + F(a + b/(c + dx)*S(2))(c + dx)S(4)/(S(4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))(c + dx), x), x, -Fablog(F)Ei(blog(F)/(c + dx)*S(2))/(S(2)d) + F*(a + b/(c + dx)*S(2))(c + dx)S(2)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))/(c + dx), x), x, -F*aEi(blog(F)/(c + dx)*S(2))/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))/(c + dx)S(3), x), x, -F(a + b/(c + dx)S(2))/(S(2)bdlog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))/(c + dx)S(5), x), x, -F(a + b/(c + dx)S(2))/(S(2)bd(c + dx)S(2)log(F)) + F*(a + b/(c + dx)*S(2))/(S(2)b*S(2)dlog(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(2))/(c + dx)S(7), x), x, -F(a + b/(c + dx)S(2))/(S(2)bd(c + dx)S(4)log(F)) + F*(a + b/(c + dx)S(2))/(bS(2)d(c + dx)S(2)log(F)S(2)) - F(a + b/(c + dx)S(2))/(bS(3)dlog(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(2))/(c + dx)S(9), x), x, -F(a + b/(c + dx)S(2))/(S(2)bd(c + dx)S(6)log(F)) + S(3)F(a + b/(c + dx)*S(2))/(S(2)b*S(2)d(c + dx)*S(4)log(F)*S(2)) - S(3)F*(a + b/(c + dx)S(2))/(bS(3)d(c + dx)S(2)log(F)*S(3)) + S(3)F*(a + b/(c + dx)S(2))/(bS(4)dlog(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)S(2))/(c + dx)S(11), x), x, -FaGamma(S(5), -blog(F)/(c + dx)S(2))/(S(2)b*S(5)dlog(F)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(2))/(c + dx)S(13), x), x, FaGamma(S(6), -blog(F)/(c + dx)S(2))/(S(2)b*S(6)dlog(F)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(2))(c + dx)S(10), x), x, Fa(-blog(F)/(c + dx)S(2))(S(11)/2)(c + dx)*S(11)Gamma(S(-11)/2, -blog(F)/(c + dx)*S(2))/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))(c + dx)S(8), x), x, Fa(-blog(F)/(c + dx)S(2))(S(9)/2)(c + dx)*S(9)Gamma(S(-9)/2, -blog(F)/(c + dx)*S(2))/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))(c + dx)S(6), x), x, -S(8)sqrt(pi)Fab*(S(7)/2)log(F)*(S(7)/2)erfi(sqrt(b)sqrt(log(F))/(c + dx))/(S(105)d) + S(8)F*(a + b/(c + dx)*S(2))b*S(3)(c + dx)log(F)*S(3)/(S(105)d) + S(4)F(a + b/(c + dx)*S(2))b*S(2)(c + dx)S(3)log(F)*S(2)/(S(105)d) + S(2)F(a + b/(c + dx)*S(2))b(c + dx)*S(5)log(F)/(S(35)d) + F(a + b/(c + dx)*S(2))(c + dx)S(7)/(S(7)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))(c + dx)S(4), x), x, -S(4)sqrt(pi)Fab*(S(5)/2)log(F)*(S(5)/2)erfi(sqrt(b)sqrt(log(F))/(c + dx))/(S(15)d) + S(4)F*(a + b/(c + dx)*S(2))b*S(2)(c + dx)log(F)*S(2)/(S(15)d) + S(2)F(a + b/(c + dx)*S(2))b(c + dx)*S(3)log(F)/(S(15)d) + F(a + b/(c + dx)*S(2))(c + dx)S(5)/(S(5)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))(c + dx)S(2), x), x, -S(2)sqrt(pi)Fab*(S(3)/2)log(F)*(S(3)/2)erfi(sqrt(b)sqrt(log(F))/(c + dx))/(S(3)d) + S(2)F*(a + b/(c + dx)*S(2))b(c + dx)log(F)/(S(3)d) + F*(a + b/(c + dx)*S(2))(c + dx)S(3)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2)), x), x, -sqrt(pi)F*asqrt(b)sqrt(log(F))erfi(sqrt(b)sqrt(log(F))/(c + dx))/d + F*(a + b/(c + dx)*S(2))(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(2))/(c + dx)*S(2), x), x, -sqrt(pi)F*aerfi(sqrt(b)sqrt(log(F))/(c + dx))/(S(2)sqrt(b)dsqrt(log(F))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(2))/(c + dx)*S(4), x), x, sqrt(pi)F*aerfi(sqrt(b)sqrt(log(F))/(c + dx))/(S(4)b(S(3)/2)dlog(F)(S(3)/2)) - F(a + b/(c + dx)*S(2))/(S(2)bd(c + dx)log(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(2))/(c + dx)*S(6), x), x, -S(3)sqrt(pi)Faerfi(sqrt(b)sqrt(log(F))/(c + dx))/(S(8)b(S(5)/2)dlog(F)(S(5)/2)) - F(a + b/(c + dx)*S(2))/(S(2)bd(c + dx)S(3)log(F)) + S(3)F(a + b/(c + dx)*S(2))/(S(4)b*S(2)d(c + dx)log(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(2))/(c + dx)*S(8), x), x, S(15)sqrt(pi)Faerfi(sqrt(b)sqrt(log(F))/(c + dx))/(S(16)b(S(7)/2)dlog(F)(S(7)/2)) - F(a + b/(c + dx)*S(2))/(S(2)bd(c + dx)S(5)log(F)) + S(5)F(a + b/(c + dx)*S(2))/(S(4)b*S(2)d(c + dx)*S(3)log(F)*S(2)) - S(15)F*(a + b/(c + dx)*S(2))/(S(8)b*S(3)d(c + dx)log(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(2))/(c + dx)*S(10), x), x, -S(105)sqrt(pi)Faerfi(sqrt(b)sqrt(log(F))/(c + dx))/(S(32)b(S(9)/2)dlog(F)(S(9)/2)) - F(a + b/(c + dx)*S(2))/(S(2)bd(c + dx)S(7)log(F)) + S(7)F(a + b/(c + dx)*S(2))/(S(4)b*S(2)d(c + dx)*S(5)log(F)*S(2)) - S(35)F*(a + b/(c + dx)*S(2))/(S(8)b*S(3)d(c + dx)*S(3)log(F)*S(3)) + S(105)F*(a + b/(c + dx)*S(2))/(S(16)b*S(4)d(c + dx)log(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(2))/(c + dx)S(12), x), x, FaGamma(S(11)/2, -blog(F)/(c + dx)S(2))/(S(2)d(-blog(F)/(c + dx)S(2))(S(11)/2)(c + dx)S(11)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(2))/(c + dx)S(14), x), x, FaGamma(S(13)/2, -blog(F)/(c + dx)S(2))/(S(2)d(-blog(F)/(c + dx)S(2))(S(13)/2)(c + dx)S(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(3))(c + dx)m, x), x, Fa(-blog(F)/(c + dx)S(3))(m/S(3) + S(1)/3)(c + dx)*(m + S(1))Gamma(-m/S(3) + S(-1)/3, -blog(F)/(c + dx)*S(3))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(3))(c + dx)S(14), x), x, -Fab*S(5)Gamma(S(-5), -blog(F)/(c + dx)*S(3))log(F)*S(5)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(3))(c + dx)S(11), x), x, Fab*S(4)Gamma(S(-4), -blog(F)/(c + dx)*S(3))log(F)*S(4)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(3))(c + dx)S(8), x), x, -Fab*S(3)log(F)*S(3)Ei(blog(F)/(c + dx)*S(3))/(S(18)d) + F*(a + b/(c + dx)*S(3))b*S(2)(c + dx)S(3)log(F)*S(2)/(S(18)d) + F*(a + b/(c + dx)*S(3))b(c + dx)*S(6)log(F)/(S(18)d) + F(a + b/(c + dx)*S(3))(c + dx)S(9)/(S(9)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(3))(c + dx)S(5), x), x, -Fab*S(2)log(F)*S(2)Ei(blog(F)/(c + dx)*S(3))/(S(6)d) + F*(a + b/(c + dx)*S(3))b(c + dx)*S(3)log(F)/(S(6)d) + F(a + b/(c + dx)*S(3))(c + dx)S(6)/(S(6)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(3))(c + dx)S(2), x), x, -Fablog(F)Ei(blog(F)/(c + dx)*S(3))/(S(3)d) + F*(a + b/(c + dx)*S(3))(c + dx)S(3)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(3))/(c + dx), x), x, -F*aEi(blog(F)/(c + dx)*S(3))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(3))/(c + dx)S(4), x), x, -F(a + b/(c + dx)S(3))/(S(3)bdlog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(3))/(c + dx)S(7), x), x, -F(a + b/(c + dx)S(3))/(S(3)bd(c + dx)S(3)log(F)) + F*(a + b/(c + dx)*S(3))/(S(3)b*S(2)dlog(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(3))/(c + dx)S(10), x), x, -F(a + b/(c + dx)S(3))/(S(3)bd(c + dx)S(6)log(F)) + S(2)F(a + b/(c + dx)*S(3))/(S(3)b*S(2)d(c + dx)*S(3)log(F)*S(2)) - S(2)F*(a + b/(c + dx)*S(3))/(S(3)b*S(3)dlog(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(3))/(c + dx)S(13), x), x, -F(a + b/(c + dx)S(3))/(S(3)bd(c + dx)S(9)log(F)) + F*(a + b/(c + dx)S(3))/(bS(2)d(c + dx)S(6)log(F)*S(2)) - S(2)F*(a + b/(c + dx)S(3))/(bS(3)d(c + dx)S(3)log(F)*S(3)) + S(2)F*(a + b/(c + dx)S(3))/(bS(4)dlog(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)S(3))/(c + dx)S(16), x), x, -FaGamma(S(5), -blog(F)/(c + dx)S(3))/(S(3)b*S(5)dlog(F)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(3))/(c + dx)S(19), x), x, FaGamma(S(6), -blog(F)/(c + dx)S(3))/(S(3)b*S(6)dlog(F)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(3))(c + dx)S(3), x), x, Fa(-blog(F)/(c + dx)S(3))(S(4)/3)(c + dx)*S(4)Gamma(S(-4)/3, -blog(F)/(c + dx)*S(3))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(3))(c + dx), x), x, Fa(-blog(F)/(c + dx)S(3))(S(2)/3)(c + dx)*S(2)Gamma(S(-2)/3, -blog(F)/(c + dx)*S(3))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)S(3)), x), x, Fa(-blog(F)/(c + dx)S(3))(S(1)/3)(c + dx)Gamma(S(-1)/3, -blog(F)/(c + dx)*S(3))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b/(c + dx)*S(3))/(c + dx)S(2), x), x, FaGamma(S(1)/3, -blog(F)/(c + dx)S(3))/(S(3)d(-blog(F)/(c + dx)S(3))(S(1)/3)(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(3))/(c + dx)S(3), x), x, FaGamma(S(2)/3, -blog(F)/(c + dx)S(3))/(S(3)d(-blog(F)/(c + dx)S(3))(S(2)/3)(c + dx)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx)*S(3))/(c + dx)S(5), x), x, FaGamma(S(4)/3, -blog(F)/(c + dx)S(3))/(S(3)d(-blog(F)/(c + dx)S(3))(S(4)/3)(c + dx)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)n)(c + dx)m, x), x, -Fa(-b(c + dx)*nlog(F))*(-(m + S(1))/n)(c + dx)(m + S(1))Gamma((m + S(1))/n, -b(c + dx)*nlog(F))/(dn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)n)(c + dx)S(3), x), x, -Fa(-b(c + dx)*nlog(F))*(-S(4)/n)(c + dx)S(4)Gamma(S(4)/n, -b(c + dx)*nlog(F))/(dn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)n)(c + dx)S(2), x), x, -Fa(-b(c + dx)*nlog(F))*(-S(3)/n)(c + dx)S(3)Gamma(S(3)/n, -b(c + dx)*nlog(F))/(dn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)n)(c + dx), x), x, -Fa(-b(c + dx)*nlog(F))*(-S(2)/n)(c + dx)S(2)Gamma(S(2)/n, -b(c + dx)*nlog(F))/(dn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)n), x), x, -Fa(-b(c + dx)*nlog(F))*(-S(1)/n)(c + dx)Gamma(S(1)/n, -b(c + dx)*nlog(F))/(dn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)n)/(c + dx), x), x, F*aEi(b(c + dx)*nlog(F))/(dn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)n)/(c + dx)S(2), x), x, -Fa(-b(c + dx)nlog(F))*(S(1)/n)Gamma(-S(1)/n, -b(c + dx)*nlog(F))/(dn(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)n)/(c + dx)S(3), x), x, -Fa(-b(c + dx)nlog(F))*(S(2)/n)Gamma(-S(2)/n, -b(c + dx)*nlog(F))/(dn(c + dx)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)n)/(c + dx)S(4), x), x, -Fa(-b(c + dx)nlog(F))*(S(3)/n)Gamma(-S(3)/n, -b(c + dx)*nlog(F))/(dn(c + dx)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)n)(c + dx)(S(6)n + S(-1)), x), x, -F*aGamma(S(6), -b(c + dx)*nlog(F))/(b*S(6)dnlog(F)S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*n)(c + dx)(S(5)n + S(-1)), x), x, F*aGamma(S(5), -b(c + dx)*nlog(F))/(b*S(5)dnlog(F)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*n)(c + dx)(S(4)n + S(-1)), x), x, F*(a + b(c + dx)n)(c + dx)(S(3)n)/(bdnlog(F)) - S(3)F*(a + b(c + dx)n)(c + dx)(S(2)n)/(b*S(2)dnlog(F)*S(2)) + S(6)F*(a + b(c + dx)n)(c + dx)n/(bS(3)dnlog(F)*S(3)) - S(6)F*(a + b(c + dx)n)/(bS(4)dnlog(F)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*n)(c + dx)(S(3)n + S(-1)), x), x, F*(a + b(c + dx)n)(c + dx)(S(2)n)/(bdnlog(F)) - S(2)F*(a + b(c + dx)n)(c + dx)n/(bS(2)dnlog(F)*S(2)) + S(2)F*(a + b(c + dx)n)/(bS(3)dnlog(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*n)(c + dx)(S(2)n + S(-1)), x), x, F*(a + b(c + dx)n)(c + dx)n/(bdnlog(F)) - F*(a + b(c + dx)n)/(bS(2)dnlog(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*n)(c + dx)(n + S(-1)), x), x, F(a + b(c + dx)n)/(bdnlog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)n)/(c + dx), x), x, F*aEi(b(c + dx)*nlog(F))/(dn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)n)(c + dx)(-n + S(-1)), x), x, Fablog(F)Ei(b(c + dx)*nlog(F))/(dn) - F(a + b(c + dx)n)(c + dx)(-n)/(dn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)n)(c + dx)(-S(2)n + S(-1)), x), x, F*ab*S(2)log(F)*S(2)Ei(b(c + dx)*nlog(F))/(S(2)dn) - F*(a + b(c + dx)n)b(c + dx)*(-n)log(F)/(S(2)dn) - F*(a + b(c + dx)n)(c + dx)(-S(2)n)/(S(2)dn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)n)(c + dx)(-S(3)n + S(-1)), x), x, F*ab*S(3)log(F)*S(3)Ei(b(c + dx)*nlog(F))/(S(6)dn) - F*(a + b(c + dx)n)b*S(2)(c + dx)(-n)log(F)*S(2)/(S(6)dn) - F(a + b(c + dx)n)b(c + dx)*(-S(2)n)log(F)/(S(6)dn) - F(a + b(c + dx)n)(c + dx)(-S(3)n)/(S(3)dn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)n)(c + dx)(-S(4)n + S(-1)), x), x, -F*ab*S(4)Gamma(S(-4), -b(c + dx)*nlog(F))log(F)S(4)/(dn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)n)(c + dx)(-S(5)n + S(-1)), x), x, F*ab*S(5)Gamma(S(-5), -b(c + dx)*nlog(F))log(F)S(5)/(dn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(c(a + bx)n)(a + bx)(n/S(2) + S(-1)), x), x, sqrt(pi)erfi(sqrt(c)(a + bx)*(n/S(2))sqrt(log(F)))/(bsqrt(c)nsqrt(log(F))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(-c(a + bx)n)(a + bx)(n/S(2) + S(-1)), x), x, sqrt(pi)erf(sqrt(c)(a + bx)*(n/S(2))sqrt(log(F)))/(bsqrt(c)nsqrt(log(F))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)S(2))(e + fx)S(5), x), x, sqrt(pi)F*a(-cf + de)*S(5)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(2)sqrt(b)dS(6)sqrt(log(F))) - S(5)sqrt(pi)F*af*S(2)(-cf + de)*S(3)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(2)b*(S(3)/2)d*S(6)log(F)*(S(3)/2)) + S(15)sqrt(pi)Faf*S(4)(-cf + de)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(8)b(S(5)/2)d*S(6)log(F)(S(5)/2)) + F(a + b(c + dx)*S(2))f*S(5)(c + dx)S(4)/(S(2)bdS(6)log(F)) + S(5)F(a + b(c + dx)S(2))f*S(4)(c + dx)S(3)(-cf + de)/(S(2)bd*S(6)log(F)) + S(5)F(a + b(c + dx)S(2))f*S(3)(c + dx)S(2)(-cf + de)*S(2)/(bd*S(6)log(F)) + S(5)F(a + b(c + dx)S(2))f*S(2)(c + dx)(-cf + de)*S(3)/(bd*S(6)log(F)) + S(5)F(a + b(c + dx)S(2))f(-cf + de)S(4)/(S(2)bdS(6)log(F)) - F*(a + b(c + dx)S(2))f*S(5)(c + dx)S(2)/(bS(2)d*S(6)log(F)*S(2)) - S(15)F*(a + b(c + dx)S(2))f*S(4)(c + dx)(-cf + de)/(S(4)bS(2)d*S(6)log(F)*S(2)) - S(5)F*(a + b(c + dx)S(2))f*S(3)(-cf + de)S(2)/(bS(2)dS(6)log(F)S(2)) + F(a + b(c + dx)*S(2))fS(5)/(bS(3)dS(6)log(F)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(2))(e + fx)S(4), x), x, sqrt(pi)F*a(-cf + de)*S(4)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(2)sqrt(b)dS(5)sqrt(log(F))) - S(3)sqrt(pi)F*af*S(2)(-cf + de)*S(2)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(2)b*(S(3)/2)d*S(5)log(F)*(S(3)/2)) + S(3)sqrt(pi)Faf*S(4)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(8)b*(S(5)/2)d*S(5)log(F)(S(5)/2)) + F(a + b(c + dx)*S(2))f*S(4)(c + dx)S(3)/(S(2)bdS(5)log(F)) + S(2)F(a + b(c + dx)S(2))f*S(3)(c + dx)S(2)(-cf + de)/(bdS(5)log(F)) + S(3)F(a + b(c + dx)S(2))f*S(2)(c + dx)(-cf + de)*S(2)/(bd*S(5)log(F)) + S(2)F(a + b(c + dx)S(2))f(-cf + de)S(3)/(bd*S(5)log(F)) - S(3)F(a + b(c + dx)S(2))f*S(4)(c + dx)/(S(4)b*S(2)d*S(5)log(F)*S(2)) - S(2)F*(a + b(c + dx)S(2))f*S(3)(-cf + de)/(b*S(2)d*S(5)log(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(2))(e + fx)S(3), x), x, sqrt(pi)F*a(-cf + de)*S(3)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(2)sqrt(b)dS(4)sqrt(log(F))) - S(3)sqrt(pi)F*af*S(2)(-cf + de)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(4)b(S(3)/2)d*S(4)log(F)(S(3)/2)) + F(a + b(c + dx)*S(2))f*S(3)(c + dx)S(2)/(S(2)bdS(4)log(F)) + S(3)F(a + b(c + dx)S(2))f*S(2)(c + dx)(-cf + de)/(S(2)bd*S(4)log(F)) + S(3)F(a + b(c + dx)S(2))f(-cf + de)S(2)/(S(2)bdS(4)log(F)) - F*(a + b(c + dx)S(2))f*S(3)/(S(2)b*S(2)d*S(4)log(F)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(2))(e + fx)S(2), x), x, sqrt(pi)F*a(-cf + de)*S(2)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(2)sqrt(b)dS(3)sqrt(log(F))) - sqrt(pi)Faf*S(2)erfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(4)b*(S(3)/2)d*S(3)log(F)(S(3)/2)) + F(a + b(c + dx)*S(2))f*S(2)(c + dx)/(S(2)bdS(3)log(F)) + F*(a + b(c + dx)S(2))f(-cf + de)/(bd*S(3)log(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)S(2))(e + fx), x), x, sqrt(pi)F*a(-cf/S(2) + de/S(2))erfi(sqrt(b)(c + dx)sqrt(log(F)))/(sqrt(b)dS(2)sqrt(log(F))) + F*(a + b(c + dx)S(2))f/(S(2)bd*S(2)log(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)S(2)), x), x, sqrt(pi)F*aerfi(sqrt(b)(c + dx)sqrt(log(F)))/(S(2)sqrt(b)dsqrt(log(F))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)S(2))/(e + fx), x), x, Integral(F*(a + b(c + dx)S(2))/(e + fx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(a + b(c + dx)S(2))/(e + fx)*S(2), x), x, sqrt(pi)F*asqrt(b)dsqrt(log(F))erfi(sqrt(b)(c + dx)sqrt(log(F)))/fS(2) - F(a + b(c + dx)*S(2))/(f(e + fx)) - S(2)bd(-cf + de)log(F)Integral(F*(a + b(c + dx)S(2))/(e + fx), x)/fS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b(c + dx)*S(2))/(e + fx)*S(3), x), x, -sqrt(pi)F*ab*(S(3)/2)d*S(2)(-cf + de)log(F)(S(3)/2)erfi(sqrt(b)(c + dx)sqrt(log(F)))/fS(4) + F(a + b(c + dx)S(2))bd(-cf + de)log(F)/(fS(3)(e + fx)) - F(a + b(c + dx)S(2))/(S(2)f(e + fx)*S(2)) + S(2)b*S(2)d*S(2)(-cf + de)*S(2)log(F)*S(2)Integral(F*(a + b(c + dx)S(2))/(e + fx), x)/f*S(4) + bd*S(2)log(F)Integral(F(a + b(c + dx)S(2))/(e + fx), x)/f*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(3)exp(e(c + dx)S(3)), x), x, -bS(3)(c + dx)*S(4)Gamma(S(4)/3, -e(c + dx)*S(3))/(S(3)d*S(4)(-e(c + dx)S(3))(S(4)/3)) - b*S(2)(-ad + bc)exp(e(c + dx)S(3))/(dS(4)e) - b(c + dx)*S(2)(-ad + bc)*S(2)Gamma(S(2)/3, -e(c + dx)S(3))/(dS(4)(-e(c + dx)S(3))(S(2)/3)) + (c + dx)(-ad + bc)S(3)Gamma(S(1)/3, -e(c + dx)*S(3))/(S(3)d*S(4)(-e(c + dx)S(3))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)exp(e(c + dx)S(3)), x), x, bS(2)exp(e(c + dx)S(3))/(S(3)d*S(3)e) + S(2)b(c + dx)S(2)(-ad + bc)Gamma(S(2)/3, -e(c + dx)S(3))/(S(3)d*S(3)(-e(c + dx)S(3))(S(2)/3)) - (c + dx)(-ad + bc)*S(2)Gamma(S(1)/3, -e(c + dx)*S(3))/(S(3)d*S(3)(-e(c + dx)S(3))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)exp(e(c + dx)*S(3)), x), x, -b(c + dx)S(2)Gamma(S(2)/3, -e(c + dx)*S(3))/(S(3)d*S(2)(-e(c + dx)S(3))(S(2)/3)) + (c + dx)(-ad/S(3) + bc/S(3))Gamma(S(1)/3, -e(c + dx)S(3))/(dS(2)(-e(c + dx)S(3))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e(c + dx)*S(3)), x), x, (-c/S(3) - dx/S(3))Gamma(S(1)/3, -e(c + dx)S(3))/(d(-e(c + dx)S(3))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e(c + dx)*S(3))/(a + bx), x), x, Integral(exp(e(c + dx)*S(3))/(a + bx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e(c + dx)*S(3))/(a + bx)*S(2), x), x, -exp(e(c + dx)S(3))/(b(a + bx)) - de(c + dx)*S(2)Gamma(S(2)/3, -e(c + dx)S(3))/(bS(2)(-e(c + dx)S(3))(S(2)/3)) + S(3)de(-ad + bc)*S(2)Integral(exp(e(c + dx)*S(3))/(a + bx), x)/b*S(3) - de(c + dx)(-ad + bc)Gamma(S(1)/3, -e(c + dx)S(3))/(bS(3)(-e(c + dx)S(3))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx))/(e + fx), x), x, -FaEi(blog(F)/(c + dx))/f + F*(a - bf/(-cf + de))Ei(bd(e + fx)log(F)/((c + dx)(-cf + de)))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx))/(e + fx)S(2), x), x, F(a + b/(c + dx))d/(f(-cf + de)) - F*(a + b/(c + dx))/(f(e + fx)) - F*(a - bf/(-cf + de))bdlog(F)Ei(bd(e + fx)log(F)/((c + dx)(-cf + de)))/(-cf + de)S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx))/(e + fx)S(3), x), x, -F(a + b/(c + dx))bdS(2)log(F)/(S(2)(-cf + de)S(3)) + F(a + b/(c + dx))bdlog(F)/(S(2)(e + fx)(-cf + de)S(2)) + F(a + b/(c + dx))d*S(2)/(S(2)f(-cf + de)S(2)) - F(a + b/(c + dx))/(S(2)f(e + fx)S(2)) + F(a - bf/(-cf + de))bS(2)d*S(2)flog(F)S(2)Ei(bd(e + fx)log(F)/((c + dx)(-cf + de)))/(S(2)(-cf + de)S(4)) - F(a - bf/(-cf + de))bd*S(2)log(F)Ei(bd(e + fx)log(F)/((c + dx)(-cf + de)))/(-cf + de)S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + b/(c + dx))/(e + fx)S(4), x), x, F(a + b/(c + dx))bS(2)d*S(3)flog(F)S(2)/(S(6)(-cf + de)S(5)) - F(a + b/(c + dx))b*S(2)d*S(2)flog(F)S(2)/(S(6)(e + fx)(-cf + de)*S(4)) - S(5)F*(a + b/(c + dx))bd*S(3)log(F)/(S(6)(-cf + de)S(4)) + S(2)F*(a + b/(c + dx))bd*S(2)log(F)/(S(3)(e + fx)(-cf + de)S(3)) + F(a + b/(c + dx))bdlog(F)/(S(6)(e + fx)S(2)(-cf + de)S(2)) + F(a + b/(c + dx))d*S(3)/(S(3)f(-cf + de)S(3)) - F(a + b/(c + dx))/(S(3)f(e + fx)S(3)) - F(a - bf/(-cf + de))bS(3)d*S(3)f*S(2)log(F)*S(3)Ei(bd(e + fx)log(F)/((c + dx)(-cf + de)))/(S(6)(-cf + de)S(6)) + F(a - bf/(-cf + de))bS(2)d*S(3)flog(F)S(2)Ei(bd(e + fx)log(F)/((c + dx)(-cf + de)))/(-cf + de)S(5) - F(a - bf/(-cf + de))bdS(3)log(F)Ei(bd(e + fx)log(F)/((c + dx)(-cf + de)))/(-cf + de)S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(4)exp(e/(c + dx)), x), x, -bS(4)e*S(5)Gamma(S(-5), -e/(c + dx))/dS(5) - S(4)b*S(3)e*S(4)(-ad + bc)Gamma(S(-4), -e/(c + dx))/dS(5) - bS(2)eS(3)(-ad + bc)*S(2)Ei(e/(c + dx))/dS(5) + bS(2)e*S(2)(c + dx)(-ad + bc)*S(2)exp(e/(c + dx))/dS(5) + bS(2)e(c + dx)*S(2)(-ad + bc)*S(2)exp(e/(c + dx))/dS(5) + S(2)b*S(2)(c + dx)S(3)(-ad + bc)*S(2)exp(e/(c + dx))/dS(5) + S(2)beS(2)(-ad + bc)*S(3)Ei(e/(c + dx))/dS(5) - S(2)be(c + dx)(-ad + bc)*S(3)exp(e/(c + dx))/dS(5) - S(2)b(c + dx)*S(2)(-ad + bc)*S(3)exp(e/(c + dx))/dS(5) - e(-ad + bc)*S(4)Ei(e/(c + dx))/dS(5) + (c + dx)(-ad + bc)S(4)exp(e/(c + dx))/dS(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(3)exp(e/(c + dx)), x), x, bS(3)e*S(4)Gamma(S(-4), -e/(c + dx))/dS(4) + bS(2)e*S(3)(-ad + bc)Ei(e/(c + dx))/(S(2)dS(4)) - bS(2)e*S(2)(c + dx)(-ad + bc)exp(e/(c + dx))/(S(2)dS(4)) - bS(2)e(c + dx)*S(2)(-ad + bc)exp(e/(c + dx))/(S(2)dS(4)) - bS(2)(c + dx)S(3)(-ad + bc)exp(e/(c + dx))/d*S(4) - S(3)beS(2)(-ad + bc)*S(2)Ei(e/(c + dx))/(S(2)d*S(4)) + S(3)be(c + dx)(-ad + bc)*S(2)exp(e/(c + dx))/(S(2)d*S(4)) + S(3)b(c + dx)*S(2)(-ad + bc)*S(2)exp(e/(c + dx))/(S(2)d*S(4)) + e(-ad + bc)*S(3)Ei(e/(c + dx))/dS(4) - (c + dx)(-ad + bc)S(3)exp(e/(c + dx))/dS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(2)exp(e/(c + dx)), x), x, -bS(2)e*S(3)Ei(e/(c + dx))/(S(6)dS(3)) + bS(2)eS(2)(c + dx)exp(e/(c + dx))/(S(6)dS(3)) + bS(2)e(c + dx)S(2)exp(e/(c + dx))/(S(6)dS(3)) + bS(2)(c + dx)*S(3)exp(e/(c + dx))/(S(3)d*S(3)) + be*S(2)(-ad + bc)Ei(e/(c + dx))/d*S(3) - be(c + dx)(-ad + bc)exp(e/(c + dx))/dS(3) - b(c + dx)S(2)(-ad + bc)exp(e/(c + dx))/d*S(3) - e(-ad + bc)*S(2)Ei(e/(c + dx))/dS(3) + (c + dx)(-ad + bc)S(2)exp(e/(c + dx))/dS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)exp(e/(c + dx)), x), x, -beS(2)Ei(e/(c + dx))/(S(2)d*S(2)) + be(c + dx)exp(e/(c + dx))/(S(2)dS(2)) + b(c + dx)S(2)exp(e/(c + dx))/(S(2)d*S(2)) + e(-ad + bc)Ei(e/(c + dx))/d*S(2) + (c + dx)(ad - bc)exp(e/(c + dx))/dS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e/(c + dx)), x), x, -eEi(e/(c + dx))/d + (c + dx)exp(e/(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e/(c + dx))/(a + bx), x), x, exp(be/(-ad + bc))Ei(-de(a + bx)/((c + dx)(-ad + bc)))/b - Ei(e/(c + dx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e/(c + dx))/(a + bx)S(2), x), x, -deexp(be/(-ad + bc))Ei(-de(a + bx)/((c + dx)(-ad + bc)))/(-ad + bc)*S(2) - dexp(e/(c + dx))/(b(-ad + bc)) - exp(e/(c + dx))/(b(a + bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e/(c + dx))/(a + bx)S(3), x), x, bd*S(2)e*S(2)exp(be/(-ad + bc))Ei(-de(a + bx)/((c + dx)(-ad + bc)))/(S(2)(-ad + bc)S(4)) + dS(2)eexp(e/(c + dx))/(S(2)(-ad + bc)S(3)) + dS(2)eexp(be/(-ad + bc))Ei(-de(a + bx)/((c + dx)(-ad + bc)))/(-ad + bc)S(3) + deexp(e/(c + dx))/(S(2)(a + bx)(-ad + bc)S(2)) + dS(2)exp(e/(c + dx))/(S(2)b(-ad + bc)S(2)) - exp(e/(c + dx))/(S(2)b(a + bx)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(3)exp(e/(c + dx)S(2)), x), x, -bS(3)e*S(2)Ei(e/(c + dx)S(2))/(S(4)dS(4)) + bS(3)e(c + dx)S(2)exp(e/(c + dx)S(2))/(S(4)dS(4)) + bS(3)(c + dx)*S(4)exp(e/(c + dx)S(2))/(S(4)d*S(4)) + S(2)sqrt(pi)bS(2)e*(S(3)/2)(-ad + bc)erfi(sqrt(e)/(c + dx))/d*S(4) - S(2)b*S(2)e(c + dx)(-ad + bc)exp(e/(c + dx)S(2))/dS(4) - bS(2)(c + dx)S(3)(-ad + bc)exp(e/(c + dx)S(2))/dS(4) - S(3)be(-ad + bc)S(2)Ei(e/(c + dx)S(2))/(S(2)d*S(4)) + S(3)b(c + dx)*S(2)(-ad + bc)*S(2)exp(e/(c + dx)S(2))/(S(2)d*S(4)) + sqrt(pi)sqrt(e)(-ad + bc)S(3)erfi(sqrt(e)/(c + dx))/dS(4) - (c + dx)(-ad + bc)S(3)exp(e/(c + dx)S(2))/dS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(2)exp(e/(c + dx)S(2)), x), x, -S(2)sqrt(pi)bS(2)e*(S(3)/2)erfi(sqrt(e)/(c + dx))/(S(3)d*S(3)) + S(2)b*S(2)e(c + dx)exp(e/(c + dx)*S(2))/(S(3)dS(3)) + bS(2)(c + dx)*S(3)exp(e/(c + dx)S(2))/(S(3)d*S(3)) + be(-ad + bc)Ei(e/(c + dx)S(2))/dS(3) - b(c + dx)S(2)(-ad + bc)exp(e/(c + dx)S(2))/dS(3) - sqrt(pi)sqrt(e)(-ad + bc)*S(2)erfi(sqrt(e)/(c + dx))/dS(3) + (c + dx)(-ad + bc)S(2)exp(e/(c + dx)S(2))/dS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)exp(e/(c + dx)*S(2)), x), x, -beEi(e/(c + dx)*S(2))/(S(2)d*S(2)) + b(c + dx)S(2)exp(e/(c + dx)S(2))/(S(2)d*S(2)) + sqrt(pi)sqrt(e)(-ad + bc)erfi(sqrt(e)/(c + dx))/dS(2) + (c + dx)(ad - bc)exp(e/(c + dx)S(2))/dS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e/(c + dx)*S(2)), x), x, -sqrt(pi)sqrt(e)erfi(sqrt(e)/(c + dx))/d + (c + dx)exp(e/(c + dx)S(2))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e/(c + dx)*S(2))/(a + bx), x), x, Integral(exp(e/(c + dx)S(2))/(a + bx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e/(c + dx)S(2))/(a + bx)*S(2), x), x, Integral(exp(e/(c + dx)*S(2))/(a + bx)*S(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e/(c + dx)*S(2))/(a + bx)*S(3), x), x, Integral(exp(e/(c + dx)*S(2))/(a + bx)*S(3), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(3)exp(e/(c + dx)S(3)), x), x, bS(3)(-e/(c + dx)S(3))(S(4)/3)(c + dx)S(4)Gamma(S(-4)/3, -e/(c + dx)S(3))/(S(3)dS(4)) + bS(2)e(-ad + bc)Ei(e/(c + dx)S(3))/dS(4) - b*S(2)(c + dx)S(3)(-ad + bc)exp(e/(c + dx)S(3))/dS(4) + b(-e/(c + dx)S(3))(S(2)/3)(c + dx)*S(2)(-ad + bc)*S(2)Gamma(S(-2)/3, -e/(c + dx)S(3))/dS(4) - (-e/(c + dx)S(3))(S(1)/3)(c + dx)(-ad + bc)S(3)Gamma(S(-1)/3, -e/(c + dx)S(3))/(S(3)d*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(2)exp(e/(c + dx)S(3)), x), x, -bS(2)eEi(e/(c + dx)*S(3))/(S(3)dS(3)) + bS(2)(c + dx)*S(3)exp(e/(c + dx)S(3))/(S(3)d*S(3)) - S(2)b(-e/(c + dx)S(3))(S(2)/3)(c + dx)*S(2)(-ad + bc)Gamma(S(-2)/3, -e/(c + dx)*S(3))/(S(3)d*S(3)) + (-e/(c + dx)S(3))(S(1)/3)(c + dx)(-ad + bc)S(2)Gamma(S(-1)/3, -e/(c + dx)S(3))/(S(3)d*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)exp(e/(c + dx)*S(3)), x), x, b(-e/(c + dx)S(3))(S(2)/3)(c + dx)S(2)Gamma(S(-2)/3, -e/(c + dx)S(3))/(S(3)d*S(2)) - (-e/(c + dx)S(3))(S(1)/3)(c + dx)(-ad/S(3) + bc/S(3))Gamma(S(-1)/3, -e/(c + dx)S(3))/dS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e/(c + dx)*S(3)), x), x, (-e/(c + dx)S(3))(S(1)/3)(c + dx)Gamma(S(-1)/3, -e/(c + dx)*S(3))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e/(c + dx)S(3))/(a + bx), x), x, Integral(exp(e/(c + dx)S(3))/(a + bx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(e/(c + dx)S(3))/(a + bx)*S(2), x), x, Integral(exp(e/(c + dx)*S(3))/(a + bx)S(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(e + f(a + bx)/(c + dx))/(g + hx), x), x, F*(e + f(-ah + bg)/(-ch + dg))Ei(f(g + hx)(ad - bc)log(F)/((c + dx)(-ch + dg)))/h - F(bf/d + e)Ei(f(ad - bc)log(F)/(d(c + dx)))/h, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(e + f(a + bx)/(c + dx))/(g + hx)S(2), x), x, -F(e + f(a + bx)/(c + dx))/(h(g + hx)) + F*(e + f(-ah + bg)/(-ch + dg))f(-ad + bc)log(F)Ei(f(g + hx)(ad - bc)log(F)/((c + dx)(-ch + dg)))/(-ch + dg)S(2) + F(bf/d + e - f(-ad + bc)/(d(c + dx)))d/(h(-ch + dg)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(e + f(a + bx)/(c + dx))/(g + hx)S(3), x), x, -F(e + f(a + bx)/(c + dx))f(-ad/S(2) + bc/S(2))log(F)/((g + hx)(-ch + dg)S(2)) - F(e + f(a + bx)/(c + dx))/(S(2)h(g + hx)S(2)) + F(e + f(-ah + bg)/(-ch + dg))df(-ad + bc)log(F)Ei(f(g + hx)(ad - bc)log(F)/((c + dx)(-ch + dg)))/(-ch + dg)S(3) + F(e + f(-ah + bg)/(-ch + dg))fS(2)h(-ad + bc)S(2)log(F)*S(2)Ei(f(g + hx)(ad - bc)log(F)/((c + dx)(-ch + dg)))/(S(2)(-ch + dg)S(4)) + F(bf/d + e - f(-ad + bc)/(d(c + dx)))d*S(2)/(S(2)h(-ch + dg)S(2)) + F(bf/d + e - f(-ad + bc)/(d(c + dx)))df(-ad + bc)log(F)/(S(2)(-ch + dg)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(e + f(a + bx)/(c + dx))/(g + hx)*S(4), x), x, -S(2)F*(e + f(a + bx)/(c + dx))df(-ad + bc)log(F)/(S(3)(g + hx)(-ch + dg)S(3)) - F(e + f(a + bx)/(c + dx))fS(2)h(-ad + bc)S(2)log(F)*S(2)/(S(6)(g + hx)(-ch + dg)S(4)) - F(e + f(a + bx)/(c + dx))f(-ad/S(6) + bc/S(6))log(F)/((g + hx)S(2)(-ch + dg)S(2)) - F(e + f(a + bx)/(c + dx))/(S(3)h(g + hx)S(3)) + F(e + f(-ah + bg)/(-ch + dg))d*S(2)f(-ad + bc)log(F)Ei(f(g + hx)(ad - bc)log(F)/((c + dx)(-ch + dg)))/(-ch + dg)S(4) + F(e + f(-ah + bg)/(-ch + dg))df*S(2)h(-ad + bc)S(2)log(F)*S(2)Ei(f(g + hx)(ad - bc)log(F)/((c + dx)(-ch + dg)))/(-ch + dg)S(5) + F(e + f(-ah + bg)/(-ch + dg))f*S(3)h*S(2)(-ad + bc)*S(3)log(F)*S(3)Ei(f(g + hx)(ad - bc)log(F)/((c + dx)(-ch + dg)))/(S(6)(-ch + dg)S(6)) + F(bf/d + e - f(-ad + bc)/(d(c + dx)))d*S(3)/(S(3)h(-ch + dg)S(3)) + S(5)F*(bf/d + e - f(-ad + bc)/(d(c + dx)))d*S(2)f(-ad + bc)log(F)/(S(6)(-ch + dg)S(4)) + F(bf/d + e - f(-ad + bc)/(d(c + dx)))dfS(2)h(-ad + bc)S(2)log(F)*S(2)/(S(6)(-ch + dg)S(5)), expand=True, _diff=True, _numerical=True) # fails 1940 and 1939 recursion assert rubi_test(rubi_integrate(f(a + bx + cx*S(2))x*S(3), x), x, -sqrt(pi)b*S(3)f(a - bS(2)/(S(4)c))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(S(16)c(S(7)/2)sqrt(log(f))) + b*S(2)f*(a + bx + cxS(2))/(S(8)c*S(3)log(f)) - bf(a + bx + cxS(2))x/(S(4)cS(2)log(f)) + S(3)sqrt(pi)bf(a - bS(2)/(S(4)c))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(S(8)c*(S(5)/2)log(f)(S(3)/2)) + f(a + bx + cx*S(2))x*S(2)/(S(2)clog(f)) - f(a + bx + cxS(2))/(S(2)c*S(2)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx + cx*S(2))x*S(2), x), x, sqrt(pi)b*S(2)f(a - bS(2)/(S(4)c))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(S(8)c(S(5)/2)sqrt(log(f))) - bf(a + bx + cxS(2))/(S(4)c*S(2)log(f)) + f*(a + bx + cxS(2))x/(S(2)clog(f)) - sqrt(pi)f(a - bS(2)/(S(4)c))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(S(4)c*(S(3)/2)log(f)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx + cx*S(2))x, x), x, -sqrt(pi)bf(a - bS(2)/(S(4)c))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(S(4)c(S(3)/2)sqrt(log(f))) + f*(a + bx + cxS(2))/(S(2)clog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx + cxS(2)), x), x, sqrt(pi)f(a - bS(2)/(S(4)c))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(S(2)sqrt(c)sqrt(log(f))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx + cxS(2))/x, x), x, Integral(f(a + bx + cxS(2))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx + cxS(2))/xS(2), x), x, blog(f)Integral(f(a + bx + cxS(2))/x, x) + sqrt(pi)sqrt(c)f(a - bS(2)/(S(4)c))sqrt(log(f))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c)) - f*(a + bx + cxS(2))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)exp(a + bx - cx*S(2)), x), x, -sqrt(pi)b*S(3)exp(a + b*S(2)/(S(4)c))erf((b/S(2) - cx)/sqrt(c))/(S(16)c(S(7)/2)) - bS(2)exp(a + bx - cx*S(2))/(S(8)c*S(3)) - bxexp(a + bx - cxS(2))/(S(4)c*S(2)) - S(3)sqrt(pi)bexp(a + b*S(2)/(S(4)c))erf((b/S(2) - cx)/sqrt(c))/(S(8)c(S(5)/2)) - xS(2)exp(a + bx - cx*S(2))/(S(2)c) - exp(a + bx - cx*S(2))/(S(2)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)exp(a + bx - cxS(2)), x), x, -sqrt(pi)b*S(2)exp(a + b*S(2)/(S(4)c))erf((b/S(2) - cx)/sqrt(c))/(S(8)c(S(5)/2)) - bexp(a + bx - cx*S(2))/(S(4)c*S(2)) - xexp(a + bx - cx*S(2))/(S(2)c) - sqrt(pi)exp(a + bS(2)/(S(4)c))erf((b/S(2) - cx)/sqrt(c))/(S(4)c(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp(a + bx - cx*S(2)), x), x, -sqrt(pi)bexp(a + bS(2)/(S(4)c))erf((b/S(2) - cx)/sqrt(c))/(S(4)c(S(3)/2)) - exp(a + bx - cxS(2))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(a + bx - cx*S(2)), x), x, -sqrt(pi)exp(a + b*S(2)/(S(4)c))erf((b/S(2) - cx)/sqrt(c))/(S(2)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(a + bx - cxS(2))/x, x), x, Integral(exp(a + bx - cxS(2))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(a + bx - cxS(2))/xS(2), x), x, bIntegral(exp(a + bx - cx*S(2))/x, x) + sqrt(pi)sqrt(c)exp(a + bS(2)/(S(4)c))erf((b/S(2) - cx)/sqrt(c)) - exp(a + bx - cxS(2))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)exp((a + bx)(c + dx)), x), x, x*S(2)exp(ac + bdxS(2) + x(ad + bc))/(S(2)bd) - x(ad/S(4) + bc/S(4))exp(ac + bdxS(2) + x(ad + bc))/(b*S(2)d*S(2)) - exp(ac + bdx*S(2) + x(ad + bc))/(S(2)bS(2)d*S(2)) + (ad + bc)S(2)exp(ac + bdxS(2) + x(ad + bc))/(S(8)bS(3)d*S(3)) + sqrt(pi)(S(3)ad/S(8) + S(3)bc/S(8))exp(-(-ad + bc)S(2)/(S(4)bd))erfi((ad/S(2) + bc/S(2) + bdx)/(sqrt(b)sqrt(d)))/(b(S(5)/2)d*(S(5)/2)) - sqrt(pi)(ad + bc)*S(3)exp(-(-ad + bc)*S(2)/(S(4)bd))erfi((ad/S(2) + bc/S(2) + bdx)/(sqrt(b)sqrt(d)))/(S(16)b*(S(7)/2)d(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)exp((a + bx)(c + dx)), x), x, xexp(ac + bdx*S(2) + x(ad + bc))/(S(2)bd) + (-ad/S(4) - bc/S(4))exp(ac + bdx*S(2) + x(ad + bc))/(b*S(2)d*S(2)) - sqrt(pi)exp(-(-ad + bc)*S(2)/(S(4)bd))erfi((ad/S(2) + bc/S(2) + bdx)/(sqrt(b)sqrt(d)))/(S(4)b*(S(3)/2)d*(S(3)/2)) + sqrt(pi)(ad + bc)*S(2)exp(-(-ad + bc)*S(2)/(S(4)bd))erfi((ad/S(2) + bc/S(2) + bdx)/(sqrt(b)sqrt(d)))/(S(8)b*(S(5)/2)d*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp((a + bx)(c + dx)), x), x, exp(ac + bdx*S(2) + x(ad + bc))/(S(2)bd) - sqrt(pi)(ad/S(4) + bc/S(4))exp(-(-ad + bc)*S(2)/(S(4)bd))erfi((ad/S(2) + bc/S(2) + bdx)/(sqrt(b)sqrt(d)))/(b(S(3)/2)d*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp((a + bx)(c + dx)), x), x, sqrt(pi)exp(-(-ad + bc)S(2)/(S(4)bd))erfi((ad/S(2) + bc/S(2) + bdx)/(sqrt(b)sqrt(d)))/(S(2)sqrt(b)sqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp((a + bx)(c + dx))/x, x), x, Integral(exp(ac + bdxS(2) + x(ad + bc))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp((a + bx)(c + dx))/xS(2), x), x, sqrt(pi)sqrt(b)sqrt(d)exp(-(-ad + bc)*S(2)/(S(4)bd))erfi((ad/S(2) + bc/S(2) + bdx)/(sqrt(b)sqrt(d))) + (ad + bc)Integral(exp(ac + bdxS(2) + x(ad + bc))/x, x) - exp(ac + bdxS(2) + x(ad + bc))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx + cxS(2))(d + ex)S(3), x), x, ef*(a + bx + cxS(2))(d + ex)S(2)/(S(2)clog(f)) - eS(3)f*(a + bx + cxS(2))/(S(2)c*S(2)log(f)*S(2)) + ef*(a + bx + cxS(2))(d + ex)(-be + S(2)cd)/(S(4)c*S(2)log(f)) + ef(a + bx + cxS(2))(-be + S(2)cd)S(2)/(S(8)c*S(3)log(f)) - S(3)sqrt(pi)e*S(2)f(a - bS(2)/(S(4)c))(-be + S(2)cd)erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(S(8)c(S(5)/2)log(f)*(S(3)/2)) + sqrt(pi)f(a - bS(2)/(S(4)c))(-be + S(2)cd)S(3)erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(S(16)c(S(7)/2)sqrt(log(f))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx + cxS(2))(d + ex)S(2), x), x, ef*(a + bx + cxS(2))(d + ex)/(S(2)clog(f)) + ef*(a + bx + cxS(2))(-be + S(2)cd)/(S(4)c*S(2)log(f)) - sqrt(pi)eS(2)f(a - bS(2)/(S(4)c))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(S(4)c(S(3)/2)log(f)*(S(3)/2)) + sqrt(pi)f(a - bS(2)/(S(4)c))(-be + S(2)cd)S(2)erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(S(8)c(S(5)/2)sqrt(log(f))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx + cxS(2))(d + ex), x), x, ef*(a + bx + cxS(2))/(S(2)clog(f)) + sqrt(pi)f(a - bS(2)/(S(4)c))(-be/S(4) + cd/S(2))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(c(S(3)/2)sqrt(log(f))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx + cxS(2))/(d + ex), x), x, Integral(f*(a + bx + cxS(2))/(d + ex), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx + cxS(2))/(d + ex)*S(2), x), x, sqrt(pi)sqrt(c)f(a - bS(2)/(S(4)c))sqrt(log(f))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/eS(2) - f(a + bx + cx*S(2))/(e(d + ex)) - (-be + S(2)cd)log(f)Integral(f*(a + bx + cxS(2))/(d + ex), x)/eS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx + cx*S(2))/(d + ex)*S(3), x), x, -sqrt(pi)sqrt(c)f(a - bS(2)/(S(4)c))(-be/S(2) + cd)log(f)*(S(3)/2)erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/e*S(4) + clog(f)Integral(f(a + bx + cxS(2))/(d + ex), x)/eS(2) - f(a + bx + cx*S(2))/(S(2)e(d + ex)S(2)) + f(a + bx + cx*S(2))(-be/S(2) + cd)log(f)/(eS(3)(d + ex)) + (-be + S(2)cd)*S(2)log(f)*S(2)Integral(f*(a + bx + cxS(2))/(d + ex), x)/(S(2)eS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx + cxS(2))(b + S(2)cx)*S(3), x), x, -S(4)cf(a + bx + cxS(2))/log(f)S(2) + f(a + bx + cxS(2))(b + S(2)cx)S(2)/log(f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx + cx*S(2))(b + S(2)cx)*S(2), x), x, -sqrt(pi)sqrt(c)f(a - bS(2)/(S(4)c))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/log(f)(S(3)/2) + f(a + bx + cxS(2))(b + S(2)cx)/log(f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx + cxS(2))(b + S(2)cx), x), x, f*(a + bx + cxS(2))/log(f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx + cxS(2))/(b + S(2)cx), x), x, f(a - bS(2)/(S(4)c))Ei((b + S(2)cx)S(2)log(f)/(S(4)c))/(S(4)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx + cxS(2))/(b + S(2)cx)S(2), x), x, -f(a + bx + cxS(2))/(S(2)c(b + S(2)cx)) + sqrt(pi)f(a - bS(2)/(S(4)c))sqrt(log(f))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(S(4)c(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx + cx*S(2))/(b + S(2)cx)S(3), x), x, -f(a + bx + cxS(2))/(S(4)c(b + S(2)cx)S(2)) + f(a - bS(2)/(S(4)c))log(f)Ei((b + S(2)cx)*S(2)log(f)/(S(4)c))/(S(16)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(bx + cx*S(2))(b + S(2)cx)*S(3), x), x, -S(4)cf(bx + cxS(2))/log(f)S(2) + f(bx + cxS(2))(b + S(2)cx)S(2)/log(f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(bx + cx*S(2))(b + S(2)cx)*S(2), x), x, -sqrt(pi)sqrt(c)f(-bS(2)/(S(4)c))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/log(f)(S(3)/2) + f(bx + cxS(2))(b + S(2)cx)/log(f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(bx + cxS(2))(b + S(2)cx), x), x, f*(bx + cxS(2))/log(f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(bx + cxS(2))/(b + S(2)cx), x), x, f(-bS(2)/(S(4)c))Ei((b + S(2)cx)S(2)log(f)/(S(4)c))/(S(4)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(bx + cxS(2))/(b + S(2)cx)S(2), x), x, -f(bx + cxS(2))/(S(2)c(b + S(2)cx)) + sqrt(pi)f(-bS(2)/(S(4)c))sqrt(log(f))erfi((b/S(2) + cx)sqrt(log(f))/sqrt(c))/(S(4)c(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(bx + cx*S(2))/(b + S(2)cx)S(3), x), x, -f(bx + cxS(2))/(S(4)c(b + S(2)cx)S(2)) + f(-bS(2)/(S(4)c))log(f)Ei((b + S(2)cx)*S(2)log(f)/(S(4)c))/(S(16)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bexp(c + dx))), x), x, Integral(S(1)/(x(a + bexp(c + dx))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bexp(c + dx)), x), x, x/a - log(a + bexp(c + dx))/(ad), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bexp(c + dx)), x), x, -xlog(aexp(-c - dx)/b + S(1))/(ad) + polylog(S(2), -aexp(-c - dx)/b)/(adS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bexp(c + dx)), x), x, -xS(2)log(aexp(-c - dx)/b + S(1))/(ad) + S(2)xpolylog(S(2), -aexp(-c - dx)/b)/(ad*S(2)) + S(2)polylog(S(3), -aexp(-c - dx)/b)/(adS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + bexp(c + dx)), x), x, -xS(3)log(aexp(-c - dx)/b + S(1))/(ad) + S(3)x*S(2)polylog(S(2), -aexp(-c - dx)/b)/(adS(2)) + S(6)xpolylog(S(3), -aexp(-c - dx)/b)/(ad*S(3)) + S(6)polylog(S(4), -aexp(-c - dx)/b)/(adS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bexp(c - dx)), x), x, x/a + log(a + bexp(c - dx))/(ad), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bexp(-c - dx)), x), x, x/a + log(a + bexp(-c - dx))/(ad), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bexp(c + dx))*S(2)), x), x, Integral(S(1)/(x(a + bexp(c + dx))*S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(c + dx))(S(-2)), x), x, S(1)/(ad(a + bexp(c + dx))) + x/aS(2) - log(a + bexp(c + dx))/(aS(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bexp(c + dx))*S(2), x), x, x/(ad(a + bexp(c + dx))) + xS(2)/(S(2)a*S(2)) - xlog(S(1) + bexp(c + dx)/a)/(a*S(2)d) - x/(a*S(2)d) + log(a + bexp(c + dx))/(a*S(2)d*S(2)) - polylog(S(2), -bexp(c + dx)/a)/(aS(2)dS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bexp(c + dx))S(2), x), x, xS(2)/(ad(a + bexp(c + dx))) + x*S(3)/(S(3)aS(2)) - xS(2)log(S(1) + bexp(c + dx)/a)/(aS(2)d) + S(2)xlog(aexp(-c - dx)/b + S(1))/(a*S(2)d*S(2)) - S(2)xpolylog(S(2), -bexp(c + dx)/a)/(aS(2)d*S(2)) - S(2)polylog(S(2), -aexp(-c - dx)/b)/(a*S(2)d*S(3)) + S(2)polylog(S(3), -bexp(c + dx)/a)/(a*S(2)dS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + bexp(c + dx))S(2), x), x, xS(3)/(ad(a + bexp(c + dx))) + x*S(4)/(S(4)aS(2)) - xS(3)log(S(1) + bexp(c + dx)/a)/(aS(2)d) + S(3)xS(2)log(aexp(-c - dx)/b + S(1))/(a*S(2)d*S(2)) - S(3)x*S(2)polylog(S(2), -bexp(c + dx)/a)/(a*S(2)d*S(2)) - S(6)xpolylog(S(2), -aexp(-c - dx)/b)/(aS(2)d*S(3)) + S(6)xpolylog(S(3), -bexp(c + dx)/a)/(aS(2)d*S(3)) - S(6)polylog(S(3), -aexp(-c - dx)/b)/(a*S(2)d*S(4)) - S(6)polylog(S(4), -bexp(c + dx)/a)/(a*S(2)d*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(c - dx))(S(-2)), x), x, -S(1)/(ad(a + bexp(c - dx))) + x/aS(2) + log(a + bexp(c - dx))/(aS(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(-c - dx))*(S(-2)), x), x, -S(1)/(ad(a + bexp(-c - dx))) + x/aS(2) + log(a + bexp(-c - dx))/(aS(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bexp(c + dx))S(3)), x), x, Integral(S(1)/(x(a + bexp(c + dx))*S(3)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(c + dx))(S(-3)), x), x, S(1)/(S(2)ad(a + bexp(c + dx))S(2)) + S(1)/(aS(2)d(a + bexp(c + dx))) + x/a*S(3) - log(a + bexp(c + dx))/(aS(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bexp(c + dx))*S(3), x), x, x/(S(2)ad(a + bexp(c + dx))S(2)) + x/(aS(2)d(a + bexp(c + dx))) - S(1)/(S(2)aS(2)d*S(2)(a + bexp(c + dx))) + x*S(2)/(S(2)a*S(3)) - xlog(S(1) + bexp(c + dx)/a)/(a*S(3)d) - S(3)x/(S(2)a*S(3)d) + S(3)log(a + bexp(c + dx))/(S(2)a*S(3)d*S(2)) - polylog(S(2), -bexp(c + dx)/a)/(aS(3)dS(2)), expand=True, _diff=True, _numerical=True) # recursion assert rubi_test(rubi_integrate(xS(2)/(a + bexp(c + dx))S(3), x), x, xS(2)/(S(2)ad(a + bexp(c + dx))S(2)) + xS(2)/(aS(2)d(a + bexp(c + dx))) - x/(aS(2)d*S(2)(a + bexp(c + dx))) + x*S(3)/(S(3)aS(3)) - xS(2)log(S(1) + bexp(c + dx)/a)/(aS(3)d) - S(3)xS(2)/(S(2)a*S(3)d) + S(3)xlog(S(1) + bexp(c + dx)/a)/(a*S(3)d*S(2)) - S(2)xpolylog(S(2), -bexp(c + dx)/a)/(aS(3)dS(2)) + x/(aS(3)dS(2)) - log(a + bexp(c + dx))/(aS(3)d*S(3)) + S(3)polylog(S(2), -bexp(c + dx)/a)/(a*S(3)d*S(3)) + S(2)polylog(S(3), -bexp(c + dx)/a)/(a*S(3)dS(3)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(2)/(a + bexp(c + dx))S(3), x), x, xS(2)/(S(2)ad(a + bexp(c + dx))S(2)) + xS(2)/(aS(2)d(a + bexp(c + dx))) - x/(aS(2)d*S(2)(a + bexp(c + dx))) + x*S(3)/(S(3)aS(3)) - xS(2)log(S(1) + bexp(c + dx)/a)/(aS(3)d) - x*S(2)/(S(2)a*S(3)d) + xlog(S(1) + bexp(c + dx)/a)/(aS(3)d*S(2)) + S(2)xlog(aexp(-c - dx)/b + S(1))/(aS(3)d*S(2)) - S(2)xpolylog(S(2), -bexp(c + dx)/a)/(aS(3)dS(2)) + x/(aS(3)dS(2)) - log(a + bexp(c + dx))/(aS(3)d*S(3)) + polylog(S(2), -bexp(c + dx)/a)/(aS(3)d*S(3)) - S(2)polylog(S(2), -aexp(-c - dx)/b)/(a*S(3)d*S(3)) + S(2)polylog(S(3), -bexp(c + dx)/a)/(a*S(3)d*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(c - dx))(S(-3)), x), x, -S(1)/(S(2)ad(a + bexp(c - dx))S(2)) - S(1)/(aS(2)d(a + bexp(c - dx))) + x/a*S(3) + log(a + bexp(c - dx))/(aS(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(-c - dx))*(S(-3)), x), x, -S(1)/(S(2)ad(a + bexp(-c - dx))S(2)) - S(1)/(aS(2)d(a + bexp(-c - dx))) + x/a*S(3) + log(a + bexp(-c - dx))/(aS(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(a + bx)/(xS(2)(c + dxS(2))), x), x, bexp(a)Ei(bx)/c - sqrt(d)exp(a - bsqrt(-c)/sqrt(d))Ei(b(sqrt(d)x + sqrt(-c))/sqrt(d))/(S(2)(-c)*(S(3)/2)) + sqrt(d)exp(a + bsqrt(-c)/sqrt(d))Ei(-b(-sqrt(d)x + sqrt(-c))/sqrt(d))/(S(2)(-c)(S(3)/2)) - exp(a + bx)/(cx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(a + bx)/(x(c + dx*S(2))), x), x, exp(a)Ei(bx)/c - exp(a - bsqrt(-c)/sqrt(d))Ei(b(sqrt(d)x + sqrt(-c))/sqrt(d))/(S(2)c) - exp(a + bsqrt(-c)/sqrt(d))Ei(-b(-sqrt(d)x + sqrt(-c))/sqrt(d))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(a + bx)/(c + dxS(2)), x), x, -exp(a - bsqrt(-c)/sqrt(d))Ei(b(sqrt(d)x + sqrt(-c))/sqrt(d))/(S(2)sqrt(d)sqrt(-c)) + exp(a + bsqrt(-c)/sqrt(d))Ei(-b(-sqrt(d)x + sqrt(-c))/sqrt(d))/(S(2)sqrt(d)sqrt(-c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp(a + bx)/(c + dx*S(2)), x), x, exp(a - bsqrt(-c)/sqrt(d))Ei(b(sqrt(d)x + sqrt(-c))/sqrt(d))/(S(2)d) + exp(a + bsqrt(-c)/sqrt(d))Ei(-b(-sqrt(d)x + sqrt(-c))/sqrt(d))/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)exp(a + bx)/(c + dx*S(2)), x), x, -sqrt(-c)exp(a - bsqrt(-c)/sqrt(d))Ei(b(sqrt(d)x + sqrt(-c))/sqrt(d))/(S(2)d(S(3)/2)) + sqrt(-c)exp(a + bsqrt(-c)/sqrt(d))Ei(-b(-sqrt(d)x + sqrt(-c))/sqrt(d))/(S(2)d(S(3)/2)) + exp(a + bx)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(d + ex)/(x*S(2)(a + bx + cx*S(2))), x), x, eexp(d)Ei(ex)/a - exp(d + ex)/(ax) - bexp(d)Ei(ex)/aS(2) + (b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))exp(d - e(b - sqrt(-S(4)ac + bS(2)))/(S(2)c))Ei(e(b + S(2)cx - sqrt(-S(4)ac + b*S(2)))/(S(2)c))/(S(2)aS(2)) + (b + (S(2)ac - bS(2))/sqrt(-S(4)ac + bS(2)))exp(d - e(b + sqrt(-S(4)ac + bS(2)))/(S(2)c))Ei(e(b + S(2)cx + sqrt(-S(4)ac + b*S(2)))/(S(2)c))/(S(2)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(d + ex)/(x(a + bx + cxS(2))), x), x, -(-b/sqrt(-S(4)ac + bS(2)) + S(1))exp(d - e(b + sqrt(-S(4)ac + bS(2)))/(S(2)c))Ei(e(b + S(2)cx + sqrt(-S(4)ac + b*S(2)))/(S(2)c))/(S(2)a) - (b/sqrt(-S(4)ac + bS(2)) + S(1))exp(d - e(b - sqrt(-S(4)ac + bS(2)))/(S(2)c))Ei(e(b + S(2)cx - sqrt(-S(4)ac + b*S(2)))/(S(2)c))/(S(2)a) + exp(d)Ei(ex)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(d + ex)/(a + bx + cx*S(2)), x), x, exp(d - e(b - sqrt(-S(4)ac + b*S(2)))/(S(2)c))Ei(e(b + S(2)cx - sqrt(-S(4)ac + b*S(2)))/(S(2)c))/sqrt(-S(4)ac + b*S(2)) - exp(d - e(b + sqrt(-S(4)ac + b*S(2)))/(S(2)c))Ei(e(b + S(2)cx + sqrt(-S(4)ac + b*S(2)))/(S(2)c))/sqrt(-S(4)ac + b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp(d + ex)/(a + bx + cxS(2)), x), x, (-b/sqrt(-S(4)ac + bS(2)) + S(1))exp(d - e(b - sqrt(-S(4)ac + bS(2)))/(S(2)c))Ei(e(b + S(2)cx - sqrt(-S(4)ac + b*S(2)))/(S(2)c))/(S(2)c) + (b/sqrt(-S(4)ac + bS(2)) + S(1))exp(d - e(b + sqrt(-S(4)ac + bS(2)))/(S(2)c))Ei(e(b + S(2)cx + sqrt(-S(4)ac + b*S(2)))/(S(2)c))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)exp(d + ex)/(a + bx + cxS(2)), x), x, exp(d + ex)/(ce) - (b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))exp(d - e(b + sqrt(-S(4)ac + bS(2)))/(S(2)c))Ei(e(b + S(2)cx + sqrt(-S(4)ac + b*S(2)))/(S(2)c))/(S(2)cS(2)) - (b + (S(2)ac - bS(2))/sqrt(-S(4)ac + bS(2)))exp(d - e(b - sqrt(-S(4)ac + bS(2)))/(S(2)c))Ei(e(b + S(2)cx - sqrt(-S(4)ac + b*S(2)))/(S(2)c))/(S(2)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)exp(d + ex)/(a + bx + cxS(2)), x), x, -bexp(d + ex)/(cS(2)e) + xexp(d + ex)/(ce) - exp(d + ex)/(ceS(2)) + (-ac + b*S(2) - b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))exp(d - e(b - sqrt(-S(4)ac + bS(2)))/(S(2)c))Ei(e(b + S(2)cx - sqrt(-S(4)ac + b*S(2)))/(S(2)c))/(S(2)cS(3)) + (-ac + b*S(2) + b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))exp(d - e(b + sqrt(-S(4)ac + bS(2)))/(S(2)c))Ei(e(b + S(2)cx + sqrt(-S(4)ac + b*S(2)))/(S(2)c))/(S(2)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(4)x/(S(2)xb + a), x), x, S(2)*x/(blog(S(2))) - alog(S(2)xb + a)/(b*S(2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)*(S(2)x)/(S(2)*xb + a), x), x, S(2)*x/(blog(S(2))) - alog(S(2)xb + a)/(b*S(2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(4)x/(-S(2)xb + a), x), x, -S(2)x/(blog(S(2))) - alog(-S(2)xb + a)/(b*S(2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)*(S(2)x)/(-S(2)*xb + a), x), x, -S(2)*x/(blog(S(2))) - alog(-S(2)xb + a)/(b*S(2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(4)x/(a + S(2)(-x)b), x), x, -S(2)xb/(a*S(2)log(S(2))) + S(2)*(S(2)x + S(-1))/(alog(S(2))) + bS(2)x/aS(3) + bS(2)log(a + S(2)(-x)b)/(a*S(3)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)*(S(2)x)/(a + S(2)*(-x)b), x), x, -S(2)*xb/(a*S(2)log(S(2))) + S(2)*(S(2)x + S(-1))/(alog(S(2))) + bS(2)x/aS(3) + bS(2)log(a + S(2)(-x)b)/(a*S(3)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(4)x/(a - S(2)(-x)b), x), x, S(2)xb/(a*S(2)log(S(2))) + S(2)*(S(2)x + S(-1))/(alog(S(2))) + bS(2)x/aS(3) + bS(2)log(a - S(2)(-x)b)/(a*S(3)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)*(S(2)x)/(a - S(2)*(-x)b), x), x, S(2)*xb/(a*S(2)log(S(2))) + S(2)*(S(2)x + S(-1))/(alog(S(2))) + bS(2)x/aS(3) + bS(2)log(a - S(2)(-x)b)/(a*S(3)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/(S(4)xb + a), x), x, atan(S(2)xsqrt(b)/sqrt(a))/(sqrt(a)sqrt(b)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/(S(2)(S(2)x)b + a), x), x, atan(S(2)*xsqrt(b)/sqrt(a))/(sqrt(a)sqrt(b)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/(-S(4)xb + a), x), x, atanh(S(2)xsqrt(b)/sqrt(a))/(sqrt(a)sqrt(b)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/(-S(2)(S(2)x)b + a), x), x, atanh(S(2)*xsqrt(b)/sqrt(a))/(sqrt(a)sqrt(b)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/(a + S(4)(-x)b), x), x, S(2)x/(alog(S(2))) - sqrt(b)atan(S(2)xsqrt(a)/sqrt(b))/(a*(S(3)/2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/(a + S(2)(-S(2)x)b), x), x, S(2)*x/(alog(S(2))) - sqrt(b)atan(S(2)xsqrt(a)/sqrt(b))/(a*(S(3)/2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/(a - S(4)(-x)b), x), x, S(2)x/(alog(S(2))) - sqrt(b)atanh(S(2)xsqrt(a)/sqrt(b))/(a*(S(3)/2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/(a - S(2)(-S(2)x)b), x), x, S(2)*x/(alog(S(2))) - sqrt(b)atanh(S(2)xsqrt(a)/sqrt(b))/(a*(S(3)/2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/sqrt(S(4)xb + a), x), x, atanh(S(2)xsqrt(b)/sqrt(S(2)*(S(2)x)b + a))/(sqrt(b)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/sqrt(S(2)(S(2)x)b + a), x), x, atanh(S(2)*xsqrt(b)/sqrt(S(2)*(S(2)x)b + a))/(sqrt(b)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/sqrt(-S(4)xb + a), x), x, atan(S(2)xsqrt(b)/sqrt(-S(2)*(S(2)x)b + a))/(sqrt(b)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/sqrt(-S(2)(S(2)x)b + a), x), x, atan(S(2)*xsqrt(b)/sqrt(-S(2)*(S(2)x)b + a))/(sqrt(b)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/sqrt(a + S(4)(-x)b), x), x, S(2)xsqrt(a + S(2)*(-S(2)x)b)/(alog(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/sqrt(a + S(2)(-S(2)x)b), x), x, S(2)*xsqrt(a + S(2)*(-S(2)x)b)/(alog(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/sqrt(a - S(4)(-x)b), x), x, S(2)xsqrt(a - S(2)*(-S(2)x)b)/(alog(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/sqrt(a - S(2)(-S(2)x)b), x), x, S(2)*xsqrt(a - S(2)*(-S(2)x)b)/(alog(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(4)x/sqrt(S(2)xb + a), x), x, -S(2)asqrt(S(2)xb + a)/(b*S(2)log(S(2))) + S(2)(S(2)xb + a)*(S(3)/2)/(S(3)b*S(2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)*(S(2)x)/sqrt(S(2)*xb + a), x), x, -S(2)asqrt(S(2)*xb + a)/(b*S(2)log(S(2))) + S(2)(S(2)xb + a)*(S(3)/2)/(S(3)b*S(2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(4)x/sqrt(-S(2)xb + a), x), x, -S(2)asqrt(-S(2)xb + a)/(b*S(2)log(S(2))) + S(2)(-S(2)xb + a)*(S(3)/2)/(S(3)b*S(2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)*(S(2)x)/sqrt(-S(2)*xb + a), x), x, -S(2)asqrt(-S(2)*xb + a)/(b*S(2)log(S(2))) + S(2)(-S(2)xb + a)*(S(3)/2)/(S(3)b*S(2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(4)x/sqrt(a + S(2)(-x)b), x), x, -S(3)S(2)*(x + S(-2))bsqrt(a + S(2)(-x)b)/(a*S(2)log(S(2))) + S(2)*(S(2)x + S(-1))sqrt(a + S(2)(-x)b)/(alog(S(2))) + S(3)b*S(2)atanh(sqrt(a + S(2)*(-x)b)/sqrt(a))/(S(4)a(S(5)/2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)*(S(2)x)/sqrt(a + S(2)*(-x)b), x), x, -S(3)S(2)(x + S(-2))bsqrt(a + S(2)(-x)b)/(a*S(2)log(S(2))) + S(2)*(S(2)x + S(-1))sqrt(a + S(2)(-x)b)/(alog(S(2))) + S(3)b*S(2)atanh(sqrt(a + S(2)*(-x)b)/sqrt(a))/(S(4)a(S(5)/2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(4)x/sqrt(a - S(2)(-x)b), x), x, S(3)S(2)*(x + S(-2))bsqrt(a - S(2)(-x)b)/(a*S(2)log(S(2))) + S(2)*(S(2)x + S(-1))sqrt(a - S(2)(-x)b)/(alog(S(2))) + S(3)b*S(2)atanh(sqrt(a - S(2)*(-x)b)/sqrt(a))/(S(4)a(S(5)/2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)*(S(2)x)/sqrt(a - S(2)*(-x)b), x), x, S(3)S(2)(x + S(-2))bsqrt(a - S(2)(-x)b)/(a*S(2)log(S(2))) + S(2)*(S(2)x + S(-1))sqrt(a - S(2)(-x)b)/(alog(S(2))) + S(3)b*S(2)atanh(sqrt(a - S(2)*(-x)b)/sqrt(a))/(S(4)a(S(5)/2)log(S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(exp(S(2)x) + S(2)exp(x) + S(1)), x), x, x - log(exp(x) + S(1)) + S(1)/(exp(x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(exp(S(2)x) + S(3)exp(x) + S(2)), x), x, x/S(2) - log(exp(x) + S(1)) + log(exp(x) + S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(exp(S(2)x) + exp(x) + S(-1)), x), x, -x + (-sqrt(S(5)) + S(5))log(S(2)exp(x) + S(1) + sqrt(S(5)))/S(10) + (sqrt(S(5)) + S(5))log(S(2)exp(x) - sqrt(S(5)) + S(1))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(exp(S(2)x) + S(3)exp(x) + S(3)), x), x, x/S(3) - log(exp(S(2)x) + S(3)exp(x) + S(3))/S(6) - sqrt(S(3))atan(sqrt(S(3))(S(2)exp(x) + S(3))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bexp(x) + cexp(S(2)x)), x), x, batanh((b + S(2)cexp(x))/sqrt(-S(4)ac + b*S(2)))/(asqrt(-S(4)ac + b*S(2))) + x/a - log(a + bexp(x) + cexp(S(2)x))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(exp(S(2)x) + S(2)exp(x) + S(1)), x), x, xS(2)/S(2) - xlog(exp(x) + S(1)) - x + x/(exp(x) + S(1)) + log(exp(x) + S(1)) - polylog(S(2), -exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(exp(S(2)x) + S(3)exp(x) + S(2)), x), x, -xlog(S(1) + exp(-x)) + xlog(S(1) + S(2)exp(-x))/S(2) - polylog(S(2), -S(2)exp(-x))/S(2) + polylog(S(2), -exp(-x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(exp(S(2)x) + exp(x) + S(-1)), x), x, S(2)sqrt(S(5))xlog(S(1) + (S(1)/2 + sqrt(S(5))/S(2))exp(-x))/(S(5)(S(1) + sqrt(S(5)))) - S(2)sqrt(S(5))xlog(S(1) + (-sqrt(S(5))/S(2) + S(1)/2)exp(-x))/(S(5)(-sqrt(S(5)) + S(1))) + S(2)sqrt(S(5))polylog(S(2), (S(-1)/2 + sqrt(S(5))/S(2))exp(-x))/(S(5)(-sqrt(S(5)) + S(1))) - S(2)sqrt(S(5))polylog(S(2), (-sqrt(S(5))/S(2) + S(-1)/2)exp(-x))/(S(5)(S(1) + sqrt(S(5)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(exp(S(2)x) + S(3)exp(x) + S(3)), x), x, -S(2)sqrt(S(3))xlog(S(1) + (S(3)/2 - sqrt(S(3))I/S(2))exp(-x))/(S(3)(sqrt(S(3)) + S(3)I)) + S(2)sqrt(S(3))xlog(S(1) + (S(3)/2 + sqrt(S(3))I/S(2))exp(-x))/(S(3)(-sqrt(S(3)) + S(3)I)) - S(2)sqrt(S(3))polylog(S(2), (S(-3)/2 - sqrt(S(3))I/S(2))exp(-x))/(S(3)(-sqrt(S(3)) + S(3)I)) + S(2)sqrt(S(3))polylog(S(2), (S(-3)/2 + sqrt(S(3))I/S(2))exp(-x))/(S(3)(sqrt(S(3)) + S(3)I)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bexp(x) + cexp(S(2)x)), x), x, S(2)cxlog(S(1) + (b/S(2) + sqrt(-S(4)ac + bS(2))/S(2))exp(-x)/c)/(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))) + S(2)cxlog(S(1) + (b/S(2) - sqrt(-S(4)ac + b*S(2))/S(2))exp(-x)/c)/(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2))) - S(2)cpolylog(S(2), (-b/S(2) - sqrt(-S(4)ac + bS(2))/S(2))exp(-x)/c)/(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))) - S(2)cpolylog(S(2), (-b/S(2) + sqrt(-S(4)ac + bS(2))/S(2))exp(-x)/c)/(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(exp(S(2)x) + S(2)exp(x) + S(1)), x), x, xS(3)/S(3) - xS(2)log(exp(x) + S(1)) + xS(2)/(exp(x) + S(1)) + S(2)xlog(S(1) + exp(-x)) - S(2)xpolylog(S(2), -exp(x)) - S(2)polylog(S(2), -exp(-x)) + S(2)polylog(S(3), -exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(exp(S(2)x) + S(3)exp(x) + S(2)), x), x, -xS(2)log(S(1) + exp(-x)) + x*S(2)log(S(1) + S(2)exp(-x))/S(2) - xpolylog(S(2), -S(2)exp(-x)) + S(2)xpolylog(S(2), -exp(-x)) - polylog(S(3), -S(2)exp(-x)) + S(2)polylog(S(3), -exp(-x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(exp(S(2)x) + exp(x) + S(-1)), x), x, S(2)sqrt(S(5))x*S(2)log(S(1) + (S(1)/2 + sqrt(S(5))/S(2))exp(-x))/(S(5)(S(1) + sqrt(S(5)))) - S(2)sqrt(S(5))x*S(2)log(S(1) + (-sqrt(S(5))/S(2) + S(1)/2)exp(-x))/(S(5)(-sqrt(S(5)) + S(1))) + S(4)sqrt(S(5))xpolylog(S(2), (S(-1)/2 + sqrt(S(5))/S(2))exp(-x))/(S(5)(-sqrt(S(5)) + S(1))) - S(4)sqrt(S(5))xpolylog(S(2), (-sqrt(S(5))/S(2) + S(-1)/2)exp(-x))/(S(5)(S(1) + sqrt(S(5)))) + S(4)sqrt(S(5))polylog(S(3), (S(-1)/2 + sqrt(S(5))/S(2))exp(-x))/(S(5)(-sqrt(S(5)) + S(1))) - S(4)sqrt(S(5))polylog(S(3), (-sqrt(S(5))/S(2) + S(-1)/2)exp(-x))/(S(5)(S(1) + sqrt(S(5)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(exp(S(2)x) + S(3)exp(x) + S(3)), x), x, -S(2)sqrt(S(3))xS(2)log(S(1) + (S(3)/2 - sqrt(S(3))I/S(2))exp(-x))/(S(3)(sqrt(S(3)) + S(3)I)) + S(2)sqrt(S(3))x*S(2)log(S(1) + (S(3)/2 + sqrt(S(3))I/S(2))exp(-x))/(S(3)(-sqrt(S(3)) + S(3)I)) - S(4)sqrt(S(3))xpolylog(S(2), (S(-3)/2 - sqrt(S(3))I/S(2))exp(-x))/(S(3)(-sqrt(S(3)) + S(3)I)) + S(4)sqrt(S(3))xpolylog(S(2), (S(-3)/2 + sqrt(S(3))I/S(2))exp(-x))/(S(3)(sqrt(S(3)) + S(3)I)) - S(4)sqrt(S(3))polylog(S(3), (S(-3)/2 - sqrt(S(3))I/S(2))exp(-x))/(S(3)(-sqrt(S(3)) + S(3)I)) + S(4)sqrt(S(3))polylog(S(3), (S(-3)/2 + sqrt(S(3))I/S(2))exp(-x))/(S(3)(sqrt(S(3)) + S(3)I)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(a + bexp(x) + cexp(S(2)x)), x), x, S(2)cx*S(2)log(S(1) + (b/S(2) + sqrt(-S(4)ac + b*S(2))/S(2))exp(-x)/c)/(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))) + S(2)cxS(2)log(S(1) + (b/S(2) - sqrt(-S(4)ac + b*S(2))/S(2))exp(-x)/c)/(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2))) - S(4)cxpolylog(S(2), (-b/S(2) - sqrt(-S(4)ac + b*S(2))/S(2))exp(-x)/c)/(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))) - S(4)cxpolylog(S(2), (-b/S(2) + sqrt(-S(4)ac + b*S(2))/S(2))exp(-x)/c)/(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2))) - S(4)cpolylog(S(3), (-b/S(2) - sqrt(-S(4)ac + bS(2))/S(2))exp(-x)/c)/(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))) - S(4)cpolylog(S(3), (-b/S(2) + sqrt(-S(4)ac + bS(2))/S(2))exp(-x)/c)/(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(2)f*(c + dx) + f*(S(2)c + S(2)dx) + S(1)), x), x, x - log(f*(c + dx) + S(1))/(dlog(f)) + S(1)/(d(f*(c + dx) + S(1))log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bf*(c + dx) + cf(S(2)c + S(2)dx)), x), x, batanh((b + S(2)cf(c + dx))/sqrt(-S(4)ac + b*S(2)))/(adsqrt(-S(4)ac + bS(2))log(f)) + x/a - log(a + bf(c + dx) + cf(S(2)c + S(2)dx))/(S(2)adlog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bf*(g + hx) + cf(S(2)g + S(2)hx)), x), x, batanh((b + S(2)cf(g + hx))/sqrt(-S(4)ac + b*S(2)))/(ahsqrt(-S(4)ac + bS(2))log(f)) + x/a - log(a + bf(g + hx) + cf(S(2)g + S(2)hx))/(S(2)ahlog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(S(2)f*(c + dx) + f*(S(2)c + S(2)dx) + S(1)), x), x, x*S(2)/S(2) - xlog(f*(c + dx) + S(1))/(dlog(f)) - x/(dlog(f)) + x/(d(f(c + dx) + S(1))log(f)) + log(f(c + dx) + S(1))/(d*S(2)log(f)S(2)) - polylog(S(2), -f(c + dx))/(dS(2)log(f)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bf*(c + dx) + cf(S(2)c + S(2)dx)), x), x, S(2)cxlog(S(1) + f(-c - dx)(b + sqrt(-S(4)ac + bS(2)))/(S(2)c))/(d(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))log(f)) - S(2)cxlog(S(1) + f(-c - dx)(b - sqrt(-S(4)ac + bS(2)))/(S(2)c))/(d(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))log(f)) - S(2)cpolylog(S(2), -f*(-c - dx)(b + sqrt(-S(4)ac + bS(2)))/(S(2)c))/(d*S(2)(b + sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))log(f)*S(2)) + S(2)cpolylog(S(2), -f(-c - dx)(b - sqrt(-S(4)ac + bS(2)))/(S(2)c))/(d*S(2)(b - sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(S(2)f(c + dx) + f*(S(2)c + S(2)dx) + S(1)), x), x, xS(3)/S(3) - xS(2)log(f(c + dx) + S(1))/(dlog(f)) + xS(2)/(d(f*(c + dx) + S(1))log(f)) + S(2)xlog(f(-c - dx) + S(1))/(d*S(2)log(f)*S(2)) - S(2)xpolylog(S(2), -f(c + dx))/(d*S(2)log(f)*S(2)) - S(2)polylog(S(2), -f*(-c - dx))/(d*S(3)log(f)*S(3)) + S(2)polylog(S(3), -f*(c + dx))/(d*S(3)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bf(c + dx) + cf(S(2)c + S(2)dx)), x), x, S(2)cx*S(2)log(S(1) + f*(-c - dx)(b + sqrt(-S(4)ac + bS(2)))/(S(2)c))/(d(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))log(f)) - S(2)cx*S(2)log(S(1) + f*(-c - dx)(b - sqrt(-S(4)ac + bS(2)))/(S(2)c))/(d(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))log(f)) - S(4)cxpolylog(S(2), -f(-c - dx)(b + sqrt(-S(4)ac + bS(2)))/(S(2)c))/(d*S(2)(b + sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))log(f)*S(2)) + S(4)cxpolylog(S(2), -f*(-c - dx)(b - sqrt(-S(4)ac + bS(2)))/(S(2)c))/(d*S(2)(b - sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))log(f)*S(2)) - S(4)cpolylog(S(3), -f(-c - dx)(b + sqrt(-S(4)ac + bS(2)))/(S(2)c))/(d*S(3)(b + sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))log(f)*S(3)) + S(4)cpolylog(S(3), -f(-c - dx)(b - sqrt(-S(4)ac + bS(2)))/(S(2)c))/(d*S(3)(b - sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))log(f)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ef*(g + hx))/(a + bf(g + hx) + cf(S(2)g + S(2)hx)), x), x, dx/a - dlog(a + bf(g + hx) + cf(S(2)g + S(2)hx))/(S(2)ahlog(f)) + (-S(2)ae + bd)atanh((b + S(2)cf(g + hx))/sqrt(-S(4)ac + b*S(2)))/(ahsqrt(-S(4)ac + bS(2))log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ef(g + hx))/(a + bf(g + hx) + cf(S(2)g + S(2)hx)), x), x, dx/a - dlog(a + bf(g + hx) + cf(S(2)g + S(2)hx))/(S(2)ahlog(f)) + (-S(2)ae + bd)atanh((b + S(2)cf(g + hx))/sqrt(-S(4)ac + b*S(2)))/(ahsqrt(-S(4)ac + bS(2))log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(exp(x) + S(2) + exp(-x)), x), x, -S(1)/(exp(x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(exp(x) + S(2) + exp(-x)), x), x, x - x/(exp(x) + S(1)) - log(exp(x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(exp(x) + S(2) + exp(-x)), x), x, -xS(2)/(exp(x) + S(1)) - S(2)xlog(S(1) + exp(-x)) + S(2)polylog(S(2), -exp(-x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(f(-c - dx) + f*(c + dx) + S(2)), x), x, -S(1)/(d(f(c + dx) + S(1))log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(f(-c - dx) + f*(c + dx) + S(2)), x), x, x/(dlog(f)) - x/(d(f*(c + dx) + S(1))log(f)) - log(f(c + dx) + S(1))/(d*S(2)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(f*(-c - dx) + f*(c + dx) + S(2)), x), x, -x*S(2)/(d(f*(c + dx) + S(1))log(f)) - S(2)xlog(f(-c - dx) + S(1))/(d*S(2)log(f)*S(2)) + S(2)polylog(S(2), -f*(-c - dx))/(d*S(3)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(3)x + S(2) + S(3)(-x)), x), x, -S(1)/((S(3)x + S(1))log(S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(2)exp(x) + S(1) - exp(-x)), x), x, log(-S(2)exp(x) + S(1))/S(3) - log(exp(x) + S(1))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bexp(-x) + cexp(x)), x), x, -S(2)atanh((a + S(2)cexp(x))/sqrt(a*S(2) - S(4)bc))/sqrt(aS(2) - S(4)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bexp(-x) + cexp(x)), x), x, xlog(S(2)cexp(x)/(a - sqrt(a*S(2) - S(4)bc)) + S(1))/sqrt(aS(2) - S(4)bc) - xlog(S(2)cexp(x)/(a + sqrt(a*S(2) - S(4)bc)) + S(1))/sqrt(aS(2) - S(4)bc) + polylog(S(2), -S(2)cexp(x)/(a - sqrt(aS(2) - S(4)bc)))/sqrt(aS(2) - S(4)bc) - polylog(S(2), -S(2)cexp(x)/(a + sqrt(aS(2) - S(4)bc)))/sqrt(aS(2) - S(4)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bexp(-x) + cexp(x)), x), x, xS(2)log(S(2)cexp(x)/(a - sqrt(a*S(2) - S(4)bc)) + S(1))/sqrt(aS(2) - S(4)bc) - xS(2)log(S(2)cexp(x)/(a + sqrt(a*S(2) - S(4)bc)) + S(1))/sqrt(aS(2) - S(4)bc) + S(2)xpolylog(S(2), -S(2)cexp(x)/(a - sqrt(aS(2) - S(4)bc)))/sqrt(aS(2) - S(4)bc) - S(2)xpolylog(S(2), -S(2)cexp(x)/(a + sqrt(aS(2) - S(4)bc)))/sqrt(aS(2) - S(4)bc) - S(2)polylog(S(3), -S(2)cexp(x)/(a - sqrt(a*S(2) - S(4)bc)))/sqrt(aS(2) - S(4)bc) + S(2)polylog(S(3), -S(2)cexp(x)/(a + sqrt(a*S(2) - S(4)bc)))/sqrt(aS(2) - S(4)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bf*(-c - dx) + cf(c + dx)), x), x, -S(2)atanh((a + S(2)cf(c + dx))/sqrt(a*S(2) - S(4)bc))/(dsqrt(a*S(2) - S(4)bc)log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bf(-c - dx) + cf(c + dx)), x), x, xlog(S(2)cf(c + dx)/(a - sqrt(a*S(2) - S(4)bc)) + S(1))/(dsqrt(a*S(2) - S(4)bc)log(f)) - xlog(S(2)cf(c + dx)/(a + sqrt(a*S(2) - S(4)bc)) + S(1))/(dsqrt(a*S(2) - S(4)bc)log(f)) + polylog(S(2), -S(2)cf*(c + dx)/(a - sqrt(a*S(2) - S(4)bc)))/(dS(2)sqrt(a*S(2) - S(4)bc)log(f)*S(2)) - polylog(S(2), -S(2)cf(c + dx)/(a + sqrt(a*S(2) - S(4)bc)))/(dS(2)sqrt(a*S(2) - S(4)bc)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bf(-c - dx) + cf(c + dx)), x), x, x*S(2)log(S(2)cf*(c + dx)/(a - sqrt(a*S(2) - S(4)bc)) + S(1))/(dsqrt(a*S(2) - S(4)bc)log(f)) - x*S(2)log(S(2)cf*(c + dx)/(a + sqrt(a*S(2) - S(4)bc)) + S(1))/(dsqrt(a*S(2) - S(4)bc)log(f)) + S(2)xpolylog(S(2), -S(2)cf*(c + dx)/(a - sqrt(a*S(2) - S(4)bc)))/(dS(2)sqrt(a*S(2) - S(4)bc)log(f)*S(2)) - S(2)xpolylog(S(2), -S(2)cf(c + dx)/(a + sqrt(a*S(2) - S(4)bc)))/(dS(2)sqrt(a*S(2) - S(4)bc)log(f)*S(2)) - S(2)polylog(S(3), -S(2)cf*(c + dx)/(a - sqrt(a*S(2) - S(4)bc)))/(dS(3)sqrt(a*S(2) - S(4)bc)log(f)*S(3)) + S(2)polylog(S(3), -S(2)cf*(c + dx)/(a + sqrt(a*S(2) - S(4)bc)))/(dS(3)sqrt(a*S(2) - S(4)bc)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((F(sqrt(-ax + S(1))/sqrt(ax + S(1))))n/(-aS(2)xS(2) + S(1)), x), x, -F(-nsqrt(-ax + S(1))/sqrt(ax + S(1)))(F(sqrt(-ax + S(1))/sqrt(ax + S(1))))nEi(nsqrt(-ax + S(1))log(F)/sqrt(ax + S(1)))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(S(3)sqrt(-ax + S(1))/sqrt(ax + S(1)))/(-a*S(2)x*S(2) + S(1)), x), x, -Ei(S(3)sqrt(-ax + S(1))log(F)/sqrt(ax + S(1)))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(S(2)sqrt(-ax + S(1))/sqrt(ax + S(1)))/(-a*S(2)x*S(2) + S(1)), x), x, -Ei(S(2)sqrt(-ax + S(1))log(F)/sqrt(ax + S(1)))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(sqrt(-ax + S(1))/sqrt(ax + S(1)))/(-aS(2)x*S(2) + S(1)), x), x, -Ei(sqrt(-ax + S(1))log(F)/sqrt(ax + S(1)))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(-sqrt(-ax + S(1))/sqrt(ax + S(1)))/(-aS(2)x*S(2) + S(1)), x), x, -Ei(-sqrt(-ax + S(1))log(F)/sqrt(ax + S(1)))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F*(-S(2)sqrt(-ax + S(1))/sqrt(ax + S(1)))/(-a*S(2)x*S(2) + S(1)), x), x, -Ei(-S(2)sqrt(-ax + S(1))log(F)/sqrt(ax + S(1)))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((F(sqrt(-cx + S(1))/sqrt(cx + S(1)))b + a)n/(-cS(2)xS(2) + S(1)), x), x, -Integral((Fxb + a)*n/x, (x, sqrt(-cx + S(1))/sqrt(cx + S(1))))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((F(sqrt(-cx + S(1))/sqrt(cx + S(1)))b + a)S(3)/(-cS(2)xS(2) + S(1)), x), x, -aS(3)log(sqrt(-cx + S(1))/sqrt(cx + S(1)))/c - S(3)aS(2)bEi(sqrt(-cx + S(1))log(F)/sqrt(cx + S(1)))/c - S(3)ab*S(2)Ei(S(2)sqrt(-cx + S(1))log(F)/sqrt(cx + S(1)))/c - b*S(3)Ei(S(3)sqrt(-cx + S(1))log(F)/sqrt(cx + S(1)))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((F*(sqrt(-cx + S(1))/sqrt(cx + S(1)))b + a)S(2)/(-cS(2)xS(2) + S(1)), x), x, -aS(2)log(sqrt(-cx + S(1))/sqrt(cx + S(1)))/c - S(2)abEi(sqrt(-cx + S(1))log(F)/sqrt(cx + S(1)))/c - b*S(2)Ei(S(2)sqrt(-cx + S(1))log(F)/sqrt(cx + S(1)))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((F*(sqrt(-cx + S(1))/sqrt(cx + S(1)))b + a)/(-c*S(2)x*S(2) + S(1)), x), x, -alog(sqrt(-cx + S(1))/sqrt(cx + S(1)))/c - bEi(sqrt(-cx + S(1))log(F)/sqrt(cx + S(1)))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((F*(sqrt(-cx + S(1))/sqrt(cx + S(1)))b + a)(-cS(2)x*S(2) + S(1))), x), x, -Integral(S(1)/(x(F*xb + a)), (x, sqrt(-cx + S(1))/sqrt(cx + S(1))))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((F*(sqrt(-cx + S(1))/sqrt(cx + S(1)))b + a)*S(2)(-c*S(2)x*S(2) + S(1))), x), x, -Integral(S(1)/(x(F*xb + a)*S(2)), (x, sqrt(-cx + S(1))/sqrt(cx + S(1))))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(axb*xxS(2), x), x, axbxx*S(2)/(log(a) + log(b)) - S(2)a*xb*xx/(log(a) + log(b))*S(2) + S(2)a*xbx/(log(a) + log(b))S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a*xb*xx, x), x, a*xb*xx/(log(a) + log(b)) - a*xbx/(log(a) + log(b))S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a*xbx, x), x, axbx/(log(a) + log(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(axb*x/x, x), x, Ei(x(log(a) + log(b))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a*xbx/xS(2), x), x, -a*xb*x/x + (log(a) + log(b))Ei(x(log(a) + log(b))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(axbx/xS(3), x), x, -a*xb*x(log(a) + log(b))/(S(2)x) - axb*x/(S(2)xS(2)) + (log(a) + log(b))S(2)Ei(x(log(a) + log(b)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a*xb*xcx, x), x, axbxcx/(log(a) + log(b) + log(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(axb(-x), x), x, axb(-x)/(log(a) - log(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(axb(-x)xS(2), x), x, axb(-x)x*S(2)/(log(a) - log(b)) - S(2)a*xb*(-x)x/(log(a) - log(b))*S(2) + S(2)a*xb(-x)/(log(a) - log(b))S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(2)x)/(a + bexp(x)), x), x, -alog(a + bexp(x))/b*S(2) + exp(x)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(2)x)/(a + bexp(x))S(2), x), x, a/(bS(2)(a + bexp(x))) + log(a + bexp(x))/b*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(2)x)/(a + bexp(x))S(3), x), x, exp(S(2)x)/(S(2)a(a + bexp(x))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(2)x)/(a + bexp(x))S(4), x), x, a/(S(3)b*S(2)(a + bexp(x))S(3)) - S(1)/(S(2)b*S(2)(a + bexp(x))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(4)x)/(a + bexp(S(2)x)), x), x, -alog(a + bexp(S(2)x))/(S(2)b*S(2)) + exp(S(2)x)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(4)x)/(a + bexp(S(2)x))*S(2), x), x, a/(S(2)b*S(2)(a + bexp(S(2)x))) + log(a + bexp(S(2)x))/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(4)x)/(a + bexp(S(2)x))*S(3), x), x, exp(S(4)x)/(S(4)a(a + bexp(S(2)x))*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(4)x)/(a + bexp(S(2)x))*S(4), x), x, a/(S(6)b*S(2)(a + bexp(S(2)x))*S(3)) - S(1)/(S(4)b*S(2)(a + bexp(S(2)x))*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(4)x)/(a + bexp(S(2)x))*(S(2)/3), x), x, -S(3)a(a + bexp(S(2)x))(S(1)/3)/(S(2)b*S(2)) + S(3)(a + bexp(S(2)x))*(S(4)/3)/(S(8)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(nx))exp(-nx), x), x, -aexp(-nx)/n + bx, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(nx))*S(2)exp(-nx), x), x, -aS(2)exp(-nx)/n + S(2)abx + b*S(2)exp(nx)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(nx))S(3)exp(-nx), x), x, -aS(3)exp(-nx)/n + S(3)a*S(2)bx + S(3)abS(2)exp(nx)/n + bS(3)exp(S(2)nx)/(S(2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(-nx)/(a + bexp(nx)), x), x, -exp(-nx)/(an) - bx/aS(2) + blog(a + bexp(nx))/(a*S(2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(-nx)/(a + bexp(nx))S(2), x), x, -b/(aS(2)n(a + bexp(nx))) - exp(-nx)/(a*S(2)n) - S(2)bx/a*S(3) + S(2)blog(a + bexp(nx))/(aS(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(-nx)/(a + bexp(nx))S(3), x), x, -b/(S(2)a*S(2)n(a + bexp(nx))S(2)) - S(2)b/(a*S(3)n(a + bexp(nx))) - exp(-nx)/(a*S(3)n) - S(3)bx/a*S(4) + S(3)blog(a + bexp(nx))/(aS(4)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + bx)/(c + df(S(2)bx + e)), x), x, f(a - e/S(2))atan(sqrt(d)f(bx + e/S(2))/sqrt(c))/(bsqrt(c)sqrt(d)log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + S(2)bx)/(c + df*(S(2)bx + e)), x), x, f(a - e)log(c + df(S(2)bx + e))/(S(2)bdlog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + S(3)bx)/(c + df*(S(2)bx + e)), x), x, -sqrt(c)f*(a - S(3)e/S(2))atan(sqrt(d)f*(bx + e/S(2))/sqrt(c))/(bd(S(3)/2)log(f)) + f*(a + bx - e)/(bdlog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + S(4)bx)/(c + df*(S(2)bx + e)), x), x, -cf*(a - S(2)e)log(c + df*(S(2)bx + e))/(S(2)bdS(2)log(f)) + f*(a + S(2)bx - e)/(S(2)bdlog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*(a + S(5)bx)/(c + df*(S(2)bx + e)), x), x, c(S(3)/2)f*(a - S(5)e/S(2))atan(sqrt(d)f*(bx + e/S(2))/sqrt(c))/(bd(S(5)/2)log(f)) - cf(a + bx - S(2)e)/(bd*S(2)log(f)) + f*(a + S(3)bx - e)/(S(3)bdlog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(exp(S(2)x) + S(1)), x), x, atan(exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(-exp(S(2)x) + S(1)), x), x, atanh(exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp(x)/(-exp(S(2)x) + S(1)), x), x, xatanh(exp(x)) + polylog(S(2), -exp(x))/S(2) - polylog(S(2), exp(x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)exp(x)/(-exp(S(2)x) + S(1)), x), x, xS(2)atanh(exp(x)) + xpolylog(S(2), -exp(x)) - xpolylog(S(2), exp(x)) - polylog(S(3), -exp(x)) + polylog(S(3), exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)exp(x)/(-exp(S(2)x) + S(1)), x), x, xS(3)atanh(exp(x)) + S(3)xS(2)polylog(S(2), -exp(x))/S(2) - S(3)xS(2)polylog(S(2), exp(x))/S(2) - S(3)xpolylog(S(3), -exp(x)) + S(3)xpolylog(S(3), exp(x)) + S(3)polylog(S(4), -exp(x)) - S(3)polylog(S(4), exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*x/(a + bf*(S(2)x)), x), x, atan(sqrt(b)fx/sqrt(a))/(sqrt(a)sqrt(b)log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(fxx/(a + bf(S(2)x)), x), x, xatan(sqrt(b)f*x/sqrt(a))/(sqrt(a)sqrt(b)log(f)) - Ipolylog(S(2), -Isqrt(b)f*x/sqrt(a))/(S(2)sqrt(a)sqrt(b)log(f)*S(2)) + Ipolylog(S(2), Isqrt(b)f*x/sqrt(a))/(S(2)sqrt(a)sqrt(b)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(fxxS(2)/(a + bf*(S(2)x)), x), x, x*S(2)atan(sqrt(b)fx/sqrt(a))/(sqrt(a)sqrt(b)log(f)) - Ixpolylog(S(2), -Isqrt(b)fx/sqrt(a))/(sqrt(a)sqrt(b)log(f)S(2)) + Ixpolylog(S(2), Isqrt(b)fx/sqrt(a))/(sqrt(a)sqrt(b)log(f)S(2)) + Ipolylog(S(3), -Isqrt(b)f*x/sqrt(a))/(sqrt(a)sqrt(b)log(f)S(3)) - Ipolylog(S(3), Isqrt(b)f*x/sqrt(a))/(sqrt(a)sqrt(b)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(fxx*S(3)/(a + bf*(S(2)x)), x), x, x*S(3)atan(sqrt(b)fx/sqrt(a))/(sqrt(a)sqrt(b)log(f)) - S(3)IxS(2)polylog(S(2), -Isqrt(b)f*x/sqrt(a))/(S(2)sqrt(a)sqrt(b)log(f)*S(2)) + S(3)IxS(2)polylog(S(2), Isqrt(b)f*x/sqrt(a))/(S(2)sqrt(a)sqrt(b)log(f)*S(2)) + S(3)Ixpolylog(S(3), -Isqrt(b)f*x/sqrt(a))/(sqrt(a)sqrt(b)log(f)S(3)) - S(3)Ixpolylog(S(3), Isqrt(b)f*x/sqrt(a))/(sqrt(a)sqrt(b)log(f)S(3)) - S(3)Ipolylog(S(4), -Isqrt(b)fx/sqrt(a))/(sqrt(a)sqrt(b)log(f)S(4)) + S(3)Ipolylog(S(4), Isqrt(b)fx/sqrt(a))/(sqrt(a)sqrt(b)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(fx/(a + bf*(S(2)x))S(2), x), x, fx/(S(2)a(a + bf(S(2)x))log(f)) + atan(sqrt(b)f*x/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(fxx/(a + bf(S(2)x))S(2), x), x, fxx/(S(2)a(a + bf*(S(2)x))log(f)) + xatan(sqrt(b)fx/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)) - atan(sqrt(b)f*x/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(2)) - Ipolylog(S(2), -Isqrt(b)f*x/sqrt(a))/(S(4)a*(S(3)/2)sqrt(b)log(f)S(2)) + Ipolylog(S(2), Isqrt(b)f*x/sqrt(a))/(S(4)a*(S(3)/2)sqrt(b)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(fxx*S(2)/(a + bf*(S(2)x))S(2), x), x, fxxS(2)/(S(2)a(a + bf*(S(2)x))log(f)) + xS(2)atan(sqrt(b)fx/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)) - xatan(sqrt(b)fx/sqrt(a))/(a(S(3)/2)sqrt(b)log(f)S(2)) - Ixpolylog(S(2), -Isqrt(b)fx/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(2)) + Ixpolylog(S(2), Isqrt(b)fx/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(2)) + Ipolylog(S(2), -Isqrt(b)f*x/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(3)) - Ipolylog(S(2), Isqrt(b)f*x/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(3)) + Ipolylog(S(3), -Isqrt(b)f*x/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(3)) - Ipolylog(S(3), Isqrt(b)f*x/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(fxx*S(3)/(a + bf*(S(2)x))S(2), x), x, fxxS(3)/(S(2)a(a + bf*(S(2)x))log(f)) + xS(3)atan(sqrt(b)fx/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)) - S(3)x*S(2)atan(sqrt(b)fx/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(2)) - S(3)IxS(2)polylog(S(2), -Isqrt(b)f*x/sqrt(a))/(S(4)a*(S(3)/2)sqrt(b)log(f)S(2)) + S(3)IxS(2)polylog(S(2), Isqrt(b)f*x/sqrt(a))/(S(4)a*(S(3)/2)sqrt(b)log(f)S(2)) + S(3)Ixpolylog(S(2), -Isqrt(b)f*x/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(3)) - S(3)Ixpolylog(S(2), Isqrt(b)f*x/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(3)) + S(3)Ixpolylog(S(3), -Isqrt(b)f*x/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(3)) - S(3)Ixpolylog(S(3), Isqrt(b)f*x/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(3)) - S(3)Ipolylog(S(3), -Isqrt(b)fx/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(4)) + S(3)Ipolylog(S(3), Isqrt(b)fx/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(4)) - S(3)Ipolylog(S(4), -Isqrt(b)fx/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(4)) + S(3)Ipolylog(S(4), Isqrt(b)fx/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(fxx/(a + bf(S(2)x))S(3), x), x, fxx/(S(4)a(a + bf*(S(2)x))*S(2)log(f)) + S(3)fxx/(S(8)aS(2)(a + bf(S(2)x))log(f)) - fx/(S(8)a*S(2)(a + bf(S(2)x))log(f)S(2)) + S(3)xatan(sqrt(b)f*x/sqrt(a))/(S(8)a*(S(5)/2)sqrt(b)log(f)) - atan(sqrt(b)f*x/sqrt(a))/(S(2)a*(S(5)/2)sqrt(b)log(f)S(2)) - S(3)Ipolylog(S(2), -Isqrt(b)fx/sqrt(a))/(S(16)a*(S(5)/2)sqrt(b)log(f)S(2)) + S(3)Ipolylog(S(2), Isqrt(b)fx/sqrt(a))/(S(16)a*(S(5)/2)sqrt(b)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(fxx*S(2)/(a + bf*(S(2)x))S(3), x), x, fxxS(2)/(S(4)a(a + bf*(S(2)x))*S(2)log(f)) + S(3)fxx*S(2)/(S(8)a*S(2)(a + bf(S(2)x))log(f)) - fxx/(S(4)aS(2)(a + bf(S(2)x))log(f)S(2)) + S(3)x*S(2)atan(sqrt(b)fx/sqrt(a))/(S(8)a*(S(5)/2)sqrt(b)log(f)) - xatan(sqrt(b)fx/sqrt(a))/(a(S(5)/2)sqrt(b)log(f)S(2)) - S(3)Ixpolylog(S(2), -Isqrt(b)f*x/sqrt(a))/(S(8)a*(S(5)/2)sqrt(b)log(f)S(2)) + S(3)Ixpolylog(S(2), Isqrt(b)f*x/sqrt(a))/(S(8)a*(S(5)/2)sqrt(b)log(f)S(2)) + atan(sqrt(b)f*x/sqrt(a))/(S(4)a*(S(5)/2)sqrt(b)log(f)S(3)) + Ipolylog(S(2), -Isqrt(b)f*x/sqrt(a))/(S(2)a*(S(5)/2)sqrt(b)log(f)S(3)) - Ipolylog(S(2), Isqrt(b)f*x/sqrt(a))/(S(2)a*(S(5)/2)sqrt(b)log(f)S(3)) + S(3)Ipolylog(S(3), -Isqrt(b)fx/sqrt(a))/(S(8)a*(S(5)/2)sqrt(b)log(f)S(3)) - S(3)Ipolylog(S(3), Isqrt(b)fx/sqrt(a))/(S(8)a*(S(5)/2)sqrt(b)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(af*x + bf*(-x)), x), x, xatan(sqrt(a)fx/sqrt(b))/(sqrt(a)sqrt(b)log(f)) - Ipolylog(S(2), -Isqrt(a)f*x/sqrt(b))/(S(2)sqrt(a)sqrt(b)log(f)*S(2)) + Ipolylog(S(2), Isqrt(a)f*x/sqrt(b))/(S(2)sqrt(a)sqrt(b)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(afx + bf(-x)), x), x, xS(2)atan(sqrt(a)f*x/sqrt(b))/(sqrt(a)sqrt(b)log(f)) - Ixpolylog(S(2), -Isqrt(a)fx/sqrt(b))/(sqrt(a)sqrt(b)log(f)S(2)) + Ixpolylog(S(2), Isqrt(a)fx/sqrt(b))/(sqrt(a)sqrt(b)log(f)S(2)) + Ipolylog(S(3), -Isqrt(a)f*x/sqrt(b))/(sqrt(a)sqrt(b)log(f)S(3)) - Ipolylog(S(3), Isqrt(a)f*x/sqrt(b))/(sqrt(a)sqrt(b)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(af*x + bf(-x)), x), x, xS(3)atan(sqrt(a)f*x/sqrt(b))/(sqrt(a)sqrt(b)log(f)) - S(3)IxS(2)polylog(S(2), -Isqrt(a)f*x/sqrt(b))/(S(2)sqrt(a)sqrt(b)log(f)*S(2)) + S(3)IxS(2)polylog(S(2), Isqrt(a)f*x/sqrt(b))/(S(2)sqrt(a)sqrt(b)log(f)*S(2)) + S(3)Ixpolylog(S(3), -Isqrt(a)f*x/sqrt(b))/(sqrt(a)sqrt(b)log(f)S(3)) - S(3)Ixpolylog(S(3), Isqrt(a)f*x/sqrt(b))/(sqrt(a)sqrt(b)log(f)S(3)) - S(3)Ipolylog(S(4), -Isqrt(a)fx/sqrt(b))/(sqrt(a)sqrt(b)log(f)S(4)) + S(3)Ipolylog(S(4), Isqrt(a)fx/sqrt(b))/(sqrt(a)sqrt(b)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(fx/(a + bf*(S(2)x))S(3), x), x, fx/(S(4)a(a + bf(S(2)x))*S(2)log(f)) + S(3)fx/(S(8)a*S(2)(a + bf(S(2)x))log(f)) + S(3)atan(sqrt(b)fx/sqrt(a))/(S(8)a*(S(5)/2)sqrt(b)log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(af*x + bf*(-x)), x), x, atan(sqrt(a)f*x/sqrt(b))/(sqrt(a)sqrt(b)log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((af*x + bf(-x))(S(-2)), x), x, -S(1)/(S(2)a(af(S(2)x) + b)log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(af*x + bf(-x))S(2), x), x, -x/(S(2)a(af(S(2)x) + b)log(f)) + x/(S(2)ablog(f)) - log(af(S(2)x) + b)/(S(4)ablog(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(af*x + bf(-x))S(2), x), x, -x*S(2)/(S(2)a(af*(S(2)x) + b)log(f)) - xlog(S(1) + bf(-S(2)x)/a)/(S(2)ablog(f)S(2)) + polylog(S(2), -bf*(-S(2)x)/a)/(S(4)ablog(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(af*x + bf(-x))S(2), x), x, -x*S(3)/(S(2)a(af*(S(2)x) + b)log(f)) - S(3)x*S(2)log(S(1) + bf(-S(2)x)/a)/(S(4)ablog(f)S(2)) + S(3)xpolylog(S(2), -bf*(-S(2)x)/a)/(S(4)ablog(f)S(3)) + S(3)polylog(S(3), -bf(-S(2)x)/a)/(S(8)ablog(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((af*x + bf(-x))(S(-3)), x), x, -f*x/(S(4)a(af*(S(2)x) + b)*S(2)log(f)) + f*x/(S(8)ab(af(S(2)x) + b)log(f)) + atan(sqrt(a)f*x/sqrt(b))/(S(8)a*(S(3)/2)b*(S(3)/2)log(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(afx + bf(-x))S(3), x), x, -f*xx/(S(4)a(af(S(2)x) + b)*S(2)log(f)) + f*xx/(S(8)ab(af*(S(2)x) + b)log(f)) + fx/(S(8)ab(af(S(2)x) + b)log(f)S(2)) + xatan(sqrt(a)fx/sqrt(b))/(S(8)a*(S(3)/2)b*(S(3)/2)log(f)) - Ipolylog(S(2), -Isqrt(a)fx/sqrt(b))/(S(16)a*(S(3)/2)b*(S(3)/2)log(f)*S(2)) + Ipolylog(S(2), Isqrt(a)f*x/sqrt(b))/(S(16)a*(S(3)/2)b*(S(3)/2)log(f)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(afx + bf(-x))S(3), x), x, -f*xx*S(2)/(S(4)a(af*(S(2)x) + b)*S(2)log(f)) + f*xx*S(2)/(S(8)ab(af(S(2)x) + b)log(f)) + fxx/(S(4)ab(af*(S(2)x) + b)log(f)S(2)) + xS(2)atan(sqrt(a)fx/sqrt(b))/(S(8)a*(S(3)/2)b*(S(3)/2)log(f)) - Ixpolylog(S(2), -Isqrt(a)f*x/sqrt(b))/(S(8)a*(S(3)/2)b*(S(3)/2)log(f)*S(2)) + Ixpolylog(S(2), Isqrt(a)fx/sqrt(b))/(S(8)a*(S(3)/2)b*(S(3)/2)log(f)*S(2)) - atan(sqrt(a)f*x/sqrt(b))/(S(4)a*(S(3)/2)b*(S(3)/2)log(f)*S(3)) + Ipolylog(S(3), -Isqrt(a)f*x/sqrt(b))/(S(8)a*(S(3)/2)b*(S(3)/2)log(f)*S(3)) - Ipolylog(S(3), Isqrt(a)f*x/sqrt(b))/(S(8)a*(S(3)/2)b*(S(3)/2)log(f)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(a + bx + cx*S(2))g*(d + ex + fxS(2)), x), x, sqrt(pi)f*ag*dexp(-(blog(f) + elog(g))*S(2)/(S(4)(clog(f) + flog(g))))erfi((blog(f)/S(2) + elog(g)/S(2) + x(clog(f) + flog(g)))/sqrt(clog(f) + flog(g)))/(S(2)sqrt(clog(f) + flog(g))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(e(c + dx))(G*(h(f + gx))b + a)n, x), x, F(e(c + dx))(G(h(f + gx))b + a)*(n + S(1))hyper((S(1), delog(F)/(ghlog(G)) + n + S(1)), (delog(F)/(ghlog(G)) + S(1),), -G*(h(f + gx))b/a)/(adelog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(e(c + dx))H*(t(r + sx))/(F(e(c + dx))b + a), x), x, H*(t(r + sx))hyper((S(1), -stlog(H)/(delog(F))), (S(1) - stlog(H)/(delog(F)),), -F*(-e(c + dx))a/b)/(bstlog(H)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(e(dx + f))H*(t(r + sx))/(F(e(c + dx))b + a), x), x, F*(-e(c - f))H(t(r + sx))hyper((S(1), -stlog(H)/(delog(F))), (S(1) - stlog(H)/(delog(F)),), -F*(-e(c + dx))a/b)/(bstlog(H)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + eexp(h + ix))(f + gx)S(3)/(a + bexp(h + ix) + cexp(S(2)h + S(2)ix)), x), x, S(6)g*S(3)(e + (be - S(2)cd)/sqrt(-S(4)ac + bS(2)))polylog(S(4), -(b + sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i*S(4)(b + sqrt(-S(4)ac + b*S(2)))) + S(6)g*S(3)(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))polylog(S(4), -(b - sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i*S(4)(b - sqrt(-S(4)ac + b*S(2)))) + S(6)g*S(2)(e + (be - S(2)cd)/sqrt(-S(4)ac + bS(2)))(f + gx)polylog(S(3), -(b + sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i*S(3)(b + sqrt(-S(4)ac + b*S(2)))) + S(6)g*S(2)(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))(f + gx)polylog(S(3), -(b - sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i*S(3)(b - sqrt(-S(4)ac + b*S(2)))) + S(3)g(e + (be - S(2)cd)/sqrt(-S(4)ac + b*S(2)))(f + gx)S(2)polylog(S(2), -(b + sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i*S(2)(b + sqrt(-S(4)ac + b*S(2)))) + S(3)g(e + (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))(f + gx)S(2)polylog(S(2), -(b - sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i*S(2)(b - sqrt(-S(4)ac + b*S(2)))) - (e + (be - S(2)cd)/sqrt(-S(4)ac + b*S(2)))(f + gx)S(3)log(S(1) + (b + sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i(b + sqrt(-S(4)ac + bS(2)))) - (e + (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))(f + gx)S(3)log(S(1) + (b - sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + eexp(h + ix))(f + gx)S(2)/(a + bexp(h + ix) + cexp(S(2)h + S(2)ix)), x), x, S(2)g*S(2)(e + (be - S(2)cd)/sqrt(-S(4)ac + bS(2)))polylog(S(3), -(b + sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i*S(3)(b + sqrt(-S(4)ac + b*S(2)))) + S(2)g*S(2)(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))polylog(S(3), -(b - sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i*S(3)(b - sqrt(-S(4)ac + b*S(2)))) + S(2)g(e + (be - S(2)cd)/sqrt(-S(4)ac + b*S(2)))(f + gx)polylog(S(2), -(b + sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i*S(2)(b + sqrt(-S(4)ac + b*S(2)))) + S(2)g(e + (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))(f + gx)polylog(S(2), -(b - sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i*S(2)(b - sqrt(-S(4)ac + b*S(2)))) - (e + (be - S(2)cd)/sqrt(-S(4)ac + b*S(2)))(f + gx)S(2)log(S(1) + (b + sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i(b + sqrt(-S(4)ac + bS(2)))) - (e + (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))(f + gx)S(2)log(S(1) + (b - sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + eexp(h + ix))(f + gx)/(a + bexp(h + ix) + cexp(S(2)h + S(2)ix)), x), x, g(e + (be - S(2)cd)/sqrt(-S(4)ac + bS(2)))polylog(S(2), -(b + sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i*S(2)(b + sqrt(-S(4)ac + b*S(2)))) + g(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))polylog(S(2), -(b - sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i*S(2)(b - sqrt(-S(4)ac + b*S(2)))) - (e + (be - S(2)cd)/sqrt(-S(4)ac + b*S(2)))(f + gx)log(S(1) + (b + sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i(b + sqrt(-S(4)ac + bS(2)))) + (e + (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))(-f - gx)log(S(1) + (b - sqrt(-S(4)ac + b*S(2)))exp(-h - ix)/(S(2)c))/(i(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + eexp(h + ix))/(a + bexp(h + ix) + cexp(S(2)h + S(2)ix)), x), x, dx/a - dlog(a + bexp(h + ix) + cexp(S(2)h + S(2)ix))/(S(2)ai) + (-S(2)ae + bd)atanh((b + S(2)cexp(h + ix))/sqrt(-S(4)ac + b*S(2)))/(aisqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate((d + eexp(h + ix))/((f + gx)(a + bexp(h + ix) + cexp(S(2)h + S(2)ix))), x), x, dIntegral(S(1)/((f + gx)(a + bexp(h + ix) + cexp(S(2)h + S(2)ix))), x) + eIntegral(exp(h + ix)/((f + gx)(a + bexp(h + ix) + cexp(S(2)h + S(2)ix))), x), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate((d + eexp(h + ix))/((f + gx)S(2)(a + bexp(h + ix) + cexp(S(2)h + S(2)ix))), x), x, dIntegral(S(1)/((f + gx)*S(2)(a + bexp(h + ix) + cexp(S(2)h + S(2)ix))), x) + eIntegral(exp(h + ix)/((f + gx)S(2)(a + bexp(h + ix) + cexp(S(2)h + S(2)ix))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-aeexp(c + dx) + be)/(-S(2)aeexp(c + dx) - beexp(S(2)c + S(2)dx) + be), x), x, -xlog(S(1) + (a - sqrt(aS(2) + bS(2)))exp(-c - dx)/b)/(S(2)d) - xlog(S(1) + (a + sqrt(aS(2) + bS(2)))exp(-c - dx)/b)/(S(2)d) + polylog(S(2), -(a - sqrt(aS(2) + bS(2)))exp(-c - dx)/b)/(S(2)dS(2)) + polylog(S(2), -(a + sqrt(aS(2) + b*S(2)))exp(-c - dx)/b)/(S(2)dS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(a + bx + cx*S(3))(b + S(3)cxS(2)), x), x, F(a + bx + cxS(3))/log(F), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(F(S(1)/(a + bx + cx*S(2)))(b + S(2)cx)/(a + bx + cxS(2))S(2), x), x, -F*(S(1)/(a + bx + cxS(2)))/log(F), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(a + bx + cxS(2))mexp(a + bx + cxS(2)), x), x, (-a - bx - cxS(2))(-m)(a + bx + cxS(2))mGamma(m + S(1), -a - bx - cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(a + bx + cxS(2))S(3)exp(a + bx + cxS(2)), x), x, (a + bx + cxS(2))S(3)exp(a + bx + cx*S(2)) - S(3)(a + bx + cxS(2))S(2)exp(a + bx + cxS(2)) + S(6)(a + bx + cx*S(2))exp(a + bx + cx*S(2)) - S(6)exp(a + bx + cx*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(a + bx + cxS(2))S(2)exp(a + bx + cxS(2)), x), x, (a + bx + cxS(2))S(2)exp(a + bx + cx*S(2)) - S(2)(a + bx + cx*S(2))exp(a + bx + cx*S(2)) + S(2)exp(a + bx + cx*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(a + bx + cx*S(2))exp(a + bx + cx*S(2)), x), x, (a + bx + cxS(2))exp(a + bx + cx*S(2)) - exp(a + bx + cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)exp(a + bx + cx*S(2)), x), x, exp(a + bx + cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)exp(a + bx + cx*S(2))/(a + bx + cxS(2)), x), x, Ei(a + bx + cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)exp(a + bx + cx*S(2))/(a + bx + cxS(2))S(2), x), x, Ei(a + bx + cxS(2)) - exp(a + bx + cxS(2))/(a + bx + cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)exp(a + bx + cx*S(2))/(a + bx + cxS(2))S(3), x), x, Ei(a + bx + cxS(2))/S(2) - exp(a + bx + cxS(2))/(S(2)(a + bx + cx*S(2))) - exp(a + bx + cxS(2))/(S(2)(a + bx + cxS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(a + bx + cxS(2))(S(7)/2)exp(a + bx + cx*S(2)), x), x, (a + bx + cxS(2))(S(7)/2)exp(a + bx + cx*S(2)) - S(7)(a + bx + cxS(2))(S(5)/2)exp(a + bx + cxS(2))/S(2) + S(35)(a + bx + cxS(2))(S(3)/2)exp(a + bx + cxS(2))/S(4) - S(105)sqrt(a + bx + cx*S(2))exp(a + bx + cx*S(2))/S(8) + S(105)sqrt(pi)erfi(sqrt(a + bx + cxS(2)))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(a + bx + cxS(2))(S(5)/2)exp(a + bx + cxS(2)), x), x, (a + bx + cxS(2))(S(5)/2)exp(a + bx + cx*S(2)) - S(5)(a + bx + cxS(2))(S(3)/2)exp(a + bx + cxS(2))/S(2) + S(15)sqrt(a + bx + cx*S(2))exp(a + bx + cx*S(2))/S(4) - S(15)sqrt(pi)erfi(sqrt(a + bx + cxS(2)))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(a + bx + cxS(2))(S(3)/2)exp(a + bx + cxS(2)), x), x, (a + bx + cxS(2))(S(3)/2)exp(a + bx + cx*S(2)) - S(3)sqrt(a + bx + cx*S(2))exp(a + bx + cx*S(2))/S(2) + S(3)sqrt(pi)erfi(sqrt(a + bx + cxS(2)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)sqrt(a + bx + cx*S(2))exp(a + bx + cx*S(2)), x), x, sqrt(a + bx + cxS(2))exp(a + bx + cx*S(2)) - sqrt(pi)erfi(sqrt(a + bx + cx*S(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)exp(a + bx + cx*S(2))/sqrt(a + bx + cxS(2)), x), x, sqrt(pi)erfi(sqrt(a + bx + cx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)exp(a + bx + cx*S(2))/(a + bx + cxS(2))(S(3)/2), x), x, S(2)sqrt(pi)erfi(sqrt(a + bx + cxS(2))) - S(2)exp(a + bx + cx*S(2))/sqrt(a + bx + cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)exp(a + bx + cx*S(2))/(a + bx + cxS(2))(S(5)/2), x), x, S(4)sqrt(pi)erfi(sqrt(a + bx + cxS(2)))/S(3) - S(4)exp(a + bx + cx*S(2))/(S(3)sqrt(a + bx + cx*S(2))) - S(2)exp(a + bx + cx*S(2))/(S(3)(a + bx + cxS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)exp(a + bx + cxS(2))/(a + bx + cxS(2))(S(7)/2), x), x, S(8)sqrt(pi)erfi(sqrt(a + bx + cxS(2)))/S(15) - S(8)exp(a + bx + cx*S(2))/(S(15)sqrt(a + bx + cx*S(2))) - S(4)exp(a + bx + cx*S(2))/(S(15)(a + bx + cxS(2))(S(3)/2)) - S(2)exp(a + bx + cxS(2))/(S(5)(a + bx + cxS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)exp(a + bx + cxS(2))/(a + bx + cxS(2))(S(9)/2), x), x, S(16)sqrt(pi)erfi(sqrt(a + bx + cxS(2)))/S(105) - S(16)exp(a + bx + cx*S(2))/(S(105)sqrt(a + bx + cx*S(2))) - S(8)exp(a + bx + cx*S(2))/(S(105)(a + bx + cxS(2))(S(3)/2)) - S(4)exp(a + bx + cxS(2))/(S(35)(a + bx + cxS(2))(S(5)/2)) - S(2)exp(a + bx + cxS(2))/(S(7)(a + bx + cxS(2))(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(-x)/sqrt(S(1) - exp(-S(2)x)), x), x, -asin(exp(-x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(exp(S(2)x) + S(4)), x), x, atan(exp(x)/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(-exp(S(2)x) + S(1)), x), x, atanh(exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(-S(4)exp(S(2)x) + S(3)), x), x, sqrt(S(3))atanh(S(2)sqrt(S(3))exp(x)/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-S(4)exp(S(2)x) + S(3))exp(x), x), x, sqrt(-S(4)exp(S(2)x) + S(3))exp(x)/S(2) + S(3)asin(S(2)sqrt(S(3))exp(x)/S(3))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)exp(xS(2)), x), x, xS(2)exp(xS(2))/S(2) - exp(xS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-exp(S(2)x) + S(1))exp(x), x), x, sqrt(-exp(S(2)x) + S(1))exp(x)/S(2) + asin(exp(x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/sqrt(exp(S(2)x) + exp(x) + S(1)), x), x, asinh(sqrt(S(3))(S(2)exp(x) + S(1))/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(exp(S(2)x) + S(-4)), x), x, -atanh(exp(x)/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp(-xS(2) + S(2)), x), x, -exp(-xS(2) + S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-xE + exp(x), x), x, -x(E + S(1))/(E + S(1)) + exp(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(S(2)x) + S(-1))/(exp(S(2)x) + S(3)), x), x, -x/S(3) + S(2)log(exp(S(2)x) + S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/sqrt(-exp(S(2)x) + S(1)), x), x, asin(exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(2)x)/(exp(S(4)x) + S(1)), x), x, atan(exp(S(2)x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(exp(S(2)x) - S(3)exp(x)), x), x, -x/S(9) + log(-exp(x) + S(3))/S(9) + exp(-x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(x) + S(-2))exp(x)/(exp(x) + S(1)), x), x, exp(x) - S(3)log(exp(x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(exp(S(2)x) + S(-1)), x), x, -atanh(exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(exp(S(2)x) + S(1)), x), x, atan(exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(x) + exp(-x))/(exp(x) - exp(-x)), x), x, log(-exp(x) + exp(-x)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((exp(x) + exp(-x))/(exp(x) - exp(-x)), x), x, -x + log(-exp(S(2)x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(x) - exp(-x))/(exp(x) + exp(-x)), x), x, log(exp(x) + exp(-x)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((exp(x) - exp(-x))/(exp(x) + exp(-x)), x), x, -x + log(exp(S(2)x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(S(2)x) + exp(-S(2)x))/(exp(S(2)x) - exp(-S(2)x)), x), x, -x + log(-exp(S(4)x) + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/sqrt(exp(S(2)x) + S(1)), x), x, asinh(exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(sqrt(x + S(4)))/sqrt(x + S(4)), x), x, S(2)exp(sqrt(x + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(exp(S(2)x*S(2)) + S(-1)), x), x, atan(sqrt(exp(S(2)x*S(2)) + S(-1)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(exp(S(2)x) + S(9))exp(x), x), x, sqrt(exp(S(2)x) + S(9))exp(x)/S(2) + S(9)asinh(exp(x)/S(3))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(exp(S(2)x) + S(1))exp(x), x), x, sqrt(exp(S(2)x) + S(1))exp(x)/S(2) + asinh(exp(x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp(xS(2))/(exp(S(2)xS(2)) + S(1)), x), x, atan(exp(xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)exp(x*(S(3)/2)), x), x, S(2)x*(S(3)/2)exp(x*(S(3)/2))/S(3) - S(2)exp(x*(S(3)/2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/sqrt(exp(S(2)x) + S(-3)), x), x, atanh(exp(x)/sqrt(exp(S(2)x) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(-exp(S(2)x) + S(16)), x), x, atanh(exp(x)/S(4))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(5)x)/(exp(S(10)x) + S(1)), x), x, atan(exp(S(5)x))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(4)x)/sqrt(exp(S(8)x) + S(16)), x), x, asinh(exp(S(4)x)/S(4))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)exp(S(4)xS(3))cos(S(7)xS(3)), x), x, S(7)exp(S(4)xS(3))sin(S(7)xS(3))/S(195) + S(4)exp(S(4)xS(3))cos(S(7)xS(3))/S(195), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp(xS(2) + S(1)), x), x, exp(xS(2) + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)exp(xS(3) + S(1)), x), x, exp(xS(3) + S(1))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(sqrt(x))/sqrt(x), x), x, S(2)exp(sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x(S(1)/3))/x(S(2)/3), x), x, S(3)exp(x(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(5) + S(2)xS(3) + S(-8))exp(S(3)x), x), x, xS(5)exp(S(3)x)/S(3) - S(5)x*S(4)exp(S(3)x)/S(9) + S(38)x*S(3)exp(S(3)x)/S(27) - S(38)x*S(2)exp(S(3)x)/S(27) + S(76)xexp(S(3)x)/S(81) - S(724)exp(S(3)x)/S(243), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + exp(x))S(2), x), x, xS(3)/S(3) + S(2)xexp(x) + exp(S(2)x)/S(2) - S(2)exp(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(S(3)x) + exp(S(2)x) + exp(x))exp(-S(4)x), x), x, -exp(-x) - exp(-S(2)x)/S(2) - exp(-S(3)x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(exp(S(2)x) + S(2)exp(x) + S(1)), x), x, -S(1)/(exp(x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(-x)cos(S(3)x), x), x, S(3)exp(-x)sin(S(3)x)/S(10) - exp(-x)cos(S(3)x)/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(2)x)/(exp(S(2)x) + S(3)exp(x) + S(2)), x), x, -log(exp(x) + S(1)) + S(2)log(exp(x) + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(2)x)/(exp(x) + S(1)), x), x, exp(x) - log(exp(x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(3)x)cos(S(5)x), x), x, S(5)exp(S(3)x)sin(S(5)x)/S(34) + S(3)exp(S(3)x)cos(S(5)x)/S(34), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)sech(exp(x)), x), x, atan(sinh(exp(x))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(-x)/(S(2)exp(x) + S(1)), x), x, -S(2)x + S(2)log(S(2)exp(x) + S(1)) - exp(-x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)cos(S(3)x + S(4)), x), x, S(3)exp(x)sin(S(3)x + S(4))/S(10) + exp(x)cos(S(3)x + S(4))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(exp(x) + exp(-x)), x), x, xexp(x) - xexp(-x) - exp(x) - exp(-x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(exp(S(2)x) + S(3)exp(x) + S(2)), x), x, -S(2)atanh(S(2)exp(x) + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(2)x)/(exp(x) + S(1))(S(1)/3), x), x, S(3)(exp(x) + S(1))*(S(5)/3)/S(5) - S(3)(exp(x) + S(1))*(S(2)/3)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(2)x)/(exp(x) + S(1))*(S(1)/4), x), x, S(4)(exp(x) + S(1))*(S(7)/4)/S(7) - S(4)(exp(x) + S(1))*(S(3)/4)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)exp(S(2)x) - exp(x))/sqrt(S(3)exp(S(2)x) - S(6)exp(x) + S(-1)), x), x, S(2)sqrt(S(3)exp(S(2)x) - S(6)exp(x) + S(-1))/S(3) - sqrt(S(3))atanh(sqrt(S(3))(-exp(x) + S(1))/sqrt(S(3)exp(S(2)x) - S(6)exp(x) + S(-1)))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) - S(5)x)exp(x), x), x, xS(2)exp(x) - S(7)xexp(x) + S(7)exp(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) - x)exp(S(3)x), x), x, xS(2)exp(S(3)x)/S(3) - S(5)xexp(S(3)x)/S(9) + S(5)exp(S(3)x)/S(27), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(2)x)(log(x) + S(1))exp(xx), x), x, (xx + S(-1))exp(xx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(S(7)x) + exp(S(5)x))/(exp(x) + exp(-x)), x), x, exp(S(6)x)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(-2) - S(1)/x)(-log(x) + S(1)), x), x, -x*(-S(1)/x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(x))S(2), x), x, aS(2)x + S(2)abexp(x) + b*S(2)exp(S(2)x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(x))S(3), x), x, aS(3)x + S(3)a*S(2)bexp(x) + S(3)abS(2)exp(S(2)x)/S(2) + bS(3)exp(S(3)x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bexp(x))S(4), x), x, aS(4)x + S(4)a*S(3)bexp(x) + S(3)a*S(2)b*S(2)exp(S(2)x) + S(4)abS(3)exp(S(3)x)/S(3) + bS(4)exp(S(4)x)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + bexp(c + dx)), x), x, -S(2)atanh(sqrt(a + bexp(c + dx))/sqrt(a))/(sqrt(a)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-a + bexp(c + dx)), x), x, S(2)atan(sqrt(-a + bexp(c + dx))/sqrt(a))/(sqrt(a)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bexp(c + dx)), x), x, -S(2)sqrt(a)atanh(sqrt(a + bexp(c + dx))/sqrt(a))/d + S(2)sqrt(a + bexp(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-a + bexp(c + dx)), x), x, -S(2)sqrt(a)atan(sqrt(-a + bexp(c + dx))/sqrt(a))/d + S(2)sqrt(-a + bexp(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(6)x)sin(S(3)x), x), x, S(2)exp(S(6)x)sin(S(3)x)/S(15) - exp(S(6)x)cos(S(3)x)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(3)x)/(exp(S(2)x) + S(1)), x), x, exp(x) - atan(exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(3)x)/(exp(S(2)x) + S(-1)), x), x, exp(x) - atanh(exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(-x)/sqrt(exp(S(2)x) + S(1)), x), x, -sqrt(exp(S(2)x) + S(1))exp(-x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(exp(S(2)x) - S(8)exp(x) + S(-1)), x), x, sqrt(S(17))atanh(sqrt(S(17))(-exp(x) + S(4))/S(17))/S(17), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)exp(S(7)x), x), x, xS(3)exp(S(7)x)/S(7) - S(3)x*S(2)exp(S(7)x)/S(49) + S(6)xexp(S(7)x)/S(343) - S(6)exp(S(7)x)/S(2401), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)exp(-S(2)x + S(8)), x), x, -xS(3)exp(-S(2)x + S(8))/S(2) - S(3)x*S(2)exp(-S(2)x + S(8))/S(4) - S(3)xexp(-S(2)x + S(8))/S(4) - S(3)exp(-S(2)x + S(8))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-exp(S(2)x) + S(9))exp(x), x), x, sqrt(-exp(S(2)x) + S(9))exp(x)/S(2) + S(9)asin(exp(x)/S(3))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-exp(S(2)x) + S(9))exp(S(6)x), x), x, -(-exp(S(2)x) + S(9))(S(7)/2)/S(7) + S(18)(-exp(S(2)x) + S(9))(S(5)/2)/S(5) - S(27)(-exp(S(2)x) + S(9))(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(6)x)/(-exp(x) + S(9))*(S(5)/2), x), x, S(2)(-exp(x) + S(9))*(S(7)/2)/S(7) - S(18)(-exp(x) + S(9))*(S(5)/2) + S(540)(-exp(x) + S(9))*(S(3)/2) - S(14580)sqrt(-exp(x) + S(9)) - S(65610)/sqrt(-exp(x) + S(9)) + S(39366)/(-exp(x) + S(9))(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(-S(7)exp(xS(4)) + S(2))S(5), x), x, S(8)xS(4) - S(16807)exp(S(5)xS(4))/S(20) + S(12005)exp(S(4)xS(4))/S(8) - S(3430)exp(S(3)xS(4))/S(3) + S(490)exp(S(2)xS(4)) - S(140)exp(x*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(-exp(S(2)xS(2)) + S(1))exp(x*S(2)), x), x, sqrt(-exp(S(2)x*S(2)) + S(1))exp(xS(2))/S(4) + asin(exp(xS(2)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(-exp(S(4)xS(3)) + S(1))S(2)exp(x*S(3)), x), x, exp(S(9)x*S(3))/S(27) - S(2)exp(S(5)xS(3))/S(15) + exp(xS(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x + exp(x)), x), x, exp(exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x + exp(x) + exp(exp(x))), x), x, exp(exp(exp(x))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(x) + exp(-x))S(2), x), x, S(2)x + exp(S(2)x)/S(2) - exp(-S(2)x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(exp(x) + exp(-x)), x), x, atan(exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(x) + exp(-x))*(S(-2)), x), x, -S(1)/(S(2)(exp(S(2)x) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(exp(x) - exp(-x)), x), x, -atanh(exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(x) - exp(-x))(S(-2)), x), x, S(1)/(S(2)(-exp(S(2)x) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(x) - exp(-x))S(2)exp(x), x), x, exp(S(3)x)/S(3) - S(2)exp(x) - exp(-x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(x) - exp(-x))*S(3)exp(x), x), x, S(3)x + exp(S(4)x)/S(4) - S(3)exp(S(2)x)/S(2) + exp(-S(2)x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x + S(1))/(S(2)x + S(1)), x), x, S(2)x/log(S(2)) + x - S(2)log(S(2)x + S(1))/log(S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x + S(1))/(S(1) + S(2)(-x)), x), x, -S(2)x/log(S(2)) + S(2)*(S(2)x + S(-1))/log(S(2)) + S(2)log(S(2)x + S(1))/log(S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-S(2)aexp((a + x)S(2))/x + exp((a + x)S(2))/xS(2), x), x, sqrt(pi)erfi(a + x) - exp((a + x)S(2))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(8) + xS(6) + xS(4))exp(-xS(2)), x), x, -xS(7)exp(-x*S(2))/S(2) - S(9)x*S(5)exp(-x*S(2))/S(4) - S(49)x*S(3)exp(-x*S(2))/S(8) - S(147)xexp(-xS(2))/S(16) + S(147)sqrt(pi)erf(x)/S(32), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(exp(S(3)x) - exp(x)), x), x, -atanh(exp(x)) + exp(-x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) + x + S(-5))exp(x)/(x + S(-1))*S(2), x), x, exp(x) - S(3)exp(x)/(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)exp(xS(2))/(xS(2) + S(1))S(2), x), x, exp(xS(2))/(S(2)(xS(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(S(3)x)/sqrt(S(16)exp(S(2)x) + S(25)), x), x, sqrt(S(16)exp(S(2)x) + S(25))exp(x)/S(32) - S(25)asinh(S(4)exp(x)/S(5))/S(128), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(x) + S(1))/sqrt(x + exp(x)), x), x, S(2)sqrt(x + exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(x) + S(1))/(x + exp(x)), x), x, log(x + exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(xS(2))/xS(2), x), x, sqrt(pi)erfi(x) - exp(xS(2))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x*S(4) + S(1))exp(xS(2))/xS(2), x), x, S(2)xexp(xS(2)) - exp(xS(2))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)sqrt(f*x), x), x, S(16)b*S(2)sqrt(fx)/log(f)S(3) - S(8)b(a + bx)sqrt(fx)/log(f)S(2) + S(2)(a + bx)*S(2)sqrt(fx)/log(f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(3)(x*S(2) + S(1))x, x), x, S(3)(xS(2) + S(1))/(S(2)log(S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)(sqrt(x))/sqrt(x), x), x, S(2)(sqrt(x) + S(1))/log(S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)(S(1)/x)/xS(2), x), x, -S(2)(S(1)/x)/log(S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x + S(2)(-x), x), x, S(2)x/log(S(2)) - S(2)(-x)/log(S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) - S(3)x + S(2))exp(-S(4)x), x), x, -x*S(2)exp(-S(4)x)/S(4) + S(5)xexp(-S(4)x)/S(8) - S(11)exp(-S(4)x)/S(32), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(k(x/S(2)) + x(sqrt(k)), x), x, S(2)k(x/S(2))/log(k) + x(sqrt(k) + S(1))/(sqrt(k) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(10)(sqrt(x))/sqrt(x), x), x, S(2)(sqrt(x) + S(1))S(5)*(sqrt(x))/log(S(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/sqrt(x + exp(x)) + S(1)/sqrt(x + exp(x)), x), x, S(2)sqrt(x + exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(exp(x) + S(1))/sqrt(x + exp(x)) + S(2)sqrt(x + exp(x)), x), x, S(2)xsqrt(x + exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp(x)/sqrt(x + exp(x)) + x/sqrt(x + exp(x)) + S(2)sqrt(x + exp(x)), x), x, S(2)xsqrt(x + exp(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(exp(x) + S(1))/sqrt(x + exp(x)), x), x, S(2)xsqrt(x + exp(x)) - S(2)Integral(sqrt(x + exp(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp(x)/sqrt(x + exp(x)) + x/sqrt(x + exp(x)), x), x, S(2)xsqrt(x + exp(x)) - S(2)Integral(sqrt(x + exp(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp(x)/sqrt(x + exp(x)), x), x, S(2)xsqrt(x + exp(x)) + S(2)sqrt(x + exp(x)) - Integral(S(1)/sqrt(x + exp(x)), x) - S(3)Integral(sqrt(x + exp(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(S(3)xS(2) + S(5)exp(x))/(S(5)sqrt(xS(3) + S(5)exp(x))) + S(4)xsqrt(x*S(3) + S(5)exp(x))/S(5), x), x, S(2)xS(2)sqrt(x*S(3) + S(5)exp(x))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)exp(x)/sqrt(x*S(3) + S(5)exp(x)), x), x, S(2)xS(2)sqrt(x*S(3) + S(5)exp(x))/S(5) - S(4)Integral(xsqrt(x*S(3) + S(5)exp(x)), x)/S(5) - S(3)Integral(xS(4)/sqrt(xS(3) + S(5)exp(x)), x)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-exp(x) + S(-1))/(x + exp(x))*(S(1)/3), x), x, -S(3)(x + exp(x))(S(2)/3)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(x + exp(x))(S(1)/3) - (x + exp(x))(S(2)/3) - S(1)/(x + exp(x))(S(1)/3), x), x, -S(3)(x + exp(x))(S(2)/3)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(x + exp(x))(S(1)/3), x), x, -S(3)(x + exp(x))(S(2)/3)/S(2) + Integral((x + exp(x))(S(-1)/3), x) + Integral((x + exp(x))*(S(2)/3), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)x + (S(2)x + S(3))exp(x))/(x + exp(x))*(S(1)/3), x), x, S(3)x(x + exp(x))(S(2)/3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)xexp(x)/(x + exp(x))(S(1)/3) + S(2)x/(x + exp(x))*(S(1)/3) + S(3)(x + exp(x))*(S(2)/3), x), x, S(3)x(x + exp(x))(S(2)/3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((exp(x) - exp(-x))(exp(x) + exp(-x))*S(2)exp(x), x), x, -x + exp(S(4)x)/S(4) + exp(S(2)x)/S(2) + exp(-S(2)x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(x + exp(x)), x), x, Integral(x/(x + exp(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(x + exp(x)), x), x, Integral(xS(2)/sqrt(x + exp(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(x + exp(x)), x), x, Integral(exp(x)/(x + exp(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)/(xS(2) + exp(x)), x), x, Integral(exp(x)/(xS(2) + exp(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(x)/(x + f(x)), x), x, x - Integral(x/(x + f(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(x)/(xS(2) + f(x)), x), x, x - Integral(xS(2)/(xS(2) + f(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(x)/(x + f(x))S(2), x), x, -Integral(x/(x + f(x))S(2), x) + Integral(S(1)/(x + f(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(x)/(xS(2) + f(x))S(2), x), x, -Integral(xS(2)/(xS(2) + f(x))S(2), x) + Integral(S(1)/(xS(2) + f(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((F(c + dx)a)m(F*(e + fx)b)n, x), x, (F(c + dx)a)m(F*(e + fx)b)n/((dm + fn)log(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(a + bxn + c + dx*n), x), x, -x(x*n(-b - d))*(-S(1)/n)Gamma(S(1)/n, x*n(-b - d))exp(a + c)/n, expand=True, _diff=True, _numerical=True) # (difference in simplify `exp(alog(f) + clog(g))` converts to `fagc` in mathematica) # failing assert rubi_test(rubi_integrate(f(a + bxn)g*(c + dxn), x), x, -fagcx(-xn(blog(f) + dlog(g)))*(-S(1)/n)Gamma(S(1)/n, -x*n(blog(f) + dlog(g)))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*mexp(xn), x), x, -x(m + S(1))(-xn)(-(m + S(1))/n)Gamma((m + S(1))/n, -xn)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f(x*n)xm, x), x, -x(m + S(1))(-xnlog(f))*(-(m + S(1))/n)Gamma((m + S(1))/n, -x*nlog(f))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)mexp((a + bx)n), x), x, -(-(a + bx)n)(-(m + S(1))/n)(a + bx)*(m + S(1))Gamma((m + S(1))/n, -(a + bx)n)/(bn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(f*((a + bx)*n)(a + bx)m, x), x, -(-(a + bx)*nlog(f))*(-(m + S(1))/n)(a + bx)(m + S(1))Gamma((m + S(1))/n, -(a + bx)nlog(f))/(bn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xexp((a + bx)S(3)), x), x, a(a + bx)Gamma(S(1)/3, -(a + bx)S(3))/(S(3)b*S(2)(-(a + bx)S(3))(S(1)/3)) - (a + bx)*S(2)Gamma(S(2)/3, -(a + bx)S(3))/(S(3)b*S(2)(-(a + bx)S(3))*(S(2)/3)), expand=True, _diff=True, _numerical=True)
177f66b9b8da0137809b8ab606e9a2ddc2672d40e376cbca4ef2bdde3f079884	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.functions import log, sqrt, exp, cos, sin, tan, sec, csc, cot,cosh, sinh, tanh, coth, csch, csch, sech from sympy.functions.elementary.hyperbolic import (acosh, acsch, asinh, atanh) from sympy.functions.elementary.trigonometric import (acos, acsc, asin, atan) from sympy.integrals.rubi.utility_function import (EllipticE, EllipticF, Int, ArcCsch, ArcCsc, Gamma, Factorial, PolyGamma , LogGamma , Subst , hypergeom, rubi_test, AppellF1, EllipticPi, Log, Sqrt, ArcTan, ArcTanh, ArcSin, ArcSinh, ArcCosh, ArcTanh, ArcCos, Hypergeometric2F1,) from sympy.core.singleton import S from sympy.core import EulerGamma from sympy.core.numbers import (E, I, pi) from sympy.core.symbol import symbols from sympy.functions.elementary.exponential import (exp, exp_polar) from sympy.functions.special.error_functions import (Chi, Ci, Ei, erf, erfi, expint, li, Shi, Si) from sympy.functions.special.hyper import HypergeometricPFQ as hyper from sympy.functions.special.zeta_functions import polylog from sympy.integrals.integrals import Integral from sympy.simplify.simplify import simplify from sympy.testing.pytest import SKIP a, b, c, d, e, f, m, n, x, u , k, p, r, s, t, i, j = symbols('a b c d e f m n x u k p r s t i j') A, B, C, D, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B C D a b c d e f g h y z m n p q u v w F', ) def test_1(): assert rubi_test(rubi_integrate((e + fx)(p + S(-1))/log(d(e + fx)p), x), x, li(d(e + fx)p)/(dfp), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((eg + fgx)*(p + S(-1))/log(d(e + fx)p), x), x, (e + fx)*(-p + S(1))(eg + fgx)(p + S(-1))li(d(e + fx)*p)/(dfp), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(g + hx)m, x), x, bfpq(g + hx)*(m + S(2))hyper((S(1), m + S(2)), (m + S(3),), f(g + hx)/(-eh + fg))/(h(m + S(1))(m + S(2))(-eh + fg)) + (a + blog(c(d(e + fx)p)q))(g + hx)(m + S(1))/(h(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(g + hx)*S(4), x), x, -bpq(g + hx)S(5)/(S(25)h) - bpq(g + hx)*S(4)(-eh + fg)/(S(20)fh) - bpq(g + hx)*S(3)(-eh + fg)*S(2)/(S(15)f*S(2)h) - bpq(g + hx)*S(2)(-eh + fg)*S(3)/(S(10)f*S(3)h) - bpqx(-eh + fg)*S(4)/(S(5)f*S(4)) - bpq(-eh + fg)*S(5)log(e + fx)/(S(5)f*S(5)h) + (a + blog(c(d(e + fx)p)q))(g + hx)*S(5)/(S(5)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(g + hx)*S(3), x), x, -bpq(g + hx)S(4)/(S(16)h) - bpq(g + hx)*S(3)(-eh + fg)/(S(12)fh) - bpq(g + hx)*S(2)(-eh + fg)*S(2)/(S(8)f*S(2)h) - bpqx(-eh + fg)*S(3)/(S(4)f*S(3)) - bpq(-eh + fg)*S(4)log(e + fx)/(S(4)f*S(4)h) + (a + blog(c(d(e + fx)p)q))(g + hx)*S(4)/(S(4)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(g + hx)*S(2), x), x, -bpq(g + hx)S(3)/(S(9)h) - bpq(g + hx)*S(2)(-eh + fg)/(S(6)fh) - bpqx(-eh + fg)*S(2)/(S(3)f*S(2)) - bpq(-eh + fg)*S(3)log(e + fx)/(S(3)f*S(3)h) + (a + blog(c(d(e + fx)p)q))(g + hx)*S(3)/(S(3)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(g + hx), x), x, -bpq(g + hx)*S(2)/(S(4)h) - bpqx(-eh + fg)/(S(2)f) - bpq(-eh + fg)*S(2)log(e + fx)/(S(2)f*S(2)h) + (a + blog(c(d(e + fx)p)q))(g + hx)*S(2)/(S(2)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a + blog(c(d(e + fx)p)q), x), x, ax - bpqx + b(e + fx)log(c(d(e + fx)p)q)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(g + hx), x), x, bpqpolylog(S(2), -h(e + fx)/(-eh + fg))/h + (a + blog(c(d(e + fx)p)q))log(f(g + hx)/(-eh + fg))/h, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(g + hx)*S(2), x), x, bfpqlog(e + fx)/(h(-eh + fg)) - bfpqlog(g + hx)/(h(-eh + fg)) + (-a - blog(c(d(e + fx)p)q))/(h(g + hx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(g + hx)*S(3), x), x, bf*S(2)pqlog(e + fx)/(S(2)h(-eh + fg)S(2)) - bf*S(2)pqlog(g + hx)/(S(2)h(-eh + fg)S(2)) + bfpq/(S(2)h(g + hx)(-eh + fg)) + (-a/S(2) - blog(c(d(e + fx)p)q)/S(2))/(h(g + hx)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(g + hx)*S(4), x), x, bf*S(3)pqlog(e + fx)/(S(3)h(-eh + fg)S(3)) - bf*S(3)pqlog(g + hx)/(S(3)h(-eh + fg)S(3)) + bf*S(2)pq/(S(3)h(g + hx)(-eh + fg)S(2)) + bfpq/(S(6)h(g + hx)S(2)(-eh + fg)) + (-a/S(3) - blog(c(d(e + fx)p)q)/S(3))/(h(g + hx)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(g + hx)*S(5), x), x, bf*S(4)pqlog(e + fx)/(S(4)h(-eh + fg)S(4)) - bf*S(4)pqlog(g + hx)/(S(4)h(-eh + fg)S(4)) + bf*S(3)pq/(S(4)h(g + hx)(-eh + fg)S(3)) + bf*S(2)pq/(S(8)h(g + hx)*S(2)(-eh + fg)*S(2)) + bfpq/(S(12)h(g + hx)S(3)(-eh + fg)) + (-a/S(4) - blog(c(d(e + fx)p)q)/S(4))/(h(g + hx)*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(2)(g + hx)m, x), x, Integral((a + blog(c(d(e + fx)p)q))S(2)(g + hx)m, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(2)(g + hx)S(3), x), x, -abpqx(-eh + fg)*S(3)/(S(2)fS(3)) + bS(2)pS(2)q*S(2)(g + hx)S(4)/(S(32)h) + S(7)bS(2)p*S(2)q*S(2)(g + hx)S(3)(-eh + fg)/(S(72)fh) + S(13)bS(2)p*S(2)q*S(2)(g + hx)S(2)(-eh + fg)*S(2)/(S(48)f*S(2)h) + S(25)bS(2)p*S(2)q*S(2)x(-eh + fg)S(3)/(S(24)fS(3)) - bS(2)pq(e + fx)(-eh + fg)S(3)log(c(d(e + fx)p)q)/(S(2)f*S(4)) + S(13)b*S(2)p*S(2)q*S(2)(-eh + fg)*S(4)log(e + fx)/(S(24)f*S(4)h) - bpq(a + blog(c(d(e + fx)p)q))(g + hx)S(4)/(S(8)h) - bpq(a + blog(c(d(e + fx)p)q))(g + hx)S(3)(-eh + fg)/(S(6)fh) - bpq(a + blog(c(d(e + fx)p)q))(g + hx)S(2)(-eh + fg)*S(2)/(S(4)f*S(2)h) + (a + blog(c(d(e + fx)p)q))*S(2)(g + hx)S(4)/(S(4)h) - (a + blog(c(d(e + fx)p)q))*S(2)(-eh + fg)*S(4)/(S(4)f*S(4)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(2)(g + hx)S(2), x), x, -S(2)abpqx(-eh + fg)S(2)/(S(3)f*S(2)) + S(2)b*S(2)p*S(2)q*S(2)(g + hx)S(3)/(S(27)h) + S(5)bS(2)p*S(2)q*S(2)(g + hx)S(2)(-eh + fg)/(S(18)fh) + S(11)bS(2)p*S(2)q*S(2)x(-eh + fg)S(2)/(S(9)f*S(2)) - S(2)b*S(2)pq(e + fx)(-eh + fg)*S(2)log(c(d(e + fx)p)q)/(S(3)f*S(3)) + S(5)b*S(2)p*S(2)q*S(2)(-eh + fg)*S(3)log(e + fx)/(S(9)f*S(3)h) - S(2)bpq(a + blog(c(d(e + fx)p)q))(g + hx)*S(3)/(S(9)h) - bpq(a + blog(c(d(e + fx)p)q))(g + hx)S(2)(-eh + fg)/(S(3)fh) + (a + blog(c(d(e + fx)p)q))*S(2)(g + hx)S(3)/(S(3)h) - (a + blog(c(d(e + fx)p)q))*S(2)(-eh + fg)*S(3)/(S(3)f*S(3)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(2)(g + hx), x), x, -S(2)abpqx(-eh + fg)/f + bS(2)ehp*S(2)q*S(2)x/(S(2)f) + bS(2)hpS(2)q*S(2)x*S(2)/S(4) + S(2)b*S(2)p*S(2)q*S(2)x(-eh + fg)/f - S(2)b*S(2)pq(e + fx)(-eh + fg)log(c(d(e + fx)p)q)/f*S(2) - bhpq(a + blog(c(d(e + fx)p)q))(e + fx)S(2)/(S(2)f*S(2)) + h(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)S(2)/(S(2)f*S(2)) + (a + blog(c(d(e + fx)p)q))S(2)(e + fx)(-eh + fg)/f*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(2), x), x, -S(2)abpqx + S(2)bS(2)p*S(2)q*S(2)x - S(2)bS(2)pq(e + fx)log(c(d(e + fx)p)q)/f + (a + blog(c(d(e + fx)p)q))S(2)(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(2)/(g + hx), x), x, -S(2)bS(2)p*S(2)q*S(2)polylog(S(3), -h(e + fx)/(-eh + fg))/h + S(2)bpq(a + blog(c(d(e + fx)p)q))polylog(S(2), -h(e + fx)/(-eh + fg))/h + (a + blog(c(d(e + fx)p)q))S(2)log(f(g + hx)/(-eh + fg))/h, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(2)/(g + hx)*S(2), x), x, -S(2)b*S(2)fpS(2)q*S(2)polylog(S(2), -h(e + fx)/(-eh + fg))/(h(-eh + fg)) - S(2)bfpq(a + blog(c(d(e + fx)p)q))log(f(g + hx)/(-eh + fg))/(h(-eh + fg)) + (a + blog(c(d(e + fx)p)q))*S(2)(e + fx)/((g + hx)(-eh + fg)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(2)/(g + hx)S(3), x), x, -bS(2)fS(2)p*S(2)q*S(2)log(e + fx)/(h(-eh + fg)S(2)) + bS(2)fS(2)p*S(2)q*S(2)log(g + hx)/(h(-eh + fg)S(2)) - bS(2)fS(2)p*S(2)q*S(2)polylog(S(2), -h(e + fx)/(-eh + fg))/(h(-eh + fg)S(2)) - bf*S(2)pq(a + blog(c(d(e + fx)p)q))log(f(g + hx)/(-eh + fg))/(h(-eh + fg)*S(2)) + bfpq(a + blog(c(d(e + fx)p)q))/(h(g + hx)(-eh + fg)) + f*S(2)(a + blog(c(d(e + fx)p)q))*S(2)/(S(2)h(-eh + fg)S(2)) - (a + blog(c(d(e + fx)p)q))S(2)/(S(2)h(g + hx)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(2)/(g + hx)S(4), x), x, -bS(2)fS(3)p*S(2)q*S(2)log(e + fx)/(h(-eh + fg)S(3)) + bS(2)fS(3)p*S(2)q*S(2)log(g + hx)/(h(-eh + fg)*S(3)) - S(2)b*S(2)f*S(3)p*S(2)q*S(2)polylog(S(2), -h(e + fx)/(-eh + fg))/(S(3)h(-eh + fg)S(3)) - bS(2)fS(2)p*S(2)q*S(2)/(S(3)h(g + hx)(-eh + fg)S(2)) - S(2)bfS(3)pq(a + blog(c(d(e + fx)p)q))log(f(g + hx)/(-eh + fg))/(S(3)h(-eh + fg)S(3)) + S(2)bfS(2)pq(a + blog(c(d(e + fx)p)q))/(S(3)h(g + hx)(-eh + fg)*S(2)) + bfpq(a + blog(c(d(e + fx)p)q))/(S(3)h(g + hx)*S(2)(-eh + fg)) + f*S(3)(a + blog(c(d(e + fx)p)q))*S(2)/(S(3)h(-eh + fg)S(3)) - (a + blog(c(d(e + fx)p)q))S(2)/(S(3)h(g + hx)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(3)(g + hx)m, x), x, Integral((a + blog(c(d(e + fx)p)q))S(3)(g + hx)m, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(3)(g + hx)S(3), x), x, S(6)abS(2)p*S(2)q*S(2)x(-eh + fg)S(3)/fS(3) - S(9)b*S(3)ehp*S(3)q*S(3)x(-eh + fg)S(2)/(S(4)f*S(3)) - S(9)b*S(3)hpS(3)q*S(3)x*S(2)(-eh + fg)*S(2)/(S(8)f*S(2)) - S(6)b*S(3)p*S(3)q*S(3)x(-eh + fg)S(3)/fS(3) - S(3)b*S(3)h*S(3)p*S(3)q*S(3)(e + fx)S(4)/(S(128)f*S(4)) - S(2)b*S(3)h*S(2)p*S(3)q*S(3)(e + fx)S(3)(-eh + fg)/(S(9)fS(4)) + S(6)b*S(3)p*S(2)q*S(2)(e + fx)(-eh + fg)*S(3)log(c(d(e + fx)p)q)/fS(4) + S(3)b*S(2)h*S(3)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))(e + fx)*S(4)/(S(32)f*S(4)) + S(2)b*S(2)h*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))(e + fx)*S(3)(-eh + fg)/(S(3)fS(4)) + S(9)b*S(2)hpS(2)q*S(2)(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)(-eh + fg)*S(2)/(S(4)f*S(4)) - S(3)bhS(3)pq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)S(4)/(S(16)f*S(4)) - bh*S(2)pq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)S(3)(-eh + fg)/f*S(4) - S(9)bhpq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)S(2)(-eh + fg)*S(2)/(S(4)f*S(4)) - S(3)bpq(a + blog(c(d(e + fx)p)q))S(2)(e + fx)(-eh + fg)S(3)/fS(4) + h*S(3)(a + blog(c(d(e + fx)p)q))*S(3)(e + fx)S(4)/(S(4)fS(4)) + hS(2)(a + blog(c(d(e + fx)p)q))S(3)(e + fx)S(3)(-eh + fg)/f*S(4) + S(3)h(a + blog(c(d(e + fx)p)q))S(3)(e + fx)S(2)(-eh + fg)*S(2)/(S(2)f*S(4)) + (a + blog(c(d(e + fx)p)q))S(3)(e + fx)(-eh + fg)S(3)/fS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(3)(g + hx)S(2), x), x, S(6)abS(2)p*S(2)q*S(2)x(-eh + fg)S(2)/fS(2) - S(3)b*S(3)ehp*S(3)q*S(3)x(-eh + fg)/(S(2)f*S(2)) - S(3)b*S(3)hpS(3)q*S(3)x*S(2)(-eh + fg)/(S(4)f) - S(6)b*S(3)p*S(3)q*S(3)x(-eh + fg)S(2)/fS(2) - S(2)b*S(3)h*S(2)p*S(3)q*S(3)(e + fx)S(3)/(S(27)f*S(3)) + S(6)b*S(3)p*S(2)q*S(2)(e + fx)(-eh + fg)*S(2)log(c(d(e + fx)p)q)/fS(3) + S(2)b*S(2)h*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))(e + fx)*S(3)/(S(9)f*S(3)) + S(3)b*S(2)hpS(2)q*S(2)(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)(-eh + fg)/(S(2)fS(3)) - bh*S(2)pq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)S(3)/(S(3)f*S(3)) - S(3)bhpq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)S(2)(-eh + fg)/(S(2)fS(3)) - S(3)bpq(a + blog(c(d(e + fx)p)q))S(2)(e + fx)(-eh + fg)S(2)/fS(3) + h*S(2)(a + blog(c(d(e + fx)p)q))*S(3)(e + fx)S(3)/(S(3)f*S(3)) + h(a + blog(c(d(e + fx)p)q))*S(3)(e + fx)S(2)(-eh + fg)/f*S(3) + (a + blog(c(d(e + fx)p)q))S(3)(e + fx)(-eh + fg)S(2)/fS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(3)(g + hx), x), x, S(6)abS(2)p*S(2)q*S(2)x(-eh + fg)/f - S(3)b*S(3)ehp*S(3)q*S(3)x/(S(4)f) - S(3)b*S(3)hpS(3)q*S(3)x*S(2)/S(8) - S(6)b*S(3)p*S(3)q*S(3)x(-eh + fg)/f + S(6)b*S(3)p*S(2)q*S(2)(e + fx)(-eh + fg)log(c(d(e + fx)p)q)/f*S(2) + S(3)b*S(2)hpS(2)q*S(2)(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)/(S(4)f*S(2)) - S(3)bhpq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)S(2)/(S(4)f*S(2)) - S(3)bpq(a + blog(c(d(e + fx)p)q))S(2)(e + fx)(-eh + fg)/f*S(2) + h(a + blog(c(d(e + fx)p)q))*S(3)(e + fx)S(2)/(S(2)f*S(2)) + (a + blog(c(d(e + fx)p)q))S(3)(e + fx)(-eh + fg)/f*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(3), x), x, S(6)abS(2)p*S(2)q*S(2)x - S(6)bS(3)p*S(3)q*S(3)x + S(6)bS(3)p*S(2)q*S(2)(e + fx)log(c(d(e + fx)p)q)/f - S(3)bpq(a + blog(c(d(e + fx)p)q))S(2)(e + fx)/f + (a + blog(c(d(e + fx)p)q))S(3)(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(3)/(g + hx), x), x, S(6)bS(3)p*S(3)q*S(3)polylog(S(4), -h(e + fx)/(-eh + fg))/h - S(6)bS(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(3), -h(e + fx)/(-eh + fg))/h + S(3)bpq(a + blog(c(d(e + fx)p)q))S(2)polylog(S(2), -h(e + fx)/(-eh + fg))/h + (a + blog(c(d(e + fx)p)q))*S(3)log(f(g + hx)/(-eh + fg))/h, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(3)/(g + hx)*S(2), x), x, S(6)b*S(3)fpS(3)q*S(3)polylog(S(3), -h(e + fx)/(-eh + fg))/(h(-eh + fg)) - S(6)b*S(2)fpS(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(2), -h(e + fx)/(-eh + fg))/(h(-eh + fg)) - S(3)bfpq(a + blog(c(d(e + fx)p)q))S(2)log(f(g + hx)/(-eh + fg))/(h(-eh + fg)) + (a + blog(c(d(e + fx)p)q))S(3)(e + fx)/((g + hx)(-eh + fg)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(3)/(g + hx)*S(3), x), x, S(3)b*S(3)f*S(2)p*S(3)q*S(3)polylog(S(2), -h(e + fx)/(-eh + fg))/(h(-eh + fg)S(2)) + S(3)b*S(3)f*S(2)p*S(3)q*S(3)polylog(S(3), -h(e + fx)/(-eh + fg))/(h(-eh + fg)S(2)) + S(3)b*S(2)f*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))log(f(g + hx)/(-eh + fg))/(h(-eh + fg)*S(2)) - S(3)b*S(2)f*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(2), -h(e + fx)/(-eh + fg))/(h(-eh + fg)*S(2)) - S(3)bfS(2)pq(a + blog(c(d(e + fx)p)q))*S(2)log(f(g + hx)/(-eh + fg))/(S(2)h(-eh + fg)*S(2)) - S(3)bfpq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)/(S(2)(g + hx)(-eh + fg)S(2)) + fS(2)(a + blog(c(d(e + fx)p)q))S(3)/(S(2)h(-eh + fg)S(2)) - (a + blog(c(d(e + fx)p)q))S(3)/(S(2)h(g + hx)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(3)/(g + hx)S(4), x), x, bS(3)fS(3)p*S(3)q*S(3)log(e + fx)/(h(-eh + fg)S(3)) - bS(3)fS(3)p*S(3)q*S(3)log(g + hx)/(h(-eh + fg)*S(3)) + S(3)b*S(3)f*S(3)p*S(3)q*S(3)polylog(S(2), -h(e + fx)/(-eh + fg))/(h(-eh + fg)S(3)) + S(2)b*S(3)f*S(3)p*S(3)q*S(3)polylog(S(3), -h(e + fx)/(-eh + fg))/(h(-eh + fg)S(3)) + S(3)b*S(2)f*S(3)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))log(f(g + hx)/(-eh + fg))/(h(-eh + fg)*S(3)) - S(2)b*S(2)f*S(3)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(2), -h(e + fx)/(-eh + fg))/(h(-eh + fg)S(3)) - bS(2)fS(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))/(h(g + hx)(-eh + fg)S(2)) - bf*S(3)pq(a + blog(c(d(e + fx)p)q))*S(2)log(f(g + hx)/(-eh + fg))/(h(-eh + fg)S(3)) - bf*S(3)pq(a + blog(c(d(e + fx)p)q))*S(2)/(S(2)h(-eh + fg)S(3)) - bf*S(2)pq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)/((g + hx)(-eh + fg)S(3)) + bfpq(a + blog(c(d(e + fx)p)q))S(2)/(S(2)h(g + hx)*S(2)(-eh + fg)) + f*S(3)(a + blog(c(d(e + fx)p)q))*S(3)/(S(3)h(-eh + fg)S(3)) - (a + blog(c(d(e + fx)p)q))S(3)/(S(3)h(g + hx)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(3)/(g + hx)*S(5), x), x, S(3)b*S(3)f*S(4)p*S(3)q*S(3)log(e + fx)/(S(2)h(-eh + fg)S(4)) - S(3)b*S(3)f*S(4)p*S(3)q*S(3)log(g + hx)/(S(2)h(-eh + fg)S(4)) + S(11)b*S(3)f*S(4)p*S(3)q*S(3)polylog(S(2), -h(e + fx)/(-eh + fg))/(S(4)h(-eh + fg)*S(4)) + S(3)b*S(3)f*S(4)p*S(3)q*S(3)polylog(S(3), -h(e + fx)/(-eh + fg))/(S(2)h(-eh + fg)S(4)) + bS(3)fS(3)p*S(3)q*S(3)/(S(4)h(g + hx)(-eh + fg)S(3)) + S(11)b*S(2)f*S(4)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))log(f(g + hx)/(-eh + fg))/(S(4)h(-eh + fg)S(4)) - S(3)b*S(2)f*S(4)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(2), -h(e + fx)/(-eh + fg))/(S(2)h(-eh + fg)S(4)) - S(5)b*S(2)f*S(3)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))/(S(4)h(g + hx)(-eh + fg)S(3)) - bS(2)fS(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))/(S(4)h(g + hx)S(2)(-eh + fg)*S(2)) - S(3)bfS(4)pq(a + blog(c(d(e + fx)p)q))*S(2)log(f(g + hx)/(-eh + fg))/(S(4)h(-eh + fg)*S(4)) - S(5)bfS(4)pq(a + blog(c(d(e + fx)p)q))*S(2)/(S(8)h(-eh + fg)S(4)) - S(3)bfS(3)pq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)/(S(4)(g + hx)(-eh + fg)*S(4)) + S(3)bfS(2)pq(a + blog(c(d(e + fx)p)q))*S(2)/(S(8)h(g + hx)*S(2)(-eh + fg)*S(2)) + bfpq(a + blog(c(d(e + fx)p)q))S(2)/(S(4)h(g + hx)*S(3)(-eh + fg)) + f*S(4)(a + blog(c(d(e + fx)p)q))*S(3)/(S(4)h(-eh + fg)S(4)) - (a + blog(c(d(e + fx)p)q))S(3)/(S(4)h(g + hx)*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(4), x), x, -S(24)abS(3)p*S(3)q*S(3)x + S(24)bS(4)p*S(4)q*S(4)x - S(24)bS(4)p*S(3)q*S(3)(e + fx)log(c(d(e + fx)p)q)/f + S(12)b*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)/f - S(4)bpq(a + blog(c(d(e + fx)p)q))S(3)(e + fx)/f + (a + blog(c(d(e + fx)p)q))S(4)(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(4)/(g + hx), x), x, -S(24)bS(4)p*S(4)q*S(4)polylog(S(5), -h(e + fx)/(-eh + fg))/h + S(24)bS(3)p*S(3)q*S(3)(a + blog(c(d(e + fx)p)q))polylog(S(4), -h(e + fx)/(-eh + fg))/h - S(12)b*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))*S(2)polylog(S(3), -h(e + fx)/(-eh + fg))/h + S(4)bpq(a + blog(c(d(e + fx)p)q))*S(3)polylog(S(2), -h(e + fx)/(-eh + fg))/h + (a + blog(c(d(e + fx)p)q))*S(4)log(f(g + hx)/(-eh + fg))/h, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(4)/(g + hx)*S(2), x), x, -S(24)b*S(4)fpS(4)q*S(4)polylog(S(4), -h(e + fx)/(-eh + fg))/(h(-eh + fg)) + S(24)b*S(3)fpS(3)q*S(3)(a + blog(c(d(e + fx)p)q))polylog(S(3), -h(e + fx)/(-eh + fg))/(h(-eh + fg)) - S(12)bS(2)fpS(2)q*S(2)(a + blog(c(d(e + fx)p)q))*S(2)polylog(S(2), -h(e + fx)/(-eh + fg))/(h(-eh + fg)) - S(4)bfpq(a + blog(c(d(e + fx)p)q))*S(3)log(f(g + hx)/(-eh + fg))/(h(-eh + fg)) + (a + blog(c(d(e + fx)p)q))S(4)(e + fx)/((g + hx)(-eh + fg)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bx), x), x, -x + (a + bx)log(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bx)*S(2), x), x, S(2)x + (a + bx)log(a + bx)S(2)/b - (S(2)a + S(2)bx)log(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bx)S(3), x), x, -S(6)x + (a + bx)log(a + bx)S(3)/b - (S(3)a + S(3)bx)log(a + bx)*S(2)/b + (S(6)a + S(6)bx)log(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bx + cx), x), x, -x + (a + x(b + c))log(a + x(b + c))/(b + c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bx + cx)S(2), x), x, S(2)x + (a + x(b + c))log(a + x(b + c))S(2)/(b + c) - (S(2)a + S(2)x(b + c))log(a + x(b + c))/(b + c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bx + cx)*S(3), x), x, -S(6)x + (a + x(b + c))log(a + x(b + c))S(3)/(b + c) - (S(3)a + S(3)x(b + c))log(a + x(b + c))*S(2)/(b + c) + (S(6)a + S(6)x(b + c))log(a + x(b + c))/(b + c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(-g(d + ex)/(-dg + ef))/(f + gx), x), x, -polylog(S(2), e(f + gx)/(-dg + ef))/g, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(bx + S(1))/x, x), x, -polylog(S(2), -bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c(a + bx)n)S(2), x), x, -S(5)aS(3)n*S(2)log(a + bx)/(S(9)bS(3)) + aS(3)log(c(a + bx)n)S(2)/(S(3)b*S(3)) + S(11)a*S(2)n*S(2)x/(S(9)bS(2)) - S(2)a*S(2)n(a + bx)log(c(a + bx)n)/(S(3)b*S(3)) - S(5)anS(2)x*S(2)/(S(18)b) + anx*S(2)log(c(a + bx)*n)/(S(3)b) + S(2)nS(2)x*S(3)/S(27) - S(2)nxS(3)log(c(a + bx)n)/S(9) + xS(3)log(c(a + bx)n)S(2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)n)S(2)/xS(4), x), x, -log(c(a + bx)n)S(2)/(S(3)x*S(3)) - bnlog(c(a + bx)n)/(S(3)axS(2)) - bS(2)n*S(2)/(S(3)a*S(2)x) + S(2)bS(2)nlog(c(a + bx)n)/(S(3)a*S(2)x) - b*S(3)n*S(2)log(x)/aS(3) + bS(3)nS(2)log(a + bx)/aS(3) + S(2)b*S(3)n*S(2)polylog(S(2), (a + bx)/a)/(S(3)a*S(3)) + S(2)b*S(3)nlog(c(a + bx)n)log(-bx/a)/(S(3)aS(3)) - bS(3)log(c(a + bx)n)S(2)/(S(3)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c(a + bx)n)S(3), x), x, S(19)a*S(3)n*S(3)log(a + bx)/(S(18)b*S(3)) - S(5)a*S(3)nlog(c(a + bx)n)S(2)/(S(6)bS(3)) + aS(3)log(c(a + bx)n)S(3)/(S(3)b*S(3)) - S(85)a*S(2)n*S(3)x/(S(18)bS(2)) + S(11)a*S(2)n*S(2)(a + bx)log(c(a + bx)*n)/(S(3)bS(3)) - aS(2)n(a + bx)log(c(a + bx)n)S(2)/b*S(3) + S(19)anS(3)x*S(2)/(S(36)b) - S(5)an*S(2)x*S(2)log(c(a + bx)*n)/(S(6)b) + anx*S(2)log(c(a + bx)n)S(2)/(S(2)b) - S(2)n*S(3)x*S(3)/S(27) + S(2)n*S(2)x*S(3)log(c(a + bx)*n)/S(9) - nx*S(3)log(c(a + bx)n)S(2)/S(3) + x*S(3)log(c(a + bx)n)S(3)/S(3), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(x*S(2)log(c(a + bx)n)S(3), x), x, -S(9)aS(2)n*S(3)x/(S(2)bS(2)) + S(6)a*S(2)n*S(2)(a + bx)log(c(a + bx)n)/bS(3) - S(3)aS(2)n(a + bx)log(c(a + bx)n)S(2)/bS(3) + aS(2)(a + bx)log(c(a + bx)n)S(3)/b*S(3) + S(3)anS(3)x*S(2)/(S(4)b) - S(3)an*S(2)(a + bx)S(2)log(c(a + bx)*n)/(S(2)b*S(3)) + S(3)an(a + bx)S(2)log(c(a + bx)n)S(2)/(S(2)bS(3)) - a(a + bx)S(2)log(c(a + bx)n)S(3)/b*S(3) - S(2)n*S(3)(a + bx)S(3)/(S(27)b*S(3)) + S(2)n*S(2)(a + bx)S(3)log(c(a + bx)*n)/(S(9)b*S(3)) - n(a + bx)S(3)log(c(a + bx)n)S(2)/(S(3)bS(3)) + (a + bx)*S(3)log(c(a + bx)n)S(3)/(S(3)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*m/(a + blog(c(d(e + fx)p)q)), x), x, Integral((g + hx)*m/(a + blog(c(d(e + fx)p)q)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*S(3)/(a + blog(c(d(e + fx)p)q)), x), x, hS(3)(c(d(e + fx)p)q)(-S(4)/(pq))(e + fx)*S(4)exp(-S(4)a/(bpq))Ei((S(4)a + S(4)blog(c(d(e + fx)p)q))/(bpq))/(bfS(4)pq) + S(3)h*S(2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)(-eh + fg)exp(-S(3)a/(bpq))Ei((S(3)a + S(3)blog(c(d(e + fx)p)q))/(bpq))/(bf*S(4)pq) + S(3)h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)*S(2)exp(-S(2)a/(bpq))Ei((S(2)a + S(2)blog(c(d(e + fx)p)q))/(bpq))/(bfS(4)pq) + (c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(3)exp(-a/(bpq))Ei((a + blog(c(d(e + fx)p)q))/(bpq))/(bf*S(4)pq), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*S(2)/(a + blog(c(d(e + fx)p)q)), x), x, hS(2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)exp(-S(3)a/(bpq))Ei((S(3)a + S(3)blog(c(d(e + fx)p)q))/(bpq))/(bfS(3)pq) + S(2)h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)exp(-S(2)a/(bpq))Ei((S(2)a + S(2)blog(c(d(e + fx)p)q))/(bpq))/(bf*S(3)pq) + (c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(2)exp(-a/(bpq))Ei((a + blog(c(d(e + fx)p)q))/(bpq))/(bf*S(3)pq), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)/(a + blog(c(d(e + fx)p)q)), x), x, h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)exp(-S(2)a/(bpq))Ei((S(2)a + S(2)blog(c(d(e + fx)p)q))/(bpq))/(bfS(2)pq) + (c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)exp(-a/(bpq))Ei((a + blog(c(d(e + fx)p)q))/(bpq))/(bf*S(2)pq), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + blog(c(d(e + fx)p)q)), x), x, (c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)exp(-a/(bpq))Ei((a + blog(c(d(e + fx)p)q))/(bpq))/(bfpq), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))(g + hx)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))(g + hx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))(g + hx)S(2)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))(g + hx)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*m/(a + blog(c(d(e + fx)p)q))S(2), x), x, Integral((g + hx)*m/(a + blog(c(d(e + fx)p)q))S(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*S(3)/(a + blog(c(d(e + fx)p)q))S(2), x), x, -(e + fx)(g + hx)*S(3)/(bfpq(a + blog(c(d(e + fx)p)q))) + S(4)h*S(3)(c(d(e + fx)p)q)(-S(4)/(pq))(e + fx)*S(4)exp(-S(4)a/(bpq))Ei((S(4)a + S(4)blog(c(d(e + fx)p)q))/(bpq))/(b*S(2)f*S(4)p*S(2)q*S(2)) + S(9)h*S(2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)(-eh + fg)exp(-S(3)a/(bpq))Ei((S(3)a + S(3)blog(c(d(e + fx)p)q))/(bpq))/(bS(2)f*S(4)p*S(2)q*S(2)) + S(6)h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)*S(2)exp(-S(2)a/(bpq))Ei((S(2)a + S(2)blog(c(d(e + fx)p)q))/(bpq))/(b*S(2)f*S(4)p*S(2)q*S(2)) + (c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(3)exp(-a/(bpq))Ei((a + blog(c(d(e + fx)p)q))/(bpq))/(bS(2)f*S(4)p*S(2)q*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*S(2)/(a + blog(c(d(e + fx)p)q))S(2), x), x, -(e + fx)(g + hx)*S(2)/(bfpq(a + blog(c(d(e + fx)p)q))) + S(3)h*S(2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)exp(-S(3)a/(bpq))Ei((S(3)a + S(3)blog(c(d(e + fx)p)q))/(bpq))/(b*S(2)f*S(3)p*S(2)q*S(2)) + S(4)h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)exp(-S(2)a/(bpq))Ei((S(2)a + S(2)blog(c(d(e + fx)p)q))/(bpq))/(bS(2)f*S(3)p*S(2)q*S(2)) + (c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(2)exp(-a/(bpq))Ei((a + blog(c(d(e + fx)p)q))/(bpq))/(bS(2)f*S(3)p*S(2)q*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)/(a + blog(c(d(e + fx)p)q))*S(2), x), x, -(e + fx)(g + hx)/(bfpq(a + blog(c(d(e + fx)p)q))) + S(2)h(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)exp(-S(2)a/(bpq))Ei((S(2)a + S(2)blog(c(d(e + fx)p)q))/(bpq))/(b*S(2)f*S(2)p*S(2)q*S(2)) + (c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)exp(-a/(bpq))Ei((a + blog(c(d(e + fx)p)q))/(bpq))/(bS(2)f*S(2)p*S(2)q*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(-2)), x), x, (-e - fx)/(bfpq(a + blog(c(d(e + fx)p)q))) + (c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)exp(-a/(bpq))Ei((a + blog(c(d(e + fx)p)q))/(bpq))/(b*S(2)fpS(2)q*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))S(2)(g + hx)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))S(2)(g + hx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))S(2)(g + hx)S(2)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))S(2)(g + hx)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*m/(a + blog(c(d(e + fx)p)q))S(3), x), x, Integral((g + hx)*m/(a + blog(c(d(e + fx)p)q))S(3), x), expand=True, _diff=True, _numerical=True) # long time in rubi_test assert rubi_test(rubi_integrate((g + hx)*S(3)/(a + blog(c(d(e + fx)p)q))S(3), x), x, -(e/S(2) + fx/S(2))(g + hx)*S(3)/(bfpq(a + blog(c(d(e + fx)p)q))S(2)) - (S(2)e + S(2)fx)(g + hx)S(3)/(bS(2)fp*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))) + (e + fx)(g + hx)S(2)(-S(3)eh/S(2) + S(3)fg/S(2))/(b*S(2)f*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))) + S(8)hS(3)(c(d(e + fx)p)q)(-S(4)/(pq))(e + fx)*S(4)exp(-S(4)a/(bpq))Ei((S(4)a + S(4)blog(c(d(e + fx)p)q))/(bpq))/(b*S(3)f*S(4)p*S(3)q*S(3)) + S(27)h*S(2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)(-eh + fg)exp(-S(3)a/(bpq))Ei((S(3)a + S(3)blog(c(d(e + fx)p)q))/(bpq))/(S(2)b*S(3)f*S(4)p*S(3)q*S(3)) + S(6)h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)*S(2)exp(-S(2)a/(bpq))Ei((S(2)a + S(2)blog(c(d(e + fx)p)q))/(bpq))/(b*S(3)f*S(4)p*S(3)q*S(3)) + (c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(3)exp(-a/(bpq))Ei((a + blog(c(d(e + fx)p)q))/(bpq))/(S(2)b*S(3)f*S(4)p*S(3)q*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*S(2)/(a + blog(c(d(e + fx)p)q))S(3), x), x, -(e/S(2) + fx/S(2))(g + hx)*S(2)/(bfpq(a + blog(c(d(e + fx)p)q))S(2)) - (S(3)e/S(2) + S(3)fx/S(2))(g + hx)S(2)/(bS(2)fp*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))) + (e + fx)(g + hx)(-eh + fg)/(b*S(2)f*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))) + S(9)hS(2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)exp(-S(3)a/(bpq))Ei((S(3)a + S(3)blog(c(d(e + fx)p)q))/(bpq))/(S(2)bS(3)f*S(3)p*S(3)q*S(3)) + S(4)h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)exp(-S(2)a/(bpq))Ei((S(2)a + S(2)blog(c(d(e + fx)p)q))/(bpq))/(bS(3)f*S(3)p*S(3)q*S(3)) + (c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(2)exp(-a/(bpq))Ei((a + blog(c(d(e + fx)p)q))/(bpq))/(S(2)b*S(3)f*S(3)p*S(3)q*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)/(a + blog(c(d(e + fx)p)q))*S(3), x), x, -(e/S(2) + fx/S(2))(g + hx)/(bfpq(a + blog(c(d(e + fx)p)q))*S(2)) - (e + fx)(g + hx)/(b*S(2)fpS(2)q*S(2)(a + blog(c(d(e + fx)p)q))) + (e + fx)(-eh/S(2) + fg/S(2))/(b*S(2)f*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))) + S(2)h(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)exp(-S(2)a/(bpq))Ei((S(2)a + S(2)blog(c(d(e + fx)p)q))/(bpq))/(b*S(3)f*S(2)p*S(3)q*S(3)) + (c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh/S(2) + fg/S(2))exp(-a/(bpq))Ei((a + blog(c(d(e + fx)p)q))/(bpq))/(bS(3)f*S(2)p*S(3)q*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(-3)), x), x, (-e/S(2) - fx/S(2))/(bfpq(a + blog(c(d(e + fx)p)q))*S(2)) + (-e/S(2) - fx/S(2))/(b*S(2)fpS(2)q*S(2)(a + blog(c(d(e + fx)p)q))) + (c(d(e + fx)p)q)(-S(1)/(pq))(e/S(2) + fx/S(2))exp(-a/(bpq))Ei((a + blog(c(d(e + fx)p)q))/(bpq))/(b*S(3)fpS(3)q*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))S(3)(g + hx)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))S(3)(g + hx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))S(3)(g + hx)S(2)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))S(3)(g + hx)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))(g + hx)m, x), x, Integral(sqrt(a + blog(c(d(e + fx)p)q))(g + hx)m, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))(g + hx)S(4), x), x, -sqrt(S(5))sqrt(pi)sqrt(b)h*S(4)sqrt(p)sqrt(q)(c(d(e + fx)p)q)(-S(5)/(pq))(e + fx)*S(5)exp(-S(5)a/(bpq))erfi(sqrt(S(5))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(50)f*S(5)) - sqrt(pi)sqrt(b)hS(3)sqrt(p)sqrt(q)(c(d(e + fx)p)q)(-S(4)/(pq))(e + fx)*S(4)(-eh + fg)exp(-S(4)a/(bpq))erfi(S(2)sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(4)fS(5)) - sqrt(S(3))sqrt(pi)sqrt(b)h*S(2)sqrt(p)sqrt(q)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)(-eh + fg)*S(2)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(3)f*S(5)) - sqrt(S(2))sqrt(pi)sqrt(b)hsqrt(p)sqrt(q)(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)*S(3)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(2)f*S(5)) - sqrt(pi)sqrt(b)sqrt(p)sqrt(q)(c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(4)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(2)fS(5)) + hS(4)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)S(5)/(S(5)fS(5)) + hS(3)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)S(4)(-eh + fg)/f*S(5) + S(2)h*S(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(3)(-eh + fg)S(2)/fS(5) + S(2)hsqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)(-eh + fg)S(3)/fS(5) + sqrt(a + blog(c(d(e + fx)p)q))(e + fx)(-eh + fg)S(4)/fS(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))(g + hx)S(3), x), x, -sqrt(pi)sqrt(b)hS(3)sqrt(p)sqrt(q)(c(d(e + fx)p)q)(-S(4)/(pq))(e + fx)*S(4)exp(-S(4)a/(bpq))erfi(S(2)sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(16)f*S(4)) - sqrt(S(3))sqrt(pi)sqrt(b)h*S(2)sqrt(p)sqrt(q)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)(-eh + fg)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(6)fS(4)) - S(3)sqrt(S(2))sqrt(pi)sqrt(b)hsqrt(p)sqrt(q)(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)*S(2)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(8)f*S(4)) - sqrt(pi)sqrt(b)sqrt(p)sqrt(q)(c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(3)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(2)fS(4)) + hS(3)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)S(4)/(S(4)fS(4)) + hS(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)S(3)(-eh + fg)/f*S(4) + S(3)hsqrt(a + blog(c(d(e + fx)p)q))(e + fx)S(2)(-eh + fg)*S(2)/(S(2)f*S(4)) + sqrt(a + blog(c(d(e + fx)p)q))(e + fx)(-eh + fg)S(3)/fS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))(g + hx)*S(2), x), x, -sqrt(S(3))sqrt(pi)sqrt(b)h*S(2)sqrt(p)sqrt(q)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(18)f*S(3)) - sqrt(S(2))sqrt(pi)sqrt(b)hsqrt(p)sqrt(q)(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(4)fS(3)) - sqrt(pi)sqrt(b)sqrt(p)sqrt(q)(c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(2)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(2)fS(3)) + hS(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)S(3)/(S(3)f*S(3)) + hsqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)(-eh + fg)/f*S(3) + sqrt(a + blog(c(d(e + fx)p)q))(e + fx)(-eh + fg)S(2)/fS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))(g + hx), x), x, -sqrt(S(2))sqrt(pi)sqrt(b)hsqrt(p)sqrt(q)(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(8)f*S(2)) - sqrt(pi)sqrt(b)sqrt(p)sqrt(q)(c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh/S(2) + fg/S(2))exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/fS(2) + hsqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)/(S(2)f*S(2)) + sqrt(a + blog(c(d(e + fx)p)q))(e + fx)(-eh + fg)/f*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q)), x), x, sqrt(pi)sqrt(b)sqrt(p)sqrt(q)(c(d(e + fx)p)q)*(-S(1)/(pq))(-e/S(2) - fx/S(2))exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/f + sqrt(a + blog(c(d(e + fx)p)q))(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))/(g + hx), x), x, Integral(sqrt(a + blog(c(d(e + fx)p)q))/(g + hx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))/(g + hx)S(2), x), x, -bfpqIntegral(S(1)/(sqrt(a + blog(c(d(e + fx)p)q))(g + hx)), x)/(S(2)(-eh + fg)) + sqrt(a + blog(c(d(e + fx)p)q))(e + fx)/((g + hx)(-eh + fg)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))/(g + hx)S(3), x), x, bfpqIntegral(S(1)/(sqrt(a + blog(c(d(e + fx)p)q))(e + fx)(g + hx)S(2)), x)/(S(4)h) - sqrt(a + blog(c(d(e + fx)p)q))/(S(2)h(g + hx)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))/(g + hx)*S(4), x), x, bfpqIntegral(S(1)/(sqrt(a + blog(c(d(e + fx)p)q))(e + fx)(g + hx)S(3)), x)/(S(6)h) - sqrt(a + blog(c(d(e + fx)p)q))/(S(3)h(g + hx)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))/(g + hx)*S(5), x), x, bfpqIntegral(S(1)/(sqrt(a + blog(c(d(e + fx)p)q))(e + fx)(g + hx)S(4)), x)/(S(8)h) - sqrt(a + blog(c(d(e + fx)p)q))/(S(4)h(g + hx)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(3)/2)(g + hx)m, x), x, Integral((a + blog(c(d(e + fx)p)q))(S(3)/2)(g + hx)m, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(3)/2)(g + hx)S(3), x), x, S(3)sqrt(pi)b(S(3)/2)h*S(3)p*(S(3)/2)q*(S(3)/2)(c(d(e + fx)p)q)(-S(4)/(pq))(e + fx)*S(4)exp(-S(4)a/(bpq))erfi(S(2)sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(128)f*S(4)) + sqrt(S(3))sqrt(pi)b(S(3)/2)h*S(2)p*(S(3)/2)q*(S(3)/2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)(-eh + fg)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(12)fS(4)) + S(9)sqrt(S(2))sqrt(pi)b*(S(3)/2)hp(S(3)/2)q*(S(3)/2)(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)*S(2)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(32)f*S(4)) + S(3)sqrt(pi)b(S(3)/2)p*(S(3)/2)q*(S(3)/2)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)(-eh + fg)S(3)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(4)f*S(4)) - S(3)bhS(3)pqsqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(4)/(S(32)f*S(4)) - bh*S(2)pqsqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(3)(-eh + fg)/(S(2)fS(4)) - S(9)bhpqsqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)(-eh + fg)*S(2)/(S(8)f*S(4)) - S(3)bpqsqrt(a + blog(c(d(e + fx)p)q))(e + fx)(-eh + fg)*S(3)/(S(2)fS(4)) + hS(3)(a + blog(c(d(e + fx)p)q))(S(3)/2)(e + fx)S(4)/(S(4)fS(4)) + hS(2)(a + blog(c(d(e + fx)p)q))(S(3)/2)(e + fx)S(3)(-eh + fg)/f*S(4) + S(3)h(a + blog(c(d(e + fx)p)q))(S(3)/2)(e + fx)S(2)(-eh + fg)*S(2)/(S(2)f*S(4)) + (a + blog(c(d(e + fx)p)q))(S(3)/2)(e + fx)(-eh + fg)S(3)/fS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*(S(3)/2)(g + hx)S(2), x), x, sqrt(S(3))sqrt(pi)b(S(3)/2)h*S(2)p*(S(3)/2)q*(S(3)/2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(36)f*S(3)) + S(3)sqrt(S(2))sqrt(pi)b*(S(3)/2)hp(S(3)/2)q*(S(3)/2)(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(16)fS(3)) + S(3)sqrt(pi)b(S(3)/2)p*(S(3)/2)q*(S(3)/2)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)(-eh + fg)S(2)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(4)f*S(3)) - bh*S(2)pqsqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(3)/(S(6)f*S(3)) - S(3)bhpqsqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)(-eh + fg)/(S(4)fS(3)) - S(3)bpqsqrt(a + blog(c(d(e + fx)p)q))(e + fx)(-eh + fg)*S(2)/(S(2)fS(3)) + hS(2)(a + blog(c(d(e + fx)p)q))(S(3)/2)(e + fx)S(3)/(S(3)f*S(3)) + h(a + blog(c(d(e + fx)p)q))*(S(3)/2)(e + fx)S(2)(-eh + fg)/f*S(3) + (a + blog(c(d(e + fx)p)q))(S(3)/2)(e + fx)(-eh + fg)S(2)/fS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*(S(3)/2)(g + hx), x), x, S(3)sqrt(S(2))sqrt(pi)b*(S(3)/2)hp(S(3)/2)q*(S(3)/2)(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(32)f*S(2)) + S(3)sqrt(pi)b(S(3)/2)p*(S(3)/2)q*(S(3)/2)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)(-eh + fg)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(4)f*S(2)) - S(3)bhpqsqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)/(S(8)f*S(2)) - S(3)bpqsqrt(a + blog(c(d(e + fx)p)q))(e + fx)(-eh + fg)/(S(2)fS(2)) + h(a + blog(c(d(e + fx)p)q))*(S(3)/2)(e + fx)S(2)/(S(2)f*S(2)) + (a + blog(c(d(e + fx)p)q))(S(3)/2)(e + fx)(-eh + fg)/f*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(3)/2), x), x, S(3)sqrt(pi)b(S(3)/2)p*(S(3)/2)q*(S(3)/2)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(4)f) - S(3)bpqsqrt(a + blog(c(d(e + fx)p)q))(e + fx)/(S(2)f) + (a + blog(c(d(e + fx)p)q))*(S(3)/2)(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(3)/2)/(g + hx), x), x, Integral((a + blog(c(d(e + fx)p)q))*(S(3)/2)/(g + hx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*(S(3)/2)/(g + hx)*S(2), x), x, -S(3)bfpqIntegral(sqrt(a + blog(c(d(e + fx)p)q))/(g + hx), x)/(S(2)(-eh + fg)) + (a + blog(c(d(e + fx)p)q))*(S(3)/2)(e + fx)/((g + hx)(-eh + fg)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(3)/2)/(g + hx)*S(3), x), x, S(3)bfpqIntegral(sqrt(a + blog(c(d(e + fx)p)q))/((e + fx)(g + hx)S(2)), x)/(S(4)h) - (a + blog(c(d(e + fx)p)q))*(S(3)/2)/(S(2)h(g + hx)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(3)/2)/(g + hx)*S(4), x), x, bfpqIntegral(sqrt(a + blog(c(d(e + fx)p)q))/((e + fx)(g + hx)*S(3)), x)/(S(2)h) - (a + blog(c(d(e + fx)p)q))*(S(3)/2)/(S(3)h(g + hx)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(5)/2)(g + hx)m, x), x, Integral((a + blog(c(d(e + fx)p)q))(S(5)/2)(g + hx)m, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(5)/2)(g + hx)S(3), x), x, -S(15)sqrt(pi)b(S(5)/2)h*S(3)p*(S(5)/2)q*(S(5)/2)(c(d(e + fx)p)q)(-S(4)/(pq))(e + fx)*S(4)exp(-S(4)a/(bpq))erfi(S(2)sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(1024)f*S(4)) - S(5)sqrt(S(3))sqrt(pi)b*(S(5)/2)h*S(2)p*(S(5)/2)q*(S(5)/2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)(-eh + fg)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(72)fS(4)) - S(45)sqrt(S(2))sqrt(pi)b*(S(5)/2)hp(S(5)/2)q*(S(5)/2)(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)*S(2)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(128)f*S(4)) - S(15)sqrt(pi)b(S(5)/2)p*(S(5)/2)q*(S(5)/2)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)(-eh + fg)S(3)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(8)f*S(4)) + S(15)b*S(2)h*S(3)p*S(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(4)/(S(256)f*S(4)) + S(5)b*S(2)h*S(2)p*S(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(3)(-eh + fg)/(S(12)fS(4)) + S(45)b*S(2)hpS(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)(-eh + fg)*S(2)/(S(32)f*S(4)) + S(15)b*S(2)p*S(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)(-eh + fg)S(3)/(S(4)f*S(4)) - S(5)bhS(3)pq(a + blog(c(d(e + fx)p)q))*(S(3)/2)(e + fx)S(4)/(S(32)f*S(4)) - S(5)bhS(2)pq(a + blog(c(d(e + fx)p)q))*(S(3)/2)(e + fx)S(3)(-eh + fg)/(S(6)fS(4)) - S(15)bhpq(a + blog(c(d(e + fx)p)q))*(S(3)/2)(e + fx)S(2)(-eh + fg)*S(2)/(S(8)f*S(4)) - S(5)bpq(a + blog(c(d(e + fx)p)q))(S(3)/2)(e + fx)(-eh + fg)*S(3)/(S(2)fS(4)) + hS(3)(a + blog(c(d(e + fx)p)q))(S(5)/2)(e + fx)S(4)/(S(4)fS(4)) + hS(2)(a + blog(c(d(e + fx)p)q))(S(5)/2)(e + fx)S(3)(-eh + fg)/f*S(4) + S(3)h(a + blog(c(d(e + fx)p)q))(S(5)/2)(e + fx)S(2)(-eh + fg)*S(2)/(S(2)f*S(4)) + (a + blog(c(d(e + fx)p)q))(S(5)/2)(e + fx)(-eh + fg)S(3)/fS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*(S(5)/2)(g + hx)S(2), x), x, -S(5)sqrt(S(3))sqrt(pi)b*(S(5)/2)h*S(2)p*(S(5)/2)q*(S(5)/2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(216)f*S(3)) - S(15)sqrt(S(2))sqrt(pi)b*(S(5)/2)hp(S(5)/2)q*(S(5)/2)(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(64)fS(3)) - S(15)sqrt(pi)b(S(5)/2)p*(S(5)/2)q*(S(5)/2)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)(-eh + fg)S(2)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(8)f*S(3)) + S(5)b*S(2)h*S(2)p*S(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(3)/(S(36)f*S(3)) + S(15)b*S(2)hpS(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)(-eh + fg)/(S(16)fS(3)) + S(15)b*S(2)p*S(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)(-eh + fg)S(2)/(S(4)f*S(3)) - S(5)bhS(2)pq(a + blog(c(d(e + fx)p)q))*(S(3)/2)(e + fx)S(3)/(S(18)f*S(3)) - S(5)bhpq(a + blog(c(d(e + fx)p)q))*(S(3)/2)(e + fx)S(2)(-eh + fg)/(S(4)fS(3)) - S(5)bpq(a + blog(c(d(e + fx)p)q))(S(3)/2)(e + fx)(-eh + fg)*S(2)/(S(2)fS(3)) + hS(2)(a + blog(c(d(e + fx)p)q))(S(5)/2)(e + fx)S(3)/(S(3)f*S(3)) + h(a + blog(c(d(e + fx)p)q))*(S(5)/2)(e + fx)S(2)(-eh + fg)/f*S(3) + (a + blog(c(d(e + fx)p)q))(S(5)/2)(e + fx)(-eh + fg)S(2)/fS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*(S(5)/2)(g + hx), x), x, -S(15)sqrt(S(2))sqrt(pi)b*(S(5)/2)hp(S(5)/2)q*(S(5)/2)(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(128)f*S(2)) - S(15)sqrt(pi)b(S(5)/2)p*(S(5)/2)q*(S(5)/2)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)(-eh + fg)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(8)f*S(2)) + S(15)b*S(2)hpS(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)/(S(32)f*S(2)) + S(15)b*S(2)p*S(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)(-eh + fg)/(S(4)f*S(2)) - S(5)bhpq(a + blog(c(d(e + fx)p)q))*(S(3)/2)(e + fx)S(2)/(S(8)f*S(2)) - S(5)bpq(a + blog(c(d(e + fx)p)q))(S(3)/2)(e + fx)(-eh + fg)/(S(2)fS(2)) + h(a + blog(c(d(e + fx)p)q))*(S(5)/2)(e + fx)S(2)/(S(2)f*S(2)) + (a + blog(c(d(e + fx)p)q))(S(5)/2)(e + fx)(-eh + fg)/f*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(5)/2), x), x, -S(15)sqrt(pi)b(S(5)/2)p*(S(5)/2)q*(S(5)/2)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(8)f) + S(15)b*S(2)p*S(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))(e + fx)/(S(4)f) - S(5)bpq(a + blog(c(d(e + fx)p)q))(S(3)/2)(e + fx)/(S(2)f) + (a + blog(c(d(e + fx)p)q))*(S(5)/2)(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(5)/2)/(g + hx), x), x, Integral((a + blog(c(d(e + fx)p)q))*(S(5)/2)/(g + hx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*(S(5)/2)/(g + hx)*S(2), x), x, -S(5)bfpqIntegral((a + blog(c(d(e + fx)p)q))*(S(3)/2)/(g + hx), x)/(S(2)(-eh + fg)) + (a + blog(c(d(e + fx)p)q))(S(5)/2)(e + fx)/((g + hx)(-eh + fg)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(5)/2)/(g + hx)*S(3), x), x, S(5)bfpqIntegral((a + blog(c(d(e + fx)p)q))*(S(3)/2)/((e + fx)(g + hx)*S(2)), x)/(S(4)h) - (a + blog(c(d(e + fx)p)q))*(S(5)/2)/(S(2)h(g + hx)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(5)/2)/(g + hx)*S(4), x), x, S(5)bfpqIntegral((a + blog(c(d(e + fx)p)q))*(S(3)/2)/((e + fx)(g + hx)*S(3)), x)/(S(6)h) - (a + blog(c(d(e + fx)p)q))*(S(5)/2)/(S(3)h(g + hx)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(5)/2)/(g + hx)*S(5), x), x, S(5)bfpqIntegral((a + blog(c(d(e + fx)p)q))*(S(3)/2)/((e + fx)(g + hx)*S(4)), x)/(S(8)h) - (a + blog(c(d(e + fx)p)q))*(S(5)/2)/(S(4)h(g + hx)*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*m/sqrt(a + blog(c(d(e + fx)p)q)), x), x, Integral((g + hx)*m/sqrt(a + blog(c(d(e + fx)p)q)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*S(3)/sqrt(a + blog(c(d(e + fx)p)q)), x), x, sqrt(pi)h*S(3)(c(d(e + fx)p)q)(-S(4)/(pq))(e + fx)*S(4)exp(-S(4)a/(bpq))erfi(S(2)sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(2)sqrt(b)fS(4)sqrt(p)sqrt(q)) + sqrt(S(3))sqrt(pi)hS(2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)(-eh + fg)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(sqrt(b)fS(4)sqrt(p)sqrt(q)) + S(3)sqrt(S(2))sqrt(pi)h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)*S(2)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(2)sqrt(b)fS(4)sqrt(p)sqrt(q)) + sqrt(pi)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)(-eh + fg)S(3)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(sqrt(b)f*S(4)sqrt(p)sqrt(q)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*S(2)/sqrt(a + blog(c(d(e + fx)p)q)), x), x, sqrt(S(3))sqrt(pi)hS(2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(3)sqrt(b)fS(3)sqrt(p)sqrt(q)) + sqrt(S(2))sqrt(pi)h(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(sqrt(b)fS(3)sqrt(p)sqrt(q)) + sqrt(pi)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)(-eh + fg)S(2)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(sqrt(b)f*S(3)sqrt(p)sqrt(q)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)/sqrt(a + blog(c(d(e + fx)p)q)), x), x, sqrt(S(2))sqrt(pi)h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(2)sqrt(b)fS(2)sqrt(p)sqrt(q)) + sqrt(pi)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)(-eh + fg)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(sqrt(b)f*S(2)sqrt(p)sqrt(q)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + blog(c(d(e + fx)p)q)), x), x, sqrt(pi)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(sqrt(b)fsqrt(p)sqrt(q)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a + blog(c(d(e + fx)p)q))(g + hx)), x), x, Integral(S(1)/(sqrt(a + blog(c(d(e + fx)p)q))(g + hx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*m/(a + blog(c(d(e + fx)p)q))(S(3)/2), x), x, Integral((g + hx)*m/(a + blog(c(d(e + fx)p)q))(S(3)/2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*S(3)/(a + blog(c(d(e + fx)p)q))(S(3)/2), x), x, -(S(2)e + S(2)fx)(g + hx)*S(3)/(bfpqsqrt(a + blog(c(d(e + fx)p)q))) + S(4)sqrt(pi)hS(3)(c(d(e + fx)p)q)(-S(4)/(pq))(e + fx)*S(4)exp(-S(4)a/(bpq))erfi(S(2)sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b(S(3)/2)f*S(4)p*(S(3)/2)q*(S(3)/2)) + S(6)sqrt(S(3))sqrt(pi)h*S(2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)(-eh + fg)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b*(S(3)/2)f*S(4)p*(S(3)/2)q*(S(3)/2)) + S(6)sqrt(S(2))sqrt(pi)h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)*S(2)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b(S(3)/2)f*S(4)p*(S(3)/2)q*(S(3)/2)) + S(2)sqrt(pi)(c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(3)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b(S(3)/2)f*S(4)p*(S(3)/2)q*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*S(2)/(a + blog(c(d(e + fx)p)q))(S(3)/2), x), x, -(S(2)e + S(2)fx)(g + hx)*S(2)/(bfpqsqrt(a + blog(c(d(e + fx)p)q))) + S(2)sqrt(S(3))sqrt(pi)h*S(2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b(S(3)/2)f*S(3)p*(S(3)/2)q*(S(3)/2)) + S(4)sqrt(S(2))sqrt(pi)h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b*(S(3)/2)f*S(3)p*(S(3)/2)q*(S(3)/2)) + S(2)sqrt(pi)(c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(2)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b(S(3)/2)f*S(3)p*(S(3)/2)q*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)/(a + blog(c(d(e + fx)p)q))*(S(3)/2), x), x, -(S(2)e + S(2)fx)(g + hx)/(bfpqsqrt(a + blog(c(d(e + fx)p)q))) + S(2)sqrt(S(2))sqrt(pi)h(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b(S(3)/2)f*S(2)p*(S(3)/2)q*(S(3)/2)) + sqrt(pi)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)(-S(2)eh + S(2)fg)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b(S(3)/2)f*S(2)p*(S(3)/2)q*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(-3)/2), x), x, -(S(2)e + S(2)fx)/(bfpqsqrt(a + blog(c(d(e + fx)p)q))) + sqrt(pi)(c(d(e + fx)p)q)*(-S(1)/(pq))(S(2)e + S(2)fx)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b*(S(3)/2)fp(S(3)/2)q*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))(S(3)/2)(g + hx)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))(S(3)/2)(g + hx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*m/(a + blog(c(d(e + fx)p)q))(S(5)/2), x), x, Integral((g + hx)*m/(a + blog(c(d(e + fx)p)q))(S(5)/2), x), expand=True, _diff=True, _numerical=True) ''' long time in rubi test assert rubi_test(rubi_integrate((g + hx)*S(3)/(a + blog(c(d(e + fx)p)q))(S(5)/2), x), x, (-S(2)e/S(3) - S(2)fx/S(3))(g + hx)*S(3)/(bfpq(a + blog(c(d(e + fx)p)q))(S(3)/2)) - (S(16)e/S(3) + S(16)fx/S(3))(g + hx)S(3)/(bS(2)fp*S(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))) + (e + fx)(g + hx)S(2)(-S(4)eh + S(4)fg)/(b*S(2)f*S(2)p*S(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))) + S(32)sqrt(pi)h*S(3)(c(d(e + fx)p)q)(-S(4)/(pq))(e + fx)*S(4)exp(-S(4)a/(bpq))erfi(S(2)sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(3)b*(S(5)/2)f*S(4)p*(S(5)/2)q*(S(5)/2)) + S(12)sqrt(S(3))sqrt(pi)h*S(2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)(-eh + fg)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b*(S(5)/2)f*S(4)p*(S(5)/2)q*(S(5)/2)) + S(8)sqrt(S(2))sqrt(pi)h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)*S(2)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b(S(5)/2)f*S(4)p*(S(5)/2)q*(S(5)/2)) + S(4)sqrt(pi)(c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(3)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(3)b*(S(5)/2)f*S(4)p*(S(5)/2)q*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)*S(2)/(a + blog(c(d(e + fx)p)q))(S(5)/2), x), x, (-S(2)e/S(3) - S(2)fx/S(3))(g + hx)*S(2)/(bfpq(a + blog(c(d(e + fx)p)q))(S(3)/2)) - (S(4)e + S(4)fx)(g + hx)S(2)/(bS(2)fp*S(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))) + (e + fx)(g + hx)(-S(8)eh/S(3) + S(8)fg/S(3))/(b*S(2)f*S(2)p*S(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))) + S(4)sqrt(S(3))sqrt(pi)hS(2)(c(d(e + fx)p)q)(-S(3)/(pq))(e + fx)*S(3)exp(-S(3)a/(bpq))erfi(sqrt(S(3))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b(S(5)/2)f*S(3)p*(S(5)/2)q*(S(5)/2)) + S(16)sqrt(S(2))sqrt(pi)h(c(d(e + fx)p)q)*(-S(2)/(pq))(e + fx)*S(2)(-eh + fg)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(3)b(S(5)/2)f*S(3)p*(S(5)/2)q*(S(5)/2)) + S(4)sqrt(pi)(c(d(e + fx)p)q)*(-S(1)/(pq))(e + fx)(-eh + fg)S(2)exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(3)b*(S(5)/2)f*S(3)p*(S(5)/2)q*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((g + hx)/(a + blog(c(d(e + fx)p)q))*(S(5)/2), x), x, (-S(2)e/S(3) - S(2)fx/S(3))(g + hx)/(bfpq(a + blog(c(d(e + fx)p)q))*(S(3)/2)) - (S(8)e/S(3) + S(8)fx/S(3))(g + hx)/(b*S(2)fpS(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))) + (e + fx)(-S(4)eh/S(3) + S(4)fg/S(3))/(b*S(2)f*S(2)p*S(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))) + S(8)sqrt(S(2))sqrt(pi)h(c(d(e + fx)p)q)(-S(2)/(pq))(e + fx)*S(2)exp(-S(2)a/(bpq))erfi(sqrt(S(2))sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(S(3)b*(S(5)/2)f*S(2)p*(S(5)/2)q*(S(5)/2)) + sqrt(pi)(c(d(e + fx)p)q)(-S(1)/(pq))(e + fx)(-S(4)eh/S(3) + S(4)fg/S(3))exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b(S(5)/2)f*S(2)p*(S(5)/2)q*(S(5)/2)), expand=True, _diff=True, _numerical=True) ''' assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(S(-5)/2), x), x, (-S(2)e/S(3) - S(2)fx/S(3))/(bfpq(a + blog(c(d(e + fx)p)q))*(S(3)/2)) - (S(4)e/S(3) + S(4)fx/S(3))/(b*S(2)fpS(2)q*S(2)sqrt(a + blog(c(d(e + fx)p)q))) + sqrt(pi)(c(d(e + fx)p)q)*(-S(1)/(pq))(S(4)e/S(3) + S(4)fx/S(3))exp(-a/(bpq))erfi(sqrt(a + blog(c(d(e + fx)p)q))/(sqrt(b)sqrt(p)sqrt(q)))/(b*(S(5)/2)fp(S(5)/2)q*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))(S(5)/2)(g + hx)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))(S(5)/2)(g + hx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(g + hx)(S(3)/2), x), x, -S(4)bpq(g + hx)*(S(5)/2)/(S(25)h) - S(4)bpq(g + hx)(S(3)/2)(-eh + fg)/(S(15)fh) - S(4)bpqsqrt(g + hx)(-eh + fg)*S(2)/(S(5)f*S(2)h) + S(4)bpq(-eh + fg)*(S(5)/2)atanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(S(5)f(S(5)/2)h) + S(2)(a + blog(c(d(e + fx)p)q))(g + hx)(S(5)/2)/(S(5)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))sqrt(g + hx), x), x, -S(4)bpq(g + hx)(S(3)/2)/(S(9)h) - S(4)bpqsqrt(g + hx)(-eh + fg)/(S(3)fh) + S(4)bpq(-eh + fg)*(S(3)/2)atanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(S(3)f(S(3)/2)h) + S(2)(a + blog(c(d(e + fx)p)q))(g + hx)(S(3)/2)/(S(3)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/sqrt(g + hx), x), x, -S(4)bpqsqrt(g + hx)/h + S(4)bpqsqrt(-eh + fg)atanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(sqrt(f)h) + (S(2)a + S(2)blog(c(d(e + fx)p)q))sqrt(g + hx)/h, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(g + hx)(S(3)/2), x), x, -S(4)bsqrt(f)pqatanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(hsqrt(-eh + fg)) - (S(2)a + S(2)blog(c(d(e + fx)p)q))/(hsqrt(g + hx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(g + hx)*(S(5)/2), x), x, -S(4)bf(S(3)/2)pqatanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(S(3)h(-eh + fg)*(S(3)/2)) + S(4)bfpq/(S(3)hsqrt(g + hx)(-eh + fg)) - (S(2)a/S(3) + S(2)blog(c(d(e + fx)p)q)/S(3))/(h(g + hx)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(g + hx)*(S(7)/2), x), x, -S(4)bf(S(5)/2)pqatanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(S(5)h(-eh + fg)*(S(5)/2)) + S(4)bfS(2)pq/(S(5)hsqrt(g + hx)(-eh + fg)S(2)) + S(4)bfpq/(S(15)h(g + hx)*(S(3)/2)(-eh + fg)) - (S(2)a/S(5) + S(2)blog(c(d(e + fx)p)q)/S(5))/(h(g + hx)*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(g + hx)*(S(9)/2), x), x, -S(4)bf(S(7)/2)pqatanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(S(7)h(-eh + fg)*(S(7)/2)) + S(4)bfS(3)pq/(S(7)hsqrt(g + hx)(-eh + fg)S(3)) + S(4)bfS(2)pq/(S(21)h(g + hx)*(S(3)/2)(-eh + fg)*S(2)) + S(4)bfpq/(S(35)h(g + hx)*(S(5)/2)(-eh + fg)) - (S(2)a/S(7) + S(2)blog(c(d(e + fx)p)q)/S(7))/(h(g + hx)*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(2)(g + hx)(S(3)/2), x), x, S(16)b*S(2)p*S(2)q*S(2)(g + hx)(S(5)/2)/(S(125)h) + S(128)bS(2)p*S(2)q*S(2)(g + hx)(S(3)/2)(-eh + fg)/(S(225)fh) + S(368)bS(2)p*S(2)q*S(2)sqrt(g + hx)(-eh + fg)*S(2)/(S(75)f*S(2)h) + S(16)bS(2)p*S(2)q*S(2)(-eh + fg)*(S(5)/2)log(S(2)/(-sqrt(f)sqrt(g + hx)/sqrt(-eh + fg) + S(1)))atanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(S(5)f*(S(5)/2)h) - S(8)bS(2)p*S(2)q*S(2)(-eh + fg)*(S(5)/2)atanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))*S(2)/(S(5)f*(S(5)/2)h) - S(368)bS(2)p*S(2)q*S(2)(-eh + fg)*(S(5)/2)atanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(S(75)f(S(5)/2)h) + S(8)bS(2)p*S(2)q*S(2)(-eh + fg)*(S(5)/2)polylog(S(2), (-sqrt(f)sqrt(g + hx) - sqrt(-eh + fg))/(-sqrt(f)sqrt(g + hx) + sqrt(-eh + fg)))/(S(5)f(S(5)/2)h) - S(8)bpq(a + blog(c(d(e + fx)p)q))(g + hx)*(S(5)/2)/(S(25)h) - S(8)bpq(a + blog(c(d(e + fx)p)q))(g + hx)*(S(3)/2)(-eh + fg)/(S(15)fh) - S(8)bpq(a + blog(c(d(e + fx)p)q))sqrt(g + hx)(-eh + fg)S(2)/(S(5)f*S(2)h) + S(8)bpq(a + blog(c(d(e + fx)p)q))(-eh + fg)(S(5)/2)atanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(S(5)f(S(5)/2)h) + S(2)(a + blog(c(d(e + fx)p)q))S(2)(g + hx)(S(5)/2)/(S(5)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(2)sqrt(g + hx), x), x, S(16)b*S(2)p*S(2)q*S(2)(g + hx)(S(3)/2)/(S(27)h) + S(64)bS(2)p*S(2)q*S(2)sqrt(g + hx)(-eh + fg)/(S(9)fh) + S(16)bS(2)p*S(2)q*S(2)(-eh + fg)*(S(3)/2)log(S(2)/(-sqrt(f)sqrt(g + hx)/sqrt(-eh + fg) + S(1)))atanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(S(3)f*(S(3)/2)h) - S(8)bS(2)p*S(2)q*S(2)(-eh + fg)*(S(3)/2)atanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))*S(2)/(S(3)f*(S(3)/2)h) - S(64)bS(2)p*S(2)q*S(2)(-eh + fg)*(S(3)/2)atanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(S(9)f(S(3)/2)h) + S(8)bS(2)p*S(2)q*S(2)(-eh + fg)*(S(3)/2)polylog(S(2), (-sqrt(f)sqrt(g + hx) - sqrt(-eh + fg))/(-sqrt(f)sqrt(g + hx) + sqrt(-eh + fg)))/(S(3)f(S(3)/2)h) - S(8)bpq(a + blog(c(d(e + fx)p)q))(g + hx)*(S(3)/2)/(S(9)h) - S(8)bpq(a + blog(c(d(e + fx)p)q))sqrt(g + hx)(-eh + fg)/(S(3)fh) + S(8)bpq(a + blog(c(d(e + fx)p)q))(-eh + fg)*(S(3)/2)atanh(sqrt(f)sqrt(g + hx)/sqrt(-eh + fg))/(S(3)f(S(3)/2)h) + S(2)(a + blog(c(d(e + fx)p)q))S(2)(g + hx)(S(3)/2)/(S(3)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))sqrt(g + hx), x), x, -bfpqIntegral((g + hx)(S(3)/2)/(sqrt(a + blog(c(d(e + fx)p)q))(e + fx)), x)/(S(3)h) + S(2)sqrt(a + blog(c(d(e + fx)p)q))(g + hx)(S(3)/2)/(S(3)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))/sqrt(g + hx), x), x, -bfpqIntegral(sqrt(g + hx)/(sqrt(a + blog(c(d(e + fx)p)q))(e + fx)), x)/h + S(2)sqrt(a + blog(c(d(e + fx)p)q))sqrt(g + hx)/h, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + blog(c(d(e + fx)p)q))/(g + hx)*(S(3)/2), x), x, bfpqIntegral(S(1)/(sqrt(a + blog(c(d(e + fx)p)q))(e + fx)sqrt(g + hx)), x)/h - S(2)sqrt(a + blog(c(d(e + fx)p)q))/(hsqrt(g + hx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(g + hx)/sqrt(a + blog(c(d(e + fx)p)q)), x), x, Integral(sqrt(g + hx)/sqrt(a + blog(c(d(e + fx)p)q)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a + blog(c(d(e + fx)p)q))sqrt(g + hx)), x), x, Integral(S(1)/(sqrt(a + blog(c(d(e + fx)p)q))sqrt(g + hx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a + blog(c(d(e + fx)p)q))(g + hx)*(S(3)/2)), x), x, Integral(S(1)/(sqrt(a + blog(c(d(e + fx)p)q))(g + hx)(S(3)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))n(g + hx)m, x), x, Integral((a + blog(c(d(e + fx)p)q))n(g + hx)m, x), expand=True, _diff=True, _numerical=True) '''long time assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))n(g + hx)S(3), x), x, S(3)S(2)*(-n + S(-1))h(c(d(e + fx)p)q)*(-S(2)/(pq))((-a - blog(c(d(e + fx)p)q))/(bpq))(-n)(a + blog(c(d(e + fx)p)q))*n(e + fx)S(2)(-eh + fg)*S(2)Gamma(n + S(1), (-S(2)a - S(2)blog(c(d(e + fx)p)q))/(bpq))exp(-S(2)a/(bpq))/fS(4) + S(4)(-n + S(-1))hS(3)(c(d(e + fx)p)q)(-S(4)/(pq))((-a - blog(c(d(e + fx)p)q))/(bpq))(-n)(a + blog(c(d(e + fx)p)q))*n(e + fx)S(4)Gamma(n + S(1), (-S(4)a - S(4)blog(c(d(e + fx)p)q))/(bpq))exp(-S(4)a/(bpq))/f*S(4) + (c(d(e + fx)p)q)*(-S(1)/(pq))((-a - blog(c(d(e + fx)p)q))/(bpq))(-n)(a + blog(c(d(e + fx)p)q))*n(e + fx)(-eh + fg)*S(3)Gamma(n + S(1), (-a - blog(c(d(e + fx)p)q))/(bpq))exp(-a/(bpq))/fS(4) + S(3)(-n)h*S(2)(c(d(e + fx)p)q)(-S(3)/(pq))((-a - blog(c(d(e + fx)p)q))/(bpq))(-n)(a + blog(c(d(e + fx)p)q))*n(e + fx)S(3)(-eh + fg)Gamma(n + S(1), (-S(3)a - S(3)blog(c(d(e + fx)p)q))/(bpq))exp(-S(3)a/(bpq))/fS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))n(g + hx)S(2), x), x, S(3)(-n + S(-1))h*S(2)(c(d(e + fx)p)q)(-S(3)/(pq))((-a - blog(c(d(e + fx)p)q))/(bpq))(-n)(a + blog(c(d(e + fx)p)q))*n(e + fx)S(3)Gamma(n + S(1), (-S(3)a - S(3)blog(c(d(e + fx)p)q))/(bpq))exp(-S(3)a/(bpq))/f*S(3) + (c(d(e + fx)p)q)*(-S(1)/(pq))((-a - blog(c(d(e + fx)p)q))/(bpq))(-n)(a + blog(c(d(e + fx)p)q))*n(e + fx)(-eh + fg)*S(2)Gamma(n + S(1), (-a - blog(c(d(e + fx)p)q))/(bpq))exp(-a/(bpq))/fS(3) + S(2)(-n)h(c(d(e + fx)p)q)*(-S(2)/(pq))((-a - blog(c(d(e + fx)p)q))/(bpq))(-n)(a + blog(c(d(e + fx)p)q))*n(e + fx)S(2)(-eh + fg)Gamma(n + S(1), (-S(2)a - S(2)blog(c(d(e + fx)p)q))/(bpq))exp(-S(2)a/(bpq))/fS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))n(g + hx), x), x, S(2)(-n + S(-1))h(c(d(e + fx)p)q)*(-S(2)/(pq))((-a - blog(c(d(e + fx)p)q))/(bpq))(-n)(a + blog(c(d(e + fx)p)q))*n(e + fx)S(2)Gamma(n + S(1), (-S(2)a - S(2)blog(c(d(e + fx)p)q))/(bpq))exp(-S(2)a/(bpq))/f*S(2) + (c(d(e + fx)p)q)*(-S(1)/(pq))((-a - blog(c(d(e + fx)p)q))/(bpq))(-n)(a + blog(c(d(e + fx)p)q))*n(e + fx)(-eh + fg)Gamma(n + S(1), (-a - blog(c(d(e + fx)p)q))/(bpq))exp(-a/(bpq))/f*S(2), expand=True, _diff=True, _numerical=True) ''' assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))n, x), x, (c(d(e + fx)p)q)*(-S(1)/(pq))((-a - blog(c(d(e + fx)p)q))/(bpq))(-n)(a + blog(c(d(e + fx)p)q))*n(e + fx)Gamma(n + S(1), (-a - blog(c(d(e + fx)p)q))/(bpq))exp(-a/(bpq))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))n/(g + hx), x), x, Integral((a + blog(c(d(e + fx)p)q))*n/(g + hx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))(i + jx)S(4)/(de + dfx), x), x, -S(4)bjx(-ej + fi)*S(3)/(df*S(4)) - bj*S(4)(e + fx)S(4)/(S(16)dfS(5)) - S(4)bjS(3)(e + fx)S(3)(-ej + fi)/(S(9)df*S(5)) - S(3)bjS(2)(e + fx)S(2)(-ej + fi)*S(2)/(S(2)dfS(5)) + jS(4)(a + blog(c(e + fx)))(e + fx)S(4)/(S(4)dfS(5)) + S(4)j*S(3)(a + blog(c(e + fx)))(e + fx)S(3)(-ej + fi)/(S(3)df*S(5)) + S(3)j*S(2)(a + blog(c(e + fx)))(e + fx)S(2)(-ej + fi)*S(2)/(df*S(5)) + S(4)j(a + blog(c(e + fx)))(e + fx)(-ej + fi)S(3)/(df*S(5)) + (a + blog(c(e + fx)))*S(2)(-ej + fi)*S(4)/(S(2)bdf*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))(i + jx)*S(3)/(de + dfx), x), x, -S(3)bjx(-ej + fi)*S(2)/(df*S(3)) - bj*S(3)(e + fx)S(3)/(S(9)dfS(4)) - S(3)bjS(2)(e + fx)S(2)(-ej + fi)/(S(4)dfS(4)) + jS(3)(a + blog(c(e + fx)))(e + fx)*S(3)/(S(3)dfS(4)) + S(3)j*S(2)(a + blog(c(e + fx)))(e + fx)S(2)(-ej + fi)/(S(2)df*S(4)) + S(3)j(a + blog(c(e + fx)))(e + fx)(-ej + fi)S(2)/(df*S(4)) + (a + blog(c(e + fx)))*S(2)(-ej + fi)*S(3)/(S(2)bdf*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))(i + jx)*S(2)/(de + dfx), x), x, -S(2)bjx(-ej + fi)/(dfS(2)) - bj*S(2)(e + fx)S(2)/(S(4)dfS(3)) + jS(2)(a + blog(c(e + fx)))(e + fx)S(2)/(S(2)dfS(3)) + S(2)j(a + blog(c(e + fx)))(e + fx)(-ej + fi)/(df*S(3)) + (a + blog(c(e + fx)))*S(2)(-ej + fi)*S(2)/(S(2)bdf*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))(i + jx)/(de + dfx), x), x, -bjx/(df) + j(a + blog(c(e + fx)))(e + fx)/(dfS(2)) + (a + blog(c(e + fx)))*S(2)(-ej/S(2) + fi/S(2))/(bdf*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))/(de + dfx), x), x, (a + blog(c(e + fx)))*S(2)/(S(2)bdf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))/((i + jx)(de + dfx)), x), x, -bpolylog(S(2), -j(e + fx)/(-ej + fi))/(d(-ej + fi)) - (a + blog(c(e + fx)))log(f(i + jx)/(-ej + fi))/(d(-ej + fi)) + (a + blog(c(e + fx)))*S(2)/(S(2)bd(-ej + fi)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))/((i + jx)*S(2)(de + dfx)), x), x, -bflog(e + fx)/(d(-ej + fi)S(2)) + bflog(i + jx)/(d(-ej + fi)S(2)) - bfpolylog(S(2), -j(e + fx)/(-ej + fi))/(d(-ej + fi)*S(2)) - f(a + blog(c(e + fx)))log(f(i + jx)/(-ej + fi))/(d(-ej + fi)S(2)) + (a + blog(c(e + fx)))/(d(i + jx)(-ej + fi)) + f(a + blog(c(e + fx)))S(2)/(S(2)bd(-ej + fi)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))/((i + jx)S(3)(de + dfx)), x), x, -S(3)bfS(2)log(e + fx)/(S(2)d(-ej + fi)S(3)) + S(3)bfS(2)log(i + jx)/(S(2)d(-ej + fi)S(3)) - bf*S(2)polylog(S(2), -j(e + fx)/(-ej + fi))/(d(-ej + fi)S(3)) - bf/(S(2)d(i + jx)(-ej + fi)S(2)) - fS(2)(a + blog(c(e + fx)))log(f(i + jx)/(-ej + fi))/(d(-ej + fi)*S(3)) + f(a + blog(c(e + fx)))/(d(i + jx)(-ej + fi)*S(2)) + (a/S(2) + blog(c(e + fx))/S(2))/(d(i + jx)*S(2)(-ej + fi)) + f*S(2)(a + blog(c(e + fx)))S(2)/(S(2)bd(-ej + fi)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))*S(2)(i + jx)S(4)/(de + dfx), x), x, S(8)bS(2)jx(-ej + fi)*S(3)/(dfS(4)) + bS(2)jS(4)(e + fx)S(4)/(S(32)dfS(5)) + S(8)b*S(2)j*S(3)(e + fx)S(3)(-ej + fi)/(S(27)df*S(5)) + S(3)b*S(2)j*S(2)(e + fx)S(2)(-ej + fi)*S(2)/(S(2)dfS(5)) - bj*S(4)(a + blog(c(e + fx)))(e + fx)S(4)/(S(8)dfS(5)) - S(8)bjS(3)(a + blog(c(e + fx)))(e + fx)S(3)(-ej + fi)/(S(9)df*S(5)) - S(3)bjS(2)(a + blog(c(e + fx)))(e + fx)S(2)(-ej + fi)*S(2)/(df*S(5)) - S(8)bj(a + blog(c(e + fx)))(e + fx)(-ej + fi)*S(3)/(dfS(5)) + jS(4)(a + blog(c(e + fx)))*S(2)(e + fx)S(4)/(S(4)dfS(5)) + S(4)j*S(3)(a + blog(c(e + fx)))S(2)(e + fx)S(3)(-ej + fi)/(S(3)df*S(5)) + S(3)j*S(2)(a + blog(c(e + fx)))S(2)(e + fx)S(2)(-ej + fi)*S(2)/(df*S(5)) + S(4)j(a + blog(c(e + fx)))*S(2)(e + fx)(-ej + fi)*S(3)/(df*S(5)) + (a + blog(c(e + fx)))*S(3)(-ej + fi)*S(4)/(S(3)bdf*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))*S(2)(i + jx)S(3)/(de + dfx), x), x, S(6)bS(2)jx(-ej + fi)*S(2)/(df*S(3)) + S(2)b*S(2)j*S(3)(e + fx)S(3)/(S(27)dfS(4)) + S(3)b*S(2)j*S(2)(e + fx)S(2)(-ej + fi)/(S(4)df*S(4)) - S(2)bjS(3)(a + blog(c(e + fx)))(e + fx)S(3)/(S(9)dfS(4)) - S(3)bjS(2)(a + blog(c(e + fx)))(e + fx)S(2)(-ej + fi)/(S(2)df*S(4)) - S(6)bj(a + blog(c(e + fx)))(e + fx)(-ej + fi)*S(2)/(dfS(4)) + jS(3)(a + blog(c(e + fx)))*S(2)(e + fx)S(3)/(S(3)dfS(4)) + S(3)j*S(2)(a + blog(c(e + fx)))S(2)(e + fx)S(2)(-ej + fi)/(S(2)df*S(4)) + S(3)j(a + blog(c(e + fx)))*S(2)(e + fx)(-ej + fi)*S(2)/(df*S(4)) + (a + blog(c(e + fx)))*S(3)(-ej + fi)*S(3)/(S(3)bdf*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))*S(2)(i + jx)S(2)/(de + dfx), x), x, S(4)bS(2)jx(-ej + fi)/(dfS(2)) + bS(2)j*S(2)(e + fx)S(2)/(S(4)dfS(3)) - bj*S(2)(a + blog(c(e + fx)))(e + fx)S(2)/(S(2)dfS(3)) - S(4)bj(a + blog(c(e + fx)))(e + fx)(-ej + fi)/(dfS(3)) + jS(2)(a + blog(c(e + fx)))S(2)(e + fx)S(2)/(S(2)dfS(3)) + S(2)j(a + blog(c(e + fx)))*S(2)(e + fx)(-ej + fi)/(dfS(3)) + (a + blog(c(e + fx)))*S(3)(-ej + fi)*S(2)/(S(3)bdf*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))*S(2)(i + jx)/(de + dfx), x), x, S(2)bS(2)jx/(df) - S(2)bj(a + blog(c(e + fx)))(e + fx)/(dfS(2)) + j(a + blog(c(e + fx)))S(2)(e + fx)/(df*S(2)) + (a + blog(c(e + fx)))*S(3)(-ej/S(3) + fi/S(3))/(bdf*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))*S(2)/(de + dfx), x), x, (a + blog(c(e + fx)))S(3)/(S(3)bdf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))S(2)/((i + jx)(de + dfx)), x), x, S(2)bS(2)polylog(S(3), -j(e + fx)/(-ej + fi))/(d(-ej + fi)) - S(2)b(a + blog(c(e + fx)))polylog(S(2), -j(e + fx)/(-ej + fi))/(d(-ej + fi)) - (a + blog(c(e + fx)))S(2)log(f(i + jx)/(-ej + fi))/(d(-ej + fi)) + (a + blog(c(e + fx)))*S(3)/(S(3)bd(-ej + fi)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))S(2)/((i + jx)*S(2)(de + dfx)), x), x, S(2)b*S(2)fpolylog(S(2), -j(e + fx)/(-ej + fi))/(d(-ej + fi)*S(2)) + S(2)b*S(2)fpolylog(S(3), -j(e + fx)/(-ej + fi))/(d(-ej + fi)*S(2)) + S(2)bf(a + blog(c(e + fx)))log(f(i + jx)/(-ej + fi))/(d(-ej + fi)S(2)) - S(2)bf(a + blog(c(e + fx)))polylog(S(2), -j(e + fx)/(-ej + fi))/(d(-ej + fi)S(2)) - f(a + blog(c(e + fx)))S(2)log(f(i + jx)/(-ej + fi))/(d(-ej + fi)S(2)) - j(a + blog(c(e + fx)))S(2)(e + fx)/(d(i + jx)(-ej + fi)*S(2)) + f(a + blog(c(e + fx)))S(3)/(S(3)bd(-ej + fi)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))*S(2)/((i + jx)*S(3)(de + dfx)), x), x, bS(2)f*S(2)log(e + fx)/(d(-ej + fi)S(3)) - bS(2)fS(2)log(i + jx)/(d(-ej + fi)*S(3)) + S(3)b*S(2)f*S(2)polylog(S(2), -j(e + fx)/(-ej + fi))/(d(-ej + fi)S(3)) + S(2)b*S(2)f*S(2)polylog(S(3), -j(e + fx)/(-ej + fi))/(d(-ej + fi)S(3)) + S(3)bfS(2)(a + blog(c(e + fx)))log(f(i + jx)/(-ej + fi))/(d(-ej + fi)S(3)) - S(2)bfS(2)(a + blog(c(e + fx)))polylog(S(2), -j(e + fx)/(-ej + fi))/(d(-ej + fi)S(3)) - bf(a + blog(c(e + fx)))/(d(i + jx)(-ej + fi)S(2)) - fS(2)(a + blog(c(e + fx)))S(2)log(f(i + jx)/(-ej + fi))/(d(-ej + fi)S(3)) - fS(2)(a + blog(c(e + fx)))S(2)/(S(2)d(-ej + fi)S(3)) - fj(a + blog(c(e + fx)))*S(2)(e + fx)/(d(i + jx)(-ej + fi)*S(3)) + (a + blog(c(e + fx)))*S(2)/(S(2)d(i + jx)*S(2)(-ej + fi)) + f*S(2)(a + blog(c(e + fx)))S(3)/(S(3)bd(-ej + fi)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((i + jx)*S(4)/((a + blog(c(e + fx)))(de + dfx)), x), x, (-ej + fi)*S(4)log(a + blog(c(e + fx)))/(bdfS(5)) + S(4)j(-ej + fi)S(3)exp(-a/b)Ei((a + blog(c(e + fx)))/b)/(bcdfS(5)) + S(6)j*S(2)(-ej + fi)*S(2)exp(-S(2)a/b)Ei((S(2)a + S(2)blog(c(e + fx)))/b)/(bc*S(2)dfS(5)) + S(4)j*S(3)(-ej + fi)exp(-S(3)a/b)Ei((S(3)a + S(3)blog(c(e + fx)))/b)/(bcS(3)dfS(5)) + jS(4)exp(-S(4)a/b)Ei((S(4)a + S(4)blog(c(e + fx)))/b)/(bc*S(4)dfS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((i + jx)*S(3)/((a + blog(c(e + fx)))(de + dfx)), x), x, (-ej + fi)*S(3)log(a + blog(c(e + fx)))/(bdfS(4)) + S(3)j(-ej + fi)S(2)exp(-a/b)Ei((a + blog(c(e + fx)))/b)/(bcdfS(4)) + S(3)j*S(2)(-ej + fi)exp(-S(2)a/b)Ei((S(2)a + S(2)blog(c(e + fx)))/b)/(bcS(2)dfS(4)) + jS(3)exp(-S(3)a/b)Ei((S(3)a + S(3)blog(c(e + fx)))/b)/(bc*S(3)dfS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((i + jx)*S(2)/((a + blog(c(e + fx)))(de + dfx)), x), x, (-ej + fi)*S(2)log(a + blog(c(e + fx)))/(bdfS(3)) + S(2)j(-ej + fi)exp(-a/b)Ei((a + blog(c(e + fx)))/b)/(bcdfS(3)) + jS(2)exp(-S(2)a/b)Ei((S(2)a + S(2)blog(c(e + fx)))/b)/(bc*S(2)dfS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((i + jx)/((a + blog(c(e + fx)))(de + dfx)), x), x, (-ej + fi)log(a + blog(c(e + fx)))/(bdfS(2)) + jexp(-a/b)Ei((a + blog(c(e + fx)))/b)/(bcdfS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(e + fx)))(de + dfx)), x), x, log(a + blog(c(e + fx)))/(bdf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(e + fx)))(i + jx)(de + dfx)), x), x, Integral(S(1)/((a + blog(c(e + fx)))(i + jx)(de + dfx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(e + fx)))(i + jx)*S(2)(de + dfx)), x), x, Integral(S(1)/((a + blog(c(e + fx)))(i + jx)*S(2)(de + dfx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d + ex)*n))(f + gx)(S(5)/2)/(d + ex), x), x, -S(4)bn(f + gx)*(S(5)/2)/(S(25)e) - S(32)bn(f + gx)*(S(3)/2)(-dg + ef)/(S(45)eS(2)) - S(92)bnsqrt(f + gx)(-dg + ef)*S(2)/(S(15)e*S(3)) - S(4)bn(-dg + ef)*(S(5)/2)log(S(2)/(-sqrt(e)sqrt(f + gx)/sqrt(-dg + ef) + S(1)))atanh(sqrt(e)sqrt(f + gx)/sqrt(-dg + ef))/e(S(7)/2) + S(2)bn(-dg + ef)*(S(5)/2)atanh(sqrt(e)sqrt(f + gx)/sqrt(-dg + ef))S(2)/e(S(7)/2) + S(92)bn(-dg + ef)(S(5)/2)atanh(sqrt(e)sqrt(f + gx)/sqrt(-dg + ef))/(S(15)e(S(7)/2)) - S(2)bn(-dg + ef)*(S(5)/2)polylog(S(2), (-sqrt(e)sqrt(f + gx) - sqrt(-dg + ef))/(-sqrt(e)sqrt(f + gx) + sqrt(-dg + ef)))/e*(S(7)/2) + S(2)(a + blog(c(d + ex)n))(f + gx)(S(5)/2)/(S(5)e) + (a + blog(c(d + ex)n))(f + gx)(S(3)/2)(-S(2)dg/S(3) + S(2)ef/S(3))/e*S(2) + S(2)(a + blog(c(d + ex)n))sqrt(f + gx)(-dg + ef)S(2)/eS(3) - S(2)(a + blog(c(d + ex)*n))(-dg + ef)*(S(5)/2)atanh(sqrt(e)sqrt(f + gx)/sqrt(-dg + ef))/e*(S(7)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d + ex)*n))(f + gx)(S(3)/2)/(d + ex), x), x, -S(4)bn(f + gx)*(S(3)/2)/(S(9)e) - S(16)bnsqrt(f + gx)(-dg + ef)/(S(3)e*S(2)) - S(4)bn(-dg + ef)*(S(3)/2)log(S(2)/(-sqrt(e)sqrt(f + gx)/sqrt(-dg + ef) + S(1)))atanh(sqrt(e)sqrt(f + gx)/sqrt(-dg + ef))/e(S(5)/2) + S(2)bn(-dg + ef)*(S(3)/2)atanh(sqrt(e)sqrt(f + gx)/sqrt(-dg + ef))S(2)/e(S(5)/2) + S(16)bn(-dg + ef)(S(3)/2)atanh(sqrt(e)sqrt(f + gx)/sqrt(-dg + ef))/(S(3)e(S(5)/2)) - S(2)bn(-dg + ef)*(S(3)/2)polylog(S(2), (-sqrt(e)sqrt(f + gx) - sqrt(-dg + ef))/(-sqrt(e)sqrt(f + gx) + sqrt(-dg + ef)))/e*(S(5)/2) + S(2)(a + blog(c(d + ex)n))(f + gx)(S(3)/2)/(S(3)e) + (a + blog(c(d + ex)n))sqrt(f + gx)(-S(2)dg + S(2)ef)/e*S(2) - S(2)(a + blog(c(d + ex)n))(-dg + ef)*(S(3)/2)atanh(sqrt(e)sqrt(f + gx)/sqrt(-dg + ef))/e*(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d + ex)*n))sqrt(f + gx)/(d + ex), x), x, -S(4)bnsqrt(f + gx)/e - S(4)bnsqrt(-dg + ef)log(S(2)/(-sqrt(e)sqrt(f + gx)/sqrt(-dg + ef) + S(1)))atanh(sqrt(e)sqrt(f + gx)/sqrt(-dg + ef))/e(S(3)/2) + S(2)bnsqrt(-dg + ef)atanh(sqrt(e)sqrt(f + gx)/sqrt(-dg + ef))S(2)/e(S(3)/2) + S(4)bnsqrt(-dg + ef)atanh(sqrt(e)sqrt(f + gx)/sqrt(-dg + ef))/e(S(3)/2) - S(2)bnsqrt(-dg + ef)polylog(S(2), (-sqrt(e)sqrt(f + gx) - sqrt(-dg + ef))/(-sqrt(e)sqrt(f + gx) + sqrt(-dg + ef)))/e(S(3)/2) + (S(2)a + S(2)blog(c(d + ex)*n))sqrt(f + gx)/e - S(2)(a + blog(c(d + ex)n))sqrt(-dg + ef)atanh(sqrt(e)sqrt(f + gx)/sqrt(-dg + ef))/e(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + ex)log(a + bx)/(a + bx), x), x, S(2)sqrt(d + ex)log(a + bx)/b - S(4)sqrt(d + ex)/b - S(2)sqrt(-ae + bd)log(a + bx)atanh(sqrt(b)sqrt(d + ex)/sqrt(-ae + bd))/b(S(3)/2) - S(4)sqrt(-ae + bd)log(S(2)/(-sqrt(b)sqrt(d + ex)/sqrt(-ae + bd) + S(1)))atanh(sqrt(b)sqrt(d + ex)/sqrt(-ae + bd))/b*(S(3)/2) + S(2)sqrt(-ae + bd)atanh(sqrt(b)sqrt(d + ex)/sqrt(-ae + bd))S(2)/b(S(3)/2) + S(4)sqrt(-ae + bd)atanh(sqrt(b)sqrt(d + ex)/sqrt(-ae + bd))/b(S(3)/2) - S(2)sqrt(-ae + bd)polylog(S(2), (-sqrt(b)sqrt(d + ex) - sqrt(-ae + bd))/(-sqrt(b)sqrt(d + ex) + sqrt(-ae + bd)))/b(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))*S(2)(i + jx)m/(de + dfx), x), x, Integral((a + blog(c(e + fx)))S(2)(i + jx)m/(de + dfx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))(i + jx)m/(de + dfx), x), x, Integral((a + blog(c(e + fx)))(i + jx)m/(de + dfx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))n(i + jx)m/(de + dfx), x), x, Integral((a + blog(c(e + fx)))n(i + jx)m/(de + dfx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))n(i + jx)S(4)/(de + dfx), x), x, S(4)S(3)(-n + S(-1))j*S(3)((-a - blog(c(e + fx)))/b)(-n)(a + blog(c(e + fx)))n(-ej + fi)Gamma(n + S(1), (-S(3)a - S(3)blog(c(e + fx)))/b)exp(-S(3)a/b)/(c*S(3)dfS(5)) + S(4)(-n + S(-1))j*S(4)((-a - blog(c(e + fx)))/b)(-n)(a + blog(c(e + fx)))nGamma(n + S(1), (-S(4)a - S(4)blog(c(e + fx)))/b)exp(-S(4)a/b)/(cS(4)dfS(5)) + S(4)j((-a - blog(c(e + fx)))/b)*(-n)(a + blog(c(e + fx)))n(-ej + fi)*S(3)Gamma(n + S(1), (-a - blog(c(e + fx)))/b)exp(-a/b)/(cdf*S(5)) + (a + blog(c(e + fx)))*(n + S(1))(-ej + fi)*S(4)/(bdfS(5)(n + S(1))) + S(3)S(2)(-n)j*S(2)((-a - blog(c(e + fx)))/b)(-n)(a + blog(c(e + fx)))n(-ej + fi)*S(2)Gamma(n + S(1), (-S(2)a - S(2)blog(c(e + fx)))/b)exp(-S(2)a/b)/(cS(2)dfS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))*n(i + jx)S(3)/(de + dfx), x), x, S(3)S(2)(-n + S(-1))j*S(2)((-a - blog(c(e + fx)))/b)(-n)(a + blog(c(e + fx)))n(-ej + fi)Gamma(n + S(1), (-S(2)a - S(2)blog(c(e + fx)))/b)exp(-S(2)a/b)/(c*S(2)dfS(4)) + S(3)(-n + S(-1))j*S(3)((-a - blog(c(e + fx)))/b)(-n)(a + blog(c(e + fx)))nGamma(n + S(1), (-S(3)a - S(3)blog(c(e + fx)))/b)exp(-S(3)a/b)/(cS(3)dfS(4)) + S(3)j((-a - blog(c(e + fx)))/b)*(-n)(a + blog(c(e + fx)))n(-ej + fi)*S(2)Gamma(n + S(1), (-a - blog(c(e + fx)))/b)exp(-a/b)/(cdf*S(4)) + (a + blog(c(e + fx)))*(n + S(1))(-ej + fi)*S(3)/(bdfS(4)(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))n(i + jx)S(2)/(de + dfx), x), x, S(2)*(-n + S(-1))j*S(2)((-a - blog(c(e + fx)))/b)(-n)(a + blog(c(e + fx)))nGamma(n + S(1), (-S(2)a - S(2)blog(c(e + fx)))/b)exp(-S(2)a/b)/(cS(2)dfS(3)) + S(2)j((-a - blog(c(e + fx)))/b)*(-n)(a + blog(c(e + fx)))n(-ej + fi)Gamma(n + S(1), (-a - blog(c(e + fx)))/b)exp(-a/b)/(cdfS(3)) + (a + blog(c(e + fx)))*(n + S(1))(-ej + fi)*S(2)/(bdfS(3)(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))n(i + jx)/(de + dfx), x), x, j((-a - blog(c(e + fx)))/b)*(-n)(a + blog(c(e + fx)))nGamma(n + S(1), (-a - blog(c(e + fx)))/b)exp(-a/b)/(cdf*S(2)) + (a + blog(c(e + fx)))*(n + S(1))(-ej + fi)/(bdf*S(2)(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))n/(de + dfx), x), x, (a + blog(c(e + fx)))(n + S(1))/(bdf(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))n/((i + jx)(de + dfx)), x), x, Integral((a + blog(c(e + fx)))n/((i + jx)(de + dfx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))n/((i + jx)*S(2)(de + dfx)), x), x, Integral((a + blog(c(e + fx)))*n/((i + jx)*S(2)(de + dfx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(e + fx)))*n/((i + jx)*S(3)(de + dfx)), x), x, Integral((a + blog(c(e + fx)))*n/((i + jx)*S(3)(de + dfx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(i + jx)S(3)/(g + hx), x), x, ajx(-gj + hi)S(2)/hS(3) - bpq(i + jx)S(3)/(S(9)h) - bpq(i + jx)*S(2)(-gj + hi)/(S(4)hS(2)) - bjpqx(-gj + hi)S(2)/hS(3) + bpq(-gj + hi)S(3)polylog(S(2), -h(e + fx)/(-eh + fg))/h*S(4) - bpq(i + jx)S(2)(-ej + fi)/(S(6)fh) - bjpqx(-ej + fi)(-gj + hi)/(S(2)fh*S(2)) + bj(e + fx)(-gj + hi)S(2)log(c(d(e + fx)p)q)/(fh*S(3)) - bjpqx(-ej + fi)*S(2)/(S(3)f*S(2)h) - bpq(-ej + fi)S(2)(-gj + hi)log(e + fx)/(S(2)fS(2)h*S(2)) - bpq(-ej + fi)*S(3)log(e + fx)/(S(3)f*S(3)h) + (a + blog(c(d(e + fx)p)q))(i + jx)*S(3)/(S(3)h) + (a + blog(c(d(e + fx)p)q))(i + jx)*S(2)(-gj/S(2) + hi/S(2))/h*S(2) + (a + blog(c(d(e + fx)p)q))(-gj + hi)*S(3)log(f(g + hx)/(-eh + fg))/h*S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(i + jx)S(2)/(g + hx), x), x, ajx(-gj + hi)/hS(2) - bpq(i + jx)S(2)/(S(4)h) - bjpqx(-gj + hi)/hS(2) + bpq(-gj + hi)*S(2)polylog(S(2), -h(e + fx)/(-eh + fg))/h*S(3) - bjpqx(-ej + fi)/(S(2)fh) + bj(e + fx)(-gj + hi)log(c(d(e + fx)p)q)/(fhS(2)) - bpq(-ej + fi)*S(2)log(e + fx)/(S(2)f*S(2)h) + (a + blog(c(d(e + fx)p)q))(i + jx)*S(2)/(S(2)h) + (a + blog(c(d(e + fx)p)q))(-gj + hi)S(2)log(f(g + hx)/(-eh + fg))/h*S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))(i + jx)/(g + hx), x), x, ajx/h - bjpqx/h + bpq(-gj + hi)polylog(S(2), -h(e + fx)/(-eh + fg))/h*S(2) + bj(e + fx)log(c(d(e + fx)p)q)/(fh) + (a + blog(c(d(e + fx)p)q))(-gj + hi)log(f(g + hx)/(-eh + fg))/hS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(g + hx), x), x, bpqpolylog(S(2), -h(e + fx)/(-eh + fg))/h + (a + blog(c(d(e + fx)p)q))log(f(g + hx)/(-eh + fg))/h, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/((g + hx)(i + jx)), x), x, bpqpolylog(S(2), -h(e + fx)/(-eh + fg))/(-gj + hi) - bpqpolylog(S(2), -j(e + fx)/(-ej + fi))/(-gj + hi) + (a + blog(c(d(e + fx)p)q))log(f(g + hx)/(-eh + fg))/(-gj + hi) - (a + blog(c(d(e + fx)p)q))log(f(i + jx)/(-ej + fi))/(-gj + hi), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/((g + hx)(i + jx)*S(2)), x), x, -bfpqlog(e + fx)/((-ej + fi)(-gj + hi)) + bfpqlog(i + jx)/((-ej + fi)(-gj + hi)) + bhpqpolylog(S(2), -h(e + fx)/(-eh + fg))/(-gj + hi)S(2) - bhpqpolylog(S(2), -j(e + fx)/(-ej + fi))/(-gj + hi)S(2) + h(a + blog(c(d(e + fx)p)q))log(f(g + hx)/(-eh + fg))/(-gj + hi)S(2) - h(a + blog(c(d(e + fx)p)q))log(f(i + jx)/(-ej + fi))/(-gj + hi)S(2) + (a + blog(c(d(e + fx)p)q))/((i + jx)(-gj + hi)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/((g + hx)(i + jx)*S(3)), x), x, -bf*S(2)pqlog(e + fx)/(S(2)(-ej + fi)*S(2)(-gj + hi)) + bfS(2)pqlog(i + jx)/(S(2)(-ej + fi)*S(2)(-gj + hi)) - bfhpqlog(e + fx)/((-ej + fi)(-gj + hi)S(2)) + bfhpqlog(i + jx)/((-ej + fi)(-gj + hi)*S(2)) - bfpq/(S(2)(i + jx)(-ej + fi)(-gj + hi)) + bhS(2)pqpolylog(S(2), -h(e + fx)/(-eh + fg))/(-gj + hi)*S(3) - bh*S(2)pqpolylog(S(2), -j(e + fx)/(-ej + fi))/(-gj + hi)S(3) + hS(2)(a + blog(c(d(e + fx)p)q))log(f(g + hx)/(-eh + fg))/(-gj + hi)S(3) - hS(2)(a + blog(c(d(e + fx)p)q))log(f(i + jx)/(-ej + fi))/(-gj + hi)*S(3) + h(a + blog(c(d(e + fx)p)q))/((i + jx)(-gj + hi)*S(2)) + (a/S(2) + blog(c(d(e + fx)p)q)/S(2))/((i + jx)*S(2)(-gj + hi)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(2)(i + jx)S(3)/(g + hx), x), x, -S(2)abjpqx(-gj + hi)S(2)/hS(3) - S(2)abjpqx(-ej + fi)(-gj + hi)/(fh*S(2)) - S(2)abjpqx(-ej + fi)*S(2)/(S(3)f*S(2)h) + b*S(2)ejS(2)p*S(2)q*S(2)x(-gj + hi)/(S(2)fhS(2)) + S(2)b*S(2)p*S(2)q*S(2)(i + jx)S(3)/(S(27)h) + b*S(2)j*S(2)p*S(2)q*S(2)x*S(2)(-gj + hi)/(S(4)hS(2)) + S(2)b*S(2)jpS(2)q*S(2)x(-gj + hi)S(2)/hS(3) - S(2)b*S(2)p*S(2)q*S(2)(-gj + hi)*S(3)polylog(S(3), -h(e + fx)/(-eh + fg))/h*S(4) + S(5)b*S(2)p*S(2)q*S(2)(i + jx)S(2)(-ej + fi)/(S(18)fh) + S(2)bS(2)jpS(2)q*S(2)x(-ej + fi)(-gj + hi)/(fhS(2)) - S(2)b*S(2)jpq(e + fx)(-gj + hi)S(2)log(c(d(e + fx)p)q)/(fh*S(3)) + S(11)b*S(2)jpS(2)q*S(2)x(-ej + fi)S(2)/(S(9)f*S(2)h) - S(2)bS(2)jpq(e + fx)(-ej + fi)(-gj + hi)log(c(d(e + fx)p)q)/(f*S(2)h*S(2)) - S(2)b*S(2)jpq(e + fx)(-ej + fi)S(2)log(c(d(e + fx)p)q)/(S(3)f*S(3)h) + S(5)bS(2)p*S(2)q*S(2)(-ej + fi)*S(3)log(e + fx)/(S(9)f*S(3)h) - S(2)bpq(a + blog(c(d(e + fx)p)q))(i + jx)*S(3)/(S(9)h) + S(2)bpq(a + blog(c(d(e + fx)p)q))(-gj + hi)S(3)polylog(S(2), -h(e + fx)/(-eh + fg))/h*S(4) - bpq(a + blog(c(d(e + fx)p)q))(i + jx)*S(2)(-ej + fi)/(S(3)fh) - bjS(2)pq(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)(-gj + hi)/(S(2)fS(2)h*S(2)) + (a + blog(c(d(e + fx)p)q))S(2)(i + jx)S(3)/(S(3)h) + (a + blog(c(d(e + fx)p)q))*S(2)(-gj + hi)*S(3)log(f(g + hx)/(-eh + fg))/h*S(4) + j(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)(-gj + hi)*S(2)/(fhS(3)) + jS(2)(a + blog(c(d(e + fx)p)q))S(2)(e + fx)S(2)(-gj + hi)/(S(2)fS(2)h*S(2)) + j(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)(-ej + fi)(-gj + hi)/(fS(2)h*S(2)) - (a + blog(c(d(e + fx)p)q))S(2)(-ej + fi)*S(3)/(S(3)f*S(3)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(2)(i + jx)S(2)/(g + hx), x), x, -S(2)abjpqx(-gj + hi)/hS(2) - S(2)abjpqx(-ej + fi)/(fh) + bS(2)ejS(2)p*S(2)q*S(2)x/(S(2)fh) + b*S(2)j*S(2)p*S(2)q*S(2)x*S(2)/(S(4)h) + S(2)bS(2)jpS(2)q*S(2)x(-gj + hi)/hS(2) - S(2)b*S(2)p*S(2)q*S(2)(-gj + hi)*S(2)polylog(S(3), -h(e + fx)/(-eh + fg))/h*S(3) + S(2)b*S(2)jpS(2)q*S(2)x(-ej + fi)/(fh) - S(2)bS(2)jpq(e + fx)(-gj + hi)log(c(d(e + fx)p)q)/(fh*S(2)) - S(2)b*S(2)jpq(e + fx)(-ej + fi)log(c(d(e + fx)p)q)/(fS(2)h) + S(2)bpq(a + blog(c(d(e + fx)p)q))(-gj + hi)S(2)polylog(S(2), -h(e + fx)/(-eh + fg))/h*S(3) - bj*S(2)pq(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)/(S(2)f*S(2)h) + (a + blog(c(d(e + fx)p)q))*S(2)(-gj + hi)*S(2)log(f(g + hx)/(-eh + fg))/h*S(3) + j(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)(-gj + hi)/(fhS(2)) + jS(2)(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)S(2)/(S(2)f*S(2)h) + j(a + blog(c(d(e + fx)p)q))S(2)(e + fx)(-ej + fi)/(f*S(2)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(2)(i + jx)/(g + hx), x), x, -S(2)abjpqx/h + S(2)bS(2)jpS(2)q*S(2)x/h - S(2)bS(2)p*S(2)q*S(2)(-gj + hi)polylog(S(3), -h(e + fx)/(-eh + fg))/hS(2) - S(2)b*S(2)jpq(e + fx)log(c(d(e + fx)p)q)/(fh) + S(2)bpq(a + blog(c(d(e + fx)p)q))(-gj + hi)polylog(S(2), -h(e + fx)/(-eh + fg))/hS(2) + (a + blog(c(d(e + fx)p)q))S(2)(-gj + hi)log(f(g + hx)/(-eh + fg))/hS(2) + j(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)/(fh), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(2)/(g + hx), x), x, -S(2)bS(2)p*S(2)q*S(2)polylog(S(3), -h(e + fx)/(-eh + fg))/h + S(2)bpq(a + blog(c(d(e + fx)p)q))polylog(S(2), -h(e + fx)/(-eh + fg))/h + (a + blog(c(d(e + fx)p)q))S(2)log(f(g + hx)/(-eh + fg))/h, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(2)/((g + hx)(i + jx)), x), x, -S(2)bS(2)p*S(2)q*S(2)polylog(S(3), -h(e + fx)/(-eh + fg))/(-gj + hi) + S(2)bS(2)p*S(2)q*S(2)polylog(S(3), -j(e + fx)/(-ej + fi))/(-gj + hi) + S(2)bpq(a + blog(c(d(e + fx)p)q))polylog(S(2), -h(e + fx)/(-eh + fg))/(-gj + hi) - S(2)bpq(a + blog(c(d(e + fx)p)q))polylog(S(2), -j(e + fx)/(-ej + fi))/(-gj + hi) + (a + blog(c(d(e + fx)p)q))*S(2)log(f(g + hx)/(-eh + fg))/(-gj + hi) - (a + blog(c(d(e + fx)p)q))*S(2)log(f(i + jx)/(-ej + fi))/(-gj + hi), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(2)/((g + hx)(i + jx)*S(2)), x), x, S(2)b*S(2)fpS(2)q*S(2)polylog(S(2), -j(e + fx)/(-ej + fi))/((-ej + fi)(-gj + hi)) - S(2)b*S(2)hpS(2)q*S(2)polylog(S(3), -h(e + fx)/(-eh + fg))/(-gj + hi)*S(2) + S(2)b*S(2)hpS(2)q*S(2)polylog(S(3), -j(e + fx)/(-ej + fi))/(-gj + hi)*S(2) + S(2)bfpq(a + blog(c(d(e + fx)p)q))log(f(i + jx)/(-ej + fi))/((-ej + fi)(-gj + hi)) + S(2)bhpq(a + blog(c(d(e + fx)p)q))polylog(S(2), -h(e + fx)/(-eh + fg))/(-gj + hi)*S(2) - S(2)bhpq(a + blog(c(d(e + fx)p)q))polylog(S(2), -j(e + fx)/(-ej + fi))/(-gj + hi)S(2) + h(a + blog(c(d(e + fx)p)q))*S(2)log(f(g + hx)/(-eh + fg))/(-gj + hi)*S(2) - h(a + blog(c(d(e + fx)p)q))*S(2)log(f(i + jx)/(-ej + fi))/(-gj + hi)*S(2) - j(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)/((i + jx)(-ej + fi)(-gj + hi)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(2)/((g + hx)(i + jx)S(3)), x), x, bS(2)fS(2)p*S(2)q*S(2)log(e + fx)/((-ej + fi)S(2)(-gj + hi)) - b*S(2)f*S(2)p*S(2)q*S(2)log(i + jx)/((-ej + fi)S(2)(-gj + hi)) + b*S(2)f*S(2)p*S(2)q*S(2)polylog(S(2), -j(e + fx)/(-ej + fi))/((-ej + fi)*S(2)(-gj + hi)) + S(2)bS(2)fhp*S(2)q*S(2)polylog(S(2), -j(e + fx)/(-ej + fi))/((-ej + fi)(-gj + hi)S(2)) - S(2)b*S(2)h*S(2)p*S(2)q*S(2)polylog(S(3), -h(e + fx)/(-eh + fg))/(-gj + hi)*S(3) + S(2)b*S(2)h*S(2)p*S(2)q*S(2)polylog(S(3), -j(e + fx)/(-ej + fi))/(-gj + hi)*S(3) + bf*S(2)pq(a + blog(c(d(e + fx)p)q))log(f(i + jx)/(-ej + fi))/((-ej + fi)S(2)(-gj + hi)) + S(2)bfhpq(a + blog(c(d(e + fx)p)q))log(f(i + jx)/(-ej + fi))/((-ej + fi)(-gj + hi)*S(2)) - bfpq(a + blog(c(d(e + fx)p)q))/((i + jx)(-ej + fi)(-gj + hi)) + S(2)bh*S(2)pq(a + blog(c(d(e + fx)p)q))polylog(S(2), -h(e + fx)/(-eh + fg))/(-gj + hi)S(3) - S(2)bhS(2)pq(a + blog(c(d(e + fx)p)q))polylog(S(2), -j(e + fx)/(-ej + fi))/(-gj + hi)S(3) - fS(2)(a + blog(c(d(e + fx)p)q))*S(2)/(S(2)(-ej + fi)*S(2)(-gj + hi)) + h*S(2)(a + blog(c(d(e + fx)p)q))*S(2)log(f(g + hx)/(-eh + fg))/(-gj + hi)S(3) - hS(2)(a + blog(c(d(e + fx)p)q))S(2)log(f(i + jx)/(-ej + fi))/(-gj + hi)*S(3) - hj(a + blog(c(d(e + fx)p)q))S(2)(e + fx)/((i + jx)(-ej + fi)(-gj + hi)*S(2)) + (a + blog(c(d(e + fx)p)q))S(2)/(S(2)(i + jx)S(2)(-gj + hi)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(3)(i + jx)S(3)/(g + hx), x), x, S(6)ab*S(2)jpS(2)q*S(2)x(-gj + hi)S(2)/hS(3) + S(6)abS(2)jpS(2)q*S(2)x(-ej + fi)(-gj + hi)/(fhS(2)) + S(6)abS(2)jpS(2)q*S(2)x(-ej + fi)S(2)/(fS(2)h) - S(3)bS(3)ejS(2)p*S(3)q*S(3)x(-gj + hi)/(S(4)fhS(2)) - S(3)b*S(3)ejS(2)p*S(3)q*S(3)x(-ej + fi)/(S(2)f*S(2)h) - S(3)bS(3)j*S(2)p*S(3)q*S(3)x*S(2)(-gj + hi)/(S(8)hS(2)) - S(6)b*S(3)jpS(3)q*S(3)x(-gj + hi)S(2)/hS(3) + S(6)b*S(3)p*S(3)q*S(3)(-gj + hi)*S(3)polylog(S(4), -h(e + fx)/(-eh + fg))/h*S(4) - S(3)b*S(3)j*S(2)p*S(3)q*S(3)x*S(2)(-ej + fi)/(S(4)fh) - S(6)bS(3)jpS(3)q*S(3)x(-ej + fi)(-gj + hi)/(fhS(2)) + S(6)b*S(3)jpS(2)q*S(2)(e + fx)(-gj + hi)*S(2)log(c(d(e + fx)p)q)/(fh*S(3)) - S(6)b*S(3)jpS(3)q*S(3)x(-ej + fi)S(2)/(fS(2)h) + S(6)bS(3)jpS(2)q*S(2)(e + fx)(-ej + fi)(-gj + hi)log(c(d(e + fx)p)q)/(fS(2)h*S(2)) - S(2)b*S(3)j*S(3)p*S(3)q*S(3)(e + fx)S(3)/(S(27)f*S(3)h) + S(6)bS(3)jpS(2)q*S(2)(e + fx)(-ej + fi)*S(2)log(c(d(e + fx)p)q)/(fS(3)h) - S(6)bS(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))(-gj + hi)S(3)polylog(S(3), -h(e + fx)/(-eh + fg))/h*S(4) + S(3)b*S(2)j*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)(-gj + hi)/(S(4)fS(2)h*S(2)) + S(2)b*S(2)j*S(3)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))(e + fx)*S(3)/(S(9)f*S(3)h) + S(3)bS(2)j*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)(-ej + fi)/(S(2)fS(3)h) + S(3)bpq(a + blog(c(d(e + fx)p)q))*S(2)(-gj + hi)*S(3)polylog(S(2), -h(e + fx)/(-eh + fg))/h*S(4) - S(3)bjpq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)(-gj + hi)*S(2)/(fh*S(3)) - S(3)bjS(2)pq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)S(2)(-gj + hi)/(S(4)fS(2)h*S(2)) - S(3)bjpq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)(-ej + fi)(-gj + hi)/(fS(2)h*S(2)) - bj*S(3)pq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)S(3)/(S(3)f*S(3)h) - S(3)bj*S(2)pq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)S(2)(-ej + fi)/(S(2)fS(3)h) - S(3)bjpq(a + blog(c(d(e + fx)p)q))S(2)(e + fx)(-ej + fi)S(2)/(fS(3)h) + (a + blog(c(d(e + fx)p)q))S(3)(-gj + hi)*S(3)log(f(g + hx)/(-eh + fg))/h*S(4) + j(a + blog(c(d(e + fx)p)q))*S(3)(e + fx)(-gj + hi)*S(2)/(fhS(3)) + jS(2)(a + blog(c(d(e + fx)p)q))S(3)(e + fx)S(2)(-gj + hi)/(S(2)fS(2)h*S(2)) + j(a + blog(c(d(e + fx)p)q))*S(3)(e + fx)(-ej + fi)(-gj + hi)/(fS(2)hS(2)) + jS(3)(a + blog(c(d(e + fx)p)q))S(3)(e + fx)S(3)/(S(3)f*S(3)h) + j*S(2)(a + blog(c(d(e + fx)p)q))*S(3)(e + fx)S(2)(-ej + fi)/(f*S(3)h) + j(a + blog(c(d(e + fx)p)q))S(3)(e + fx)(-ej + fi)S(2)/(fS(3)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))S(3)(i + jx)S(2)/(g + hx), x), x, S(6)ab*S(2)jpS(2)q*S(2)x(-gj + hi)/hS(2) + S(6)abS(2)jpS(2)q*S(2)x(-ej + fi)/(fh) - S(3)bS(3)ejS(2)p*S(3)q*S(3)x/(S(4)fh) - S(3)bS(3)j*S(2)p*S(3)q*S(3)x*S(2)/(S(8)h) - S(6)bS(3)jpS(3)q*S(3)x(-gj + hi)/hS(2) + S(6)b*S(3)p*S(3)q*S(3)(-gj + hi)*S(2)polylog(S(4), -h(e + fx)/(-eh + fg))/h*S(3) - S(6)b*S(3)jpS(3)q*S(3)x(-ej + fi)/(fh) + S(6)bS(3)jpS(2)q*S(2)(e + fx)(-gj + hi)log(c(d(e + fx)p)q)/(fhS(2)) + S(6)b*S(3)jpS(2)q*S(2)(e + fx)(-ej + fi)log(c(d(e + fx)p)q)/(f*S(2)h) - S(6)bS(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))(-gj + hi)S(2)polylog(S(3), -h(e + fx)/(-eh + fg))/h*S(3) + S(3)b*S(2)j*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))(e + fx)*S(2)/(S(4)f*S(2)h) + S(3)bpq(a + blog(c(d(e + fx)p)q))*S(2)(-gj + hi)*S(2)polylog(S(2), -h(e + fx)/(-eh + fg))/h*S(3) - S(3)bjpq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)(-gj + hi)/(fhS(2)) - S(3)bjS(2)pq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)S(2)/(S(4)f*S(2)h) - S(3)bjpq(a + blog(c(d(e + fx)p)q))S(2)(e + fx)(-ej + fi)/(f*S(2)h) + (a + blog(c(d(e + fx)p)q))*S(3)(-gj + hi)*S(2)log(f(g + hx)/(-eh + fg))/h*S(3) + j(a + blog(c(d(e + fx)p)q))*S(3)(e + fx)(-gj + hi)/(fhS(2)) + jS(2)(a + blog(c(d(e + fx)p)q))*S(3)(e + fx)S(2)/(S(2)f*S(2)h) + j(a + blog(c(d(e + fx)p)q))S(3)(e + fx)(-ej + fi)/(f*S(2)h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(3)(i + jx)/(g + hx), x), x, S(6)ab*S(2)jpS(2)q*S(2)x/h - S(6)bS(3)jpS(3)q*S(3)x/h + S(6)bS(3)p*S(3)q*S(3)(-gj + hi)polylog(S(4), -h(e + fx)/(-eh + fg))/hS(2) + S(6)b*S(3)jpS(2)q*S(2)(e + fx)log(c(d(e + fx)p)q)/(fh) - S(6)bS(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))(-gj + hi)polylog(S(3), -h(e + fx)/(-eh + fg))/h*S(2) + S(3)bpq(a + blog(c(d(e + fx)p)q))S(2)(-gj + hi)polylog(S(2), -h(e + fx)/(-eh + fg))/hS(2) - S(3)bjpq(a + blog(c(d(e + fx)p)q))*S(2)(e + fx)/(fh) + (a + blog(c(d(e + fx)p)q))*S(3)(-gj + hi)log(f(g + hx)/(-eh + fg))/hS(2) + j(a + blog(c(d(e + fx)p)q))*S(3)(e + fx)/(fh), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(3)/(g + hx), x), x, S(6)bS(3)p*S(3)q*S(3)polylog(S(4), -h(e + fx)/(-eh + fg))/h - S(6)bS(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(3), -h(e + fx)/(-eh + fg))/h + S(3)bpq(a + blog(c(d(e + fx)p)q))S(2)polylog(S(2), -h(e + fx)/(-eh + fg))/h + (a + blog(c(d(e + fx)p)q))*S(3)log(f(g + hx)/(-eh + fg))/h, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(3)/((g + hx)(i + jx)), x), x, S(6)bS(3)p*S(3)q*S(3)polylog(S(4), -h(e + fx)/(-eh + fg))/(-gj + hi) - S(6)bS(3)p*S(3)q*S(3)polylog(S(4), -j(e + fx)/(-ej + fi))/(-gj + hi) - S(6)bS(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(3), -h(e + fx)/(-eh + fg))/(-gj + hi) + S(6)b*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(3), -j(e + fx)/(-ej + fi))/(-gj + hi) + S(3)bpq(a + blog(c(d(e + fx)p)q))S(2)polylog(S(2), -h(e + fx)/(-eh + fg))/(-gj + hi) - S(3)bpq(a + blog(c(d(e + fx)p)q))*S(2)polylog(S(2), -j(e + fx)/(-ej + fi))/(-gj + hi) + (a + blog(c(d(e + fx)p)q))*S(3)log(f(g + hx)/(-eh + fg))/(-gj + hi) - (a + blog(c(d(e + fx)p)q))*S(3)log(f(i + jx)/(-ej + fi))/(-gj + hi), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(3)/((g + hx)(i + jx)*S(2)), x), x, -S(6)b*S(3)fpS(3)q*S(3)polylog(S(3), -j(e + fx)/(-ej + fi))/((-ej + fi)(-gj + hi)) + S(6)b*S(3)hpS(3)q*S(3)polylog(S(4), -h(e + fx)/(-eh + fg))/(-gj + hi)*S(2) - S(6)b*S(3)hpS(3)q*S(3)polylog(S(4), -j(e + fx)/(-ej + fi))/(-gj + hi)*S(2) + S(6)b*S(2)fpS(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(2), -j(e + fx)/(-ej + fi))/((-ej + fi)(-gj + hi)) - S(6)bS(2)hpS(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(3), -h(e + fx)/(-eh + fg))/(-gj + hi)S(2) + S(6)b*S(2)hpS(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(3), -j(e + fx)/(-ej + fi))/(-gj + hi)S(2) + S(3)bfpq(a + blog(c(d(e + fx)p)q))*S(2)log(f(i + jx)/(-ej + fi))/((-ej + fi)(-gj + hi)) + S(3)bhpq(a + blog(c(d(e + fx)p)q))*S(2)polylog(S(2), -h(e + fx)/(-eh + fg))/(-gj + hi)*S(2) - S(3)bhpq(a + blog(c(d(e + fx)p)q))*S(2)polylog(S(2), -j(e + fx)/(-ej + fi))/(-gj + hi)*S(2) + h(a + blog(c(d(e + fx)p)q))*S(3)log(f(g + hx)/(-eh + fg))/(-gj + hi)*S(2) - h(a + blog(c(d(e + fx)p)q))*S(3)log(f(i + jx)/(-ej + fi))/(-gj + hi)*S(2) - j(a + blog(c(d(e + fx)p)q))*S(3)(e + fx)/((i + jx)(-ej + fi)(-gj + hi)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))*S(3)/((g + hx)(i + jx)*S(3)), x), x, -S(3)b*S(3)f*S(2)p*S(3)q*S(3)polylog(S(2), -j(e + fx)/(-ej + fi))/((-ej + fi)*S(2)(-gj + hi)) - S(3)bS(3)f*S(2)p*S(3)q*S(3)polylog(S(3), -j(e + fx)/(-ej + fi))/((-ej + fi)*S(2)(-gj + hi)) - S(6)bS(3)fhp*S(3)q*S(3)polylog(S(3), -j(e + fx)/(-ej + fi))/((-ej + fi)(-gj + hi)S(2)) + S(6)b*S(3)h*S(2)p*S(3)q*S(3)polylog(S(4), -h(e + fx)/(-eh + fg))/(-gj + hi)*S(3) - S(6)b*S(3)h*S(2)p*S(3)q*S(3)polylog(S(4), -j(e + fx)/(-ej + fi))/(-gj + hi)*S(3) - S(3)b*S(2)f*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))log(f(i + jx)/(-ej + fi))/((-ej + fi)S(2)(-gj + hi)) + S(3)bS(2)f*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(2), -j(e + fx)/(-ej + fi))/((-ej + fi)S(2)(-gj + hi)) + S(6)bS(2)fhp*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(2), -j(e + fx)/(-ej + fi))/((-ej + fi)(-gj + hi)*S(2)) - S(6)b*S(2)h*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(3), -h(e + fx)/(-eh + fg))/(-gj + hi)S(3) + S(6)b*S(2)h*S(2)p*S(2)q*S(2)(a + blog(c(d(e + fx)p)q))polylog(S(3), -j(e + fx)/(-ej + fi))/(-gj + hi)S(3) + S(3)bfS(2)pq(a + blog(c(d(e + fx)p)q))*S(2)log(f(i + jx)/(-ej + fi))/(S(2)(-ej + fi)S(2)(-gj + hi)) + S(3)bfhpq(a + blog(c(d(e + fx)p)q))*S(2)log(f(i + jx)/(-ej + fi))/((-ej + fi)(-gj + hi)S(2)) + S(3)bfjpq(a + blog(c(d(e + fx)p)q))S(2)(e + fx)/(S(2)(i + jx)(-ej + fi)*S(2)(-gj + hi)) + S(3)bh*S(2)pq(a + blog(c(d(e + fx)p)q))*S(2)polylog(S(2), -h(e + fx)/(-eh + fg))/(-gj + hi)*S(3) - S(3)bhS(2)pq(a + blog(c(d(e + fx)p)q))*S(2)polylog(S(2), -j(e + fx)/(-ej + fi))/(-gj + hi)S(3) - fS(2)(a + blog(c(d(e + fx)p)q))S(3)/(S(2)(-ej + fi)*S(2)(-gj + hi)) + h*S(2)(a + blog(c(d(e + fx)p)q))*S(3)log(f(g + hx)/(-eh + fg))/(-gj + hi)S(3) - hS(2)(a + blog(c(d(e + fx)p)q))S(3)log(f(i + jx)/(-ej + fi))/(-gj + hi)*S(3) - hj(a + blog(c(d(e + fx)p)q))S(3)(e + fx)/((i + jx)(-ej + fi)(-gj + hi)*S(2)) + (a + blog(c(d(e + fx)p)q))S(3)/(S(2)(i + jx)S(2)(-gj + hi)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((i + jx)/((a + blog(c(d(e + fx)p)q))(g + hx)), x), x, Integral((i + jx)/((a + blog(c(d(e + fx)p)q))(g + hx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))(g + hx)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))(g + hx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))(g + hx)(i + jx)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))(g + hx)(i + jx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))(g + hx)(i + jx)*S(2)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))(g + hx)(i + jx)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((i + jx)/((a + blog(c(d(e + fx)p)q))*S(2)(g + hx)), x), x, Integral((i + jx)/((a + blog(c(d(e + fx)p)q))*S(2)(g + hx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))S(2)(g + hx)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))S(2)(g + hx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))S(2)(g + hx)(i + jx)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))S(2)(g + hx)(i + jx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(c(d(e + fx)p)q))S(2)(g + hx)(i + jx)S(2)), x), x, Integral(S(1)/((a + blog(c(d(e + fx)p)q))S(2)(g + hx)(i + jx)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(g + hx*S(2)), x), x, -bpqpolylog(S(2), sqrt(h)(-e - fx)/(-esqrt(h) + fsqrt(-g)))/(S(2)sqrt(h)sqrt(-g)) + bpqpolylog(S(2), sqrt(h)(e + fx)/(esqrt(h) + fsqrt(-g)))/(S(2)sqrt(h)sqrt(-g)) - (a/S(2) + blog(c(d(e + fx)p)q)/S(2))log(f(sqrt(h)x + sqrt(-g))/(-esqrt(h) + fsqrt(-g)))/(sqrt(h)sqrt(-g)) + (a/S(2) + blog(c(d(e + fx)p)q)/S(2))log(f(-sqrt(h)x + sqrt(-g))/(esqrt(h) + fsqrt(-g)))/(sqrt(h)sqrt(-g)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/sqrt(hx*S(2) + S(2)), x), x, -bpqlog(sqrt(S(2))fexp(asinh(sqrt(S(2))sqrt(h)x/S(2)))/(esqrt(h) - sqrt(eS(2)h + S(2)fS(2))) + S(1))asinh(sqrt(S(2))sqrt(h)x/S(2))/sqrt(h) - bpqlog(sqrt(S(2))fexp(asinh(sqrt(S(2))sqrt(h)x/S(2)))/(esqrt(h) + sqrt(e*S(2)h + S(2)fS(2))) + S(1))asinh(sqrt(S(2))sqrt(h)x/S(2))/sqrt(h) + bpqasinh(sqrt(S(2))sqrt(h)x/S(2))S(2)/(S(2)sqrt(h)) - bpqpolylog(S(2), -sqrt(S(2))fexp(asinh(sqrt(S(2))sqrt(h)x/S(2)))/(esqrt(h) - sqrt(e*S(2)h + S(2)fS(2))))/sqrt(h) - bpqpolylog(S(2), -sqrt(S(2))fexp(asinh(sqrt(S(2))sqrt(h)x/S(2)))/(esqrt(h) + sqrt(eS(2)h + S(2)fS(2))))/sqrt(h) + (a + blog(c(d(e + fx)p)q))asinh(sqrt(S(2))sqrt(h)x/S(2))/sqrt(h), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/sqrt(g + hxS(2)), x), x, -bsqrt(g)pqsqrt(S(1) + hx*S(2)/g)log(fsqrt(g)exp(asinh(sqrt(h)x/sqrt(g)))/(esqrt(h) - sqrt(e*S(2)h + f*S(2)g)) + S(1))asinh(sqrt(h)x/sqrt(g))/(sqrt(h)sqrt(g + hx*S(2))) - bsqrt(g)pqsqrt(S(1) + hx*S(2)/g)log(fsqrt(g)exp(asinh(sqrt(h)x/sqrt(g)))/(esqrt(h) + sqrt(e*S(2)h + f*S(2)g)) + S(1))asinh(sqrt(h)x/sqrt(g))/(sqrt(h)sqrt(g + hx*S(2))) + bsqrt(g)pqsqrt(S(1) + hx*S(2)/g)asinh(sqrt(h)x/sqrt(g))S(2)/(S(2)sqrt(h)sqrt(g + hx*S(2))) - bsqrt(g)pqsqrt(S(1) + hx*S(2)/g)polylog(S(2), -fsqrt(g)exp(asinh(sqrt(h)x/sqrt(g)))/(esqrt(h) - sqrt(e*S(2)h + f*S(2)g)))/(sqrt(h)sqrt(g + hx*S(2))) - bsqrt(g)pqsqrt(S(1) + hx*S(2)/g)polylog(S(2), -fsqrt(g)exp(asinh(sqrt(h)x/sqrt(g)))/(esqrt(h) + sqrt(e*S(2)h + f*S(2)g)))/(sqrt(h)sqrt(g + hx*S(2))) + sqrt(g)sqrt(S(1) + hxS(2)/g)(a + blog(c(d(e + fx)p)q))asinh(sqrt(h)x/sqrt(g))/(sqrt(h)sqrt(g + hx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(sqrt(-hx + S(2))sqrt(hx + S(2))), x), x, -bpqlog(S(2)fexp(Iasin(hx/S(2)))/(Ieh - sqrt(-eS(2)h*S(2) + S(4)f*S(2))) + S(1))asin(hx/S(2))/h - bpqlog(S(2)fexp(Iasin(hx/S(2)))/(Ieh + sqrt(-e*S(2)h*S(2) + S(4)f*S(2))) + S(1))asin(hx/S(2))/h + Ibpqasin(hx/S(2))*S(2)/(S(2)h) + Ibpqpolylog(S(2), -S(2)fexp(Iasin(hx/S(2)))/(Ieh - sqrt(-e*S(2)h*S(2) + S(4)f*S(2))))/h + Ibpqpolylog(S(2), -S(2)fexp(Iasin(hx/S(2)))/(Ieh + sqrt(-eS(2)h*S(2) + S(4)f*S(2))))/h + (a + blog(c(d(e + fx)p)q))asin(hx/S(2))/h, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d(e + fx)p)q))/(sqrt(g - hx)sqrt(g + hx)), x), x, -bgpqsqrt(S(1) - h*S(2)xS(2)/gS(2))log(fgexp(Iasin(hx/g))/(Ieh - sqrt(-eS(2)hS(2) + fS(2)gS(2))) + S(1))asin(hx/g)/(hsqrt(g - hx)sqrt(g + hx)) - bgpqsqrt(S(1) - hS(2)xS(2)/gS(2))log(fgexp(Iasin(hx/g))/(Ieh + sqrt(-eS(2)hS(2) + fS(2)gS(2))) + S(1))asin(hx/g)/(hsqrt(g - hx)sqrt(g + hx)) + Ibgpqsqrt(S(1) - h*S(2)xS(2)/gS(2))asin(hx/g)*S(2)/(S(2)hsqrt(g - hx)sqrt(g + hx)) + Ibgpqsqrt(S(1) - hS(2)xS(2)/gS(2))polylog(S(2), -fgexp(Iasin(hx/g))/(Ieh - sqrt(-eS(2)hS(2) + fS(2)gS(2))))/(hsqrt(g - hx)sqrt(g + hx)) + Ibgpqsqrt(S(1) - h*S(2)xS(2)/gS(2))polylog(S(2), -fgexp(Iasin(hx/g))/(Ieh + sqrt(-eS(2)hS(2) + fS(2)gS(2))))/(hsqrt(g - hx)sqrt(g + hx)) + gsqrt(S(1) - h*S(2)xS(2)/gS(2))(a + blog(c(d(e + fx)p)q))asin(hx/g)/(hsqrt(g - hx)sqrt(g + hx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(2)e/(e + fx))/(eS(2) - fS(2)x*S(2)), x), x, polylog(S(2), (-e + fx)/(e + fx))/(S(2)ef), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(S(2)e/(e + fx)))/(eS(2) - fS(2)xS(2)), x), x, aatanh(fx/e)/(ef) + bpolylog(S(2), (-e + fx)/(e + fx))/(S(2)ef), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e/(e + fx))/(eS(2) - fS(2)xS(2)), x), x, -log(S(2))atanh(fx/e)/(ef) + polylog(S(2), (-e + fx)/(e + fx))/(S(2)ef), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(e/(e + fx)))/(eS(2) - fS(2)xS(2)), x), x, bpolylog(S(2), (-e + fx)/(e + fx))/(S(2)ef) + (a - blog(S(2)))atanh(fx/e)/(ef), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bx)/(c + d/xS(2)), x), x, -sqrt(d)log(b(sqrt(d) - xsqrt(-c))/(asqrt(-c) + bsqrt(d)))log(a + bx)/(S(2)(-c)(S(3)/2)) + sqrt(d)log(-b(sqrt(d) + xsqrt(-c))/(asqrt(-c) - bsqrt(d)))log(a + bx)/(S(2)(-c)(S(3)/2)) + sqrt(d)polylog(S(2), sqrt(-c)(a + bx)/(asqrt(-c) - bsqrt(d)))/(S(2)(-c)(S(3)/2)) - sqrt(d)polylog(S(2), sqrt(-c)(a + bx)/(asqrt(-c) + bsqrt(d)))/(S(2)(-c)(S(3)/2)) - x/c + (a + bx)log(a + bx)/(bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)n)S(3)/(d + ex*S(2)), x), x, -S(3)n*S(3)polylog(S(4), sqrt(e)(-a - bx)/(-asqrt(e) + bsqrt(-d)))/(sqrt(e)sqrt(-d)) + S(3)n*S(3)polylog(S(4), sqrt(e)(a + bx)/(asqrt(e) + bsqrt(-d)))/(sqrt(e)sqrt(-d)) + S(3)n*S(2)log(c(a + bx)*n)polylog(S(3), sqrt(e)(-a - bx)/(-asqrt(e) + bsqrt(-d)))/(sqrt(e)sqrt(-d)) - S(3)n*S(2)log(c(a + bx)*n)polylog(S(3), sqrt(e)(a + bx)/(asqrt(e) + bsqrt(-d)))/(sqrt(e)sqrt(-d)) - S(3)nlog(c(a + bx)n)S(2)polylog(S(2), sqrt(e)(-a - bx)/(-asqrt(e) + bsqrt(-d)))/(S(2)sqrt(e)sqrt(-d)) + S(3)nlog(c(a + bx)n)S(2)polylog(S(2), sqrt(e)(a + bx)/(asqrt(e) + bsqrt(-d)))/(S(2)sqrt(e)sqrt(-d)) - log(c(a + bx)n)S(3)log(b(sqrt(e)x + sqrt(-d))/(-asqrt(e) + bsqrt(-d)))/(S(2)sqrt(e)sqrt(-d)) + log(c(a + bx)n)S(3)log(b(-sqrt(e)x + sqrt(-d))/(asqrt(e) + bsqrt(-d)))/(S(2)sqrt(e)sqrt(-d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)n)S(2)/(d + exS(2)), x), x, nS(2)polylog(S(3), sqrt(e)(-a - bx)/(-asqrt(e) + bsqrt(-d)))/(sqrt(e)sqrt(-d)) - n*S(2)polylog(S(3), sqrt(e)(a + bx)/(asqrt(e) + bsqrt(-d)))/(sqrt(e)sqrt(-d)) - nlog(c(a + bx)*n)polylog(S(2), sqrt(e)(-a - bx)/(-asqrt(e) + bsqrt(-d)))/(sqrt(e)sqrt(-d)) + nlog(c(a + bx)*n)polylog(S(2), sqrt(e)(a + bx)/(asqrt(e) + bsqrt(-d)))/(sqrt(e)sqrt(-d)) - log(c(a + bx)n)S(2)log(b(sqrt(e)x + sqrt(-d))/(-asqrt(e) + bsqrt(-d)))/(S(2)sqrt(e)sqrt(-d)) + log(c(a + bx)n)S(2)log(b(-sqrt(e)x + sqrt(-d))/(asqrt(e) + bsqrt(-d)))/(S(2)sqrt(e)sqrt(-d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)n)/(d + ex*S(2)), x), x, -npolylog(S(2), sqrt(e)(-a - bx)/(-asqrt(e) + bsqrt(-d)))/(S(2)sqrt(e)sqrt(-d)) + npolylog(S(2), sqrt(e)(a + bx)/(asqrt(e) + bsqrt(-d)))/(S(2)sqrt(e)sqrt(-d)) - log(c(a + bx)n)log(b(sqrt(e)x + sqrt(-d))/(-asqrt(e) + bsqrt(-d)))/(S(2)sqrt(e)sqrt(-d)) + log(c(a + bx)*n)log(b(-sqrt(e)x + sqrt(-d))/(asqrt(e) + bsqrt(-d)))/(S(2)sqrt(e)sqrt(-d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + exS(2))log(c(a + bx)*n)), x), x, Integral(S(1)/((d + ex*S(2))log(c(a + bx)n)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)log(c + dx)/(a + bxS(2)), x), x, aS(2)log(-d(sqrt(b)x + sqrt(-a))/(sqrt(b)c - dsqrt(-a)))log(c + dx)/(S(2)bS(3)) + aS(2)log(d(-sqrt(b)x + sqrt(-a))/(sqrt(b)c + dsqrt(-a)))log(c + dx)/(S(2)bS(3)) + aS(2)polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c - dsqrt(-a)))/(S(2)bS(3)) + aS(2)polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c + dsqrt(-a)))/(S(2)bS(3)) + ac*S(2)log(c + dx)/(S(2)b*S(2)d*S(2)) - acx/(S(2)b*S(2)d) - axS(2)log(c + dx)/(S(2)b*S(2)) + ax*S(2)/(S(4)bS(2)) - cS(4)log(c + dx)/(S(4)bdS(4)) + cS(3)x/(S(4)bdS(3)) - cS(2)x*S(2)/(S(8)bdS(2)) + cx*S(3)/(S(12)bd) + xS(4)log(c + dx)/(S(4)b) - x*S(4)/(S(16)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)log(c + dx)/(a + bx*S(2)), x), x, ax/b*S(2) - a(c + dx)log(c + dx)/(bS(2)d) + c*S(3)log(c + dx)/(S(3)bdS(3)) - cS(2)x/(S(3)bd*S(2)) + cx*S(2)/(S(6)bd) + xS(3)log(c + dx)/(S(3)b) - x*S(3)/(S(9)b) - (-a)*(S(3)/2)log(-d(sqrt(b)x + sqrt(-a))/(sqrt(b)c - dsqrt(-a)))log(c + dx)/(S(2)b(S(5)/2)) + (-a)(S(3)/2)log(d(-sqrt(b)x + sqrt(-a))/(sqrt(b)c + dsqrt(-a)))log(c + dx)/(S(2)b(S(5)/2)) - (-a)(S(3)/2)polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c - dsqrt(-a)))/(S(2)b(S(5)/2)) + (-a)(S(3)/2)polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c + dsqrt(-a)))/(S(2)b(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(c + dx)/(a + bx*S(2)), x), x, -alog(-d(sqrt(b)x + sqrt(-a))/(sqrt(b)c - dsqrt(-a)))log(c + dx)/(S(2)bS(2)) - alog(d(-sqrt(b)x + sqrt(-a))/(sqrt(b)c + dsqrt(-a)))log(c + dx)/(S(2)bS(2)) - apolylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c - dsqrt(-a)))/(S(2)bS(2)) - apolylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c + dsqrt(-a)))/(S(2)bS(2)) - cS(2)log(c + dx)/(S(2)bdS(2)) + cx/(S(2)bd) + x*S(2)log(c + dx)/(S(2)b) - x*S(2)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)log(c + dx)/(a + bx*S(2)), x), x, -x/b + (c + dx)log(c + dx)/(bd) - sqrt(-a)log(-d(sqrt(b)x + sqrt(-a))/(sqrt(b)c - dsqrt(-a)))log(c + dx)/(S(2)b(S(3)/2)) + sqrt(-a)log(d(-sqrt(b)x + sqrt(-a))/(sqrt(b)c + dsqrt(-a)))log(c + dx)/(S(2)b(S(3)/2)) - sqrt(-a)polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c - dsqrt(-a)))/(S(2)b(S(3)/2)) + sqrt(-a)polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c + dsqrt(-a)))/(S(2)b(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c + dx)/(a + bx*S(2)), x), x, log(-d(sqrt(b)x + sqrt(-a))/(sqrt(b)c - dsqrt(-a)))log(c + dx)/(S(2)b) + log(d(-sqrt(b)x + sqrt(-a))/(sqrt(b)c + dsqrt(-a)))log(c + dx)/(S(2)b) + polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c - dsqrt(-a)))/(S(2)b) + polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c + dsqrt(-a)))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c + dx)/(a + bxS(2)), x), x, -log(-d(sqrt(b)x + sqrt(-a))/(sqrt(b)c - dsqrt(-a)))log(c + dx)/(S(2)sqrt(b)sqrt(-a)) + log(d(-sqrt(b)x + sqrt(-a))/(sqrt(b)c + dsqrt(-a)))log(c + dx)/(S(2)sqrt(b)sqrt(-a)) - polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c - dsqrt(-a)))/(S(2)sqrt(b)sqrt(-a)) + polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c + dsqrt(-a)))/(S(2)sqrt(b)sqrt(-a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c + dx)/(x(a + bx*S(2))), x), x, log(-dx/c)log(c + dx)/a - log(-d(sqrt(b)x + sqrt(-a))/(sqrt(b)c - dsqrt(-a)))log(c + dx)/(S(2)a) - log(d(-sqrt(b)x + sqrt(-a))/(sqrt(b)c + dsqrt(-a)))log(c + dx)/(S(2)a) + polylog(S(2), (c + dx)/c)/a - polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c - dsqrt(-a)))/(S(2)a) - polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c + dsqrt(-a)))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c + dx)/(x*S(2)(a + bxS(2))), x), x, -sqrt(b)log(-d(sqrt(b)x + sqrt(-a))/(sqrt(b)c - dsqrt(-a)))log(c + dx)/(S(2)(-a)(S(3)/2)) + sqrt(b)log(d(-sqrt(b)x + sqrt(-a))/(sqrt(b)c + dsqrt(-a)))log(c + dx)/(S(2)(-a)(S(3)/2)) - sqrt(b)polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c - dsqrt(-a)))/(S(2)(-a)(S(3)/2)) + sqrt(b)polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c + dsqrt(-a)))/(S(2)(-a)(S(3)/2)) - log(c + dx)/(ax) + dlog(x)/(ac) - dlog(c + dx)/(ac), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c + dx)/(xS(3)(a + bxS(2))), x), x, -log(c + dx)/(S(2)ax*S(2)) - d/(S(2)acx) - d*S(2)log(x)/(S(2)acS(2)) + dS(2)log(c + dx)/(S(2)ac*S(2)) - blog(-dx/c)log(c + dx)/aS(2) + blog(-d(sqrt(b)x + sqrt(-a))/(sqrt(b)c - dsqrt(-a)))log(c + dx)/(S(2)aS(2)) + blog(d(-sqrt(b)x + sqrt(-a))/(sqrt(b)c + dsqrt(-a)))log(c + dx)/(S(2)aS(2)) - bpolylog(S(2), (c + dx)/c)/aS(2) + bpolylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c - dsqrt(-a)))/(S(2)aS(2)) + bpolylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c + dsqrt(-a)))/(S(2)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)log(c + dx)/(a + bx*S(3)), x), x, -alog(-d(a(S(1)/3) + b(S(1)/3)x)/(-a*(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)bS(2)) - alog(-d((S(-1))(S(2)/3)a(S(1)/3) + b(S(1)/3)x)/(-(S(-1))(S(2)/3)a*(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)bS(2)) - alog((S(-1))*(S(1)/3)d(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/((S(-1))*(S(1)/3)a*(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)bS(2)) - apolylog(S(2), b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + b*(S(1)/3)c))/(S(3)bS(2)) - apolylog(S(2), b*(S(1)/3)(c + dx)/((S(-1))(S(1)/3)a*(S(1)/3)d + b*(S(1)/3)c))/(S(3)bS(2)) - apolylog(S(2), b*(S(1)/3)(c + dx)/(-(S(-1))(S(2)/3)a*(S(1)/3)d + b*(S(1)/3)c))/(S(3)bS(2)) + cS(3)log(c + dx)/(S(3)bdS(3)) - cS(2)x/(S(3)bd*S(2)) + cx*S(2)/(S(6)bd) + xS(3)log(c + dx)/(S(3)b) - x*S(3)/(S(9)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)log(c + dx)/(a + bxS(3)), x), x, a(S(2)/3)log(-d(a(S(1)/3) + b(S(1)/3)x)/(-a(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)b(S(5)/3)) - (S(-1))(S(1)/3)a*(S(2)/3)log(d(a(S(1)/3) - (S(-1))(S(1)/3)b*(S(1)/3)x)/(a*(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)c))log(c + dx)/(S(3)b(S(5)/3)) + (S(-1))(S(2)/3)a*(S(2)/3)log(-d(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/(-a*(S(1)/3)d + (S(-1))*(S(2)/3)b*(S(1)/3)c))log(c + dx)/(S(3)b(S(5)/3)) + a(S(2)/3)polylog(S(2), b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + b*(S(1)/3)c))/(S(3)b(S(5)/3)) - (S(-1))(S(1)/3)a*(S(2)/3)polylog(S(2), (S(-1))*(S(1)/3)b*(S(1)/3)(c + dx)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)c))/(S(3)b(S(5)/3)) + (S(-1))(S(2)/3)a*(S(2)/3)polylog(S(2), (S(-1))*(S(2)/3)b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + (S(-1))*(S(2)/3)b*(S(1)/3)c))/(S(3)b(S(5)/3)) - cS(2)log(c + dx)/(S(2)bdS(2)) + cx/(S(2)bd) + x*S(2)log(c + dx)/(S(2)b) - x*S(2)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)log(c + dx)/(a + bxS(3)), x), x, -a(S(1)/3)log(-d(a(S(1)/3) + b(S(1)/3)x)/(-a(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)b(S(4)/3)) - (S(-1))(S(2)/3)a*(S(1)/3)log(d(a(S(1)/3) - (S(-1))(S(1)/3)b*(S(1)/3)x)/(a*(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)c))log(c + dx)/(S(3)b(S(4)/3)) + (S(-1))(S(1)/3)a*(S(1)/3)log(-d(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/(-a*(S(1)/3)d + (S(-1))*(S(2)/3)b*(S(1)/3)c))log(c + dx)/(S(3)b(S(4)/3)) - a(S(1)/3)polylog(S(2), b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + b*(S(1)/3)c))/(S(3)b(S(4)/3)) - (S(-1))(S(2)/3)a*(S(1)/3)polylog(S(2), (S(-1))*(S(1)/3)b*(S(1)/3)(c + dx)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)c))/(S(3)b(S(4)/3)) + (S(-1))(S(1)/3)a*(S(1)/3)polylog(S(2), (S(-1))*(S(2)/3)b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + (S(-1))*(S(2)/3)b*(S(1)/3)c))/(S(3)b(S(4)/3)) - x/b + (c + dx)log(c + dx)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c + dx)/(a + bx*S(3)), x), x, log(-d(a(S(1)/3) + b(S(1)/3)x)/(-a(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)b) + log(-d((S(-1))*(S(2)/3)a(S(1)/3) + b(S(1)/3)x)/(-(S(-1))(S(2)/3)a*(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)b) + log((S(-1))(S(1)/3)d(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/((S(-1))*(S(1)/3)a*(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)b) + polylog(S(2), b(S(1)/3)(c + dx)/(-a(S(1)/3)d + b*(S(1)/3)c))/(S(3)b) + polylog(S(2), b(S(1)/3)(c + dx)/((S(-1))(S(1)/3)a*(S(1)/3)d + b*(S(1)/3)c))/(S(3)b) + polylog(S(2), b(S(1)/3)(c + dx)/(-(S(-1))(S(2)/3)a*(S(1)/3)d + b*(S(1)/3)c))/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c + dx)/(a + bx*S(3)), x), x, -log(-d(a(S(1)/3) + b(S(1)/3)x)/(-a(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)a(S(1)/3)b(S(2)/3)) + (S(-1))(S(1)/3)log(d(a(S(1)/3) - (S(-1))(S(1)/3)b(S(1)/3)x)/(a*(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)c))log(c + dx)/(S(3)a(S(1)/3)b(S(2)/3)) - (S(-1))(S(2)/3)log(-d(a(S(1)/3) + (S(-1))(S(2)/3)b(S(1)/3)x)/(-a*(S(1)/3)d + (S(-1))*(S(2)/3)b*(S(1)/3)c))log(c + dx)/(S(3)a(S(1)/3)b(S(2)/3)) - polylog(S(2), b(S(1)/3)(c + dx)/(-a*(S(1)/3)d + b*(S(1)/3)c))/(S(3)a(S(1)/3)b(S(2)/3)) + (S(-1))(S(1)/3)polylog(S(2), (S(-1))(S(1)/3)b*(S(1)/3)(c + dx)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)c))/(S(3)a(S(1)/3)b(S(2)/3)) - (S(-1))(S(2)/3)polylog(S(2), (S(-1))(S(2)/3)b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + (S(-1))*(S(2)/3)b*(S(1)/3)c))/(S(3)a(S(1)/3)b*(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c + dx)/(a + bxS(3)), x), x, log(-d(a(S(1)/3) + b(S(1)/3)x)/(-a(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)a(S(2)/3)b(S(1)/3)) + (S(-1))(S(2)/3)log(d(a(S(1)/3) - (S(-1))(S(1)/3)b(S(1)/3)x)/(a*(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)c))log(c + dx)/(S(3)a(S(2)/3)b(S(1)/3)) - (S(-1))(S(1)/3)log(-d(a(S(1)/3) + (S(-1))(S(2)/3)b(S(1)/3)x)/(-a*(S(1)/3)d + (S(-1))*(S(2)/3)b*(S(1)/3)c))log(c + dx)/(S(3)a(S(2)/3)b(S(1)/3)) + polylog(S(2), b(S(1)/3)(c + dx)/(-a*(S(1)/3)d + b*(S(1)/3)c))/(S(3)a(S(2)/3)b(S(1)/3)) + (S(-1))(S(2)/3)polylog(S(2), (S(-1))(S(1)/3)b*(S(1)/3)(c + dx)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)c))/(S(3)a(S(2)/3)b(S(1)/3)) - (S(-1))(S(1)/3)polylog(S(2), (S(-1))(S(2)/3)b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + (S(-1))*(S(2)/3)b*(S(1)/3)c))/(S(3)a(S(2)/3)b*(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c + dx)/(x(a + bx*S(3))), x), x, log(-dx/c)log(c + dx)/a - log(-d(a(S(1)/3) + b(S(1)/3)x)/(-a*(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)a) - log(-d((S(-1))*(S(2)/3)a(S(1)/3) + b(S(1)/3)x)/(-(S(-1))(S(2)/3)a*(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)a) - log((S(-1))(S(1)/3)d(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/((S(-1))*(S(1)/3)a*(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)a) + polylog(S(2), (c + dx)/c)/a - polylog(S(2), b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + b*(S(1)/3)c))/(S(3)a) - polylog(S(2), b(S(1)/3)(c + dx)/((S(-1))(S(1)/3)a*(S(1)/3)d + b*(S(1)/3)c))/(S(3)a) - polylog(S(2), b(S(1)/3)(c + dx)/(-(S(-1))(S(2)/3)a*(S(1)/3)d + b*(S(1)/3)c))/(S(3)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c + dx)/(x*S(2)(a + bxS(3))), x), x, -log(c + dx)/(ax) + dlog(x)/(ac) - dlog(c + dx)/(ac) + b*(S(1)/3)log(-d(a(S(1)/3) + b(S(1)/3)x)/(-a*(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)a(S(4)/3)) - (S(-1))(S(1)/3)b*(S(1)/3)log(d(a(S(1)/3) - (S(-1))(S(1)/3)b*(S(1)/3)x)/(a*(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)c))log(c + dx)/(S(3)a(S(4)/3)) + (S(-1))(S(2)/3)b*(S(1)/3)log(-d(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/(-a*(S(1)/3)d + (S(-1))*(S(2)/3)b*(S(1)/3)c))log(c + dx)/(S(3)a(S(4)/3)) + b(S(1)/3)polylog(S(2), b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + b*(S(1)/3)c))/(S(3)a(S(4)/3)) - (S(-1))(S(1)/3)b*(S(1)/3)polylog(S(2), (S(-1))*(S(1)/3)b*(S(1)/3)(c + dx)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)c))/(S(3)a(S(4)/3)) + (S(-1))(S(2)/3)b*(S(1)/3)polylog(S(2), (S(-1))*(S(2)/3)b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + (S(-1))*(S(2)/3)b*(S(1)/3)c))/(S(3)a(S(4)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c + dx)/(x*S(3)(a + bxS(3))), x), x, -log(c + dx)/(S(2)ax*S(2)) - d/(S(2)acx) - d*S(2)log(x)/(S(2)acS(2)) + dS(2)log(c + dx)/(S(2)acS(2)) - b(S(2)/3)log(-d(a(S(1)/3) + b(S(1)/3)x)/(-a(S(1)/3)d + b*(S(1)/3)c))log(c + dx)/(S(3)a(S(5)/3)) - (S(-1))(S(2)/3)b*(S(2)/3)log(d(a(S(1)/3) - (S(-1))(S(1)/3)b*(S(1)/3)x)/(a*(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)c))log(c + dx)/(S(3)a(S(5)/3)) + (S(-1))(S(1)/3)b*(S(2)/3)log(-d(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/(-a*(S(1)/3)d + (S(-1))*(S(2)/3)b*(S(1)/3)c))log(c + dx)/(S(3)a(S(5)/3)) - b(S(2)/3)polylog(S(2), b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + b*(S(1)/3)c))/(S(3)a(S(5)/3)) - (S(-1))(S(2)/3)b*(S(2)/3)polylog(S(2), (S(-1))*(S(1)/3)b*(S(1)/3)(c + dx)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)c))/(S(3)a(S(5)/3)) + (S(-1))(S(1)/3)b*(S(2)/3)polylog(S(2), (S(-1))*(S(2)/3)b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + (S(-1))*(S(2)/3)b*(S(1)/3)c))/(S(3)a(S(5)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)log(c + dx)/(a + bx*S(4)), x), x, -x/b + (c + dx)log(c + dx)/(bd) - (-a)(S(1)/4)log(-d(b(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c - d(-a)*(S(1)/4)))log(c + dx)/(S(4)b(S(5)/4)) + (-a)(S(1)/4)log(d(-b*(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c + d(-a)*(S(1)/4)))log(c + dx)/(S(4)b(S(5)/4)) - (-a)(S(1)/4)polylog(S(2), b(S(1)/4)(c + dx)/(b(S(1)/4)c - d(-a)(S(1)/4)))/(S(4)b(S(5)/4)) + (-a)(S(1)/4)polylog(S(2), b(S(1)/4)(c + dx)/(b(S(1)/4)c + d(-a)(S(1)/4)))/(S(4)b*(S(5)/4)) - sqrt(-sqrt(-a))log(-d(b(S(1)/4)x + sqrt(-sqrt(-a)))/(b*(S(1)/4)c - dsqrt(-sqrt(-a))))log(c + dx)/(S(4)b*(S(5)/4)) + sqrt(-sqrt(-a))log(d(-b(S(1)/4)x + sqrt(-sqrt(-a)))/(b*(S(1)/4)c + dsqrt(-sqrt(-a))))log(c + dx)/(S(4)b*(S(5)/4)) - sqrt(-sqrt(-a))polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c - dsqrt(-sqrt(-a))))/(S(4)b*(S(5)/4)) + sqrt(-sqrt(-a))polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c + dsqrt(-sqrt(-a))))/(S(4)b(S(5)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(c + dx)/(a + bxS(4)), x), x, log(-d(b*(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c - d(-a)*(S(1)/4)))log(c + dx)/(S(4)b) + log(d(-b(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c + d(-a)*(S(1)/4)))log(c + dx)/(S(4)b) + log(-d(b(S(1)/4)x + I(-a)(S(1)/4))/(b(S(1)/4)c - Id(-a)*(S(1)/4)))log(c + dx)/(S(4)b) + log(d(-b(S(1)/4)x + I(-a)(S(1)/4))/(b(S(1)/4)c + Id(-a)*(S(1)/4)))log(c + dx)/(S(4)b) + polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c - d(-a)(S(1)/4)))/(S(4)b) + polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c + d(-a)(S(1)/4)))/(S(4)b) + polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c - Id(-a)*(S(1)/4)))/(S(4)b) + polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c + Id(-a)*(S(1)/4)))/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)log(c + dx)/(a + bx*S(4)), x), x, -log(-d(b*(S(1)/4)x + sqrt(-sqrt(-a)))/(b*(S(1)/4)c - dsqrt(-sqrt(-a))))log(c + dx)/(S(4)b*(S(3)/4)sqrt(-sqrt(-a))) + log(d(-b(S(1)/4)x + sqrt(-sqrt(-a)))/(b*(S(1)/4)c + dsqrt(-sqrt(-a))))log(c + dx)/(S(4)b*(S(3)/4)sqrt(-sqrt(-a))) - polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c - dsqrt(-sqrt(-a))))/(S(4)b*(S(3)/4)sqrt(-sqrt(-a))) + polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c + dsqrt(-sqrt(-a))))/(S(4)b*(S(3)/4)sqrt(-sqrt(-a))) - log(-d(b(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c - d(-a)*(S(1)/4)))log(c + dx)/(S(4)b*(S(3)/4)(-a)*(S(1)/4)) + log(d(-b*(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c + d(-a)*(S(1)/4)))log(c + dx)/(S(4)b*(S(3)/4)(-a)(S(1)/4)) - polylog(S(2), b(S(1)/4)(c + dx)/(b*(S(1)/4)c - d(-a)(S(1)/4)))/(S(4)b*(S(3)/4)(-a)(S(1)/4)) + polylog(S(2), b(S(1)/4)(c + dx)/(b*(S(1)/4)c + d(-a)(S(1)/4)))/(S(4)b*(S(3)/4)(-a)*(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c + dx)/(a + bx*S(4)), x), x, log(-d(b*(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c - d(-a)*(S(1)/4)))log(c + dx)/(S(4)sqrt(b)sqrt(-a)) + log(d(-b*(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c + d(-a)*(S(1)/4)))log(c + dx)/(S(4)sqrt(b)sqrt(-a)) - log(-d(b*(S(1)/4)x + I(-a)(S(1)/4))/(b(S(1)/4)c - Id(-a)*(S(1)/4)))log(c + dx)/(S(4)sqrt(b)sqrt(-a)) - log(d(-b*(S(1)/4)x + I(-a)(S(1)/4))/(b(S(1)/4)c + Id(-a)*(S(1)/4)))log(c + dx)/(S(4)sqrt(b)sqrt(-a)) + polylog(S(2), b(S(1)/4)(c + dx)/(b(S(1)/4)c - d(-a)(S(1)/4)))/(S(4)sqrt(b)sqrt(-a)) + polylog(S(2), b(S(1)/4)(c + dx)/(b(S(1)/4)c + d(-a)(S(1)/4)))/(S(4)sqrt(b)sqrt(-a)) - polylog(S(2), b(S(1)/4)(c + dx)/(b(S(1)/4)c - Id(-a)*(S(1)/4)))/(S(4)sqrt(b)sqrt(-a)) - polylog(S(2), b(S(1)/4)(c + dx)/(b(S(1)/4)c + Id(-a)*(S(1)/4)))/(S(4)sqrt(b)sqrt(-a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c + dx)/(a + bxS(4)), x), x, -log(-d(b*(S(1)/4)x + sqrt(-sqrt(-a)))/(b*(S(1)/4)c - dsqrt(-sqrt(-a))))log(c + dx)/(S(4)b*(S(1)/4)(-sqrt(-a))*(S(3)/2)) + log(d(-b*(S(1)/4)x + sqrt(-sqrt(-a)))/(b*(S(1)/4)c + dsqrt(-sqrt(-a))))log(c + dx)/(S(4)b*(S(1)/4)(-sqrt(-a))(S(3)/2)) - polylog(S(2), b(S(1)/4)(c + dx)/(b*(S(1)/4)c - dsqrt(-sqrt(-a))))/(S(4)b*(S(1)/4)(-sqrt(-a))(S(3)/2)) + polylog(S(2), b(S(1)/4)(c + dx)/(b*(S(1)/4)c + dsqrt(-sqrt(-a))))/(S(4)b*(S(1)/4)(-sqrt(-a))*(S(3)/2)) - log(-d(b*(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c - d(-a)*(S(1)/4)))log(c + dx)/(S(4)b*(S(1)/4)(-a)*(S(3)/4)) + log(d(-b*(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c + d(-a)*(S(1)/4)))log(c + dx)/(S(4)b*(S(1)/4)(-a)(S(3)/4)) - polylog(S(2), b(S(1)/4)(c + dx)/(b*(S(1)/4)c - d(-a)(S(1)/4)))/(S(4)b*(S(1)/4)(-a)(S(3)/4)) + polylog(S(2), b(S(1)/4)(c + dx)/(b*(S(1)/4)c + d(-a)(S(1)/4)))/(S(4)b*(S(1)/4)(-a)*(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c + dx)/(x(a + bx*S(4))), x), x, log(-dx/c)log(c + dx)/a - log(-d(b(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c - d(-a)*(S(1)/4)))log(c + dx)/(S(4)a) - log(d(-b(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c + d(-a)*(S(1)/4)))log(c + dx)/(S(4)a) - log(-d(b(S(1)/4)x + I(-a)(S(1)/4))/(b(S(1)/4)c - Id(-a)*(S(1)/4)))log(c + dx)/(S(4)a) - log(d(-b(S(1)/4)x + I(-a)(S(1)/4))/(b(S(1)/4)c + Id(-a)*(S(1)/4)))log(c + dx)/(S(4)a) + polylog(S(2), (c + dx)/c)/a - polylog(S(2), b(S(1)/4)(c + dx)/(b(S(1)/4)c - d(-a)(S(1)/4)))/(S(4)a) - polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c + d(-a)(S(1)/4)))/(S(4)a) - polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c - Id(-a)*(S(1)/4)))/(S(4)a) - polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c + Id(-a)*(S(1)/4)))/(S(4)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c + dx)/(xS(2)(a + bxS(4))), x), x, -b(S(1)/4)log(-d(b(S(1)/4)x + sqrt(-sqrt(-a)))/(b*(S(1)/4)c - dsqrt(-sqrt(-a))))log(c + dx)/(S(4)(-sqrt(-a))(S(5)/2)) + b(S(1)/4)log(d(-b*(S(1)/4)x + sqrt(-sqrt(-a)))/(b*(S(1)/4)c + dsqrt(-sqrt(-a))))log(c + dx)/(S(4)(-sqrt(-a))(S(5)/2)) - b(S(1)/4)polylog(S(2), b(S(1)/4)(c + dx)/(b(S(1)/4)c - dsqrt(-sqrt(-a))))/(S(4)(-sqrt(-a))(S(5)/2)) + b(S(1)/4)polylog(S(2), b(S(1)/4)(c + dx)/(b(S(1)/4)c + dsqrt(-sqrt(-a))))/(S(4)(-sqrt(-a))(S(5)/2)) - b(S(1)/4)log(-d(b*(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c - d(-a)*(S(1)/4)))log(c + dx)/(S(4)(-a)(S(5)/4)) + b(S(1)/4)log(d(-b*(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c + d(-a)*(S(1)/4)))log(c + dx)/(S(4)(-a)(S(5)/4)) - b(S(1)/4)polylog(S(2), b(S(1)/4)(c + dx)/(b(S(1)/4)c - d(-a)(S(1)/4)))/(S(4)(-a)(S(5)/4)) + b(S(1)/4)polylog(S(2), b(S(1)/4)(c + dx)/(b(S(1)/4)c + d(-a)(S(1)/4)))/(S(4)(-a)*(S(5)/4)) - log(c + dx)/(ax) + dlog(x)/(ac) - dlog(c + dx)/(ac), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c + dx)/(xS(3)(a + bxS(4))), x), x, sqrt(b)log(-d(b(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c - d(-a)*(S(1)/4)))log(c + dx)/(S(4)(-a)*(S(3)/2)) + sqrt(b)log(d(-b(S(1)/4)x + (-a)(S(1)/4))/(b(S(1)/4)c + d(-a)*(S(1)/4)))log(c + dx)/(S(4)(-a)*(S(3)/2)) - sqrt(b)log(-d(b(S(1)/4)x + I(-a)(S(1)/4))/(b(S(1)/4)c - Id(-a)*(S(1)/4)))log(c + dx)/(S(4)(-a)*(S(3)/2)) - sqrt(b)log(d(-b(S(1)/4)x + I(-a)(S(1)/4))/(b(S(1)/4)c + Id(-a)*(S(1)/4)))log(c + dx)/(S(4)(-a)*(S(3)/2)) + sqrt(b)polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c - d(-a)(S(1)/4)))/(S(4)(-a)*(S(3)/2)) + sqrt(b)polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c + d(-a)(S(1)/4)))/(S(4)(-a)*(S(3)/2)) - sqrt(b)polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c - Id(-a)*(S(1)/4)))/(S(4)(-a)*(S(3)/2)) - sqrt(b)polylog(S(2), b*(S(1)/4)(c + dx)/(b(S(1)/4)c + Id(-a)*(S(1)/4)))/(S(4)(-a)*(S(3)/2)) - log(c + dx)/(S(2)ax*S(2)) - d/(S(2)acx) - d*S(2)log(x)/(S(2)acS(2)) + dS(2)log(c + dx)/(S(2)ac*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)n)S(3)/(dx + exS(2)), x), x, S(6)n*S(3)polylog(S(4), (a + bx)/a)/d - S(6)n*S(3)polylog(S(4), -e(a + bx)/(-ae + bd))/d - S(6)nS(2)log(c(a + bx)*n)polylog(S(3), (a + bx)/a)/d + S(6)n*S(2)log(c(a + bx)*n)polylog(S(3), -e(a + bx)/(-ae + bd))/d + S(3)nlog(c(a + bx)n)S(2)polylog(S(2), (a + bx)/a)/d - S(3)nlog(c(a + bx)n)S(2)polylog(S(2), -e(a + bx)/(-ae + bd))/d + log(c(a + bx)n)S(3)log(-bx/a)/d - log(c(a + bx)n)S(3)log(b(d + ex)/(-ae + bd))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)n)S(2)/(dx + ex*S(2)), x), x, -S(2)n*S(2)polylog(S(3), (a + bx)/a)/d + S(2)n*S(2)polylog(S(3), -e(a + bx)/(-ae + bd))/d + S(2)nlog(c(a + bx)*n)polylog(S(2), (a + bx)/a)/d - S(2)nlog(c(a + bx)n)polylog(S(2), -e(a + bx)/(-ae + bd))/d + log(c(a + bx)n)S(2)log(-bx/a)/d - log(c(a + bx)n)S(2)log(b(d + ex)/(-ae + bd))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)n)/(dx + exS(2)), x), x, npolylog(S(2), (a + bx)/a)/d - npolylog(S(2), -e(a + bx)/(-ae + bd))/d + log(c(a + bx)*n)log(-bx/a)/d - log(c(a + bx)n)log(b(d + ex)/(-ae + bd))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx + ex*S(2))log(c(a + bx)*n)), x), x, Integral(S(1)/(x(d + ex)log(c(a + bx)*n)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)n)S(3)/(d + ex + fxS(2)), x), x, S(6)n*S(3)polylog(S(4), S(2)f(a + bx)/(S(2)af - b(e - sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) - S(6)n*S(3)polylog(S(4), S(2)f(a + bx)/(S(2)af - b(e + sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) - S(6)n*S(2)log(c(a + bx)*n)polylog(S(3), S(2)f(a + bx)/(S(2)af - b(e - sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) + S(6)n*S(2)log(c(a + bx)*n)polylog(S(3), S(2)f(a + bx)/(S(2)af - b(e + sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) + S(3)nlog(c(a + bx)n)S(2)polylog(S(2), S(2)f(a + bx)/(S(2)af - b(e - sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) - S(3)nlog(c(a + bx)n)S(2)polylog(S(2), S(2)f(a + bx)/(S(2)af - b(e + sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) + log(c(a + bx)n)S(3)log(-b(e + S(2)fx - sqrt(-S(4)df + eS(2)))/(S(2)af - b(e - sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) - log(c(a + bx)n)S(3)log(-b(e + S(2)fx + sqrt(-S(4)df + eS(2)))/(S(2)af - b(e + sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)n)S(2)/(d + ex + fxS(2)), x), x, -S(2)n*S(2)polylog(S(3), S(2)f(a + bx)/(S(2)af - b(e - sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) + S(2)n*S(2)polylog(S(3), S(2)f(a + bx)/(S(2)af - b(e + sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) + S(2)nlog(c(a + bx)n)polylog(S(2), S(2)f(a + bx)/(S(2)af - b(e - sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) - S(2)nlog(c(a + bx)n)polylog(S(2), S(2)f(a + bx)/(S(2)af - b(e + sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) + log(c(a + bx)n)S(2)log(-b(e + S(2)fx - sqrt(-S(4)df + eS(2)))/(S(2)af - b(e - sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) - log(c(a + bx)n)S(2)log(-b(e + S(2)fx + sqrt(-S(4)df + eS(2)))/(S(2)af - b(e + sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)n)/(d + ex + fxS(2)), x), x, npolylog(S(2), S(2)f(a + bx)/(S(2)af - b(e - sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) - npolylog(S(2), S(2)f(a + bx)/(S(2)af - b(e + sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) + log(c(a + bx)n)log(-b(e + S(2)fx - sqrt(-S(4)df + eS(2)))/(S(2)af - b(e - sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)) - log(c(a + bx)n)log(-b(e + S(2)fx + sqrt(-S(4)df + eS(2)))/(S(2)af - b(e + sqrt(-S(4)df + e*S(2)))))/sqrt(-S(4)df + eS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex + fxS(2))log(c(a + bx)*n)), x), x, Integral(S(1)/((d + ex + fxS(2))log(c(a + bx)n)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(x)/(a + bx + cxS(2)), x), x, -bxlog(x)/cS(2) + bx/cS(2) + xS(2)log(x)/(S(2)c) - x*S(2)/(S(4)c) + (-ac + bS(2) - b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(x)log((b + S(2)cx - sqrt(-S(4)ac + bS(2)))/(b - sqrt(-S(4)ac + bS(2))))/(S(2)c*S(3)) + (-ac + b*S(2) - b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))polylog(S(2), -S(2)cx/(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*S(3)) + (-ac + b*S(2) + b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(x)log((b + S(2)cx + sqrt(-S(4)ac + bS(2)))/(b + sqrt(-S(4)ac + bS(2))))/(S(2)c*S(3)) + (-ac + b*S(2) + b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))polylog(S(2), -S(2)cx/(b + sqrt(-S(4)ac + b*S(2))))/(S(2)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(x)/(a + bx + cxS(2)), x), x, xlog(x)/c - x/c - (b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(x)log((b + S(2)cx + sqrt(-S(4)ac + bS(2)))/(b + sqrt(-S(4)ac + bS(2))))/(S(2)c*S(2)) - (b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))polylog(S(2), -S(2)cx/(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*S(2)) - (b + (S(2)ac - bS(2))/sqrt(-S(4)ac + bS(2)))log(x)log((b + S(2)cx - sqrt(-S(4)ac + bS(2)))/(b - sqrt(-S(4)ac + bS(2))))/(S(2)c*S(2)) - (b + (S(2)ac - bS(2))/sqrt(-S(4)ac + bS(2)))polylog(S(2), -S(2)cx/(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(x)/(a + bx + cx*S(2)), x), x, (-b/sqrt(-S(4)ac + bS(2)) + S(1))log(x)log((b + S(2)cx - sqrt(-S(4)ac + bS(2)))/(b - sqrt(-S(4)ac + bS(2))))/(S(2)c) + (-b/sqrt(-S(4)ac + b*S(2)) + S(1))polylog(S(2), -S(2)cx/(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c) + (b/sqrt(-S(4)ac + b*S(2)) + S(1))log(x)log((b + S(2)cx + sqrt(-S(4)ac + bS(2)))/(b + sqrt(-S(4)ac + bS(2))))/(S(2)c) + (b/sqrt(-S(4)ac + b*S(2)) + S(1))polylog(S(2), -S(2)cx/(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/(a + bx + cx*S(2)), x), x, log(x)log((b + S(2)cx - sqrt(-S(4)ac + b*S(2)))/(b - sqrt(-S(4)ac + bS(2))))/sqrt(-S(4)ac + bS(2)) - log(x)log((b + S(2)cx + sqrt(-S(4)ac + b*S(2)))/(b + sqrt(-S(4)ac + bS(2))))/sqrt(-S(4)ac + bS(2)) + polylog(S(2), -S(2)cx/(b - sqrt(-S(4)ac + bS(2))))/sqrt(-S(4)ac + bS(2)) - polylog(S(2), -S(2)cx/(b + sqrt(-S(4)ac + bS(2))))/sqrt(-S(4)ac + bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/(x(a + bx + cx*S(2))), x), x, -(-b/sqrt(-S(4)ac + bS(2)) + S(1))log(x)log((b + S(2)cx + sqrt(-S(4)ac + bS(2)))/(b + sqrt(-S(4)ac + bS(2))))/(S(2)a) - (-b/sqrt(-S(4)ac + b*S(2)) + S(1))polylog(S(2), -S(2)cx/(b + sqrt(-S(4)ac + b*S(2))))/(S(2)a) - (b/sqrt(-S(4)ac + b*S(2)) + S(1))log(x)log((b + S(2)cx - sqrt(-S(4)ac + bS(2)))/(b - sqrt(-S(4)ac + bS(2))))/(S(2)a) - (b/sqrt(-S(4)ac + b*S(2)) + S(1))polylog(S(2), -S(2)cx/(b - sqrt(-S(4)ac + b*S(2))))/(S(2)a) + log(x)*S(2)/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/(x*S(2)(a + bx + cx*S(2))), x), x, -log(x)/(ax) - S(1)/(ax) - blog(x)*S(2)/(S(2)a*S(2)) + (b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))log(x)log((b + S(2)cx - sqrt(-S(4)ac + bS(2)))/(b - sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)) + (b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))polylog(S(2), -S(2)cx/(b - sqrt(-S(4)ac + b*S(2))))/(S(2)a*S(2)) + (b + (S(2)ac - bS(2))/sqrt(-S(4)ac + bS(2)))log(x)log((b + S(2)cx + sqrt(-S(4)ac + bS(2)))/(b + sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)) + (b + (S(2)ac - bS(2))/sqrt(-S(4)ac + bS(2)))polylog(S(2), -S(2)cx/(b + sqrt(-S(4)ac + b*S(2))))/(S(2)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/(xS(3)(a + bx + cxS(2))), x), x, -log(x)/(S(2)axS(2)) - S(1)/(S(4)axS(2)) + blog(x)/(a*S(2)x) + b/(a*S(2)x) + (-ac/S(2) + bS(2)/S(2))log(x)S(2)/aS(3) - (-ac + bS(2) - b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(x)log((b + S(2)cx + sqrt(-S(4)ac + bS(2)))/(b + sqrt(-S(4)ac + bS(2))))/(S(2)a*S(3)) - (-ac + b*S(2) - b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))polylog(S(2), -S(2)cx/(b + sqrt(-S(4)ac + b*S(2))))/(S(2)a*S(3)) - (-ac + b*S(2) + b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(x)log((b + S(2)cx - sqrt(-S(4)ac + bS(2)))/(b - sqrt(-S(4)ac + bS(2))))/(S(2)a*S(3)) - (-ac + b*S(2) + b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))polylog(S(2), -S(2)cx/(b - sqrt(-S(4)ac + b*S(2))))/(S(2)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d + e/(f + gx))p))S(4), x), x, -S(24)bS(4)epS(4)polylog(S(4), (d + e/(f + gx))/d)/(dg) + S(24)bS(3)epS(3)(a + blog(c(d + e/(f + gx))p))polylog(S(3), (d + e/(f + gx))/d)/(dg) - S(12)bS(2)epS(2)(a + blog(c(d + e/(f + gx))p))S(2)polylog(S(2), (d + e/(f + gx))/d)/(dg) - S(4)bep(a + blog(c(d + e/(f + gx))p))S(3)log(-e/(d(f + gx)))/(dg) + (a + blog(c(d + e/(f + gx))p))S(4)(d(f + gx) + e)/(dg), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d + e/(f + gx))p))S(3), x), x, S(6)b*S(3)epS(3)polylog(S(3), (d + e/(f + gx))/d)/(dg) - S(6)bS(2)epS(2)(a + blog(c(d + e/(f + gx))p))polylog(S(2), (d + e/(f + gx))/d)/(dg) - S(3)bep(a + blog(c(d + e/(f + gx))p))S(2)log(-e/(d(f + gx)))/(dg) + (a + blog(c(d + e/(f + gx))p))S(3)(d(f + gx) + e)/(dg), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d + e/(f + gx))p))S(2), x), x, -S(2)b*S(2)epS(2)polylog(S(2), (d + e/(f + gx))/d)/(dg) - S(2)bep(a + blog(c(d + e/(f + gx))p))log(-e/(d(f + gx)))/(dg) + (a + blog(c(d + e/(f + gx))p))S(2)(d(f + gx) + e)/(dg), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a + blog(c(d + e/(f + gx))p), x), x, ax + b(f + gx)log(c(d + e/(f + gx))p)/g + beplog(d(f + gx) + e)/(dg), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + blog(c(d + e/(f + gx))*p)), x), x, Integral(S(1)/(a + blog(c(d + e/x)p)), (x, f + gx))/g, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(c(d + e/(f + gx))p))(S(-2)), x), x, Integral((a + blog(c(d + e/x)p))(S(-2)), (x, f + gx))/g, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(e(f + gx)p)q), x), x, -pqx + (f + gx)log(c(e(f + gx)p)q)/g, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(d + e(f + gx)p)q), x), x, -pqx + pq(f + gx)hyper((S(1), S(1)/p), (S(1) + S(1)/p,), -e(f + gx)p/d)/g + (f + gx)log(c(d + e(f + gx)p)q)/g, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(d + e(f + gx)S(3))q), x), x, d(S(1)/3)qlog(d(S(1)/3) + e(S(1)/3)(f + gx))/(e(S(1)/3)g) - d*(S(1)/3)qlog(d(S(2)/3) - d(S(1)/3)e*(S(1)/3)(f + gx) + e(S(2)/3)(f + gx)S(2))/(S(2)e*(S(1)/3)g) - sqrt(S(3))d(S(1)/3)qatan(sqrt(S(3))(d*(S(1)/3) - S(2)e*(S(1)/3)(f + gx))/(S(3)d(S(1)/3)))/(e(S(1)/3)g) - S(3)qx + (f + gx)log(c(d + e(f + gx)S(3))q)/g, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(d + e(f + gx)S(2))q), x), x, S(2)sqrt(d)qatan(sqrt(e)(f + gx)/sqrt(d))/(sqrt(e)g) - S(2)qx + (f + gx)log(c(d + e(f + gx)S(2))q)/g, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(d + e(f + gx))q), x), x, -qx + (d + ef + egx)log(c(d + ef + egx)*q)/(eg), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(d + e/(f + gx))*q), x), x, (f + gx)log(c(d + e/(f + gx))q)/g + eqlog(d(f + gx) + e)/(dg), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(d + e/(f + gx)S(2))q), x), x, (f + gx)log(c(d + e/(f + gx)S(2))q)/g + S(2)sqrt(e)qatan(sqrt(d)(f + gx)/sqrt(e))/(sqrt(d)g), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(d + e/(f + gx)S(3))q), x), x, (f + gx)log(c(d + e/(f + gx)S(3))q)/g + e*(S(1)/3)qlog(d(S(1)/3)(f + gx) + e(S(1)/3))/(d(S(1)/3)g) - e*(S(1)/3)qlog(d(S(2)/3)(f + gx)S(2) - d(S(1)/3)e*(S(1)/3)(f + gx) + e(S(2)/3))/(S(2)d*(S(1)/3)g) - sqrt(S(3))e(S(1)/3)qatan(sqrt(S(3))(-S(2)d(S(1)/3)(f + gx) + e(S(1)/3))/(S(3)e(S(1)/3)))/(d(S(1)/3)g), expand=True, _diff=True, _numerical=True) def test_2(): assert rubi_test(rubi_integrate(xmlog(axn), x), x, -nx(m + S(1))/(m + S(1))S(2) + x*(m + S(1))log(axn)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))log(axn), x), x, xnlog(axn)/n - xn/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(axn), x), x, -nxS(4)/S(16) + xS(4)log(axn)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(ax*n), x), x, -nxS(3)/S(9) + xS(3)log(ax*n)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(axn), x), x, -nxS(2)/S(4) + xS(2)log(ax*n)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax*n), x), x, -nx + xlog(ax*n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax*n)/x, x), x, log(axn)S(2)/(S(2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)/xS(2), x), x, -n/x - log(axn)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)/xS(3), x), x, -n/(S(4)xS(2)) - log(ax*n)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(axn)S(2), x), x, S(2)nS(2)x(m + S(1))/(m + S(1))S(3) - S(2)nx*(m + S(1))log(axn)/(m + S(1))S(2) + x(m + S(1))log(axn)S(2)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))log(axn)S(2), x), x, xnlog(axn)S(2)/n - S(2)x*nlog(axn)/n + S(2)xn/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(axn)S(2), x), x, n*S(2)x*S(4)/S(32) - nx*S(4)log(axn)/S(8) + xS(4)log(axn)S(2)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(axn)S(2), x), x, S(2)n*S(2)x*S(3)/S(27) - S(2)nxS(3)log(axn)/S(9) + xS(3)log(axn)S(2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(axn)S(2), x), x, nS(2)x*S(2)/S(4) - nx*S(2)log(axn)/S(2) + xS(2)log(axn)S(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)S(2), x), x, S(2)nS(2)x - S(2)nxlog(ax*n) + xlog(axn)S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)S(2)/x, x), x, log(axn)S(3)/(S(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)S(2)/xS(2), x), x, -S(2)n*S(2)/x - S(2)nlog(ax*n)/x - log(axn)S(2)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)S(2)/xS(3), x), x, -nS(2)/(S(4)x*S(2)) - nlog(axn)/(S(2)x*S(2)) - log(axn)S(2)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(axn)S(3), x), x, -S(6)n*S(3)x(m + S(1))/(m + S(1))S(4) + S(6)nS(2)x*(m + S(1))log(axn)/(m + S(1))S(3) - S(3)nx(m + S(1))log(axn)S(2)/(m + S(1))S(2) + x(m + S(1))log(axn)S(3)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))log(axn)S(3), x), x, xnlog(axn)S(3)/n - S(3)x*nlog(axn)S(2)/n + S(6)x*nlog(axn)/n - S(6)xn/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(axn)S(3), x), x, -S(3)nS(3)x*S(4)/S(128) + S(3)n*S(2)x*S(4)log(axn)/S(32) - S(3)nxS(4)log(axn)S(2)/S(16) + xS(4)log(axn)S(3)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(axn)S(3), x), x, -S(2)n*S(3)x*S(3)/S(27) + S(2)n*S(2)x*S(3)log(axn)/S(9) - nx*S(3)log(axn)S(2)/S(3) + xS(3)log(axn)S(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(axn)S(3), x), x, -S(3)n*S(3)x*S(2)/S(8) + S(3)n*S(2)x*S(2)log(axn)/S(4) - S(3)nxS(2)log(axn)S(2)/S(4) + xS(2)log(axn)S(3)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)S(3), x), x, -S(6)nS(3)x + S(6)nS(2)xlog(ax*n) - S(3)nxlog(axn)S(2) + xlog(axn)S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)S(3)/x, x), x, log(axn)S(4)/(S(4)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)S(3)/xS(2), x), x, -S(6)n*S(3)/x - S(6)n*S(2)log(axn)/x - S(3)nlog(axn)S(2)/x - log(axn)S(3)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)S(3)/x*S(3), x), x, -S(3)n*S(3)/(S(8)x*S(2)) - S(3)n*S(2)log(axn)/(S(4)x*S(2)) - S(3)nlog(axn)S(2)/(S(4)xS(2)) - log(axn)S(3)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)log(ax), x), x, S(2)x*(S(7)/2)log(ax)/S(7) - S(4)x(S(7)/2)/S(49), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)log(ax), x), x, S(2)x(S(5)/2)log(ax)/S(5) - S(4)x*(S(5)/2)/S(25), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)log(ax), x), x, S(2)x*(S(3)/2)log(ax)/S(3) - S(4)x*(S(3)/2)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax)/sqrt(x), x), x, S(2)sqrt(x)log(ax) - S(4)sqrt(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax)/x(S(3)/2), x), x, -S(2)log(ax)/sqrt(x) - S(4)/sqrt(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax)/x*(S(5)/2), x), x, -S(2)log(ax)/(S(3)x*(S(3)/2)) - S(4)/(S(9)x(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/log(axn), x), x, x(m + S(1))(axn)(-(m + S(1))/n)Ei((m + S(1))log(axn)/n)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))/log(axn), x), x, li(ax*n)/(an), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/log(axn), x), x, xS(4)(axn)(-S(4)/n)Ei(S(4)log(axn)/n)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/log(axn), x), x, xS(3)(axn)(-S(3)/n)Ei(S(3)log(axn)/n)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(axn), x), x, xS(2)(axn)(-S(2)/n)Ei(S(2)log(axn)/n)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/log(ax*n), x), x, x(axn)(-S(1)/n)Ei(log(axn)/n)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(axn)), x), x, log(log(axn))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)log(ax*n)), x), x, (axn)(S(1)/n)Ei(-log(ax*n)/n)/(nx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)log(axn)), x), x, (axn)(S(2)/n)Ei(-S(2)log(axn)/n)/(nxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/log(axn)S(2), x), x, -x(m + S(1))/(nlog(axn)) + x(m + S(1))(axn)(-(m + S(1))/n)(m + S(1))Ei((m + S(1))log(axn)/n)/nS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))/log(axn)S(2), x), x, -x*n/(nlog(axn)) + li(ax*n)/(an), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/log(axn)S(2), x), x, -x*S(4)/(nlog(axn)) + S(4)x*S(4)(axn)(-S(4)/n)Ei(S(4)log(axn)/n)/nS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/log(axn)S(2), x), x, -x*S(3)/(nlog(axn)) + S(3)x*S(3)(axn)(-S(3)/n)Ei(S(3)log(axn)/n)/nS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(axn)S(2), x), x, -xS(2)/(nlog(axn)) + S(2)x*S(2)(axn)(-S(2)/n)Ei(S(2)log(axn)/n)/nS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)(S(-2)), x), x, -x/(nlog(axn)) + x(axn)(-S(1)/n)Ei(log(axn)/n)/nS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(axn)S(2)), x), x, -S(1)/(nlog(axn)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)log(axn)S(2)), x), x, -S(1)/(nxlog(ax*n)) - (axn)(S(1)/n)Ei(-log(axn)/n)/(nS(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)log(axn)S(2)), x), x, -S(1)/(nx*S(2)log(axn)) - S(2)(axn)(S(2)/n)Ei(-S(2)log(axn)/n)/(nS(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/log(axn)S(3), x), x, -x*(m + S(1))/(S(2)nlog(axn)S(2)) - x*(m + S(1))(m/S(2) + S(1)/2)/(n*S(2)log(axn)) + x(m + S(1))(axn)(-(m + S(1))/n)(m + S(1))*S(2)Ei((m + S(1))log(ax*n)/n)/(S(2)nS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))/log(axn)S(3), x), x, -xn/(S(2)nlog(axn)) - xn/(S(2)nlog(axn)S(2)) + li(ax*n)/(S(2)an), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/log(axn)S(3), x), x, -x*S(4)/(S(2)nlog(axn)S(2)) - S(2)xS(4)/(nS(2)log(axn)) + S(8)x*S(4)(axn)(-S(4)/n)Ei(S(4)log(axn)/n)/nS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/log(axn)S(3), x), x, -x*S(3)/(S(2)nlog(axn)S(2)) - S(3)xS(3)/(S(2)n*S(2)log(axn)) + S(9)x*S(3)(axn)(-S(3)/n)Ei(S(3)log(ax*n)/n)/(S(2)n*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(axn)S(3), x), x, -x*S(2)/(S(2)nlog(axn)S(2)) - xS(2)/(nS(2)log(ax*n)) + S(2)x*S(2)(axn)(-S(2)/n)Ei(S(2)log(axn)/n)/nS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)(S(-3)), x), x, -x/(S(2)nlog(axn)S(2)) - x/(S(2)nS(2)log(axn)) + x(axn)(-S(1)/n)Ei(log(axn)/n)/(S(2)n*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(axn)S(3)), x), x, -S(1)/(S(2)nlog(axn)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)log(axn)S(3)), x), x, -S(1)/(S(2)nxlog(axn)S(2)) + S(1)/(S(2)n*S(2)xlog(ax*n)) + (axn)(S(1)/n)Ei(-log(ax*n)/n)/(S(2)n*S(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)log(axn)S(3)), x), x, -S(1)/(S(2)nxS(2)log(axn)S(2)) + S(1)/(nS(2)x*S(2)log(axn)) + S(2)(axn)(S(2)/n)Ei(-S(2)log(axn)/n)/(nS(3)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/log(ax), x), x, x*(m + S(1))(ax)(-m + S(-1))Ei((m + S(1))log(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/log(ax), x), x, Ei(S(4)log(ax))/aS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/log(ax), x), x, Ei(S(3)log(ax))/aS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(ax), x), x, Ei(S(2)log(ax))/a*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/log(ax), x), x, li(ax)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(ax)), x), x, log(log(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)log(ax)), x), x, aEi(-log(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)log(ax)), x), x, aS(2)Ei(-S(2)log(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*m/log(ax)S(2), x), x, x(m + S(1))(ax)*(-m + S(-1))(m + S(1))Ei((m + S(1))log(ax)) - x(m + S(1))/log(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/log(ax)S(2), x), x, -xS(4)/log(ax) + S(4)Ei(S(4)log(ax))/aS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/log(ax)S(2), x), x, -xS(3)/log(ax) + S(3)Ei(S(3)log(ax))/aS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(ax)S(2), x), x, -xS(2)/log(ax) + S(2)Ei(S(2)log(ax))/a*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax)*(S(-2)), x), x, -x/log(ax) + li(ax)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(ax)S(2)), x), x, -S(1)/log(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)log(ax)S(2)), x), x, -aEi(-log(ax)) - S(1)/(xlog(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)log(ax)S(2)), x), x, -S(2)a*S(2)Ei(-S(2)log(ax)) - S(1)/(x*S(2)log(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/log(ax)S(3), x), x, x(m + S(1))(ax)*(-m + S(-1))(m + S(1))*S(2)Ei((m + S(1))log(ax))/S(2) - x*(m + S(1))(m/S(2) + S(1)/2)/log(ax) - x(m + S(1))/(S(2)log(ax)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/log(ax)*S(3), x), x, -S(2)x*S(4)/log(ax) - x*S(4)/(S(2)log(ax)S(2)) + S(8)Ei(S(4)log(ax))/aS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/log(ax)S(3), x), x, -S(3)x*S(3)/(S(2)log(ax)) - xS(3)/(S(2)log(ax)S(2)) + S(9)Ei(S(3)log(ax))/(S(2)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(ax)S(3), x), x, -xS(2)/log(ax) - xS(2)/(S(2)log(ax)S(2)) + S(2)Ei(S(2)log(ax))/a*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax)*(S(-3)), x), x, -x/(S(2)log(ax)) - x/(S(2)log(ax)S(2)) + li(ax)/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(ax)S(3)), x), x, -S(1)/(S(2)log(ax)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)log(ax)S(3)), x), x, aEi(-log(ax))/S(2) + S(1)/(S(2)xlog(ax)) - S(1)/(S(2)xlog(ax)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)log(ax)S(3)), x), x, S(2)a*S(2)Ei(-S(2)log(ax)) + S(1)/(x*S(2)log(ax)) - S(1)/(S(2)x*S(2)log(ax)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmsqrt(log(axn)), x), x, -sqrt(pi)sqrt(n)x(m + S(1))(axn)(-(m + S(1))/n)erfi(sqrt(m + S(1))sqrt(log(ax*n))/sqrt(n))/(S(2)(m + S(1))(S(3)/2)) + x(m + S(1))sqrt(log(axn))/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(log(ax*n)), x), x, -sqrt(pi)sqrt(n)xS(4)(axn)(-S(4)/n)erfi(S(2)sqrt(log(axn))/sqrt(n))/S(16) + xS(4)sqrt(log(axn))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(log(ax*n)), x), x, -sqrt(S(3))sqrt(pi)sqrt(n)x*S(3)(axn)(-S(3)/n)erfi(sqrt(S(3))sqrt(log(axn))/sqrt(n))/S(18) + xS(3)sqrt(log(ax*n))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(log(axn)), x), x, -sqrt(S(2))sqrt(pi)sqrt(n)x*S(2)(axn)(-S(2)/n)erfi(sqrt(S(2))sqrt(log(axn))/sqrt(n))/S(8) + xS(2)sqrt(log(ax*n))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(log(ax*n)), x), x, -sqrt(pi)sqrt(n)x(axn)(-S(1)/n)erfi(sqrt(log(axn))/sqrt(n))/S(2) + xsqrt(log(axn)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(log(ax*n))/x, x), x, S(2)log(axn)(S(3)/2)/(S(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(log(axn))/xS(2), x), x, sqrt(pi)sqrt(n)(axn)(S(1)/n)erf(sqrt(log(ax*n))/sqrt(n))/(S(2)x) - sqrt(log(axn))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(log(axn))/xS(3), x), x, sqrt(S(2))sqrt(pi)sqrt(n)(axn)(S(2)/n)erf(sqrt(S(2))sqrt(log(axn))/sqrt(n))/(S(8)x*S(2)) - sqrt(log(ax*n))/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(axn)(S(3)/2), x), x, S(3)sqrt(pi)n*(S(3)/2)x*(m + S(1))(axn)(-(m + S(1))/n)erfi(sqrt(m + S(1))sqrt(log(ax*n))/sqrt(n))/(S(4)(m + S(1))*(S(5)/2)) - S(3)nx(m + S(1))sqrt(log(axn))/(S(2)(m + S(1))S(2)) + x(m + S(1))log(axn)(S(3)/2)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)log(axn)(S(3)/2), x), x, S(3)sqrt(pi)n(S(3)/2)x*S(4)(axn)(-S(4)/n)erfi(S(2)sqrt(log(ax*n))/sqrt(n))/S(128) - S(3)nxS(4)sqrt(log(axn))/S(32) + xS(4)log(axn)(S(3)/2)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(axn)(S(3)/2), x), x, sqrt(S(3))sqrt(pi)n(S(3)/2)x*S(3)(axn)(-S(3)/n)erfi(sqrt(S(3))sqrt(log(ax*n))/sqrt(n))/S(36) - nx*S(3)sqrt(log(axn))/S(6) + xS(3)log(axn)(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(axn)(S(3)/2), x), x, S(3)sqrt(S(2))sqrt(pi)n*(S(3)/2)x*S(2)(axn)(-S(2)/n)erfi(sqrt(S(2))sqrt(log(ax*n))/sqrt(n))/S(32) - S(3)nxS(2)sqrt(log(axn))/S(8) + xS(2)log(axn)(S(3)/2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)(S(3)/2), x), x, S(3)sqrt(pi)n*(S(3)/2)x(axn)(-S(1)/n)erfi(sqrt(log(ax*n))/sqrt(n))/S(4) - S(3)nxsqrt(log(axn))/S(2) + xlog(axn)(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)(S(3)/2)/x, x), x, S(2)log(axn)(S(5)/2)/(S(5)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)(S(3)/2)/x*S(2), x), x, S(3)sqrt(pi)n(S(3)/2)(axn)(S(1)/n)erf(sqrt(log(axn))/sqrt(n))/(S(4)x) - S(3)nsqrt(log(axn))/(S(2)x) - log(axn)(S(3)/2)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)(S(3)/2)/x*S(3), x), x, S(3)sqrt(S(2))sqrt(pi)n*(S(3)/2)(axn)(S(2)/n)erf(sqrt(S(2))sqrt(log(ax*n))/sqrt(n))/(S(32)x*S(2)) - S(3)nsqrt(log(ax*n))/(S(8)x*S(2)) - log(axn)(S(3)/2)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/sqrt(log(ax*n)), x), x, sqrt(pi)x*(m + S(1))(axn)(-(m + S(1))/n)erfi(sqrt(m + S(1))sqrt(log(ax*n))/sqrt(n))/(sqrt(n)sqrt(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/sqrt(log(ax*n)), x), x, sqrt(pi)x*S(4)(axn)(-S(4)/n)erfi(S(2)sqrt(log(ax*n))/sqrt(n))/(S(2)sqrt(n)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/sqrt(log(ax*n)), x), x, sqrt(S(3))sqrt(pi)xS(3)(axn)(-S(3)/n)erfi(sqrt(S(3))sqrt(log(ax*n))/sqrt(n))/(S(3)sqrt(n)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(log(axn)), x), x, sqrt(S(2))sqrt(pi)xS(2)(axn)(-S(2)/n)erfi(sqrt(S(2))sqrt(log(ax*n))/sqrt(n))/(S(2)sqrt(n)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(log(axn)), x), x, sqrt(pi)x(axn)(-S(1)/n)erfi(sqrt(log(ax*n))/sqrt(n))/sqrt(n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(log(axn))), x), x, S(2)sqrt(log(axn))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(log(axn))), x), x, sqrt(pi)(axn)(S(1)/n)erf(sqrt(log(axn))/sqrt(n))/(sqrt(n)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)sqrt(log(axn))), x), x, sqrt(S(2))sqrt(pi)(axn)(S(2)/n)erf(sqrt(S(2))sqrt(log(axn))/sqrt(n))/(S(2)sqrt(n)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/log(axn)(S(3)/2), x), x, -S(2)x(m + S(1))/(nsqrt(log(axn))) + S(2)sqrt(pi)x(m + S(1))(axn)(-(m + S(1))/n)sqrt(m + S(1))erfi(sqrt(m + S(1))sqrt(log(axn))/sqrt(n))/n(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/log(axn)(S(3)/2), x), x, -S(2)xS(4)/(nsqrt(log(axn))) + S(4)sqrt(pi)xS(4)(axn)(-S(4)/n)erfi(S(2)sqrt(log(axn))/sqrt(n))/n(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/log(axn)(S(3)/2), x), x, -S(2)xS(3)/(nsqrt(log(axn))) + S(2)sqrt(S(3))sqrt(pi)x*S(3)(axn)(-S(3)/n)erfi(sqrt(S(3))sqrt(log(axn))/sqrt(n))/n(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(axn)(S(3)/2), x), x, -S(2)x*S(2)/(nsqrt(log(axn))) + S(2)sqrt(S(2))sqrt(pi)x*S(2)(axn)(-S(2)/n)erfi(sqrt(S(2))sqrt(log(axn))/sqrt(n))/n(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)(S(-3)/2), x), x, -S(2)x/(nsqrt(log(ax*n))) + S(2)sqrt(pi)x(axn)(-S(1)/n)erfi(sqrt(log(axn))/sqrt(n))/n(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(axn)(S(3)/2)), x), x, -S(2)/(nsqrt(log(axn))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)log(axn)(S(3)/2)), x), x, -S(2)/(nxsqrt(log(ax*n))) - S(2)sqrt(pi)(axn)(S(1)/n)erf(sqrt(log(axn))/sqrt(n))/(n(S(3)/2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)log(axn)(S(3)/2)), x), x, -S(2)/(nx*S(2)sqrt(log(axn))) - S(2)sqrt(S(2))sqrt(pi)(axn)(S(2)/n)erf(sqrt(S(2))sqrt(log(axn))/sqrt(n))/(n(S(3)/2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/log(axn)(S(5)/2), x), x, -S(2)x(m + S(1))/(S(3)nlog(axn)(S(3)/2)) - x*(m + S(1))(S(4)m/S(3) + S(4)/3)/(nS(2)sqrt(log(axn))) + S(4)sqrt(pi)x(m + S(1))(axn)(-(m + S(1))/n)(m + S(1))*(S(3)/2)erfi(sqrt(m + S(1))sqrt(log(ax*n))/sqrt(n))/(S(3)n(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/log(axn)(S(5)/2), x), x, -S(2)x*S(4)/(S(3)nlog(axn)(S(3)/2)) - S(16)xS(4)/(S(3)n*S(2)sqrt(log(axn))) + S(32)sqrt(pi)xS(4)(axn)(-S(4)/n)erfi(S(2)sqrt(log(ax*n))/sqrt(n))/(S(3)n(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/log(axn)(S(5)/2), x), x, -S(2)x*S(3)/(S(3)nlog(axn)(S(3)/2)) - S(4)xS(3)/(nS(2)sqrt(log(axn))) + S(4)sqrt(S(3))sqrt(pi)x*S(3)(axn)(-S(3)/n)erfi(sqrt(S(3))sqrt(log(axn))/sqrt(n))/n(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(axn)(S(5)/2), x), x, -S(2)x*S(2)/(S(3)nlog(axn)(S(3)/2)) - S(8)xS(2)/(S(3)n*S(2)sqrt(log(axn))) + S(8)sqrt(S(2))sqrt(pi)x*S(2)(axn)(-S(2)/n)erfi(sqrt(S(2))sqrt(log(ax*n))/sqrt(n))/(S(3)n*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)(S(-5)/2), x), x, -S(2)x/(S(3)nlog(axn)(S(3)/2)) - S(4)x/(S(3)n*S(2)sqrt(log(axn))) + S(4)sqrt(pi)x(axn)(-S(1)/n)erfi(sqrt(log(axn))/sqrt(n))/(S(3)n*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(axn)(S(5)/2)), x), x, -S(2)/(S(3)nlog(axn)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)log(axn)(S(5)/2)), x), x, -S(2)/(S(3)nxlog(axn)(S(3)/2)) + S(4)/(S(3)n*S(2)xsqrt(log(ax*n))) + S(4)sqrt(pi)(axn)(S(1)/n)erf(sqrt(log(ax*n))/sqrt(n))/(S(3)n*(S(5)/2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)log(axn)(S(5)/2)), x), x, -S(2)/(S(3)nxS(2)log(axn)(S(3)/2)) + S(8)/(S(3)n*S(2)x*S(2)sqrt(log(axn))) + S(8)sqrt(S(2))sqrt(pi)(axn)(S(2)/n)erf(sqrt(S(2))sqrt(log(ax*n))/sqrt(n))/(S(3)n*(S(5)/2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(ax)p, x), x, x(m + S(1))(ax)*(-m + S(-1))((-m + S(-1))log(ax))*(-p)Gamma(p + S(1), (-m + S(-1))log(ax))log(ax)p/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(ax)p, x), x, S(4)(-p + S(-1))(-log(ax))*(-p)Gamma(p + S(1), -S(4)log(ax))log(ax)p/aS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)log(ax)p, x), x, S(3)(-p + S(-1))(-log(ax))(-p)Gamma(p + S(1), -S(3)log(ax))log(ax)p/aS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(ax)p, x), x, S(2)(-p + S(-1))(-log(ax))*(-p)Gamma(p + S(1), -S(2)log(ax))log(ax)p/aS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax)p, x), x, (-log(ax))*(-p)Gamma(p + S(1), -log(ax))log(ax)p/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax)*p/x, x), x, log(ax)*(p + S(1))/(p + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax)p/xS(2), x), x, -aGamma(p + S(1), log(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax)p/xS(3), x), x, -S(2)(-p + S(-1))a*S(2)Gamma(p + S(1), S(2)log(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*mlog(axn)p, x), x, x(m + S(1))(axn)(-(m + S(1))/n)((-m + S(-1))log(axn)/n)(-p)Gamma(p + S(1), (-m + S(-1))log(axn)/n)log(axn)p/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))log(axn)p, x), x, (-log(axn))(-p)Gamma(p + S(1), -log(ax*n))log(axn)p/(an), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)log(axn)p, x), x, S(4)(-p + S(-1))x*S(4)(axn)(-S(4)/n)(-log(axn)/n)(-p)Gamma(p + S(1), -S(4)log(ax*n)/n)log(axn)p, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(axn)p, x), x, S(3)(-p + S(-1))x*S(3)(axn)(-S(3)/n)(-log(axn)/n)(-p)Gamma(p + S(1), -S(3)log(ax*n)/n)log(axn)p, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(axn)p, x), x, S(2)(-p + S(-1))x*S(2)(axn)(-S(2)/n)(-log(axn)/n)(-p)Gamma(p + S(1), -S(2)log(ax*n)/n)log(axn)p, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)p, x), x, x(axn)(-S(1)/n)(-log(axn)/n)(-p)Gamma(p + S(1), -log(ax*n)/n)log(axn)p, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)p/x, x), x, log(axn)(p + S(1))/(n(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)p/xS(2), x), x, -(axn)(S(1)/n)(log(axn)/n)(-p)Gamma(p + S(1), log(ax*n)/n)log(axn)p/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(axn)p/xS(3), x), x, -S(2)(-p + S(-1))(axn)(S(2)/n)(log(axn)/n)(-p)Gamma(p + S(1), S(2)log(axn)/n)log(axn)p/xS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(c(bxn)p), x), x, -npx(m + S(1))/(m + S(1))S(2) + x*(m + S(1))log(c(bxn)p)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)log(c(bxn)p), x), x, -npxS(3)/S(9) + xS(3)log(c(bxn)p)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(bxn)p), x), x, -npxS(2)/S(4) + xS(2)log(c(bxn)p)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(bxn)p), x), x, -npx + xlog(c(bxn)p), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(bxn)p)/x, x), x, log(c(bxn)p)*S(2)/(S(2)np), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(bxn)p)/xS(2), x), x, -np/x - log(c(bxn)p)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(bxn)p)/x*S(3), x), x, -np/(S(4)xS(2)) - log(c(bxn)p)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(bxn)p)/xS(4), x), x, -np/(S(9)xS(3)) - log(c(bxn)p)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(c(bxn)p)S(2), x), x, S(2)n*S(2)p*S(2)x(m + S(1))/(m + S(1))S(3) - S(2)npx(m + S(1))log(c(bxn)p)/(m + S(1))S(2) + x(m + S(1))log(c(bxn)p)S(2)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c(bxn)p)*S(2), x), x, S(2)n*S(2)p*S(2)x*S(3)/S(27) - S(2)npx*S(3)log(c(bxn)p)/S(9) + x*S(3)log(c(bxn)p)*S(2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(bxn)p)S(2), x), x, nS(2)pS(2)x*S(2)/S(4) - npxS(2)log(c(bxn)p)/S(2) + x*S(2)log(c(bxn)p)*S(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(bxn)p)S(2), x), x, S(2)n*S(2)p*S(2)x - S(2)npxlog(c(bxn)p) + xlog(c(bxn)p)S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(bxn)p)S(2)/x, x), x, log(c(bxn)p)S(3)/(S(3)np), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(bxn)p)S(2)/xS(2), x), x, -S(2)n*S(2)p*S(2)/x - S(2)nplog(c(bxn)p)/x - log(c(bxn)p)*S(2)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(bxn)p)S(2)/xS(3), x), x, -nS(2)p*S(2)/(S(4)x*S(2)) - nplog(c(bxn)p)/(S(2)x*S(2)) - log(c(bxn)p)S(2)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(bxn)p)S(2)/xS(4), x), x, -S(2)n*S(2)p*S(2)/(S(27)x*S(3)) - S(2)nplog(c(bxn)p)/(S(9)xS(3)) - log(c(bxn)p)S(2)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/log(c(bxn)p), x), x, x*(m + S(1))(c(bxn)p)*(-(m + S(1))/(np))Ei((m + S(1))log(c(bxn)p)/(np))/(np), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*m/log(c(bxn)p)S(2), x), x, -x(m + S(1))/(nplog(c(bxn)p)) + x(m + S(1))(c(bxn)p)*(-(m + S(1))/(np))(m + S(1))Ei((m + S(1))log(c(bxn)p)/(np))/(n*S(2)pS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(c(bxn)p)q, x), x, x(m + S(1))(c(bxn)p)*(-(m + S(1))/(np))((-m + S(-1))log(c(bxn)p)/(np))(-q)Gamma(q + S(1), (-m + S(-1))log(c(bxn)p)/(np))log(c(bxn)p)q/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*mlog(cx), x), x, (a + bx)*(m + S(1))log(cx)/(b(m + S(1))) + (a + bx)(m + S(2))hyper((S(1), m + S(2)), (m + S(3),), S(1) + bx/a)/(ab(mS(2) + S(3)m + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(3)log(cx), x), x, -aS(4)log(x)/(S(4)b) - aS(3)x - S(3)aS(2)bxS(2)/S(4) - ab*S(2)xS(3)/S(3) - bS(3)xS(4)/S(16) + (a + bx)*S(4)log(cx)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)log(cx), x), x, -aS(3)log(x)/(S(3)b) - aS(2)x - abxS(2)/S(2) - bS(2)xS(3)/S(9) + (a + bx)*S(3)log(cx)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)log(cx), x), x, -ax - bxS(2)/S(4) + x(S(2)a + bx)log(cx)/S(2), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + bx)log(cx), x), x, -aS(2)log(x)/(S(2)b) - ax - bxS(2)/S(4) + (a + bx)*S(2)log(cx)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cx)/(a + bx), x), x, log((a + bx)/a)log(cx)/b + polylog(S(2), -bx/a)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cx)/(a + bx)*S(2), x), x, -log(cx)/(b(a + bx)) + log(x)/(ab) - log(a + bx)/(ab), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cx)/(a + bx)S(3), x), x, -log(cx)/(S(2)b(a + bx)S(2)) + S(1)/(S(2)ab(a + bx)) + log(x)/(S(2)a*S(2)b) - log(a + bx)/(S(2)a*S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cx)/(a + bx)*S(4), x), x, -log(cx)/(S(3)b(a + bx)S(3)) + S(1)/(S(6)ab(a + bx)S(2)) + S(1)/(S(3)a*S(2)b(a + bx)) + log(x)/(S(3)aS(3)b) - log(a + bx)/(S(3)a*S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)mlog(cxn), x), x, (a + bx)*(m + S(1))log(cxn)/(b(m + S(1))) + n(a + bx)*(m + S(2))hyper((S(1), m + S(2)), (m + S(3),), S(1) + bx/a)/(ab(mS(2) + S(3)m + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(3)log(cxn), x), x, -aS(4)nlog(x)/(S(4)b) - a*S(3)nx - S(3)a*S(2)bnx*S(2)/S(4) - ab*S(2)nxS(3)/S(3) - bS(3)nxS(4)/S(16) + (a + bx)*S(4)log(cxn)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)log(cxn), x), x, -aS(3)nlog(x)/(S(3)b) - a*S(2)nx - abnxS(2)/S(2) - bS(2)nx*S(3)/S(9) + (a + bx)*S(3)log(cxn)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)log(cxn), x), x, -aS(2)nlog(x)/(S(2)b) - anx - bnx*S(2)/S(4) + (a + bx)*S(2)log(cxn)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cxn)/(a + bx), x), x, npolylog(S(2), -bx/a)/b + log((a + bx)/a)log(cxn)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cx*n)/(a + bx)*S(2), x), x, -log(cx*n)/(b(a + bx)) + nlog(x)/(ab) - nlog(a + bx)/(ab), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cxn)/(a + bx)*S(3), x), x, -log(cx*n)/(S(2)b(a + bx)*S(2)) + n/(S(2)ab(a + bx)) + nlog(x)/(S(2)aS(2)b) - nlog(a + bx)/(S(2)aS(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cxn)/(a + bx)*S(4), x), x, -log(cx*n)/(S(3)b(a + bx)*S(3)) + n/(S(6)ab(a + bx)S(2)) + n/(S(3)a*S(2)b(a + bx)) + nlog(x)/(S(3)a*S(3)b) - nlog(a + bx)/(S(3)aS(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cxn)/(S(4)x + S(2))*S(2), x), x, nlog(x)/S(8) - nlog(S(2)x + S(1))/S(8) - log(cxn)/(S(8)(S(2)x + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax)/(-ax + S(1)), x), x, polylog(S(2), -ax + S(1))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x/a)/(a - x), x), x, polylog(S(2), (a - x)/a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(ax*S(2))/(-ax*S(2) + S(1)), x), x, polylog(S(2), -ax*S(2) + S(1))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(xS(2)/a)/(a - xS(2)), x), x, polylog(S(2), (a - xS(2))/a)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))log(axn)/(-ax*n + S(1)), x), x, polylog(S(2), -ax*n + S(1))/(an), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(n + S(-1))log(xn/a)/(a - xn), x), x, polylog(S(2), (a - x*n)/a)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a/x)/(ax - xS(2)), x), x, polylog(S(2), -a/x + S(1))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a/xS(2))/(ax - xS(3)), x), x, polylog(S(2), (-a + xS(2))/xS(2))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax(-n + S(1)))/(ax - x*n), x), x, -polylog(S(2), -ax*(-n + S(1)) + S(1))/(a(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(-ax(-m)(-c + S(1))/b + c)/(x(a + bxm)), x), x, polylog(S(2), x(-m)(a + bx*m)(-c + S(1))/b)/(am), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x(-m)(ac - a + bcxm)/b)/(x(a + bxm)), x), x, polylog(S(2), x(-m)(a + bxm)(-c + S(1))/b)/(am), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + x*(-m)(acd - d)/(ce)))/(x(d + exm)), x), x, polylog(S(2), x(-m)(d + exm)(-ac + S(1))/e)/(dm), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x*(-m)(acd + acexm - d)/e)/(x(d + exm)), x), x, polylog(S(2), x(-m)(d + exm)(-ac + S(1))/e)/(dm), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(2)a/(a + bx))/(aS(2) - bS(2)xS(2)), x), x, polylog(S(2), (-a + bx)/(a + bx))/(S(2)ab), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(2)a/(a + bx))/((a - bx)(a + bx)), x), x, polylog(S(2), (-a + bx)/(a + bx))/(S(2)ab), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((a(-c + S(1)) + bx(c + S(1)))/(a + bx))/(aS(2) - bS(2)xS(2)), x), x, polylog(S(2), c(a - bx)/(a + bx))/(S(2)ab), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((a(-c + S(1)) + bx(c + S(1)))/(a + bx))/((a - bx)(a + bx)), x), x, polylog(S(2), c(a - bx)/(a + bx))/(S(2)ab), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(-c(a - bx)/(a + bx) + S(1))/(aS(2) - bS(2)x*S(2)), x), x, polylog(S(2), c(a - bx)/(a + bx))/(S(2)ab), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(-c(a - bx)/(a + bx) + S(1))/((a - bx)(a + bx)), x), x, polylog(S(2), c(a - bx)/(a + bx))/(S(2)ab), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(cxn))/(d + ex*S(2)), x), x, -Ibnpolylog(S(2), -Isqrt(e)x/sqrt(d))/(S(2)sqrt(d)sqrt(e)) + Ibnpolylog(S(2), Isqrt(e)x/sqrt(d))/(S(2)sqrt(d)sqrt(e)) + (a + blog(cxn))atan(sqrt(e)x/sqrt(d))/(sqrt(d)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(cx*n))/(d + ex + fxS(2)), x), x, bnpolylog(S(2), -S(2)fx/(e - sqrt(-S(4)df + eS(2))))/sqrt(-S(4)df + eS(2)) - bnpolylog(S(2), -S(2)fx/(e + sqrt(-S(4)df + eS(2))))/sqrt(-S(4)df + eS(2)) + (a + blog(cxn))log((e + S(2)fx - sqrt(-S(4)df + e*S(2)))/(e - sqrt(-S(4)df + eS(2))))/sqrt(-S(4)df + eS(2)) - (a + blog(cxn))log((e + S(2)fx + sqrt(-S(4)df + e*S(2)))/(e + sqrt(-S(4)df + eS(2))))/sqrt(-S(4)df + eS(2)), expand=True, _diff=True, _numerical=True) # same result as in mathematica but fails assert rubi_test(rubi_integrate((d + ex)*mlog(cx)/x, x), x, (d + ex)*m(d/(ex) + S(1))(-m)log(cx)hyper((-m, -m), (-m + S(1),), -d/(ex))/m - (d + ex)*m(d/(ex) + S(1))(-m)hyper((-m, -m, -m), (-m + S(1), -m + S(1)), -d/(ex))/mS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(cxn))S(3), x), x, S(6)abS(2)n*S(2)x - S(6)bS(3)n*S(3)x + S(6)bS(3)n*S(2)xlog(cx*n) - S(3)bnx(a + blog(cxn))S(2) + x(a + blog(cxn))S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(cxn))S(2), x), x, -S(2)abnx + S(2)bS(2)n*S(2)x - S(2)bS(2)nxlog(cxn) + x(a + blog(cxn))S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a + blog(cx*n), x), x, ax - bnx + bxlog(cxn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + blog(cxn)), x), x, x(cxn)(-S(1)/n)exp(-a/(bn))Ei((a + blog(cx*n))/(bn))/(bn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(cxn))(S(-2)), x), x, -x/(bn(a + blog(cxn))) + x(cxn)(-S(1)/n)exp(-a/(bn))Ei((a + blog(cx*n))/(bn))/(b*S(2)n*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(cxn))(S(-3)), x), x, -x/(S(2)bn(a + blog(cxn))S(2)) - x/(S(2)bS(2)n*S(2)(a + blog(cx*n))) + x(cxn)(-S(1)/n)exp(-a/(bn))Ei((a + blog(cx*n))/(bn))/(S(2)bS(3)n*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(cxn))m, x), x, x(cxn)(-S(1)/n)((-a - blog(cx*n))/(bn))*(-m)(a + blog(cxn))mGamma(m + S(1), (-a - blog(cxn))/(bn))exp(-a/(bn)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*m/(a + blog(cxn)), x), x, x(m + S(1))(cxn)(-(m + S(1))/n)exp(-a(m + S(1))/(bn))Ei((a + blog(cxn))(m + S(1))/(bn))/(bn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*m/(a + blog(cxn))S(2), x), x, -x(m + S(1))/(bn(a + blog(cxn))) + x(m + S(1))(cxn)(-(m + S(1))/n)(m + S(1))exp(-a(m + S(1))/(bn))Ei((a + blog(cx*n))(m + S(1))/(bn))/(bS(2)nS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(a + blog(cxn))p, x), x, x(m + S(1))(cxn)(-(m + S(1))/n)((a + blog(cx*n))(-m + S(-1))/(bn))(-p)(a + blog(cxn))pGamma(p + S(1), (a + blog(cxn))(-m + S(-1))/(bn))exp(-a(m + S(1))/(bn))/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(n + S(-1))log(-bxn/a)/(a + bx*n), x), x, -polylog(S(2), (a + bx*n)/a)/(bn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*mlog(c(a + bxS(2))p), x), x, x*(m + S(1))log(c(a + bxS(2))p)/(m + S(1)) - S(2)bpx(m + S(3))hyper((S(1), m/S(2) + S(3)/2), (m/S(2) + S(5)/2,), -bxS(2)/a)/(a(m*S(2) + S(4)m + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)log(c(a + bxS(2))p), x), x, S(2)a(S(5)/2)patan(sqrt(b)x/sqrt(a))/(S(5)b(S(5)/2)) - S(2)a*S(2)px/(S(5)b*S(2)) + S(2)apx*S(3)/(S(15)b) - S(2)pxS(5)/S(25) + xS(5)log(c(a + bxS(2))p)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(c(a + bxS(2))p), x), x, -a*S(2)plog(a + bx*S(2))/(S(4)b*S(2)) + apxS(2)/(S(4)b) - pxS(4)/S(8) + xS(4)log(c(a + bxS(2))p)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)log(c(a + bxS(2))p), x), x, -S(2)a(S(3)/2)patan(sqrt(b)x/sqrt(a))/(S(3)b(S(3)/2)) + S(2)apx/(S(3)b) - S(2)pxS(3)/S(9) + xS(3)log(c(a + bxS(2))p)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + bxS(2))p), x), x, -px*S(2)/S(2) + (a/S(2) + bx*S(2)/S(2))log(c(a + bxS(2))p)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p), x), x, S(2)sqrt(a)patan(sqrt(b)x/sqrt(a))/sqrt(b) - S(2)px + xlog(c(a + bxS(2))p), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/x, x), x, ppolylog(S(2), (a + bxS(2))/a)/S(2) + log(c(a + bxS(2))p)log(-bxS(2)/a)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/xS(2), x), x, -log(c(a + bxS(2))p)/x + S(2)sqrt(b)patan(sqrt(b)x/sqrt(a))/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/xS(3), x), x, bplog(x)/a - (a/S(2) + bx*S(2)/S(2))log(c(a + bxS(2))p)/(axS(2)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(log(c(a + bxS(2))p)/xS(3), x), x, -log(c(a + bxS(2))p)/(S(2)x*S(2)) + bplog(x)/a - bplog(a + bx*S(2))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/x*S(4), x), x, -log(c(a + bxS(2))p)/(S(3)x*S(3)) - S(2)bp/(S(3)ax) - S(2)b*(S(3)/2)patan(sqrt(b)x/sqrt(a))/(S(3)a(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/xS(5), x), x, -log(c(a + bxS(2))p)/(S(4)x*S(4)) - bp/(S(4)axS(2)) - bS(2)plog(x)/(S(2)aS(2)) + bS(2)plog(a + bx*S(2))/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/xS(6), x), x, -log(c(a + bxS(2))p)/(S(5)x*S(5)) - S(2)bp/(S(15)axS(3)) + S(2)b*S(2)p/(S(5)aS(2)x) + S(2)b(S(5)/2)patan(sqrt(b)x/sqrt(a))/(S(5)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/xS(7), x), x, -log(c(a + bxS(2))p)/(S(6)x*S(6)) - bp/(S(12)axS(4)) + bS(2)p/(S(6)a*S(2)xS(2)) + bS(3)plog(x)/(S(3)aS(3)) - bS(3)plog(a + bx*S(2))/(S(6)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(c(a + bxS(3))p), x), x, x(m + S(1))log(c(a + bxS(3))p)/(m + S(1)) - S(3)bpx(m + S(4))hyper((S(1), m/S(3) + S(4)/3), (m/S(3) + S(7)/3,), -bxS(3)/a)/(a(m*S(2) + S(5)m + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)log(c(a + bxS(3))p), x), x, -a*S(2)plog(a + bx*S(3))/(S(6)b*S(2)) + apxS(3)/(S(6)b) - pxS(6)/S(12) + xS(6)log(c(a + bxS(3))p)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)log(c(a + bxS(3))p), x), x, a*(S(5)/3)plog(a(S(1)/3) + b(S(1)/3)x)/(S(5)b(S(5)/3)) - a(S(5)/3)plog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(10)b*(S(5)/3)) + sqrt(S(3))a*(S(5)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(5)b*(S(5)/3)) + S(3)apx*S(2)/(S(10)b) - S(3)pxS(5)/S(25) + xS(5)log(c(a + bxS(3))p)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(c(a + bxS(3))p), x), x, -a*(S(4)/3)plog(a(S(1)/3) + b(S(1)/3)x)/(S(4)b(S(4)/3)) + a(S(4)/3)plog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(8)b*(S(4)/3)) + sqrt(S(3))a*(S(4)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(4)b*(S(4)/3)) + S(3)apx/(S(4)b) - S(3)pxS(4)/S(16) + xS(4)log(c(a + bxS(3))p)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)log(c(a + bxS(3))p), x), x, -pxS(3)/S(3) + (a/S(3) + bx*S(3)/S(3))log(c(a + bxS(3))p)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + bxS(3))p), x), x, -a(S(2)/3)plog(a(S(1)/3) + b(S(1)/3)x)/(S(2)b(S(2)/3)) + a(S(2)/3)plog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(4)b*(S(2)/3)) - sqrt(S(3))a*(S(2)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(2)b*(S(2)/3)) - S(3)pxS(2)/S(4) + xS(2)log(c(a + bxS(3))p)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p), x), x, a*(S(1)/3)plog(a(S(1)/3) + b(S(1)/3)x)/b(S(1)/3) - a(S(1)/3)plog(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(2)b*(S(1)/3)) - sqrt(S(3))a*(S(1)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/b(S(1)/3) - S(3)px + xlog(c(a + bxS(3))p), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/x, x), x, ppolylog(S(2), (a + bx*S(3))/a)/S(3) + log(c(a + bxS(3))p)log(-bxS(3)/a)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/xS(2), x), x, -log(c(a + bxS(3))p)/x - b(S(1)/3)plog(a(S(1)/3) + b(S(1)/3)x)/a(S(1)/3) + b(S(1)/3)plog(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(2)a*(S(1)/3)) - sqrt(S(3))b*(S(1)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/a(S(1)/3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/xS(3), x), x, -log(c(a + bxS(3))p)/(S(2)xS(2)) + b(S(2)/3)plog(a(S(1)/3) + b(S(1)/3)x)/(S(2)a(S(2)/3)) - b(S(2)/3)plog(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(4)a*(S(2)/3)) - sqrt(S(3))b*(S(2)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(2)a*(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/xS(4), x), x, -log(c(a + bxS(3))p)/(S(3)x*S(3)) + bplog(x)/a - bplog(a + bx*S(3))/(S(3)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/x*S(5), x), x, -log(c(a + bxS(3))p)/(S(4)x*S(4)) - S(3)bp/(S(4)ax) + b(S(4)/3)plog(a(S(1)/3) + b(S(1)/3)x)/(S(4)a(S(4)/3)) - b(S(4)/3)plog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(8)a*(S(4)/3)) + sqrt(S(3))b*(S(4)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(4)a*(S(4)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/xS(6), x), x, -log(c(a + bxS(3))p)/(S(5)x*S(5)) - S(3)bp/(S(10)axS(2)) - b(S(5)/3)plog(a(S(1)/3) + b(S(1)/3)x)/(S(5)a(S(5)/3)) + b(S(5)/3)plog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(10)a*(S(5)/3)) + sqrt(S(3))b*(S(5)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(5)a*(S(5)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/xS(7), x), x, -log(c(a + bxS(3))p)/(S(6)x*S(6)) - bp/(S(6)axS(3)) - bS(2)plog(x)/(S(2)aS(2)) + bS(2)plog(a + bx*S(3))/(S(6)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(c(a + bsqrt(x))p), x), x, x(m + S(1))log(c(a + bsqrt(x))*p)/(m + S(1)) - bpx(m + S(3)/2)hyper((S(1), S(2)m + S(3)), (S(2)m + S(4),), -bsqrt(x)/a)/(a(S(2)mS(2) + S(5)m + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)log(c(a + bsqrt(x))p), x), x, -aS(8)plog(a + bsqrt(x))/(S(4)bS(8)) + aS(7)psqrt(x)/(S(4)bS(7)) - aS(6)px/(S(8)bS(6)) + aS(5)px*(S(3)/2)/(S(12)bS(5)) - aS(4)px*S(2)/(S(16)bS(4)) + aS(3)px*(S(5)/2)/(S(20)bS(3)) - aS(2)px*S(3)/(S(24)b*S(2)) + apx(S(7)/2)/(S(28)b) - pxS(4)/S(32) + xS(4)log(c(a + bsqrt(x))p)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c(a + bsqrt(x))p), x), x, -aS(6)plog(a + bsqrt(x))/(S(3)bS(6)) + aS(5)psqrt(x)/(S(3)bS(5)) - aS(4)px/(S(6)bS(4)) + aS(3)px(S(3)/2)/(S(9)bS(3)) - aS(2)px*S(2)/(S(12)b*S(2)) + apx(S(5)/2)/(S(15)b) - pxS(3)/S(18) + xS(3)log(c(a + bsqrt(x))*p)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + bsqrt(x))p), x), x, -aS(4)plog(a + bsqrt(x))/(S(2)bS(4)) + aS(3)psqrt(x)/(S(2)bS(3)) - aS(2)px/(S(4)b*S(2)) + apx(S(3)/2)/(S(6)b) - pxS(2)/S(8) + xS(2)log(c(a + bsqrt(x))*p)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bsqrt(x))p), x), x, -aS(2)plog(a + bsqrt(x))/b*S(2) + apsqrt(x)/b - px/S(2) + xlog(c(a + bsqrt(x))p), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bsqrt(x))p)/x, x), x, S(2)ppolylog(S(2), (a + bsqrt(x))/a) + S(2)log(c(a + bsqrt(x))p)log(-bsqrt(x)/a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bsqrt(x))p)/xS(2), x), x, -log(c(a + bsqrt(x))p)/x - bp/(asqrt(x)) - bS(2)plog(x)/(S(2)aS(2)) + bS(2)plog(a + bsqrt(x))/aS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bsqrt(x))p)/xS(3), x), x, -log(c(a + bsqrt(x))p)/(S(2)x*S(2)) - bp/(S(6)ax(S(3)/2)) + bS(2)p/(S(4)a*S(2)x) - b*S(3)p/(S(2)aS(3)sqrt(x)) - b*S(4)plog(x)/(S(4)aS(4)) + bS(4)plog(a + bsqrt(x))/(S(2)a*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bsqrt(x))p)/xS(4), x), x, -log(c(a + bsqrt(x))p)/(S(3)x*S(3)) - bp/(S(15)ax(S(5)/2)) + bS(2)p/(S(12)a*S(2)xS(2)) - bS(3)p/(S(9)a*S(3)x(S(3)/2)) + bS(4)p/(S(6)a*S(4)x) - b*S(5)p/(S(3)aS(5)sqrt(x)) - b*S(6)plog(x)/(S(6)aS(6)) + bS(6)plog(a + bsqrt(x))/(S(3)a*S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bsqrt(x))/sqrt(x), x), x, -S(2)sqrt(x) + S(2)(a + bsqrt(x))log(a + bsqrt(x))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(c(a + b/x)p), x), x, px*(m + S(1))hyper((S(1), m + S(1)), (m + S(2),), -ax/b)/(m + S(1))S(2) + x(m + S(1))log(c(a + b/x)p)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)log(c(a + b/x)p), x), x, xS(5)log(c(a + b/x)p)/S(5) + bpxS(4)/(S(20)a) - b*S(2)pxS(3)/(S(15)aS(2)) + bS(3)px*S(2)/(S(10)aS(3)) - bS(4)px/(S(5)aS(4)) + bS(5)plog(ax + b)/(S(5)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(c(a + b/x)p), x), x, xS(4)log(c(a + b/x)p)/S(4) + bpxS(3)/(S(12)a) - b*S(2)pxS(2)/(S(8)aS(2)) + bS(3)px/(S(4)aS(3)) - bS(4)plog(ax + b)/(S(4)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c(a + b/x)p), x), x, xS(3)log(c(a + b/x)p)/S(3) + bpxS(2)/(S(6)a) - b*S(2)px/(S(3)aS(2)) + bS(3)plog(ax + b)/(S(3)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + b/x)p), x), x, xS(2)log(c(a + b/x)p)/S(2) + bpx/(S(2)a) - b*S(2)plog(ax + b)/(S(2)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/x)*p), x), x, xlog(c(a + b/x)p) + bplog(ax + b)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/x)p)/x, x), x, -ppolylog(S(2), (a + b/x)/a) - log(c(a + b/x)p)log(-b/(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/x)p)/xS(2), x), x, p/x - (a + b/x)log(c(a + b/x)*p)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/x)p)/xS(3), x), x, -a*S(2)plog(x)/(S(2)bS(2)) + aS(2)plog(ax + b)/(S(2)b*S(2)) - ap/(S(2)bx) + p/(S(4)xS(2)) - log(c(a + b/x)*p)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/x)p)/xS(4), x), x, a*S(3)plog(x)/(S(3)bS(3)) - aS(3)plog(ax + b)/(S(3)bS(3)) + aS(2)p/(S(3)b*S(2)x) - ap/(S(6)bxS(2)) + p/(S(9)x*S(3)) - log(c(a + b/x)*p)/(S(3)x*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/x)p)/xS(5), x), x, -a*S(4)plog(x)/(S(4)bS(4)) + aS(4)plog(ax + b)/(S(4)bS(4)) - aS(3)p/(S(4)b*S(3)x) + a*S(2)p/(S(8)bS(2)x*S(2)) - ap/(S(12)bx*S(3)) + p/(S(16)x*S(4)) - log(c(a + b/x)*p)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(b/x + S(1))/x, x), x, polylog(S(2), -b/x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(c(a + b/xS(2))p), x), x, S(2)px*(m + S(1))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -axS(2)/b)/(m + S(1))S(2) + x(m + S(1))log(c(a + b/xS(2))p)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)log(c(a + b/xS(2))p), x), x, xS(5)log(c(a + b/xS(2))p)/S(5) + S(2)bpx*S(3)/(S(15)a) - S(2)bS(2)px/(S(5)a*S(2)) + S(2)b*(S(5)/2)patan(sqrt(a)x/sqrt(b))/(S(5)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(c(a + b/xS(2))p), x), x, xS(4)log(c(a + b/xS(2))p)/S(4) + bpxS(2)/(S(4)a) - b*S(2)plog(ax*S(2) + b)/(S(4)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c(a + b/xS(2))p), x), x, x*S(3)log(c(a + b/xS(2))p)/S(3) + S(2)bpx/(S(3)a) - S(2)b*(S(3)/2)patan(sqrt(a)x/sqrt(b))/(S(3)a(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + b/xS(2))p), x), x, xS(2)log(c(a + b/xS(2))p)/S(2) + bplog(ax*S(2) + b)/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(2))p), x), x, xlog(c(a + b/xS(2))p) + S(2)sqrt(b)patan(sqrt(a)x/sqrt(b))/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(2))p)/x, x), x, -ppolylog(S(2), (a + b/xS(2))/a)/S(2) - log(c(a + b/xS(2))p)log(-b/(ax*S(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(2))p)/x*S(2), x), x, S(2)sqrt(a)patan(sqrt(a)x/sqrt(b))/sqrt(b) + S(2)p/x - log(c(a + b/xS(2))p)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(2))p)/x*S(3), x), x, p/(S(2)x*S(2)) - (a/S(2) + b/(S(2)x*S(2)))log(c(a + b/xS(2))p)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(2))p)/x*S(4), x), x, -S(2)a*(S(3)/2)patan(sqrt(a)x/sqrt(b))/(S(3)b(S(3)/2)) - S(2)ap/(S(3)bx) + S(2)p/(S(9)xS(3)) - log(c(a + b/xS(2))p)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(c(a + bxn)p), x), x, x*(m + S(1))log(c(a + bxn)p)/(m + S(1)) - bnpx(m + n + S(1))hyper((S(1), (m + n + S(1))/n), ((m + S(2)n + S(1))/n,), -bx*n/a)/(a(m + S(1))(m + n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c(a + bxn)p), x), x, x*S(3)log(c(a + bxn)p)/S(3) - bnpx(n + S(3))hyper((S(1), (n + S(3))/n), (S(2) + S(3)/n,), -bxn/a)/(S(3)a(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + bxn)p), x), x, x*S(2)log(c(a + bxn)p)/S(2) - bnpx(n + S(2))hyper((S(1), (n + S(2))/n), (S(2) + S(2)/n,), -bxn/a)/(S(2)a(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxn)p), x), x, xlog(c(a + bxn)p) - bnpx(n + S(1))hyper((S(1), S(1) + S(1)/n), (S(2) + S(1)/n,), -bxn/a)/(a(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxn)p)/x, x), x, ppolylog(S(2), (a + bx*n)/a)/n + log(c(a + bxn)p)log(-bxn/a)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxn)p)/xS(2), x), x, -log(c(a + bxn)p)/x - bnpx*(n + S(-1))hyper((S(1), (n + S(-1))/n), (S(2) - S(1)/n,), -bxn/a)/(a(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxn)p)/x*S(3), x), x, -log(c(a + bxn)p)/(S(2)x*S(2)) - bnpx*(n + S(-2))hyper((S(1), (n + S(-2))/n), (S(2) - S(2)/n,), -bxn/a)/(S(2)a(-n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxn)p)/xS(4), x), x, -log(c(a + bxn)p)/(S(3)x*S(3)) - bnpx*(n + S(-3))hyper((S(1), (n + S(-3))/n), (S(2) - S(3)/n,), -bxn/a)/(S(3)a(-n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*mlog(c(a + bx)*p), x), x, bp(d + ex)*(m + S(2))hyper((S(1), m + S(2)), (m + S(3),), b(d + ex)/(-ae + bd))/(e(m + S(1))(m + S(2))(-ae + bd)) + (d + ex)*(m + S(1))log(c(a + bx)*p)/(e(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)S(3)log(c(a + bx)*p), x), x, -p(d + ex)S(4)/(S(16)e) + (d + ex)S(4)log(c(a + bx)*p)/(S(4)e) - p(d + ex)*S(3)(-ae/S(12) + bd/S(12))/(be) - p(d + ex)S(2)(-ae + bd)*S(2)/(S(8)b*S(2)e) - px(-ae + bd)*S(3)/(S(4)b*S(3)) - p(-ae + bd)*S(4)log(a + bx)/(S(4)b*S(4)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)S(2)log(c(a + bx)*p), x), x, -p(d + ex)S(3)/(S(9)e) + (d + ex)S(3)log(c(a + bx)*p)/(S(3)e) - p(d + ex)*S(2)(-ae/S(6) + bd/S(6))/(be) - px(-ae + bd)S(2)/(S(3)b*S(2)) - p(-ae + bd)*S(3)log(a + bx)/(S(3)b*S(3)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)log(c(a + bx)*p), x), x, -p(d + ex)S(2)/(S(4)e) + (d + ex)S(2)log(c(a + bx)*p)/(S(2)e) + px(ae/S(2) - bd/S(2))/b - p(-ae + bd)S(2)log(a + bx)/(S(2)b*S(2)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)*p), x), x, -px + (a + bx)log(c(a + bx)*p)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)p)/(d + ex), x), x, ppolylog(S(2), -e(a + bx)/(-ae + bd))/e + log(c(a + bx)p)log(b(d + ex)/(-ae + bd))/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)*p)/(d + ex)*S(2), x), x, bplog(a + bx)/(e(-ae + bd)) - bplog(d + ex)/(e(-ae + bd)) - log(c(a + bx)p)/(e(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)p)/(d + ex)S(3), x), x, bS(2)plog(a + bx)/(S(2)e(-ae + bd)S(2)) - bS(2)plog(d + ex)/(S(2)e(-ae + bd)*S(2)) + bp/(S(2)e(d + ex)(-ae + bd)) - log(c(a + bx)*p)/(S(2)e(d + ex)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)p)/(d + ex)S(4), x), x, bS(3)plog(a + bx)/(S(3)e(-ae + bd)S(3)) - bS(3)plog(d + ex)/(S(3)e(-ae + bd)S(3)) + bS(2)p/(S(3)e(d + ex)(-ae + bd)S(2)) + bp/(S(6)e(d + ex)S(2)(-ae + bd)) - log(c(a + bx)*p)/(S(3)e(d + ex)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*mlog(c(a + bxS(2))p), x), x, sqrt(b)p(d + ex)(m + S(2))hyper((S(1), m + S(2)), (m + S(3),), sqrt(b)(d + ex)/(sqrt(b)d + esqrt(-a)))/(e(m + S(1))(m + S(2))(sqrt(b)d + esqrt(-a))) + sqrt(b)p(d + ex)*(m + S(2))hyper((S(1), m + S(2)), (m + S(3),), sqrt(b)(d + ex)/(sqrt(b)d - esqrt(-a)))/(e(m + S(1))(m + S(2))(sqrt(b)d - esqrt(-a))) + (d + ex)*(m + S(1))log(c(a + bxS(2))p)/(e(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*S(3)log(c(a + bxS(2))p), x), x, S(2)sqrt(a)dp(-aeS(2) + bd*S(2))atan(sqrt(b)x/sqrt(a))/b(S(3)/2) - S(2)deS(2)pxS(3)/S(3) - eS(3)pxS(4)/S(8) + (d + ex)*S(4)log(c(a + bxS(2))p)/(S(4)e) - S(2)dpx(-ae*S(2) + bd*S(2))/b - epxS(2)(-aeS(2) + S(6)bdS(2))/(S(4)b) - p(aS(2)e*S(4)/S(4) - S(3)abd*S(2)eS(2)/S(2) + bS(2)dS(4)/S(4))log(a + bxS(2))/(bS(2)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)S(2)log(c(a + bxS(2))p), x), x, sqrt(a)p(-S(2)ae*S(2)/S(3) + S(2)bdS(2))atan(sqrt(b)x/sqrt(a))/b(S(3)/2) - depx*S(2) - S(2)e*S(2)pxS(3)/S(9) + (d + ex)*S(3)log(c(a + bxS(2))p)/(S(3)e) - dp(-S(3)aeS(2) + bd*S(2))log(a + bxS(2))/(S(3)be) + px(S(2)aeS(2)/S(3) - S(2)bdS(2))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)log(c(a + bxS(2))p), x), x, S(2)sqrt(a)dpatan(sqrt(b)x/sqrt(a))/sqrt(b) - S(2)dpx - epxS(2)/S(2) + (d + ex)*S(2)log(c(a + bxS(2))p)/(S(2)e) - p(-aeS(2)/S(2) + bd*S(2)/S(2))log(a + bxS(2))/(be), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p), x), x, S(2)sqrt(a)patan(sqrt(b)x/sqrt(a))/sqrt(b) - S(2)px + xlog(c(a + bxS(2))p), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/(d + ex), x), x, -plog(-e(sqrt(b)x + sqrt(-a))/(sqrt(b)d - esqrt(-a)))log(d + ex)/e - plog(e(-sqrt(b)x + sqrt(-a))/(sqrt(b)d + esqrt(-a)))log(d + ex)/e - ppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d - esqrt(-a)))/e - ppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d + esqrt(-a)))/e + log(c(a + bxS(2))p)log(d + ex)/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/(d + ex)S(2), x), x, S(2)sqrt(a)sqrt(b)patan(sqrt(b)x/sqrt(a))/(aeS(2) + bd*S(2)) + bdplog(a + bxS(2))/(e(aeS(2) + bd*S(2))) - S(2)bdplog(d + ex)/(e(ae*S(2) + bd*S(2))) - log(c(a + bxS(2))p)/(e(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/(d + ex)*S(3), x), x, S(2)sqrt(a)b(S(3)/2)dpatan(sqrt(b)x/sqrt(a))/(ae*S(2) + bdS(2))S(2) + bdp/(e(d + ex)(ae*S(2) + bd*S(2))) + bp(-ae*S(2) + bd*S(2))log(a + bxS(2))/(S(2)e(ae*S(2) + bdS(2))S(2)) - bp(-aeS(2) + bd*S(2))log(d + ex)/(e(aeS(2) + bdS(2))S(2)) - log(c(a + bxS(2))p)/(S(2)e(d + ex)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*mlog(c(a + bxS(3))p), x), x, b*(S(1)/3)p(d + ex)*(m + S(2))hyper((S(1), m + S(2)), (m + S(3),), b*(S(1)/3)(d + ex)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))/(e(m + S(1))(m + S(2))(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d)) + b*(S(1)/3)p(d + ex)*(m + S(2))hyper((S(1), m + S(2)), (m + S(3),), b*(S(1)/3)(d + ex)/((S(-1))(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))/(e(m + S(1))(m + S(2))((S(-1))(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d)) + b*(S(1)/3)p(d + ex)*(m + S(2))hyper((S(1), m + S(2)), (m + S(3),), b*(S(1)/3)(d + ex)/(-a(S(1)/3)e + b*(S(1)/3)d))/(e(m + S(1))(m + S(2))(-a(S(1)/3)e + b*(S(1)/3)d)) + (d + ex)(m + S(1))log(c(a + bxS(3))p)/(e(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*S(3)log(c(a + bxS(3))p), x), x, a*(S(1)/3)p(-S(6)a*(S(1)/3)b*(S(2)/3)d*S(2)e - aeS(3) + S(4)bdS(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(4)b(S(4)/3)) - a(S(1)/3)p(-S(6)a(S(1)/3)b*(S(2)/3)d*S(2)e - aeS(3) + S(4)bdS(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(8)b*(S(4)/3)) - sqrt(S(3))a*(S(1)/3)p(S(6)a*(S(1)/3)b*(S(2)/3)d*S(2)e - aeS(3) + S(4)bdS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(4)b*(S(4)/3)) - S(9)d*S(2)epx*S(2)/S(4) - de*S(2)pxS(3) - S(3)e*S(3)pxS(4)/S(16) + (d + ex)*S(4)log(c(a + bxS(3))p)/(S(4)e) - dp(-S(4)aeS(3) + bd*S(3))log(a + bxS(3))/(S(4)be) + px(S(3)aeS(3)/S(4) - S(3)bdS(3))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*S(2)log(c(a + bxS(3))p), x), x, a*(S(1)/3)dp(-a*(S(1)/3)e + b*(S(1)/3)d)log(a(S(1)/3) + b(S(1)/3)x)/b(S(2)/3) - a(S(1)/3)dp(-a(S(1)/3)e + b*(S(1)/3)d)log(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(2)b*(S(2)/3)) - sqrt(S(3))a*(S(1)/3)dp(a*(S(1)/3)e + b*(S(1)/3)d)atan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/b(S(2)/3) - S(3)d*S(2)px - S(3)depxS(2)/S(2) - eS(2)pxS(3)/S(3) + (d + ex)*S(3)log(c(a + bxS(3))p)/(S(3)e) - p(-aeS(3)/S(3) + bd*S(3)/S(3))log(a + bxS(3))/(be), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)log(c(a + bxS(3))p), x), x, a*(S(1)/3)p(-a(S(1)/3)e + S(2)b(S(1)/3)d)log(a(S(1)/3) + b(S(1)/3)x)/(S(2)b(S(2)/3)) - a(S(1)/3)p(-a(S(1)/3)e + S(2)b(S(1)/3)d)log(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(4)b*(S(2)/3)) - sqrt(S(3))a*(S(1)/3)p(a(S(1)/3)e + S(2)b(S(1)/3)d)atan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(2)b(S(2)/3)) - dS(2)plog(a + bxS(3))/(S(2)e) - S(3)dpx - S(3)epx*S(2)/S(4) + (d + ex)*S(2)log(c(a + bxS(3))p)/(S(2)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p), x), x, a(S(1)/3)plog(a(S(1)/3) + b(S(1)/3)x)/b(S(1)/3) - a(S(1)/3)plog(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(2)b*(S(1)/3)) - sqrt(S(3))a*(S(1)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/b(S(1)/3) - S(3)px + xlog(c(a + bxS(3))p), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/(d + ex), x), x, -plog(-e(a(S(1)/3) + b(S(1)/3)x)/(-a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/e - plog(-e((S(-1))*(S(2)/3)a(S(1)/3) + b(S(1)/3)x)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/e - plog((S(-1))(S(1)/3)e(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/((S(-1))*(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/e - ppolylog(S(2), b(S(1)/3)(d + ex)/(-a(S(1)/3)e + b*(S(1)/3)d))/e - ppolylog(S(2), b(S(1)/3)(d + ex)/((S(-1))(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))/e - ppolylog(S(2), b(S(1)/3)(d + ex)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))/e + log(c(a + bxS(3))p)log(d + ex)/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/(d + ex)S(2), x), x, a(S(1)/3)b*(S(1)/3)p(a(S(1)/3)e + b*(S(1)/3)d)log(a(S(1)/3) + b(S(1)/3)x)/(-aeS(3) + bdS(3)) - a(S(1)/3)b(S(1)/3)p(a(S(1)/3)e + b*(S(1)/3)d)log(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(2)(-aeS(3) + bd*S(3))) - sqrt(S(3))a*(S(1)/3)b*(S(1)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(a(S(2)/3)eS(2) + a(S(1)/3)b(S(1)/3)de + b(S(2)/3)d*S(2)) + bd*S(2)plog(a + bx*S(3))/(e(-aeS(3) + bd*S(3))) - S(3)bdS(2)plog(d + ex)/(e(-ae*S(3) + bd*S(3))) - log(c(a + bxS(3))p)/(e(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/(d + ex)*S(3), x), x, -sqrt(S(3))a*(S(1)/3)b*(S(2)/3)p(-S(3)a*(S(1)/3)b*(S(2)/3)d*S(2)e + aeS(3) + S(2)bdS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(2)(-aeS(3) + bdS(3))S(2)) + a*(S(1)/3)b*(S(2)/3)p(S(3)a*(S(1)/3)b*(S(2)/3)d*S(2)e + aeS(3) + S(2)bdS(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(2)(-aeS(3) + bdS(3))S(2)) - a*(S(1)/3)b*(S(2)/3)p(S(3)a*(S(1)/3)b*(S(2)/3)d*S(2)e + aeS(3) + S(2)bdS(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(4)(-aeS(3) + bdS(3))S(2)) + S(3)bd*S(2)p/(S(2)e(d + ex)(-aeS(3) + bd*S(3))) + bdp(S(2)ae*S(3) + bd*S(3))log(a + bxS(3))/(S(2)e(-ae*S(3) + bdS(3))S(2)) - S(3)bdp(S(2)ae*S(3) + bd*S(3))log(d + ex)/(S(2)e(-ae*S(3) + bdS(3))S(2)) - log(c(a + bxS(3))p)/(S(2)e(d + ex)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + b/x)/(c + dx), x), x, log(-dx/c)log(c + dx)/d - log(-d(ax + b)/(ac - bd))log(c + dx)/d + log(a + b/x)log(c + dx)/d + polylog(S(2), (c + dx)/c)/d - polylog(S(2), a(c + dx)/(ac - bd))/d, expand=True, _diff=True, _numerical=True) # recursion sympy and mathematica assert rubi_test(rubi_integrate(log(a + bxn)/(c + dx), x), x, -bnIntegral(x*(n + S(-1))log(c + dx)/(a + bx*n), x)/d + log(a + bx*n)log(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax)/(c + dx), x), x, log(ax)log((c + dx)/c)/d + polylog(S(2), -dx/c)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a/x)/(c + dx), x), x, log(a/x)log((c + dx)/c)/d - polylog(S(2), -dx/c)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(ax*n)/(c + dx), x), x, npolylog(S(2), -dx/c)/d + log(axn)log((c + dx)/c)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(xn)/(a + bx), x), x, npolylog(S(2), -bx/a)/b + log(x*n)log((a + bx)/a)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(c(a + bx)*p)/(d + ex), x), x, a*S(3)plog(a + bx)/(S(3)bS(3)e) + a*S(2)dplog(a + bx)/(S(2)b*S(2)eS(2)) - aS(2)px/(S(3)bS(2)e) - adpx/(S(2)beS(2)) + apxS(2)/(S(6)be) - dS(3)ppolylog(S(2), -e(a + bx)/(-ae + bd))/eS(4) - dS(3)log(c(a + bx)*p)log(b(d + ex)/(-ae + bd))/eS(4) - dS(2)px/e*S(3) + dpxS(2)/(S(4)e*S(2)) - dx*S(2)log(c(a + bx)*p)/(S(2)e*S(2)) - px*S(3)/(S(9)e) + x*S(3)log(c(a + bx)*p)/(S(3)e) + d*S(2)(a + bx)log(c(a + bx)*p)/(beS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c(a + bx)p)/(d + ex), x), x, -a*S(2)plog(a + bx)/(S(2)bS(2)e) + apx/(S(2)be) + d*S(2)ppolylog(S(2), -e(a + bx)/(-ae + bd))/eS(3) + dS(2)log(c(a + bx)*p)log(b(d + ex)/(-ae + bd))/e*S(3) + dpx/eS(2) - px*S(2)/(S(4)e) + x*S(2)log(c(a + bx)*p)/(S(2)e) - d(a + bx)log(c(a + bx)p)/(be*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + bx)*p)/(d + ex), x), x, -dppolylog(S(2), -e(a + bx)/(-ae + bd))/e*S(2) - dlog(c(a + bx)*p)log(b(d + ex)/(-ae + bd))/e*S(2) - px/e + (a + bx)log(c(a + bx)*p)/(be), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)*p)/(d + ex), x), x, ppolylog(S(2), -e(a + bx)/(-ae + bd))/e + log(c(a + bx)p)log(b(d + ex)/(-ae + bd))/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)*p)/(x(d + ex)), x), x, ppolylog(S(2), (a + bx)/a)/d - ppolylog(S(2), -e(a + bx)/(-ae + bd))/d + log(c(a + bx)*p)log(-bx/a)/d - log(c(a + bx)p)log(b(d + ex)/(-ae + bd))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)p)/(xS(2)(d + ex)), x), x, -log(c(a + bx)*p)/(dx) - eppolylog(S(2), (a + bx)/a)/dS(2) + eppolylog(S(2), -e(a + bx)/(-ae + bd))/dS(2) - elog(c(a + bx)*p)log(-bx/a)/dS(2) + elog(c(a + bx)*p)log(b(d + ex)/(-ae + bd))/d*S(2) + bplog(x)/(ad) - bplog(a + bx)/(ad), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bx)p)/(xS(3)(d + ex)), x), x, -log(c(a + bx)*p)/(S(2)dxS(2)) + elog(c(a + bx)p)/(dS(2)x) + eS(2)ppolylog(S(2), (a + bx)/a)/dS(3) - eS(2)ppolylog(S(2), -e(a + bx)/(-ae + bd))/dS(3) + eS(2)log(c(a + bx)p)log(-bx/a)/dS(3) - eS(2)log(c(a + bx)*p)log(b(d + ex)/(-ae + bd))/d*S(3) - bp/(S(2)adx) - beplog(x)/(adS(2)) + beplog(a + bx)/(adS(2)) - bS(2)plog(x)/(S(2)aS(2)d) + b*S(2)plog(a + bx)/(S(2)aS(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)log(c(a + bxS(2))p)/(d + ex), x), x, -S(2)a*(S(3)/2)patan(sqrt(b)x/sqrt(a))/(S(3)b(S(3)/2)e) + S(2)sqrt(a)d*S(2)patan(sqrt(b)x/sqrt(a))/(sqrt(b)eS(3)) + S(2)apx/(S(3)be) + d*S(3)plog(-e(sqrt(b)x + sqrt(-a))/(sqrt(b)d - esqrt(-a)))log(d + ex)/eS(4) + dS(3)plog(e(-sqrt(b)x + sqrt(-a))/(sqrt(b)d + esqrt(-a)))log(d + ex)/eS(4) + dS(3)ppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d - esqrt(-a)))/eS(4) + dS(3)ppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d + esqrt(-a)))/eS(4) - dS(3)log(c(a + bxS(2))p)log(d + ex)/e*S(4) - S(2)d*S(2)px/eS(3) + dS(2)xlog(c(a + bxS(2))p)/eS(3) + dpxS(2)/(S(2)e*S(2)) - S(2)pxS(3)/(S(9)e) + x*S(3)log(c(a + bxS(2))p)/(S(3)e) - d(a + bxS(2))log(c(a + bxS(2))p)/(S(2)beS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c(a + bxS(2))p)/(d + ex), x), x, -S(2)sqrt(a)dpatan(sqrt(b)x/sqrt(a))/(sqrt(b)eS(2)) - dS(2)plog(-e(sqrt(b)x + sqrt(-a))/(sqrt(b)d - esqrt(-a)))log(d + ex)/eS(3) - dS(2)plog(e(-sqrt(b)x + sqrt(-a))/(sqrt(b)d + esqrt(-a)))log(d + ex)/eS(3) - dS(2)ppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d - esqrt(-a)))/eS(3) - dS(2)ppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d + esqrt(-a)))/eS(3) + dS(2)log(c(a + bxS(2))p)log(d + ex)/eS(3) + S(2)dpx/e*S(2) - dxlog(c(a + bxS(2))p)/eS(2) - px*S(2)/(S(2)e) + (a/S(2) + bxS(2)/S(2))log(c(a + bxS(2))p)/(be), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + bxS(2))p)/(d + ex), x), x, S(2)sqrt(a)patan(sqrt(b)x/sqrt(a))/(sqrt(b)e) + dplog(-e(sqrt(b)x + sqrt(-a))/(sqrt(b)d - esqrt(-a)))log(d + ex)/e*S(2) + dplog(e(-sqrt(b)x + sqrt(-a))/(sqrt(b)d + esqrt(-a)))log(d + ex)/eS(2) + dppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d - esqrt(-a)))/eS(2) + dppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d + esqrt(-a)))/eS(2) - dlog(c(a + bxS(2))p)log(d + ex)/e*S(2) - S(2)px/e + xlog(c(a + bxS(2))p)/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/(d + ex), x), x, -plog(-e(sqrt(b)x + sqrt(-a))/(sqrt(b)d - esqrt(-a)))log(d + ex)/e - plog(e(-sqrt(b)x + sqrt(-a))/(sqrt(b)d + esqrt(-a)))log(d + ex)/e - ppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d - esqrt(-a)))/e - ppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d + esqrt(-a)))/e + log(c(a + bxS(2))p)log(d + ex)/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/(x(d + ex)), x), x, plog(-e(sqrt(b)x + sqrt(-a))/(sqrt(b)d - esqrt(-a)))log(d + ex)/d + plog(e(-sqrt(b)x + sqrt(-a))/(sqrt(b)d + esqrt(-a)))log(d + ex)/d + ppolylog(S(2), (a + bxS(2))/a)/(S(2)d) + ppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d - esqrt(-a)))/d + ppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d + esqrt(-a)))/d + log(c(a + bxS(2))p)log(-bx*S(2)/a)/(S(2)d) - log(c(a + bxS(2))p)log(d + ex)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/(x*S(2)(d + ex)), x), x, -log(c(a + bxS(2))p)/(dx) - eplog(-e(sqrt(b)x + sqrt(-a))/(sqrt(b)d - esqrt(-a)))log(d + ex)/d*S(2) - eplog(e(-sqrt(b)x + sqrt(-a))/(sqrt(b)d + esqrt(-a)))log(d + ex)/dS(2) - eppolylog(S(2), (a + bx*S(2))/a)/(S(2)d*S(2)) - eppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d - esqrt(-a)))/dS(2) - eppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d + esqrt(-a)))/dS(2) - elog(c(a + bxS(2))p)log(-bx*S(2)/a)/(S(2)d*S(2)) + elog(c(a + bxS(2))p)log(d + ex)/d*S(2) + S(2)sqrt(b)patan(sqrt(b)x/sqrt(a))/(sqrt(a)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))p)/(x*S(3)(d + ex)), x), x, -log(c(a + bxS(2))p)/(S(2)dxS(2)) + elog(c(a + bxS(2))p)/(d*S(2)x) + e*S(2)plog(-e(sqrt(b)x + sqrt(-a))/(sqrt(b)d - esqrt(-a)))log(d + ex)/dS(3) + eS(2)plog(e(-sqrt(b)x + sqrt(-a))/(sqrt(b)d + esqrt(-a)))log(d + ex)/dS(3) + eS(2)ppolylog(S(2), (a + bx*S(2))/a)/(S(2)dS(3)) + eS(2)ppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d - esqrt(-a)))/dS(3) + eS(2)ppolylog(S(2), sqrt(b)(d + ex)/(sqrt(b)d + esqrt(-a)))/dS(3) + eS(2)log(c(a + bxS(2))p)log(-bxS(2)/a)/(S(2)dS(3)) - eS(2)log(c(a + bxS(2))p)log(d + ex)/dS(3) + bplog(x)/(ad) - bplog(a + bxS(2))/(S(2)ad) - S(2)sqrt(b)epatan(sqrt(b)x/sqrt(a))/(sqrt(a)dS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(c(a + bxS(3))p)/(d + ex), x), x, a(S(2)/3)dplog(a(S(1)/3) + b(S(1)/3)x)/(S(2)b*(S(2)/3)eS(2)) - a(S(2)/3)dplog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(4)b*(S(2)/3)e*S(2)) + sqrt(S(3))a*(S(2)/3)dpatan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(2)b*(S(2)/3)eS(2)) + a(S(1)/3)dS(2)plog(a(S(1)/3) + b(S(1)/3)x)/(b*(S(1)/3)eS(3)) - a(S(1)/3)dS(2)plog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(2)b*(S(1)/3)e*S(3)) - sqrt(S(3))a*(S(1)/3)d*S(2)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(b(S(1)/3)eS(3)) + dS(3)plog(-e(a(S(1)/3) + b(S(1)/3)x)/(-a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/eS(4) + dS(3)plog(-e((S(-1))(S(2)/3)a(S(1)/3) + b(S(1)/3)x)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/eS(4) + dS(3)plog((S(-1))*(S(1)/3)e(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/((S(-1))*(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/eS(4) + dS(3)ppolylog(S(2), b*(S(1)/3)(d + ex)/(-a(S(1)/3)e + b*(S(1)/3)d))/eS(4) + dS(3)ppolylog(S(2), b*(S(1)/3)(d + ex)/((S(-1))(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))/eS(4) + dS(3)ppolylog(S(2), b*(S(1)/3)(d + ex)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))/eS(4) - dS(3)log(c(a + bxS(3))p)log(d + ex)/eS(4) - S(3)d*S(2)px/eS(3) + dS(2)xlog(c(a + bxS(3))p)/eS(3) + S(3)dpx*S(2)/(S(4)e*S(2)) - dx*S(2)log(c(a + bxS(3))p)/(S(2)eS(2)) - px*S(3)/(S(3)e) + (a/S(3) + bxS(3)/S(3))log(c(a + bxS(3))p)/(be), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c(a + bxS(3))p)/(d + ex), x), x, -a(S(2)/3)plog(a(S(1)/3) + b(S(1)/3)x)/(S(2)b(S(2)/3)e) + a*(S(2)/3)plog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(4)b*(S(2)/3)e) - sqrt(S(3))a(S(2)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(2)b*(S(2)/3)e) - a*(S(1)/3)dplog(a(S(1)/3) + b(S(1)/3)x)/(b(S(1)/3)eS(2)) + a(S(1)/3)dplog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(2)b*(S(1)/3)e*S(2)) + sqrt(S(3))a*(S(1)/3)dpatan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(b(S(1)/3)eS(2)) - dS(2)plog(-e(a(S(1)/3) + b(S(1)/3)x)/(-a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/eS(3) - dS(2)plog(-e((S(-1))(S(2)/3)a(S(1)/3) + b(S(1)/3)x)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/eS(3) - dS(2)plog((S(-1))*(S(1)/3)e(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/((S(-1))*(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/eS(3) - dS(2)ppolylog(S(2), b*(S(1)/3)(d + ex)/(-a(S(1)/3)e + b*(S(1)/3)d))/eS(3) - dS(2)ppolylog(S(2), b*(S(1)/3)(d + ex)/((S(-1))(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))/eS(3) - dS(2)ppolylog(S(2), b*(S(1)/3)(d + ex)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))/eS(3) + dS(2)log(c(a + bxS(3))p)log(d + ex)/eS(3) + S(3)dpx/e*S(2) - dxlog(c(a + bxS(3))p)/eS(2) - S(3)pxS(2)/(S(4)e) + x*S(2)log(c(a + bxS(3))p)/(S(2)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + bxS(3))p)/(d + ex), x), x, a(S(1)/3)plog(a(S(1)/3) + b(S(1)/3)x)/(b*(S(1)/3)e) - a*(S(1)/3)plog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(2)b*(S(1)/3)e) - sqrt(S(3))a(S(1)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(b(S(1)/3)e) + dplog(-e(a(S(1)/3) + b(S(1)/3)x)/(-a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/e*S(2) + dplog(-e((S(-1))*(S(2)/3)a(S(1)/3) + b(S(1)/3)x)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/e*S(2) + dplog((S(-1))(S(1)/3)e(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/((S(-1))*(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/e*S(2) + dppolylog(S(2), b(S(1)/3)(d + ex)/(-a(S(1)/3)e + b*(S(1)/3)d))/e*S(2) + dppolylog(S(2), b(S(1)/3)(d + ex)/((S(-1))(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))/e*S(2) + dppolylog(S(2), b(S(1)/3)(d + ex)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))/e*S(2) - dlog(c(a + bxS(3))p)log(d + ex)/e*S(2) - S(3)px/e + xlog(c(a + bxS(3))p)/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/(d + ex), x), x, -plog(-e(a(S(1)/3) + b(S(1)/3)x)/(-a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/e - plog(-e((S(-1))*(S(2)/3)a(S(1)/3) + b(S(1)/3)x)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/e - plog((S(-1))(S(1)/3)e(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/((S(-1))*(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/e - ppolylog(S(2), b(S(1)/3)(d + ex)/(-a(S(1)/3)e + b*(S(1)/3)d))/e - ppolylog(S(2), b(S(1)/3)(d + ex)/((S(-1))(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))/e - ppolylog(S(2), b(S(1)/3)(d + ex)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))/e + log(c(a + bxS(3))p)log(d + ex)/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/(x(d + ex)), x), x, plog(-e(a(S(1)/3) + b(S(1)/3)x)/(-a(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/d + plog(-e((S(-1))*(S(2)/3)a(S(1)/3) + b(S(1)/3)x)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/d + plog((S(-1))(S(1)/3)e(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/((S(-1))*(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/d + ppolylog(S(2), (a + bx*S(3))/a)/(S(3)d) + ppolylog(S(2), b(S(1)/3)(d + ex)/(-a(S(1)/3)e + b*(S(1)/3)d))/d + ppolylog(S(2), b(S(1)/3)(d + ex)/((S(-1))(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))/d + ppolylog(S(2), b(S(1)/3)(d + ex)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))/d + log(c(a + bxS(3))p)log(-bx*S(3)/a)/(S(3)d) - log(c(a + bxS(3))p)log(d + ex)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/(x*S(2)(d + ex)), x), x, -log(c(a + bxS(3))p)/(dx) - eplog(-e(a(S(1)/3) + b(S(1)/3)x)/(-a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/d*S(2) - eplog(-e((S(-1))*(S(2)/3)a(S(1)/3) + b(S(1)/3)x)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/d*S(2) - eplog((S(-1))(S(1)/3)e(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/((S(-1))*(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/d*S(2) - eppolylog(S(2), (a + bx*S(3))/a)/(S(3)d*S(2)) - eppolylog(S(2), b(S(1)/3)(d + ex)/(-a(S(1)/3)e + b*(S(1)/3)d))/d*S(2) - eppolylog(S(2), b(S(1)/3)(d + ex)/((S(-1))(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))/d*S(2) - eppolylog(S(2), b(S(1)/3)(d + ex)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))/d*S(2) - elog(c(a + bxS(3))p)log(-bx*S(3)/a)/(S(3)d*S(2)) + elog(c(a + bxS(3))p)log(d + ex)/dS(2) - b(S(1)/3)plog(a(S(1)/3) + b(S(1)/3)x)/(a(S(1)/3)d) + b*(S(1)/3)plog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(2)a*(S(1)/3)d) - sqrt(S(3))b(S(1)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(a(S(1)/3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(3))p)/(x*S(3)(d + ex)), x), x, -log(c(a + bxS(3))p)/(S(2)dxS(2)) + elog(c(a + bxS(3))p)/(d*S(2)x) + e*S(2)plog(-e(a(S(1)/3) + b(S(1)/3)x)/(-a(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/dS(3) + eS(2)plog(-e((S(-1))(S(2)/3)a(S(1)/3) + b(S(1)/3)x)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/dS(3) + eS(2)plog((S(-1))*(S(1)/3)e(a(S(1)/3) + (S(-1))(S(2)/3)b*(S(1)/3)x)/((S(-1))*(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))log(d + ex)/dS(3) + eS(2)ppolylog(S(2), (a + bxS(3))/a)/(S(3)dS(3)) + eS(2)ppolylog(S(2), b*(S(1)/3)(d + ex)/(-a(S(1)/3)e + b*(S(1)/3)d))/dS(3) + eS(2)ppolylog(S(2), b*(S(1)/3)(d + ex)/((S(-1))(S(1)/3)a*(S(1)/3)e + b*(S(1)/3)d))/dS(3) + eS(2)ppolylog(S(2), b*(S(1)/3)(d + ex)/(-(S(-1))(S(2)/3)a*(S(1)/3)e + b*(S(1)/3)d))/dS(3) + eS(2)log(c(a + bxS(3))p)log(-bxS(3)/a)/(S(3)dS(3)) - eS(2)log(c(a + bxS(3))p)log(d + ex)/dS(3) + b(S(1)/3)eplog(a(S(1)/3) + b(S(1)/3)x)/(a(S(1)/3)dS(2)) - b(S(1)/3)eplog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(2)a*(S(1)/3)d*S(2)) + sqrt(S(3))b*(S(1)/3)epatan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(a(S(1)/3)dS(2)) + b(S(2)/3)plog(a(S(1)/3) + b(S(1)/3)x)/(S(2)a*(S(2)/3)d) - b*(S(2)/3)plog(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(4)a*(S(2)/3)d) - sqrt(S(3))b(S(2)/3)patan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(2)a*(S(2)/3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)log(c(a + b/x)p)/(d + ex), x), x, -d*S(3)plog(-ex/d)log(d + ex)/eS(4) + dS(3)plog(-e(ax + b)/(ad - be))log(d + ex)/eS(4) - dS(3)ppolylog(S(2), (d + ex)/d)/eS(4) + dS(3)ppolylog(S(2), a(d + ex)/(ad - be))/eS(4) - dS(3)log(c(a + b/x)p)log(d + ex)/eS(4) + dS(2)xlog(c(a + b/x)p)/eS(3) - dxS(2)log(c(a + b/x)p)/(S(2)eS(2)) + xS(3)log(c(a + b/x)*p)/(S(3)e) + bdS(2)plog(ax + b)/(aeS(3)) - bdpx/(S(2)ae*S(2)) + bpxS(2)/(S(6)ae) + bS(2)dplog(ax + b)/(S(2)a*S(2)eS(2)) - bS(2)px/(S(3)aS(2)e) + b*S(3)plog(ax + b)/(S(3)aS(3)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)log(c(a + b/x)p)/(d + ex), x), x, d*S(2)plog(-ex/d)log(d + ex)/eS(3) - dS(2)plog(-e(ax + b)/(ad - be))log(d + ex)/eS(3) + dS(2)ppolylog(S(2), (d + ex)/d)/eS(3) - dS(2)ppolylog(S(2), a(d + ex)/(ad - be))/eS(3) + dS(2)log(c(a + b/x)p)log(d + ex)/eS(3) - dxlog(c(a + b/x)p)/eS(2) + x*S(2)log(c(a + b/x)p)/(S(2)e) - bdplog(ax + b)/(aeS(2)) + bpx/(S(2)ae) - bS(2)plog(ax + b)/(S(2)aS(2)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + b/x)*p)/(d + ex), x), x, -dplog(-ex/d)log(d + ex)/eS(2) + dplog(-e(ax + b)/(ad - be))log(d + ex)/eS(2) - dppolylog(S(2), (d + ex)/d)/e*S(2) + dppolylog(S(2), a(d + ex)/(ad - be))/eS(2) - dlog(c(a + b/x)p)log(d + ex)/eS(2) + xlog(c(a + b/x)p)/e + bplog(ax + b)/(ae), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/x)*p)/(d + ex), x), x, plog(-ex/d)log(d + ex)/e - plog(-e(ax + b)/(ad - be))log(d + ex)/e + ppolylog(S(2), (d + ex)/d)/e - ppolylog(S(2), a(d + ex)/(ad - be))/e + log(c(a + b/x)p)log(d + ex)/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/x)*p)/(x(d + ex)), x), x, -plog(-ex/d)log(d + ex)/d + plog(-e(ax + b)/(ad - be))log(d + ex)/d - ppolylog(S(2), (a + b/x)/a)/d - ppolylog(S(2), (d + ex)/d)/d + ppolylog(S(2), a(d + ex)/(ad - be))/d - log(c(a + b/x)p)log(-b/(ax))/d - log(c(a + b/x)*p)log(d + ex)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/x)p)/(xS(2)(d + ex)), x), x, p/(dx) + eplog(-ex/d)log(d + ex)/d*S(2) - eplog(-e(ax + b)/(ad - be))log(d + ex)/dS(2) + eppolylog(S(2), (a + b/x)/a)/dS(2) + eppolylog(S(2), (d + ex)/d)/d*S(2) - eppolylog(S(2), a(d + ex)/(ad - be))/dS(2) + elog(c(a + b/x)p)log(-b/(ax))/dS(2) + elog(c(a + b/x)p)log(d + ex)/dS(2) - (a + b/x)log(c(a + b/x)p)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/x)p)/(xS(3)(d + ex)), x), x, -aS(2)plog(x)/(S(2)b*S(2)d) + a*S(2)plog(ax + b)/(S(2)bS(2)d) - ap/(S(2)bdx) + p/(S(4)dx*S(2)) - log(c(a + b/x)*p)/(S(2)dxS(2)) - ep/(d*S(2)x) - e*S(2)plog(-ex/d)log(d + ex)/dS(3) + eS(2)plog(-e(ax + b)/(ad - be))log(d + ex)/dS(3) - eS(2)ppolylog(S(2), (a + b/x)/a)/dS(3) - eS(2)ppolylog(S(2), (d + ex)/d)/dS(3) + eS(2)ppolylog(S(2), a(d + ex)/(ad - be))/dS(3) - eS(2)log(c(a + b/x)p)log(-b/(ax))/dS(3) - eS(2)log(c(a + b/x)p)log(d + ex)/dS(3) + e(a + b/x)log(c(a + b/x)*p)/(bdS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(c(a + b/xS(2))p)/(d + ex), x), x, -S(2)d*S(3)plog(-ex/d)log(d + ex)/eS(4) + dS(3)plog(e(sqrt(b) - xsqrt(-a))/(sqrt(b)e + dsqrt(-a)))log(d + ex)/eS(4) + dS(3)plog(-e(sqrt(b) + xsqrt(-a))/(-sqrt(b)e + dsqrt(-a)))log(d + ex)/e*S(4) - S(2)d*S(3)ppolylog(S(2), (d + ex)/d)/eS(4) + dS(3)ppolylog(S(2), sqrt(-a)(d + ex)/(-sqrt(b)e + dsqrt(-a)))/eS(4) + dS(3)ppolylog(S(2), sqrt(-a)(d + ex)/(sqrt(b)e + dsqrt(-a)))/eS(4) - dS(3)log(c(a + b/xS(2))p)log(d + ex)/eS(4) + dS(2)xlog(c(a + b/xS(2))p)/eS(3) - dx*S(2)log(c(a + b/xS(2))p)/(S(2)eS(2)) + xS(3)log(c(a + b/xS(2))p)/(S(3)e) - bdplog(axS(2) + b)/(S(2)aeS(2)) + S(2)bpx/(S(3)ae) + S(2)sqrt(b)d*S(2)patan(sqrt(a)x/sqrt(b))/(sqrt(a)eS(3)) - S(2)b*(S(3)/2)patan(sqrt(a)x/sqrt(b))/(S(3)a(S(3)/2)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)log(c(a + b/xS(2))p)/(d + ex), x), x, S(2)dS(2)plog(-ex/d)log(d + ex)/eS(3) - dS(2)plog(e(sqrt(b) - xsqrt(-a))/(sqrt(b)e + dsqrt(-a)))log(d + ex)/eS(3) - dS(2)plog(-e(sqrt(b) + xsqrt(-a))/(-sqrt(b)e + dsqrt(-a)))log(d + ex)/e*S(3) + S(2)d*S(2)ppolylog(S(2), (d + ex)/d)/eS(3) - dS(2)ppolylog(S(2), sqrt(-a)(d + ex)/(-sqrt(b)e + dsqrt(-a)))/eS(3) - dS(2)ppolylog(S(2), sqrt(-a)(d + ex)/(sqrt(b)e + dsqrt(-a)))/eS(3) + dS(2)log(c(a + b/xS(2))p)log(d + ex)/e*S(3) - dxlog(c(a + b/xS(2))p)/eS(2) + xS(2)log(c(a + b/xS(2))p)/(S(2)e) + bplog(ax*S(2) + b)/(S(2)ae) - S(2)sqrt(b)dpatan(sqrt(a)x/sqrt(b))/(sqrt(a)eS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + b/xS(2))p)/(d + ex), x), x, -S(2)dplog(-ex/d)log(d + ex)/e*S(2) + dplog(e(sqrt(b) - xsqrt(-a))/(sqrt(b)e + dsqrt(-a)))log(d + ex)/eS(2) + dplog(-e(sqrt(b) + xsqrt(-a))/(-sqrt(b)e + dsqrt(-a)))log(d + ex)/eS(2) - S(2)dppolylog(S(2), (d + ex)/d)/eS(2) + dppolylog(S(2), sqrt(-a)(d + ex)/(-sqrt(b)e + dsqrt(-a)))/eS(2) + dppolylog(S(2), sqrt(-a)(d + ex)/(sqrt(b)e + dsqrt(-a)))/eS(2) - dlog(c(a + b/xS(2))p)log(d + ex)/eS(2) + xlog(c(a + b/xS(2))p)/e + S(2)sqrt(b)patan(sqrt(a)x/sqrt(b))/(sqrt(a)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(2))p)/(d + ex), x), x, S(2)plog(-ex/d)log(d + ex)/e - plog(e(sqrt(b) - xsqrt(-a))/(sqrt(b)e + dsqrt(-a)))log(d + ex)/e - plog(-e(sqrt(b) + xsqrt(-a))/(-sqrt(b)e + dsqrt(-a)))log(d + ex)/e + S(2)ppolylog(S(2), (d + ex)/d)/e - ppolylog(S(2), sqrt(-a)(d + ex)/(-sqrt(b)e + dsqrt(-a)))/e - ppolylog(S(2), sqrt(-a)(d + ex)/(sqrt(b)e + dsqrt(-a)))/e + log(c(a + b/xS(2))p)log(d + ex)/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(2))p)/(x(d + ex)), x), x, -S(2)plog(-ex/d)log(d + ex)/d + plog(e(sqrt(b) - xsqrt(-a))/(sqrt(b)e + dsqrt(-a)))log(d + ex)/d + plog(-e(sqrt(b) + xsqrt(-a))/(-sqrt(b)e + dsqrt(-a)))log(d + ex)/d - ppolylog(S(2), (a + b/x*S(2))/a)/(S(2)d) - S(2)ppolylog(S(2), (d + ex)/d)/d + ppolylog(S(2), sqrt(-a)(d + ex)/(-sqrt(b)e + dsqrt(-a)))/d + ppolylog(S(2), sqrt(-a)(d + ex)/(sqrt(b)e + dsqrt(-a)))/d - log(c(a + b/xS(2))p)log(-b/(ax*S(2)))/(S(2)d) - log(c(a + b/xS(2))p)log(d + ex)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(2))p)/(x*S(2)(d + ex)), x), x, S(2)sqrt(a)patan(sqrt(a)x/sqrt(b))/(sqrt(b)d) + S(2)p/(dx) - log(c(a + b/xS(2))p)/(dx) + S(2)eplog(-ex/d)log(d + ex)/d*S(2) - eplog(e(sqrt(b) - xsqrt(-a))/(sqrt(b)e + dsqrt(-a)))log(d + ex)/dS(2) - eplog(-e(sqrt(b) + xsqrt(-a))/(-sqrt(b)e + dsqrt(-a)))log(d + ex)/dS(2) + eppolylog(S(2), (a + b/xS(2))/a)/(S(2)d*S(2)) + S(2)eppolylog(S(2), (d + ex)/d)/dS(2) - eppolylog(S(2), sqrt(-a)(d + ex)/(-sqrt(b)e + dsqrt(-a)))/dS(2) - eppolylog(S(2), sqrt(-a)(d + ex)/(sqrt(b)e + dsqrt(-a)))/dS(2) + elog(c(a + b/xS(2))p)log(-b/(axS(2)))/(S(2)d*S(2)) + elog(c(a + b/xS(2))p)log(d + ex)/dS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(2))p)/(x*S(3)(d + ex)), x), x, -S(2)sqrt(a)epatan(sqrt(a)x/sqrt(b))/(sqrt(b)dS(2)) + p/(S(2)dxS(2)) - S(2)ep/(dS(2)x) + elog(c(a + b/xS(2))p)/(d*S(2)x) - S(2)eS(2)plog(-ex/d)log(d + ex)/dS(3) + eS(2)plog(e(sqrt(b) - xsqrt(-a))/(sqrt(b)e + dsqrt(-a)))log(d + ex)/dS(3) + eS(2)plog(-e(sqrt(b) + xsqrt(-a))/(-sqrt(b)e + dsqrt(-a)))log(d + ex)/dS(3) - eS(2)ppolylog(S(2), (a + b/x*S(2))/a)/(S(2)d*S(3)) - S(2)e*S(2)ppolylog(S(2), (d + ex)/d)/dS(3) + eS(2)ppolylog(S(2), sqrt(-a)(d + ex)/(-sqrt(b)e + dsqrt(-a)))/dS(3) + eS(2)ppolylog(S(2), sqrt(-a)(d + ex)/(sqrt(b)e + dsqrt(-a)))/dS(3) - eS(2)log(c(a + b/xS(2))p)log(-b/(ax*S(2)))/(S(2)dS(3)) - eS(2)log(c(a + b/xS(2))p)log(d + ex)/d*S(3) - (a/S(2) + b/(S(2)x*S(2)))log(c(a + b/xS(2))p)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)log(c(a + b/xS(3))p)/(d + ex), x), x, -S(3)dS(3)plog(-ex/d)log(d + ex)/eS(4) + dS(3)plog(-e(a(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d - b(S(1)/3)e))log(d + ex)/eS(4) + dS(3)plog(-e(a(S(1)/3)x + (S(-1))*(S(2)/3)b(S(1)/3))/(a(S(1)/3)d - (S(-1))(S(2)/3)b*(S(1)/3)e))log(d + ex)/eS(4) + dS(3)plog((S(-1))*(S(1)/3)e((S(-1))(S(2)/3)a*(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d + (S(-1))(S(1)/3)b*(S(1)/3)e))log(d + ex)/e*S(4) - S(3)d*S(3)ppolylog(S(2), (d + ex)/d)/eS(4) + dS(3)ppolylog(S(2), a*(S(1)/3)(d + ex)/(a(S(1)/3)d - b*(S(1)/3)e))/eS(4) + dS(3)ppolylog(S(2), a*(S(1)/3)(d + ex)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)e))/eS(4) + dS(3)ppolylog(S(2), a*(S(1)/3)(d + ex)/(a(S(1)/3)d - (S(-1))*(S(2)/3)b*(S(1)/3)e))/eS(4) - dS(3)log(c(a + b/xS(3))p)log(d + ex)/eS(4) + dS(2)xlog(c(a + b/xS(3))p)/eS(3) - dx*S(2)log(c(a + b/xS(3))p)/(S(2)eS(2)) + xS(3)log(c(a + b/xS(3))p)/(S(3)e) + bplog(ax*S(3) + b)/(S(3)ae) + b(S(1)/3)d*S(2)plog(a(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)eS(3)) - b(S(1)/3)d*S(2)plog(a(S(2)/3)xS(2) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3))/(S(2)a*(S(1)/3)e*S(3)) - sqrt(S(3))b*(S(1)/3)d*S(2)patan(sqrt(S(3))(-S(2)a(S(1)/3)x + b*(S(1)/3))/(S(3)b(S(1)/3)))/(a(S(1)/3)eS(3)) + b(S(2)/3)dplog(a*(S(1)/3)x + b*(S(1)/3))/(S(2)a*(S(2)/3)eS(2)) - b(S(2)/3)dplog(a(S(2)/3)xS(2) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3))/(S(4)a*(S(2)/3)e*S(2)) + sqrt(S(3))b*(S(2)/3)dpatan(sqrt(S(3))(-S(2)a*(S(1)/3)x + b*(S(1)/3))/(S(3)b*(S(1)/3)))/(S(2)a*(S(2)/3)eS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c(a + b/xS(3))p)/(d + ex), x), x, S(3)d*S(2)plog(-ex/d)log(d + ex)/eS(3) - dS(2)plog(-e(a(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d - b(S(1)/3)e))log(d + ex)/eS(3) - dS(2)plog(-e(a(S(1)/3)x + (S(-1))*(S(2)/3)b(S(1)/3))/(a(S(1)/3)d - (S(-1))(S(2)/3)b*(S(1)/3)e))log(d + ex)/eS(3) - dS(2)plog((S(-1))*(S(1)/3)e((S(-1))(S(2)/3)a*(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d + (S(-1))(S(1)/3)b*(S(1)/3)e))log(d + ex)/e*S(3) + S(3)d*S(2)ppolylog(S(2), (d + ex)/d)/eS(3) - dS(2)ppolylog(S(2), a*(S(1)/3)(d + ex)/(a(S(1)/3)d - b*(S(1)/3)e))/eS(3) - dS(2)ppolylog(S(2), a*(S(1)/3)(d + ex)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)e))/eS(3) - dS(2)ppolylog(S(2), a*(S(1)/3)(d + ex)/(a(S(1)/3)d - (S(-1))*(S(2)/3)b*(S(1)/3)e))/eS(3) + dS(2)log(c(a + b/xS(3))p)log(d + ex)/e*S(3) - dxlog(c(a + b/xS(3))p)/eS(2) + xS(2)log(c(a + b/xS(3))p)/(S(2)e) - b(S(1)/3)dplog(a*(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)eS(2)) + b(S(1)/3)dplog(a*(S(2)/3)xS(2) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3))/(S(2)a*(S(1)/3)e*S(2)) + sqrt(S(3))b*(S(1)/3)dpatan(sqrt(S(3))(-S(2)a*(S(1)/3)x + b*(S(1)/3))/(S(3)b(S(1)/3)))/(a(S(1)/3)eS(2)) - b(S(2)/3)plog(a(S(1)/3)x + b*(S(1)/3))/(S(2)a*(S(2)/3)e) + b*(S(2)/3)plog(a(S(2)/3)xS(2) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3))/(S(4)a*(S(2)/3)e) - sqrt(S(3))b(S(2)/3)patan(sqrt(S(3))(-S(2)a(S(1)/3)x + b*(S(1)/3))/(S(3)b*(S(1)/3)))/(S(2)a*(S(2)/3)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + b/xS(3))p)/(d + ex), x), x, -S(3)dplog(-ex/d)log(d + ex)/eS(2) + dplog(-e(a*(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d - b(S(1)/3)e))log(d + ex)/e*S(2) + dplog(-e(a*(S(1)/3)x + (S(-1))*(S(2)/3)b(S(1)/3))/(a(S(1)/3)d - (S(-1))(S(2)/3)b*(S(1)/3)e))log(d + ex)/e*S(2) + dplog((S(-1))(S(1)/3)e((S(-1))(S(2)/3)a*(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d + (S(-1))(S(1)/3)b*(S(1)/3)e))log(d + ex)/e*S(2) - S(3)dppolylog(S(2), (d + ex)/d)/eS(2) + dppolylog(S(2), a(S(1)/3)(d + ex)/(a(S(1)/3)d - b*(S(1)/3)e))/e*S(2) + dppolylog(S(2), a(S(1)/3)(d + ex)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)e))/e*S(2) + dppolylog(S(2), a(S(1)/3)(d + ex)/(a(S(1)/3)d - (S(-1))*(S(2)/3)b*(S(1)/3)e))/e*S(2) - dlog(c(a + b/xS(3))p)log(d + ex)/eS(2) + xlog(c(a + b/xS(3))p)/e + b(S(1)/3)plog(a(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)e) - b(S(1)/3)plog(a(S(2)/3)xS(2) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3))/(S(2)a*(S(1)/3)e) - sqrt(S(3))b(S(1)/3)patan(sqrt(S(3))(-S(2)a(S(1)/3)x + b*(S(1)/3))/(S(3)b(S(1)/3)))/(a(S(1)/3)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(3))p)/(d + ex), x), x, S(3)plog(-ex/d)log(d + ex)/e - plog(-e(a*(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d - b(S(1)/3)e))log(d + ex)/e - plog(-e(a*(S(1)/3)x + (S(-1))*(S(2)/3)b(S(1)/3))/(a(S(1)/3)d - (S(-1))(S(2)/3)b*(S(1)/3)e))log(d + ex)/e - plog((S(-1))(S(1)/3)e((S(-1))(S(2)/3)a*(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d + (S(-1))(S(1)/3)b*(S(1)/3)e))log(d + ex)/e + S(3)ppolylog(S(2), (d + ex)/d)/e - ppolylog(S(2), a*(S(1)/3)(d + ex)/(a(S(1)/3)d - b*(S(1)/3)e))/e - ppolylog(S(2), a(S(1)/3)(d + ex)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)e))/e - ppolylog(S(2), a(S(1)/3)(d + ex)/(a(S(1)/3)d - (S(-1))*(S(2)/3)b*(S(1)/3)e))/e + log(c(a + b/xS(3))p)log(d + ex)/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(3))p)/(x(d + ex)), x), x, -S(3)plog(-ex/d)log(d + ex)/d + plog(-e(a(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d - b(S(1)/3)e))log(d + ex)/d + plog(-e(a*(S(1)/3)x + (S(-1))*(S(2)/3)b(S(1)/3))/(a(S(1)/3)d - (S(-1))(S(2)/3)b*(S(1)/3)e))log(d + ex)/d + plog((S(-1))(S(1)/3)e((S(-1))(S(2)/3)a*(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d + (S(-1))(S(1)/3)b*(S(1)/3)e))log(d + ex)/d - ppolylog(S(2), (a + b/xS(3))/a)/(S(3)d) - S(3)ppolylog(S(2), (d + ex)/d)/d + ppolylog(S(2), a*(S(1)/3)(d + ex)/(a(S(1)/3)d - b*(S(1)/3)e))/d + ppolylog(S(2), a(S(1)/3)(d + ex)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)e))/d + ppolylog(S(2), a(S(1)/3)(d + ex)/(a(S(1)/3)d - (S(-1))*(S(2)/3)b*(S(1)/3)e))/d - log(c(a + b/xS(3))p)log(-b/(axS(3)))/(S(3)d) - log(c(a + b/xS(3))p)log(d + ex)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(3))p)/(x*S(2)(d + ex)), x), x, -a(S(1)/3)plog(a(S(1)/3)x + b(S(1)/3))/(b(S(1)/3)d) + a(S(1)/3)plog(a(S(2)/3)xS(2) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3))/(S(2)b*(S(1)/3)d) - sqrt(S(3))a(S(1)/3)patan(sqrt(S(3))(-S(2)a(S(1)/3)x + b*(S(1)/3))/(S(3)b(S(1)/3)))/(b(S(1)/3)d) + S(3)p/(dx) - log(c(a + b/xS(3))p)/(dx) + S(3)eplog(-ex/d)log(d + ex)/dS(2) - eplog(-e(a*(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d - b(S(1)/3)e))log(d + ex)/d*S(2) - eplog(-e(a*(S(1)/3)x + (S(-1))*(S(2)/3)b(S(1)/3))/(a(S(1)/3)d - (S(-1))(S(2)/3)b*(S(1)/3)e))log(d + ex)/d*S(2) - eplog((S(-1))(S(1)/3)e((S(-1))(S(2)/3)a*(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d + (S(-1))(S(1)/3)b*(S(1)/3)e))log(d + ex)/d*S(2) + eppolylog(S(2), (a + b/xS(3))/a)/(S(3)d*S(2)) + S(3)eppolylog(S(2), (d + ex)/d)/dS(2) - eppolylog(S(2), a(S(1)/3)(d + ex)/(a(S(1)/3)d - b*(S(1)/3)e))/d*S(2) - eppolylog(S(2), a(S(1)/3)(d + ex)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)e))/d*S(2) - eppolylog(S(2), a(S(1)/3)(d + ex)/(a(S(1)/3)d - (S(-1))*(S(2)/3)b*(S(1)/3)e))/d*S(2) + elog(c(a + b/xS(3))p)log(-b/(axS(3)))/(S(3)d*S(2)) + elog(c(a + b/xS(3))p)log(d + ex)/dS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + b/xS(3))p)/(x*S(3)(d + ex)), x), x, a(S(2)/3)plog(a(S(1)/3)x + b*(S(1)/3))/(S(2)b*(S(2)/3)d) - a*(S(2)/3)plog(a(S(2)/3)xS(2) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3))/(S(4)b*(S(2)/3)d) - sqrt(S(3))a(S(2)/3)patan(sqrt(S(3))(-S(2)a(S(1)/3)x + b*(S(1)/3))/(S(3)b*(S(1)/3)))/(S(2)b*(S(2)/3)d) + a*(S(1)/3)eplog(a*(S(1)/3)x + b(S(1)/3))/(b(S(1)/3)dS(2)) - a(S(1)/3)eplog(a*(S(2)/3)xS(2) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3))/(S(2)b*(S(1)/3)d*S(2)) + sqrt(S(3))a*(S(1)/3)epatan(sqrt(S(3))(-S(2)a*(S(1)/3)x + b*(S(1)/3))/(S(3)b(S(1)/3)))/(b(S(1)/3)dS(2)) + S(3)p/(S(4)dx*S(2)) - log(c(a + b/xS(3))p)/(S(2)dx*S(2)) - S(3)ep/(dS(2)x) + elog(c(a + b/xS(3))p)/(d*S(2)x) - S(3)eS(2)plog(-ex/d)log(d + ex)/dS(3) + eS(2)plog(-e(a(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d - b(S(1)/3)e))log(d + ex)/dS(3) + eS(2)plog(-e(a(S(1)/3)x + (S(-1))*(S(2)/3)b(S(1)/3))/(a(S(1)/3)d - (S(-1))(S(2)/3)b*(S(1)/3)e))log(d + ex)/dS(3) + eS(2)plog((S(-1))*(S(1)/3)e((S(-1))(S(2)/3)a*(S(1)/3)x + b(S(1)/3))/(a(S(1)/3)d + (S(-1))(S(1)/3)b*(S(1)/3)e))log(d + ex)/dS(3) - eS(2)ppolylog(S(2), (a + b/x*S(3))/a)/(S(3)d*S(3)) - S(3)e*S(2)ppolylog(S(2), (d + ex)/d)/dS(3) + eS(2)ppolylog(S(2), a*(S(1)/3)(d + ex)/(a(S(1)/3)d - b*(S(1)/3)e))/dS(3) + eS(2)ppolylog(S(2), a*(S(1)/3)(d + ex)/(a(S(1)/3)d + (S(-1))*(S(1)/3)b*(S(1)/3)e))/dS(3) + eS(2)ppolylog(S(2), a*(S(1)/3)(d + ex)/(a(S(1)/3)d - (S(-1))*(S(2)/3)b*(S(1)/3)e))/dS(3) - eS(2)log(c(a + b/xS(3))p)log(-b/(ax*S(3)))/(S(3)dS(3)) - eS(2)log(c(a + b/xS(3))p)log(d + ex)/d*S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d + exS(2))/(-xS(2) + S(1)), x), x, log((-sqrt(e)x + sqrt(-d))/(-sqrt(e) + sqrt(-d)))log(-x + S(1))/S(2) - log((sqrt(e)x + sqrt(-d))/(-sqrt(e) + sqrt(-d)))log(x + S(1))/S(2) - log((-sqrt(e)x + sqrt(-d))/(sqrt(e) + sqrt(-d)))log(x + S(1))/S(2) + log((sqrt(e)x + sqrt(-d))/(sqrt(e) + sqrt(-d)))log(-x + S(1))/S(2) + log(d + exS(2))atanh(x) - polylog(S(2), sqrt(e)(-x + S(-1))/(-sqrt(e) + sqrt(-d)))/S(2) + polylog(S(2), sqrt(e)(x + S(-1))/(-sqrt(e) + sqrt(-d)))/S(2) + polylog(S(2), sqrt(e)(-x + S(1))/(sqrt(e) + sqrt(-d)))/S(2) - polylog(S(2), sqrt(e)(x + S(1))/(sqrt(e) + sqrt(-d)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d + exS(2))/(a + bx*S(2)), x), x, Ilog(sqrt(b)(-sqrt(e)x + sqrt(-d))/(-Isqrt(a)sqrt(e) + sqrt(b)sqrt(-d)))log(S(1) + Isqrt(b)x/sqrt(a))/(S(2)sqrt(a)sqrt(b)) - Ilog(sqrt(b)(-sqrt(e)x + sqrt(-d))/(Isqrt(a)sqrt(e) + sqrt(b)sqrt(-d)))log(S(1) - Isqrt(b)x/sqrt(a))/(S(2)sqrt(a)sqrt(b)) - Ilog(sqrt(b)(sqrt(e)x + sqrt(-d))/(-Isqrt(a)sqrt(e) + sqrt(b)sqrt(-d)))log(S(1) - Isqrt(b)x/sqrt(a))/(S(2)sqrt(a)sqrt(b)) + Ilog(sqrt(b)(sqrt(e)x + sqrt(-d))/(Isqrt(a)sqrt(e) + sqrt(b)sqrt(-d)))log(S(1) + Isqrt(b)x/sqrt(a))/(S(2)sqrt(a)sqrt(b)) + log(d + ex*S(2))atan(sqrt(b)x/sqrt(a))/(sqrt(a)sqrt(b)) + Ipolylog(S(2), sqrt(e)(-sqrt(a) - Isqrt(b)x)/(-sqrt(a)sqrt(e) + Isqrt(b)sqrt(-d)))/(S(2)sqrt(a)sqrt(b)) - Ipolylog(S(2), sqrt(e)(-sqrt(a) + Isqrt(b)x)/(-sqrt(a)sqrt(e) + Isqrt(b)sqrt(-d)))/(S(2)sqrt(a)sqrt(b)) - Ipolylog(S(2), sqrt(e)(sqrt(a) - Isqrt(b)x)/(sqrt(a)sqrt(e) + Isqrt(b)sqrt(-d)))/(S(2)sqrt(a)sqrt(b)) + Ipolylog(S(2), sqrt(e)(sqrt(a) + Isqrt(b)x)/(sqrt(a)sqrt(e) + Isqrt(b)sqrt(-d)))/(S(2)sqrt(a)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(-xS(2) + S(1))/(-xS(2) + S(2)), x), x, sqrt(S(2))log(-xS(2) + S(1))atanh(sqrt(S(2))x/S(2))/S(2) - sqrt(S(2))log(-S(2)sqrt(S(2)) + S(3))atanh(x)/S(2) + sqrt(S(2))polylog(S(2), sqrt(S(2))(-x + S(-1))/(-sqrt(S(2)) + S(2)))/S(4) - sqrt(S(2))polylog(S(2), sqrt(S(2))(x + S(-1))/(-sqrt(S(2)) + S(2)))/S(4) + sqrt(S(2))polylog(S(2), sqrt(S(2))(-x + S(1))/(sqrt(S(2)) + S(2)))/S(4) - sqrt(S(2))polylog(S(2), sqrt(S(2))(x + S(1))/(sqrt(S(2)) + S(2)))/S(4), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(log(-xS(2) + S(1))/(-xS(2) + S(2)), x), x, -sqrt(S(2))log(-x + S(1))atanh(sqrt(S(2))/S(2))/S(2) + sqrt(S(2))log(x + S(1))atanh(sqrt(S(2))/S(2))/S(2) + sqrt(S(2))log(-xS(2) + S(1))atanh(sqrt(S(2))x/S(2))/S(2) + sqrt(S(2))polylog(S(2), sqrt(S(2))(-x + S(-1))/(-sqrt(S(2)) + S(2)))/S(4) - sqrt(S(2))polylog(S(2), sqrt(S(2))(x + S(-1))/(-sqrt(S(2)) + S(2)))/S(4) + sqrt(S(2))polylog(S(2), sqrt(S(2))(-x + S(1))/(sqrt(S(2)) + S(2)))/S(4) - sqrt(S(2))polylog(S(2), sqrt(S(2))(x + S(1))/(sqrt(S(2)) + S(2)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cx*S(2))log(d + exS(2))/xS(2), x), x, -alog(d + exS(2))/x + cxlog(d + ex*S(2)) - S(2)cx + (S(2)ae + S(2)cd)atan(sqrt(e)x/sqrt(d))/(sqrt(d)sqrt(e)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + cxS(2))log(d + exS(2))/xS(2), x), x, -alog(d + exS(2))/x + S(2)asqrt(e)atan(sqrt(e)x/sqrt(d))/sqrt(d) + S(2)csqrt(d)atan(sqrt(e)x/sqrt(d))/sqrt(e) + cxlog(d + ex*S(2)) - S(2)cx, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)log(c(a + bxS(2))n)*S(2), x), x, -S(5)a*S(3)n*S(2)log(a + bxS(2))/(S(18)bS(3)) + aS(3)log(c(a + bxS(2))n)S(2)/(S(6)b*S(3)) + S(11)a*S(2)n*S(2)x*S(2)/(S(18)bS(2)) - aS(2)n(a + bxS(2))log(c(a + bxS(2))n)/(S(3)bS(3)) - S(5)anS(2)x*S(4)/(S(36)b) + anx*S(4)log(c(a + bxS(2))n)/(S(6)b) + nS(2)x*S(6)/S(27) - nx*S(6)log(c(a + bxS(2))n)/S(9) + x*S(6)log(c(a + bxS(2))n)S(2)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(c(a + bxS(2))n)S(2), x), x, -S(3)anS(2)x*S(2)/(S(4)b) + an(a + bxS(2))log(c(a + bxS(2))n)/b*S(2) - a(a + bxS(2))log(c(a + bxS(2))n)*S(2)/(S(2)bS(2)) + nS(2)xS(4)/S(8) - n(a + bxS(2))S(2)log(c(a + bxS(2))n)/(S(4)bS(2)) + (a + bxS(2))S(2)log(c(a + bxS(2))n)S(2)/(S(4)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + bxS(2))n)S(2), x), x, nS(2)xS(2) - n(a + bxS(2))log(c(a + bxS(2))n)/b + (a/S(2) + bxS(2)/S(2))log(c(a + bxS(2))n)*S(2)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(2)/x, x), x, -nS(2)polylog(S(3), (a + bxS(2))/a) + nlog(c(a + bxS(2))n)polylog(S(2), (a + bx*S(2))/a) + log(c(a + bxS(2))n)S(2)log(-bxS(2)/a)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(2)/xS(3), x), x, bn*S(2)polylog(S(2), (a + bxS(2))/a)/a + bnlog(c(a + bxS(2))n)log(-bxS(2)/a)/a - (a/S(2) + bx*S(2)/S(2))log(c(a + bxS(2))n)*S(2)/(ax*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(2)/xS(5), x), x, -log(c(a + bxS(2))n)S(2)/(S(4)x*S(4)) - bnlog(c(a + bxS(2))n)/(S(2)axS(2)) + bS(2)n*S(2)log(x)/aS(2) - bS(2)nS(2)log(a + bxS(2))/(S(2)aS(2)) - bS(2)nS(2)polylog(S(2), (a + bxS(2))/a)/(S(2)aS(2)) - bS(2)nlog(c(a + bxS(2))n)log(-bx*S(2)/a)/(S(2)aS(2)) + bS(2)log(c(a + bxS(2))n)S(2)/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(2)/xS(7), x), x, -log(c(a + bxS(2))n)S(2)/(S(6)x*S(6)) - bnlog(c(a + bxS(2))n)/(S(6)axS(4)) - bS(2)n*S(2)/(S(6)a*S(2)xS(2)) + bS(2)nlog(c(a + bxS(2))n)/(S(3)aS(2)xS(2)) - bS(3)nS(2)log(x)/aS(3) + bS(3)nS(2)log(a + bxS(2))/(S(2)aS(3)) + bS(3)nS(2)polylog(S(2), (a + bxS(2))/a)/(S(3)aS(3)) + bS(3)nlog(c(a + bxS(2))n)log(-bx*S(2)/a)/(S(3)aS(3)) - bS(3)log(c(a + bxS(2))n)S(2)/(S(6)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)log(c(a + bxS(2))n)S(2), x), x, S(8)a*(S(5)/2)n*S(2)log(S(2)Isqrt(a)/(Isqrt(a) - sqrt(b)x))atan(sqrt(b)x/sqrt(a))/(S(5)b(S(5)/2)) + S(4)Ia(S(5)/2)n*S(2)atan(sqrt(b)x/sqrt(a))S(2)/(S(5)b*(S(5)/2)) - S(184)a*(S(5)/2)n*S(2)atan(sqrt(b)x/sqrt(a))/(S(75)b*(S(5)/2)) + S(4)Ia(S(5)/2)n*S(2)polylog(S(2), (-sqrt(a) + Isqrt(b)x)/(sqrt(a) + Isqrt(b)x))/(S(5)b(S(5)/2)) + S(4)a*(S(5)/2)nlog(c(a + bxS(2))n)atan(sqrt(b)x/sqrt(a))/(S(5)b*(S(5)/2)) + S(184)a*S(2)n*S(2)x/(S(75)bS(2)) - S(4)a*S(2)nxlog(c(a + bxS(2))n)/(S(5)bS(2)) - S(64)anS(2)x*S(3)/(S(225)b) + S(4)anxS(3)log(c(a + bxS(2))n)/(S(15)b) + S(8)n*S(2)x*S(5)/S(125) - S(4)nxS(5)log(c(a + bxS(2))n)/S(25) + x*S(5)log(c(a + bxS(2))n)S(2)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(c(a + bxS(2))n)S(2), x), x, -S(8)a*(S(3)/2)n*S(2)log(S(2)Isqrt(a)/(Isqrt(a) - sqrt(b)x))atan(sqrt(b)x/sqrt(a))/(S(3)b(S(3)/2)) - S(4)Ia(S(3)/2)n*S(2)atan(sqrt(b)x/sqrt(a))S(2)/(S(3)b*(S(3)/2)) + S(32)a*(S(3)/2)n*S(2)atan(sqrt(b)x/sqrt(a))/(S(9)b*(S(3)/2)) - S(4)Ia(S(3)/2)n*S(2)polylog(S(2), (-sqrt(a) + Isqrt(b)x)/(sqrt(a) + Isqrt(b)x))/(S(3)b(S(3)/2)) - S(4)a*(S(3)/2)nlog(c(a + bxS(2))n)atan(sqrt(b)x/sqrt(a))/(S(3)b*(S(3)/2)) - S(32)anS(2)x/(S(9)b) + S(4)anxlog(c(a + bxS(2))n)/(S(3)b) + S(8)nS(2)x*S(3)/S(27) - S(4)nxS(3)log(c(a + bxS(2))n)/S(9) + x*S(3)log(c(a + bxS(2))n)*S(2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(2), x), x, S(8)sqrt(a)nS(2)log(S(2)Isqrt(a)/(Isqrt(a) - sqrt(b)x))atan(sqrt(b)x/sqrt(a))/sqrt(b) + S(4)Isqrt(a)nS(2)atan(sqrt(b)x/sqrt(a))S(2)/sqrt(b) - S(8)sqrt(a)nS(2)atan(sqrt(b)x/sqrt(a))/sqrt(b) + S(4)Isqrt(a)n*S(2)polylog(S(2), (-sqrt(a) + Isqrt(b)x)/(sqrt(a) + Isqrt(b)x))/sqrt(b) + S(4)sqrt(a)nlog(c(a + bxS(2))n)atan(sqrt(b)x/sqrt(a))/sqrt(b) + S(8)n*S(2)x - S(4)nxlog(c(a + bxS(2))n) + xlog(c(a + bxS(2))n)*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(2)/xS(2), x), x, -log(c(a + bxS(2))n)S(2)/x + S(8)sqrt(b)nS(2)log(S(2)Isqrt(a)/(Isqrt(a) - sqrt(b)x))atan(sqrt(b)x/sqrt(a))/sqrt(a) + S(4)Isqrt(b)nS(2)atan(sqrt(b)x/sqrt(a))S(2)/sqrt(a) + S(4)Isqrt(b)n*S(2)polylog(S(2), (-sqrt(a) + Isqrt(b)x)/(sqrt(a) + Isqrt(b)x))/sqrt(a) + S(4)sqrt(b)nlog(c(a + bxS(2))n)atan(sqrt(b)x/sqrt(a))/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(2)/xS(4), x), x, -log(c(a + bxS(2))n)S(2)/(S(3)x*S(3)) - S(4)bnlog(c(a + bxS(2))n)/(S(3)ax) - S(8)b(S(3)/2)n*S(2)log(S(2)Isqrt(a)/(Isqrt(a) - sqrt(b)x))atan(sqrt(b)x/sqrt(a))/(S(3)a(S(3)/2)) - S(4)Ib(S(3)/2)n*S(2)atan(sqrt(b)x/sqrt(a))S(2)/(S(3)a*(S(3)/2)) + S(8)b*(S(3)/2)n*S(2)atan(sqrt(b)x/sqrt(a))/(S(3)a*(S(3)/2)) - S(4)Ib(S(3)/2)n*S(2)polylog(S(2), (-sqrt(a) + Isqrt(b)x)/(sqrt(a) + Isqrt(b)x))/(S(3)a(S(3)/2)) - S(4)b*(S(3)/2)nlog(c(a + bxS(2))n)atan(sqrt(b)x/sqrt(a))/(S(3)a*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(2)/xS(6), x), x, -log(c(a + bxS(2))n)S(2)/(S(5)x*S(5)) - S(4)bnlog(c(a + bxS(2))n)/(S(15)ax*S(3)) - S(8)b*S(2)n*S(2)/(S(15)a*S(2)x) + S(4)bS(2)nlog(c(a + bxS(2))n)/(S(5)a*S(2)x) + S(8)b(S(5)/2)n*S(2)log(S(2)Isqrt(a)/(Isqrt(a) - sqrt(b)x))atan(sqrt(b)x/sqrt(a))/(S(5)a(S(5)/2)) + S(4)Ib(S(5)/2)n*S(2)atan(sqrt(b)x/sqrt(a))S(2)/(S(5)a*(S(5)/2)) - S(32)b*(S(5)/2)n*S(2)atan(sqrt(b)x/sqrt(a))/(S(15)a*(S(5)/2)) + S(4)Ib(S(5)/2)n*S(2)polylog(S(2), (-sqrt(a) + Isqrt(b)x)/(sqrt(a) + Isqrt(b)x))/(S(5)a(S(5)/2)) + S(4)b*(S(5)/2)nlog(c(a + bxS(2))n)atan(sqrt(b)x/sqrt(a))/(S(5)a*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(2)/xS(8), x), x, -log(c(a + bxS(2))n)S(2)/(S(7)x*S(7)) - S(4)bnlog(c(a + bxS(2))n)/(S(35)ax*S(5)) - S(8)b*S(2)n*S(2)/(S(105)a*S(2)x*S(3)) + S(4)b*S(2)nlog(c(a + bxS(2))n)/(S(21)a*S(2)x*S(3)) + S(64)b*S(3)n*S(2)/(S(105)a*S(3)x) - S(4)bS(3)nlog(c(a + bxS(2))n)/(S(7)a*S(3)x) - S(8)b(S(7)/2)n*S(2)log(S(2)Isqrt(a)/(Isqrt(a) - sqrt(b)x))atan(sqrt(b)x/sqrt(a))/(S(7)a(S(7)/2)) - S(4)Ib(S(7)/2)n*S(2)atan(sqrt(b)x/sqrt(a))S(2)/(S(7)a*(S(7)/2)) + S(184)b*(S(7)/2)n*S(2)atan(sqrt(b)x/sqrt(a))/(S(105)a*(S(7)/2)) - S(4)Ib(S(7)/2)n*S(2)polylog(S(2), (-sqrt(a) + Isqrt(b)x)/(sqrt(a) + Isqrt(b)x))/(S(7)a(S(7)/2)) - S(4)b*(S(7)/2)nlog(c(a + bxS(2))n)atan(sqrt(b)x/sqrt(a))/(S(7)a(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)log(c(a + bxS(2))n)S(3), x), x, -S(9)a*S(2)n*S(3)x*S(2)/(S(4)b*S(2)) + S(3)a*S(2)n*S(2)(a + bxS(2))log(c(a + bxS(2))n)/b*S(3) - S(3)a*S(2)n(a + bx*S(2))log(c(a + bxS(2))n)*S(2)/(S(2)bS(3)) + aS(2)(a + bx*S(2))log(c(a + bxS(2))n)*S(3)/(S(2)b*S(3)) + S(3)anS(3)x*S(4)/(S(8)b) - S(3)an*S(2)(a + bxS(2))S(2)log(c(a + bxS(2))n)/(S(4)bS(3)) + S(3)an(a + bxS(2))S(2)log(c(a + bxS(2))n)*S(2)/(S(4)b*S(3)) - a(a + bxS(2))S(2)log(c(a + bxS(2))n)*S(3)/(S(2)bS(3)) - nS(3)(a + bxS(2))S(3)/(S(27)bS(3)) + nS(2)(a + bxS(2))S(3)log(c(a + bxS(2))n)/(S(9)bS(3)) - n(a + bxS(2))S(3)log(c(a + bxS(2))n)*S(2)/(S(6)b*S(3)) + (a + bxS(2))S(3)log(c(a + bxS(2))n)S(3)/(S(6)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(c(a + bxS(2))n)S(3), x), x, S(21)anS(3)x*S(2)/(S(8)b) - S(3)an*S(2)(a + bxS(2))log(c(a + bxS(2))n)/b*S(2) + S(3)an(a + bxS(2))log(c(a + bxS(2))n)*S(2)/(S(2)b*S(2)) - a(a + bxS(2))log(c(a + bxS(2))n)*S(3)/(S(2)b*S(2)) - S(3)n*S(3)x*S(4)/S(16) + S(3)n*S(2)(a + bxS(2))S(2)log(c(a + bxS(2))n)/(S(8)bS(2)) - S(3)n(a + bxS(2))S(2)log(c(a + bxS(2))n)S(2)/(S(8)b*S(2)) + (a + bxS(2))S(2)log(c(a + bxS(2))n)S(3)/(S(4)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c(a + bxS(2))n)*S(3), x), x, -S(3)n*S(3)x*S(2) + S(3)n*S(2)(a + bxS(2))log(c(a + bxS(2))n)/b - S(3)n(a + bxS(2))log(c(a + bxS(2))n)*S(2)/(S(2)b) + (a/S(2) + bxS(2)/S(2))log(c(a + bxS(2))n)*S(3)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(3)/x, x), x, S(3)n*S(3)polylog(S(4), (a + bxS(2))/a) - S(3)n*S(2)log(c(a + bxS(2))n)polylog(S(3), (a + bx*S(2))/a) + S(3)nlog(c(a + bxS(2))n)S(2)polylog(S(2), (a + bxS(2))/a)/S(2) + log(c(a + bxS(2))n)S(3)log(-bxS(2)/a)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(3)/xS(3), x), x, -S(3)bnS(3)polylog(S(3), (a + bxS(2))/a)/a + S(3)bnS(2)log(c(a + bxS(2))n)polylog(S(2), (a + bx*S(2))/a)/a + S(3)bnlog(c(a + bxS(2))n)*S(2)log(-bxS(2)/a)/(S(2)a) - (a/S(2) + bxS(2)/S(2))log(c(a + bxS(2))n)*S(3)/(ax*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(3)/xS(5), x), x, -log(c(a + bxS(2))n)S(3)/(S(4)x*S(4)) + S(3)b*S(2)n*S(3)polylog(S(2), (a + bxS(2))/a)/(S(2)a*S(2)) + S(3)b*S(2)n*S(3)polylog(S(3), (a + bxS(2))/a)/(S(2)a*S(2)) + S(3)b*S(2)n*S(2)log(c(a + bxS(2))n)log(-bx*S(2)/a)/(S(2)a*S(2)) - S(3)b*S(2)n*S(2)log(c(a + bxS(2))n)polylog(S(2), (a + bx*S(2))/a)/(S(2)a*S(2)) - S(3)b*S(2)nlog(c(a + bxS(2))n)S(2)log(-bxS(2)/a)/(S(4)aS(2)) + bS(2)log(c(a + bxS(2))n)S(3)/(S(4)a*S(2)) - S(3)bn(a + bxS(2))log(c(a + bxS(2))n)*S(2)/(S(4)a*S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)S(3)/xS(7), x), x, -log(c(a + bxS(2))n)S(3)/(S(6)x*S(6)) - bnlog(c(a + bxS(2))n)S(2)/(S(4)axS(4)) - bS(2)n*S(2)log(c(a + bxS(2))n)/(S(2)aS(2)xS(2)) + bS(3)nS(3)log(x)/aS(3) - bS(3)nS(3)log(a + bxS(2))/(S(2)a*S(3)) - S(3)b*S(3)n*S(3)polylog(S(2), (a + bxS(2))/a)/(S(2)aS(3)) - bS(3)nS(3)polylog(S(3), (a + bxS(2))/a)/aS(3) - S(3)b*S(3)n*S(2)log(c(a + bxS(2))n)log(-bx*S(2)/a)/(S(2)aS(3)) + bS(3)nS(2)log(c(a + bxS(2))n)polylog(S(2), (a + bxS(2))/a)/aS(3) + b*S(3)nlog(c(a + bxS(2))n)S(2)log(-bxS(2)/a)/(S(2)aS(3)) + bS(3)nlog(c(a + bxS(2))n)*S(2)/(S(4)aS(3)) - bS(3)log(c(a + bxS(2))n)S(3)/(S(6)aS(3)) + bS(2)n(a + bxS(2))log(c(a + bxS(2))n)*S(2)/(S(2)a*S(3)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/log(c(a + bxS(2))n), x), x, -a(c(a + bxS(2))n)(-S(1)/n)(a + bxS(2))Ei(log(c(a + bxS(2))n)/n)/(S(2)bS(2)n) + (c(a + bxS(2))n)*(-S(2)/n)(a + bxS(2))S(2)Ei(S(2)log(c(a + bxS(2))n)/n)/(S(2)b*S(2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(c(a + bxS(2))n), x), x, (c(a + bxS(2))n)*(-S(1)/n)(a/S(2) + bxS(2)/S(2))Ei(log(c(a + bxS(2))n)/n)/(bn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(c(a + bxS(2))n)), x), x, Integral(S(1)/(xlog(c(a + bx)n)), (x, xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)log(c(a + bxS(2))n)), x), x, Integral(S(1)/(x*S(2)log(c(a + bx)n)), (x, xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/log(c(a + bxS(2))n)S(2), x), x, -a(c(a + bxS(2))n)*(-S(1)/n)(a + bxS(2))Ei(log(c(a + bxS(2))n)/n)/(S(2)bS(2)nS(2)) - xS(2)(a + bx*S(2))/(S(2)bnlog(c(a + bxS(2))n)) + (c(a + bxS(2))n)*(-S(2)/n)(a + bxS(2))S(2)Ei(S(2)log(c(a + bxS(2))n)/n)/(bS(2)n*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(c(a + bxS(2))n)S(2), x), x, (-a/S(2) - bx*S(2)/S(2))/(bnlog(c(a + bxS(2))n)) + (c(a + bxS(2))n)(-S(1)/n)(a/S(2) + bxS(2)/S(2))Ei(log(c(a + bxS(2))n)/n)/(bnS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(c(a + bxS(2))n)*S(2)), x), x, Integral(S(1)/(xlog(c(a + bx)n)S(2)), (x, xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)log(c(a + bxS(2))n)S(2)), x), x, Integral(S(1)/(xS(2)log(c(a + bx)n)S(2)), (x, xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/log(c(a + bxS(2))n)*S(3), x), x, -a(a + bxS(2))/(S(4)b*S(2)n*S(2)log(c(a + bxS(2))n)) - a(c(a + bxS(2))n)(-S(1)/n)(a + bxS(2))Ei(log(c(a + bxS(2))n)/n)/(S(4)bS(2)nS(3)) - xS(2)(a + bx*S(2))/(S(4)bnlog(c(a + bxS(2))n)S(2)) - xS(2)(a + bx*S(2))/(S(2)bnS(2)log(c(a + bxS(2))n)) + (c(a + bxS(2))n)*(-S(2)/n)(a + bxS(2))S(2)Ei(S(2)log(c(a + bxS(2))n)/n)/(bS(2)n*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(c(a + bxS(2))n)S(3), x), x, (-a/S(4) - bx*S(2)/S(4))/(bnlog(c(a + bxS(2))n)S(2)) + (-a/S(4) - bx*S(2)/S(4))/(bn*S(2)log(c(a + bxS(2))n)) + (c(a + bxS(2))n)*(-S(1)/n)(a/S(4) + bxS(2)/S(4))Ei(log(c(a + bxS(2))n)/n)/(bnS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(c(a + bxS(2))n)*S(3)), x), x, Integral(S(1)/(xlog(c(a + bx)n)S(3)), (x, xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)log(c(a + bxS(2))n)S(3)), x), x, Integral(S(1)/(xS(2)log(c(a + bx)n)S(3)), (x, xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/log(c(a + bxS(2))), x), x, Integral(xm/log(c(a + bxS(2))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/log(c(a + bx*S(2))), x), x, -ali(ac + bcxS(2))/(S(2)b*S(2)c) + Ei(S(2)log(ac + bcx*S(2)))/(S(2)b*S(2)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/log(c(a + bxS(2))), x), x, Integral(xS(2)/log(c(a + bx*S(2))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(c(a + bxS(2))), x), x, li(c(a + bxS(2)))/(S(2)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/log(c(a + bxS(2))), x), x, Integral(S(1)/log(c(a + bxS(2))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(c(a + bx*S(2)))), x), x, Integral(S(1)/(xlog(ac + bcx)), (x, xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)log(c(a + bxS(2)))), x), x, Integral(S(1)/(xS(2)log(c(a + bxS(2)))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)log(c(a + bxS(2)))), x), x, Integral(S(1)/(xS(2)log(ac + bcx)), (x, xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/log(c(a + bxS(2)))S(2), x), x, Integral(x*m/log(c(a + bxS(2)))S(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/log(c(a + bxS(2)))S(2), x), x, -ali(ac + bcxS(2))/(S(2)b*S(2)c) - x*S(2)(a + bxS(2))/(S(2)blog(ac + bcx*S(2))) + Ei(S(2)log(ac + bcxS(2)))/(bS(2)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/log(c(a + bxS(2)))S(2), x), x, Integral(x*S(2)/log(c(a + bxS(2)))S(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(c(a + bxS(2)))S(2), x), x, (-a/S(2) - bx*S(2)/S(2))/(blog(c(a + bx*S(2)))) + li(c(a + bxS(2)))/(S(2)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2)))(S(-2)), x), x, Integral(log(c(a + bxS(2)))(S(-2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(c(a + bxS(2)))S(2)), x), x, Integral(S(1)/(xlog(ac + bcx)S(2)), (x, xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)log(c(a + bxS(2)))S(2)), x), x, Integral(S(1)/(x*S(2)log(c(a + bxS(2)))S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)log(c(a + bxS(2)))S(2)), x), x, Integral(S(1)/(x*S(2)log(ac + bcx)S(2)), (x, xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/log(c(a + bxS(2)))S(3), x), x, Integral(xm/log(c(a + bxS(2)))S(3), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/log(c(a + bxS(2)))S(3), x), x, -a(a + bxS(2))/(S(4)b*S(2)log(ac + bcxS(2))) - ali(ac + bcxS(2))/(S(4)b*S(2)c) - x*S(2)(a + bxS(2))/(S(2)blog(ac + bcxS(2))) - xS(2)(a + bx*S(2))/(S(4)blog(ac + bcxS(2))S(2)) + Ei(S(2)log(ac + bcxS(2)))/(bS(2)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/log(c(a + bxS(2)))S(3), x), x, Integral(xS(2)/log(c(a + bxS(2)))S(3), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/log(c(a + bxS(2)))S(3), x), x, (-a/S(4) - bx*S(2)/S(4))/(blog(c(a + bx*S(2)))) + (-a/S(4) - bx*S(2)/S(4))/(blog(c(a + bxS(2)))S(2)) + li(c(a + bx*S(2)))/(S(4)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2)))(S(-3)), x), x, Integral(log(c(a + bxS(2)))(S(-3)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(c(a + bxS(2)))S(3)), x), x, Integral(S(1)/(xlog(ac + bcx)S(3)), (x, xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)log(c(a + bxS(2)))S(3)), x), x, Integral(S(1)/(x*S(2)log(c(a + bxS(2)))S(3)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)log(c(a + bxS(2)))S(3)), x), x, Integral(S(1)/(x*S(2)log(ac + bcx)S(3)), (x, xS(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(bx*m + S(1))/x, x), x, -polylog(S(2), -bx*m)/m, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(bx*m + S(2))/x, x), x, log(S(2))log(x) - polylog(S(2), -bxm/S(2))/m, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(2)bxm + S(6))/x, x), x, log(S(6))log(x) - polylog(S(2), -bxm/S(3))/m, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxm))/x, x), x, log(c(a + bxm))log(-bxm/a)/m + polylog(S(2), (a + bx*m)/a)/m, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxm)n)/x, x), x, npolylog(S(2), (a + bxm)/a)/m + log(c(a + bxm)n)log(-bxm/a)/m, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxm)n)S(2)/x, x), x, -S(2)n*S(2)polylog(S(3), (a + bxm)/a)/m + S(2)nlog(c(a + bxm)n)polylog(S(2), (a + bxm)/a)/m + log(c(a + bxm)n)S(2)log(-bxm/a)/m, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxm)n)S(3)/x, x), x, S(6)n*S(3)polylog(S(4), (a + bxm)/a)/m - S(6)n*S(2)log(c(a + bxm)n)polylog(S(3), (a + bx*m)/a)/m + S(3)nlog(c(a + bxm)n)S(2)polylog(S(2), (a + bxm)/a)/m + log(c(a + bxm)n)S(3)log(-bxm/a)/m, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(d(bx + cxS(2))n), x), x, nx*(m + S(1))hyper((S(1), m + S(1)), (m + S(2),), -cx/b)/(m + S(1))S(2) - S(2)nx(m + S(1))/(m + S(1))S(2) + x(m + S(1))log(d(bx + cxS(2))n)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)log(d(bx + cxS(2))n), x), x, bS(5)nlog(b + cx)/(S(5)cS(5)) - bS(4)nx/(S(5)cS(4)) + bS(3)nx*S(2)/(S(10)cS(3)) - bS(2)nx*S(3)/(S(15)c*S(2)) + bnxS(4)/(S(20)c) - S(2)nxS(5)/S(25) + xS(5)log(d(bx + cxS(2))n)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)log(d(bx + cxS(2))n), x), x, -bS(4)nlog(b + cx)/(S(4)cS(4)) + bS(3)nx/(S(4)cS(3)) - bS(2)nx*S(2)/(S(8)c*S(2)) + bnxS(3)/(S(12)c) - nxS(4)/S(8) + xS(4)log(d(bx + cxS(2))n)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(d(bx + cxS(2))n), x), x, bS(3)nlog(b + cx)/(S(3)cS(3)) - bS(2)nx/(S(3)c*S(2)) + bnxS(2)/(S(6)c) - S(2)nxS(3)/S(9) + xS(3)log(d(bx + cxS(2))n)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(d(bx + cxS(2))n), x), x, -b*S(2)nlog(b + cx)/(S(2)cS(2)) + bnx/(S(2)c) - nxS(2)/S(2) + xS(2)log(d(bx + cxS(2))n)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(bx + cxS(2))n), x), x, bnlog(b + cx)/c - S(2)nx + xlog(d(bx + cxS(2))n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(bx + cxS(2))n)/x, x), x, -nlog(x)S(2)/S(2) - nlog(x)log((b + cx)/b) - npolylog(S(2), -cx/b) + log(x)log(d(bx + cxS(2))n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(bx + cxS(2))n)/xS(2), x), x, -n/x - log(d(bx + cxS(2))n)/x + cnlog(x)/b - cnlog(b + cx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(bx + cxS(2))n)/x*S(3), x), x, -n/(S(4)x*S(2)) - log(d(bx + cxS(2))n)/(S(2)xS(2)) - cn/(S(2)bx) - c*S(2)nlog(x)/(S(2)bS(2)) + cS(2)nlog(b + cx)/(S(2)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(bx + cxS(2))n)/x*S(4), x), x, -n/(S(9)x*S(3)) - log(d(bx + cxS(2))n)/(S(3)xS(3)) - cn/(S(6)bxS(2)) + cS(2)n/(S(3)b*S(2)x) + c*S(3)nlog(x)/(S(3)bS(3)) - cS(3)nlog(b + cx)/(S(3)b*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(bx + cxS(2))n)/x*S(5), x), x, -n/(S(16)x*S(4)) - log(d(bx + cxS(2))n)/(S(4)xS(4)) - cn/(S(12)bxS(3)) + cS(2)n/(S(8)b*S(2)xS(2)) - cS(3)n/(S(4)b*S(3)x) - c*S(4)nlog(x)/(S(4)bS(4)) + cS(4)nlog(b + cx)/(S(4)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(d(a + bx + cxS(2))n), x), x, -S(2)cnx(m + S(2))hyper((S(1), m + S(2)), (m + S(3),), -S(2)cx/(b + sqrt(-S(4)ac + b*S(2))))/((b + sqrt(-S(4)ac + bS(2)))(m + S(1))(m + S(2))) - S(2)cnx*(m + S(2))hyper((S(1), m + S(2)), (m + S(3),), -S(2)cx/(b - sqrt(-S(4)ac + b*S(2))))/((b - sqrt(-S(4)ac + bS(2)))(m + S(1))(m + S(2))) + x(m + S(1))log(d(a + bx + cxS(2))n)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)log(d(a + bx + cxS(2))n), x), x, bnxS(4)/(S(20)c) + bnx*S(2)(-S(3)ac + b*S(2))/(S(10)c*S(3)) + bn(S(5)a*S(2)c*S(2) - S(5)abS(2)c + b*S(4))log(a + bx + cx*S(2))/(S(10)c*S(5)) - S(2)nxS(5)/S(25) + xS(5)log(d(a + bx + cxS(2))n)/S(5) - nx*S(3)(-S(2)ac/S(15) + bS(2)/S(15))/cS(2) + nx(-S(2)aS(2)c*S(2)/S(5) + S(4)abS(2)c/S(5) - bS(4)/S(5))/cS(4) + nsqrt(-S(4)ac + bS(2))(a*S(2)c*S(2)/S(5) - S(3)abS(2)c/S(5) + b*S(4)/S(5))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/cS(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)log(d(a + bx + cxS(2))n), x), x, bnxS(3)/(S(12)c) + bnx(-S(3)ac + bS(2))/(S(4)c*S(3)) - bnsqrt(-S(4)ac + bS(2))(-S(2)ac + b*S(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(S(4)c*S(4)) - nxS(4)/S(8) + xS(4)log(d(a + bx + cxS(2))n)/S(4) - nxS(2)(-ac/S(4) + bS(2)/S(8))/cS(2) - n(a*S(2)c*S(2)/S(4) - ab*S(2)c/S(2) + b*S(4)/S(8))log(a + bx + cxS(2))/cS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)log(d(a + bx + cxS(2))n), x), x, bnxS(2)/(S(6)c) + bn(-S(3)ac + b*S(2))log(a + bx + cx*S(2))/(S(6)c*S(3)) - S(2)nxS(3)/S(9) + xS(3)log(d(a + bx + cxS(2))n)/S(3) + nx(S(2)ac/S(3) - bS(2)/S(3))/cS(2) + nsqrt(-S(4)ac + b*S(2))(-ac/S(3) + bS(2)/S(3))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/cS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(d(a + bx + cxS(2))n), x), x, bnx/(S(2)c) - bnsqrt(-S(4)ac + bS(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(S(2)c*S(2)) - nxS(2)/S(2) + xS(2)log(d(a + bx + cxS(2))n)/S(2) - n(-ac/S(2) + b*S(2)/S(4))log(a + bx + cxS(2))/cS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n), x), x, bnlog(a + bx + cxS(2))/(S(2)c) - S(2)nx + xlog(d(a + bx + cxS(2))n) + nsqrt(-S(4)ac + bS(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n)/x, x), x, -nlog(x)log((b + S(2)cx - sqrt(-S(4)ac + b*S(2)))/(b - sqrt(-S(4)ac + bS(2)))) - nlog(x)log((b + S(2)cx + sqrt(-S(4)ac + bS(2)))/(b + sqrt(-S(4)ac + bS(2)))) - npolylog(S(2), -S(2)cx/(b - sqrt(-S(4)ac + b*S(2)))) - npolylog(S(2), -S(2)cx/(b + sqrt(-S(4)ac + b*S(2)))) + log(x)log(d(a + bx + cxS(2))n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n)/x*S(2), x), x, -log(d(a + bx + cxS(2))n)/x + bnlog(x)/a - bnlog(a + bx + cx*S(2))/(S(2)a) + nsqrt(-S(4)ac + bS(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n)/x*S(3), x), x, -log(d(a + bx + cxS(2))n)/(S(2)xS(2)) - bn/(S(2)ax) - bnsqrt(-S(4)ac + b*S(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(S(2)a*S(2)) - n(-ac + bS(2)/S(2))log(x)/a*S(2) + n(-ac/S(2) + bS(2)/S(4))log(a + bx + cxS(2))/aS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n)/xS(4), x), x, -log(d(a + bx + cxS(2))n)/(S(3)xS(3)) - bn/(S(6)ax*S(2)) + n(-S(2)ac/S(3) + bS(2)/S(3))/(aS(2)x) + bn(-S(3)ac + bS(2))log(x)/(S(3)aS(3)) - bn(-S(3)ac + bS(2))log(a + bx + cx*S(2))/(S(6)a*S(3)) + nsqrt(-S(4)ac + b*S(2))(-ac/S(3) + bS(2)/S(3))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/aS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n)/xS(5), x), x, -log(d(a + bx + cxS(2))n)/(S(4)xS(4)) - bn/(S(12)ax*S(3)) + n(-ac/S(4) + bS(2)/S(8))/(aS(2)x*S(2)) - bn(-S(3)ac + bS(2))/(S(4)a*S(3)x) - bnsqrt(-S(4)ac + b*S(2))(-S(2)ac + b*S(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(S(4)a*S(4)) + n(a*S(2)c*S(2)/S(4) - ab*S(2)c/S(2) + b*S(4)/S(8))log(a + bx + cxS(2))/aS(4) - n(aS(2)c*S(2)/S(2) - ab*S(2)c + b*S(4)/S(4))log(x)/aS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(xS(2) + x + S(1)), x), x, xlog(xS(2) + x + S(1)) - S(2)x + log(x*S(2) + x + S(1))/S(2) + sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3)), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate((d + ex)S(4)log(d(a + bx + cxS(2))n), x), x, -S(2)e*S(4)nxS(5)/S(25) + (d + ex)*S(5)log(d(a + bx + cxS(2))n)/(S(5)e) - e*S(3)nxS(4)(-be + S(10)cd)/(S(20)c) - e*S(2)nxS(3)(b*S(2)e*S(2) + S(20)c*S(2)d*S(2) - ce(S(2)ae + S(5)bd))/(S(15)c*S(2)) - enxS(2)(-b*S(3)e*S(3) + bceS(2)(S(3)ae + S(5)bd) + S(20)cS(3)d*S(3) - S(10)c*S(2)de(ae + bd))/(S(10)cS(3)) + nx(-bS(4)eS(4)/S(5) + bS(2)ce*S(3)(S(4)ae + S(5)bd)/S(5) - S(2)cS(4)d*S(4) + S(2)c*S(3)d*S(2)e(S(2)ae + bd) - c*S(2)e*S(2)(S(2)aS(2)e*S(2) + S(15)abde + S(10)b*S(2)dS(2))/S(5))/cS(4) + nsqrt(-S(4)ac + bS(2))(b*S(4)eS(4)/S(5) - bS(2)ce*S(3)(S(3)ae + S(5)bd)/S(5) + c*S(4)d*S(4) - S(2)c*S(3)d*S(2)e(ae + bd) + cS(2)e*S(2)(a*S(2)e*S(2) + S(10)abde + S(10)b*S(2)d*S(2))/S(5))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/cS(5) - n(-be/S(10) + cd/S(5))(b*S(4)eS(4) - bS(2)ce*S(3)(S(5)ae + S(3)bd) + c*S(4)d*S(4) - S(2)c*S(3)d*S(2)e(S(5)ae + bd) + c*S(2)e*S(2)(S(5)aS(2)e*S(2) + S(10)abde + S(4)b*S(2)d*S(2)))log(a + bx + cxS(2))/(cS(5)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*S(3)log(d(a + bx + cxS(2))n), x), x, -eS(3)nxS(4)/S(8) + (d + ex)*S(4)log(d(a + bx + cxS(2))n)/(S(4)e) - e*S(2)nxS(3)(-be + S(8)cd)/(S(12)c) - enx*S(2)(b*S(2)e*S(2) + S(12)c*S(2)d*S(2) - S(2)ce(ae + S(2)bd))/(S(8)c*S(2)) + nx(bS(3)e*S(3)/S(4) - bceS(2)(S(3)ae + S(4)bd)/S(4) - S(2)cS(3)dS(3) + cS(2)de(S(4)ae + S(3)bd)/S(2))/cS(3) + nsqrt(-S(4)ac + b*S(2))(-be/S(4) + cd/S(2))(bS(2)e*S(2) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/cS(4) - n(b*S(4)eS(4)/S(8) - bS(2)ce*S(3)(ae + bd)/S(2) + c*S(4)dS(4)/S(4) - cS(3)dS(2)e(S(3)ae + bd)/S(2) + c*S(2)e*S(2)(a*S(2)e*S(2) + S(6)abde + S(3)b*S(2)d*S(2))/S(4))log(a + bx + cxS(2))/(cS(4)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*S(2)log(d(a + bx + cxS(2))n), x), x, -S(2)e*S(2)nxS(3)/S(9) + (d + ex)*S(3)log(d(a + bx + cxS(2))n)/(S(3)e) - enx*S(2)(-be + S(6)cd)/(S(6)c) + nx(-b*S(2)e*S(2)/S(3) - S(2)c*S(2)d*S(2) + ce(S(2)ae + S(3)bd)/S(3))/cS(2) + nsqrt(-S(4)ac + b*S(2))(b*S(2)eS(2)/S(3) + cS(2)dS(2) - ce(ae + S(3)bd)/S(3))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/cS(3) - n(-be/S(6) + cd/S(3))(bS(2)eS(2) + cS(2)dS(2) - ce(S(3)ae + bd))log(a + bx + cxS(2))/(cS(3)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)log(d(a + bx + cxS(2))n), x), x, -enxS(2)/S(2) + nx(be/(S(2)c) - S(2)d) + (d + ex)S(2)log(d(a + bx + cxS(2))n)/(S(2)e) + nsqrt(-S(4)ac + bS(2))(-be/S(2) + cd)atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/cS(2) - n(b*S(2)eS(2)/S(4) + cS(2)dS(2)/S(2) - ce(ae + bd)/S(2))log(a + bx + cxS(2))/(cS(2)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n), x), x, bnlog(a + bx + cx*S(2))/(S(2)c) - S(2)nx + xlog(d(a + bx + cxS(2))n) + nsqrt(-S(4)ac + bS(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n)/(d + ex), x), x, -nlog(-e(b + S(2)cx - sqrt(-S(4)ac + bS(2)))/(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2)))))log(d + ex)/e - nlog(-e(b + S(2)cx + sqrt(-S(4)ac + bS(2)))/(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))log(d + ex)/e - npolylog(S(2), S(2)c(d + ex)/(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/e - npolylog(S(2), S(2)c(d + ex)/(-be + S(2)cd + esqrt(-S(4)ac + bS(2))))/e + log(d(a + bx + cxS(2))n)log(d + ex)/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n)/(d + ex)*S(2), x), x, nsqrt(-S(4)ac + b*S(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(ae*S(2) - bde + cd*S(2)) + n(-be/S(2) + cd)log(a + bx + cxS(2))/(e(aeS(2) - bde + cd*S(2))) + n(be - S(2)cd)log(d + ex)/(e(aeS(2) - bde + cd*S(2))) - log(d(a + bx + cxS(2))n)/(e(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n)/(d + ex)*S(3), x), x, nsqrt(-S(4)ac + b*S(2))(-be/S(2) + cd)atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(ae*S(2) - bde + cdS(2))S(2) + n(bS(2)eS(2)/S(4) + cS(2)dS(2)/S(2) - ce(ae + bd)/S(2))log(a + bx + cx*S(2))/(e(aeS(2) - bde + cdS(2))S(2)) - n(bS(2)eS(2)/S(2) + cS(2)dS(2) - ce(ae + bd))log(d + ex)/(e(aeS(2) - bde + cdS(2))S(2)) + n(-be/S(2) + cd)/(e(d + ex)(aeS(2) - bde + cd*S(2))) - log(d(a + bx + cxS(2))n)/(S(2)e(d + ex)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n)/(d + ex)S(4), x), x, nsqrt(-S(4)ac + b*S(2))(b*S(2)eS(2)/S(3) + cS(2)dS(2) - ce(ae + S(3)bd)/S(3))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(ae*S(2) - bde + cdS(2))S(3) - n(-be/S(3) + S(2)cd/S(3))(bS(2)eS(2) + cS(2)dS(2) - ce(S(3)ae + bd))log(d + ex)/(e(ae*S(2) - bde + cdS(2))S(3)) + n(-be/S(6) + cd/S(3))(b*S(2)eS(2) + cS(2)dS(2) - ce(S(3)ae + bd))log(a + bx + cxS(2))/(e(aeS(2) - bde + cdS(2))S(3)) + n(bS(2)e*S(2)/S(3) + S(2)c*S(2)d*S(2)/S(3) - S(2)ce(ae + bd)/S(3))/(e(d + ex)(ae*S(2) - bde + cdS(2))S(2)) + n(-be/S(6) + cd/S(3))/(e(d + ex)S(2)(aeS(2) - bde + cd*S(2))) - log(d(a + bx + cxS(2))n)/(S(3)e(d + ex)S(3)), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n)/(d + ex)S(5), x), x, nsqrt(-S(4)ac + b*S(2))(-be/S(4) + cd/S(2))(bS(2)e*S(2) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(ae*S(2) - bde + cdS(2))S(4) + n(bS(4)eS(4)/S(8) - bS(2)ce*S(3)(ae + bd)/S(2) + c*S(4)dS(4)/S(4) - cS(3)dS(2)e(S(3)ae + bd)/S(2) + c*S(2)e*S(2)(a*S(2)e*S(2) + S(6)abde + S(3)b*S(2)d*S(2))/S(4))log(a + bx + cx*S(2))/(e(aeS(2) - bde + cdS(2))S(4)) - n(bS(4)eS(4)/S(4) - bS(2)ce*S(3)(ae + bd) + c*S(4)dS(4)/S(2) - cS(3)dS(2)e(S(3)ae + bd) + c*S(2)e*S(2)(a*S(2)e*S(2) + S(6)abde + S(3)b*S(2)d*S(2))/S(2))log(d + ex)/(e(aeS(2) - bde + cdS(2))S(4)) + n(-be/S(4) + cd/S(2))(b*S(2)eS(2) + cS(2)dS(2) - ce(S(3)ae + bd))/(e(d + ex)(ae*S(2) - bde + cdS(2))S(3)) + n(bS(2)eS(2)/S(8) + cS(2)dS(2)/S(4) - ce(ae + bd)/S(4))/(e(d + ex)S(2)(aeS(2) - bde + cdS(2))S(2)) + n(-be/S(12) + cd/S(6))/(e(d + ex)S(3)(aeS(2) - bde + cd*S(2))) - log(d(a + bx + cxS(2))n)/(S(4)e(d + ex)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + cxS(2))n)/(ae + cex*S(2)), x), x, S(2)nlog(S(2)Isqrt(a)/(Isqrt(a) - sqrt(c)x))atan(sqrt(c)x/sqrt(a))/(sqrt(a)sqrt(c)e) + Inatan(sqrt(c)x/sqrt(a))*S(2)/(sqrt(a)sqrt(c)e) + Inpolylog(S(2), (-sqrt(a) + Isqrt(c)x)/(sqrt(a) + Isqrt(c)x))/(sqrt(a)sqrt(c)e) + log(d(a + cxS(2))n)atan(sqrt(c)x/sqrt(a))/(sqrt(a)sqrt(c)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n)/(ae + bex + cexS(2)), x), x, -S(4)nlog(S(2)/(-b/sqrt(-S(4)ac + bS(2)) - S(2)cx/sqrt(-S(4)ac + bS(2)) + S(1)))atanh(b/sqrt(-S(4)ac + b*S(2)) + S(2)cx/sqrt(-S(4)ac + bS(2)))/(esqrt(-S(4)ac + b*S(2))) + S(2)natanh(b/sqrt(-S(4)ac + bS(2)) + S(2)cx/sqrt(-S(4)ac + bS(2)))S(2)/(esqrt(-S(4)ac + b*S(2))) - S(2)npolylog(S(2), (-b/sqrt(-S(4)ac + bS(2)) - S(2)cx/sqrt(-S(4)ac + bS(2)) + S(-1))/(-b/sqrt(-S(4)ac + bS(2)) - S(2)cx/sqrt(-S(4)ac + bS(2)) + S(1)))/(esqrt(-S(4)ac + b*S(2))) - S(2)log(d(a + bx + cxS(2))n)atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(esqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(g(a + bx + cxS(2))n)/(d + exS(2)), x), x, nlog(sqrt(e)(-b - S(2)cx + sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-d) - sqrt(e)(b - sqrt(-S(4)ac + b*S(2)))))log(sqrt(e)x + sqrt(-d))/(S(2)sqrt(e)sqrt(-d)) - nlog(sqrt(e)(b + S(2)cx - sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-d) + sqrt(e)(b - sqrt(-S(4)ac + b*S(2)))))log(-sqrt(e)x + sqrt(-d))/(S(2)sqrt(e)sqrt(-d)) + nlog(sqrt(e)(-b - S(2)cx - sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-d) - sqrt(e)(b + sqrt(-S(4)ac + b*S(2)))))log(sqrt(e)x + sqrt(-d))/(S(2)sqrt(e)sqrt(-d)) - nlog(sqrt(e)(b + S(2)cx + sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-d) + sqrt(e)(b + sqrt(-S(4)ac + b*S(2)))))log(-sqrt(e)x + sqrt(-d))/(S(2)sqrt(e)sqrt(-d)) + npolylog(S(2), S(2)c(sqrt(e)x + sqrt(-d))/(S(2)csqrt(-d) - sqrt(e)(b - sqrt(-S(4)ac + b*S(2)))))/(S(2)sqrt(e)sqrt(-d)) - npolylog(S(2), S(2)c(-sqrt(e)x + sqrt(-d))/(S(2)csqrt(-d) + sqrt(e)(b - sqrt(-S(4)ac + b*S(2)))))/(S(2)sqrt(e)sqrt(-d)) + npolylog(S(2), S(2)c(sqrt(e)x + sqrt(-d))/(S(2)csqrt(-d) - sqrt(e)(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)sqrt(e)sqrt(-d)) - npolylog(S(2), S(2)c(-sqrt(e)x + sqrt(-d))/(S(2)csqrt(-d) + sqrt(e)(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)sqrt(e)sqrt(-d)) + log(g(a + bx + cxS(2))n)log(-sqrt(e)x + sqrt(-d))/(S(2)sqrt(e)sqrt(-d)) - log(g(a + bx + cxS(2))n)log(sqrt(e)x + sqrt(-d))/(S(2)sqrt(e)sqrt(-d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(g(a + bx + cxS(2))n)/(d + ex + fx*S(2)), x), x, -nlog(f(b + S(2)cx + sqrt(-S(4)ac + bS(2)))/(-c(e - sqrt(-S(4)df + e*S(2))) + f(b + sqrt(-S(4)ac + b*S(2)))))log(e + S(2)fx - sqrt(-S(4)df + e*S(2)))/sqrt(-S(4)df + eS(2)) + nlog(f(b + S(2)cx - sqrt(-S(4)ac + bS(2)))/(-c(e + sqrt(-S(4)df + e*S(2))) + f(b - sqrt(-S(4)ac + b*S(2)))))log(e + S(2)fx + sqrt(-S(4)df + e*S(2)))/sqrt(-S(4)df + eS(2)) + nlog(f(b + S(2)cx + sqrt(-S(4)ac + bS(2)))/(-c(e + sqrt(-S(4)df + e*S(2))) + f(b + sqrt(-S(4)ac + b*S(2)))))log(e + S(2)fx + sqrt(-S(4)df + e*S(2)))/sqrt(-S(4)df + eS(2)) - nlog(-f(b + S(2)cx - sqrt(-S(4)ac + bS(2)))/(-bf + ce - csqrt(-S(4)df + e*S(2)) + fsqrt(-S(4)ac + b*S(2))))log(e + S(2)fx - sqrt(-S(4)df + e*S(2)))/sqrt(-S(4)df + eS(2)) - npolylog(S(2), -c(e + S(2)fx - sqrt(-S(4)df + eS(2)))/(-c(e - sqrt(-S(4)df + e*S(2))) + f(b - sqrt(-S(4)ac + b*S(2)))))/sqrt(-S(4)df + eS(2)) - npolylog(S(2), -c(e + S(2)fx - sqrt(-S(4)df + eS(2)))/(-c(e - sqrt(-S(4)df + e*S(2))) + f(b + sqrt(-S(4)ac + b*S(2)))))/sqrt(-S(4)df + eS(2)) + npolylog(S(2), -c(e + S(2)fx + sqrt(-S(4)df + eS(2)))/(-c(e + sqrt(-S(4)df + e*S(2))) + f(b - sqrt(-S(4)ac + b*S(2)))))/sqrt(-S(4)df + eS(2)) + npolylog(S(2), -c(e + S(2)fx + sqrt(-S(4)df + eS(2)))/(-c(e + sqrt(-S(4)df + e*S(2))) + f(b + sqrt(-S(4)ac + b*S(2)))))/sqrt(-S(4)df + eS(2)) + log(g(a + bx + cxS(2))n)log(e + S(2)fx - sqrt(-S(4)df + eS(2)))/sqrt(-S(4)df + eS(2)) - log(g(a + bx + cxS(2))n)log(e + S(2)fx + sqrt(-S(4)df + eS(2)))/sqrt(-S(4)df + eS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(bx + cxS(2))n)*S(2), x), x, -S(2)bnS(2)log(-cx/b)log(b + cx)/c - bn*S(2)log(b + cx)S(2)/c - S(4)bnS(2)log(b + cx)/c - S(2)bnS(2)polylog(S(2), (b + cx)/b)/c + S(2)bnlog(d(bx + cxS(2))n)log(b + cx)/c + S(8)n*S(2)x - S(4)nxlog(d(bx + cxS(2))n) + xlog(d(bx + cxS(2))n)*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx + cxS(2))n)*S(2), x), x, -S(2)bnS(2)log(a + bx + cx*S(2))/c + S(8)n*S(2)x - S(4)nxlog(d(a + bx + cxS(2))n) + xlog(d(a + bx + cxS(2))n)S(2) - nS(2)(b - sqrt(-S(4)ac + bS(2)))log((b/S(2) + cx + sqrt(-S(4)ac + bS(2))/S(2))/sqrt(-S(4)ac + bS(2)))log(b + S(2)cx - sqrt(-S(4)ac + bS(2)))/c - nS(2)(b - sqrt(-S(4)ac + bS(2)))log(b + S(2)cx - sqrt(-S(4)ac + bS(2)))S(2)/(S(2)c) - nS(2)(b - sqrt(-S(4)ac + b*S(2)))polylog(S(2), (-b/S(2) - cx + sqrt(-S(4)ac + bS(2))/S(2))/sqrt(-S(4)ac + bS(2)))/c - nS(2)(b + sqrt(-S(4)ac + b*S(2)))log((-b/S(2) - cx + sqrt(-S(4)ac + bS(2))/S(2))/sqrt(-S(4)ac + bS(2)))log(b + S(2)cx + sqrt(-S(4)ac + bS(2)))/c - nS(2)(b + sqrt(-S(4)ac + bS(2)))log(b + S(2)cx + sqrt(-S(4)ac + bS(2)))S(2)/(S(2)c) - nS(2)(b + sqrt(-S(4)ac + b*S(2)))polylog(S(2), (b/S(2) + cx + sqrt(-S(4)ac + bS(2))/S(2))/sqrt(-S(4)ac + bS(2)))/c - S(4)n*S(2)sqrt(-S(4)ac + b*S(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/c + n(b - sqrt(-S(4)ac + b*S(2)))log(d(a + bx + cxS(2))n)log(b + S(2)cx - sqrt(-S(4)ac + b*S(2)))/c + n(b + sqrt(-S(4)ac + b*S(2)))log(d(a + bx + cxS(2))n)log(b + S(2)cx + sqrt(-S(4)ac + bS(2)))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(xS(2) + x + S(1))/(xS(2) + S(3)x + S(2)), x), x, xlog(xS(2) + x + S(1)) - S(2)x - log((-S(2)x + S(-1) - sqrt(S(3))I)/(S(1) - sqrt(S(3))I))log(S(2)x + S(2)) - log((-S(2)x + S(-1) + sqrt(S(3))I)/(S(1) + sqrt(S(3))I))log(S(2)x + S(2)) + S(4)log((-S(2)x + S(-1) - sqrt(S(3))I)/(S(3) - sqrt(S(3))I))log(S(2)x + S(4)) + S(4)log((-S(2)x + S(-1) + sqrt(S(3))I)/(S(3) + sqrt(S(3))I))log(S(2)x + S(4)) + log(S(2)x + S(2))log(x*S(2) + x + S(1)) - S(4)log(S(2)x + S(4))log(xS(2) + x + S(1)) + log(xS(2) + x + S(1))/S(2) + sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3)) - polylog(S(2), (S(2)x + S(2))/(S(1) - sqrt(S(3))I)) - polylog(S(2), (S(2)x + S(2))/(S(1) + sqrt(S(3))I)) + S(4)polylog(S(2), (S(2)x + S(4))/(S(3) - sqrt(S(3))I)) + S(4)polylog(S(2), (S(2)x + S(4))/(S(3) + sqrt(S(3))I)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(xS(2) + x + S(1))S(2), x), x, xlog(xS(2) + x + S(1))S(2) - S(4)xlog(x*S(2) + x + S(1)) + S(8)x - (S(1) + sqrt(S(3))I)log(sqrt(S(3))I(S(2)x + S(1) - sqrt(S(3))I)/S(6))log(S(2)x + S(1) + sqrt(S(3))I) - (S(1) - sqrt(S(3))I)log(-sqrt(S(3))I(S(2)x + S(1) + sqrt(S(3))I)/S(6))log(S(2)x + S(1) - sqrt(S(3))I) - (S(1) - sqrt(S(3))I)log(S(2)x + S(1) - sqrt(S(3))I)*S(2)/S(2) + (S(1) - sqrt(S(3))I)log(S(2)x + S(1) - sqrt(S(3))I)log(x*S(2) + x + S(1)) - (S(1) + sqrt(S(3))I)log(S(2)x + S(1) + sqrt(S(3))I)S(2)/S(2) + (S(1) + sqrt(S(3))I)log(S(2)x + S(1) + sqrt(S(3))I)log(x*S(2) + x + S(1)) - S(2)log(x*S(2) + x + S(1)) - S(4)sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3)) - (S(1) - sqrt(S(3))I)polylog(S(2), sqrt(S(3))I(S(2)x + S(1) - sqrt(S(3))I)/S(6)) - (S(1) + sqrt(S(3))I)polylog(S(2), -sqrt(S(3))I(S(2)x + S(1) + sqrt(S(3))I)/S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(xS(2) + x + S(-1))S(2)/xS(3), x), x, S(3)log(x)log((S(2)x + S(1) + sqrt(S(5)))/(S(1) + sqrt(S(5)))) - S(3)log(x)log(x*S(2) + x + S(-1)) + log(x) - (sqrt(S(5)) + S(3))log(sqrt(S(5))(x + S(1)/2 + sqrt(S(5))/S(2))/S(5))log(S(2)x - sqrt(S(5)) + S(1))/S(2) - (-sqrt(S(5)) + S(3))log(S(2)x + S(1) + sqrt(S(5)))S(2)/S(4) + (-sqrt(S(5)) + S(3))log(S(2)x + S(1) + sqrt(S(5)))log(x*S(2) + x + S(-1))/S(2) - (-sqrt(S(5)) + S(1))log(S(2)x + S(1) + sqrt(S(5)))/S(2) - (sqrt(S(5)) + S(3))log(S(2)x - sqrt(S(5)) + S(1))S(2)/S(4) + (sqrt(S(5)) + S(3))log(S(2)x - sqrt(S(5)) + S(1))log(x*S(2) + x + S(-1))/S(2) - (S(1) + sqrt(S(5)))log(S(2)x - sqrt(S(5)) + S(1))/S(2) + S(3)log(S(-1)/2 + sqrt(S(5))/S(2))log(S(2)x - sqrt(S(5)) + S(1)) - (-sqrt(S(5)) + S(3))log(S(2)sqrt(S(5)))log(S(2)x - sqrt(S(5)) + S(1))/S(2) - (sqrt(S(5)) + S(3))polylog(S(2), sqrt(S(5))(-x + S(-1)/2 + sqrt(S(5))/S(2))/S(5))/S(2) + (-sqrt(S(5)) + S(3))polylog(S(2), sqrt(S(5))(-x + S(-1)/2 + sqrt(S(5))/S(2))/S(5))/S(2) + S(3)polylog(S(2), -S(2)x/(S(1) + sqrt(S(5)))) - S(3)polylog(S(2), (S(2)x - sqrt(S(5)) + S(1))/(-sqrt(S(5)) + S(1))) + log(xS(2) + x + S(-1))/x - log(xS(2) + x + S(-1))*S(2)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(S(4)x + S(4)sqrt(x(x + S(-1))) + S(-1)), x), x, x*S(4)log(S(4)x + S(4)sqrt(xS(2) - x) + S(-1))/S(4) - xS(4)/S(32) + xS(3)/S(192) - xS(2)/S(1024) - x(xS(2) - x)(S(3)/2)/S(32) + x/S(4096) + (-S(149)x/S(1024) + S(149)/2048)sqrt(xS(2) - x) - (xS(2) - x)(S(3)/2)/S(12) - S(683)sqrt(x*S(2) - x)/S(4096) - log(S(8)x + S(1))/S(32768) - S(1537)atanh(x/sqrt(xS(2) - x))/S(16384) + atanh((-S(5)x/S(3) + S(1)/6)/sqrt(xS(2) - x))/S(32768), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(S(4)x + S(4)sqrt(x(x + S(-1))) + S(-1)), x), x, x*S(3)log(S(4)x + S(4)sqrt(xS(2) - x) + S(-1))/S(3) - xS(3)/S(18) + x*S(2)/S(96) - x/S(384) + (-S(5)x/S(32) + S(5)/64)sqrt(xS(2) - x) - (xS(2) - x)(S(3)/2)/S(18) - S(85)sqrt(x*S(2) - x)/S(384) + log(S(8)x + S(1))/S(3072) - S(223)atanh(x/sqrt(xS(2) - x))/S(1536) - atanh((-S(5)x/S(3) + S(1)/6)/sqrt(x*S(2) - x))/S(3072), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(S(4)x + S(4)sqrt(x(x + S(-1))) + S(-1)), x), x, xS(2)log(S(4)x + S(4)sqrt(xS(2) - x) + S(-1))/S(2) - xS(2)/S(8) + x/S(32) + (-x/S(8) + S(1)/16)sqrt(xS(2) - x) - S(11)sqrt(x*S(2) - x)/S(32) - log(S(8)x + S(1))/S(256) - S(33)atanh(x/sqrt(xS(2) - x))/S(128) + atanh((-S(5)x/S(3) + S(1)/6)/sqrt(x*S(2) - x))/S(256), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(4)x + S(4)sqrt(x(x + S(-1))) + S(-1)), x), x, xlog(S(4)x + S(4)sqrt(xS(2) - x) + S(-1)) - x/S(2) - sqrt(xS(2) - x)/S(2) + log(S(8)x + S(1))/S(16) - S(7)atanh(x/sqrt(xS(2) - x))/S(8) - atanh((-S(5)x/S(3) + S(1)/6)/sqrt(x*S(2) - x))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(4)x + S(4)sqrt(x(x + S(-1))) + S(-1))/x, x), x, Integral(log(S(4)x + S(4)sqrt(x*S(2) - x) + S(-1))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(4)x + S(4)sqrt(x(x + S(-1))) + S(-1))/x*S(2), x), x, S(4)log(x) - S(4)log(S(8)x + S(1)) + S(4)atanh((-S(5)x/S(3) + S(1)/6)/sqrt(x*S(2) - x)) + S(4)sqrt(x*S(2) - x)/x - log(S(4)x + S(4)sqrt(xS(2) - x) + S(-1))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(4)x + S(4)sqrt(x(x + S(-1))) + S(-1))/x*S(3), x), x, -S(16)log(x) + S(16)log(S(8)x + S(1)) - S(16)atanh((-S(5)x/S(3) + S(1)/6)/sqrt(x*S(2) - x)) - S(10)sqrt(x*S(2) - x)/x - S(2)/x - log(S(4)x + S(4)sqrt(xS(2) - x) + S(-1))/(S(2)x*S(2)) - S(2)(xS(2) - x)(S(3)/2)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)log(S(4)x + S(4)sqrt(x(x + S(-1))) + S(-1)), x), x, S(2)x*(S(5)/2)log(S(4)x + S(4)sqrt(x*S(2) - x) + S(-1))/S(5) - S(2)x(S(5)/2)/S(25) + x(S(3)/2)/S(60) - sqrt(x)/S(160) + sqrt(S(2))atan(S(2)sqrt(S(2))sqrt(x))/S(640) - S(2)(xS(2) - x)(S(3)/2)/(S(25)sqrt(x)) - S(127)sqrt(x*S(2) - x)/(S(480)sqrt(x)) - sqrt(S(2))sqrt(xS(2) - x)atan(S(2)sqrt(S(2))sqrt(x + S(-1))/S(3))/(S(640)sqrt(x)sqrt(x + S(-1))) - (-S(2)x/S(15) + S(2)/15)sqrt(x*S(2) - x)/(sqrt(x)(sqrt(x) + S(1))) - (-S(2)x/S(15) + S(2)/15)sqrt(x*S(2) - x)/(sqrt(x)(-sqrt(x) + S(1))) - S(71)(xS(2) - x)(S(3)/2)/(S(300)x*(S(3)/2)), expand=True, _diff=True, _numerical=True) # failing due to apart assert rubi_test(rubi_integrate(sqrt(x)log(S(4)x + S(4)sqrt(x(x + S(-1))) + S(-1)), x), x, S(2)x*(S(3)/2)log(S(4)x + S(4)sqrt(x*S(2) - x) + S(-1))/S(3) - S(2)x*(S(3)/2)/S(9) + sqrt(x)/S(12) - sqrt(S(2))atan(S(2)sqrt(S(2))sqrt(x))/S(48) - S(17)sqrt(xS(2) - x)/(S(36)sqrt(x)) + sqrt(S(2))sqrt(xS(2) - x)atan(S(2)sqrt(S(2))sqrt(x + S(-1))/S(3))/(S(48)sqrt(x)sqrt(x + S(-1))) - (-S(2)x/S(9) + S(2)/9)sqrt(x*S(2) - x)/(sqrt(x)(sqrt(x) + S(1))) - (-S(2)x/S(9) + S(2)/9)sqrt(x*S(2) - x)/(sqrt(x)(-sqrt(x) + S(1))) - S(2)(xS(2) - x)(S(3)/2)/(S(9)x*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(4)x + S(4)sqrt(x(x + S(-1))) + S(-1))/sqrt(x), x), x, S(2)sqrt(x)log(S(4)x + S(4)sqrt(x*S(2) - x) + S(-1)) - S(2)sqrt(x) + sqrt(S(2))atan(S(2)sqrt(S(2))sqrt(x))/S(2) - S(2)sqrt(x*S(2) - x)/(S(3)sqrt(x)) - sqrt(S(2))sqrt(xS(2) - x)atan(S(2)sqrt(S(2))sqrt(x + S(-1))/S(3))/(S(2)sqrt(x)sqrt(x + S(-1))) - (-S(2)x/S(3) + S(2)/3)sqrt(x*S(2) - x)/(sqrt(x)(sqrt(x) + S(1))) - (-S(2)x/S(3) + S(2)/3)sqrt(x*S(2) - x)/(sqrt(x)(-sqrt(x) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(4)x + S(4)sqrt(x(x + S(-1))) + S(-1))/x(S(3)/2), x), x, S(4)sqrt(S(2))atan(S(2)sqrt(S(2))sqrt(x)) - S(8)atan(sqrt(x)/sqrt(x*S(2) - x)) - S(4)sqrt(x*S(2) - x)/(S(3)sqrt(x)) - S(2)log(S(4)x + S(4)sqrt(xS(2) - x) + S(-1))/sqrt(x) - S(4)sqrt(S(2))sqrt(xS(2) - x)atan(S(2)sqrt(S(2))sqrt(x + S(-1))/S(3))/(sqrt(x)sqrt(x + S(-1))) + (-S(2)x/S(3) + S(2)/3)sqrt(xS(2) - x)/(sqrt(x)(sqrt(x) + S(1))) + (-S(2)x/S(3) + S(2)/3)sqrt(x*S(2) - x)/(sqrt(x)(-sqrt(x) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(4)x + S(4)sqrt(x(x + S(-1))) + S(-1))/x(S(5)/2), x), x, -S(32)sqrt(S(2))atan(S(2)sqrt(S(2))sqrt(x))/S(3) + S(44)atan(sqrt(x)/sqrt(x*S(2) - x))/S(3) - S(4)sqrt(x*S(2) - x)/(S(9)sqrt(x)) - S(16)/(S(3)sqrt(x)) + S(32)sqrt(S(2))sqrt(xS(2) - x)atan(S(2)sqrt(S(2))sqrt(x + S(-1))/S(3))/(S(3)sqrt(x)sqrt(x + S(-1))) + (-S(2)x/S(9) + S(2)/9)sqrt(x*S(2) - x)/(sqrt(x)(sqrt(x) + S(1))) + (-S(2)x/S(9) + S(2)/9)sqrt(x*S(2) - x)/(sqrt(x)(-sqrt(x) + S(1))) + S(4)sqrt(xS(2) - x)/(S(3)x*(S(3)/2)) - S(2)log(S(4)x + S(4)sqrt(x*S(2) - x) + S(-1))/(S(3)x(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((a + x)/x)/x, x), x, polylog(S(2), -a/x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((a + xS(2))/xS(2))/x, x), x, polylog(S(2), -a/xS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x*(-n)(a + x*n))/x, x), x, polylog(S(2), -ax*(-n))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((a + bx)/x)/x, x), x, -log(-a/(bx))log(a/x + b) - polylog(S(2), (a/x + b)/b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((a + bxS(2))/xS(2))/x, x), x, -log(-a/(bx*S(2)))log(a/xS(2) + b)/S(2) - polylog(S(2), (a/xS(2) + b)/b)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x*(-n)(a + bxn))/x, x), x, -log(-ax*(-n)/b)log(ax(-n) + b)/n - polylog(S(2), (ax*(-n) + b)/b)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((a + bx)/x)/(c + dx), x), x, log((a + bx)/x)log(c + dx)/d + log(-dx/c)log(c + dx)/d - log(-d(a + bx)/(-ad + bc))log(c + dx)/d + polylog(S(2), (c + dx)/c)/d - polylog(S(2), b(c + dx)/(-ad + bc))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((a + bxS(2))/xS(2))/(c + dx), x), x, S(2)log(-dx/c)log(c + dx)/d - log(-d(sqrt(b)x + sqrt(-a))/(sqrt(b)c - dsqrt(-a)))log(c + dx)/d - log(d(-sqrt(b)x + sqrt(-a))/(sqrt(b)c + dsqrt(-a)))log(c + dx)/d + log(c + dx)log(a/x*S(2) + b)/d + S(2)polylog(S(2), (c + dx)/c)/d - polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c - dsqrt(-a)))/d - polylog(S(2), sqrt(b)(c + dx)/(sqrt(b)c + dsqrt(-a)))/d, expand=True, _diff=True, _numerical=True) # recursion sympy and mathematica assert rubi_test(rubi_integrate(log(x(-n)(a + bxn))/(c + dx), x), x, anIntegral(log(c + dx)/(x(a + bxn)), x)/d + log(c + dx)log(ax*(-n) + b)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(4), x), x, (a + bx)log(e((a + bx)/(c + dx))n)S(4)/b + nS(4)(-S(24)ad + S(24)bc)polylog(S(4), d(a + bx)/(b(c + dx)))/(bd) - n*S(3)(-S(24)ad + S(24)bc)log(e((a + bx)/(c + dx))*n)polylog(S(3), d(a + bx)/(b(c + dx)))/(bd) + nS(2)(-S(12)ad + S(12)bc)log(e((a + bx)/(c + dx))n)S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/(bd) + n(-S(4)ad + S(4)bc)log(e((a + bx)/(c + dx))n)S(3)log((-ad + bc)/(b(c + dx)))/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(3), x), x, (a + bx)log(e((a + bx)/(c + dx))n)S(3)/b - nS(3)(-S(6)ad + S(6)bc)polylog(S(3), d(a + bx)/(b(c + dx)))/(bd) + n*S(2)(-S(6)ad + S(6)bc)log(e((a + bx)/(c + dx))*n)polylog(S(2), d(a + bx)/(b(c + dx)))/(bd) + n(-S(3)ad + S(3)bc)log(e((a + bx)/(c + dx))n)S(2)log((-ad + bc)/(b(c + dx)))/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(2), x), x, (a + bx)log(e((a + bx)/(c + dx))n)S(2)/b + n*S(2)(-S(2)ad + S(2)bc)polylog(S(2), d(a + bx)/(b(c + dx)))/(bd) + n(-S(2)ad + S(2)bc)log(e((a + bx)/(c + dx))n)log((-ad + bc)/(b(c + dx)))/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))*n), x), x, (a + bx)log(e((a + bx)/(c + dx))*n)/b - n(-ad + bc)log(c + dx)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/log(e((a + bx)/(c + dx))*n), x), x, Integral(S(1)/log(e((a + bx)/(c + dx))*n), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)(S(-2)), x), x, Integral(log(e((a + bx)/(c + dx))n)(S(-2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(3)/x, x), x, S(6)nS(3)polylog(S(4), c(a + bx)/(a(c + dx))) - S(6)nS(3)polylog(S(4), d(a + bx)/(b(c + dx))) - S(6)nS(2)log(e((a + bx)/(c + dx))n)polylog(S(3), c(a + bx)/(a(c + dx))) + S(6)nS(2)log(e((a + bx)/(c + dx))n)polylog(S(3), d(a + bx)/(b(c + dx))) + S(3)nlog(e((a + bx)/(c + dx))n)S(2)polylog(S(2), c(a + bx)/(a(c + dx))) - S(3)nlog(e((a + bx)/(c + dx))n)S(2)polylog(S(2), d(a + bx)/(b(c + dx))) - log(e((a + bx)/(c + dx))n)S(3)log((-ad + bc)/(b(c + dx))) + log(e((a + bx)/(c + dx))n)S(3)log(x(ad - bc)/(a(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(2)/x, x), x, -S(2)nS(2)polylog(S(3), c(a + bx)/(a(c + dx))) + S(2)nS(2)polylog(S(3), d(a + bx)/(b(c + dx))) + S(2)nlog(e((a + bx)/(c + dx))n)polylog(S(2), c(a + bx)/(a(c + dx))) - S(2)nlog(e((a + bx)/(c + dx))n)polylog(S(2), d(a + bx)/(b(c + dx))) - log(e((a + bx)/(c + dx))n)S(2)log((-ad + bc)/(b(c + dx))) + log(e((a + bx)/(c + dx))n)S(2)log(x(ad - bc)/(a(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))*n)/x, x), x, -nlog(x)log((a + bx)/a) + nlog(x)log((c + dx)/c) - npolylog(S(2), -bx/a) + npolylog(S(2), -dx/c) + log(x)log(e((a + bx)/(c + dx))n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(e((a + bx)/(c + dx))n)), x), x, Integral(S(1)/(xlog(e((a + bx)/(c + dx))n)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(e((a + bx)/(c + dx))n)S(2)), x), x, Integral(S(1)/(xlog(e((a + bx)/(c + dx))n)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(3)log(e(a + bx)/(c + dx)), x), x, -x(-ad + bc)*S(3)/(S(4)d*S(3)) + (a + bx)*S(4)log(e(a + bx)/(c + dx))/(S(4)b) - (a + bx)S(3)(-ad/S(12) + bc/S(12))/(bd) + (a + bx)*S(2)(-ad + bc)*S(2)/(S(8)bdS(2)) + (-ad + bc)S(4)log(c + dx)/(S(4)bdS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(2)log(e(a + bx)/(c + dx)), x), x, x(-ad + bc)*S(2)/(S(3)d*S(2)) + (a + bx)*S(3)log(e(a + bx)/(c + dx))/(S(3)b) - (a + bx)S(2)(-ad/S(6) + bc/S(6))/(bd) - (-ad + bc)S(3)log(c + dx)/(S(3)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)log(e(a + bx)/(c + dx)), x), x, x(ad/S(2) - bc/S(2))/d + (a + bx)*S(2)log(e(a + bx)/(c + dx))/(S(2)b) + (-ad + bc)*S(2)log(c + dx)/(S(2)bdS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx)), x), x, (a + bx)log(e(a + bx)/(c + dx))/b - (-ad + bc)log(c + dx)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))/(a + bx), x), x, -log((ad - bc)/(d(a + bx)))log(e(a + bx)/(c + dx))/b + polylog(S(2), b(c + dx)/(d(a + bx)))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))/(a + bx)*S(2), x), x, -(c + dx)log(e(a + bx)/(c + dx))/((a + bx)(-ad + bc)) - S(1)/(b(a + bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))/(a + bx)S(3), x), x, dS(2)log(a + bx)/(S(2)b(-ad + bc)S(2)) - dS(2)log(c + dx)/(S(2)b(-ad + bc)*S(2)) + d/(S(2)b(a + bx)(-ad + bc)) - log(e(a + bx)/(c + dx))/(S(2)b(a + bx)S(2)) - S(1)/(S(4)b(a + bx)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(3)log(e(a + bx)/(c + dx))S(2), x), x, -S(5)x(-ad + bc)S(3)/(S(12)d*S(3)) + (a + bx)*S(4)log(e(a + bx)/(c + dx))S(2)/(S(4)b) - (a + bx)S(3)(-ad/S(6) + bc/S(6))log(e(a + bx)/(c + dx))/(bd) + (a + bx)*S(2)(-ad + bc)*S(2)log(e(a + bx)/(c + dx))/(S(4)bdS(2)) + (a + bx)*S(2)(-ad + bc)*S(2)/(S(12)bdS(2)) - (a + bx)(-ad + bc)S(3)log(e(a + bx)/(c + dx))/(S(2)bdS(3)) - (-ad + bc)S(4)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))/(S(2)bd*S(4)) + S(11)(-ad + bc)*S(4)log(c + dx)/(S(12)bdS(4)) - (-ad + bc)S(4)polylog(S(2), d(a + bx)/(b(c + dx)))/(S(2)bd*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(2)log(e(a + bx)/(c + dx))S(2), x), x, x(-ad + bc)*S(2)/(S(3)d*S(2)) + (a + bx)*S(3)log(e(a + bx)/(c + dx))S(2)/(S(3)b) - (a + bx)S(2)(-ad/S(3) + bc/S(3))log(e(a + bx)/(c + dx))/(bd) + S(2)(a + bx)(-ad + bc)*S(2)log(e(a + bx)/(c + dx))/(S(3)bdS(2)) + S(2)(-ad + bc)*S(3)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))/(S(3)bd*S(3)) - (-ad + bc)S(3)log(c + dx)/(bd*S(3)) + S(2)(-ad + bc)*S(3)polylog(S(2), d(a + bx)/(b(c + dx)))/(S(3)bd*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)log(e(a + bx)/(c + dx))*S(2), x), x, (a + bx)*S(2)log(e(a + bx)/(c + dx))S(2)/(S(2)b) + (a + bx)(ad - bc)log(e(a + bx)/(c + dx))/(bd) - (-ad + bc)S(2)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))/(bdS(2)) + (-ad + bc)S(2)log(c + dx)/(bd*S(2)) - (-ad + bc)S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/(bdS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))*S(2), x), x, (a + bx)log(e(a + bx)/(c + dx))*S(2)/b + (-S(2)ad + S(2)bc)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))/(bd) + (-S(2)ad + S(2)bc)polylog(S(2), d(a + bx)/(b(c + dx)))/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))*S(2)/(a + bx), x), x, -log((ad - bc)/(d(a + bx)))log(e(a + bx)/(c + dx))*S(2)/b + S(2)log(e(a + bx)/(c + dx))polylog(S(2), b(c + dx)/(d(a + bx)))/b + S(2)polylog(S(3), b(c + dx)/(d(a + bx)))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))*S(2)/(a + bx)*S(2), x), x, -(c + dx)log(e(a + bx)/(c + dx))*S(2)/((a + bx)(-ad + bc)) - (S(2)c + S(2)dx)log(e(a + bx)/(c + dx))/((a + bx)(-ad + bc)) - S(2)/(b(a + bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))S(2)/(a + bx)*S(3), x), x, -b(c + dx)S(2)log(e(a + bx)/(c + dx))S(2)/(S(2)(a + bx)S(2)(-ad + bc)*S(2)) - b(c + dx)S(2)log(e(a + bx)/(c + dx))/(S(2)(a + bx)S(2)(-ad + bc)*S(2)) - b(c + dx)S(2)/(S(4)(a + bx)S(2)(-ad + bc)*S(2)) + d(c + dx)log(e(a + bx)/(c + dx))S(2)/((a + bx)(-ad + bc)S(2)) + S(2)d(c + dx)log(e(a + bx)/(c + dx))/((a + bx)(-ad + bc)*S(2)) + S(2)d/(b(a + bx)(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(2)log(e(a + bx)/(c + dx))S(3), x), x, (a + bx)*S(3)log(e(a + bx)/(c + dx))S(3)/(S(3)b) - (a + bx)S(2)(-ad/S(2) + bc/S(2))log(e(a + bx)/(c + dx))*S(2)/(bd) + (a + bx)(-ad + bc)*S(2)log(e(a + bx)/(c + dx))S(2)/(bd*S(2)) + (a + bx)(-ad + bc)S(2)log(e(a + bx)/(c + dx))/(bd*S(2)) + (-ad + bc)S(3)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))*S(2)/(bd*S(3)) + S(3)(-ad + bc)*S(3)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))/(bdS(3)) + S(2)(-ad + bc)*S(3)log(e(a + bx)/(c + dx))polylog(S(2), d(a + bx)/(b(c + dx)))/(bdS(3)) - (-ad + bc)S(3)log(c + dx)/(bd*S(3)) + S(3)(-ad + bc)*S(3)polylog(S(2), d(a + bx)/(b(c + dx)))/(bdS(3)) - S(2)(-ad + bc)*S(3)polylog(S(3), d(a + bx)/(b(c + dx)))/(bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)log(e(a + bx)/(c + dx))*S(3), x), x, (a + bx)*S(2)log(e(a + bx)/(c + dx))S(3)/(S(2)b) - (a + bx)(-S(3)ad/S(2) + S(3)bc/S(2))log(e(a + bx)/(c + dx))*S(2)/(bd) - S(3)(-ad + bc)S(2)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))*S(2)/(S(2)bdS(2)) - S(3)(-ad + bc)*S(2)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))/(bdS(2)) - S(3)(-ad + bc)*S(2)log(e(a + bx)/(c + dx))polylog(S(2), d(a + bx)/(b(c + dx)))/(bdS(2)) - S(3)(-ad + bc)*S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/(bdS(2)) + S(3)(-ad + bc)*S(2)polylog(S(3), d(a + bx)/(b(c + dx)))/(bdS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))*S(3), x), x, (a + bx)log(e(a + bx)/(c + dx))*S(3)/b + (-S(6)ad + S(6)bc)log(e(a + bx)/(c + dx))polylog(S(2), d(a + bx)/(b(c + dx)))/(bd) - (-S(6)ad + S(6)bc)polylog(S(3), d(a + bx)/(b(c + dx)))/(bd) + (-S(3)ad + S(3)bc)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))*S(2)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))S(3)/(a + bx), x), x, -log((ad - bc)/(d(a + bx)))log(e(a + bx)/(c + dx))*S(3)/b + S(3)log(e(a + bx)/(c + dx))S(2)polylog(S(2), b(c + dx)/(d(a + bx)))/b + S(6)log(e(a + bx)/(c + dx))polylog(S(3), b(c + dx)/(d(a + bx)))/b + S(6)polylog(S(4), b(c + dx)/(d(a + bx)))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))S(3)/(a + bx)*S(2), x), x, -(c + dx)log(e(a + bx)/(c + dx))*S(3)/((a + bx)(-ad + bc)) - (S(3)c + S(3)dx)log(e(a + bx)/(c + dx))*S(2)/((a + bx)(-ad + bc)) - (S(6)c + S(6)dx)log(e(a + bx)/(c + dx))/((a + bx)(-ad + bc)) - S(6)/(b(a + bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))S(3)/(a + bx)*S(3), x), x, -b(c + dx)S(2)log(e(a + bx)/(c + dx))S(3)/(S(2)(a + bx)S(2)(-ad + bc)*S(2)) - S(3)b(c + dx)*S(2)log(e(a + bx)/(c + dx))S(2)/(S(4)(a + bx)S(2)(-ad + bc)*S(2)) - S(3)b(c + dx)*S(2)log(e(a + bx)/(c + dx))/(S(4)(a + bx)S(2)(-ad + bc)*S(2)) - S(3)b(c + dx)*S(2)/(S(8)(a + bx)S(2)(-ad + bc)*S(2)) + d(c + dx)log(e(a + bx)/(c + dx))S(3)/((a + bx)(-ad + bc)S(2)) + S(3)d(c + dx)log(e(a + bx)/(c + dx))*S(2)/((a + bx)(-ad + bc)S(2)) + S(6)d(c + dx)log(e(a + bx)/(c + dx))/((a + bx)(-ad + bc)*S(2)) + S(6)d/(b(a + bx)(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)log(e((a + bx)/(c + dx))n), x), x, (c + dx)*S(4)log(e((a + bx)/(c + dx))n)/(S(4)d) - n(c + dx)*S(3)(-ad/S(12) + bc/S(12))/(bd) - n(c + dx)S(2)(-ad + bc)*S(2)/(S(8)b*S(2)d) - nx(-ad + bc)*S(3)/(S(4)b*S(3)) - n(-ad + bc)*S(4)log(a + bx)/(S(4)b*S(4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)S(2)log(e((a + bx)/(c + dx))n), x), x, (c + dx)*S(3)log(e((a + bx)/(c + dx))n)/(S(3)d) - n(c + dx)*S(2)(-ad/S(6) + bc/S(6))/(bd) - nx(-ad + bc)S(2)/(S(3)b*S(2)) - n(-ad + bc)*S(3)log(a + bx)/(S(3)b*S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)log(e((a + bx)/(c + dx))n), x), x, (c + dx)*S(2)log(e((a + bx)/(c + dx))n)/(S(2)d) + nx(ad/S(2) - bc/S(2))/b - n(-ad + bc)S(2)log(a + bx)/(S(2)b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n), x), x, (a + bx)log(e((a + bx)/(c + dx))*n)/b - n(-ad + bc)log(c + dx)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))*n)/(c + dx), x), x, -npolylog(S(2), d(a + bx)/(b(c + dx)))/d - log(e((a + bx)/(c + dx))*n)log((-ad + bc)/(b(c + dx)))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)/(c + dx)*S(2), x), x, (a + bx)log(e((a + bx)/(c + dx))*n)/((c + dx)(-ad + bc)) + n/(d(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))*n)/(c + dx)S(3), x), x, bS(2)nlog(a + bx)/(S(2)d(-ad + bc)S(2)) - bS(2)nlog(c + dx)/(S(2)d(-ad + bc)*S(2)) + bn/(S(2)d(c + dx)(-ad + bc)) + n/(S(4)d(c + dx)S(2)) - log(e((a + bx)/(c + dx))*n)/(S(2)d(c + dx)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))*n)/(c + dx)S(4), x), x, bS(3)nlog(a + bx)/(S(3)d(-ad + bc)S(3)) - bS(3)nlog(c + dx)/(S(3)d(-ad + bc)S(3)) + bS(2)n/(S(3)d(c + dx)(-ad + bc)S(2)) + bn/(S(6)d(c + dx)S(2)(-ad + bc)) + n/(S(9)d(c + dx)S(3)) - log(e((a + bx)/(c + dx))*n)/(S(3)d(c + dx)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)log(e(a + bx)/(c + dx))S(2), x), x, (c + dx)*S(4)log(e(a + bx)/(c + dx))S(2)/(S(4)d) - (c + dx)S(3)(-ad/S(6) + bc/S(6))log(e(a + bx)/(c + dx))/(bd) - (c + dx)*S(2)(-ad + bc)*S(2)log(e(a + bx)/(c + dx))/(S(4)b*S(2)d) + (c + dx)S(2)(-ad + bc)*S(2)/(S(12)b*S(2)d) + S(5)x(-ad + bc)*S(3)/(S(12)b*S(3)) - (a + bx)(-ad + bc)S(3)log(e(a + bx)/(c + dx))/(S(2)b*S(4)) + (-ad + bc)S(4)log((ad - bc)/(d(a + bx)))log(e(a + bx)/(c + dx))/(S(2)bS(4)d) + S(5)(-ad + bc)S(4)log(a + bx)/(S(12)b*S(4)d) + (-ad + bc)*S(4)log(c + dx)/(S(2)b*S(4)d) - (-ad + bc)*S(4)polylog(S(2), b(c + dx)/(d(a + bx)))/(S(2)bS(4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)S(2)log(e(a + bx)/(c + dx))S(2), x), x, (c + dx)*S(3)log(e(a + bx)/(c + dx))S(2)/(S(3)d) - (c + dx)S(2)(-ad/S(3) + bc/S(3))log(e(a + bx)/(c + dx))/(bd) + x(-ad + bc)*S(2)/(S(3)b*S(2)) - S(2)(a + bx)(-ad + bc)*S(2)log(e(a + bx)/(c + dx))/(S(3)b*S(3)) + S(2)(-ad + bc)*S(3)log((ad - bc)/(d(a + bx)))log(e(a + bx)/(c + dx))/(S(3)bS(3)d) + (-ad + bc)*S(3)log(a + bx)/(S(3)b*S(3)d) + S(2)(-ad + bc)S(3)log(c + dx)/(S(3)b*S(3)d) - S(2)(-ad + bc)S(3)polylog(S(2), b(c + dx)/(d(a + bx)))/(S(3)bS(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)log(e(a + bx)/(c + dx))S(2), x), x, (c + dx)*S(2)log(e(a + bx)/(c + dx))S(2)/(S(2)d) + (a + bx)(ad - bc)log(e(a + bx)/(c + dx))/b*S(2) + (-ad + bc)S(2)log((ad - bc)/(d(a + bx)))log(e(a + bx)/(c + dx))/(b*S(2)d) + (-ad + bc)*S(2)log(c + dx)/(bS(2)d) - (-ad + bc)*S(2)polylog(S(2), b(c + dx)/(d(a + bx)))/(b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))S(2), x), x, (a + bx)log(e(a + bx)/(c + dx))*S(2)/b + (-S(2)ad + S(2)bc)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))/(bd) + (-S(2)ad + S(2)bc)polylog(S(2), d(a + bx)/(b(c + dx)))/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))*S(2)/(c + dx), x), x, -log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))*S(2)/d - S(2)log(e(a + bx)/(c + dx))polylog(S(2), d(a + bx)/(b(c + dx)))/d + S(2)polylog(S(3), d(a + bx)/(b(c + dx)))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))*S(2)/(c + dx)*S(2), x), x, (a + bx)log(e(a + bx)/(c + dx))*S(2)/((c + dx)(-ad + bc)) - (S(2)a + S(2)bx)log(e(a + bx)/(c + dx))/((c + dx)(-ad + bc)) - S(2)/(d(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))S(2)/(c + dx)*S(3), x), x, b(a + bx)log(e(a + bx)/(c + dx))S(2)/((c + dx)(-ad + bc)S(2)) - S(2)b(a + bx)log(e(a + bx)/(c + dx))/((c + dx)(-ad + bc)*S(2)) - S(2)b/(d(c + dx)(-ad + bc)) - d(a + bx)S(2)log(e(a + bx)/(c + dx))S(2)/(S(2)(c + dx)S(2)(-ad + bc)*S(2)) + d(a + bx)S(2)log(e(a + bx)/(c + dx))/(S(2)(c + dx)S(2)(-ad + bc)*S(2)) - d(a + bx)S(2)/(S(4)(c + dx)S(2)(-ad + bc)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(2)log(e(a + bx)/(c + dx))S(3), x), x, (c + dx)*S(3)log(e(a + bx)/(c + dx))S(3)/(S(3)d) - (c + dx)S(2)(-ad/S(2) + bc/S(2))log(e(a + bx)/(c + dx))*S(2)/(bd) - (a + bx)(-ad + bc)*S(2)log(e(a + bx)/(c + dx))S(2)/bS(3) + (a + bx)(-ad + bc)S(2)log(e(a + bx)/(c + dx))/bS(3) - S(2)(-ad + bc)*S(3)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))/(b*S(3)d) + (-ad + bc)*S(3)log((ad - bc)/(d(a + bx)))log(e(a + bx)/(c + dx))S(2)/(bS(3)d) - (-ad + bc)S(3)log((ad - bc)/(d(a + bx)))log(e(a + bx)/(c + dx))/(b*S(3)d) - S(2)(-ad + bc)S(3)log(e(a + bx)/(c + dx))polylog(S(2), b(c + dx)/(d(a + bx)))/(b*S(3)d) - (-ad + bc)*S(3)log(c + dx)/(bS(3)d) - S(2)(-ad + bc)S(3)polylog(S(2), d(a + bx)/(b(c + dx)))/(b*S(3)d) + (-ad + bc)*S(3)polylog(S(2), b(c + dx)/(d(a + bx)))/(b*S(3)d) - S(2)(-ad + bc)S(3)polylog(S(3), b(c + dx)/(d(a + bx)))/(b*S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)log(e(a + bx)/(c + dx))S(3), x), x, (c + dx)*S(2)log(e(a + bx)/(c + dx))S(3)/(S(2)d) - (a + bx)(-S(3)ad/S(2) + S(3)bc/S(2))log(e(a + bx)/(c + dx))S(2)/bS(2) - S(3)(-ad + bc)S(2)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))/(b*S(2)d) + S(3)(-ad + bc)S(2)log((ad - bc)/(d(a + bx)))log(e(a + bx)/(c + dx))*S(2)/(S(2)b*S(2)d) - S(3)(-ad + bc)S(2)log(e(a + bx)/(c + dx))polylog(S(2), b(c + dx)/(d(a + bx)))/(b*S(2)d) - S(3)(-ad + bc)S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/(b*S(2)d) - S(3)(-ad + bc)S(2)polylog(S(3), b(c + dx)/(d(a + bx)))/(b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))S(3), x), x, (a + bx)log(e(a + bx)/(c + dx))*S(3)/b + (-S(6)ad + S(6)bc)log(e(a + bx)/(c + dx))polylog(S(2), d(a + bx)/(b(c + dx)))/(bd) - (-S(6)ad + S(6)bc)polylog(S(3), d(a + bx)/(b(c + dx)))/(bd) + (-S(3)ad + S(3)bc)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))*S(2)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))S(3)/(c + dx), x), x, -log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))*S(3)/d - S(3)log(e(a + bx)/(c + dx))S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/d + S(6)log(e(a + bx)/(c + dx))polylog(S(3), d(a + bx)/(b(c + dx)))/d - S(6)polylog(S(4), d(a + bx)/(b(c + dx)))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))S(3)/(c + dx)*S(2), x), x, (a + bx)log(e(a + bx)/(c + dx))*S(3)/((c + dx)(-ad + bc)) - (S(3)a + S(3)bx)log(e(a + bx)/(c + dx))*S(2)/((c + dx)(-ad + bc)) + (S(6)a + S(6)bx)log(e(a + bx)/(c + dx))/((c + dx)(-ad + bc)) + S(6)/(d(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))S(3)/(c + dx)*S(3), x), x, b(a + bx)log(e(a + bx)/(c + dx))S(3)/((c + dx)(-ad + bc)S(2)) - S(3)b(a + bx)log(e(a + bx)/(c + dx))*S(2)/((c + dx)(-ad + bc)S(2)) + S(6)b(a + bx)log(e(a + bx)/(c + dx))/((c + dx)(-ad + bc)*S(2)) + S(6)b/(d(c + dx)(-ad + bc)) - d(a + bx)S(2)log(e(a + bx)/(c + dx))S(3)/(S(2)(c + dx)S(2)(-ad + bc)*S(2)) + S(3)d(a + bx)*S(2)log(e(a + bx)/(c + dx))S(2)/(S(4)(c + dx)S(2)(-ad + bc)*S(2)) - S(3)d(a + bx)*S(2)log(e(a + bx)/(c + dx))/(S(4)(c + dx)S(2)(-ad + bc)*S(2)) + S(3)d(a + bx)*S(2)/(S(8)(c + dx)S(2)(-ad + bc)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))*S(4), x), x, (a + bx)log(e(a + bx)/(c + dx))*S(4)/b - (-S(24)ad + S(24)bc)log(e(a + bx)/(c + dx))polylog(S(3), d(a + bx)/(b(c + dx)))/(bd) + (-S(24)ad + S(24)bc)polylog(S(4), d(a + bx)/(b(c + dx)))/(bd) + (-S(12)ad + S(12)bc)log(e(a + bx)/(c + dx))S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/(bd) + (-S(4)ad + S(4)bc)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))*S(3)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(a + bx)/(c + dx))S(5), x), x, (a + bx)log(e(a + bx)/(c + dx))*S(5)/b + (-S(120)ad + S(120)bc)log(e(a + bx)/(c + dx))polylog(S(4), d(a + bx)/(b(c + dx)))/(bd) - (-S(120)ad + S(120)bc)polylog(S(5), d(a + bx)/(b(c + dx)))/(bd) - (-S(60)ad + S(60)bc)log(e(a + bx)/(c + dx))S(2)polylog(S(3), d(a + bx)/(b(c + dx)))/(bd) + (-S(20)ad + S(20)bc)log(e(a + bx)/(c + dx))S(3)polylog(S(2), d(a + bx)/(b(c + dx)))/(bd) + (-S(5)ad + S(5)bc)log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))*S(4)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d(a + bx)/(b(c + dx)))/(cf + dfx), x), x, polylog(S(2), (-ad + bc)/(b(c + dx)))/(df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(1) + S(1)/(a + bx))/(a + bx), x), x, polylog(S(2), -S(1)/(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(1) - S(1)/(a + bx))/(a + bx), x), x, polylog(S(2), S(1)/(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((f + gx)S(3)log(e((a + bx)/(c + dx))n), x), x, (f + gx)*S(4)log(e((a + bx)/(c + dx))n)/(S(4)g) + n(-cg + df)S(4)log(c + dx)/(S(4)d*S(4)g) - g*S(3)nxS(3)(-ad/S(12) + bc/S(12))/(bd) - gS(2)nxS(2)(-ad/S(8) + bc/S(8))(-adg - bcg + S(4)bdf)/(b*S(2)d*S(2)) + gnx(ad/S(4) - bc/S(4))(aS(2)d*S(2)g*S(2) - abdg(-cg + S(4)df) + b*S(2)(c*S(2)g*S(2) - S(4)cdfg + S(6)d*S(2)fS(2)))/(bS(3)dS(3)) - n(-ag + bf)*S(4)log(a + bx)/(S(4)b*S(4)g), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((f + gx)S(2)log(e((a + bx)/(c + dx))n), x), x, (f + gx)*S(3)log(e((a + bx)/(c + dx))n)/(S(3)g) + n(-cg + df)S(3)log(c + dx)/(S(3)d*S(3)g) - g*S(2)nxS(2)(-ad/S(6) + bc/S(6))/(bd) + gnx(ad/S(3) - bc/S(3))(-adg - bcg + S(3)bdf)/(b*S(2)d*S(2)) - n(-ag + bf)*S(3)log(a + bx)/(S(3)b*S(3)g), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((f + gx)log(e((a + bx)/(c + dx))n), x), x, (f + gx)*S(2)log(e((a + bx)/(c + dx))n)/(S(2)g) + n(-cg + df)S(2)log(c + dx)/(S(2)d*S(2)g) + gnx(ad/S(2) - bc/S(2))/(bd) - n(-ag + bf)S(2)log(a + bx)/(S(2)b*S(2)g), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n), x), x, (a + bx)log(e((a + bx)/(c + dx))*n)/b - n(-ad + bc)log(c + dx)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))*n)/(f + gx), x), x, -nlog(-g(a + bx)/(-ag + bf))log(f + gx)/g + nlog(-g(c + dx)/(-cg + df))log(f + gx)/g - npolylog(S(2), b(f + gx)/(-ag + bf))/g + npolylog(S(2), d(f + gx)/(-cg + df))/g + log(e((a + bx)/(c + dx))n)log(f + gx)/g, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))*n)/(f + gx)*S(2), x), x, -n(-ad + bc)log(c + dx)/((-ag + bf)(-cg + df)) + n(-ad + bc)log(f + gx)/((-ag + bf)(-cg + df)) + (a + bx)log(e((a + bx)/(c + dx))*n)/((f + gx)(-ag + bf)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))*n)/(f + gx)S(3), x), x, bS(2)nlog(a + bx)/(S(2)g(-ag + bf)S(2)) - dS(2)nlog(c + dx)/(S(2)g(-cg + df)*S(2)) + n(-ad/S(2) + bc/S(2))(-adg - bcg + S(2)bdf)log(f + gx)/((-ag + bf)*S(2)(-cg + df)*S(2)) + n(ad/S(2) - bc/S(2))/((f + gx)(-ag + bf)(-cg + df)) - log(e((a + bx)/(c + dx))*n)/(S(2)g(f + gx)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))*n)/(f + gx)S(4), x), x, bS(3)nlog(a + bx)/(S(3)g(-ag + bf)S(3)) - dS(3)nlog(c + dx)/(S(3)g(-cg + df)*S(3)) + n(-ad/S(3) + bc/S(3))(aS(2)d*S(2)g*S(2) - abdg(-cg + S(3)df) + b*S(2)(c*S(2)g*S(2) - S(3)cdfg + S(3)d*S(2)f*S(2)))log(f + gx)/((-ag + bf)S(3)(-cg + df)*S(3)) - n(-ad/S(3) + bc/S(3))(-adg - bcg + S(2)bdf)/((f + gx)(-ag + bf)*S(2)(-cg + df)*S(2)) + n(ad/S(6) - bc/S(6))/((f + gx)S(2)(-ag + bf)(-cg + df)) - log(e((a + bx)/(c + dx))*n)/(S(3)g(f + gx)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((f + gx)*S(3)log(e((a + bx)/(c + dx))n)S(2), x), x, -aS(3)g*S(3)n*S(2)(-ad + bc)log(a + bx)/(S(6)bS(4)d) + a*S(2)g*S(2)n*S(2)(-ad + bc)(-adg - bcg + S(4)bdf)log(a + bx)/(S(4)bS(4)d*S(2)) + (f + gx)*S(4)log(e((a + bx)/(c + dx))n)S(2)/(S(4)g) - n*S(2)(-cg + df)*S(4)polylog(S(2), d(a + bx)/(b(c + dx)))/(S(2)dS(4)g) - n(-cg + df)S(4)log(e((a + bx)/(c + dx))n)log((-ad + bc)/(b(c + dx)))/(S(2)dS(4)g) + c*S(3)g*S(3)n*S(2)(-ad + bc)log(c + dx)/(S(6)bdS(4)) - gS(3)nx*S(3)(-ad/S(6) + bc/S(6))log(e((a + bx)/(c + dx))*n)/(bd) - c*S(2)g*S(2)n*S(2)(-ad + bc)(-adg - bcg + S(4)bdf)log(c + dx)/(S(4)bS(2)dS(4)) + gS(3)nS(2)x*S(2)(-ad + bc)*S(2)/(S(12)b*S(2)dS(2)) - gS(2)nx*S(2)(-ad/S(4) + bc/S(4))(-adg - bcg + S(4)bdf)log(e((a + bx)/(c + dx))n)/(bS(2)dS(2)) - gS(3)n*S(2)x(-ad + bc)S(2)(ad + bc)/(S(6)bS(3)dS(3)) + gS(2)nS(2)x(-ad + bc)S(2)(-adg - bcg + S(4)bdf)/(S(4)b*S(3)dS(3)) - nS(2)(-ag + bf)S(4)polylog(S(2), b(c + dx)/(d(a + bx)))/(S(2)bS(4)g) + n(-ag + bf)S(4)log(e((a + bx)/(c + dx))n)log((ad - bc)/(d(a + bx)))/(S(2)bS(4)g) - gn(a + bx)(-ad/S(2) + bc/S(2))(aS(2)d*S(2)g*S(2) - abdg(-cg + S(4)df) + b*S(2)(c*S(2)g*S(2) - S(4)cdfg + S(6)d*S(2)f*S(2)))log(e((a + bx)/(c + dx))n)/(bS(4)d*S(3)) + gn*S(2)(-ad + bc)*S(2)(a*S(2)d*S(2)g*S(2) - abdg(-cg + S(4)df) + b*S(2)(c*S(2)g*S(2) - S(4)cdfg + S(6)d*S(2)f*S(2)))log(c + dx)/(S(2)b*S(4)d*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((f + gx)*S(2)log(e((a + bx)/(c + dx))n)S(2), x), x, aS(2)g*S(2)n*S(2)(-ad + bc)log(a + bx)/(S(3)bS(3)d) + (f + gx)S(3)log(e((a + bx)/(c + dx))n)S(2)/(S(3)g) - S(2)nS(2)(-cg + df)*S(3)polylog(S(2), d(a + bx)/(b(c + dx)))/(S(3)dS(3)g) - S(2)n(-cg + df)*S(3)log(e((a + bx)/(c + dx))n)log((-ad + bc)/(b(c + dx)))/(S(3)dS(3)g) - c*S(2)g*S(2)n*S(2)(-ad + bc)log(c + dx)/(S(3)bdS(3)) - gS(2)nx*S(2)(-ad/S(3) + bc/S(3))log(e((a + bx)/(c + dx))*n)/(bd) + g*S(2)n*S(2)x(-ad + bc)S(2)/(S(3)b*S(2)d*S(2)) - S(2)n*S(2)(-ag + bf)*S(3)polylog(S(2), b(c + dx)/(d(a + bx)))/(S(3)bS(3)g) + S(2)n(-ag + bf)*S(3)log(e((a + bx)/(c + dx))n)log((ad - bc)/(d(a + bx)))/(S(3)bS(3)g) - gn(a + bx)(-S(2)ad/S(3) + S(2)bc/S(3))(-adg - bcg + S(3)bdf)log(e((a + bx)/(c + dx))n)/(bS(3)dS(2)) + S(2)gnS(2)(-ad + bc)*S(2)(-adg - bcg + S(3)bdf)log(c + dx)/(S(3)b*S(3)d*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((f + gx)log(e((a + bx)/(c + dx))n)S(2), x), x, (f + gx)S(2)log(e((a + bx)/(c + dx))n)S(2)/(S(2)g) - n*S(2)(-cg + df)*S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/(d*S(2)g) - n(-cg + df)S(2)log(e((a + bx)/(c + dx))n)log((-ad + bc)/(b(c + dx)))/(d*S(2)g) - n*S(2)(-ag + bf)*S(2)polylog(S(2), b(c + dx)/(d(a + bx)))/(b*S(2)g) + n(-ag + bf)S(2)log(e((a + bx)/(c + dx))n)log((ad - bc)/(d(a + bx)))/(b*S(2)g) + gn(a + bx)(ad - bc)log(e((a + bx)/(c + dx))n)/(bS(2)d) + gn*S(2)(-ad + bc)*S(2)log(c + dx)/(bS(2)d*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(2), x), x, (a + bx)log(e((a + bx)/(c + dx))n)S(2)/b + nS(2)(-S(2)ad + S(2)bc)polylog(S(2), d(a + bx)/(b(c + dx)))/(bd) + n(-S(2)ad + S(2)bc)log(e((a + bx)/(c + dx))n)log((-ad + bc)/(b(c + dx)))/(bd), expand=True, _diff=True, _numerical=True) # taking long time in rubi_test assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(2)/(f + gx), x), x, -S(2)n*S(2)polylog(S(3), (a(-cg + df) - bcgx + bdfx)/((c + dx)(-ag + bf)))/g + S(2)n*S(2)polylog(S(3), d(a + bx)/(b(c + dx)))/g + S(2)nlog(e((a + bx)/(c + dx))n)polylog(S(2), (a(-cg + df) - bcgx + bdfx)/((c + dx)(-ag + bf)))/g - S(2)nlog(e((a + bx)/(c + dx))*n)polylog(S(2), d(a + bx)/(b(c + dx)))/g - log(e((a + bx)/(c + dx))n)S(2)log((-ad + bc)/(b(c + dx)))/g + log(e((a + bx)/(c + dx))n)S(2)log((f + gx)(-ad + bc)/((c + dx)(-ag + bf)))/g, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(2)/(f + gx)S(2), x), x, nS(2)(-S(2)ad + S(2)bc)polylog(S(2), (a + bx)(-cg + df)/((c + dx)(-ag + bf)))/((-ag + bf)(-cg + df)) + n(-S(2)ad + S(2)bc)log(e((a + bx)/(c + dx))*n)log((f + gx)(-ad + bc)/((c + dx)(-ag + bf)))/((-ag + bf)(-cg + df)) + (a + bx)log(e((a + bx)/(c + dx))n)S(2)/((f + gx)(-ag + bf)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(2)/(f + gx)S(3), x), x, bS(2)nS(2)polylog(S(2), b(c + dx)/(d(a + bx)))/(g(-ag + bf)S(2)) - bS(2)nlog(e((a + bx)/(c + dx))*n)log((ad - bc)/(d(a + bx)))/(g(-ag + bf)S(2)) + dS(2)n*S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/(g(-cg + df)S(2)) + dS(2)nlog(e((a + bx)/(c + dx))*n)log((-ad + bc)/(b(c + dx)))/(g(-cg + df)S(2)) - gn*S(2)(-ad + bc)*S(2)log(c + dx)/((-ag + bf)S(2)(-cg + df)*S(2)) + gn*S(2)(-ad + bc)*S(2)log(f + gx)/((-ag + bf)S(2)(-cg + df)*S(2)) + gn(a + bx)(-ad + bc)log(e((a + bx)/(c + dx))n)/((f + gx)(-ag + bf)S(2)(-cg + df)) - n*S(2)(-ad + bc)(-adg - bcg + S(2)bdf)log(-g(a + bx)/(-ag + bf))log(f + gx)/((-ag + bf)S(2)(-cg + df)S(2)) + nS(2)(-ad + bc)(-adg - bcg + S(2)bdf)log(-g(c + dx)/(-cg + df))log(f + gx)/((-ag + bf)*S(2)(-cg + df)S(2)) - nS(2)(-ad + bc)(-adg - bcg + S(2)bdf)polylog(S(2), b(f + gx)/(-ag + bf))/((-ag + bf)*S(2)(-cg + df)S(2)) + nS(2)(-ad + bc)(-adg - bcg + S(2)bdf)polylog(S(2), d(f + gx)/(-cg + df))/((-ag + bf)*S(2)(-cg + df)*S(2)) + n(-ad + bc)(-adg - bcg + S(2)bdf)log(e((a + bx)/(c + dx))*n)log(f + gx)/((-ag + bf)S(2)(-cg + df)*S(2)) - log(e((a + bx)/(c + dx))n)S(2)/(S(2)g(f + gx)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((f + gx)*S(2)log(e((a + bx)/(c + dx))n)S(3), x), x, aS(2)g*S(2)n*S(3)(-ad + bc)polylog(S(2), b(c + dx)/(d(a + bx)))/(bS(3)d) - a*S(2)g*S(2)n*S(2)(-ad + bc)log(e((a + bx)/(c + dx))*n)log((ad - bc)/(d(a + bx)))/(b*S(3)d) + (f + gx)S(3)log(e((a + bx)/(c + dx))n)S(3)/(S(3)g) + S(2)nS(3)(-cg + df)*S(3)polylog(S(3), d(a + bx)/(b(c + dx)))/(d*S(3)g) - S(2)nS(2)(-cg + df)*S(3)log(e((a + bx)/(c + dx))n)polylog(S(2), d(a + bx)/(b(c + dx)))/(d*S(3)g) - n(-cg + df)S(3)log(e((a + bx)/(c + dx))n)S(2)log((-ad + bc)/(b(c + dx)))/(d*S(3)g) + c*S(2)g*S(2)n*S(3)(-ad + bc)polylog(S(2), d(a + bx)/(b(c + dx)))/(bdS(3)) + cS(2)gS(2)n*S(2)(-ad + bc)log(e((a + bx)/(c + dx))*n)log((-ad + bc)/(b(c + dx)))/(bdS(3)) - gS(2)nxS(2)(-ad/S(2) + bc/S(2))log(e((a + bx)/(c + dx))n)S(2)/(bd) - S(2)n*S(3)(-ag + bf)*S(3)polylog(S(3), b(c + dx)/(d(a + bx)))/(b*S(3)g) - S(2)nS(2)(-ag + bf)*S(3)log(e((a + bx)/(c + dx))n)polylog(S(2), b(c + dx)/(d(a + bx)))/(b*S(3)g) + n(-ag + bf)S(3)log(e((a + bx)/(c + dx))n)S(2)log((ad - bc)/(d(a + bx)))/(b*S(3)g) + g*S(2)n*S(2)(a + bx)(-ad + bc)*S(2)log(e((a + bx)/(c + dx))n)/(bS(3)d*S(2)) - gn(a + bx)(-ad + bc)(-adg - bcg + S(3)bdf)log(e((a + bx)/(c + dx))n)S(2)/(bS(3)dS(2)) - gS(2)nS(3)(-ad + bc)*S(3)log(c + dx)/(bS(3)d*S(3)) - S(2)gnS(3)(-ad + bc)*S(2)(-adg - bcg + S(3)bdf)polylog(S(2), d(a + bx)/(b(c + dx)))/(b*S(3)d*S(3)) - S(2)gnS(2)(-ad + bc)*S(2)(-adg - bcg + S(3)bdf)log(e((a + bx)/(c + dx))n)log((-ad + bc)/(b(c + dx)))/(b*S(3)d*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((f + gx)log(e((a + bx)/(c + dx))n)S(3), x), x, (f + gx)S(2)log(e((a + bx)/(c + dx))n)S(3)/(S(2)g) + S(3)nS(3)(-cg + df)*S(2)polylog(S(3), d(a + bx)/(b(c + dx)))/(d*S(2)g) - S(3)nS(2)(-cg + df)*S(2)log(e((a + bx)/(c + dx))n)polylog(S(2), d(a + bx)/(b(c + dx)))/(d*S(2)g) - S(3)n(-cg + df)*S(2)log(e((a + bx)/(c + dx))n)S(2)log((-ad + bc)/(b(c + dx)))/(S(2)dS(2)g) - S(3)nS(3)(-ag + bf)*S(2)polylog(S(3), b(c + dx)/(d(a + bx)))/(b*S(2)g) - S(3)nS(2)(-ag + bf)*S(2)log(e((a + bx)/(c + dx))n)polylog(S(2), b(c + dx)/(d(a + bx)))/(b*S(2)g) + S(3)n(-ag + bf)*S(2)log(e((a + bx)/(c + dx))n)S(2)log((ad - bc)/(d(a + bx)))/(S(2)bS(2)g) + gn(a + bx)(S(3)ad/S(2) - S(3)bc/S(2))log(e((a + bx)/(c + dx))n)S(2)/(b*S(2)d) - S(3)gn*S(3)(-ad + bc)*S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/(b*S(2)d*S(2)) - S(3)gnS(2)(-ad + bc)*S(2)log(e((a + bx)/(c + dx))n)log((-ad + bc)/(b(c + dx)))/(b*S(2)d*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(3), x), x, (a + bx)log(e((a + bx)/(c + dx))n)S(3)/b - nS(3)(-S(6)ad + S(6)bc)polylog(S(3), d(a + bx)/(b(c + dx)))/(bd) + n*S(2)(-S(6)ad + S(6)bc)log(e((a + bx)/(c + dx))*n)polylog(S(2), d(a + bx)/(b(c + dx)))/(bd) + n(-S(3)ad + S(3)bc)log(e((a + bx)/(c + dx))n)S(2)log((-ad + bc)/(b(c + dx)))/(bd), expand=True, _diff=True, _numerical=True) # takes long time in test assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(3)/(f + gx), x), x, -S(6)nS(3)polylog(S(4), d(a + bx)/(b(c + dx)))/g + S(6)nS(3)polylog(S(4), (a + bx)(-cg + df)/((c + dx)(-ag + bf)))/g - S(6)nS(2)log(e((a + bx)/(c + dx))n)polylog(S(3), (a(-cg + df) - bcgx + bdfx)/((c + dx)(-ag + bf)))/g + S(6)n*S(2)log(e((a + bx)/(c + dx))n)polylog(S(3), d(a + bx)/(b(c + dx)))/g + S(3)nlog(e((a + bx)/(c + dx))n)S(2)polylog(S(2), (a(-cg + df) - bcgx + bdfx)/((c + dx)(-ag + bf)))/g - S(3)nlog(e((a + bx)/(c + dx))n)S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/g - log(e((a + bx)/(c + dx))n)S(3)log((-ad + bc)/(b(c + dx)))/g + log(e((a + bx)/(c + dx))n)S(3)log((f + gx)(-ad + bc)/((c + dx)(-ag + bf)))/g, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(3)/(f + gx)S(2), x), x, -nS(3)(-S(6)ad + S(6)bc)polylog(S(3), (a(-cg + df) - bcgx + bdfx)/((c + dx)(-ag + bf)))/((-ag + bf)(-cg + df)) + nS(2)(-S(6)ad + S(6)bc)log(e((a + bx)/(c + dx))*n)polylog(S(2), (a(-cg + df) - bcgx + bdfx)/((c + dx)(-ag + bf)))/((-ag + bf)(-cg + df)) + n(-S(3)ad + S(3)bc)log(e((a + bx)/(c + dx))n)S(2)log((f + gx)(-ad + bc)/((c + dx)(-ag + bf)))/((-ag + bf)(-cg + df)) + (a + bx)log(e((a + bx)/(c + dx))n)S(3)/((f + gx)(-ag + bf)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(3)/(f + gx)*S(3), x), x, S(3)b*S(2)n*S(3)polylog(S(3), b(c + dx)/(d(a + bx)))/(g(-ag + bf)S(2)) + S(3)b*S(2)n*S(2)log(e((a + bx)/(c + dx))n)polylog(S(2), b(c + dx)/(d(a + bx)))/(g(-ag + bf)S(2)) - S(3)b*S(2)nlog(e((a + bx)/(c + dx))n)S(2)log((ad - bc)/(d(a + bx)))/(S(2)g(-ag + bf)S(2)) - S(3)d*S(2)n*S(3)polylog(S(3), d(a + bx)/(b(c + dx)))/(g(-cg + df)S(2)) + S(3)d*S(2)n*S(2)log(e((a + bx)/(c + dx))n)polylog(S(2), d(a + bx)/(b(c + dx)))/(g(-cg + df)S(2)) + S(3)d*S(2)nlog(e((a + bx)/(c + dx))n)S(2)log((-ad + bc)/(b(c + dx)))/(S(2)g(-cg + df)S(2)) + S(3)gnS(3)(-ad + bc)*S(2)polylog(S(2), (a + bx)(-cg + df)/((c + dx)(-ag + bf)))/((-ag + bf)*S(2)(-cg + df)*S(2)) + S(3)gnS(2)(-ad + bc)*S(2)log(e((a + bx)/(c + dx))n)log((f + gx)(-ad + bc)/((c + dx)(-ag + bf)))/((-ag + bf)*S(2)(-cg + df)*S(2)) + gn(a + bx)(-S(3)ad/S(2) + S(3)bc/S(2))log(e((a + bx)/(c + dx))n)S(2)/((f + gx)(-ag + bf)S(2)(-cg + df)) - n*S(3)(-S(3)ad + S(3)bc)(-adg - bcg + S(2)bdf)polylog(S(3), (a(-cg + df) - bcgx + bdfx)/((c + dx)(-ag + bf)))/((-ag + bf)*S(2)(-cg + df)S(2)) + nS(3)(-S(3)ad + S(3)bc)(-adg - bcg + S(2)bdf)polylog(S(3), d(a + bx)/(b(c + dx)))/((-ag + bf)*S(2)(-cg + df)S(2)) + nS(2)(-S(3)ad + S(3)bc)(-adg - bcg + S(2)bdf)log(e((a + bx)/(c + dx))n)polylog(S(2), (a(-cg + df) - bcgx + bdfx)/((c + dx)(-ag + bf)))/((-ag + bf)S(2)(-cg + df)S(2)) - nS(2)(-S(3)ad + S(3)bc)(-adg - bcg + S(2)bdf)log(e((a + bx)/(c + dx))n)polylog(S(2), d(a + bx)/(b(c + dx)))/((-ag + bf)*S(2)(-cg + df)*S(2)) - n(-S(3)ad/S(2) + S(3)bc/S(2))(-adg - bcg + S(2)bdf)log(e((a + bx)/(c + dx))n)S(2)log((-ad + bc)/(b(c + dx)))/((-ag + bf)S(2)(-cg + df)*S(2)) + n(-S(3)ad/S(2) + S(3)bc/S(2))(-adg - bcg + S(2)bdf)log(e((a + bx)/(c + dx))n)S(2)log((f + gx)(-ad + bc)/((c + dx)(-ag + bf)))/((-ag + bf)S(2)(-cg + df)*S(2)) - log(e((a + bx)/(c + dx))n)S(3)/(S(2)g(f + gx)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(4), x), x, (a + bx)log(e((a + bx)/(c + dx))n)S(4)/b + nS(4)(-S(24)ad + S(24)bc)polylog(S(4), d(a + bx)/(b(c + dx)))/(bd) - n*S(3)(-S(24)ad + S(24)bc)log(e((a + bx)/(c + dx))*n)polylog(S(3), d(a + bx)/(b(c + dx)))/(bd) + nS(2)(-S(12)ad + S(12)bc)log(e((a + bx)/(c + dx))n)S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/(bd) + n(-S(4)ad + S(4)bc)log(e((a + bx)/(c + dx))n)S(3)log((-ad + bc)/(b(c + dx)))/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(5), x), x, (a + bx)log(e((a + bx)/(c + dx))n)S(5)/b - nS(5)(-S(120)ad + S(120)bc)polylog(S(5), d(a + bx)/(b(c + dx)))/(bd) + n*S(4)(-S(120)ad + S(120)bc)log(e((a + bx)/(c + dx))*n)polylog(S(4), d(a + bx)/(b(c + dx)))/(bd) - nS(3)(-S(60)ad + S(60)bc)log(e((a + bx)/(c + dx))n)S(2)polylog(S(3), d(a + bx)/(b(c + dx)))/(bd) + n*S(2)(-S(20)ad + S(20)bc)log(e((a + bx)/(c + dx))n)S(3)polylog(S(2), d(a + bx)/(b(c + dx)))/(bd) + n(-S(5)ad + S(5)bc)log(e((a + bx)/(c + dx))n)S(4)log((-ad + bc)/(b(c + dx)))/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*m(c + dx)(-m + S(-2))/log(e((a + bx)/(c + dx))*n), x), x, (e((a + bx)/(c + dx))n)(-(m + S(1))/n)(a + bx)*(m + S(1))(c + dx)(-m + S(-1))Ei((m + S(1))log(e((a + bx)/(c + dx))*n)/n)/(n(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(3)/((c + dx)*S(5)log(e((a + bx)/(c + dx))n)), x), x, (e((a + bx)/(c + dx))n)(-S(4)/n)(a + bx)*S(4)Ei(S(4)log(e((a + bx)/(c + dx))*n)/n)/(n(c + dx)S(4)(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)/((c + dx)*S(4)log(e((a + bx)/(c + dx))n)), x), x, (e((a + bx)/(c + dx))n)(-S(3)/n)(a + bx)*S(3)Ei(S(3)log(e((a + bx)/(c + dx))*n)/n)/(n(c + dx)S(3)(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/((c + dx)*S(3)log(e((a + bx)/(c + dx))n)), x), x, (e((a + bx)/(c + dx))n)(-S(2)/n)(a + bx)*S(2)Ei(S(2)log(e((a + bx)/(c + dx))*n)/n)/(n(c + dx)S(2)(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((c + dx)S(2)log(e((a + bx)/(c + dx))n)), x), x, (e((a + bx)/(c + dx))n)(-S(1)/n)(a + bx)Ei(log(e((a + bx)/(c + dx))*n)/n)/(n(c + dx)(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bx)(c + dx)log(e((a + bx)/(c + dx))n)), x), x, log(log(e((a + bx)/(c + dx))*n))/(n(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bx)S(2)log(e((a + bx)/(c + dx))n)), x), x, (e((a + bx)/(c + dx))n)(S(1)/n)(c + dx)Ei(-log(e((a + bx)/(c + dx))*n)/n)/(n(a + bx)(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)/((a + bx)*S(3)log(e((a + bx)/(c + dx))n)), x), x, (e((a + bx)/(c + dx))n)(S(2)/n)(c + dx)*S(2)Ei(-S(2)log(e((a + bx)/(c + dx))*n)/n)/(n(a + bx)S(2)(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)S(2)/((a + bx)*S(4)log(e((a + bx)/(c + dx))n)), x), x, (e((a + bx)/(c + dx))n)(S(3)/n)(c + dx)*S(3)Ei(-S(3)log(e((a + bx)/(c + dx))*n)/n)/(n(a + bx)S(3)(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)p/((a + bx)(c + dx)), x), x, log(e((a + bx)/(c + dx))n)(p + S(1))/(n(p + S(1))(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)p/(ac + bdxS(2) + x(ad + bc)), x), x, log(e((a + bx)/(c + dx))n)(p + S(1))/(n(p + S(1))(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cx/(a + bx))/(a + bx), x), x, -log(a/(a + bx))log(cx/(a + bx))/b - polylog(S(2), bx/(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cx/(a + bx))*S(2)/(x(a + bx)), x), x, log(cx/(a + bx))S(3)/(S(3)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a/(a + bx))log(cx/(a + bx))*S(2)/(x(a + bx)), x), x, -log(cx/(a + bx))S(2)polylog(S(2), bx/(a + bx))/a + S(2)log(cx/(a + bx))polylog(S(3), bx/(a + bx))/a - S(2)polylog(S(4), bx/(a + bx))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))*n)/((c + dx)(f + gx)), x), x, -npolylog(S(2), (a + bx)(-cg + df)/((c + dx)(-ag + bf)))/(-cg + df) - log(e((a + bx)/(c + dx))*n)log((f + gx)(-ad + bc)/((c + dx)(-ag + bf)))/(-cg + df), expand=True, _diff=True, _numerical=True) # long time in test assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(2)/((c + dx)(f + gx)), x), x, S(2)nS(2)polylog(S(3), (a(-cg + df) - bcgx + bdfx)/((c + dx)(-ag + bf)))/(-cg + df) - S(2)nlog(e((a + bx)/(c + dx))*n)polylog(S(2), (a(-cg + df) - bcgx + bdfx)/((c + dx)(-ag + bf)))/(-cg + df) - log(e((a + bx)/(c + dx))n)S(2)log((f + gx)(-ad + bc)/((c + dx)(-ag + bf)))/(-cg + df), expand=True, _diff=True, _numerical=True) # \|\| assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(3)/((c + dx)(f + gx)), x), x, -S(6)n*S(3)polylog(S(4), (a + bx)(-cg + df)/((c + dx)(-ag + bf)))/(-cg + df) + S(6)nS(2)log(e((a + bx)/(c + dx))n)polylog(S(3), (a(-cg + df) - bcgx + bdfx)/((c + dx)(-ag + bf)))/(-cg + df) - S(3)nlog(e((a + bx)/(c + dx))n)S(2)polylog(S(2), (a(-cg + df) - bcgx + bdfx)/((c + dx)(-ag + bf)))/(-cg + df) - log(e((a + bx)/(c + dx))n)S(3)log((f + gx)(-ad + bc)/((c + dx)(-ag + bf)))/(-cg + df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((-ad + bc)/(b(c + dx)))log(e(a + bx)/(c + dx))*S(2)/((c + dx)(ag + bgx)), x), x, -log(e(a + bx)/(c + dx))S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/(g(-ad + bc)) + S(2)log(e(a + bx)/(c + dx))polylog(S(3), d(a + bx)/(b(c + dx)))/(g(-ad + bc)) - S(2)polylog(S(4), d(a + bx)/(b(c + dx)))/(g(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)S(2)log((-ad + bc)/(b(c + dx)))/((c + dx)(ag + bgx)), x), x, -S(2)nS(2)polylog(S(4), d(a + bx)/(b(c + dx)))/(g(-ad + bc)) + S(2)nlog(e((a + bx)/(c + dx))*n)polylog(S(3), d(a + bx)/(b(c + dx)))/(g(-ad + bc)) - log(e((a + bx)/(c + dx))n)S(2)polylog(S(2), d(a + bx)/(b(c + dx)))/(g(-ad + bc)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(ax + b)/x), x), x, blog(x)/a + (ax + b)log(c(ax + b)/x)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(ax + b)/x)S(2), x), x, -S(2)blog(-b/(ax))log(c(ax + b)/x)/a - S(2)bpolylog(S(2), S(1) + b/(ax))/a + (ax + b)log(c(ax + b)/x)*S(2)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(ax + b)/x)S(3), x), x, -S(3)blog(-b/(ax))log(c(ax + b)/x)S(2)/a - S(6)blog(c(ax + b)/x)polylog(S(2), (ax + b)/(ax))/a + S(6)bpolylog(S(3), (ax + b)/(ax))/a + (ax + b)log(c(ax + b)/x)*S(3)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(ax + b)S(2)/xS(2)), x), x, xlog(c(ax + b)S(2)/xS(2)) + S(2)blog(ax + b)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(ax + b)S(2)/xS(2))S(2), x), x, xlog(c(ax + b)S(2)/xS(2))*S(2) - S(4)blog(b/(ax + b))log(c(ax + b)S(2)/xS(2))/a + S(8)bpolylog(S(2), ax/(ax + b))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(ax + b)S(2)/xS(2))S(3), x), x, xlog(c(ax + b)S(2)/xS(2))*S(3) - S(6)blog(b/(ax + b))log(c(ax + b)S(2)/xS(2))S(2)/a + S(24)blog(c(ax + b)S(2)/xS(2))polylog(S(2), ax/(ax + b))/a + S(48)bpolylog(S(3), ax/(ax + b))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cxS(2)/(ax + b)*S(2)), x), x, xlog(cxS(2)/(ax + b)*S(2)) - S(2)blog(ax + b)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cxS(2)/(ax + b)S(2))S(2), x), x, xlog(cx*S(2)/(ax + b)S(2))S(2) + S(4)blog(b/(ax + b))log(cxS(2)/(ax + b)*S(2))/a + S(8)bpolylog(S(2), ax/(ax + b))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cx*S(2)/(ax + b)S(2))S(3), x), x, xlog(cx*S(2)/(ax + b)S(2))S(3) + S(6)blog(b/(ax + b))log(cxS(2)/(ax + b)S(2))S(2)/a + S(24)blog(cxS(2)/(ax + b)*S(2))polylog(S(2), ax/(ax + b))/a - S(48)bpolylog(S(3), ax/(ax + b))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + b/x)/(d + exS(2)), x), x, -Ilog(sqrt(e)(-ax - b)/(Iasqrt(d) - bsqrt(e)))log(S(1) - Isqrt(e)x/sqrt(d))/(S(2)sqrt(d)sqrt(e)) + Ilog(sqrt(e)(ax + b)/(Iasqrt(d) + bsqrt(e)))log(S(1) + Isqrt(e)x/sqrt(d))/(S(2)sqrt(d)sqrt(e)) + log(a + b/x)atan(sqrt(e)x/sqrt(d))/(sqrt(d)sqrt(e)) - Ipolylog(S(2), a(sqrt(d) - Isqrt(e)x)/(asqrt(d) + Ibsqrt(e)))/(S(2)sqrt(d)sqrt(e)) + Ipolylog(S(2), a(sqrt(d) + Isqrt(e)x)/(asqrt(d) - Ibsqrt(e)))/(S(2)sqrt(d)sqrt(e)) + Ipolylog(S(2), -Isqrt(e)x/sqrt(d))/(S(2)sqrt(d)sqrt(e)) - Ipolylog(S(2), Isqrt(e)x/sqrt(d))/(S(2)sqrt(d)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)/(f + gx*S(2)), x), x, -Inlog(sqrt(g)(-a - bx)/(-asqrt(g) + Ibsqrt(f)))log(S(1) - Isqrt(g)x/sqrt(f))/(S(2)sqrt(f)sqrt(g)) + Inlog(sqrt(g)(a + bx)/(asqrt(g) + Ibsqrt(f)))log(S(1) + Isqrt(g)x/sqrt(f))/(S(2)sqrt(f)sqrt(g)) + Inlog(sqrt(g)(-c - dx)/(-csqrt(g) + Idsqrt(f)))log(S(1) - Isqrt(g)x/sqrt(f))/(S(2)sqrt(f)sqrt(g)) - Inlog(sqrt(g)(c + dx)/(csqrt(g) + Idsqrt(f)))log(S(1) + Isqrt(g)x/sqrt(f))/(S(2)sqrt(f)sqrt(g)) - Inpolylog(S(2), b(sqrt(f) - Isqrt(g)x)/(Iasqrt(g) + bsqrt(f)))/(S(2)sqrt(f)sqrt(g)) + Inpolylog(S(2), b(sqrt(f) + Isqrt(g)x)/(-Iasqrt(g) + bsqrt(f)))/(S(2)sqrt(f)sqrt(g)) + Inpolylog(S(2), d(sqrt(f) - Isqrt(g)x)/(Icsqrt(g) + dsqrt(f)))/(S(2)sqrt(f)sqrt(g)) - Inpolylog(S(2), d(sqrt(f) + Isqrt(g)x)/(-Icsqrt(g) + dsqrt(f)))/(S(2)sqrt(f)sqrt(g)) + log(e((a + bx)/(c + dx))*n)atan(sqrt(g)x/sqrt(f))/(sqrt(f)sqrt(g)), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate(log(e((a + bx)/(c + dx))n)/(f + gx + hxS(2)), x), x, nlog((S(2)ah - bg + b(g + S(2)hx))/(S(2)ah - b(g + sqrt(-S(4)fh + gS(2)))))log(g/sqrt(-S(4)fh + g*S(2)) + S(2)hx/sqrt(-S(4)fh + gS(2)) + S(1))/sqrt(-S(4)fh + gS(2)) - nlog((S(2)ch - dg + d(g + S(2)hx))/(S(2)ch - d(g + sqrt(-S(4)fh + gS(2)))))log(g/sqrt(-S(4)fh + g*S(2)) + S(2)hx/sqrt(-S(4)fh + gS(2)) + S(1))/sqrt(-S(4)fh + gS(2)) - nlog((-S(2)ah + bg - b(g + S(2)hx))/(-S(2)ah + bg - bsqrt(-S(4)fh + g*S(2))))log(-g/sqrt(-S(4)fh + g*S(2)) - S(2)hx/sqrt(-S(4)fh + gS(2)) + S(1))/sqrt(-S(4)fh + gS(2)) + nlog((-S(2)ch + dg - d(g + S(2)hx))/(-S(2)ch + dg - dsqrt(-S(4)fh + g*S(2))))log(-g/sqrt(-S(4)fh + g*S(2)) - S(2)hx/sqrt(-S(4)fh + gS(2)) + S(1))/sqrt(-S(4)fh + gS(2)) - npolylog(S(2), bsqrt(-S(4)fh + gS(2))(-g/sqrt(-S(4)fh + g*S(2)) - S(2)hx/sqrt(-S(4)fh + gS(2)) + S(1))/(S(2)ah - b(g - sqrt(-S(4)fh + g*S(2)))))/sqrt(-S(4)fh + gS(2)) + npolylog(S(2), -bsqrt(-S(4)fh + gS(2))(g/sqrt(-S(4)fh + g*S(2)) + S(2)hx/sqrt(-S(4)fh + gS(2)) + S(1))/(S(2)ah - b(g + sqrt(-S(4)fh + g*S(2)))))/sqrt(-S(4)fh + gS(2)) + npolylog(S(2), dsqrt(-S(4)fh + gS(2))(-g/sqrt(-S(4)fh + g*S(2)) - S(2)hx/sqrt(-S(4)fh + gS(2)) + S(1))/(S(2)ch - d(g - sqrt(-S(4)fh + g*S(2)))))/sqrt(-S(4)fh + gS(2)) - npolylog(S(2), -dsqrt(-S(4)fh + gS(2))(g/sqrt(-S(4)fh + g*S(2)) + S(2)hx/sqrt(-S(4)fh + gS(2)) + S(1))/(S(2)ch - d(g + sqrt(-S(4)fh + g*S(2)))))/sqrt(-S(4)fh + gS(2)) - S(2)log(e((a + bx)/(c + dx))n)atanh((g + S(2)hx)/sqrt(-S(4)fh + g*S(2)))/sqrt(-S(4)fh + gS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))n/(-cS(2)xS(2) + S(1)), x), x, -(a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))*(n + S(1))/(bc(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))S(3)/(-cS(2)xS(2) + S(1)), x), x, -(a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))*S(4)/(S(4)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))S(2)/(-cS(2)xS(2) + S(1)), x), x, -(a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))*S(3)/(S(3)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))/(-c*S(2)x*S(2) + S(1)), x), x, -(a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))*S(2)/(S(2)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))(-cS(2)x*S(2) + S(1))), x), x, -log(a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))/(bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))*S(2)(-c*S(2)x*S(2) + S(1))), x), x, S(1)/(bc(a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))S(3)(-c*S(2)x*S(2) + S(1))), x), x, S(1)/(S(2)bc(a + blog(sqrt(-cx + S(1))/sqrt(cx + S(1))))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sqrt(-ax + S(1))/sqrt(ax + S(1)))/(-aS(2)x*S(2) + S(1)), x), x, -log(sqrt(-ax + S(1))/sqrt(ax + S(1)))S(2)/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)log(a + bexp(x)), x), x, -xS(4)log(S(1) + bexp(x)/a)/S(4) + xS(4)log(a + bexp(x))/S(4) - xS(3)polylog(S(2), -bexp(x)/a) + S(3)x*S(2)polylog(S(3), -bexp(x)/a) - S(6)xpolylog(S(4), -bexp(x)/a) + S(6)polylog(S(5), -bexp(x)/a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)log(a + bexp(x)), x), x, -xS(3)log(S(1) + bexp(x)/a)/S(3) + xS(3)log(a + bexp(x))/S(3) - xS(2)polylog(S(2), -bexp(x)/a) + S(2)xpolylog(S(3), -bexp(x)/a) - S(2)polylog(S(4), -bexp(x)/a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(a + bexp(x)), x), x, -x*S(2)log(S(1) + bexp(x)/a)/S(2) + xS(2)log(a + bexp(x))/S(2) - xpolylog(S(2), -bexp(x)/a) + polylog(S(3), -bexp(x)/a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bexp(x)), x), x, -xlog(S(1) + bexp(x)/a) + xlog(a + bexp(x)) - polylog(S(2), -bexp(x)/a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bexp(x))/x, x), x, Integral(log(a + bexp(x))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)log(e(f(c(a + bx)))n + S(1)), x), x, -xS(3)polylog(S(2), -e(f(c(a + bx)))n)/(bcnlog(f)) + S(3)xS(2)polylog(S(3), -e(f(c(a + bx)))n)/(bS(2)c*S(2)n*S(2)log(f)*S(2)) - S(6)xpolylog(S(4), -e(f*(c(a + bx)))n)/(bS(3)c*S(3)n*S(3)log(f)*S(3)) + S(6)polylog(S(5), -e(f(c(a + bx)))n)/(bS(4)c*S(4)n*S(4)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(e(f*(c(a + bx)))n + S(1)), x), x, -xS(2)polylog(S(2), -e(f(c(a + bx)))n)/(bcnlog(f)) + S(2)xpolylog(S(3), -e(f(c(a + bx)))n)/(bS(2)c*S(2)n*S(2)log(f)*S(2)) - S(2)polylog(S(4), -e(f(c(a + bx)))n)/(bS(3)c*S(3)n*S(3)log(f)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(e(f(c(a + bx)))n + S(1)), x), x, -xpolylog(S(2), -e(f(c(a + bx)))n)/(bcnlog(f)) + polylog(S(3), -e(f(c(a + bx)))n)/(bS(2)c*S(2)n*S(2)log(f)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(f*(c(a + bx)))n + S(1)), x), x, -polylog(S(2), -e(f*(c(a + bx)))n)/(bcnlog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(e(f(c(a + bx)))n + S(1))/x, x), x, Integral(log(e(f*(c(a + bx)))n + S(1))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log(d + e(f(c(a + bx)))n), x), x, -xS(4)log(S(1) + e(f(c(a + bx)))n/d)/S(4) + xS(4)log(d + e(f(c(a + bx)))n)/S(4) - xS(3)polylog(S(2), -e(f(c(a + bx)))n/d)/(bcnlog(f)) + S(3)xS(2)polylog(S(3), -e(f(c(a + bx)))n/d)/(bS(2)c*S(2)n*S(2)log(f)*S(2)) - S(6)xpolylog(S(4), -e(f*(c(a + bx)))n/d)/(bS(3)c*S(3)n*S(3)log(f)*S(3)) + S(6)polylog(S(5), -e(f(c(a + bx)))n/d)/(bS(4)c*S(4)n*S(4)log(f)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)log(d + e(f*(c(a + bx)))n), x), x, -xS(3)log(S(1) + e(f(c(a + bx)))n/d)/S(3) + xS(3)log(d + e(f(c(a + bx)))n)/S(3) - xS(2)polylog(S(2), -e(f(c(a + bx)))n/d)/(bcnlog(f)) + S(2)xpolylog(S(3), -e(f(c(a + bx)))n/d)/(bS(2)c*S(2)n*S(2)log(f)*S(2)) - S(2)polylog(S(4), -e(f(c(a + bx)))n/d)/(bS(3)c*S(3)n*S(3)log(f)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(d + e(f(c(a + bx)))n), x), x, -xS(2)log(S(1) + e(f(c(a + bx)))n/d)/S(2) + xS(2)log(d + e(f(c(a + bx)))n)/S(2) - xpolylog(S(2), -e(f(c(a + bx)))n/d)/(bcnlog(f)) + polylog(S(3), -e(f(c(a + bx)))n/d)/(bS(2)c*S(2)n*S(2)log(f)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d + e(f*(c(a + bx)))n), x), x, -xlog(S(1) + e(f(c(a + bx)))n/d) + xlog(d + e(f(c(a + bx)))n) - polylog(S(2), -e(f*(c(a + bx)))n/d)/(bcnlog(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(d + e(f(c(a + bx)))n)/x, x), x, Integral(log(d + e(f*(c(a + bx)))n)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(b(F*(e(c + dx)))n + pi), x), x, xlog(pi) - polylog(S(2), -b(F(e(c + dx)))n/pi)/(denlog(F)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)sin(a + bx), x), x, -log(x)cos(a + bx)/b - sin(a)Si(bx)/b + cos(a)Ci(bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)sin(a + bx)*S(2), x), x, xlog(x)/S(2) - x/S(2) - log(x)sin(a + bx)cos(a + bx)/(S(2)b) + sin(S(2)a)Ci(S(2)bx)/(S(4)b) + cos(S(2)a)Si(S(2)bx)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)sin(a + bx)S(3), x), x, log(x)cos(a + bx)S(3)/(S(3)b) - log(x)cos(a + bx)/b - S(3)sin(a)Si(bx)/(S(4)b) + sin(S(3)a)Si(S(3)bx)/(S(12)b) + S(3)cos(a)Ci(bx)/(S(4)b) - cos(S(3)a)Ci(S(3)bx)/(S(12)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)cos(a + bx), x), x, log(x)sin(a + bx)/b - sin(a)Ci(bx)/b - cos(a)Si(bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)cos(a + bx)*S(2), x), x, xlog(x)/S(2) - x/S(2) + log(x)sin(a + bx)cos(a + bx)/(S(2)b) - sin(S(2)a)Ci(S(2)bx)/(S(4)b) - cos(S(2)a)Si(S(2)bx)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)cos(a + bx)S(3), x), x, -log(x)sin(a + bx)S(3)/(S(3)b) + log(x)sin(a + bx)/b - S(3)sin(a)Ci(bx)/(S(4)b) - sin(S(3)a)Ci(S(3)bx)/(S(12)b) - S(3)cos(a)Si(bx)/(S(4)b) - cos(S(3)a)Si(S(3)bx)/(S(12)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)cos(x) + sin(x)/x, x), x, log(x)sin(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(asin(x)), x), x, Ix*S(2)/S(2) + xlog(asin(x)) - xlog(-exp(S(2)Ix) + S(1)) + Ipolylog(S(2), exp(S(2)Ix))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(asin(x)*S(2)), x), x, Ix*S(2) + xlog(asin(x)S(2)) - S(2)xlog(-exp(S(2)Ix) + S(1)) + Ipolylog(S(2), exp(S(2)Ix)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(asin(x)n), x), x, InxS(2)/S(2) - nxlog(-exp(S(2)Ix) + S(1)) + Inpolylog(S(2), exp(S(2)Ix))/S(2) + xlog(asin(x)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acos(x)), x), x, IxS(2)/S(2) + xlog(acos(x)) - xlog(exp(S(2)Ix) + S(1)) + Ipolylog(S(2), -exp(S(2)Ix))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acos(x)*S(2)), x), x, Ix*S(2) + xlog(acos(x)S(2)) - S(2)xlog(exp(S(2)Ix) + S(1)) + Ipolylog(S(2), -exp(S(2)Ix)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acos(x)n), x), x, InxS(2)/S(2) - nxlog(exp(S(2)Ix) + S(1)) + Inpolylog(S(2), -exp(S(2)Ix))/S(2) + xlog(acos(x)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(atan(x)), x), x, xlog(atan(x)) + S(2)xatanh(exp(S(2)Ix)) - Ipolylog(S(2), -exp(S(2)Ix))/S(2) + Ipolylog(S(2), exp(S(2)Ix))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(atan(x)S(2)), x), x, xlog(atan(x)S(2)) + S(4)xatanh(exp(S(2)Ix)) - Ipolylog(S(2), -exp(S(2)Ix)) + Ipolylog(S(2), exp(S(2)Ix)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(atan(x)*n), x), x, S(2)nxatanh(exp(S(2)Ix)) - Inpolylog(S(2), -exp(S(2)Ix))/S(2) + Inpolylog(S(2), exp(S(2)Ix))/S(2) + xlog(atan(x)*n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acot(x)), x), x, xlog(acot(x)) - S(2)xatanh(exp(S(2)Ix)) + Ipolylog(S(2), -exp(S(2)Ix))/S(2) - Ipolylog(S(2), exp(S(2)Ix))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acot(x)S(2)), x), x, xlog(acot(x)S(2)) - S(4)xatanh(exp(S(2)Ix)) + Ipolylog(S(2), -exp(S(2)Ix)) - Ipolylog(S(2), exp(S(2)Ix)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acot(x)*n), x), x, -S(2)nxatanh(exp(S(2)Ix)) + Inpolylog(S(2), -exp(S(2)Ix))/S(2) - Inpolylog(S(2), exp(S(2)Ix))/S(2) + xlog(acot(x)*n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(asec(x)), x), x, -IxS(2)/S(2) + xlog(asec(x)) + xlog(exp(S(2)Ix) + S(1)) - Ipolylog(S(2), -exp(S(2)Ix))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(asec(x)*S(2)), x), x, -Ix*S(2) + xlog(asec(x)S(2)) + S(2)xlog(exp(S(2)Ix) + S(1)) - Ipolylog(S(2), -exp(S(2)Ix)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(asec(x)n), x), x, -InxS(2)/S(2) + nxlog(exp(S(2)Ix) + S(1)) - Inpolylog(S(2), -exp(S(2)Ix))/S(2) + xlog(asec(x)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acsc(x)), x), x, -IxS(2)/S(2) + xlog(acsc(x)) + xlog(-exp(S(2)Ix) + S(1)) - Ipolylog(S(2), exp(S(2)Ix))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acsc(x)*S(2)), x), x, -Ix*S(2) + xlog(acsc(x)S(2)) + S(2)xlog(-exp(S(2)Ix) + S(1)) - Ipolylog(S(2), exp(S(2)Ix)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acsc(x)n), x), x, -InxS(2)/S(2) + nxlog(-exp(S(2)Ix) + S(1)) - Inpolylog(S(2), exp(S(2)Ix))/S(2) + xlog(acsc(x)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(-cos(S(2)x)/S(2) + S(1)/2)cos(x), x), x, log(-cos(S(2)x)/S(2) + S(1)/2)sin(x) - S(2)sin(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cot(x)/log(Esin(x)), x), x, log(log(Esin(x))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(cot(x)/log(Esin(x)), x), x, log(log(sin(x)) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cot(x)/log(exp(sin(x))), x), x, log(log(exp(sin(x))))/(-log(exp(sin(x))) + sin(x)) - log(sin(x))/(-log(exp(sin(x))) + sin(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cos(x))sec(x)*S(2), x), x, -x + log(cos(x))tan(x) + tan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sin(x))cot(x), x), x, log(sin(x))S(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sin(x))sin(x)*S(2)cos(x), x), x, log(sin(x))sin(x)S(3)/S(3) - sin(x)S(3)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sin(a/S(2) + bx/S(2))cos(a/S(2) + bx/S(2)))cos(a + bx), x), x, log(sin(a/S(2) + bx/S(2))cos(a/S(2) + bx/S(2)))sin(a + bx)/b - sin(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(x)/log(cos(x)), x), x, -log(log(cos(x))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cos(x))tan(x), x), x, -log(cos(x))S(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cos(x))sin(x), x), x, -log(cos(x))cos(x) + cos(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cos(x))cos(x), x), x, log(cos(x))sin(x) - sin(x) + atanh(sin(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sin(x))cos(x), x), x, log(sin(x))sin(x) - sin(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sin(x))sin(x)*S(2), x), x, Ix*S(2)/S(4) - xlog(-exp(S(2)Ix) + S(1))/S(2) + xlog(sin(x))/S(2) + x/S(4) - log(sin(x))sin(x)cos(x)/S(2) + sin(x)cos(x)/S(4) + Ipolylog(S(2), exp(S(2)Ix))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sin(x))sin(x)*S(3), x), x, log(sin(x))cos(x)*S(3)/S(3) - log(sin(x))cos(x) - cos(x)*S(3)/S(9) + S(2)cos(x)/S(3) - S(2)atanh(cos(x))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sin(sqrt(x))), x), x, Ix*(S(3)/2)/S(3) + Isqrt(x)polylog(S(2), exp(S(2)Isqrt(x))) - xlog(-exp(S(2)Isqrt(x)) + S(1)) + xlog(sin(sqrt(x))) - polylog(S(3), exp(S(2)Isqrt(x)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sin(x))csc(x)*S(2), x), x, -x - log(sin(x))cot(x) - cot(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)sinh(a + bx), x), x, log(x)cosh(a + bx)/b - sinh(a)Shi(bx)/b - cosh(a)Chi(bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)sinh(a + bx)*S(2), x), x, -xlog(x)/S(2) + x/S(2) + log(x)sinh(a + bx)cosh(a + bx)/(S(2)b) - sinh(S(2)a)Chi(S(2)bx)/(S(4)b) - cosh(S(2)a)Shi(S(2)bx)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)sinh(a + bx)S(3), x), x, log(x)cosh(a + bx)S(3)/(S(3)b) - log(x)cosh(a + bx)/b + S(3)sinh(a)Shi(bx)/(S(4)b) - sinh(S(3)a)Shi(S(3)bx)/(S(12)b) + S(3)cosh(a)Chi(bx)/(S(4)b) - cosh(S(3)a)Chi(S(3)bx)/(S(12)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)cosh(a + bx), x), x, log(x)sinh(a + bx)/b - sinh(a)Chi(bx)/b - cosh(a)Shi(bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)cosh(a + bx)*S(2), x), x, xlog(x)/S(2) - x/S(2) + log(x)sinh(a + bx)cosh(a + bx)/(S(2)b) - sinh(S(2)a)Chi(S(2)bx)/(S(4)b) - cosh(S(2)a)Shi(S(2)bx)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)cosh(a + bx)S(3), x), x, log(x)sinh(a + bx)S(3)/(S(3)b) + log(x)sinh(a + bx)/b - S(3)sinh(a)Chi(bx)/(S(4)b) - sinh(S(3)a)Chi(S(3)bx)/(S(12)b) - S(3)cosh(a)Shi(bx)/(S(4)b) - cosh(S(3)a)Shi(S(3)bx)/(S(12)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(asinh(x)), x), x, xS(2)/S(2) + xlog(asinh(x)) - xlog(-exp(S(2)x) + S(1)) - polylog(S(2), exp(S(2)x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(asinh(x)S(2)), x), x, xS(2) + xlog(asinh(x)S(2)) - S(2)xlog(-exp(S(2)x) + S(1)) - polylog(S(2), exp(S(2)x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(asinh(x)*n), x), x, nx*S(2)/S(2) - nxlog(-exp(S(2)x) + S(1)) - npolylog(S(2), exp(S(2)x))/S(2) + xlog(asinh(x)*n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acosh(x)), x), x, x*S(2)/S(2) + xlog(acosh(x)) - xlog(exp(S(2)x) + S(1)) - polylog(S(2), -exp(S(2)x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acosh(x)S(2)), x), x, xS(2) + xlog(acosh(x)S(2)) - S(2)xlog(exp(S(2)x) + S(1)) - polylog(S(2), -exp(S(2)x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acosh(x)*n), x), x, nx*S(2)/S(2) - nxlog(exp(S(2)x) + S(1)) - npolylog(S(2), -exp(S(2)x))/S(2) + xlog(acosh(x)*n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(tanh(x)), x), x, xlog(tanh(x)) + S(2)xatanh(exp(S(2)x)) + polylog(S(2), -exp(S(2)x))/S(2) - polylog(S(2), exp(S(2)x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(atanh(x)), x), x, xlog(atanh(x)) + S(2)xatanh(exp(S(2)x)) + polylog(S(2), -exp(S(2)x))/S(2) - polylog(S(2), exp(S(2)x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(atanh(x)*S(2)), x), x, xlog(atanh(x)S(2)) + S(4)xatanh(exp(S(2)x)) + polylog(S(2), -exp(S(2)x)) - polylog(S(2), exp(S(2)x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(atanh(x)n), x), x, S(2)nxatanh(exp(S(2)x)) + npolylog(S(2), -exp(S(2)x))/S(2) - npolylog(S(2), exp(S(2)x))/S(2) + xlog(atanh(x)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(coth(x)), x), x, xlog(coth(x)) - S(2)xatanh(exp(S(2)x)) - polylog(S(2), -exp(S(2)x))/S(2) + polylog(S(2), exp(S(2)x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acoth(x)), x), x, xlog(acoth(x)) - S(2)xatanh(exp(S(2)x)) - polylog(S(2), -exp(S(2)x))/S(2) + polylog(S(2), exp(S(2)x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acoth(x)*S(2)), x), x, xlog(acoth(x)S(2)) - S(4)xatanh(exp(S(2)x)) - polylog(S(2), -exp(S(2)x)) + polylog(S(2), exp(S(2)x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acoth(x)n), x), x, -S(2)nxatanh(exp(S(2)x)) - npolylog(S(2), -exp(S(2)x))/S(2) + npolylog(S(2), exp(S(2)x))/S(2) + xlog(acoth(x)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(asech(x)), x), x, -x*S(2)/S(2) + xlog(asech(x)) + xlog(exp(S(2)x) + S(1)) + polylog(S(2), -exp(S(2)x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(asech(x)S(2)), x), x, -xS(2) + xlog(asech(x)S(2)) + S(2)xlog(exp(S(2)x) + S(1)) + polylog(S(2), -exp(S(2)x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(asech(x)*n), x), x, -nx*S(2)/S(2) + nxlog(exp(S(2)x) + S(1)) + npolylog(S(2), -exp(S(2)x))/S(2) + xlog(asech(x)*n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acsch(x)), x), x, -x*S(2)/S(2) + xlog(acsch(x)) + xlog(-exp(S(2)x) + S(1)) + polylog(S(2), exp(S(2)x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acsch(x)S(2)), x), x, -xS(2) + xlog(acsch(x)S(2)) + S(2)xlog(-exp(S(2)x) + S(1)) + polylog(S(2), exp(S(2)x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(acsch(x)*n), x), x, -nx*S(2)/S(2) + nxlog(-exp(S(2)x) + S(1)) + npolylog(S(2), exp(S(2)x))/S(2) + xlog(acsch(x)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cosh(x)S(2))sinh(x), x), x, log(cosh(x)S(2))cosh(x) - S(2)cosh(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/sqrt(x), x), x, S(2)sqrt(x)log(x) - S(4)sqrt(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(-S(3)xS(2) + S(2)), x), x, -xS(2)/S(2) - (-x*S(2)/S(2) + S(1)/3)log(-S(3)xS(2) + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(-log(x)*S(2) + S(1))), x), x, asin(log(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(16)x*S(3)log(x)*S(2), x), x, S(4)x*S(4)log(x)*S(2) - S(2)x*S(4)log(x) + x*S(4)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sqrt(a + bx)), x), x, -x/S(2) + (a + bx)log(sqrt(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(sqrt(x + S(2))), x), x, x*S(2)log(sqrt(x + S(2)))/S(2) - x*S(2)/S(8) + x/S(2) - log(x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog((S(3)x + S(1))(S(1)/3)), x), x, xS(2)log((S(3)x + S(1))(S(1)/3))/S(2) - xS(2)/S(12) + x/S(18) - log(S(3)x + S(1))/S(54), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(xS(3) + x), x), x, xS(2)log(x*S(3) + x)/S(2) - S(3)xS(2)/S(4) + log(xS(2) + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x + sqrt(x*S(2) + S(1))), x), x, xlog(x + sqrt(xS(2) + S(1))) - sqrt(xS(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x + sqrt(x*S(2) + S(-1))), x), x, xlog(x + sqrt(xS(2) + S(-1))) - sqrt(xS(2) + S(-1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x - sqrt(x*S(2) + S(-1))), x), x, xlog(x - sqrt(xS(2) + S(-1))) + sqrt(xS(2) + S(-1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sqrt(x) + sqrt(x + S(1))), x), x, -sqrt(x)sqrt(x + S(1))/S(2) + xlog(sqrt(x) + sqrt(x + S(1))) + asinh(sqrt(x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(1)/3)log(x), x), x, S(3)x(S(4)/3)log(x)/S(4) - S(9)x(S(4)/3)/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)log(x), x), x, x(log(S(2)) + S(1))/(log(S(2)) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-log(x) + S(1))/xS(2), x), x, log(x)/x, expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((-log(x) + S(1))/xS(2), x), x, (log(x) + S(-1))/x + S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x + sqrt(x + S(1)) + S(1)), x), x, xlog(x + sqrt(x + S(1)) + S(1)) - x + sqrt(x + S(1)) + log(x + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x*S(3) + x), x), x, xlog(x*S(3) + x) - S(3)x + S(2)atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)log(S(7)x + S(-8)), x), x, (S(7)x + S(-8))(log(S(2)) + S(1))/(S(7)(log(S(2)) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((S(5)x + S(-11))/(S(76)x + S(5))), x), x, (x + S(-11)/5)log((S(5)x + S(-11))/(S(76)x + S(5))) - S(861)log(S(76)x + S(5))/S(380), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((x + S(1))/(x + S(-1)))/xS(2), x), x, S(2)log(x) - S(2)log(-x + S(1)) - (x + S(1))log((-x + S(-1))/(-x + S(1)))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(1)/(x + S(13))), x), x, x + (x + S(13))log(S(1)/(x + S(13))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog((x + S(1))/xS(2)), x), x, xS(2)log((x + S(1))/xS(2))/S(2) + xS(2)/S(4) + x/S(2) - log(x + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)log((S(5)x + S(7))/xS(2)), x), x, xS(4)log((S(5)x + S(7))/xS(2))/S(4) + xS(4)/S(16) + S(7)x*S(3)/S(60) - S(49)x*S(2)/S(200) + S(343)x/S(500) - S(2401)log(S(5)x + S(7))/S(2500), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)log(a + bx), x), x, -ax/S(2) - bxS(2)/S(4) + (a + bx)*S(2)log(a + bx)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)log(a + bx), x), x, (a + bx)*S(3)log(a + bx)/(S(3)b) - (a + bx)S(3)/(S(9)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bx)/(a + bx), x), x, log(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bx)/(a + bx)*S(2), x), x, -log(a + bx)/(b(a + bx)) - S(1)/(b(a + bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)nlog(a + bx), x), x, (a + bx)*(n + S(1))log(a + bx)/(b(n + S(1))) - (a + bx)(n + S(1))/(b(n + S(1))*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(alog(bx)p), x), x, xlog(alog(bx)*p) - pli(bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(alog(bxn)p), x), x, -px(bxn)(-S(1)/n)Ei(log(bx*n)/n) + xlog(alog(bxn)p), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(alog(bx)*p)/x, x), x, -(p - log(alog(bx)p))log(bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(alog(bxn)p)/x, x), x, -(p - log(alog(bxn)p))log(bxn)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(alog(bx)*p), x), x, -px*(m + S(1))(bx)(-m + S(-1))Ei((m + S(1))log(bx))/(m + S(1)) + x*(m + S(1))log(alog(bx)p)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmlog(alog(bxn)p), x), x, -px*(m + S(1))(bxn)(-(m + S(1))/n)Ei((m + S(1))log(bxn)/n)/(m + S(1)) + x(m + S(1))log(alog(bxn)p)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/sqrt(a + blog(x)), x), x, -sqrt(pi)aexp(-a/b)erfi(sqrt(a + blog(x))/sqrt(b))/b*(S(3)/2) + xsqrt(a + blog(x))/b - sqrt(pi)exp(-a/b)erfi(sqrt(a + blog(x))/sqrt(b))/(S(2)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/sqrt(a - blog(x)), x), x, -sqrt(pi)aexp(a/b)erf(sqrt(a - blog(x))/sqrt(b))/b*(S(3)/2) - xsqrt(a - blog(x))/b + sqrt(pi)exp(a/b)erf(sqrt(a - blog(x))/sqrt(b))/(S(2)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Blog(x))/sqrt(a + blog(x)), x), x, Bxsqrt(a + blog(x))/b - sqrt(pi)Bexp(-a/b)erfi(sqrt(a + blog(x))/sqrt(b))/(S(2)sqrt(b)) + sqrt(pi)(Ab - Ba)exp(-a/b)erfi(sqrt(a + blog(x))/sqrt(b))/b(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Blog(x))/sqrt(a - blog(x)), x), x, -Bxsqrt(a - blog(x))/b + sqrt(pi)Bexp(a/b)erf(sqrt(a - blog(x))/sqrt(b))/(S(2)sqrt(b)) + sqrt(pi)(-Ab - Ba)exp(a/b)erf(sqrt(a - blog(x))/sqrt(b))/b(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(x)/sqrt(xS(2) + S(-1)), x), x, sqrt(xS(2) + S(-1))log(x) - sqrt(xS(2) + S(-1)) + atan(sqrt(xS(2) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(x*S(2) + S(4))log(x), x), x, (xS(2) + S(4))(S(3)/2)log(x)/S(3) - (xS(2) + S(4))(S(3)/2)/S(9) - S(4)sqrt(x*S(2) + S(4))/S(3) + S(8)atanh(sqrt(x*S(2) + S(4))/S(2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ax + bxlog(cxn)), x), x, log(a + blog(cxn))/(bn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ax + bxlog(cxn)S(2)), x), x, atan(sqrt(b)log(cx*n)/sqrt(a))/(sqrt(a)sqrt(b)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ax + bxlog(cxn)S(3)), x), x, log(a(S(1)/3) + b(S(1)/3)log(cxn))/(S(3)a*(S(2)/3)b*(S(1)/3)n) - log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)log(cxn) + b(S(2)/3)log(cxn)S(2))/(S(6)a*(S(2)/3)b*(S(1)/3)n) - sqrt(S(3))atan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)log(cxn))/(S(3)a*(S(1)/3)))/(S(3)a*(S(2)/3)b*(S(1)/3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ax + bxlog(cxn)S(4)), x), x, -sqrt(S(2))log(-sqrt(S(2))a*(S(1)/4)b*(S(1)/4)log(cxn) + sqrt(a) + sqrt(b)log(cxn)S(2))/(S(8)a*(S(3)/4)b*(S(1)/4)n) + sqrt(S(2))log(sqrt(S(2))a*(S(1)/4)b*(S(1)/4)log(cxn) + sqrt(a) + sqrt(b)log(cxn)S(2))/(S(8)a*(S(3)/4)b*(S(1)/4)n) - sqrt(S(2))atan(S(1) - sqrt(S(2))b*(S(1)/4)log(cxn)/a(S(1)/4))/(S(4)a*(S(3)/4)b*(S(1)/4)n) + sqrt(S(2))atan(S(1) + sqrt(S(2))b*(S(1)/4)log(cxn)/a(S(1)/4))/(S(4)a*(S(3)/4)b*(S(1)/4)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ax + bx/log(cxn)), x), x, log(x)/a - blog(alog(cxn) + b)/(aS(2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ax + bx/log(cxn)S(2)), x), x, log(x)/a - sqrt(b)atan(sqrt(a)log(cxn)/sqrt(b))/(a(S(3)/2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ax + bx/log(cxn)S(3)), x), x, log(x)/a - b(S(1)/3)log(a*(S(1)/3)log(cxn) + b(S(1)/3))/(S(3)a*(S(4)/3)n) + b*(S(1)/3)log(a*(S(2)/3)log(cxn)S(2) - a(S(1)/3)b*(S(1)/3)log(cxn) + b(S(2)/3))/(S(6)a*(S(4)/3)n) + sqrt(S(3))b(S(1)/3)atan(sqrt(S(3))(-S(2)a*(S(1)/3)log(cxn) + b(S(1)/3))/(S(3)b*(S(1)/3)))/(S(3)a*(S(4)/3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ax + bx/log(cxn)S(4)), x), x, log(x)/a + sqrt(S(2))b*(S(1)/4)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)log(cxn) + sqrt(a)log(cxn)S(2) + sqrt(b))/(S(8)a*(S(5)/4)n) - sqrt(S(2))b(S(1)/4)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)log(cxn) + sqrt(a)log(cxn)S(2) + sqrt(b))/(S(8)a*(S(5)/4)n) - sqrt(S(2))b(S(1)/4)atan(sqrt(S(2))a(S(1)/4)log(cxn)/b(S(1)/4) + S(-1))/(S(4)a*(S(5)/4)n) - sqrt(S(2))b(S(1)/4)atan(sqrt(S(2))a(S(1)/4)log(cxn)/b(S(1)/4) + S(1))/(S(4)a*(S(5)/4)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/(S(4)xlog(x)*S(2) + x), x), x, log(S(4)log(x)*S(2) + S(1))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(S(7)x)S(2) + xlog(S(7)x) + x), x), x, S(2)sqrt(S(3))atan(sqrt(S(3))(S(2)log(S(7)x) + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((log(S(3)x) + S(-1))/(x(log(S(3)x)S(2) - log(S(3)x) + S(1))), x), x, log(log(S(3)x)S(2) - log(S(3)x) + S(1))/S(2) + sqrt(S(3))atan(sqrt(S(3))(-S(2)log(S(3)x) + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((log(S(3)x)S(2) + S(-1))/(xlog(S(3)x)S(3) + x), x), x, log(log(S(3)x)*S(2) - log(S(3)x) + S(1))/S(2) + sqrt(S(3))atan(sqrt(S(3))(-S(2)log(S(3)x) + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((log(S(3)x)S(2) + S(-1))/(xlog(S(3)x)S(2) + xlog(S(3)x) + x), x), x, log(x) - log(log(S(3)x)*S(2) + log(S(3)x) + S(1))/S(2) - sqrt(S(3))atan(sqrt(S(3))(S(2)log(S(3)x) + S(1))/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(log(x) + S(3))), x), x, log(log(x) + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(log(x) + S(1))/x, x), x, S(2)(log(x) + S(1))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((log(x) + S(1))S(5)/x, x), x, (log(x) + S(1))*S(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(log(x))), x), x, S(2)sqrt(log(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(log(x)*S(2) + S(1))), x), x, atan(log(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(log(x)S(2) + S(-3))), x), x, atanh(log(x)/sqrt(log(x)S(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(-S(9)log(x)*S(2) + S(4))), x), x, asin(S(3)log(x)/S(2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(log(x)S(2) + S(4))), x), x, asinh(log(x)/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(3)log(S(6)x)S(3) + S(2))), x), x, S(2)(S(1)/3)S(3)(S(2)/3)log(S(3)*(S(1)/3)log(S(6)x) + S(2)(S(1)/3))/S(18) - S(2)(S(1)/3)S(3)*(S(2)/3)log(S(3)*(S(2)/3)log(S(6)x)S(2) - S(6)(S(1)/3)log(S(6)x) + S(2)(S(2)/3))/S(36) - S(2)(S(1)/3)S(3)*(S(1)/6)atan(sqrt(S(3))(-S(2)(S(2)/3)S(3)*(S(1)/3)log(S(6)x) + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(log(S(6)x))/(xlog(S(6)x)), x), x, log(log(S(6)x))S(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)log(x)/x, x), x, S(2)log(x)/log(S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(log(x))S(2)/x, x), x, log(x)/S(2) - sin(log(x))cos(log(x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-log(x) + S(7))/(x(log(x) + S(3))), x), x, -log(x) + S(10)log(log(x) + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-log(x) + S(2))(log(x) + S(3))S(2)/x, x), x, -log(x)S(4)/S(4) - S(4)log(x)*S(3)/S(3) + S(3)log(x)*S(2)/S(2) + S(18)log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(log(x)*S(2) + S(1))log(x)S(2)/x, x), x, sqrt(log(x)S(2) + S(1))log(x)S(3)/S(4) + sqrt(log(x)S(2) + S(1))log(x)/S(8) - asinh(log(x))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((log(x) + S(1))/(x(S(2)log(x) + S(3))*S(2)), x), x, log(S(2)log(x) + S(3))/S(4) + S(1)/(S(4)(S(2)log(x) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/(xsqrt(log(x) + S(1))), x), x, S(2)(log(x) + S(1))*(S(3)/2)/S(3) - S(2)sqrt(log(x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/(xsqrt(S(4)log(x) + S(-1))), x), x, (S(4)log(x) + S(-1))(S(3)/2)/S(24) + sqrt(S(4)log(x) + S(-1))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(log(x) + S(1))/(xlog(x)), x), x, S(2)sqrt(log(x) + S(1)) - S(2)atanh(sqrt(log(x) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((log(x)S(2) - S(4)log(x) + S(1))/(x(log(x) + S(-1))S(4)), x), x, (log(x) + S(-1))(S(-2)) + S(1)/(-log(x) + S(1)) - S(2)/(S(3)(-log(x) + S(1))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(S(1)/x)S(2)/xS(5), x), x, -log(S(1)/x)S(2)/(S(4)xS(4)) + log(S(1)/x)/(S(8)x*S(4)) - S(1)/(S(32)x*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((log(axn)S(2))*p/x, x), x, (log(axn)S(2))*plog(axn)/(n(S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((log(axn)m)*p/x, x), x, (log(axn)m)*plog(axn)/(n(mp + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(log(axn)S(2))/x, x), x, sqrt(log(axn)S(2))log(axn)/(S(2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((blog(axn)m)*p/x, x), x, (blog(axn)m)plog(axn)/(n(mp + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-log(ax*S(2))), x), x, -sqrt(S(2))sqrt(pi)xerf(sqrt(S(2))sqrt(-log(ax*S(2)))/S(2))/(S(2)sqrt(axS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-log(a/xS(2))), x), x, sqrt(S(2))sqrt(pi)xsqrt(a/x*S(2))erfi(sqrt(S(2))sqrt(-log(a/xS(2)))/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-log(ax*n)), x), x, -sqrt(pi)x(axn)(-S(1)/n)erf(sqrt(-log(ax*n))/sqrt(n))/sqrt(n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sqrt(x) - x + S(1))/x, x), x, -S(2)log(sqrt(x))log((-S(2)sqrt(x) - sqrt(S(5)) + S(1))/(-sqrt(S(5)) + S(1))) + S(2)log(sqrt(x))log(sqrt(x) - x + S(1)) - S(2)log(S(1)/2 + sqrt(S(5))/S(2))log(-S(2)sqrt(x) + S(1) + sqrt(S(5))) - S(2)polylog(S(2), S(2)sqrt(x)/(-sqrt(S(5)) + S(1))) + S(2)polylog(S(2), (-S(2)sqrt(x) + S(1) + sqrt(S(5)))/(S(1) + sqrt(S(5)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(c + dx)/(a + bx), x), x, -alog(-d(a + bx)/(-ad + bc))log(c + dx)/bS(2) - apolylog(S(2), b(c + dx)/(-ad + bc))/b*S(2) - x/b + (c + dx)log(c + dx)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/(x + S(-1)), x), x, -polylog(S(2), -x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xlog(-a - bx + S(1))/(a + bx), x), x, apolylog(S(2), a + bx)/b*S(2) - x/b - (-a - bx + S(1))log(-a - bx + S(1))/b*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)log(x)/(x(b + cx)), x), x, log(x)*S(2)/S(2) + log(x)log((b + cx)/b) + polylog(S(2), -cx/b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)sin(xlog(x)) + sin(xlog(x)), x), x, -cos(xlog(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((-(x + S(-1))S(2) + S(1))/((x + S(-1))S(2) + S(1)))/xS(2), x), x, log(x)/S(2) + log(-x + S(2))/S(2) - log(xS(2) - S(2)x + S(2))/S(2) - atan(x + S(-1)) - log((-(-x + S(1))S(2) + S(1))/((x + S(-1))S(2) + S(1)))/x - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(sqrt(x) + x), x), x, sqrt(x) + xlog(sqrt(x) + x) - x - log(sqrt(x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(-x/(x + S(1))), x), x, xlog(-x/(x + S(1))) - log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((x + S(-1))/(x + S(1))), x), x, (x + S(-1))log((x + S(-1))/(x + S(1))) - S(2)log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log((-xS(2) + S(1))/(xS(2) + S(1)))/(x + S(1))S(2), x), x, log(-xS(2) + S(1))/S(2) - log(xS(2) + S(1))/S(2) - atan(x) - log((-xS(2) + S(1))/(xS(2) + S(1)))/(x + S(1)) - S(1)/(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/(-xS(2) + S(1)), x), x, log(x)atanh(x) + polylog(S(2), -x)/S(2) - polylog(S(2), x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(x)/(x*S(2) + S(1)), x), x, log(x)atan(x) - Ipolylog(S(2), -Ix)/S(2) + Ipolylog(S(2), Ix)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(xS(2) + S(1))n)/(xS(2) + S(1)), x), x, S(2)nlog(S(2)I/(-x + I))atan(x) + Inatan(x)S(2) + Inpolylog(S(2), (-x - I)/(-x + I)) + log(c(xS(2) + S(1))n)atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(xS(2)/(xS(2) + S(1)))/(xS(2) + S(1)), x), x, -S(2)log(S(2)x/(x + I))atan(x) + log(xS(2)/(xS(2) + S(1)))atan(x) + Iatan(x)*S(2) + Ipolylog(S(2), (-x + I)/(x + I)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cxn)/(a + bx*S(2)), x), x, -Inpolylog(S(2), -Isqrt(b)x/sqrt(a))/(S(2)sqrt(a)sqrt(b)) + Inpolylog(S(2), Isqrt(b)x/sqrt(a))/(S(2)sqrt(a)sqrt(b)) + log(cx*n)atan(sqrt(b)x/sqrt(a))/(sqrt(a)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(c(a + bxS(2))n)/(a + bxS(2)), x), x, S(2)nlog(S(2)Isqrt(a)/(Isqrt(a) - sqrt(b)x))atan(sqrt(b)x/sqrt(a))/(sqrt(a)sqrt(b)) + Inatan(sqrt(b)x/sqrt(a))S(2)/(sqrt(a)sqrt(b)) + Inpolylog(S(2), (-sqrt(a) + Isqrt(b)x)/(sqrt(a) + Isqrt(b)x))/(sqrt(a)sqrt(b)) + log(c(a + bxS(2))n)atan(sqrt(b)x/sqrt(a))/(sqrt(a)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(cxS(2)/(a + bx*S(2)))/(a + bx*S(2)), x), x, -S(2)log(S(2)sqrt(b)x/(Isqrt(a) + sqrt(b)x))atan(sqrt(b)x/sqrt(a))/(sqrt(a)sqrt(b)) + log(cx*S(2)/(a + bx*S(2)))atan(sqrt(b)x/sqrt(a))/(sqrt(a)sqrt(b)) + Iatan(sqrt(b)x/sqrt(a))*S(2)/(sqrt(a)sqrt(b)) + Ipolylog(S(2), (sqrt(a) + Isqrt(b)x)/(sqrt(a) - Isqrt(b)x))/(sqrt(a)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(Isqrt(-ax + S(1))/sqrt(ax + S(1)) + S(1))/(-aS(2)x*S(2) + S(1)), x), x, polylog(S(2), -Isqrt(-ax + S(1))/sqrt(ax + S(1)))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(-Isqrt(-ax + S(1))/sqrt(ax + S(1)) + S(1))/(-aS(2)x*S(2) + S(1)), x), x, polylog(S(2), Isqrt(-ax + S(1))/sqrt(ax + S(1)))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(exp(a + bx)), x), x, log(exp(a + bx))*S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(exp(a + bxn)), x), x, -bnx(n + S(1))/(n + S(1)) + xlog(exp(a + bxn)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(x)log(a + bexp(x)), x), x, -exp(x) + (a + bexp(x))log(a + bexp(x))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xlog(exp(x))), x), x, -log(x)/(x - log(exp(x))) + log(log(exp(x)))/(x - log(exp(x))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(exp(a + bx)log(x), x), x, -exp(a)Ei(bx)/b + exp(a + bx)log(x)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(x + log(x)), x), x, Integral(xS(2)/(x + log(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(x + log(x)), x), x, Integral(x/(x + log(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x + log(x)), x), x, Integral(S(1)/(x + log(x)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(x + log(x))), x), x, Integral(S(1)/(x(x + log(x))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(x + log(x))), x), x, Integral(S(1)/(x*S(2)(x + log(x))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-log(x) + S(1))/(x(x + log(x))), x), x, log(S(1) + log(x)/x), expand=True, _diff=True, _numerical=True) ''' apart # apart assert rubi_test(rubi_integrate((x + S(1))/((x + log(x))log(x)), x), x, -log(x + log(x)) + log(log(x)) + li(x), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((x + S(1))/((x + log(x))log(x)), x), x, -log(x + log(x)) + log(log(x)) + Ei(log(x)), expand=True, _diff=True, _numerical=True) # assert rubi_test(rubi_integrate(log(sqrt((x + S(1))/x) + S(2)), x), x, xlog(sqrt((x + S(1))/x) + S(2)) - log(-sqrt((x + S(1))/x) + S(1))/S(6) + log(sqrt((x + S(1))/x) + S(1))/S(2) - log(sqrt((x + S(1))/x) + S(2))/S(3), expand=True, _diff=True, _numerical=True) # assert rubi_test(rubi_integrate(log(sqrt((x + S(1))/x) + S(1)), x), x, xlog(sqrt((x + S(1))/x) + S(1)) + atanh(sqrt((x + S(1))/x))/S(2) - S(1)/(S(2)(sqrt((x + S(1))/x) + S(1))), expand=True, _diff=True, _numerical=True) # assert rubi_test(rubi_integrate(log(sqrt((x + S(1))/x)), x), x, (x + S(1))log(sqrt((x + S(1))/x)) + log(x)/S(2), expand=True, _diff=True, _numerical=True) # assert rubi_test(rubi_integrate(log(sqrt((x + S(1))/x) + S(-1)), x), x, xlog(sqrt((x + S(1))/x) + S(-1)) - atanh(sqrt(S(1) + S(1)/x))/S(2) - S(1)/(S(2)(-sqrt(S(1) + S(1)/x) + S(1))), expand=True, _diff=True, _numerical=True) # assert rubi_test(rubi_integrate(log(sqrt((x + S(1))/x) + S(-2)), x), x, xlog(sqrt((x + S(1))/x) + S(-2)) + log(-sqrt(S(1) + S(1)/x) + S(1))/S(2) - log(-sqrt(S(1) + S(1)/x) + S(2))/S(3) - log(sqrt(S(1) + S(1)/x) + S(1))/S(6), expand=True, _diff=True, _numerical=True) ''' assert rubi_test(rubi_integrate(x*(ax)log(x) + x(ax), x), x, x*(ax)/a, expand=True, _diff=True, _numerical=True) # fails in mathematica too assert rubi_test(rubi_integrate((log(x)m)p, x), x, (-log(x))*(-mp)(log(x)m)pGamma(mp + S(1), -log(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(log(a + bx + csqrt(d + ex))/(f + gxS(2)), x), x, -log((ae - bd + b(d + ex) + cesqrt(d + ex))/e)log(-g(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) - esqrt(-f)))/(S(2)sqrt(g)sqrt(-f)) + log((ae - bd + b(d + ex) + cesqrt(d + ex))/e)log(-g(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) + esqrt(-f)))/(S(2)sqrt(g)sqrt(-f)) - log((ae - bd + b(d + ex) + cesqrt(d + ex))/e)log(g(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) - esqrt(-f)))/(S(2)sqrt(g)sqrt(-f)) + log((ae - bd + b(d + ex) + cesqrt(d + ex))/e)log(g(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) + esqrt(-f)))/(S(2)sqrt(g)sqrt(-f)) + log(-g(S(1)/4)(S(2)bsqrt(d + ex) + ce - sqrt(-S(4)abe + S(4)b*S(2)d + c*S(2)e*S(2)))/(S(2)bsqrt(dsqrt(g) - esqrt(-f)) - g(S(1)/4)(ce - sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))log(g*(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) - esqrt(-f)))/(S(2)sqrt(g)sqrt(-f)) + log(g(S(1)/4)(S(2)bsqrt(d + ex) + ce - sqrt(-S(4)abe + S(4)b*S(2)d + c*S(2)e*S(2)))/(S(2)bsqrt(dsqrt(g) - esqrt(-f)) + g(S(1)/4)(ce - sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))log(-g*(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) - esqrt(-f)))/(S(2)sqrt(g)sqrt(-f)) + log(-g(S(1)/4)(S(2)bsqrt(d + ex) + ce + sqrt(-S(4)abe + S(4)b*S(2)d + c*S(2)e*S(2)))/(S(2)bsqrt(dsqrt(g) - esqrt(-f)) - g(S(1)/4)(ce + sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))log(g*(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) - esqrt(-f)))/(S(2)sqrt(g)sqrt(-f)) + log(g(S(1)/4)(S(2)bsqrt(d + ex) + ce + sqrt(-S(4)abe + S(4)b*S(2)d + c*S(2)e*S(2)))/(S(2)bsqrt(dsqrt(g) - esqrt(-f)) + g(S(1)/4)(ce + sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))log(-g*(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) - esqrt(-f)))/(S(2)sqrt(g)sqrt(-f)) - log(-g(S(1)/4)(S(2)bsqrt(d + ex) + ce - sqrt(-S(4)abe + S(4)b*S(2)d + c*S(2)e*S(2)))/(S(2)bsqrt(dsqrt(g) + esqrt(-f)) - g(S(1)/4)(ce - sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))log(g*(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) + esqrt(-f)))/(S(2)sqrt(g)sqrt(-f)) - log(g(S(1)/4)(S(2)bsqrt(d + ex) + ce - sqrt(-S(4)abe + S(4)b*S(2)d + c*S(2)e*S(2)))/(S(2)bsqrt(dsqrt(g) + esqrt(-f)) + g(S(1)/4)(ce - sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))log(-g*(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) + esqrt(-f)))/(S(2)sqrt(g)sqrt(-f)) - log(-g(S(1)/4)(S(2)bsqrt(d + ex) + ce + sqrt(-S(4)abe + S(4)b*S(2)d + c*S(2)e*S(2)))/(S(2)bsqrt(dsqrt(g) + esqrt(-f)) - g(S(1)/4)(ce + sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))log(g*(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) + esqrt(-f)))/(S(2)sqrt(g)sqrt(-f)) - log(g(S(1)/4)(S(2)bsqrt(d + ex) + ce + sqrt(-S(4)abe + S(4)b*S(2)d + c*S(2)e*S(2)))/(S(2)bsqrt(dsqrt(g) + esqrt(-f)) + g(S(1)/4)(ce + sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))log(-g*(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) + esqrt(-f)))/(S(2)sqrt(g)sqrt(-f)) + polylog(S(2), S(2)b(g(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) - esqrt(-f)))/(S(2)bsqrt(dsqrt(g) - esqrt(-f)) - g(S(1)/4)(ce - sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))/(S(2)sqrt(g)sqrt(-f)) + polylog(S(2), S(2)b(-g(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) - esqrt(-f)))/(S(2)bsqrt(dsqrt(g) - esqrt(-f)) + g(S(1)/4)(ce - sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))/(S(2)sqrt(g)sqrt(-f)) + polylog(S(2), S(2)b(g(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) - esqrt(-f)))/(S(2)bsqrt(dsqrt(g) - esqrt(-f)) - g(S(1)/4)(ce + sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))/(S(2)sqrt(g)sqrt(-f)) + polylog(S(2), S(2)b(-g(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) - esqrt(-f)))/(S(2)bsqrt(dsqrt(g) - esqrt(-f)) + g(S(1)/4)(ce + sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))/(S(2)sqrt(g)sqrt(-f)) - polylog(S(2), S(2)b(g(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) + esqrt(-f)))/(S(2)bsqrt(dsqrt(g) + esqrt(-f)) - g(S(1)/4)(ce - sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))/(S(2)sqrt(g)sqrt(-f)) - polylog(S(2), S(2)b(-g(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) + esqrt(-f)))/(S(2)bsqrt(dsqrt(g) + esqrt(-f)) + g(S(1)/4)(ce - sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))/(S(2)sqrt(g)sqrt(-f)) - polylog(S(2), S(2)b(g(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) + esqrt(-f)))/(S(2)bsqrt(dsqrt(g) + esqrt(-f)) - g(S(1)/4)(ce + sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))/(S(2)sqrt(g)sqrt(-f)) - polylog(S(2), S(2)b(-g(S(1)/4)sqrt(d + ex) + sqrt(dsqrt(g) + esqrt(-f)))/(S(2)bsqrt(dsqrt(g) + esqrt(-f)) + g(S(1)/4)(ce + sqrt(-S(4)abe + S(4)bS(2)d + c*S(2)e*S(2)))))/(S(2)sqrt(g)*sqrt(-f)), expand=True, _diff=True, _numerical=True)
4c23aa0161f7f5b11e4c7e042c306e5e3f198871dea6af6713f050df44d2b0fd	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.utility_function import ( sympy_op_factory, Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ, ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ, PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ, NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest, SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient, Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan, ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec, ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less, Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ, InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ, EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, ExpandLinearProduct, GCD, ContentFactor, NumericFactor, NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, ExpandExpression, Apart, SmartApart, MatchQ, PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, RationalFunctionFactors, NonrationalFunctionFactors, Reverse, RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, GeneralizedBinomialParts, GeneralizedTrinomialDegree, GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, MinimumMonomialExponent, MonomialExponent, LinearMatchQ, PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, AlgebraicFunctionFactors, NonalgebraicFunctionFactors, QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, CombineExponents, FactorInteger, FactorAbsurdNumber, SubstForInverseFunction, SubstForFractionalPower, SubstForFractionalPowerOfQuotientOfLinears, FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, NormalizeIntegrandAux, NormalizeIntegrandFactor, NormalizeIntegrandFactorBase, NormalizeTogether, NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand, SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, KnownTangentIntegrandQ, KnownCotangentIntegrandQ, KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, FunctionOfLog, PowerVariableExpn, PowerVariableDegree, PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, ConstantFactor, SameQ, ReplacePart, CommonFactors, MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, _SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify, _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor, PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral, CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial, Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ ) from sympy.core.add import Add from sympy.core.mod import Mod from sympy.core.mul import Mul from sympy.core.numbers import (Float, I, Integer) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.functions.elementary.complexes import Abs from sympy.functions.elementary.miscellaneous import sqrt from sympy.integrals.integrals import Integral from sympy.logic.boolalg import (And, Or) from sympy.simplify.simplify import simplify from sympy.integrals.rubi.symbol import WC from sympy.core.symbol import symbols, Symbol from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf, exp, log) from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch) from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec) from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.core.numbers import pi as Pi a, b, c, d, e, f, m, n, x, u , k, p, r, s, t, i, j= symbols('a b c d e f m n x u k p r s t i j') A, B, C, D, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B C D a b c d e f g h y z m n p q u v w F', ) def test_1(): assert rubi_test(rubi_integrate(tan(c + dx), x), x, -log(cos(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(c + dx)S(2), x), x, -x + tan(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(c + dx)S(3), x), x, log(cos(c + dx))/d + tan(c + dx)S(2)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(c + dx)S(4), x), x, x + tan(c + dx)*S(3)/(S(3)d) - tan(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(c + dx)*S(5), x), x, -log(cos(c + dx))/d + tan(c + dx)S(4)/(S(4)d) - tan(c + dx)S(2)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(c + dx)S(6), x), x, -x + tan(c + dx)*S(5)/(S(5)d) - tan(c + dx)S(3)/(S(3)d) + tan(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(c + dx)*S(7), x), x, log(cos(c + dx))/d + tan(c + dx)S(6)/(S(6)d) - tan(c + dx)S(4)/(S(4)d) + tan(c + dx)S(2)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(c + dx)S(8), x), x, x + tan(c + dx)*S(7)/(S(7)d) - tan(c + dx)S(5)/(S(5)d) + tan(c + dx)S(3)/(S(3)d) - tan(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))(S(7)/2), x), x, -sqrt(S(2))b*(S(7)/2)ArcTan(S(1) - sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)d) + sqrt(S(2))b(S(7)/2)ArcTan(S(1) + sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)d) - sqrt(S(2))b(S(7)/2)log(sqrt(b)tan(c + dx) + sqrt(b) - sqrt(S(2))sqrt(btan(c + dx)))/(S(4)d) + sqrt(S(2))b(S(7)/2)log(sqrt(b)tan(c + dx) + sqrt(b) + sqrt(S(2))sqrt(btan(c + dx)))/(S(4)d) - S(2)bS(3)sqrt(btan(c + dx))/d + S(2)b(btan(c + dx))*(S(5)/2)/(S(5)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))*(S(5)/2), x), x, sqrt(S(2))b*(S(5)/2)ArcTan(S(1) - sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)d) - sqrt(S(2))b(S(5)/2)ArcTan(S(1) + sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)d) - sqrt(S(2))b(S(5)/2)log(sqrt(b)tan(c + dx) + sqrt(b) - sqrt(S(2))sqrt(btan(c + dx)))/(S(4)d) + sqrt(S(2))b(S(5)/2)log(sqrt(b)tan(c + dx) + sqrt(b) + sqrt(S(2))sqrt(btan(c + dx)))/(S(4)d) + S(2)b(btan(c + dx))*(S(3)/2)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))*(S(3)/2), x), x, sqrt(S(2))b*(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)d) - sqrt(S(2))b(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)d) + sqrt(S(2))b(S(3)/2)log(sqrt(b)tan(c + dx) + sqrt(b) - sqrt(S(2))sqrt(btan(c + dx)))/(S(4)d) - sqrt(S(2))b(S(3)/2)log(sqrt(b)tan(c + dx) + sqrt(b) + sqrt(S(2))sqrt(btan(c + dx)))/(S(4)d) + S(2)bsqrt(btan(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(c + dx)), x), x, -sqrt(S(2))sqrt(b)ArcTan(S(1) - sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)d) + sqrt(S(2))sqrt(b)ArcTan(S(1) + sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)d) + sqrt(S(2))sqrt(b)log(sqrt(b)tan(c + dx) + sqrt(b) - sqrt(S(2))sqrt(btan(c + dx)))/(S(4)d) - sqrt(S(2))sqrt(b)log(sqrt(b)tan(c + dx) + sqrt(b) + sqrt(S(2))sqrt(btan(c + dx)))/(S(4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(btan(c + dx)), x), x, -sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)sqrt(b)d) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)sqrt(b)d) - sqrt(S(2))log(sqrt(b)tan(c + dx) + sqrt(b) - sqrt(S(2))sqrt(btan(c + dx)))/(S(4)sqrt(b)d) + sqrt(S(2))log(sqrt(b)tan(c + dx) + sqrt(b) + sqrt(S(2))sqrt(btan(c + dx)))/(S(4)sqrt(b)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))*(S(-3)/2), x), x, -S(2)/(bdsqrt(btan(c + dx))) + sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)b*(S(3)/2)d) - sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)b(S(3)/2)d) - sqrt(S(2))log(sqrt(b)tan(c + dx) + sqrt(b) - sqrt(S(2))sqrt(btan(c + dx)))/(S(4)b(S(3)/2)d) + sqrt(S(2))log(sqrt(b)tan(c + dx) + sqrt(b) + sqrt(S(2))sqrt(btan(c + dx)))/(S(4)b(S(3)/2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))*(S(-5)/2), x), x, -S(2)/(S(3)bd(btan(c + dx))*(S(3)/2)) + sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)b*(S(5)/2)d) - sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)b(S(5)/2)d) + sqrt(S(2))log(sqrt(b)tan(c + dx) + sqrt(b) - sqrt(S(2))sqrt(btan(c + dx)))/(S(4)b(S(5)/2)d) - sqrt(S(2))log(sqrt(b)tan(c + dx) + sqrt(b) + sqrt(S(2))sqrt(btan(c + dx)))/(S(4)b(S(5)/2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))*(S(-7)/2), x), x, -S(2)/(S(5)bd(btan(c + dx))(S(5)/2)) + S(2)/(bS(3)dsqrt(btan(c + dx))) - sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)b(S(7)/2)d) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(btan(c + dx))/sqrt(b))/(S(2)b(S(7)/2)d) + sqrt(S(2))log(sqrt(b)tan(c + dx) + sqrt(b) - sqrt(S(2))sqrt(btan(c + dx)))/(S(4)b(S(7)/2)d) - sqrt(S(2))log(sqrt(b)tan(c + dx) + sqrt(b) + sqrt(S(2))sqrt(btan(c + dx)))/(S(4)b(S(7)/2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))(S(4)/3), x), x, -b(S(4)/3)ArcTan((btan(c + dx))(S(1)/3)/b(S(1)/3))/d + b(S(4)/3)ArcTan((sqrt(S(3))b(S(1)/3) - S(2)(btan(c + dx))(S(1)/3))/b(S(1)/3))/(S(2)d) - b(S(4)/3)ArcTan((sqrt(S(3))b(S(1)/3) + S(2)(btan(c + dx))(S(1)/3))/b(S(1)/3))/(S(2)d) + sqrt(S(3))b*(S(4)/3)log(b*(S(2)/3) - sqrt(S(3))b*(S(1)/3)(btan(c + dx))*(S(1)/3) + (btan(c + dx))(S(2)/3))/(S(4)d) - sqrt(S(3))b(S(4)/3)log(b*(S(2)/3) + sqrt(S(3))b*(S(1)/3)(btan(c + dx))*(S(1)/3) + (btan(c + dx))(S(2)/3))/(S(4)d) + S(3)b(btan(c + dx))*(S(1)/3)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))(S(2)/3), x), x, b(S(2)/3)ArcTan((btan(c + dx))(S(1)/3)/b(S(1)/3))/d - b*(S(2)/3)ArcTan((sqrt(S(3))b(S(1)/3) - S(2)(btan(c + dx))(S(1)/3))/b(S(1)/3))/(S(2)d) + b(S(2)/3)ArcTan((sqrt(S(3))b(S(1)/3) + S(2)(btan(c + dx))(S(1)/3))/b(S(1)/3))/(S(2)d) + sqrt(S(3))b*(S(2)/3)log(b*(S(2)/3) - sqrt(S(3))b*(S(1)/3)(btan(c + dx))*(S(1)/3) + (btan(c + dx))(S(2)/3))/(S(4)d) - sqrt(S(3))b(S(2)/3)log(b*(S(2)/3) + sqrt(S(3))b*(S(1)/3)(btan(c + dx))*(S(1)/3) + (btan(c + dx))(S(2)/3))/(S(4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))*(S(1)/3), x), x, -sqrt(S(3))b*(S(1)/3)ArcTan(sqrt(S(3))(b(S(2)/3) - S(2)(btan(c + dx))*(S(2)/3))/(S(3)b*(S(2)/3)))/(S(2)d) - b*(S(1)/3)log(b*(S(2)/3) + (btan(c + dx))(S(2)/3))/(S(2)d) + b*(S(1)/3)log(b(S(4)/3) - b(S(2)/3)(btan(c + dx))(S(2)/3) + (btan(c + dx))(S(4)/3))/(S(4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))*(S(-1)/3), x), x, -sqrt(S(3))ArcTan(sqrt(S(3))(b(S(2)/3) - S(2)(btan(c + dx))*(S(2)/3))/(S(3)b*(S(2)/3)))/(S(2)b*(S(1)/3)d) + log(b*(S(2)/3) + (btan(c + dx))(S(2)/3))/(S(2)b*(S(1)/3)d) - log(b(S(4)/3) - b(S(2)/3)(btan(c + dx))(S(2)/3) + (btan(c + dx))(S(4)/3))/(S(4)b*(S(1)/3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))*(S(-2)/3), x), x, ArcTan((btan(c + dx))(S(1)/3)/b(S(1)/3))/(b(S(2)/3)d) - ArcTan((sqrt(S(3))b(S(1)/3) - S(2)(btan(c + dx))(S(1)/3))/b(S(1)/3))/(S(2)b(S(2)/3)d) + ArcTan((sqrt(S(3))b(S(1)/3) + S(2)(btan(c + dx))(S(1)/3))/b(S(1)/3))/(S(2)b(S(2)/3)d) - sqrt(S(3))log(b(S(2)/3) - sqrt(S(3))b*(S(1)/3)(btan(c + dx))*(S(1)/3) + (btan(c + dx))(S(2)/3))/(S(4)b*(S(2)/3)d) + sqrt(S(3))log(b(S(2)/3) + sqrt(S(3))b*(S(1)/3)(btan(c + dx))*(S(1)/3) + (btan(c + dx))(S(2)/3))/(S(4)b*(S(2)/3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))*(S(-4)/3), x), x, -S(3)/(bd(btan(c + dx))(S(1)/3)) - ArcTan((btan(c + dx))(S(1)/3)/b(S(1)/3))/(b(S(4)/3)d) + ArcTan((sqrt(S(3))b(S(1)/3) - S(2)(btan(c + dx))(S(1)/3))/b(S(1)/3))/(S(2)b(S(4)/3)d) - ArcTan((sqrt(S(3))b(S(1)/3) + S(2)(btan(c + dx))(S(1)/3))/b(S(1)/3))/(S(2)b(S(4)/3)d) - sqrt(S(3))log(b(S(2)/3) - sqrt(S(3))b*(S(1)/3)(btan(c + dx))*(S(1)/3) + (btan(c + dx))(S(2)/3))/(S(4)b*(S(4)/3)d) + sqrt(S(3))log(b(S(2)/3) + sqrt(S(3))b*(S(1)/3)(btan(c + dx))*(S(1)/3) + (btan(c + dx))(S(2)/3))/(S(4)b*(S(4)/3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx))*n, x), x, (btan(c + dx))(n + S(1))Hypergeometric2F1(S(1), n/S(2) + S(1)/2, n/S(2) + S(3)/2, -tan(c + dx)S(2))/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(2))(S(5)/2), x), x, -bS(2)sqrt(btan(c + dx)*S(2))log(cos(c + dx))cot(c + dx)/d + bS(2)sqrt(btan(c + dx)*S(2))tan(c + dx)S(3)/(S(4)d) - b*S(2)sqrt(btan(c + dx)*S(2))tan(c + dx)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(2))(S(3)/2), x), x, bsqrt(btan(c + dx)S(2))log(cos(c + dx))cot(c + dx)/d + bsqrt(btan(c + dx)*S(2))tan(c + dx)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(c + dx)*S(2)), x), x, -sqrt(btan(c + dx)S(2))log(cos(c + dx))cot(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(btan(c + dx)S(2)), x), x, log(sin(c + dx))tan(c + dx)/(dsqrt(btan(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(2))(S(-3)/2), x), x, -log(sin(c + dx))tan(c + dx)/(bdsqrt(btan(c + dx)*S(2))) - cot(c + dx)/(S(2)bdsqrt(btan(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(2))(S(-5)/2), x), x, log(sin(c + dx))tan(c + dx)/(b*S(2)dsqrt(btan(c + dx)S(2))) - cot(c + dx)*S(3)/(S(4)b*S(2)dsqrt(btan(c + dx)S(2))) + cot(c + dx)/(S(2)bS(2)dsqrt(btan(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(3))(S(5)/2), x), x, -sqrt(S(2))b*S(2)sqrt(btan(c + dx)*S(3))ArcTan(-sqrt(S(2))sqrt(tan(c + dx)) + S(1))/(S(2)dtan(c + dx)(S(3)/2)) + sqrt(S(2))b*S(2)sqrt(btan(c + dx)*S(3))ArcTan(sqrt(S(2))sqrt(tan(c + dx)) + S(1))/(S(2)dtan(c + dx)(S(3)/2)) - sqrt(S(2))b*S(2)sqrt(btan(c + dx)*S(3))log(-sqrt(S(2))sqrt(tan(c + dx)) + tan(c + dx) + S(1))/(S(4)dtan(c + dx)*(S(3)/2)) + sqrt(S(2))b*S(2)sqrt(btan(c + dx)*S(3))log(sqrt(S(2))sqrt(tan(c + dx)) + tan(c + dx) + S(1))/(S(4)dtan(c + dx)*(S(3)/2)) + S(2)b*S(2)sqrt(btan(c + dx)*S(3))tan(c + dx)S(5)/(S(13)d) - S(2)bS(2)sqrt(btan(c + dx)*S(3))tan(c + dx)S(3)/(S(9)d) + S(2)bS(2)sqrt(btan(c + dx)*S(3))tan(c + dx)/(S(5)d) - S(2)bS(2)sqrt(btan(c + dx)*S(3))cot(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(3))(S(3)/2), x), x, -sqrt(S(2))bsqrt(btan(c + dx)S(3))ArcTan(-sqrt(S(2))sqrt(tan(c + dx)) + S(1))/(S(2)dtan(c + dx)(S(3)/2)) + sqrt(S(2))bsqrt(btan(c + dx)S(3))ArcTan(sqrt(S(2))sqrt(tan(c + dx)) + S(1))/(S(2)dtan(c + dx)(S(3)/2)) + sqrt(S(2))bsqrt(btan(c + dx)S(3))log(-sqrt(S(2))sqrt(tan(c + dx)) + tan(c + dx) + S(1))/(S(4)dtan(c + dx)*(S(3)/2)) - sqrt(S(2))bsqrt(btan(c + dx)S(3))log(sqrt(S(2))sqrt(tan(c + dx)) + tan(c + dx) + S(1))/(S(4)dtan(c + dx)*(S(3)/2)) + S(2)bsqrt(btan(c + dx)S(3))tan(c + dx)S(2)/(S(7)d) - S(2)bsqrt(btan(c + dx)*S(3))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(c + dx)*S(3)), x), x, sqrt(S(2))sqrt(btan(c + dx)*S(3))ArcTan(-sqrt(S(2))sqrt(tan(c + dx)) + S(1))/(S(2)dtan(c + dx)(S(3)/2)) - sqrt(S(2))sqrt(btan(c + dx)*S(3))ArcTan(sqrt(S(2))sqrt(tan(c + dx)) + S(1))/(S(2)dtan(c + dx)(S(3)/2)) + sqrt(S(2))sqrt(btan(c + dx)*S(3))log(-sqrt(S(2))sqrt(tan(c + dx)) + tan(c + dx) + S(1))/(S(4)dtan(c + dx)*(S(3)/2)) - sqrt(S(2))sqrt(btan(c + dx)*S(3))log(sqrt(S(2))sqrt(tan(c + dx)) + tan(c + dx) + S(1))/(S(4)dtan(c + dx)*(S(3)/2)) + S(2)sqrt(btan(c + dx)*S(3))cot(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(btan(c + dx)S(3)), x), x, sqrt(S(2))ArcTan(-sqrt(S(2))sqrt(tan(c + dx)) + S(1))tan(c + dx)*(S(3)/2)/(S(2)dsqrt(btan(c + dx)S(3))) - sqrt(S(2))ArcTan(sqrt(S(2))sqrt(tan(c + dx)) + S(1))tan(c + dx)*(S(3)/2)/(S(2)dsqrt(btan(c + dx)S(3))) - sqrt(S(2))log(-sqrt(S(2))sqrt(tan(c + dx)) + tan(c + dx) + S(1))tan(c + dx)(S(3)/2)/(S(4)dsqrt(btan(c + dx)S(3))) + sqrt(S(2))log(sqrt(S(2))sqrt(tan(c + dx)) + tan(c + dx) + S(1))tan(c + dx)(S(3)/2)/(S(4)dsqrt(btan(c + dx)S(3))) - S(2)tan(c + dx)/(dsqrt(btan(c + dx)*S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(3))(S(-3)/2), x), x, -sqrt(S(2))ArcTan(-sqrt(S(2))sqrt(tan(c + dx)) + S(1))tan(c + dx)*(S(3)/2)/(S(2)bdsqrt(btan(c + dx)*S(3))) + sqrt(S(2))ArcTan(sqrt(S(2))sqrt(tan(c + dx)) + S(1))tan(c + dx)*(S(3)/2)/(S(2)bdsqrt(btan(c + dx)*S(3))) - sqrt(S(2))log(-sqrt(S(2))sqrt(tan(c + dx)) + tan(c + dx) + S(1))tan(c + dx)(S(3)/2)/(S(4)bdsqrt(btan(c + dx)*S(3))) + sqrt(S(2))log(sqrt(S(2))sqrt(tan(c + dx)) + tan(c + dx) + S(1))tan(c + dx)(S(3)/2)/(S(4)bdsqrt(btan(c + dx)*S(3))) - S(2)cot(c + dx)S(2)/(S(7)bdsqrt(btan(c + dx)*S(3))) + S(2)/(S(3)bdsqrt(btan(c + dx)*S(3))), expand=True, _diff=True, _numerical=True) # taking a long time assert rubi_test(rubi_integrate((btan(c + dx)S(3))(S(-5)/2), x), x, -sqrt(S(2))ArcTan(-sqrt(S(2))sqrt(tan(c + dx)) + S(1))tan(c + dx)*(S(3)/2)/(S(2)b*S(2)dsqrt(btan(c + dx)S(3))) + sqrt(S(2))ArcTan(sqrt(S(2))sqrt(tan(c + dx)) + S(1))tan(c + dx)*(S(3)/2)/(S(2)b*S(2)dsqrt(btan(c + dx)S(3))) + sqrt(S(2))log(-sqrt(S(2))sqrt(tan(c + dx)) + tan(c + dx) + S(1))tan(c + dx)(S(3)/2)/(S(4)b*S(2)dsqrt(btan(c + dx)S(3))) - sqrt(S(2))log(sqrt(S(2))sqrt(tan(c + dx)) + tan(c + dx) + S(1))tan(c + dx)(S(3)/2)/(S(4)b*S(2)dsqrt(btan(c + dx)S(3))) + S(2)tan(c + dx)/(bS(2)dsqrt(btan(c + dx)S(3))) - S(2)cot(c + dx)S(5)/(S(13)b*S(2)dsqrt(btan(c + dx)S(3))) + S(2)cot(c + dx)S(3)/(S(9)b*S(2)dsqrt(btan(c + dx)S(3))) - S(2)cot(c + dx)/(S(5)b*S(2)dsqrt(btan(c + dx)S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(4))(S(5)/2), x), x, -bS(2)xsqrt(btan(c + dx)S(4))cot(c + dx)S(2) + bS(2)sqrt(btan(c + dx)*S(4))tan(c + dx)S(7)/(S(9)d) - b*S(2)sqrt(btan(c + dx)*S(4))tan(c + dx)S(5)/(S(7)d) + b*S(2)sqrt(btan(c + dx)*S(4))tan(c + dx)S(3)/(S(5)d) - b*S(2)sqrt(btan(c + dx)*S(4))tan(c + dx)/(S(3)d) + b*S(2)sqrt(btan(c + dx)*S(4))cot(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(4))(S(3)/2), x), x, -bxsqrt(btan(c + dx)S(4))cot(c + dx)S(2) + bsqrt(btan(c + dx)*S(4))tan(c + dx)S(3)/(S(5)d) - bsqrt(btan(c + dx)S(4))tan(c + dx)/(S(3)d) + bsqrt(btan(c + dx)S(4))cot(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(c + dx)S(4)), x), x, -xsqrt(btan(c + dx)*S(4))cot(c + dx)S(2) + sqrt(btan(c + dx)S(4))cot(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(btan(c + dx)S(4)), x), x, -xtan(c + dx)S(2)/sqrt(btan(c + dx)S(4)) - tan(c + dx)/(dsqrt(btan(c + dx)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(4))(S(-3)/2), x), x, -xtan(c + dx)S(2)/(bsqrt(btan(c + dx)*S(4))) - tan(c + dx)/(bdsqrt(btan(c + dx)*S(4))) - cot(c + dx)*S(3)/(S(5)bdsqrt(btan(c + dx)*S(4))) + cot(c + dx)/(S(3)bdsqrt(btan(c + dx)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(4))(S(-5)/2), x), x, -xtan(c + dx)S(2)/(bS(2)sqrt(btan(c + dx)*S(4))) - tan(c + dx)/(b*S(2)dsqrt(btan(c + dx)S(4))) - cot(c + dx)*S(7)/(S(9)b*S(2)dsqrt(btan(c + dx)S(4))) + cot(c + dx)*S(5)/(S(7)b*S(2)dsqrt(btan(c + dx)S(4))) - cot(c + dx)*S(3)/(S(5)b*S(2)dsqrt(btan(c + dx)S(4))) + cot(c + dx)/(S(3)bS(2)dsqrt(btan(c + dx)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)p)n, x), x, (btan(c + dx)p)nHypergeometric2F1(S(1), np/S(2) + S(1)/2, np/S(2) + S(3)/2, -tan(c + dx)S(2))tan(c + dx)/(d(np + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(2))n, x), x, (btan(c + dx)S(2))nHypergeometric2F1(S(1), n + S(1)/2, n + S(3)/2, -tan(c + dx)S(2))tan(c + dx)/(d(S(2)n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(3))n, x), x, (btan(c + dx)S(3))nHypergeometric2F1(S(1), S(3)n/S(2) + S(1)/2, S(3)n/S(2) + S(3)/2, -tan(c + dx)S(2))tan(c + dx)/(d(S(3)n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)S(4))n, x), x, (btan(c + dx)S(4))nHypergeometric2F1(S(1), S(2)n + S(1)/2, S(2)n + S(3)/2, -tan(c + dx)S(2))tan(c + dx)/(d(S(4)n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)p)(S(5)/2), x), x, S(2)b*S(2)sqrt(btan(c + dx)*p)Hypergeometric2F1(S(1), S(5)p/S(4) + S(1)/2, S(5)p/S(4) + S(3)/2, -tan(c + dx)S(2))tan(c + dx)(S(2)p + S(1))/(d(S(5)p + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)p)(S(3)/2), x), x, S(2)bsqrt(btan(c + dx)*p)Hypergeometric2F1(S(1), S(3)p/S(4) + S(1)/2, S(3)p/S(4) + S(3)/2, -tan(c + dx)S(2))tan(c + dx)(p + S(1))/(d(S(3)p + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(c + dx)p), x), x, S(2)sqrt(btan(c + dx)*p)Hypergeometric2F1(S(1), p/S(4) + S(1)/2, p/S(4) + S(3)/2, -tan(c + dx)S(2))tan(c + dx)/(d(p + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(btan(c + dx)*p), x), x, S(2)Hypergeometric2F1(S(1), -p/S(4) + S(1)/2, -p/S(4) + S(3)/2, -tan(c + dx)S(2))tan(c + dx)/(dsqrt(btan(c + dx)*p)(-p + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)p)(S(-3)/2), x), x, S(2)Hypergeometric2F1(S(1), -S(3)p/S(4) + S(1)/2, -S(3)p/S(4) + S(3)/2, -tan(c + dx)*S(2))tan(c + dx)(-p + S(1))/(bdsqrt(btan(c + dx)p)(-S(3)p + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)p)(S(-5)/2), x), x, S(2)Hypergeometric2F1(S(1), -S(5)p/S(4) + S(1)/2, -S(5)p/S(4) + S(3)/2, -tan(c + dx)S(2))tan(c + dx)(-S(2)p + S(1))/(b*S(2)dsqrt(btan(c + dx)p)(-S(5)p + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(c + dx)p)(S(1)/p), x), x, -(btan(c + dx)p)(S(1)/p)log(cos(c + dx))cot(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a(btan(c + dx))p)n, x), x, (a(btan(c + dx))p)nHypergeometric2F1(S(1), np/S(2) + S(1)/2, np/S(2) + S(3)/2, -tan(c + dx)S(2))tan(c + dx)/(d(np + S(1))), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate(sqrt(dtan(a + bx))sin(a + bx)S(4), x), x, -S(21)sqrt(S(2))sqrt(d)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(64)b) + S(21)sqrt(S(2))sqrt(d)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(64)b) + S(21)sqrt(S(2))sqrt(d)log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(128)b) - S(21)sqrt(S(2))sqrt(d)log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(128)b) - S(7)(dtan(a + bx))*(S(3)/2)cos(a + bx)S(2)/(S(16)bd) - (dtan(a + bx))(S(7)/2)cos(a + bx)S(4)/(S(4)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(a + bx))sin(a + bx)S(2), x), x, -S(3)sqrt(S(2))sqrt(d)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)b) + S(3)sqrt(S(2))sqrt(d)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)b) + S(3)sqrt(S(2))sqrt(d)log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)b) - S(3)sqrt(S(2))sqrt(d)log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)b) - (dtan(a + bx))(S(3)/2)cos(a + bx)S(2)/(S(2)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(a + bx))csc(a + bx)S(2), x), x, -S(2)d/(bsqrt(dtan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(a + bx))csc(a + bx)S(4), x), x, -S(2)d*S(3)/(S(5)b(dtan(a + bx))(S(5)/2)) - S(2)d/(bsqrt(dtan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(a + bx))csc(a + bx)S(6), x), x, -S(2)d*S(5)/(S(9)b(dtan(a + bx))(S(9)/2)) - S(4)d*S(3)/(S(5)b(dtan(a + bx))(S(5)/2)) - S(2)d/(bsqrt(dtan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(a + bx))sin(a + bx)S(3), x), x, S(5)dEllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(12)bsqrt(dtan(a + bx))) - S(5)sqrt(dtan(a + bx))cos(a + bx)/(S(6)b) - (dtan(a + bx))*(S(5)/2)cos(a + bx)S(3)/(S(3)bdS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(a + bx))sin(a + bx), x), x, dEllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(2)bsqrt(dtan(a + bx))) - sqrt(dtan(a + bx))cos(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(a + bx))csc(a + bx), x), x, dEllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(bsqrt(dtan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(a + bx))csc(a + bx)*S(3), x), x, -S(2)d*S(2)sec(a + bx)/(S(3)b(dtan(a + bx))(S(3)/2)) + S(2)dEllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(3)bsqrt(dtan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(a + bx))csc(a + bx)*S(5), x), x, -S(2)d*S(4)sec(a + bx)S(3)/(S(7)b(dtan(a + bx))(S(7)/2)) - S(4)d*S(2)sec(a + bx)/(S(7)b(dtan(a + bx))(S(3)/2)) + S(4)dEllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(7)bsqrt(dtan(a + bx))), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate((dtan(a + bx))*(S(3)/2)sin(a + bx)S(4), x), x, S(45)sqrt(S(2))d(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(64)b) - S(45)sqrt(S(2))d*(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(64)b) + S(45)sqrt(S(2))d*(S(3)/2)log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(128)b) - S(45)sqrt(S(2))d*(S(3)/2)log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(128)b) + S(45)dsqrt(dtan(a + bx))/(S(16)b) - S(9)(dtan(a + bx))*(S(5)/2)cos(a + bx)S(2)/(S(16)bd) - (dtan(a + bx))(S(9)/2)cos(a + bx)S(4)/(S(4)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)sin(a + bx)S(2), x), x, S(5)sqrt(S(2))d(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)b) - S(5)sqrt(S(2))d*(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)b) + S(5)sqrt(S(2))d*(S(3)/2)log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)b) - S(5)sqrt(S(2))d*(S(3)/2)log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)b) + S(5)dsqrt(dtan(a + bx))/(S(2)b) - (dtan(a + bx))(S(5)/2)cos(a + bx)S(2)/(S(2)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)csc(a + bx)S(2), x), x, S(2)dsqrt(dtan(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)csc(a + bx)S(4), x), x, -S(2)d*S(3)/(S(3)b(dtan(a + bx))(S(3)/2)) + S(2)dsqrt(dtan(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)csc(a + bx)S(6), x), x, -S(2)d*S(5)/(S(7)b(dtan(a + bx))(S(7)/2)) - S(4)d*S(3)/(S(3)b(dtan(a + bx))(S(3)/2)) + S(2)dsqrt(dtan(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)sin(a + bx)S(3), x), x, -S(7)dsqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(S(2)bsqrt(sin(S(2)a + S(2)bx))) + S(7)(dtan(a + bx))(S(3)/2)cos(a + bx)/(S(3)b) - (dtan(a + bx))*(S(7)/2)cos(a + bx)S(3)/(S(3)bdS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)sin(a + bx), x), x, -S(3)dsqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(bsqrt(sin(S(2)a + S(2)bx))) + S(2)(dtan(a + bx))*(S(3)/2)cos(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)csc(a + bx), x), x, -S(2)dsqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(bsqrt(sin(S(2)a + S(2)bx))) + S(2)(dtan(a + bx))*(S(3)/2)cos(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)csc(a + bx)S(3), x), x, -S(2)d*S(2)sec(a + bx)/(bsqrt(dtan(a + bx))) - S(4)dsqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(bsqrt(sin(S(2)a + S(2)bx))) + S(4)(dtan(a + bx))(S(3)/2)cos(a + bx)/b, expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate((dtan(a + bx))(S(5)/2)sin(a + bx)S(4), x), x, S(77)sqrt(S(2))d(S(5)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(64)b) - S(77)sqrt(S(2))d*(S(5)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(64)b) - S(77)sqrt(S(2))d*(S(5)/2)log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(128)b) + S(77)sqrt(S(2))d*(S(5)/2)log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(128)b) + S(77)d(dtan(a + bx))*(S(3)/2)/(S(48)b) - S(11)(dtan(a + bx))(S(7)/2)cos(a + bx)S(2)/(S(16)bd) - (dtan(a + bx))(S(11)/2)cos(a + bx)S(4)/(S(4)bdS(3)), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate((dtan(a + bx))(S(5)/2)sin(a + bx)S(2), x), x, S(7)sqrt(S(2))d(S(5)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)b) - S(7)sqrt(S(2))d*(S(5)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)b) - S(7)sqrt(S(2))d*(S(5)/2)log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)b) + S(7)sqrt(S(2))d*(S(5)/2)log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)b) + S(7)d(dtan(a + bx))*(S(3)/2)/(S(6)b) - (dtan(a + bx))*(S(7)/2)cos(a + bx)S(2)/(S(2)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(5)/2)csc(a + bx)S(2), x), x, S(2)d(dtan(a + bx))(S(3)/2)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))*(S(5)/2)csc(a + bx)S(4), x), x, -S(2)d*S(3)/(bsqrt(dtan(a + bx))) + S(2)d(dtan(a + bx))*(S(3)/2)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))*(S(5)/2)csc(a + bx)S(6), x), x, -S(2)d*S(5)/(S(5)b(dtan(a + bx))(S(5)/2)) - S(4)d*S(3)/(bsqrt(dtan(a + bx))) + S(2)d(dtan(a + bx))*(S(3)/2)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))*(S(5)/2)sin(a + bx)S(3), x), x, -S(5)d*S(3)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(4)bsqrt(dtan(a + bx))) + S(5)d*S(2)sqrt(dtan(a + bx))cos(a + bx)/(S(2)b) + (dtan(a + bx))(S(5)/2)cos(a + bx)/b - (dtan(a + bx))(S(9)/2)cos(a + bx)S(3)/(S(3)bdS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(5)/2)sin(a + bx), x), x, -S(5)d*S(3)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(6)bsqrt(dtan(a + bx))) + S(5)d*S(2)sqrt(dtan(a + bx))cos(a + bx)/(S(3)b) + S(2)(dtan(a + bx))*(S(5)/2)cos(a + bx)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))*(S(5)/2)csc(a + bx), x), x, -dS(3)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(3)bsqrt(dtan(a + bx))) + S(2)d*S(2)sqrt(dtan(a + bx))sec(a + bx)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(5)/2)csc(a + bx)S(3), x), x, S(2)d*S(3)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(3)bsqrt(dtan(a + bx))) + S(2)d*S(2)sqrt(dtan(a + bx))sec(a + bx)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(5)/2)csc(a + bx)S(5), x), x, -S(2)d*S(4)sec(a + bx)S(3)/(S(3)b(dtan(a + bx))(S(3)/2)) + S(4)d*S(3)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(3)bsqrt(dtan(a + bx))) + S(4)d*S(2)sqrt(dtan(a + bx))sec(a + bx)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(5)/2)csc(a + bx)S(7), x), x, -S(2)d*S(6)sec(a + bx)S(5)/(S(7)b(dtan(a + bx))(S(7)/2)) - S(20)d*S(4)sec(a + bx)S(3)/(S(21)b(dtan(a + bx))(S(3)/2)) + S(40)d*S(3)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(21)bsqrt(dtan(a + bx))) + S(40)d*S(2)sqrt(dtan(a + bx))sec(a + bx)/(S(21)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(4)/sqrt(dtan(a + bx)), x), x, -S(5)sqrt(dtan(a + bx))cos(a + bx)*S(2)/(S(16)bd) - (dtan(a + bx))(S(5)/2)cos(a + bx)S(4)/(S(4)bdS(3)) - S(5)sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(64)bsqrt(d)) + S(5)sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(64)bsqrt(d)) - S(5)sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(128)bsqrt(d)) + S(5)sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(128)bsqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(2)/sqrt(dtan(a + bx)), x), x, -sqrt(dtan(a + bx))cos(a + bx)S(2)/(S(2)bd) - sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)bsqrt(d)) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)bsqrt(d)) - sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)bsqrt(d)) + sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)bsqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(2)/sqrt(dtan(a + bx)), x), x, -S(2)d/(S(3)b(dtan(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(4)/sqrt(dtan(a + bx)), x), x, -S(2)d*S(3)/(S(7)b(dtan(a + bx))(S(7)/2)) - S(2)d/(S(3)b(dtan(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(6)/sqrt(dtan(a + bx)), x), x, -S(2)d*S(5)/(S(11)b(dtan(a + bx))(S(11)/2)) - S(4)d*S(3)/(S(7)b(dtan(a + bx))(S(7)/2)) - S(2)d/(S(3)b(dtan(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(3)/sqrt(dtan(a + bx)), x), x, sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(S(2)bdsqrt(sin(S(2)a + S(2)bx))) - (dtan(a + bx))(S(3)/2)cos(a + bx)S(3)/(S(3)bdS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)/sqrt(dtan(a + bx)), x), x, sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(bdsqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)/sqrt(dtan(a + bx)), x), x, -S(2)cos(a + bx)/(bsqrt(dtan(a + bx))) - S(2)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(bdsqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(3)/sqrt(dtan(a + bx)), x), x, -S(2)d*S(2)sec(a + bx)/(S(5)b(dtan(a + bx))(S(5)/2)) - S(4)cos(a + bx)/(S(5)bsqrt(dtan(a + bx))) - S(4)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(S(5)bdsqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)/(dtan(a + bx))(S(3)/2), x), x, -(dtan(a + bx))(S(3)/2)cos(a + bx)S(4)/(S(4)bdS(3)) + S(3)(dtan(a + bx))*(S(3)/2)cos(a + bx)S(2)/(S(16)bdS(3)) - S(3)sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(64)bd*(S(3)/2)) + S(3)sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(64)bd*(S(3)/2)) + S(3)sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(128)bd*(S(3)/2)) - S(3)sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(128)bd*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(2)/(dtan(a + bx))(S(3)/2), x), x, (dtan(a + bx))(S(3)/2)cos(a + bx)S(2)/(S(2)bdS(3)) - sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)bd(S(3)/2)) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)bd(S(3)/2)) + sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)bd(S(3)/2)) - sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)bd(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(2)/(dtan(a + bx))(S(3)/2), x), x, -S(2)d/(S(5)b(dtan(a + bx))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(4)/(dtan(a + bx))(S(3)/2), x), x, -S(2)d*S(3)/(S(9)b(dtan(a + bx))(S(9)/2)) - S(2)d/(S(5)b(dtan(a + bx))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(6)/(dtan(a + bx))(S(3)/2), x), x, -S(2)d*S(5)/(S(13)b(dtan(a + bx))(S(13)/2)) - S(4)d*S(3)/(S(9)b(dtan(a + bx))(S(9)/2)) - S(2)d/(S(5)b(dtan(a + bx))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(3)/(dtan(a + bx))(S(3)/2), x), x, EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(12)bdsqrt(dtan(a + bx))) - sqrt(dtan(a + bx))cos(a + bx)S(3)/(S(3)bdS(2)) + sqrt(dtan(a + bx))cos(a + bx)/(S(6)bdS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)/(dtan(a + bx))*(S(3)/2), x), x, EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(2)bdsqrt(dtan(a + bx))) + sqrt(dtan(a + bx))cos(a + bx)/(bd*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)/(dtan(a + bx))*(S(3)/2), x), x, -S(2)sec(a + bx)/(S(3)b(dtan(a + bx))(S(3)/2)) - EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(3)bdsqrt(dtan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(3)/(dtan(a + bx))(S(3)/2), x), x, -S(2)d*S(2)sec(a + bx)/(S(7)b(dtan(a + bx))(S(7)/2)) - S(4)sec(a + bx)/(S(21)b(dtan(a + bx))(S(3)/2)) - S(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(21)bdsqrt(dtan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)/(dtan(a + bx))(S(5)/2), x), x, -sqrt(dtan(a + bx))cos(a + bx)S(4)/(S(4)bdS(3)) + sqrt(dtan(a + bx))cos(a + bx)S(2)/(S(16)bdS(3)) - S(3)sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(64)bd*(S(5)/2)) + S(3)sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(64)bd*(S(5)/2)) - S(3)sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(128)bd*(S(5)/2)) + S(3)sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(128)bd*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(2)/(dtan(a + bx))(S(5)/2), x), x, sqrt(dtan(a + bx))cos(a + bx)S(2)/(S(2)bdS(3)) - S(3)sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)bd*(S(5)/2)) + S(3)sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)bd*(S(5)/2)) - S(3)sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)bd*(S(5)/2)) + S(3)sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)bd*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(2)/(dtan(a + bx))(S(5)/2), x), x, -S(2)d/(S(7)b(dtan(a + bx))*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(4)/(dtan(a + bx))(S(5)/2), x), x, -S(2)d*S(3)/(S(11)b(dtan(a + bx))(S(11)/2)) - S(2)d/(S(7)b(dtan(a + bx))*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(6)/(dtan(a + bx))(S(5)/2), x), x, -S(2)d*S(5)/(S(15)b(dtan(a + bx))(S(15)/2)) - S(4)d*S(3)/(S(11)b(dtan(a + bx))(S(11)/2)) - S(2)d/(S(7)b(dtan(a + bx))*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(5)/(dtan(a + bx))(S(5)/2), x), x, S(3)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(S(20)bd*S(3)sqrt(sin(S(2)a + S(2)bx))) - (dtan(a + bx))(S(3)/2)cos(a + bx)S(5)/(S(5)bdS(4)) + (dtan(a + bx))(S(3)/2)cos(a + bx)S(3)/(S(10)bdS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(3)/(dtan(a + bx))(S(5)/2), x), x, sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(S(2)bdS(3)sqrt(sin(S(2)a + S(2)bx))) + (dtan(a + bx))(S(3)/2)cos(a + bx)S(3)/(S(3)bdS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)/(dtan(a + bx))*(S(5)/2), x), x, -S(2)cos(a + bx)/(bd*S(2)sqrt(dtan(a + bx))) - S(3)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(bd*S(3)sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)/(dtan(a + bx))*(S(5)/2), x), x, -S(2)sec(a + bx)/(S(5)b(dtan(a + bx))(S(5)/2)) + S(6)cos(a + bx)/(S(5)bdS(2)sqrt(dtan(a + bx))) + S(6)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(S(5)bdS(3)sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(3)/(dtan(a + bx))(S(5)/2), x), x, -S(2)d*S(2)sec(a + bx)/(S(9)b(dtan(a + bx))(S(9)/2)) - S(4)sec(a + bx)/(S(45)b(dtan(a + bx))(S(5)/2)) + S(4)cos(a + bx)/(S(15)bdS(2)sqrt(dtan(a + bx))) + S(4)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(S(15)bdS(3)sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(3)/2)(dtan(e + fx))*(S(3)/2), x), x, S(8)b*S(2)dsqrt(dtan(e + fx))/(S(3)fsqrt(bsin(e + fx))) - S(2)d(bsin(e + fx))(S(3)/2)sqrt(dtan(e + fx))/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(x)(S(5)/2)/tan(x)(S(3)/2), x), x, -S(2)sin(x)*(S(5)/2)/(S(5)tan(x)*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(4)/3)sqrt(dtan(e + fx)), x), x, S(6)(bsin(e + fx))(S(4)/3)(dtan(e + fx))*(S(3)/2)(cos(e + fx)S(2))(S(3)/4)Hypergeometric2F1(S(3)/4, S(17)/12, S(29)/12, sin(e + fx)S(2))/(S(17)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(1)/3)sqrt(dtan(e + fx)), x), x, S(6)(bsin(e + fx))(S(1)/3)(dtan(e + fx))*(S(3)/2)(cos(e + fx)S(2))(S(3)/4)Hypergeometric2F1(S(3)/4, S(11)/12, S(23)/12, sin(e + fx)S(2))/(S(11)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))/(bsin(e + fx))(S(1)/3), x), x, S(6)(dtan(e + fx))*(S(3)/2)(cos(e + fx)S(2))(S(3)/4)Hypergeometric2F1(S(7)/12, S(3)/4, S(19)/12, sin(e + fx)S(2))/(S(7)df(bsin(e + fx))*(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))/(bsin(e + fx))(S(4)/3), x), x, S(6)(dtan(e + fx))*(S(3)/2)(cos(e + fx)S(2))(S(3)/4)Hypergeometric2F1(S(1)/12, S(3)/4, S(13)/12, sin(e + fx)S(2))/(df(bsin(e + fx))(S(4)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(4)/3)(dtan(e + fx))*(S(3)/2), x), x, S(6)d(bsin(e + fx))(S(10)/3)sqrt(dtan(e + fx))(cos(e + fx)S(2))(S(1)/4)Hypergeometric2F1(S(5)/4, S(23)/12, S(35)/12, sin(e + fx)*S(2))/(S(23)b*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))*(S(1)/3)(dtan(e + fx))*(S(3)/2), x), x, S(6)d(bsin(e + fx))(S(7)/3)sqrt(dtan(e + fx))(cos(e + fx)S(2))(S(1)/4)Hypergeometric2F1(S(5)/4, S(17)/12, S(29)/12, sin(e + fx)*S(2))/(S(17)b*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(e + fx))*(S(3)/2)/(bsin(e + fx))(S(1)/3), x), x, S(6)d(bsin(e + fx))(S(5)/3)sqrt(dtan(e + fx))(cos(e + fx)S(2))(S(1)/4)Hypergeometric2F1(S(13)/12, S(5)/4, S(25)/12, sin(e + fx)*S(2))/(S(13)b*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(e + fx))*(S(3)/2)/(bsin(e + fx))(S(4)/3), x), x, S(6)d(bsin(e + fx))(S(2)/3)sqrt(dtan(e + fx))(cos(e + fx)S(2))(S(1)/4)Hypergeometric2F1(S(7)/12, S(5)/4, S(19)/12, sin(e + fx)*S(2))/(S(7)b*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsin(e + fx))(dtan(e + fx))(S(4)/3), x), x, S(6)d(bsin(e + fx))(S(5)/2)(dtan(e + fx))*(S(1)/3)(cos(e + fx)S(2))(S(1)/6)Hypergeometric2F1(S(7)/6, S(17)/12, S(29)/12, sin(e + fx)S(2))/(S(17)b*S(2)f), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt(bsin(e + fx))(dtan(e + fx))(S(4)/3), x), x, -S(3)dsqrt(bsin(e + fx))(dtan(e + fx))*(S(1)/3)(sec(e + fx)S(2))(S(1)/4)Hypergeometric2F1(S(5)/12, S(5)/4, S(17)/12, -tan(e + fx)S(2))/f + S(3)dsqrt(bsin(e + fx))(dtan(e + fx))*(S(1)/3)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsin(e + fx))(dtan(e + fx))*(S(1)/3), x), x, S(6)sqrt(bsin(e + fx))(dtan(e + fx))(S(4)/3)(cos(e + fx)S(2))(S(2)/3)Hypergeometric2F1(S(2)/3, S(11)/12, S(23)/12, sin(e + fx)S(2))/(S(11)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsin(e + fx))/(dtan(e + fx))(S(1)/3), x), x, S(6)sqrt(bsin(e + fx))(dtan(e + fx))(S(2)/3)(cos(e + fx)S(2))(S(1)/3)Hypergeometric2F1(S(1)/3, S(7)/12, S(19)/12, sin(e + fx)S(2))/(S(7)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsin(e + fx))/(dtan(e + fx))(S(4)/3), x), x, S(6)sqrt(bsin(e + fx))Hypergeometric2F1(S(-1)/6, S(1)/12, S(13)/12, sin(e + fx)*S(2))/(df(dtan(e + fx))(S(1)/3)(cos(e + fx)S(2))(S(1)/6)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt(bsin(e + fx))/(dtan(e + fx))(S(4)/3), x), x, S(4)sqrt(bsin(e + fx))(sec(e + fx)S(2))(S(1)/4)Hypergeometric2F1(S(1)/12, S(1)/4, S(13)/12, -tan(e + fx)*S(2))/(df(dtan(e + fx))(S(1)/3)) + S(2)sqrt(bsin(e + fx))/(df(dtan(e + fx))*(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(3)/2)(dtan(e + fx))*(S(4)/3), x), x, S(6)d(bsin(e + fx))(S(7)/2)(dtan(e + fx))*(S(1)/3)(cos(e + fx)S(2))(S(1)/6)Hypergeometric2F1(S(7)/6, S(23)/12, S(35)/12, sin(e + fx)S(2))/(S(23)b*S(2)f), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((bsin(e + fx))*(S(3)/2)(dtan(e + fx))*(S(4)/3), x), x, -S(3)d(bsin(e + fx))(S(3)/2)(dtan(e + fx))*(S(1)/3)(sec(e + fx)S(2))(S(3)/4)Hypergeometric2F1(S(11)/12, S(7)/4, S(23)/12, -tan(e + fx)S(2))/f + S(3)d(bsin(e + fx))(S(3)/2)(dtan(e + fx))*(S(1)/3)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(3)/2)(dtan(e + fx))*(S(1)/3), x), x, S(6)(bsin(e + fx))*(S(3)/2)(dtan(e + fx))*(S(4)/3)(cos(e + fx)S(2))(S(2)/3)Hypergeometric2F1(S(2)/3, S(17)/12, S(29)/12, sin(e + fx)S(2))/(S(17)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(3)/2)/(dtan(e + fx))(S(1)/3), x), x, S(6)(bsin(e + fx))*(S(3)/2)(dtan(e + fx))*(S(2)/3)(cos(e + fx)S(2))(S(1)/3)Hypergeometric2F1(S(1)/3, S(13)/12, S(25)/12, sin(e + fx)S(2))/(S(13)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(3)/2)/(dtan(e + fx))(S(4)/3), x), x, S(6)(bsin(e + fx))*(S(3)/2)Hypergeometric2F1(S(-1)/6, S(7)/12, S(19)/12, sin(e + fx)S(2))/(S(7)df(dtan(e + fx))*(S(1)/3)(cos(e + fx)S(2))(S(1)/6)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((bsin(e + fx))(S(3)/2)/(dtan(e + fx))(S(4)/3), x), x, S(4)(bsin(e + fx))*(S(3)/2)(sec(e + fx)S(2))(S(3)/4)Hypergeometric2F1(S(7)/12, S(3)/4, S(19)/12, -tan(e + fx)S(2))/(S(21)df(dtan(e + fx))*(S(1)/3)) + S(2)(bsin(e + fx))*(S(3)/2)/(S(3)df(dtan(e + fx))*(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))mtan(e + fx)S(3), x), x, (bsin(e + fx))(m + S(4))Hypergeometric2F1(S(2), m/S(2) + S(2), m/S(2) + S(3), sin(e + fx)S(2))/(bS(4)f(m + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))mtan(e + fx), x), x, (bsin(e + fx))(m + S(2))Hypergeometric2F1(S(1), m/S(2) + S(1), m/S(2) + S(2), sin(e + fx)S(2))/(bS(2)f(m + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))mcot(e + fx), x), x, (bsin(e + fx))m/(fm), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))*mcot(e + fx)S(3), x), x, -bS(2)(bsin(e + fx))*(m + S(-2))/(f(-m + S(2))) - (bsin(e + fx))*m/(fm), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))*mcot(e + fx)S(5), x), x, -bS(4)(bsin(e + fx))*(m + S(-4))/(f(-m + S(4))) + S(2)bS(2)(bsin(e + fx))*(m + S(-2))/(f(-m + S(2))) + (bsin(e + fx))*m/(fm), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))*mtan(e + fx)S(4), x), x, (bsin(e + fx))(m + S(5))sqrt(cos(e + fx)S(2))Hypergeometric2F1(S(5)/2, m/S(2) + S(5)/2, m/S(2) + S(7)/2, sin(e + fx)S(2))sec(e + fx)/(bS(5)f(m + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))mtan(e + fx)S(2), x), x, (bsin(e + fx))(m + S(3))sqrt(cos(e + fx)S(2))Hypergeometric2F1(S(3)/2, m/S(2) + S(3)/2, m/S(2) + S(5)/2, sin(e + fx)S(2))sec(e + fx)/(bS(3)f(m + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))mcot(e + fx)S(2), x), x, -b(bsin(e + fx))*(m + S(-1))Hypergeometric2F1(S(-1)/2, m/S(2) + S(-1)/2, m/S(2) + S(1)/2, sin(e + fx)S(2))cos(e + fx)/(f(-m + S(1))sqrt(cos(e + fx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))mcot(e + fx)S(4), x), x, -bS(3)(bsin(e + fx))*(m + S(-3))Hypergeometric2F1(S(-3)/2, m/S(2) + S(-3)/2, m/S(2) + S(-1)/2, sin(e + fx)S(2))cos(e + fx)/(f(-m + S(3))sqrt(cos(e + fx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))m(dtan(e + fx))*(S(3)/2), x), x, S(2)d(bsin(e + fx))(m + S(2))sqrt(dtan(e + fx))(cos(e + fx)S(2))(S(1)/4)Hypergeometric2F1(S(5)/4, m/S(2) + S(5)/4, m/S(2) + S(9)/4, sin(e + fx)S(2))/(bS(2)f(S(2)m + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))msqrt(dtan(e + fx)), x), x, S(2)(bsin(e + fx))m(dtan(e + fx))*(S(3)/2)(cos(e + fx)S(2))(S(3)/4)Hypergeometric2F1(S(3)/4, m/S(2) + S(3)/4, m/S(2) + S(7)/4, sin(e + fx)S(2))/(df(S(2)m + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))*m/sqrt(dtan(e + fx)), x), x, S(2)(bsin(e + fx))*msqrt(dtan(e + fx))(cos(e + fx)S(2))(S(1)/4)Hypergeometric2F1(S(1)/4, m/S(2) + S(1)/4, m/S(2) + S(5)/4, sin(e + fx)*S(2))/(df(S(2)m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))*m/(dtan(e + fx))(S(3)/2), x), x, -S(2)(bsin(e + fx))*mHypergeometric2F1(S(-1)/4, m/S(2) + S(-1)/4, m/S(2) + S(3)/4, sin(e + fx)S(2))/(dfsqrt(dtan(e + fx))(-S(2)m + S(1))(cos(e + fx)S(2))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(e + fx))m(btan(e + fx))*n, x), x, (asin(e + fx))m(btan(e + fx))*(n + S(1))(cos(e + fx)S(2))(n/S(2) + S(1)/2)Hypergeometric2F1(n/S(2) + S(1)/2, m/S(2) + n/S(2) + S(1)/2, m/S(2) + n/S(2) + S(3)/2, sin(e + fx)S(2))/(bf(m + n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))nsin(e + fx)S(4), x), x, (btan(e + fx))(n + S(5))Hypergeometric2F1(S(3), n/S(2) + S(5)/2, n/S(2) + S(7)/2, -tan(e + fx)S(2))/(bS(5)f(n + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))nsin(e + fx)S(2), x), x, (btan(e + fx))(n + S(3))Hypergeometric2F1(S(2), n/S(2) + S(3)/2, n/S(2) + S(5)/2, -tan(e + fx)S(2))/(bS(3)f(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))ncsc(e + fx)S(2), x), x, -b(btan(e + fx))*(n + S(-1))/(f(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))*ncsc(e + fx)S(4), x), x, -bS(3)(btan(e + fx))*(n + S(-3))/(f(-n + S(3))) - b(btan(e + fx))(n + S(-1))/(f(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))*ncsc(e + fx)S(6), x), x, -bS(5)(btan(e + fx))*(n + S(-5))/(f(-n + S(5))) - S(2)bS(3)(btan(e + fx))*(n + S(-3))/(f(-n + S(3))) - b(btan(e + fx))(n + S(-1))/(f(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))*nsin(e + fx)S(3), x), x, (btan(e + fx))(n + S(4))(cos(e + fx)S(2))(n/S(2) + S(1)/2)Hypergeometric2F1(n/S(2) + S(1)/2, n/S(2) + S(2), n/S(2) + S(3), sin(e + fx)S(2))cos(e + fx)S(3)/(bS(4)f(n + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))nsin(e + fx), x), x, (btan(e + fx))(n + S(2))(cos(e + fx)S(2))(n/S(2) + S(1)/2)Hypergeometric2F1(n/S(2) + S(1)/2, n/S(2) + S(1), n/S(2) + S(2), sin(e + fx)S(2))cos(e + fx)/(bS(2)f(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))ncsc(e + fx), x), x, (btan(e + fx))n(cos(e + fx)S(2))(n/S(2) + S(1)/2)Hypergeometric2F1(n/S(2), n/S(2) + S(1)/2, n/S(2) + S(1), sin(e + fx)S(2))sec(e + fx)/(fn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))*ncsc(e + fx)S(3), x), x, -bS(2)(btan(e + fx))*(n + S(-2))(cos(e + fx)S(2))(n/S(2) + S(1)/2)Hypergeometric2F1(n/S(2) + S(-1), n/S(2) + S(1)/2, n/S(2), sin(e + fx)S(2))sec(e + fx)S(3)/(f(-n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((acos(e + fx))*m(btan(e + fx))*n, x), x, (acos(e + fx))m(btan(e + fx))*(n + S(1))(cos(e + fx)S(2))(-m/S(2) + n/S(2) + S(1)/2)Hypergeometric2F1(n/S(2) + S(1)/2, -m/S(2) + n/S(2) + S(1)/2, n/S(2) + S(3)/2, sin(e + fx)S(2))/(bf(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((atan(e + fx))m(btan(e + fx))*n, x), x, (atan(e + fx))(m + S(1))(btan(e + fx))*nHypergeometric2F1(S(1), m/S(2) + n/S(2) + S(1)/2, m/S(2) + n/S(2) + S(3)/2, -tan(e + fx)S(2))/(af(m + n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcot(e + fx))tan(e + fx)S(4), x), x, sqrt(S(2))sqrt(d)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))sqrt(d)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + S(2)d*S(3)/(S(5)f(dcot(e + fx))(S(5)/2)) - S(2)d/(fsqrt(dcot(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcot(e + fx))tan(e + fx)S(3), x), x, -sqrt(S(2))sqrt(d)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))sqrt(d)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + S(2)d*S(2)/(S(3)f(dcot(e + fx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcot(e + fx))tan(e + fx)S(2), x), x, -sqrt(S(2))sqrt(d)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))sqrt(d)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) - sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + S(2)d/(fsqrt(dcot(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcot(e + fx))tan(e + fx), x), x, sqrt(S(2))sqrt(d)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))sqrt(d)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) - sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcot(e + fx)), x), x, sqrt(S(2))sqrt(d)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))sqrt(d)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcot(e + fx))cot(e + fx), x), x, -sqrt(S(2))sqrt(d)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))sqrt(d)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) - S(2)sqrt(dcot(e + fx))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcot(e + fx))cot(e + fx)*S(2), x), x, -sqrt(S(2))sqrt(d)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))sqrt(d)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) - sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) - S(2)(dcot(e + fx))*(S(3)/2)/(S(3)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcot(e + fx))cot(e + fx)S(3), x), x, sqrt(S(2))sqrt(d)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))sqrt(d)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) - sqrt(S(2))sqrt(d)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + S(2)sqrt(dcot(e + fx))/f - S(2)(dcot(e + fx))(S(5)/2)/(S(5)d*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcot(e + fx))*(S(3)/2)tan(e + fx)S(5), x), x, sqrt(S(2))d*(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))d(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + S(2)dS(4)/(S(5)f(dcot(e + fx))(S(5)/2)) - S(2)d*S(2)/(fsqrt(dcot(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcot(e + fx))*(S(3)/2)tan(e + fx)S(4), x), x, -sqrt(S(2))d*(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))d(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + S(2)dS(3)/(S(3)f(dcot(e + fx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcot(e + fx))(S(3)/2)tan(e + fx)S(3), x), x, -sqrt(S(2))d*(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))d(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) - sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + S(2)dS(2)/(fsqrt(dcot(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcot(e + fx))*(S(3)/2)tan(e + fx)S(2), x), x, sqrt(S(2))d*(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))d(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) - sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcot(e + fx))*(S(3)/2)tan(e + fx), x), x, sqrt(S(2))d*(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))d(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcot(e + fx))*(S(3)/2), x), x, -sqrt(S(2))d*(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))d(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) - S(2)dsqrt(dcot(e + fx))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcot(e + fx))*(S(3)/2)cot(e + fx), x), x, -sqrt(S(2))d*(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))d(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) - sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) - S(2)(dcot(e + fx))(S(3)/2)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcot(e + fx))*(S(3)/2)cot(e + fx)S(2), x), x, sqrt(S(2))d*(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))d(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) - sqrt(S(2))d(S(3)/2)log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)f) + S(2)dsqrt(dcot(e + fx))/f - S(2)(dcot(e + fx))(S(5)/2)/(S(5)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)*S(3)/sqrt(dcot(e + fx)), x), x, S(2)d*S(2)/(S(5)f(dcot(e + fx))(S(5)/2)) - S(2)/(fsqrt(dcot(e + fx))) + sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) - sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(2)/sqrt(dcot(e + fx)), x), x, S(2)d/(S(3)f(dcot(e + fx))*(S(3)/2)) - sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)/sqrt(dcot(e + fx)), x), x, S(2)/(fsqrt(dcot(e + fx))) - sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(dcot(e + fx)), x), x, sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) - sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cot(e + fx)/sqrt(dcot(e + fx)), x), x, sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) - sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cot(e + fx)S(2)/sqrt(dcot(e + fx)), x), x, -S(2)sqrt(dcot(e + fx))/(df) - sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cot(e + fx)*S(3)/sqrt(dcot(e + fx)), x), x, -S(2)(dcot(e + fx))*(S(3)/2)/(S(3)d*S(2)f) - sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)sqrt(d)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)sqrt(d)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(2)/(dcot(e + fx))(S(3)/2), x), x, S(2)d/(S(5)f(dcot(e + fx))*(S(5)/2)) - S(2)/(dfsqrt(dcot(e + fx))) + sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d*(S(3)/2)f) - sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d(S(3)/2)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)/(dcot(e + fx))(S(3)/2), x), x, S(2)/(S(3)f(dcot(e + fx))(S(3)/2)) - sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d*(S(3)/2)f) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d(S(3)/2)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcot(e + fx))*(S(-3)/2), x), x, S(2)/(dfsqrt(dcot(e + fx))) - sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d*(S(3)/2)f) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d(S(3)/2)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cot(e + fx)/(dcot(e + fx))(S(3)/2), x), x, sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d*(S(3)/2)f) - sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d(S(3)/2)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cot(e + fx)S(2)/(dcot(e + fx))(S(3)/2), x), x, sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d*(S(3)/2)f) - sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d(S(3)/2)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cot(e + fx)S(3)/(dcot(e + fx))(S(3)/2), x), x, -S(2)sqrt(dcot(e + fx))/(d*S(2)f) - sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d(S(3)/2)f) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d(S(3)/2)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cot(e + fx)S(4)/(dcot(e + fx))(S(3)/2), x), x, -S(2)(dcot(e + fx))*(S(3)/2)/(S(3)d*S(3)f) - sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d(S(3)/2)f) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d(S(3)/2)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cot(e + fx)S(5)/(dcot(e + fx))(S(3)/2), x), x, S(2)sqrt(dcot(e + fx))/(d*S(2)f) - S(2)(dcot(e + fx))(S(5)/2)/(S(5)d*S(4)f) + sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d(S(3)/2)f) - sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dcot(e + fx))/sqrt(d))/(S(2)d(S(3)/2)f) + sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f) - sqrt(S(2))log(sqrt(d)cot(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dcot(e + fx)))/(S(4)d(S(3)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)ncot(e + fx)m, x), x, Hypergeometric2F1(S(1), -m/S(2) + n/S(2) + S(1)/2, -m/S(2) + n/S(2) + S(3)/2, -tan(e + fx)*S(2))tan(e + fx)(n + S(1))cot(e + fx)m/(f(-m + n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))*ncot(e + fx)m, x), x, (btan(e + fx))(n + S(1))Hypergeometric2F1(S(1), -m/S(2) + n/S(2) + S(1)/2, -m/S(2) + n/S(2) + S(3)/2, -tan(e + fx)S(2))cot(e + fx)m/(bf(-m + n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((acot(e + fx))mtan(e + fx)n, x), x, (acot(e + fx))mHypergeometric2F1(S(1), -m/S(2) + n/S(2) + S(1)/2, -m/S(2) + n/S(2) + S(3)/2, -tan(e + fx)S(2))tan(e + fx)(n + S(1))/(f(-m + n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((acot(e + fx))*m(btan(e + fx))*n, x), x, (acot(e + fx))m(btan(e + fx))*(n + S(1))Hypergeometric2F1(S(1), -m/S(2) + n/S(2) + S(1)/2, -m/S(2) + n/S(2) + S(3)/2, -tan(e + fx)S(2))/(bf(-m + n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))sec(e + fx)S(6), x), x, S(2)(dtan(e + fx))*(S(3)/2)/(S(3)df) + S(4)(dtan(e + fx))*(S(7)/2)/(S(7)d*S(3)f) + S(2)(dtan(e + fx))(S(11)/2)/(S(11)d*S(5)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))sec(e + fx)*S(4), x), x, S(2)(dtan(e + fx))*(S(3)/2)/(S(3)df) + S(2)(dtan(e + fx))*(S(7)/2)/(S(7)d*S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))sec(e + fx)*S(2), x), x, S(2)(dtan(e + fx))*(S(3)/2)/(S(3)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx)), x), x, -sqrt(S(2))sqrt(d)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))sqrt(d)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(e + fx))/sqrt(d))/(S(2)f) + sqrt(S(2))sqrt(d)log(sqrt(d)tan(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dtan(e + fx)))/(S(4)f) - sqrt(S(2))sqrt(d)log(sqrt(d)tan(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dtan(e + fx)))/(S(4)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))cos(e + fx)S(2), x), x, -sqrt(S(2))sqrt(d)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(e + fx))/sqrt(d))/(S(8)f) + sqrt(S(2))sqrt(d)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(e + fx))/sqrt(d))/(S(8)f) + sqrt(S(2))sqrt(d)log(sqrt(d)tan(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dtan(e + fx)))/(S(16)f) - sqrt(S(2))sqrt(d)log(sqrt(d)tan(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dtan(e + fx)))/(S(16)f) + (dtan(e + fx))(S(3)/2)cos(e + fx)S(2)/(S(2)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))sec(e + fx)S(3), x), x, -S(4)sqrt(dtan(e + fx))EllipticE(-Pi/S(4) + e + fx, S(2))cos(e + fx)/(S(5)fsqrt(sin(S(2)e + S(2)fx))) + S(4)(dtan(e + fx))*(S(3)/2)cos(e + fx)/(S(5)df) + S(2)(dtan(e + fx))*(S(3)/2)sec(e + fx)/(S(5)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))sec(e + fx), x), x, -S(2)sqrt(dtan(e + fx))EllipticE(-Pi/S(4) + e + fx, S(2))cos(e + fx)/(fsqrt(sin(S(2)e + S(2)fx))) + S(2)(dtan(e + fx))(S(3)/2)cos(e + fx)/(df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))cos(e + fx), x), x, sqrt(dtan(e + fx))EllipticE(-Pi/S(4) + e + fx, S(2))cos(e + fx)/(fsqrt(sin(S(2)e + S(2)fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))cos(e + fx)*S(3), x), x, sqrt(dtan(e + fx))EllipticE(-Pi/S(4) + e + fx, S(2))cos(e + fx)/(S(2)fsqrt(sin(S(2)e + S(2)fx))) + (dtan(e + fx))*(S(3)/2)cos(e + fx)S(3)/(S(3)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))cos(e + fx)S(5), x), x, S(7)sqrt(dtan(e + fx))EllipticE(-Pi/S(4) + e + fx, S(2))cos(e + fx)/(S(20)fsqrt(sin(S(2)e + S(2)fx))) + (dtan(e + fx))(S(3)/2)cos(e + fx)S(5)/(S(5)df) + S(7)(dtan(e + fx))*(S(3)/2)cos(e + fx)S(3)/(S(30)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)sec(a + bx)S(6), x), x, S(2)(dtan(a + bx))*(S(5)/2)/(S(5)bd) + S(4)(dtan(a + bx))*(S(9)/2)/(S(9)bdS(3)) + S(2)(dtan(a + bx))*(S(13)/2)/(S(13)bdS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)sec(a + bx)S(4), x), x, S(2)(dtan(a + bx))*(S(5)/2)/(S(5)bd) + S(2)(dtan(a + bx))*(S(9)/2)/(S(9)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)sec(a + bx)S(2), x), x, S(2)(dtan(a + bx))*(S(5)/2)/(S(5)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2), x), x, sqrt(S(2))d*(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(2)b) - sqrt(S(2))d(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(2)b) + sqrt(S(2))d(S(3)/2)log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(4)b) - sqrt(S(2))d(S(3)/2)log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(4)b) + S(2)dsqrt(dtan(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))*(S(3)/2)cos(a + bx)S(2), x), x, -sqrt(S(2))d*(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)b) + sqrt(S(2))d(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)b) - sqrt(S(2))d(S(3)/2)log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)b) + sqrt(S(2))d(S(3)/2)log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)b) - dsqrt(dtan(a + bx))cos(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))*(S(3)/2)sec(a + bx)S(5), x), x, -S(4)d*S(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(77)bsqrt(dtan(a + bx))) + S(2)dsqrt(dtan(a + bx))sec(a + bx)S(5)/(S(11)b) - S(2)dsqrt(dtan(a + bx))sec(a + bx)*S(3)/(S(77)b) - S(4)dsqrt(dtan(a + bx))sec(a + bx)/(S(77)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)sec(a + bx)S(3), x), x, -S(2)d*S(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(21)bsqrt(dtan(a + bx))) + S(2)dsqrt(dtan(a + bx))sec(a + bx)S(3)/(S(7)b) - S(2)dsqrt(dtan(a + bx))sec(a + bx)/(S(21)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(3)/2)sec(a + bx), x), x, -dS(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(3)bsqrt(dtan(a + bx))) + S(2)dsqrt(dtan(a + bx))sec(a + bx)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))*(S(3)/2)cos(a + bx), x), x, dS(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(2)bsqrt(dtan(a + bx))) - dsqrt(dtan(a + bx))cos(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))*(S(3)/2)cos(a + bx)S(3), x), x, dS(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(12)bsqrt(dtan(a + bx))) - dsqrt(dtan(a + bx))cos(a + bx)*S(3)/(S(3)b) + dsqrt(dtan(a + bx))cos(a + bx)/(S(6)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))*(S(3)/2)cos(a + bx)S(5), x), x, dS(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(24)bsqrt(dtan(a + bx))) - dsqrt(dtan(a + bx))cos(a + bx)*S(5)/(S(5)b) + dsqrt(dtan(a + bx))cos(a + bx)S(3)/(S(30)b) + dsqrt(dtan(a + bx))cos(a + bx)/(S(12)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(e + fx))*(S(5)/2)sec(e + fx)S(6), x), x, S(2)(dtan(e + fx))*(S(7)/2)/(S(7)df) + S(4)(dtan(e + fx))*(S(11)/2)/(S(11)d*S(3)f) + S(2)(dtan(e + fx))(S(15)/2)/(S(15)d*S(5)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(e + fx))*(S(5)/2)sec(e + fx)S(4), x), x, S(2)(dtan(e + fx))*(S(7)/2)/(S(7)df) + S(2)(dtan(e + fx))*(S(11)/2)/(S(11)d*S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(e + fx))*(S(5)/2)sec(e + fx)S(2), x), x, S(2)(dtan(e + fx))*(S(7)/2)/(S(7)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(e + fx))(S(5)/2), x), x, sqrt(S(2))d*(S(5)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))d(S(5)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(e + fx))/sqrt(d))/(S(2)f) - sqrt(S(2))d(S(5)/2)log(sqrt(d)tan(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dtan(e + fx)))/(S(4)f) + sqrt(S(2))d(S(5)/2)log(sqrt(d)tan(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dtan(e + fx)))/(S(4)f) + S(2)d(dtan(e + fx))*(S(3)/2)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(e + fx))*(S(5)/2)cos(e + fx)S(2), x), x, -S(3)sqrt(S(2))d(S(5)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(e + fx))/sqrt(d))/(S(8)f) + S(3)sqrt(S(2))d*(S(5)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(e + fx))/sqrt(d))/(S(8)f) + S(3)sqrt(S(2))d*(S(5)/2)log(sqrt(d)tan(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dtan(e + fx)))/(S(16)f) - S(3)sqrt(S(2))d*(S(5)/2)log(sqrt(d)tan(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dtan(e + fx)))/(S(16)f) - d(dtan(e + fx))(S(3)/2)cos(e + fx)S(2)/(S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(e + fx))*(S(5)/2)cos(e + fx)S(4), x), x, -S(3)sqrt(S(2))d(S(5)/2)ArcTan(S(1) - sqrt(S(2))sqrt(dtan(e + fx))/sqrt(d))/(S(64)f) + S(3)sqrt(S(2))d*(S(5)/2)ArcTan(S(1) + sqrt(S(2))sqrt(dtan(e + fx))/sqrt(d))/(S(64)f) + S(3)sqrt(S(2))d*(S(5)/2)log(sqrt(d)tan(e + fx) + sqrt(d) - sqrt(S(2))sqrt(dtan(e + fx)))/(S(128)f) - S(3)sqrt(S(2))d*(S(5)/2)log(sqrt(d)tan(e + fx) + sqrt(d) + sqrt(S(2))sqrt(dtan(e + fx)))/(S(128)f) - d(dtan(e + fx))(S(3)/2)cos(e + fx)S(4)/(S(4)f) + S(3)d(dtan(e + fx))*(S(3)/2)cos(e + fx)S(2)/(S(16)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(e + fx)S(5)/sqrt(dtan(e + fx)), x), x, S(4)EllipticF(-Pi/S(4) + e + fx, S(2))sqrt(sin(S(2)e + S(2)fx))sec(e + fx)/(S(7)fsqrt(dtan(e + fx))) + S(2)sqrt(dtan(e + fx))sec(e + fx)*S(3)/(S(7)df) + S(4)sqrt(dtan(e + fx))sec(e + fx)/(S(7)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(e + fx)S(3)/sqrt(dtan(e + fx)), x), x, S(2)EllipticF(-Pi/S(4) + e + fx, S(2))sqrt(sin(S(2)e + S(2)fx))sec(e + fx)/(S(3)fsqrt(dtan(e + fx))) + S(2)sqrt(dtan(e + fx))sec(e + fx)/(S(3)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(e + fx)/sqrt(dtan(e + fx)), x), x, EllipticF(-Pi/S(4) + e + fx, S(2))sqrt(sin(S(2)e + S(2)fx))sec(e + fx)/(fsqrt(dtan(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(e + fx)/sqrt(dtan(e + fx)), x), x, EllipticF(-Pi/S(4) + e + fx, S(2))sqrt(sin(S(2)e + S(2)fx))sec(e + fx)/(S(2)fsqrt(dtan(e + fx))) + sqrt(dtan(e + fx))cos(e + fx)/(df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(e + fx)S(3)/sqrt(dtan(e + fx)), x), x, S(5)EllipticF(-Pi/S(4) + e + fx, S(2))sqrt(sin(S(2)e + S(2)fx))sec(e + fx)/(S(12)fsqrt(dtan(e + fx))) + sqrt(dtan(e + fx))cos(e + fx)S(3)/(S(3)df) + S(5)sqrt(dtan(e + fx))cos(e + fx)/(S(6)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(6)/(dtan(a + bx))(S(3)/2), x), x, -S(2)/(bdsqrt(dtan(a + bx))) + S(4)(dtan(a + bx))*(S(3)/2)/(S(3)bdS(3)) + S(2)(dtan(a + bx))*(S(7)/2)/(S(7)bdS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(4)/(dtan(a + bx))(S(3)/2), x), x, -S(2)/(bdsqrt(dtan(a + bx))) + S(2)(dtan(a + bx))*(S(3)/2)/(S(3)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(2)/(dtan(a + bx))(S(3)/2), x), x, -S(2)/(bdsqrt(dtan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))(S(-3)/2), x), x, -S(2)/(bdsqrt(dtan(a + bx))) + sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(2)bd(S(3)/2)) - sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(2)bd(S(3)/2)) - sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(4)bd(S(3)/2)) + sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(4)bd(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(2)/(dtan(a + bx))(S(3)/2), x), x, cos(a + bx)*S(2)/(S(2)bdsqrt(dtan(a + bx))) - S(5)/(S(2)bdsqrt(dtan(a + bx))) + S(5)sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)bd*(S(3)/2)) - S(5)sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(dtan(a + bx))/sqrt(d))/(S(8)bd*(S(3)/2)) - S(5)sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) - sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)bd*(S(3)/2)) + S(5)sqrt(S(2))log(sqrt(d)tan(a + bx) + sqrt(d) + sqrt(S(2))sqrt(dtan(a + bx)))/(S(16)bd*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(5)/(dtan(a + bx))(S(3)/2), x), x, -S(2)sec(a + bx)S(3)/(bdsqrt(dtan(a + bx))) - S(24)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(S(5)bd*S(2)sqrt(sin(S(2)a + S(2)bx))) + S(24)(dtan(a + bx))*(S(3)/2)cos(a + bx)/(S(5)bdS(3)) + S(12)(dtan(a + bx))*(S(3)/2)sec(a + bx)/(S(5)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(3)/(dtan(a + bx))(S(3)/2), x), x, -S(2)sec(a + bx)/(bdsqrt(dtan(a + bx))) - S(4)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(bdS(2)sqrt(sin(S(2)a + S(2)bx))) + S(4)(dtan(a + bx))*(S(3)/2)cos(a + bx)/(bd*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)/(dtan(a + bx))*(S(3)/2), x), x, -S(2)cos(a + bx)/(bdsqrt(dtan(a + bx))) - S(2)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(bdS(2)sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)/(dtan(a + bx))*(S(3)/2), x), x, -S(2)cos(a + bx)/(bdsqrt(dtan(a + bx))) - S(3)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(bdS(2)sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(3)/(dtan(a + bx))(S(3)/2), x), x, -S(2)cos(a + bx)S(3)/(bdsqrt(dtan(a + bx))) - S(7)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(S(2)bd*S(2)sqrt(sin(S(2)a + S(2)bx))) - S(7)(dtan(a + bx))*(S(3)/2)cos(a + bx)S(3)/(S(3)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(5)/(dtan(a + bx))(S(3)/2), x), x, -S(2)cos(a + bx)S(5)/(bdsqrt(dtan(a + bx))) - S(77)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(S(20)bd*S(2)sqrt(sin(S(2)a + S(2)bx))) - S(11)(dtan(a + bx))*(S(3)/2)cos(a + bx)S(5)/(S(5)bdS(3)) - S(77)(dtan(a + bx))*(S(3)/2)cos(a + bx)S(3)/(S(30)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)/(dtan(a + bx))*(S(5)/2), x), x, -S(2)sec(a + bx)/(S(3)bd(dtan(a + bx))*(S(3)/2)) - EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))sec(a + bx)/(S(3)bd*S(2)sqrt(dtan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(3)/(dtan(a + bx))(S(7)/2), x), x, -S(2)sec(a + bx)/(S(5)bd(dtan(a + bx))*(S(5)/2)) - S(4)cos(a + bx)/(S(5)bdS(3)sqrt(dtan(a + bx))) - S(4)sqrt(dtan(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))cos(a + bx)/(S(5)bdS(4)sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)*S(2)sec(e + fx)(S(4)/3), x), x, (cos(e + fx)S(2))(S(1)/6)Hypergeometric2F1(S(3)/2, S(13)/6, S(5)/2, sin(e + fx)*S(2))sin(e + fx)S(3)sec(e + fx)(S(1)/3)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(2)sec(e + fx)(S(2)/3), x), x, (cos(e + fx)S(2))(S(5)/6)Hypergeometric2F1(S(3)/2, S(11)/6, S(5)/2, sin(e + fx)*S(2))sin(e + fx)S(3)sec(e + fx)(S(5)/3)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(2)sec(e + fx)(S(1)/3), x), x, (cos(e + fx)S(2))(S(2)/3)Hypergeometric2F1(S(3)/2, S(5)/3, S(5)/2, sin(e + fx)*S(2))sin(e + fx)S(3)sec(e + fx)(S(4)/3)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(2)/sec(e + fx)*(S(1)/3), x), x, (cos(e + fx)S(2))(S(1)/3)Hypergeometric2F1(S(4)/3, S(3)/2, S(5)/2, sin(e + fx)*S(2))sin(e + fx)S(3)sec(e + fx)(S(2)/3)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(2)/sec(e + fx)*(S(2)/3), x), x, (cos(e + fx)S(2))(S(1)/6)Hypergeometric2F1(S(7)/6, S(3)/2, S(5)/2, sin(e + fx)*S(2))sin(e + fx)S(3)sec(e + fx)(S(1)/3)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(4)sec(e + fx)(S(4)/3), x), x, (cos(e + fx)S(2))(S(1)/6)Hypergeometric2F1(S(5)/2, S(19)/6, S(7)/2, sin(e + fx)*S(2))sin(e + fx)S(5)sec(e + fx)(S(1)/3)/(S(5)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(4)sec(e + fx)(S(2)/3), x), x, (cos(e + fx)S(2))(S(5)/6)Hypergeometric2F1(S(5)/2, S(17)/6, S(7)/2, sin(e + fx)*S(2))sin(e + fx)S(5)sec(e + fx)(S(5)/3)/(S(5)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(4)sec(e + fx)(S(1)/3), x), x, (cos(e + fx)S(2))(S(2)/3)Hypergeometric2F1(S(5)/2, S(8)/3, S(7)/2, sin(e + fx)*S(2))sin(e + fx)S(5)sec(e + fx)(S(4)/3)/(S(5)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(4)/sec(e + fx)*(S(1)/3), x), x, (cos(e + fx)S(2))(S(1)/3)Hypergeometric2F1(S(7)/3, S(5)/2, S(7)/2, sin(e + fx)*S(2))sin(e + fx)S(5)sec(e + fx)(S(2)/3)/(S(5)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(4)/sec(e + fx)*(S(2)/3), x), x, (cos(e + fx)S(2))(S(1)/6)Hypergeometric2F1(S(13)/6, S(5)/2, S(7)/2, sin(e + fx)*S(2))sin(e + fx)S(5)sec(e + fx)(S(1)/3)/(S(5)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))*(S(4)/3)tan(e + fx)S(2), x), x, (dsec(e + fx))(S(4)/3)(cos(e + fx)S(2))(S(13)/6)Hypergeometric2F1(S(3)/2, S(13)/6, S(5)/2, sin(e + fx)S(2))tan(e + fx)S(3)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))*(S(2)/3)tan(e + fx)S(2), x), x, (dsec(e + fx))(S(2)/3)(cos(e + fx)S(2))(S(11)/6)Hypergeometric2F1(S(3)/2, S(11)/6, S(5)/2, sin(e + fx)S(2))tan(e + fx)S(3)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))*(S(1)/3)tan(e + fx)S(2), x), x, (dsec(e + fx))(S(1)/3)(cos(e + fx)S(2))(S(5)/3)Hypergeometric2F1(S(3)/2, S(5)/3, S(5)/2, sin(e + fx)S(2))tan(e + fx)S(3)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(2)/(dsec(e + fx))(S(1)/3), x), x, (cos(e + fx)S(2))(S(4)/3)Hypergeometric2F1(S(4)/3, S(3)/2, S(5)/2, sin(e + fx)*S(2))tan(e + fx)S(3)/(S(3)f(dsec(e + fx))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)*S(2)/(dsec(e + fx))(S(2)/3), x), x, (cos(e + fx)S(2))(S(7)/6)Hypergeometric2F1(S(7)/6, S(3)/2, S(5)/2, sin(e + fx)*S(2))tan(e + fx)S(3)/(S(3)f(dsec(e + fx))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))(S(4)/3)tan(e + fx)S(4), x), x, (dsec(e + fx))(S(4)/3)(cos(e + fx)S(2))(S(19)/6)Hypergeometric2F1(S(5)/2, S(19)/6, S(7)/2, sin(e + fx)S(2))tan(e + fx)S(5)/(S(5)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))*(S(2)/3)tan(e + fx)S(4), x), x, (dsec(e + fx))(S(2)/3)(cos(e + fx)S(2))(S(17)/6)Hypergeometric2F1(S(5)/2, S(17)/6, S(7)/2, sin(e + fx)S(2))tan(e + fx)S(5)/(S(5)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))*(S(1)/3)tan(e + fx)S(4), x), x, (dsec(e + fx))(S(1)/3)(cos(e + fx)S(2))(S(8)/3)Hypergeometric2F1(S(5)/2, S(8)/3, S(7)/2, sin(e + fx)S(2))tan(e + fx)S(5)/(S(5)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)S(4)/(dsec(e + fx))(S(1)/3), x), x, (cos(e + fx)S(2))(S(7)/3)Hypergeometric2F1(S(7)/3, S(5)/2, S(7)/2, sin(e + fx)*S(2))tan(e + fx)S(5)/(S(5)f(dsec(e + fx))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(tan(e + fx)*S(4)/(dsec(e + fx))(S(2)/3), x), x, (cos(e + fx)S(2))(S(13)/6)Hypergeometric2F1(S(13)/6, S(5)/2, S(7)/2, sin(e + fx)*S(2))tan(e + fx)S(5)/(S(5)f(dsec(e + fx))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(e + fx))(dsec(e + fx))*(S(5)/2), x), x, -sqrt(b)d*S(3)sqrt(btan(e + fx))ArcTan(sqrt(bsin(e + fx))/sqrt(b))/(S(4)fsqrt(bsin(e + fx))sqrt(dsec(e + fx))) + sqrt(b)dS(3)sqrt(btan(e + fx))atanh(sqrt(bsin(e + fx))/sqrt(b))/(S(4)fsqrt(bsin(e + fx))sqrt(dsec(e + fx))) + d*S(2)(btan(e + fx))*(S(3)/2)sqrt(dsec(e + fx))/(S(2)bf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(e + fx))(dsec(e + fx))(S(3)/2), x), x, -dS(2)sqrt(btan(e + fx))EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(fsqrt(dsec(e + fx))sqrt(sin(e + fx))) + dS(2)(btan(e + fx))*(S(3)/2)/(bfsqrt(dsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(e + fx))sqrt(dsec(e + fx)), x), x, -sqrt(b)dsqrt(btan(e + fx))ArcTan(sqrt(bsin(e + fx))/sqrt(b))/(fsqrt(bsin(e + fx))sqrt(dsec(e + fx))) + sqrt(b)dsqrt(btan(e + fx))atanh(sqrt(bsin(e + fx))/sqrt(b))/(fsqrt(bsin(e + fx))sqrt(dsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(e + fx))/sqrt(dsec(e + fx)), x), x, S(2)sqrt(btan(e + fx))EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(fsqrt(dsec(e + fx))sqrt(sin(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(e + fx))/(dsec(e + fx))*(S(3)/2), x), x, S(2)(btan(e + fx))*(S(3)/2)/(S(3)bf(dsec(e + fx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(e + fx))/(dsec(e + fx))(S(5)/2), x), x, S(4)sqrt(btan(e + fx))EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(5)dS(2)fsqrt(dsec(e + fx))sqrt(sin(e + fx))) + S(2)(btan(e + fx))*(S(3)/2)/(S(5)bf(dsec(e + fx))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(e + fx))/(dsec(e + fx))(S(7)/2), x), x, S(2)(btan(e + fx))*(S(3)/2)/(S(7)bf(dsec(e + fx))*(S(7)/2)) + S(8)(btan(e + fx))*(S(3)/2)/(S(21)bdS(2)f(dsec(e + fx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(btan(e + fx))/(dsec(e + fx))(S(9)/2), x), x, S(8)sqrt(btan(e + fx))EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(15)dS(4)fsqrt(dsec(e + fx))sqrt(sin(e + fx))) + S(2)(btan(e + fx))*(S(3)/2)/(S(9)bf(dsec(e + fx))*(S(9)/2)) + S(4)(btan(e + fx))*(S(3)/2)/(S(15)bdS(2)f(dsec(e + fx))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))(S(3)/2)(dsec(e + fx))(S(5)/2), x), x, -bS(2)dS(2)sqrt(dsec(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(6)fsqrt(btan(e + fx))) - bdS(2)sqrt(btan(e + fx))sqrt(dsec(e + fx))/(S(6)f) + bsqrt(btan(e + fx))(dsec(e + fx))*(S(5)/2)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))*(S(3)/2)(dsec(e + fx))(S(3)/2), x), x, -b(S(3)/2)dsqrt(bsin(e + fx))sqrt(dsec(e + fx))ArcTan(sqrt(bsin(e + fx))/sqrt(b))/(S(4)fsqrt(btan(e + fx))) - b*(S(3)/2)dsqrt(bsin(e + fx))sqrt(dsec(e + fx))atanh(sqrt(bsin(e + fx))/sqrt(b))/(S(4)fsqrt(btan(e + fx))) + bsqrt(btan(e + fx))(dsec(e + fx))(S(3)/2)/(S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))*(S(3)/2)sqrt(dsec(e + fx)), x), x, -b*S(2)sqrt(dsec(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(fsqrt(btan(e + fx))) + bsqrt(btan(e + fx))sqrt(dsec(e + fx))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))(S(3)/2)/sqrt(dsec(e + fx)), x), x, b(S(3)/2)d(btan(e + fx))(S(3)/2)ArcTan(sqrt(bsin(e + fx))/sqrt(b))/(f(bsin(e + fx))(S(3)/2)(dsec(e + fx))(S(3)/2)) + b(S(3)/2)d(btan(e + fx))*(S(3)/2)atanh(sqrt(bsin(e + fx))/sqrt(b))/(f(bsin(e + fx))(S(3)/2)(dsec(e + fx))*(S(3)/2)) - S(2)d(btan(e + fx))(S(3)/2)csc(e + fx)/(f(dsec(e + fx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))(S(3)/2)/(dsec(e + fx))(S(3)/2), x), x, S(2)b*S(2)sqrt(dsec(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)dS(2)fsqrt(btan(e + fx))) - S(2)bsqrt(btan(e + fx))/(S(3)f(dsec(e + fx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))(S(3)/2)/(dsec(e + fx))(S(5)/2), x), x, S(2)(btan(e + fx))*(S(5)/2)/(S(5)bf(dsec(e + fx))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))(S(3)/2)/(dsec(e + fx))(S(7)/2), x), x, S(4)b*S(2)sqrt(dsec(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(21)dS(4)fsqrt(btan(e + fx))) - S(2)bsqrt(btan(e + fx))/(S(7)f(dsec(e + fx))(S(7)/2)) + S(2)bsqrt(btan(e + fx))/(S(21)d*S(2)f(dsec(e + fx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))(S(3)/2)/(dsec(e + fx))(S(9)/2), x), x, -S(2)bsqrt(btan(e + fx))/(S(9)f(dsec(e + fx))(S(9)/2)) + S(2)bsqrt(btan(e + fx))/(S(45)d*S(2)f(dsec(e + fx))(S(5)/2)) + S(8)bsqrt(btan(e + fx))/(S(45)d*S(4)fsqrt(dsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))(S(5)/2)(dsec(e + fx))*(S(5)/2), x), x, S(3)b*(S(5)/2)d*S(3)sqrt(btan(e + fx))ArcTan(sqrt(bsin(e + fx))/sqrt(b))/(S(32)fsqrt(bsin(e + fx))sqrt(dsec(e + fx))) - S(3)b(S(5)/2)d*S(3)sqrt(btan(e + fx))atanh(sqrt(bsin(e + fx))/sqrt(b))/(S(32)fsqrt(bsin(e + fx))sqrt(dsec(e + fx))) - S(3)bd*S(2)(btan(e + fx))*(S(3)/2)sqrt(dsec(e + fx))/(S(16)f) + b(btan(e + fx))*(S(3)/2)(dsec(e + fx))*(S(5)/2)/(S(4)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))*(S(5)/2)(dsec(e + fx))(S(3)/2), x), x, bS(2)dS(2)sqrt(btan(e + fx))EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(2)fsqrt(dsec(e + fx))sqrt(sin(e + fx))) - bdS(2)(btan(e + fx))*(S(3)/2)/(S(2)fsqrt(dsec(e + fx))) + b(btan(e + fx))*(S(3)/2)(dsec(e + fx))*(S(3)/2)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))*(S(5)/2)sqrt(dsec(e + fx)), x), x, S(3)b(S(5)/2)dsqrt(btan(e + fx))ArcTan(sqrt(bsin(e + fx))/sqrt(b))/(S(4)fsqrt(bsin(e + fx))sqrt(dsec(e + fx))) - S(3)b*(S(5)/2)dsqrt(btan(e + fx))atanh(sqrt(bsin(e + fx))/sqrt(b))/(S(4)fsqrt(bsin(e + fx))sqrt(dsec(e + fx))) + b(btan(e + fx))*(S(3)/2)sqrt(dsec(e + fx))/(S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))(S(5)/2)/sqrt(dsec(e + fx)), x), x, -S(3)b*S(2)sqrt(btan(e + fx))EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(fsqrt(dsec(e + fx))sqrt(sin(e + fx))) + b(btan(e + fx))*(S(3)/2)/(fsqrt(dsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))*(S(5)/2)/(dsec(e + fx))(S(3)/2), x), x, -b(S(5)/2)sqrt(btan(e + fx))ArcTan(sqrt(bsin(e + fx))/sqrt(b))/(dfsqrt(bsin(e + fx))sqrt(dsec(e + fx))) + b*(S(5)/2)sqrt(btan(e + fx))atanh(sqrt(bsin(e + fx))/sqrt(b))/(dfsqrt(bsin(e + fx))sqrt(dsec(e + fx))) - S(2)b(btan(e + fx))*(S(3)/2)/(S(3)f(dsec(e + fx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))(S(5)/2)/(dsec(e + fx))(S(5)/2), x), x, S(6)b*S(2)sqrt(btan(e + fx))EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(5)dS(2)fsqrt(dsec(e + fx))sqrt(sin(e + fx))) - S(2)b(btan(e + fx))(S(3)/2)/(S(5)f(dsec(e + fx))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))(S(5)/2)/(dsec(e + fx))(S(7)/2), x), x, S(2)(btan(e + fx))*(S(7)/2)/(S(7)bf(dsec(e + fx))*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((btan(e + fx))(S(5)/2)/(dsec(e + fx))(S(9)/2), x), x, S(4)b*S(2)sqrt(btan(e + fx))EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(15)dS(4)fsqrt(dsec(e + fx))sqrt(sin(e + fx))) - S(2)b(btan(e + fx))(S(3)/2)/(S(9)f(dsec(e + fx))(S(9)/2)) + S(2)b(btan(e + fx))(S(3)/2)/(S(15)d*S(2)f(dsec(e + fx))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))(S(7)/2)/sqrt(btan(e + fx)), x), x, dS(2)sqrt(btan(e + fx))(dsec(e + fx))(S(3)/2)/(S(2)bf) + S(3)d*S(3)sqrt(bsin(e + fx))sqrt(dsec(e + fx))ArcTan(sqrt(bsin(e + fx))/sqrt(b))/(S(4)sqrt(b)fsqrt(btan(e + fx))) + S(3)d*S(3)sqrt(bsin(e + fx))sqrt(dsec(e + fx))atanh(sqrt(bsin(e + fx))/sqrt(b))/(S(4)sqrt(b)fsqrt(btan(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))(S(5)/2)/sqrt(btan(e + fx)), x), x, dS(2)sqrt(dsec(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(fsqrt(btan(e + fx))) + dS(2)sqrt(btan(e + fx))sqrt(dsec(e + fx))/(bf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))*(S(3)/2)/sqrt(btan(e + fx)), x), x, dsqrt(bsin(e + fx))sqrt(dsec(e + fx))ArcTan(sqrt(bsin(e + fx))/sqrt(b))/(sqrt(b)fsqrt(btan(e + fx))) + dsqrt(bsin(e + fx))sqrt(dsec(e + fx))atanh(sqrt(bsin(e + fx))/sqrt(b))/(sqrt(b)fsqrt(btan(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dsec(e + fx))/sqrt(btan(e + fx)), x), x, S(2)sqrt(dsec(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(fsqrt(btan(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(btan(e + fx))sqrt(dsec(e + fx))), x), x, S(2)sqrt(btan(e + fx))/(bfsqrt(dsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(btan(e + fx))(dsec(e + fx))*(S(3)/2)), x), x, S(4)sqrt(dsec(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)dS(2)fsqrt(btan(e + fx))) + S(2)sqrt(btan(e + fx))/(S(3)bf(dsec(e + fx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(btan(e + fx))(dsec(e + fx))*(S(5)/2)), x), x, S(2)sqrt(btan(e + fx))/(S(5)bf(dsec(e + fx))(S(5)/2)) + S(8)sqrt(btan(e + fx))/(S(5)bd*S(2)fsqrt(dsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))(S(5)/2)/(btan(e + fx))(S(3)/2), x), x, -S(2)d*S(2)sqrt(dsec(e + fx))/(bfsqrt(btan(e + fx))) - d*S(3)sqrt(btan(e + fx))ArcTan(sqrt(bsin(e + fx))/sqrt(b))/(b(S(3)/2)fsqrt(bsin(e + fx))sqrt(dsec(e + fx))) + d*S(3)sqrt(btan(e + fx))atanh(sqrt(bsin(e + fx))/sqrt(b))/(b(S(3)/2)fsqrt(bsin(e + fx))sqrt(dsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))*(S(3)/2)/(btan(e + fx))(S(3)/2), x), x, -S(2)d*S(2)/(bfsqrt(btan(e + fx))sqrt(dsec(e + fx))) - S(2)dS(2)sqrt(btan(e + fx))EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(b*S(2)fsqrt(dsec(e + fx))sqrt(sin(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dsec(e + fx))/(btan(e + fx))(S(3)/2), x), x, -S(2)sqrt(dsec(e + fx))/(bfsqrt(btan(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((btan(e + fx))*(S(3)/2)sqrt(dsec(e + fx))), x), x, -S(2)/(bfsqrt(btan(e + fx))sqrt(dsec(e + fx))) - S(4)sqrt(btan(e + fx))EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(b*S(2)fsqrt(dsec(e + fx))sqrt(sin(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((btan(e + fx))(S(3)/2)(dsec(e + fx))*(S(3)/2)), x), x, S(2)/(S(3)bfsqrt(btan(e + fx))(dsec(e + fx))(S(3)/2)) - S(8)sqrt(dsec(e + fx))/(S(3)bd*S(2)fsqrt(btan(e + fx))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/((btan(e + fx))(S(3)/2)(dsec(e + fx))*(S(3)/2)), x), x, -S(2)/(bfsqrt(btan(e + fx))(dsec(e + fx))*(S(3)/2)) - S(8)(btan(e + fx))*(S(3)/2)/(S(3)b*S(3)f(dsec(e + fx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((btan(e + fx))(S(3)/2)(dsec(e + fx))*(S(5)/2)), x), x, -S(2)/(bfsqrt(btan(e + fx))(dsec(e + fx))*(S(5)/2)) - S(24)sqrt(btan(e + fx))EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(5)bS(2)d*S(2)fsqrt(dsec(e + fx))sqrt(sin(e + fx))) - S(12)(btan(e + fx))*(S(3)/2)/(S(5)b*S(3)f(dsec(e + fx))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))(S(7)/2)/(btan(e + fx))(S(5)/2), x), x, -S(2)d*S(2)(dsec(e + fx))*(S(3)/2)/(S(3)bf(btan(e + fx))(S(3)/2)) + dS(3)sqrt(bsin(e + fx))sqrt(dsec(e + fx))ArcTan(sqrt(bsin(e + fx))/sqrt(b))/(b(S(5)/2)fsqrt(btan(e + fx))) + dS(3)sqrt(bsin(e + fx))sqrt(dsec(e + fx))atanh(sqrt(bsin(e + fx))/sqrt(b))/(b*(S(5)/2)fsqrt(btan(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))(S(5)/2)/(btan(e + fx))(S(5)/2), x), x, -S(2)d*S(2)sqrt(dsec(e + fx))/(S(3)bf(btan(e + fx))(S(3)/2)) + S(2)d*S(2)sqrt(dsec(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)bS(2)fsqrt(btan(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(e + fx))(S(3)/2)/(btan(e + fx))(S(5)/2), x), x, -S(2)(dsec(e + fx))*(S(3)/2)/(S(3)bf(btan(e + fx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dsec(e + fx))/(btan(e + fx))(S(5)/2), x), x, -S(2)sqrt(dsec(e + fx))/(S(3)bf(btan(e + fx))(S(3)/2)) - S(4)sqrt(dsec(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)bS(2)fsqrt(btan(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((btan(e + fx))(S(5)/2)sqrt(dsec(e + fx))), x), x, -S(2)/(S(3)bf(btan(e + fx))(S(3)/2)sqrt(dsec(e + fx))) - S(8)sqrt(btan(e + fx))/(S(3)b*S(3)fsqrt(dsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((btan(e + fx))(S(5)/2)(dsec(e + fx))*(S(3)/2)), x), x, -S(2)/(S(3)bf(btan(e + fx))*(S(3)/2)(dsec(e + fx))*(S(3)/2)) - S(8)sqrt(dsec(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)bS(2)d*S(2)fsqrt(btan(e + fx))) - S(4)sqrt(btan(e + fx))/(S(3)bS(3)f(dsec(e + fx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((btan(e + fx))(S(5)/2)(dsec(e + fx))*(S(5)/2)), x), x, -S(2)/(S(3)bf(btan(e + fx))*(S(3)/2)(dsec(e + fx))*(S(5)/2)) - S(16)sqrt(btan(e + fx))/(S(15)bS(3)f(dsec(e + fx))(S(5)/2)) - S(64)sqrt(btan(e + fx))/(S(15)bS(3)d*S(2)fsqrt(dsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(4)/3)sqrt(dtan(e + fx)), x), x, S(2)(bsec(e + fx))(S(4)/3)(dtan(e + fx))*(S(3)/2)(cos(e + fx)S(2))(S(17)/12)Hypergeometric2F1(S(3)/4, S(17)/12, S(7)/4, sin(e + fx)S(2))/(S(3)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(1)/3)sqrt(dtan(e + fx)), x), x, S(2)(bsec(e + fx))(S(1)/3)(dtan(e + fx))*(S(3)/2)(cos(e + fx)S(2))(S(11)/12)Hypergeometric2F1(S(3)/4, S(11)/12, S(7)/4, sin(e + fx)S(2))/(S(3)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))/(bsec(e + fx))(S(1)/3), x), x, S(2)(dtan(e + fx))*(S(3)/2)(cos(e + fx)S(2))(S(7)/12)Hypergeometric2F1(S(7)/12, S(3)/4, S(7)/4, sin(e + fx)S(2))/(S(3)df(bsec(e + fx))*(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dtan(e + fx))/(bsec(e + fx))(S(4)/3), x), x, S(2)(dtan(e + fx))*(S(3)/2)(cos(e + fx)S(2))(S(1)/12)Hypergeometric2F1(S(1)/12, S(3)/4, S(7)/4, sin(e + fx)S(2))/(S(3)df(bsec(e + fx))*(S(4)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(4)/3)(dtan(e + fx))*(S(3)/2), x), x, S(2)(bsec(e + fx))*(S(4)/3)(dtan(e + fx))*(S(5)/2)(cos(e + fx)S(2))(S(23)/12)Hypergeometric2F1(S(5)/4, S(23)/12, S(9)/4, sin(e + fx)S(2))/(S(5)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(1)/3)(dtan(e + fx))*(S(3)/2), x), x, S(2)(bsec(e + fx))*(S(1)/3)(dtan(e + fx))*(S(5)/2)(cos(e + fx)S(2))(S(17)/12)Hypergeometric2F1(S(5)/4, S(17)/12, S(9)/4, sin(e + fx)S(2))/(S(5)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(e + fx))(S(3)/2)/(bsec(e + fx))(S(1)/3), x), x, S(2)(dtan(e + fx))*(S(5)/2)(cos(e + fx)S(2))(S(13)/12)Hypergeometric2F1(S(13)/12, S(5)/4, S(9)/4, sin(e + fx)S(2))/(S(5)df(bsec(e + fx))*(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(e + fx))(S(3)/2)/(bsec(e + fx))(S(4)/3), x), x, S(2)(dtan(e + fx))*(S(5)/2)(cos(e + fx)S(2))(S(7)/12)Hypergeometric2F1(S(7)/12, S(5)/4, S(9)/4, sin(e + fx)S(2))/(S(5)df(bsec(e + fx))*(S(4)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))(dtan(e + fx))*(S(4)/3), x), x, S(3)sqrt(bsec(e + fx))(dtan(e + fx))(S(7)/3)(cos(e + fx)S(2))(S(17)/12)Hypergeometric2F1(S(7)/6, S(17)/12, S(13)/6, sin(e + fx)S(2))/(S(7)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))(dtan(e + fx))*(S(1)/3), x), x, S(3)sqrt(bsec(e + fx))(dtan(e + fx))(S(4)/3)(cos(e + fx)S(2))(S(11)/12)Hypergeometric2F1(S(2)/3, S(11)/12, S(5)/3, sin(e + fx)S(2))/(S(4)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))/(dtan(e + fx))(S(1)/3), x), x, S(3)sqrt(bsec(e + fx))(dtan(e + fx))(S(2)/3)(cos(e + fx)S(2))(S(7)/12)Hypergeometric2F1(S(1)/3, S(7)/12, S(4)/3, sin(e + fx)S(2))/(S(2)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))/(dtan(e + fx))(S(4)/3), x), x, -S(3)sqrt(bsec(e + fx))(cos(e + fx)S(2))(S(1)/12)Hypergeometric2F1(S(-1)/6, S(1)/12, S(5)/6, sin(e + fx)*S(2))/(df(dtan(e + fx))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(3)/2)(dtan(e + fx))*(S(4)/3), x), x, S(3)(bsec(e + fx))*(S(3)/2)(dtan(e + fx))*(S(7)/3)(cos(e + fx)S(2))(S(23)/12)Hypergeometric2F1(S(7)/6, S(23)/12, S(13)/6, sin(e + fx)S(2))/(S(7)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(3)/2)(dtan(e + fx))*(S(1)/3), x), x, S(3)(bsec(e + fx))*(S(3)/2)(dtan(e + fx))*(S(4)/3)(cos(e + fx)S(2))(S(17)/12)Hypergeometric2F1(S(2)/3, S(17)/12, S(5)/3, sin(e + fx)S(2))/(S(4)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(3)/2)/(dtan(e + fx))(S(1)/3), x), x, S(3)(bsec(e + fx))*(S(3)/2)(dtan(e + fx))*(S(2)/3)(cos(e + fx)S(2))(S(13)/12)Hypergeometric2F1(S(1)/3, S(13)/12, S(4)/3, sin(e + fx)S(2))/(S(2)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(3)/2)/(dtan(e + fx))(S(4)/3), x), x, -S(3)(bsec(e + fx))*(S(3)/2)(cos(e + fx)S(2))(S(7)/12)Hypergeometric2F1(S(-1)/6, S(7)/12, S(5)/6, sin(e + fx)S(2))/(df(dtan(e + fx))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))mtan(e + fx)S(5), x), x, (bsec(e + fx))m/(fm) - S(2)(bsec(e + fx))(m + S(2))/(bS(2)f(m + S(2))) + (bsec(e + fx))(m + S(4))/(bS(4)f(m + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))mtan(e + fx)S(3), x), x, -(bsec(e + fx))m/(fm) + (bsec(e + fx))(m + S(2))/(bS(2)f(m + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*mtan(e + fx), x), x, (bsec(e + fx))m/(fm), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*mcot(e + fx), x), x, -(bsec(e + fx))mHypergeometric2F1(S(1), m/S(2), m/S(2) + S(1), sec(e + fx)S(2))/(fm), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*mcot(e + fx)S(3), x), x, (bsec(e + fx))mHypergeometric2F1(S(2), m/S(2), m/S(2) + S(1), sec(e + fx)S(2))/(fm), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*mcot(e + fx)S(5), x), x, -(bsec(e + fx))mHypergeometric2F1(S(3), m/S(2), m/S(2) + S(1), sec(e + fx)S(2))/(fm), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*mtan(e + fx)S(4), x), x, (bsec(e + fx))m(cos(e + fx)S(2))(m/S(2) + S(5)/2)Hypergeometric2F1(S(5)/2, m/S(2) + S(5)/2, S(7)/2, sin(e + fx)S(2))tan(e + fx)S(5)/(S(5)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*mtan(e + fx)S(2), x), x, (bsec(e + fx))m(cos(e + fx)S(2))(m/S(2) + S(3)/2)Hypergeometric2F1(S(3)/2, m/S(2) + S(3)/2, S(5)/2, sin(e + fx)S(2))tan(e + fx)S(3)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*mcot(e + fx)S(2), x), x, -(bsec(e + fx))m(cos(e + fx)S(2))(m/S(2) + S(-1)/2)Hypergeometric2F1(S(-1)/2, m/S(2) + S(-1)/2, S(1)/2, sin(e + fx)S(2))cot(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))mcot(e + fx)S(4), x), x, -(bsec(e + fx))m(cos(e + fx)S(2))(m/S(2) + S(-3)/2)Hypergeometric2F1(S(-3)/2, m/S(2) + S(-3)/2, S(-1)/2, sin(e + fx)S(2))cot(e + fx)S(3)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*mcot(e + fx)S(6), x), x, -(bsec(e + fx))m(cos(e + fx)S(2))(m/S(2) + S(-5)/2)Hypergeometric2F1(S(-5)/2, m/S(2) + S(-5)/2, S(-3)/2, sin(e + fx)S(2))cot(e + fx)S(5)/(S(5)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(e + fx))*m(btan(e + fx))*n, x), x, (asec(e + fx))m(btan(e + fx))*(n + S(1))(cos(e + fx)S(2))(m/S(2) + n/S(2) + S(1)/2)Hypergeometric2F1(n/S(2) + S(1)/2, m/S(2) + n/S(2) + S(1)/2, n/S(2) + S(3)/2, sin(e + fx)S(2))/(bf(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))nsec(a + bx)S(6), x), x, (dtan(a + bx))(n + S(1))/(bd(n + S(1))) + S(2)(dtan(a + bx))*(n + S(3))/(bd*S(3)(n + S(3))) + (dtan(a + bx))*(n + S(5))/(bd*S(5)(n + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))*nsec(a + bx)S(4), x), x, (dtan(a + bx))(n + S(1))/(bd(n + S(1))) + (dtan(a + bx))(n + S(3))/(bd*S(3)(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))*nsec(a + bx)S(2), x), x, (dtan(a + bx))(n + S(1))/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))n, x), x, (dtan(a + bx))(n + S(1))Hypergeometric2F1(S(1), n/S(2) + S(1)/2, n/S(2) + S(3)/2, -tan(a + bx)S(2))/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))ncos(a + bx)S(2), x), x, (dtan(a + bx))(n + S(1))Hypergeometric2F1(S(2), n/S(2) + S(1)/2, n/S(2) + S(3)/2, -tan(a + bx)S(2))/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))ncos(a + bx)S(4), x), x, (dtan(a + bx))(n + S(1))Hypergeometric2F1(S(3), n/S(2) + S(1)/2, n/S(2) + S(3)/2, -tan(a + bx)S(2))/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))nsec(a + bx)S(5), x), x, (dtan(a + bx))(n + S(1))(cos(a + bx)S(2))(n/S(2) + S(3))Hypergeometric2F1(n/S(2) + S(1)/2, n/S(2) + S(3), n/S(2) + S(3)/2, sin(a + bx)S(2))sec(a + bx)S(5)/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))nsec(a + bx)S(3), x), x, (dtan(a + bx))(n + S(1))(cos(a + bx)S(2))(n/S(2) + S(2))Hypergeometric2F1(n/S(2) + S(1)/2, n/S(2) + S(2), n/S(2) + S(3)/2, sin(a + bx)S(2))sec(a + bx)S(3)/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))nsec(a + bx), x), x, (dtan(a + bx))(n + S(1))(cos(a + bx)S(2))(n/S(2) + S(1))Hypergeometric2F1(n/S(2) + S(1)/2, n/S(2) + S(1), n/S(2) + S(3)/2, sin(a + bx)S(2))sec(a + bx)/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))ncos(a + bx), x), x, (dtan(a + bx))(n + S(1))(cos(a + bx)S(2))(n/S(2))Hypergeometric2F1(n/S(2), n/S(2) + S(1)/2, n/S(2) + S(3)/2, sin(a + bx)S(2))cos(a + bx)/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dtan(a + bx))ncos(a + bx)S(3), x), x, (dtan(a + bx))(n + S(1))(cos(a + bx)S(2))(n/S(2) + S(-1))Hypergeometric2F1(n/S(2) + S(-1), n/S(2) + S(1)/2, n/S(2) + S(3)/2, sin(a + bx)S(2))cos(a + bx)S(3)/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate((bcsc(e + fx))mtan(e + fx)S(3), x), x, -(bcsc(e + fx))mHypergeometric2F1(S(2), m/S(2), m/S(2) + S(1), csc(e + fx)S(2))/(fm), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bcsc(e + fx))*m(dtan(e + fx))*(S(3)/2), x), x, (bcsc(e + fx))m(dtan(e + fx))*(S(5)/2)(cos(e + fx)S(2))(S(5)/4)Hypergeometric2F1(S(5)/4, -m/S(2) + S(5)/4, -m/S(2) + S(9)/4, sin(e + fx)S(2))/(df(-m + S(5)/2)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((bcsc(e + fx))m(dtan(e + fx))*(S(3)/2), x), x, S(2)d(bcsc(e + fx))msqrt(dtan(e + fx))(cos(e + fx)S(2))(S(1)/4)Hypergeometric2F1(S(5)/4, -m/S(2) + S(5)/4, -m/S(2) + S(9)/4, sin(e + fx)*S(2))sin(e + fx)S(2)/(f(-S(2)m + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bcsc(e + fx))msqrt(dtan(e + fx)), x), x, S(2)(bcsc(e + fx))m(dtan(e + fx))*(S(3)/2)(cos(e + fx)S(2))(S(3)/4)Hypergeometric2F1(S(3)/4, -m/S(2) + S(3)/4, -m/S(2) + S(7)/4, sin(e + fx)S(2))/(df(-S(2)m + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bcsc(e + fx))*m/sqrt(dtan(e + fx)), x), x, S(2)(bcsc(e + fx))*msqrt(dtan(e + fx))(cos(e + fx)S(2))(S(1)/4)Hypergeometric2F1(S(1)/4, -m/S(2) + S(1)/4, -m/S(2) + S(5)/4, sin(e + fx)*S(2))/(df(-S(2)m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bcsc(e + fx))*m/(dtan(e + fx))(S(3)/2), x), x, -S(2)(bcsc(e + fx))*mHypergeometric2F1(S(-1)/4, -m/S(2) + S(-1)/4, -m/S(2) + S(3)/4, sin(e + fx)S(2))/(dfsqrt(dtan(e + fx))(S(2)m + S(1))(cos(e + fx)S(2))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((acsc(e + fx))m(btan(e + fx))*n, x), x, (acsc(e + fx))m(btan(e + fx))*(n + S(1))(cos(e + fx)S(2))(n/S(2) + S(1)/2)Hypergeometric2F1(n/S(2) + S(1)/2, -m/S(2) + n/S(2) + S(1)/2, -m/S(2) + n/S(2) + S(3)/2, sin(e + fx)S(2))/(bf(-m + n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-tan(e + fx)S(2))nsin(e + fx), x), x, -Hypergeometric2F1(S(-1)/2, -n, S(1)/2, sec(e + fx)S(2))cos(e + f*x)/f, expand=True, _diff=True, _numerical=True)
a0433854e4c7fc2bc255d58396f97cddf203524e79ece3c10a762440045d7eb9	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.functions import log, sqrt, exp, cos, sin, tan, sec, csc, cot from sympy.functions.elementary.hyperbolic import atanh, asinh, acosh from sympy.functions.elementary.hyperbolic import atanh as arctanh from sympy.functions.elementary.hyperbolic import asinh as arcsinh from sympy.functions.elementary.hyperbolic import acosh as arccosh from sympy.functions.elementary.trigonometric import atan, asin, acos from sympy.functions.elementary.trigonometric import atan as arctan from sympy.functions.elementary.trigonometric import asin as arcsin from sympy.functions.elementary.trigonometric import acos as arccos from sympy.integrals.rubi.utility_function import (EllipticE, EllipticF, hypergeom, rubi_test, AppellF1, EllipticPi, Log, Sqrt, ArcTan, ArcTanh, ArcSin, Hypergeometric2F1) from sympy.core.mod import Mod from sympy.core.numbers import (I, pi as Pi) from sympy.core.singleton import S from sympy.core.symbol import symbols from sympy.functions.elementary.exponential import exp_polar from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.special.elliptic_integrals import (elliptic_e, elliptic_f, elliptic_pi as Pi) from sympy.functions.special.hyper import hyper from sympy.simplify.simplify import simplify from sympy.testing.pytest import SKIP a, b, c, d, e, f, m, n, x, u , k, p, j, l , i= symbols('a b c d e f m n x u k p j l i1') A, B, C, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B C a b c d e f g h y z m n p q u v w F', real=True, imaginary=False) def test_1(): assert rubi_test(rubi_integrate(x*m(bxS(2) + cx*S(4)), x), x, bx*(m + S(3))/(m + S(3)) + cx(m + S(5))/(m + S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(bx*S(2) + cx*S(4)), x), x, bx*S(5)/S(5) + cx*S(7)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(bxS(2) + cx*S(4)), x), x, bx*S(4)/S(4) + cx*S(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(bx*S(2) + cx*S(4), x), x, bx*S(3)/S(3) + cx*S(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cx*S(4))/x, x), x, bx*S(2)/S(2) + cx*S(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))/xS(2), x), x, bx + cx*S(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))/xS(3), x), x, blog(x) + cx*S(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))/xS(4), x), x, -b/x + cx, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))/xS(5), x), x, -b/(S(2)xS(2)) + clog(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + cxS(4))/xS(6), x), x, -b/(S(3)xS(3)) - c/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))/xS(7), x), x, -b/(S(4)xS(4)) - c/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))/xS(8), x), x, -b/(S(5)xS(5)) - c/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(bx*S(2) + cxS(4))S(2), x), x, b*S(2)x*(m + S(5))/(m + S(5)) + S(2)bcx(m + S(7))/(m + S(7)) + cS(2)x(m + S(9))/(m + S(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2), x), x, b*S(2)x*S(5)/S(5) + S(2)bcxS(7)/S(7) + cS(2)xS(9)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/x, x), x, b*S(2)x*S(4)/S(4) + bcxS(6)/S(3) + cS(2)x*S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/xS(2), x), x, bS(2)xS(3)/S(3) + S(2)bcxS(5)/S(5) + cS(2)xS(7)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/x*S(3), x), x, (b + cxS(2))S(3)/(S(6)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/xS(4), x), x, bS(2)x + S(2)bcxS(3)/S(3) + cS(2)xS(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/xS(5), x), x, bS(2)log(x) + bcxS(2) + cS(2)x*S(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/xS(6), x), x, -bS(2)/x + S(2)bcx + cS(2)x*S(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/xS(7), x), x, -bS(2)/(S(2)xS(2)) + S(2)bclog(x) + c*S(2)x*S(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/xS(8), x), x, -bS(2)/(S(3)xS(3)) - S(2)bc/x + cS(2)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + cxS(4))S(2)/xS(9), x), x, -bS(2)/(S(4)xS(4)) - bc/xS(2) + cS(2)log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/xS(10), x), x, -bS(2)/(S(5)xS(5)) - S(2)bc/(S(3)xS(3)) - cS(2)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + cxS(4))S(2)/x*S(11), x), x, -(b + cxS(2))S(3)/(S(6)bx*S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/xS(12), x), x, -bS(2)/(S(7)xS(7)) - S(2)bc/(S(5)xS(5)) - cS(2)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(bxS(2) + cxS(4))S(3), x), x, b*S(3)x*(m + S(7))/(m + S(7)) + S(3)b*S(2)cx(m + S(9))/(m + S(9)) + S(3)bcS(2)x(m + S(11))/(m + S(11)) + cS(3)x(m + S(13))/(m + S(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/xS(2), x), x, bS(3)xS(5)/S(5) + S(3)b*S(2)cxS(7)/S(7) + bc*S(2)xS(9)/S(3) + cS(3)xS(11)/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/x*S(3), x), x, -b(b + cxS(2))S(4)/(S(8)c*S(2)) + (b + cxS(2))S(5)/(S(10)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/xS(4), x), x, bS(3)xS(3)/S(3) + S(3)b*S(2)cxS(5)/S(5) + S(3)bcS(2)xS(7)/S(7) + cS(3)xS(9)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/x*S(5), x), x, (b + cxS(2))S(4)/(S(8)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/xS(6), x), x, bS(3)x + bS(2)cxS(3) + S(3)bcS(2)xS(5)/S(5) + cS(3)xS(7)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/xS(7), x), x, bS(3)log(x) + S(3)b*S(2)cxS(2)/S(2) + S(3)bcS(2)xS(4)/S(4) + cS(3)xS(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/xS(8), x), x, -bS(3)/x + S(3)bS(2)cx + bc*S(2)xS(3) + cS(3)xS(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/xS(9), x), x, -bS(3)/(S(2)xS(2)) + S(3)b*S(2)clog(x) + S(3)bcS(2)xS(2)/S(2) + cS(3)xS(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/xS(10), x), x, -bS(3)/(S(3)xS(3)) - S(3)b*S(2)c/x + S(3)bc*S(2)x + c*S(3)x*S(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/xS(11), x), x, -bS(3)/(S(4)xS(4)) - S(3)b*S(2)c/(S(2)xS(2)) + S(3)bcS(2)log(x) + c*S(3)x*S(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/xS(12), x), x, -bS(3)/(S(5)xS(5)) - bS(2)c/x*S(3) - S(3)bcS(2)/x + cS(3)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + cxS(4))S(3)/xS(13), x), x, -bS(3)/(S(6)xS(6)) - S(3)b*S(2)c/(S(4)xS(4)) - S(3)bcS(2)/(S(2)xS(2)) + cS(3)log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/xS(14), x), x, -bS(3)/(S(7)xS(7)) - S(3)b*S(2)c/(S(5)xS(5)) - bcS(2)/xS(3) - c*S(3)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/x*S(15), x), x, -(b + cxS(2))S(4)/(S(8)bx*S(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/xS(16), x), x, -bS(3)/(S(9)xS(9)) - S(3)b*S(2)c/(S(7)xS(7)) - S(3)bcS(2)/(S(5)xS(5)) - cS(3)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/xS(17), x), x, -bS(3)/(S(10)xS(10)) - S(3)b*S(2)c/(S(8)xS(8)) - bc*S(2)/(S(2)xS(6)) - cS(3)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(10)/(bx*S(2) + cxS(4)), x), x, b(S(7)/2)atan(sqrt(c)x/sqrt(b))/c(S(9)/2) - bS(3)x/cS(4) + bS(2)x*S(3)/(S(3)c*S(3)) - bx*S(5)/(S(5)cS(2)) + xS(7)/(S(7)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(bx*S(2) + cxS(4)), x), x, -bS(3)log(b + cx*S(2))/(S(2)cS(4)) + bS(2)xS(2)/(S(2)c*S(3)) - bx*S(4)/(S(4)cS(2)) + xS(6)/(S(6)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(bx*S(2) + cxS(4)), x), x, -b(S(5)/2)atan(sqrt(c)x/sqrt(b))/c(S(7)/2) + bS(2)x/cS(3) - bx*S(3)/(S(3)cS(2)) + xS(5)/(S(5)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(bx*S(2) + cxS(4)), x), x, bS(2)log(b + cx*S(2))/(S(2)c*S(3)) - bx*S(2)/(S(2)cS(2)) + xS(4)/(S(4)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(bx*S(2) + cxS(4)), x), x, b(S(3)/2)atan(sqrt(c)x/sqrt(b))/c*(S(5)/2) - bx/cS(2) + xS(3)/(S(3)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(bx*S(2) + cx*S(4)), x), x, -blog(b + cxS(2))/(S(2)cS(2)) + xS(2)/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(bx*S(2) + cx*S(4)), x), x, -sqrt(b)atan(sqrt(c)x/sqrt(b))/c(S(3)/2) + x/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(bx*S(2) + cx*S(4)), x), x, log(b + cx*S(2))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(bx*S(2) + cx*S(4)), x), x, atan(sqrt(c)x/sqrt(b))/(sqrt(b)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(bx*S(2) + cx*S(4)), x), x, log(x)/b - log(b + cx*S(2))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(bxS(2) + cx*S(4)), x), x, -S(1)/(bx) - sqrt(c)atan(sqrt(c)x/sqrt(b))/b*(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(bxS(2) + cx*S(4))), x), x, -S(1)/(S(2)bxS(2)) - clog(x)/b*S(2) + clog(b + cxS(2))/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(bx*S(2) + cx*S(4))), x), x, -S(1)/(S(3)bxS(3)) + c/(bS(2)x) + c*(S(3)/2)atan(sqrt(c)x/sqrt(b))/b(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(bxS(2) + cx*S(4))), x), x, -S(1)/(S(4)bxS(4)) + c/(S(2)b*S(2)xS(2)) + cS(2)log(x)/bS(3) - cS(2)log(b + cxS(2))/(S(2)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(bx*S(2) + cx*S(4))), x), x, -S(1)/(S(5)bxS(5)) + c/(S(3)b*S(2)xS(3)) - cS(2)/(b*S(3)x) - c*(S(5)/2)atan(sqrt(c)x/sqrt(b))/b(S(7)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(bxS(2) + cx*S(4))), x), x, -S(1)/(S(6)bxS(6)) + c/(S(4)b*S(2)xS(4)) - cS(2)/(S(2)bS(3)xS(2)) - cS(3)log(x)/bS(4) + cS(3)log(b + cxS(2))/(S(2)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(12)/(bxS(2) + cxS(4))S(2), x), x, -S(7)b(S(5)/2)atan(sqrt(c)x/sqrt(b))/(S(2)c*(S(9)/2)) + S(7)b*S(2)x/(S(2)cS(4)) - S(7)bxS(3)/(S(6)cS(3)) - xS(7)/(S(2)c(b + cxS(2))) + S(7)x*S(5)/(S(10)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(bxS(2) + cxS(4))S(2), x), x, b*S(3)/(S(2)c*S(4)(b + cxS(2))) + S(3)b*S(2)log(b + cxS(2))/(S(2)c*S(4)) - bxS(2)/cS(3) + x*S(4)/(S(4)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(10)/(bxS(2) + cxS(4))S(2), x), x, S(5)b(S(3)/2)atan(sqrt(c)x/sqrt(b))/(S(2)c*(S(7)/2)) - S(5)bx/(S(2)cS(3)) - xS(5)/(S(2)c(b + cxS(2))) + S(5)x*S(3)/(S(6)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(bxS(2) + cxS(4))S(2), x), x, -b*S(2)/(S(2)c*S(3)(b + cxS(2))) - blog(b + cxS(2))/cS(3) + xS(2)/(S(2)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(bxS(2) + cxS(4))S(2), x), x, -S(3)sqrt(b)atan(sqrt(c)x/sqrt(b))/(S(2)c(S(5)/2)) - xS(3)/(S(2)c(b + cxS(2))) + S(3)x/(S(2)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(bx*S(2) + cxS(4))S(2), x), x, b/(S(2)cS(2)(b + cxS(2))) + log(b + cx*S(2))/(S(2)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(bxS(2) + cxS(4))S(2), x), x, -x/(S(2)c(b + cxS(2))) + atan(sqrt(c)x/sqrt(b))/(S(2)sqrt(b)c(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(bxS(2) + cxS(4))S(2), x), x, -S(1)/(S(2)c(b + cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(bx*S(2) + cxS(4))S(2), x), x, x/(S(2)b(b + cxS(2))) + atan(sqrt(c)x/sqrt(b))/(S(2)b(S(3)/2)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/(bx*S(2) + cxS(4))S(2), x), x, S(1)/(S(2)b(b + cxS(2))) + log(x)/bS(2) - log(b + cx*S(2))/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(bxS(2) + cxS(4))S(2), x), x, S(1)/(S(2)bx(b + cx*S(2))) - S(3)/(S(2)b*S(2)x) - S(3)sqrt(c)atan(sqrt(c)x/sqrt(b))/(S(2)b*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(bx*S(2) + cxS(4))S(2), x), x, -c/(S(2)bS(2)(b + cxS(2))) - S(1)/(S(2)b*S(2)x*S(2)) - S(2)clog(x)/bS(3) + clog(b + cxS(2))/bS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(-2)), x), x, S(1)/(S(2)bx*S(3)(b + cxS(2))) - S(5)/(S(6)b*S(2)x*S(3)) + S(5)c/(S(2)bS(3)x) + S(5)c(S(3)/2)atan(sqrt(c)x/sqrt(b))/(S(2)b*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(bxS(2) + cxS(4))S(2)), x), x, -S(1)/(S(4)bS(2)xS(4)) + cS(2)/(S(2)bS(3)(b + cxS(2))) + c/(bS(3)x*S(2)) + S(3)c*S(2)log(x)/b*S(4) - S(3)c*S(2)log(b + cxS(2))/(S(2)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(bx*S(2) + cxS(4))S(2)), x), x, S(1)/(S(2)bx*S(5)(b + cxS(2))) - S(7)/(S(10)b*S(2)x*S(5)) + S(7)c/(S(6)bS(3)x*S(3)) - S(7)c*S(2)/(S(2)b*S(4)x) - S(7)c(S(5)/2)atan(sqrt(c)x/sqrt(b))/(S(2)b(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(14)/(bxS(2) + cxS(4))S(3), x), x, S(35)b(S(3)/2)atan(sqrt(c)x/sqrt(b))/(S(8)c*(S(9)/2)) - S(35)bx/(S(8)cS(4)) - xS(7)/(S(4)c(b + cxS(2))S(2)) - S(7)x*S(5)/(S(8)c*S(2)(b + cxS(2))) + S(35)x*S(3)/(S(24)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(13)/(bxS(2) + cxS(4))S(3), x), x, b*S(3)/(S(4)c*S(4)(b + cxS(2))S(2)) - S(3)b*S(2)/(S(2)c*S(4)(b + cxS(2))) - S(3)blog(b + cx*S(2))/(S(2)cS(4)) + xS(2)/(S(2)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(12)/(bx*S(2) + cxS(4))S(3), x), x, -S(15)sqrt(b)atan(sqrt(c)x/sqrt(b))/(S(8)c(S(7)/2)) - xS(5)/(S(4)c(b + cxS(2))S(2)) - S(5)x*S(3)/(S(8)c*S(2)(b + cxS(2))) + S(15)x/(S(8)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(bx*S(2) + cxS(4))S(3), x), x, -b*S(2)/(S(4)c*S(3)(b + cxS(2))S(2)) + b/(cS(3)(b + cxS(2))) + log(b + cx*S(2))/(S(2)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(10)/(bxS(2) + cxS(4))S(3), x), x, -x*S(3)/(S(4)c(b + cxS(2))S(2)) - S(3)x/(S(8)c*S(2)(b + cxS(2))) + S(3)atan(sqrt(c)x/sqrt(b))/(S(8)sqrt(b)c(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(bx*S(2) + cxS(4))S(3), x), x, x*S(4)/(S(4)b(b + cxS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(8)/(bx*S(2) + cxS(4))S(3), x), x, -x/(S(4)c(b + cxS(2))S(2)) + x/(S(8)bc(b + cxS(2))) + atan(sqrt(c)x/sqrt(b))/(S(8)b(S(3)/2)c(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(bxS(2) + cxS(4))S(3), x), x, -S(1)/(S(4)c(b + cxS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(bx*S(2) + cxS(4))S(3), x), x, x/(S(4)b(b + cxS(2))S(2)) + S(3)x/(S(8)bS(2)(b + cxS(2))) + S(3)atan(sqrt(c)x/sqrt(b))/(S(8)b*(S(5)/2)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)/(bx*S(2) + cxS(4))S(3), x), x, S(1)/(S(4)b(b + cxS(2))S(2)) + S(1)/(S(2)b*S(2)(b + cxS(2))) + log(x)/bS(3) - log(b + cx*S(2))/(S(2)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(bxS(2) + cxS(4))S(3), x), x, S(1)/(S(4)bx(b + cxS(2))S(2)) + S(5)/(S(8)bS(2)x(b + cx*S(2))) - S(15)/(S(8)b*S(3)x) - S(15)sqrt(c)atan(sqrt(c)x/sqrt(b))/(S(8)b(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(bxS(2) + cxS(4))S(3), x), x, -c/(S(4)bS(2)(b + cxS(2))S(2)) - c/(bS(3)(b + cxS(2))) - S(1)/(S(2)b*S(3)x*S(2)) - S(3)clog(x)/bS(4) + S(3)clog(b + cx*S(2))/(S(2)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(bxS(2) + cxS(4))S(3), x), x, S(1)/(S(4)bx*S(3)(b + cxS(2))S(2)) + S(7)/(S(8)b*S(2)x*S(3)(b + cxS(2))) - S(35)/(S(24)b*S(3)x*S(3)) + S(35)c/(S(8)bS(4)x) + S(35)c(S(3)/2)atan(sqrt(c)x/sqrt(b))/(S(8)b*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(bx*S(2) + cxS(4))S(3), x), x, c*S(2)/(S(4)b*S(3)(b + cxS(2))S(2)) - S(1)/(S(4)b*S(3)x*S(4)) + S(3)c*S(2)/(S(2)b*S(4)(b + cxS(2))) + S(3)c/(S(2)bS(4)x*S(2)) + S(6)c*S(2)log(x)/b*S(5) - S(3)c*S(2)log(b + cxS(2))/bS(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(-3)), x), x, S(1)/(S(4)bx*S(5)(b + cxS(2))S(2)) + S(9)/(S(8)b*S(2)x*S(5)(b + cxS(2))) - S(63)/(S(40)b*S(3)x*S(5)) + S(21)c/(S(8)bS(4)x*S(3)) - S(63)c*S(2)/(S(8)b*S(5)x) - S(63)c(S(5)/2)atan(sqrt(c)x/sqrt(b))/(S(8)b*(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(bxS(2) + cxS(4))S(3)), x), x, -S(1)/(S(6)bS(3)xS(6)) - cS(3)/(S(4)bS(4)(b + cxS(2))S(2)) + S(3)c/(S(4)bS(4)x*S(4)) - S(2)cS(3)/(bS(5)(b + cx*S(2))) - S(3)cS(2)/(bS(5)xS(2)) - S(10)c*S(3)log(x)/b*S(6) + S(5)c*S(3)log(b + cxS(2))/bS(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)sqrt(bxS(2) + cx*S(4)), x), x, -S(5)b*S(4)atanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/(S(128)c*(S(7)/2)) + S(5)b*S(2)(b + S(2)cx*S(2))sqrt(bxS(2) + cx*S(4))/(S(128)c*S(3)) - S(5)b(bx*S(2) + cxS(4))(S(3)/2)/(S(48)cS(2)) + xS(2)(bxS(2) + cxS(4))(S(3)/2)/(S(8)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(bxS(2) + cxS(4)), x), x, bS(3)atanh(sqrt(c)x*S(2)/sqrt(bx*S(2) + cx*S(4)))/(S(16)c*(S(5)/2)) - b(b + S(2)cx*S(2))sqrt(bxS(2) + cx*S(4))/(S(16)c*S(2)) + (bx*S(2) + cxS(4))(S(3)/2)/(S(6)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(bxS(2) + cxS(4)), x), x, -bS(2)atanh(sqrt(c)x*S(2)/sqrt(bx*S(2) + cx*S(4)))/(S(8)c*(S(3)/2)) + (b + S(2)cxS(2))sqrt(bxS(2) + cx*S(4))/(S(8)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bxS(2) + cx*S(4))/x, x), x, batanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/(S(2)sqrt(c)) + sqrt(bxS(2) + cx*S(4))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/xS(3), x), x, sqrt(c)atanh(sqrt(c)x*S(2)/sqrt(bx*S(2) + cx*S(4))) - sqrt(bx*S(2) + cxS(4))/xS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bxS(2) + cxS(4))/xS(5), x), x, -(bxS(2) + cxS(4))(S(3)/2)/(S(3)bx*S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/xS(7), x), x, -(bxS(2) + cxS(4))(S(3)/2)/(S(5)bx*S(8)) + S(2)c(bx*S(2) + cxS(4))(S(3)/2)/(S(15)bS(2)x*S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/xS(9), x), x, -(bxS(2) + cxS(4))(S(3)/2)/(S(7)bx*S(10)) + S(4)c(bx*S(2) + cxS(4))(S(3)/2)/(S(35)bS(2)x*S(8)) - S(8)c*S(2)(bxS(2) + cxS(4))(S(3)/2)/(S(105)bS(3)x*S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/xS(11), x), x, -(bxS(2) + cxS(4))(S(3)/2)/(S(9)bx*S(12)) + S(2)c(bx*S(2) + cxS(4))(S(3)/2)/(S(21)bS(2)x*S(10)) - S(8)c*S(2)(bxS(2) + cxS(4))(S(3)/2)/(S(105)bS(3)x*S(8)) + S(16)c*S(3)(bxS(2) + cxS(4))(S(3)/2)/(S(315)bS(4)x*S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/xS(13), x), x, -(bxS(2) + cxS(4))(S(3)/2)/(S(11)bx*S(14)) + S(8)c(bx*S(2) + cxS(4))(S(3)/2)/(S(99)bS(2)x*S(12)) - S(16)c*S(2)(bxS(2) + cxS(4))(S(3)/2)/(S(231)bS(3)x*S(10)) + S(64)c*S(3)(bxS(2) + cxS(4))(S(3)/2)/(S(1155)bS(4)x*S(8)) - S(128)c*S(4)(bxS(2) + cxS(4))(S(3)/2)/(S(3465)bS(5)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)sqrt(bx*S(2) + cx*S(4)), x), x, S(8)b*S(2)(bxS(2) + cxS(4))(S(3)/2)/(S(105)cS(3)x*S(3)) - S(4)b(bx*S(2) + cxS(4))(S(3)/2)/(S(35)cS(2)x) + x(bx*S(2) + cxS(4))(S(3)/2)/(S(7)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(bxS(2) + cx*S(4)), x), x, -S(2)b(bx*S(2) + cxS(4))(S(3)/2)/(S(15)cS(2)x*S(3)) + (bx*S(2) + cxS(4))(S(3)/2)/(S(5)cx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bxS(2) + cx*S(4)), x), x, (bx*S(2) + cxS(4))(S(3)/2)/(S(3)cx*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/xS(2), x), x, -sqrt(b)atanh(sqrt(b)x/sqrt(bxS(2) + cx*S(4))) + sqrt(bx*S(2) + cx*S(4))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/xS(4), x), x, -sqrt(bxS(2) + cx*S(4))/(S(2)x*S(3)) - catanh(sqrt(b)x/sqrt(bx*S(2) + cx*S(4)))/(S(2)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bxS(2) + cxS(4))/xS(6), x), x, -sqrt(bxS(2) + cx*S(4))/(S(4)x*S(5)) - csqrt(bxS(2) + cx*S(4))/(S(8)bxS(3)) + cS(2)atanh(sqrt(b)x/sqrt(bx*S(2) + cx*S(4)))/(S(8)b*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/xS(8), x), x, -sqrt(bxS(2) + cx*S(4))/(S(6)x*S(7)) - csqrt(bxS(2) + cx*S(4))/(S(24)bxS(5)) + cS(2)sqrt(bxS(2) + cx*S(4))/(S(16)b*S(2)xS(3)) - cS(3)atanh(sqrt(b)x/sqrt(bxS(2) + cx*S(4)))/(S(16)b(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(bx*S(2) + cxS(4))(S(3)/2), x), x, -S(3)bS(5)atanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/(S(256)c*(S(7)/2)) + S(3)b*S(3)(b + S(2)cx*S(2))sqrt(bxS(2) + cx*S(4))/(S(256)c*S(3)) - b(b + S(2)cx*S(2))(bxS(2) + cxS(4))(S(3)/2)/(S(32)cS(2)) + (bx*S(2) + cxS(4))(S(5)/2)/(S(10)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(bxS(2) + cxS(4))(S(3)/2), x), x, S(3)bS(4)atanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/(S(128)c*(S(5)/2)) - S(3)b*S(2)(b + S(2)cx*S(2))sqrt(bxS(2) + cx*S(4))/(S(128)c*S(2)) + (b + S(2)cxS(2))(bxS(2) + cxS(4))(S(3)/2)/(S(16)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x, x), x, -b*S(3)atanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/(S(16)c*(S(3)/2)) + b(b + S(2)cx*S(2))sqrt(bxS(2) + cx*S(4))/(S(16)c) + (bxS(2) + cxS(4))(S(3)/2)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + cxS(4))(S(3)/2)/x*S(3), x), x, S(3)b*S(2)atanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/(S(8)sqrt(c)) + S(3)bsqrt(bxS(2) + cx*S(4))/S(8) + (bx*S(2) + cxS(4))(S(3)/2)/(S(4)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*S(5), x), x, S(3)bsqrt(c)atanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/S(2) + S(3)csqrt(bx*S(2) + cx*S(4))/S(2) - (bx*S(2) + cxS(4))(S(3)/2)/x*S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/xS(7), x), x, c(S(3)/2)atanh(sqrt(c)x*S(2)/sqrt(bx*S(2) + cx*S(4))) - csqrt(bxS(2) + cxS(4))/xS(2) - (bxS(2) + cxS(4))(S(3)/2)/(S(3)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*S(9), x), x, -(bx*S(2) + cxS(4))(S(5)/2)/(S(5)bx*S(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*S(11), x), x, -(bx*S(2) + cxS(4))(S(5)/2)/(S(7)bx*S(12)) + S(2)c(bx*S(2) + cxS(4))(S(5)/2)/(S(35)bS(2)x*S(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*S(13), x), x, -(bx*S(2) + cxS(4))(S(5)/2)/(S(9)bx*S(14)) + S(4)c(bx*S(2) + cxS(4))(S(5)/2)/(S(63)bS(2)x*S(12)) - S(8)c*S(2)(bxS(2) + cxS(4))(S(5)/2)/(S(315)bS(3)x*S(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*S(15), x), x, -(bx*S(2) + cxS(4))(S(5)/2)/(S(11)bx*S(16)) + S(2)c(bx*S(2) + cxS(4))(S(5)/2)/(S(33)bS(2)x*S(14)) - S(8)c*S(2)(bxS(2) + cxS(4))(S(5)/2)/(S(231)bS(3)x*S(12)) + S(16)c*S(3)(bxS(2) + cxS(4))(S(5)/2)/(S(1155)bS(4)x*S(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*S(17), x), x, -(bx*S(2) + cxS(4))(S(5)/2)/(S(13)bx*S(18)) + S(8)c(bx*S(2) + cxS(4))(S(5)/2)/(S(143)bS(2)x*S(16)) - S(16)c*S(2)(bxS(2) + cxS(4))(S(5)/2)/(S(429)bS(3)x*S(14)) + S(64)c*S(3)(bxS(2) + cxS(4))(S(5)/2)/(S(3003)bS(4)x*S(12)) - S(128)c*S(4)(bxS(2) + cxS(4))(S(5)/2)/(S(15015)bS(5)xS(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)(bx*S(2) + cxS(4))(S(3)/2), x), x, S(128)bS(4)(bxS(2) + cxS(4))(S(5)/2)/(S(15015)cS(5)x*S(5)) - S(64)b*S(3)(bxS(2) + cxS(4))(S(5)/2)/(S(3003)cS(4)x*S(3)) + S(16)b*S(2)(bxS(2) + cxS(4))(S(5)/2)/(S(429)cS(3)x) - S(8)bx(bx*S(2) + cxS(4))(S(5)/2)/(S(143)cS(2)) + xS(3)(bxS(2) + cxS(4))(S(5)/2)/(S(13)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(bxS(2) + cxS(4))(S(3)/2), x), x, -S(16)bS(3)(bxS(2) + cxS(4))(S(5)/2)/(S(1155)cS(4)x*S(5)) + S(8)b*S(2)(bxS(2) + cxS(4))(S(5)/2)/(S(231)cS(3)x*S(3)) - S(2)b(bx*S(2) + cxS(4))(S(5)/2)/(S(33)cS(2)x) + x(bx*S(2) + cxS(4))(S(5)/2)/(S(11)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(bxS(2) + cxS(4))(S(3)/2), x), x, S(8)bS(2)(bxS(2) + cxS(4))(S(5)/2)/(S(315)cS(3)x*S(5)) - S(4)b(bx*S(2) + cxS(4))(S(5)/2)/(S(63)cS(2)x*S(3)) + (bx*S(2) + cxS(4))(S(5)/2)/(S(9)cx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + cxS(4))(S(3)/2), x), x, -S(2)b(bxS(2) + cxS(4))(S(5)/2)/(S(35)cS(2)x*S(5)) + (bx*S(2) + cxS(4))(S(5)/2)/(S(7)cx*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*S(2), x), x, (bx*S(2) + cxS(4))(S(5)/2)/(S(5)cx*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/xS(4), x), x, -b(S(3)/2)atanh(sqrt(b)x/sqrt(bxS(2) + cx*S(4))) + bsqrt(bxS(2) + cx*S(4))/x + (bx*S(2) + cxS(4))(S(3)/2)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*S(6), x), x, -S(3)sqrt(b)catanh(sqrt(b)x/sqrt(bx*S(2) + cx*S(4)))/S(2) + S(3)csqrt(bx*S(2) + cx*S(4))/(S(2)x) - (bxS(2) + cxS(4))(S(3)/2)/(S(2)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*S(8), x), x, -S(3)csqrt(bx*S(2) + cx*S(4))/(S(8)x*S(3)) - (bx*S(2) + cxS(4))(S(3)/2)/(S(4)xS(7)) - S(3)c*S(2)atanh(sqrt(b)x/sqrt(bx*S(2) + cx*S(4)))/(S(8)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + cxS(4))(S(3)/2)/x*S(10), x), x, -csqrt(bxS(2) + cx*S(4))/(S(8)x*S(5)) - (bx*S(2) + cxS(4))(S(3)/2)/(S(6)xS(9)) - cS(2)sqrt(bxS(2) + cx*S(4))/(S(16)bxS(3)) + cS(3)atanh(sqrt(b)x/sqrt(bx*S(2) + cx*S(4)))/(S(16)b*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*S(12), x), x, -csqrt(bxS(2) + cx*S(4))/(S(16)x*S(7)) - (bx*S(2) + cxS(4))(S(3)/2)/(S(8)xS(11)) - cS(2)sqrt(bxS(2) + cx*S(4))/(S(64)bxS(5)) + S(3)c*S(3)sqrt(bxS(2) + cx*S(4))/(S(128)b*S(2)x*S(3)) - S(3)c*S(4)atanh(sqrt(b)x/sqrt(bx*S(2) + cx*S(4)))/(S(128)b*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*S(14), x), x, -S(3)csqrt(bx*S(2) + cx*S(4))/(S(80)x*S(9)) - (bx*S(2) + cxS(4))(S(3)/2)/(S(10)xS(13)) - cS(2)sqrt(bxS(2) + cx*S(4))/(S(160)bxS(7)) + cS(3)sqrt(bxS(2) + cx*S(4))/(S(128)b*S(2)x*S(5)) - S(3)c*S(4)sqrt(bxS(2) + cx*S(4))/(S(256)b*S(3)x*S(3)) + S(3)c*S(5)atanh(sqrt(b)x/sqrt(bx*S(2) + cx*S(4)))/(S(256)b(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/sqrt(bxS(2) + cx*S(4)), x), x, -S(5)b*S(3)atanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/(S(16)c*(S(7)/2)) + S(5)b*S(2)sqrt(bxS(2) + cx*S(4))/(S(16)c*S(3)) - S(5)bxS(2)sqrt(bxS(2) + cx*S(4))/(S(24)cS(2)) + xS(4)sqrt(bx*S(2) + cx*S(4))/(S(6)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)/sqrt(bx*S(2) + cx*S(4)), x), x, S(3)b*S(2)atanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/(S(8)c*(S(5)/2)) - S(3)bsqrt(bx*S(2) + cx*S(4))/(S(8)cS(2)) + xS(2)sqrt(bx*S(2) + cx*S(4))/(S(4)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/sqrt(bx*S(2) + cx*S(4)), x), x, -batanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/(S(2)c*(S(3)/2)) + sqrt(bx*S(2) + cx*S(4))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(bxS(2) + cx*S(4)), x), x, atanh(sqrt(c)x*S(2)/sqrt(bx*S(2) + cx*S(4)))/sqrt(c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(bxS(2) + cx*S(4))), x), x, -sqrt(bx*S(2) + cx*S(4))/(bxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)sqrt(bx*S(2) + cx*S(4))), x), x, -sqrt(bx*S(2) + cx*S(4))/(S(3)bxS(4)) + S(2)csqrt(bx*S(2) + cx*S(4))/(S(3)b*S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)sqrt(bx*S(2) + cx*S(4))), x), x, -sqrt(bx*S(2) + cx*S(4))/(S(5)bxS(6)) + S(4)csqrt(bx*S(2) + cx*S(4))/(S(15)b*S(2)x*S(4)) - S(8)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(15)b*S(3)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)sqrt(bx*S(2) + cx*S(4))), x), x, -sqrt(bx*S(2) + cx*S(4))/(S(7)bxS(8)) + S(6)csqrt(bx*S(2) + cx*S(4))/(S(35)b*S(2)x*S(6)) - S(8)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(35)b*S(3)x*S(4)) + S(16)c*S(3)sqrt(bxS(2) + cx*S(4))/(S(35)b*S(4)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/sqrt(bxS(2) + cx*S(4)), x), x, -S(2)bsqrt(bx*S(2) + cx*S(4))/(S(3)c*S(2)x) + xsqrt(bx*S(2) + cx*S(4))/(S(3)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/sqrt(bx*S(2) + cx*S(4)), x), x, sqrt(bx*S(2) + cx*S(4))/(cx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(bxS(2) + cx*S(4)), x), x, -atanh(sqrt(b)x/sqrt(bxS(2) + cxS(4)))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(bx*S(2) + cx*S(4))), x), x, -sqrt(bx*S(2) + cx*S(4))/(S(2)bxS(3)) + catanh(sqrt(b)x/sqrt(bx*S(2) + cx*S(4)))/(S(2)b(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)sqrt(bx*S(2) + cx*S(4))), x), x, -sqrt(bx*S(2) + cx*S(4))/(S(4)bxS(5)) + S(3)csqrt(bx*S(2) + cx*S(4))/(S(8)b*S(2)x*S(3)) - S(3)c*S(2)atanh(sqrt(b)x/sqrt(bx*S(2) + cx*S(4)))/(S(8)b(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(bxS(2) + cxS(4))(S(3)/2), x), x, S(15)bS(2)atanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/(S(8)c*(S(7)/2)) - S(15)bsqrt(bx*S(2) + cx*S(4))/(S(8)cS(3)) - xS(6)/(csqrt(bx*S(2) + cx*S(4))) + S(5)x*S(2)sqrt(bxS(2) + cx*S(4))/(S(4)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(bxS(2) + cxS(4))(S(3)/2), x), x, -S(3)batanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/(S(2)c(S(5)/2)) - xS(4)/(csqrt(bx*S(2) + cx*S(4))) + S(3)sqrt(bxS(2) + cx*S(4))/(S(2)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(bxS(2) + cxS(4))(S(3)/2), x), x, -x*S(2)/(csqrt(bxS(2) + cx*S(4))) + atanh(sqrt(c)x*S(2)/sqrt(bx*S(2) + cxS(4)))/c(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/(bx*S(2) + cxS(4))(S(3)/2), x), x, x*S(2)/(bsqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(bx*S(2) + cxS(4))(S(3)/2), x), x, -(b + S(2)cxS(2))/(bS(2)sqrt(bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(bxS(2) + cxS(4))(S(3)/2)), x), x, S(1)/(bxS(2)sqrt(bxS(2) + cx*S(4))) - S(4)sqrt(bxS(2) + cx*S(4))/(S(3)b*S(2)x*S(4)) + S(8)csqrt(bx*S(2) + cx*S(4))/(S(3)b*S(3)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(bx*S(2) + cxS(4))(S(3)/2)), x), x, S(1)/(bxS(4)sqrt(bxS(2) + cx*S(4))) - S(6)sqrt(bxS(2) + cx*S(4))/(S(5)b*S(2)x*S(6)) + S(8)csqrt(bx*S(2) + cx*S(4))/(S(5)b*S(3)x*S(4)) - S(16)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(5)b*S(4)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(bx*S(2) + cxS(4))(S(3)/2)), x), x, S(1)/(bxS(6)sqrt(bxS(2) + cx*S(4))) - S(8)sqrt(bxS(2) + cx*S(4))/(S(7)b*S(2)x*S(8)) + S(48)csqrt(bx*S(2) + cx*S(4))/(S(35)b*S(3)x*S(6)) - S(64)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(35)b*S(4)x*S(4)) + S(128)c*S(3)sqrt(bxS(2) + cx*S(4))/(S(35)b*S(5)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(bxS(2) + cxS(4))(S(3)/2), x), x, -x*S(3)/(csqrt(bxS(2) + cx*S(4))) + S(2)sqrt(bxS(2) + cxS(4))/(cS(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(bx*S(2) + cxS(4))(S(3)/2), x), x, -x/(csqrt(bx*S(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(bxS(2) + cxS(4))(S(3)/2), x), x, x/(bsqrt(bx*S(2) + cx*S(4))) - atanh(sqrt(b)x/sqrt(bxS(2) + cxS(4)))/b(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + cxS(4))(S(-3)/2), x), x, S(1)/(bxsqrt(bxS(2) + cx*S(4))) - S(3)sqrt(bxS(2) + cx*S(4))/(S(2)b*S(2)x*S(3)) + S(3)catanh(sqrt(b)x/sqrt(bxS(2) + cx*S(4)))/(S(2)b(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(bx*S(2) + cxS(4))(S(3)/2)), x), x, S(1)/(bxS(3)sqrt(bxS(2) + cx*S(4))) - S(5)sqrt(bxS(2) + cx*S(4))/(S(4)b*S(2)x*S(5)) + S(15)csqrt(bx*S(2) + cx*S(4))/(S(8)b*S(3)x*S(3)) - S(15)c*S(2)atanh(sqrt(b)x/sqrt(bx*S(2) + cx*S(4)))/(S(8)b(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(-S(4)xS(4) + S(3)x*S(2)), x), x, -sqrt(-S(4)x*S(4) + S(3)x*S(2))/S(8) + S(3)asin(S(8)xS(2)/S(3) + S(-1))/S(32), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(-S(4)x*S(4) - S(3)x*S(2)), x), x, -sqrt(-S(4)x*S(4) - S(3)x*S(2))/S(8) - S(3)asin(S(8)xS(2)/S(3) + S(1))/S(32), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(S(4)x*S(4) + S(3)x*S(2)), x), x, sqrt(S(4)x*S(4) + S(3)x*S(2))/S(8) - S(3)atanh(S(2)xS(2)/sqrt(S(4)x*S(4) + S(3)xS(2)))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(S(4)xS(4) - S(3)x*S(2)), x), x, sqrt(S(4)x*S(4) - S(3)x*S(2))/S(8) + S(3)atanh(S(2)xS(2)/sqrt(S(4)x*S(4) - S(3)xS(2)))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(axS(2) + bx*S(4)), x), x, -aatanh(sqrt(b)xS(2)/sqrt(ax*S(2) + bx*S(4)))/(S(2)b*(S(3)/2)) + sqrt(ax*S(2) + bx*S(4))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/sqrt(ax*S(2) - bx*S(4)), x), x, aatan(sqrt(b)xS(2)/sqrt(ax*S(2) - bx*S(4)))/(S(2)b*(S(3)/2)) - sqrt(ax*S(2) - bx*S(4))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(7)/2)(bxS(2) + cx*S(4)), x), x, S(2)bx(S(13)/2)/S(13) + S(2)cx(S(17)/2)/S(17), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)(bxS(2) + cx*S(4)), x), x, S(2)bx(S(11)/2)/S(11) + S(2)cx(S(15)/2)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)(bxS(2) + cx*S(4)), x), x, S(2)bx(S(9)/2)/S(9) + S(2)cx(S(13)/2)/S(13), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)(bxS(2) + cx*S(4)), x), x, S(2)bx(S(7)/2)/S(7) + S(2)cx(S(11)/2)/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cx*S(4))/sqrt(x), x), x, S(2)bx(S(5)/2)/S(5) + S(2)cx(S(9)/2)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))/x(S(3)/2), x), x, S(2)bx*(S(3)/2)/S(3) + S(2)cx(S(7)/2)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))/x(S(5)/2), x), x, S(2)bsqrt(x) + S(2)cx*(S(5)/2)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))/x(S(7)/2), x), x, -S(2)b/sqrt(x) + S(2)cx(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(7)/2)(bxS(2) + cxS(4))S(2), x), x, S(2)bS(2)x*(S(17)/2)/S(17) + S(4)bcx*(S(21)/2)/S(21) + S(2)c*S(2)x(S(25)/2)/S(25), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)(bx*S(2) + cxS(4))S(2), x), x, S(2)bS(2)x*(S(15)/2)/S(15) + S(4)bcx*(S(19)/2)/S(19) + S(2)c*S(2)x(S(23)/2)/S(23), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)(bx*S(2) + cxS(4))S(2), x), x, S(2)bS(2)x*(S(13)/2)/S(13) + S(4)bcx*(S(17)/2)/S(17) + S(2)c*S(2)x*(S(21)/2)/S(21), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)(bxS(2) + cxS(4))S(2), x), x, S(2)bS(2)x*(S(11)/2)/S(11) + S(4)bcx*(S(15)/2)/S(15) + S(2)c*S(2)x*(S(19)/2)/S(19), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/sqrt(x), x), x, S(2)bS(2)x*(S(9)/2)/S(9) + S(4)bcx*(S(13)/2)/S(13) + S(2)c*S(2)x*(S(17)/2)/S(17), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/x*(S(3)/2), x), x, S(2)b*S(2)x*(S(7)/2)/S(7) + S(4)bcx*(S(11)/2)/S(11) + S(2)c*S(2)x*(S(15)/2)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/x*(S(5)/2), x), x, S(2)b*S(2)x*(S(5)/2)/S(5) + S(4)bcx*(S(9)/2)/S(9) + S(2)c*S(2)x*(S(13)/2)/S(13), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(2)/x*(S(7)/2), x), x, S(2)b*S(2)x*(S(3)/2)/S(3) + S(4)bcx*(S(7)/2)/S(7) + S(2)c*S(2)x(S(11)/2)/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(7)/2)(bx*S(2) + cxS(4))S(3), x), x, S(2)bS(3)x*(S(21)/2)/S(21) + S(6)b*S(2)cx(S(25)/2)/S(25) + S(6)bcS(2)x*(S(29)/2)/S(29) + S(2)c*S(3)x(S(33)/2)/S(33), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)(bx*S(2) + cxS(4))S(3), x), x, S(2)bS(3)x*(S(19)/2)/S(19) + S(6)b*S(2)cx(S(23)/2)/S(23) + S(2)bcS(2)x*(S(27)/2)/S(9) + S(2)c*S(3)x(S(31)/2)/S(31), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)(bx*S(2) + cxS(4))S(3), x), x, S(2)bS(3)x*(S(17)/2)/S(17) + S(2)b*S(2)cx(S(21)/2)/S(7) + S(6)bcS(2)x*(S(25)/2)/S(25) + S(2)c*S(3)x*(S(29)/2)/S(29), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)(bxS(2) + cxS(4))S(3), x), x, S(2)bS(3)x*(S(15)/2)/S(15) + S(6)b*S(2)cx(S(19)/2)/S(19) + S(6)bcS(2)x*(S(23)/2)/S(23) + S(2)c*S(3)x*(S(27)/2)/S(27), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/sqrt(x), x), x, S(2)bS(3)x*(S(13)/2)/S(13) + S(6)b*S(2)cx(S(17)/2)/S(17) + S(2)bcS(2)x*(S(21)/2)/S(7) + S(2)c*S(3)x*(S(25)/2)/S(25), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/x*(S(3)/2), x), x, S(2)b*S(3)x*(S(11)/2)/S(11) + S(2)b*S(2)cx(S(15)/2)/S(5) + S(6)bcS(2)x*(S(19)/2)/S(19) + S(2)c*S(3)x*(S(23)/2)/S(23), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/x*(S(5)/2), x), x, S(2)b*S(3)x*(S(9)/2)/S(9) + S(6)b*S(2)cx(S(13)/2)/S(13) + S(6)bcS(2)x*(S(17)/2)/S(17) + S(2)c*S(3)x*(S(21)/2)/S(21), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))S(3)/x*(S(7)/2), x), x, S(2)b*S(3)x*(S(7)/2)/S(7) + S(6)b*S(2)cx(S(11)/2)/S(11) + S(2)bcS(2)x*(S(15)/2)/S(5) + S(2)c*S(3)x(S(19)/2)/S(19), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(13)/2)/(bxS(2) + cx*S(4)), x), x, sqrt(S(2))b*(S(7)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)c*(S(11)/4)) - sqrt(S(2))b*(S(7)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)c*(S(11)/4)) - sqrt(S(2))b*(S(7)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)c*(S(11)/4)) + sqrt(S(2))b*(S(7)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)c*(S(11)/4)) - S(2)bx(S(3)/2)/(S(3)c*S(2)) + S(2)x*(S(7)/2)/(S(7)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(11)/2)/(bx*S(2) + cx*S(4)), x), x, -sqrt(S(2))b*(S(5)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)c*(S(9)/4)) + sqrt(S(2))b*(S(5)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)c*(S(9)/4)) - sqrt(S(2))b*(S(5)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)c*(S(9)/4)) + sqrt(S(2))b*(S(5)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)c*(S(9)/4)) - S(2)bsqrt(x)/cS(2) + S(2)x*(S(5)/2)/(S(5)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(9)/2)/(bx*S(2) + cx*S(4)), x), x, -sqrt(S(2))b*(S(3)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)c*(S(7)/4)) + sqrt(S(2))b*(S(3)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)c*(S(7)/4)) + sqrt(S(2))b*(S(3)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)c*(S(7)/4)) - sqrt(S(2))b*(S(3)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)c*(S(7)/4)) + S(2)x*(S(3)/2)/(S(3)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(7)/2)/(bx*S(2) + cx*S(4)), x), x, sqrt(S(2))b*(S(1)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)c*(S(5)/4)) - sqrt(S(2))b*(S(1)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)c*(S(5)/4)) + sqrt(S(2))b*(S(1)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)c*(S(5)/4)) - sqrt(S(2))b*(S(1)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)c*(S(5)/4)) + S(2)sqrt(x)/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(5)/2)/(bx*S(2) + cx*S(4)), x), x, sqrt(S(2))log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(1)/4)c*(S(3)/4)) - sqrt(S(2))log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(1)/4)c*(S(3)/4)) - sqrt(S(2))atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b*(S(1)/4)c*(S(3)/4)) + sqrt(S(2))atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b*(S(1)/4)c(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)/(bxS(2) + cx*S(4)), x), x, -sqrt(S(2))log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(3)/4)c*(S(1)/4)) + sqrt(S(2))log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(3)/4)c*(S(1)/4)) - sqrt(S(2))atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b*(S(3)/4)c*(S(1)/4)) + sqrt(S(2))atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b*(S(3)/4)c*(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(bx*S(2) + cx*S(4)), x), x, -S(2)/(bsqrt(x)) - sqrt(S(2))c(S(1)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(5)/4)) + sqrt(S(2))c*(S(1)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(5)/4)) + sqrt(S(2))c*(S(1)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b*(S(5)/4)) - sqrt(S(2))c*(S(1)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b*(S(5)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)(bxS(2) + cx*S(4))), x), x, -S(2)/(S(3)bx(S(3)/2)) + sqrt(S(2))c*(S(3)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(7)/4)) - sqrt(S(2))c*(S(3)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(7)/4)) + sqrt(S(2))c*(S(3)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b*(S(7)/4)) - sqrt(S(2))c*(S(3)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b(S(7)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(3)/2)(bx*S(2) + cx*S(4))), x), x, -S(2)/(S(5)bx(S(5)/2)) + S(2)c/(b*S(2)sqrt(x)) + sqrt(S(2))c(S(5)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(9)/4)) - sqrt(S(2))c*(S(5)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(9)/4)) - sqrt(S(2))c*(S(5)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b*(S(9)/4)) + sqrt(S(2))c*(S(5)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b(S(9)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(5)/2)(bx*S(2) + cx*S(4))), x), x, -S(2)/(S(7)bx(S(7)/2)) + S(2)c/(S(3)bS(2)x*(S(3)/2)) - sqrt(S(2))c*(S(7)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(11)/4)) + sqrt(S(2))c*(S(7)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(11)/4)) - sqrt(S(2))c*(S(7)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b*(S(11)/4)) + sqrt(S(2))c*(S(7)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b(S(11)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(7)/2)(bx*S(2) + cx*S(4))), x), x, -S(2)/(S(9)bx(S(9)/2)) + S(2)c/(S(5)bS(2)x*(S(5)/2)) - S(2)cS(2)/(bS(3)sqrt(x)) - sqrt(S(2))c*(S(9)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(13)/4)) + sqrt(S(2))c*(S(9)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(4)b*(S(13)/4)) + sqrt(S(2))c*(S(9)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b*(S(13)/4)) - sqrt(S(2))c*(S(9)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(2)b(S(13)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(19)/2)/(bxS(2) + cxS(4))S(2), x), x, -S(9)sqrt(S(2))b*(S(5)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)c*(S(13)/4)) + S(9)sqrt(S(2))b(S(5)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)c*(S(13)/4)) - S(9)sqrt(S(2))b(S(5)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)c*(S(13)/4)) + S(9)sqrt(S(2))b(S(5)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)c*(S(13)/4)) - S(9)bsqrt(x)/(S(2)cS(3)) - x(S(9)/2)/(S(2)c(b + cxS(2))) + S(9)x*(S(5)/2)/(S(10)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(17)/2)/(bxS(2) + cxS(4))S(2), x), x, -S(7)sqrt(S(2))b*(S(3)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)c*(S(11)/4)) + S(7)sqrt(S(2))b(S(3)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)c*(S(11)/4)) + S(7)sqrt(S(2))b(S(3)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)c*(S(11)/4)) - S(7)sqrt(S(2))b(S(3)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)c(S(11)/4)) - x(S(7)/2)/(S(2)c(b + cxS(2))) + S(7)x*(S(3)/2)/(S(6)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(15)/2)/(bxS(2) + cxS(4))S(2), x), x, S(5)sqrt(S(2))b*(S(1)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)c*(S(9)/4)) - S(5)sqrt(S(2))b(S(1)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)c*(S(9)/4)) + S(5)sqrt(S(2))b(S(1)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)c*(S(9)/4)) - S(5)sqrt(S(2))b(S(1)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)c(S(9)/4)) - x(S(5)/2)/(S(2)c(b + cxS(2))) + S(5)sqrt(x)/(S(2)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(13)/2)/(bx*S(2) + cxS(4))S(2), x), x, -x*(S(3)/2)/(S(2)c(b + cx*S(2))) + S(3)sqrt(S(2))log(-sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(1)/4)c*(S(7)/4)) - S(3)sqrt(S(2))log(sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(1)/4)c*(S(7)/4)) - S(3)sqrt(S(2))atan(S(1) - sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(1)/4)c*(S(7)/4)) + S(3)sqrt(S(2))atan(S(1) + sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(1)/4)c(S(7)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(11)/2)/(bxS(2) + cxS(4))S(2), x), x, -sqrt(x)/(S(2)c(b + cxS(2))) - sqrt(S(2))log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(3)/4)c*(S(5)/4)) + sqrt(S(2))log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(3)/4)c*(S(5)/4)) - sqrt(S(2))atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(3)/4)c*(S(5)/4)) + sqrt(S(2))atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(3)/4)c(S(5)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(9)/2)/(bxS(2) + cxS(4))S(2), x), x, x*(S(3)/2)/(S(2)b(b + cx*S(2))) + sqrt(S(2))log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(5)/4)c*(S(3)/4)) - sqrt(S(2))log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(5)/4)c*(S(3)/4)) - sqrt(S(2))atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(5)/4)c*(S(3)/4)) + sqrt(S(2))atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(5)/4)c(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(7)/2)/(bxS(2) + cxS(4))S(2), x), x, sqrt(x)/(S(2)b(b + cxS(2))) - S(3)sqrt(S(2))log(-sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(7)/4)c*(S(1)/4)) + S(3)sqrt(S(2))log(sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(7)/4)c*(S(1)/4)) - S(3)sqrt(S(2))atan(S(1) - sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(7)/4)c*(S(1)/4)) + S(3)sqrt(S(2))atan(S(1) + sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(7)/4)c(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)/(bxS(2) + cxS(4))S(2), x), x, S(1)/(S(2)bsqrt(x)(b + cx*S(2))) - S(5)/(S(2)b*S(2)sqrt(x)) - S(5)sqrt(S(2))c*(S(1)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(9)/4)) + S(5)sqrt(S(2))c(S(1)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(9)/4)) + S(5)sqrt(S(2))c(S(1)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(9)/4)) - S(5)sqrt(S(2))c(S(1)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b(S(9)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)/(bxS(2) + cxS(4))S(2), x), x, S(1)/(S(2)bx*(S(3)/2)(b + cxS(2))) - S(7)/(S(6)b*S(2)x*(S(3)/2)) + S(7)sqrt(S(2))c(S(3)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(11)/4)) - S(7)sqrt(S(2))c(S(3)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(11)/4)) + S(7)sqrt(S(2))c(S(3)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(11)/4)) - S(7)sqrt(S(2))c(S(3)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(11)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(bx*S(2) + cxS(4))S(2), x), x, S(1)/(S(2)bx*(S(5)/2)(b + cxS(2))) - S(9)/(S(10)b*S(2)x*(S(5)/2)) + S(9)c/(S(2)bS(3)sqrt(x)) + S(9)sqrt(S(2))c*(S(5)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(13)/4)) - S(9)sqrt(S(2))c(S(5)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(13)/4)) - S(9)sqrt(S(2))c(S(5)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(13)/4)) + S(9)sqrt(S(2))c(S(5)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(13)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)(bxS(2) + cxS(4))S(2)), x), x, S(1)/(S(2)bx*(S(7)/2)(b + cxS(2))) - S(11)/(S(14)b*S(2)x*(S(7)/2)) + S(11)c/(S(6)bS(3)x*(S(3)/2)) - S(11)sqrt(S(2))c(S(7)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(15)/4)) + S(11)sqrt(S(2))c(S(7)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(15)/4)) - S(11)sqrt(S(2))c(S(7)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(15)/4)) + S(11)sqrt(S(2))c(S(7)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b(S(15)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(3)/2)(bx*S(2) + cxS(4))S(2)), x), x, S(1)/(S(2)bx*(S(9)/2)(b + cxS(2))) - S(13)/(S(18)b*S(2)x*(S(9)/2)) + S(13)c/(S(10)bS(3)x*(S(5)/2)) - S(13)c*S(2)/(S(2)b*S(4)sqrt(x)) - S(13)sqrt(S(2))c*(S(9)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(17)/4)) + S(13)sqrt(S(2))c(S(9)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(16)b*(S(17)/4)) + S(13)sqrt(S(2))c(S(9)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b*(S(17)/4)) - S(13)sqrt(S(2))c(S(9)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(8)b(S(17)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(23)/2)/(bxS(2) + cxS(4))S(3), x), x, S(45)sqrt(S(2))b*(S(1)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)c*(S(13)/4)) - S(45)sqrt(S(2))b(S(1)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)c*(S(13)/4)) + S(45)sqrt(S(2))b(S(1)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)c*(S(13)/4)) - S(45)sqrt(S(2))b(S(1)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)c(S(13)/4)) - x(S(9)/2)/(S(4)c(b + cxS(2))S(2)) - S(9)x*(S(5)/2)/(S(16)c*S(2)(b + cxS(2))) + S(45)sqrt(x)/(S(16)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(21)/2)/(bx*S(2) + cxS(4))S(3), x), x, -x*(S(7)/2)/(S(4)c(b + cxS(2))S(2)) - S(7)x(S(3)/2)/(S(16)c*S(2)(b + cxS(2))) + S(21)sqrt(S(2))log(-sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(1)/4)c*(S(11)/4)) - S(21)sqrt(S(2))log(sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(1)/4)c*(S(11)/4)) - S(21)sqrt(S(2))atan(S(1) - sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(1)/4)c*(S(11)/4)) + S(21)sqrt(S(2))atan(S(1) + sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(1)/4)c(S(11)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(19)/2)/(bxS(2) + cxS(4))S(3), x), x, -x*(S(5)/2)/(S(4)c(b + cxS(2))S(2)) - S(5)sqrt(x)/(S(16)c*S(2)(b + cxS(2))) - S(5)sqrt(S(2))log(-sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(3)/4)c*(S(9)/4)) + S(5)sqrt(S(2))log(sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(3)/4)c*(S(9)/4)) - S(5)sqrt(S(2))atan(S(1) - sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(3)/4)c*(S(9)/4)) + S(5)sqrt(S(2))atan(S(1) + sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(3)/4)c(S(9)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(17)/2)/(bxS(2) + cxS(4))S(3), x), x, -x*(S(3)/2)/(S(4)c(b + cxS(2))S(2)) + S(3)x(S(3)/2)/(S(16)bc(b + cxS(2))) + S(3)sqrt(S(2))log(-sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(5)/4)c*(S(7)/4)) - S(3)sqrt(S(2))log(sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(5)/4)c*(S(7)/4)) - S(3)sqrt(S(2))atan(S(1) - sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(5)/4)c*(S(7)/4)) + S(3)sqrt(S(2))atan(S(1) + sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(5)/4)c(S(7)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(15)/2)/(bxS(2) + cxS(4))S(3), x), x, -sqrt(x)/(S(4)c(b + cxS(2))S(2)) + sqrt(x)/(S(16)bc(b + cxS(2))) - S(3)sqrt(S(2))log(-sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(7)/4)c*(S(5)/4)) + S(3)sqrt(S(2))log(sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(7)/4)c*(S(5)/4)) - S(3)sqrt(S(2))atan(S(1) - sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(7)/4)c*(S(5)/4)) + S(3)sqrt(S(2))atan(S(1) + sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(7)/4)c(S(5)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(13)/2)/(bxS(2) + cxS(4))S(3), x), x, x*(S(3)/2)/(S(4)b(b + cxS(2))S(2)) + S(5)x(S(3)/2)/(S(16)b*S(2)(b + cxS(2))) + S(5)sqrt(S(2))log(-sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(9)/4)c*(S(3)/4)) - S(5)sqrt(S(2))log(sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(9)/4)c*(S(3)/4)) - S(5)sqrt(S(2))atan(S(1) - sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(9)/4)c*(S(3)/4)) + S(5)sqrt(S(2))atan(S(1) + sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(9)/4)c(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(11)/2)/(bxS(2) + cxS(4))S(3), x), x, sqrt(x)/(S(4)b(b + cxS(2))S(2)) + S(7)sqrt(x)/(S(16)bS(2)(b + cxS(2))) - S(21)sqrt(S(2))log(-sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(11)/4)c*(S(1)/4)) + S(21)sqrt(S(2))log(sqrt(S(2))b*(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(11)/4)c*(S(1)/4)) - S(21)sqrt(S(2))atan(S(1) - sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(11)/4)c*(S(1)/4)) + S(21)sqrt(S(2))atan(S(1) + sqrt(S(2))c*(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(11)/4)c(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(9)/2)/(bxS(2) + cxS(4))S(3), x), x, S(1)/(S(4)bsqrt(x)(b + cxS(2))S(2)) + S(9)/(S(16)bS(2)sqrt(x)(b + cx*S(2))) - S(45)/(S(16)b*S(3)sqrt(x)) - S(45)sqrt(S(2))c*(S(1)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(13)/4)) + S(45)sqrt(S(2))c(S(1)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(13)/4)) + S(45)sqrt(S(2))c(S(1)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(13)/4)) - S(45)sqrt(S(2))c(S(1)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b(S(13)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(7)/2)/(bxS(2) + cxS(4))S(3), x), x, S(1)/(S(4)bx*(S(3)/2)(b + cxS(2))S(2)) + S(11)/(S(16)b*S(2)x*(S(3)/2)(b + cxS(2))) - S(77)/(S(48)b*S(3)x*(S(3)/2)) + S(77)sqrt(S(2))c(S(3)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(15)/4)) - S(77)sqrt(S(2))c(S(3)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(15)/4)) + S(77)sqrt(S(2))c(S(3)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(15)/4)) - S(77)sqrt(S(2))c(S(3)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b(S(15)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)/(bxS(2) + cxS(4))S(3), x), x, S(1)/(S(4)bx*(S(5)/2)(b + cxS(2))S(2)) + S(13)/(S(16)b*S(2)x*(S(5)/2)(b + cxS(2))) - S(117)/(S(80)b*S(3)x*(S(5)/2)) + S(117)c/(S(16)bS(4)sqrt(x)) + S(117)sqrt(S(2))c*(S(5)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(17)/4)) - S(117)sqrt(S(2))c(S(5)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(17)/4)) - S(117)sqrt(S(2))c(S(5)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(17)/4)) + S(117)sqrt(S(2))c(S(5)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b(S(17)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)/(bxS(2) + cxS(4))S(3), x), x, S(1)/(S(4)bx*(S(7)/2)(b + cxS(2))S(2)) + S(15)/(S(16)b*S(2)x*(S(7)/2)(b + cxS(2))) - S(165)/(S(112)b*S(3)x*(S(7)/2)) + S(55)c/(S(16)bS(4)x*(S(3)/2)) - S(165)sqrt(S(2))c(S(7)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(19)/4)) + S(165)sqrt(S(2))c(S(7)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(19)/4)) - S(165)sqrt(S(2))c(S(7)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(19)/4)) + S(165)sqrt(S(2))c(S(7)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(19)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(bx*S(2) + cxS(4))S(3), x), x, S(1)/(S(4)bx*(S(9)/2)(b + cxS(2))S(2)) + S(17)/(S(16)b*S(2)x*(S(9)/2)(b + cxS(2))) - S(221)/(S(144)b*S(3)x*(S(9)/2)) + S(221)c/(S(80)bS(4)x*(S(5)/2)) - S(221)c*S(2)/(S(16)b*S(5)sqrt(x)) - S(221)sqrt(S(2))c*(S(9)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(21)/4)) + S(221)sqrt(S(2))c(S(9)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(21)/4)) + S(221)sqrt(S(2))c(S(9)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(21)/4)) - S(221)sqrt(S(2))c(S(9)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(21)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)(bxS(2) + cxS(4))S(3)), x), x, S(1)/(S(4)bx*(S(11)/2)(b + cxS(2))S(2)) + S(19)/(S(16)b*S(2)x*(S(11)/2)(b + cxS(2))) - S(285)/(S(176)b*S(3)x*(S(11)/2)) + S(285)c/(S(112)bS(4)x*(S(7)/2)) - S(95)c*S(2)/(S(16)b*S(5)x*(S(3)/2)) + S(285)sqrt(S(2))c(S(11)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(23)/4)) - S(285)sqrt(S(2))c(S(11)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)sqrt(x) + sqrt(b) + sqrt(c)x)/(S(128)b*(S(23)/4)) + S(285)sqrt(S(2))c(S(11)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b*(S(23)/4)) - S(285)sqrt(S(2))c(S(11)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)sqrt(x)/b*(S(1)/4))/(S(64)b(S(23)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(7)/2)sqrt(bx*S(2) + cx*S(4)), x), x, -S(28)b*(S(13)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(195)c*(S(11)/4)sqrt(bxS(2) + cx*S(4))) + S(14)b*(S(13)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(195)c*(S(11)/4)sqrt(bxS(2) + cx*S(4))) + S(28)b*S(3)x*(S(3)/2)(b + cxS(2))/(S(195)c*(S(5)/2)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) - S(28)b*S(2)sqrt(x)sqrt(bx*S(2) + cx*S(4))/(S(585)c*S(2)) + S(4)bx(S(5)/2)sqrt(bxS(2) + cx*S(4))/(S(117)c) + S(2)x(S(9)/2)sqrt(bxS(2) + cxS(4))/S(13), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)sqrt(bx*S(2) + cx*S(4)), x), x, S(10)b*(S(11)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(231)c*(S(9)/4)sqrt(bxS(2) + cx*S(4))) - S(20)b*S(2)sqrt(bxS(2) + cx*S(4))/(S(231)c*S(2)sqrt(x)) + S(4)bx*(S(3)/2)sqrt(bxS(2) + cx*S(4))/(S(77)c) + S(2)x(S(7)/2)sqrt(bxS(2) + cxS(4))/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)sqrt(bx*S(2) + cx*S(4)), x), x, S(4)b*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)c*(S(7)/4)sqrt(bxS(2) + cx*S(4))) - S(2)b*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)c*(S(7)/4)sqrt(bxS(2) + cx*S(4))) - S(4)b*S(2)x*(S(3)/2)(b + cxS(2))/(S(15)c*(S(3)/2)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) + S(4)bsqrt(x)sqrt(bxS(2) + cx*S(4))/(S(45)c) + S(2)x(S(5)/2)sqrt(bxS(2) + cx*S(4))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)sqrt(bxS(2) + cx*S(4)), x), x, -S(2)b*(S(7)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(21)c*(S(5)/4)sqrt(bxS(2) + cx*S(4))) + S(4)bsqrt(bx*S(2) + cx*S(4))/(S(21)csqrt(x)) + S(2)x*(S(3)/2)sqrt(bxS(2) + cx*S(4))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cx*S(4))/sqrt(x), x), x, -S(4)b*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)c*(S(3)/4)sqrt(bxS(2) + cx*S(4))) + S(2)b*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)c*(S(3)/4)sqrt(bxS(2) + cx*S(4))) + S(4)bx(S(3)/2)(b + cxS(2))/(S(5)sqrt(c)(sqrt(b) + sqrt(c)x)sqrt(bx*S(2) + cx*S(4))) + S(2)sqrt(x)sqrt(bx*S(2) + cx*S(4))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/x(S(3)/2), x), x, S(2)b(S(3)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(3)c*(S(1)/4)sqrt(bxS(2) + cx*S(4))) + S(2)sqrt(bxS(2) + cx*S(4))/(S(3)sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bxS(2) + cxS(4))/x(S(5)/2), x), x, -S(4)b(S(1)/4)c*(S(1)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/sqrt(bx*S(2) + cx*S(4)) + S(2)b*(S(1)/4)c*(S(1)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/sqrt(bx*S(2) + cx*S(4)) + S(4)sqrt(c)x(S(3)/2)(b + cxS(2))/((sqrt(b) + sqrt(c)x)sqrt(bx*S(2) + cx*S(4))) - S(2)sqrt(bxS(2) + cxS(4))/x(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bxS(2) + cxS(4))/x(S(7)/2), x), x, -S(2)sqrt(bx*S(2) + cx*S(4))/(S(3)x*(S(5)/2)) + S(2)c*(S(3)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(3)b*(S(1)/4)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/x(S(9)/2), x), x, -S(2)sqrt(bx*S(2) + cx*S(4))/(S(5)x*(S(7)/2)) + S(4)c*(S(3)/2)x*(S(3)/2)(b + cxS(2))/(S(5)b(sqrt(b) + sqrt(c)x)sqrt(bx*S(2) + cx*S(4))) - S(4)csqrt(bx*S(2) + cx*S(4))/(S(5)bx(S(3)/2)) - S(4)c*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)b*(S(3)/4)sqrt(bxS(2) + cx*S(4))) + S(2)c*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)b*(S(3)/4)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/x(S(11)/2), x), x, -S(2)sqrt(bx*S(2) + cx*S(4))/(S(7)x*(S(9)/2)) - S(4)csqrt(bx*S(2) + cx*S(4))/(S(21)bx(S(5)/2)) - S(2)c*(S(7)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(21)b*(S(5)/4)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/x(S(13)/2), x), x, -S(2)sqrt(bx*S(2) + cx*S(4))/(S(9)x*(S(11)/2)) - S(4)csqrt(bx*S(2) + cx*S(4))/(S(45)bx(S(7)/2)) - S(4)c*(S(5)/2)x*(S(3)/2)(b + cxS(2))/(S(15)b*S(2)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) + S(4)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(15)b*S(2)x*(S(3)/2)) + S(4)c*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)b*(S(7)/4)sqrt(bxS(2) + cx*S(4))) - S(2)c*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)b*(S(7)/4)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx*S(2) + cxS(4))/x(S(15)/2), x), x, -S(2)sqrt(bx*S(2) + cx*S(4))/(S(11)x*(S(13)/2)) - S(4)csqrt(bx*S(2) + cx*S(4))/(S(77)bx(S(9)/2)) + S(20)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(231)b*S(2)x*(S(5)/2)) + S(10)c*(S(11)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(231)b*(S(9)/4)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)(bx*S(2) + cxS(4))(S(3)/2), x), x, -S(56)b(S(17)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(1105)c*(S(11)/4)sqrt(bxS(2) + cx*S(4))) + S(28)b*(S(17)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(1105)c*(S(11)/4)sqrt(bxS(2) + cx*S(4))) + S(56)b*S(4)x*(S(3)/2)(b + cxS(2))/(S(1105)c*(S(5)/2)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) - S(56)b*S(3)sqrt(x)sqrt(bx*S(2) + cx*S(4))/(S(3315)c*S(2)) + S(8)b*S(2)x*(S(5)/2)sqrt(bxS(2) + cx*S(4))/(S(663)c) + S(12)bx*(S(9)/2)sqrt(bxS(2) + cx*S(4))/S(221) + S(2)x*(S(5)/2)(bxS(2) + cxS(4))(S(3)/2)/S(17), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)(bx*S(2) + cxS(4))(S(3)/2), x), x, S(4)b(S(15)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(231)c*(S(9)/4)sqrt(bxS(2) + cx*S(4))) - S(8)b*S(3)sqrt(bxS(2) + cx*S(4))/(S(231)c*S(2)sqrt(x)) + S(8)bS(2)x*(S(3)/2)sqrt(bxS(2) + cx*S(4))/(S(385)c) + S(4)bx*(S(7)/2)sqrt(bxS(2) + cx*S(4))/S(55) + S(2)x*(S(3)/2)(bxS(2) + cxS(4))(S(3)/2)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + cxS(4))(S(3)/2)/sqrt(x), x), x, S(8)b(S(13)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(65)c*(S(7)/4)sqrt(bxS(2) + cx*S(4))) - S(4)b*(S(13)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(65)c*(S(7)/4)sqrt(bxS(2) + cx*S(4))) - S(8)b*S(3)x*(S(3)/2)(b + cxS(2))/(S(65)c*(S(3)/2)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) + S(8)b*S(2)sqrt(x)sqrt(bx*S(2) + cx*S(4))/(S(195)c) + S(4)bx*(S(5)/2)sqrt(bxS(2) + cx*S(4))/S(39) + S(2)sqrt(x)(bx*S(2) + cxS(4))(S(3)/2)/S(13), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + cxS(4))(S(3)/2)/x*(S(3)/2), x), x, -S(4)b*(S(11)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(77)c*(S(5)/4)sqrt(bxS(2) + cx*S(4))) + S(8)b*S(2)sqrt(bxS(2) + cx*S(4))/(S(77)csqrt(x)) + S(12)bx(S(3)/2)sqrt(bxS(2) + cx*S(4))/S(77) + S(2)(bxS(2) + cxS(4))(S(3)/2)/(S(11)sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*(S(5)/2), x), x, -S(8)b*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)c*(S(3)/4)sqrt(bxS(2) + cx*S(4))) + S(4)b*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)c*(S(3)/4)sqrt(bxS(2) + cx*S(4))) + S(8)b*S(2)x*(S(3)/2)(b + cxS(2))/(S(15)sqrt(c)(sqrt(b) + sqrt(c)x)sqrt(bx*S(2) + cx*S(4))) + S(4)bsqrt(x)sqrt(bxS(2) + cx*S(4))/S(15) + S(2)(bxS(2) + cxS(4))(S(3)/2)/(S(9)x(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*(S(7)/2), x), x, S(4)b*(S(7)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(7)c*(S(1)/4)sqrt(bxS(2) + cx*S(4))) + S(4)bsqrt(bx*S(2) + cx*S(4))/(S(7)sqrt(x)) + S(2)(bx*S(2) + cxS(4))(S(3)/2)/(S(7)x(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*(S(9)/2), x), x, -S(24)b*(S(5)/4)c*(S(1)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)sqrt(bxS(2) + cx*S(4))) + S(12)b*(S(5)/4)c*(S(1)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)sqrt(bxS(2) + cx*S(4))) + S(24)bsqrt(c)x*(S(3)/2)(b + cxS(2))/((S(5)sqrt(b) + S(5)sqrt(c)x)sqrt(bx*S(2) + cx*S(4))) + S(12)csqrt(x)sqrt(bxS(2) + cx*S(4))/S(5) - S(2)(bxS(2) + cxS(4))(S(3)/2)/x*(S(7)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*(S(11)/2), x), x, S(4)b*(S(3)/4)c*(S(3)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(3)sqrt(bxS(2) + cx*S(4))) + S(4)csqrt(bx*S(2) + cx*S(4))/(S(3)sqrt(x)) - S(2)(bx*S(2) + cxS(4))(S(3)/2)/(S(3)x(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*(S(13)/2), x), x, -S(24)b*(S(1)/4)c*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)sqrt(bxS(2) + cx*S(4))) + S(12)b*(S(1)/4)c*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)sqrt(bxS(2) + cx*S(4))) + S(24)c*(S(3)/2)x*(S(3)/2)(b + cxS(2))/((S(5)sqrt(b) + S(5)sqrt(c)x)sqrt(bx*S(2) + cx*S(4))) - S(12)csqrt(bx*S(2) + cx*S(4))/(S(5)x*(S(3)/2)) - S(2)(bxS(2) + cxS(4))(S(3)/2)/(S(5)x(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*(S(15)/2), x), x, -S(4)csqrt(bx*S(2) + cx*S(4))/(S(7)x*(S(5)/2)) - S(2)(bxS(2) + cxS(4))(S(3)/2)/(S(7)x(S(13)/2)) + S(4)c*(S(7)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(7)b*(S(1)/4)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*(S(17)/2), x), x, -S(4)csqrt(bx*S(2) + cx*S(4))/(S(15)x*(S(7)/2)) - S(2)(bxS(2) + cxS(4))(S(3)/2)/(S(9)x(S(15)/2)) + S(8)c*(S(5)/2)x*(S(3)/2)(b + cxS(2))/(S(15)b(sqrt(b) + sqrt(c)x)sqrt(bx*S(2) + cx*S(4))) - S(8)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(15)bx(S(3)/2)) - S(8)c*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)b*(S(3)/4)sqrt(bxS(2) + cx*S(4))) + S(4)c*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)b*(S(3)/4)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*(S(19)/2), x), x, -S(12)csqrt(bx*S(2) + cx*S(4))/(S(77)x*(S(9)/2)) - S(2)(bxS(2) + cxS(4))(S(3)/2)/(S(11)x(S(17)/2)) - S(8)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(77)bx(S(5)/2)) - S(4)c*(S(11)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(77)b*(S(5)/4)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*(S(21)/2), x), x, -S(4)csqrt(bx*S(2) + cx*S(4))/(S(39)x*(S(11)/2)) - S(2)(bxS(2) + cxS(4))(S(3)/2)/(S(13)x(S(19)/2)) - S(8)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(195)bx(S(7)/2)) - S(8)c*(S(7)/2)x*(S(3)/2)(b + cxS(2))/(S(65)b*S(2)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) + S(8)c*S(3)sqrt(bxS(2) + cx*S(4))/(S(65)b*S(2)x*(S(3)/2)) + S(8)c*(S(13)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(65)b*(S(7)/4)sqrt(bxS(2) + cx*S(4))) - S(4)c*(S(13)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(65)b*(S(7)/4)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx*S(2) + cxS(4))(S(3)/2)/x*(S(23)/2), x), x, -S(4)csqrt(bx*S(2) + cx*S(4))/(S(55)x*(S(13)/2)) - S(2)(bxS(2) + cxS(4))(S(3)/2)/(S(15)x(S(21)/2)) - S(8)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(385)bx(S(9)/2)) + S(8)c*S(3)sqrt(bxS(2) + cx*S(4))/(S(231)b*S(2)x*(S(5)/2)) + S(4)c*(S(15)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(231)b*(S(9)/4)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(13)/2)/sqrt(bxS(2) + cx*S(4)), x), x, -S(15)b*(S(11)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(77)c*(S(13)/4)sqrt(bxS(2) + cx*S(4))) + S(30)b*S(2)sqrt(bxS(2) + cx*S(4))/(S(77)c*S(3)sqrt(x)) - S(18)bx*(S(3)/2)sqrt(bxS(2) + cx*S(4))/(S(77)c*S(2)) + S(2)x*(S(7)/2)sqrt(bxS(2) + cx*S(4))/(S(11)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(11)/2)/sqrt(bx*S(2) + cx*S(4)), x), x, -S(14)b*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)c*(S(11)/4)sqrt(bxS(2) + cx*S(4))) + S(7)b*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)c*(S(11)/4)sqrt(bxS(2) + cx*S(4))) + S(14)b*S(2)x*(S(3)/2)(b + cxS(2))/(S(15)c*(S(5)/2)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) - S(14)bsqrt(x)sqrt(bxS(2) + cx*S(4))/(S(45)c*S(2)) + S(2)x*(S(5)/2)sqrt(bxS(2) + cx*S(4))/(S(9)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(9)/2)/sqrt(bx*S(2) + cx*S(4)), x), x, S(5)b*(S(7)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(21)c*(S(9)/4)sqrt(bxS(2) + cx*S(4))) - S(10)bsqrt(bx*S(2) + cx*S(4))/(S(21)c*S(2)sqrt(x)) + S(2)x(S(3)/2)sqrt(bxS(2) + cx*S(4))/(S(7)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(7)/2)/sqrt(bx*S(2) + cx*S(4)), x), x, S(6)b*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)c*(S(7)/4)sqrt(bxS(2) + cx*S(4))) - S(3)b*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)c*(S(7)/4)sqrt(bxS(2) + cx*S(4))) - S(6)bx(S(3)/2)(b + cxS(2))/(S(5)c*(S(3)/2)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) + S(2)sqrt(x)sqrt(bx*S(2) + cx*S(4))/(S(5)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(5)/2)/sqrt(bx*S(2) + cxS(4)), x), x, -b(S(3)/4)xsqrt((b + cxS(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(3)c*(S(5)/4)sqrt(bxS(2) + cx*S(4))) + S(2)sqrt(bxS(2) + cx*S(4))/(S(3)csqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)/sqrt(bx*S(2) + cx*S(4)), x), x, -S(2)b*(S(1)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b(S(1)/4)), S(1)/2)/(c(S(3)/4)sqrt(bx*S(2) + cxS(4))) + b(S(1)/4)xsqrt((b + cxS(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b(S(1)/4)), S(1)/2)/(c(S(3)/4)sqrt(bx*S(2) + cx*S(4))) + S(2)x*(S(3)/2)(b + cxS(2))/(sqrt(c)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/sqrt(bx*S(2) + cx*S(4)), x), x, xsqrt((b + cxS(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b(S(1)/4)), S(1)/2)/(b(S(1)/4)c(S(1)/4)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)sqrt(bxS(2) + cx*S(4))), x), x, S(2)sqrt(c)x(S(3)/2)(b + cxS(2))/(b(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) - S(2)sqrt(bxS(2) + cx*S(4))/(bx*(S(3)/2)) - S(2)c*(S(1)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b(S(1)/4)), S(1)/2)/(b(S(3)/4)sqrt(bx*S(2) + cxS(4))) + c(S(1)/4)xsqrt((b + cxS(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b(S(1)/4)), S(1)/2)/(b(S(3)/4)sqrt(bx*S(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(3)/2)sqrt(bx*S(2) + cx*S(4))), x), x, -S(2)sqrt(bxS(2) + cx*S(4))/(S(3)bx(S(5)/2)) - c(S(3)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(3)b*(S(5)/4)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(5)/2)sqrt(bx*S(2) + cx*S(4))), x), x, -S(2)sqrt(bxS(2) + cx*S(4))/(S(5)bx(S(7)/2)) - S(6)c*(S(3)/2)x*(S(3)/2)(b + cxS(2))/(S(5)b*S(2)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) + S(6)csqrt(bx*S(2) + cx*S(4))/(S(5)b*S(2)x*(S(3)/2)) + S(6)c*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)b*(S(7)/4)sqrt(bxS(2) + cx*S(4))) - S(3)c*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)b*(S(7)/4)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(7)/2)sqrt(bx*S(2) + cx*S(4))), x), x, -S(2)sqrt(bxS(2) + cx*S(4))/(S(7)bx(S(9)/2)) + S(10)csqrt(bx*S(2) + cx*S(4))/(S(21)b*S(2)x*(S(5)/2)) + S(5)c*(S(7)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(21)b*(S(9)/4)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(9)/2)sqrt(bx*S(2) + cx*S(4))), x), x, -S(2)sqrt(bxS(2) + cx*S(4))/(S(9)bx(S(11)/2)) + S(14)csqrt(bx*S(2) + cx*S(4))/(S(45)b*S(2)x*(S(7)/2)) + S(14)c*(S(5)/2)x*(S(3)/2)(b + cxS(2))/(S(15)b*S(3)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) - S(14)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(15)b*S(3)x*(S(3)/2)) - S(14)c*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)b*(S(11)/4)sqrt(bxS(2) + cx*S(4))) + S(7)c*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)b*(S(11)/4)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(11)/2)sqrt(bx*S(2) + cx*S(4))), x), x, -S(2)sqrt(bxS(2) + cx*S(4))/(S(11)bx(S(13)/2)) + S(18)csqrt(bx*S(2) + cx*S(4))/(S(77)b*S(2)x*(S(9)/2)) - S(30)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(77)b*S(3)x*(S(5)/2)) - S(15)c*(S(11)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(77)b*(S(13)/4)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(17)/2)/(bxS(2) + cxS(4))(S(3)/2), x), x, S(15)b(S(7)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(14)c*(S(13)/4)sqrt(bxS(2) + cx*S(4))) - S(15)bsqrt(bx*S(2) + cx*S(4))/(S(7)c*S(3)sqrt(x)) - x*(S(11)/2)/(csqrt(bxS(2) + cx*S(4))) + S(9)x*(S(3)/2)sqrt(bxS(2) + cx*S(4))/(S(7)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(15)/2)/(bxS(2) + cxS(4))(S(3)/2), x), x, S(21)b(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)c*(S(11)/4)sqrt(bxS(2) + cx*S(4))) - S(21)b*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(10)c*(S(11)/4)sqrt(bxS(2) + cx*S(4))) - S(21)bx(S(3)/2)(b + cxS(2))/(S(5)c*(S(5)/2)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cxS(4))) - x(S(9)/2)/(csqrt(bx*S(2) + cx*S(4))) + S(7)sqrt(x)sqrt(bx*S(2) + cx*S(4))/(S(5)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(13)/2)/(bxS(2) + cxS(4))(S(3)/2), x), x, -S(5)b(S(3)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(6)c*(S(9)/4)sqrt(bxS(2) + cxS(4))) - x(S(7)/2)/(csqrt(bx*S(2) + cx*S(4))) + S(5)sqrt(bxS(2) + cx*S(4))/(S(3)c*S(2)sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(11)/2)/(bx*S(2) + cxS(4))(S(3)/2), x), x, -S(3)b(S(1)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b(S(1)/4)), S(1)/2)/(c(S(7)/4)sqrt(bx*S(2) + cx*S(4))) + S(3)b*(S(1)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(2)c*(S(7)/4)sqrt(bxS(2) + cxS(4))) - x(S(5)/2)/(csqrt(bx*S(2) + cx*S(4))) + S(3)x*(S(3)/2)(b + cxS(2))/(c(S(3)/2)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(9)/2)/(bxS(2) + cxS(4))(S(3)/2), x), x, -x*(S(3)/2)/(csqrt(bxS(2) + cx*S(4))) + xsqrt((b + cxS(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(2)b*(S(1)/4)c*(S(5)/4)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(7)/2)/(bxS(2) + cxS(4))(S(3)/2), x), x, x*(S(5)/2)/(bsqrt(bxS(2) + cxS(4))) - x(S(3)/2)(b + cx*S(2))/(bsqrt(c)(sqrt(b) + sqrt(c)x)sqrt(bx*S(2) + cx*S(4))) + xsqrt((b + cxS(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b(S(1)/4)), S(1)/2)/(b(S(3)/4)c(S(3)/4)sqrt(bxS(2) + cx*S(4))) - xsqrt((b + cxS(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(2)b*(S(3)/4)c*(S(3)/4)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)/(bxS(2) + cxS(4))(S(3)/2), x), x, x*(S(3)/2)/(bsqrt(bxS(2) + cx*S(4))) + xsqrt((b + cxS(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(2)b*(S(5)/4)c*(S(1)/4)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)/(bxS(2) + cxS(4))(S(3)/2), x), x, sqrt(x)/(bsqrt(bx*S(2) + cx*S(4))) + S(3)sqrt(c)x(S(3)/2)(b + cxS(2))/(bS(2)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) - S(3)sqrt(bxS(2) + cxS(4))/(bS(2)x(S(3)/2)) - S(3)c*(S(1)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b(S(1)/4)), S(1)/2)/(b(S(7)/4)sqrt(bx*S(2) + cx*S(4))) + S(3)c*(S(1)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(2)b*(S(7)/4)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(bx*S(2) + cxS(4))(S(3)/2), x), x, S(1)/(bsqrt(x)sqrt(bxS(2) + cx*S(4))) - S(5)sqrt(bxS(2) + cx*S(4))/(S(3)b*S(2)x*(S(5)/2)) - S(5)c*(S(3)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(6)b*(S(9)/4)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)(bxS(2) + cxS(4))(S(3)/2)), x), x, S(1)/(bx(S(3)/2)sqrt(bxS(2) + cx*S(4))) - S(7)sqrt(bxS(2) + cx*S(4))/(S(5)b*S(2)x*(S(7)/2)) - S(21)c*(S(3)/2)x*(S(3)/2)(b + cxS(2))/(S(5)b*S(3)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) + S(21)csqrt(bx*S(2) + cx*S(4))/(S(5)b*S(3)x*(S(3)/2)) + S(21)c*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(5)b*(S(11)/4)sqrt(bxS(2) + cx*S(4))) - S(21)c*(S(5)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(10)b*(S(11)/4)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(3)/2)(bx*S(2) + cxS(4))(S(3)/2)), x), x, S(1)/(bx(S(5)/2)sqrt(bxS(2) + cx*S(4))) - S(9)sqrt(bxS(2) + cx*S(4))/(S(7)b*S(2)x*(S(9)/2)) + S(15)csqrt(bx*S(2) + cx*S(4))/(S(7)b*S(3)x*(S(5)/2)) + S(15)c*(S(7)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(14)b*(S(13)/4)sqrt(bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(5)/2)(bx*S(2) + cxS(4))(S(3)/2)), x), x, S(1)/(bx(S(7)/2)sqrt(bxS(2) + cx*S(4))) - S(11)sqrt(bxS(2) + cx*S(4))/(S(9)b*S(2)x*(S(11)/2)) + S(77)csqrt(bx*S(2) + cx*S(4))/(S(45)b*S(3)x*(S(7)/2)) + S(77)c*(S(5)/2)x*(S(3)/2)(b + cxS(2))/(S(15)b*S(4)(sqrt(b) + sqrt(c)x)sqrt(bxS(2) + cx*S(4))) - S(77)c*S(2)sqrt(bxS(2) + cx*S(4))/(S(15)b*S(4)x*(S(3)/2)) - S(77)c*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_e(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(15)b*(S(15)/4)sqrt(bxS(2) + cx*S(4))) + S(77)c*(S(9)/4)xsqrt((b + cx*S(2))/(sqrt(b) + sqrt(c)x)*S(2))(sqrt(b) + sqrt(c)x)elliptic_f(S(2)atan(c(S(1)/4)sqrt(x)/b*(S(1)/4)), S(1)/2)/(S(30)b*(S(15)/4)sqrt(bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, aS(2)(dx)(m + S(1))/(d(m + S(1))) + S(2)ab(dx)(m + S(3))/(dS(3)(m + S(3))) + bS(2)(dx)(m + S(5))/(dS(5)(m + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, aS(2)x*S(4)/S(4) + abxS(6)/S(3) + bS(2)xS(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, aS(2)x*S(3)/S(3) + S(2)abxS(5)/S(5) + bS(2)xS(7)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, aS(2)x*S(2)/S(2) + abxS(4)/S(2) + bS(2)xS(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(aS(2) + S(2)abxS(2) + bS(2)xS(4), x), x, aS(2)x + S(2)abxS(3)/S(3) + bS(2)xS(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))/x, x), x, aS(2)log(x) + abxS(2) + bS(2)xS(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(2), x), x, -aS(2)/x + S(2)abx + b*S(2)xS(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(3), x), x, -a*S(2)/(S(2)x*S(2)) + S(2)ablog(x) + b*S(2)xS(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(4), x), x, -a*S(2)/(S(3)x*S(3)) - S(2)ab/x + bS(2)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))/xS(5), x), x, -aS(2)/(S(4)x*S(4)) - ab/xS(2) + bS(2)log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(6), x), x, -aS(2)/(S(5)x*S(5)) - S(2)ab/(S(3)xS(3)) - bS(2)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))/xS(7), x), x, -aS(2)/(S(6)x*S(6)) - ab/(S(2)xS(4)) - bS(2)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(8), x), x, -a*S(2)/(S(7)x*S(7)) - S(2)ab/(S(5)xS(5)) - bS(2)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, aS(4)(dx)(m + S(1))/(d(m + S(1))) + S(4)aS(3)b(dx)(m + S(3))/(dS(3)(m + S(3))) + S(6)a*S(2)b*S(2)(dx)(m + S(5))/(dS(5)(m + S(5))) + S(4)ab*S(3)(dx)(m + S(7))/(dS(7)(m + S(7))) + b*S(4)(dx)(m + S(9))/(dS(9)(m + S(9))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(6)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, aS(4)x*S(7)/S(7) + S(4)a*S(3)bxS(9)/S(9) + S(6)a*S(2)b*S(2)x*S(11)/S(11) + S(4)abS(3)xS(13)/S(13) + bS(4)xS(15)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, aS(4)xS(6)/S(6) + aS(3)bx*S(8)/S(2) + S(3)a*S(2)b*S(2)x*S(10)/S(5) + ab*S(3)xS(12)/S(3) + bS(4)xS(14)/S(14), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, aS(4)x*S(5)/S(5) + S(4)a*S(3)bxS(7)/S(7) + S(2)a*S(2)b*S(2)x*S(9)/S(3) + S(4)abS(3)xS(11)/S(11) + bS(4)xS(13)/S(13), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -a(a + bxS(2))S(5)/(S(10)b*S(2)) + (a + bxS(2))S(6)/(S(12)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, aS(4)x*S(3)/S(3) + S(4)a*S(3)bxS(5)/S(5) + S(6)a*S(2)b*S(2)x*S(7)/S(7) + S(4)abS(3)xS(9)/S(9) + bS(4)xS(11)/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, (a + bxS(2))S(5)/(S(10)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, aS(4)x + S(4)aS(3)bxS(3)/S(3) + S(6)a*S(2)b*S(2)x*S(5)/S(5) + S(4)abS(3)xS(7)/S(7) + bS(4)xS(9)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/x, x), x, aS(4)log(x) + S(2)aS(3)bxS(2) + S(3)a*S(2)b*S(2)x*S(4)/S(2) + S(2)abS(3)xS(6)/S(3) + bS(4)xS(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(2), x), x, -aS(4)/x + S(4)a*S(3)bx + S(2)a*S(2)b*S(2)x*S(3) + S(4)abS(3)xS(5)/S(5) + bS(4)xS(7)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(3), x), x, -aS(4)/(S(2)x*S(2)) + S(4)a*S(3)blog(x) + S(3)a*S(2)b*S(2)x*S(2) + ab*S(3)xS(4) + bS(4)xS(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(4), x), x, -aS(4)/(S(3)x*S(3)) - S(4)a*S(3)b/x + S(6)aS(2)b*S(2)x + S(4)ab*S(3)xS(3)/S(3) + bS(4)xS(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(5), x), x, -aS(4)/(S(4)x*S(4)) - S(2)a*S(3)b/x*S(2) + S(6)a*S(2)b*S(2)log(x) + S(2)ab*S(3)xS(2) + bS(4)xS(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(6), x), x, -aS(4)/(S(5)x*S(5)) - S(4)a*S(3)b/(S(3)xS(3)) - S(6)a*S(2)b*S(2)/x + S(4)abS(3)x + b*S(4)xS(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(7), x), x, -aS(4)/(S(6)xS(6)) - aS(3)b/x*S(4) - S(3)a*S(2)bS(2)/xS(2) + S(4)ab*S(3)log(x) + b*S(4)xS(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(8), x), x, -aS(4)/(S(7)xS(7)) - S(4)a*S(3)b/(S(5)xS(5)) - S(2)a*S(2)bS(2)/xS(3) - S(4)abS(3)/x + bS(4)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(9), x), x, -aS(4)/(S(8)x*S(8)) - S(2)a*S(3)b/(S(3)xS(6)) - S(3)a*S(2)b*S(2)/(S(2)x*S(4)) - S(2)abS(3)/xS(2) + bS(4)log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(10), x), x, -aS(4)/(S(9)x*S(9)) - S(4)a*S(3)b/(S(7)xS(7)) - S(6)a*S(2)b*S(2)/(S(5)x*S(5)) - S(4)abS(3)/(S(3)xS(3)) - bS(4)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(11), x), x, -(a + bxS(2))S(5)/(S(10)axS(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(12), x), x, -aS(4)/(S(11)xS(11)) - S(4)a*S(3)b/(S(9)xS(9)) - S(6)a*S(2)b*S(2)/(S(7)x*S(7)) - S(4)abS(3)/(S(5)xS(5)) - bS(4)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(13), x), x, -(a + bxS(2))S(5)/(S(12)ax*S(12)) + b(a + bxS(2))S(5)/(S(60)a*S(2)xS(10)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(13), x), x, -aS(4)/(S(12)xS(12)) - S(2)a*S(3)b/(S(5)xS(10)) - S(3)a*S(2)b*S(2)/(S(4)x*S(8)) - S(2)abS(3)/(S(3)xS(6)) - bS(4)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(14), x), x, -aS(4)/(S(13)x*S(13)) - S(4)a*S(3)b/(S(11)xS(11)) - S(2)a*S(2)b*S(2)/(S(3)x*S(9)) - S(4)abS(3)/(S(7)xS(7)) - bS(4)/(S(5)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(15), x), x, -aS(4)/(S(14)xS(14)) - aS(3)b/(S(3)x*S(12)) - S(3)a*S(2)b*S(2)/(S(5)x*S(10)) - ab*S(3)/(S(2)xS(8)) - bS(4)/(S(6)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/xS(16), x), x, -aS(4)/(S(15)x*S(15)) - S(4)a*S(3)b/(S(13)xS(13)) - S(6)a*S(2)b*S(2)/(S(11)x*S(11)) - S(4)abS(3)/(S(9)xS(9)) - bS(4)/(S(7)xS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, aS(6)(dx)(m + S(1))/(d(m + S(1))) + S(6)aS(5)b(dx)(m + S(3))/(dS(3)(m + S(3))) + S(15)a*S(4)b*S(2)(dx)(m + S(5))/(dS(5)(m + S(5))) + S(20)aS(3)b*S(3)(dx)(m + S(7))/(dS(7)(m + S(7))) + S(15)aS(2)b*S(4)(dx)(m + S(9))/(dS(9)(m + S(9))) + S(6)ab*S(5)(dx)(m + S(11))/(dS(11)(m + S(11))) + b*S(6)(dx)(m + S(13))/(dS(13)(m + S(13))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(8)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, aS(6)x*S(9)/S(9) + S(6)a*S(5)bxS(11)/S(11) + S(15)a*S(4)b*S(2)x*S(13)/S(13) + S(4)a*S(3)b*S(3)x*S(15)/S(3) + S(15)a*S(2)b*S(4)x*S(17)/S(17) + S(6)abS(5)xS(19)/S(19) + bS(6)xS(21)/S(21), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, aS(6)x*S(8)/S(8) + S(3)a*S(5)bxS(10)/S(5) + S(5)a*S(4)b*S(2)x*S(12)/S(4) + S(10)a*S(3)b*S(3)x*S(14)/S(7) + S(15)a*S(2)b*S(4)x*S(16)/S(16) + ab*S(5)xS(18)/S(3) + bS(6)xS(20)/S(20), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, aS(6)x*S(7)/S(7) + S(2)a*S(5)bxS(9)/S(3) + S(15)a*S(4)b*S(2)x*S(11)/S(11) + S(20)a*S(3)b*S(3)xS(13)/S(13) + aS(2)bS(4)x*S(15) + S(6)abS(5)xS(17)/S(17) + bS(6)xS(19)/S(19), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, aS(2)(a + bxS(2))S(7)/(S(14)b*S(3)) - a(a + bxS(2))S(8)/(S(8)b*S(3)) + (a + bxS(2))S(9)/(S(18)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, aS(6)x*S(5)/S(5) + S(6)a*S(5)bxS(7)/S(7) + S(5)a*S(4)b*S(2)x*S(9)/S(3) + S(20)a*S(3)b*S(3)x*S(11)/S(11) + S(15)a*S(2)b*S(4)x*S(13)/S(13) + S(2)abS(5)xS(15)/S(5) + bS(6)xS(17)/S(17), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -a(a + bxS(2))S(7)/(S(14)b*S(2)) + (a + bxS(2))S(8)/(S(16)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, aS(6)x*S(3)/S(3) + S(6)a*S(5)bxS(5)/S(5) + S(15)a*S(4)b*S(2)x*S(7)/S(7) + S(20)a*S(3)b*S(3)x*S(9)/S(9) + S(15)a*S(2)b*S(4)x*S(11)/S(11) + S(6)abS(5)xS(13)/S(13) + bS(6)xS(15)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, (a + bxS(2))S(7)/(S(14)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, aS(6)x + S(2)aS(5)bxS(3) + S(3)a*S(4)b*S(2)x*S(5) + S(20)a*S(3)b*S(3)x*S(7)/S(7) + S(5)a*S(2)b*S(4)x*S(9)/S(3) + S(6)abS(5)xS(11)/S(11) + bS(6)xS(13)/S(13), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/x, x), x, aS(6)log(x) + S(3)aS(5)bxS(2) + S(15)a*S(4)b*S(2)x*S(4)/S(4) + S(10)a*S(3)b*S(3)x*S(6)/S(3) + S(15)a*S(2)b*S(4)x*S(8)/S(8) + S(3)abS(5)xS(10)/S(5) + bS(6)xS(12)/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(2), x), x, -aS(6)/x + S(6)a*S(5)bx + S(5)a*S(4)b*S(2)x*S(3) + S(4)a*S(3)b*S(3)x*S(5) + S(15)a*S(2)b*S(4)x*S(7)/S(7) + S(2)abS(5)xS(9)/S(3) + bS(6)xS(11)/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(3), x), x, -aS(6)/(S(2)x*S(2)) + S(6)a*S(5)blog(x) + S(15)a*S(4)b*S(2)x*S(2)/S(2) + S(5)a*S(3)b*S(3)x*S(4) + S(5)a*S(2)b*S(4)x*S(6)/S(2) + S(3)abS(5)xS(8)/S(4) + bS(6)xS(10)/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(4), x), x, -aS(6)/(S(3)x*S(3)) - S(6)a*S(5)b/x + S(15)aS(4)b*S(2)x + S(20)aS(3)b*S(3)x*S(3)/S(3) + S(3)a*S(2)b*S(4)x*S(5) + S(6)abS(5)xS(7)/S(7) + bS(6)xS(9)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(5), x), x, -aS(6)/(S(4)x*S(4)) - S(3)a*S(5)b/x*S(2) + S(15)a*S(4)b*S(2)log(x) + S(10)aS(3)b*S(3)x*S(2) + S(15)a*S(2)b*S(4)x*S(4)/S(4) + ab*S(5)xS(6) + bS(6)xS(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(6), x), x, -aS(6)/(S(5)x*S(5)) - S(2)a*S(5)b/x*S(3) - S(15)a*S(4)b*S(2)/x + S(20)a*S(3)b*S(3)x + S(5)aS(2)b*S(4)x*S(3) + S(6)abS(5)xS(5)/S(5) + bS(6)xS(7)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(7), x), x, -aS(6)/(S(6)x*S(6)) - S(3)a*S(5)b/(S(2)xS(4)) - S(15)a*S(4)b*S(2)/(S(2)x*S(2)) + S(20)a*S(3)b*S(3)log(x) + S(15)aS(2)b*S(4)x*S(2)/S(2) + S(3)abS(5)xS(4)/S(2) + bS(6)xS(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(8), x), x, -aS(6)/(S(7)x*S(7)) - S(6)a*S(5)b/(S(5)xS(5)) - S(5)a*S(4)bS(2)/xS(3) - S(20)aS(3)b*S(3)/x + S(15)a*S(2)b*S(4)x + S(2)ab*S(5)xS(3) + bS(6)xS(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(9), x), x, -aS(6)/(S(8)xS(8)) - aS(5)b/xS(6) - S(15)a*S(4)b*S(2)/(S(4)x*S(4)) - S(10)a*S(3)bS(3)/xS(2) + S(15)aS(2)b*S(4)log(x) + S(3)ab*S(5)xS(2) + bS(6)xS(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(10), x), x, -aS(6)/(S(9)x*S(9)) - S(6)a*S(5)b/(S(7)xS(7)) - S(3)a*S(4)bS(2)/xS(5) - S(20)aS(3)b*S(3)/(S(3)x*S(3)) - S(15)a*S(2)b*S(4)/x + S(6)abS(5)x + b*S(6)xS(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(11), x), x, -aS(6)/(S(10)xS(10)) - S(3)a*S(5)b/(S(4)xS(8)) - S(5)a*S(4)b*S(2)/(S(2)x*S(6)) - S(5)a*S(3)bS(3)/xS(4) - S(15)aS(2)b*S(4)/(S(2)x*S(2)) + S(6)abS(5)log(x) + b*S(6)xS(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(12), x), x, -aS(6)/(S(11)xS(11)) - S(2)a*S(5)b/(S(3)xS(9)) - S(15)a*S(4)b*S(2)/(S(7)x*S(7)) - S(4)a*S(3)bS(3)/xS(5) - S(5)aS(2)bS(4)/xS(3) - S(6)abS(5)/x + bS(6)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(13), x), x, -aS(6)/(S(12)x*S(12)) - S(3)a*S(5)b/(S(5)xS(10)) - S(15)a*S(4)b*S(2)/(S(8)x*S(8)) - S(10)a*S(3)b*S(3)/(S(3)x*S(6)) - S(15)a*S(2)b*S(4)/(S(4)x*S(4)) - S(3)abS(5)/xS(2) + bS(6)log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(14), x), x, -aS(6)/(S(13)x*S(13)) - S(6)a*S(5)b/(S(11)xS(11)) - S(5)a*S(4)b*S(2)/(S(3)x*S(9)) - S(20)a*S(3)b*S(3)/(S(7)x*S(7)) - S(3)a*S(2)bS(4)/xS(5) - S(2)abS(5)/xS(3) - bS(6)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/x*S(15), x), x, -(a + bxS(2))S(7)/(S(14)axS(14)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(16), x), x, -aS(6)/(S(15)xS(15)) - S(6)a*S(5)b/(S(13)xS(13)) - S(15)a*S(4)b*S(2)/(S(11)x*S(11)) - S(20)a*S(3)b*S(3)/(S(9)x*S(9)) - S(15)a*S(2)b*S(4)/(S(7)x*S(7)) - S(6)abS(5)/(S(5)xS(5)) - bS(6)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(17), x), x, -(a + bxS(2))S(7)/(S(16)ax*S(16)) + b(a + bxS(2))S(7)/(S(112)a*S(2)xS(14)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(18), x), x, -aS(6)/(S(17)xS(17)) - S(2)a*S(5)b/(S(5)xS(15)) - S(15)a*S(4)b*S(2)/(S(13)x*S(13)) - S(20)a*S(3)b*S(3)/(S(11)x*S(11)) - S(5)a*S(2)b*S(4)/(S(3)x*S(9)) - S(6)abS(5)/(S(7)xS(7)) - bS(6)/(S(5)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(19), x), x, -(a + bxS(2))S(7)/(S(18)ax*S(18)) + b(a + bxS(2))S(7)/(S(72)a*S(2)xS(16)) - bS(2)(a + bxS(2))S(7)/(S(504)aS(3)xS(14)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(20), x), x, -aS(6)/(S(19)xS(19)) - S(6)a*S(5)b/(S(17)xS(17)) - aS(4)bS(2)/xS(15) - S(20)aS(3)b*S(3)/(S(13)x*S(13)) - S(15)a*S(2)b*S(4)/(S(11)x*S(11)) - S(2)abS(5)/(S(3)xS(9)) - bS(6)/(S(7)xS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(21), x), x, -aS(6)/(S(20)xS(20)) - aS(5)b/(S(3)x*S(18)) - S(15)a*S(4)b*S(2)/(S(16)x*S(16)) - S(10)a*S(3)b*S(3)/(S(7)x*S(14)) - S(5)a*S(2)b*S(4)/(S(4)x*S(12)) - S(3)abS(5)/(S(5)xS(10)) - bS(6)/(S(8)xS(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/xS(22), x), x, -aS(6)/(S(21)x*S(21)) - S(6)a*S(5)b/(S(19)xS(19)) - S(15)a*S(4)b*S(2)/(S(17)x*S(17)) - S(4)a*S(3)b*S(3)/(S(3)x*S(15)) - S(15)a*S(2)b*S(4)/(S(13)x*S(13)) - S(6)abS(5)/(S(11)xS(11)) - bS(6)/(S(9)xS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, (dx)*(m + S(1))hyper((S(2), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -bxS(2)/a)/(aS(2)d(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, aS(5)/(S(2)b*S(6)(a + bxS(2))) + S(5)a*S(4)log(a + bxS(2))/(S(2)b*S(6)) - S(2)a*S(3)xS(2)/bS(5) + S(3)aS(2)x*S(4)/(S(4)b*S(4)) - ax*S(6)/(S(3)bS(3)) + xS(8)/(S(8)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, -aS(4)/(S(2)b*S(5)(a + bxS(2))) - S(2)a*S(3)log(a + bxS(2))/bS(5) + S(3)a*S(2)x*S(2)/(S(2)b*S(4)) - ax*S(4)/(S(2)bS(3)) + xS(6)/(S(6)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, aS(3)/(S(2)b*S(4)(a + bxS(2))) + S(3)a*S(2)log(a + bxS(2))/(S(2)b*S(4)) - axS(2)/bS(3) + x*S(4)/(S(4)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, -aS(2)/(S(2)b*S(3)(a + bxS(2))) - alog(a + bxS(2))/bS(3) + xS(2)/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, a/(S(2)b*S(2)(a + bxS(2))) + log(a + bx*S(2))/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, -S(1)/(S(2)b(a + bx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, S(1)/(S(2)a(a + bxS(2))) + log(x)/aS(2) - log(a + bxS(2))/(S(2)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(aS(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, -b/(S(2)a*S(2)(a + bxS(2))) - S(1)/(S(2)a*S(2)x*S(2)) - S(2)blog(x)/aS(3) + blog(a + bxS(2))/aS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, -S(1)/(S(4)a*S(2)xS(4)) + bS(2)/(S(2)aS(3)(a + bxS(2))) + b/(aS(3)x*S(2)) + S(3)b*S(2)log(x)/a*S(4) - S(3)b*S(2)log(a + bxS(2))/(S(2)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(10)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, S(9)a*(S(7)/2)atan(sqrt(b)x/sqrt(a))/(S(2)b*(S(11)/2)) - S(9)a*S(3)x/(S(2)bS(5)) + S(3)a*S(2)x*S(3)/(S(2)b*S(4)) - S(9)axS(5)/(S(10)bS(3)) - xS(9)/(S(2)b(a + bxS(2))) + S(9)x*S(7)/(S(14)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, -S(7)a*(S(5)/2)atan(sqrt(b)x/sqrt(a))/(S(2)b*(S(9)/2)) + S(7)a*S(2)x/(S(2)bS(4)) - S(7)axS(3)/(S(6)bS(3)) - xS(7)/(S(2)b(a + bxS(2))) + S(7)x*S(5)/(S(10)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, S(5)a*(S(3)/2)atan(sqrt(b)x/sqrt(a))/(S(2)b*(S(7)/2)) - S(5)ax/(S(2)bS(3)) - xS(5)/(S(2)b(a + bxS(2))) + S(5)x*S(3)/(S(6)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, -S(3)sqrt(a)atan(sqrt(b)x/sqrt(a))/(S(2)b(S(5)/2)) - xS(3)/(S(2)b(a + bx*S(2))) + S(3)x/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, -x/(S(2)b(a + bx*S(2))) + atan(sqrt(b)x/sqrt(a))/(S(2)sqrt(a)b(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, x/(S(2)a(a + bx*S(2))) + atan(sqrt(b)x/sqrt(a))/(S(2)a(S(3)/2)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, S(1)/(S(2)ax(a + bxS(2))) - S(3)/(S(2)a*S(2)x) - S(3)sqrt(b)atan(sqrt(b)x/sqrt(a))/(S(2)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(aS(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, S(1)/(S(2)axS(3)(a + bxS(2))) - S(5)/(S(6)a*S(2)x*S(3)) + S(5)b/(S(2)aS(3)x) + S(5)b(S(3)/2)atan(sqrt(b)x/sqrt(a))/(S(2)a(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(aS(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, S(1)/(S(2)axS(5)(a + bxS(2))) - S(7)/(S(10)a*S(2)x*S(5)) + S(7)b/(S(6)aS(3)x*S(3)) - S(7)b*S(2)/(S(2)a*S(4)x) - S(7)b(S(5)/2)atan(sqrt(b)x/sqrt(a))/(S(2)a*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, (dx)(m + S(1))hyper((S(4), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -bxS(2)/a)/(aS(4)d(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, aS(5)/(S(6)b*S(6)(a + bxS(2))S(3)) - S(5)a*S(4)/(S(4)b*S(6)(a + bxS(2))S(2)) + S(5)aS(3)/(bS(6)(a + bx*S(2))) + S(5)a*S(2)log(a + bxS(2))/bS(6) - S(2)axS(2)/bS(5) + xS(4)/(S(4)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -aS(4)/(S(6)b*S(5)(a + bxS(2))S(3)) + aS(3)/(bS(5)(a + bxS(2))S(2)) - S(3)aS(2)/(bS(5)(a + bx*S(2))) - S(2)alog(a + bxS(2))/bS(5) + x*S(2)/(S(2)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, aS(3)/(S(6)b*S(4)(a + bxS(2))S(3)) - S(3)a*S(2)/(S(4)b*S(4)(a + bxS(2))S(2)) + S(3)a/(S(2)bS(4)(a + bxS(2))) + log(a + bx*S(2))/(S(2)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, xS(6)/(S(6)a(a + bxS(2))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, a/(S(6)bS(2)(a + bxS(2))S(3)) - S(1)/(S(4)b*S(2)(a + bxS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -S(1)/(S(6)b(a + bxS(2))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)), x), x, S(1)/(S(6)a(a + bxS(2))S(3)) + S(1)/(S(4)aS(2)(a + bxS(2))S(2)) + S(1)/(S(2)a*S(3)(a + bxS(2))) + log(x)/aS(4) - log(a + bx*S(2))/(S(2)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)), x), x, -b/(S(6)a*S(2)(a + bxS(2))S(3)) - b/(S(2)a*S(3)(a + bxS(2))S(2)) - S(3)b/(S(2)aS(4)(a + bxS(2))) - S(1)/(S(2)a*S(4)x*S(2)) - S(4)blog(x)/aS(5) + S(2)blog(a + bxS(2))/aS(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2)), x), x, bS(2)/(S(6)a*S(3)(a + bxS(2))S(3)) + S(3)b*S(2)/(S(4)a*S(4)(a + bxS(2))S(2)) - S(1)/(S(4)a*S(4)x*S(4)) + S(3)bS(2)/(aS(5)(a + bx*S(2))) + S(2)b/(a*S(5)x*S(2)) + S(10)b*S(2)log(x)/a*S(6) - S(5)b*S(2)log(a + bxS(2))/aS(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(12)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -S(231)a*(S(5)/2)atan(sqrt(b)x/sqrt(a))/(S(16)b*(S(13)/2)) + S(231)a*S(2)x/(S(16)bS(6)) - S(77)axS(3)/(S(16)bS(5)) - xS(11)/(S(6)b(a + bxS(2))S(3)) - S(11)x*S(9)/(S(24)b*S(2)(a + bxS(2))S(2)) - S(33)x*S(7)/(S(16)b*S(3)(a + bxS(2))) + S(231)x*S(5)/(S(80)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(10)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, S(105)a*(S(3)/2)atan(sqrt(b)x/sqrt(a))/(S(16)b*(S(11)/2)) - S(105)ax/(S(16)bS(5)) - xS(9)/(S(6)b(a + bxS(2))S(3)) - S(3)x*S(7)/(S(8)b*S(2)(a + bxS(2))S(2)) - S(21)x*S(5)/(S(16)b*S(3)(a + bxS(2))) + S(35)x*S(3)/(S(16)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -S(35)sqrt(a)atan(sqrt(b)x/sqrt(a))/(S(16)b(S(9)/2)) - xS(7)/(S(6)b(a + bxS(2))S(3)) - S(7)xS(5)/(S(24)b*S(2)(a + bxS(2))S(2)) - S(35)x*S(3)/(S(48)b*S(3)(a + bxS(2))) + S(35)x/(S(16)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -xS(5)/(S(6)b(a + bxS(2))S(3)) - S(5)xS(3)/(S(24)b*S(2)(a + bxS(2))S(2)) - S(5)x/(S(16)bS(3)(a + bxS(2))) + S(5)atan(sqrt(b)x/sqrt(a))/(S(16)sqrt(a)b(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -xS(3)/(S(6)b(a + bxS(2))S(3)) - x/(S(8)bS(2)(a + bxS(2))S(2)) + x/(S(16)abS(2)(a + bxS(2))) + atan(sqrt(b)x/sqrt(a))/(S(16)a(S(3)/2)b(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -x/(S(6)b(a + bxS(2))S(3)) + x/(S(24)ab(a + bxS(2))S(2)) + x/(S(16)aS(2)b(a + bx*S(2))) + atan(sqrt(b)x/sqrt(a))/(S(16)a(S(5)/2)b(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(-2)), x), x, x/(S(6)a(a + bxS(2))S(3)) + S(5)x/(S(24)aS(2)(a + bxS(2))S(2)) + S(5)x/(S(16)aS(3)(a + bxS(2))) + S(5)atan(sqrt(b)x/sqrt(a))/(S(16)a*(S(7)/2)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2)), x), x, S(1)/(S(6)ax(a + bxS(2))S(3)) + S(7)/(S(24)a*S(2)x(a + bxS(2))S(2)) + S(35)/(S(48)aS(3)x(a + bx*S(2))) - S(35)/(S(16)a*S(4)x) - S(35)sqrt(b)atan(sqrt(b)x/sqrt(a))/(S(16)a(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)), x), x, S(1)/(S(6)axS(3)(a + bxS(2))S(3)) + S(3)/(S(8)a*S(2)x*S(3)(a + bxS(2))S(2)) + S(21)/(S(16)a*S(3)x*S(3)(a + bxS(2))) - S(35)/(S(16)a*S(4)x*S(3)) + S(105)b/(S(16)aS(5)x) + S(105)b(S(3)/2)atan(sqrt(b)x/sqrt(a))/(S(16)a(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)), x), x, S(1)/(S(6)axS(5)(a + bxS(2))S(3)) + S(11)/(S(24)a*S(2)x*S(5)(a + bxS(2))S(2)) + S(33)/(S(16)a*S(3)x*S(5)(a + bxS(2))) - S(231)/(S(80)a*S(4)x*S(5)) + S(77)b/(S(16)aS(5)x*S(3)) - S(231)b*S(2)/(S(16)a*S(6)x) - S(231)b(S(5)/2)atan(sqrt(b)x/sqrt(a))/(S(16)a*(S(13)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, (dx)(m + S(1))hyper((S(6), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -bxS(2)/a)/(aS(6)d(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(15)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, aS(7)/(S(10)b*S(8)(a + bxS(2))S(5)) - S(7)a*S(6)/(S(8)b*S(8)(a + bxS(2))S(4)) + S(7)a*S(5)/(S(2)b*S(8)(a + bxS(2))S(3)) - S(35)a*S(4)/(S(4)b*S(8)(a + bxS(2))S(2)) + S(35)a*S(3)/(S(2)b*S(8)(a + bxS(2))) + S(21)a*S(2)log(a + bxS(2))/(S(2)b*S(8)) - S(3)axS(2)/bS(7) + xS(4)/(S(4)bS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(13)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -aS(6)/(S(10)b*S(7)(a + bxS(2))S(5)) + S(3)a*S(5)/(S(4)b*S(7)(a + bxS(2))S(4)) - S(5)a*S(4)/(S(2)b*S(7)(a + bxS(2))S(3)) + S(5)aS(3)/(bS(7)(a + bxS(2))S(2)) - S(15)aS(2)/(S(2)b*S(7)(a + bxS(2))) - S(3)alog(a + bxS(2))/bS(7) + x*S(2)/(S(2)bS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, aS(5)/(S(10)b*S(6)(a + bxS(2))S(5)) - S(5)a*S(4)/(S(8)b*S(6)(a + bxS(2))S(4)) + S(5)a*S(3)/(S(3)b*S(6)(a + bxS(2))S(3)) - S(5)a*S(2)/(S(2)b*S(6)(a + bxS(2))S(2)) + S(5)a/(S(2)bS(6)(a + bxS(2))) + log(a + bx*S(2))/(S(2)bS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, xS(10)/(S(10)a(a + bxS(2))S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, x*S(8)/(S(10)a(a + bxS(2))S(5)) + x*S(8)/(S(40)a*S(2)(a + bxS(2))S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -aS(2)/(S(10)b*S(3)(a + bxS(2))S(5)) + a/(S(4)b*S(3)(a + bxS(2))S(4)) - S(1)/(S(6)b*S(3)(a + bxS(2))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, a/(S(10)b*S(2)(a + bxS(2))S(5)) - S(1)/(S(8)b*S(2)(a + bxS(2))S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -S(1)/(S(10)b(a + bxS(2))S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)), x), x, S(1)/(S(10)a(a + bxS(2))S(5)) + S(1)/(S(8)aS(2)(a + bxS(2))S(4)) + S(1)/(S(6)a*S(3)(a + bxS(2))S(3)) + S(1)/(S(4)a*S(4)(a + bxS(2))S(2)) + S(1)/(S(2)a*S(5)(a + bxS(2))) + log(x)/aS(6) - log(a + bx*S(2))/(S(2)aS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)), x), x, -b/(S(10)a*S(2)(a + bxS(2))S(5)) - b/(S(4)a*S(3)(a + bxS(2))S(4)) - b/(S(2)a*S(4)(a + bxS(2))S(3)) - b/(aS(5)(a + bxS(2))S(2)) - S(5)b/(S(2)aS(6)(a + bxS(2))) - S(1)/(S(2)a*S(6)x*S(2)) - S(6)blog(x)/aS(7) + S(3)blog(a + bxS(2))/aS(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3)), x), x, bS(2)/(S(10)a*S(3)(a + bxS(2))S(5)) + S(3)b*S(2)/(S(8)a*S(4)(a + bxS(2))S(4)) + bS(2)/(aS(5)(a + bxS(2))S(3)) + S(5)b*S(2)/(S(2)a*S(6)(a + bxS(2))S(2)) - S(1)/(S(4)a*S(6)x*S(4)) + S(15)b*S(2)/(S(2)a*S(7)(a + bxS(2))) + S(3)b/(a*S(7)x*S(2)) + S(21)b*S(2)log(x)/a*S(8) - S(21)b*S(2)log(a + bxS(2))/(S(2)aS(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(16)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -S(9009)a*(S(5)/2)atan(sqrt(b)x/sqrt(a))/(S(256)b*(S(17)/2)) + S(9009)a*S(2)x/(S(256)bS(8)) - S(3003)axS(3)/(S(256)bS(7)) - xS(15)/(S(10)b(a + bxS(2))S(5)) - S(3)x*S(13)/(S(16)b*S(2)(a + bxS(2))S(4)) - S(13)x*S(11)/(S(32)b*S(3)(a + bxS(2))S(3)) - S(143)x*S(9)/(S(128)b*S(4)(a + bxS(2))S(2)) - S(1287)x*S(7)/(S(256)b*S(5)(a + bxS(2))) + S(9009)x*S(5)/(S(1280)bS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(14)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, S(3003)a*(S(3)/2)atan(sqrt(b)x/sqrt(a))/(S(256)b*(S(15)/2)) - S(3003)ax/(S(256)bS(7)) - xS(13)/(S(10)b(a + bxS(2))S(5)) - S(13)x*S(11)/(S(80)b*S(2)(a + bxS(2))S(4)) - S(143)x*S(9)/(S(480)b*S(3)(a + bxS(2))S(3)) - S(429)x*S(7)/(S(640)b*S(4)(a + bxS(2))S(2)) - S(3003)x*S(5)/(S(1280)b*S(5)(a + bxS(2))) + S(1001)x*S(3)/(S(256)bS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(12)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -S(693)sqrt(a)atan(sqrt(b)x/sqrt(a))/(S(256)b(S(13)/2)) - xS(11)/(S(10)b(a + bxS(2))S(5)) - S(11)xS(9)/(S(80)b*S(2)(a + bxS(2))S(4)) - S(33)x*S(7)/(S(160)b*S(3)(a + bxS(2))S(3)) - S(231)x*S(5)/(S(640)b*S(4)(a + bxS(2))S(2)) - S(231)x*S(3)/(S(256)b*S(5)(a + bxS(2))) + S(693)x/(S(256)bS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(10)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -xS(9)/(S(10)b(a + bxS(2))S(5)) - S(9)xS(7)/(S(80)b*S(2)(a + bxS(2))S(4)) - S(21)x*S(5)/(S(160)b*S(3)(a + bxS(2))S(3)) - S(21)x*S(3)/(S(128)b*S(4)(a + bxS(2))S(2)) - S(63)x/(S(256)bS(5)(a + bxS(2))) + S(63)atan(sqrt(b)x/sqrt(a))/(S(256)sqrt(a)b(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -xS(7)/(S(10)b(a + bxS(2))S(5)) - S(7)xS(5)/(S(80)b*S(2)(a + bxS(2))S(4)) - S(7)x*S(3)/(S(96)b*S(3)(a + bxS(2))S(3)) - S(7)x/(S(128)bS(4)(a + bxS(2))S(2)) + S(7)x/(S(256)ab*S(4)(a + bxS(2))) + S(7)atan(sqrt(b)x/sqrt(a))/(S(256)a*(S(3)/2)b(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -xS(5)/(S(10)b(a + bxS(2))S(5)) - x*S(3)/(S(16)b*S(2)(a + bxS(2))S(4)) - x/(S(32)b*S(3)(a + bxS(2))S(3)) + x/(S(128)abS(3)(a + bxS(2))S(2)) + S(3)x/(S(256)aS(2)b*S(3)(a + bxS(2))) + S(3)atan(sqrt(b)x/sqrt(a))/(S(256)a*(S(5)/2)b(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -xS(3)/(S(10)b(a + bxS(2))S(5)) - S(3)x/(S(80)b*S(2)(a + bxS(2))S(4)) + x/(S(160)abS(2)(a + bxS(2))S(3)) + x/(S(128)a*S(2)b*S(2)(a + bxS(2))S(2)) + S(3)x/(S(256)aS(3)b*S(2)(a + bxS(2))) + S(3)atan(sqrt(b)x/sqrt(a))/(S(256)a*(S(7)/2)b(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -x/(S(10)b(a + bxS(2))S(5)) + x/(S(80)ab(a + bxS(2))S(4)) + S(7)x/(S(480)a*S(2)b(a + bxS(2))S(3)) + S(7)x/(S(384)a*S(3)b(a + bxS(2))S(2)) + S(7)x/(S(256)a*S(4)b(a + bx*S(2))) + S(7)atan(sqrt(b)x/sqrt(a))/(S(256)a*(S(9)/2)b(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(-3)), x), x, x/(S(10)a(a + bxS(2))S(5)) + S(9)x/(S(80)aS(2)(a + bxS(2))S(4)) + S(21)x/(S(160)aS(3)(a + bxS(2))S(3)) + S(21)x/(S(128)aS(4)(a + bxS(2))S(2)) + S(63)x/(S(256)aS(5)(a + bxS(2))) + S(63)atan(sqrt(b)x/sqrt(a))/(S(256)a*(S(11)/2)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3)), x), x, S(1)/(S(10)ax(a + bxS(2))S(5)) + S(11)/(S(80)a*S(2)x(a + bxS(2))S(4)) + S(33)/(S(160)aS(3)x(a + bxS(2))S(3)) + S(231)/(S(640)aS(4)x(a + bxS(2))S(2)) + S(231)/(S(256)aS(5)x(a + bx*S(2))) - S(693)/(S(256)a*S(6)x) - S(693)sqrt(b)atan(sqrt(b)x/sqrt(a))/(S(256)a(S(13)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)), x), x, S(1)/(S(10)axS(3)(a + bxS(2))S(5)) + S(13)/(S(80)a*S(2)x*S(3)(a + bxS(2))S(4)) + S(143)/(S(480)a*S(3)x*S(3)(a + bxS(2))S(3)) + S(429)/(S(640)a*S(4)x*S(3)(a + bxS(2))S(2)) + S(3003)/(S(1280)a*S(5)x*S(3)(a + bxS(2))) - S(1001)/(S(256)a*S(6)x*S(3)) + S(3003)b/(S(256)aS(7)x) + S(3003)b(S(3)/2)atan(sqrt(b)x/sqrt(a))/(S(256)a(S(15)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)), x), x, S(1)/(S(10)axS(5)(a + bxS(2))S(5)) + S(3)/(S(16)a*S(2)x*S(5)(a + bxS(2))S(4)) + S(13)/(S(32)a*S(3)x*S(5)(a + bxS(2))S(3)) + S(143)/(S(128)a*S(4)x*S(5)(a + bxS(2))S(2)) + S(1287)/(S(256)a*S(5)x*S(5)(a + bxS(2))) - S(9009)/(S(1280)a*S(6)x*S(5)) + S(3003)b/(S(256)aS(7)x*S(3)) - S(9009)b*S(2)/(S(256)a*S(8)x) - S(9009)b(S(5)/2)atan(sqrt(b)x/sqrt(a))/(S(256)a(S(17)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4) + S(2)xS(2) + S(1)), x), x, x/(S(2)xS(2) + S(2)) + atan(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(4) + S(2)xS(2) + S(1)), x), x, -S(1)/(S(2)xS(2) + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(x*S(4) + S(2)x*S(2) + S(1)), x), x, -x/(S(2)xS(2) + S(2)) + atan(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(x*S(4) + S(2)xS(2) + S(1)), x), x, log(xS(2) + S(1))/S(2) + S(1)/(S(2)xS(2) + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(4) - S(18)x*S(2) + S(81)), x), x, S(1)/(-S(2)xS(2) + S(18)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(x*S(4) - S(8)xS(2) + S(16)), x), x, log(-xS(2) + S(4))/S(2) + S(2)/(-x*S(2) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*msqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, S(2)a(dx)*(m + S(1))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(d(a + bxS(2))(m*S(2) + S(4)m + S(3))) + (dx)(m + S(1))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(d(m + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, ax*S(6)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(24)a + S(24)bxS(2)) + xS(6)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, S(2)axS(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(35)a + S(35)bxS(2)) + xS(5)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, ax*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(12)a + S(12)bxS(2)) + xS(4)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, S(2)axS(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(15)a + S(15)bxS(2)) + xS(3)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, (a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, S(2)axsqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(3)a + S(3)bx*S(2)) + xsqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/x, x), x, asqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))log(x)/(a + bxS(2)) + sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(2), x), x, -S(2)asqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(x(a + bxS(2))) + sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(3), x), x, -asqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2)x*S(2)(a + bxS(2))) + bsqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))log(x)/(a + bxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(4), x), x, S(2)asqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(3)x*S(3)(a + bxS(2))) - sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(5), x), x, -(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4)axS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(6), x), x, S(2)asqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(15)x*S(5)(a + bxS(2))) - sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(3)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(7), x), x, asqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(12)x*S(6)(a + bxS(2))) - sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(8), x), x, S(2)asqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(35)x*S(7)(a + bxS(2))) - sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5)xS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(9), x), x, asqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(24)x*S(8)(a + bxS(2))) - sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(6)xS(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(10), x), x, S(2)asqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(63)x*S(9)(a + bxS(2))) - sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(7)xS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/xS(11), x), x, asqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(40)x*S(10)(a + bxS(2))) - sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(8)xS(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, aS(3)x*S(10)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(560)a + S(560)bxS(2)) + aS(2)xS(10)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(112) + S(3)axS(10)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(112) + xS(10)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, S(16)a*S(3)x*S(9)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(6435)a + S(6435)bx*S(2)) + S(8)a*S(2)x*S(9)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(715) + S(2)axS(9)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(65) + xS(9)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, aS(3)x*S(8)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(280)a + S(280)bxS(2)) + aS(2)xS(8)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(70) + ax*S(8)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(28) + xS(8)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(14), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, S(16)a*S(3)x*S(7)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(3003)a + S(3003)bx*S(2)) + S(8)a*S(2)x*S(7)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(429) + S(6)axS(7)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(143) + xS(7)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(13), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, aS(2)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(24)b*S(3)) - a(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(30)bS(3)) + xS(4)(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(12)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, S(16)a*S(3)x*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(1155)a + S(1155)bx*S(2)) + S(8)a*S(2)x*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(231) + S(2)axS(5)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(33) + xS(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, -a(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(8)bS(2)) + (aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(10)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, S(16)a*S(3)x*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(315)a + S(315)bx*S(2)) + S(8)a*S(2)x*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(105) + S(2)axS(3)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(21) + xS(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, (a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, S(16)a*S(3)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(35)a + S(35)bx*S(2)) + S(8)a*S(2)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/S(35) + S(6)ax(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(35) + x(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/x, x), x, aS(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))log(x)/(a + bxS(2)) + aS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(2) + a(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(4) + (aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/xS(2), x), x, -S(16)a*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5)x(a + bx*S(2))) + S(8)a*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5)x) + S(2)a(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5)x) + (a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(5)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/xS(3), x), x, S(3)a*S(2)bsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))log(x)/(a + bxS(2)) + S(3)absqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(2) - S(3)a(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4)xS(2)) + (aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(4)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/xS(4), x), x, S(16)abS(2)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(3)a + S(3)bx*S(2)) + S(2)a(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/xS(3) + S(8)b*S(2)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/S(3) - S(7)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/x*S(5), x), x, S(3)abS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))log(x)/(a + bxS(2)) + S(3)a(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4)x*S(4)) + S(3)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(2) - (aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/xS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/xS(6), x), x, -S(16)abS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5)x(a + bx*S(2))) + S(2)a(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5)x*S(5)) + S(8)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5)x) - S(3)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(5)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/x*S(7), x), x, -ab*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2)x*S(2)(a + bxS(2))) + a(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4)xS(6)) + bS(3)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))log(x)/(a + bxS(2)) - S(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(12)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/x*S(8), x), x, S(16)abS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(35)x*S(3)(a + bxS(2))) + S(6)a(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(35)x*S(7)) - S(24)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(35)x*S(3)) - S(11)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(35)xS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/x*S(9), x), x, -(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(8)axS(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/xS(10), x), x, S(16)abS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(315)x*S(5)(a + bxS(2))) + S(2)a(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(21)x*S(9)) - S(8)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(63)x*S(5)) - S(13)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(63)xS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/x*S(11), x), x, -(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(8)axS(10)) + (aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(40)a*S(2)xS(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/x*S(12), x), x, S(16)abS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(1155)x*S(7)(a + bxS(2))) + S(2)a(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(33)x*S(11)) - S(8)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(165)x*S(7)) - S(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(33)xS(11)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/x*S(13), x), x, -(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(12)axS(12)) + b(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(24)a*S(2)x*S(10)) - b(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(120)a*S(3)xS(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/x*S(14), x), x, S(16)abS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(3003)x*S(9)(a + bxS(2))) + S(6)a(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(143)x*S(13)) - S(24)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(1001)x*S(9)) - S(17)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(143)xS(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/x*S(15), x), x, ab*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(280)x*S(10)(a + bxS(2))) + a(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(28)xS(14)) - bS(2)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(56)x*S(10)) - S(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(28)xS(14)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/x*S(16), x), x, S(16)abS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(6435)x*S(11)(a + bxS(2))) + S(2)a(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(65)x*S(15)) - S(8)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(585)x*S(11)) - S(19)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(195)xS(15)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/x*S(17), x), x, ab*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(560)x*S(12)(a + bxS(2))) + S(3)a(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(112)x*S(16)) - S(3)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(280)x*S(12)) - S(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(56)xS(16)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(13)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, aS(5)x*S(14)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(11088)a + S(11088)bxS(2)) + aS(4)xS(14)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(1584) + aS(3)x*S(14)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(396) + aS(2)x*S(14)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(132) + S(5)axS(14)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(264) + xS(14)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/S(24), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(12)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, S(256)a*S(5)x*S(13)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2028117)a + S(2028117)bx*S(2)) + S(128)a*S(4)x*S(13)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(156009) + S(160)a*S(3)x*S(13)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(52003) + S(80)a*S(2)x*S(13)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(9177) + S(10)axS(13)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(483) + xS(13)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/S(23), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, aS(5)x*S(12)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5544)a + S(5544)bxS(2)) + aS(4)xS(12)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(924) + aS(3)x*S(12)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(264) + aS(2)x*S(12)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(99) + ax*S(12)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(44) + xS(12)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/S(22), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(10)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, S(256)a*S(5)x*S(11)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(969969)a + S(969969)bx*S(2)) + S(128)a*S(4)x*S(11)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(88179) + S(32)a*S(3)x*S(11)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(6783) + S(80)a*S(2)x*S(11)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(6783) + S(10)axS(11)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(399) + xS(11)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/S(21), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, aS(4)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(360)bS(5)) - aS(3)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(7)/2)/(S(420)bS(5)) + aS(2)xS(4)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(120)b*S(3)) - ax*S(6)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(45)bS(2)) + xS(8)(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(20)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(8)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, S(256)a*S(5)x*S(9)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(415701)a + S(415701)bx*S(2)) + S(128)a*S(4)x*S(9)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(46189) + S(32)a*S(3)x*S(9)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(4199) + S(16)a*S(2)x*S(9)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(969) + S(10)axS(9)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(323) + xS(9)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/S(19), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, -aS(3)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(144)bS(4)) + aS(2)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(7)/2)/(S(168)b*S(4)) - ax*S(4)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(48)bS(2)) + xS(6)(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(18)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(6)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, S(256)a*S(5)x*S(7)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(153153)a + S(153153)bx*S(2)) + S(128)a*S(4)x*S(7)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(21879) + S(32)a*S(3)x*S(7)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(2431) + S(16)a*S(2)x*S(7)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(663) + S(2)axS(7)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(51) + xS(7)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/S(17), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, aS(2)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(48)b*S(3)) - a(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(7)/2)/(S(56)bS(3)) + xS(4)(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(16)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, S(256)a*S(5)x*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(45045)a + S(45045)bx*S(2)) + S(128)a*S(4)x*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(9009) + S(32)a*S(3)x*S(5)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(1287) + S(16)a*S(2)x*S(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(429) + S(2)axS(5)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(39) + xS(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, -a(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(12)bS(2)) + (aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(7)/2)/(S(14)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, S(256)a*S(5)x*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(9009)a + S(9009)bx*S(2)) + S(128)a*S(4)x*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(3003) + S(160)a*S(3)x*S(3)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(3003) + S(80)a*S(2)x*S(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(1287) + S(10)axS(3)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(143) + xS(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/S(13), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, (a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(12)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, S(256)a*S(5)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(693)a + S(693)bx*S(2)) + S(128)a*S(4)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/S(693) + S(32)a*S(3)x(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(231) + S(80)a*S(2)x(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(693) + S(10)ax(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(99) + x(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x, x), x, aS(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))log(x)/(a + bxS(2)) + aS(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(2) + aS(3)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(4) + aS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(6) + a(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(8) + (aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/xS(2), x), x, -S(256)a*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(63)x(a + bx*S(2))) + S(128)a*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(63)x) + S(32)aS(3)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(63)x) + S(16)aS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(63)x) + S(10)a(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(63)x) + (a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(9)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/xS(3), x), x, S(5)a*S(4)bsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))log(x)/(a + bxS(2)) + S(5)a*S(3)bsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/S(2) + S(5)a*S(2)b(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(4) + S(5)ab(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(6) - S(5)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(8)xS(2)) + (aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(8)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/xS(4), x), x, S(256)a*S(3)b*S(2)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(21)a + S(21)bx*S(2)) + S(128)a*S(2)b*S(2)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/S(21) + S(32)abS(2)x(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(7) + S(10)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(3)x*S(3)) + S(80)b*S(2)x(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(21) - S(11)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(5), x), x, S(10)a*S(3)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))log(x)/(a + bxS(2)) + S(5)a*S(2)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)) + S(5)abS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(2) + S(5)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(4)x*S(4)) + S(5)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/S(3) - S(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(2)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(6), x), x, -S(256)a*S(3)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(15)x(a + bx*S(2))) + S(128)a*S(2)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(15)x) + S(32)ab*S(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(15)x) + S(2)a(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(3)x*S(5)) + S(16)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(15)x) - S(13)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(15)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(7), x), x, S(10)a*S(2)b*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))log(x)/(a + bxS(2)) + S(5)abS(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)) - S(5)abS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2)x*S(2)) + S(5)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(12)x*S(6)) + S(5)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(6)x*S(2)) - S(7)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(12)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(8), x), x, S(256)abS(4)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(21)a + S(21)bx*S(2)) + S(32)abS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(7)x*S(3)) + S(2)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(7)x*S(7)) + S(128)b*S(4)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/S(21) - S(16)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(3)x*S(3)) - S(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(7)xS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(9), x), x, S(5)abS(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))log(x)/(a + bxS(2)) + S(5)abS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4)x*S(4)) + S(5)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(24)x*S(8)) + S(5)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/S(2) - S(5)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(3)xS(4)) - (aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(3)xS(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/xS(10), x), x, -S(256)abS(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(63)x(a + bx*S(2))) + S(32)abS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(63)x*S(5)) + S(10)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(63)x*S(9)) + S(128)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(63)x) - S(16)bS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(21)x*S(5)) - S(17)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(63)xS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(11), x), x, -ab*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2)x*S(2)(a + bxS(2))) + ab*S(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4)x*S(6)) + a(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(8)xS(10)) + bS(5)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))log(x)/(a + bxS(2)) - S(5)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(12)x*S(6)) - S(9)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(40)xS(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(12), x), x, S(256)abS(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(693)x*S(3)(a + bxS(2))) + S(32)abS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(231)x*S(7)) + S(10)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(99)x*S(11)) - S(128)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(231)x*S(3)) - S(16)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(63)x*S(7)) - S(19)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(99)xS(11)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(13), x), x, -(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(12)axS(12)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/xS(14), x), x, S(256)abS(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(9009)x*S(5)(a + bxS(2))) + S(160)abS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(3003)x*S(9)) + S(10)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(143)x*S(13)) - S(640)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(9009)x*S(5)) - S(80)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(693)x*S(9)) - S(21)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(143)xS(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(15), x), x, -(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(12)axS(14)) + (aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(7)/2)/(S(84)a*S(2)xS(14)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(16), x), x, S(256)abS(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(45045)x*S(7)(a + bxS(2))) + S(32)abS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(1287)x*S(11)) + S(2)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(39)x*S(15)) - S(128)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(6435)x*S(7)) - S(80)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(1287)x*S(11)) - S(23)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(195)xS(15)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(17), x), x, -(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(16)axS(16)) + b(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(48)a*S(2)x*S(14)) - b(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(7)/2)/(S(336)a*S(3)xS(14)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(18), x), x, S(256)abS(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(153153)x*S(9)(a + bxS(2))) + S(32)abS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2431)x*S(13)) + S(2)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(51)x*S(17)) - S(128)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(17017)x*S(9)) - S(16)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(429)x*S(13)) - S(5)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(51)xS(17)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(19), x), x, -(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(18)axS(18)) + b(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(48)a*S(2)xS(16)) - bS(2)(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(144)a*S(3)xS(14)) + bS(2)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(7)/2)/(S(1008)a*S(4)xS(14)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(20), x), x, S(256)abS(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(415701)x*S(11)(a + bxS(2))) + S(32)abS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4199)x*S(15)) + S(10)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(323)x*S(19)) - S(128)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(37791)x*S(11)) - S(16)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(663)x*S(15)) - S(27)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(323)xS(19)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(21), x), x, ab*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2520)x*S(12)(a + bxS(2))) + ab*S(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(168)x*S(16)) + a(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(36)xS(20)) - bS(4)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(420)x*S(12)) - S(5)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(252)x*S(16)) - S(7)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(90)xS(20)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(22), x), x, S(256)abS(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(969969)x*S(13)(a + bxS(2))) + S(32)abS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(6783)x*S(17)) + S(10)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(399)x*S(21)) - S(128)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(74613)x*S(13)) - S(16)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(969)x*S(17)) - S(29)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(399)xS(21)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(23), x), x, ab*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5544)x*S(14)(a + bxS(2))) + ab*S(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(264)x*S(18)) + a(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(44)xS(22)) - bS(4)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(792)xS(14)) - bS(2)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(72)x*S(18)) - S(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(44)xS(22)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(24), x), x, S(256)abS(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2028117)x*S(15)(a + bxS(2))) + S(160)abS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(52003)x*S(19)) + S(10)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(483)x*S(23)) - S(640)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(676039)x*S(15)) - S(80)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(6783)x*S(19)) - S(31)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(483)xS(23)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/x*S(25), x), x, ab*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(11088)x*S(16)(a + bxS(2))) + ab*S(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(396)x*S(20)) + S(5)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(264)xS(24)) - bS(4)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(1386)xS(16)) - bS(2)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(99)x*S(20)) - S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(33)x*S(24)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/sqrt(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, (dx)*(m + S(1))(a + bxS(2))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -bxS(2)/a)/(ad(m + S(1))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, a(S(3)/2)(a + bxS(2))atan(sqrt(b)x/sqrt(a))/(b(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - ax(a + bxS(2))/(bS(2)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + xS(3)(a + bxS(2))/(S(3)bsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, -a(a + bxS(2))log(a + bxS(2))/(S(2)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, -sqrt(a)(a + bxS(2))atan(sqrt(b)x/sqrt(a))/(b(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + x(a + bxS(2))/(bsqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, (a + bx*S(2))log(a + bxS(2))/(S(2)bsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, (a + bx*S(2))atan(sqrt(b)x/sqrt(a))/(sqrt(a)sqrt(b)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, (a + bx*S(2))log(x)/(asqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - (a + bx*S(2))log(a + bxS(2))/(S(2)asqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, -(a + bx*S(2))/(axsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(b)(a + bxS(2))atan(sqrt(b)x/sqrt(a))/(a(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, -b(a + bxS(2))log(x)/(a*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + b(a + bxS(2))log(a + bxS(2))/(S(2)a*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2)a*S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, -(a + bx*S(2))/(S(3)axS(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + b(a + bxS(2))/(aS(2)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + b(S(3)/2)(a + bxS(2))atan(sqrt(b)x/sqrt(a))/(a(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, (dx)(m + S(1))(a + bxS(2))hyper((S(3), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -bxS(2)/a)/(aS(3)d(m + S(1))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, ax(a + bx*S(2))/(S(4)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(5)x/(S(8)bS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (S(3)a + S(3)bx*S(2))atan(sqrt(b)x/sqrt(a))/(S(8)sqrt(a)b(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, xS(4)(a + bxS(2))/(S(4)a(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, xS(3)(a + bxS(2))/(S(4)a(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - x/(S(8)absqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (a + bx*S(2))atan(sqrt(b)x/sqrt(a))/(S(8)a*(S(3)/2)b*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, -(a + bx*S(2))/(S(4)b(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(-3)/2), x), x, x(a + bxS(2))/(S(4)a(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(3)x/(S(8)aS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (S(3)a + S(3)bx*S(2))atan(sqrt(b)x/sqrt(a))/(S(8)a*(S(5)/2)sqrt(b)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), x), x, (a + bx*S(2))/(S(4)a(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(1)/(S(2)a*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (a + bx*S(2))log(x)/(a*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - (a + bx*S(2))log(a + bxS(2))/(S(2)a*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), x), x, (a + bx*S(2))/(S(4)ax(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(5)/(S(8)a*S(2)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - (S(15)a + S(15)bx*S(2))/(S(8)a*S(3)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - S(15)sqrt(b)(a + bx*S(2))atan(sqrt(b)x/sqrt(a))/(S(8)a*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), x), x, (a + bx*S(2))/(S(4)axS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(3)/(S(4)a*S(2)x*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(3)b(a + bx*S(2))log(x)/(a*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(3)b(a + bx*S(2))log(a + bxS(2))/(S(2)a*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2)a*S(4)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), x), x, (a + bx*S(2))/(S(4)axS(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(7)/(S(8)a*S(2)x*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - (S(35)a + S(35)bx*S(2))/(S(24)a*S(3)x*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(35)b(a + bx*S(2))/(S(8)a*S(4)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + S(35)b*(S(3)/2)(a + bxS(2))atan(sqrt(b)x/sqrt(a))/(S(8)a*(S(9)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, (dx)(m + S(1))(a + bxS(2))hyper((S(5), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -bxS(2)/a)/(aS(5)d(m + S(1))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, ax*S(3)(a + bxS(2))/(S(8)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) - S(11)x*S(3)/(S(48)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(5)x*S(3)(a + bxS(2))/(S(64)abS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(5)x/(S(128)ab*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (S(5)a + S(5)bx*S(2))atan(sqrt(b)x/sqrt(a))/(S(128)a*(S(3)/2)b*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, xS(6)(a + bxS(2))/(S(8)a(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + xS(6)/(S(24)a*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, ax(a + bx*S(2))/(S(8)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) - S(3)x/(S(16)bS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + x(a + bxS(2))/(S(64)abS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(3)x/(S(128)aS(2)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (S(3)a + S(3)bx*S(2))atan(sqrt(b)x/sqrt(a))/(S(128)a*(S(5)/2)b*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, a(a + bxS(2))/(S(8)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) - S(1)/(S(6)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, xS(3)(a + bxS(2))/(S(8)a(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) - S(5)x/(S(48)ab(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(5)x(a + bx*S(2))/(S(192)a*S(2)b(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(5)x/(S(128)aS(3)bsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + (S(5)a + S(5)bx*S(2))atan(sqrt(b)x/sqrt(a))/(S(128)a*(S(7)/2)b*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, -(a + bx*S(2))/(S(8)b(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(-5)/2), x), x, x(a + bxS(2))/(S(8)a(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(7)x/(S(48)aS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(35)x(a + bx*S(2))/(S(192)a*S(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(35)x/(S(128)aS(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (S(35)a + S(35)bx*S(2))atan(sqrt(b)x/sqrt(a))/(S(128)a*(S(9)/2)sqrt(b)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)), x), x, (a + bx*S(2))/(S(8)a(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(1)/(S(6)a*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + (a + bx*S(2))/(S(4)a*S(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(1)/(S(2)a*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (a + bx*S(2))log(x)/(a*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - (a + bx*S(2))log(a + bxS(2))/(S(2)a*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)), x), x, (a + bx*S(2))/(S(8)ax(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(3)/(S(16)a*S(2)x(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + (S(21)a + S(21)bx*S(2))/(S(64)a*S(3)x(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(105)/(S(128)a*S(4)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - (S(315)a + S(315)bx*S(2))/(S(128)a*S(5)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - S(315)sqrt(b)(a + bx*S(2))atan(sqrt(b)x/sqrt(a))/(S(128)a*(S(11)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)), x), x, (a + bx*S(2))/(S(8)axS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(5)/(S(24)a*S(2)x*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + (S(5)a + S(5)bx*S(2))/(S(12)a*S(3)x*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(5)/(S(4)a*S(4)x*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(5)b(a + bx*S(2))log(x)/(a*S(6)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(5)b(a + bx*S(2))log(a + bxS(2))/(S(2)a*S(6)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2)a*S(6)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)), x), x, (a + bx*S(2))/(S(8)axS(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(11)/(S(48)a*S(2)x*S(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + (S(33)a + S(33)bx*S(2))/(S(64)a*S(3)x*S(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(231)/(S(128)a*S(4)x*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - (S(385)a + S(385)bx*S(2))/(S(128)a*S(5)x*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(1155)b(a + bx*S(2))/(S(128)a*S(6)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + S(1155)b*(S(3)/2)(a + bxS(2))atan(sqrt(b)x/sqrt(a))/(S(128)a*(S(13)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, S(2)a*S(2)(dx)(S(7)/2)/(S(7)d) + S(4)ab(dx)*(S(11)/2)/(S(11)d*S(3)) + S(2)b*S(2)(dx)(S(15)/2)/(S(15)d*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, S(2)a*S(2)(dx)(S(5)/2)/(S(5)d) + S(4)ab(dx)*(S(9)/2)/(S(9)d*S(3)) + S(2)b*S(2)(dx)(S(13)/2)/(S(13)d*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, S(2)a*S(2)(dx)(S(3)/2)/(S(3)d) + S(4)ab(dx)*(S(7)/2)/(S(7)d*S(3)) + S(2)b*S(2)(dx)(S(11)/2)/(S(11)dS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)x*S(4))/sqrt(dx), x), x, S(2)aS(2)sqrt(dx)/d + S(4)ab(dx)(S(5)/2)/(S(5)d*S(3)) + S(2)b*S(2)(dx)(S(9)/2)/(S(9)dS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)x*S(4))/(dx)*(S(3)/2), x), x, -S(2)a*S(2)/(dsqrt(dx)) + S(4)ab(dx)(S(3)/2)/(S(3)d*S(3)) + S(2)b*S(2)(dx)(S(7)/2)/(S(7)dS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)x*S(4))/(dx)*(S(5)/2), x), x, -S(2)a*S(2)/(S(3)d(dx)*(S(3)/2)) + S(4)absqrt(dx)/dS(3) + S(2)b*S(2)(dx)(S(5)/2)/(S(5)dS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)x*S(4))/(dx)*(S(7)/2), x), x, -S(2)a*S(2)/(S(5)d(dx)*(S(5)/2)) - S(4)ab/(dS(3)sqrt(dx)) + S(2)b*S(2)(dx)(S(3)/2)/(S(3)d*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, S(2)a*S(4)(dx)(S(7)/2)/(S(7)d) + S(8)aS(3)b(dx)*(S(11)/2)/(S(11)d*S(3)) + S(4)a*S(2)b*S(2)(dx)(S(15)/2)/(S(5)d*S(5)) + S(8)abS(3)(dx)(S(19)/2)/(S(19)d*S(7)) + S(2)b*S(4)(dx)(S(23)/2)/(S(23)d*S(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, S(2)a*S(4)(dx)(S(5)/2)/(S(5)d) + S(8)aS(3)b(dx)*(S(9)/2)/(S(9)d*S(3)) + S(12)a*S(2)b*S(2)(dx)(S(13)/2)/(S(13)d*S(5)) + S(8)abS(3)(dx)(S(17)/2)/(S(17)d*S(7)) + S(2)b*S(4)(dx)(S(21)/2)/(S(21)d*S(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, S(2)a*S(4)(dx)(S(3)/2)/(S(3)d) + S(8)aS(3)b(dx)*(S(7)/2)/(S(7)d*S(3)) + S(12)a*S(2)b*S(2)(dx)(S(11)/2)/(S(11)d*S(5)) + S(8)abS(3)(dx)(S(15)/2)/(S(15)d*S(7)) + S(2)b*S(4)(dx)(S(19)/2)/(S(19)dS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/sqrt(dx), x), x, S(2)a*S(4)sqrt(dx)/d + S(8)a*S(3)b(dx)*(S(5)/2)/(S(5)d*S(3)) + S(4)a*S(2)b*S(2)(dx)(S(9)/2)/(S(3)d*S(5)) + S(8)abS(3)(dx)(S(13)/2)/(S(13)d*S(7)) + S(2)b*S(4)(dx)(S(17)/2)/(S(17)dS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/(dx)(S(3)/2), x), x, -S(2)a*S(4)/(dsqrt(dx)) + S(8)a*S(3)b(dx)*(S(3)/2)/(S(3)d*S(3)) + S(12)a*S(2)b*S(2)(dx)(S(7)/2)/(S(7)d*S(5)) + S(8)abS(3)(dx)(S(11)/2)/(S(11)d*S(7)) + S(2)b*S(4)(dx)(S(15)/2)/(S(15)dS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/(dx)(S(5)/2), x), x, -S(2)a*S(4)/(S(3)d(dx)*(S(3)/2)) + S(8)a*S(3)bsqrt(dx)/d*S(3) + S(12)a*S(2)b*S(2)(dx)(S(5)/2)/(S(5)d*S(5)) + S(8)abS(3)(dx)(S(9)/2)/(S(9)d*S(7)) + S(2)b*S(4)(dx)(S(13)/2)/(S(13)dS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)/(dx)(S(7)/2), x), x, -S(2)a*S(4)/(S(5)d(dx)*(S(5)/2)) - S(8)a*S(3)b/(d*S(3)sqrt(dx)) + S(4)a*S(2)b*S(2)(dx)(S(3)/2)/dS(5) + S(8)abS(3)(dx)(S(7)/2)/(S(7)d*S(7)) + S(2)b*S(4)(dx)(S(11)/2)/(S(11)d*S(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, S(2)a*S(6)(dx)(S(7)/2)/(S(7)d) + S(12)aS(5)b(dx)*(S(11)/2)/(S(11)d*S(3)) + S(2)a*S(4)b*S(2)(dx)(S(15)/2)/dS(5) + S(40)a*S(3)b*S(3)(dx)(S(19)/2)/(S(19)d*S(7)) + S(30)a*S(2)b*S(4)(dx)(S(23)/2)/(S(23)d*S(9)) + S(4)abS(5)(dx)(S(27)/2)/(S(9)d*S(11)) + S(2)b*S(6)(dx)(S(31)/2)/(S(31)d*S(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, S(2)a*S(6)(dx)(S(5)/2)/(S(5)d) + S(4)aS(5)b(dx)*(S(9)/2)/(S(3)d*S(3)) + S(30)a*S(4)b*S(2)(dx)(S(13)/2)/(S(13)d*S(5)) + S(40)a*S(3)b*S(3)(dx)(S(17)/2)/(S(17)d*S(7)) + S(10)a*S(2)b*S(4)(dx)(S(21)/2)/(S(7)d*S(9)) + S(12)abS(5)(dx)(S(25)/2)/(S(25)d*S(11)) + S(2)b*S(6)(dx)(S(29)/2)/(S(29)d*S(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, S(2)a*S(6)(dx)(S(3)/2)/(S(3)d) + S(12)aS(5)b(dx)*(S(7)/2)/(S(7)d*S(3)) + S(30)a*S(4)b*S(2)(dx)(S(11)/2)/(S(11)d*S(5)) + S(8)a*S(3)b*S(3)(dx)(S(15)/2)/(S(3)d*S(7)) + S(30)a*S(2)b*S(4)(dx)(S(19)/2)/(S(19)d*S(9)) + S(12)abS(5)(dx)(S(23)/2)/(S(23)d*S(11)) + S(2)b*S(6)(dx)(S(27)/2)/(S(27)dS(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/sqrt(dx), x), x, S(2)a*S(6)sqrt(dx)/d + S(12)a*S(5)b(dx)*(S(5)/2)/(S(5)d*S(3)) + S(10)a*S(4)b*S(2)(dx)(S(9)/2)/(S(3)d*S(5)) + S(40)a*S(3)b*S(3)(dx)(S(13)/2)/(S(13)d*S(7)) + S(30)a*S(2)b*S(4)(dx)(S(17)/2)/(S(17)d*S(9)) + S(4)abS(5)(dx)(S(21)/2)/(S(7)d*S(11)) + S(2)b*S(6)(dx)(S(25)/2)/(S(25)dS(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/(dx)(S(3)/2), x), x, -S(2)a*S(6)/(dsqrt(dx)) + S(4)a*S(5)b(dx)(S(3)/2)/dS(3) + S(30)aS(4)b*S(2)(dx)(S(7)/2)/(S(7)d*S(5)) + S(40)a*S(3)b*S(3)(dx)(S(11)/2)/(S(11)d*S(7)) + S(2)a*S(2)b*S(4)(dx)(S(15)/2)/dS(9) + S(12)abS(5)(dx)(S(19)/2)/(S(19)d*S(11)) + S(2)b*S(6)(dx)(S(23)/2)/(S(23)dS(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/(dx)(S(5)/2), x), x, -S(2)a*S(6)/(S(3)d(dx)*(S(3)/2)) + S(12)a*S(5)bsqrt(dx)/d*S(3) + S(6)a*S(4)b*S(2)(dx)(S(5)/2)/dS(5) + S(40)a*S(3)b*S(3)(dx)(S(9)/2)/(S(9)d*S(7)) + S(30)a*S(2)b*S(4)(dx)(S(13)/2)/(S(13)d*S(9)) + S(12)abS(5)(dx)(S(17)/2)/(S(17)d*S(11)) + S(2)b*S(6)(dx)(S(21)/2)/(S(21)dS(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)/(dx)(S(7)/2), x), x, -S(2)a*S(6)/(S(5)d(dx)*(S(5)/2)) - S(12)a*S(5)b/(d*S(3)sqrt(dx)) + S(10)a*S(4)b*S(2)(dx)(S(3)/2)/dS(5) + S(40)a*S(3)b*S(3)(dx)(S(7)/2)/(S(7)d*S(7)) + S(30)a*S(2)b*S(4)(dx)(S(11)/2)/(S(11)d*S(9)) + S(4)abS(5)(dx)(S(15)/2)/(S(5)d*S(11)) + S(2)b*S(6)(dx)(S(19)/2)/(S(19)d*S(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(11)/2)/(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, -S(9)sqrt(S(2))a(S(5)/4)d*(S(11)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)b(S(13)/4)) + S(9)sqrt(S(2))a(S(5)/4)d*(S(11)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)b(S(13)/4)) - S(9)sqrt(S(2))a(S(5)/4)d*(S(11)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)b(S(13)/4)) + S(9)sqrt(S(2))a(S(5)/4)d*(S(11)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)b(S(13)/4)) - S(9)adS(5)sqrt(dx)/(S(2)b*S(3)) - d(dx)(S(9)/2)/(S(2)b(a + bx*S(2))) + S(9)d*S(3)(dx)(S(5)/2)/(S(10)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(9)/2)/(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, -S(7)sqrt(S(2))a(S(3)/4)d*(S(9)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)b(S(11)/4)) + S(7)sqrt(S(2))a(S(3)/4)d*(S(9)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)b(S(11)/4)) + S(7)sqrt(S(2))a(S(3)/4)d*(S(9)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)b(S(11)/4)) - S(7)sqrt(S(2))a(S(3)/4)d*(S(9)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)b(S(11)/4)) - d(dx)(S(7)/2)/(S(2)b(a + bx*S(2))) + S(7)d*S(3)(dx)(S(3)/2)/(S(6)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(7)/2)/(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, S(5)sqrt(S(2))a(S(1)/4)d*(S(7)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)b(S(9)/4)) - S(5)sqrt(S(2))a(S(1)/4)d*(S(7)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)b(S(9)/4)) + S(5)sqrt(S(2))a(S(1)/4)d*(S(7)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)b(S(9)/4)) - S(5)sqrt(S(2))a(S(1)/4)d*(S(7)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)b(S(9)/4)) - d(dx)(S(5)/2)/(S(2)b(a + bx*S(2))) + S(5)d*S(3)sqrt(dx)/(S(2)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(5)/2)/(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, -d(dx)(S(3)/2)/(S(2)b(a + bx*S(2))) + S(3)sqrt(S(2))d(S(5)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(1)/4)b*(S(7)/4)) - S(3)sqrt(S(2))d(S(5)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(1)/4)b*(S(7)/4)) - S(3)sqrt(S(2))d(S(5)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(1)/4)b*(S(7)/4)) + S(3)sqrt(S(2))d(S(5)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(1)/4)b*(S(7)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(3)/2)/(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, -dsqrt(dx)/(S(2)b(a + bx*S(2))) - sqrt(S(2))d*(S(3)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(3)/4)b*(S(5)/4)) + sqrt(S(2))d*(S(3)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(3)/4)b*(S(5)/4)) - sqrt(S(2))d*(S(3)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(3)/4)b*(S(5)/4)) + sqrt(S(2))d*(S(3)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(3)/4)b*(S(5)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, (dx)*(S(3)/2)/(S(2)ad(a + bxS(2))) + sqrt(S(2))sqrt(d)log(-sqrt(S(2))a*(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(5)/4)b*(S(3)/4)) - sqrt(S(2))sqrt(d)log(sqrt(S(2))a*(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(5)/4)b*(S(3)/4)) - sqrt(S(2))sqrt(d)atan(S(1) - sqrt(S(2))b*(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(5)/4)b*(S(3)/4)) + sqrt(S(2))sqrt(d)atan(S(1) + sqrt(S(2))b*(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(5)/4)b*(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(dx)(aS(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, sqrt(dx)/(S(2)ad(a + bx*S(2))) - S(3)sqrt(S(2))log(-sqrt(S(2))a*(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(7)/4)b*(S(1)/4)sqrt(d)) + S(3)sqrt(S(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(7)/4)b*(S(1)/4)sqrt(d)) - S(3)sqrt(S(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(7)/4)b*(S(1)/4)sqrt(d)) + S(3)sqrt(S(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(7)/4)b*(S(1)/4)sqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, S(1)/(S(2)adsqrt(dx)(a + bxS(2))) - S(5)/(S(2)a*S(2)dsqrt(dx)) - S(5)sqrt(S(2))b*(S(1)/4)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(9)/4)d*(S(3)/2)) + S(5)sqrt(S(2))b(S(1)/4)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(9)/4)d*(S(3)/2)) + S(5)sqrt(S(2))b(S(1)/4)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(9)/4)d*(S(3)/2)) - S(5)sqrt(S(2))b(S(1)/4)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(9)/4)d*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, S(1)/(S(2)ad(dx)(S(3)/2)(a + bxS(2))) - S(7)/(S(6)a*S(2)d(dx)*(S(3)/2)) + S(7)sqrt(S(2))b(S(3)/4)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(11)/4)d*(S(5)/2)) - S(7)sqrt(S(2))b(S(3)/4)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(11)/4)d*(S(5)/2)) + S(7)sqrt(S(2))b(S(3)/4)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(11)/4)d*(S(5)/2)) - S(7)sqrt(S(2))b(S(3)/4)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(11)/4)d*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(7)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, S(1)/(S(2)ad(dx)(S(5)/2)(a + bxS(2))) - S(9)/(S(10)a*S(2)d(dx)*(S(5)/2)) + S(9)b/(S(2)aS(3)d*S(3)sqrt(dx)) + S(9)sqrt(S(2))b(S(5)/4)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(13)/4)d*(S(7)/2)) - S(9)sqrt(S(2))b(S(5)/4)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(16)a(S(13)/4)d*(S(7)/2)) - S(9)sqrt(S(2))b(S(5)/4)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(13)/4)d*(S(7)/2)) + S(9)sqrt(S(2))b(S(5)/4)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(8)a(S(13)/4)d*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(19)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -S(663)sqrt(S(2))a*(S(5)/4)d*(S(19)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)b(S(21)/4)) + S(663)sqrt(S(2))a(S(5)/4)d*(S(19)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)b(S(21)/4)) - S(663)sqrt(S(2))a(S(5)/4)d*(S(19)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)b(S(21)/4)) + S(663)sqrt(S(2))a(S(5)/4)d*(S(19)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)b(S(21)/4)) - S(663)adS(9)sqrt(dx)/(S(64)b*S(5)) - d(dx)(S(17)/2)/(S(6)b(a + bxS(2))S(3)) - S(17)dS(3)(dx)(S(13)/2)/(S(48)b*S(2)(a + bxS(2))S(2)) - S(221)d*S(5)(dx)(S(9)/2)/(S(192)b*S(3)(a + bxS(2))) + S(663)d*S(7)(dx)(S(5)/2)/(S(320)b*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(17)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -S(385)sqrt(S(2))a*(S(3)/4)d*(S(17)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)b(S(19)/4)) + S(385)sqrt(S(2))a(S(3)/4)d*(S(17)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)b(S(19)/4)) + S(385)sqrt(S(2))a(S(3)/4)d*(S(17)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)b(S(19)/4)) - S(385)sqrt(S(2))a(S(3)/4)d*(S(17)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)b(S(19)/4)) - d(dx)(S(15)/2)/(S(6)b(a + bxS(2))S(3)) - S(5)dS(3)(dx)(S(11)/2)/(S(16)b*S(2)(a + bxS(2))S(2)) - S(55)d*S(5)(dx)(S(7)/2)/(S(64)b*S(3)(a + bxS(2))) + S(385)d*S(7)(dx)(S(3)/2)/(S(192)b*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(15)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, S(195)sqrt(S(2))a*(S(1)/4)d*(S(15)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)b(S(17)/4)) - S(195)sqrt(S(2))a(S(1)/4)d*(S(15)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)b(S(17)/4)) + S(195)sqrt(S(2))a(S(1)/4)d*(S(15)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)b(S(17)/4)) - S(195)sqrt(S(2))a(S(1)/4)d*(S(15)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)b(S(17)/4)) - d(dx)(S(13)/2)/(S(6)b(a + bxS(2))S(3)) - S(13)dS(3)(dx)(S(9)/2)/(S(48)b*S(2)(a + bxS(2))S(2)) - S(39)d*S(5)(dx)(S(5)/2)/(S(64)b*S(3)(a + bxS(2))) + S(195)d*S(7)sqrt(dx)/(S(64)b*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(13)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -d(dx)*(S(11)/2)/(S(6)b(a + bxS(2))S(3)) - S(11)dS(3)(dx)(S(7)/2)/(S(48)b*S(2)(a + bxS(2))S(2)) - S(77)d*S(5)(dx)(S(3)/2)/(S(192)b*S(3)(a + bxS(2))) + S(77)sqrt(S(2))d(S(13)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(1)/4)b*(S(15)/4)) - S(77)sqrt(S(2))d(S(13)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(1)/4)b*(S(15)/4)) - S(77)sqrt(S(2))d(S(13)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(1)/4)b*(S(15)/4)) + S(77)sqrt(S(2))d(S(13)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(1)/4)b*(S(15)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(11)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -d(dx)*(S(9)/2)/(S(6)b(a + bxS(2))S(3)) - S(3)dS(3)(dx)(S(5)/2)/(S(16)b*S(2)(a + bxS(2))S(2)) - S(15)d*S(5)sqrt(dx)/(S(64)b*S(3)(a + bxS(2))) - S(15)sqrt(S(2))d(S(11)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(3)/4)b*(S(13)/4)) + S(15)sqrt(S(2))d(S(11)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(3)/4)b*(S(13)/4)) - S(15)sqrt(S(2))d(S(11)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(3)/4)b*(S(13)/4)) + S(15)sqrt(S(2))d(S(11)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(3)/4)b*(S(13)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(9)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -d(dx)*(S(7)/2)/(S(6)b(a + bxS(2))S(3)) - S(7)dS(3)(dx)(S(3)/2)/(S(48)b*S(2)(a + bxS(2))S(2)) + S(7)d*S(3)(dx)(S(3)/2)/(S(64)abS(2)(a + bxS(2))) + S(7)sqrt(S(2))d(S(9)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(5)/4)b*(S(11)/4)) - S(7)sqrt(S(2))d(S(9)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(5)/4)b*(S(11)/4)) - S(7)sqrt(S(2))d(S(9)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(5)/4)b*(S(11)/4)) + S(7)sqrt(S(2))d(S(9)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(5)/4)b*(S(11)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(7)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -d(dx)*(S(5)/2)/(S(6)b(a + bxS(2))S(3)) - S(5)dS(3)sqrt(dx)/(S(48)b*S(2)(a + bxS(2))S(2)) + S(5)d*S(3)sqrt(dx)/(S(192)abS(2)(a + bxS(2))) - S(5)sqrt(S(2))d(S(7)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(7)/4)b*(S(9)/4)) + S(5)sqrt(S(2))d(S(7)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(7)/4)b*(S(9)/4)) - S(5)sqrt(S(2))d(S(7)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(7)/4)b*(S(9)/4)) + S(5)sqrt(S(2))d(S(7)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(7)/4)b*(S(9)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(5)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -d(dx)*(S(3)/2)/(S(6)b(a + bxS(2))S(3)) + d(dx)*(S(3)/2)/(S(16)ab(a + bxS(2))S(2)) + S(5)d(dx)*(S(3)/2)/(S(64)a*S(2)b(a + bx*S(2))) + S(5)sqrt(S(2))d(S(5)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(9)/4)b*(S(7)/4)) - S(5)sqrt(S(2))d(S(5)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(9)/4)b*(S(7)/4)) - S(5)sqrt(S(2))d(S(5)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(9)/4)b*(S(7)/4)) + S(5)sqrt(S(2))d(S(5)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(9)/4)b*(S(7)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(3)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, -dsqrt(dx)/(S(6)b(a + bxS(2))S(3)) + dsqrt(dx)/(S(48)ab(a + bxS(2))S(2)) + S(7)dsqrt(dx)/(S(192)aS(2)b(a + bx*S(2))) - S(7)sqrt(S(2))d(S(3)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(11)/4)b*(S(5)/4)) + S(7)sqrt(S(2))d(S(3)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(11)/4)b*(S(5)/4)) - S(7)sqrt(S(2))d(S(3)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(11)/4)b*(S(5)/4)) + S(7)sqrt(S(2))d(S(3)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(11)/4)b*(S(5)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2), x), x, (dx)*(S(3)/2)/(S(6)ad(a + bxS(2))S(3)) + S(3)(dx)(S(3)/2)/(S(16)a*S(2)d(a + bxS(2))S(2)) + S(15)(dx)*(S(3)/2)/(S(64)a*S(3)d(a + bx*S(2))) + S(15)sqrt(S(2))sqrt(d)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(13)/4)b*(S(3)/4)) - S(15)sqrt(S(2))sqrt(d)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(13)/4)b*(S(3)/4)) - S(15)sqrt(S(2))sqrt(d)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(13)/4)b*(S(3)/4)) + S(15)sqrt(S(2))sqrt(d)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(13)/4)b*(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(dx)(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(2)), x), x, sqrt(dx)/(S(6)ad(a + bxS(2))S(3)) + S(11)sqrt(dx)/(S(48)aS(2)d(a + bxS(2))S(2)) + S(77)sqrt(dx)/(S(192)aS(3)d(a + bx*S(2))) - S(77)sqrt(S(2))log(-sqrt(S(2))a*(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(15)/4)b*(S(1)/4)sqrt(d)) + S(77)sqrt(S(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(15)/4)b*(S(1)/4)sqrt(d)) - S(77)sqrt(S(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(15)/4)b*(S(1)/4)sqrt(d)) + S(77)sqrt(S(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(15)/4)b*(S(1)/4)sqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2)), x), x, S(1)/(S(6)adsqrt(dx)(a + bxS(2))S(3)) + S(13)/(S(48)a*S(2)dsqrt(dx)(a + bxS(2))S(2)) + S(39)/(S(64)aS(3)dsqrt(dx)(a + bx*S(2))) - S(195)/(S(64)a*S(4)dsqrt(dx)) - S(195)sqrt(S(2))b*(S(1)/4)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(17)/4)d*(S(3)/2)) + S(195)sqrt(S(2))b(S(1)/4)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(17)/4)d*(S(3)/2)) + S(195)sqrt(S(2))b(S(1)/4)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(17)/4)d*(S(3)/2)) - S(195)sqrt(S(2))b(S(1)/4)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(17)/4)d*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2)), x), x, S(1)/(S(6)ad(dx)(S(3)/2)(a + bxS(2))S(3)) + S(5)/(S(16)a*S(2)d(dx)*(S(3)/2)(a + bxS(2))S(2)) + S(55)/(S(64)a*S(3)d(dx)*(S(3)/2)(a + bxS(2))) - S(385)/(S(192)a*S(4)d(dx)*(S(3)/2)) + S(385)sqrt(S(2))b(S(3)/4)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(19)/4)d*(S(5)/2)) - S(385)sqrt(S(2))b(S(3)/4)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(19)/4)d*(S(5)/2)) + S(385)sqrt(S(2))b(S(3)/4)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(19)/4)d*(S(5)/2)) - S(385)sqrt(S(2))b(S(3)/4)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(19)/4)d*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(7)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(2)), x), x, S(1)/(S(6)ad(dx)(S(5)/2)(a + bxS(2))S(3)) + S(17)/(S(48)a*S(2)d(dx)*(S(5)/2)(a + bxS(2))S(2)) + S(221)/(S(192)a*S(3)d(dx)*(S(5)/2)(a + bxS(2))) - S(663)/(S(320)a*S(4)d(dx)*(S(5)/2)) + S(663)b/(S(64)aS(5)d*S(3)sqrt(dx)) + S(663)sqrt(S(2))b(S(5)/4)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(21)/4)d*(S(7)/2)) - S(663)sqrt(S(2))b(S(5)/4)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(512)a(S(21)/4)d*(S(7)/2)) - S(663)sqrt(S(2))b(S(5)/4)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(21)/4)d*(S(7)/2)) + S(663)sqrt(S(2))b(S(5)/4)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(256)a(S(21)/4)d*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(27)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -S(69615)sqrt(S(2))a*(S(5)/4)d*(S(27)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)b(S(29)/4)) + S(69615)sqrt(S(2))a(S(5)/4)d*(S(27)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)b(S(29)/4)) - S(69615)sqrt(S(2))a(S(5)/4)d*(S(27)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)b(S(29)/4)) + S(69615)sqrt(S(2))a(S(5)/4)d*(S(27)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)b(S(29)/4)) - S(69615)adS(13)sqrt(dx)/(S(4096)b*S(7)) - d(dx)(S(25)/2)/(S(10)b(a + bxS(2))S(5)) - S(5)dS(3)(dx)(S(21)/2)/(S(32)b*S(2)(a + bxS(2))S(4)) - S(35)d*S(5)(dx)(S(17)/2)/(S(128)b*S(3)(a + bxS(2))S(3)) - S(595)d*S(7)(dx)(S(13)/2)/(S(1024)b*S(4)(a + bxS(2))S(2)) - S(7735)d*S(9)(dx)(S(9)/2)/(S(4096)b*S(5)(a + bxS(2))) + S(13923)d*S(11)(dx)(S(5)/2)/(S(4096)b*S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(25)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -S(33649)sqrt(S(2))a*(S(3)/4)d*(S(25)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)b(S(27)/4)) + S(33649)sqrt(S(2))a(S(3)/4)d*(S(25)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)b(S(27)/4)) + S(33649)sqrt(S(2))a(S(3)/4)d*(S(25)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)b(S(27)/4)) - S(33649)sqrt(S(2))a(S(3)/4)d*(S(25)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)b(S(27)/4)) - d(dx)(S(23)/2)/(S(10)b(a + bxS(2))S(5)) - S(23)dS(3)(dx)(S(19)/2)/(S(160)b*S(2)(a + bxS(2))S(4)) - S(437)d*S(5)(dx)(S(15)/2)/(S(1920)b*S(3)(a + bxS(2))S(3)) - S(437)d*S(7)(dx)(S(11)/2)/(S(1024)b*S(4)(a + bxS(2))S(2)) - S(4807)d*S(9)(dx)(S(7)/2)/(S(4096)b*S(5)(a + bxS(2))) + S(33649)d*S(11)(dx)(S(3)/2)/(S(12288)b*S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(23)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, S(13923)sqrt(S(2))a*(S(1)/4)d*(S(23)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)b(S(25)/4)) - S(13923)sqrt(S(2))a(S(1)/4)d*(S(23)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)b(S(25)/4)) + S(13923)sqrt(S(2))a(S(1)/4)d*(S(23)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)b(S(25)/4)) - S(13923)sqrt(S(2))a(S(1)/4)d*(S(23)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)b(S(25)/4)) - d(dx)(S(21)/2)/(S(10)b(a + bxS(2))S(5)) - S(21)dS(3)(dx)(S(17)/2)/(S(160)b*S(2)(a + bxS(2))S(4)) - S(119)d*S(5)(dx)(S(13)/2)/(S(640)b*S(3)(a + bxS(2))S(3)) - S(1547)d*S(7)(dx)(S(9)/2)/(S(5120)b*S(4)(a + bxS(2))S(2)) - S(13923)d*S(9)(dx)(S(5)/2)/(S(20480)b*S(5)(a + bxS(2))) + S(13923)d*S(11)sqrt(dx)/(S(4096)b*S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(21)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -d(dx)*(S(19)/2)/(S(10)b(a + bxS(2))S(5)) - S(19)dS(3)(dx)(S(15)/2)/(S(160)b*S(2)(a + bxS(2))S(4)) - S(19)d*S(5)(dx)(S(11)/2)/(S(128)b*S(3)(a + bxS(2))S(3)) - S(209)d*S(7)(dx)(S(7)/2)/(S(1024)b*S(4)(a + bxS(2))S(2)) - S(1463)d*S(9)(dx)(S(3)/2)/(S(4096)b*S(5)(a + bxS(2))) + S(4389)sqrt(S(2))d(S(21)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(1)/4)b*(S(23)/4)) - S(4389)sqrt(S(2))d(S(21)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(1)/4)b*(S(23)/4)) - S(4389)sqrt(S(2))d(S(21)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(1)/4)b*(S(23)/4)) + S(4389)sqrt(S(2))d(S(21)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(1)/4)b*(S(23)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(19)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -d(dx)*(S(17)/2)/(S(10)b(a + bxS(2))S(5)) - S(17)dS(3)(dx)(S(13)/2)/(S(160)b*S(2)(a + bxS(2))S(4)) - S(221)d*S(5)(dx)(S(9)/2)/(S(1920)b*S(3)(a + bxS(2))S(3)) - S(663)d*S(7)(dx)(S(5)/2)/(S(5120)b*S(4)(a + bxS(2))S(2)) - S(663)d*S(9)sqrt(dx)/(S(4096)b*S(5)(a + bxS(2))) - S(663)sqrt(S(2))d(S(19)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(3)/4)b*(S(21)/4)) + S(663)sqrt(S(2))d(S(19)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(3)/4)b*(S(21)/4)) - S(663)sqrt(S(2))d(S(19)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(3)/4)b*(S(21)/4)) + S(663)sqrt(S(2))d(S(19)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(3)/4)b*(S(21)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(17)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -d(dx)*(S(15)/2)/(S(10)b(a + bxS(2))S(5)) - S(3)dS(3)(dx)(S(11)/2)/(S(32)b*S(2)(a + bxS(2))S(4)) - S(11)d*S(5)(dx)(S(7)/2)/(S(128)b*S(3)(a + bxS(2))S(3)) - S(77)d*S(7)(dx)(S(3)/2)/(S(1024)b*S(4)(a + bxS(2))S(2)) + S(231)d*S(7)(dx)(S(3)/2)/(S(4096)abS(4)(a + bxS(2))) + S(231)sqrt(S(2))d(S(17)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(5)/4)b*(S(19)/4)) - S(231)sqrt(S(2))d(S(17)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(5)/4)b*(S(19)/4)) - S(231)sqrt(S(2))d(S(17)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(5)/4)b*(S(19)/4)) + S(231)sqrt(S(2))d(S(17)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(5)/4)b*(S(19)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(15)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -d(dx)*(S(13)/2)/(S(10)b(a + bxS(2))S(5)) - S(13)dS(3)(dx)(S(9)/2)/(S(160)b*S(2)(a + bxS(2))S(4)) - S(39)d*S(5)(dx)(S(5)/2)/(S(640)b*S(3)(a + bxS(2))S(3)) - S(39)d*S(7)sqrt(dx)/(S(1024)b*S(4)(a + bxS(2))S(2)) + S(39)d*S(7)sqrt(dx)/(S(4096)abS(4)(a + bxS(2))) - S(117)sqrt(S(2))d(S(15)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(7)/4)b*(S(17)/4)) + S(117)sqrt(S(2))d(S(15)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(7)/4)b*(S(17)/4)) - S(117)sqrt(S(2))d(S(15)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(7)/4)b*(S(17)/4)) + S(117)sqrt(S(2))d(S(15)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(7)/4)b*(S(17)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(13)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -d(dx)*(S(11)/2)/(S(10)b(a + bxS(2))S(5)) - S(11)dS(3)(dx)(S(7)/2)/(S(160)b*S(2)(a + bxS(2))S(4)) - S(77)d*S(5)(dx)(S(3)/2)/(S(1920)b*S(3)(a + bxS(2))S(3)) + S(77)d*S(5)(dx)(S(3)/2)/(S(5120)abS(3)(a + bxS(2))S(2)) + S(77)d*S(5)(dx)(S(3)/2)/(S(4096)a*S(2)b*S(3)(a + bxS(2))) + S(77)sqrt(S(2))d(S(13)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(9)/4)b*(S(15)/4)) - S(77)sqrt(S(2))d(S(13)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(9)/4)b*(S(15)/4)) - S(77)sqrt(S(2))d(S(13)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(9)/4)b*(S(15)/4)) + S(77)sqrt(S(2))d(S(13)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(9)/4)b*(S(15)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(11)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -d(dx)*(S(9)/2)/(S(10)b(a + bxS(2))S(5)) - S(9)dS(3)(dx)(S(5)/2)/(S(160)b*S(2)(a + bxS(2))S(4)) - S(3)d*S(5)sqrt(dx)/(S(128)b*S(3)(a + bxS(2))S(3)) + S(3)d*S(5)sqrt(dx)/(S(1024)abS(3)(a + bxS(2))S(2)) + S(21)d*S(5)sqrt(dx)/(S(4096)a*S(2)b*S(3)(a + bxS(2))) - S(63)sqrt(S(2))d(S(11)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(11)/4)b*(S(13)/4)) + S(63)sqrt(S(2))d(S(11)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(11)/4)b*(S(13)/4)) - S(63)sqrt(S(2))d(S(11)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(11)/4)b*(S(13)/4)) + S(63)sqrt(S(2))d(S(11)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(11)/4)b*(S(13)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(9)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -d(dx)*(S(7)/2)/(S(10)b(a + bxS(2))S(5)) - S(7)dS(3)(dx)(S(3)/2)/(S(160)b*S(2)(a + bxS(2))S(4)) + S(7)d*S(3)(dx)(S(3)/2)/(S(640)abS(2)(a + bxS(2))S(3)) + S(63)d*S(3)(dx)(S(3)/2)/(S(5120)a*S(2)b*S(2)(a + bxS(2))S(2)) + S(63)d*S(3)(dx)(S(3)/2)/(S(4096)a*S(3)b*S(2)(a + bxS(2))) + S(63)sqrt(S(2))d(S(9)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(13)/4)b*(S(11)/4)) - S(63)sqrt(S(2))d(S(9)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(13)/4)b*(S(11)/4)) - S(63)sqrt(S(2))d(S(9)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(13)/4)b*(S(11)/4)) + S(63)sqrt(S(2))d(S(9)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(13)/4)b*(S(11)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(7)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -d(dx)*(S(5)/2)/(S(10)b(a + bxS(2))S(5)) - d*S(3)sqrt(dx)/(S(32)b*S(2)(a + bxS(2))S(4)) + dS(3)sqrt(dx)/(S(384)abS(2)(a + bxS(2))S(3)) + S(11)d*S(3)sqrt(dx)/(S(3072)a*S(2)b*S(2)(a + bxS(2))S(2)) + S(77)d*S(3)sqrt(dx)/(S(12288)a*S(3)b*S(2)(a + bxS(2))) - S(77)sqrt(S(2))d(S(7)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(15)/4)b*(S(9)/4)) + S(77)sqrt(S(2))d(S(7)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(15)/4)b*(S(9)/4)) - S(77)sqrt(S(2))d(S(7)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(15)/4)b*(S(9)/4)) + S(77)sqrt(S(2))d(S(7)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(15)/4)b*(S(9)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(5)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -d(dx)*(S(3)/2)/(S(10)b(a + bxS(2))S(5)) + S(3)d(dx)(S(3)/2)/(S(160)ab(a + bxS(2))S(4)) + S(13)d(dx)*(S(3)/2)/(S(640)a*S(2)b(a + bxS(2))S(3)) + S(117)d(dx)(S(3)/2)/(S(5120)a*S(3)b(a + bxS(2))S(2)) + S(117)d(dx)(S(3)/2)/(S(4096)a*S(4)b(a + bx*S(2))) + S(117)sqrt(S(2))d(S(5)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(17)/4)b*(S(7)/4)) - S(117)sqrt(S(2))d(S(5)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(17)/4)b*(S(7)/4)) - S(117)sqrt(S(2))d(S(5)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(17)/4)b*(S(7)/4)) + S(117)sqrt(S(2))d(S(5)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(17)/4)b*(S(7)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(3)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, -dsqrt(dx)/(S(10)b(a + bxS(2))S(5)) + dsqrt(dx)/(S(160)ab(a + bxS(2))S(4)) + dsqrt(dx)/(S(128)a*S(2)b(a + bxS(2))S(3)) + S(11)dsqrt(dx)/(S(1024)a*S(3)b(a + bxS(2))S(2)) + S(77)dsqrt(dx)/(S(4096)a*S(4)b(a + bx*S(2))) - S(231)sqrt(S(2))d(S(3)/2)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(19)/4)b*(S(5)/4)) + S(231)sqrt(S(2))d(S(3)/2)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(19)/4)b*(S(5)/4)) - S(231)sqrt(S(2))d(S(3)/2)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(19)/4)b*(S(5)/4)) + S(231)sqrt(S(2))d(S(3)/2)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(19)/4)b*(S(5)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3), x), x, (dx)*(S(3)/2)/(S(10)ad(a + bxS(2))S(5)) + S(17)(dx)(S(3)/2)/(S(160)a*S(2)d(a + bxS(2))S(4)) + S(221)(dx)*(S(3)/2)/(S(1920)a*S(3)d(a + bxS(2))S(3)) + S(663)(dx)*(S(3)/2)/(S(5120)a*S(4)d(a + bxS(2))S(2)) + S(663)(dx)*(S(3)/2)/(S(4096)a*S(5)d(a + bx*S(2))) + S(663)sqrt(S(2))sqrt(d)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(21)/4)b*(S(3)/4)) - S(663)sqrt(S(2))sqrt(d)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(21)/4)b*(S(3)/4)) - S(663)sqrt(S(2))sqrt(d)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(21)/4)b*(S(3)/4)) + S(663)sqrt(S(2))sqrt(d)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(21)/4)b*(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(dx)(aS(2) + S(2)abxS(2) + bS(2)xS(4))S(3)), x), x, sqrt(dx)/(S(10)ad(a + bxS(2))S(5)) + S(19)sqrt(dx)/(S(160)aS(2)d(a + bxS(2))S(4)) + S(19)sqrt(dx)/(S(128)aS(3)d(a + bxS(2))S(3)) + S(209)sqrt(dx)/(S(1024)aS(4)d(a + bxS(2))S(2)) + S(1463)sqrt(dx)/(S(4096)aS(5)d(a + bx*S(2))) - S(4389)sqrt(S(2))log(-sqrt(S(2))a*(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(23)/4)b*(S(1)/4)sqrt(d)) + S(4389)sqrt(S(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(23)/4)b*(S(1)/4)sqrt(d)) - S(4389)sqrt(S(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(23)/4)b*(S(1)/4)sqrt(d)) + S(4389)sqrt(S(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(23)/4)b*(S(1)/4)sqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3)), x), x, S(1)/(S(10)adsqrt(dx)(a + bxS(2))S(5)) + S(21)/(S(160)a*S(2)dsqrt(dx)(a + bxS(2))S(4)) + S(119)/(S(640)aS(3)dsqrt(dx)(a + bxS(2))S(3)) + S(1547)/(S(5120)aS(4)dsqrt(dx)(a + bxS(2))S(2)) + S(13923)/(S(20480)aS(5)dsqrt(dx)(a + bx*S(2))) - S(13923)/(S(4096)a*S(6)dsqrt(dx)) - S(13923)sqrt(S(2))b*(S(1)/4)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(25)/4)d*(S(3)/2)) + S(13923)sqrt(S(2))b(S(1)/4)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(25)/4)d*(S(3)/2)) + S(13923)sqrt(S(2))b(S(1)/4)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(25)/4)d*(S(3)/2)) - S(13923)sqrt(S(2))b(S(1)/4)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(25)/4)d*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3)), x), x, S(1)/(S(10)ad(dx)(S(3)/2)(a + bxS(2))S(5)) + S(23)/(S(160)a*S(2)d(dx)*(S(3)/2)(a + bxS(2))S(4)) + S(437)/(S(1920)a*S(3)d(dx)*(S(3)/2)(a + bxS(2))S(3)) + S(437)/(S(1024)a*S(4)d(dx)*(S(3)/2)(a + bxS(2))S(2)) + S(4807)/(S(4096)a*S(5)d(dx)*(S(3)/2)(a + bxS(2))) - S(33649)/(S(12288)a*S(6)d(dx)*(S(3)/2)) + S(33649)sqrt(S(2))b(S(3)/4)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(27)/4)d*(S(5)/2)) - S(33649)sqrt(S(2))b(S(3)/4)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(27)/4)d*(S(5)/2)) + S(33649)sqrt(S(2))b(S(3)/4)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(27)/4)d*(S(5)/2)) - S(33649)sqrt(S(2))b(S(3)/4)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(27)/4)d*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(7)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))S(3)), x), x, S(1)/(S(10)ad(dx)(S(5)/2)(a + bxS(2))S(5)) + S(5)/(S(32)a*S(2)d(dx)*(S(5)/2)(a + bxS(2))S(4)) + S(35)/(S(128)a*S(3)d(dx)*(S(5)/2)(a + bxS(2))S(3)) + S(595)/(S(1024)a*S(4)d(dx)*(S(5)/2)(a + bxS(2))S(2)) + S(7735)/(S(4096)a*S(5)d(dx)*(S(5)/2)(a + bxS(2))) - S(13923)/(S(4096)a*S(6)d(dx)*(S(5)/2)) + S(69615)b/(S(4096)aS(7)d*S(3)sqrt(dx)) + S(69615)sqrt(S(2))b(S(5)/4)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(29)/4)d*(S(7)/2)) - S(69615)sqrt(S(2))b(S(5)/4)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(32768)a(S(29)/4)d*(S(7)/2)) - S(69615)sqrt(S(2))b(S(5)/4)atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(29)/4)d*(S(7)/2)) + S(69615)sqrt(S(2))b(S(5)/4)atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(16384)a(S(29)/4)d*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, S(8)a(dx)*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(77)d(a + bx*S(2))) + S(2)(dx)(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(11)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, S(8)a(dx)*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(45)d(a + bx*S(2))) + S(2)(dx)(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(9)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, S(8)a(dx)*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(21)d(a + bx*S(2))) + S(2)(dx)(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(7)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/sqrt(dx), x), x, S(8)asqrt(dx)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5)d(a + bx*S(2))) + S(2)sqrt(dx)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(dx)*(S(3)/2), x), x, -S(8)asqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(3)dsqrt(dx)(a + bx*S(2))) + S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(3)dsqrt(dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(dx)*(S(5)/2), x), x, -S(8)asqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(3)d(dx)*(S(3)/2)(a + bxS(2))) + S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(d(dx)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(dx)*(S(7)/2), x), x, S(8)asqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5)d(dx)*(S(5)/2)(a + bxS(2))) - S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(d(dx)(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, S(256)a*S(3)(dx)(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(7315)d(a + bx*S(2))) + S(64)a*S(2)(dx)(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(1045)d) + S(8)a(dx)(S(7)/2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(95)d) + S(2)(dx)*(S(7)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(19)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, S(256)a*S(3)(dx)(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(3315)d(a + bx*S(2))) + S(64)a*S(2)(dx)(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(663)d) + S(24)a(dx)(S(5)/2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(221)d) + S(2)(dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(17)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, S(256)a*S(3)(dx)(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(1155)d(a + bx*S(2))) + S(64)a*S(2)(dx)(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(385)d) + S(8)a(dx)(S(3)/2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(55)d) + S(2)(dx)*(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(15)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/sqrt(dx), x), x, S(256)aS(3)sqrt(dx)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(195)d(a + bx*S(2))) + S(64)a*S(2)sqrt(dx)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(195)d) + S(8)asqrt(dx)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(39)d) + S(2)sqrt(dx)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(13)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(dx)*(S(3)/2), x), x, -S(256)a*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(77)dsqrt(dx)(a + bx*S(2))) + S(64)a*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(77)dsqrt(dx)) + S(24)a(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(77)dsqrt(dx)) + S(2)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(11)dsqrt(dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(dx)*(S(5)/2), x), x, -S(256)a*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(45)d(dx)*(S(3)/2)(a + bxS(2))) + S(64)a*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(15)d(dx)*(S(3)/2)) + S(8)a(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(15)d(dx)*(S(3)/2)) + S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(9)d(dx)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(dx)(S(7)/2), x), x, S(256)a*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(35)d(dx)*(S(5)/2)(a + bxS(2))) - S(64)a*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(7)d(dx)*(S(5)/2)) + S(8)a(a + bx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(7)d(dx)*(S(5)/2)) + S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(7)d(dx)*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, S(16384)a*S(5)(dx)(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(908523)d(a + bx*S(2))) + S(4096)a*S(4)(dx)(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(129789)d) + S(512)aS(3)(dx)(S(7)/2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(11799)d) + S(640)aS(2)(dx)(S(7)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(11799)d) + S(40)a(dx)(S(7)/2)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(621)d) + S(2)(dx)*(S(7)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(27)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, S(16384)a*S(5)(dx)(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(348075)d(a + bx*S(2))) + S(4096)a*S(4)(dx)(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(69615)d) + S(512)aS(3)(dx)(S(5)/2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(7735)d) + S(128)aS(2)(dx)(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(1785)d) + S(8)a(dx)(S(5)/2)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(105)d) + S(2)(dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(25)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, S(16384)a*S(5)(dx)(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(100947)d(a + bx*S(2))) + S(4096)a*S(4)(dx)(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(33649)d) + S(512)aS(3)(dx)(S(3)/2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4807)d) + S(128)aS(2)(dx)(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(1311)d) + S(40)a(dx)(S(3)/2)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(437)d) + S(2)(dx)*(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(23)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/sqrt(dx), x), x, S(16384)aS(5)sqrt(dx)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(13923)d(a + bx*S(2))) + S(4096)a*S(4)sqrt(dx)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(13923)d) + S(2560)aS(3)sqrt(dx)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(13923)d) + S(640)aS(2)sqrt(dx)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(4641)d) + S(40)asqrt(dx)(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(357)d) + S(2)sqrt(dx)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(21)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(dx)*(S(3)/2), x), x, -S(16384)a*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4389)dsqrt(dx)(a + bx*S(2))) + S(4096)a*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4389)dsqrt(dx)) + S(512)aS(3)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(1463)dsqrt(dx)) + S(128)aS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(627)dsqrt(dx)) + S(8)a(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(57)dsqrt(dx)) + S(2)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(19)dsqrt(dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(dx)*(S(5)/2), x), x, -S(16384)a*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(1989)d(dx)*(S(3)/2)(a + bxS(2))) + S(4096)a*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(663)d(dx)*(S(3)/2)) + S(512)a*S(3)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(663)d(dx)*(S(3)/2)) + S(640)a*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(1989)d(dx)*(S(3)/2)) + S(40)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(221)d(dx)*(S(3)/2)) + S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(17)d(dx)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(dx)(S(7)/2), x), x, S(16384)a*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(1155)d(dx)*(S(5)/2)(a + bxS(2))) - S(4096)a*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(231)d(dx)*(S(5)/2)) + S(512)a*S(3)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(231)d(dx)*(S(5)/2)) + S(128)a*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(231)d(dx)*(S(5)/2)) + S(8)a(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)/(S(33)d(dx)*(S(5)/2)) + S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)/(S(15)d(dx)*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(7)/2)/sqrt(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, -sqrt(S(2))a*(S(5)/4)d*(S(7)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)b(S(9)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))a*(S(5)/4)d*(S(7)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)b(S(9)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))a*(S(5)/4)d*(S(7)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)b(S(9)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))a*(S(5)/4)d*(S(7)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)b(S(9)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(2)adS(3)sqrt(dx)(a + bxS(2))/(bS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(2)d(dx)*(S(5)/2)(a + bxS(2))/(S(5)bsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(5)/2)/sqrt(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, -sqrt(S(2))a*(S(3)/4)d*(S(5)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)b(S(7)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))a*(S(3)/4)d*(S(5)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)b(S(7)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))a*(S(3)/4)d*(S(5)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)b(S(7)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))a*(S(3)/4)d*(S(5)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)b(S(7)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(2)d(dx)*(S(3)/2)(a + bxS(2))/(S(3)bsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(3)/2)/sqrt(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, sqrt(S(2))a*(S(1)/4)d*(S(3)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)b(S(5)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))a*(S(1)/4)d*(S(3)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)b(S(5)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))a*(S(1)/4)d*(S(3)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)b(S(5)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))a*(S(1)/4)d*(S(3)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)b(S(5)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(2)dsqrt(dx)(a + bx*S(2))/(bsqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)/sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, sqrt(S(2))sqrt(d)(a + bx*S(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)a(S(1)/4)b*(S(3)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))sqrt(d)(a + bx*S(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)a(S(1)/4)b*(S(3)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))sqrt(d)(a + bx*S(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)a(S(1)/4)b*(S(3)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))sqrt(d)(a + bx*S(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)a(S(1)/4)b*(S(3)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(dx)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, -sqrt(S(2))(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)a(S(3)/4)b*(S(1)/4)sqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)a(S(3)/4)b*(S(1)/4)sqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)a(S(3)/4)b*(S(1)/4)sqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)a(S(3)/4)b*(S(1)/4)sqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, (-S(2)a - S(2)bx*S(2))/(adsqrt(dx)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))b*(S(1)/4)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)a(S(5)/4)d*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))b*(S(1)/4)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)a(S(5)/4)d*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))b*(S(1)/4)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)a(S(5)/4)d*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))b*(S(1)/4)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)a(S(5)/4)d*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, (-S(2)a - S(2)bx*S(2))/(S(3)ad(dx)(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))b*(S(3)/4)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)a(S(7)/4)d*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))b*(S(3)/4)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)a(S(7)/4)d*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))b*(S(3)/4)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)a(S(7)/4)d*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))b*(S(3)/4)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)a(S(7)/4)d*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, (-S(2)a - S(2)bx*S(2))/(S(5)ad(dx)(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(2)b(a + bxS(2))/(aS(2)dS(3)sqrt(dx)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))b*(S(5)/4)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)a(S(9)/4)d*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))b*(S(5)/4)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(4)a(S(9)/4)d*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))b*(S(5)/4)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)a(S(9)/4)d*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))b*(S(5)/4)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(2)a(S(9)/4)d*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(15)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, -S(117)sqrt(S(2))a*(S(5)/4)d*(S(15)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)b(S(17)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(117)sqrt(S(2))a(S(5)/4)d*(S(15)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)b(S(17)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(117)sqrt(S(2))a(S(5)/4)d*(S(15)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)b(S(17)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(117)sqrt(S(2))a(S(5)/4)d*(S(15)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)b(S(17)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + ad*S(3)(dx)(S(9)/2)(a + bxS(2))/(S(4)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(117)adS(7)sqrt(dx)(a + bxS(2))/(S(16)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(17)d*S(3)(dx)(S(9)/2)/(S(16)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(117)d*S(5)(dx)(S(5)/2)(a + bxS(2))/(S(80)b*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(13)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, -S(77)sqrt(S(2))a*(S(3)/4)d*(S(13)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)b(S(15)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(77)sqrt(S(2))a(S(3)/4)d*(S(13)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)b(S(15)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(77)sqrt(S(2))a(S(3)/4)d*(S(13)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)b(S(15)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(77)sqrt(S(2))a(S(3)/4)d*(S(13)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)b(S(15)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + ad*S(3)(dx)(S(7)/2)(a + bxS(2))/(S(4)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(15)d*S(3)(dx)(S(7)/2)/(S(16)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(77)d*S(5)(dx)(S(3)/2)(a + bxS(2))/(S(48)b*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(11)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, S(45)sqrt(S(2))a*(S(1)/4)d*(S(11)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)b(S(13)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(45)sqrt(S(2))a(S(1)/4)d*(S(11)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)b(S(13)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(45)sqrt(S(2))a(S(1)/4)d*(S(11)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)b(S(13)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(45)sqrt(S(2))a(S(1)/4)d*(S(11)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)b(S(13)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + ad*S(3)(dx)(S(5)/2)(a + bxS(2))/(S(4)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(13)d*S(3)(dx)(S(5)/2)/(S(16)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(45)d*S(5)sqrt(dx)(a + bxS(2))/(S(16)b*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(9)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, adS(3)(dx)(S(3)/2)(a + bxS(2))/(S(4)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(11)d*S(3)(dx)(S(3)/2)/(S(16)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(21)sqrt(S(2))d(S(9)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(1)/4)b*(S(11)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(21)sqrt(S(2))d(S(9)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(1)/4)b*(S(11)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(21)sqrt(S(2))d(S(9)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(1)/4)b*(S(11)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(21)sqrt(S(2))d(S(9)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(1)/4)b*(S(11)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(7)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, adS(3)sqrt(dx)(a + bxS(2))/(S(4)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(9)d*S(3)sqrt(dx)/(S(16)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(5)sqrt(S(2))d(S(7)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(3)/4)b*(S(9)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(5)sqrt(S(2))d(S(7)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(3)/4)b*(S(9)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(5)sqrt(S(2))d(S(7)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(3)/4)b*(S(9)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(5)sqrt(S(2))d(S(7)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(3)/4)b*(S(9)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(5)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, (dx)(S(7)/2)(a + bxS(2))/(S(4)ad(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - d(dx)(S(3)/2)/(S(16)absqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(3)sqrt(S(2))d(S(5)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(5)/4)b*(S(7)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(3)sqrt(S(2))d(S(5)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(5)/4)b*(S(7)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(3)sqrt(S(2))d(S(5)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(5)/4)b*(S(7)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(3)sqrt(S(2))d(S(5)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(5)/4)b*(S(7)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(3)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, (dx)(S(5)/2)(a + bxS(2))/(S(4)ad(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(3)dsqrt(dx)/(S(16)absqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - S(3)sqrt(S(2))d(S(3)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(7)/4)b*(S(5)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(3)sqrt(S(2))d(S(3)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(7)/4)b*(S(5)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(3)sqrt(S(2))d(S(3)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(7)/4)b*(S(5)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(3)sqrt(S(2))d(S(3)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(7)/4)b*(S(5)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, (dx)*(S(3)/2)(a + bxS(2))/(S(4)ad(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(5)(dx)(S(3)/2)/(S(16)a*S(2)dsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + S(5)sqrt(S(2))sqrt(d)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(9)/4)b*(S(3)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(5)sqrt(S(2))sqrt(d)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(9)/4)b*(S(3)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(5)sqrt(S(2))sqrt(d)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(9)/4)b*(S(3)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(5)sqrt(S(2))sqrt(d)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(9)/4)b*(S(3)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(dx)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), x), x, sqrt(dx)(a + bx*S(2))/(S(4)ad(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(7)sqrt(dx)/(S(16)a*S(2)dsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))(S(21)a + S(21)bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(11)/4)b*(S(1)/4)sqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))(S(21)a + S(21)bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(11)/4)b*(S(1)/4)sqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))(S(21)a + S(21)bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(11)/4)b*(S(1)/4)sqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))(S(21)a + S(21)bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(11)/4)b*(S(1)/4)sqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), x), x, (a + bx*S(2))/(S(4)adsqrt(dx)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(9)/(S(16)a*S(2)dsqrt(dx)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - (S(45)a + S(45)bx*S(2))/(S(16)a*S(3)dsqrt(dx)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - S(45)sqrt(S(2))b(S(1)/4)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(13)/4)d*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(45)sqrt(S(2))b(S(1)/4)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(13)/4)d*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(45)sqrt(S(2))b(S(1)/4)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(13)/4)d*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(45)sqrt(S(2))b(S(1)/4)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(13)/4)d*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), x), x, (a + bx*S(2))/(S(4)ad(dx)(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(11)/(S(16)a*S(2)d(dx)*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - (S(77)a + S(77)bx*S(2))/(S(48)a*S(3)d(dx)*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(77)sqrt(S(2))b(S(3)/4)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(15)/4)d*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(77)sqrt(S(2))b(S(3)/4)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(15)/4)d*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(77)sqrt(S(2))b(S(3)/4)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(15)/4)d*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(77)sqrt(S(2))b(S(3)/4)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(15)/4)d*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(7)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), x), x, (a + bx*S(2))/(S(4)ad(dx)(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(13)/(S(16)a*S(2)d(dx)*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - (S(117)a + S(117)bx*S(2))/(S(80)a*S(3)d(dx)*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(117)b(a + bx*S(2))/(S(16)a*S(4)d*S(3)sqrt(dx)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(117)sqrt(S(2))b(S(5)/4)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(17)/4)d*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(117)sqrt(S(2))b(S(5)/4)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(128)a(S(17)/4)d*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(117)sqrt(S(2))b(S(5)/4)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(17)/4)d*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(117)sqrt(S(2))b(S(5)/4)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(64)a(S(17)/4)d*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(23)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, -S(13923)sqrt(S(2))a*(S(5)/4)d*(S(23)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)b(S(25)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(13923)sqrt(S(2))a(S(5)/4)d*(S(23)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)b(S(25)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(13923)sqrt(S(2))a(S(5)/4)d*(S(23)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)b(S(25)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(13923)sqrt(S(2))a(S(5)/4)d*(S(23)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)b(S(25)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + ad*S(3)(dx)(S(17)/2)(a + bxS(2))/(S(8)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(119)adS(7)(dx)(S(9)/2)(a + bxS(2))/(S(256)b*S(4)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(13923)adS(11)sqrt(dx)(a + bxS(2))/(S(1024)b*S(6)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(11)d*S(3)(dx)(S(17)/2)/(S(32)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(2023)d*S(7)(dx)(S(9)/2)/(S(1024)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(13923)d*S(9)(dx)(S(5)/2)(a + bxS(2))/(S(5120)b*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(21)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, -S(7315)sqrt(S(2))a*(S(3)/4)d*(S(21)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)b(S(23)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(7315)sqrt(S(2))a(S(3)/4)d*(S(21)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)b(S(23)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(7315)sqrt(S(2))a(S(3)/4)d*(S(21)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)b(S(23)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(7315)sqrt(S(2))a(S(3)/4)d*(S(21)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)b(S(23)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + ad*S(3)(dx)(S(15)/2)(a + bxS(2))/(S(8)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(95)adS(7)(dx)(S(7)/2)(a + bxS(2))/(S(256)b*S(4)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(31)d*S(3)(dx)(S(15)/2)/(S(96)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(1425)d*S(7)(dx)(S(7)/2)/(S(1024)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(7315)d*S(9)(dx)(S(3)/2)(a + bxS(2))/(S(3072)b*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(19)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, S(3315)sqrt(S(2))a*(S(1)/4)d*(S(19)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)b(S(21)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(3315)sqrt(S(2))a(S(1)/4)d*(S(19)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)b(S(21)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(3315)sqrt(S(2))a(S(1)/4)d*(S(19)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)b(S(21)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(3315)sqrt(S(2))a(S(1)/4)d*(S(19)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)b(S(21)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + ad*S(3)(dx)(S(13)/2)(a + bxS(2))/(S(8)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(221)adS(7)(dx)(S(5)/2)(a + bxS(2))/(S(768)b*S(4)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(29)d*S(3)(dx)(S(13)/2)/(S(96)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(2873)d*S(7)(dx)(S(5)/2)/(S(3072)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(3315)d*S(9)sqrt(dx)(a + bxS(2))/(S(1024)b*S(5)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(17)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, adS(3)(dx)(S(11)/2)(a + bxS(2))/(S(8)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(55)adS(7)(dx)(S(3)/2)(a + bxS(2))/(S(256)b*S(4)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(9)d*S(3)(dx)(S(11)/2)/(S(32)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(605)d*S(7)(dx)(S(3)/2)/(S(1024)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(1155)sqrt(S(2))d(S(17)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(1)/4)b*(S(19)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(1155)sqrt(S(2))d(S(17)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(1)/4)b*(S(19)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(1155)sqrt(S(2))d(S(17)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(1)/4)b*(S(19)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(1155)sqrt(S(2))d(S(17)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(1)/4)b*(S(19)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(15)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, adS(3)(dx)(S(9)/2)(a + bxS(2))/(S(8)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(39)adS(7)sqrt(dx)(a + bxS(2))/(S(256)b*S(4)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(25)d*S(3)(dx)(S(9)/2)/(S(96)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(351)d*S(7)sqrt(dx)/(S(1024)b*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(195)sqrt(S(2))d(S(15)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(3)/4)b*(S(17)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(195)sqrt(S(2))d(S(15)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(3)/4)b*(S(17)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(195)sqrt(S(2))d(S(15)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(3)/4)b*(S(17)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(195)sqrt(S(2))d(S(15)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(3)/4)b*(S(17)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(13)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, adS(3)(dx)(S(7)/2)(a + bxS(2))/(S(8)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) - S(23)d*S(3)(dx)(S(7)/2)/(S(96)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(77)d*S(3)(dx)(S(7)/2)(a + bxS(2))/(S(768)abS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(77)d*S(5)(dx)(S(3)/2)/(S(3072)abS(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(77)sqrt(S(2))d(S(13)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(5)/4)b*(S(15)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(77)sqrt(S(2))d(S(13)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(5)/4)b*(S(15)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(77)sqrt(S(2))d(S(13)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(5)/4)b*(S(15)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(77)sqrt(S(2))d(S(13)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(5)/4)b*(S(15)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(11)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, adS(3)(dx)(S(5)/2)(a + bxS(2))/(S(8)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) - S(7)d*S(3)(dx)(S(5)/2)/(S(32)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(15)d*S(3)(dx)(S(5)/2)(a + bxS(2))/(S(256)abS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) - S(45)d*S(5)sqrt(dx)/(S(1024)abS(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(45)sqrt(S(2))d(S(11)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(7)/4)b*(S(13)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(45)sqrt(S(2))d(S(11)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(7)/4)b*(S(13)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(45)sqrt(S(2))d(S(11)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(7)/4)b*(S(13)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(45)sqrt(S(2))d(S(11)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(7)/4)b*(S(13)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(9)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, adS(3)(dx)(S(3)/2)(a + bxS(2))/(S(8)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) - S(19)d*S(3)(dx)(S(3)/2)/(S(96)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(7)d*S(3)(dx)(S(3)/2)(a + bxS(2))/(S(256)abS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(35)d*S(3)(dx)(S(3)/2)/(S(1024)a*S(2)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(35)sqrt(S(2))d(S(9)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(9)/4)b*(S(11)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(35)sqrt(S(2))d(S(9)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(9)/4)b*(S(11)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(35)sqrt(S(2))d(S(9)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(9)/4)b*(S(11)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(35)sqrt(S(2))d(S(9)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(9)/4)b*(S(11)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(7)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, adS(3)sqrt(dx)(a + bxS(2))/(S(8)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) - S(17)d*S(3)sqrt(dx)/(S(96)b*S(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(5)d*S(3)sqrt(dx)(a + bxS(2))/(S(768)abS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(35)d*S(3)sqrt(dx)/(S(3072)a*S(2)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(35)sqrt(S(2))d(S(7)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(11)/4)b*(S(9)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(35)sqrt(S(2))d(S(7)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(11)/4)b*(S(9)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(35)sqrt(S(2))d(S(7)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(11)/4)b*(S(9)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(35)sqrt(S(2))d(S(7)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(11)/4)b*(S(9)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(5)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, (dx)(S(7)/2)(a + bxS(2))/(S(8)ad(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) - S(3)d(dx)*(S(3)/2)/(S(32)ab(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(9)d(dx)*(S(3)/2)(a + bxS(2))/(S(256)a*S(2)b(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(45)d(dx)*(S(3)/2)/(S(1024)a*S(3)bsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + S(45)sqrt(S(2))d(S(5)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(13)/4)b*(S(7)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(45)sqrt(S(2))d(S(5)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(13)/4)b*(S(7)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(45)sqrt(S(2))d(S(5)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(13)/4)b*(S(7)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(45)sqrt(S(2))d(S(5)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(13)/4)b*(S(7)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(3)/2)/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, (dx)(S(5)/2)(a + bxS(2))/(S(8)ad(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) - S(11)dsqrt(dx)/(S(96)ab(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(11)dsqrt(dx)(a + bx*S(2))/(S(768)a*S(2)b(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(77)dsqrt(dx)/(S(3072)aS(3)bsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - S(77)sqrt(S(2))d(S(3)/2)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(15)/4)b*(S(5)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(77)sqrt(S(2))d(S(3)/2)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(15)/4)b*(S(5)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(77)sqrt(S(2))d(S(3)/2)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(15)/4)b*(S(5)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(77)sqrt(S(2))d(S(3)/2)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(15)/4)b*(S(5)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)/(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2), x), x, (dx)*(S(3)/2)(a + bxS(2))/(S(8)ad(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(13)(dx)(S(3)/2)/(S(96)a*S(2)d(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(39)(dx)(S(3)/2)(a + bxS(2))/(S(256)a*S(3)d(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(195)(dx)(S(3)/2)/(S(1024)a*S(4)dsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + S(195)sqrt(S(2))sqrt(d)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(17)/4)b*(S(3)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(195)sqrt(S(2))sqrt(d)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(17)/4)b*(S(3)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(195)sqrt(S(2))sqrt(d)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(17)/4)b*(S(3)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(195)sqrt(S(2))sqrt(d)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(17)/4)b*(S(3)/4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(dx)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)), x), x, sqrt(dx)(a + bx*S(2))/(S(8)ad(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(5)sqrt(dx)/(S(32)a*S(2)d(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(55)sqrt(dx)(a + bxS(2))/(S(256)a*S(3)d(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(385)sqrt(dx)/(S(1024)a*S(4)dsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))(S(1155)a + S(1155)bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(19)/4)b*(S(1)/4)sqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))(S(1155)a + S(1155)bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(19)/4)b*(S(1)/4)sqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - sqrt(S(2))(S(1155)a + S(1155)bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(19)/4)b*(S(1)/4)sqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + sqrt(S(2))(S(1155)a + S(1155)bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(19)/4)b*(S(1)/4)sqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)), x), x, (a + bx*S(2))/(S(8)adsqrt(dx)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(17)/(S(96)a*S(2)dsqrt(dx)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + (S(221)a + S(221)bx*S(2))/(S(768)a*S(3)dsqrt(dx)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(663)/(S(1024)a*S(4)dsqrt(dx)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - (S(3315)a + S(3315)bx*S(2))/(S(1024)a*S(5)dsqrt(dx)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - S(3315)sqrt(S(2))b(S(1)/4)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(21)/4)d*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(3315)sqrt(S(2))b(S(1)/4)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(21)/4)d*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(3315)sqrt(S(2))b(S(1)/4)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(21)/4)d*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(3315)sqrt(S(2))b(S(1)/4)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(21)/4)d*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)), x), x, (a + bx*S(2))/(S(8)ad(dx)(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(19)/(S(96)a*S(2)d(dx)*(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + (S(95)a + S(95)bx*S(2))/(S(256)a*S(3)d(dx)*(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(1045)/(S(1024)a*S(4)d(dx)*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - (S(7315)a + S(7315)bx*S(2))/(S(3072)a*S(5)d(dx)*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(7315)sqrt(S(2))b(S(3)/4)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(23)/4)d*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(7315)sqrt(S(2))b(S(3)/4)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(23)/4)d*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(7315)sqrt(S(2))b(S(3)/4)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(23)/4)d*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(7315)sqrt(S(2))b(S(3)/4)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(23)/4)d*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(7)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)), x), x, (a + bx*S(2))/(S(8)ad(dx)(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(5)/2)) + S(7)/(S(32)a*S(2)d(dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + (S(119)a + S(119)bx*S(2))/(S(256)a*S(3)d(dx)*(S(5)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)) + S(1547)/(S(1024)a*S(4)d(dx)*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - (S(13923)a + S(13923)bx*S(2))/(S(5120)a*S(5)d(dx)*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(13923)b(a + bx*S(2))/(S(1024)a*S(6)d*S(3)sqrt(dx)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(13923)sqrt(S(2))b(S(5)/4)(a + bxS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(25)/4)d*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(13923)sqrt(S(2))b(S(5)/4)(a + bxS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)sqrt(dx) + sqrt(a)sqrt(d) + sqrt(b)sqrt(d)x)/(S(8192)a(S(25)/4)d*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - S(13923)sqrt(S(2))b(S(5)/4)(a + bxS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(25)/4)d*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + S(13923)sqrt(S(2))b(S(5)/4)(a + bxS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)sqrt(dx)/(a(S(1)/4)sqrt(d)))/(S(4096)a(S(25)/4)d*(S(7)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(4)/3)), x), x, S(3)/(S(10)ax(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(1)/3)) + S(39)/(S(40)a*S(2)x(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(1)/3)) - (S(91)a + S(91)bx*S(2))/(S(40)a*S(3)x(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(1)/3)) + S(91)S(3)*(S(3)/4)sqrt((a*(S(2)/3)b(S(2)/3) + a(S(1)/3)b(S(1)/3)(ab + bS(2)xS(2))(S(1)/3) + (ab + bS(2)xS(2))(S(2)/3))/(a*(S(1)/3)b*(S(1)/3)(-sqrt(S(3)) + S(1)) - (ab + bS(2)xS(2))(S(1)/3))*S(2))sqrt(-sqrt(S(3)) + S(2))(a(S(1)/3)b*(S(1)/3) - (ab + b*S(2)xS(2))(S(1)/3))(ab + b*S(2)xS(2))(S(2)/3)elliptic_f(asin((a(S(1)/3)b*(S(1)/3)(S(1) + sqrt(S(3))) - (ab + bS(2)xS(2))(S(1)/3))/(a*(S(1)/3)b*(S(1)/3)(-sqrt(S(3)) + S(1)) - (ab + bS(2)xS(2))(S(1)/3))), S(-7) + S(4)sqrt(S(3)))/(S(120)a*S(3)bxsqrt(-a*(S(1)/3)b*(S(1)/3)(a*(S(1)/3)b*(S(1)/3) - (ab + b*S(2)xS(2))(S(1)/3))/(a*(S(1)/3)b*(S(1)/3)(-sqrt(S(3)) + S(1)) - (ab + bS(2)xS(2))(S(1)/3))*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a*S(2) + S(2)abxS(2) + bS(2)xS(4))p, x), x, (dx)*(m + S(1))(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(1), m/S(2) + S(2)p + S(3)/2), (m/S(2) + S(3)/2,), -bx*S(2)/a)/(ad(m + S(1))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((dx)*m(a*S(2) + S(2)abxS(2) + bS(2)xS(4))p, x), x, (dx)*(m + S(1))(S(1) + bxS(2)/a)(-S(2)p)(aS(2) + S(2)abxS(2) + bS(2)xS(4))phyper((m/S(2) + S(1)/2, -S(2)p), (m/S(2) + S(3)/2,), -bx*S(2)/a)/(d(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))p, x), x, -a(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))p/(S(2)b*S(2)(S(2)p + S(1))) + (aS(2) + S(2)abxS(2) + bS(2)xS(4))(p + S(1))/(S(4)b*S(2)(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(aS(2) + S(2)abxS(2) + bS(2)xS(4))p, x), x, (a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))p/(S(2)b(S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))p/x, x), x, -(a + bx*S(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(1), S(2)p + S(1)), (S(2)p + S(2),), S(1) + bxS(2)/a)/(S(2)a(S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))p/xS(3), x), x, b(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(2), S(2)p + S(1)), (S(2)p + S(2),), S(1) + bxS(2)/a)/(S(2)a*S(2)(S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))p, x), x, xS(5)(S(1) + bxS(2)/a)(-S(2)p)(aS(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(5)/2, -S(2)p), (S(7)/2,), -bxS(2)/a)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(aS(2) + S(2)abxS(2) + bS(2)xS(4))p, x), x, xS(3)(S(1) + bxS(2)/a)(-S(2)p)(aS(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(3)/2, -S(2)p), (S(5)/2,), -bxS(2)/a)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))p, x), x, x(S(1) + bxS(2)/a)(-S(2)p)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(1)/2, -S(2)p), (S(3)/2,), -bxS(2)/a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))p/x*S(2), x), x, -(S(1) + bxS(2)/a)(-S(2)p)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(-1)/2, -S(2)p), (S(1)/2,), -bxS(2)/a)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))p/x*S(4), x), x, -(S(1) + bxS(2)/a)(-S(2)p)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(-3)/2, -S(2)p), (S(-1)/2,), -bx*S(2)/a)/(S(3)x*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(3)/2)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))p, x), x, S(2)(dx)(S(5)/2)(S(1) + bxS(2)/a)(-S(2)p)(aS(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(5)/4, -S(2)p), (S(9)/4,), -bx*S(2)/a)/(S(5)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))p, x), x, S(2)(dx)(S(3)/2)(S(1) + bxS(2)/a)(-S(2)p)(aS(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(3)/4, -S(2)p), (S(7)/4,), -bx*S(2)/a)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(2) + bS(2)xS(4))p/sqrt(dx), x), x, S(2)sqrt(dx)(S(1) + bxS(2)/a)(-S(2)p)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(1)/4, -S(2)p), (S(5)/4,), -bxS(2)/a)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))p/(dx)(S(3)/2), x), x, -S(2)(S(1) + bxS(2)/a)(-S(2)p)(aS(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(-1)/4, -S(2)p), (S(3)/4,), -bx*S(2)/a)/(dsqrt(dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(2) + bS(2)xS(4))p/(dx)*(S(5)/2), x), x, -S(2)(S(1) + bxS(2)/a)(-S(2)p)(aS(2) + S(2)abxS(2) + bS(2)xS(4))phyper((S(-3)/4, -S(2)p), (S(1)/4,), -bx*S(2)/a)/(S(3)d(dx)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx*S(2) + cx*S(4)), x), x, ax*S(3)/S(3) + bx*S(5)/S(5) + cx*S(7)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxS(2) + cx*S(4)), x), x, ax*S(2)/S(2) + bx*S(4)/S(4) + cx*S(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a + bx*S(2) + cx*S(4), x), x, ax + bxS(3)/S(3) + cx*S(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/x, x), x, alog(x) + bxS(2)/S(2) + cx*S(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/xS(2), x), x, -a/x + bx + cx*S(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/xS(3), x), x, -a/(S(2)xS(2)) + blog(x) + cxS(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/xS(4), x), x, -a/(S(3)xS(3)) - b/x + cx, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))/xS(5), x), x, -a/(S(4)xS(4)) - b/(S(2)x*S(2)) + clog(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))/xS(6), x), x, -a/(S(5)xS(5)) - b/(S(3)x*S(3)) - c/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/xS(7), x), x, -a/(S(6)xS(6)) - b/(S(4)x*S(4)) - c/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/xS(8), x), x, -a/(S(7)xS(7)) - b/(S(5)x*S(5)) - c/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx*S(2) + cxS(4))S(2), x), x, a*S(2)x*S(3)/S(3) + S(2)abx*S(5)/S(5) + S(2)bcxS(9)/S(9) + cS(2)xS(11)/S(11) + xS(7)(S(2)ac/S(7) + b*S(2)/S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxS(2) + cxS(4))S(2), x), x, a*S(2)x*S(2)/S(2) + abxS(4)/S(2) + bcxS(8)/S(4) + cS(2)xS(10)/S(10) + xS(6)(ac/S(3) + b*S(2)/S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2), x), x, a*S(2)x + S(2)abxS(3)/S(3) + S(2)bcxS(7)/S(7) + cS(2)xS(9)/S(9) + xS(5)(S(2)ac/S(5) + b*S(2)/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/x, x), x, a*S(2)log(x) + abx*S(2) + bcxS(6)/S(3) + cS(2)xS(8)/S(8) + xS(4)(ac/S(2) + b*S(2)/S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/xS(2), x), x, -aS(2)/x + S(2)abx + S(2)bcxS(5)/S(5) + cS(2)xS(7)/S(7) + xS(3)(S(2)ac/S(3) + b*S(2)/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/xS(3), x), x, -aS(2)/(S(2)xS(2)) + S(2)ablog(x) + bcxS(4)/S(2) + cS(2)xS(6)/S(6) + xS(2)(ac + bS(2)/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/xS(4), x), x, -aS(2)/(S(3)xS(3)) - S(2)ab/x + S(2)bcxS(3)/S(3) + cS(2)xS(5)/S(5) + x(S(2)ac + b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/xS(5), x), x, -aS(2)/(S(4)xS(4)) - ab/x*S(2) + bcxS(2) + cS(2)x*S(4)/S(4) + (S(2)ac + bS(2))log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))S(2)/xS(6), x), x, -aS(2)/(S(5)xS(5)) - S(2)ab/(S(3)x*S(3)) + S(2)bcx + c*S(2)x*S(3)/S(3) - (S(2)ac + bS(2))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/xS(7), x), x, -aS(2)/(S(6)xS(6)) - ab/(S(2)xS(4)) + S(2)bclog(x) + c*S(2)x*S(2)/S(2) - (S(2)ac + bS(2))/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/xS(8), x), x, -aS(2)/(S(7)xS(7)) - S(2)ab/(S(5)x*S(5)) - S(2)bc/x + cS(2)x - (S(2)ac + b*S(2))/(S(3)x*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/xS(9), x), x, -aS(2)/(S(8)xS(8)) - ab/(S(3)xS(6)) - bc/xS(2) + cS(2)log(x) - (S(2)ac + bS(2))/(S(4)x*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/xS(10), x), x, -aS(2)/(S(9)xS(9)) - S(2)ab/(S(7)x*S(7)) - S(2)bc/(S(3)xS(3)) - cS(2)/x - (S(2)ac + b*S(2))/(S(5)x*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/xS(11), x), x, -aS(2)/(S(10)xS(10)) - ab/(S(4)xS(8)) - bc/(S(2)xS(4)) - cS(2)/(S(2)x*S(2)) - (S(2)ac + bS(2))/(S(6)x*S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/xS(12), x), x, -aS(2)/(S(11)xS(11)) - S(2)ab/(S(9)x*S(9)) - S(2)bc/(S(5)xS(5)) - cS(2)/(S(3)xS(3)) - (S(2)ac + bS(2))/(S(7)x*S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/xS(13), x), x, -aS(2)/(S(12)xS(12)) - ab/(S(5)xS(10)) - bc/(S(3)xS(6)) - cS(2)/(S(4)x*S(4)) - (S(2)ac + bS(2))/(S(8)xS(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx*S(2) + cxS(4))S(3), x), x, a*S(3)x*S(3)/S(3) + S(3)a*S(2)bxS(5)/S(5) + S(3)axS(7)(ac + bS(2))/S(7) + S(3)bcS(2)x*S(13)/S(13) + bx*S(9)(S(6)ac + bS(2))/S(9) + cS(3)xS(15)/S(15) + S(3)cxS(11)(ac + bS(2))/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxS(2) + cxS(4))S(3), x), x, a*S(3)x*S(2)/S(2) + S(3)a*S(2)bxS(4)/S(4) + ax*S(6)(ac + bS(2))/S(2) + bc*S(2)x*S(12)/S(4) + bx*S(8)(S(6)ac + bS(2))/S(8) + cS(3)xS(14)/S(14) + S(3)cxS(10)(ac + bS(2))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(3), x), x, a*S(3)x + a*S(2)bxS(3) + S(3)axS(5)(ac + bS(2))/S(5) + S(3)bcS(2)x*S(11)/S(11) + bx*S(7)(S(6)ac + bS(2))/S(7) + cS(3)xS(13)/S(13) + cx*S(9)(ac + bS(2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(3)/x, x), x, a*S(3)log(x) + S(3)aS(2)bxS(2)/S(2) + S(3)axS(4)(ac + bS(2))/S(4) + S(3)bcS(2)x*S(10)/S(10) + bx*S(6)(S(6)ac + bS(2))/S(6) + cS(3)xS(12)/S(12) + S(3)cxS(8)(ac + bS(2))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(3)/xS(2), x), x, -aS(3)/x + S(3)aS(2)bx + ax*S(3)(ac + bS(2)) + bc*S(2)x*S(9)/S(3) + bx*S(5)(S(6)ac + bS(2))/S(5) + cS(3)xS(11)/S(11) + S(3)cxS(7)(ac + bS(2))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(3)/xS(3), x), x, -aS(3)/(S(2)xS(2)) + S(3)a*S(2)blog(x) + S(3)axS(2)(ac + bS(2))/S(2) + S(3)bcS(2)x*S(8)/S(8) + bx*S(4)(S(6)ac + bS(2))/S(4) + cS(3)xS(10)/S(10) + cx*S(6)(ac + bS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(3)/xS(4), x), x, -aS(3)/(S(3)xS(3)) - S(3)a*S(2)b/x + S(3)ax(ac + b*S(2)) + S(3)bcS(2)x*S(7)/S(7) + bx*S(3)(S(6)ac + bS(2))/S(3) + cS(3)xS(9)/S(9) + S(3)cxS(5)(ac + bS(2))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(a + bx*S(2) + cx*S(4)), x), x, -bx*S(2)/(S(2)c*S(2)) + b(-S(3)ac + b*S(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(3)sqrt(-S(4)ac + bS(2))) + xS(4)/(S(4)c) + (-ac + b*S(2))log(a + bxS(2) + cx*S(4))/(S(4)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(a + bxS(2) + cx*S(4)), x), x, -blog(a + bxS(2) + cx*S(4))/(S(4)cS(2)) + xS(2)/(S(2)c) - (-S(2)ac + bS(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + bxS(2) + cx*S(4)), x), x, batanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-S(4)ac + bS(2))) + log(a + bx*S(2) + cx*S(4))/(S(4)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bxS(2) + cx*S(4)), x), x, -atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bxS(2) + cx*S(4))), x), x, batanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)asqrt(-S(4)ac + bS(2))) + log(x)/a - log(a + bx*S(2) + cx*S(4))/(S(4)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)(a + bxS(2) + cx*S(4))), x), x, -S(1)/(S(2)axS(2)) - blog(x)/a*S(2) + blog(a + bxS(2) + cx*S(4))/(S(4)a*S(2)) - (-S(2)ac + bS(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(a + bx*S(2) + cx*S(4))), x), x, -S(1)/(S(4)axS(4)) + b/(S(2)a*S(2)x*S(2)) + b(-S(3)ac + b*S(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(3)sqrt(-S(4)ac + b*S(2))) + (-ac + b*S(2))log(x)/a*S(3) - (-ac + b*S(2))log(a + bxS(2) + cx*S(4))/(S(4)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(a + bxS(2) + cx*S(4)), x), x, -bx/cS(2) + xS(3)/(S(3)c) + sqrt(S(2))(-ac + bS(2) + b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) + sqrt(S(2))(-ac + bS(2) - b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(a + bxS(2) + cx*S(4)), x), x, x/c - sqrt(S(2))(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))) - sqrt(S(2))(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bxS(2) + cx*S(4)), x), x, -sqrt(S(2))sqrt(b - sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(-S(4)ac + bS(2))) + sqrt(S(2))sqrt(b + sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bx*S(2) + cx*S(4)), x), x, -sqrt(S(2))sqrt(c)atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + bS(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + bS(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + bx*S(2) + cx*S(4))), x), x, -sqrt(S(2))sqrt(c)(-b/sqrt(-S(4)ac + bS(2)) + S(1))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(b + sqrt(-S(4)ac + bS(2)))) - sqrt(S(2))sqrt(c)(b/sqrt(-S(4)ac + bS(2)) + S(1))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(b - sqrt(-S(4)ac + bS(2)))) - S(1)/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(4)(a + bxS(2) + cx*S(4))), x), x, -S(1)/(S(3)axS(3)) + b/(aS(2)x) + sqrt(S(2))sqrt(c)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) + sqrt(S(2))sqrt(c)(b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)a*S(2)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(a + bxS(2) + cxS(4))S(2), x), x, -bxS(2)/(S(2)c(-S(4)ac + bS(2))) + b(-S(6)ac + b*S(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)(-S(4)ac + bS(2))(S(3)/2)) + x*S(4)(S(2)a + bx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) + log(a + bx*S(2) + cx*S(4))/(S(4)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(a + bxS(2) + cxS(4))S(2), x), x, S(2)aatanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + xS(2)(S(2)a + bx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + bxS(2) + cxS(4))S(2), x), x, -batanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + (S(2)a + bxS(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bx*S(2) + cxS(4))S(2), x), x, S(2)catanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) - (b + S(2)cxS(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bxS(2) + cxS(4))S(2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))) + b(-S(6)ac + b*S(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)(-S(4)ac + bS(2))(S(3)/2)) + log(x)/a*S(2) - log(a + bx*S(2) + cx*S(4))/(S(4)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a + bx*S(2) + cxS(4))S(2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(S(2)axS(2)(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) - (-S(3)ac + bS(2))/(aS(2)x*S(2)(-S(4)ac + b*S(2))) - S(2)blog(x)/aS(3) + blog(a + bxS(2) + cx*S(4))/(S(2)a*S(3)) - (S(6)a*S(2)c*S(2) - S(6)abS(2)c + b*S(4))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(aS(3)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(8)/(a + bx*S(2) + cxS(4))S(2), x), x, -bxS(3)/(S(2)c(-S(4)ac + bS(2))) + xS(5)(S(2)a + bx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) + x(-S(10)ac + S(3)bS(2))/(S(2)c*S(2)(-S(4)ac + b*S(2))) - sqrt(S(2))(-S(13)abc + S(3)b*S(3) + (S(20)a*S(2)c*S(2) - S(19)abS(2)c + S(3)bS(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(5)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + b*S(2))) - sqrt(S(2))(-S(13)abc + S(3)b*S(3) - (S(20)a*S(2)c*S(2) - S(19)abS(2)c + S(3)bS(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(5)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(a + bxS(2) + cxS(4))S(2), x), x, -bx/(S(2)c(-S(4)ac + bS(2))) + xS(3)(S(2)a + bx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) + sqrt(S(2))(-S(6)ac + b*S(2) + b(-S(8)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))(-S(6)ac + b*S(2) - b(-S(8)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(a + bxS(2) + cxS(4))S(2), x), x, x(S(2)a + bxS(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) + sqrt(S(2))(b - (S(4)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))(S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(a + bx*S(2) + cxS(4))S(2), x), x, -sqrt(S(2))sqrt(c)(S(2)b + sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))sqrt(c)(S(2)b - sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - x(b + S(2)cxS(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(-2)), x), x, -sqrt(S(2))sqrt(c)(-S(12)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))sqrt(c)(-S(12)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + x(-S(2)ac + bS(2) + bcxS(2))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + bx*S(2) + cxS(4))S(2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(S(2)ax(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) + sqrt(S(2))sqrt(c)(-S(16)abc + S(3)bS(3) - (-S(10)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - sqrt(S(2))sqrt(c)(-S(16)abc + S(3)b*S(3) + (-S(10)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - (-S(10)ac + S(3)bS(2))/(S(2)a*S(2)x(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(a + bx*S(2) + cxS(4))S(3), x), x, -bxS(2)(-S(7)ac + b*S(2))/(S(2)c*S(2)(-S(4)ac + bS(2))S(2)) + b(S(30)a*S(2)c*S(2) - S(10)abS(2)c + b*S(4))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(3)(-S(4)ac + bS(2))(S(5)/2)) + x*S(8)(S(2)a + bx*S(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + x*S(4)(a(-S(16)ac + bS(2)) + bx*S(2)(-S(10)ac + b*S(2)))/(S(4)c(-S(4)ac + bS(2))S(2)(a + bxS(2) + cx*S(4))) + log(a + bx*S(2) + cx*S(4))/(S(4)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(a + bxS(2) + cxS(4))S(3), x), x, -S(6)aS(2)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) - S(3)axS(2)(S(2)a + bx*S(2))/(S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cxS(4))) + xS(6)(S(2)a + bxS(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(7)/(a + bx*S(2) + cxS(4))S(3), x), x, S(3)abatanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) + S(3)bxS(2)(S(2)a + bx*S(2))/(S(4)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cxS(4))) - xS(6)(b + S(2)cxS(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)/(a + bx*S(2) + cxS(4))S(3), x), x, x*S(2)(S(2)a + bx*S(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + (S(3)ab + x*S(2)(S(2)ac + b*S(2)))/(S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) - (S(2)ac + bS(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + bx*S(2) + cxS(4))S(3), x), x, S(3)bcatanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) - S(3)b(b + S(2)cxS(2))/(S(4)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) + (S(2)a + bxS(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bxS(2) + cxS(4))S(3), x), x, -S(6)cS(2)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) + S(3)c(b + S(2)cxS(2))/(S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) - (b + S(2)cxS(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bx*S(2) + cxS(4))S(3)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(S(4)a(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))S(2)) + (S(16)aS(2)c*S(2) - S(15)abS(2)c + S(2)bS(4) + S(2)bcx*S(2)(-S(7)ac + b*S(2)))/(S(4)a*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) + b(S(30)aS(2)c*S(2) - S(10)abS(2)c + b*S(4))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(3)(-S(4)ac + bS(2))(S(5)/2)) + log(x)/a*S(3) - log(a + bx*S(2) + cx*S(4))/(S(4)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a + bx*S(2) + cxS(4))S(3)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(S(4)axS(2)(-S(4)ac + b*S(2))(a + bxS(2) + cxS(4))S(2)) + (S(20)aS(2)c*S(2) - S(20)abS(2)c + S(3)bS(4) + S(3)bcx*S(2)(-S(6)ac + b*S(2)))/(S(4)a*S(2)x*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) - (S(30)a*S(2)c*S(2) - S(21)abS(2)c + S(3)bS(4))/(S(2)a*S(3)x*S(2)(-S(4)ac + bS(2))S(2)) - S(3)blog(x)/a*S(4) + S(3)blog(a + bx*S(2) + cx*S(4))/(S(4)a*S(4)) - (-S(60)a*S(3)c*S(3) + S(90)a*S(2)b*S(2)c*S(2) - S(30)abS(4)c + S(3)bS(6))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(4)(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(10)/(a + bx*S(2) + cxS(4))S(3), x), x, -S(3)bx(-S(8)ac + bS(2))/(S(8)c*S(2)(-S(4)ac + bS(2))S(2)) + x*S(7)(S(2)a + bx*S(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + x*S(5)(S(12)ab - x*S(2)(-S(28)ac + b*S(2)))/(S(8)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cxS(4))) + xS(3)(-S(28)ac + bS(2))/(S(8)c(-S(4)ac + bS(2))S(2)) + sqrt(S(2))(S(84)aS(2)c*S(2) - S(27)abS(2)c + S(3)bS(4) + S(3)(S(44)aS(2)bcS(2) - S(11)abS(3)c + b*S(5))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)c*(S(5)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + sqrt(S(2))(S(84)a*S(2)c*S(2) - S(27)abS(2)c + S(3)bS(4) - S(3)b(S(44)a*S(2)c*S(2) - S(11)abS(2)c + b*S(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)c*(S(5)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(8)/(a + bx*S(2) + cxS(4))S(3), x), x, x*S(5)(S(2)a + bx*S(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + x*S(3)(S(12)ab + x*S(2)(S(20)ac + b*S(2)))/(S(8)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) - x(S(20)ac + b*S(2))/(S(8)c(-S(4)ac + bS(2))S(2)) + sqrt(S(2))(-S(16)abc + bS(3) + (-S(40)a*S(2)c*S(2) - S(18)abS(2)c + b*S(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + sqrt(S(2))(-S(16)abc + b*S(3) - (-S(40)a*S(2)c*S(2) - S(18)abS(2)c + b*S(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(6)/(a + bx*S(2) + cxS(4))S(3), x), x, x*S(3)(S(2)a + bx*S(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + S(3)x(S(4)ab + x*S(2)(S(4)ac + b*S(2)))/(S(8)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) + sqrt(S(2))(S(12)ac + S(3)bS(2) + S(3)b(S(12)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))S(2)) + sqrt(S(2))(S(12)ac + S(3)b*S(2) - S(3)b(S(12)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)/(a + bx*S(2) + cxS(4))S(3), x), x, -S(3)sqrt(S(2))sqrt(c)(S(4)ac + S(3)b*S(2) + S(2)bsqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(8)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + S(3)sqrt(S(2))sqrt(c)(S(4)ac + S(3)b*S(2) - S(2)bsqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(8)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + x(S(2)a + bxS(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) - x(-S(4)ac + S(7)b*S(2) + S(12)bcx*S(2))/(S(8)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bxS(2) + cxS(4))S(3), x), x, -x(b + S(2)cxS(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + sqrt(S(2))sqrt(c)(S(20)ac + b*S(2) - b(-S(52)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))S(2)) + sqrt(S(2))sqrt(c)(S(20)ac + b*S(2) + b(-S(52)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))S(2)) + x(b(S(8)ac + b*S(2)) + cx*S(2)(S(20)ac + b*S(2)))/(S(8)a(-S(4)ac + bS(2))S(2)(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(-3)), x), x, x(-S(2)ac + bS(2) + bcxS(2))/(S(4)a(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))S(2)) + S(3)sqrt(S(2))sqrt(c)(-S(8)abc + b*S(3) - (S(56)a*S(2)c*S(2) - S(10)abS(2)c + b*S(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + S(3)sqrt(S(2))sqrt(c)(S(56)a*S(2)c*S(2) - S(10)abS(2)c + b*S(4) + b(-S(8)ac + b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + x(S(3)bcx*S(2)(-S(8)ac + b*S(2)) + (-S(7)ac + bS(2))(-S(4)ac + S(3)bS(2)))/(S(8)a*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + bx*S(2) + cxS(4))S(3)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(S(4)ax(-S(4)ac + b*S(2))(a + bxS(2) + cxS(4))S(2)) + (S(36)aS(2)c*S(2) - S(35)abS(2)c + S(5)bS(4) + bcxS(2)(-S(32)ac + S(5)bS(2)))/(S(8)a*S(2)x(-S(4)ac + bS(2))S(2)(a + bxS(2) + cx*S(4))) - S(3)sqrt(S(2))sqrt(c)((-S(12)ac + S(5)bS(2))(-S(5)ac + b*S(2)) - (S(124)a*S(2)bcS(2) - S(47)abS(3)c + S(5)bS(5))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(3)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) - S(3)sqrt(S(2))sqrt(c)(b(S(124)aS(2)c*S(2) - S(47)abS(2)c + S(5)bS(4))/sqrt(-S(4)ac + bS(2)) + (-S(12)ac + S(5)b*S(2))(-S(5)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(3)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) - (-S(36)ac + S(15)bS(2))(-S(5)ac + b*S(2))/(S(8)a*S(3)x(-S(4)ac + bS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(a - bx*S(2) + cx*S(4)), x), x, blog(a - bxS(2) + cx*S(4))/(S(4)cS(2)) + xS(2)/(S(2)c) + (-S(2)ac + bS(2))atanh((b - S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a - bxS(2) + cx*S(4)), x), x, batanh((b - S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-S(4)ac + bS(2))) + log(a - bx*S(2) + cx*S(4))/(S(4)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a - bxS(2) + cx*S(4)), x), x, atanh((b - S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a - bxS(2) + cx*S(4))), x), x, batanh((b - S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)asqrt(-S(4)ac + bS(2))) + log(x)/a - log(a - bx*S(2) + cx*S(4))/(S(4)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)(a - bxS(2) + cx*S(4))), x), x, -S(1)/(S(2)axS(2)) + blog(x)/a*S(2) - blog(a - bxS(2) + cx*S(4))/(S(4)a*S(2)) + (-S(2)ac + bS(2))atanh((b - S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(a - bxS(2) + cx*S(4)), x), x, x/c - sqrt(S(2))(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))) - sqrt(S(2))(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a - bxS(2) + cx*S(4)), x), x, sqrt(S(2))sqrt(b - sqrt(-S(4)ac + b*S(2)))atanh(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(-S(4)ac + bS(2))) - sqrt(S(2))sqrt(b + sqrt(-S(4)ac + b*S(2)))atanh(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a - bx*S(2) + cx*S(4)), x), x, -sqrt(S(2))sqrt(c)atanh(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + bS(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)atanh(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + bS(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a - bx*S(2) + cx*S(4))), x), x, sqrt(S(2))sqrt(c)(-b/sqrt(-S(4)ac + bS(2)) + S(1))atanh(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(b + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))sqrt(c)(b/sqrt(-S(4)ac + bS(2)) + S(1))atanh(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(b - sqrt(-S(4)ac + bS(2)))) - S(1)/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)/(ax*S(4) + S(2)axS(2) + a + b), x), x, xS(2)/(S(2)a) - log(axS(4) + S(2)axS(2) + a + b)/(S(2)a) + (a - b)atan(sqrt(a)(x*S(2) + S(1))/sqrt(b))/(S(2)a*(S(3)/2)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/(ax*S(4) + S(2)axS(2) + a + b), x), x, log(ax*S(4) + S(2)axS(2) + a + b)/(S(4)a) - atan(sqrt(a)(xS(2) + S(1))/sqrt(b))/(S(2)sqrt(a)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(ax*S(4) + S(2)axS(2) + a + b), x), x, atan(sqrt(a)(x*S(2) + S(1))/sqrt(b))/(S(2)sqrt(a)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(axS(4) + S(2)axS(2) + a + b)), x), x, -sqrt(a)atan(sqrt(a)(xS(2) + S(1))/sqrt(b))/(S(2)sqrt(b)(a + b)) - log(ax*S(4) + S(2)axS(2) + a + b)/(S(4)a + S(4)b) + log(x)/(a + b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(axS(4) + S(2)axS(2) + a + b)), x), x, sqrt(a)(a - b)atan(sqrt(a)(x*S(2) + S(1))/sqrt(b))/(S(2)sqrt(b)(a + b)S(2)) - S(2)alog(x)/(a + b)S(2) + alog(axS(4) + S(2)axS(2) + a + b)/(S(2)(a + b)S(2)) - S(1)/(xS(2)(S(2)a + S(2)b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(ax*S(4) + S(2)axS(2) + a + b), x), x, (S(2)sqrt(-a) + (a - b)/sqrt(b))atan(x(-a)*(S(1)/4)/sqrt(-sqrt(b) + sqrt(-a)))/(S(2)(-a)*(S(5)/4)sqrt(-sqrt(b) + sqrt(-a))) - (a - S(2)sqrt(b)sqrt(-a) - b)atan(x(-a)*(S(1)/4)/sqrt(sqrt(b) + sqrt(-a)))/(S(2)sqrt(b)(-a)(S(5)/4)sqrt(sqrt(b) + sqrt(-a))) + x/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(ax*S(4) + S(2)axS(2) + a + b), x), x, sqrt(-sqrt(b) + sqrt(-a))atan(x(-a)(S(1)/4)/sqrt(-sqrt(b) + sqrt(-a)))/(S(2)sqrt(b)(-a)(S(3)/4)) - sqrt(sqrt(b) + sqrt(-a))atan(x(-a)(S(1)/4)/sqrt(sqrt(b) + sqrt(-a)))/(S(2)sqrt(b)(-a)(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ax*S(4) + S(2)axS(2) + a + b), x), x, atan(x(-a)*(S(1)/4)/sqrt(sqrt(b) + sqrt(-a)))/(S(2)sqrt(b)(-a)(S(1)/4)sqrt(sqrt(b) + sqrt(-a))) - atan(x(-a)(S(1)/4)/sqrt(-sqrt(b) + sqrt(-a)))/(S(2)sqrt(b)(-a)(S(1)/4)sqrt(-sqrt(b) + sqrt(-a))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(axS(4) + S(2)axS(2) + a + b)), x), x, -S(1)/(x(a + b)) + (-a)*(S(1)/4)(-sqrt(b) + sqrt(-a))atan(x(-a)*(S(1)/4)/sqrt(sqrt(b) + sqrt(-a)))/(S(2)sqrt(b)(a + b)sqrt(sqrt(b) + sqrt(-a))) - (-a)*(S(1)/4)(sqrt(b) + sqrt(-a))atan(x(-a)*(S(1)/4)/sqrt(-sqrt(b) + sqrt(-a)))/(S(2)sqrt(b)(a + b)sqrt(-sqrt(b) + sqrt(-a))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a*S(2) + S(2)axS(2) + b + xS(4)), x), x, -atan(x/sqrt(a + sqrt(-b)))/(S(2)sqrt(-b)sqrt(a + sqrt(-b))) + atan(x/sqrt(a - sqrt(-b)))/(S(2)sqrt(-b)sqrt(a - sqrt(-b))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(aS(2) + S(2)axS(2) + xS(4) + S(-1)), x), x, -atan(x/sqrt(a + S(1)))/(S(2)sqrt(a + S(1))) - atanh(x/sqrt(-a + S(1)))/(S(2)sqrt(-a + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(aS(2) + S(2)axS(2) + xS(4) + S(1)), x), x, -sqrt(S(2))atan((-sqrt(S(2))x + sqrt(-a + sqrt(aS(2) + S(1))))/sqrt(a + sqrt(aS(2) + S(1))))/(S(4)sqrt(a + sqrt(a*S(2) + S(1)))sqrt(a*S(2) + S(1))) + sqrt(S(2))atan((sqrt(S(2))x + sqrt(-a + sqrt(aS(2) + S(1))))/sqrt(a + sqrt(aS(2) + S(1))))/(S(4)sqrt(a + sqrt(a*S(2) + S(1)))sqrt(a*S(2) + S(1))) - sqrt(S(2))log(x*S(2) - sqrt(S(2))xsqrt(-a + sqrt(aS(2) + S(1))) + sqrt(aS(2) + S(1)))/(S(8)sqrt(-a + sqrt(a*S(2) + S(1)))sqrt(a*S(2) + S(1))) + sqrt(S(2))log(x*S(2) + sqrt(S(2))xsqrt(-a + sqrt(aS(2) + S(1))) + sqrt(aS(2) + S(1)))/(S(8)sqrt(-a + sqrt(a*S(2) + S(1)))sqrt(aS(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4) - S(5)xS(2) + S(4)), x), x, -atanh(x/S(2))/S(6) + atanh(x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4) + S(4)x*S(2) + S(3)), x), x, atan(x)/S(2) - sqrt(S(3))atan(sqrt(S(3))x/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4) + S(5)xS(2) + S(9)), x), x, -log(xS(2) - x + S(3))/S(12) + log(x*S(2) + x + S(3))/S(12) - sqrt(S(11))atan(sqrt(S(11))(-S(2)x + S(1))/S(11))/S(66) + sqrt(S(11))atan(sqrt(S(11))(S(2)x + S(1))/S(11))/S(66), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4) - xS(2) + S(1)), x), x, -sqrt(S(3))log(x*S(2) - sqrt(S(3))x + S(1))/S(12) + sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(12) + atan(S(2)x - sqrt(S(3)))/S(2) + atan(S(2)x + sqrt(S(3)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(4) + S(2)xS(2) + S(2)), x), x, -log(xS(2) - xsqrt(S(-2) + S(2)sqrt(S(2))) + sqrt(S(2)))/(S(8)sqrt(S(-1) + sqrt(S(2)))) + log(xS(2) + xsqrt(S(-2) + S(2)sqrt(S(2))) + sqrt(S(2)))/(S(8)sqrt(S(-1) + sqrt(S(2)))) - sqrt(S(-1) + sqrt(S(2)))atan((-S(2)x + sqrt(S(-2) + S(2)sqrt(S(2))))/sqrt(S(2) + S(2)sqrt(S(2))))/S(4) + sqrt(S(-1) + sqrt(S(2)))atan((S(2)x + sqrt(S(-2) + S(2)sqrt(S(2))))/sqrt(S(2) + S(2)sqrt(S(2))))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(4) + xS(2) + S(1)), x), x, sqrt(S(3))atan(sqrt(S(3))(S(2)xS(2) + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(4) + S(2)xS(2) + S(10)), x), x, atan(xS(2)/S(3) + S(1)/3)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(xS(4) + S(9)xS(2) + S(20)), x), x, -S(2)atan(x/S(2)) + sqrt(S(5))atan(sqrt(S(5))x/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(xS(4) - x*S(2) + S(1)), x), x, sqrt(S(3))log(x*S(2) - sqrt(S(3))x + S(1))/S(12) - sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(12) + atan(S(2)x - sqrt(S(3)))/S(2) + atan(S(2)x + sqrt(S(3)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(xS(4) - S(2)xS(2) + S(2)), x), x, log(xS(2) - xsqrt(S(2) + S(2)sqrt(S(2))) + sqrt(S(2)))/(S(4)sqrt(S(2) + S(2)sqrt(S(2)))) - log(xS(2) + xsqrt(S(2) + S(2)sqrt(S(2))) + sqrt(S(2)))/(S(4)sqrt(S(2) + S(2)sqrt(S(2)))) - sqrt(S(1)/2 + sqrt(S(2))/S(2))atan((-S(2)x + sqrt(S(2) + S(2)sqrt(S(2))))/sqrt(S(-2) + S(2)sqrt(S(2))))/S(2) + sqrt(S(1)/2 + sqrt(S(2))/S(2))atan((S(2)x + sqrt(S(2) + S(2)sqrt(S(2))))/sqrt(S(-2) + S(2)sqrt(S(2))))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)sqrt(a + bxS(2) + cx*S(4)), x), x, -b(b + S(2)cx*S(2))(-S(12)ac + S(7)bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(256)c*S(4)) + b(-S(12)ac + S(7)bS(2))(-S(4)ac + b*S(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(512)c(S(9)/2)) + xS(4)(a + bx*S(2) + cxS(4))(S(3)/2)/(S(10)c) + (a + bx*S(2) + cxS(4))(S(3)/2)(-S(32)ac + S(35)b*S(2) - S(42)bcx*S(2))/(S(480)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)sqrt(a + bx*S(2) + cx*S(4)), x), x, -S(5)b(a + bx*S(2) + cxS(4))(S(3)/2)/(S(48)cS(2)) + xS(2)(a + bxS(2) + cxS(4))(S(3)/2)/(S(8)c) + (b + S(2)cxS(2))(-S(4)ac + S(5)bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(128)c*S(3)) - (-S(4)ac + bS(2))(-S(4)ac + S(5)bS(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(256)c(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(a + bx*S(2) + cx*S(4)), x), x, -b(b + S(2)cx*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(16)c*S(2)) + b(-S(4)ac + b*S(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)c*(S(5)/2)) + (a + bx*S(2) + cxS(4))(S(3)/2)/(S(6)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(a + bxS(2) + cx*S(4)), x), x, (b + S(2)cxS(2))sqrt(a + bxS(2) + cx*S(4))/(S(8)c) - (-S(4)ac + b*S(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)c*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cx*S(4))/x, x), x, -sqrt(a)atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/S(2) + batanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)sqrt(c)) + sqrt(a + bxS(2) + cx*S(4))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cxS(4))/xS(3), x), x, sqrt(c)atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/S(2) - sqrt(a + bx*S(2) + cx*S(4))/(S(2)x*S(2)) - batanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bxS(2) + cxS(4))/xS(5), x), x, -(S(2)a + bx*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(8)axS(4)) + (-S(4)ac + bS(2))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)a*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cxS(4))/xS(7), x), x, -(a + bxS(2) + cxS(4))(S(3)/2)/(S(6)ax*S(6)) + b(S(2)a + bx*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(16)a*S(2)x*S(4)) - b(-S(4)ac + b*S(2))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)a*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cxS(4))/xS(9), x), x, -(a + bxS(2) + cxS(4))(S(3)/2)/(S(8)ax*S(8)) + S(5)b(a + bx*S(2) + cxS(4))(S(3)/2)/(S(48)aS(2)x*S(6)) - (S(2)a + bxS(2))(-S(4)ac + S(5)bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(128)a*S(3)x*S(4)) + (-S(4)ac + bS(2))(-S(4)ac + S(5)bS(2))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(256)a*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cxS(4))/xS(11), x), x, -(a + bxS(2) + cxS(4))(S(3)/2)/(S(10)ax*S(10)) + S(7)b(a + bx*S(2) + cxS(4))(S(3)/2)/(S(80)aS(2)x*S(8)) - (-S(32)ac + S(35)b*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(480)aS(3)x*S(6)) + b(S(2)a + bx*S(2))(-S(12)ac + S(7)bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(256)a*S(4)x*S(4)) - b(-S(12)ac + S(7)bS(2))(-S(4)ac + b*S(2))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(512)a(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)sqrt(a + bx*S(2) + cxS(4)), x), x, -a(S(1)/4)bsqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(29)ac + S(8)bS(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(105)c*(S(11)/4)sqrt(a + bxS(2) + cxS(4))) + a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(2)sqrt(a)sqrt(c)(-S(5)ac + S(2)b*S(2)) - S(29)abc + S(8)bS(3))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(210)c*(S(11)/4)sqrt(a + bxS(2) + cx*S(4))) + bx(-S(29)ac + S(8)b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(105)c*(S(5)/2)(sqrt(a) + sqrt(c)xS(2))) + xS(3)(b + S(5)cx*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(35)c) - x(-S(10)ac + S(4)b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(105)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(a + bx*S(2) + cx*S(4)), x), x, S(2)a*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(3)ac + b*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(15)c*(S(7)/4)sqrt(a + bxS(2) + cxS(4))) - a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(sqrt(a)bsqrt(c) - S(6)ac + S(2)bS(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(30)c*(S(7)/4)sqrt(a + bxS(2) + cx*S(4))) + x(b + S(3)cx*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(15)c) - x(-S(6)ac + S(2)b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(15)c*(S(3)/2)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cxS(4)), x), x, -a(S(1)/4)bsqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(3)c*(S(3)/4)sqrt(a + bxS(2) + cxS(4))) + a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(2)sqrt(a)sqrt(c) + b)elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(6)c*(S(3)/4)sqrt(a + bxS(2) + cx*S(4))) + bxsqrt(a + bx*S(2) + cx*S(4))/(S(3)sqrt(c)(sqrt(a) + sqrt(c)x*S(2))) + xsqrt(a + bxS(2) + cx*S(4))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cxS(4))/xS(2), x), x, -S(2)a(S(1)/4)c*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/sqrt(a + bx*S(2) + cx*S(4)) + S(2)sqrt(c)xsqrt(a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)x*S(2)) - sqrt(a + bx*S(2) + cx*S(4))/x + sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(2)sqrt(a)sqrt(c) + b)elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(1)/4)c*(S(1)/4)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cxS(4))/xS(4), x), x, -sqrt(a + bxS(2) + cx*S(4))/(S(3)x*S(3)) + bsqrt(c)xsqrt(a + bxS(2) + cx*S(4))/(S(3)a(sqrt(a) + sqrt(c)x*S(2))) - bsqrt(a + bxS(2) + cx*S(4))/(S(3)ax) - bc*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(3)a*(S(3)/4)sqrt(a + bxS(2) + cxS(4))) + c(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(2)sqrt(a)sqrt(c) + b)elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(6)a*(S(3)/4)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cxS(4))/xS(6), x), x, -sqrt(a + bxS(2) + cx*S(4))/(S(5)x*S(5)) - bsqrt(a + bxS(2) + cx*S(4))/(S(15)axS(3)) - S(2)sqrt(c)x(-S(3)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(15)a*S(2)(sqrt(a) + sqrt(c)xS(2))) + (-S(6)ac + S(2)b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(15)a*S(2)x) + S(2)c(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(3)ac + b*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(15)a*(S(7)/4)sqrt(a + bxS(2) + cxS(4))) - c(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(sqrt(a)bsqrt(c) - S(6)ac + S(2)bS(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(30)a*(S(7)/4)sqrt(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)(a + bx*S(2) + cxS(4))(S(3)/2), x), x, -b(b + S(2)cxS(2))(-S(4)ac + S(3)bS(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(256)cS(4)) + S(3)b(b + S(2)cxS(2))(-S(4)ac + b*S(2))(-S(4)ac + S(3)bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(2048)c*S(5)) - S(3)b(-S(4)ac + bS(2))S(2)(-S(4)ac + S(3)bS(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4096)c(S(11)/2)) + xS(4)(a + bx*S(2) + cxS(4))(S(5)/2)/(S(14)c) + (a + bx*S(2) + cxS(4))(S(5)/2)(-S(16)ac + S(21)b*S(2) - S(30)bcx*S(2))/(S(560)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(a + bx*S(2) + cxS(4))(S(3)/2), x), x, -S(7)b(a + bxS(2) + cxS(4))(S(5)/2)/(S(120)cS(2)) + xS(2)(a + bxS(2) + cxS(4))(S(5)/2)/(S(12)c) + (b + S(2)cxS(2))(-S(4)ac + S(7)bS(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(384)cS(3)) - (b + S(2)cxS(2))(-S(4)ac + b*S(2))(-S(4)ac + S(7)bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(1024)c*S(4)) + (-S(4)ac + bS(2))S(2)(-S(4)ac + S(7)bS(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(2048)c(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a + bx*S(2) + cxS(4))(S(3)/2), x), x, -b(b + S(2)cxS(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(32)cS(2)) + S(3)b(b + S(2)cxS(2))(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(256)c*S(3)) - S(3)b(-S(4)ac + bS(2))S(2)atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(512)c*(S(7)/2)) + (a + bx*S(2) + cxS(4))(S(5)/2)/(S(10)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxS(2) + cxS(4))(S(3)/2), x), x, (b + S(2)cx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(16)c) - (b + S(2)cxS(2))(-S(12)ac + S(3)bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(128)c*S(2)) + S(3)(-S(4)ac + bS(2))S(2)atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(256)c*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(3)/2)/x, x), x, -a*(S(3)/2)atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/S(2) - b(-S(12)ac + b*S(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)c*(S(3)/2)) + (a + bx*S(2) + cxS(4))(S(3)/2)/S(6) + sqrt(a + bxS(2) + cx*S(4))(S(8)ac + b*S(2) + S(2)bcx*S(2))/(S(16)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))(S(3)/2)/x*S(3), x), x, -S(3)sqrt(a)batanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/S(4) + (S(9)b/S(8) + S(3)cx*S(2)/S(4))sqrt(a + bxS(2) + cx*S(4)) - (a + bx*S(2) + cxS(4))(S(3)/2)/(S(2)xS(2)) + (S(12)ac + S(3)b*S(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))(S(3)/2)/x*S(5), x), x, S(3)bsqrt(c)atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/S(4) - (S(3)b - S(6)cx*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(8)x*S(2)) - (a + bx*S(2) + cxS(4))(S(3)/2)/(S(4)xS(4)) - (S(12)ac + S(3)b*S(2))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))(S(3)/2)/xS(7), x), x, c(S(3)/2)atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/S(2) - (a + bx*S(2) + cxS(4))(S(3)/2)/(S(6)xS(6)) - (S(2)ab + xS(2)(S(8)ac + b*S(2)))sqrt(a + bxS(2) + cx*S(4))/(S(16)axS(4)) + b(-S(12)ac + b*S(2))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)a*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(3)/2)/x*S(9), x), x, -(S(2)a + bxS(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(16)ax*S(8)) + (S(2)a + bxS(2))(-S(12)ac + S(3)bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(128)a*S(2)x*S(4)) - S(3)(-S(4)ac + bS(2))S(2)atanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(256)a*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(3)/2)/x*S(11), x), x, -(a + bx*S(2) + cxS(4))(S(5)/2)/(S(10)ax*S(10)) + b(S(2)a + bx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(32)aS(2)x*S(8)) - S(3)b(S(2)a + bxS(2))(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(256)a*S(3)x*S(4)) + S(3)b(-S(4)ac + bS(2))S(2)atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(512)a*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(3)/2)/x*S(13), x), x, -(a + bx*S(2) + cxS(4))(S(5)/2)/(S(12)ax*S(12)) + S(7)b(a + bx*S(2) + cxS(4))(S(5)/2)/(S(120)aS(2)x*S(10)) - (S(2)a + bxS(2))(-S(4)ac + S(7)bS(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(384)aS(3)x*S(8)) + (S(2)a + bxS(2))(-S(4)ac + b*S(2))(-S(4)ac + S(7)bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(1024)a*S(4)x*S(4)) - (-S(4)ac + bS(2))S(2)(-S(4)ac + S(7)bS(2))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2048)a(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(a + bx*S(2) + cxS(4))(S(3)/2), x), x, S(8)a(S(1)/4)bsqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(9)ac + S(2)bS(2))(-S(3)ac + b*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(1155)c*(S(15)/4)sqrt(a + bxS(2) + cxS(4))) - a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(sqrt(a)sqrt(c)(S(60)aS(2)c*S(2) - S(51)abS(2)c + S(8)bS(4)) + S(8)b(-S(9)ac + S(2)b*S(2))(-S(3)ac + b*S(2)))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2310)c*(S(15)/4)sqrt(a + bxS(2) + cx*S(4))) - S(8)bx(-S(9)ac + S(2)bS(2))(-S(3)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(1155)c*(S(7)/2)(sqrt(a) + sqrt(c)xS(2))) + xS(3)(b + S(3)cx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(33)c) - xS(3)(b(ac + S(2)bS(2)) + S(10)cxS(2)(-S(3)ac + b*S(2)))sqrt(a + bxS(2) + cx*S(4))/(S(385)c*S(2)) + xsqrt(a + bxS(2) + cx*S(4))(S(60)aS(2)c*S(2) - S(51)abS(2)c + S(8)bS(4))/(S(1155)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx*S(2) + cxS(4))(S(3)/2), x), x, -a*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(84)aS(2)c*S(2) - S(57)abS(2)c + S(8)bS(4))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(315)c*(S(11)/4)sqrt(a + bxS(2) + cxS(4))) + a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(4)sqrt(a)bsqrt(c)(-S(6)ac + b*S(2)) + S(84)a*S(2)c*S(2) - S(57)abS(2)c + S(8)bS(4))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(630)c*(S(11)/4)sqrt(a + bxS(2) + cx*S(4))) + x(S(3)b + S(7)cxS(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(63)c) - x(b(-S(9)ac + S(4)b*S(2)) + S(6)cxS(2)(-S(7)ac + S(2)bS(2)))sqrt(a + bxS(2) + cx*S(4))/(S(315)c*S(2)) + xsqrt(a + bxS(2) + cx*S(4))(S(84)aS(2)c*S(2) - S(57)abS(2)c + S(8)bS(4))/(S(315)c*(S(5)/2)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(3)/2), x), x, S(2)a(S(1)/4)bsqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(8)ac + b*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(35)c*(S(7)/4)sqrt(a + bxS(2) + cxS(4))) - a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(sqrt(a)sqrt(c)(-S(20)ac + b*S(2)) + S(2)b(-S(8)ac + bS(2)))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(70)c*(S(7)/4)sqrt(a + bxS(2) + cx*S(4))) - S(2)bx(-S(8)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(35)c*(S(3)/2)(sqrt(a) + sqrt(c)xS(2))) + x(a + bxS(2) + cxS(4))(S(3)/2)/S(7) + xsqrt(a + bx*S(2) + cx*S(4))(S(10)ac + b*S(2) + S(3)bcx*S(2))/(S(35)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))(S(3)/2)/xS(2), x), x, -a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(12)ac + b*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(5)c*(S(3)/4)sqrt(a + bxS(2) + cxS(4))) + a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(8)sqrt(a)bsqrt(c) + S(12)ac + bS(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(10)c*(S(3)/4)sqrt(a + bxS(2) + cx*S(4))) + x(S(7)b + S(6)cxS(2))sqrt(a + bxS(2) + cx*S(4))/S(5) - (a + bx*S(2) + cxS(4))(S(3)/2)/x + x(S(12)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(5)sqrt(c)(sqrt(a) + sqrt(c)x*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(3)/2)/x*S(4), x), x, -S(8)a*(S(1)/4)bc(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(3)sqrt(a + bxS(2) + cx*S(4))) + S(8)bsqrt(c)xsqrt(a + bx*S(2) + cx*S(4))/(S(3)sqrt(a) + S(3)sqrt(c)x*S(2)) - (S(3)b - S(2)cx*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(3)x) - (a + bxS(2) + cxS(4))(S(3)/2)/(S(3)xS(3)) + sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(8)sqrt(a)bsqrt(c) + S(4)ac + S(3)b*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(6)a*(S(1)/4)c*(S(1)/4)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(3)/2)/x*S(6), x), x, -(b - S(6)cxS(2))sqrt(a + bxS(2) + cx*S(4))/(S(5)x*S(3)) - (a + bx*S(2) + cxS(4))(S(3)/2)/(S(5)xS(5)) + sqrt(c)x(S(12)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(5)a(sqrt(a) + sqrt(c)x*S(2))) - (S(12)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(5)ax) - c(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(12)ac + b*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(5)a*(S(3)/4)sqrt(a + bxS(2) + cxS(4))) + c(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(8)sqrt(a)bsqrt(c) + S(12)ac + bS(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(10)a*(S(3)/4)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(3)/2)/x*S(8), x), x, -(S(3)b + S(30)cx*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(35)x*S(5)) - (a + bx*S(2) + cxS(4))(S(3)/2)/(S(7)xS(7)) - (-S(20)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(35)axS(3)) - S(2)bsqrt(c)x(-S(8)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(35)a*S(2)(sqrt(a) + sqrt(c)xS(2))) + S(2)b(-S(8)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(35)a*S(2)x) + S(2)bc*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(8)ac + b*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(35)a*(S(7)/4)sqrt(a + bxS(2) + cxS(4))) - c(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(sqrt(a)sqrt(c)(-S(20)ac + b*S(2)) + S(2)b(-S(8)ac + bS(2)))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(70)a*(S(7)/4)sqrt(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-xS(4) - S(2)xS(2) + S(3)), x), x, xsqrt(-x*S(4) - S(2)x*S(2) + S(3))/S(3) - S(2)sqrt(S(3))elliptic_e(asin(x), S(-1)/3)/S(3) + S(4)sqrt(S(3))elliptic_f(asin(x), S(-1)/3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/sqrt(a + bx*S(2) + cx*S(4)), x), x, -b(-S(12)ac + S(5)bS(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)c(S(7)/2)) + xS(4)sqrt(a + bx*S(2) + cx*S(4))/(S(6)c) + sqrt(a + bxS(2) + cx*S(4))(-S(16)ac + S(15)bS(2) - S(10)bcx*S(2))/(S(48)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/sqrt(a + bxS(2) + cx*S(4)), x), x, -S(3)bsqrt(a + bx*S(2) + cx*S(4))/(S(8)cS(2)) + xS(2)sqrt(a + bx*S(2) + cx*S(4))/(S(4)c) + (-S(4)ac + S(3)bS(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)c(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(a + bxS(2) + cx*S(4)), x), x, -batanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)c*(S(3)/2)) + sqrt(a + bx*S(2) + cx*S(4))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + bxS(2) + cx*S(4)), x), x, atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + bx*S(2) + cx*S(4))), x), x, -atanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)sqrt(a + bxS(2) + cx*S(4))), x), x, -sqrt(a + bx*S(2) + cx*S(4))/(S(2)axS(2)) + batanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)a(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)sqrt(a + bx*S(2) + cx*S(4))), x), x, -sqrt(a + bx*S(2) + cx*S(4))/(S(4)axS(4)) + S(3)bsqrt(a + bx*S(2) + cx*S(4))/(S(8)a*S(2)x*S(2)) - (-S(4)ac + S(3)b*S(2))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)sqrt(a + bx*S(2) + cx*S(4))), x), x, -sqrt(a + bx*S(2) + cx*S(4))/(S(6)axS(6)) + S(5)bsqrt(a + bx*S(2) + cx*S(4))/(S(24)a*S(2)x*S(4)) - (-S(16)ac + S(15)b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(48)a*S(3)x*S(2)) + b(-S(12)ac + S(5)bS(2))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)a(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/sqrt(a + bxS(2) + cx*S(4)), x), x, S(2)a*(S(1)/4)bsqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(3)c*(S(7)/4)sqrt(a + bxS(2) + cxS(4))) - a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(sqrt(a)sqrt(c) + S(2)b)elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(6)c*(S(7)/4)sqrt(a + bxS(2) + cx*S(4))) - S(2)bxsqrt(a + bxS(2) + cx*S(4))/(S(3)c*(S(3)/2)(sqrt(a) + sqrt(c)xS(2))) + xsqrt(a + bxS(2) + cx*S(4))/(S(3)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/sqrt(a + bx*S(2) + cxS(4)), x), x, -a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(c(S(3)/4)sqrt(a + bxS(2) + cxS(4))) + a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)c*(S(3)/4)sqrt(a + bxS(2) + cx*S(4))) + xsqrt(a + bxS(2) + cx*S(4))/(sqrt(c)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + bx*S(2) + cx*S(4)), x), x, sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(1)/4)c*(S(1)/4)sqrt(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(a + bx*S(2) + cx*S(4))), x), x, sqrt(c)xsqrt(a + bx*S(2) + cx*S(4))/(a(sqrt(a) + sqrt(c)xS(2))) - sqrt(a + bx*S(2) + cx*S(4))/(ax) - c*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(a(S(3)/4)sqrt(a + bxS(2) + cxS(4))) + c(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(3)/4)sqrt(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)sqrt(a + bx*S(2) + cx*S(4))), x), x, -sqrt(a + bx*S(2) + cx*S(4))/(S(3)axS(3)) - S(2)bsqrt(c)xsqrt(a + bx*S(2) + cx*S(4))/(S(3)a*S(2)(sqrt(a) + sqrt(c)xS(2))) + S(2)bsqrt(a + bx*S(2) + cx*S(4))/(S(3)a*S(2)x) + S(2)bc*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(3)a*(S(7)/4)sqrt(a + bxS(2) + cxS(4))) - c(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(sqrt(a)sqrt(c) + S(2)b)elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(6)a*(S(7)/4)sqrt(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/sqrt(a + bxS(2) - cx*S(4)), x), x, -b(S(12)ac + S(5)bS(2))atan((b - S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) - cx*S(4))))/(S(32)c(S(7)/2)) - xS(4)sqrt(a + bx*S(2) - cx*S(4))/(S(6)c) - sqrt(a + bxS(2) - cx*S(4))(S(16)ac + S(15)bS(2) + S(10)bcx*S(2))/(S(48)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/sqrt(a + bxS(2) - cx*S(4)), x), x, -S(3)bsqrt(a + bx*S(2) - cx*S(4))/(S(8)cS(2)) - xS(2)sqrt(a + bx*S(2) - cx*S(4))/(S(4)c) - (S(4)ac + S(3)bS(2))atan((b - S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) - cx*S(4))))/(S(16)c(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(a + bxS(2) - cx*S(4)), x), x, -batan((b - S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) - cx*S(4))))/(S(4)c*(S(3)/2)) - sqrt(a + bx*S(2) - cx*S(4))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + bxS(2) - cx*S(4)), x), x, -atan((b - S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) - cx*S(4))))/(S(2)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(-a + bx*S(2) + cx*S(4))), x), x, -atan((S(2)a - bxS(2))/(S(2)sqrt(a)sqrt(-a + bx*S(2) + cx*S(4))))/(S(2)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)sqrt(-a + bxS(2) + cx*S(4))), x), x, sqrt(-a + bx*S(2) + cx*S(4))/(S(2)axS(2)) - batan((S(2)a - bx*S(2))/(S(2)sqrt(a)sqrt(-a + bx*S(2) + cx*S(4))))/(S(4)a(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)sqrt(-a + bx*S(2) + cx*S(4))), x), x, sqrt(-a + bx*S(2) + cx*S(4))/(S(4)axS(4)) + S(3)bsqrt(-a + bx*S(2) + cx*S(4))/(S(8)a*S(2)x*S(2)) - (S(4)ac + S(3)b*S(2))atan((S(2)a - bx*S(2))/(S(2)sqrt(a)sqrt(-a + bx*S(2) + cx*S(4))))/(S(16)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)sqrt(-a + bx*S(2) + cx*S(4))), x), x, sqrt(-a + bx*S(2) + cx*S(4))/(S(6)axS(6)) + S(5)bsqrt(-a + bx*S(2) + cx*S(4))/(S(24)a*S(2)x*S(4)) + (S(16)ac + S(15)b*S(2))sqrt(-a + bxS(2) + cx*S(4))/(S(48)a*S(3)x*S(2)) - b(S(12)ac + S(5)bS(2))atan((S(2)a - bx*S(2))/(S(2)sqrt(a)sqrt(-a + bx*S(2) + cx*S(4))))/(S(32)a(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/sqrt(a + bxS(2) - cx*S(4)), x), x, -sqrt(S(2))b(b - sqrt(S(4)ac + bS(2)))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_e(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(6)c*(S(5)/2)sqrt(a + bxS(2) - cx*S(4))) - xsqrt(a + bxS(2) - cx*S(4))/(S(3)c) + sqrt(S(2))sqrt(b + sqrt(S(4)ac + bS(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))(ac + bS(2) - bsqrt(S(4)ac + b*S(2)))elliptic_f(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(6)c*(S(5)/2)sqrt(a + bxS(2) - cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(a + bxS(2) - cx*S(4)), x), x, -sqrt(S(2))(b - sqrt(S(4)ac + b*S(2)))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_e(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(4)c*(S(3)/2)sqrt(a + bxS(2) - cx*S(4))) + sqrt(S(2))(b - sqrt(S(4)ac + b*S(2)))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_f(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(4)c*(S(3)/2)sqrt(a + bxS(2) - cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + bx*S(2) - cx*S(4)), x), x, sqrt(S(2))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_f(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(2)sqrt(c)sqrt(a + bx*S(2) - cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(a + bx*S(2) - cx*S(4))), x), x, -sqrt(a + bx*S(2) - cx*S(4))/(ax) + sqrt(S(2))(b - sqrt(S(4)ac + bS(2)))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_e(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(4)asqrt(c)sqrt(a + bxS(2) - cx*S(4))) - sqrt(S(2))(b - sqrt(S(4)ac + b*S(2)))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_f(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(4)asqrt(c)sqrt(a + bxS(2) - cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)sqrt(a + bx*S(2) - cx*S(4))), x), x, -sqrt(a + bx*S(2) - cx*S(4))/(S(3)axS(3)) + S(2)bsqrt(a + bx*S(2) - cx*S(4))/(S(3)a*S(2)x) - sqrt(S(2))b(b - sqrt(S(4)ac + b*S(2)))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_e(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(6)a*S(2)sqrt(c)sqrt(a + bx*S(2) - cx*S(4))) + sqrt(S(2))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))(ac + bS(2) - bsqrt(S(4)ac + b*S(2)))elliptic_f(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(6)a*S(2)sqrt(c)sqrt(a + bx*S(2) - cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(a + bxS(2) + cxS(4))(S(3)/2), x), x, -bxS(4)sqrt(a + bxS(2) + cx*S(4))/(c(-S(4)ac + bS(2))) + xS(6)(S(2)a + bxS(2))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))) - (b(-S(52)ac + S(15)bS(2)) - S(2)cxS(2)(-S(12)ac + S(5)bS(2)))sqrt(a + bxS(2) + cx*S(4))/(S(8)c*S(3)(-S(4)ac + b*S(2))) + (-S(12)ac + S(15)b*S(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)c(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(a + bxS(2) + cxS(4))(S(3)/2), x), x, -S(3)batanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)c(S(5)/2)) + xS(4)(S(2)a + bxS(2))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))) + sqrt(a + bx*S(2) + cx*S(4))(-S(8)ac + S(3)bS(2) - S(2)bcx*S(2))/(S(2)c*S(2)(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(a + bxS(2) + cxS(4))(S(3)/2), x), x, -bsqrt(a + bx*S(2) + cx*S(4))/(c(-S(4)ac + bS(2))) + xS(2)(S(2)a + bxS(2))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))) + atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)c(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + bxS(2) + cxS(4))(S(3)/2), x), x, (S(2)a + bx*S(2))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, -(b + S(2)cx*S(2))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bxS(2) + cxS(4))(S(3)/2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(a(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) - atanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)a(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a + bx*S(2) + cxS(4))(S(3)/2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(ax*S(2)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) - (-S(8)ac + S(3)b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(2)a*S(2)x*S(2)(-S(4)ac + b*S(2))) + S(3)batanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(a + bx*S(2) + cxS(4))(S(3)/2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(ax*S(4)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) - (-S(12)ac + S(5)b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(4)a*S(2)x*S(4)(-S(4)ac + b*S(2))) + b(-S(52)ac + S(15)bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(8)a*S(3)x*S(2)(-S(4)ac + b*S(2))) - (-S(12)ac + S(15)b*S(2))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)a(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(a + bxS(2) + cxS(4))(S(3)/2), x), x, -S(2)a(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(3)ac + b*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(c(S(7)/4)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cxS(4))) + a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(sqrt(a)bsqrt(c) - S(6)ac + S(2)bS(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)c*(S(7)/4)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) - bxsqrt(a + bx*S(2) + cx*S(4))/(c(-S(4)ac + bS(2))) + xS(3)(S(2)a + bxS(2))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))) + x(-S(6)ac + S(2)bS(2))sqrt(a + bxS(2) + cxS(4))/(c(S(3)/2)(sqrt(a) + sqrt(c)x*S(2))(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(a + bxS(2) + cxS(4))(S(3)/2), x), x, a*(S(1)/4)bsqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(c(S(3)/4)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cxS(4))) - a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(c(S(3)/4)(-S(4)sqrt(a)sqrt(c) + S(2)b)sqrt(a + bxS(2) + cx*S(4))) - bxsqrt(a + bx*S(2) + cx*S(4))/(sqrt(c)(sqrt(a) + sqrt(c)xS(2))(-S(4)ac + b*S(2))) + x(S(2)a + bx*S(2))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bxS(2) + cxS(4))(S(3)/2), x), x, -S(2)a(S(1)/4)c*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))) + S(2)sqrt(c)xsqrt(a + bxS(2) + cx*S(4))/((sqrt(a) + sqrt(c)x*S(2))(-S(4)ac + b*S(2))) - x(b + S(2)cx*S(2))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))) + sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(1)/4)c*(S(1)/4)(-S(2)sqrt(a)sqrt(c) + b)sqrt(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(-3)/2), x), x, -bsqrt(c)xsqrt(a + bx*S(2) + cx*S(4))/(a(sqrt(a) + sqrt(c)xS(2))(-S(4)ac + b*S(2))) + x(-S(2)ac + b*S(2) + bcxS(2))/(a(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) + bc*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(a(S(3)/4)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cxS(4))) - c(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(3)/4)(-S(2)sqrt(a)sqrt(c) + b)sqrt(a + bx*S(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + bx*S(2) + cxS(4))(S(3)/2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(ax(-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))) + S(2)sqrt(c)x(-S(3)ac + b*S(2))sqrt(a + bxS(2) + cxS(4))/(aS(2)(sqrt(a) + sqrt(c)x*S(2))(-S(4)ac + b*S(2))) - (-S(6)ac + S(2)b*S(2))sqrt(a + bxS(2) + cxS(4))/(aS(2)x(-S(4)ac + b*S(2))) - S(2)c*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(3)ac + b*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(a(S(7)/4)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cxS(4))) + c(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(sqrt(a)bsqrt(c) - S(6)ac + S(2)bS(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(7)/4)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)x*S(4) + S(5)x*S(2) + S(2)), x), x, elliptic_f(asin(sqrt(S(2))x/S(2)), S(-6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)xS(4) + S(4)x*S(2) + S(2)), x), x, sqrt(S(1)/3 + sqrt(S(10))/S(6))elliptic_f(asin(xsqrt(S(-1) + sqrt(S(10))/S(2))), S(-7)/3 - S(2)sqrt(S(10))/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)xS(4) + S(3)x*S(2) + S(2)), x), x, sqrt(S(2))elliptic_f(asin(sqrt(S(6))x/sqrt(S(3) + sqrt(S(33)))), S(-7)/4 - sqrt(S(33))/S(4))/sqrt(S(-3) + sqrt(S(33))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)x*S(4) + S(2)x*S(2) + S(2)), x), x, elliptic_f(asin(sqrt(S(3))x/sqrt(S(1) + sqrt(S(7)))), S(-4)/3 - sqrt(S(7))/S(3))/sqrt(S(-1) + sqrt(S(7))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)xS(4) + xS(2) + S(2)), x), x, sqrt(S(2))elliptic_f(asin(x), S(-3)/2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)xS(4) + S(2)), x), x, S(6)(S(3)/4)elliptic_f(asin(S(2)*(S(3)/4)S(3)*(S(1)/4)x/S(2)), S(-1))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)xS(4) - xS(2) + S(2)), x), x, sqrt(S(3))elliptic_f(asin(sqrt(S(6))x/S(2)), S(-2)/3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)x*S(4) - S(2)x*S(2) + S(2)), x), x, elliptic_f(asin(sqrt(S(3))x/sqrt(S(-1) + sqrt(S(7)))), S(-4)/3 + sqrt(S(7))/S(3))/sqrt(S(1) + sqrt(S(7))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)xS(4) - S(3)x*S(2) + S(2)), x), x, sqrt(S(2))elliptic_f(asin(sqrt(S(6))x/sqrt(S(-3) + sqrt(S(33)))), S(-7)/4 + sqrt(S(33))/S(4))/sqrt(S(3) + sqrt(S(33))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)x*S(4) - S(4)x*S(2) + S(2)), x), x, sqrt(S(-1)/3 + sqrt(S(10))/S(6))elliptic_f(asin(xsqrt(S(1) + sqrt(S(10))/S(2))), S(-7)/3 + S(2)sqrt(S(10))/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)xS(4) - S(5)x*S(2) + S(2)), x), x, sqrt(S(6))elliptic_f(asin(sqrt(S(3))x), S(-1)/6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) + S(7)x*S(2) + S(3)), x), x, sqrt(S(2))elliptic_f(asin(S(2)x/sqrt(S(7) + sqrt(S(73)))), S(-61)/12 - S(7)sqrt(S(73))/S(12))/sqrt(S(-7) + sqrt(S(73))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) + S(6)x*S(2) + S(3)), x), x, sqrt(S(1)/2 + sqrt(S(15))/S(6))elliptic_f(asin(xsqrt(S(-1) + sqrt(S(15))/S(3))), S(-4) - sqrt(S(15))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) + S(5)x*S(2) + S(3)), x), x, elliptic_f(asin(sqrt(S(3))x/S(3)), S(-6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) + S(4)x*S(2) + S(3)), x), x, elliptic_f(asin(sqrt(S(2))x/sqrt(S(2) + sqrt(S(10)))), S(-7)/3 - S(2)sqrt(S(10))/S(3))/sqrt(S(-2) + sqrt(S(10))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) + S(3)x*S(2) + S(3)), x), x, sqrt(S(2))elliptic_f(asin(S(2)x/sqrt(S(3) + sqrt(S(33)))), S(-7)/4 - sqrt(S(33))/S(4))/sqrt(S(-3) + sqrt(S(33))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) + S(2)x*S(2) + S(3)), x), x, elliptic_f(asin(sqrt(S(2))x/sqrt(S(1) + sqrt(S(7)))), S(-4)/3 - sqrt(S(7))/S(3))/sqrt(S(-1) + sqrt(S(7))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) + xS(2) + S(3)), x), x, sqrt(S(2))elliptic_f(asin(sqrt(S(6))x/S(3)), S(-3)/2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) + S(3)), x), x, S(6)(S(3)/4)elliptic_f(asin(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), S(-1))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) - xS(2) + S(3)), x), x, sqrt(S(3))elliptic_f(asin(x), S(-2)/3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) - S(2)x*S(2) + S(3)), x), x, elliptic_f(asin(sqrt(S(2))x/sqrt(S(-1) + sqrt(S(7)))), S(-4)/3 + sqrt(S(7))/S(3))/sqrt(S(1) + sqrt(S(7))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) - S(3)x*S(2) + S(3)), x), x, sqrt(S(2))elliptic_f(asin(S(2)x/sqrt(S(-3) + sqrt(S(33)))), S(-7)/4 + sqrt(S(33))/S(4))/sqrt(S(3) + sqrt(S(33))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) - S(4)x*S(2) + S(3)), x), x, elliptic_f(asin(sqrt(S(2))x/sqrt(S(-2) + sqrt(S(10)))), S(-7)/3 + S(2)sqrt(S(10))/S(3))/sqrt(S(2) + sqrt(S(10))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) - S(5)x*S(2) + S(3)), x), x, sqrt(S(6))elliptic_f(asin(sqrt(S(2))x), S(-1)/6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) - S(6)x*S(2) + S(3)), x), x, sqrt(S(-1)/2 + sqrt(S(15))/S(6))elliptic_f(asin(xsqrt(S(1) + sqrt(S(15))/S(3))), S(-4) + sqrt(S(15))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) - S(7)x*S(2) + S(3)), x), x, sqrt(S(2))elliptic_f(asin(S(2)x/sqrt(S(-7) + sqrt(S(73)))), S(-61)/12 + S(7)sqrt(S(73))/S(12))/sqrt(S(7) + sqrt(S(73))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) + S(5)x*S(2) + S(-2)), x), x, sqrt(S(7))sqrt(x*S(2) + S(2))sqrt(S(3)xS(2) + S(-1))elliptic_f(asin(sqrt(S(14))x/(S(2)sqrt(S(3)xS(2) + S(-1)))), S(6)/7)/(S(7)sqrt(S(3)xS(4) + S(5)x*S(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)x*S(4) + S(4)xS(2) + S(-2)), x), x, S(10)(S(3)/4)sqrt((-xS(2)(-sqrt(S(10)) + S(2)) + S(2))/(-x*S(2)(S(2) + sqrt(S(10))) + S(2)))sqrt(xS(2)(S(2) + sqrt(S(10))) + S(-2))elliptic_f(asin(S(2)(S(3)/4)S(5)*(S(1)/4)x/sqrt(x*S(2)(S(2) + sqrt(S(10))) + S(-2))), sqrt(S(10))/S(10) + S(1)/2)/(S(20)sqrt(S(3)x*S(4) + S(4)x*S(2) + S(-2))sqrt(S(1)/(-x*S(2)(S(2) + sqrt(S(10))) + S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) + S(3)x*S(2) + S(-2)), x), x, sqrt(S(2))S(33)*(S(3)/4)sqrt((-x*S(2)(-sqrt(S(33)) + S(3)) + S(4))/(-x*S(2)(S(3) + sqrt(S(33))) + S(4)))sqrt(xS(2)(S(3) + sqrt(S(33))) + S(-4))elliptic_f(asin(sqrt(S(2))S(33)*(S(1)/4)x/sqrt(x*S(2)(S(3) + sqrt(S(33))) + S(-4))), sqrt(S(33))/S(22) + S(1)/2)/(S(132)sqrt(S(3)x*S(4) + S(3)x*S(2) + S(-2))sqrt(S(1)/(-x*S(2)(S(3) + sqrt(S(33))) + S(4)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) + S(2)xS(2) + S(-2)), x), x, S(7)(S(3)/4)sqrt((-xS(2)(-sqrt(S(7)) + S(1)) + S(2))/(-x*S(2)(S(1) + sqrt(S(7))) + S(2)))sqrt(xS(2)(S(1) + sqrt(S(7))) + S(-2))elliptic_f(asin(sqrt(S(2))S(7)*(S(1)/4)x/sqrt(x*S(2)(S(1) + sqrt(S(7))) + S(-2))), sqrt(S(7))/S(14) + S(1)/2)/(S(14)sqrt(S(3)x*S(4) + S(2)x*S(2) + S(-2))sqrt(S(1)/(-x*S(2)(S(1) + sqrt(S(7))) + S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) + xS(2) + S(-2)), x), x, sqrt(S(5))sqrt(x*S(2) + S(1))sqrt(S(3)xS(2) + S(-2))elliptic_f(asin(sqrt(S(5))x/sqrt(S(3)x*S(2) + S(-2))), S(3)/5)/(S(5)sqrt(S(3)xS(4) + xS(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) + S(-2)), x), x, S(6)(S(3)/4)sqrt((sqrt(S(6))x*S(2) + S(2))/(-sqrt(S(6))x*S(2) + S(2)))sqrt(sqrt(S(6))xS(2) + S(-2))elliptic_f(asin(S(2)*(S(3)/4)S(3)*(S(1)/4)x/sqrt(sqrt(S(6))xS(2) + S(-2))), S(1)/2)/(S(12)sqrt(S(3)xS(4) + S(-2))sqrt(S(1)/(-sqrt(S(6))xS(2) + S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) - xS(2) + S(-2)), x), x, sqrt(S(5))sqrt(xS(2) + S(-1))sqrt(S(3)xS(2) + S(2))elliptic_f(asin(sqrt(S(10))x/(S(2)sqrt(x*S(2) + S(-1)))), S(2)/5)/(S(5)sqrt(S(3)xS(4) - xS(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)x*S(4) - S(2)xS(2) + S(-2)), x), x, S(7)(S(3)/4)sqrt((xS(2)(S(1) + sqrt(S(7))) + S(2))/(x*S(2)(-sqrt(S(7)) + S(1)) + S(2)))sqrt(-xS(2)(-sqrt(S(7)) + S(1)) + S(-2))elliptic_f(asin(sqrt(S(2))S(7)*(S(1)/4)x/sqrt(-x*S(2)(-sqrt(S(7)) + S(1)) + S(-2))), -sqrt(S(7))/S(14) + S(1)/2)/(S(14)sqrt(S(3)x*S(4) - S(2)x*S(2) + S(-2))sqrt(S(1)/(x*S(2)(-sqrt(S(7)) + S(1)) + S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) - S(3)x*S(2) + S(-2)), x), x, sqrt(S(2))S(33)*(S(3)/4)sqrt((x*S(2)(S(3) + sqrt(S(33))) + S(4))/(x*S(2)(-sqrt(S(33)) + S(3)) + S(4)))sqrt(-xS(2)(-sqrt(S(33)) + S(3)) + S(-4))elliptic_f(asin(sqrt(S(2))S(33)*(S(1)/4)x/sqrt(-x*S(2)(-sqrt(S(33)) + S(3)) + S(-4))), -sqrt(S(33))/S(22) + S(1)/2)/(S(132)sqrt(S(3)x*S(4) - S(3)x*S(2) + S(-2))sqrt(S(1)/(x*S(2)(-sqrt(S(33)) + S(3)) + S(4)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) - S(4)xS(2) + S(-2)), x), x, S(10)(S(3)/4)sqrt((xS(2)(S(2) + sqrt(S(10))) + S(2))/(x*S(2)(-sqrt(S(10)) + S(2)) + S(2)))sqrt(-xS(2)(-sqrt(S(10)) + S(2)) + S(-2))elliptic_f(asin(S(2)(S(3)/4)S(5)*(S(1)/4)x/sqrt(-x*S(2)(-sqrt(S(10)) + S(2)) + S(-2))), -sqrt(S(10))/S(10) + S(1)/2)/(S(20)sqrt(S(3)x*S(4) - S(4)x*S(2) + S(-2))sqrt(S(1)/(x*S(2)(-sqrt(S(10)) + S(2)) + S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) - S(5)x*S(2) + S(-2)), x), x, sqrt(S(7))sqrt(x*S(2) + S(-2))sqrt(S(3)xS(2) + S(1))elliptic_f(asin(sqrt(S(7))x/sqrt(xS(2) + S(-2))), S(1)/7)/(S(7)sqrt(S(3)xS(4) - S(5)x*S(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) + S(7)x*S(2) + S(-3)), x), x, sqrt(S(3))S(73)*(S(3)/4)sqrt((-x*S(2)(-sqrt(S(73)) + S(7)) + S(6))/(-x*S(2)(S(7) + sqrt(S(73))) + S(6)))sqrt(xS(2)(S(7) + sqrt(S(73))) + S(-6))elliptic_f(asin(sqrt(S(2))S(73)*(S(1)/4)x/sqrt(x*S(2)(S(7) + sqrt(S(73))) + S(-6))), S(7)sqrt(S(73))/S(146) + S(1)/2)/(S(438)sqrt(S(2)xS(4) + S(7)x*S(2) + S(-3))sqrt(S(1)/(-x*S(2)(S(7) + sqrt(S(73))) + S(6)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) + S(6)x*S(2) + S(-3)), x), x, sqrt(S(2))S(3)*(S(1)/4)S(5)*(S(3)/4)sqrt((-x*S(2)(-sqrt(S(15)) + S(3)) + S(3))/(-x*S(2)(S(3) + sqrt(S(15))) + S(3)))sqrt(xS(2)(S(3) + sqrt(S(15))) + S(-3))elliptic_f(asin(S(15)(S(1)/4)sqrt(S(2))x/sqrt(xS(2)(S(3) + sqrt(S(15))) + S(-3))), sqrt(S(15))/S(10) + S(1)/2)/(S(30)sqrt(S(2)x*S(4) + S(6)x*S(2) + S(-3))sqrt(S(1)/(-x*S(2)(S(3) + sqrt(S(15))) + S(3)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) + S(5)x*S(2) + S(-3)), x), x, sqrt(S(7))sqrt(x*S(2) + S(3))sqrt(S(2)xS(2) + S(-1))elliptic_f(asin(sqrt(S(21))x/(S(3)sqrt(S(2)xS(2) + S(-1)))), S(6)/7)/(S(7)sqrt(S(2)xS(4) + S(5)x*S(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) + S(4)xS(2) + S(-3)), x), x, S(2)(S(1)/4)sqrt(S(3))S(5)*(S(3)/4)sqrt((-x*S(2)(-sqrt(S(10)) + S(2)) + S(3))/(-x*S(2)(S(2) + sqrt(S(10))) + S(3)))sqrt(xS(2)(S(2) + sqrt(S(10))) + S(-3))elliptic_f(asin(S(2)(S(3)/4)S(5)*(S(1)/4)x/sqrt(x*S(2)(S(2) + sqrt(S(10))) + S(-3))), sqrt(S(10))/S(10) + S(1)/2)/(S(30)sqrt(S(2)x*S(4) + S(4)x*S(2) + S(-3))sqrt(S(1)/(-x*S(2)(S(2) + sqrt(S(10))) + S(3)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) + S(3)xS(2) + S(-3)), x), x, S(11)(S(3)/4)S(3)(S(1)/4)sqrt((-x*S(2)(-sqrt(S(33)) + S(3)) + S(6))/(-x*S(2)(S(3) + sqrt(S(33))) + S(6)))sqrt(xS(2)(S(3) + sqrt(S(33))) + S(-6))elliptic_f(asin(sqrt(S(2))S(33)*(S(1)/4)x/sqrt(x*S(2)(S(3) + sqrt(S(33))) + S(-6))), sqrt(S(33))/S(22) + S(1)/2)/(S(66)sqrt(S(2)x*S(4) + S(3)x*S(2) + S(-3))sqrt(S(1)/(-x*S(2)(S(3) + sqrt(S(33))) + S(6)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) + S(2)x*S(2) + S(-3)), x), x, sqrt(S(6))S(7)*(S(3)/4)sqrt((-x*S(2)(-sqrt(S(7)) + S(1)) + S(3))/(-x*S(2)(S(1) + sqrt(S(7))) + S(3)))sqrt(xS(2)(S(1) + sqrt(S(7))) + S(-3))elliptic_f(asin(sqrt(S(2))S(7)*(S(1)/4)x/sqrt(x*S(2)(S(1) + sqrt(S(7))) + S(-3))), sqrt(S(7))/S(14) + S(1)/2)/(S(42)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(-3))sqrt(S(1)/(-x*S(2)(S(1) + sqrt(S(7))) + S(3)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) + xS(2) + S(-3)), x), x, sqrt(S(5))sqrt(x*S(2) + S(-1))sqrt(S(2)xS(2) + S(3))elliptic_f(asin(sqrt(S(15))x/(S(3)sqrt(x*S(2) + S(-1)))), S(3)/5)/(S(5)sqrt(S(2)xS(4) + xS(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) + S(-3)), x), x, S(6)(S(1)/4)sqrt((sqrt(S(6))x*S(2) + S(3))/(-sqrt(S(6))x*S(2) + S(3)))sqrt(sqrt(S(6))xS(2) + S(-3))elliptic_f(asin(S(2)*(S(3)/4)S(3)*(S(1)/4)x/sqrt(sqrt(S(6))xS(2) + S(-3))), S(1)/2)/(S(6)sqrt(S(2)xS(4) + S(-3))sqrt(S(1)/(-sqrt(S(6))xS(2) + S(3)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) - xS(2) + S(-3)), x), x, sqrt(S(5))sqrt(xS(2) + S(1))sqrt(S(2)xS(2) + S(-3))elliptic_f(asin(sqrt(S(5))x/sqrt(S(2)x*S(2) + S(-3))), S(2)/5)/(S(5)sqrt(S(2)xS(4) - xS(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) - S(2)x*S(2) + S(-3)), x), x, sqrt(S(6))S(7)*(S(3)/4)sqrt((x*S(2)(S(1) + sqrt(S(7))) + S(3))/(x*S(2)(-sqrt(S(7)) + S(1)) + S(3)))sqrt(-xS(2)(-sqrt(S(7)) + S(1)) + S(-3))elliptic_f(asin(sqrt(S(2))S(7)*(S(1)/4)x/sqrt(-x*S(2)(-sqrt(S(7)) + S(1)) + S(-3))), -sqrt(S(7))/S(14) + S(1)/2)/(S(42)sqrt(S(2)x*S(4) - S(2)x*S(2) + S(-3))sqrt(S(1)/(x*S(2)(-sqrt(S(7)) + S(1)) + S(3)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) - S(3)xS(2) + S(-3)), x), x, S(11)(S(3)/4)S(3)(S(1)/4)sqrt((x*S(2)(S(3) + sqrt(S(33))) + S(6))/(x*S(2)(-sqrt(S(33)) + S(3)) + S(6)))sqrt(-xS(2)(-sqrt(S(33)) + S(3)) + S(-6))elliptic_f(asin(sqrt(S(2))S(33)*(S(1)/4)x/sqrt(-x*S(2)(-sqrt(S(33)) + S(3)) + S(-6))), -sqrt(S(33))/S(22) + S(1)/2)/(S(66)sqrt(S(2)x*S(4) - S(3)x*S(2) + S(-3))sqrt(S(1)/(x*S(2)(-sqrt(S(33)) + S(3)) + S(6)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) - S(4)xS(2) + S(-3)), x), x, S(2)(S(1)/4)sqrt(S(3))S(5)*(S(3)/4)sqrt((x*S(2)(S(2) + sqrt(S(10))) + S(3))/(x*S(2)(-sqrt(S(10)) + S(2)) + S(3)))sqrt(-xS(2)(-sqrt(S(10)) + S(2)) + S(-3))elliptic_f(asin(S(2)(S(3)/4)S(5)*(S(1)/4)x/sqrt(-x*S(2)(-sqrt(S(10)) + S(2)) + S(-3))), -sqrt(S(10))/S(10) + S(1)/2)/(S(30)sqrt(S(2)x*S(4) - S(4)x*S(2) + S(-3))sqrt(S(1)/(x*S(2)(-sqrt(S(10)) + S(2)) + S(3)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) - S(5)x*S(2) + S(-3)), x), x, sqrt(S(7))sqrt(x*S(2) + S(-3))sqrt(S(2)xS(2) + S(1))elliptic_f(asin(sqrt(S(7))x/sqrt(xS(2) + S(-3))), S(1)/7)/(S(7)sqrt(S(2)xS(4) - S(5)x*S(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)x*S(4) + S(5)x*S(2) + S(2)), x), x, sqrt(S(2))sqrt((S(3)xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(-1)/2)/(S(2)sqrt(S(3)x*S(4) + S(5)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)x*S(4) + S(4)xS(2) + S(2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) + S(4)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), -sqrt(S(6))/S(6) + S(1)/2)/(S(12)sqrt(S(3)x*S(4) + S(4)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)x*S(4) + S(3)xS(2) + S(2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) + S(3)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), -sqrt(S(6))/S(8) + S(1)/2)/(S(12)sqrt(S(3)x*S(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)x*S(4) + S(2)xS(2) + S(2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) + S(2)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), -sqrt(S(6))/S(12) + S(1)/2)/(S(12)sqrt(S(3)x*S(4) + S(2)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) + xS(2) + S(2)), x), x, S(6)*(S(3)/4)sqrt((S(3)xS(4) + xS(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), -sqrt(S(6))/S(24) + S(1)/2)/(S(12)sqrt(S(3)xS(4) + xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) + S(2)), x), x, S(6)(S(3)/4)sqrt((S(3)xS(4) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), S(1)/2)/(S(12)sqrt(S(3)x*S(4) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) - xS(2) + S(2)), x), x, S(6)*(S(3)/4)sqrt((S(3)xS(4) - xS(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), sqrt(S(6))/S(24) + S(1)/2)/(S(12)sqrt(S(3)xS(4) - xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)xS(4) - S(2)xS(2) + S(2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) - S(2)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), sqrt(S(6))/S(12) + S(1)/2)/(S(12)sqrt(S(3)x*S(4) - S(2)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)x*S(4) - S(3)xS(2) + S(2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) - S(3)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), sqrt(S(6))/S(8) + S(1)/2)/(S(12)sqrt(S(3)x*S(4) - S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)x*S(4) - S(4)xS(2) + S(2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) - S(4)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), sqrt(S(6))/S(6) + S(1)/2)/(S(12)sqrt(S(3)x*S(4) - S(4)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)x*S(4) - S(5)xS(2) + S(2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) - S(5)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), S(1)/2 + S(5)sqrt(S(6))/S(24))/(S(12)sqrt(S(3)xS(4) - S(5)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)x*S(4) - S(6)xS(2) + S(2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) - S(6)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), S(1)/2 + sqrt(S(6))/S(4))/(S(12)sqrt(S(3)x*S(4) - S(6)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) + S(9)xS(2) + S(3)), x), x, sqrt((xS(2)(-sqrt(S(57)) + S(9)) + S(6))/(xS(2)(sqrt(S(57)) + S(9)) + S(6)))(xS(2)(sqrt(S(57)) + S(9)) + S(6))elliptic_f(atan(xsqrt(sqrt(S(57))/S(6) + S(3)/2)), S(-19)/4 + S(3)sqrt(S(57))/S(4))/(sqrt(S(6)sqrt(S(57)) + S(54))sqrt(S(2)x*S(4) + S(9)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) + S(8)xS(2) + S(3)), x), x, sqrt((xS(2)(-sqrt(S(10)) + S(4)) + S(3))/(xS(2)(sqrt(S(10)) + S(4)) + S(3)))(xS(2)(sqrt(S(10)) + S(4)) + S(3))elliptic_f(atan(xsqrt(sqrt(S(10))/S(3) + S(4)/3)), S(-10)/3 + S(4)sqrt(S(10))/S(3))/(sqrt(S(3)sqrt(S(10)) + S(12))sqrt(S(2)x*S(4) + S(8)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) + S(7)x*S(2) + S(3)), x), x, sqrt(S(6))sqrt((x*S(2) + S(3))/(S(2)x*S(2) + S(1)))(S(2)xS(2) + S(1))elliptic_f(atan(sqrt(S(2))x), S(5)/6)/(S(6)sqrt(S(2)xS(4) + S(7)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) + S(6)xS(2) + S(3)), x), x, sqrt((xS(2)(-sqrt(S(3)) + S(3)) + S(3))/(xS(2)(sqrt(S(3)) + S(3)) + S(3)))(xS(2)(sqrt(S(3)) + S(3)) + S(3))elliptic_f(atan(xsqrt(sqrt(S(3))/S(3) + S(1))), S(-1) + sqrt(S(3)))/(sqrt(S(3)sqrt(S(3)) + S(9))sqrt(S(2)xS(4) + S(6)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) + S(5)x*S(2) + S(3)), x), x, sqrt(S(3))sqrt((S(2)xS(2) + S(3))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/3)/(S(3)sqrt(S(2)x*S(4) + S(5)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) + S(4)xS(2) + S(3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) + S(4)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), -sqrt(S(6))/S(6) + S(1)/2)/(S(12)sqrt(S(2)x*S(4) + S(4)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) + S(3)xS(2) + S(3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) + S(3)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), -sqrt(S(6))/S(8) + S(1)/2)/(S(12)sqrt(S(2)x*S(4) + S(3)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) + S(2)xS(2) + S(3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), -sqrt(S(6))/S(12) + S(1)/2)/(S(12)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) + xS(2) + S(3)), x), x, S(6)*(S(3)/4)sqrt((S(2)xS(4) + xS(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), -sqrt(S(6))/S(24) + S(1)/2)/(S(12)sqrt(S(2)xS(4) + xS(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) + S(3)), x), x, S(6)(S(3)/4)sqrt((S(2)xS(4) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), S(1)/2)/(S(12)sqrt(S(2)x*S(4) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) - xS(2) + S(3)), x), x, S(6)*(S(3)/4)sqrt((S(2)xS(4) - xS(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), sqrt(S(6))/S(24) + S(1)/2)/(S(12)sqrt(S(2)xS(4) - xS(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)xS(4) - S(2)xS(2) + S(3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) - S(2)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), sqrt(S(6))/S(12) + S(1)/2)/(S(12)sqrt(S(2)x*S(4) - S(2)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) - S(3)xS(2) + S(3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) - S(3)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), sqrt(S(6))/S(8) + S(1)/2)/(S(12)sqrt(S(2)x*S(4) - S(3)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) - S(4)xS(2) + S(3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) - S(4)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), sqrt(S(6))/S(6) + S(1)/2)/(S(12)sqrt(S(2)x*S(4) - S(4)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) - S(5)xS(2) + S(3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) - S(5)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), S(1)/2 + S(5)sqrt(S(6))/S(24))/(S(12)sqrt(S(2)xS(4) - S(5)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) - S(6)xS(2) + S(3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) - S(6)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), S(1)/2 + sqrt(S(6))/S(4))/(S(12)sqrt(S(2)x*S(4) - S(6)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) - S(7)xS(2) + S(3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) - S(7)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), S(1)/2 + S(7)sqrt(S(6))/S(24))/(S(12)sqrt(S(2)xS(4) - S(7)x*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) + S(7)x*S(2) + S(-3)), x), x, -sqrt(S(5))elliptic_f(acos(sqrt(S(3))x/S(3)), S(6)/5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) + S(6)x*S(2) + S(-3)), x), x, -sqrt(S(2))S(3)*(S(3)/4)elliptic_f(acos(xsqrt(-sqrt(S(3))/S(3) + S(1))), S(1)/2 + sqrt(S(3))/S(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) + S(5)x*S(2) + S(-3)), x), x, -elliptic_f(acos(sqrt(S(6))x/S(3)), S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) + S(4)xS(2) + S(-3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) - S(4)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), sqrt(S(6))/S(6) + S(1)/2)/(S(12)sqrt(-S(2)x*S(4) + S(4)x*S(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) + S(3)xS(2) + S(-3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) - S(3)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), sqrt(S(6))/S(8) + S(1)/2)/(S(12)sqrt(-S(2)x*S(4) + S(3)x*S(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) + S(2)xS(2) + S(-3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) - S(2)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), sqrt(S(6))/S(12) + S(1)/2)/(S(12)sqrt(-S(2)x*S(4) + S(2)x*S(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) + xS(2) + S(-3)), x), x, S(6)*(S(3)/4)sqrt((S(2)xS(4) - xS(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), sqrt(S(6))/S(24) + S(1)/2)/(S(12)sqrt(-S(2)xS(4) + xS(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) + S(-3)), x), x, S(6)(S(3)/4)sqrt((S(2)xS(4) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), S(1)/2)/(S(12)sqrt(-S(2)x*S(4) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) - xS(2) + S(-3)), x), x, S(6)*(S(3)/4)sqrt((S(2)xS(4) + xS(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), -sqrt(S(6))/S(24) + S(1)/2)/(S(12)sqrt(-S(2)xS(4) - xS(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) - S(2)xS(2) + S(-3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), -sqrt(S(6))/S(12) + S(1)/2)/(S(12)sqrt(-S(2)x*S(4) - S(2)x*S(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) - S(3)xS(2) + S(-3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) + S(3)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), -sqrt(S(6))/S(8) + S(1)/2)/(S(12)sqrt(-S(2)x*S(4) - S(3)x*S(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) - S(4)xS(2) + S(-3)), x), x, S(6)(S(3)/4)sqrt((S(2)x*S(4) + S(4)x*S(2) + S(3))/(sqrt(S(6))xS(2) + S(3))S(2))(sqrt(S(6))x*S(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)S(3)*(S(3)/4)x/S(3)), -sqrt(S(6))/S(6) + S(1)/2)/(S(12)sqrt(-S(2)x*S(4) - S(4)x*S(2) + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)x*S(4) - S(5)x*S(2) + S(-3)), x), x, sqrt(S(3))sqrt(S(2)xS(2) + S(3))elliptic_f(atan(x), S(1)/3)/(S(3)sqrt((S(2)xS(2) + S(3))/(xS(2) + S(1)))sqrt(-xS(2) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)x*S(4) + S(6)x*S(2) + S(-2)), x), x, -sqrt(S(2))S(3)*(S(3)/4)elliptic_f(acos(sqrt(S(3))x/sqrt(sqrt(S(3)) + S(3))), S(1)/2 + sqrt(S(3))/S(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)x*S(4) + S(5)x*S(2) + S(-2)), x), x, -elliptic_f(acos(x), S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)x*S(4) + S(4)xS(2) + S(-2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) - S(4)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), sqrt(S(6))/S(6) + S(1)/2)/(S(12)sqrt(-S(3)x*S(4) + S(4)x*S(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)x*S(4) + S(3)xS(2) + S(-2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) - S(3)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), sqrt(S(6))/S(8) + S(1)/2)/(S(12)sqrt(-S(3)x*S(4) + S(3)x*S(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)x*S(4) + S(2)xS(2) + S(-2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) - S(2)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), sqrt(S(6))/S(12) + S(1)/2)/(S(12)sqrt(-S(3)x*S(4) + S(2)x*S(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)xS(4) + xS(2) + S(-2)), x), x, S(6)*(S(3)/4)sqrt((S(3)xS(4) - xS(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), sqrt(S(6))/S(24) + S(1)/2)/(S(12)sqrt(-S(3)xS(4) + xS(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)xS(4) + S(-2)), x), x, S(6)(S(3)/4)sqrt((S(3)xS(4) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), S(1)/2)/(S(12)sqrt(-S(3)x*S(4) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)xS(4) - xS(2) + S(-2)), x), x, S(6)*(S(3)/4)sqrt((S(3)xS(4) + xS(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), -sqrt(S(6))/S(24) + S(1)/2)/(S(12)sqrt(-S(3)xS(4) - xS(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)xS(4) - S(2)xS(2) + S(-2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) + S(2)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), -sqrt(S(6))/S(12) + S(1)/2)/(S(12)sqrt(-S(3)x*S(4) - S(2)x*S(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)x*S(4) - S(3)xS(2) + S(-2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) + S(3)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), -sqrt(S(6))/S(8) + S(1)/2)/(S(12)sqrt(-S(3)x*S(4) - S(3)x*S(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)x*S(4) - S(4)xS(2) + S(-2)), x), x, S(6)(S(3)/4)sqrt((S(3)x*S(4) + S(4)x*S(2) + S(2))/(sqrt(S(6))xS(2) + S(2))S(2))(sqrt(S(6))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(3)*(S(1)/4)x/S(2)), -sqrt(S(6))/S(6) + S(1)/2)/(S(12)sqrt(-S(3)x*S(4) - S(4)x*S(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)x*S(4) - S(5)x*S(2) + S(-2)), x), x, -sqrt(S(2))sqrt(-S(3)xS(2) + S(-2))elliptic_f(atan(x), S(-1)/2)/(S(2)sqrt((S(3)xS(2) + S(2))/(xS(2) + S(1)))sqrt(xS(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(5)x*S(4) + S(5)xS(2) + S(2)), x), x, S(10)(S(3)/4)sqrt((S(5)x*S(4) + S(5)x*S(2) + S(2))/(sqrt(S(10))xS(2) + S(2))S(2))(sqrt(S(10))x*S(2) + S(2))elliptic_f(S(2)atan(S(2)(S(3)/4)S(5)*(S(1)/4)x/S(2)), -sqrt(S(10))/S(8) + S(1)/2)/(S(20)sqrt(S(5)x*S(4) + S(5)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(4)x*S(4) + S(5)xS(2) + S(2)), x), x, S(2)(S(1)/4)sqrt((S(4)x*S(4) + S(5)x*S(2) + S(2))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -S(5)sqrt(S(2))/S(16) + S(1)/2)/(S(4)sqrt(S(4)xS(4) + S(5)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)x*S(4) + S(5)x*S(2) + S(2)), x), x, sqrt(S(2))sqrt((S(3)xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(-1)/2)/(S(2)sqrt(S(3)x*S(4) + S(5)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x*S(4) + S(5)xS(2) + S(2)), x), x, sqrt((xS(2) + S(2))/(S(2)xS(2) + S(1)))(S(2)xS(2) + S(1))elliptic_f(atan(sqrt(S(2))x), S(3)/4)/(S(2)sqrt(S(2)xS(4) + S(5)xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(xS(4) + S(5)xS(2) + S(2)), x), x, sqrt((xS(2)(-sqrt(S(17)) + S(5)) + S(4))/(x*S(2)(sqrt(S(17)) + S(5)) + S(4)))(xS(2)(sqrt(S(17)) + S(5)) + S(4))elliptic_f(atan(xsqrt(sqrt(S(17)) + S(5))/S(2)), S(-17)/4 + S(5)sqrt(S(17))/S(4))/(S(2)sqrt(sqrt(S(17)) + S(5))sqrt(xS(4) + S(5)xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-xS(4) + S(5)xS(2) + S(2)), x), x, sqrt(S(2))elliptic_f(asin(sqrt(S(2))x/sqrt(S(5) + sqrt(S(33)))), S(-29)/4 - S(5)sqrt(S(33))/S(4))/sqrt(S(-5) + sqrt(S(33))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(2)xS(4) + S(5)x*S(2) + S(2)), x), x, sqrt(S(2))elliptic_f(asin(S(2)x/sqrt(S(5) + sqrt(S(41)))), S(-33)/8 - S(5)sqrt(S(41))/S(8))/sqrt(S(-5) + sqrt(S(41))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(3)xS(4) + S(5)x*S(2) + S(2)), x), x, elliptic_f(asin(sqrt(S(2))x/S(2)), S(-6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(4)xS(4) + S(5)x*S(2) + S(2)), x), x, sqrt(S(2))elliptic_f(asin(S(2)sqrt(S(2))x/sqrt(S(5) + sqrt(S(57)))), S(-41)/16 - S(5)sqrt(S(57))/S(16))/sqrt(S(-5) + sqrt(S(57))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(5)x*S(4) + S(5)x*S(2) + S(2)), x), x, sqrt(S(2))elliptic_f(asin(sqrt(S(10))x/sqrt(S(5) + sqrt(S(65)))), S(-9)/4 - sqrt(S(65))/S(4))/sqrt(S(-5) + sqrt(S(65))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(6)x*S(4) + S(5)x*S(2) + S(2)), x), x, sqrt(S(2))elliptic_f(asin(S(2)sqrt(S(3))x/sqrt(S(5) + sqrt(S(73)))), S(-49)/24 - S(5)sqrt(S(73))/S(24))/sqrt(S(-5) + sqrt(S(73))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(7)x*S(4) + S(5)x*S(2) + S(2)), x), x, sqrt(S(2))elliptic_f(asin(x), S(-7)/2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(8)xS(4) + S(5)x*S(2) + S(2)), x), x, sqrt(S(2))elliptic_f(asin(S(4)x/sqrt(S(5) + sqrt(S(89)))), S(-57)/32 - S(5)sqrt(S(89))/S(32))/sqrt(S(-5) + sqrt(S(89))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(9)xS(4) + S(5)x*S(2) + S(2)), x), x, sqrt(S(2))elliptic_f(asin(S(3)sqrt(S(2))x/sqrt(S(5) + sqrt(S(97)))), S(-61)/36 - S(5)sqrt(S(97))/S(36))/sqrt(S(-5) + sqrt(S(97))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/sqrt(bx*S(2) + cx*S(4)), x), x, -S(2)bsqrt(bx*S(2) + cx*S(4))/(S(3)c*S(2)x) + xsqrt(bx*S(2) + cx*S(4))/(S(3)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/sqrt(bx*S(2) + cx*S(4)), x), x, -batanh(sqrt(c)xS(2)/sqrt(bx*S(2) + cx*S(4)))/(S(2)c*(S(3)/2)) + sqrt(bx*S(2) + cx*S(4))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/sqrt(bx*S(2) + cx*S(4)), x), x, sqrt(bx*S(2) + cx*S(4))/(cx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(bxS(2) + cx*S(4)), x), x, atanh(sqrt(c)x*S(2)/sqrt(bx*S(2) + cx*S(4)))/sqrt(c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(bx*S(2) + cx*S(4)), x), x, -atanh(sqrt(b)x/sqrt(bxS(2) + cx*S(4)))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(bxS(2) + cx*S(4))), x), x, -sqrt(bx*S(2) + cx*S(4))/(bxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(bx*S(2) + cx*S(4))), x), x, -sqrt(bx*S(2) + cx*S(4))/(S(2)bxS(3)) + catanh(sqrt(b)x/sqrt(bx*S(2) + cx*S(4)))/(S(2)b(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)sqrt(bx*S(2) + cx*S(4))), x), x, -sqrt(bx*S(2) + cx*S(4))/(S(3)bxS(4)) + S(2)csqrt(bx*S(2) + cx*S(4))/(S(3)b*S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)sqrt(bx*S(2) + cx*S(4))), x), x, -sqrt(bx*S(2) + cx*S(4))/(S(4)bxS(5)) + S(3)csqrt(bx*S(2) + cx*S(4))/(S(8)b*S(2)x*S(3)) - S(3)c*S(2)atanh(sqrt(b)x/sqrt(bx*S(2) + cx*S(4)))/(S(8)b(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/sqrt(a + cxS(4)), x), x, -a(S(3)/4)sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(6)c*(S(5)/4)sqrt(a + cxS(4))) + xsqrt(a + cxS(4))/(S(3)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/sqrt(a + cx*S(4)), x), x, sqrt(a + cx*S(4))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/sqrt(a + cxS(4)), x), x, -a(S(1)/4)sqrt((a + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a(S(1)/4)), S(1)/2)/(c(S(3)/4)sqrt(a + cxS(4))) + a(S(1)/4)sqrt((a + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(2)c*(S(3)/4)sqrt(a + cxS(4))) + xsqrt(a + cxS(4))/(sqrt(c)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + cx*S(4)), x), x, atanh(sqrt(c)x*S(2)/sqrt(a + cx*S(4)))/(S(2)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + cxS(4)), x), x, sqrt((a + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(2)a*(S(1)/4)c*(S(1)/4)sqrt(a + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + cxS(4))), x), x, -atanh(sqrt(a + cx*S(4))/sqrt(a))/(S(2)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)sqrt(a + cxS(4))), x), x, sqrt(c)xsqrt(a + cx*S(4))/(a(sqrt(a) + sqrt(c)xS(2))) - sqrt(a + cx*S(4))/(ax) - c*(S(1)/4)sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a(S(1)/4)), S(1)/2)/(a(S(3)/4)sqrt(a + cxS(4))) + c(S(1)/4)sqrt((a + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(2)a*(S(3)/4)sqrt(a + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)sqrt(a + cxS(4))), x), x, -sqrt(a + cx*S(4))/(S(2)axS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)sqrt(a + cxS(4))), x), x, -sqrt(a + cx*S(4))/(S(3)axS(3)) - c(S(3)/4)sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(6)a*(S(5)/4)sqrt(a + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/sqrt(a + bx*S(2)), x), x, S(3)a*S(2)atanh(sqrt(b)x/sqrt(a + bx*S(2)))/(S(8)b*(S(5)/2)) - S(3)axsqrt(a + bxS(2))/(S(8)bS(2)) + xS(3)sqrt(a + bx*S(2))/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/sqrt(a + bx*S(2)), x), x, -asqrt(a + bxS(2))/bS(2) + (a + bxS(2))(S(3)/2)/(S(3)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(a + bx*S(2)), x), x, -aatanh(sqrt(b)x/sqrt(a + bx*S(2)))/(S(2)b*(S(3)/2)) + xsqrt(a + bxS(2))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + bxS(2)), x), x, sqrt(a + bx*S(2))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + bx*S(2)), x), x, atanh(sqrt(b)x/sqrt(a + bxS(2)))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + bxS(2))), x), x, -atanh(sqrt(a + bxS(2))/sqrt(a))/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(a + bx*S(2))), x), x, -sqrt(a + bx*S(2))/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)sqrt(a + bxS(2))), x), x, -sqrt(a + bx*S(2))/(S(2)axS(2)) + batanh(sqrt(a + bxS(2))/sqrt(a))/(S(2)a(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)sqrt(a + bx*S(2))), x), x, -sqrt(a + bx*S(2))/(S(3)axS(3)) + S(2)bsqrt(a + bx*S(2))/(S(3)a*S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)/sqrt(cx*S(4)), x), x, xsqrt(cxS(4))/(S(3)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/sqrt(cx*S(4)), x), x, sqrt(cx*S(4))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/sqrt(cxS(4)), x), x, xS(3)/sqrt(cxS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(cxS(4)), x), x, xS(2)log(x)/sqrt(cx*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(cx*S(4)), x), x, -x/sqrt(cx*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(cxS(4))), x), x, -S(1)/(S(2)sqrt(cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(cxS(4))), x), x, -S(1)/(S(3)xsqrt(cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)sqrt(cx*S(4))), x), x, -S(1)/(S(4)x*S(2)sqrt(cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)sqrt(cxS(4))), x), x, -S(1)/(S(5)x*S(3)sqrt(cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/sqrt(a), x), x, xS(5)/(S(5)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(a), x), x, xS(4)/(S(4)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(a), x), x, xS(3)/(S(3)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a), x), x, x*S(2)/(S(2)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a), x), x, x/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a)x), x), x, log(x)/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a)x*S(2)), x), x, -S(1)/(sqrt(a)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a)xS(3)), x), x, -S(1)/(S(2)sqrt(a)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a)x*S(4)), x), x, -S(1)/(S(3)sqrt(a)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-xS(4) - S(2)x*S(2) + S(3)), x), x, sqrt(S(3))elliptic_f(asin(x), S(-1)/3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-x*S(4) + S(5)xS(2) + S(-1)), x), x, -S(21)(S(3)/4)elliptic_f(acos(sqrt(S(2))x/sqrt(sqrt(S(21)) + S(5))), S(1)/2 + S(5)sqrt(S(21))/S(42))/S(21), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)(a + bxS(2) + cx*S(4)), x), x, S(2)ax(S(7)/2)/S(7) + S(2)bx(S(11)/2)/S(11) + S(2)cx(S(15)/2)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)(a + bxS(2) + cx*S(4)), x), x, S(2)ax(S(5)/2)/S(5) + S(2)bx(S(9)/2)/S(9) + S(2)cx(S(13)/2)/S(13), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)(a + bxS(2) + cx*S(4)), x), x, S(2)ax(S(3)/2)/S(3) + S(2)bx(S(7)/2)/S(7) + S(2)cx(S(11)/2)/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/sqrt(x), x), x, S(2)asqrt(x) + S(2)bx(S(5)/2)/S(5) + S(2)cx(S(9)/2)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/x(S(3)/2), x), x, -S(2)a/sqrt(x) + S(2)bx(S(3)/2)/S(3) + S(2)cx(S(7)/2)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/x(S(5)/2), x), x, -S(2)a/(S(3)x*(S(3)/2)) + S(2)bsqrt(x) + S(2)cx(S(5)/2)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/x(S(7)/2), x), x, -S(2)a/(S(5)x*(S(5)/2)) - S(2)b/sqrt(x) + S(2)cx(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)(a + bx*S(2) + cxS(4))S(2), x), x, S(2)aS(2)x*(S(7)/2)/S(7) + S(4)abx*(S(11)/2)/S(11) + S(4)bcx*(S(19)/2)/S(19) + S(2)c*S(2)x(S(23)/2)/S(23) + x(S(15)/2)(S(4)ac/S(15) + S(2)bS(2)/S(15)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)(a + bx*S(2) + cxS(4))S(2), x), x, S(2)aS(2)x*(S(5)/2)/S(5) + S(4)abx*(S(9)/2)/S(9) + S(4)bcx*(S(17)/2)/S(17) + S(2)c*S(2)x(S(21)/2)/S(21) + x(S(13)/2)(S(4)ac/S(13) + S(2)b*S(2)/S(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)(a + bxS(2) + cxS(4))S(2), x), x, S(2)aS(2)x*(S(3)/2)/S(3) + S(4)abx*(S(7)/2)/S(7) + S(4)bcx*(S(15)/2)/S(15) + S(2)c*S(2)x(S(19)/2)/S(19) + x(S(11)/2)(S(4)ac/S(11) + S(2)b*S(2)/S(11)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/sqrt(x), x), x, S(2)aS(2)sqrt(x) + S(4)abx(S(5)/2)/S(5) + S(4)bcx*(S(13)/2)/S(13) + S(2)c*S(2)x(S(17)/2)/S(17) + x(S(9)/2)(S(4)ac/S(9) + S(2)b*S(2)/S(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/x*(S(3)/2), x), x, -S(2)a*S(2)/sqrt(x) + S(4)abx*(S(3)/2)/S(3) + S(4)bcx*(S(11)/2)/S(11) + S(2)c*S(2)x(S(15)/2)/S(15) + x(S(7)/2)(S(4)ac/S(7) + S(2)b*S(2)/S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/x*(S(5)/2), x), x, -S(2)a*S(2)/(S(3)x*(S(3)/2)) + S(4)absqrt(x) + S(4)bcx(S(9)/2)/S(9) + S(2)c*S(2)x(S(13)/2)/S(13) + x(S(5)/2)(S(4)ac/S(5) + S(2)b*S(2)/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/x*(S(7)/2), x), x, -S(2)a*S(2)/(S(5)x*(S(5)/2)) - S(4)ab/sqrt(x) + S(4)bcx*(S(7)/2)/S(7) + S(2)c*S(2)x(S(11)/2)/S(11) + x(S(3)/2)(S(4)ac/S(3) + S(2)bS(2)/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)(a + bx*S(2) + cxS(4))S(3), x), x, S(2)aS(3)x*(S(7)/2)/S(7) + S(6)a*S(2)bx(S(11)/2)/S(11) + S(2)ax(S(15)/2)(ac + bS(2))/S(5) + S(2)bcS(2)x*(S(27)/2)/S(9) + S(2)bx(S(19)/2)(S(6)ac + b*S(2))/S(19) + S(2)c*S(3)x*(S(31)/2)/S(31) + S(6)cx(S(23)/2)(ac + bS(2))/S(23), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)(a + bxS(2) + cxS(4))S(3), x), x, S(2)aS(3)x*(S(5)/2)/S(5) + S(2)a*S(2)bx(S(9)/2)/S(3) + S(6)ax(S(13)/2)(ac + bS(2))/S(13) + S(6)bcS(2)x*(S(25)/2)/S(25) + S(2)bx(S(17)/2)(S(6)ac + b*S(2))/S(17) + S(2)c*S(3)x*(S(29)/2)/S(29) + S(2)cx(S(21)/2)(ac + bS(2))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)(a + bxS(2) + cxS(4))S(3), x), x, S(2)aS(3)x*(S(3)/2)/S(3) + S(6)a*S(2)bx(S(7)/2)/S(7) + S(6)ax(S(11)/2)(ac + bS(2))/S(11) + S(6)bcS(2)x*(S(23)/2)/S(23) + S(2)bx(S(15)/2)(S(6)ac + b*S(2))/S(15) + S(2)c*S(3)x*(S(27)/2)/S(27) + S(6)cx(S(19)/2)(ac + bS(2))/S(19), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(3)/sqrt(x), x), x, S(2)aS(3)sqrt(x) + S(6)aS(2)bx(S(5)/2)/S(5) + S(2)ax(S(9)/2)(ac + bS(2))/S(3) + S(2)bcS(2)x*(S(21)/2)/S(7) + S(2)bx(S(13)/2)(S(6)ac + b*S(2))/S(13) + S(2)c*S(3)x*(S(25)/2)/S(25) + S(6)cx(S(17)/2)(ac + bS(2))/S(17), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(3)/x*(S(3)/2), x), x, -S(2)a*S(3)/sqrt(x) + S(2)a*S(2)bx(S(3)/2) + S(6)ax(S(7)/2)(ac + bS(2))/S(7) + S(6)bcS(2)x*(S(19)/2)/S(19) + S(2)bx(S(11)/2)(S(6)ac + b*S(2))/S(11) + S(2)c*S(3)x*(S(23)/2)/S(23) + S(2)cx(S(15)/2)(ac + bS(2))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(3)/x*(S(5)/2), x), x, -S(2)a*S(3)/(S(3)x*(S(3)/2)) + S(6)a*S(2)bsqrt(x) + S(6)ax(S(5)/2)(ac + bS(2))/S(5) + S(6)bcS(2)x*(S(17)/2)/S(17) + S(2)bx(S(9)/2)(S(6)ac + b*S(2))/S(9) + S(2)c*S(3)x*(S(21)/2)/S(21) + S(6)cx(S(13)/2)(ac + bS(2))/S(13), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(3)/x*(S(7)/2), x), x, -S(2)a*S(3)/(S(5)x*(S(5)/2)) - S(6)a*S(2)b/sqrt(x) + S(2)ax*(S(3)/2)(ac + bS(2)) + S(2)bcS(2)x*(S(15)/2)/S(5) + S(2)bx(S(7)/2)(S(6)ac + b*S(2))/S(7) + S(2)c*S(3)x*(S(19)/2)/S(19) + S(6)cx(S(11)/2)(ac + bS(2))/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(9)/2)/(a + bx*S(2) + cx*S(4)), x), x, S(2)x*(S(3)/2)/(S(3)c) - S(2)*(S(1)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(7)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) + S(2)*(S(1)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(7)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) - S(2)*(S(1)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(7)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)) + S(2)*(S(1)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(7)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(7)/2)/(a + bx*S(2) + cx*S(4)), x), x, S(2)sqrt(x)/c + S(2)*(S(3)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(5)/2)/(a + bx*S(2) + cxS(4)), x), x, -S(2)(S(1)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(3)/4)sqrt(-S(4)ac + bS(2))) + S(2)(S(1)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(3)/4)sqrt(-S(4)ac + bS(2))) + S(2)(S(1)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(3)/4)sqrt(-S(4)ac + bS(2))) - S(2)(S(1)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(3)/4)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)/(a + bxS(2) + cxS(4)), x), x, S(2)(S(3)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(1)/4)sqrt(-S(4)ac + bS(2))) + S(2)(S(3)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(1)/4)sqrt(-S(4)ac + bS(2))) - S(2)(S(3)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(1)/4)sqrt(-S(4)ac + bS(2))) - S(2)(S(3)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)c(S(1)/4)sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(a + bx*S(2) + cxS(4)), x), x, S(2)(S(1)/4)c(S(1)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/((-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)sqrt(-S(4)ac + bS(2))) - S(2)(S(1)/4)c*(S(1)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/((-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)sqrt(-S(4)ac + bS(2))) - S(2)(S(1)/4)c*(S(1)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/((-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)sqrt(-S(4)ac + bS(2))) + S(2)(S(1)/4)c*(S(1)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/((-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)(a + bxS(2) + cxS(4))), x), x, -S(2)(S(3)/4)c(S(3)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/((-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))) - S(2)(S(3)/4)c*(S(3)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/((-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))) + S(2)(S(3)/4)c*(S(3)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/((-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))) + S(2)(S(3)/4)c*(S(3)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/((-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(3)/2)(a + bxS(2) + cxS(4))), x), x, -S(2)(S(1)/4)c(S(1)/4)(b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)a(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) + S(2)*(S(1)/4)c*(S(1)/4)(b/sqrt(-S(4)ac + b*S(2)) + S(1))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)a(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) - S(2)*(S(1)/4)c*(S(1)/4)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)a(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)) + S(2)*(S(1)/4)c*(S(1)/4)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)a(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)) - S(2)/(asqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(5)/2)(a + bxS(2) + cxS(4))), x), x, S(2)(S(3)/4)c(S(3)/4)(b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)a(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)c*(S(3)/4)(b/sqrt(-S(4)ac + b*S(2)) + S(1))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)a(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)c*(S(3)/4)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)a(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)c*(S(3)/4)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)a(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) - S(2)/(S(3)ax(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(7)/2)(a + bx*S(2) + cx*S(4))), x), x, -S(2)/(S(5)ax(S(5)/2)) + S(2)b/(a*S(2)sqrt(x)) + S(2)*(S(1)/4)c*(S(1)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)aS(2)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) - S(2)*(S(1)/4)c*(S(1)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)aS(2)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) + S(2)*(S(1)/4)c*(S(1)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)aS(2)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)) - S(2)*(S(1)/4)c*(S(1)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)aS(2)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(13)/2)/(a + bx*S(2) + cxS(4))S(2), x), x, -bx(S(3)/2)/(S(2)c(-S(4)ac + bS(2))) + x(S(7)/2)(S(2)a + bx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cxS(4))) - S(2)(S(1)/4)(-S(20)abc + S(3)bS(3) - (-S(14)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(7)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)) + S(2)(S(1)/4)(-S(20)abc + S(3)b*S(3) - (-S(14)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(7)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)) + S(2)(S(1)/4)(-S(20)abc + S(3)b*S(3) + (-S(14)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(7)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)) - S(2)(S(1)/4)(-S(20)abc + S(3)b*S(3) + (-S(14)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(7)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(11)/2)/(a + bx*S(2) + cxS(4))S(2), x), x, -bsqrt(x)/(S(2)c(-S(4)ac + bS(2))) + x(S(5)/2)(S(2)a + bx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cxS(4))) - S(2)(S(3)/4)(-S(10)ac + bS(2) - b(-S(12)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))) - S(2)(S(3)/4)(-S(10)ac + b*S(2) - b(-S(12)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))) - S(2)(S(3)/4)(-S(10)ac + b*S(2) + b(-S(12)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))) - S(2)(S(3)/4)(-S(10)ac + b*S(2) + b(-S(12)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(9)/2)/(a + bx*S(2) + cxS(4))S(2), x), x, x*(S(3)/2)(S(2)a + bx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cxS(4))) + S(2)(S(1)/4)(b - (S(12)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(3)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))) - S(2)(S(1)/4)(b - (S(12)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(3)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))) + S(2)(S(1)/4)(S(12)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(3)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)) - S(2)(S(1)/4)(S(12)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(3)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(7)/2)/(a + bx*S(2) + cxS(4))S(2), x), x, sqrt(x)(S(2)a + bxS(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cxS(4))) + S(2)(S(3)/4)(S(4)ac + S(3)b*S(2) - S(3)bsqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(1)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(3)/2)) + S(2)(S(3)/4)(S(4)ac + S(3)bS(2) - S(3)bsqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(1)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(3)/2)) - S(2)(S(3)/4)(S(4)ac + S(3)bS(2) + S(3)bsqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(1)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(3)/2)) - S(2)(S(3)/4)(S(4)ac + S(3)bS(2) + S(3)bsqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)c(S(1)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)/(a + bx*S(2) + cxS(4))S(2), x), x, S(2)*(S(1)/4)c*(S(1)/4)(S(4)b - sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)) - S(2)*(S(1)/4)c*(S(1)/4)(S(4)b - sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)) - S(2)*(S(1)/4)c*(S(1)/4)(S(4)b + sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)) + S(2)*(S(1)/4)c*(S(1)/4)(S(4)b + sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)) - x*(S(3)/2)(b + S(2)cx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)/(a + bxS(2) + cxS(4))S(2), x), x, S(2)*(S(3)/4)c*(S(3)/4)(-S(4)b/sqrt(-S(4)ac + bS(2)) + S(3))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))) + S(2)(S(3)/4)c(S(3)/4)(-S(4)b/sqrt(-S(4)ac + bS(2)) + S(3))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))) + S(2)(S(3)/4)c(S(3)/4)(S(4)b/sqrt(-S(4)ac + bS(2)) + S(3))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))) + S(2)(S(3)/4)c(S(3)/4)(S(4)b/sqrt(-S(4)ac + bS(2)) + S(3))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + b*S(2))) - sqrt(x)(b + S(2)cx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(a + bx*S(2) + cxS(4))S(2), x), x, S(2)*(S(1)/4)c*(S(1)/4)(b + (-S(20)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)a(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))) - S(2)(S(1)/4)c*(S(1)/4)(b + (-S(20)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)a(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))) + S(2)(S(1)/4)c*(S(1)/4)(b - (-S(20)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)a(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))) - S(2)(S(1)/4)c*(S(1)/4)(b - (-S(20)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)a(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))) + x(S(3)/2)(-S(2)ac + b*S(2) + bcxS(2))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)(a + bxS(2) + cxS(4))S(2)), x), x, -S(2)*(S(3)/4)c*(S(3)/4)(-S(28)ac + S(3)bS(2) + S(3)bsqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)a(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(3)/2)) - S(2)(S(3)/4)c*(S(3)/4)(-S(28)ac + S(3)bS(2) + S(3)bsqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)a(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(3)/2)) + S(2)(S(3)/4)c*(S(3)/4)(-S(28)ac + S(3)bS(2) - S(3)bsqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)a(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(3)/2)) + S(2)(S(3)/4)c*(S(3)/4)(-S(28)ac + S(3)bS(2) - S(3)bsqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)a(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(3)/2)) + sqrt(x)(-S(2)ac + b*S(2) + bcxS(2))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(3)/2)(a + bx*S(2) + cxS(4))S(2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(S(2)asqrt(x)(-S(4)ac + b*S(2))(a + bxS(2) + cxS(4))) - S(2)(S(1)/4)c(S(1)/4)(-S(28)abc + S(5)b*S(3) + (-S(18)ac + S(5)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)aS(2)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)) + S(2)(S(1)/4)c*(S(1)/4)(-S(28)abc + S(5)b*S(3) + (-S(18)ac + S(5)b*S(2))sqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)aS(2)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)) + S(2)(S(1)/4)c*(S(1)/4)(-S(28)abc + S(5)b*S(3) - (-S(18)ac + S(5)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)aS(2)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)) - S(2)(S(1)/4)c*(S(1)/4)(-S(28)abc + S(5)b*S(3) - (-S(18)ac + S(5)b*S(2))sqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(8)aS(2)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(3)/2)) - (-S(18)ac + S(5)b*S(2))/(S(2)a*S(2)sqrt(x)(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(15)/2)/(a + bx*S(2) + cxS(4))S(3), x), x, x*(S(9)/2)(S(2)a + bx*S(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + S(3)x(S(5)/2)(S(8)ab + x*S(2)(S(12)ac + b*S(2)))/(S(16)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) - sqrt(x)(S(36)ac + S(3)bS(2))/(S(16)c(-S(4)ac + bS(2))S(2)) - S(2)(S(3)/4)(-S(84)abc + S(3)b*S(3) - S(3)(-S(24)aS(2)c*S(2) - S(30)abS(2)c + b*S(4))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))S(2)) - S(2)(S(3)/4)(-S(84)abc + S(3)b*S(3) - S(3)(-S(24)aS(2)c*S(2) - S(30)abS(2)c + b*S(4))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))S(2)) - S(2)(S(3)/4)(-S(84)abc + S(3)b*S(3) + S(3)(-S(24)aS(2)c*S(2) - S(30)abS(2)c + b*S(4))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))S(2)) - S(2)(S(3)/4)(-S(84)abc + S(3)b*S(3) + S(3)(-S(24)aS(2)c*S(2) - S(30)abS(2)c + b*S(4))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(13)/2)/(a + bx*S(2) + cxS(4))S(3), x), x, x*(S(7)/2)(S(2)a + bx*S(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + x*(S(3)/2)(S(24)ab + x*S(2)(S(28)ac + S(5)bS(2)))/(S(16)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cxS(4))) + S(2)(S(1)/4)(S(28)ac + S(5)b*S(2) - (S(172)abc + S(5)bS(3))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)c(S(3)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))S(2)) - S(2)(S(1)/4)(S(28)ac + S(5)bS(2) - (S(172)abc + S(5)bS(3))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)c(S(3)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))S(2)) + S(2)(S(1)/4)(S(172)abc + S(5)b*S(3) + sqrt(-S(4)ac + bS(2))(S(28)ac + S(5)bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)c(S(3)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(5)/2)) - S(2)(S(1)/4)(S(172)abc + S(5)b*S(3) + sqrt(-S(4)ac + bS(2))(S(28)ac + S(5)bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)c(S(3)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(11)/2)/(a + bx*S(2) + cxS(4))S(3), x), x, x*(S(5)/2)(S(2)a + bx*S(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + sqrt(x)(S(24)ab + xS(2)(S(20)ac + S(7)bS(2)))/(S(16)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cxS(4))) - S(2)(S(3)/4)(S(60)ac + S(21)b*S(2) - S(3)(S(36)abc + S(7)b*S(3))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)c(S(1)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))S(2)) - S(2)(S(3)/4)(S(60)ac + S(21)bS(2) - S(3)(S(36)abc + S(7)b*S(3))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)c(S(1)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))S(2)) - S(2)(S(3)/4)(S(108)abc + S(21)b*S(3) + S(3)sqrt(-S(4)ac + b*S(2))(S(20)ac + S(7)bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)c(S(1)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(5)/2)) - S(2)(S(3)/4)(S(108)abc + S(21)b*S(3) + S(3)sqrt(-S(4)ac + b*S(2))(S(20)ac + S(7)bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)c(S(1)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(9)/2)/(a + bx*S(2) + cxS(4))S(3), x), x, S(3)S(2)(S(1)/4)c*(S(1)/4)(S(20)ac + S(11)bS(2) - S(4)bsqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(32)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(5)/2)) - S(3)S(2)(S(1)/4)c*(S(1)/4)(S(20)ac + S(11)bS(2) - S(4)bsqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(32)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(5)/2)) - S(3)S(2)(S(1)/4)c*(S(1)/4)(S(20)ac + S(11)bS(2) + S(4)bsqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(32)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(5)/2)) + S(3)S(2)(S(1)/4)c*(S(1)/4)(S(20)ac + S(11)bS(2) + S(4)bsqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(32)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(5)/2)) + x*(S(3)/2)(S(2)a + bx*S(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) - S(3)x(S(3)/2)(-S(4)ac + S(5)bS(2) + S(8)bcx*S(2))/(S(16)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(7)/2)/(a + bxS(2) + cxS(4))S(3), x), x, -S(2)*(S(3)/4)c*(S(3)/4)(S(28)ac + S(41)bS(2) - S(36)bsqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(32)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(5)/2)) - S(2)*(S(3)/4)c*(S(3)/4)(S(28)ac + S(41)bS(2) - S(36)bsqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(32)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(5)/2)) + S(2)*(S(3)/4)c*(S(3)/4)(S(28)ac + S(41)bS(2) + S(36)bsqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(32)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(5)/2)) + S(2)*(S(3)/4)c*(S(3)/4)(S(28)ac + S(41)bS(2) + S(36)bsqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(32)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(5)/2)) + sqrt(x)(S(2)a + bxS(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) - sqrt(x)(-S(4)ac + S(13)b*S(2) + S(24)bcx*S(2))/(S(16)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(5)/2)/(a + bxS(2) + cxS(4))S(3), x), x, -x*(S(3)/2)(b + S(2)cx*S(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + S(3)S(2)(S(1)/4)c*(S(1)/4)(-S(68)abc + bS(3) + sqrt(-S(4)ac + bS(2))(S(12)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)a(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(5)/2)) - S(3)S(2)*(S(1)/4)c*(S(1)/4)(-S(68)abc + bS(3) + sqrt(-S(4)ac + bS(2))(S(12)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)a(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(5)/2)) + S(3)S(2)*(S(1)/4)c*(S(1)/4)(S(68)abc/sqrt(-S(4)ac + bS(2)) + S(12)ac - bS(3)/sqrt(-S(4)ac + bS(2)) + bS(2))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)a(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))S(2)) - S(3)S(2)*(S(1)/4)c*(S(1)/4)(S(68)abc/sqrt(-S(4)ac + bS(2)) + S(12)ac - bS(3)/sqrt(-S(4)ac + bS(2)) + bS(2))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)a(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))S(2)) + S(3)x*(S(3)/2)(b(S(4)ac + bS(2)) + cx*S(2)(S(12)ac + b*S(2)))/(S(16)a(-S(4)ac + bS(2))S(2)(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)/(a + bxS(2) + cxS(4))S(3), x), x, -sqrt(x)(b + S(2)cxS(2))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) - S(3)S(2)(S(3)/4)c*(S(3)/4)(-S(68)abc + bS(3) + sqrt(-S(4)ac + bS(2))(S(44)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)a(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(5)/2)) - S(3)S(2)*(S(3)/4)c*(S(3)/4)(-S(68)abc + bS(3) + sqrt(-S(4)ac + bS(2))(S(44)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)a(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(5)/2)) - S(3)S(2)*(S(3)/4)c*(S(3)/4)(S(68)abc/sqrt(-S(4)ac + bS(2)) + S(44)ac - bS(3)/sqrt(-S(4)ac + bS(2)) + bS(2))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)a(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))S(2)) - S(3)S(2)*(S(3)/4)c*(S(3)/4)(S(68)abc/sqrt(-S(4)ac + bS(2)) + S(44)ac - bS(3)/sqrt(-S(4)ac + bS(2)) + bS(2))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)a(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))S(2)) + sqrt(x)(b(S(20)ac + bS(2)) + cx*S(2)(S(44)ac + b*S(2)))/(S(16)a(-S(4)ac + bS(2))S(2)(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(a + bx*S(2) + cxS(4))S(3), x), x, x*(S(3)/2)(-S(2)ac + b*S(2) + bcxS(2))/(S(4)a(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))S(2)) + S(2)*(S(1)/4)c*(S(1)/4)(S(520)aS(2)c*S(2) - S(54)abS(2)c + S(5)bS(4) + b(-S(44)ac + S(5)bS(2))sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)aS(2)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(5)/2)) - S(2)(S(1)/4)c*(S(1)/4)(S(520)aS(2)c*S(2) - S(54)abS(2)c + S(5)bS(4) + b(-S(44)ac + S(5)bS(2))sqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)aS(2)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(5)/2)) - S(2)(S(1)/4)c*(S(1)/4)(S(520)aS(2)c*S(2) - S(54)abS(2)c + S(5)bS(4) - b(-S(44)ac + S(5)bS(2))sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)aS(2)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(5)/2)) + S(2)(S(1)/4)c*(S(1)/4)(S(520)aS(2)c*S(2) - S(54)abS(2)c + S(5)bS(4) - b(-S(44)ac + S(5)bS(2))sqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)aS(2)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)(-S(4)ac + bS(2))(S(5)/2)) + x(S(3)/2)(S(52)aS(2)c*S(2) - S(45)abS(2)c + S(5)bS(4) + bcxS(2)(-S(44)ac + S(5)bS(2)))/(S(16)a*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)(a + bxS(2) + cxS(4))S(3)), x), x, sqrt(x)(-S(2)ac + bS(2) + bcxS(2))/(S(4)a(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))S(2)) - S(3)S(2)(S(3)/4)c*(S(3)/4)(S(280)aS(2)c*S(2) - S(66)abS(2)c + S(7)bS(4) + b(-S(52)ac + S(7)bS(2))sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)aS(2)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(5)/2)) - S(3)S(2)*(S(3)/4)c*(S(3)/4)(S(280)aS(2)c*S(2) - S(66)abS(2)c + S(7)bS(4) + b(-S(52)ac + S(7)bS(2))sqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)aS(2)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(5)/2)) + S(3)S(2)*(S(3)/4)c*(S(3)/4)(S(280)aS(2)c*S(2) - S(66)abS(2)c + S(7)bS(4) - b(-S(52)ac + S(7)bS(2))sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)aS(2)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(5)/2)) + S(3)S(2)*(S(3)/4)c*(S(3)/4)(S(280)aS(2)c*S(2) - S(66)abS(2)c + S(7)bS(4) - b(-S(52)ac + S(7)bS(2))sqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(x)/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(64)aS(2)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)(-S(4)ac + bS(2))(S(5)/2)) + sqrt(x)(S(60)aS(2)c*S(2) - S(55)abS(2)c + S(7)bS(4) + bcxS(2)(-S(52)ac + S(7)bS(2)))/(S(16)a*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(3)/2)sqrt(a + bxS(2) + cx*S(4)), x), x, S(2)(dx)(S(5)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(5)/4, S(-1)/2, S(-1)/2, S(9)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)dsqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)sqrt(a + bx*S(2) + cx*S(4)), x), x, S(2)(dx)(S(3)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(3)/4, S(-1)/2, S(-1)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)dsqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cx*S(4))/sqrt(dx), x), x, S(2)sqrt(dx)sqrt(a + bx*S(2) + cx*S(4))AppellF1(S(1)/4, S(-1)/2, S(-1)/2, S(5)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(dsqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cx*S(4))/(dx)*(S(3)/2), x), x, -S(2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(-1)/4, S(-1)/2, S(-1)/2, S(3)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(dsqrt(dx)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(3)/2)(a + bxS(2) + cxS(4))(S(3)/2), x), x, S(2)a(dx)(S(5)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(5)/4, S(-3)/2, S(-3)/2, S(9)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)dsqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)(a + bx*S(2) + cxS(4))(S(3)/2), x), x, S(2)a(dx)(S(3)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(3)/4, S(-3)/2, S(-3)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)dsqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(3)/2)/sqrt(dx), x), x, S(2)asqrt(dx)sqrt(a + bx*S(2) + cx*S(4))AppellF1(S(1)/4, S(-3)/2, S(-3)/2, S(5)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(dsqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(3)/2)/(dx)(S(3)/2), x), x, -S(2)asqrt(a + bx*S(2) + cx*S(4))AppellF1(S(-1)/4, S(-3)/2, S(-3)/2, S(3)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(dsqrt(dx)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(3)/2)/sqrt(a + bx*S(2) + cx*S(4)), x), x, S(2)(dx)(S(5)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(5)/4, S(1)/2, S(1)/2, S(9)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)dsqrt(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)/sqrt(a + bxS(2) + cx*S(4)), x), x, S(2)(dx)(S(3)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(3)/4, S(1)/2, S(1)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)dsqrt(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(dx)sqrt(a + bx*S(2) + cx*S(4))), x), x, S(2)sqrt(dx)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/4, S(1)/2, S(1)/2, S(5)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(dsqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(3)/2)sqrt(a + bxS(2) + cx*S(4))), x), x, -S(2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(-1)/4, S(1)/2, S(1)/2, S(3)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(dsqrt(dx)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(3)/2)/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, S(2)(dx)*(S(5)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(5)/4, S(3)/2, S(3)/2, S(9)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)adsqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)/(a + bxS(2) + cxS(4))(S(3)/2), x), x, S(2)(dx)*(S(3)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(3)/4, S(3)/2, S(3)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)adsqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(dx)(a + bx*S(2) + cxS(4))(S(3)/2)), x), x, S(2)sqrt(dx)sqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/4, S(3)/2, S(3)/2, S(5)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(adsqrt(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(3)/2)(a + bxS(2) + cxS(4))(S(3)/2)), x), x, -S(2)sqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(-1)/4, S(3)/2, S(3)/2, S(3)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(adsqrt(dx)sqrt(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a + bxS(2) + cxS(4))S(3), x), x, a*S(3)(dx)(m + S(1))/(d(m + S(1))) + S(3)aS(2)b(dx)(m + S(3))/(dS(3)(m + S(3))) + S(3)a(dx)*(m + S(5))(ac + bS(2))/(dS(5)(m + S(5))) + S(3)bc*S(2)(dx)(m + S(11))/(dS(11)(m + S(11))) + b(dx)*(m + S(7))(S(6)ac + bS(2))/(dS(7)(m + S(7))) + cS(3)(dx)(m + S(13))/(dS(13)(m + S(13))) + S(3)c(dx)(m + S(9))(ac + bS(2))/(dS(9)(m + S(9))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m(a + bxS(2) + cxS(4))S(2), x), x, a*S(2)(dx)(m + S(1))/(d(m + S(1))) + S(2)ab(dx)(m + S(3))/(dS(3)(m + S(3))) + S(2)bc(dx)(m + S(7))/(dS(7)(m + S(7))) + c*S(2)(dx)(m + S(9))/(dS(9)(m + S(9))) + (dx)(m + S(5))(S(2)ac + bS(2))/(dS(5)(m + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a + bxS(2) + cx*S(4)), x), x, a(dx)(m + S(1))/(d(m + S(1))) + b(dx)(m + S(3))/(dS(3)(m + S(3))) + c(dx)(m + S(5))/(dS(5)(m + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/(a + bx*S(2) + cx*S(4)), x), x, -S(2)c(dx)*(m + S(1))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))))/(d(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))sqrt(-S(4)ac + bS(2))) + S(2)c(dx)*(m + S(1))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))))/(d(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m/(a + bx*S(2) + cxS(4))S(2), x), x, -c(dx)*(m + S(1))(-S(4)ac(-m + S(3)) + bS(2)(-m + S(1)) - b(-m + S(1))sqrt(-S(4)ac + b*S(2)))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)ad(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))(S(3)/2)) + c(dx)(m + S(1))(-S(4)ac(-m + S(3)) + bS(2)(-m + S(1)) + b(-m + S(1))sqrt(-S(4)ac + b*S(2)))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))))/(S(2)ad(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))(S(3)/2)) + (dx)*(m + S(1))(-S(2)ac + b*S(2) + bcxS(2))/(S(2)ad(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))p/x, x), x, S(4)*(p + S(-1))((b + S(2)cx*S(2) - sqrt(-S(4)ac + bS(2)))/(cxS(2)))(-p)((b + S(2)cxS(2) + sqrt(-S(4)ac + bS(2)))/(cxS(2)))(-p)(a + bx*S(2) + cxS(4))pAppellF1(-S(2)p, -p, -p, -S(2)p + S(1), -(b - sqrt(-S(4)ac + bS(2)))/(S(2)cxS(2)), -(b + sqrt(-S(4)ac + bS(2)))/(S(2)cxS(2)))/p, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))p/xS(3), x), x, -S(2)(S(2)p + S(-1))((b + S(2)cx*S(2) - sqrt(-S(4)ac + bS(2)))/(cxS(2)))(-p)((b + S(2)cxS(2) + sqrt(-S(4)ac + bS(2)))/(cxS(2)))(-p)(a + bx*S(2) + cxS(4))pAppellF1(-S(2)p + S(1), -p, -p, -S(2)p + S(2), -(b - sqrt(-S(4)ac + bS(2)))/(S(2)cxS(2)), -(b + sqrt(-S(4)ac + bS(2)))/(S(2)cxS(2)))/(xS(2)(-S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))p/xS(5), x), x, -S(4)(p + S(-1))((b + S(2)cxS(2) - sqrt(-S(4)ac + bS(2)))/(cxS(2)))(-p)((b + S(2)cxS(2) + sqrt(-S(4)ac + bS(2)))/(cxS(2)))(-p)(a + bx*S(2) + cxS(4))pAppellF1(-S(2)p + S(2), -p, -p, -S(2)p + S(3), -(b - sqrt(-S(4)ac + bS(2)))/(S(2)cxS(2)), -(b + sqrt(-S(4)ac + bS(2)))/(S(2)cxS(2)))/(xS(4)(-p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)(a + bxS(2) + cxS(4))p, x), x, x*S(5)(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(2) + cxS(4))pAppellF1(S(5)/2, -p, -p, S(7)/2, -S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bxS(2) + cxS(4))p, x), x, x*S(3)(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(2) + cxS(4))pAppellF1(S(3)/2, -p, -p, S(5)/2, -S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))p, x), x, x(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(2) + cxS(4))pAppellF1(S(1)/2, -p, -p, S(3)/2, -S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))p/x*S(2), x), x, -(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(2) + cxS(4))pAppellF1(S(-1)/2, -p, -p, S(1)/2, -S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))p/x*S(4), x), x, -(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(2) + cxS(4))pAppellF1(S(-3)/2, -p, -p, S(-1)/2, -S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)x*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a + bxS(2) + cxS(4))(S(3)/2), x), x, a(dx)*(m + S(1))sqrt(a + bxS(2) + cx*S(4))AppellF1(m/S(2) + S(1)/2, S(-3)/2, S(-3)/2, m/S(2) + S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(d(m + S(1))sqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*msqrt(a + bxS(2) + cx*S(4)), x), x, (dx)*(m + S(1))sqrt(a + bxS(2) + cx*S(4))AppellF1(m/S(2) + S(1)/2, S(-1)/2, S(-1)/2, m/S(2) + S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(d(m + S(1))sqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m/sqrt(a + bx*S(2) + cx*S(4)), x), x, (dx)*(m + S(1))sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(m/S(2) + S(1)/2, S(1)/2, S(1)/2, m/S(2) + S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(d(m + S(1))sqrt(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, (dx)(m + S(1))sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(m/S(2) + S(1)/2, S(3)/2, S(3)/2, m/S(2) + S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(ad(m + S(1))sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a + bxS(2) + cxS(4))p, x), x, (dx)(m + S(1))(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(2) + cxS(4))pAppellF1(m/S(2) + S(1)/2, -p, -p, m/S(2) + S(3)/2, -S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(d(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a + bxS(2) + cxS(4))p, x), x, S(2)*(p + S(-1))b(-(b + S(2)cxS(2) - sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)))(-p + S(-1))(a + bxS(2) + cxS(4))(p + S(1))hyper((-p, p + S(1)), (p + S(2),), (b + S(2)cxS(2) + sqrt(-S(4)ac + bS(2)))/(S(2)sqrt(-S(4)ac + b*S(2))))/(c(p + S(1))sqrt(-S(4)ac + bS(2))) + (a + bx*S(2) + cxS(4))(p + S(1))/(S(4)c(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx*S(2) + cxS(4))p, x), x, -S(2)*p(-(b + S(2)cx*S(2) - sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)))(-p + S(-1))(a + bxS(2) + cxS(4))(p + S(1))hyper((-p, p + S(1)), (p + S(2),), (b + S(2)cxS(2) + sqrt(-S(4)ac + bS(2)))/(S(2)sqrt(-S(4)ac + b*S(2))))/((p + S(1))sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(3) + bxS(6))(S(5)/3), x), x, -S(3)a(axS(3) + bxS(6))(S(8)/3)/(S(88)bS(2)x*S(8)) + (ax*S(3) + bxS(6))(S(8)/3)/(S(11)bx*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(3) + bxS(6))(S(2)/3), x), x, (axS(3) + bxS(6))(S(5)/3)/(S(5)bx*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(3) + bxS(6))(S(-2)/3), x), x, -(axS(3) + bxS(6))(S(1)/3)/(axS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(3) + bxS(6))(S(-5)/3), x), x, S(1)/(S(2)ax*S(2)(axS(3) + bxS(6))(S(2)/3)) - S(3)(ax*S(3) + bxS(6))(S(1)/3)/(S(4)aS(2)x*S(5)) + S(9)b(ax*S(3) + bxS(6))(S(1)/3)/(S(4)aS(3)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6) - xS(3)), x), x, log(-x + S(1))/S(3) - log(xS(2) + x + S(1))/S(6) - sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(3) + S(1)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, S(3)ax(m + S(1))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/((a + bx*S(3))(m*S(2) + S(5)m + S(4))) + x*(m + S(1))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(m + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, ax*S(6)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(18)a + S(18)bxS(3)) + xS(6)sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, S(3)axS(5)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(40)a + S(40)bxS(3)) + xS(5)sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, S(3)axS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(28)a + S(28)bxS(3)) + xS(4)sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, (a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(6)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, S(3)axS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(10)a + S(10)bxS(3)) + xS(2)sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, S(3)axsqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(4)a + S(4)bx*S(3)) + xsqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/x, x), x, asqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))log(x)/(a + bxS(3)) + sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/xS(2), x), x, -S(3)asqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(2)x(a + bxS(3))) + sqrt(aS(2) + S(2)abxS(3) + bS(2)x*S(6))/(S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/xS(3), x), x, -S(3)asqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(2)x*S(2)(a + bxS(3))) + sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/xS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/xS(4), x), x, -asqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(3)x*S(3)(a + bxS(3))) + bsqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))log(x)/(a + bxS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/xS(5), x), x, S(3)asqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(4)x*S(4)(a + bxS(3))) - sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/xS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/xS(6), x), x, S(3)asqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(10)x*S(5)(a + bxS(3))) - sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(2)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/xS(7), x), x, -(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(6)axS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/xS(8), x), x, S(3)asqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(28)x*S(7)(a + bxS(3))) - sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(4)xS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/xS(9), x), x, S(3)asqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(40)x*S(8)(a + bxS(3))) - sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(5)xS(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/xS(10), x), x, asqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(18)x*S(9)(a + bxS(3))) - sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(6)xS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/xS(11), x), x, S(3)asqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(70)x*S(10)(a + bxS(3))) - sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(7)xS(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, S(162)a*S(3)x*(m + S(1))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/((a + bx*S(3))(m + S(7))(m + S(10))(m*S(2) + S(5)m + S(4))) + S(54)aS(2)x*(m + S(1))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/((m + S(4))(m + S(7))(m + S(10))) + S(9)ax(m + S(1))(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(mS(2) + S(17)m + S(70)) + x*(m + S(1))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(m + S(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, S(81)a*S(3)x*S(10)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(19760)a + S(19760)bx*S(3)) + S(27)a*S(2)x*S(10)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(1976) + S(9)axS(10)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(304) + xS(10)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(19), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, aS(2)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(36)b*S(3)) - a(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(45)bS(3)) + xS(6)(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(18)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(7)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, S(81)a*S(3)x*S(8)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(10472)a + S(10472)bx*S(3)) + S(27)a*S(2)x*S(8)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(1309) + S(9)axS(8)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(238) + xS(8)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(17), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, S(81)a*S(3)x*S(7)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(7280)a + S(7280)bx*S(3)) + S(27)a*S(2)x*S(7)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(1040) + S(9)axS(7)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(208) + xS(7)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, -a(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(12)bS(2)) + (aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(15)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, S(81)a*S(3)x*S(5)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(3080)a + S(3080)bx*S(3)) + S(27)a*S(2)x*S(5)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(616) + S(9)axS(5)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(154) + xS(5)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(14), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, S(81)a*S(3)x*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(1820)a + S(1820)bx*S(3)) + S(27)a*S(2)x*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(455) + S(9)axS(4)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(130) + xS(4)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(13), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, (a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(12)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, S(81)a*S(3)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(440)a + S(440)bx*S(3)) + S(27)a*S(2)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(220) + S(9)axS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(88) + xS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, S(81)a*S(3)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(140)a + S(140)bx*S(3)) + S(27)a*S(2)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/S(140) + S(9)ax(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(70) + x(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/x, x), x, aS(3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))log(x)/(a + bxS(3)) + aS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(3) + a(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(6) + (aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/xS(2), x), x, -S(81)a*S(3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(40)x(a + bx*S(3))) + S(27)a*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(40)x) + S(9)a(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(40)x) + (a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(8)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/xS(3), x), x, -S(81)a*S(3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(28)x*S(2)(a + bxS(3))) + S(27)a*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(14)x*S(2)) + S(9)a(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(28)xS(2)) + (aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(7)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/xS(4), x), x, S(3)a*S(2)bsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))log(x)/(a + bxS(3)) + absqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6)) - a(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(2)xS(3)) + (aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(6)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/xS(5), x), x, S(81)abS(2)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(20)a + S(20)bx*S(3)) + S(9)a(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(4)x*S(4)) + S(27)b*S(2)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(10) - S(5)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(2)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/x*S(6), x), x, S(81)abS(2)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(20)a + S(20)bx*S(3)) + S(9)a(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(10)x*S(5)) + S(27)b*S(2)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/S(20) - S(11)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(10)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/x*S(7), x), x, S(3)abS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))log(x)/(a + bxS(3)) + a(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(2)xS(6)) + bS(2)sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6)) - S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(3)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/x*S(8), x), x, -S(81)abS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(28)x(a + bx*S(3))) + S(9)a(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(28)x*S(7)) + S(27)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(28)x) - S(13)(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(28)xS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/x*S(9), x), x, -S(81)abS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(40)x*S(2)(a + bxS(3))) + S(9)a(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(40)x*S(8)) + S(27)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(20)x*S(2)) - S(7)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(20)xS(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/x*S(10), x), x, -ab*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(3)x*S(3)(a + bxS(3))) + a(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(6)xS(9)) + bS(3)sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))log(x)/(a + bxS(3)) - S(5)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(18)xS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/x*S(11), x), x, S(81)abS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(140)x*S(4)(a + bxS(3))) + S(9)a(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(70)x*S(10)) - S(27)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(35)x*S(4)) - S(8)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(35)xS(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/x*S(12), x), x, S(81)abS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(440)x*S(5)(a + bxS(3))) + S(9)a(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(88)x*S(11)) - S(27)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(88)x*S(5)) - S(17)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(88)xS(11)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/x*S(13), x), x, -(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(12)axS(12)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/xS(14), x), x, S(81)abS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(1820)x*S(7)(a + bxS(3))) + S(9)a(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(130)x*S(13)) - S(27)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(260)x*S(7)) - S(19)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(130)xS(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/x*S(15), x), x, S(81)abS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(3080)x*S(8)(a + bxS(3))) + S(9)a(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(154)x*S(14)) - S(27)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(385)x*S(8)) - S(10)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(77)xS(14)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/x*S(16), x), x, -(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(12)axS(15)) + (aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(60)a*S(2)xS(15)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/x*S(17), x), x, S(81)abS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(7280)x*S(10)(a + bxS(3))) + S(9)a(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(208)x*S(16)) - S(27)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(728)x*S(10)) - S(11)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(104)xS(16)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, S(29160)a*S(5)x*(m + S(1))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/((a + bx*S(3))(m + S(7))(m + S(10))(m + S(13))(m + S(16))(m*S(2) + S(5)m + S(4))) + S(9720)aS(4)x*(m + S(1))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/((m + S(4))(m + S(7))(m + S(10))(m + S(13))(m + S(16))) + S(1620)a*S(3)x*(m + S(1))(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/((m + S(13))(m + S(16))(mS(2) + S(17)m + S(70))) + S(180)aS(2)x*(m + S(1))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/((m + S(10))(m + S(13))(m + S(16))) + S(15)ax(m + S(1))(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(mS(2) + S(29)m + S(208)) + x*(m + S(1))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(m + S(16)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(13)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, S(729)a*S(5)x*S(14)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(2063698)a + S(2063698)bx*S(3)) + S(243)a*S(4)x*S(14)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(147407) + S(81)a*S(3)x*S(14)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(17342) + S(90)a*S(2)x*S(14)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(8671) + S(15)axS(14)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(754) + xS(14)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/S(29), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(12)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, S(729)a*S(5)x*S(13)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(1521520)a + S(1521520)bx*S(3)) + S(243)a*S(4)x*S(13)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(117040) + S(81)a*S(3)x*S(13)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(14630) + S(9)a*S(2)x*S(13)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(770) + S(3)axS(13)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(140) + xS(13)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/S(28), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, -aS(3)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(216)bS(4)) + aS(2)(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(7)/2)/(S(252)b*S(4)) - ax*S(6)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(72)bS(2)) + xS(9)(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(27)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(10)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, S(729)a*S(5)x*S(11)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(782782)a + S(782782)bx*S(3)) + S(243)a*S(4)x*S(11)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(71162) + S(81)a*S(3)x*S(11)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(10166) + S(9)a*S(2)x*S(11)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(598) + S(15)axS(11)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(598) + xS(11)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/S(26), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, S(729)a*S(5)x*S(10)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(543400)a + S(543400)bx*S(3)) + S(243)a*S(4)x*S(10)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(54340) + S(81)a*S(3)x*S(10)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(8360) + S(18)a*S(2)x*S(10)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(1045) + S(3)axS(10)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(110) + xS(10)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/S(25), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, aS(2)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(72)b*S(3)) - a(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(7)/2)/(S(84)bS(3)) + xS(6)(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(24)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(7)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, S(729)a*S(5)x*S(8)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(240856)a + S(240856)bx*S(3)) + S(243)a*S(4)x*S(8)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(30107) + S(81)a*S(3)x*S(8)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(5474) + S(9)a*S(2)x*S(8)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(391) + S(3)axS(8)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(92) + xS(8)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/S(23), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, S(729)a*S(5)x*S(7)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(152152)a + S(152152)bx*S(3)) + S(243)a*S(4)x*S(7)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(21736) + S(405)a*S(3)x*S(7)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(21736) + S(45)a*S(2)x*S(7)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(1672) + S(15)axS(7)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(418) + xS(7)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/S(22), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, -a(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(18)bS(2)) + (aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(7)/2)/(S(21)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, S(729)a*S(5)x*S(5)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(52360)a + S(52360)bx*S(3)) + S(243)a*S(4)x*S(5)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(10472) + S(81)a*S(3)x*S(5)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(2618) + S(9)a*S(2)x*S(5)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(238) + S(3)axS(5)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(68) + xS(5)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/S(20), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, S(729)a*S(5)x*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(27664)a + S(27664)bx*S(3)) + S(243)a*S(4)x*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(6916) + S(81)a*S(3)x*S(4)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(1976) + S(45)a*S(2)x*S(4)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(988) + S(15)axS(4)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(304) + xS(4)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/S(19), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, (a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(18)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, S(729)a*S(5)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(5236)a + S(5236)bx*S(3)) + S(243)a*S(4)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(2618) + S(405)a*S(3)x*S(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(5236) + S(90)a*S(2)x*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(1309) + S(15)axS(2)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(238) + xS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/S(17), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, S(729)a*S(5)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(1456)a + S(1456)bx*S(3)) + S(243)a*S(4)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/S(1456) + S(81)a*S(3)x(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(728) + S(9)a*S(2)x(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(104) + S(15)ax(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(208) + x(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x, x), x, aS(5)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))log(x)/(a + bxS(3)) + aS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(3) + aS(3)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(6) + aS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(9) + a(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(12) + (aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/xS(2), x), x, -S(729)a*S(5)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(308)x(a + bx*S(3))) + S(243)a*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(308)x) + S(81)aS(3)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(308)x) + S(45)aS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(308)x) + S(15)a(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(154)x) + (a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(14)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/xS(3), x), x, -S(729)a*S(5)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(182)x*S(2)(a + bxS(3))) + S(243)a*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(91)x*S(2)) + S(81)a*S(3)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(182)x*S(2)) + S(18)a*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(91)x*S(2)) + S(3)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(26)xS(2)) + (aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(13)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/xS(4), x), x, S(5)a*S(4)bsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))log(x)/(a + bxS(3)) + S(5)a*S(3)bsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/S(3) + S(5)a*S(2)b(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(6) + S(5)ab(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(9) - S(5)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(12)xS(3)) + (aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(12)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/xS(5), x), x, S(729)a*S(3)b*S(2)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(88)a + S(88)bx*S(3)) + S(243)a*S(2)b*S(2)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(44) + S(405)abS(2)x*S(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(88) + S(15)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(4)x*S(4)) + S(45)b*S(2)x*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(11) - S(4)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/xS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/xS(6), x), x, S(729)a*S(3)b*S(2)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(70)a + S(70)bx*S(3)) + S(243)a*S(2)b*S(2)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/S(70) + S(81)abS(2)x(a + bx*S(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(35) + S(3)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(2)x*S(5)) + S(9)b*S(2)x(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(5) - S(17)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(10)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(7), x), x, S(10)a*S(3)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))log(x)/(a + bxS(3)) + S(10)a*S(2)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(3) + S(5)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(3) + S(5)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(6)x*S(6)) + S(10)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/S(9) - (aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/xS(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/xS(8), x), x, -S(729)a*S(3)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(56)x(a + bx*S(3))) + S(243)a*S(2)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(56)x) + S(81)ab*S(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(56)x) + S(15)a(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(28)x*S(7)) + S(45)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(56)x) - S(19)(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(28)xS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(9), x), x, -S(729)a*S(3)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(56)x*S(2)(a + bxS(3))) + S(243)a*S(2)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(28)x*S(2)) + S(81)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(56)x*S(2)) + S(3)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(8)x*S(8)) + S(9)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(14)xS(2)) - (aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(2)xS(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/xS(10), x), x, S(10)a*S(2)b*S(3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))log(x)/(a + bxS(3)) + S(10)abS(3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(3) - S(5)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(3)x*S(3)) + S(5)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(18)x*S(9)) + S(5)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(9)x*S(3)) - S(7)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(18)xS(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(11), x), x, S(729)abS(4)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(70)a + S(70)bx*S(3)) + S(81)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(14)x*S(4)) + S(3)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(14)x*S(10)) + S(243)b*S(4)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(35) - S(45)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(7)x*S(4)) - S(11)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(35)xS(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(12), x), x, S(729)abS(4)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(88)a + S(88)bx*S(3)) + S(81)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(44)x*S(5)) + S(15)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(88)x*S(11)) + S(243)b*S(4)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/S(88) - S(9)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(4)x*S(5)) - S(23)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(88)xS(11)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(13), x), x, S(5)abS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))log(x)/(a + bxS(3)) + S(5)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(6)x*S(6)) + S(5)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(36)x*S(12)) + S(5)b*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/S(3) - S(10)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(9)x*S(6)) - S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(9)xS(12)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(14), x), x, -S(729)abS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(182)x(a + bx*S(3))) + S(81)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(182)x*S(7)) + S(3)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(26)x*S(13)) + S(243)b*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(182)x) - S(9)bS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(14)x*S(7)) - S(5)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(26)xS(13)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(15), x), x, -S(729)abS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(308)x*S(2)(a + bxS(3))) + S(81)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(308)x*S(8)) + S(15)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(154)x*S(14)) + S(243)b*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(154)x*S(2)) - S(9)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(22)x*S(8)) - S(13)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(77)xS(14)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(16), x), x, -ab*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(3)x*S(3)(a + bxS(3))) + ab*S(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(6)x*S(9)) + a(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(12)xS(15)) + bS(5)sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))log(x)/(a + bxS(3)) - S(5)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(18)x*S(9)) - S(3)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(20)xS(15)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(17), x), x, S(729)abS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(1456)x*S(4)(a + bxS(3))) + S(81)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(728)x*S(10)) + S(15)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(208)x*S(16)) - S(243)b*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(364)x*S(4)) - S(18)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(91)x*S(10)) - S(7)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(52)xS(16)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(18), x), x, S(729)abS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(5236)x*S(5)(a + bxS(3))) + S(405)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(5236)x*S(11)) + S(15)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(238)x*S(17)) - S(1215)b*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(5236)x*S(5)) - S(45)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(308)x*S(11)) - S(29)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(238)xS(17)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(19), x), x, -(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(18)axS(18)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/xS(20), x), x, S(729)abS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(27664)x*S(7)(a + bxS(3))) + S(81)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(1976)x*S(13)) + S(15)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(304)x*S(19)) - S(243)b*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(3952)x*S(7)) - S(9)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(104)x*S(13)) - S(31)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(304)xS(19)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(21), x), x, S(729)abS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(52360)x*S(8)(a + bxS(3))) + S(81)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(2618)x*S(14)) + S(3)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(68)x*S(20)) - S(243)b*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(6545)x*S(8)) - S(90)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(1309)x*S(14)) - S(8)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(85)xS(20)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(22), x), x, -(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(18)axS(21)) + (aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(7)/2)/(S(126)a*S(2)xS(21)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(23), x), x, S(729)abS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(152152)x*S(10)(a + bxS(3))) + S(405)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(21736)x*S(16)) + S(15)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(418)x*S(22)) - S(1215)b*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(76076)x*S(10)) - S(45)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(988)x*S(16)) - S(17)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(209)xS(22)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(24), x), x, S(729)abS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(240856)x*S(11)(a + bxS(3))) + S(81)abS(2)(a + bxS(3))sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(5474)x*S(17)) + S(3)a(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(92)x*S(23)) - S(243)b*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(21896)x*S(11)) - S(9)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)/(S(238)x*S(17)) - S(7)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(92)xS(23)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/x*S(25), x), x, -(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(24)axS(24)) + b(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)/(S(72)a*S(2)x*S(21)) - b(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(7)/2)/(S(504)a*S(3)xS(21)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, x(m + S(1))(a + bxS(3))hyper((S(1), m/S(3) + S(1)/3), (m/S(3) + S(4)/3,), -bxS(3)/a)/(a(m + S(1))sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, a(S(2)/3)(a + bxS(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(3)b*(S(5)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - a(S(2)/3)(a + bxS(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(6)b*(S(5)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + sqrt(S(3))a*(S(2)/3)(a + bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(3)b*(S(5)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + xS(2)(a + bxS(3))/(S(2)bsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, -a(S(1)/3)(a + bxS(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(3)b*(S(4)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + a(S(1)/3)(a + bxS(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(6)b*(S(4)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + sqrt(S(3))a*(S(1)/3)(a + bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(3)b*(S(4)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + x(a + bxS(3))/(bsqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, (a + bx*S(3))log(a + bxS(3))/(S(3)bsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, -(a + bx*S(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(3)a*(S(1)/3)b*(S(2)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + (a + bx*S(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(6)a*(S(1)/3)b*(S(2)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(S(3))(a + bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(3)a*(S(1)/3)b*(S(2)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6)), x), x, (a + bx*S(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(3)a*(S(2)/3)b*(S(1)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - (a + bx*S(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(6)a*(S(2)/3)b*(S(1)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(S(3))(a + bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(3)a*(S(2)/3)b*(S(1)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), x), x, (a + bx*S(3))log(x)/(asqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))) - (a + bx*S(3))log(a + bxS(3))/(S(3)asqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), x), x, -(a + bx*S(3))/(axsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))) + b(S(1)/3)(a + bxS(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(3)a*(S(4)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - b(S(1)/3)(a + bxS(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(6)a*(S(4)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + sqrt(S(3))b*(S(1)/3)(a + bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(3)a*(S(4)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), x), x, (-a - bx*S(3))/(S(2)axS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - b(S(2)/3)(a + bxS(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(3)a*(S(5)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + b(S(2)/3)(a + bxS(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(6)a*(S(5)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + sqrt(S(3))b*(S(2)/3)(a + bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(3)a*(S(5)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), x), x, -b(a + bxS(3))log(x)/(a*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + b(a + bxS(3))log(a + bxS(3))/(S(3)a*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(S(3)a*S(2)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, x(m + S(1))(a + bxS(3))hyper((S(3), m/S(3) + S(1)/3), (m/S(3) + S(4)/3,), -bxS(3)/a)/(aS(3)(m + S(1))sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, xS(5)(a + bxS(3))/(S(6)a(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) - xS(2)/(S(18)absqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - (a + bx*S(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(27)a*(S(4)/3)b*(S(5)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + (a + bx*S(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(54)a*(S(4)/3)b*(S(5)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(S(3))(a + bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(27)a*(S(4)/3)b*(S(5)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, xS(4)(a + bxS(3))/(S(6)a(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) - x/(S(9)absqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + (a + bx*S(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(27)a*(S(5)/3)b*(S(4)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - (a + bx*S(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(54)a*(S(5)/3)b*(S(4)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(S(3))(a + bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(27)a*(S(5)/3)b*(S(4)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, (-a - bx*S(3))/(S(6)b(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2), x), x, xS(2)(a + bxS(3))/(S(6)a(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(2)x*S(2)/(S(9)a*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + (a + bx*S(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(27)a*(S(7)/3)b*(S(2)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - (S(2)a + S(2)bx*S(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(27)a*(S(7)/3)b*(S(2)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(S(3))(S(2)a + S(2)bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(27)a*(S(7)/3)b*(S(2)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(-3)/2), x), x, x(a + bxS(3))/(S(6)a(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(5)x/(S(18)aS(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + (S(5)a + S(5)bx*S(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(27)a*(S(8)/3)b*(S(1)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - (S(5)a + S(5)bx*S(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(54)a*(S(8)/3)b*(S(1)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(S(3))(S(5)a + S(5)bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(27)a*(S(8)/3)b*(S(1)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)), x), x, (a + bx*S(3))/(S(6)a(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(1)/(S(3)a*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + (a + bx*S(3))log(x)/(a*S(3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - (a + bx*S(3))log(a + bxS(3))/(S(3)a*S(3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)), x), x, (a + bx*S(3))/(S(6)ax(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(7)/(S(18)a*S(2)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))) - (S(14)a + S(14)bx*S(3))/(S(9)a*S(3)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))) + S(14)b*(S(1)/3)(a + bxS(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(27)a*(S(10)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - S(7)b*(S(1)/3)(a + bxS(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(27)a*(S(10)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + S(14)sqrt(S(3))b(S(1)/3)(a + bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(27)a*(S(10)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)), x), x, (a + bx*S(3))/(S(6)axS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(4)/(S(9)a*S(2)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - (S(10)a + S(10)bx*S(3))/(S(9)a*S(3)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - S(20)b*(S(2)/3)(a + bxS(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(27)a*(S(11)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + S(10)b*(S(2)/3)(a + bxS(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(27)a*(S(11)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + S(20)sqrt(S(3))b(S(2)/3)(a + bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(27)a*(S(11)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)), x), x, (a + bx*S(3))/(S(6)axS(3)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(1)/(S(2)a*S(2)x*S(3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - S(3)b(a + bx*S(3))log(x)/(a*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + b(a + bxS(3))log(a + bxS(3))/(aS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))/(aS(4)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, x(m + S(1))(a + bxS(3))hyper((S(5), m/S(3) + S(1)/3), (m/S(3) + S(4)/3,), -bxS(3)/a)/(aS(5)(m + S(1))sqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, ax(a + bx*S(3))/(S(12)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)) - S(13)x/(S(108)bS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + x(a + bxS(3))/(S(162)abS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(5)x/(S(486)aS(2)b*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + (S(5)a + S(5)bx*S(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(729)a*(S(8)/3)b*(S(7)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - (S(5)a + S(5)bx*S(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(1458)a*(S(8)/3)b*(S(7)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(S(3))(S(5)a + S(5)bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(729)a*(S(8)/3)b*(S(7)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, a(a + bxS(3))/(S(12)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)) - S(1)/(S(9)b*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, xS(5)(a + bxS(3))/(S(12)a(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)) - S(7)x*S(2)/(S(108)ab(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(7)x*S(2)(a + bxS(3))/(S(324)a*S(2)b(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(7)x*S(2)/(S(243)a*S(3)bsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))) - (S(7)a + S(7)bx*S(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(729)a*(S(10)/3)b*(S(5)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + (S(7)a + S(7)bx*S(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(1458)a*(S(10)/3)b*(S(5)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(S(3))(S(7)a + S(7)bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(729)a*(S(10)/3)b*(S(5)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, xS(4)(a + bxS(3))/(S(12)a(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)) - S(2)x/(S(27)ab(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + x(a + bxS(3))/(S(81)a*S(2)b(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(5)x/(S(243)aS(3)bsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))) - (S(5)a + S(5)bx*S(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(729)a*(S(11)/3)b*(S(4)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + (S(10)a + S(10)bx*S(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(729)a*(S(11)/3)b*(S(4)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(S(3))(S(10)a + S(10)bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(729)a*(S(11)/3)b*(S(4)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, (-a - bx*S(3))/(S(12)b(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2), x), x, xS(2)(a + bxS(3))/(S(12)a(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)) + S(5)x*S(2)/(S(54)a*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(35)x*S(2)(a + bxS(3))/(S(324)a*S(3)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(35)x*S(2)/(S(243)a*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - (S(35)a + S(35)bx*S(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(729)a*(S(13)/3)b*(S(2)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + (S(35)a + S(35)bx*S(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(1458)a*(S(13)/3)b*(S(2)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(S(3))(S(35)a + S(35)bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(729)a*(S(13)/3)b*(S(2)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(-5)/2), x), x, x(a + bxS(3))/(S(12)a(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)) + S(11)x/(S(108)aS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(11)x(a + bx*S(3))/(S(81)a*S(3)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(55)x/(S(243)aS(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - (S(55)a + S(55)bx*S(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(729)a*(S(14)/3)b*(S(1)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + (S(110)a + S(110)bx*S(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(729)a*(S(14)/3)b*(S(1)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - sqrt(S(3))(S(110)a + S(110)bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(729)a*(S(14)/3)b*(S(1)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)), x), x, (a + bx*S(3))/(S(12)a(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)) + S(1)/(S(9)a*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + (a + bx*S(3))/(S(6)a*S(3)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(1)/(S(3)a*S(4)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + (a + bx*S(3))log(x)/(a*S(5)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - (a + bx*S(3))log(a + bxS(3))/(S(3)a*S(5)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)), x), x, (a + bx*S(3))/(S(12)ax(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)) + S(13)/(S(108)a*S(2)x(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + (S(65)a + S(65)bx*S(3))/(S(324)a*S(3)x(aS(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(455)/(S(972)a*S(4)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))) - (S(455)a + S(455)bx*S(3))/(S(243)a*S(5)xsqrt(aS(2) + S(2)abxS(3) + bS(2)xS(6))) + S(455)b*(S(1)/3)(a + bxS(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(729)a*(S(16)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - S(455)b*(S(1)/3)(a + bxS(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(1458)a*(S(16)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + S(455)sqrt(S(3))b(S(1)/3)(a + bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(729)a*(S(16)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)), x), x, (a + bx*S(3))/(S(12)axS(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)) + S(7)/(S(54)a*S(2)x*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + (S(77)a + S(77)bx*S(3))/(S(324)a*S(3)x*S(2)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(154)/(S(243)a*S(4)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - (S(385)a + S(385)bx*S(3))/(S(243)a*S(5)x*S(2)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - S(770)b*(S(2)/3)(a + bxS(3))log(a(S(1)/3) + b(S(1)/3)x)/(S(729)a*(S(17)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + S(385)b*(S(2)/3)(a + bxS(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(729)a*(S(17)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + S(770)sqrt(S(3))b(S(2)/3)(a + bxS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(729)a*(S(17)/3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)), x), x, (a + bx*S(3))/(S(12)axS(3)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(5)/2)) + S(5)/(S(36)a*S(2)x*S(3)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + (S(5)a + S(5)bx*S(3))/(S(18)a*S(3)x*S(3)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))(S(3)/2)) + S(5)/(S(6)a*S(4)x*S(3)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - S(5)b(a + bx*S(3))log(x)/(a*S(6)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) + S(5)b(a + bx*S(3))log(a + bxS(3))/(S(3)a*S(6)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))) - S(5)sqrt(a*S(2) + S(2)abxS(3) + bS(2)xS(6))/(S(3)a*S(6)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(aS(2) + S(2)abxS(3) + bS(2)xS(6))p, x), x, x(m + S(1))(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))phyper((S(1), m/S(3) + S(2)p + S(4)/3), (m/S(3) + S(4)/3,), -bx*S(3)/a)/(a(m + S(1))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(x*m(a*S(2) + S(2)abxS(3) + bS(2)xS(6))p, x), x, x(m + S(1))(S(1) + bxS(3)/a)(-S(2)p)(aS(2) + S(2)abxS(3) + bS(2)xS(6))phyper((m/S(3) + S(1)/3, -S(2)p), (m/S(3) + S(4)/3,), -bxS(3)/a)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(aS(2) + S(2)abxS(3) + bS(2)xS(6))p, x), x, -a(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))p/(S(3)b*S(2)(S(2)p + S(1))) + (aS(2) + S(2)abxS(3) + bS(2)xS(6))(p + S(1))/(S(6)b*S(2)(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))p, x), x, xS(5)(S(1) + bxS(3)/a)(-S(2)p)(aS(2) + S(2)abxS(3) + bS(2)xS(6))phyper((S(5)/3, -S(2)p), (S(8)/3,), -bxS(3)/a)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(aS(2) + S(2)abxS(3) + bS(2)xS(6))p, x), x, xS(4)(S(1) + bxS(3)/a)(-S(2)p)(aS(2) + S(2)abxS(3) + bS(2)xS(6))phyper((S(4)/3, -S(2)p), (S(7)/3,), -bxS(3)/a)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(aS(2) + S(2)abxS(3) + bS(2)xS(6))p, x), x, (a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))p/(S(3)b(S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(aS(2) + S(2)abxS(3) + bS(2)xS(6))p, x), x, xS(2)(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))phyper((S(1), S(2)p + S(5)/3), (S(5)/3,), -bx*S(3)/a)/(S(2)a), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(x(aS(2) + S(2)abxS(3) + bS(2)xS(6))p, x), x, xS(2)(S(1) + bxS(3)/a)(-S(2)p)(aS(2) + S(2)abxS(3) + bS(2)xS(6))phyper((S(2)/3, -S(2)p), (S(5)/3,), -bxS(3)/a)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))p, x), x, x(a + bx*S(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))phyper((S(1), S(2)p + S(4)/3), (S(4)/3,), -bxS(3)/a)/a, expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))p, x), x, x(S(1) + bxS(3)/a)(-S(2)p)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))phyper((S(1)/3, -S(2)p), (S(4)/3,), -bxS(3)/a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))p/x, x), x, -(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))phyper((S(1), S(2)p + S(1)), (S(2)p + S(2),), S(1) + bxS(3)/a)/(S(3)a(S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxS(3) + bS(2)xS(6))p/xS(2), x), x, -(S(1) + bxS(3)/a)(-S(2)p)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))phyper((S(-1)/3, -S(2)p), (S(2)/3,), -bxS(3)/a)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))p/x*S(3), x), x, -(S(1) + bxS(3)/a)(-S(2)p)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))phyper((S(-2)/3, -S(2)p), (S(1)/3,), -bx*S(3)/a)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))p/x*S(4), x), x, b(a + bxS(3))(a*S(2) + S(2)abxS(3) + bS(2)xS(6))phyper((S(2), S(2)p + S(1)), (S(2)p + S(2),), S(1) + bxS(3)/a)/(S(3)a*S(2)(S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxS(3) + bS(2)xS(6))p/xS(5), x), x, -(S(1) + bxS(3)/a)(-S(2)p)(a*S(2) + S(2)abxS(3) + bS(2)xS(6))phyper((S(-4)/3, -S(2)p), (S(-1)/3,), -bx*S(3)/a)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(a + bxS(3) + cx*S(6)), x), x, -blog(a + bxS(3) + cx*S(6))/(S(6)cS(2)) + xS(3)/(S(3)c) - (-S(2)ac + bS(2))atanh((b + S(2)cx*S(3))/sqrt(-S(4)ac + bS(2)))/(S(3)c*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(a + bxS(3) + cx*S(6)), x), x, batanh((b + S(2)cx*S(3))/sqrt(-S(4)ac + bS(2)))/(S(3)csqrt(-S(4)ac + bS(2))) + log(a + bx*S(3) + cx*S(6))/(S(6)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(a + bx*S(3) + cx*S(6)), x), x, -S(2)atanh((b + S(2)cx*S(3))/sqrt(-S(4)ac + bS(2)))/(S(3)sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bxS(3) + cx*S(6))), x), x, batanh((b + S(2)cx*S(3))/sqrt(-S(4)ac + bS(2)))/(S(3)asqrt(-S(4)ac + bS(2))) + log(x)/a - log(a + bx*S(3) + cx*S(6))/(S(6)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(4)(a + bxS(3) + cx*S(6))), x), x, -S(1)/(S(3)axS(3)) - blog(x)/a*S(2) + blog(a + bxS(3) + cx*S(6))/(S(6)a*S(2)) - (-S(2)ac + bS(2))atanh((b + S(2)cx*S(3))/sqrt(-S(4)ac + bS(2)))/(S(3)a*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(a + bxS(3) + cxS(6)), x), x, xS(2)/(S(2)c) + S(2)(S(1)/3)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(5)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(5)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)*(S(1)/3)sqrt(S(3))(b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(5)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)*(S(1)/3)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(5)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(5)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)*(S(1)/3)sqrt(S(3))(b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(5)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(6)/(a + bx*S(3) + cxS(6)), x), x, x/c - S(2)(S(2)/3)(b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(4)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(4)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)sqrt(S(3))(b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(4)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(4)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(4)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)sqrt(S(3))(b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(4)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)/(a + bx*S(3) + cxS(6)), x), x, S(2)(S(1)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(2)/3)sqrt(-S(4)ac + bS(2))) - S(2)(S(1)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(2)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(1)/3)sqrt(S(3))(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)atan(sqrt(S(3))(-S(2)S(2)(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(2)/3)sqrt(-S(4)ac + bS(2))) - S(2)(S(1)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(2)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(1)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(2)/3)sqrt(-S(4)ac + bS(2))) - S(2)(S(1)/3)sqrt(S(3))(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)atan(sqrt(S(3))(-S(2)S(2)(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(2)/3)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + bxS(3) + cxS(6)), x), x, -S(2)(S(2)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(1)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(2)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(1)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(2)/3)sqrt(S(3))(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)atan(sqrt(S(3))(-S(2)S(2)(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(1)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(2)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(1)/3)sqrt(-S(4)ac + bS(2))) - S(2)(S(2)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(1)/3)sqrt(-S(4)ac + bS(2))) - S(2)(S(2)/3)sqrt(S(3))(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)atan(sqrt(S(3))(-S(2)S(2)(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(1)/3)sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bx*S(3) + cxS(6)), x), x, S(2)(S(1)/3)c(S(1)/3)log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)sqrt(-S(4)ac + bS(2))) - S(2)(S(1)/3)c(S(1)/3)log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(6)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(1)/3)sqrt(S(3))c(S(1)/3)atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)sqrt(-S(4)ac + bS(2))) - S(2)(S(1)/3)c(S(1)/3)log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(1)/3)c(S(1)/3)log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(6)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)sqrt(-S(4)ac + bS(2))) - S(2)(S(1)/3)sqrt(S(3))c(S(1)/3)atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bx*S(3) + cxS(6)), x), x, -S(2)(S(2)/3)c(S(2)/3)log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(2)/3)c(S(2)/3)log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(6)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(2)/3)sqrt(S(3))c(S(2)/3)atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(2)/3)c(S(2)/3)log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)sqrt(-S(4)ac + bS(2))) - S(2)(S(2)/3)c(S(2)/3)log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(6)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)sqrt(-S(4)ac + bS(2))) - S(2)(S(2)/3)sqrt(S(3))c(S(2)/3)atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + bx*S(3) + cxS(6))), x), x, S(2)(S(1)/3)c(S(1)/3)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)a(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)c*(S(1)/3)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)a(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)(S(1)/3)sqrt(S(3))c(S(1)/3)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)a(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)*(S(1)/3)c*(S(1)/3)(b/sqrt(-S(4)ac + b*S(2)) + S(1))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)a(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)c*(S(1)/3)(b/sqrt(-S(4)ac + b*S(2)) + S(1))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)a(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)(S(1)/3)sqrt(S(3))c(S(1)/3)(b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)a(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(1)/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a + bxS(3) + cxS(6))), x), x, -S(2)(S(2)/3)c(S(2)/3)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)a(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)c*(S(2)/3)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)a(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)(S(2)/3)sqrt(S(3))c(S(2)/3)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)a(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)c*(S(2)/3)(b/sqrt(-S(4)ac + b*S(2)) + S(1))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)a(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)c*(S(2)/3)(b/sqrt(-S(4)ac + b*S(2)) + S(1))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)a(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)(S(2)/3)sqrt(S(3))c(S(2)/3)(b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)a(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(1)/(S(2)axS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(x*S(6) + S(4)xS(3) + S(3)), x), x, xS(6)/S(6) - S(4)xS(3)/S(3) - log(xS(3) + S(1))/S(6) + S(9)log(xS(3) + S(3))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(x*S(6) + S(4)xS(3) + S(3)), x), x, xS(3)/S(3) + log(x*S(3) + S(1))/S(6) - S(3)log(xS(3) + S(3))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(x*S(6) + S(4)xS(3) + S(3)), x), x, -log(xS(3) + S(1))/S(6) + log(xS(3) + S(3))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(x*S(6) + S(4)xS(3) + S(3)), x), x, -atanh(xS(3) + S(2))/S(3), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(2)/(xS(6) + S(4)xS(3) + S(3)), x), x, log(xS(3) + S(1))/S(6) - log(xS(3) + S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(x*S(6) + S(4)xS(3) + S(3))), x), x, log(x)/S(3) - log(xS(3) + S(1))/S(6) + log(xS(3) + S(3))/S(18), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(xS(6) + S(4)x*S(3) + S(3))), x), x, -S(4)log(x)/S(9) + log(xS(3) + S(1))/S(6) - log(xS(3) + S(3))/S(54) - S(1)/(S(9)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)(x*S(6) + S(4)x*S(3) + S(3))), x), x, S(13)log(x)/S(27) - log(xS(3) + S(1))/S(6) + log(xS(3) + S(3))/S(162) + S(4)/(S(27)xS(3)) - S(1)/(S(18)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(10)/(x*S(6) + S(4)xS(3) + S(3)), x), x, xS(5)/S(5) - S(2)xS(2) + log(x + S(1))/S(6) - S(3)S(3)*(S(2)/3)log(x + S(3)(S(1)/3))/S(2) - log(xS(2) - x + S(1))/S(12) + S(3)S(3)(S(2)/3)log(xS(2) - S(3)(S(1)/3)x + S(3)(S(2)/3))/S(4) - S(9)S(3)*(S(1)/6)atan(S(3)*(S(1)/6)(-S(2)x + S(3)(S(1)/3))/S(3))/S(2) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(xS(6) + S(4)xS(3) + S(3)), x), x, xS(4)/S(4) - S(4)x - log(x + S(1))/S(6) + S(3)S(3)(S(1)/3)log(x + S(3)(S(1)/3))/S(2) + log(xS(2) - x + S(1))/S(12) - S(3)S(3)(S(1)/3)log(xS(2) - S(3)(S(1)/3)x + S(3)(S(2)/3))/S(4) - S(3)S(3)*(S(5)/6)atan(S(3)*(S(1)/6)(-S(2)x + S(3)(S(1)/3))/S(3))/S(2) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(xS(6) + S(4)xS(3) + S(3)), x), x, xS(2)/S(2) - log(x + S(1))/S(6) + S(3)(S(2)/3)log(x + S(3)(S(1)/3))/S(2) + log(xS(2) - x + S(1))/S(12) - S(3)*(S(2)/3)log(xS(2) - S(3)(S(1)/3)x + S(3)(S(2)/3))/S(4) + S(3)S(3)*(S(1)/6)atan(S(3)*(S(1)/6)(-S(2)x + S(3)(S(1)/3))/S(3))/S(2) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(xS(6) + S(4)xS(3) + S(3)), x), x, x + log(x + S(1))/S(6) - S(3)(S(1)/3)log(x + S(3)(S(1)/3))/S(2) - log(xS(2) - x + S(1))/S(12) + S(3)*(S(1)/3)log(xS(2) - S(3)(S(1)/3)x + S(3)(S(2)/3))/S(4) + S(3)(S(5)/6)atan(S(3)*(S(1)/6)(-S(2)x + S(3)(S(1)/3))/S(3))/S(2) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(xS(6) + S(4)xS(3) + S(3)), x), x, log(x + S(1))/S(6) - S(3)(S(2)/3)log(x + S(3)(S(1)/3))/S(6) - log(xS(2) - x + S(1))/S(12) + S(3)*(S(2)/3)log(xS(2) - S(3)(S(1)/3)x + S(3)(S(2)/3))/S(12) - S(3)(S(1)/6)atan(S(3)*(S(1)/6)(-S(2)x + S(3)(S(1)/3))/S(3))/S(2) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(xS(6) + S(4)xS(3) + S(3)), x), x, -log(x + S(1))/S(6) + S(3)(S(1)/3)log(x + S(3)(S(1)/3))/S(6) + log(xS(2) - x + S(1))/S(12) - S(3)*(S(1)/3)log(xS(2) - S(3)(S(1)/3)x + S(3)(S(2)/3))/S(12) - S(3)(S(5)/6)atan(S(3)*(S(1)/6)(-S(2)x + S(3)(S(1)/3))/S(3))/S(6) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(x*S(6) + S(4)xS(3) + S(3)), x), x, -log(x + S(1))/S(6) + S(3)(S(2)/3)log(x + S(3)(S(1)/3))/S(18) + log(xS(2) - x + S(1))/S(12) - S(3)(S(2)/3)log(xS(2) - S(3)(S(1)/3)x + S(3)(S(2)/3))/S(36) + S(3)(S(1)/6)atan(S(3)*(S(1)/6)(-S(2)x + S(3)(S(1)/3))/S(3))/S(6) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(6) + S(4)xS(3) + S(3)), x), x, log(x + S(1))/S(6) - S(3)(S(1)/3)log(x + S(3)(S(1)/3))/S(18) - log(xS(2) - x + S(1))/S(12) + S(3)(S(1)/3)log(xS(2) - S(3)(S(1)/3)x + S(3)(S(2)/3))/S(36) + S(3)(S(5)/6)atan(S(3)*(S(1)/6)(-S(2)x + S(3)(S(1)/3))/S(3))/S(18) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(x*S(6) + S(4)xS(3) + S(3))), x), x, log(x + S(1))/S(6) - S(3)(S(2)/3)log(x + S(3)(S(1)/3))/S(54) - log(xS(2) - x + S(1))/S(12) + S(3)(S(2)/3)log(xS(2) - S(3)(S(1)/3)x + S(3)(S(2)/3))/S(108) - S(3)(S(1)/6)atan(S(3)*(S(1)/6)(-S(2)x + S(3)(S(1)/3))/S(3))/S(18) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) - S(1)/(S(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(x*S(6) + S(4)xS(3) + S(3))), x), x, -log(x + S(1))/S(6) + S(3)(S(1)/3)log(x + S(3)(S(1)/3))/S(54) + log(xS(2) - x + S(1))/S(12) - S(3)(S(1)/3)log(xS(2) - S(3)(S(1)/3)x + S(3)(S(2)/3))/S(108) - S(3)(S(5)/6)atan(S(3)*(S(1)/6)(-S(2)x + S(3)(S(1)/3))/S(3))/S(54) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) - S(1)/(S(6)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(x*S(6) + S(4)xS(3) + S(3))), x), x, -log(x + S(1))/S(6) + S(3)(S(2)/3)log(x + S(3)(S(1)/3))/S(162) + log(xS(2) - x + S(1))/S(12) - S(3)(S(2)/3)log(xS(2) - S(3)(S(1)/3)x + S(3)(S(2)/3))/S(324) + S(3)(S(1)/6)atan(S(3)*(S(1)/6)(-S(2)x + S(3)(S(1)/3))/S(3))/S(54) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) + S(4)/(S(9)x) - S(1)/(S(12)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(xS(6) + S(4)xS(3) + S(3))), x), x, log(x + S(1))/S(6) - S(3)(S(1)/3)log(x + S(3)(S(1)/3))/S(162) - log(xS(2) - x + S(1))/S(12) + S(3)(S(1)/3)log(xS(2) - S(3)(S(1)/3)x + S(3)(S(2)/3))/S(324) + S(3)(S(5)/6)atan(S(3)*(S(1)/6)(-S(2)x + S(3)(S(1)/3))/S(3))/S(162) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) + S(2)/(S(9)xS(2)) - S(1)/(S(15)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(xS(6) - xS(3) + S(1)), x), x, x + S(2)*(S(2)/3)(S(3) - sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(S(3) + sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(S(3) - sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) - sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(S(3) + sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) + sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(-sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) - sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) + sqrt(S(3))I)(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(xS(6) - xS(3) + S(1)), x), x, log(xS(6) - xS(3) + S(1))/S(6) - sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(3) + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(xS(6) - xS(3) + S(1)), x), x, S(2)*(S(1)/3)(S(3) + sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(S(3) - sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(1)/3)) - S(2)(S(1)/3)(S(3) + sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) - sqrt(S(3))I)(S(1)/3)) - S(2)(S(1)/3)(S(3) - sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) + sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) - sqrt(S(3))I)(S(1)/3)) - S(2)(S(1)/3)(-sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) + sqrt(S(3))I)(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(xS(6) - xS(3) + S(1)), x), x, S(2)(S(2)/3)(S(3) + sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(S(3) - sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(S(3) + sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) - sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(S(3) - sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) + sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) - sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(-sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) + sqrt(S(3))I)(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(xS(6) - xS(3) + S(1)), x), x, -S(2)sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(3) + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(6) - xS(3) + S(1)), x), x, sqrt(S(3))Ilog(-S(2)(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(9)(S(1)/2 - sqrt(S(3))I/S(2))(S(1)/3)) - sqrt(S(3))Ilog(-S(2)(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(9)(S(1)/2 + sqrt(S(3))I/S(2))(S(1)/3)) - S(2)(S(1)/3)sqrt(S(3))Ilog(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)sqrt(S(3))Ilog(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(1)/3)) + Iatan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))(S(1)/3) + S(1))/S(3))/(S(3)(S(1)/2 - sqrt(S(3))I/S(2))(S(1)/3)) - Iatan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))(S(1)/3) + S(1))/S(3))/(S(3)(S(1)/2 + sqrt(S(3))I/S(2))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6) - xS(3) + S(1)), x), x, -(S(-1))(S(5)/18)sqrt(S(3))(S(3)log(-x + (S(-1))(S(1)/9)) + log(S(2)))/S(27) + (S(-1))(S(13)/18)sqrt(S(3))log(-S(2)*(S(1)/3)(x + (S(-1))(S(8)/9)))/S(9) - (S(-1))(S(13)/18)sqrt(S(3))log(-S(2)*(S(2)/3)(x(-x + (S(-1))(S(8)/9)) + (S(-1))(S(7)/9)))/S(18) + (S(-1))(S(5)/18)sqrt(S(3))log(S(2)(S(2)/3)(x(x + (S(-1))(S(1)/9)) + (S(-1))(S(2)/9)))/S(18) - (S(-1))(S(13)/18)atan(sqrt(S(3))(S(2)(S(-1))*(S(1)/9)x + S(1))/S(3))/S(3) + (S(-1))*(S(5)/18)atan(sqrt(S(3))(-S(2)(S(-1))*(S(8)/9)x + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/(xS(6) - xS(3) + S(1)), x), x, sqrt(S(3))Ilog(-S(2)*(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(9)(S(1)/2 - sqrt(S(3))I/S(2))(S(2)/3)) - sqrt(S(3))Ilog(-S(2)(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(9)(S(1)/2 + sqrt(S(3))I/S(2))(S(2)/3)) - S(2)(S(2)/3)sqrt(S(3))Ilog(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)sqrt(S(3))Ilog(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(2)/3)) - Iatan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))(S(1)/3) + S(1))/S(3))/(S(3)(S(1)/2 - sqrt(S(3))I/S(2))(S(2)/3)) + Iatan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))(S(1)/3) + S(1))/S(3))/(S(3)(S(1)/2 + sqrt(S(3))I/S(2))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(xS(6) - xS(3) + S(1))), x), x, log(x) - log(xS(6) - xS(3) + S(1))/S(6) - sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(3) + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(xS(6) - xS(3) + S(1))), x), x, -S(2)*(S(1)/3)(S(3) - sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(1)/3)) - S(2)(S(1)/3)(S(3) + sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(S(3) - sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) - sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(S(3) + sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) + sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(-sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) - sqrt(S(3))I)(S(1)/3)) - S(2)(S(1)/3)(sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) + sqrt(S(3))I)(S(1)/3)) - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(xS(6) - xS(3) + S(1))), x), x, -S(2)*(S(2)/3)(S(3) - sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(S(3) + sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(S(3) - sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) - sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(S(3) + sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) + sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(-sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) - sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) + sqrt(S(3))I)(S(2)/3)) - S(1)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(xS(6) - xS(3) + S(1))), x), x, log(x) - log(xS(6) - xS(3) + S(1))/S(6) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x*S(3) + S(1))/S(3))/S(9) - S(1)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(xS(6) - xS(3) + S(1))), x), x, -S(2)(S(1)/3)(S(3) + sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(1)/3)) - S(2)(S(1)/3)(S(3) - sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(S(3) + sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) - sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(S(3) - sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) + sqrt(S(3))I)(S(1)/3)) - S(2)(S(1)/3)(sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) - sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(-sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) + sqrt(S(3))I)(S(1)/3)) - S(1)/x - S(1)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6) + x*S(3) + S(2)), x), x, -sqrt(S(7))Ilog(S(2)(S(1)/3)x + (S(1) - sqrt(S(7))I)(S(1)/3))/(S(21)(S(1)/2 - sqrt(S(7))I/S(2))(S(2)/3)) + sqrt(S(7))Ilog(S(2)(S(1)/3)x + (S(1) + sqrt(S(7))I)(S(1)/3))/(S(21)(S(1)/2 + sqrt(S(7))I/S(2))(S(2)/3)) + S(2)(S(2)/3)sqrt(S(7))Ilog(S(2)*(S(2)/3)x*S(2) - x(S(2) - S(2)sqrt(S(7))I)*(S(1)/3) + (S(1) - sqrt(S(7))I)*(S(2)/3))/(S(42)(S(1) - sqrt(S(7))I)(S(2)/3)) - S(2)(S(2)/3)sqrt(S(7))Ilog(S(2)*(S(2)/3)x*S(2) - x(S(2) + S(2)sqrt(S(7))I)*(S(1)/3) + (S(1) + sqrt(S(7))I)*(S(2)/3))/(S(42)(S(1) + sqrt(S(7))I)(S(2)/3)) + sqrt(S(21))Iatan(sqrt(S(3))(-S(2)x/(S(1)/2 - sqrt(S(7))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(21)(S(1)/2 - sqrt(S(7))I/S(2))(S(2)/3)) - sqrt(S(21))Iatan(sqrt(S(3))(-S(2)x/(S(1)/2 + sqrt(S(7))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(21)(S(1)/2 + sqrt(S(7))I/S(2))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(xS(6) + xS(3) + S(2)), x), x, S(2)sqrt(S(7))atan(sqrt(S(7))(S(2)xS(3) + S(1))/S(7))/S(21), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(xS(6) + xS(3) + S(2)), x), x, S(2)(S(2)/3)(S(7) + sqrt(S(7))I)log(S(2)*(S(1)/3)x + (S(1) - sqrt(S(7))I)(S(1)/3))/(S(42)(S(1) - sqrt(S(7))I)(S(2)/3)) + S(2)(S(2)/3)(S(7) - sqrt(S(7))I)log(S(2)*(S(1)/3)x + (S(1) + sqrt(S(7))I)(S(1)/3))/(S(42)(S(1) + sqrt(S(7))I)(S(2)/3)) - S(2)(S(2)/3)(S(7) + sqrt(S(7))I)log(S(2)*(S(2)/3)x*S(2) - x(S(2) - S(2)sqrt(S(7))I)*(S(1)/3) + (S(1) - sqrt(S(7))I)*(S(2)/3))/(S(84)(S(1) - sqrt(S(7))I)(S(2)/3)) - S(2)(S(2)/3)(S(7) - sqrt(S(7))I)log(S(2)*(S(2)/3)x*S(2) - x(S(2) + S(2)sqrt(S(7))I)*(S(1)/3) + (S(1) + sqrt(S(7))I)*(S(2)/3))/(S(84)(S(1) + sqrt(S(7))I)(S(2)/3)) - sqrt(S(21))I(S(1)/2 - sqrt(S(7))I/S(2))*(S(1)/3)atan(sqrt(S(3))(-S(2)x/(S(1)/2 - sqrt(S(7))I/S(2))(S(1)/3) + S(1))/S(3))/S(21) + sqrt(S(21))I(S(1)/2 + sqrt(S(7))I/S(2))*(S(1)/3)atan(sqrt(S(3))(-S(2)x/(S(1)/2 + sqrt(S(7))I/S(2))(S(1)/3) + S(1))/S(3))/S(21), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(14)sqrt(a + bxS(3) + cx*S(6)), x), x, -bx*S(6)(a + bxS(3) + cxS(6))(S(3)/2)/(S(20)cS(2)) + xS(9)(a + bxS(3) + cxS(6))(S(3)/2)/(S(18)c) - (S(7)b(-S(28)ac + S(15)b*S(2)) - S(6)cxS(3)(-S(20)ac + S(21)bS(2)))(a + bxS(3) + cxS(6))(S(3)/2)/(S(2880)cS(4)) + (b + S(2)cxS(3))sqrt(a + bxS(3) + cx*S(6))(S(16)aS(2)c*S(2) - S(56)abS(2)c + S(21)bS(4))/(S(1536)c*S(5)) - (-S(4)ac + bS(2))(S(16)aS(2)c*S(2) - S(56)abS(2)c + S(21)bS(4))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(3072)c(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)sqrt(a + bx*S(3) + cx*S(6)), x), x, -b(b + S(2)cx*S(3))(-S(12)ac + S(7)bS(2))sqrt(a + bxS(3) + cx*S(6))/(S(384)c*S(4)) + b(-S(12)ac + S(7)bS(2))(-S(4)ac + b*S(2))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(768)c(S(9)/2)) + xS(6)(a + bx*S(3) + cxS(6))(S(3)/2)/(S(15)c) + (a + bx*S(3) + cxS(6))(S(3)/2)(-S(32)ac + S(35)b*S(2) - S(42)bcx*S(3))/(S(720)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)sqrt(a + bx*S(3) + cx*S(6)), x), x, -S(5)b(a + bx*S(3) + cxS(6))(S(3)/2)/(S(72)cS(2)) + xS(3)(a + bxS(3) + cxS(6))(S(3)/2)/(S(12)c) + (b + S(2)cxS(3))(-S(4)ac + S(5)bS(2))sqrt(a + bxS(3) + cx*S(6))/(S(192)c*S(3)) - (-S(4)ac + bS(2))(-S(4)ac + S(5)bS(2))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(384)c(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)sqrt(a + bx*S(3) + cx*S(6)), x), x, -b(b + S(2)cx*S(3))sqrt(a + bxS(3) + cx*S(6))/(S(24)c*S(2)) + b(-S(4)ac + b*S(2))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(48)c*(S(5)/2)) + (a + bx*S(3) + cxS(6))(S(3)/2)/(S(9)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(a + bxS(3) + cx*S(6)), x), x, (b + S(2)cxS(3))sqrt(a + bxS(3) + cx*S(6))/(S(12)c) - (-S(4)ac + b*S(2))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(24)c*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(3) + cx*S(6))/x, x), x, -sqrt(a)atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/S(3) + batanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(6)sqrt(c)) + sqrt(a + bxS(3) + cx*S(6))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(3) + cxS(6))/xS(4), x), x, sqrt(c)atanh((b + S(2)cxS(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/S(3) - sqrt(a + bx*S(3) + cx*S(6))/(S(3)x*S(3)) - batanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(6)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bxS(3) + cxS(6))/xS(7), x), x, -(S(2)a + bx*S(3))sqrt(a + bxS(3) + cx*S(6))/(S(12)axS(6)) + (-S(4)ac + bS(2))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(24)a*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(3) + cxS(6))/xS(10), x), x, -(a + bxS(3) + cxS(6))(S(3)/2)/(S(9)ax*S(9)) + b(S(2)a + bx*S(3))sqrt(a + bxS(3) + cx*S(6))/(S(24)a*S(2)x*S(6)) - b(-S(4)ac + b*S(2))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(48)a*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(3) + cxS(6))/xS(13), x), x, -(a + bxS(3) + cxS(6))(S(3)/2)/(S(12)ax*S(12)) + S(5)b(a + bx*S(3) + cxS(6))(S(3)/2)/(S(72)aS(2)x*S(9)) - (S(2)a + bxS(3))(-S(4)ac + S(5)bS(2))sqrt(a + bxS(3) + cx*S(6))/(S(192)a*S(3)x*S(6)) + (-S(4)ac + bS(2))(-S(4)ac + S(5)bS(2))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(384)a*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(3) + cxS(6))/xS(16), x), x, -(a + bxS(3) + cxS(6))(S(3)/2)/(S(15)ax*S(15)) + S(7)b(a + bx*S(3) + cxS(6))(S(3)/2)/(S(120)aS(2)x*S(12)) - (-S(32)ac + S(35)b*S(2))(a + bxS(3) + cxS(6))(S(3)/2)/(S(720)aS(3)x*S(9)) + b(S(2)a + bx*S(3))(-S(12)ac + S(7)bS(2))sqrt(a + bxS(3) + cx*S(6))/(S(384)a*S(4)x*S(6)) - b(-S(12)ac + S(7)bS(2))(-S(4)ac + b*S(2))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(768)a(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(a + bx*S(3) + cxS(6)), x), x, xS(4)sqrt(a + bx*S(3) + cx*S(6))AppellF1(S(4)/3, S(-1)/2, S(-1)/2, S(7)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(a + bxS(3) + cxS(6)), x), x, xS(2)sqrt(a + bx*S(3) + cx*S(6))AppellF1(S(2)/3, S(-1)/2, S(-1)/2, S(5)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)sqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(3) + cx*S(6)), x), x, xsqrt(a + bxS(3) + cx*S(6))AppellF1(S(1)/3, S(-1)/2, S(-1)/2, S(4)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(sqrt(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(3) + cxS(6))/xS(2), x), x, -sqrt(a + bxS(3) + cx*S(6))AppellF1(S(-1)/3, S(-1)/2, S(-1)/2, S(2)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(xsqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(3) + cxS(6))/xS(3), x), x, -sqrt(a + bxS(3) + cx*S(6))AppellF1(S(-2)/3, S(-1)/2, S(-1)/2, S(1)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)x*S(2)sqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a + bxS(3) + cxS(6))(S(3)/2), x), x, axS(4)sqrt(a + bxS(3) + cx*S(6))AppellF1(S(4)/3, S(-3)/2, S(-3)/2, S(7)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxS(3) + cxS(6))(S(3)/2), x), x, axS(2)sqrt(a + bxS(3) + cx*S(6))AppellF1(S(2)/3, S(-3)/2, S(-3)/2, S(5)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)sqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))(S(3)/2), x), x, axsqrt(a + bxS(3) + cx*S(6))AppellF1(S(1)/3, S(-3)/2, S(-3)/2, S(4)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(sqrt(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))(S(3)/2)/x*S(2), x), x, -asqrt(a + bxS(3) + cx*S(6))AppellF1(S(-1)/3, S(-3)/2, S(-3)/2, S(2)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(xsqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))(S(3)/2)/x*S(3), x), x, -asqrt(a + bxS(3) + cx*S(6))AppellF1(S(-2)/3, S(-3)/2, S(-3)/2, S(1)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)x*S(2)sqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(a + bx*S(3) + cxS(6)), x), x, xS(4)sqrt(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(4)/3, S(1)/2, S(1)/2, S(7)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(a + bxS(3) + cx*S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + bx*S(3) + cxS(6)), x), x, xS(2)sqrt(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(2)/3, S(1)/2, S(1)/2, S(5)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)sqrt(a + bxS(3) + cx*S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + bx*S(3) + cx*S(6)), x), x, xsqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/3, S(1)/2, S(1)/2, S(4)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/sqrt(a + bx*S(3) + cxS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(a + bx*S(3) + cx*S(6))), x), x, -sqrt(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(-1)/3, S(1)/2, S(1)/2, S(2)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(xsqrt(a + bxS(3) + cxS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)sqrt(a + bx*S(3) + cx*S(6))), x), x, -sqrt(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(-2)/3, S(1)/2, S(1)/2, S(1)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)x*S(2)sqrt(a + bxS(3) + cxS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + bxS(3) + cxS(6))(S(3)/2), x), x, x*S(4)sqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(4)/3, S(3)/2, S(3)/2, S(7)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(4)asqrt(a + bx*S(3) + cx*S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bx*S(3) + cxS(6))(S(3)/2), x), x, x*S(2)sqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(2)/3, S(3)/2, S(3)/2, S(5)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)asqrt(a + bx*S(3) + cx*S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))(S(-3)/2), x), x, xsqrt(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/3, S(3)/2, S(3)/2, S(4)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(asqrt(a + bxS(3) + cxS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + bx*S(3) + cxS(6))(S(3)/2)), x), x, -sqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(-1)/3, S(3)/2, S(3)/2, S(2)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(axsqrt(a + bx*S(3) + cxS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a + bx*S(3) + cxS(6))(S(3)/2)), x), x, -sqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(-2)/3, S(3)/2, S(3)/2, S(1)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)axS(2)sqrt(a + bxS(3) + cx*S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a + bxS(3) + cxS(6))(S(3)/2), x), x, a(dx)*(m + S(1))sqrt(a + bxS(3) + cx*S(6))AppellF1(m/S(3) + S(1)/3, S(-3)/2, S(-3)/2, m/S(3) + S(4)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(d(m + S(1))sqrt(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*msqrt(a + bxS(3) + cx*S(6)), x), x, (dx)*(m + S(1))sqrt(a + bxS(3) + cx*S(6))AppellF1(m/S(3) + S(1)/3, S(-1)/2, S(-1)/2, m/S(3) + S(4)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(d(m + S(1))sqrt(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m/sqrt(a + bx*S(3) + cx*S(6)), x), x, (dx)*(m + S(1))sqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(m/S(3) + S(1)/3, S(1)/2, S(1)/2, m/S(3) + S(4)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(d(m + S(1))sqrt(a + bx*S(3) + cx*S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m/(a + bx*S(3) + cxS(6))(S(3)/2), x), x, (dx)(m + S(1))sqrt(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(m/S(3) + S(1)/3, S(3)/2, S(3)/2, m/S(3) + S(4)/3, -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(ad(m + S(1))sqrt(a + bxS(3) + cx*S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a + bxS(3) + cxS(6))p, x), x, (dx)(m + S(1))(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(3) + cxS(6))pAppellF1(m/S(3) + S(1)/3, -p, -p, m/S(3) + S(4)/3, -S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(d(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)(a + bxS(3) + cxS(6))p, x), x, x*S(5)(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(3) + cxS(6))pAppellF1(S(5)/3, -p, -p, S(8)/3, -S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a + bxS(3) + cxS(6))p, x), x, x*S(4)(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(3) + cxS(6))pAppellF1(S(4)/3, -p, -p, S(7)/3, -S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxS(3) + cxS(6))p, x), x, x*S(2)(S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(3) + cxS(6))pAppellF1(S(2)/3, -p, -p, S(5)/3, -S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))p, x), x, x(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(3) + cxS(6))pAppellF1(S(1)/3, -p, -p, S(4)/3, -S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))p/x, x), x, S(2)*(S(2)p + S(-1))((b + S(2)cxS(3) - sqrt(-S(4)ac + bS(2)))/(cxS(3)))(-p)((b + S(2)cxS(3) + sqrt(-S(4)ac + bS(2)))/(cxS(3)))(-p)(a + bx*S(3) + cxS(6))pAppellF1(-S(2)p, -p, -p, -S(2)p + S(1), -(b - sqrt(-S(4)ac + bS(2)))/(S(2)cxS(3)), -(b + sqrt(-S(4)ac + bS(2)))/(S(2)cxS(3)))/(S(3)p), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(3) + cxS(6))p/x*S(2), x), x, -(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(3) + cxS(6))pAppellF1(S(-1)/3, -p, -p, S(2)/3, -S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))p/x*S(3), x), x, -(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(3) + cxS(6))pAppellF1(S(-2)/3, -p, -p, S(1)/3, -S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))p/xS(4), x), x, -S(4)p((b + S(2)cxS(3) - sqrt(-S(4)ac + bS(2)))/(cxS(3)))(-p)((b + S(2)cxS(3) + sqrt(-S(4)ac + bS(2)))/(cxS(3)))(-p)(a + bx*S(3) + cxS(6))pAppellF1(-S(2)p + S(1), -p, -p, -S(2)p + S(2), -(b - sqrt(-S(4)ac + bS(2)))/(S(2)cxS(3)), -(b + sqrt(-S(4)ac + bS(2)))/(S(2)cxS(3)))/(xS(3)(-S(6)p + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))p/x*S(5), x), x, -(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(3) + cxS(6))pAppellF1(S(-4)/3, -p, -p, S(-1)/3, -S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(4)x*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))p/x*S(6), x), x, -(S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxS(3) + cxS(6))pAppellF1(S(-5)/3, -p, -p, S(-2)/3, -S(2)cxS(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)x*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))p/xS(7), x), x, -S(2)(S(2)p + S(-1))((b + S(2)cx*S(3) - sqrt(-S(4)ac + bS(2)))/(cxS(3)))(-p)((b + S(2)cxS(3) + sqrt(-S(4)ac + bS(2)))/(cxS(3)))(-p)(a + bx*S(3) + cxS(6))pAppellF1(-S(2)p + S(2), -p, -p, -S(2)p + S(3), -(b - sqrt(-S(4)ac + bS(2)))/(S(2)cxS(3)), -(b + sqrt(-S(4)ac + bS(2)))/(S(2)cxS(3)))/(xS(6)(-S(3)p + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(14)(a + bxS(3) + cxS(6))(S(3)/2), x), x, -S(11)bx*S(6)(a + bxS(3) + cxS(6))(S(5)/2)/(S(336)cS(2)) + xS(9)(a + bxS(3) + cxS(6))(S(5)/2)/(S(24)c) - (S(3)b(-S(124)ac + S(77)b*S(2)) - S(10)cxS(3)(-S(28)ac + S(33)bS(2)))(a + bxS(3) + cxS(6))(S(5)/2)/(S(13440)cS(4)) + (b + S(2)cxS(3))(a + bxS(3) + cxS(6))(S(3)/2)(S(16)a*S(2)c*S(2) - S(72)abS(2)c + S(33)bS(4))/(S(6144)c*S(5)) - (b + S(2)cxS(3))(-S(4)ac + b*S(2))sqrt(a + bxS(3) + cx*S(6))(S(16)aS(2)c*S(2) - S(72)abS(2)c + S(33)bS(4))/(S(16384)c*S(6)) + (-S(4)ac + bS(2))S(2)(S(16)aS(2)c*S(2) - S(72)abS(2)c + S(33)bS(4))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(32768)c(S(13)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)(a + bx*S(3) + cxS(6))(S(3)/2), x), x, -b(b + S(2)cxS(3))(-S(4)ac + S(3)bS(2))(a + bxS(3) + cxS(6))(S(3)/2)/(S(384)cS(4)) + b(b + S(2)cx*S(3))(-S(4)ac + b*S(2))(-S(4)ac + S(3)bS(2))sqrt(a + bxS(3) + cx*S(6))/(S(1024)c*S(5)) - b(-S(4)ac + bS(2))S(2)(-S(4)ac + S(3)b*S(2))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(2048)c(S(11)/2)) + xS(6)(a + bx*S(3) + cxS(6))(S(5)/2)/(S(21)c) + (a + bx*S(3) + cxS(6))(S(5)/2)(-S(16)ac + S(21)b*S(2) - S(30)bcx*S(3))/(S(840)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)(a + bx*S(3) + cxS(6))(S(3)/2), x), x, -S(7)b(a + bxS(3) + cxS(6))(S(5)/2)/(S(180)cS(2)) + xS(3)(a + bxS(3) + cxS(6))(S(5)/2)/(S(18)c) + (b + S(2)cxS(3))(-S(4)ac + S(7)bS(2))(a + bxS(3) + cxS(6))(S(3)/2)/(S(576)cS(3)) - (b + S(2)cxS(3))(-S(4)ac + b*S(2))(-S(4)ac + S(7)bS(2))sqrt(a + bxS(3) + cx*S(6))/(S(1536)c*S(4)) + (-S(4)ac + bS(2))S(2)(-S(4)ac + S(7)bS(2))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(3072)c(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(a + bx*S(3) + cxS(6))(S(3)/2), x), x, -b(b + S(2)cxS(3))(a + bxS(3) + cxS(6))(S(3)/2)/(S(48)cS(2)) + b(b + S(2)cx*S(3))(-S(4)ac + b*S(2))sqrt(a + bxS(3) + cx*S(6))/(S(128)c*S(3)) - b(-S(4)ac + bS(2))S(2)atanh((b + S(2)cxS(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(256)c*(S(7)/2)) + (a + bx*S(3) + cxS(6))(S(5)/2)/(S(15)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bxS(3) + cxS(6))(S(3)/2), x), x, (b + S(2)cx*S(3))(a + bxS(3) + cxS(6))(S(3)/2)/(S(24)c) - (b + S(2)cxS(3))(-S(4)ac + b*S(2))sqrt(a + bxS(3) + cx*S(6))/(S(64)c*S(2)) + (-S(4)ac + bS(2))S(2)atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(128)c*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))(S(3)/2)/x, x), x, -a*(S(3)/2)atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/S(3) - b(-S(12)ac + b*S(2))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(48)c*(S(3)/2)) + (a + bx*S(3) + cxS(6))(S(3)/2)/S(9) + sqrt(a + bxS(3) + cx*S(6))(S(8)ac + b*S(2) + S(2)bcx*S(3))/(S(24)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(3) + cxS(6))(S(3)/2)/x*S(4), x), x, -sqrt(a)batanh((S(2)a + bxS(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/S(2) + (S(3)b/S(4) + cxS(3)/S(2))sqrt(a + bxS(3) + cx*S(6)) - (a + bx*S(3) + cxS(6))(S(3)/2)/(S(3)xS(3)) + (S(4)ac + bS(2))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(8)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(3) + cxS(6))(S(3)/2)/x*S(7), x), x, bsqrt(c)atanh((b + S(2)cxS(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/S(2) - (b - S(2)cxS(3))sqrt(a + bxS(3) + cx*S(6))/(S(4)x*S(3)) - (a + bx*S(3) + cxS(6))(S(3)/2)/(S(6)xS(6)) - (S(4)ac + bS(2))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(8)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(3) + cxS(6))(S(3)/2)/xS(10), x), x, c(S(3)/2)atanh((b + S(2)cxS(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/S(3) - (a + bx*S(3) + cxS(6))(S(3)/2)/(S(9)xS(9)) - (S(2)ab + xS(3)(S(8)ac + b*S(2)))sqrt(a + bxS(3) + cx*S(6))/(S(24)axS(6)) + b(-S(12)ac + b*S(2))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(48)a*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))(S(3)/2)/x*S(13), x), x, -(S(2)a + bxS(3))(a + bxS(3) + cxS(6))(S(3)/2)/(S(24)ax*S(12)) + (S(2)a + bxS(3))(-S(4)ac + b*S(2))sqrt(a + bxS(3) + cx*S(6))/(S(64)a*S(2)x*S(6)) - (-S(4)ac + bS(2))S(2)atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(128)a*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))(S(3)/2)/x*S(16), x), x, -(a + bx*S(3) + cxS(6))(S(5)/2)/(S(15)ax*S(15)) + b(S(2)a + bx*S(3))(a + bxS(3) + cxS(6))(S(3)/2)/(S(48)aS(2)x*S(12)) - b(S(2)a + bx*S(3))(-S(4)ac + b*S(2))sqrt(a + bxS(3) + cx*S(6))/(S(128)a*S(3)x*S(6)) + b(-S(4)ac + bS(2))S(2)atanh((S(2)a + bxS(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(256)a*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))(S(3)/2)/x*S(19), x), x, -(a + bx*S(3) + cxS(6))(S(5)/2)/(S(18)ax*S(18)) + S(7)b(a + bx*S(3) + cxS(6))(S(5)/2)/(S(180)aS(2)x*S(15)) - (S(2)a + bxS(3))(-S(4)ac + S(7)bS(2))(a + bxS(3) + cxS(6))(S(3)/2)/(S(576)aS(3)x*S(12)) + (S(2)a + bxS(3))(-S(4)ac + b*S(2))(-S(4)ac + S(7)bS(2))sqrt(a + bxS(3) + cx*S(6))/(S(1536)a*S(4)x*S(6)) - (-S(4)ac + bS(2))S(2)(-S(4)ac + S(7)bS(2))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(3072)a*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cxS(6))(S(3)/2)/x*S(22), x), x, -(a + bx*S(3) + cxS(6))(S(5)/2)/(S(21)ax*S(21)) + b(a + bxS(3) + cxS(6))(S(5)/2)/(S(28)aS(2)x*S(18)) - (-S(16)ac + S(21)b*S(2))(a + bxS(3) + cxS(6))(S(5)/2)/(S(840)aS(3)x*S(15)) + b(S(2)a + bx*S(3))(-S(4)ac + S(3)bS(2))(a + bxS(3) + cxS(6))(S(3)/2)/(S(384)aS(4)x*S(12)) - b(S(2)a + bx*S(3))(-S(4)ac + b*S(2))(-S(4)ac + S(3)bS(2))sqrt(a + bxS(3) + cx*S(6))/(S(1024)a*S(5)x*S(6)) + b(-S(4)ac + bS(2))S(2)(-S(4)ac + S(3)b*S(2))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(2048)a(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/(x*S(8) + S(2)xS(4) + S(1)), x), x, x(m + S(1))hyper((S(2), m/S(4) + S(1)/4), (m/S(4) + S(5)/4,), -xS(4))/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(14)/sqrt(a + bx*S(3) + cx*S(6)), x), x, -S(7)bxS(6)sqrt(a + bxS(3) + cx*S(6))/(S(72)cS(2)) + xS(9)sqrt(a + bx*S(3) + cx*S(6))/(S(12)c) - (S(5)b(-S(44)ac + S(21)bS(2)) - S(2)cxS(3)(-S(36)ac + S(35)bS(2)))sqrt(a + bxS(3) + cx*S(6))/(S(576)c*S(4)) + (S(48)a*S(2)c*S(2) - S(120)abS(2)c + S(35)bS(4))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(384)c(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/sqrt(a + bxS(3) + cx*S(6)), x), x, -b(-S(12)ac + S(5)bS(2))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(48)c(S(7)/2)) + xS(6)sqrt(a + bx*S(3) + cx*S(6))/(S(9)c) + sqrt(a + bxS(3) + cx*S(6))(-S(16)ac + S(15)bS(2) - S(10)bcx*S(3))/(S(72)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/sqrt(a + bxS(3) + cx*S(6)), x), x, -bsqrt(a + bxS(3) + cx*S(6))/(S(4)cS(2)) + xS(3)sqrt(a + bx*S(3) + cx*S(6))/(S(6)c) + (-S(4)ac + S(3)bS(2))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(24)c(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/sqrt(a + bxS(3) + cx*S(6)), x), x, -batanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(6)c*(S(3)/2)) + sqrt(a + bx*S(3) + cx*S(6))/(S(3)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/sqrt(a + bx*S(3) + cx*S(6)), x), x, atanh((b + S(2)cxS(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(3)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + bx*S(3) + cx*S(6))), x), x, -atanh((S(2)a + bxS(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(3)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(4)sqrt(a + bxS(3) + cx*S(6))), x), x, -sqrt(a + bx*S(3) + cx*S(6))/(S(3)axS(3)) + batanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(6)a(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)sqrt(a + bx*S(3) + cx*S(6))), x), x, -sqrt(a + bx*S(3) + cx*S(6))/(S(6)axS(6)) + bsqrt(a + bxS(3) + cx*S(6))/(S(4)a*S(2)x*S(3)) - (-S(4)ac + S(3)b*S(2))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(24)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(10)sqrt(a + bx*S(3) + cx*S(6))), x), x, -sqrt(a + bx*S(3) + cx*S(6))/(S(9)axS(9)) + S(5)bsqrt(a + bx*S(3) + cx*S(6))/(S(36)a*S(2)x*S(6)) - (-S(16)ac + S(15)b*S(2))sqrt(a + bxS(3) + cx*S(6))/(S(72)a*S(3)x*S(3)) + b(-S(12)ac + S(5)bS(2))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(48)a(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(13)sqrt(a + bx*S(3) + cx*S(6))), x), x, -sqrt(a + bx*S(3) + cx*S(6))/(S(12)axS(12)) + S(7)bsqrt(a + bx*S(3) + cx*S(6))/(S(72)a*S(2)x*S(9)) - (-S(36)ac + S(35)b*S(2))sqrt(a + bxS(3) + cx*S(6))/(S(288)a*S(3)x*S(6)) + S(5)b(-S(44)ac + S(21)b*S(2))sqrt(a + bxS(3) + cx*S(6))/(S(576)a*S(4)x*S(3)) - (S(48)a*S(2)c*S(2) - S(120)abS(2)c + S(35)bS(4))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(384)a(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(14)/(a + bxS(3) + cxS(6))(S(3)/2), x), x, -S(2)bx*S(6)sqrt(a + bxS(3) + cx*S(6))/(S(3)c(-S(4)ac + bS(2))) + S(2)x*S(9)(S(2)a + bx*S(3))/((-S(12)ac + S(3)b*S(2))sqrt(a + bxS(3) + cx*S(6))) - (b(-S(52)ac + S(15)bS(2)) - S(2)cxS(3)(-S(12)ac + S(5)bS(2)))sqrt(a + bxS(3) + cx*S(6))/(S(12)c*S(3)(-S(4)ac + b*S(2))) + (-S(4)ac + S(5)b*S(2))atanh((b + S(2)cx*S(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(8)c(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(a + bxS(3) + cxS(6))(S(3)/2), x), x, -batanh((b + S(2)cxS(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(2)c*(S(5)/2)) + S(2)x*S(6)(S(2)a + bx*S(3))/((-S(12)ac + S(3)b*S(2))sqrt(a + bxS(3) + cx*S(6))) + sqrt(a + bx*S(3) + cx*S(6))(-S(8)ac + S(3)bS(2) - S(2)bcx*S(3))/(S(3)c*S(2)(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(a + bxS(3) + cxS(6))(S(3)/2), x), x, -S(2)bsqrt(a + bxS(3) + cx*S(6))/(S(3)c(-S(4)ac + bS(2))) + S(2)x*S(3)(S(2)a + bx*S(3))/((-S(12)ac + S(3)b*S(2))sqrt(a + bxS(3) + cx*S(6))) + atanh((b + S(2)cxS(3))/(S(2)sqrt(c)sqrt(a + bx*S(3) + cx*S(6))))/(S(3)c(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(a + bxS(3) + cxS(6))(S(3)/2), x), x, (S(4)a + S(2)bxS(3))/((-S(12)ac + S(3)b*S(2))sqrt(a + bxS(3) + cxS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bxS(3) + cxS(6))(S(3)/2), x), x, -(S(2)b + S(4)cxS(3))/((-S(12)ac + S(3)b*S(2))sqrt(a + bxS(3) + cx*S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bxS(3) + cxS(6))(S(3)/2)), x), x, (-S(4)ac + S(2)bS(2) + S(2)bcx*S(3))/(S(3)a(-S(4)ac + bS(2))sqrt(a + bxS(3) + cx*S(6))) - atanh((S(2)a + bxS(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(3)a(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(a + bx*S(3) + cxS(6))(S(3)/2)), x), x, (-S(4)ac + S(2)bS(2) + S(2)bcx*S(3))/(S(3)axS(3)(-S(4)ac + b*S(2))sqrt(a + bxS(3) + cx*S(6))) - (-S(8)ac + S(3)b*S(2))sqrt(a + bxS(3) + cx*S(6))/(S(3)a*S(2)x*S(3)(-S(4)ac + b*S(2))) + batanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(2)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)(a + bx*S(3) + cxS(6))(S(3)/2)), x), x, (-S(4)ac + S(2)bS(2) + S(2)bcx*S(3))/(S(3)axS(6)(-S(4)ac + b*S(2))sqrt(a + bxS(3) + cx*S(6))) - (-S(12)ac + S(5)b*S(2))sqrt(a + bxS(3) + cx*S(6))/(S(6)a*S(2)x*S(6)(-S(4)ac + b*S(2))) + b(-S(52)ac + S(15)bS(2))sqrt(a + bxS(3) + cx*S(6))/(S(12)a*S(3)x*S(3)(-S(4)ac + b*S(2))) - (-S(4)ac + S(5)b*S(2))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(8)a(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(10)(a + bx*S(3) + cxS(6))(S(3)/2)), x), x, (-S(4)ac + S(2)bS(2) + S(2)bcx*S(3))/(S(3)axS(9)(-S(4)ac + b*S(2))sqrt(a + bxS(3) + cx*S(6))) - (-S(16)ac + S(7)b*S(2))sqrt(a + bxS(3) + cx*S(6))/(S(9)a*S(2)x*S(9)(-S(4)ac + b*S(2))) + b(-S(116)ac + S(35)bS(2))sqrt(a + bxS(3) + cx*S(6))/(S(36)a*S(3)x*S(6)(-S(4)ac + b*S(2))) - sqrt(a + bx*S(3) + cx*S(6))(S(256)aS(2)c*S(2) - S(460)abS(2)c + S(105)bS(4))/(S(72)a*S(4)x*S(3)(-S(4)ac + b*S(2))) + S(5)b(-S(12)ac + S(7)b*S(2))atanh((S(2)a + bx*S(3))/(S(2)sqrt(a)sqrt(a + bx*S(3) + cx*S(6))))/(S(48)a*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a + bxS(3) + cxS(6))S(2), x), x, a*S(2)(dx)(m + S(1))/(d(m + S(1))) + S(2)ab(dx)(m + S(4))/(dS(4)(m + S(4))) + S(2)bc(dx)(m + S(10))/(dS(10)(m + S(10))) + c*S(2)(dx)(m + S(13))/(dS(13)(m + S(13))) + (dx)(m + S(7))(S(2)ac + bS(2))/(dS(7)(m + S(7))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a + bxS(3) + cx*S(6)), x), x, a(dx)(m + S(1))/(d(m + S(1))) + b(dx)(m + S(4))/(dS(4)(m + S(4))) + c(dx)(m + S(7))/(dS(7)(m + S(7))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/(a + bx*S(3) + cx*S(6)), x), x, -S(2)c(dx)*(m + S(1))hyper((S(1), m/S(3) + S(1)/3), (m/S(3) + S(4)/3,), -S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))))/(d(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))sqrt(-S(4)ac + bS(2))) + S(2)c(dx)*(m + S(1))hyper((S(1), m/S(3) + S(1)/3), (m/S(3) + S(4)/3,), -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))))/(d(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m/(a + bx*S(3) + cxS(6))S(2), x), x, -c(dx)*(m + S(1))(-S(4)ac(-m + S(5)) + bS(2)(-m + S(2)) - b(-m + S(2))sqrt(-S(4)ac + b*S(2)))hyper((S(1), m/S(3) + S(1)/3), (m/S(3) + S(4)/3,), -S(2)cx*S(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)ad(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))(S(3)/2)) + c(dx)(m + S(1))(-S(4)ac(-m + S(5)) + bS(2)(-m + S(2)) + b(-m + S(2))sqrt(-S(4)ac + b*S(2)))hyper((S(1), m/S(3) + S(1)/3), (m/S(3) + S(4)/3,), -S(2)cx*S(3)/(b - sqrt(-S(4)ac + bS(2))))/(S(3)ad(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))(S(3)/2)) + (dx)*(m + S(1))(-S(2)ac + b*S(2) + bcxS(3))/(S(3)ad(-S(4)ac + b*S(2))(a + bxS(3) + cxS(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(a + bx*S(3) + cxS(6))p, x), x, S(2)*pb(-(b + S(2)cxS(3) - sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)))(-p + S(-1))(a + bxS(3) + cxS(6))(p + S(1))hyper((-p, p + S(1)), (p + S(2),), (b + S(2)cxS(3) + sqrt(-S(4)ac + bS(2)))/(S(2)sqrt(-S(4)ac + b*S(2))))/(S(3)c(p + S(1))sqrt(-S(4)ac + b*S(2))) + (a + bx*S(3) + cxS(6))(p + S(1))/(S(6)c(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(a + bxS(3) + cxS(6))p, x), x, -S(2)*(p + S(1))(-(b + S(2)cx*S(3) - sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)))(-p + S(-1))(a + bxS(3) + cxS(6))(p + S(1))hyper((-p, p + S(1)), (p + S(2),), (b + S(2)cxS(3) + sqrt(-S(4)ac + bS(2)))/(S(2)sqrt(-S(4)ac + b*S(2))))/(S(3)(p + S(1))sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(xS(8) + S(2)xS(4) + S(1)), x), x, -xS(6)/(S(4)xS(4) + S(4)) + S(3)x*S(2)/S(4) - S(3)atan(xS(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(x*S(8) + S(2)xS(4) + S(1)), x), x, log(xS(4) + S(1))/S(4) + S(1)/(S(4)xS(4) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(xS(8) + S(2)xS(4) + S(1)), x), x, -xS(2)/(S(4)xS(4) + S(4)) + atan(xS(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(xS(8) + S(2)x*S(4) + S(1)), x), x, -S(1)/(S(4)xS(4) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(8) + S(2)xS(4) + S(1)), x), x, xS(2)/(S(4)xS(4) + S(4)) + atan(xS(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(xS(8) + S(2)xS(4) + S(1))), x), x, log(x) - log(xS(4) + S(1))/S(4) + S(1)/(S(4)xS(4) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(x*S(8) + S(2)x*S(4) + S(1))), x), x, -S(3)atan(x*S(2))/S(4) - S(3)/(S(4)x*S(2)) + S(1)/(S(4)x*S(2)(xS(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(xS(8) + S(2)x*S(4) + S(1))), x), x, -S(2)log(x) + log(x*S(4) + S(1))/S(2) - S(1)/(S(4)x*S(4) + S(4)) - S(1)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)(xS(8) + S(2)x*S(4) + S(1))), x), x, S(5)atan(x*S(2))/S(4) + S(5)/(S(4)x*S(2)) - S(5)/(S(12)x*S(6)) + S(1)/(S(4)x*S(6)(xS(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(x*S(8) + S(2)xS(4) + S(1)), x), x, -xS(5)/(S(4)xS(4) + S(4)) + S(5)x/S(4) + S(5)sqrt(S(2))log(x*S(2) - sqrt(S(2))x + S(1))/S(32) - S(5)sqrt(S(2))log(x*S(2) + sqrt(S(2))x + S(1))/S(32) - S(5)sqrt(S(2))atan(sqrt(S(2))x + S(-1))/S(16) - S(5)sqrt(S(2))atan(sqrt(S(2))x + S(1))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(xS(8) + S(2)xS(4) + S(1)), x), x, -xS(3)/(S(4)x*S(4) + S(4)) + S(3)sqrt(S(2))log(xS(2) - sqrt(S(2))x + S(1))/S(32) - S(3)sqrt(S(2))log(x*S(2) + sqrt(S(2))x + S(1))/S(32) + S(3)sqrt(S(2))atan(sqrt(S(2))x + S(-1))/S(16) + S(3)sqrt(S(2))atan(sqrt(S(2))x + S(1))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(xS(8) + S(2)xS(4) + S(1)), x), x, -x/(S(4)x*S(4) + S(4)) - sqrt(S(2))log(x*S(2) - sqrt(S(2))x + S(1))/S(32) + sqrt(S(2))log(xS(2) + sqrt(S(2))x + S(1))/S(32) + sqrt(S(2))atan(sqrt(S(2))x + S(-1))/S(16) + sqrt(S(2))atan(sqrt(S(2))x + S(1))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(xS(8) + S(2)xS(4) + S(1)), x), x, xS(3)/(S(4)x*S(4) + S(4)) + sqrt(S(2))log(x*S(2) - sqrt(S(2))x + S(1))/S(32) - sqrt(S(2))log(xS(2) + sqrt(S(2))x + S(1))/S(32) + sqrt(S(2))atan(sqrt(S(2))x + S(-1))/S(16) + sqrt(S(2))atan(sqrt(S(2))x + S(1))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(8) + S(2)x*S(4) + S(1)), x), x, x/(S(4)x*S(4) + S(4)) - S(3)sqrt(S(2))log(xS(2) - sqrt(S(2))x + S(1))/S(32) + S(3)sqrt(S(2))log(x*S(2) + sqrt(S(2))x + S(1))/S(32) + S(3)sqrt(S(2))atan(sqrt(S(2))x + S(-1))/S(16) + S(3)sqrt(S(2))atan(sqrt(S(2))x + S(1))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(x*S(8) + S(2)x*S(4) + S(1))), x), x, -S(5)sqrt(S(2))log(xS(2) - sqrt(S(2))x + S(1))/S(32) + S(5)sqrt(S(2))log(x*S(2) + sqrt(S(2))x + S(1))/S(32) - S(5)sqrt(S(2))atan(sqrt(S(2))x + S(-1))/S(16) - S(5)sqrt(S(2))atan(sqrt(S(2))x + S(1))/S(16) - S(5)/(S(4)x) + S(1)/(S(4)x(xS(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(x*S(8) + S(2)x*S(4) + S(1))), x), x, S(7)sqrt(S(2))log(xS(2) - sqrt(S(2))x + S(1))/S(32) - S(7)sqrt(S(2))log(x*S(2) + sqrt(S(2))x + S(1))/S(32) - S(7)sqrt(S(2))atan(sqrt(S(2))x + S(-1))/S(16) - S(7)sqrt(S(2))atan(sqrt(S(2))x + S(1))/S(16) - S(7)/(S(12)xS(3)) + S(1)/(S(4)x*S(3)(xS(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(xS(8) + S(2)x*S(4) + S(1))), x), x, S(9)sqrt(S(2))log(xS(2) - sqrt(S(2))x + S(1))/S(32) - S(9)sqrt(S(2))log(x*S(2) + sqrt(S(2))x + S(1))/S(32) + S(9)sqrt(S(2))atan(sqrt(S(2))x + S(-1))/S(16) + S(9)sqrt(S(2))atan(sqrt(S(2))x + S(1))/S(16) + S(9)/(S(4)x) - S(9)/(S(20)x*S(5)) + S(1)/(S(4)x*S(5)(xS(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(8)(xS(8) + S(2)x*S(4) + S(1))), x), x, -S(11)sqrt(S(2))log(xS(2) - sqrt(S(2))x + S(1))/S(32) + S(11)sqrt(S(2))log(x*S(2) + sqrt(S(2))x + S(1))/S(32) + S(11)sqrt(S(2))atan(sqrt(S(2))x + S(-1))/S(16) + S(11)sqrt(S(2))atan(sqrt(S(2))x + S(1))/S(16) + S(11)/(S(12)xS(3)) - S(11)/(S(28)x*S(7)) + S(1)/(S(4)x*S(7)(xS(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/(x*S(8) - S(2)xS(4) + S(1)), x), x, x(m + S(1))hyper((S(2), m/S(4) + S(1)/4), (m/S(4) + S(5)/4,), xS(4))/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(xS(8) - S(2)xS(4) + S(1)), x), x, xS(6)/(-S(4)xS(4) + S(4)) + S(3)x*S(2)/S(4) - S(3)atanh(xS(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(x*S(8) - S(2)xS(4) + S(1)), x), x, log(-xS(4) + S(1))/S(4) + S(1)/(-S(4)xS(4) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(xS(8) - S(2)xS(4) + S(1)), x), x, xS(2)/(-S(4)xS(4) + S(4)) - atanh(xS(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(xS(8) - S(2)x*S(4) + S(1)), x), x, S(1)/(-S(4)xS(4) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(8) - S(2)xS(4) + S(1)), x), x, xS(2)/(-S(4)xS(4) + S(4)) + atanh(xS(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(xS(8) - S(2)xS(4) + S(1))), x), x, log(x) - log(-xS(4) + S(1))/S(4) + S(1)/(-S(4)xS(4) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(x*S(8) - S(2)x*S(4) + S(1))), x), x, S(3)atanh(x*S(2))/S(4) - S(3)/(S(4)x*S(2)) + S(1)/(S(4)x*S(2)(-xS(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(xS(8) - S(2)x*S(4) + S(1))), x), x, S(2)log(x) - log(-x*S(4) + S(1))/S(2) + S(1)/(-S(4)x*S(4) + S(4)) - S(1)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)(xS(8) - S(2)x*S(4) + S(1))), x), x, S(5)atanh(x*S(2))/S(4) - S(5)/(S(4)x*S(2)) - S(5)/(S(12)x*S(6)) + S(1)/(S(4)x*S(6)(-xS(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(x*S(8) - S(2)xS(4) + S(1)), x), x, xS(5)/(-S(4)xS(4) + S(4)) + S(5)x/S(4) - S(5)atan(x)/S(8) - S(5)atanh(x)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(xS(8) - S(2)xS(4) + S(1)), x), x, xS(3)/(-S(4)x*S(4) + S(4)) + S(3)atan(x)/S(8) - S(3)atanh(x)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(xS(8) - S(2)x*S(4) + S(1)), x), x, x/(-S(4)xS(4) + S(4)) - atan(x)/S(8) - atanh(x)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(x*S(8) - S(2)xS(4) + S(1)), x), x, xS(3)/(-S(4)xS(4) + S(4)) - atan(x)/S(8) + atanh(x)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(8) - S(2)x*S(4) + S(1)), x), x, x/(-S(4)x*S(4) + S(4)) + S(3)atan(x)/S(8) + S(3)atanh(x)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(x*S(8) - S(2)x*S(4) + S(1))), x), x, -S(5)atan(x)/S(8) + S(5)atanh(x)/S(8) - S(5)/(S(4)x) + S(1)/(S(4)x(-xS(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(xS(8) - S(2)x*S(4) + S(1))), x), x, S(7)atan(x)/S(8) + S(7)atanh(x)/S(8) - S(7)/(S(12)x*S(3)) + S(1)/(S(4)x*S(3)(-xS(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(xS(8) - S(2)x*S(4) + S(1))), x), x, -S(9)atan(x)/S(8) + S(9)atanh(x)/S(8) - S(9)/(S(4)x) - S(9)/(S(20)xS(5)) + S(1)/(S(4)x*S(5)(-xS(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(8)(xS(8) - S(2)x*S(4) + S(1))), x), x, S(11)atan(x)/S(8) + S(11)atanh(x)/S(8) - S(11)/(S(12)x*S(3)) - S(11)/(S(28)x*S(7)) + S(1)/(S(4)x*S(7)(-xS(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/(a + bxS(4) + cx*S(8)), x), x, -S(2)cx(m + S(1))hyper((S(1), m/S(4) + S(1)/4), (m/S(4) + S(5)/4,), -S(2)cx*S(4)/(b + sqrt(-S(4)ac + bS(2))))/((b + sqrt(-S(4)ac + bS(2)))(m + S(1))sqrt(-S(4)ac + bS(2))) + S(2)cx(m + S(1))hyper((S(1), m/S(4) + S(1)/4), (m/S(4) + S(5)/4,), -S(2)cx*S(4)/(b - sqrt(-S(4)ac + bS(2))))/((b - sqrt(-S(4)ac + bS(2)))(m + S(1))sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(a + bx*S(4) + cx*S(8)), x), x, -blog(a + bxS(4) + cx*S(8))/(S(8)cS(2)) + xS(4)/(S(4)c) - (-S(2)ac + bS(2))atanh((b + S(2)cx*S(4))/sqrt(-S(4)ac + bS(2)))/(S(4)c*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(a + bxS(4) + cxS(8)), x), x, xS(2)/(S(2)c) - sqrt(S(2))(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x*S(2)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(4)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))) - sqrt(S(2))(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x*S(2)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(4)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(a + bxS(4) + cx*S(8)), x), x, batanh((b + S(2)cx*S(4))/sqrt(-S(4)ac + bS(2)))/(S(4)csqrt(-S(4)ac + bS(2))) + log(a + bx*S(4) + cx*S(8))/(S(8)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)/(a + bx*S(4) + cx*S(8)), x), x, -sqrt(S(2))sqrt(b - sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x*S(2)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(c)sqrt(-S(4)ac + bS(2))) + sqrt(S(2))sqrt(b + sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x*S(2)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(c)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + bx*S(4) + cx*S(8)), x), x, -atanh((b + S(2)cxS(4))/sqrt(-S(4)ac + bS(2)))/(S(2)sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bx*S(4) + cx*S(8)), x), x, -sqrt(S(2))sqrt(c)atan(sqrt(S(2))sqrt(c)xS(2)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)atan(sqrt(S(2))sqrt(c)xS(2)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bxS(4) + cx*S(8))), x), x, batanh((b + S(2)cx*S(4))/sqrt(-S(4)ac + bS(2)))/(S(4)asqrt(-S(4)ac + bS(2))) + log(x)/a - log(a + bx*S(4) + cx*S(8))/(S(8)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)(a + bxS(4) + cx*S(8))), x), x, -sqrt(S(2))sqrt(c)(-b/sqrt(-S(4)ac + bS(2)) + S(1))atan(sqrt(S(2))sqrt(c)x*S(2)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))) - sqrt(S(2))sqrt(c)(b/sqrt(-S(4)ac + bS(2)) + S(1))atan(sqrt(S(2))sqrt(c)x*S(2)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))) - S(1)/(S(2)axS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(a + bxS(4) + cx*S(8))), x), x, -S(1)/(S(4)axS(4)) - blog(x)/a*S(2) + blog(a + bxS(4) + cx*S(8))/(S(8)a*S(2)) - (-S(2)ac + bS(2))atanh((b + S(2)cx*S(4))/sqrt(-S(4)ac + bS(2)))/(S(4)a*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(10)/(a + bxS(4) + cxS(8)), x), x, xS(3)/(S(3)c) - S(2)(S(1)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(7)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) + S(2)*(S(1)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(7)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) - S(2)*(S(1)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(7)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)) + S(2)*(S(1)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(7)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(8)/(a + bx*S(4) + cxS(8)), x), x, x/c + S(2)(S(3)/4)(b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(6)/(a + bx*S(4) + cxS(8)), x), x, -S(2)(S(1)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(3)/4)sqrt(-S(4)ac + bS(2))) + S(2)(S(1)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(3)/4)sqrt(-S(4)ac + bS(2))) + S(2)(S(1)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(3)/4)sqrt(-S(4)ac + bS(2))) - S(2)(S(1)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(3)/4)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(a + bxS(4) + cxS(8)), x), x, S(2)(S(3)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(1)/4)sqrt(-S(4)ac + bS(2))) + S(2)(S(3)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(1)/4)sqrt(-S(4)ac + bS(2))) - S(2)(S(3)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(1)/4)sqrt(-S(4)ac + bS(2))) - S(2)(S(3)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(1)/4)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bxS(4) + cxS(8)), x), x, S(2)(S(1)/4)c(S(1)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)sqrt(-S(4)ac + bS(2))) - S(2)(S(1)/4)c(S(1)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)sqrt(-S(4)ac + bS(2))) - S(2)(S(1)/4)c(S(1)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)sqrt(-S(4)ac + bS(2))) + S(2)(S(1)/4)c(S(1)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bx*S(4) + cxS(8)), x), x, -S(2)(S(3)/4)c(S(3)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))) - S(2)(S(3)/4)c(S(3)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))) + S(2)(S(3)/4)c(S(3)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))) + S(2)(S(3)/4)c(S(3)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(2)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + bx*S(4) + cxS(8))), x), x, -S(2)(S(1)/4)c(S(1)/4)(b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) + S(2)*(S(1)/4)c*(S(1)/4)(b/sqrt(-S(4)ac + b*S(2)) + S(1))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) - S(2)*(S(1)/4)c*(S(1)/4)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)) + S(2)*(S(1)/4)c*(S(1)/4)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)) - S(1)/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(a + bxS(4) + cxS(8))), x), x, S(2)(S(3)/4)c(S(3)/4)(b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)c*(S(3)/4)(b/sqrt(-S(4)ac + b*S(2)) + S(1))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)c*(S(3)/4)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)c*(S(3)/4)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) - S(1)/(S(3)axS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/(xS(8) + xS(4) + S(1)), x), x, S(2)sqrt(S(3))x*(m + S(1))hyper((S(1), m/S(4) + S(1)/4), (m/S(4) + S(5)/4,), -S(2)xS(4)/(S(1) - sqrt(S(3))I))/(S(3)(sqrt(S(3)) + I)(m + S(1))) - S(2)sqrt(S(3))x*(m + S(1))hyper((S(1), m/S(4) + S(1)/4), (m/S(4) + S(5)/4,), -S(2)xS(4)/(S(1) + sqrt(S(3))I))/(S(3)(-sqrt(S(3)) + I)(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(xS(8) + xS(4) + S(1)), x), x, xS(4)/S(4) - log(xS(8) + xS(4) + S(1))/S(8) - sqrt(S(3))atan(sqrt(S(3))(S(2)xS(4) + S(1))/S(3))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(xS(8) + xS(4) + S(1)), x), x, xS(2)/S(2) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x*S(2) + S(1))/S(3))/S(6) - sqrt(S(3))atan(sqrt(S(3))(S(2)xS(2) + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(xS(8) + xS(4) + S(1)), x), x, log(xS(8) + xS(4) + S(1))/S(8) - sqrt(S(3))atan(sqrt(S(3))(S(2)xS(4) + S(1))/S(3))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(xS(8) + xS(4) + S(1)), x), x, log(xS(4) - xS(2) + S(1))/S(8) - log(xS(4) + xS(2) + S(1))/S(8) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x*S(2) + S(1))/S(3))/S(12) + sqrt(S(3))atan(sqrt(S(3))(S(2)xS(2) + S(1))/S(3))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(xS(8) + xS(4) + S(1)), x), x, sqrt(S(3))atan(sqrt(S(3))(S(2)xS(4) + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(8) + xS(4) + S(1)), x), x, -log(xS(4) - xS(2) + S(1))/S(8) + log(xS(4) + xS(2) + S(1))/S(8) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x*S(2) + S(1))/S(3))/S(12) + sqrt(S(3))atan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(xS(8) + xS(4) + S(1))), x), x, log(x) - log(xS(8) + xS(4) + S(1))/S(8) - sqrt(S(3))atan(sqrt(S(3))(S(2)xS(4) + S(1))/S(3))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(xS(8) + xS(4) + S(1))), x), x, sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(2) + S(1))/S(3))/S(6) - sqrt(S(3))atan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(6) - S(1)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(xS(8) + xS(4) + S(1))), x), x, -log(x) + log(xS(8) + xS(4) + S(1))/S(8) - sqrt(S(3))atan(sqrt(S(3))(S(2)x*S(4) + S(1))/S(3))/S(12) - S(1)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)(xS(8) + xS(4) + S(1))), x), x, log(xS(4) - xS(2) + S(1))/S(8) - log(xS(4) + xS(2) + S(1))/S(8) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x*S(2) + S(1))/S(3))/S(12) + sqrt(S(3))atan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(12) + S(1)/(S(2)x*S(2)) - S(1)/(S(6)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(xS(8) + xS(4) + S(1)), x), x, x + log(xS(2) - x + S(1))/S(8) - log(xS(2) + x + S(1))/S(8) + sqrt(S(3))log(xS(2) - sqrt(S(3))x + S(1))/S(24) - sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(24) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(12) - sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(12) - atan(S(2)x - sqrt(S(3)))/S(4) - atan(S(2)x + sqrt(S(3)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(xS(8) + x*S(4) + S(1)), x), x, sqrt(S(3))log(x*S(2) - sqrt(S(3))x + S(1))/S(12) - sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(12) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) + sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(xS(8) + xS(4) + S(1)), x), x, -log(xS(2) - x + S(1))/S(8) + log(x*S(2) + x + S(1))/S(8) + sqrt(S(3))log(x*S(2) - sqrt(S(3))x + S(1))/S(24) - sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(24) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(12) - sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(12) + atan(S(2)x - sqrt(S(3)))/S(4) + atan(S(2)x + sqrt(S(3)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(xS(8) + xS(4) + S(1)), x), x, log(xS(2) - x + S(1))/S(8) - log(x*S(2) + x + S(1))/S(8) - sqrt(S(3))log(x*S(2) - sqrt(S(3))x + S(1))/S(24) + sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(24) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(12) - sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(12) + atan(S(2)x - sqrt(S(3)))/S(4) + atan(S(2)x + sqrt(S(3)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(8) + xS(4) + S(1)), x), x, -sqrt(S(3))log(xS(2) - sqrt(S(3))x + S(1))/S(12) + sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(12) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) + sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(xS(8) + xS(4) + S(1))), x), x, -log(xS(2) - x + S(1))/S(8) + log(xS(2) + x + S(1))/S(8) - sqrt(S(3))log(xS(2) - sqrt(S(3))x + S(1))/S(24) + sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(24) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(12) - sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(12) - atan(S(2)x - sqrt(S(3)))/S(4) - atan(S(2)x + sqrt(S(3)))/S(4) - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(4)(xS(8) + xS(4) + S(1))), x), x, log(xS(2) - x + S(1))/S(8) - log(xS(2) + x + S(1))/S(8) + sqrt(S(3))log(xS(2) - sqrt(S(3))x + S(1))/S(24) - sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(24) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(12) - sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(12) - atan(S(2)x - sqrt(S(3)))/S(4) - atan(S(2)x + sqrt(S(3)))/S(4) - S(1)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(xS(8) + xS(4) + S(1))), x), x, sqrt(S(3))log(xS(2) - sqrt(S(3))x + S(1))/S(12) - sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(12) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) + sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(6) + S(1)/x - S(1)/(S(5)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(8)(xS(8) + xS(4) + S(1))), x), x, -log(xS(2) - x + S(1))/S(8) + log(xS(2) + x + S(1))/S(8) + sqrt(S(3))log(xS(2) - sqrt(S(3))x + S(1))/S(24) - sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(24) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(12) - sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(12) + atan(S(2)x - sqrt(S(3)))/S(4) + atan(S(2)x + sqrt(S(3)))/S(4) + S(1)/(S(3)xS(3)) - S(1)/(S(7)xS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/(xS(8) - xS(4) + S(1)), x), x, S(2)sqrt(S(3))x*(m + S(1))hyper((S(1), m/S(4) + S(1)/4), (m/S(4) + S(5)/4,), S(2)xS(4)/(S(1) - sqrt(S(3))I))/(S(3)(sqrt(S(3)) + I)(m + S(1))) - S(2)sqrt(S(3))x*(m + S(1))hyper((S(1), m/S(4) + S(1)/4), (m/S(4) + S(5)/4,), S(2)xS(4)/(S(1) + sqrt(S(3))I))/(S(3)(-sqrt(S(3)) + I)(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(xS(8) - xS(4) + S(1)), x), x, xS(4)/S(4) + log(xS(8) - xS(4) + S(1))/S(8) + sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(4) + S(1))/S(3))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(xS(8) - xS(4) + S(1)), x), x, xS(2)/S(2) + sqrt(S(3))log(x*S(4) - sqrt(S(3))x*S(2) + S(1))/S(12) - sqrt(S(3))log(x*S(4) + sqrt(S(3))xS(2) + S(1))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(xS(8) - xS(4) + S(1)), x), x, log(xS(8) - xS(4) + S(1))/S(8) - sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(4) + S(1))/S(3))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(xS(8) - xS(4) + S(1)), x), x, sqrt(S(3))log(x*S(4) - sqrt(S(3))x*S(2) + S(1))/S(24) - sqrt(S(3))log(x*S(4) + sqrt(S(3))x*S(2) + S(1))/S(24) + atan(S(2)x*S(2) - sqrt(S(3)))/S(4) + atan(S(2)xS(2) + sqrt(S(3)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(xS(8) - xS(4) + S(1)), x), x, -sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(4) + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(8) - xS(4) + S(1)), x), x, -sqrt(S(3))log(x*S(4) - sqrt(S(3))x*S(2) + S(1))/S(24) + sqrt(S(3))log(x*S(4) + sqrt(S(3))x*S(2) + S(1))/S(24) + atan(S(2)x*S(2) - sqrt(S(3)))/S(4) + atan(S(2)x*S(2) + sqrt(S(3)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(xS(8) - xS(4) + S(1))), x), x, log(x) - log(xS(8) - xS(4) + S(1))/S(8) - sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(4) + S(1))/S(3))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(xS(8) - xS(4) + S(1))), x), x, -sqrt(S(3))log(xS(4) - sqrt(S(3))x*S(2) + S(1))/S(12) + sqrt(S(3))log(x*S(4) + sqrt(S(3))x*S(2) + S(1))/S(12) - S(1)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(xS(8) - xS(4) + S(1))), x), x, log(x) - log(xS(8) - xS(4) + S(1))/S(8) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x*S(4) + S(1))/S(3))/S(12) - S(1)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)(xS(8) - xS(4) + S(1))), x), x, -sqrt(S(3))log(x*S(4) - sqrt(S(3))x*S(2) + S(1))/S(24) + sqrt(S(3))log(x*S(4) + sqrt(S(3))x*S(2) + S(1))/S(24) - atan(S(2)x*S(2) - sqrt(S(3)))/S(4) - atan(S(2)x*S(2) + sqrt(S(3)))/S(4) - S(1)/(S(2)x*S(2)) - S(1)/(S(6)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(xS(8) - xS(4) + S(1)), x), x, x - sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) + sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) + sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) - atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/(S(4)sqrt(S(3)sqrt(S(3)) + S(6))) + atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/(S(4)sqrt(S(3)sqrt(S(3)) + S(6))) + atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/(S(4)sqrt(-S(3)sqrt(S(3)) + S(6))) - atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/(S(4)sqrt(-S(3)sqrt(S(3)) + S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(xS(8) - x*S(4) + S(1)), x), x, sqrt(S(6))log(x*S(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(24) - sqrt(S(6))log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(24) + sqrt(S(6))log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(24) - sqrt(S(6))log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(24) - sqrt(S(6))atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(12) + sqrt(S(6))atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(12) - sqrt(S(6))atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(12) + sqrt(S(6))atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(xS(8) - xS(4) + S(1)), x), x, -log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/(S(8)sqrt(-S(3)sqrt(S(3)) + S(6))) + log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/(S(8)sqrt(-S(3)sqrt(S(3)) + S(6))) + log(x*S(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/(S(8)sqrt(S(3)sqrt(S(3)) + S(6))) - log(x*S(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/(S(8)sqrt(S(3)sqrt(S(3)) + S(6))) - atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/(S(4)sqrt(-S(3)sqrt(S(3)) + S(6))) + atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/(S(4)sqrt(-S(3)sqrt(S(3)) + S(6))) + atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/(S(4)sqrt(S(3)sqrt(S(3)) + S(6))) - atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/(S(4)sqrt(S(3)sqrt(S(3)) + S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(xS(8) - xS(4) + S(1)), x), x, log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/(S(8)sqrt(-S(3)sqrt(S(3)) + S(6))) - log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/(S(8)sqrt(-S(3)sqrt(S(3)) + S(6))) - log(x*S(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/(S(8)sqrt(S(3)sqrt(S(3)) + S(6))) + log(x*S(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/(S(8)sqrt(S(3)sqrt(S(3)) + S(6))) - sqrt(sqrt(S(3))/S(3) + S(2)/3)atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(4) + sqrt(sqrt(S(3))/S(3) + S(2)/3)atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(4) + sqrt(-sqrt(S(3))/S(3) + S(2)/3)atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(4) - sqrt(-sqrt(S(3))/S(3) + S(2)/3)atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(8) - xS(4) + S(1)), x), x, -sqrt(S(6))log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(24) + sqrt(S(6))log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(24) - sqrt(S(6))log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(24) + sqrt(S(6))log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(24) - sqrt(S(6))atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(12) + sqrt(S(6))atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(12) - sqrt(S(6))atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(12) + sqrt(S(6))atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(xS(8) - xS(4) + S(1))), x), x, sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) + sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) - atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/(S(4)sqrt(S(3)sqrt(S(3)) + S(6))) + atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/(S(4)sqrt(S(3)sqrt(S(3)) + S(6))) + atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/(S(4)sqrt(-S(3)sqrt(S(3)) + S(6))) - atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/(S(4)sqrt(-S(3)sqrt(S(3)) + S(6))) - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(4)(xS(8) - xS(4) + S(1))), x), x, sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) + sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) + sqrt(-sqrt(S(3))/S(3) + S(2)/3)atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(4) - sqrt(-sqrt(S(3))/S(3) + S(2)/3)atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(4) - sqrt(sqrt(S(3))/S(3) + S(2)/3)atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(4) + sqrt(sqrt(S(3))/S(3) + S(2)/3)atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(4) - S(1)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(xS(8) - xS(4) + S(1))), x), x, -sqrt(S(6))log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(24) + sqrt(S(6))log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(24) - sqrt(S(6))log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(24) + sqrt(S(6))log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(24) + sqrt(S(6))atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(12) - sqrt(S(6))atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(12) + sqrt(S(6))atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(12) - sqrt(S(6))atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(12) - S(1)/x - S(1)/(S(5)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(8)(xS(8) - xS(4) + S(1))), x), x, sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) + sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) + sqrt(sqrt(S(3))/S(3) + S(2)/3)atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(4) - sqrt(sqrt(S(3))/S(3) + S(2)/3)atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(4) - sqrt(-sqrt(S(3))/S(3) + S(2)/3)atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(4) + sqrt(-sqrt(S(3))/S(3) + S(2)/3)atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(4) - S(1)/(S(3)xS(3)) - S(1)/(S(7)xS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/(x*S(8) + S(3)x*S(4) + S(1)), x), x, S(2)sqrt(S(5))x(m + S(1))hyper((S(1), m/S(4) + S(1)/4), (m/S(4) + S(5)/4,), -S(2)xS(4)/(-sqrt(S(5)) + S(3)))/(S(5)(-sqrt(S(5)) + S(3))(m + S(1))) - S(2)sqrt(S(5))x(m + S(1))hyper((S(1), m/S(4) + S(1)/4), (m/S(4) + S(5)/4,), -S(2)xS(4)/(sqrt(S(5)) + S(3)))/(S(5)(sqrt(S(5)) + S(3))(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(xS(8) + S(3)xS(4) + S(1)), x), x, xS(4)/S(4) - (-S(7)sqrt(S(5))/S(40) + S(3)/8)log(S(2)xS(4) - sqrt(S(5)) + S(3)) - (S(3)/8 + S(7)sqrt(S(5))/S(40))log(S(2)xS(4) + sqrt(S(5)) + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(x*S(8) + S(3)xS(4) + S(1)), x), x, xS(2)/S(2) + sqrt(-S(4)sqrt(S(5))/S(5) + S(9)/5)atan(x*S(2)sqrt(sqrt(S(5))/S(2) + S(3)/2))/S(2) - sqrt(S(4)sqrt(S(5))/S(5) + S(9)/5)atan(sqrt(S(2))xS(2)/sqrt(sqrt(S(5)) + S(3)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(xS(8) + S(3)x*S(4) + S(1)), x), x, (-S(3)sqrt(S(5))/S(40) + S(1)/8)log(S(2)x*S(4) - sqrt(S(5)) + S(3)) + (S(1)/8 + S(3)sqrt(S(5))/S(40))log(S(2)xS(4) + sqrt(S(5)) + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(x*S(8) + S(3)x*S(4) + S(1)), x), x, -sqrt(-sqrt(S(5))/S(10) + S(3)/10)atan(x*S(2)sqrt(sqrt(S(5))/S(2) + S(3)/2))/S(2) + sqrt(sqrt(S(5))/S(10) + S(3)/10)atan(sqrt(S(2))xS(2)/sqrt(sqrt(S(5)) + S(3)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(x*S(8) + S(3)x*S(4) + S(1)), x), x, -sqrt(S(5))atanh(sqrt(S(5))(S(2)xS(4) + S(3))/S(5))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(8) + S(3)xS(4) + S(1)), x), x, sqrt(sqrt(S(5))/S(10) + S(3)/10)atan(x*S(2)sqrt(sqrt(S(5))/S(2) + S(3)/2))/S(2) - atan(sqrt(S(2))xS(2)/sqrt(sqrt(S(5)) + S(3)))/sqrt(S(10)sqrt(S(5)) + S(30)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(xS(8) + S(3)x*S(4) + S(1))), x), x, log(x) - (S(1)/8 + S(3)sqrt(S(5))/S(40))log(S(2)x*S(4) - sqrt(S(5)) + S(3)) - (-S(3)sqrt(S(5))/S(40) + S(1)/8)log(S(2)xS(4) + sqrt(S(5)) + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(xS(8) + S(3)x*S(4) + S(1))), x), x, -sqrt(S(10))(sqrt(S(5)) + S(3))*(S(3)/2)atan(x*S(2)sqrt(sqrt(S(5))/S(2) + S(3)/2))/S(40) + sqrt(-S(4)sqrt(S(5))/S(5) + S(9)/5)atan(sqrt(S(2))xS(2)/sqrt(sqrt(S(5)) + S(3)))/S(2) - S(1)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(xS(8) + S(3)x*S(4) + S(1))), x), x, -S(3)log(x) + (S(3)/8 + S(7)sqrt(S(5))/S(40))log(S(2)xS(4) - sqrt(S(5)) + S(3)) + (-S(7)sqrt(S(5))/S(40) + S(3)/8)log(S(2)x*S(4) + sqrt(S(5)) + S(3)) - S(1)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)(xS(8) + S(3)x*S(4) + S(1))), x), x, sqrt(S(11)sqrt(S(5))/S(2) + S(123)/10)atan(xS(2)sqrt(sqrt(S(5))/S(2) + S(3)/2))/S(2) - sqrt(-S(11)sqrt(S(5))/S(2) + S(123)/10)atan(sqrt(S(2))xS(2)/sqrt(sqrt(S(5)) + S(3)))/S(2) + S(3)/(S(2)x*S(2)) - S(1)/(S(6)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/(x*S(8) - S(3)x*S(4) + S(1)), x), x, S(2)sqrt(S(5))x(m + S(1))hyper((S(1), m/S(4) + S(1)/4), (m/S(4) + S(5)/4,), S(2)xS(4)/(-sqrt(S(5)) + S(3)))/(S(5)(-sqrt(S(5)) + S(3))(m + S(1))) - S(2)sqrt(S(5))x(m + S(1))hyper((S(1), m/S(4) + S(1)/4), (m/S(4) + S(5)/4,), S(2)xS(4)/(sqrt(S(5)) + S(3)))/(S(5)(sqrt(S(5)) + S(3))(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(xS(8) - S(3)xS(4) + S(1)), x), x, xS(4)/S(4) + (-S(7)sqrt(S(5))/S(40) + S(3)/8)log(-S(2)xS(4) - sqrt(S(5)) + S(3)) + (S(3)/8 + S(7)sqrt(S(5))/S(40))log(-S(2)xS(4) + sqrt(S(5)) + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(x*S(8) - S(3)xS(4) + S(1)), x), x, xS(2)/S(2) + sqrt(-S(4)sqrt(S(5))/S(5) + S(9)/5)atanh(x*S(2)sqrt(sqrt(S(5))/S(2) + S(3)/2))/S(2) - sqrt(S(4)sqrt(S(5))/S(5) + S(9)/5)atanh(sqrt(S(2))xS(2)/sqrt(sqrt(S(5)) + S(3)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(xS(8) - S(3)x*S(4) + S(1)), x), x, (-S(3)sqrt(S(5))/S(40) + S(1)/8)log(-S(2)x*S(4) - sqrt(S(5)) + S(3)) + (S(1)/8 + S(3)sqrt(S(5))/S(40))log(-S(2)xS(4) + sqrt(S(5)) + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(x*S(8) - S(3)x*S(4) + S(1)), x), x, sqrt(-sqrt(S(5))/S(10) + S(3)/10)atanh(x*S(2)sqrt(sqrt(S(5))/S(2) + S(3)/2))/S(2) - sqrt(sqrt(S(5))/S(10) + S(3)/10)atanh(sqrt(S(2))xS(2)/sqrt(sqrt(S(5)) + S(3)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(x*S(8) - S(3)x*S(4) + S(1)), x), x, sqrt(S(5))atanh(sqrt(S(5))(-S(2)xS(4) + S(3))/S(5))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(8) - S(3)xS(4) + S(1)), x), x, sqrt(sqrt(S(5))/S(10) + S(3)/10)atanh(x*S(2)sqrt(sqrt(S(5))/S(2) + S(3)/2))/S(2) - atanh(sqrt(S(2))xS(2)/sqrt(sqrt(S(5)) + S(3)))/sqrt(S(10)sqrt(S(5)) + S(30)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(xS(8) - S(3)x*S(4) + S(1))), x), x, log(x) - (S(1)/8 + S(3)sqrt(S(5))/S(40))log(-S(2)x*S(4) - sqrt(S(5)) + S(3)) - (-S(3)sqrt(S(5))/S(40) + S(1)/8)log(-S(2)xS(4) + sqrt(S(5)) + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(xS(8) - S(3)x*S(4) + S(1))), x), x, sqrt(S(10))(sqrt(S(5)) + S(3))*(S(3)/2)atanh(x*S(2)sqrt(sqrt(S(5))/S(2) + S(3)/2))/S(40) - sqrt(-S(4)sqrt(S(5))/S(5) + S(9)/5)atanh(sqrt(S(2))xS(2)/sqrt(sqrt(S(5)) + S(3)))/S(2) - S(1)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(xS(8) - S(3)x*S(4) + S(1))), x), x, S(3)log(x) - (S(3)/8 + S(7)sqrt(S(5))/S(40))log(-S(2)xS(4) - sqrt(S(5)) + S(3)) - (-S(7)sqrt(S(5))/S(40) + S(3)/8)log(-S(2)x*S(4) + sqrt(S(5)) + S(3)) - S(1)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)(xS(8) - S(3)x*S(4) + S(1))), x), x, sqrt(S(11)sqrt(S(5))/S(2) + S(123)/10)atanh(xS(2)sqrt(sqrt(S(5))/S(2) + S(3)/2))/S(2) - sqrt(-S(11)sqrt(S(5))/S(2) + S(123)/10)atanh(sqrt(S(2))xS(2)/sqrt(sqrt(S(5)) + S(3)))/S(2) - S(3)/(S(2)x*S(2)) - S(1)/(S(6)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(x*S(8) - S(3)x*S(4) + S(1)), x), x, x + sqrt(S(5))(-S(440)sqrt(S(5)) + S(984))(S(1)/4)atan(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(20) - sqrt(S(5))(S(55)sqrt(S(5))/S(2) + S(123)/2)(S(1)/4)atan(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))*(S(1)/4))/S(10) + sqrt(S(5))(-S(440)sqrt(S(5)) + S(984))(S(1)/4)atanh(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(20) - sqrt(S(5))(S(55)sqrt(S(5))/S(2) + S(123)/2)(S(1)/4)atanh(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))(S(1)/4))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(x*S(8) - S(3)x*S(4) + S(1)), x), x, -sqrt(S(5))(-S(64)sqrt(S(5)) + S(144))(S(1)/4)atan(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(20) + S(2)(S(1)/4)sqrt(S(5))(sqrt(S(5)) + S(3))(S(3)/4)atan(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))*(S(1)/4))/S(20) + sqrt(S(5))(-S(64)sqrt(S(5)) + S(144))(S(1)/4)atanh(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(20) - S(2)(S(1)/4)sqrt(S(5))(sqrt(S(5)) + S(3))(S(3)/4)atanh(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))(S(1)/4))/S(20), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(x*S(8) - S(3)x*S(4) + S(1)), x), x, sqrt(S(5))(-sqrt(S(5))/S(2) + S(3)/2)*(S(1)/4)atan(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(10) - sqrt(S(5))(sqrt(S(5))/S(2) + S(3)/2)*(S(1)/4)atan(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))*(S(1)/4))/S(10) + sqrt(S(5))(-sqrt(S(5))/S(2) + S(3)/2)*(S(1)/4)atanh(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(10) - sqrt(S(5))(sqrt(S(5))/S(2) + S(3)/2)*(S(1)/4)atanh(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))(S(1)/4))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(x*S(8) - S(3)x*S(4) + S(1)), x), x, sqrt(S(-10) + S(10)sqrt(S(5)))atan(xsqrt(S(-2) + S(2)sqrt(S(5)))/S(2))/S(20) - sqrt(S(10) + S(10)sqrt(S(5)))atan(xsqrt(S(2) + S(2)sqrt(S(5)))/S(2))/S(20) - sqrt(S(-10) + S(10)sqrt(S(5)))atanh(xsqrt(S(-2) + S(2)sqrt(S(5)))/S(2))/S(20) + sqrt(S(10) + S(10)sqrt(S(5)))atanh(xsqrt(S(2) + S(2)sqrt(S(5)))/S(2))/S(20), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(2)/(xS(8) - S(3)x*S(4) + S(1)), x), x, -sqrt(S(5))(sqrt(S(5))/S(2) + S(3)/2)*(S(1)/4)atan(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(10) + S(2)(S(1)/4)sqrt(S(5))atan(S(2)(S(1)/4)x/(sqrt(S(5)) + S(3))*(S(1)/4))/(S(10)(sqrt(S(5)) + S(3))*(S(1)/4)) + sqrt(S(5))(sqrt(S(5))/S(2) + S(3)/2)*(S(1)/4)atanh(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(10) - S(2)(S(1)/4)sqrt(S(5))atanh(S(2)(S(1)/4)x/(sqrt(S(5)) + S(3))*(S(1)/4))/(S(10)(sqrt(S(5)) + S(3))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(xS(8) - S(3)xS(4) + S(1))), x), x, -S(2)(S(3)/4)sqrt(S(5))(sqrt(S(5)) + S(3))*(S(5)/4)atan(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(40) + sqrt(S(5))(-S(440)sqrt(S(5)) + S(984))(S(1)/4)atan(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))(S(1)/4))/S(20) + S(2)(S(3)/4)sqrt(S(5))(sqrt(S(5)) + S(3))*(S(5)/4)atanh(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(40) - sqrt(S(5))(-S(440)sqrt(S(5)) + S(984))(S(1)/4)atanh(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))(S(1)/4))/S(20) - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(xS(8) - S(3)x*S(4) + S(1))), x), x, -sqrt(S(5))(S(1292)sqrt(S(5)) + S(2889))(S(1)/4)atan(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(10) + sqrt(S(5))(-S(1292)sqrt(S(5)) + S(2889))(S(1)/4)atan(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))*(S(1)/4))/S(10) + sqrt(S(5))(S(1292)sqrt(S(5)) + S(2889))(S(1)/4)atanh(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(10) - sqrt(S(5))(-S(1292)sqrt(S(5)) + S(2889))(S(1)/4)atanh(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))*(S(1)/4))/S(10) - S(3)/x - S(1)/(S(5)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(8)(xS(8) - S(3)x*S(4) + S(1))), x), x, sqrt(S(5))(S(17711)sqrt(S(5))/S(2) + S(39603)/2)(S(1)/4)atan(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(10) - sqrt(S(5))(-S(17711)sqrt(S(5))/S(2) + S(39603)/2)(S(1)/4)atan(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))*(S(1)/4))/S(10) + sqrt(S(5))(S(17711)sqrt(S(5))/S(2) + S(39603)/2)(S(1)/4)atanh(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(10) - sqrt(S(5))(-S(17711)sqrt(S(5))/S(2) + S(39603)/2)(S(1)/4)atanh(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))(S(1)/4))/S(10) - S(1)/xS(3) - S(1)/(S(7)xS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(xS(8) + S(3)x*S(4) + S(2)), x), x, -atanh(S(2)xS(4) + S(3))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(x*S(8) + S(3)xS(4) + S(2)), x), x, xS(4)/S(4) + log(xS(4) + S(1))/S(4) - log(xS(4) + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(xS(10) + xS(5) + S(2)), x), x, log(xS(10) + x*S(5) + S(2))/S(10) - sqrt(S(7))atan(sqrt(S(7))(S(2)xS(5) + S(1))/S(7))/S(35), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(xS(10) + xS(5) + S(2)), x), x, S(2)sqrt(S(7))atan(sqrt(S(7))(S(2)x*S(5) + S(1))/S(7))/S(35), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(xS(10) + xS(5) + S(1))), x), x, log(x) - log(xS(10) + xS(5) + S(1))/S(10) - sqrt(S(3))atan(sqrt(S(3))(S(2)xS(5) + S(1))/S(3))/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(xS(10) + xS(5) + S(1))), x), x, -log(x) + log(xS(10) + xS(5) + S(1))/S(10) - sqrt(S(3))atan(sqrt(S(3))(S(2)xS(5) + S(1))/S(3))/S(15) - S(1)/(S(5)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(11) + xS(6) + x), x), x, log(x) - log(xS(10) + x*S(5) + S(1))/S(10) - sqrt(S(3))atan(sqrt(S(3))(S(2)xS(5) + S(1))/S(3))/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a/x*S(2) + b/x + c), x), x, -bx*S(3)/(S(3)c*S(2)) - bx(-S(2)ac + bS(2))/cS(4) + b(S(5)aS(2)c*S(2) - S(5)abS(2)c + b*S(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(5)sqrt(-S(4)ac + bS(2))) + xS(4)/(S(4)c) + x*S(2)(-ac + bS(2))/(S(2)cS(3)) + (aS(2)cS(2) - S(3)abS(2)c + b*S(4))log(a + bx + cx*S(2))/(S(2)cS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a/x*S(2) + b/x + c), x), x, -bx*S(2)/(S(2)c*S(2)) - b(-S(2)ac + b*S(2))log(a + bx + cx*S(2))/(S(2)cS(4)) + xS(3)/(S(3)c) + x(-ac + bS(2))/cS(3) - (S(2)a*S(2)c*S(2) - S(4)abS(2)c + b*S(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(4)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a/xS(2) + b/x + c), x), x, -bx/c*S(2) + b(-S(3)ac + b*S(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(3)sqrt(-S(4)ac + bS(2))) + xS(2)/(S(2)c) + (-ac + bS(2))log(a + bx + cx*S(2))/(S(2)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a/xS(2) + b/x + c), x), x, -blog(a + bx + cxS(2))/(S(2)c*S(2)) + x/c - (-S(2)ac + bS(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a/x*S(2) + b/x + c)), x), x, batanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(csqrt(-S(4)ac + b*S(2))) + log(a + bx + cxS(2))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(a/x*S(2) + b/x + c)), x), x, S(2)atanh((S(2)a/x + b)/sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a/x*S(2) + b/x + c)), x), x, batanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(asqrt(-S(4)ac + b*S(2))) + log(x)/a - log(a + bx + cxS(2))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(4)(a/x*S(2) + b/x + c)), x), x, -S(1)/(ax) - blog(x)/aS(2) + blog(a + bx + cx*S(2))/(S(2)a*S(2)) - (-S(2)ac + bS(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(a/x*S(2) + b/x + c)), x), x, -S(1)/(S(2)axS(2)) + b/(aS(2)x) + b(-S(3)ac + bS(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(3)sqrt(-S(4)ac + bS(2))) + (-ac + b*S(2))log(x)/a*S(3) - (-ac + b*S(2))log(a + bx + cx*S(2))/(S(2)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(a/xS(2) + b/x + c)), x), x, -S(1)/(S(3)axS(3)) + b/(S(2)a*S(2)x*S(2)) - (-ac + bS(2))/(aS(3)x) - b(-S(2)ac + b*S(2))log(x)/a*S(4) + b(-S(2)ac + b*S(2))log(a + bx + cx*S(2))/(S(2)a*S(4)) - (S(2)a*S(2)c*S(2) - S(4)abS(2)c + b*S(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(4)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a/xS(2) + b/x + c)S(2), x), x, -bx*S(3)/(c(-S(4)ac + b*S(2))) - bx(-S(11)ac + S(3)bS(2))/(cS(3)(-S(4)ac + bS(2))) + b(S(30)aS(2)c*S(2) - S(20)abS(2)c + S(3)bS(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(4)(-S(4)ac + bS(2))(S(3)/2)) + xS(4)(S(2)a + bx)/((-S(4)ac + b*S(2))(a + bx + cxS(2))) + xS(2)(-S(8)ac + S(3)b*S(2))/(S(2)c*S(2)(-S(4)ac + b*S(2))) + (-S(2)ac + S(3)b*S(2))log(a + bx + cx*S(2))/(S(2)cS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a/xS(2) + b/x + c)*(S(-2)), x), x, -bx*S(2)/(c(-S(4)ac + b*S(2))) - blog(a + bx + cxS(2))/cS(3) + x*S(3)(S(2)a + bx)/((-S(4)ac + b*S(2))(a + bx + cx*S(2))) + x(-S(6)ac + S(2)bS(2))/(cS(2)(-S(4)ac + b*S(2))) - (S(12)a*S(2)c*S(2) - S(12)abS(2)c + S(2)bS(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(3)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a/xS(2) + b/x + c)S(2)), x), x, -bx/(c(-S(4)ac + b*S(2))) + b(-S(6)ac + b*S(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(2)(-S(4)ac + bS(2))(S(3)/2)) + xS(2)(S(2)a + bx)/((-S(4)ac + b*S(2))(a + bx + cx*S(2))) + log(a + bx + cxS(2))/(S(2)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a/xS(2) + b/x + c)S(2)), x), x, -S(4)aatanh((S(2)a/x + b)/sqrt(-S(4)ac + b*S(2)))/(-S(4)ac + bS(2))(S(3)/2) + (S(2)a/x + b)/((-S(4)ac + b*S(2))(a/xS(2) + b/x + c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a/xS(2) + b/x + c)S(2)), x), x, -S(2)batanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + (S(2)a + bx)/((-S(4)ac + bS(2))(a + bx + cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(a/xS(2) + b/x + c)S(2)), x), x, S(4)catanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) - (b + S(2)cx)/((-S(4)ac + bS(2))(a + bx + cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(a/xS(2) + b/x + c)S(2)), x), x, (-S(2)ac + bS(2) + bcx)/(a(-S(4)ac + b*S(2))(a + bx + cx*S(2))) + b(-S(6)ac + b*S(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(2)(-S(4)ac + bS(2))(S(3)/2)) + log(x)/aS(2) - log(a + bx + cxS(2))/(S(2)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(a/xS(2) + b/x + c)S(2)), x), x, (-S(2)ac + bS(2) + bcx)/(ax(-S(4)ac + bS(2))(a + bx + cx*S(2))) + (S(6)ac - S(2)bS(2))/(aS(2)x(-S(4)ac + b*S(2))) - S(2)blog(x)/aS(3) + blog(a + bx + cxS(2))/aS(3) - (S(12)aS(2)c*S(2) - S(12)abS(2)c + S(2)bS(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(3)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(7)(a/xS(2) + b/x + c)S(2)), x), x, (-S(2)ac + b*S(2) + bcx)/(ax*S(2)(-S(4)ac + b*S(2))(a + bx + cx*S(2))) - (-S(8)ac + S(3)b*S(2))/(S(2)a*S(2)x*S(2)(-S(4)ac + b*S(2))) + b(-S(11)ac + S(3)bS(2))/(aS(3)x(-S(4)ac + bS(2))) + b(S(30)aS(2)c*S(2) - S(20)abS(2)c + S(3)bS(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(4)(-S(4)ac + bS(2))(S(3)/2)) + (-S(2)ac + S(3)b*S(2))log(x)/a*S(4) - (-S(2)ac + S(3)b*S(2))log(a + bx + cx*S(2))/(S(2)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a/xS(2) + b/x + c)*(S(-3)), x), x, -S(3)bxS(2)(-S(6)ac + b*S(2))/(S(2)c*S(2)(-S(4)ac + bS(2))S(2)) - S(3)blog(a + bx + cx*S(2))/(S(2)cS(4)) + xS(5)(S(2)a + bx)/((-S(8)ac + S(2)b*S(2))(a + bx + cxS(2))S(2)) + x*S(3)(a(-S(10)ac + bS(2)) + bx(-S(7)ac + bS(2)))/(c(-S(4)ac + bS(2))S(2)(a + bx + cxS(2))) + x(S(30)aS(2)c*S(2) - S(21)abS(2)c + S(3)bS(4))/(cS(3)(-S(4)ac + bS(2))S(2)) - (-S(60)aS(3)c*S(3) + S(90)a*S(2)b*S(2)c*S(2) - S(30)abS(4)c + S(3)bS(6))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(4)(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a/xS(2) + b/x + c)S(3)), x), x, -bx(-S(7)ac + bS(2))/(cS(2)(-S(4)ac + bS(2))S(2)) + b(S(30)aS(2)c*S(2) - S(10)abS(2)c + b*S(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(3)(-S(4)ac + bS(2))(S(5)/2)) + xS(4)(S(2)a + bx)/((-S(8)ac + S(2)bS(2))(a + bx + cxS(2))S(2)) + x*S(2)(a(-S(16)ac + bS(2)) + bx(-S(10)ac + bS(2)))/(S(2)c(-S(4)ac + bS(2))S(2)(a + bx + cx*S(2))) + log(a + bx + cxS(2))/(S(2)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a/xS(2) + b/x + c)S(3)), x), x, S(12)a*S(2)atanh((S(2)a/x + b)/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) - S(3)a(S(2)a/x + b)/((-S(4)ac + bS(2))S(2)(a/xS(2) + b/x + c)) + (S(2)a/x + b)/((-S(8)ac + S(2)bS(2))(a/xS(2) + b/x + c)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)(a/xS(2) + b/x + c)S(3)), x), x, S(6)abatanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) + S(3)bx(S(2)a + bx)/(S(2)(-S(4)ac + bS(2))S(2)(a + bx + cxS(2))) - xS(3)(b + S(2)cx)/((-S(8)ac + S(2)b*S(2))(a + bx + cxS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(4)(a/xS(2) + b/x + c)S(3)), x), x, x(S(2)a + bx)/((-S(8)ac + S(2)b*S(2))(a + bx + cxS(2))S(2)) + (S(3)ab + x(S(2)ac + bS(2)))/((-S(4)ac + bS(2))S(2)(a + bx + cx*S(2))) - (S(4)ac + S(2)b*S(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(-S(4)ac + bS(2))(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(a/xS(2) + b/x + c)S(3)), x), x, S(6)bcatanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) - S(3)b(b + S(2)cx)/(S(2)(-S(4)ac + bS(2))S(2)(a + bx + cxS(2))) + (S(2)a + bx)/((-S(8)ac + S(2)b*S(2))(a + bx + cxS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(6)(a/xS(2) + b/x + c)S(3)), x), x, -S(12)cS(2)atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(-S(4)ac + bS(2))(S(5)/2) + S(3)c(b + S(2)cx)/((-S(4)ac + bS(2))S(2)(a + bx + cx*S(2))) + (-b - S(2)cx)/((-S(8)ac + S(2)b*S(2))(a + bx + cxS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(7)(a/xS(2) + b/x + c)S(3)), x), x, (-S(2)ac + b*S(2) + bcx)/(S(2)a(-S(4)ac + bS(2))(a + bx + cxS(2))S(2)) + (S(16)aS(2)c*S(2) - S(15)abS(2)c + S(2)bS(4) + S(2)bcx(-S(7)ac + bS(2)))/(S(2)a*S(2)(-S(4)ac + bS(2))S(2)(a + bx + cxS(2))) + b(S(30)aS(2)c*S(2) - S(10)abS(2)c + b*S(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(3)(-S(4)ac + bS(2))(S(5)/2)) + log(x)/aS(3) - log(a + bx + cxS(2))/(S(2)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(8)(a/xS(2) + b/x + c)S(3)), x), x, (-S(2)ac + bS(2) + bcx)/(S(2)ax(-S(4)ac + b*S(2))(a + bx + cxS(2))S(2)) + (S(20)aS(2)c*S(2) - S(20)abS(2)c + S(3)bS(4) + S(3)bcx(-S(6)ac + bS(2)))/(S(2)a*S(2)x(-S(4)ac + bS(2))S(2)(a + bx + cx*S(2))) - (S(30)a*S(2)c*S(2) - S(21)abS(2)c + S(3)bS(4))/(aS(3)x(-S(4)ac + bS(2))S(2)) - S(3)blog(x)/aS(4) + S(3)blog(a + bx + cxS(2))/(S(2)a*S(4)) - (-S(60)a*S(3)c*S(3) + S(90)a*S(2)b*S(2)c*S(2) - S(30)abS(4)c + S(3)bS(6))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(4)(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(S(15) + S(13)/x + S(2)/xS(2)), x), x, xS(3)/S(45) - S(13)x*S(2)/S(450) + S(139)x/S(3375) - S(16)log(S(3)x + S(2))/S(567) + log(S(5)x + S(1))/S(4375), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(S(15) + S(13)/x + S(2)/xS(2)), x), x, xS(2)/S(30) - S(13)x/S(225) + S(8)log(S(3)x + S(2))/S(189) - log(S(5)x + S(1))/S(875), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(15) + S(13)/x + S(2)/xS(2)), x), x, x/S(15) - S(4)log(S(3)x + S(2))/S(63) + log(S(5)x + S(1))/S(175), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(15) + S(13)/x + S(2)/xS(2))), x), x, S(2)log(S(3)x + S(2))/S(21) - log(S(5)x + S(1))/S(35), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(S(15) + S(13)/x + S(2)/xS(2))), x), x, -log(S(3) + S(2)/x)/S(7) + log(S(5) + S(1)/x)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(S(15) + S(13)/x + S(2)/xS(2))), x), x, log(x)/S(2) + S(3)log(S(3)x + S(2))/S(14) - S(5)log(S(5)x + S(1))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(S(15) + S(13)/x + S(2)/x*S(2))), x), x, -S(13)log(x)/S(4) - S(9)log(S(3)x + S(2))/S(28) + S(25)log(S(5)x + S(1))/S(7) - S(1)/(S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(S(15) + S(13)/x + S(2)/x*S(2))), x), x, S(139)log(x)/S(8) + S(27)log(S(3)x + S(2))/S(56) - S(125)log(S(5)x + S(1))/S(7) + S(13)/(S(4)x) - S(1)/(S(4)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)(S(15) + S(13)/x + S(2)/xS(2))), x), x, -S(1417)log(x)/S(16) - S(81)log(S(3)x + S(2))/S(112) + S(625)log(S(5)x + S(1))/S(7) - S(139)/(S(8)x) + S(13)/(S(8)x*S(2)) - S(1)/(S(6)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b/x + c/xS(2))*(S(5)/2), x), x, S(5)a*(S(3)/2)batanh((S(2)a + b/x)/(S(2)sqrt(a)sqrt(a + b/x + c/x*S(2))))/S(2) + x(a + b/x + c/xS(2))(S(5)/2) - S(5)(S(7)b + S(6)c/x)(a + b/x + c/xS(2))(S(3)/2)/S(24) - S(5)(b(S(44)ac + b*S(2)) + S(2)c(S(12)ac + bS(2))/x)sqrt(a + b/x + c/x*S(2))/(S(64)c) + (-S(240)aS(2)c*S(2) - S(120)abS(2)c + S(5)bS(4))atanh((b + S(2)c/x)/(S(2)sqrt(c)sqrt(a + b/x + c/xS(2))))/(S(128)c(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b/x + c/xS(2))*(S(3)/2), x), x, S(3)sqrt(a)batanh((S(2)a + b/x)/(S(2)sqrt(a)sqrt(a + b/x + c/xS(2))))/S(2) + x(a + b/x + c/xS(2))(S(3)/2) - S(3)(S(3)b + S(2)c/x)sqrt(a + b/x + c/x*S(2))/S(4) - (S(12)ac + S(3)b*S(2))atanh((b + S(2)c/x)/(S(2)sqrt(c)sqrt(a + b/x + c/xS(2))))/(S(8)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + b/x + c/x*S(2)), x), x, -sqrt(c)atanh((b + S(2)c/x)/(S(2)sqrt(c)sqrt(a + b/x + c/xS(2)))) + xsqrt(a + b/x + c/x*S(2)) + batanh((S(2)a + b/x)/(S(2)sqrt(a)sqrt(a + b/x + c/xS(2))))/(S(2)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + b/x + c/x*S(2)), x), x, xsqrt(a + b/x + c/x*S(2))/a - batanh((S(2)a + b/x)/(S(2)sqrt(a)sqrt(a + b/x + c/xS(2))))/(S(2)a(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b/x + c/xS(2))*(S(-3)/2), x), x, -x(-S(4)ac + S(2)bS(2) + S(2)bc/x)/(a(-S(4)ac + b*S(2))sqrt(a + b/x + c/x*S(2))) + x(-S(8)ac + S(3)bS(2))sqrt(a + b/x + c/xS(2))/(aS(2)(-S(4)ac + bS(2))) - S(3)batanh((S(2)a + b/x)/(S(2)sqrt(a)sqrt(a + b/x + c/x*S(2))))/(S(2)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b/x + c/xS(2))*(S(-5)/2), x), x, -x(-S(4)ac + S(2)bS(2) + S(2)bc/x)/(S(3)a(-S(4)ac + bS(2))(a + b/x + c/xS(2))(S(3)/2)) - x(S(64)a*S(2)c*S(2) - S(64)abS(2)c + S(10)bS(4) + S(2)bc(-S(28)ac + S(5)bS(2))/x)/(S(3)a*S(2)(-S(4)ac + bS(2))S(2)sqrt(a + b/x + c/xS(2))) + xsqrt(a + b/x + c/x*S(2))(S(128)aS(2)c*S(2) - S(100)abS(2)c + S(15)bS(4))/(S(3)a*S(3)(-S(4)ac + bS(2))S(2)) - S(5)batanh((S(2)a + b/x)/(S(2)sqrt(a)sqrt(a + b/x + c/xS(2))))/(S(2)a(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)ab/x + bS(2)/xS(2)), x), x, axsqrt(a*S(2) + S(2)ab/x + bS(2)/xS(2))/(a + b/x) + bsqrt(a*S(2) + S(2)ab/x + bS(2)/xS(2))log(x)/(a + b/x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a/xS(4) + b/xS(2) + c), x), x, x/c - sqrt(S(2))(b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))) - sqrt(S(2))(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a/xS(6) + b/xS(3) + c), x), x, x/c - S(2)(S(2)/3)(b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(4)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(4)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)sqrt(S(3))(b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(4)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(4)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(4)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)sqrt(S(3))(b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(4)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a/xS(8) + b/xS(4) + c), x), x, x/c + S(2)*(S(3)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(x) + cx)/x, x), x, -S(2)sqrt(a)atanh((S(2)a + bsqrt(x))/(S(2)sqrt(a)sqrt(a + bsqrt(x) + cx))) + batanh((b + S(2)csqrt(x))/(S(2)sqrt(c)sqrt(a + bsqrt(x) + cx)))/sqrt(c) + S(2)sqrt(a + bsqrt(x) + cx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b*S(2)/(S(4)c) + bsqrt(x) + cx)*S(2), x), x, -b(b + S(2)csqrt(x))*S(5)/(S(160)c*S(4)) + (b + S(2)csqrt(x))S(6)/(S(192)cS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(aS(2) + S(2)absqrt(x) + bS(2)x), x), x, -S(2)a(a + bsqrt(x))log(a + bsqrt(x))/(bS(2)sqrt(a*S(2) + S(2)absqrt(x) + b*S(2)x)) + S(2)sqrt(aS(2) + S(2)absqrt(x) + b*S(2)x)/b*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p, x), x, x(dx)m(S(1) + bx(S(1)/3)/a)(-S(2)p)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))phyper((S(3)m + S(3), -S(2)p), (S(3)m + S(4),), -bx(S(1)/3)/a)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p, x), x, S(3)a*S(8)(a + bx(S(1)/3))(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(9)(S(2)p + S(1))) - S(12)a*S(7)(ab + bS(2)x(S(1)/3))S(2)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(11)(p + S(1))) + S(84)aS(6)(ab + bS(2)x(S(1)/3))S(3)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(12)(S(2)p + S(3))) - S(84)a*S(5)(ab + bS(2)x(S(1)/3))S(4)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(13)(p + S(2))) + S(210)aS(4)(ab + bS(2)x(S(1)/3))S(5)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(14)(S(2)p + S(5))) - S(84)a*S(3)(ab + bS(2)x(S(1)/3))S(6)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(15)(p + S(3))) + S(84)aS(2)(ab + bS(2)x(S(1)/3))S(7)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(16)(S(2)p + S(7))) - S(12)a(ab + b*S(2)x(S(1)/3))S(8)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(17)(p + S(4))) + S(3)(ab + b*S(2)x(S(1)/3))S(9)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(18)(S(2)p + S(9))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p, x), x, -S(3)a*S(5)(a + bx(S(1)/3))(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(6)(S(2)p + S(1))) + S(15)a*S(4)(ab + bS(2)x(S(1)/3))S(2)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(S(2)b*S(8)(p + S(1))) - S(30)aS(3)(ab + bS(2)x(S(1)/3))S(3)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(9)(S(2)p + S(3))) + S(15)a*S(2)(ab + bS(2)x(S(1)/3))S(4)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(10)(p + S(2))) - S(15)a(ab + bS(2)x(S(1)/3))S(5)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(bS(11)(S(2)p + S(5))) + S(3)(ab + bS(2)x(S(1)/3))S(6)(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(S(2)b*S(12)(p + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/x, x), x, -(S(3)a + S(3)bx*(S(1)/3))(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))phyper((S(1), S(2)p + S(1)), (S(2)p + S(2),), S(1) + bx(S(1)/3)/a)/(a(S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/xS(2), x), x, (S(3)abS(3) + S(3)b*S(4)x*(S(1)/3))(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))phyper((S(4), S(2)p + S(1)), (S(2)p + S(2),), S(1) + bx(S(1)/3)/a)/(aS(4)(S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/xS(2) - S(2)b*S(3)p(-S(2)p + S(1))(-p + S(1))(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(S(3)a*S(3)x), x), x, -(a + bx(S(1)/3))(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(ax) + b(a + bx*(S(1)/3))(-p + S(1))(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(aS(2)x(S(2)/3)) - bS(2)(a + bx*(S(1)/3))(-S(2)p + S(1))(-p + S(1))(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))p/(aS(3)x(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abx(S(1)/4) + bS(2)sqrt(x))*(S(-3)/2), x), x, -S(12)a(a + bx*(S(1)/4))log(a + bx(S(1)/4))/(bS(4)sqrt(a*S(2) + S(2)abx(S(1)/4) + bS(2)sqrt(x))) - x(S(3)/4)(S(2)a + S(2)bx(S(1)/4))/(b(a*S(2) + S(2)abx(S(1)/4) + bS(2)sqrt(x))(S(3)/2)) - S(6)sqrt(x)/(b*S(2)sqrt(a*S(2) + S(2)abx(S(1)/4) + bS(2)sqrt(x))) + S(12)sqrt(a*S(2) + S(2)abx(S(1)/4) + bS(2)sqrt(x))/bS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abx(S(1)/6) + bS(2)x(S(1)/3))(S(-5)/2), x), x, -S(30)a(a + bx*(S(1)/6))log(a + bx(S(1)/6))/(bS(6)sqrt(a*S(2) + S(2)abx(S(1)/6) + bS(2)x(S(1)/3))) - x(S(5)/6)(S(3)a + S(3)bx(S(1)/6))/(S(2)b(aS(2) + S(2)abx(S(1)/6) + bS(2)x(S(1)/3))(S(5)/2)) - S(5)x*(S(2)/3)/(S(2)b*S(2)(a*S(2) + S(2)abx(S(1)/6) + bS(2)x(S(1)/3))(S(3)/2)) - sqrt(x)(S(5)a + S(5)bx(S(1)/6))/(bS(3)(a*S(2) + S(2)abx(S(1)/6) + bS(2)x(S(1)/3))(S(3)/2)) - S(15)x(S(1)/3)/(bS(4)sqrt(aS(2) + S(2)abx(S(1)/6) + bS(2)x(S(1)/3))) + S(30)sqrt(a*S(2) + S(2)abx(S(1)/6) + bS(2)x(S(1)/3))/bS(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)ab/sqrt(x) + bS(2)/x)(S(3)/2), x), x, -S(6)abS(2)sqrt(a*S(2) + S(2)ab/sqrt(x) + bS(2)/x)log(S(1)/sqrt(x))/(a + b/sqrt(x)) - S(6)bS(2)sqrt(a*S(2) + S(2)ab/sqrt(x) + bS(2)/x) + S(3)bsqrt(x)(a + b/sqrt(x))sqrt(aS(2) + S(2)ab/sqrt(x) + bS(2)/x) + x(a*S(2) + S(2)ab/sqrt(x) + bS(2)/x)(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))(S(7)/2), x), x, -S(105)a*S(4)b*S(3)sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))log(x(S(-1)/3))/(a + b/x(S(1)/3)) - S(105)aS(3)b*S(3)sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3)) - S(105)a*S(2)b*S(3)(a + b/x*(S(1)/3))sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))/S(2) - S(35)abS(3)(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))(S(3)/2) - S(105)b*S(3)(a + b/x*(S(1)/3))(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))(S(3)/2)/S(4) + S(21)b*S(2)x*(S(1)/3)(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))(S(5)/2) + S(7)bx(S(2)/3)(a + b/x*(S(1)/3))(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))(S(5)/2)/S(2) + x(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))(S(7)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))(S(5)/2), x), x, -S(30)a*S(2)b*S(3)sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))log(x(S(-1)/3))/(a + b/x(S(1)/3)) - S(30)ab*S(3)sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3)) - S(15)b*S(3)(a + b/x*(S(1)/3))sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3)) + S(10)b*S(2)x*(S(1)/3)(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))(S(3)/2) + S(5)bx(S(2)/3)(a + b/x*(S(1)/3))(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))(S(3)/2)/S(2) + x(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))(S(3)/2), x), x, S(3)abS(2)x*(S(1)/3)sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))/(a + b/x(S(1)/3)) - S(3)b*S(3)sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))log(x(S(-1)/3))/(a + b/x(S(1)/3)) + S(3)bx*(S(2)/3)(a + b/x*(S(1)/3))sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))/S(2) + x(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3)), x), x, -axsqrt(aS(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))/(S(2)a + S(2)b/x(S(1)/3)) + S(3)xsqrt(aS(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(aS(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3)), x), x, x(a + b/x*(S(1)/3))/(asqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))) - S(3)bx(S(2)/3)(a + b/x*(S(1)/3))/(S(2)a*S(2)sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))) + S(3)b*S(2)x*(S(1)/3)(a + b/x(S(1)/3))/(aS(3)sqrt(aS(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))) - S(3)b*S(3)(a + b/x*(S(1)/3))log(ax(S(1)/3) + b)/(aS(4)sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/sqrt(aS(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3)), x), x, x(a + b/x*(S(1)/3))/(asqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))) - S(3)bx(S(2)/3)(a + b/x*(S(1)/3))/(S(2)a*S(2)sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))) + S(3)b*S(3)(a + b/x*(S(1)/3))log(x(S(-1)/3))/(aS(4)sqrt(aS(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))) - S(3)b*S(3)(a + b/x*(S(1)/3))log(a + b/x(S(1)/3))/(aS(4)sqrt(aS(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))) + S(3)b*S(2)x*(S(1)/3)sqrt(a*S(2) + S(2)ab/x(S(1)/3) + bS(2)/x(S(2)/3))/aS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(-3)/2), x), x, -x(S(2)/3)(S(3)a + S(3)bx(S(1)/3))/(S(2)b(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(3)/2)) - S(3)x(S(1)/3)/(bS(2)sqrt(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))) + (S(3)a + S(3)bx*(S(1)/3))log(a + bx(S(1)/3))/(bS(3)sqrt(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(-5)/2), x), x, x(S(3)a + S(3)bx(S(1)/3))/(S(4)a(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(5)/2)) + x/(S(4)a*S(2)(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(-7)/2), x), x, -x(S(2)/3)(a + bx(S(1)/3))/(S(2)b(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(7)/2)) - x(S(1)/3)/(S(5)b*S(2)(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(5)/2)) - (a + bx*(S(1)/3))/(S(20)b*S(3)(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(-9)/2), x), x, -x(S(2)/3)(S(3)a + S(3)bx(S(1)/3))/(S(8)b(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(9)/2)) - S(3)x*(S(1)/3)/(S(28)b*S(2)(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(7)/2)) - (a + bx*(S(1)/3))/(S(56)b*S(3)(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(-11)/2), x), x, -x(S(2)/3)(S(3)a + S(3)bx(S(1)/3))/(S(10)b(aS(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(11)/2)) - x(S(1)/3)/(S(15)b*S(2)(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(9)/2)) - (a + bx*(S(1)/3))/(S(120)b*S(3)(a*S(2) + S(2)abx(S(1)/3) + bS(2)x(S(2)/3))(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)ab/x(S(1)/4) + bS(2)/sqrt(x))(S(5)/2), x), x, -S(20)abS(4)sqrt(a*S(2) + S(2)ab/x(S(1)/4) + bS(2)/sqrt(x))log(x(S(-1)/4))/(a + b/x(S(1)/4)) - S(20)bS(4)sqrt(a*S(2) + S(2)ab/x(S(1)/4) + bS(2)/sqrt(x)) + S(10)b*S(3)x*(S(1)/4)(a + b/x*(S(1)/4))sqrt(a*S(2) + S(2)ab/x(S(1)/4) + bS(2)/sqrt(x)) + S(10)b*S(2)sqrt(x)(aS(2) + S(2)ab/x(S(1)/4) + bS(2)/sqrt(x))(S(3)/2)/S(3) + S(5)bx(S(3)/4)(a + b/x*(S(1)/4))(a*S(2) + S(2)ab/x(S(1)/4) + bS(2)/sqrt(x))(S(3)/2)/S(3) + x(a*S(2) + S(2)ab/x(S(1)/4) + bS(2)/sqrt(x))(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)ab/x(S(1)/5) + bS(2)/x(S(2)/5))(S(5)/2), x), x, S(5)abS(4)x*(S(1)/5)sqrt(a*S(2) + S(2)ab/x(S(1)/5) + bS(2)/x(S(2)/5))/(a + b/x(S(1)/5)) - S(5)b*S(5)sqrt(a*S(2) + S(2)ab/x(S(1)/5) + bS(2)/x(S(2)/5))log(x(S(-1)/5))/(a + b/x(S(1)/5)) + S(5)bS(3)x*(S(2)/5)(a + b/x*(S(1)/5))sqrt(a*S(2) + S(2)ab/x(S(1)/5) + bS(2)/x(S(2)/5))/S(2) + S(5)b*S(2)x*(S(3)/5)(a*S(2) + S(2)ab/x(S(1)/5) + bS(2)/x(S(2)/5))(S(3)/2)/S(3) + S(5)bx(S(4)/5)(a + b/x*(S(1)/5))(a*S(2) + S(2)ab/x(S(1)/5) + bS(2)/x(S(2)/5))(S(3)/2)/S(4) + x(a*S(2) + S(2)ab/x(S(1)/5) + bS(2)/x(S(2)/5))(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abx(S(1)/5) + bS(2)x(S(2)/5))(S(-5)/2), x), x, -x(S(4)/5)(S(5)a + S(5)bx(S(1)/5))/(S(4)b(aS(2) + S(2)abx(S(1)/5) + bS(2)x(S(2)/5))(S(5)/2)) - S(5)x*(S(3)/5)/(S(3)b*S(2)(a*S(2) + S(2)abx(S(1)/5) + bS(2)x(S(2)/5))(S(3)/2)) - x(S(2)/5)(S(5)a + S(5)bx(S(1)/5))/(S(2)b*S(3)(a*S(2) + S(2)abx(S(1)/5) + bS(2)x(S(2)/5))(S(3)/2)) - S(5)x(S(1)/5)/(bS(4)sqrt(aS(2) + S(2)abx(S(1)/5) + bS(2)x(S(2)/5))) + (S(5)a + S(5)bx*(S(1)/5))log(a + bx(S(1)/5))/(bS(5)sqrt(a*S(2) + S(2)abx(S(1)/5) + bS(2)x(S(2)/5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)ab/x(S(1)/6) + bS(2)/x(S(1)/3))(S(7)/2), x), x, -S(42)abS(6)sqrt(a*S(2) + S(2)ab/x(S(1)/6) + bS(2)/x(S(1)/3))log(x(S(-1)/6))/(a + b/x(S(1)/6)) - S(42)bS(6)sqrt(a*S(2) + S(2)ab/x(S(1)/6) + bS(2)/x(S(1)/3)) + S(21)b*S(5)x*(S(1)/6)(a + b/x*(S(1)/6))sqrt(a*S(2) + S(2)ab/x(S(1)/6) + bS(2)/x(S(1)/3)) + S(7)b*S(4)x*(S(1)/3)(a*S(2) + S(2)ab/x(S(1)/6) + bS(2)/x(S(1)/3))(S(3)/2) + S(7)b*S(3)sqrt(x)(a + b/x(S(1)/6))(a*S(2) + S(2)ab/x(S(1)/6) + bS(2)/x(S(1)/3))(S(3)/2)/S(2) + S(21)b*S(2)x*(S(2)/3)(a*S(2) + S(2)ab/x(S(1)/6) + bS(2)/x(S(1)/3))(S(5)/2)/S(10) + S(7)bx(S(5)/6)(a + b/x*(S(1)/6))(a*S(2) + S(2)ab/x(S(1)/6) + bS(2)/x(S(1)/3))(S(5)/2)/S(5) + x(a*S(2) + S(2)ab/x(S(1)/6) + bS(2)/x(S(1)/3))(S(7)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(4)n + S(-1))/(bxn + cx*(S(2)n)), x), x, b*S(2)log(b + cxn)/(cS(3)n) - bxn/(cS(2)n) + x*(S(2)n)/(S(2)cn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(3)n + S(-1))/(bxn + cx*(S(2)n)), x), x, -blog(b + cxn)/(cS(2)n) + xn/(cn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(2)n + S(-1))/(bxn + cx*(S(2)n)), x), x, log(b + cxn)/(cn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(n + S(-1))/(bx*n + cx*(S(2)n)), x), x, log(x)/b - log(b + cxn)/(bn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(-n + S(-1))/(bx*n + cx*(S(2)n)), x), x, -x*(-S(2)n)/(S(2)bn) + cx(-n)/(bS(2)n) + c*S(2)log(x)/bS(3) - cS(2)log(b + cxn)/(bS(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-S(2)n + S(-1))/(bxn + cx*(S(2)n)), x), x, -x*(-S(3)n)/(S(3)bn) + cx(-S(2)n)/(S(2)bS(2)n) - c*S(2)x(-n)/(bS(3)n) - cS(3)log(x)/bS(4) + cS(3)log(b + cxn)/(bS(4)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-S(3)n + S(-1))/(bxn + cx*(S(2)n)), x), x, -x*(-S(4)n)/(S(4)bn) + cx(-S(3)n)/(S(3)bS(2)n) - c*S(2)x*(-S(2)n)/(S(2)bS(3)n) + c*S(3)x(-n)/(bS(4)n) + cS(4)log(x)/bS(5) - cS(4)log(b + cxn)/(bS(5)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n/S(4) + S(-1))/(bx*n + cx*(S(2)n)), x), x, -S(4)x(-S(3)n/S(4))/(S(3)bn) + sqrt(S(2))c(S(3)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)x*(n/S(4)) + sqrt(b) + sqrt(c)x*(n/S(2)))/(S(2)b*(S(7)/4)n) - sqrt(S(2))c(S(3)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)x*(n/S(4)) + sqrt(b) + sqrt(c)x*(n/S(2)))/(S(2)b*(S(7)/4)n) + sqrt(S(2))c(S(3)/4)atan(S(1) - sqrt(S(2))c(S(1)/4)x(n/S(4))/b(S(1)/4))/(b*(S(7)/4)n) - sqrt(S(2))c(S(3)/4)atan(S(1) + sqrt(S(2))c(S(1)/4)x(n/S(4))/b(S(1)/4))/(b*(S(7)/4)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(n/S(3) + S(-1))/(bx*n + cx*(S(2)n)), x), x, -S(3)x(-S(2)n/S(3))/(S(2)bn) - c*(S(2)/3)log(b(S(1)/3) + c(S(1)/3)x(n/S(3)))/(b(S(5)/3)n) + c*(S(2)/3)log(b(S(2)/3) - b(S(1)/3)c(S(1)/3)x(n/S(3)) + c(S(2)/3)x(S(2)n/S(3)))/(S(2)b(S(5)/3)n) + sqrt(S(3))c(S(2)/3)atan(sqrt(S(3))(b(S(1)/3) - S(2)c*(S(1)/3)x*(n/S(3)))/(S(3)b(S(1)/3)))/(b(S(5)/3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n/S(2) + S(-1))/(bx*n + cx*(S(2)n)), x), x, -S(2)x(-n/S(2))/(bn) + S(2)sqrt(c)atan(sqrt(b)x(-n/S(2))/sqrt(c))/(b(S(3)/2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(-n/S(2) + S(-1))/(bx*n + cx*(S(2)n)), x), x, -S(2)x(-S(3)n/S(2))/(S(3)bn) + S(2)cx(-n/S(2))/(bS(2)n) - S(2)c*(S(3)/2)atan(sqrt(b)x(-n/S(2))/sqrt(c))/(b(S(5)/2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(-n/S(3) + S(-1))/(bx*n + cx*(S(2)n)), x), x, -S(3)x(-S(4)n/S(3))/(S(4)bn) + S(3)cx(-n/S(3))/(bS(2)n) - c(S(4)/3)log(b*(S(1)/3)x(-n/S(3)) + c(S(1)/3))/(b*(S(7)/3)n) + c*(S(4)/3)log(b*(S(2)/3)x*(-S(2)n/S(3)) - b*(S(1)/3)c*(S(1)/3)x(-n/S(3)) + c(S(2)/3))/(S(2)b(S(7)/3)n) + sqrt(S(3))c(S(4)/3)atan(sqrt(S(3))(-S(2)b*(S(1)/3)x(-n/S(3)) + c(S(1)/3))/(S(3)c(S(1)/3)))/(b(S(7)/3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(-n/S(4) + S(-1))/(bx*n + cx*(S(2)n)), x), x, -S(4)x(-S(5)n/S(4))/(S(5)bn) + S(4)cx(-n/S(4))/(bS(2)n) + sqrt(S(2))c*(S(5)/4)log(-sqrt(S(2))b(S(1)/4)c*(S(1)/4)x*(-n/S(4)) + sqrt(b)x*(-n/S(2)) + sqrt(c))/(S(2)b*(S(9)/4)n) - sqrt(S(2))c(S(5)/4)log(sqrt(S(2))b(S(1)/4)c*(S(1)/4)x*(-n/S(4)) + sqrt(b)x*(-n/S(2)) + sqrt(c))/(S(2)b*(S(9)/4)n) - sqrt(S(2))c(S(5)/4)atan(sqrt(S(2))b(S(1)/4)x(-n/S(4))/c(S(1)/4) + S(-1))/(b*(S(9)/4)n) - sqrt(S(2))c(S(5)/4)atan(sqrt(S(2))b(S(1)/4)x(-n/S(4))/c(S(1)/4) + S(1))/(b*(S(9)/4)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(-n(p + S(-1)) + S(-1))(bx*n + cx*(S(2)n))p, x), x, x(-n(p + S(1)))(bxn + cx*(S(2)n))*(p + S(1))/(cn(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-n(S(2)p + S(1)) + S(-1))(bxn + cx*(S(2)n))p, x), x, -x(-S(2)n(p + S(1)))(bx*n + cx*(S(2)n))*(p + S(1))/(bn(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)n + S(-1))(aS(2) + S(2)abxn + bS(2)x(S(2)n))*p, x), x, -a(a + bxn)(a*S(2) + S(2)abxn + bS(2)x(S(2)n))p/(bS(2)n(S(2)p + S(1))) + (aS(2) + S(2)abxn + bS(2)x(S(2)n))*(p + S(1))/(S(2)b*S(2)n(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)n + S(-1))(aS(2) + S(2)abxn + bS(2)x(S(2)n))*(S(5)/2), x), x, -a(a + bxn)(a*S(2) + S(2)abxn + bS(2)x(S(2)n))*(S(5)/2)/(S(6)b*S(2)n) + (a*S(2) + S(2)abxn + bS(2)x(S(2)n))*(S(7)/2)/(S(7)b*S(2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(2)n + S(-1))(aS(2) + S(2)abxn + bS(2)x(S(2)n))*(S(3)/2), x), x, -a(a + bxn)(a*S(2) + S(2)abxn + bS(2)x(S(2)n))*(S(3)/2)/(S(4)b*S(2)n) + (a*S(2) + S(2)abxn + bS(2)x(S(2)n))*(S(5)/2)/(S(5)b*S(2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(2)n + S(-1))sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n)), x), x, ax(S(2)n)sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/(S(6)n(a + bxn)) + x(S(2)n)sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/(S(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)n + S(-1))/sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n)), x), x, -a(a + bx*n)log(a + bxn)/(bS(2)nsqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))) + sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(b*S(2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(2)n + S(-1))/(a*S(2) + S(2)abxn + bS(2)x(S(2)n))(S(3)/2), x), x, x(S(2)n)(a + bxn)/(S(2)an(a*S(2) + S(2)abxn + bS(2)x(S(2)n))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)n + S(-1))/(aS(2) + S(2)abxn + bS(2)x(S(2)n))*(S(5)/2), x), x, a(a + bxn)/(S(4)b*S(2)n(aS(2) + S(2)abxn + bS(2)x(S(2)n))*(S(5)/2)) - S(1)/(S(3)b*S(2)n(aS(2) + S(2)abxn + bS(2)x(S(2)n))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)n + S(-1))/(aS(2) + S(2)abxn + bS(2)x(S(2)n))*(S(7)/2), x), x, a(a + bxn)/(S(6)b*S(2)n(aS(2) + S(2)abxn + bS(2)x(S(2)n))*(S(7)/2)) - S(1)/(S(5)b*S(2)n(aS(2) + S(2)abxn + bS(2)x(S(2)n))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*msqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n)), x), x, an(dx)(m + S(1))sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(d(a + bx*n)(m + S(1))(m + n + S(1))) + (dx)*(m + S(1))sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(d(m + n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n)), x), x, anx*S(3)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/((a + bxn)(S(3)n + S(9))) + xS(3)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(n + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n)), x), x, anx*S(2)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/((a + bxn)(S(2)n + S(4))) + xS(2)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(n + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n)), x), x, anxsqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/((a + bxn)(n + S(1))) + xsqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/x, x), x, asqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))log(x)/(a + bxn) + sqrt(aS(2) + S(2)abxn + bS(2)x*(S(2)n))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/x*S(2), x), x, ansqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/(x(a + bx*n)(-n + S(1))) - sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(x(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/x*S(3), x), x, ansqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/(x*S(2)(a + bxn)(-S(2)n + S(4))) - sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/(x*S(2)(-n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m(a*S(2) + S(2)abxn + bS(2)x(S(2)n))(S(3)/2), x), x, aS(3)(dx)*(m + S(1))sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(d(a + bx*n)(m + S(1))) + S(3)aS(2)b*S(2)x*(n + S(1))(dx)msqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/((ab + bS(2)x*n)(m + n + S(1))) + S(3)ab*S(3)x*(S(2)n + S(1))(dx)*msqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/((ab + bS(2)x*n)(m + S(2)n + S(1))) + bS(4)x*(S(3)n + S(1))(dx)*msqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/((ab + bS(2)x*n)(m + S(3)n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a*S(2) + S(2)abxn + bS(2)x(S(2)n))(S(3)/2), x), x, aS(3)xS(3)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(S(3)a + S(3)bxn) + S(3)a*S(2)b*S(2)x*(n + S(3))sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/((n + S(3))(ab + b*S(2)x*n)) + S(3)abS(3)x*(S(2)n + S(3))sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/((S(2)n + S(3))(ab + bS(2)xn)) + bS(4)x(S(3)n + S(3))sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/((S(3)n + S(3))(ab + bS(2)x*n)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a*S(2) + S(2)abxn + bS(2)x(S(2)n))(S(3)/2), x), x, aS(3)xS(2)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(S(2)a + S(2)bxn) + S(3)a*S(2)b*S(2)x*(n + S(2))sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/((n + S(2))(ab + b*S(2)x*n)) + S(3)abS(3)x*(S(2)n + S(2))sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/((S(2)n + S(2))(ab + bS(2)xn)) + bS(4)x(S(3)n + S(2))sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/((S(3)n + S(2))(ab + bS(2)xn)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxn + bS(2)x*(S(2)n))*(S(3)/2), x), x, S(6)a*S(3)n*S(3)xsqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/((a + bxn)(S(6)nS(3) + S(11)n*S(2) + S(6)n + S(1))) + S(6)aS(2)n*S(2)xsqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/(S(6)nS(3) + S(11)n*S(2) + S(6)n + S(1)) + S(3)nx(aS(2) + abxn)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(S(6)nS(2) + S(5)n + S(1)) + x(aS(2) + S(2)abxn + bS(2)x(S(2)n))*(S(3)/2)/(S(3)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxn + bS(2)x(S(2)n))(S(3)/2)/x, x), x, aS(3)sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))log(x)/(a + bxn) + aS(2)sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/n + (a*S(2) + abxn)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(S(2)n) + (aS(2) + S(2)abxn + bS(2)x(S(2)n))*(S(3)/2)/(S(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxn + bS(2)x(S(2)n))(S(3)/2)/xS(2), x), x, -a*S(3)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(x(a + bx*n)) - S(3)a*S(2)b*S(2)x*(n + S(-1))sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/((-n + S(1))(ab + b*S(2)x*n)) - S(3)abS(3)x*(S(2)n + S(-1))sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/((-S(2)n + S(1))(ab + bS(2)xn)) - bS(4)x(S(3)n + S(-1))sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/((-S(3)n + S(1))(ab + bS(2)xn)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxn + bS(2)x*(S(2)n))(S(3)/2)/xS(3), x), x, -a*S(3)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/(S(2)xS(2)(a + bxn)) - S(3)a*S(2)b*S(2)x*(n + S(-2))sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))/((-n + S(2))(ab + b*S(2)x*n)) - S(3)abS(3)x*(S(2)n + S(-2))sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/((-S(2)n + S(2))(ab + bS(2)xn)) - bS(4)x(S(3)n + S(-2))sqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))/((-S(3)n + S(2))(ab + bS(2)x*n)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/sqrt(aS(2) + S(2)abxn + bS(2)x*(S(2)n)), x), x, (dx)(m + S(1))(a + bxn)hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -bxn/a)/(ad(m + S(1))sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(aS(2) + S(2)abxn + bS(2)x*(S(2)n)), x), x, x*S(3)(a + bxn)hyper((S(1), S(3)/n), ((n + S(3))/n,), -bxn/a)/(S(3)asqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n)), x), x, x*S(2)(a + bxn)hyper((S(1), S(2)/n), ((n + S(2))/n,), -bxn/a)/(S(2)asqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n)), x), x, x(a + bx*n)hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -bxn/a)/(asqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))), x), x, (a + bxn)log(x)/(asqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))) - (a + bxn)log(a + bxn)/(ansqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))), x), x, -(a + bxn)hyper((S(1), -S(1)/n), (-(-n + S(1))/n,), -bxn/a)/(axsqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))), x), x, -(a + bxn)hyper((S(1), -S(2)/n), (-(-n + S(2))/n,), -bxn/a)/(S(2)axS(2)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/(aS(2) + S(2)abxn + bS(2)x(S(2)n))*(S(3)/2), x), x, (dx)*(m + S(1))(a + bxn)hyper((S(3), (m + S(1))/n), ((m + n + S(1))/n,), -bxn/a)/(aS(3)d(m + S(1))sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(aS(2) + S(2)abxn + bS(2)x*(S(2)n))(S(3)/2), x), x, xS(3)(a + bx*n)hyper((S(3), S(3)/n), ((n + S(3))/n,), -bxn/a)/(S(3)a*S(3)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a*S(2) + S(2)abxn + bS(2)x(S(2)n))(S(3)/2), x), x, xS(2)(a + bx*n)hyper((S(3), S(2)/n), ((n + S(2))/n,), -bxn/a)/(S(2)a*S(3)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abxn + bS(2)x(S(2)n))*(S(-3)/2), x), x, x(a + bxn)hyper((S(3), S(1)/n), (S(1) + S(1)/n,), -bxn/a)/(aS(3)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(aS(2) + S(2)abxn + bS(2)x(S(2)n))*(S(3)/2)), x), x, (a + bx*n)/(S(2)an(a*S(2) + S(2)abxn + bS(2)x(S(2)n))(S(3)/2)) + S(1)/(aS(2)nsqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))) + (a + bxn)log(x)/(a*S(3)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))) - (a + bxn)log(a + bxn)/(aS(3)nsqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(a*S(2) + S(2)abxn + bS(2)x(S(2)n))*(S(3)/2)), x), x, -(a + bx*n)hyper((S(3), -S(1)/n), (-(-n + S(1))/n,), -bxn/a)/(aS(3)xsqrt(aS(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)(a*S(2) + S(2)abxn + bS(2)x(S(2)n))*(S(3)/2)), x), x, -(a + bx*n)hyper((S(3), -S(2)/n), (-(-n + S(2))/n,), -bxn/a)/(S(2)a*S(3)x*S(2)sqrt(a*S(2) + S(2)abxn + bS(2)x(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abx*(-S(1)/(S(2)p + S(1))) + b*S(2)x*(-S(2)/(S(2)p + S(1))))*p, x), x, x(a + bx(S(1)/(-S(2)p + S(-1))))(aS(2) + S(2)abx*(S(1)/(-S(2)p + S(-1))) + b*S(2)x*(-S(2)/(S(2)p + S(1))))p/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxn + bS(2)x*(S(2)n))*((-n + S(-1))/(S(2)n)), x), x, x(a + bx*n)(a*S(2) + S(2)abxn + bS(2)x(S(2)n))*(-(n + S(1))/(S(2)n))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(2) + S(2)abx*(-S(1)/(S(2)p + S(2))) + b*S(2)x(-S(1)/(p + S(1))))p, x), x, x(a + bx*(-S(1)/(S(2)p + S(2))))(S(2)p + S(2))(aS(2) + S(2)abx*(-S(1)/(S(2)p + S(2))) + b*S(2)x(-S(1)/(p + S(1))))p/(a(S(2)p + S(1))) - x(aS(2) + S(2)abx*(-S(1)/(S(2)p + S(2))) + b*S(2)x(-S(1)/(p + S(1))))(p + S(1))/(a*S(2)(S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2) + S(2)abxn + bS(2)x(S(2)n))*((-S(2)n + S(-1))/(S(2)n)), x), x, x(a + bxn)(a*S(2) + S(2)abxn + bS(2)x(S(2)n))*(S(-1) - S(1)/(S(2)n))/(a(n + S(1))) + nx(aS(2) + S(2)abxn + bS(2)x(S(2)n))*(-S(1)/(S(2)n))/(a*S(2)(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(4)n + S(-1))/(a + bxn + cx*(S(2)n)), x), x, -bxn/(cS(2)n) + b(-S(3)ac + bS(2))atanh((b + S(2)cx*n)/sqrt(-S(4)ac + bS(2)))/(cS(3)nsqrt(-S(4)ac + bS(2))) + x(S(2)n)/(S(2)cn) + (-ac + bS(2))log(a + bxn + cx*(S(2)n))/(S(2)cS(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(3)n + S(-1))/(a + bxn + cx*(S(2)n)), x), x, -blog(a + bx*n + cx*(S(2)n))/(S(2)cS(2)n) + x*n/(cn) - (-S(2)ac + b*S(2))atanh((b + S(2)cx*n)/sqrt(-S(4)ac + bS(2)))/(cS(2)nsqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)n + S(-1))/(a + bxn + cx*(S(2)n)), x), x, batanh((b + S(2)cxn)/sqrt(-S(4)ac + bS(2)))/(cnsqrt(-S(4)ac + bS(2))) + log(a + bx*n + cx*(S(2)n))/(S(2)cn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(n + S(-1))/(a + bx*n + cx*(S(2)n)), x), x, -S(2)atanh((b + S(2)cxn)/sqrt(-S(4)ac + bS(2)))/(nsqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-n + S(-1))/(a + bxn + cx*(S(2)n)), x), x, -x*(-n)/(an) - blog(x)/aS(2) + blog(a + bxn + cx*(S(2)n))/(S(2)aS(2)n) - (-S(2)ac + b*S(2))atanh((b + S(2)cx*n)/sqrt(-S(4)ac + bS(2)))/(aS(2)nsqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-S(2)n + S(-1))/(a + bxn + cx*(S(2)n)), x), x, -x*(-S(2)n)/(S(2)an) + bx(-n)/(aS(2)n) + b(-S(3)ac + bS(2))atanh((b + S(2)cx*n)/sqrt(-S(4)ac + bS(2)))/(aS(3)nsqrt(-S(4)ac + bS(2))) + (-ac + b*S(2))log(x)/a*S(3) - (-ac + b*S(2))log(a + bxn + cx*(S(2)n))/(S(2)aS(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(-S(3)n + S(-1))/(a + bxn + cx*(S(2)n)), x), x, -x*(-S(3)n)/(S(3)an) + bx(-S(2)n)/(S(2)aS(2)n) - x*(-n)(-ac + bS(2))/(aS(3)n) - b(-S(2)ac + bS(2))log(x)/a*S(4) + b(-S(2)ac + b*S(2))log(a + bxn + cx*(S(2)n))/(S(2)aS(4)n) - (S(2)aS(2)c*S(2) - S(4)abS(2)c + b*S(4))atanh((b + S(2)cx*n)/sqrt(-S(4)ac + bS(2)))/(aS(4)nsqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n/S(4) + S(-1))/(a + bx*n + cx*(S(2)n)), x), x, -S(2)S(2)(S(3)/4)c*(S(3)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)x*(n/S(4))/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(n(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))) - S(2)S(2)*(S(3)/4)c*(S(3)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)x*(n/S(4))/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(n(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))) + S(2)S(2)*(S(3)/4)c*(S(3)/4)atan(S(2)*(S(1)/4)c*(S(1)/4)x*(n/S(4))/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(n(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))) + S(2)S(2)*(S(3)/4)c*(S(3)/4)atanh(S(2)*(S(1)/4)c*(S(1)/4)x*(n/S(4))/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(n(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n/S(3) + S(-1))/(a + bx*n + cx*(S(2)n)), x), x, -S(2)*(S(2)/3)c*(S(2)/3)log(S(2)*(S(1)/3)c*(S(1)/3)x*(n/S(3)) + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(n(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(2)/3)c*(S(2)/3)log(S(2)*(S(2)/3)c*(S(2)/3)x*(S(2)n/S(3)) - S(2)*(S(1)/3)c*(S(1)/3)x*(n/S(3))(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(2)n(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(2)/3)sqrt(S(3))c(S(2)/3)atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x*(n/S(3))/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(n(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)sqrt(-S(4)ac + bS(2))) + S(2)(S(2)/3)c*(S(2)/3)log(S(2)*(S(1)/3)c*(S(1)/3)x*(n/S(3)) + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(n(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)sqrt(-S(4)ac + bS(2))) - S(2)(S(2)/3)c*(S(2)/3)log(S(2)*(S(2)/3)c*(S(2)/3)x*(S(2)n/S(3)) - S(2)*(S(1)/3)c*(S(1)/3)x*(n/S(3))(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(2)n(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)sqrt(-S(4)ac + bS(2))) - S(2)(S(2)/3)sqrt(S(3))c(S(2)/3)atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x*(n/S(3))/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(n(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n/S(2) + S(-1))/(a + bx*n + cx*(S(2)n)), x), x, -S(2)sqrt(S(2))sqrt(c)atan(sqrt(S(2))sqrt(c)x(n/S(2))/sqrt(b + sqrt(-S(4)ac + bS(2))))/(nsqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))) + S(2)sqrt(S(2))sqrt(c)atan(sqrt(S(2))sqrt(c)x*(n/S(2))/sqrt(b - sqrt(-S(4)ac + bS(2))))/(nsqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-n/S(2) + S(-1))/(a + bxn + cx*(S(2)n)), x), x, -S(2)x(-n/S(2))/(an) + sqrt(S(2))(b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(a)x*(-n/S(2))/sqrt(b - sqrt(-S(4)ac + bS(2))))/(a(S(3)/2)nsqrt(b - sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(a)x*(-n/S(2))/sqrt(b + sqrt(-S(4)ac + bS(2))))/(a(S(3)/2)nsqrt(b + sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-n/S(3) + S(-1))/(a + bx*n + cx*(S(2)n)), x), x, -S(3)x(-n/S(3))/(an) + S(2)*(S(2)/3)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)a*(S(1)/3)x*(-n/S(3)) + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(2)a*(S(4)/3)n(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)(S(2)/3)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)a*(S(2)/3)x*(-S(2)n/S(3)) - S(2)*(S(1)/3)a*(S(1)/3)x*(-n/S(3))(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(4)a(S(4)/3)n(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)(S(2)/3)sqrt(S(3))(b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)a*(S(1)/3)x*(-n/S(3))/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(2)a*(S(4)/3)n(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)(S(2)/3)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)a*(S(1)/3)x*(-n/S(3)) + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(2)a*(S(4)/3)n(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)(S(2)/3)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)a*(S(2)/3)x*(-S(2)n/S(3)) - S(2)*(S(1)/3)a*(S(1)/3)x*(-n/S(3))(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(4)a(S(4)/3)n(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)(S(2)/3)sqrt(S(3))(b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)a*(S(1)/3)x*(-n/S(3))/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(2)a*(S(4)/3)n(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-n/S(4) + S(-1))/(a + bx*n + cx*(S(2)n)), x), x, -S(4)x(-n/S(4))/(an) - S(2)*(S(3)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)a*(S(1)/4)x*(-n/S(4))/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(a(S(5)/4)n(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) - S(2)(S(3)/4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)a*(S(1)/4)x*(-n/S(4))/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(a(S(5)/4)n(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) - S(2)(S(3)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)a*(S(1)/4)x*(-n/S(4))/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(a(S(5)/4)n(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) - S(2)(S(3)/4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)a*(S(1)/4)x*(-n/S(4))/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(a(S(5)/4)n(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bx*n + cx*(S(2)n)), x), x, -S(2)cx*S(3)hyper((S(1), S(3)/n), ((n + S(3))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(-S(12)ac + S(3)b*S(2) + S(3)bsqrt(-S(4)ac + bS(2))) - S(2)cxS(3)hyper((S(1), S(3)/n), ((n + S(3))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(-S(12)ac + S(3)b*S(2) - S(3)bsqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bx*n + cx*(S(2)n)), x), x, -cxS(2)hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - cx*S(2)hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bx*n + cx*(S(2)n)), x), x, -S(2)cxhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - S(2)cxhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bxn + cx*(S(2)n))), x), x, batanh((b + S(2)cxn)/sqrt(-S(4)ac + bS(2)))/(ansqrt(-S(4)ac + bS(2))) + log(x)/a - log(a + bx*n + cx*(S(2)n))/(S(2)an), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(a + bxn + cx*(S(2)n))), x), x, S(2)chyper((S(1), -S(1)/n), (-(-n + S(1))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(x(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) + S(2)chyper((S(1), -S(1)/n), (-(-n + S(1))/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(x(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a + bx*n + cx*(S(2)n))), x), x, chyper((S(1), -S(2)/n), (-(-n + S(2))/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(xS(2)(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) + chyper((S(1), -S(2)/n), (-(-n + S(2))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(xS(2)(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(a + bx*n + cx*(S(2)n)), x), x, x*S(4)sqrt(a + bxn + cx*(S(2)n))AppellF1(S(4)/n, S(-1)/2, S(-1)/2, (n + S(4))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(a + bxn + cx*(S(2)n)), x), x, x*S(3)sqrt(a + bxn + cx*(S(2)n))AppellF1(S(3)/n, S(-1)/2, S(-1)/2, (n + S(3))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(3)sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(a + bxn + cx*(S(2)n)), x), x, x*S(2)sqrt(a + bxn + cx*(S(2)n))AppellF1(S(2)/n, S(-1)/2, S(-1)/2, (n + S(2))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(2)sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*n + cx*(S(2)n)), x), x, xsqrt(a + bx*n + cx*(S(2)n))AppellF1(S(1)/n, S(-1)/2, S(-1)/2, S(1) + S(1)/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(sqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*n + cx*(S(2)n))/x*S(2), x), x, -sqrt(a + bx*n + cx*(S(2)n))AppellF1(-S(1)/n, S(-1)/2, S(-1)/2, -(-n + S(1))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(xsqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*n + cx*(S(2)n))/x*S(3), x), x, -sqrt(a + bx*n + cx*(S(2)n))AppellF1(-S(2)/n, S(-1)/2, S(-1)/2, -(-n + S(2))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(2)x*S(2)sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a + bxn + cx*(S(2)n))*(S(3)/2), x), x, ax*S(4)sqrt(a + bxn + cx*(S(2)n))AppellF1(S(4)/n, S(-3)/2, S(-3)/2, (n + S(4))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bxn + cx*(S(2)n))*(S(3)/2), x), x, ax*S(3)sqrt(a + bxn + cx*(S(2)n))AppellF1(S(3)/n, S(-3)/2, S(-3)/2, (n + S(3))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(3)sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxn + cx*(S(2)n))*(S(3)/2), x), x, ax*S(2)sqrt(a + bxn + cx*(S(2)n))AppellF1(S(2)/n, S(-3)/2, S(-3)/2, (n + S(2))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(2)sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*n + cx*(S(2)n))*(S(3)/2), x), x, axsqrt(a + bx*n + cx*(S(2)n))AppellF1(S(1)/n, S(-3)/2, S(-3)/2, S(1) + S(1)/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(sqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*n + cx*(S(2)n))(S(3)/2)/xS(2), x), x, -asqrt(a + bx*n + cx*(S(2)n))AppellF1(-S(1)/n, S(-3)/2, S(-3)/2, -(-n + S(1))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(xsqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*n + cx*(S(2)n))(S(3)/2)/xS(3), x), x, -asqrt(a + bx*n + cx*(S(2)n))AppellF1(-S(2)/n, S(-3)/2, S(-3)/2, -(-n + S(2))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(2)x*S(2)sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(a + bx*n + cx*(S(2)n)), x), x, x*S(4)sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(4)/n, S(1)/2, S(1)/2, (n + S(4))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/sqrt(a + bx*n + cx*(S(2)n)), x), x, x*S(3)sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(3)/n, S(1)/2, S(1)/2, (n + S(3))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(3)sqrt(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + bxn + cx*(S(2)n)), x), x, x*S(2)sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(2)/n, S(1)/2, S(1)/2, (n + S(2))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(2)sqrt(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + bxn + cx*(S(2)n)), x), x, xsqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/n, S(1)/2, S(1)/2, S(1) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/sqrt(a + bx*n + cx*(S(2)n)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(a + bx*n + cx*(S(2)n))(S(3)/2), x), x, xS(3)sqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(3)/n, S(3)/2, S(3)/2, (n + S(3))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(3)asqrt(a + bx*n + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bxn + cx*(S(2)n))(S(3)/2), x), x, xS(2)sqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(2)/n, S(3)/2, S(3)/2, (n + S(2))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(2)asqrt(a + bx*n + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxn + cx*(S(2)n))*(S(-3)/2), x), x, xsqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/n, S(3)/2, S(3)/2, S(1) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(asqrt(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bx*n + cx*(S(2)n))*(S(3)/2)), x), x, (-S(4)ac + S(2)b*S(2) + S(2)bcx*n)/(an(-S(4)ac + bS(2))sqrt(a + bxn + cx*(S(2)n))) - atanh((S(2)a + bx*n)/(S(2)sqrt(a)sqrt(a + bx*n + cx*(S(2)n))))/(a*(S(3)/2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(a + bxn + cx*(S(2)n))*(S(3)/2)), x), x, -sqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(-S(1)/n, S(3)/2, S(3)/2, -(-n + S(1))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(axsqrt(a + bx*n + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)(a + bxn + cx*(S(2)n))*(S(3)/2)), x), x, -sqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(-S(2)/n, S(3)/2, S(3)/2, -(-n + S(2))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(2)axS(2)sqrt(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)sqrt(a + bxn + cx*(S(2)n))), x), x, -sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(-S(1)/n, S(1)/2, S(1)/2, -(-n + S(1))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(xsqrt(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)sqrt(a + bxn + cx*(S(2)n))), x), x, -sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(-S(2)/n, S(1)/2, S(1)/2, -(-n + S(2))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(2)x*S(2)sqrt(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/(a + bx*n + cx*(S(2)n))(S(3)/2), x), x, xS(4)sqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(4)/n, S(3)/2, S(3)/2, (n + S(4))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(4)asqrt(a + bx*n + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m(a + bxn + cx*(S(2)n))*(S(3)/2), x), x, a(dx)(m + S(1))sqrt(a + bxn + cx*(S(2)n))AppellF1((m + S(1))/n, S(-3)/2, S(-3)/2, (m + n + S(1))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(d(m + S(1))sqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*msqrt(a + bxn + cx*(S(2)n)), x), x, (dx)(m + S(1))sqrt(a + bxn + cx*(S(2)n))AppellF1((m + S(1))/n, S(-1)/2, S(-1)/2, (m + n + S(1))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(d(m + S(1))sqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m/sqrt(a + bx*n + cx*(S(2)n)), x), x, (dx)(m + S(1))sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1((m + S(1))/n, S(1)/2, S(1)/2, (m + n + S(1))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(d(m + S(1))sqrt(a + bx*n + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/(a + bx*n + cx*(S(2)n))*(S(3)/2), x), x, (dx)*(m + S(1))sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1((m + S(1))/n, S(3)/2, S(3)/2, (m + n + S(1))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(ad(m + S(1))sqrt(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m(a + bxn + cx*(S(2)n))*p, x), x, (dx)*(m + S(1))(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1((m + S(1))/n, -p, -p, (m + n + S(1))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(d(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + b(d + ex)*S(3) + c(d + ex)S(6)), x), x, -d(d + ex)sqrt(S(2)c(d + ex)S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)c(d + ex)S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/3, S(1)/2, S(1)/2, S(4)/3, -S(2)c(d + ex)S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)c(d + ex)*S(3)/(b + sqrt(-S(4)ac + bS(2))))/(eS(2)sqrt(a + b(d + ex)*S(3) + c(d + ex)S(6))) + (d + ex)*S(2)sqrt(S(2)c(d + ex)S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)c(d + ex)S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(2)/3, S(1)/2, S(1)/2, S(5)/3, -S(2)c(d + ex)S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)c(d + ex)*S(3)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)e*S(2)sqrt(a + b(d + ex)*S(3) + c(d + ex)S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(a + b(d + ex)S(3) + c(d + ex)S(6)), x), x, dS(2)(d + ex)sqrt(S(2)c(d + ex)S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)c(d + ex)S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/3, S(1)/2, S(1)/2, S(4)/3, -S(2)c(d + ex)S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)c(d + ex)*S(3)/(b + sqrt(-S(4)ac + bS(2))))/(eS(3)sqrt(a + b(d + ex)*S(3) + c(d + ex)S(6))) - d(d + ex)S(2)sqrt(S(2)c(d + ex)S(3)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)c(d + ex)S(3)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(2)/3, S(1)/2, S(1)/2, S(5)/3, -S(2)c(d + ex)S(3)/(b - sqrt(-S(4)ac + bS(2))), -S(2)c(d + ex)*S(3)/(b + sqrt(-S(4)ac + bS(2))))/(eS(3)sqrt(a + b(d + ex)*S(3) + c(d + ex)S(6))) + atanh((b + S(2)c(d + ex)*S(3))/(S(2)sqrt(c)sqrt(a + b(d + ex)S(3) + c(d + ex)S(6))))/(S(3)sqrt(c)eS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*n + cx*(S(2)n))/x, x), x, -sqrt(a)atanh((S(2)a + bxn)/(S(2)sqrt(a)sqrt(a + bx*n + cx*(S(2)n))))/n + batanh((b + S(2)cxn)/(S(2)sqrt(c)sqrt(a + bx*n + cx*(S(2)n))))/(S(2)sqrt(c)n) + sqrt(a + bxn + cx*(S(2)n))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxn + cx*(S(2)n))(S(3)/2)/x, x), x, -a(S(3)/2)atanh((S(2)a + bxn)/(S(2)sqrt(a)sqrt(a + bx*n + cx*(S(2)n))))/n - b(-S(12)ac + bS(2))atanh((b + S(2)cx*n)/(S(2)sqrt(c)sqrt(a + bx*n + cx*(S(2)n))))/(S(16)c(S(3)/2)n) + (a + bxn + cx*(S(2)n))*(S(3)/2)/(S(3)n) + sqrt(a + bxn + cx*(S(2)n))(S(8)ac + bS(2) + S(2)bcx*n)/(S(8)cn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + bxn + cx*(S(2)n))), x), x, -atanh((S(2)a + bx*n)/(S(2)sqrt(a)sqrt(a + bx*n + cx*(S(2)n))))/(sqrt(a)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m(a + bxn + cx*(S(2)n))S(3), x), x, aS(3)(dx)*(m + S(1))/(d(m + S(1))) + S(3)aS(2)bx(n + S(1))(dx)m/(m + n + S(1)) + S(3)ax(S(2)n + S(1))(dx)*m(ac + bS(2))/(m + S(2)n + S(1)) + S(3)bc*S(2)x*(S(5)n + S(1))(dx)*m/(m + S(5)n + S(1)) + bx(S(3)n + S(1))(dx)*m(S(6)ac + b*S(2))/(m + S(3)n + S(1)) + c*S(3)x*(S(6)n + S(1))(dx)*m/(m + S(6)n + S(1)) + S(3)cx*(S(4)n + S(1))(dx)*m(ac + bS(2))/(m + S(4)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m(a + bxn + cx*(S(2)n))S(2), x), x, aS(2)(dx)*(m + S(1))/(d(m + S(1))) + S(2)abx(n + S(1))(dx)m/(m + n + S(1)) + S(2)bcx*(S(3)n + S(1))(dx)*m/(m + S(3)n + S(1)) + c*S(2)x*(S(4)n + S(1))(dx)*m/(m + S(4)n + S(1)) + x*(S(2)n + S(1))(dx)*m(S(2)ac + b*S(2))/(m + S(2)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m(a + bxn + cx*(S(2)n)), x), x, a(dx)*(m + S(1))/(d(m + S(1))) + bx(n + S(1))(dx)m/(m + n + S(1)) + cx*(S(2)n + S(1))(dx)*m/(m + S(2)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/(a + bx*n + cx*(S(2)n)), x), x, -S(2)c(dx)(m + S(1))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(d(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))sqrt(-S(4)ac + bS(2))) + S(2)c(dx)*(m + S(1))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(d(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*m/(a + bx*n + cx*(S(2)n))*S(2), x), x, -c(dx)(m + S(1))(S(4)ac(m - S(2)n + S(1)) - b*S(2)(m - n + S(1)) + bsqrt(-S(4)ac + bS(2))(m - n + S(1)))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(adn(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))(S(3)/2)) + c(dx)(m + S(1))(-b(m - n + S(1)) + (S(4)ac(m - S(2)n + S(1)) - bS(2)(m - n + S(1)))/sqrt(-S(4)ac + b*S(2)))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(adn(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))) + (dx)*(m + S(1))(-S(2)ac + b*S(2) + bcxn)/(adn(-S(4)ac + b*S(2))(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)m/(a + bx*n + cx*(S(2)n))*S(3), x), x, (dx)*(m + S(1))(-S(2)ac + b*S(2) + bcxn)/(S(2)adn(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))*S(2)) - c(dx)(m + S(1))(S(8)aS(2)c*S(2)(m*S(2) + m(-S(6)n + S(2)) + S(8)n*S(2) - S(6)n + S(1)) - S(6)ab*S(2)c(mS(2) + m(-S(4)n + S(2)) + S(3)n*S(2) - S(4)n + S(1)) + b*S(4)(m*S(2) + m(-S(3)n + S(2)) + S(2)n*S(2) - S(3)n + S(1)) + bsqrt(-S(4)ac + bS(2))(S(2)ac(S(2)m - S(7)n + S(2)) - bS(2)(m - S(2)n + S(1)))(m - n + S(1)))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)dnS(2)(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))(S(5)/2)) - c(dx)(m + S(1))(-S(8)aS(2)c*S(2)(m*S(2) + m(-S(6)n + S(2)) + S(8)n*S(2) - S(6)n + S(1)) + S(6)ab*S(2)c(mS(2) + m(-S(4)n + S(2)) + S(3)n*S(2) - S(4)n + S(1)) - b*S(4)(m*S(2) + m(-S(3)n + S(2)) + S(2)n*S(2) - S(3)n + S(1)) + bsqrt(-S(4)ac + bS(2))(S(2)ac(S(2)m - S(7)n + S(2)) - bS(2)(m - S(2)n + S(1)))(m - n + S(1)))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)dnS(2)(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))(S(5)/2)) - (dx)*(m + S(1))(S(4)aS(2)c*S(2)(m - S(4)n + S(1)) - S(5)abS(2)c(m - S(3)n + S(1)) + b*S(4)(m - S(2)n + S(1)) - bcxn(S(2)ac(S(2)m - S(7)n + S(2)) - bS(2)(m - S(2)n + S(1))))/(S(2)a*S(2)dnS(2)(-S(4)ac + bS(2))S(2)(a + bx*n + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)S(3)(a + b(d + ex)*S(2) + c(d + ex)S(4)), x), x, a(d + ex)S(4)/(S(4)e) + b(d + ex)*S(6)/(S(6)e) + c(d + ex)*S(8)/(S(8)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)S(3)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2), x), x, aS(2)(d + ex)S(4)/(S(4)e) + ab(d + ex)S(6)/(S(3)e) + bc(d + ex)S(10)/(S(5)e) + c*S(2)(d + ex)S(12)/(S(12)e) + (d + ex)S(8)(S(2)ac + b*S(2))/(S(8)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)S(3)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(3), x), x, aS(3)(d + ex)S(4)/(S(4)e) + a*S(2)b(d + ex)*S(6)/(S(2)e) + S(3)a(d + ex)S(8)(ac + bS(2))/(S(8)e) + S(3)bc*S(2)(d + ex)S(14)/(S(14)e) + b(d + ex)*S(10)(S(6)ac + b*S(2))/(S(10)e) + c*S(3)(d + ex)S(16)/(S(16)e) + c(d + ex)*S(12)(ac + bS(2))/(S(4)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)S(3)(a + b(d + ex)*S(2) + c(d + ex)S(4)), x), x, af*S(3)(d + ex)S(4)/(S(4)e) + bfS(3)(d + ex)S(6)/(S(6)e) + cfS(3)(d + ex)S(8)/(S(8)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)S(3)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2), x), x, aS(2)f*S(3)(d + ex)S(4)/(S(4)e) + abf*S(3)(d + ex)S(6)/(S(3)e) + bcf*S(3)(d + ex)S(10)/(S(5)e) + c*S(2)f*S(3)(d + ex)S(12)/(S(12)e) + f*S(3)(d + ex)S(8)(S(2)ac + b*S(2))/(S(8)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)S(3)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(3), x), x, aS(3)f*S(3)(d + ex)S(4)/(S(4)e) + a*S(2)bfS(3)(d + ex)S(6)/(S(2)e) + S(3)af*S(3)(d + ex)S(8)(ac + bS(2))/(S(8)e) + S(3)bc*S(2)f*S(3)(d + ex)S(14)/(S(14)e) + bfS(3)(d + ex)S(10)(S(6)ac + b*S(2))/(S(10)e) + c*S(3)f*S(3)(d + ex)S(16)/(S(16)e) + cfS(3)(d + ex)S(12)(ac + bS(2))/(S(4)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)S(4)/(a + b(d + ex)S(2) + c(d + ex)S(4)), x), x, x/c - sqrt(S(2))(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(2)c*(S(3)/2)esqrt(b - sqrt(-S(4)ac + bS(2)))) - sqrt(S(2))(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(2)c*(S(3)/2)esqrt(b + sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*S(3)/(a + b(d + ex)S(2) + c(d + ex)S(4)), x), x, batanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)cesqrt(-S(4)ac + b*S(2))) + log(a + b(d + ex)S(2) + c(d + ex)S(4))/(S(4)ce), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*S(2)/(a + b(d + ex)S(2) + c(d + ex)S(4)), x), x, -sqrt(S(2))sqrt(b - sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(2)sqrt(c)esqrt(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(b + sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(2)sqrt(c)esqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(a + b(d + ex)*S(2) + c(d + ex)S(4)), x), x, -atanh((b + S(2)c(d + ex)*S(2))/sqrt(-S(4)ac + bS(2)))/(esqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)(a + b(d + ex)S(2) + c(d + ex)S(4))), x), x, batanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)aesqrt(-S(4)ac + b*S(2))) + log(d + ex)/(ae) - log(a + b(d + ex)S(2) + c(d + ex)S(4))/(S(4)ae), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)*S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))), x), x, -sqrt(S(2))sqrt(c)(-b/sqrt(-S(4)ac + bS(2)) + S(1))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(2)aesqrt(b + sqrt(-S(4)ac + b*S(2)))) - sqrt(S(2))sqrt(c)(b/sqrt(-S(4)ac + bS(2)) + S(1))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(2)aesqrt(b - sqrt(-S(4)ac + b*S(2)))) - S(1)/(ae(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)S(3)(a + b(d + ex)*S(2) + c(d + ex)S(4))), x), x, -S(1)/(S(2)ae(d + ex)S(2)) - blog(d + ex)/(aS(2)e) + blog(a + b(d + ex)S(2) + c(d + ex)S(4))/(S(4)a*S(2)e) - (-S(2)ac + b*S(2))atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)esqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)*S(4)(a + b(d + ex)*S(2) + c(d + ex)S(4))), x), x, -S(1)/(S(3)ae(d + ex)S(3)) + b/(aS(2)e(d + ex)) + sqrt(S(2))sqrt(c)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)esqrt(b + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))sqrt(c)(b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)esqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*S(4)/(a + b(d + ex)S(2) + c(d + ex)S(4))S(2), x), x, (S(2)a + b(d + ex)*S(2))(d + ex)/(e(-S(8)ac + S(2)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))) + sqrt(S(2))(b - (S(4)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(c)esqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))(S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(c)esqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)S(3)/(a + b(d + ex)S(2) + c(d + ex)S(4))S(2), x), x, -batanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(e(-S(4)ac + bS(2))(S(3)/2)) + (S(2)a + b(d + ex)S(2))/(e(-S(8)ac + S(2)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*S(2)/(a + b(d + ex)S(2) + c(d + ex)S(4))S(2), x), x, -sqrt(S(2))sqrt(c)(S(2)b + sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(2)esqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))sqrt(c)(S(2)b - sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(2)esqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) - (b + S(2)c(d + ex)S(2))(d + ex)/(e(-S(8)ac + S(2)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2), x), x, S(2)catanh((b + S(2)c(d + ex)*S(2))/sqrt(-S(4)ac + bS(2)))/(e(-S(4)ac + bS(2))(S(3)/2)) + (-b - S(2)c(d + ex)S(2))/(e(-S(8)ac + S(2)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(d + ex)S(2) + c(d + ex)S(4))(S(-2)), x), x, -sqrt(S(2))sqrt(c)(-S(12)ac + bS(2) - bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(4)aesqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))sqrt(c)(-S(12)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(4)aesqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) + (d + ex)(-S(2)ac + b*S(2) + bc(d + ex)*S(2))/(S(2)ae(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)(a + b(d + ex)S(2) + c(d + ex)S(4))S(2)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(2)ae(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))) + b(-S(6)ac + b*S(2))atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)e(-S(4)ac + bS(2))(S(3)/2)) + log(d + ex)/(a*S(2)e) - log(a + b(d + ex)*S(2) + c(d + ex)S(4))/(S(4)a*S(2)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(2)ae(d + ex)(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))) + sqrt(S(2))sqrt(c)(-S(16)abc + S(3)bS(3) - (-S(10)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(4)a*S(2)esqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) - sqrt(S(2))sqrt(c)(-S(16)abc + S(3)b*S(3) + (-S(10)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(4)a*S(2)esqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) - (-S(10)ac + S(3)bS(2))/(S(2)a*S(2)e(d + ex)(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)*S(3)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(2)ae(d + ex)S(2)(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))) - (-S(3)ac + bS(2))/(aS(2)e(d + ex)*S(2)(-S(4)ac + b*S(2))) - S(2)blog(d + ex)/(a*S(3)e) + blog(a + b(d + ex)S(2) + c(d + ex)S(4))/(S(2)a*S(3)e) - (S(6)aS(2)c*S(2) - S(6)abS(2)c + b*S(4))atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(aS(3)e(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)*S(4)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(2)ae(d + ex)S(3)(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))) - (-S(14)ac + S(5)b*S(2))/(S(6)a*S(2)e(d + ex)*S(3)(-S(4)ac + b*S(2))) + b(-S(19)ac + S(5)bS(2))/(S(2)a*S(3)e(d + ex)(-S(4)ac + bS(2))) - sqrt(S(2))sqrt(c)(S(28)a*S(2)c*S(2) - S(29)abS(2)c + S(5)bS(4) - b(-S(19)ac + S(5)bS(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(4)a*S(3)esqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))sqrt(c)(S(28)aS(2)c*S(2) - S(29)abS(2)c + S(5)bS(4) + b(-S(19)ac + S(5)bS(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(4)a*S(3)esqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)S(4)/(a + b(d + ex)S(2) + c(d + ex)S(4))S(3), x), x, -S(3)sqrt(S(2))sqrt(c)(S(4)ac + S(3)bS(2) + S(2)bsqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(8)esqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(5)/2)) + S(3)sqrt(S(2))sqrt(c)(S(4)ac + S(3)b*S(2) - S(2)bsqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(8)esqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(5)/2)) + (S(2)a + b(d + ex)S(2))(d + ex)/(e(-S(16)ac + S(4)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)) - (d + ex)(-S(4)ac + S(7)b*S(2) + S(12)bc(d + ex)S(2))/(S(8)e(-S(4)ac + bS(2))S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*S(3)/(a + b(d + ex)S(2) + c(d + ex)S(4))S(3), x), x, S(3)bcatanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(e(-S(4)ac + bS(2))(S(5)/2)) - S(3)b(b + S(2)c(d + ex)S(2))/(S(4)e(-S(4)ac + bS(2))S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))) + (S(2)a + b(d + ex)*S(2))/(e(-S(16)ac + S(4)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*S(2)/(a + b(d + ex)S(2) + c(d + ex)S(4))S(3), x), x, -(b + S(2)c(d + ex)*S(2))(d + ex)/(e(-S(16)ac + S(4)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)) + sqrt(S(2))sqrt(c)(S(20)ac + bS(2) - b(-S(52)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(16)aesqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + sqrt(S(2))sqrt(c)(S(20)ac + b*S(2) + b(-S(52)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(16)aesqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + (d + ex)(b(S(8)ac + bS(2)) + c(d + ex)S(2)(S(20)ac + b*S(2)))/(S(8)ae(-S(4)ac + bS(2))S(2)(a + b(d + ex)S(2) + c(d + ex)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(a + b(d + ex)*S(2) + c(d + ex)S(4))S(3), x), x, -S(6)c*S(2)atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(e(-S(4)ac + bS(2))(S(5)/2)) + S(3)c(b + S(2)c(d + ex)S(2))/(S(2)e(-S(4)ac + bS(2))S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))) + (-b - S(2)c(d + ex)*S(2))/(e(-S(16)ac + S(4)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(d + ex)S(2) + c(d + ex)S(4))(S(-3)), x), x, (d + ex)(-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(4)ae(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)) + S(3)sqrt(S(2))sqrt(c)(-S(8)abc + bS(3) - (S(56)a*S(2)c*S(2) - S(10)abS(2)c + b*S(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(16)a*S(2)esqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))S(2)) + S(3)sqrt(S(2))sqrt(c)(S(56)a*S(2)c*S(2) - S(10)abS(2)c + b*S(4) + b(-S(8)ac + b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(16)a*S(2)esqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(5)/2)) + (d + ex)(S(3)bc(d + ex)*S(2)(-S(8)ac + b*S(2)) + (-S(7)ac + bS(2))(-S(4)ac + S(3)bS(2)))/(S(8)a*S(2)e(-S(4)ac + bS(2))S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)(a + b(d + ex)S(2) + c(d + ex)S(4))S(3)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(4)ae(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)) + (S(16)a*S(2)c*S(2) - S(15)abS(2)c + S(2)bS(4) + S(2)bc(d + ex)S(2)(-S(7)ac + b*S(2)))/(S(4)a*S(2)e(-S(4)ac + bS(2))S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))) + b(S(30)aS(2)c*S(2) - S(10)abS(2)c + b*S(4))atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(3)e(-S(4)ac + bS(2))(S(5)/2)) + log(d + ex)/(a*S(3)e) - log(a + b(d + ex)*S(2) + c(d + ex)S(4))/(S(4)a*S(3)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(3)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(4)ae(d + ex)(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)) + (S(36)a*S(2)c*S(2) - S(35)abS(2)c + S(5)bS(4) + bc(d + ex)*S(2)(-S(32)ac + S(5)bS(2)))/(S(8)a*S(2)e(d + ex)(-S(4)ac + bS(2))S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))) - S(3)sqrt(S(2))sqrt(c)((-S(12)ac + S(5)bS(2))(-S(5)ac + b*S(2)) - (S(124)a*S(2)bcS(2) - S(47)abS(3)c + S(5)bS(5))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(16)a*S(3)esqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))S(2)) - S(3)sqrt(S(2))sqrt(c)(b(S(124)aS(2)c*S(2) - S(47)abS(2)c + S(5)bS(4))/sqrt(-S(4)ac + bS(2)) + (-S(12)ac + S(5)b*S(2))(-S(5)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(16)a*S(3)esqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))S(2)) - (-S(36)ac + S(15)bS(2))(-S(5)ac + b*S(2))/(S(8)a*S(3)e(d + ex)(-S(4)ac + bS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)*S(3)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(3)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(4)ae(d + ex)S(2)(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)) + (S(20)a*S(2)c*S(2) - S(20)abS(2)c + S(3)bS(4) + S(3)bc(d + ex)S(2)(-S(6)ac + b*S(2)))/(S(4)a*S(2)e(d + ex)*S(2)(-S(4)ac + bS(2))S(2)(a + b(d + ex)S(2) + c(d + ex)S(4))) - (S(30)a*S(2)c*S(2) - S(21)abS(2)c + S(3)bS(4))/(S(2)a*S(3)e(d + ex)*S(2)(-S(4)ac + bS(2))S(2)) - S(3)blog(d + ex)/(aS(4)e) + S(3)blog(a + b(d + ex)*S(2) + c(d + ex)S(4))/(S(4)a*S(4)e) - (-S(60)aS(3)c*S(3) + S(90)a*S(2)b*S(2)c*S(2) - S(30)abS(4)c + S(3)bS(6))atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(4)e(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)*S(4)/(a + b(d + ex)S(2) + c(d + ex)S(4)), x), x, fS(4)x/c - sqrt(S(2))fS(4)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(2)c*(S(3)/2)esqrt(b - sqrt(-S(4)ac + bS(2)))) - sqrt(S(2))f*S(4)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(2)c*(S(3)/2)esqrt(b + sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)*S(3)/(a + b(d + ex)S(2) + c(d + ex)S(4)), x), x, bf*S(3)atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)cesqrt(-S(4)ac + bS(2))) + fS(3)log(a + b(d + ex)S(2) + c(d + ex)S(4))/(S(4)ce), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)*S(2)/(a + b(d + ex)S(2) + c(d + ex)S(4)), x), x, -sqrt(S(2))f*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(2)sqrt(c)esqrt(-S(4)ac + b*S(2))) + sqrt(S(2))f*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(2)sqrt(c)esqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)/(a + b(d + ex)*S(2) + c(d + ex)S(4)), x), x, -fatanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(esqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((df + efx)(a + b(d + ex)S(2) + c(d + ex)S(4))), x), x, batanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)aefsqrt(-S(4)ac + bS(2))) + log(d + ex)/(aef) - log(a + b(d + ex)*S(2) + c(d + ex)S(4))/(S(4)aef), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((df + efx)S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))), x), x, -sqrt(S(2))sqrt(c)(-b/sqrt(-S(4)ac + bS(2)) + S(1))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(2)aef*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) - sqrt(S(2))sqrt(c)(b/sqrt(-S(4)ac + bS(2)) + S(1))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(2)aef*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))) - S(1)/(aefS(2)(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((df + efx)*S(3)(a + b(d + ex)*S(2) + c(d + ex)S(4))), x), x, -S(1)/(S(2)aef*S(3)(d + ex)S(2)) - blog(d + ex)/(aS(2)efS(3)) + blog(a + b(d + ex)*S(2) + c(d + ex)S(4))/(S(4)a*S(2)efS(3)) - (-S(2)ac + bS(2))atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)efS(3)sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((df + efx)*S(4)(a + b(d + ex)*S(2) + c(d + ex)S(4))), x), x, -S(1)/(S(3)aef*S(4)(d + ex)S(3)) + b/(aS(2)efS(4)(d + ex)) + sqrt(S(2))sqrt(c)(b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)efS(4)sqrt(b + sqrt(-S(4)ac + b*S(2)))) + sqrt(S(2))sqrt(c)(b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)efS(4)sqrt(b - sqrt(-S(4)ac + b*S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)*S(4)/(a + b(d + ex)S(2) + c(d + ex)S(4))S(2), x), x, fS(4)(S(2)a + b(d + ex)S(2))(d + ex)/(e(-S(8)ac + S(2)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))) + sqrt(S(2))f*S(4)(b - (S(4)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(c)esqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))f*S(4)(S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(c)esqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)S(3)/(a + b(d + ex)S(2) + c(d + ex)S(4))S(2), x), x, -bf*S(3)atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(e(-S(4)ac + bS(2))(S(3)/2)) + f*S(3)(S(2)a + b(d + ex)S(2))/(e(-S(8)ac + S(2)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)*S(2)/(a + b(d + ex)S(2) + c(d + ex)S(4))S(2), x), x, -sqrt(S(2))sqrt(c)fS(2)(S(2)b + sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(2)esqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))sqrt(c)f*S(2)(S(2)b - sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(2)esqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) - f*S(2)(b + S(2)c(d + ex)S(2))(d + ex)/(e(-S(8)ac + S(2)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)/(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2), x), x, S(2)cfatanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(e(-S(4)ac + bS(2))(S(3)/2)) - f(b + S(2)c(d + ex)*S(2))/(e(-S(8)ac + S(2)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((df + efx)(a + b(d + ex)S(2) + c(d + ex)S(4))S(2)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(2)aef(-S(4)ac + bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))) + b(-S(6)ac + b*S(2))atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)ef(-S(4)ac + bS(2))(S(3)/2)) + log(d + ex)/(aS(2)ef) - log(a + b(d + ex)S(2) + c(d + ex)S(4))/(S(4)a*S(2)ef), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((df + efx)*S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(2)aef*S(2)(d + ex)(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))) + sqrt(S(2))sqrt(c)(-S(16)abc + S(3)bS(3) - (-S(10)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(4)a*S(2)efS(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - sqrt(S(2))sqrt(c)(-S(16)abc + S(3)b*S(3) + (-S(10)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(4)a*S(2)efS(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - (-S(10)ac + S(3)bS(2))/(S(2)a*S(2)efS(2)(d + ex)(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((df + efx)*S(3)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(2)aef*S(3)(d + ex)S(2)(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))) - (-S(3)ac + bS(2))/(aS(2)efS(3)(d + ex)S(2)(-S(4)ac + b*S(2))) - S(2)blog(d + ex)/(a*S(3)efS(3)) + blog(a + b(d + ex)*S(2) + c(d + ex)S(4))/(S(2)a*S(3)efS(3)) - (S(6)a*S(2)c*S(2) - S(6)abS(2)c + b*S(4))atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(aS(3)efS(3)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((df + efx)S(4)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(2)aef*S(4)(d + ex)S(3)(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))) - (-S(14)ac + S(5)b*S(2))/(S(6)a*S(2)efS(4)(d + ex)S(3)(-S(4)ac + b*S(2))) + b(-S(19)ac + S(5)bS(2))/(S(2)a*S(3)efS(4)(d + ex)(-S(4)ac + b*S(2))) - sqrt(S(2))sqrt(c)(S(28)a*S(2)c*S(2) - S(29)abS(2)c + S(5)bS(4) - b(-S(19)ac + S(5)bS(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(4)a*S(3)efS(4)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))sqrt(c)(S(28)aS(2)c*S(2) - S(29)abS(2)c + S(5)bS(4) + b(-S(19)ac + S(5)bS(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(4)a*S(3)efS(4)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)S(4)/(a + b(d + ex)S(2) + c(d + ex)S(4))S(3), x), x, -S(3)sqrt(S(2))sqrt(c)f*S(4)(S(4)ac + S(3)bS(2) + S(2)bsqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(8)esqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(5)/2)) + S(3)sqrt(S(2))sqrt(c)fS(4)(S(4)ac + S(3)bS(2) - S(2)bsqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(8)esqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(5)/2)) + f*S(4)(S(2)a + b(d + ex)S(2))(d + ex)/(e(-S(16)ac + S(4)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)) - fS(4)(d + ex)(-S(4)ac + S(7)bS(2) + S(12)bc(d + ex)S(2))/(S(8)e(-S(4)ac + bS(2))S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)*S(3)/(a + b(d + ex)S(2) + c(d + ex)S(4))S(3), x), x, S(3)bcf*S(3)atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(e(-S(4)ac + bS(2))(S(5)/2)) - S(3)bf*S(3)(b + S(2)c(d + ex)S(2))/(S(4)e(-S(4)ac + bS(2))S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))) + fS(3)(S(2)a + b(d + ex)S(2))/(e(-S(16)ac + S(4)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)*S(2)/(a + b(d + ex)S(2) + c(d + ex)S(4))S(3), x), x, -fS(2)(b + S(2)c(d + ex)S(2))(d + ex)/(e(-S(16)ac + S(4)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)) + sqrt(S(2))sqrt(c)fS(2)(S(20)ac + b*S(2) - b(-S(52)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(16)aesqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + sqrt(S(2))sqrt(c)f*S(2)(S(20)ac + b*S(2) + b(-S(52)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(16)aesqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + f*S(2)(d + ex)(b(S(8)ac + bS(2)) + c(d + ex)S(2)(S(20)ac + b*S(2)))/(S(8)ae(-S(4)ac + bS(2))S(2)(a + b(d + ex)S(2) + c(d + ex)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((df + efx)/(a + b(d + ex)*S(2) + c(d + ex)S(4))S(3), x), x, -S(6)c*S(2)fatanh((b + S(2)c(d + ex)*S(2))/sqrt(-S(4)ac + bS(2)))/(e(-S(4)ac + bS(2))(S(5)/2)) + S(3)cf(b + S(2)c(d + ex)*S(2))/(S(2)e(-S(4)ac + bS(2))S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))) - f(b + S(2)c(d + ex)S(2))/(e(-S(16)ac + S(4)bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((df + efx)(a + b(d + ex)S(2) + c(d + ex)S(4))S(3)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(4)aef(-S(4)ac + bS(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)) + (S(16)a*S(2)c*S(2) - S(15)abS(2)c + S(2)bS(4) + S(2)bc(d + ex)S(2)(-S(7)ac + b*S(2)))/(S(4)a*S(2)ef(-S(4)ac + bS(2))S(2)(a + b(d + ex)S(2) + c(d + ex)S(4))) + b(S(30)aS(2)c*S(2) - S(10)abS(2)c + b*S(4))atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(3)ef(-S(4)ac + bS(2))(S(5)/2)) + log(d + ex)/(aS(3)ef) - log(a + b(d + ex)S(2) + c(d + ex)S(4))/(S(4)a*S(3)ef), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((df + efx)*S(2)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(3)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(4)aef*S(2)(d + ex)(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)) + (S(36)a*S(2)c*S(2) - S(35)abS(2)c + S(5)bS(4) + bc(d + ex)*S(2)(-S(32)ac + S(5)bS(2)))/(S(8)a*S(2)efS(2)(d + ex)(-S(4)ac + bS(2))S(2)(a + b(d + ex)S(2) + c(d + ex)S(4))) - S(3)sqrt(S(2))sqrt(c)((-S(12)ac + S(5)bS(2))(-S(5)ac + b*S(2)) - (S(124)a*S(2)bcS(2) - S(47)abS(3)c + S(5)bS(5))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(16)a*S(3)efS(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) - S(3)sqrt(S(2))sqrt(c)(b(S(124)aS(2)c*S(2) - S(47)abS(2)c + S(5)bS(4))/sqrt(-S(4)ac + bS(2)) + (-S(12)ac + S(5)b*S(2))(-S(5)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)(d + ex)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(16)a*S(3)efS(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) - (-S(36)ac + S(15)bS(2))(-S(5)ac + b*S(2))/(S(8)a*S(3)efS(2)(d + ex)(-S(4)ac + bS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((df + efx)S(3)(a + b(d + ex)*S(2) + c(d + ex)S(4))S(3)), x), x, (-S(2)ac + bS(2) + bc(d + ex)*S(2))/(S(4)aef*S(3)(d + ex)S(2)(-S(4)ac + b*S(2))(a + b(d + ex)*S(2) + c(d + ex)S(4))S(2)) + (S(20)a*S(2)c*S(2) - S(20)abS(2)c + S(3)bS(4) + S(3)bc(d + ex)S(2)(-S(6)ac + b*S(2)))/(S(4)a*S(2)efS(3)(d + ex)S(2)(-S(4)ac + bS(2))S(2)(a + b(d + ex)S(2) + c(d + ex)S(4))) - (S(30)a*S(2)c*S(2) - S(21)abS(2)c + S(3)bS(4))/(S(2)a*S(3)efS(3)(d + ex)S(2)(-S(4)ac + bS(2))S(2)) - S(3)blog(d + ex)/(aS(4)efS(3)) + S(3)blog(a + b(d + ex)S(2) + c(d + ex)S(4))/(S(4)a*S(4)efS(3)) - (-S(60)a*S(3)c*S(3) + S(90)a*S(2)b*S(2)c*S(2) - S(30)abS(4)c + S(3)bS(6))atanh((b + S(2)c(d + ex)S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(4)efS(3)(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x + S(2))S(6)((S(3)x + S(2))S(14) + (S(3)x + S(2))*S(7) + S(1)), x), x, (S(3)x + S(2))*S(21)/S(63) + (S(3)x + S(2))*S(14)/S(42) + (S(3)x + S(2))*S(7)/S(21), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x + S(2))*S(6)((S(3)x + S(2))S(14) + (S(3)x + S(2))S(7) + S(1))S(2), x), x, (S(3)x + S(2))S(35)/S(105) + (S(3)x + S(2))*S(28)/S(42) + (S(3)x + S(2))*S(21)/S(21) + (S(3)x + S(2))*S(14)/S(21) + (S(3)x + S(2))*S(7)/S(21), expand=True, _diff=True, _numerical=True) def test_2(): assert rubi_test(rubi_integrate((c + dx*S(2))/(a + bx*S(4)), x), x, -sqrt(S(2))(-sqrt(a)d + sqrt(b)c)log(-sqrt(S(2))a*(S(1)/4)b*(S(1)/4)x + sqrt(a) + sqrt(b)xS(2))/(S(8)a*(S(3)/4)b*(S(3)/4)) + sqrt(S(2))(-sqrt(a)d + sqrt(b)c)log(sqrt(S(2))a*(S(1)/4)b*(S(1)/4)x + sqrt(a) + sqrt(b)xS(2))/(S(8)a*(S(3)/4)b*(S(3)/4)) - sqrt(S(2))(sqrt(a)d + sqrt(b)c)atan(S(1) - sqrt(S(2))b*(S(1)/4)x/a*(S(1)/4))/(S(4)a*(S(3)/4)b*(S(3)/4)) + sqrt(S(2))(sqrt(a)d + sqrt(b)c)atan(S(1) + sqrt(S(2))b*(S(1)/4)x/a*(S(1)/4))/(S(4)a*(S(3)/4)b*(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c - dx*S(2))/(a + bx*S(4)), x), x, -sqrt(S(2))(-sqrt(a)d + sqrt(b)c)atan(S(1) - sqrt(S(2))b*(S(1)/4)x/a*(S(1)/4))/(S(4)a*(S(3)/4)b*(S(3)/4)) + sqrt(S(2))(-sqrt(a)d + sqrt(b)c)atan(S(1) + sqrt(S(2))b*(S(1)/4)x/a*(S(1)/4))/(S(4)a*(S(3)/4)b*(S(3)/4)) - sqrt(S(2))(sqrt(a)d + sqrt(b)c)log(-sqrt(S(2))a*(S(1)/4)b*(S(1)/4)x + sqrt(a) + sqrt(b)xS(2))/(S(8)a*(S(3)/4)b*(S(3)/4)) + sqrt(S(2))(sqrt(a)d + sqrt(b)c)log(sqrt(S(2))a*(S(1)/4)b*(S(1)/4)x + sqrt(a) + sqrt(b)xS(2))/(S(8)a*(S(3)/4)b*(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx*S(2))/(a - bx*S(4)), x), x, (-sqrt(a)d + sqrt(b)c)atan(b*(S(1)/4)x/a*(S(1)/4))/(S(2)a*(S(3)/4)b*(S(3)/4)) + (sqrt(a)d + sqrt(b)c)atanh(b*(S(1)/4)x/a*(S(1)/4))/(S(2)a*(S(3)/4)b*(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c - dx*S(2))/(a - bx*S(4)), x), x, (-sqrt(a)d + sqrt(b)c)atanh(b*(S(1)/4)x/a*(S(1)/4))/(S(2)a*(S(3)/4)b*(S(3)/4)) + (sqrt(a)d + sqrt(b)c)atan(b*(S(1)/4)x/a*(S(1)/4))/(S(2)a*(S(3)/4)b*(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x*S(2) + S(2))/(S(9)x*S(4) + S(4)), x), x, sqrt(S(3))atan(sqrt(S(3))x + S(-1))/S(6) + sqrt(S(3))atan(sqrt(S(3))x + S(1))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(3)x*S(2) + S(2))/(S(9)x*S(4) + S(4)), x), x, -sqrt(S(3))log(S(3)xS(2) - S(2)sqrt(S(3))x + S(2))/S(12) + sqrt(S(3))log(S(3)xS(2) + S(2)sqrt(S(3))x + S(2))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x*S(2) + S(2))/(-S(9)x*S(4) + S(4)), x), x, sqrt(S(6))atanh(sqrt(S(6))x/S(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(3)x*S(2) + S(2))/(-S(9)x*S(4) + S(4)), x), x, sqrt(S(6))atan(sqrt(S(6))x/S(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(a)sqrt(b) + bxS(2))/(a + bx*S(4)), x), x, -sqrt(S(2))b*(S(1)/4)atan(S(1) - sqrt(S(2))b(S(1)/4)x/a*(S(1)/4))/(S(2)a*(S(1)/4)) + sqrt(S(2))b*(S(1)/4)atan(S(1) + sqrt(S(2))b(S(1)/4)x/a*(S(1)/4))/(S(2)a*(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(a)sqrt(b) - bxS(2))/(a + bx*S(4)), x), x, -sqrt(S(2))b*(S(1)/4)log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)x + sqrt(a) + sqrt(b)xS(2))/(S(4)a*(S(1)/4)) + sqrt(S(2))b*(S(1)/4)log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)x + sqrt(a) + sqrt(b)xS(2))/(S(4)a*(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/(dS(2) + e*S(2)x*S(4)), x), x, -sqrt(S(2))atan(S(1) - sqrt(S(2))sqrt(e)x/sqrt(d))/(S(2)sqrt(d)sqrt(e)) + sqrt(S(2))atan(S(1) + sqrt(S(2))sqrt(e)x/sqrt(d))/(S(2)sqrt(d)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d - exS(2))/(dS(2) + e*S(2)x*S(4)), x), x, -sqrt(S(2))log(-sqrt(S(2))sqrt(d)sqrt(e)x + d + ex*S(2))/(S(4)sqrt(d)sqrt(e)) + sqrt(S(2))log(sqrt(S(2))sqrt(d)sqrt(e)x + d + ex*S(2))/(S(4)sqrt(d)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(2) + S(5))/(xS(4) + S(-1)), x), x, -S(3)atan(x)/S(2) - S(7)atanh(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(3)/sqrt(a + cx*S(4)), x), x, -S(3)a*(S(1)/4)esqrt((a + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-aeS(2) + S(5)cdS(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(5)c*(S(7)/4)sqrt(a + cxS(4))) + a(S(1)/4)sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(3)ae*S(3) + S(15)cdS(2)e + S(5)sqrt(c)d(-ae*S(2) + cd*S(2))/sqrt(a))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(10)c*(S(7)/4)sqrt(a + cxS(4))) + de*S(2)xsqrt(a + cxS(4))/c + eS(3)xS(3)sqrt(a + cxS(4))/(S(5)c) + S(3)exsqrt(a + cx*S(4))(-aeS(2) + S(5)cdS(2))/(S(5)c*(S(3)/2)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)/sqrt(a + cxS(4)), x), x, -S(2)a*(S(1)/4)desqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a(S(1)/4)), S(1)/2)/(c(S(3)/4)sqrt(a + cxS(4))) + eS(2)xsqrt(a + cxS(4))/(S(3)c) + S(2)dexsqrt(a + cxS(4))/(sqrt(c)(sqrt(a) + sqrt(c)xS(2))) + sqrt((a + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(6)sqrt(a)sqrt(c)de - aeS(2) + S(3)cdS(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(6)a*(S(1)/4)c*(S(5)/4)sqrt(a + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/sqrt(a + cxS(4)), x), x, -a(S(1)/4)esqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a(S(1)/4)), S(1)/2)/(c(S(3)/4)sqrt(a + cxS(4))) + a(S(1)/4)sqrt((a + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(e + sqrt(c)d/sqrt(a))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(2)c*(S(3)/4)sqrt(a + cxS(4))) + exsqrt(a + cx*S(4))/(sqrt(c)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a + cx*S(4))(d + exS(2))), x), x, atan(xsqrt(ae/d + cd/e)/sqrt(a + cxS(4)))/(S(2)dsqrt(ae/d + cd/e)) + c(S(1)/4)sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(2)a*(S(1)/4)sqrt(a + cxS(4))(-sqrt(a)e + sqrt(c)d)) - sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(e + sqrt(c)d/sqrt(a))elliptic_pi(-sqrt(a)(-e + sqrt(c)d/sqrt(a))*S(2)/(S(4)sqrt(c)de), S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(4)a*(S(1)/4)c*(S(1)/4)dsqrt(a + cx*S(4))(-e + sqrt(c)d/sqrt(a))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a + cx*S(4))(d + exS(2))S(2)), x), x, a(S(1)/4)c*(S(1)/4)esqrt((a + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(2)dsqrt(a + cx*S(4))(aeS(2) + cdS(2))) - a(S(1)/4)c(S(1)/4)sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(e + sqrt(c)d/sqrt(a))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(4)dsqrt(a + cx*S(4))(aeS(2) + cdS(2))) - a(S(1)/4)sqrt((a + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(e + sqrt(c)d/sqrt(a))(aeS(2) + S(3)cdS(2))elliptic_pi(-(-sqrt(a)e + sqrt(c)d)*S(2)/(S(4)sqrt(a)sqrt(c)de), S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(8)c*(S(1)/4)d*S(2)sqrt(a + cxS(4))(-sqrt(a)e + sqrt(c)d)(ae*S(2) + cd*S(2))) - sqrt(c)exsqrt(a + cxS(4))/(S(2)d(sqrt(a) + sqrt(c)x*S(2))(aeS(2) + cdS(2))) + eS(2)xsqrt(a + cxS(4))/(S(2)d(d + ex*S(2))(aeS(2) + cd*S(2))) + (ae*S(2) + S(3)cdS(2))atan(xsqrt(ae/d + cd/e)/sqrt(a + cx*S(4)))/(S(4)d*S(3)e(ae/d + cd/e)(S(3)/2)) + c(S(1)/4)sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(aeS(2) + S(3)cdS(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(4)a*(S(1)/4)dsqrt(a + cx*S(4))(-sqrt(a)e + sqrt(c)d)(ae*S(2) + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(3)/sqrt(a - cxS(4)), x), x, S(3)a*(S(3)/4)esqrt(S(1) - cx*S(4)/a)(aeS(2) + S(5)cdS(2))elliptic_e(asin(c*(S(1)/4)x/a*(S(1)/4)), S(-1))/(S(5)c*(S(7)/4)sqrt(a - cxS(4))) - a(S(3)/4)sqrt(S(1) - cxS(4)/a)(S(3)ae*S(3) + S(15)cdS(2)e - S(5)sqrt(c)d(ae*S(2) + cd*S(2))/sqrt(a))elliptic_f(asin(c*(S(1)/4)x/a*(S(1)/4)), S(-1))/(S(5)c*(S(7)/4)sqrt(a - cxS(4))) - de*S(2)xsqrt(a - cxS(4))/c - eS(3)xS(3)sqrt(a - cxS(4))/(S(5)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)/sqrt(a - cx*S(4)), x), x, S(2)a*(S(3)/4)desqrt(S(1) - cxS(4)/a)elliptic_e(asin(c*(S(1)/4)x/a(S(1)/4)), S(-1))/(c(S(3)/4)sqrt(a - cxS(4))) + a(S(1)/4)sqrt(S(1) - cx*S(4)/a)(-S(6)sqrt(a)sqrt(c)de + aeS(2) + S(3)cdS(2))elliptic_f(asin(c*(S(1)/4)x/a*(S(1)/4)), S(-1))/(S(3)c*(S(5)/4)sqrt(a - cxS(4))) - eS(2)xsqrt(a - cx*S(4))/(S(3)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/sqrt(a - cxS(4)), x), x, a(S(3)/4)esqrt(S(1) - cxS(4)/a)elliptic_e(asin(c*(S(1)/4)x/a(S(1)/4)), S(-1))/(c(S(3)/4)sqrt(a - cxS(4))) + a(S(3)/4)sqrt(S(1) - cx*S(4)/a)(-e + sqrt(c)d/sqrt(a))elliptic_f(asin(c*(S(1)/4)x/a(S(1)/4)), S(-1))/(c(S(3)/4)sqrt(a - cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a - cx*S(4))(d + exS(2))), x), x, a(S(1)/4)sqrt(S(1) - cxS(4)/a)elliptic_pi(-sqrt(a)e/(sqrt(c)d), asin(c*(S(1)/4)x/a(S(1)/4)), S(-1))/(c(S(1)/4)dsqrt(a - cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a - cx*S(4))(d + exS(2))S(2)), x), x, -a(S(3)/4)c*(S(1)/4)esqrt(S(1) - cx*S(4)/a)elliptic_e(asin(c*(S(1)/4)x/a*(S(1)/4)), S(-1))/(S(2)dsqrt(a - cx*S(4))(-aeS(2) + cdS(2))) - a(S(1)/4)c(S(1)/4)sqrt(S(1) - cxS(4)/a)elliptic_f(asin(c*(S(1)/4)x/a*(S(1)/4)), S(-1))/(S(2)dsqrt(a - cx*S(4))(sqrt(a)e + sqrt(c)d)) + a*(S(1)/4)sqrt(S(1) - cxS(4)/a)(-aeS(2) + S(3)cdS(2))elliptic_pi(-sqrt(a)e/(sqrt(c)d), asin(c*(S(1)/4)x/a*(S(1)/4)), S(-1))/(S(2)c*(S(1)/4)d*S(2)sqrt(a - cxS(4))(-aeS(2) + cdS(2))) - eS(2)xsqrt(a - cxS(4))/(S(2)d(d + ex*S(2))(-aeS(2) + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/sqrt(-a + cxS(4)), x), x, a(S(3)/4)esqrt(S(1) - cxS(4)/a)elliptic_e(asin(c*(S(1)/4)x/a(S(1)/4)), S(-1))/(c(S(3)/4)sqrt(-a + cxS(4))) + a(S(3)/4)sqrt(S(1) - cx*S(4)/a)(-e + sqrt(c)d/sqrt(a))elliptic_f(asin(c*(S(1)/4)x/a(S(1)/4)), S(-1))/(c(S(3)/4)sqrt(-a + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(-a + cx*S(4))(d + exS(2))), x), x, a(S(1)/4)sqrt(S(1) - cxS(4)/a)elliptic_pi(-sqrt(a)e/(sqrt(c)d), asin(c*(S(1)/4)x/a(S(1)/4)), S(-1))/(c(S(1)/4)dsqrt(-a + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(a) + sqrt(c)x*S(2))/sqrt(-a + cxS(4)), x), x, a(S(3)/4)sqrt(S(1) - cx*S(4)/a)elliptic_e(asin(c*(S(1)/4)x/a(S(1)/4)), S(-1))/(c(S(1)/4)sqrt(-a + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2)sqrt(c/a) + S(1))/sqrt(-a + cx*S(4)), x), x, sqrt(S(1) - cx*S(4)/a)elliptic_e(asin(x(c/a)(S(1)/4)), S(-1))/((c/a)(S(1)/4)sqrt(-a + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/sqrt(-a - cxS(4)), x), x, -a(S(1)/4)esqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a(S(1)/4)), S(1)/2)/(c(S(3)/4)sqrt(-a - cxS(4))) + a(S(1)/4)sqrt((a + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(e + sqrt(c)d/sqrt(a))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(2)c*(S(3)/4)sqrt(-a - cxS(4))) - exsqrt(-a - cx*S(4))/(sqrt(c)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(-a - cx*S(4))(d + exS(2))), x), x, atan(xsqrt(-ae/d - cd/e)/sqrt(-a - cxS(4)))/(S(2)dsqrt(-ae/d - cd/e)) + c(S(1)/4)sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(2)a*(S(1)/4)sqrt(-a - cxS(4))(-sqrt(a)e + sqrt(c)d)) - sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(e + sqrt(c)d/sqrt(a))elliptic_pi(-sqrt(a)(-e + sqrt(c)d/sqrt(a))*S(2)/(S(4)sqrt(c)de), S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(4)a*(S(1)/4)c*(S(1)/4)dsqrt(-a - cx*S(4))(-e + sqrt(c)d/sqrt(a))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bx*S(2))sqrt(-S(5)xS(4) + S(4))), x), x, sqrt(S(2))S(5)*(S(3)/4)elliptic_pi(-S(2)sqrt(S(5))b/(S(5)a), asin(sqrt(S(2))S(5)*(S(1)/4)x/S(2)), S(-1))/(S(10)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bx*S(2))sqrt(S(5)xS(4) + S(4))), x), x, sqrt(S(2))S(5)*(S(1)/4)sqrt((S(5)xS(4) + S(4))/(sqrt(S(5))xS(2) + S(2))S(2))(sqrt(S(5))x*S(2) + S(2))elliptic_f(S(2)atan(sqrt(S(2))S(5)*(S(1)/4)x/S(2)), S(1)/2)/(S(4)sqrt(S(5)x*S(4) + S(4))(sqrt(S(5))a - S(2)b)) - sqrt(S(2))S(5)(S(3)/4)sqrt((S(5)xS(4) + S(4))/(sqrt(S(5))xS(2) + S(2))S(2))(sqrt(S(5))a + S(2)b)(sqrt(S(5))xS(2) + S(2))elliptic_pi(-sqrt(S(5))(sqrt(S(5))a - S(2)b)S(2)/(S(40)ab), S(2)atan(sqrt(S(2))S(5)(S(1)/4)x/S(2)), S(1)/2)/(S(40)asqrt(S(5)xS(4) + S(4))(sqrt(S(5))a - S(2)b)) + atan(xsqrt(S(5)a/b + S(4)b/a)/sqrt(S(5)x*S(4) + S(4)))/(S(2)asqrt(S(5)a/b + S(4)b/a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bx*S(2))sqrt(-dxS(4) + S(4))), x), x, sqrt(S(2))elliptic_pi(-S(2)b/(asqrt(d)), asin(sqrt(S(2))d(S(1)/4)x/S(2)), S(-1))/(S(2)ad*(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bx*S(2))sqrt(dxS(4) + S(4))), x), x, -sqrt(S(2))d*(S(1)/4)sqrt((dxS(4) + S(4))/(sqrt(d)xS(2) + S(2))S(2))(sqrt(d)x*S(2) + S(2))elliptic_f(S(2)atan(sqrt(S(2))d*(S(1)/4)x/S(2)), S(1)/2)/(S(4)(-asqrt(d) + S(2)b)sqrt(dxS(4) + S(4))) + atan(xsqrt(ad/b + S(4)b/a)/sqrt(dxS(4) + S(4)))/(S(2)asqrt(ad/b + S(4)b/a)) + sqrt(S(2))sqrt((dxS(4) + S(4))/(sqrt(d)xS(2) + S(2))S(2))(asqrt(d) + S(2)b)(sqrt(d)xS(2) + S(2))elliptic_pi(-(-asqrt(d) + S(2)b)*S(2)/(S(8)absqrt(d)), S(2)atan(sqrt(S(2))d*(S(1)/4)x/S(2)), S(1)/2)/(S(8)ad*(S(1)/4)(-asqrt(d) + S(2)b)sqrt(dxS(4) + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cS(2)xS(2) + S(1))/sqrt(-cS(2)x*S(2) + S(1)), x), x, elliptic_e(asin(cx), S(-1))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c*S(2)xS(2) + S(1))/sqrt(-cS(4)xS(4) + S(1)), x), x, elliptic_e(asin(cx), S(-1))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-c*S(2)xS(2) + S(1))/sqrt(cS(2)xS(2) + S(1)), x), x, -elliptic_e(asin(cx), S(-1))/c + S(2)elliptic_f(asin(cx), S(-1))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-c*S(2)xS(2) + S(1))/sqrt(-cS(4)xS(4) + S(1)), x), x, -elliptic_e(asin(cx), S(-1))/c + S(2)elliptic_f(asin(cx), S(-1))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + S(1))/sqrt(-bS(2)x*S(4) + S(1)), x), x, elliptic_e(asin(sqrt(b)x), S(-1))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-bxS(2) + S(1))/sqrt(-bS(2)x*S(4) + S(1)), x), x, -elliptic_e(asin(sqrt(b)x), S(-1))/sqrt(b) + S(2)elliptic_f(asin(sqrt(b)x), S(-1))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + S(1))/sqrt(bS(2)xS(4) + S(-1)), x), x, sqrt(-bS(2)xS(4) + S(1))elliptic_e(asin(sqrt(b)x), S(-1))/(sqrt(b)sqrt(b*S(2)x*S(4) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-bxS(2) + S(1))/sqrt(bS(2)xS(4) + S(-1)), x), x, -sqrt(-bS(2)x*S(4) + S(1))elliptic_e(asin(sqrt(b)x), S(-1))/(sqrt(b)sqrt(b*S(2)x*S(4) + S(-1))) + S(2)sqrt(-b*S(2)x*S(4) + S(1))elliptic_f(asin(sqrt(b)x), S(-1))/(sqrt(b)sqrt(b*S(2)x*S(4) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-bxS(2) + S(1))/sqrt(bS(2)xS(4) + S(1)), x), x, -xsqrt(b*S(2)x*S(4) + S(1))/(bxS(2) + S(1)) + sqrt((bS(2)xS(4) + S(1))/(bxS(2) + S(1))S(2))(bx*S(2) + S(1))elliptic_e(S(2)atan(sqrt(b)x), S(1)/2)/(sqrt(b)sqrt(bS(2)x*S(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + S(1))/sqrt(bS(2)xS(4) + S(1)), x), x, xsqrt(b*S(2)x*S(4) + S(1))/(bxS(2) + S(1)) - sqrt((bS(2)xS(4) + S(1))/(bxS(2) + S(1))S(2))(bx*S(2) + S(1))elliptic_e(S(2)atan(sqrt(b)x), S(1)/2)/(sqrt(b)sqrt(bS(2)xS(4) + S(1))) + sqrt((bS(2)xS(4) + S(1))/(bxS(2) + S(1))S(2))(bx*S(2) + S(1))elliptic_f(S(2)atan(sqrt(b)x), S(1)/2)/(sqrt(b)sqrt(bS(2)x*S(4) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-bxS(2) + S(1))/sqrt(-bS(2)xS(4) + S(-1)), x), x, xsqrt(-b*S(2)x*S(4) + S(-1))/(bxS(2) + S(1)) + sqrt((bS(2)xS(4) + S(1))/(bxS(2) + S(1))S(2))(bx*S(2) + S(1))elliptic_e(S(2)atan(sqrt(b)x), S(1)/2)/(sqrt(b)sqrt(-bS(2)x*S(4) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bxS(2) + S(1))/sqrt(-bS(2)xS(4) + S(-1)), x), x, -xsqrt(-b*S(2)x*S(4) + S(-1))/(bxS(2) + S(1)) - sqrt((bS(2)xS(4) + S(1))/(bxS(2) + S(1))S(2))(bx*S(2) + S(1))elliptic_e(S(2)atan(sqrt(b)x), S(1)/2)/(sqrt(b)sqrt(-bS(2)xS(4) + S(-1))) + sqrt((bS(2)xS(4) + S(1))/(bxS(2) + S(1))S(2))(bx*S(2) + S(1))elliptic_f(S(2)atan(sqrt(b)x), S(1)/2)/(sqrt(b)sqrt(-bS(2)x*S(4) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(4)/(-dS(2) + eS(2)xS(4)), x), x, -S(8)d*(S(5)/2)atanh(sqrt(e)x/sqrt(d))/sqrt(e) + S(7)d*S(2)x + S(4)dexS(3)/S(3) + eS(2)x*S(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(3)/(-dS(2) + eS(2)xS(4)), x), x, -S(4)d*(S(3)/2)atanh(sqrt(e)x/sqrt(d))/sqrt(e) + S(3)dx + ex*S(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)/(-dS(2) + eS(2)xS(4)), x), x, -S(2)sqrt(d)atanh(sqrt(e)x/sqrt(d))/sqrt(e) + x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/(-dS(2) + eS(2)x*S(4)), x), x, -atanh(sqrt(e)x/sqrt(d))/(sqrt(d)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex*S(2))(-dS(2) + eS(2)xS(4))), x), x, -x/(S(4)d*S(2)(d + exS(2))) - atan(sqrt(e)x/sqrt(d))/(S(2)d(S(5)/2)sqrt(e)) - atanh(sqrt(e)x/sqrt(d))/(S(4)d*(S(5)/2)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + exS(2))S(2)(-dS(2) + eS(2)xS(4))), x), x, -x/(S(8)d*S(2)(d + exS(2))S(2)) - S(5)x/(S(16)dS(3)(d + exS(2))) - S(7)atan(sqrt(e)x/sqrt(d))/(S(16)d*(S(7)/2)sqrt(e)) - atanh(sqrt(e)x/sqrt(d))/(S(8)d*(S(7)/2)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))(S(3)/2)/(-dS(2) + eS(2)x*S(4)), x), x, atanh(sqrt(e)x/sqrt(d + exS(2)))/sqrt(e) - sqrt(S(2))atanh(sqrt(S(2))sqrt(e)x/sqrt(d + exS(2)))/sqrt(e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + exS(2))/(-dS(2) + e*S(2)x*S(4)), x), x, -sqrt(S(2))atanh(sqrt(S(2))sqrt(e)x/sqrt(d + exS(2)))/(S(2)dsqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(d + ex*S(2))(-dS(2) + eS(2)xS(4))), x), x, -x/(S(2)d*S(2)sqrt(d + exS(2))) - sqrt(S(2))atanh(sqrt(S(2))sqrt(e)x/sqrt(d + exS(2)))/(S(4)d*S(2)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + exS(2))(S(3)/2)(-dS(2) + eS(2)xS(4))), x), x, -x/(S(6)d*S(2)(d + exS(2))(S(3)/2)) - S(7)x/(S(12)dS(3)sqrt(d + exS(2))) - sqrt(S(2))atanh(sqrt(S(2))sqrt(e)x/sqrt(d + exS(2)))/(S(8)d*S(3)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(4)/(bde + be*S(2)x*S(2) - cd*S(2) + ce*S(2)xS(4)), x), x, eS(2)xS(5)/(S(5)c) + exS(3)(-be + S(4)cd)/(S(3)c*S(2)) + x(b*S(2)e*S(2) - S(5)bcde + S(7)c*S(2)dS(2))/cS(3) - (-be + S(2)cd)S(3)atanh(sqrt(c)sqrt(e)x/sqrt(-be + cd))/(c*(S(7)/2)sqrt(e)sqrt(-be + cd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(3)/(bde + beS(2)x*S(2) - cd*S(2) + ce*S(2)x*S(4)), x), x, ex*S(3)/(S(3)c) + x(-be + S(3)cd)/c*S(2) - (-be + S(2)cd)*S(2)atanh(sqrt(c)sqrt(e)x/sqrt(-be + cd))/(c*(S(5)/2)sqrt(e)sqrt(-be + cd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)/(bde + beS(2)x*S(2) - cd*S(2) + ce*S(2)x*S(4)), x), x, x/c - (-be + S(2)cd)atanh(sqrt(c)sqrt(e)x/sqrt(-be + cd))/(c(S(3)/2)sqrt(e)sqrt(-be + cd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/(bde + be*S(2)x*S(2) - cd*S(2) + ce*S(2)x*S(4)), x), x, -atanh(sqrt(c)sqrt(e)x/sqrt(-be + cd))/(sqrt(c)sqrt(e)sqrt(-be + cd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex*S(2))(bde + beS(2)x*S(2) - cd*S(2) + ce*S(2)xS(4))), x), x, -c(S(3)/2)atanh(sqrt(c)sqrt(e)x/sqrt(-be + cd))/(sqrt(e)sqrt(-be + cd)(-be + S(2)cd)*S(2)) - x/(S(2)d(d + ex*S(2))(-be + S(2)cd)) - (-be + S(4)cd)atan(sqrt(e)x/sqrt(d))/(S(2)d(S(3)/2)sqrt(e)(-be + S(2)cd)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + exS(2))S(2)(bde + be*S(2)x*S(2) - cd*S(2) + ce*S(2)xS(4))), x), x, -c(S(5)/2)atanh(sqrt(c)sqrt(e)x/sqrt(-be + cd))/(sqrt(e)sqrt(-be + cd)(-be + S(2)cd)*S(3)) - x/(S(4)d(d + exS(2))S(2)(-be + S(2)cd)) - x(-S(3)be + S(10)cd)/(S(8)d*S(2)(d + exS(2))(-be + S(2)cd)S(2)) - (S(3)b*S(2)e*S(2) - S(16)bcde + S(28)c*S(2)d*S(2))atan(sqrt(e)x/sqrt(d))/(S(8)d*(S(5)/2)sqrt(e)(-be + S(2)cd)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))(S(5)/2)/(bde + beS(2)x*S(2) - cd*S(2) + ce*S(2)x*S(4)), x), x, xsqrt(d + exS(2))/(S(2)c) + (-S(2)be + S(5)cd)atanh(sqrt(e)x/sqrt(d + exS(2)))/(S(2)c*S(2)sqrt(e)) - (-be + S(2)cd)(S(3)/2)atanh(sqrt(e)xsqrt(-be + S(2)cd)/(sqrt(d + ex*S(2))sqrt(-be + cd)))/(c*S(2)sqrt(e)sqrt(-be + cd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))(S(3)/2)/(bde + beS(2)x*S(2) - cd*S(2) + ce*S(2)x*S(4)), x), x, atanh(sqrt(e)x/sqrt(d + exS(2)))/(csqrt(e)) - sqrt(-be + S(2)cd)atanh(sqrt(e)xsqrt(-be + S(2)cd)/(sqrt(d + ex*S(2))sqrt(-be + cd)))/(csqrt(e)sqrt(-be + cd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + exS(2))/(bde + be*S(2)x*S(2) - cd*S(2) + ce*S(2)x*S(4)), x), x, -atanh(sqrt(e)xsqrt(-be + S(2)cd)/(sqrt(d + exS(2))sqrt(-be + cd)))/(sqrt(e)sqrt(-be + cd)sqrt(-be + S(2)cd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(d + ex*S(2))(bde + beS(2)x*S(2) - cd*S(2) + ce*S(2)x*S(4))), x), x, -catanh(sqrt(e)xsqrt(-be + S(2)cd)/(sqrt(d + ex*S(2))sqrt(-be + cd)))/(sqrt(e)sqrt(-be + cd)(-be + S(2)cd)(S(3)/2)) - x/(dsqrt(d + exS(2))(-be + S(2)cd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + exS(2))(S(3)/2)(bde + be*S(2)x*S(2) - cd*S(2) + ce*S(2)xS(4))), x), x, -cS(2)atanh(sqrt(e)xsqrt(-be + S(2)cd)/(sqrt(d + exS(2))sqrt(-be + cd)))/(sqrt(e)sqrt(-be + cd)(-be + S(2)cd)(S(5)/2)) - x/(S(3)d(d + exS(2))(S(3)/2)(-be + S(2)cd)) - x(-S(2)be + S(7)cd)/(S(3)d*S(2)sqrt(d + exS(2))(-be + S(2)cd)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/(bxS(2) + dS(2) + e*S(2)x*S(4)), x), x, -atan((-S(2)ex + sqrt(-b + S(2)de))/sqrt(b + S(2)de))/sqrt(b + S(2)de) + atan((S(2)ex + sqrt(-b + S(2)de))/sqrt(b + S(2)de))/sqrt(b + S(2)de), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/(-bxS(2) + dS(2) + e*S(2)x*S(4)), x), x, atanh((-S(2)ex + sqrt(b + S(2)de))/sqrt(b - S(2)de))/sqrt(b - S(2)de) - atanh((S(2)ex + sqrt(b + S(2)de))/sqrt(b - S(2)de))/sqrt(b - S(2)de), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/(dS(2) + e*S(2)x*S(4) + fx*S(2)), x), x, -atan((-S(2)ex + sqrt(S(2)de - f))/sqrt(S(2)de + f))/sqrt(S(2)de + f) + atan((S(2)ex + sqrt(S(2)de - f))/sqrt(S(2)de + f))/sqrt(S(2)de + f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/(dS(2) + e*S(2)x*S(4) - fx*S(2)), x), x, -atan((-S(2)ex + sqrt(S(2)de + f))/sqrt(S(2)de - f))/sqrt(S(2)de - f) + atan((S(2)ex + sqrt(S(2)de + f))/sqrt(S(2)de - f))/sqrt(S(2)de - f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d - ex*S(2))/(bxS(2) + dS(2) + e*S(2)x*S(4)), x), x, -log(d + ex*S(2) - xsqrt(-b + S(2)de))/(S(2)sqrt(-b + S(2)de)) + log(d + ex*S(2) + xsqrt(-b + S(2)de))/(S(2)sqrt(-b + S(2)de)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d - ex*S(2))/(-bxS(2) + dS(2) + e*S(2)x*S(4)), x), x, -log(d + ex*S(2) - xsqrt(b + S(2)de))/(S(2)sqrt(b + S(2)de)) + log(d + ex*S(2) + xsqrt(b + S(2)de))/(S(2)sqrt(b + S(2)de)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d - exS(2))/(dS(2) + e*S(2)x*S(4) + fx*S(2)), x), x, -log(d + ex*S(2) - xsqrt(S(2)de - f))/(S(2)sqrt(S(2)de - f)) + log(d + ex*S(2) + xsqrt(S(2)de - f))/(S(2)sqrt(S(2)de - f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d - exS(2))/(dS(2) + e*S(2)x*S(4) - fx*S(2)), x), x, -log(d + ex*S(2) - xsqrt(S(2)de + f))/(S(2)sqrt(S(2)de + f)) + log(d + ex*S(2) + xsqrt(S(2)de + f))/(S(2)sqrt(S(2)de + f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d - ex*S(2))/(bx*S(2) + cdS(2)/eS(2) + cxS(4)), x), x, -e(S(3)/2)log(sqrt(c)d + sqrt(c)exS(2) - sqrt(e)xsqrt(-be + S(2)cd))/(S(2)sqrt(c)sqrt(-be + S(2)cd)) + e(S(3)/2)log(sqrt(c)d + sqrt(c)exS(2) + sqrt(e)xsqrt(-be + S(2)cd))/(S(2)sqrt(c)sqrt(-be + S(2)cd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/(bx*S(2) + cdS(2)/eS(2) + cxS(4)), x), x, -e(S(3)/2)atan((-S(2)sqrt(c)sqrt(e)x + sqrt(-be + S(2)cd))/sqrt(be + S(2)cd))/(sqrt(c)sqrt(be + S(2)cd)) + e(S(3)/2)atan((S(2)sqrt(c)sqrt(e)x + sqrt(-be + S(2)cd))/sqrt(be + S(2)cd))/(sqrt(c)sqrt(be + S(2)cd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/(bx*S(2) + c(dS(2)/eS(2) + xS(4))), x), x, -e(S(3)/2)atan((-S(2)sqrt(c)sqrt(e)x + sqrt(-be + S(2)cd))/sqrt(be + S(2)cd))/(sqrt(c)sqrt(be + S(2)cd)) + e*(S(3)/2)atan((S(2)sqrt(c)sqrt(e)x + sqrt(-be + S(2)cd))/sqrt(be + S(2)cd))/(sqrt(c)sqrt(be + S(2)cd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a - bxS(2))/(aS(2) + b*S(2)xS(4) + xS(2)(S(2)ab + S(-1))), x), x, -log(a + bx*S(2) - x)/S(2) + log(a + bx*S(2) + x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(aS(2) + b*S(2)xS(4) + xS(2)(S(2)ab + S(-1))), x), x, atanh((-S(2)bx + S(1))/sqrt(-S(4)ab + S(1)))/sqrt(-S(4)ab + S(1)) - atanh((S(2)bx + S(1))/sqrt(-S(4)ab + S(1)))/sqrt(-S(4)ab + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(bx*S(2) + S(4)x*S(4) + S(1)), x), x, -atan((-S(4)x + sqrt(-b + S(4)))/sqrt(b + S(4)))/sqrt(b + S(4)) + atan((S(4)x + sqrt(-b + S(4)))/sqrt(b + S(4)))/sqrt(b + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(-bx*S(2) + S(4)x*S(4) + S(1)), x), x, -atan((-S(4)x + sqrt(b + S(4)))/sqrt(-b + S(4)))/sqrt(-b + S(4)) + atan((S(4)x + sqrt(b + S(4)))/sqrt(-b + S(4)))/sqrt(-b + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(S(4)x*S(4) + S(6)x*S(2) + S(1)), x), x, sqrt(S(10))atan(S(2)x/sqrt(-sqrt(S(5)) + S(3)))/S(10) + sqrt(S(10))atan(S(2)x/sqrt(sqrt(S(5)) + S(3)))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(S(4)x*S(4) + S(5)x*S(2) + S(1)), x), x, atan(x)/S(3) + atan(S(2)x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(2) + S(1))/(S(4)x*S(4) + S(4)x*S(2) + S(1)), x), x, sqrt(S(2))atan(sqrt(S(2))x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(S(4)x*S(4) + S(3)x*S(2) + S(1)), x), x, -sqrt(S(7))atan(sqrt(S(7))(-S(4)x + S(1))/S(7))/S(7) + sqrt(S(7))atan(sqrt(S(7))(S(4)x + S(1))/S(7))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(S(4)x*S(4) + S(2)x*S(2) + S(1)), x), x, -sqrt(S(6))atan(sqrt(S(6))(-S(4)x + sqrt(S(2)))/S(6))/S(6) + sqrt(S(6))atan(sqrt(S(6))(S(4)x + sqrt(S(2)))/S(6))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(S(4)xS(4) + xS(2) + S(1)), x), x, -sqrt(S(5))atan(sqrt(S(5))(-S(4)x + sqrt(S(3)))/S(5))/S(5) + sqrt(S(5))atan(sqrt(S(5))(S(4)x + sqrt(S(3)))/S(5))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(2) + S(1))/(S(4)x*S(4) + S(1)), x), x, atan(S(2)x + S(-1))/S(2) + atan(S(2)x + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(S(4)xS(4) - xS(2) + S(1)), x), x, -sqrt(S(3))atan(sqrt(S(3))(-S(4)x + sqrt(S(5)))/S(3))/S(3) + sqrt(S(3))atan(sqrt(S(3))(S(4)x + sqrt(S(5)))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(2) + S(1))/(S(4)x*S(4) - S(2)x*S(2) + S(1)), x), x, -sqrt(S(2))atan(sqrt(S(2))(-S(4)x + sqrt(S(6)))/S(2))/S(2) + sqrt(S(2))atan(sqrt(S(2))(S(4)x + sqrt(S(6)))/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(S(4)x*S(4) - S(3)x*S(2) + S(1)), x), x, atan(S(4)x - sqrt(S(7))) + atan(S(4)x + sqrt(S(7))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(S(4)x*S(4) - S(4)x*S(2) + S(1)), x), x, x/(-S(2)x*S(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(S(4)x*S(4) - S(5)x*S(2) + S(1)), x), x, -log(-S(2)x*S(2) - x + S(1))/S(2) + log(-S(2)x*S(2) + x + S(1))/S(2), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(S(4)x*S(4) - S(5)x*S(2) + S(1)), x), x, -log(-S(2)x + S(1))/S(2) + log(-x + S(1))/S(2) - log(x + S(1))/S(2) + log(S(2)x + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) + S(1))/(S(4)x*S(4) - S(6)x*S(2) + S(1)), x), x, -sqrt(S(10))log(S(2)xS(2) - sqrt(S(10))x + S(1))/S(20) + sqrt(S(10))log(S(2)x*S(2) + sqrt(S(10))x + S(1))/S(20), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((S(2)xS(2) + S(1))/(S(4)x*S(4) - S(6)x*S(2) + S(1)), x), x, sqrt(S(2))atanh(sqrt(S(2))(-S(4)x + sqrt(S(10)))/S(2))/S(2) - sqrt(S(2))atanh(sqrt(S(2))(S(4)x + sqrt(S(10)))/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)x*S(2) + S(1))/(bx*S(2) + S(4)x*S(4) + S(1)), x), x, -log(S(2)x*S(2) - xsqrt(-b + S(4)) + S(1))/(S(2)sqrt(-b + S(4))) + log(S(2)x*S(2) + xsqrt(-b + S(4)) + S(1))/(S(2)sqrt(-b + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)x*S(2) + S(1))/(S(4)x*S(4) + S(6)x*S(2) + S(1)), x), x, sqrt(S(2))atan(S(2)x/sqrt(-sqrt(S(5)) + S(3)))/S(2) - sqrt(S(2))atan(S(2)x/sqrt(sqrt(S(5)) + S(3)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)x*S(2) + S(1))/(S(4)x*S(4) + S(5)x*S(2) + S(1)), x), x, -atan(x) + atan(S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)xS(2) + S(1))/(S(4)x*S(4) + S(4)x*S(2) + S(1)), x), x, x/(S(2)x*S(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)x*S(2) + S(1))/(S(4)x*S(4) + S(3)x*S(2) + S(1)), x), x, -log(S(2)x*S(2) - x + S(1))/S(2) + log(S(2)x*S(2) + x + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)x*S(2) + S(1))/(S(4)x*S(4) + S(2)x*S(2) + S(1)), x), x, -sqrt(S(2))log(S(2)xS(2) - sqrt(S(2))x + S(1))/S(4) + sqrt(S(2))log(S(2)x*S(2) + sqrt(S(2))x + S(1))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)xS(2) + S(1))/(S(4)xS(4) + xS(2) + S(1)), x), x, -sqrt(S(3))log(S(2)x*S(2) - sqrt(S(3))x + S(1))/S(6) + sqrt(S(3))log(S(2)x*S(2) + sqrt(S(3))x + S(1))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)xS(2) + S(1))/(S(4)x*S(4) + S(1)), x), x, -log(S(2)x*S(2) - S(2)x + S(1))/S(4) + log(S(2)xS(2) + S(2)x + S(1))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)xS(2) + S(1))/(S(4)xS(4) - xS(2) + S(1)), x), x, -sqrt(S(5))log(S(2)x*S(2) - sqrt(S(5))x + S(1))/S(10) + sqrt(S(5))log(S(2)x*S(2) + sqrt(S(5))x + S(1))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)xS(2) + S(1))/(S(4)x*S(4) - S(2)x*S(2) + S(1)), x), x, -sqrt(S(6))log(S(2)xS(2) - sqrt(S(6))x + S(1))/S(12) + sqrt(S(6))log(S(2)x*S(2) + sqrt(S(6))x + S(1))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)xS(2) + S(1))/(S(4)x*S(4) - S(3)x*S(2) + S(1)), x), x, -sqrt(S(7))log(S(2)xS(2) - sqrt(S(7))x + S(1))/S(14) + sqrt(S(7))log(S(2)x*S(2) + sqrt(S(7))x + S(1))/S(14), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)xS(2) + S(1))/(S(4)x*S(4) - S(4)x*S(2) + S(1)), x), x, sqrt(S(2))atanh(sqrt(S(2))x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)x*S(2) + S(1))/(S(4)x*S(4) - S(5)x*S(2) + S(1)), x), x, -log(S(2)x*S(2) - S(3)x + S(1))/S(6) + log(S(2)xS(2) + S(3)x + S(1))/S(6), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((-S(2)xS(2) + S(1))/(S(4)x*S(4) - S(5)x*S(2) + S(1)), x), x, atanh(x)/S(3) + atanh(S(2)x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)xS(2) + S(1))/(S(4)x*S(4) - S(6)x*S(2) + S(1)), x), x, -sqrt(S(10))log(S(2)xS(2) - sqrt(S(10))x + S(1))/S(20) + sqrt(S(10))log(S(2)x*S(2) + sqrt(S(10))x + S(1))/S(20), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((-S(2)xS(2) + S(1))/(S(4)x*S(4) - S(6)x*S(2) + S(1)), x), x, sqrt(S(10))atanh(S(2)x/sqrt(-sqrt(S(5)) + S(3)))/S(10) + sqrt(S(10))atanh(S(2)x/sqrt(sqrt(S(5)) + S(3)))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(bxS(2) + xS(4) + S(1)), x), x, -atan((-S(2)x + sqrt(-b + S(2)))/sqrt(b + S(2)))/sqrt(b + S(2)) + atan((S(2)x + sqrt(-b + S(2)))/sqrt(b + S(2)))/sqrt(b + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(xS(4) + S(5)xS(2) + S(1)), x), x, sqrt(S(7))atan(xsqrt(sqrt(S(21))/S(2) + S(5)/2))/S(7) + sqrt(S(7))atan(sqrt(S(2))x/sqrt(sqrt(S(21)) + S(5)))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(xS(4) + S(4)x*S(2) + S(1)), x), x, sqrt(S(6))atan(x/sqrt(-sqrt(S(3)) + S(2)))/S(6) + sqrt(S(6))atan(x/sqrt(sqrt(S(3)) + S(2)))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(xS(4) + S(3)x*S(2) + S(1)), x), x, sqrt(S(5))atan(xsqrt(sqrt(S(5))/S(2) + S(3)/2))/S(5) + sqrt(S(5))atan(sqrt(S(2))x/sqrt(sqrt(S(5)) + S(3)))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(xS(4) + S(2)xS(2) + S(1)), x), x, atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(xS(4) + xS(2) + S(1)), x), x, -sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(3) + sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(xS(4) + S(1)), x), x, sqrt(S(2))atan(sqrt(S(2))x + S(-1))/S(2) + sqrt(S(2))atan(sqrt(S(2))x + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(xS(4) - x*S(2) + S(1)), x), x, atan(S(2)x - sqrt(S(3))) + atan(S(2)x + sqrt(S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(xS(4) - S(2)xS(2) + S(1)), x), x, x/(-xS(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(xS(4) - S(3)xS(2) + S(1)), x), x, atanh(-S(2)x + sqrt(S(5))) - atanh(S(2)x + sqrt(S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(xS(4) - S(4)x*S(2) + S(1)), x), x, sqrt(S(2))atanh(-sqrt(S(2))x + sqrt(S(3)))/S(2) - sqrt(S(2))atanh(sqrt(S(2))x + sqrt(S(3)))/S(2), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((xS(2) + S(1))/(xS(4) - S(4)x*S(2) + S(1)), x), x, sqrt(S(2))atanh(sqrt(S(2))(-S(2)x + sqrt(S(6)))/S(2))/S(2) - sqrt(S(2))atanh(sqrt(S(2))(S(2)x + sqrt(S(6)))/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(xS(4) - S(5)x*S(2) + S(1)), x), x, sqrt(S(3))atanh(sqrt(S(3))(-S(2)x + sqrt(S(7)))/S(3))/S(3) - sqrt(S(3))atanh(sqrt(S(3))(S(2)x + sqrt(S(7)))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(bxS(2) + xS(4) + S(1)), x), x, -log(x*S(2) - xsqrt(-b + S(2)) + S(1))/(S(2)sqrt(-b + S(2))) + log(xS(2) + xsqrt(-b + S(2)) + S(1))/(S(2)sqrt(-b + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) + S(5)x*S(2) + S(1)), x), x, sqrt(S(3))atan(xsqrt(sqrt(S(21))/S(2) + S(5)/2))/S(3) - sqrt(S(3))atan(sqrt(S(2))x/sqrt(sqrt(S(21)) + S(5)))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) + S(4)x*S(2) + S(1)), x), x, sqrt(S(2))atan(x/sqrt(-sqrt(S(3)) + S(2)))/S(2) - sqrt(S(2))atan(x/sqrt(sqrt(S(3)) + S(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) + S(3)x*S(2) + S(1)), x), x, atan(xsqrt(sqrt(S(5))/S(2) + S(3)/2)) - atan(sqrt(S(2))x/sqrt(sqrt(S(5)) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) + S(2)xS(2) + S(1)), x), x, x/(xS(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) + xS(2) + S(1)), x), x, -log(xS(2) - x + S(1))/S(2) + log(xS(2) + x + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(x*S(4) + S(1)), x), x, -sqrt(S(2))log(x*S(2) - sqrt(S(2))x + S(1))/S(4) + sqrt(S(2))log(xS(2) + sqrt(S(2))x + S(1))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) - x*S(2) + S(1)), x), x, -sqrt(S(3))log(x*S(2) - sqrt(S(3))x + S(1))/S(6) + sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) - S(2)xS(2) + S(1)), x), x, atanh(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) - S(3)x*S(2) + S(1)), x), x, -sqrt(S(5))log(x*S(2) - sqrt(S(5))x + S(1))/S(10) + sqrt(S(5))log(xS(2) + sqrt(S(5))x + S(1))/S(10), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) - S(3)xS(2) + S(1)), x), x, sqrt(S(5))atanh(xsqrt(sqrt(S(5))/S(2) + S(3)/2))/S(5) + sqrt(S(5))atanh(sqrt(S(2))x/sqrt(sqrt(S(5)) + S(3)))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) - S(4)x*S(2) + S(1)), x), x, -sqrt(S(6))log(x*S(2) - sqrt(S(6))x + S(1))/S(12) + sqrt(S(6))log(xS(2) + sqrt(S(6))x + S(1))/S(12), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) - S(4)xS(2) + S(1)), x), x, sqrt(S(6))atanh(x/sqrt(-sqrt(S(3)) + S(2)))/S(6) + sqrt(S(6))atanh(x/sqrt(sqrt(S(3)) + S(2)))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) - S(5)x*S(2) + S(1)), x), x, -sqrt(S(7))log(x*S(2) - sqrt(S(7))x + S(1))/S(14) + sqrt(S(7))log(xS(2) + sqrt(S(7))x + S(1))/S(14), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((-xS(2) + S(1))/(xS(4) - S(5)xS(2) + S(1)), x), x, sqrt(S(7))atanh(xsqrt(sqrt(S(21))/S(2) + S(5)/2))/S(7) + sqrt(S(7))atanh(sqrt(S(2))x/sqrt(sqrt(S(21)) + S(5)))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(4)(a + bx*S(2) + cx*S(4)), x), x, ad*S(4)x + ceS(4)xS(13)/S(13) + dS(3)xS(3)(S(4)ae + bd)/S(3) + dS(2)x*S(5)(S(6)ae*S(2) + S(4)bde + cdS(2))/S(5) + S(2)dex*S(7)(S(2)cd*S(2) + e(S(2)ae + S(3)bd))/S(7) + e*S(3)x*S(11)(be + S(4)cd)/S(11) + eS(2)x*S(9)(S(6)cd*S(2) + e(ae + S(4)bd))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(3)(a + bx*S(2) + cx*S(4)), x), x, ad*S(3)x + ceS(3)xS(11)/S(11) + dS(2)xS(3)(S(3)ae + bd)/S(3) + dx*S(5)(cdS(2) + S(3)e(ae + bd))/S(5) + eS(2)x*S(9)(be + S(3)cd)/S(9) + ex*S(7)(S(3)cd*S(2) + e(ae + S(3)bd))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)(a + bx*S(2) + cx*S(4)), x), x, ad*S(2)x + ceS(2)x*S(9)/S(9) + dx*S(3)(S(2)ae + bd)/S(3) + ex*S(7)(be + S(2)cd)/S(7) + xS(5)(cdS(2) + e(ae + S(2)bd))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))(a + bxS(2) + cx*S(4)), x), x, adx + cexS(7)/S(7) + xS(5)(be + cd)/S(5) + x*S(3)(ae + bd)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cx*S(4))/(d + ex*S(2)), x), x, cx*S(3)/(S(3)e) - x(-be + cd)/eS(2) + (ae*S(2) - bde + cd*S(2))atan(sqrt(e)x/sqrt(d))/(sqrt(d)e*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(d + exS(2))S(2), x), x, cx/eS(2) + x(aeS(2) - bde + cd*S(2))/(S(2)deS(2)(d + exS(2))) - (S(3)cdS(2) - e(ae + bd))atan(sqrt(e)x/sqrt(d))/(S(2)d(S(3)/2)e*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(d + exS(2))S(3), x), x, x(ae*S(2) - bde + cd*S(2))/(S(4)deS(2)(d + exS(2))S(2)) - x(S(5)cd*S(2) - e(S(3)ae + bd))/(S(8)d*S(2)e*S(2)(d + exS(2))) + (S(3)cdS(2) + e(S(3)ae + bd))atan(sqrt(e)x/sqrt(d))/(S(8)d*(S(5)/2)e*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(d + exS(2))S(4), x), x, x(ae*S(2) - bde + cd*S(2))/(S(6)deS(2)(d + exS(2))S(3)) - x(S(7)cd*S(2) - e(S(5)ae + bd))/(S(24)d*S(2)e*S(2)(d + exS(2))S(2)) + x(cdS(2) + e(S(5)ae + bd))/(S(16)d*S(3)e*S(2)(d + exS(2))) + (cd*S(2) + e(S(5)ae + bd))atan(sqrt(e)x/sqrt(d))/(S(16)d*(S(7)/2)e*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(3)(a + bx*S(2) + cxS(4))S(2), x), x, a*S(2)d*S(3)x + adS(2)x*S(3)(S(3)ae + S(2)bd)/S(3) + c*S(2)e*S(3)x*S(15)/S(15) + ce*S(2)x*S(13)(S(2)be + S(3)cd)/S(13) + dxS(5)(S(6)abde + a(S(3)aeS(2) + S(2)cdS(2)) + bS(2)d*S(2))/S(5) + ex*S(11)(b*S(2)e*S(2) + S(3)c*S(2)d*S(2) + S(2)ce(ae + S(3)bd))/S(11) + xS(9)(beS(2)(S(2)ae + S(3)bd)/S(9) + c*S(2)d*S(3)/S(9) + S(2)cde(ae + bd)/S(3)) + xS(7)(a*S(2)e*S(3)/S(7) + S(6)abdeS(2)/S(7) + S(6)acd*S(2)e/S(7) + S(3)bS(2)d*S(2)e/S(7) + S(2)bcdS(3)/S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)(a + bx*S(2) + cxS(4))S(2), x), x, a*S(2)d*S(2)x + S(2)adxS(3)(ae + bd)/S(3) + c*S(2)e*S(2)x*S(13)/S(13) + S(2)cex*S(11)(be + cd)/S(11) + x*S(9)(b*S(2)eS(2)/S(9) + cS(2)dS(2)/S(9) + S(2)ce(ae + S(2)bd)/S(9)) + xS(7)(S(2)abeS(2)/S(7) + S(4)acde/S(7) + S(2)b*S(2)de/S(7) + S(2)bcdS(2)/S(7)) + xS(5)(S(4)abde/S(5) + a(aeS(2) + S(2)cdS(2))/S(5) + bS(2)d*S(2)/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))(a + bxS(2) + cxS(4))S(2), x), x, a*S(2)dx + ax*S(3)(ae + S(2)bd)/S(3) + cS(2)exS(11)/S(11) + cx*S(9)(S(2)be + cd)/S(9) + xS(7)(S(2)ace/S(7) + bS(2)e/S(7) + S(2)bcd/S(7)) + xS(5)(S(2)abe/S(5) + S(2)acd/S(5) + b*S(2)d/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))S(2), x), x, a*S(2)x + S(2)abxS(3)/S(3) + S(2)bcxS(7)/S(7) + cS(2)xS(9)/S(9) + xS(5)(S(2)ac/S(5) + b*S(2)/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/(d + exS(2)), x), x, cS(2)x*S(7)/(S(7)e) - cxS(5)(-S(2)be + cd)/(S(5)eS(2)) + xS(3)(bS(2)eS(2) + cS(2)dS(2) - S(2)ce(-ae + bd))/(S(3)eS(3)) - x(-be + cd)(cd*S(2) - e(-S(2)ae + bd))/eS(4) + (ae*S(2) - bde + cdS(2))S(2)atan(sqrt(e)x/sqrt(d))/(sqrt(d)e(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/(d + exS(2))S(2), x), x, cS(2)x*S(7)/(S(5)e(d + ex*S(2))) - cx*S(3)(-S(10)be + S(7)cd)/(S(15)eS(3)) + x(S(5)bS(2)e*S(2) + S(14)c*S(2)d*S(2) - S(10)ce(-ae + S(2)bd))/(S(5)e*S(4)) + x(S(7)cS(2)d*S(4) - S(10)cdS(2)e(-ae + bd) + S(5)e*S(2)(-ae + bd)*S(2))/(S(10)deS(4)(d + exS(2))) - (S(7)cdS(2) - e(ae + S(3)bd))(aeS(2) - bde + cd*S(2))atan(sqrt(e)x/sqrt(d))/(S(2)d*(S(3)/2)e*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/(d + exS(2))S(3), x), x, cS(2)x*S(7)/(S(3)e(d + exS(2))S(2)) - cx(-S(6)be + S(7)cd)/(S(3)eS(4)) + x(S(7)cS(2)d*S(4) - S(6)cdS(2)e(-ae + bd) + S(3)e*S(2)(-ae + bd)*S(2))/(S(12)deS(4)(d + exS(2))S(2)) - x(S(21)cS(2)d*S(4) - S(2)cdS(2)e(-S(5)ae + S(9)bd) + eS(2)(-S(3)aS(2)e*S(2) - S(2)abde + S(5)b*S(2)d*S(2)))/(S(8)d*S(2)e*S(4)(d + exS(2))) + (S(35)c*S(2)d*S(4) - S(6)cdS(2)e(-ae + S(5)bd) + e*S(2)(S(3)aS(2)e*S(2) + S(2)abde + S(3)b*S(2)d*S(2)))atan(sqrt(e)x/sqrt(d))/(S(8)d*(S(5)/2)e*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/(d + exS(2))S(4), x), x, cS(2)x*S(7)/(e(d + exS(2))S(3)) + x(S(7)cS(2)d*S(4) - S(2)cdS(2)e(-ae + bd) + eS(2)(-ae + bd)*S(2))/(S(6)deS(4)(d + exS(2))S(3)) - x(S(91)cS(2)d*S(4) - S(2)cdS(2)e(-S(7)ae + S(13)bd) + eS(2)(-S(5)aS(2)e*S(2) - S(2)abde + S(7)b*S(2)d*S(2)))/(S(24)d*S(2)e*S(4)(d + exS(2))S(2)) + x(S(77)cS(2)d*S(4) - S(2)cdS(2)e(-ae + S(11)bd) + e*S(2)(S(5)aS(2)e*S(2) + S(2)abde + bS(2)d*S(2)))/(S(16)d*S(3)e*S(4)(d + exS(2))) - (S(35)c*S(2)d*S(4) - S(2)cdS(2)e(ae + S(5)bd) - e*S(2)(S(5)aS(2)e*S(2) + S(2)abde + bS(2)d*S(2)))atan(sqrt(e)x/sqrt(d))/(S(16)d*(S(7)/2)e*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)/(d + exS(2))S(5), x), x, -cS(2)x*S(7)/(e(d + exS(2))S(4)) - x(S(7)cS(2)d*S(4) + S(2)cdS(2)e(-ae + bd) - eS(2)(-ae + bd)*S(2))/(S(8)deS(4)(d + exS(2))S(4)) + x(S(119)cS(2)d*S(4) + S(2)cdS(2)e(-S(9)ae + S(17)bd) - eS(2)(-S(7)aS(2)e*S(2) - S(2)abde + S(9)b*S(2)d*S(2)))/(S(48)d*S(2)e*S(4)(d + exS(2))S(3)) - x(S(413)cS(2)d*S(4) + S(2)cdS(2)e(-S(3)ae + S(59)bd) - eS(2)(S(35)aS(2)e*S(2) + S(10)abde + S(3)b*S(2)d*S(2)))/(S(192)d*S(3)e*S(4)(d + exS(2))S(2)) + x(S(35)cS(2)d*S(4) + S(2)cdS(2)e(S(3)ae + S(5)bd) + eS(2)(S(35)aS(2)e*S(2) + S(10)abde + S(3)b*S(2)d*S(2)))/(S(128)d*S(4)e*S(4)(d + exS(2))) + (S(35)c*S(2)d*S(4) + S(2)cdS(2)e(S(3)ae + S(5)bd) + eS(2)(S(35)aS(2)e*S(2) + S(10)abde + S(3)b*S(2)d*S(2)))atan(sqrt(e)x/sqrt(d))/(S(128)d*(S(9)/2)e*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(d + exS(2))S(2), x), x, cx/eS(2) + x(aeS(2) - bde + cd*S(2))/(S(2)deS(2)(d + exS(2))) - (S(3)cdS(2) - e(ae + bd))atan(sqrt(e)x/sqrt(d))/(S(2)d(S(3)/2)e(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + xS(2)(b + cx*S(2)))/(d + exS(2))S(2), x), x, cx/eS(2) + x(aeS(2) - bde + cd*S(2))/(S(2)deS(2)(d + exS(2))) - (S(3)cdS(2) - e(ae + bd))atan(sqrt(e)x/sqrt(d))/(S(2)d(S(3)/2)e*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(4)/(a + bxS(2) + cxS(4)), x), x, eS(4)xS(5)/(S(5)c) + e*S(3)x*S(3)(-be + S(4)cd)/(S(3)cS(2)) + eS(2)x(b*S(2)e*S(2) + S(6)c*S(2)d*S(2) - ce(ae + S(4)bd))/c*S(3) + sqrt(S(2))(e(-be + S(2)cd)(bS(2)e*S(2) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd)) - (b*S(4)e*S(4) - S(4)b*S(2)ceS(3)(ae + bd) + S(2)cS(4)d*S(4) - S(4)c*S(3)d*S(2)e(S(3)ae + bd) + S(2)cS(2)e*S(2)(a*S(2)e*S(2) + S(6)abde + S(3)b*S(2)d*S(2)))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(7)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) + sqrt(S(2))(e(-be + S(2)cd)(bS(2)e*S(2) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd)) + (b*S(4)e*S(4) - S(4)b*S(2)ceS(3)(ae + bd) + S(2)cS(4)d*S(4) - S(4)c*S(3)d*S(2)e(S(3)ae + bd) + S(2)cS(2)e*S(2)(a*S(2)e*S(2) + S(6)abde + S(3)b*S(2)d*S(2)))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(7)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(3)/(a + bxS(2) + cxS(4)), x), x, eS(3)xS(3)/(S(3)c) + e*S(2)x(-be + S(3)cd)/c*S(2) + sqrt(S(2))(e(bS(2)e*S(2) + S(3)c*S(2)d*S(2) - ce(ae + S(3)bd)) - (-be + S(2)cd)(b*S(2)eS(2) + cS(2)dS(2) - ce(S(3)ae + bd))/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) + sqrt(S(2))(e(bS(2)e*S(2) + S(3)c*S(2)d*S(2) - ce(ae + S(3)bd)) + (-be + S(2)cd)(b*S(2)eS(2) + cS(2)dS(2) - ce(S(3)ae + bd))/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)/(a + bxS(2) + cxS(4)), x), x, eS(2)x/c + sqrt(S(2))(e(-be + S(2)cd) - (b*S(2)e*S(2) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd))/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) + sqrt(S(2))(e(-be + S(2)cd) + (b*S(2)e*S(2) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd))/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/(a + bx*S(2) + cx*S(4)), x), x, sqrt(S(2))(e - (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bx*S(2) + cx*S(4)), x), x, -sqrt(S(2))sqrt(c)atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + bS(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + bS(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex*S(2))(a + bxS(2) + cx*S(4))), x), x, -sqrt(S(2))sqrt(c)(e + (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cd*S(2))) - sqrt(S(2))sqrt(c)(e - (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))) + e(S(3)/2)atan(sqrt(e)x/sqrt(d))/(sqrt(d)(ae*S(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + exS(2))S(2)(a + bx*S(2) + cx*S(4))), x), x, -sqrt(S(2))sqrt(c)(be*S(2)(b - sqrt(-S(4)ac + b*S(2))) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd - dsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cdS(2))S(2)) + sqrt(S(2))sqrt(c)(beS(2)(b + sqrt(-S(4)ac + b*S(2))) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd + dsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cdS(2))S(2)) + e*S(2)x/(S(2)d(d + exS(2))(aeS(2) - bde + cdS(2))) + e(S(3)/2)(-be + S(2)cd)atan(sqrt(e)x/sqrt(d))/(sqrt(d)(ae*S(2) - bde + cdS(2))S(2)) + e*(S(3)/2)atan(sqrt(e)x/sqrt(d))/(S(2)d*(S(3)/2)(aeS(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(3)/(a + bxS(2) + cxS(4))S(2), x), x, x(-abe(aeS(2) + S(3)cdS(2)) - S(2)acd(-S(3)aeS(2) + cdS(2)) + bS(2)cdS(3) - xS(2)(ab*S(2)e*S(3) + S(2)ace(-ae*S(2) + S(3)cdS(2)) - bcd(S(3)ae*S(2) + cd*S(2))))/(S(2)ac(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) + sqrt(S(2))(abS(3)e*S(3) + S(6)ac(aeS(2) + cd*S(2))(S(2)cd - esqrt(-S(4)ac + bS(2))) - bS(2)(-S(3)acde*S(2) - ae*S(3)sqrt(-S(4)ac + bS(2)) + cS(2)dS(3)) + bc(ae*S(2)(-S(8)ae + S(3)dsqrt(-S(4)ac + b*S(2))) + cd*S(2)(-S(12)ae + dsqrt(-S(4)ac + bS(2)))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)ac(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - sqrt(S(2))(ab*S(3)e*S(3) + S(6)ac(aeS(2) + cd*S(2))(S(2)cd + esqrt(-S(4)ac + bS(2))) - bS(2)(-S(3)acde*S(2) + ae*S(3)sqrt(-S(4)ac + bS(2)) + cS(2)dS(3)) - bc(ae*S(2)(S(8)ae + S(3)dsqrt(-S(4)ac + b*S(2))) + cd*S(2)(S(12)ae + dsqrt(-S(4)ac + bS(2)))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)ac(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)/(a + bx*S(2) + cxS(4))S(2), x), x, x(-S(2)abde - S(2)a(-ae*S(2) + cdS(2)) + bS(2)dS(2) + xS(2)(abe*S(2) - S(4)acde + bcdS(2)))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))) - sqrt(S(2))(-S(4)ac(S(3)cdS(2) - e(-ae + dsqrt(-S(4)ac + bS(2)))) + bS(2)(-ae*S(2) + cd*S(2)) - b(aeS(2)sqrt(-S(4)ac + b*S(2)) + cd(-S(8)ae + dsqrt(-S(4)ac + b*S(2)))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(c)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))(-S(4)ac(S(3)cd*S(2) + e(ae + dsqrt(-S(4)ac + bS(2)))) + bS(2)(-ae*S(2) + cd*S(2)) + b(aeS(2)sqrt(-S(4)ac + b*S(2)) + cd(S(8)ae + dsqrt(-S(4)ac + b*S(2)))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(c)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/(a + bx*S(2) + cxS(4))S(2), x), x, sqrt(S(2))sqrt(c)(-S(2)ae + bd - (S(4)abe - S(12)acd + bS(2)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)(-S(2)ae + bd + (S(4)abe - S(12)acd + b*S(2)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + x(-abe - S(2)acd + bS(2)d + cxS(2)(-S(2)ae + bd))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(-2)), x), x, -sqrt(S(2))sqrt(c)(-S(12)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))sqrt(c)(-S(12)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + x(-S(2)ac + bS(2) + bcxS(2))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex*S(2))(a + bxS(2) + cxS(4))S(2)), x), x, -sqrt(S(2))sqrt(c)e*S(2)(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(2)) - sqrt(S(2))sqrt(c)e*S(2)(e - (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(2)) + e*(S(7)/2)atan(sqrt(e)x/sqrt(d))/(sqrt(d)(aeS(2) - bde + cdS(2))S(2)) + sqrt(S(2))sqrt(c)(S(2)ace - bS(2)e + bcd - (S(8)abce - S(12)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))(aeS(2) - bde + cd*S(2))) + sqrt(S(2))sqrt(c)(S(2)ace - b*S(2)e + bcd + (S(8)abce - S(12)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))(aeS(2) - bde + cd*S(2))) + x(S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd + cx*S(2)(S(2)ace - bS(2)e + bcd))/(S(2)a(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))(aeS(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + exS(2))S(2)(a + bx*S(2) + cxS(4))S(2)), x), x, -sqrt(S(2))sqrt(c)e*S(2)(beS(2)(b - sqrt(-S(4)ac + b*S(2))) + S(3)c*S(2)d*S(2) - ce(ae + S(3)bd - S(2)dsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cdS(2))S(3)) + sqrt(S(2))sqrt(c)e*S(2)(beS(2)(b + sqrt(-S(4)ac + b*S(2))) + S(3)c*S(2)d*S(2) - ce(ae + S(3)bd + S(2)dsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cdS(2))S(3)) + e*S(4)x/(S(2)d(d + exS(2))(aeS(2) - bde + cdS(2))S(2)) + S(2)e(S(7)/2)(-be + S(2)cd)atan(sqrt(e)x/sqrt(d))/(sqrt(d)(aeS(2) - bde + cdS(2))S(3)) + e*(S(7)/2)atan(sqrt(e)x/sqrt(d))/(S(2)d*(S(3)/2)(aeS(2) - bde + cdS(2))S(2)) - sqrt(S(2))sqrt(c)(-S(4)ac*S(2)(S(3)cd*S(2) + e(-S(3)ae + dsqrt(-S(4)ac + bS(2)))) + bS(4)eS(2) - bS(3)e(S(2)cd + esqrt(-S(4)ac + bS(2))) + bS(2)c(cd*S(2) + e(-S(9)ae + S(2)dsqrt(-S(4)ac + b*S(2)))) + bc(S(3)aeS(2)sqrt(-S(4)ac + b*S(2)) - cd(-S(16)ae + dsqrt(-S(4)ac + b*S(2)))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)(ae*S(2) - bde + cdS(2))S(2)) + sqrt(S(2))sqrt(c)(-S(4)ac*S(2)(S(3)cd*S(2) - e(S(3)ae + dsqrt(-S(4)ac + bS(2)))) + bS(4)eS(2) - bS(3)e(S(2)cd - esqrt(-S(4)ac + bS(2))) + bS(2)c(cd*S(2) - e(S(9)ae + S(2)dsqrt(-S(4)ac + b*S(2)))) - bc(S(3)aeS(2)sqrt(-S(4)ac + b*S(2)) - cd(S(16)ae + dsqrt(-S(4)ac + b*S(2)))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)(ae*S(2) - bde + cdS(2))S(2)) - x(-S(6)abc*S(2)de + S(2)acS(2)(-aeS(2) + cdS(2)) - bS(4)eS(2) + S(2)b*S(3)cde - b*S(2)c(-S(4)aeS(2) + cd*S(2)) + cx*S(2)(-S(4)ac*S(2)de - bS(3)e*S(2) + S(2)b*S(2)cde - bc(-S(3)ae*S(2) + cd*S(2))))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))(aeS(2) - bde + cdS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(2) + S(3))/(xS(4) - S(2)x*S(2) + S(1)), x), x, S(5)x/(-S(2)xS(2) + S(2)) + atanh(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x*S(2) + S(2))/(S(3)x*S(4) - S(8)x*S(2) + S(5)), x), x, S(5)atanh(x)/S(2) - S(7)sqrt(S(15))atanh(sqrt(S(15))x/S(5))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/(S(3)x*S(4) - S(8)x*S(2) + S(5)), x), x, (d/S(2) + e/S(2))atanh(x) - sqrt(S(15))(S(3)d + S(5)e)atanh(sqrt(S(15))x/S(5))/S(30), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(3))/(xS(4) + S(3)x*S(2) + S(1)), x), x, sqrt(S(10))(sqrt(S(5)) + S(3))*(S(3)/2)atan(xsqrt(sqrt(S(5))/S(2) + S(3)/2))/S(20) - sqrt(-S(80)sqrt(S(5)) + S(180))atan(sqrt(S(2))x/sqrt(sqrt(S(5)) + S(3)))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(4) + xS(2) + S(1)), x), x, -(a/S(4) - b/S(4))log(x*S(2) - x + S(1)) + (a/S(4) - b/S(4))log(x*S(2) + x + S(1)) - sqrt(S(3))(a + b)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) + sqrt(S(3))(a + b)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(4) + xS(2) + S(1))S(2), x), x, x(a + b - xS(2)(a - S(2)b))/(S(6)x*S(4) + S(6)x*S(2) + S(6)) - (a/S(4) - b/S(8))log(x*S(2) - x + S(1)) + (a/S(4) - b/S(8))log(x*S(2) + x + S(1)) - sqrt(S(3))(S(4)a + b)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(36) + sqrt(S(3))(S(4)a + b)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(36), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(4) + x*S(2) + S(2)), x), x, -(a - sqrt(S(2))b)log(xS(2) - xsqrt(S(-1) + S(2)sqrt(S(2))) + sqrt(S(2)))/(S(4)sqrt(S(-2) + S(4)sqrt(S(2)))) + (a - sqrt(S(2))b)log(xS(2) + xsqrt(S(-1) + S(2)sqrt(S(2))) + sqrt(S(2)))/(S(4)sqrt(S(-2) + S(4)sqrt(S(2)))) - sqrt(S(-1)/14 + sqrt(S(2))/S(7))(a + sqrt(S(2))b)atan((-S(2)x + sqrt(S(-1) + S(2)sqrt(S(2))))/sqrt(S(1) + S(2)sqrt(S(2))))/S(2) + sqrt(S(-1)/14 + sqrt(S(2))/S(7))(a + sqrt(S(2))b)atan((S(2)x + sqrt(S(-1) + S(2)sqrt(S(2))))/sqrt(S(1) + S(2)sqrt(S(2))))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(4) + xS(2) + S(2))S(2), x), x, x(S(3)a + S(2)b - xS(2)(a - S(4)b))/(S(28)x*S(4) + S(28)x*S(2) + S(56)) - sqrt(S(-1)/14 + sqrt(S(2))/S(7))(a(-sqrt(S(2)) + S(11)) - b(-S(4)sqrt(S(2)) + S(2)))atan((-S(2)x + sqrt(S(-1) + S(2)sqrt(S(2))))/sqrt(S(1) + S(2)sqrt(S(2))))/S(56) + sqrt(S(-1)/14 + sqrt(S(2))/S(7))(a(-sqrt(S(2)) + S(11)) - b(-S(4)sqrt(S(2)) + S(2)))atan((S(2)x + sqrt(S(-1) + S(2)sqrt(S(2))))/sqrt(S(1) + S(2)sqrt(S(2))))/S(56) - (S(11)a - S(2)b + sqrt(S(2))(a - S(4)b))log(x*S(2) - xsqrt(S(-1) + S(2)sqrt(S(2))) + sqrt(S(2)))/(S(112)sqrt(S(-2) + S(4)sqrt(S(2)))) + (a(sqrt(S(2)) + S(11)) - S(4)sqrt(S(2))b - S(2)b)log(x*S(2) + xsqrt(S(-1) + S(2)sqrt(S(2))) + sqrt(S(2)))/(S(112)sqrt(S(-2) + S(4)sqrt(S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + sqrt(S(2)))/(xS(4) - sqrt(S(2))x*S(2) + S(1)), x), x, -sqrt(sqrt(S(2))/S(2) + S(1))log(x*S(2) - xsqrt(sqrt(S(2)) + S(2)) + S(1))/S(4) + sqrt(sqrt(S(2))/S(2) + S(1))log(xS(2) + xsqrt(sqrt(S(2)) + S(2)) + S(1))/S(4) - atan((-S(2)x + sqrt(sqrt(S(2)) + S(2)))/sqrt(-sqrt(S(2)) + S(2)))/(S(2)sqrt(sqrt(S(2)) + S(2))) + atan((S(2)x + sqrt(sqrt(S(2)) + S(2)))/sqrt(-sqrt(S(2)) + S(2)))/(S(2)sqrt(sqrt(S(2)) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + sqrt(S(2)))/(xS(4) + sqrt(S(2))xS(2) + S(1)), x), x, -sqrt(-sqrt(S(2))/S(2) + S(1))log(x*S(2) - xsqrt(-sqrt(S(2)) + S(2)) + S(1))/S(4) + sqrt(-sqrt(S(2))/S(2) + S(1))log(xS(2) + xsqrt(-sqrt(S(2)) + S(2)) + S(1))/S(4) - atan((-S(2)x + sqrt(-sqrt(S(2)) + S(2)))/sqrt(sqrt(S(2)) + S(2)))/(S(2)sqrt(-sqrt(S(2)) + S(2))) + atan((S(2)x + sqrt(-sqrt(S(2)) + S(2)))/sqrt(sqrt(S(2)) + S(2)))/(S(2)sqrt(-sqrt(S(2)) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-x*S(2) + sqrt(S(2)))/(bxS(2) + xS(4) + S(1)), x), x, (-sqrt(S(2)) + S(1))atan((-S(2)x + sqrt(-b + S(2)))/sqrt(b + S(2)))/(S(2)sqrt(b + S(2))) - (-sqrt(S(2)) + S(1))atan((S(2)x + sqrt(-b + S(2)))/sqrt(b + S(2)))/(S(2)sqrt(b + S(2))) - (S(1) + sqrt(S(2)))log(xS(2) - xsqrt(-b + S(2)) + S(1))/(S(4)sqrt(-b + S(2))) + (S(1) + sqrt(S(2)))log(x*S(2) + xsqrt(-b + S(2)) + S(1))/(S(4)sqrt(-b + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + sqrt(S(2)))/(bxS(2) + xS(4) + S(1)), x), x, -(S(1) + sqrt(S(2)))atan((-S(2)x + sqrt(-b + S(2)))/sqrt(b + S(2)))/(S(2)sqrt(b + S(2))) + (S(1) + sqrt(S(2)))atan((S(2)x + sqrt(-b + S(2)))/sqrt(b + S(2)))/(S(2)sqrt(b + S(2))) + (-sqrt(S(2)) + S(1))log(xS(2) - xsqrt(-b + S(2)) + S(1))/(S(4)sqrt(-b + S(2))) - (-sqrt(S(2)) + S(1))log(x*S(2) + xsqrt(-b + S(2)) + S(1))/(S(4)sqrt(-b + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)a - xS(2))/(aS(2) - axS(2) + xS(4)), x), x, -sqrt(S(3))log(-sqrt(S(3))sqrt(a)x + a + x*S(2))/(S(4)sqrt(a)) + sqrt(S(3))log(sqrt(S(3))sqrt(a)x + a + xS(2))/(S(4)sqrt(a)) - atan((sqrt(S(3))sqrt(a) - S(2)x)/sqrt(a))/(S(2)sqrt(a)) + atan((sqrt(S(3))sqrt(a) + S(2)x)/sqrt(a))/(S(2)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)sqrt(a) - xS(2))/(-sqrt(a)xS(2) + a + xS(4)), x), x, -sqrt(S(3))log(-sqrt(S(3))a*(S(1)/4)x + sqrt(a) + x*S(2))/(S(4)a*(S(1)/4)) + sqrt(S(3))log(sqrt(S(3))a(S(1)/4)x + sqrt(a) + x*S(2))/(S(4)a*(S(1)/4)) - atan((sqrt(S(3))a*(S(1)/4) - S(2)x)/a*(S(1)/4))/(S(2)a*(S(1)/4)) + atan((sqrt(S(3))a*(S(1)/4) + S(2)x)/a*(S(1)/4))/(S(2)a*(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)b(S(2)/3) + xS(2))/(b(S(4)/3) + b(S(2)/3)xS(2) + xS(4)), x), x, -log(b(S(2)/3) - b(S(1)/3)x + x*S(2))/(S(4)b(S(1)/3)) + log(b(S(2)/3) + b*(S(1)/3)x + x*S(2))/(S(4)b*(S(1)/3)) - sqrt(S(3))atan(sqrt(S(3))(b(S(1)/3) - S(2)x)/(S(3)b(S(1)/3)))/(S(2)b*(S(1)/3)) + sqrt(S(3))atan(sqrt(S(3))(b(S(1)/3) + S(2)x)/(S(3)b(S(1)/3)))/(S(2)b*(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(aS(2) - axS(2) + xS(4)), x), x, -sqrt(S(3))(A - Ba)log(-sqrt(S(3))sqrt(a)x + a + x*S(2))/(S(12)a*(S(3)/2)) + sqrt(S(3))(A - Ba)log(sqrt(S(3))sqrt(a)x + a + x*S(2))/(S(12)a*(S(3)/2)) - (A + Ba)atan((sqrt(S(3))sqrt(a) - S(2)x)/sqrt(a))/(S(2)a*(S(3)/2)) + (A + Ba)atan((sqrt(S(3))sqrt(a) + S(2)x)/sqrt(a))/(S(2)a*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))/(-sqrt(a)xS(2) + a + xS(4)), x), x, -sqrt(S(3))(A - Bsqrt(a))log(-sqrt(S(3))a*(S(1)/4)x + sqrt(a) + x*S(2))/(S(12)a*(S(3)/4)) + sqrt(S(3))(A - Bsqrt(a))log(sqrt(S(3))a(S(1)/4)x + sqrt(a) + x*S(2))/(S(12)a*(S(3)/4)) - (A + Bsqrt(a))atan((sqrt(S(3))a*(S(1)/4) - S(2)x)/a*(S(1)/4))/(S(2)a*(S(3)/4)) + (A + Bsqrt(a))atan((sqrt(S(3))a*(S(1)/4) + S(2)x)/a*(S(1)/4))/(S(2)a*(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))/(a + cxS(4) - xS(2)sqrt(ac)), x), x, -(A - Bsqrt(a)/sqrt(c))log(sqrt(a) + sqrt(c)xS(2) - xsqrt(S(2)sqrt(a)sqrt(c) + sqrt(ac)))/(S(4)sqrt(a)sqrt(S(2)sqrt(a)sqrt(c) + sqrt(ac))) + (A - Bsqrt(a)/sqrt(c))log(sqrt(a) + sqrt(c)xS(2) + xsqrt(S(2)sqrt(a)sqrt(c) + sqrt(ac)))/(S(4)sqrt(a)sqrt(S(2)sqrt(a)sqrt(c) + sqrt(ac))) - (Asqrt(c) + Bsqrt(a))atan((-S(2)sqrt(c)x + sqrt(S(2)sqrt(a)sqrt(c) + sqrt(ac)))/sqrt(S(2)sqrt(a)sqrt(c) - sqrt(ac)))/(S(2)sqrt(a)sqrt(c)sqrt(S(2)sqrt(a)sqrt(c) - sqrt(ac))) + (Asqrt(c) + Bsqrt(a))atan((S(2)sqrt(c)x + sqrt(S(2)sqrt(a)sqrt(c) + sqrt(ac)))/sqrt(S(2)sqrt(a)sqrt(c) - sqrt(ac)))/(S(2)sqrt(a)sqrt(c)sqrt(S(2)sqrt(a)sqrt(c) - sqrt(ac))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(-sqrt(a)sqrt(c)xS(2) + a + cx*S(4)), x), x, -sqrt(S(3))(A - Bsqrt(a)/sqrt(c))log(-sqrt(S(3))a(S(1)/4)c*(S(1)/4)x + sqrt(a) + sqrt(c)xS(2))/(S(12)a*(S(3)/4)c*(S(1)/4)) + sqrt(S(3))(A - Bsqrt(a)/sqrt(c))log(sqrt(S(3))a(S(1)/4)c*(S(1)/4)x + sqrt(a) + sqrt(c)xS(2))/(S(12)a*(S(3)/4)c*(S(1)/4)) - (Asqrt(c) + Bsqrt(a))atan((sqrt(S(3))a(S(1)/4) - S(2)c*(S(1)/4)x)/a*(S(1)/4))/(S(2)a*(S(3)/4)c*(S(3)/4)) + (Asqrt(c) + Bsqrt(a))atan((sqrt(S(3))a(S(1)/4) + S(2)c*(S(1)/4)x)/a*(S(1)/4))/(S(2)a*(S(3)/4)c*(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)xS(2) + S(7))S(3)sqrt(xS(4) + S(3)x*S(2) + S(2)), x), x, S(125)x*S(3)(x*S(4) + S(3)xS(2) + S(2))(S(3)/2)/S(9) + S(577)x(x*S(2) + S(2))/(S(3)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + x(S(757)xS(2) + S(2608))sqrt(x*S(4) + S(3)x*S(2) + S(2))/S(21) + S(275)x(xS(4) + S(3)xS(2) + S(2))(S(3)/2)/S(7) - S(577)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(3)sqrt(xS(4) + S(3)x*S(2) + S(2))) + S(2945)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(21)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)xS(2) + S(7))S(2)sqrt(xS(4) + S(3)x*S(2) + S(2)), x), x, S(31)x(xS(2) + S(2))/sqrt(xS(4) + S(3)x*S(2) + S(2)) + x(S(114)xS(2) + S(407))sqrt(x*S(4) + S(3)x*S(2) + S(2))/S(21) + S(25)x(xS(4) + S(3)xS(2) + S(2))(S(3)/2)/S(7) - S(31)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_e(atan(x), S(1)/2)/sqrt(x*S(4) + S(3)x*S(2) + S(2)) + S(472)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(21)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)x*S(2) + S(7))sqrt(x*S(4) + S(3)x*S(2) + S(2)), x), x, S(5)x(xS(2) + S(2))/sqrt(xS(4) + S(3)x*S(2) + S(2)) + x(S(3)xS(2) + S(10))sqrt(x*S(4) + S(3)x*S(2) + S(2))/S(3) - S(5)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_e(atan(x), S(1)/2)/sqrt(x*S(4) + S(3)x*S(2) + S(2)) + S(11)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(3)sqrt(xS(4) + S(3)xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xS(4) + S(3)xS(2) + S(2)), x), x, x(xS(2) + S(2))/sqrt(xS(4) + S(3)xS(2) + S(2)) + xsqrt(x*S(4) + S(3)x*S(2) + S(2))/S(3) - sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_e(atan(x), S(1)/2)/sqrt(x*S(4) + S(3)x*S(2) + S(2)) + S(2)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(3)sqrt(xS(4) + S(3)xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xS(4) + S(3)xS(2) + S(2))/(S(5)x*S(2) + S(7)), x), x, x(x*S(2) + S(2))/(S(5)sqrt(x*S(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(5)sqrt(xS(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(3)x*S(2) + S(3))elliptic_pi(S(2)/7, atan(x), S(1)/2)/(S(70)sqrt(xS(4) + S(3)xS(2) + S(2))) + sqrt((xS(2) + S(2))/(S(2)xS(2) + S(2)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(5)sqrt(xS(4) + S(3)xS(2) + S(2))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt(xS(4) + S(3)xS(2) + S(2))/(S(5)x*S(2) + S(7)), x), x, x(x*S(2) + S(2))/(S(5)sqrt(x*S(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(5)sqrt(xS(4) + S(3)x*S(2) + S(2))) + S(4)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(25)sqrt(xS(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(3)x*S(2) + S(3))elliptic_f(atan(x), S(1)/2)/(S(50)sqrt(xS(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))(S(3)xS(2) + S(6))elliptic_pi(S(2)/7, atan(x), S(1)/2)/(S(70)sqrt((xS(2) + S(2))/(xS(2) + S(1)))sqrt(x*S(4) + S(3)xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xS(4) + S(3)xS(2) + S(2))/(S(5)xS(2) + S(7))S(2), x), x, xsqrt(xS(4) + S(3)x*S(2) + S(2))/(S(70)x*S(2) + S(98)) - xsqrt(x*S(4) + S(3)x*S(2) + S(2))/(S(70)x*S(2) + S(70)) + sqrt(S(2))sqrt(x*S(4) + S(3)x*S(2) + S(2))elliptic_e(atan(x), S(1)/2)/(S(70)sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))) + S(3)sqrt(S(2))sqrt(xS(4) + S(3)x*S(2) + S(2))elliptic_f(atan(x), S(1)/2)/(S(280)sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))) - sqrt(S(2))sqrt(x*S(4) + S(3)x*S(2) + S(2))elliptic_pi(S(2)/7, atan(x), S(1)/2)/(S(1960)sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)xS(2) + S(7))S(3)(xS(4) + S(3)xS(2) + S(2))(S(3)/2), x), x, S(125)xS(3)(x*S(4) + S(3)xS(2) + S(2))(S(5)/2)/S(13) + S(20884)x(x*S(2) + S(2))/(S(65)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + x(S(65345)xS(2) + S(208212))(x*S(4) + S(3)xS(2) + S(2))(S(3)/2)/S(3003) + x(S(297911)x*S(2) + S(1032541))sqrt(x*S(4) + S(3)x*S(2) + S(2))/S(5005) + S(3825)x(xS(4) + S(3)xS(2) + S(2))(S(5)/2)/S(143) - S(20884)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(65)sqrt(xS(4) + S(3)x*S(2) + S(2))) + S(1171349)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(5005)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)xS(2) + S(7))S(2)(xS(4) + S(3)xS(2) + S(2))(S(3)/2), x), x, S(742)x(x*S(2) + S(2))/(S(15)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + x(S(2240)xS(2) + S(7281))(x*S(4) + S(3)xS(2) + S(2))(S(3)/2)/S(693) + x(S(10643)x*S(2) + S(36783))sqrt(x*S(4) + S(3)x*S(2) + S(2))/S(1155) + S(25)x(xS(4) + S(3)xS(2) + S(2))(S(5)/2)/S(11) - S(742)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(15)sqrt(xS(4) + S(3)x*S(2) + S(2))) + S(13879)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(385)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)x*S(2) + S(7))(x*S(4) + S(3)xS(2) + S(2))(S(3)/2), x), x, S(116)x(x*S(2) + S(2))/(S(15)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + x(S(35)xS(2) + S(108))(x*S(4) + S(3)xS(2) + S(2))(S(3)/2)/S(63) + x(S(149)x*S(2) + S(519))sqrt(x*S(4) + S(3)x*S(2) + S(2))/S(105) - S(116)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(15)sqrt(xS(4) + S(3)x*S(2) + S(2))) + S(197)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(35)sqrt(xS(4) + S(3)xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + S(3)xS(2) + S(2))(S(3)/2), x), x, S(6)x(xS(2) + S(2))/(S(5)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + x(S(9)xS(2) + S(29))sqrt(x*S(4) + S(3)x*S(2) + S(2))/S(35) + x(x*S(4) + S(3)xS(2) + S(2))(S(3)/2)/S(7) - S(6)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(5)sqrt(xS(4) + S(3)x*S(2) + S(2))) + S(31)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(35)sqrt(xS(4) + S(3)xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + S(3)xS(2) + S(2))(S(3)/2)/(S(5)xS(2) + S(7)), x), x, xS(3)sqrt(xS(4) + S(3)x*S(2) + S(2))/S(25) + S(24)x(xS(2) + S(2))/(S(125)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + S(11)xsqrt(xS(4) + S(3)x*S(2) + S(2))/S(75) - S(24)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(125)sqrt(xS(4) + S(3)x*S(2) + S(2))) + S(47)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(375)sqrt(xS(4) + S(3)x*S(2) + S(2))) + (S(24)x*S(2) + S(24))elliptic_pi(S(-3)/7, atan(sqrt(S(2))x/S(2)), S(-1))/(S(875)sqrt((xS(2) + S(1))/(xS(2) + S(2)))sqrt(xS(4) + S(3)xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + S(3)xS(2) + S(2))(S(3)/2)/(S(5)xS(2) + S(7))S(2), x), x, xsqrt(xS(4) + S(3)x*S(2) + S(2))/S(75) - S(3)xsqrt(xS(4) + S(3)x*S(2) + S(2))/(S(875)x*S(2) + S(1225)) + S(9)xsqrt(xS(4) + S(3)x*S(2) + S(2))/(S(175)x*S(2) + S(175)) + S(8)sqrt(x*S(4) + S(3)x*S(2) + S(2))elliptic_pi(S(-3)/7, atan(sqrt(S(2))x/S(2)), S(-1))/(S(875)sqrt((xS(2) + S(1))/(xS(2) + S(2)))(xS(2) + S(2))) - S(9)sqrt(S(2))sqrt(xS(4) + S(3)x*S(2) + S(2))elliptic_e(atan(x), S(1)/2)/(sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(175)x*S(2) + S(175))) + S(211)sqrt(S(2))sqrt(xS(4) + S(3)x*S(2) + S(2))elliptic_f(atan(x), S(1)/2)/(S(10500)sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))) + S(129)sqrt(S(2))sqrt(xS(4) + S(3)x*S(2) + S(2))elliptic_pi(S(2)/7, atan(x), S(1)/2)/(S(24500)sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(3)/sqrt(a + bxS(2) + cxS(4)), x), x, a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(9)ae*S(3)/c + S(8)b*S(2)eS(3)/cS(2) - S(30)bdeS(2)/c + S(45)d*S(2)e + (S(4)abeS(3) - S(15)acdeS(2) + S(15)c*S(2)d*S(3))/(sqrt(a)c*(S(3)/2)))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(30)c*(S(3)/4)sqrt(a + bxS(2) + cxS(4))) - a(S(1)/4)esqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(8)bS(2)e*S(2) + S(45)c*S(2)d*S(2) - S(3)ce(S(3)ae + S(10)bd))elliptic_e(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(15)c*(S(11)/4)sqrt(a + bxS(2) + cxS(4))) + eS(3)xS(3)sqrt(a + bxS(2) + cx*S(4))/(S(5)c) + e*S(2)x(-S(4)be + S(15)cd)sqrt(a + bxS(2) + cx*S(4))/(S(15)c*S(2)) + exsqrt(a + bx*S(2) + cx*S(4))(S(8)bS(2)e*S(2) + S(45)c*S(2)d*S(2) - S(3)ce(S(3)ae + S(10)bd))/(S(15)c(S(5)/2)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)/sqrt(a + bxS(2) + cxS(4)), x), x, a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(2)be*S(2)/c + S(6)de + (-ae*S(2) + S(3)cdS(2))/(sqrt(a)sqrt(c)))elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(6)c*(S(3)/4)sqrt(a + bxS(2) + cx*S(4))) - S(2)a*(S(1)/4)esqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-be + S(3)cd)elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(3)c*(S(7)/4)sqrt(a + bxS(2) + cxS(4))) + eS(2)xsqrt(a + bxS(2) + cx*S(4))/(S(3)c) + S(2)ex(-be + S(3)cd)sqrt(a + bx*S(2) + cx*S(4))/(S(3)c*(S(3)/2)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/sqrt(a + bx*S(2) + cxS(4)), x), x, -a(S(1)/4)esqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(c(S(3)/4)sqrt(a + bxS(2) + cxS(4))) + a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(e + sqrt(c)d/sqrt(a))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)c*(S(3)/4)sqrt(a + bxS(2) + cx*S(4))) + exsqrt(a + bx*S(2) + cx*S(4))/(sqrt(c)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex*S(2))sqrt(a + bxS(2) + cx*S(4))), x), x, atan(xsqrt(ae/d - b + cd/e)/sqrt(a + bxS(2) + cx*S(4)))/(S(2)dsqrt(ae/d - b + cd/e)) + c(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(1)/4)(-sqrt(a)e + sqrt(c)d)sqrt(a + bx*S(2) + cx*S(4))) - sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(e + sqrt(c)d/sqrt(a))elliptic_pi(-sqrt(a)(-e + sqrt(c)d/sqrt(a))*S(2)/(S(4)sqrt(c)de), S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(4)a*(S(1)/4)c*(S(1)/4)d(-e + sqrt(c)d/sqrt(a))sqrt(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + exS(2))S(2)sqrt(a + bx*S(2) + cxS(4))), x), x, a(S(1)/4)c(S(1)/4)esqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)dsqrt(a + bx*S(2) + cx*S(4))(aeS(2) - bde + cdS(2))) - a(S(1)/4)c(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(e + sqrt(c)d/sqrt(a))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(4)dsqrt(a + bx*S(2) + cx*S(4))(aeS(2) - bde + cdS(2))) - a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(e + sqrt(c)d/sqrt(a))(S(3)cd*S(2) - e(-ae + S(2)bd))elliptic_pi(-(-sqrt(a)e + sqrt(c)d)*S(2)/(S(4)sqrt(a)sqrt(c)de), S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(8)c*(S(1)/4)d*S(2)(-sqrt(a)e + sqrt(c)d)sqrt(a + bx*S(2) + cx*S(4))(aeS(2) - bde + cd*S(2))) - sqrt(c)exsqrt(a + bxS(2) + cx*S(4))/(S(2)d(sqrt(a) + sqrt(c)x*S(2))(aeS(2) - bde + cdS(2))) + eS(2)xsqrt(a + bxS(2) + cx*S(4))/(S(2)d(d + ex*S(2))(aeS(2) - bde + cd*S(2))) + (S(3)cdS(2) - e(-ae + S(2)bd))atan(xsqrt(ae/d - b + cd/e)/sqrt(a + bx*S(2) + cx*S(4)))/(S(4)d*S(3)e(ae/d - b + cd/e)(S(3)/2)) + c(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(3)cd*S(2) - e(-ae + S(2)bd))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(4)a*(S(1)/4)d(-sqrt(a)e + sqrt(c)d)sqrt(a + bxS(2) + cx*S(4))(aeS(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(3)/sqrt(a + bxS(2) - cxS(4)), x), x, -eS(3)xS(3)sqrt(a + bxS(2) - cx*S(4))/(S(5)c) - e*S(2)x(S(4)be + S(15)cd)sqrt(a + bxS(2) - cx*S(4))/(S(15)c*S(2)) + sqrt(S(2))(b - sqrt(S(4)ac + b*S(2)))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))(S(9)ae*S(3)/c + S(8)b*S(2)eS(3)/cS(2) + S(30)bdeS(2)/c + S(45)d*S(2)e + (S(8)abeS(3) + S(30)acdeS(2) + S(30)c*S(2)d*S(3))/(bc - csqrt(S(4)ac + bS(2))))elliptic_f(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(60)c*(S(3)/2)sqrt(a + bxS(2) - cx*S(4))) - sqrt(S(2))e(b - sqrt(S(4)ac + bS(2)))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))(S(8)bS(2)e*S(2) + S(45)c*S(2)d*S(2) + S(3)ce(S(3)ae + S(10)bd))elliptic_e(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + bS(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(60)c*(S(7)/2)sqrt(a + bxS(2) - cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)/sqrt(a + bxS(2) - cxS(4)), x), x, -eS(2)xsqrt(a + bxS(2) - cx*S(4))/(S(3)c) - sqrt(S(2))e(b - sqrt(S(4)ac + b*S(2)))sqrt(b + sqrt(S(4)ac + b*S(2)))(be + S(3)cd)sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_e(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(6)c*(S(5)/2)sqrt(a + bxS(2) - cx*S(4))) + sqrt(S(2))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))(beS(2)(b - sqrt(S(4)ac + b*S(2))) + S(3)c*S(2)d*S(2) + ce(ae + S(3)bd - S(3)dsqrt(S(4)ac + b*S(2))))elliptic_f(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(6)c*(S(5)/2)sqrt(a + bxS(2) - cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/sqrt(a + bx*S(2) - cx*S(4)), x), x, -sqrt(S(2))e(b - sqrt(S(4)ac + bS(2)))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_e(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(4)c*(S(3)/2)sqrt(a + bxS(2) - cx*S(4))) + sqrt(S(2))sqrt(b + sqrt(S(4)ac + b*S(2)))(S(2)cd + e(b - sqrt(S(4)ac + bS(2))))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_f(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(4)c*(S(3)/2)sqrt(a + bxS(2) - cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex*S(2))sqrt(a + bxS(2) - cx*S(4))), x), x, sqrt(S(2))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_pi(-e(b + sqrt(S(4)ac + bS(2)))/(S(2)cd), asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + bS(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(2)sqrt(c)dsqrt(a + bxS(2) - cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + exS(2))S(2)sqrt(a + bx*S(2) - cxS(4))), x), x, -eS(2)xsqrt(a + bxS(2) - cx*S(4))/(S(2)d(d + ex*S(2))(-aeS(2) + bde + cd*S(2))) + sqrt(S(2))e(b - sqrt(S(4)ac + bS(2)))sqrt(b + sqrt(S(4)ac + b*S(2)))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_e(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(8)sqrt(c)d(cdS(2) + e(-ae + bd))sqrt(a + bx*S(2) - cx*S(4))) - sqrt(S(2))sqrt(b + sqrt(S(4)ac + b*S(2)))(S(2)cd + e(b - sqrt(S(4)ac + bS(2))))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_f(asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(8)sqrt(c)d(cdS(2) + e(-ae + bd))sqrt(a + bx*S(2) - cx*S(4))) + sqrt(S(2))sqrt(b + sqrt(S(4)ac + b*S(2)))(S(3)cd*S(2) + e(-ae + S(2)bd))sqrt(-S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(-S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_pi(-e(b + sqrt(S(4)ac + bS(2)))/(S(2)cd), asin(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + bS(2)))), (b + sqrt(S(4)ac + bS(2)))/(b - sqrt(S(4)ac + bS(2))))/(S(4)sqrt(c)dS(2)(cdS(2) + e(-ae + bd))sqrt(a + bx*S(2) - cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/sqrt(-a + bx*S(2) + cx*S(4)), x), x, ex(b - sqrt(S(4)ac + bS(2)))(S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))/(S(2)csqrt(-a + bx*S(2) + cx*S(4))) + sqrt(S(2))dsqrt(b + sqrt(S(4)ac + bS(2)))(S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))elliptic_f(atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), -S(2)sqrt(S(4)ac + b*S(2))/(b - sqrt(S(4)ac + bS(2))))/(S(2)sqrt(c)sqrt((S(2)cxS(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))/(S(2)cxS(2)/(b + sqrt(S(4)ac + bS(2))) + S(1)))sqrt(-a + bxS(2) + cx*S(4))) - sqrt(S(2))e(b - sqrt(S(4)ac + bS(2)))sqrt(b + sqrt(S(4)ac + b*S(2)))(S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))elliptic_e(atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), -S(2)sqrt(S(4)ac + b*S(2))/(b - sqrt(S(4)ac + bS(2))))/(S(4)c*(S(3)/2)sqrt((S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))/(S(2)cxS(2)/(b + sqrt(S(4)ac + bS(2))) + S(1)))sqrt(-a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex*S(2))sqrt(-a + bxS(2) + cx*S(4))), x), x, sqrt(S(2))sqrt(-b + sqrt(S(4)ac + b*S(2)))sqrt(S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(S(4)ac + bS(2))) + S(1))elliptic_pi(e(b - sqrt(S(4)ac + bS(2)))/(S(2)cd), asin(sqrt(S(2))sqrt(c)x/sqrt(-b + sqrt(S(4)ac + bS(2)))), (b - sqrt(S(4)ac + bS(2)))/(b + sqrt(S(4)ac + bS(2))))/(S(2)sqrt(c)dsqrt(-a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/((d + ex*S(2))sqrt(-a + bxS(2) + cx*S(4))), x), x, sqrt(S(2))sqrt(c)sqrt(b + sqrt(S(4)ac + bS(2)))(S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))elliptic_f(atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + b*S(2)))), -S(2)sqrt(S(4)ac + b*S(2))/(b - sqrt(S(4)ac + bS(2))))/(sqrt((S(2)cxS(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))/(S(2)cxS(2)/(b + sqrt(S(4)ac + bS(2))) + S(1)))(S(2)cd - e(b + sqrt(S(4)ac + bS(2))))sqrt(-a + bxS(2) + cx*S(4))) - sqrt(S(2))e(b + sqrt(S(4)ac + bS(2)))(S(3)/2)(S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))elliptic_pi(S(1) - e(b + sqrt(S(4)ac + bS(2)))/(S(2)cd), atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(S(4)ac + bS(2)))), -S(2)sqrt(S(4)ac + b*S(2))/(b - sqrt(S(4)ac + bS(2))))/(S(2)sqrt(c)dsqrt((S(2)cx*S(2)/(b - sqrt(S(4)ac + bS(2))) + S(1))/(S(2)cxS(2)/(b + sqrt(S(4)ac + bS(2))) + S(1)))(S(2)cd - e(b + sqrt(S(4)ac + bS(2))))sqrt(-a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/sqrt(-a + bx*S(2) - cxS(4)), x), x, -a(S(1)/4)esqrt((a - bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 + b/(S(4)sqrt(a)sqrt(c)))/(c(S(3)/4)sqrt(-a + bxS(2) - cxS(4))) + a(S(1)/4)sqrt((a - bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(e + sqrt(c)d/sqrt(a))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 + b/(S(4)sqrt(a)sqrt(c)))/(S(2)c*(S(3)/4)sqrt(-a + bxS(2) - cx*S(4))) - exsqrt(-a + bx*S(2) - cx*S(4))/(sqrt(c)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex*S(2))sqrt(-a + bxS(2) - cx*S(4))), x), x, atan(xsqrt(-ae/d - b - cd/e)/sqrt(-a + bxS(2) - cx*S(4)))/(S(2)dsqrt(-ae/d - b - cd/e)) + c(S(1)/4)sqrt((a - bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 + b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(1)/4)(-sqrt(a)e + sqrt(c)d)sqrt(-a + bx*S(2) - cx*S(4))) - sqrt((a - bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(e + sqrt(c)d/sqrt(a))elliptic_pi(-sqrt(a)(-e + sqrt(c)d/sqrt(a))*S(2)/(S(4)sqrt(c)de), S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 + b/(S(4)sqrt(a)sqrt(c)))/(S(4)a*(S(1)/4)c*(S(1)/4)d(-e + sqrt(c)d/sqrt(a))sqrt(-a + bx*S(2) - cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(3)/sqrt(x*S(4) + S(3)xS(2) + S(2)), x), x, eS(3)xS(3)sqrt(x*S(4) + S(3)xS(2) + S(2))/S(5) + eS(2)x(d - S(4)e/S(5))sqrt(x*S(4) + S(3)x*S(2) + S(2)) + S(3)ex(x*S(2) + S(2))(S(5)dS(2) - S(10)de + S(6)e*S(2))/(S(5)sqrt(x*S(4) + S(3)x*S(2) + S(2))) - S(3)sqrt(S(2))esqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))(S(5)dS(2) - S(10)de + S(6)e*S(2))elliptic_e(atan(x), S(1)/2)/(S(5)sqrt(xS(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))(S(5)dS(3) - S(10)deS(2) + S(8)e*S(3))elliptic_f(atan(x), S(1)/2)/(S(10)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)/sqrt(x*S(4) + S(3)xS(2) + S(2)), x), x, eS(2)xsqrt(x*S(4) + S(3)x*S(2) + S(2))/S(3) + ex(S(2)d - S(2)e)(xS(2) + S(2))/sqrt(xS(4) + S(3)xS(2) + S(2)) - S(2)sqrt(S(2))esqrt((xS(2) + S(2))/(xS(2) + S(1)))(d - e)(x*S(2) + S(1))elliptic_e(atan(x), S(1)/2)/sqrt(x*S(4) + S(3)x*S(2) + S(2)) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(3)d*S(2) - S(2)e*S(2))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(6)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/sqrt(xS(4) + S(3)xS(2) + S(2)), x), x, sqrt(S(2))dsqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(2)sqrt(xS(4) + S(3)x*S(2) + S(2))) + ex(xS(2) + S(2))/sqrt(xS(4) + S(3)x*S(2) + S(2)) - sqrt(S(2))esqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_e(atan(x), S(1)/2)/sqrt(x*S(4) + S(3)x*S(2) + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex*S(2))sqrt(x*S(4) + S(3)x*S(2) + S(2))), x), x, sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(2)(d - e)sqrt(x*S(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))esqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_pi(S(1) - e/d, atan(x), S(1)/2)/(S(2)d(d - e)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/((d + ex*S(2))sqrt(x*S(4) + S(3)x*S(2) + S(2))), x), x, sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(2)(d - e)sqrt(x*S(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))e(xS(2) + S(2))elliptic_pi(S(1) - e/d, atan(x), S(1)/2)/(S(2)dsqrt((xS(2) + S(2))/(xS(2) + S(1)))(d - e)sqrt(x*S(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + exS(2))S(2)sqrt(xS(4) + S(3)xS(2) + S(2))), x), x, eS(2)xsqrt(x*S(4) + S(3)x*S(2) + S(2))/(S(2)d(d + ex*S(2))(d*S(2) - S(3)de + S(2)e*S(2))) - ex(xS(2) + S(2))/(S(2)d(dS(2) - S(3)de + S(2)e*S(2))sqrt(x*S(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))esqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(2)d(d - S(2)e)(d - e)sqrt(xS(4) + S(3)xS(2) + S(2))) + sqrt((xS(2) + S(2))/(S(2)xS(2) + S(2)))(S(2)d - e)(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(2)d(d - e)*S(2)sqrt(x*S(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))e(xS(2) + S(2))(S(3)dS(2) - S(6)de + S(2)e*S(2))elliptic_pi(S(1) - e/d, atan(x), S(1)/2)/(S(4)dS(2)sqrt((xS(2) + S(2))/(xS(2) + S(1)))(d - S(2)e)(d - e)S(2)sqrt(x*S(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/((d + exS(2))S(2)sqrt(xS(4) + S(3)x*S(2) + S(2))), x), x, -sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(4)(d - S(2)e)(d - e)sqrt(x*S(4) + S(3)xS(2) + S(2))) + eS(2)xsqrt(x*S(4) + S(3)x*S(2) + S(2))/(S(2)d(d + ex*S(2))(d*S(2) - S(3)de + S(2)e*S(2))) - ex(xS(2) + S(2))/(S(2)d(dS(2) - S(3)de + S(2)e*S(2))sqrt(x*S(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))esqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(2)d(d - S(2)e)(d - e)sqrt(xS(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))(S(3)dS(2) - S(6)de + S(2)e*S(2))elliptic_f(atan(x), S(1)/2)/(S(4)d(d - S(2)e)(d - e)*S(2)sqrt(x*S(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))e(xS(2) + S(2))(S(3)dS(2) - S(6)de + S(2)e*S(2))elliptic_pi(S(1) - e/d, atan(x), S(1)/2)/(S(4)dS(2)sqrt((xS(2) + S(2))/(xS(2) + S(1)))(d - S(2)e)(d - e)S(2)sqrt(x*S(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)xS(2) + S(7))S(3)/sqrt(x*S(4) + S(3)x*S(2) + S(2)), x), x, S(25)x*S(3)sqrt(x*S(4) + S(3)x*S(2) + S(2)) + S(135)x(xS(2) + S(2))/sqrt(xS(4) + S(3)x*S(2) + S(2)) + S(75)xsqrt(xS(4) + S(3)x*S(2) + S(2)) - S(135)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_e(atan(x), S(1)/2)/sqrt(x*S(4) + S(3)x*S(2) + S(2)) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(193)x*S(2) + S(193))elliptic_f(atan(x), S(1)/2)/(S(2)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)xS(2) + S(7))S(2)/sqrt(x*S(4) + S(3)x*S(2) + S(2)), x), x, S(20)x(xS(2) + S(2))/sqrt(xS(4) + S(3)x*S(2) + S(2)) + S(25)xsqrt(xS(4) + S(3)x*S(2) + S(2))/S(3) - S(20)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_e(atan(x), S(1)/2)/sqrt(x*S(4) + S(3)x*S(2) + S(2)) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(97)x*S(2) + S(97))elliptic_f(atan(x), S(1)/2)/(S(6)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)xS(2) + S(7))/sqrt(xS(4) + S(3)xS(2) + S(2)), x), x, S(5)x(xS(2) + S(2))/sqrt(xS(4) + S(3)x*S(2) + S(2)) - S(5)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_e(atan(x), S(1)/2)/sqrt(x*S(4) + S(3)x*S(2) + S(2)) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(7)x*S(2) + S(7))elliptic_f(atan(x), S(1)/2)/(S(2)sqrt(xS(4) + S(3)xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(xS(4) + S(3)xS(2) + S(2)), x), x, sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(2)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((S(5)x*S(2) + S(7))sqrt(x*S(4) + S(3)x*S(2) + S(2))), x), x, sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(4)sqrt(xS(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))(S(5)xS(2) + S(10))elliptic_pi(S(2)/7, atan(x), S(1)/2)/(S(28)sqrt((xS(2) + S(2))/(xS(2) + S(1)))sqrt(x*S(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((S(5)xS(2) + S(7))S(2)sqrt(xS(4) + S(3)x*S(2) + S(2))), x), x, S(5)x(xS(2) + S(2))/(S(84)sqrt(x*S(4) + S(3)x*S(2) + S(2))) - S(25)xsqrt(xS(4) + S(3)x*S(2) + S(2))/(S(420)x*S(2) + S(588)) - sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(5)x*S(2) + S(5))elliptic_e(atan(x), S(1)/2)/(S(84)sqrt(xS(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(9)x*S(2) + S(9))elliptic_f(atan(x), S(1)/2)/(S(112)sqrt(xS(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))(S(65)xS(2) + S(130))elliptic_pi(S(2)/7, atan(x), S(1)/2)/(S(2352)sqrt((xS(2) + S(2))/(xS(2) + S(1)))sqrt(x*S(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)xS(2) + S(7))S(3)/(x*S(4) + S(3)xS(2) + S(2))(S(3)/2), x), x, x(-S(11)x*S(2) + S(5))/(S(2)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + S(261)x(xS(2) + S(2))/(S(2)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(169)x*S(2) + S(169))elliptic_f(atan(x), S(1)/2)/(S(2)sqrt(xS(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(261)x*S(2) + S(261))elliptic_e(atan(x), S(1)/2)/(S(2)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)xS(2) + S(7))S(2)/(x*S(4) + S(3)xS(2) + S(2))(S(3)/2), x), x, -S(17)x(x*S(2) + S(2))/(S(2)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + x(S(17)xS(2) + S(25))/(S(2)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + S(6)sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(x*S(2) + S(1))elliptic_f(atan(x), S(1)/2)/sqrt(x*S(4) + S(3)x*S(2) + S(2)) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(17)x*S(2) + S(17))elliptic_e(atan(x), S(1)/2)/(S(2)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)xS(2) + S(7))/(xS(4) + S(3)xS(2) + S(2))(S(3)/2), x), x, -x(x*S(2) + S(2))/(S(2)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + x(x*S(2) + S(5))/(S(2)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(2)sqrt(xS(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_f(atan(x), S(1)/2)/(S(2)sqrt(xS(4) + S(3)xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + S(3)xS(2) + S(2))(S(-3)/2), x), x, -S(3)x(xS(2) + S(2))/(S(2)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + x(S(3)xS(2) + S(5))/(S(2)sqrt(x*S(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_f(atan(x), S(1)/2)/sqrt(x*S(4) + S(3)x*S(2) + S(2)) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(3)x*S(2) + S(3))elliptic_e(atan(x), S(1)/2)/(S(2)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((S(5)x*S(2) + S(7))(x*S(4) + S(3)xS(2) + S(2))(S(3)/2)), x), x, x/(S(6)sqrt(xS(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(3)sqrt(xS(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(125)x*S(2) + S(125))elliptic_pi(S(2)/7, atan(x), S(1)/2)/(S(168)sqrt(xS(4) + S(3)xS(2) + S(2))) - sqrt((xS(2) + S(2))/(S(2)xS(2) + S(2)))(S(9)xS(2) + S(9))elliptic_f(atan(x), S(1)/2)/(S(4)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/((S(5)x*S(2) + S(7))(x*S(4) + S(3)xS(2) + S(2))(S(3)/2)), x), x, -x(xS(2) + S(2))/(S(3)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + x(S(2)xS(2) + S(5))/(S(6)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(xS(2) + S(1))elliptic_e(atan(x), S(1)/2)/(S(3)sqrt(xS(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(9)x*S(2) + S(9))elliptic_f(atan(x), S(1)/2)/(S(8)sqrt(xS(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))(S(125)xS(2) + S(250))elliptic_pi(S(2)/7, atan(x), S(1)/2)/(S(168)sqrt((xS(2) + S(2))/(xS(2) + S(1)))sqrt(x*S(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((S(5)xS(2) + S(7))S(2)(xS(4) + S(3)xS(2) + S(2))(S(3)/2)), x), x, S(625)x(x*S(2) + S(1))(x*S(2) + S(2))/((S(2520)x*S(2) + S(3528))sqrt(x*S(4) + S(3)x*S(2) + S(2))) - S(125)x(xS(2) + S(2))/(S(504)sqrt(x*S(4) + S(3)x*S(2) + S(2))) - x/(S(18)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(31)x*S(2) + S(31))elliptic_e(atan(x), S(1)/2)/(S(56)sqrt(xS(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))sqrt((xS(2) + S(2))/(xS(2) + S(1)))(S(375)x*S(2) + S(375))elliptic_pi(S(2)/7, atan(x), S(1)/2)/(S(1568)sqrt(xS(4) + S(3)xS(2) + S(2))) - sqrt((xS(2) + S(2))/(S(2)xS(2) + S(2)))(S(463)xS(2) + S(463))elliptic_f(atan(x), S(1)/2)/(S(336)sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/((S(5)xS(2) + S(7))S(2)(xS(4) + S(3)xS(2) + S(2))(S(3)/2)), x), x, S(625)x(x*S(2) + S(1))(x*S(2) + S(2))/((S(2520)x*S(2) + S(3528))sqrt(x*S(4) + S(3)x*S(2) + S(2))) - S(125)x(xS(2) + S(2))/(S(504)sqrt(x*S(4) + S(3)x*S(2) + S(2))) - x/(S(18)sqrt(x*S(4) + S(3)x*S(2) + S(2))) + (S(125)x*S(2) + S(125))elliptic_pi(S(-3)/7, atan(sqrt(S(2))x/S(2)), S(-1))/(S(189)sqrt((xS(2) + S(1))/(xS(2) + S(2)))sqrt(xS(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))(S(31)xS(2) + S(62))elliptic_e(atan(x), S(1)/2)/(S(56)sqrt((xS(2) + S(2))/(xS(2) + S(1)))sqrt(x*S(4) + S(3)x*S(2) + S(2))) + sqrt(S(2))(S(6875)xS(2) + S(13750))elliptic_pi(S(2)/7, atan(x), S(1)/2)/(S(14112)sqrt((xS(2) + S(2))/(xS(2) + S(1)))sqrt(x*S(4) + S(3)x*S(2) + S(2))) - sqrt(S(2))(S(7667)xS(2) + S(15334))elliptic_f(atan(x), S(1)/2)/(S(6048)sqrt((xS(2) + S(2))/(xS(2) + S(1)))sqrt(x*S(4) + S(3)xS(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(3))/sqrt(-xS(4) + xS(2) + S(3)), x), x, -sqrt(S(-1)/2 + sqrt(S(13))/S(2))elliptic_e(asin(sqrt(S(2))x/sqrt(S(1) + sqrt(S(13)))), S(-7)/6 - sqrt(S(13))/S(6)) + sqrt(S(7) + S(2)sqrt(S(13)))elliptic_f(asin(sqrt(S(2))x/sqrt(S(1) + sqrt(S(13)))), S(-7)/6 - sqrt(S(13))/S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(3))/sqrt(-xS(4) + S(2)x*S(2) + S(3)), x), x, -elliptic_e(asin(sqrt(S(3))x/S(3)), S(-3)) + S(4)elliptic_f(asin(sqrt(S(3))x/S(3)), S(-3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(3))/sqrt(-xS(4) + S(3)xS(2) + S(3)), x), x, -sqrt(S(-3)/2 + sqrt(S(21))/S(2))elliptic_e(asin(sqrt(S(2))x/sqrt(S(3) + sqrt(S(21)))), S(-5)/2 - sqrt(S(21))/S(2)) + sqrt(S(9) + S(2)sqrt(S(21)))elliptic_f(asin(sqrt(S(2))x/sqrt(S(3) + sqrt(S(21)))), S(-5)/2 - sqrt(S(21))/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(3))/sqrt(-xS(4) - x*S(2) + S(3)), x), x, -sqrt(S(1)/2 + sqrt(S(13))/S(2))elliptic_e(asin(sqrt(S(2))x/sqrt(S(-1) + sqrt(S(13)))), S(-7)/6 + sqrt(S(13))/S(6)) + sqrt(S(5) + S(2)sqrt(S(13)))elliptic_f(asin(sqrt(S(2))x/sqrt(S(-1) + sqrt(S(13)))), S(-7)/6 + sqrt(S(13))/S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(3))/sqrt(-xS(4) - S(2)xS(2) + S(3)), x), x, -sqrt(S(3))elliptic_e(asin(x), S(-1)/3) + S(2)sqrt(S(3))elliptic_f(asin(x), S(-1)/3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(3))/sqrt(-xS(4) - S(3)xS(2) + S(3)), x), x, -sqrt(S(3)/2 + sqrt(S(21))/S(2))elliptic_e(asin(sqrt(S(2))x/sqrt(S(-3) + sqrt(S(21)))), S(-5)/2 + sqrt(S(21))/S(2)) + sqrt(S(3) + S(2)sqrt(S(21)))elliptic_f(asin(sqrt(S(2))x/sqrt(S(-3) + sqrt(S(21)))), S(-5)/2 + sqrt(S(21))/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx*S(2) - sqrt(-S(4)ac + bS(2)))/sqrt(a + bx*S(2) + cx*S(4)), x), x, -S(2)a*(S(1)/4)c*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/sqrt(a + bx*S(2) + cx*S(4)) + S(2)sqrt(c)xsqrt(a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)x*S(2)) + sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(2)sqrt(a)sqrt(c) + b - sqrt(-S(4)ac + b*S(2)))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(1)/4)c*(S(1)/4)sqrt(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(2))/((x*S(2) + S(1))sqrt(x*S(4) + S(3)x*S(2) + S(2))), x), x, sqrt(S(2))(x*S(2) + S(2))elliptic_e(atan(x), S(1)/2)/(sqrt((xS(2) + S(2))/(xS(2) + S(1)))sqrt(xS(4) + S(3)x*S(2) + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))(S(5)/2)(a + bx*S(2) + cx*S(4)), x), x, cx*S(3)(d + exS(2))(S(7)/2)/(S(10)e) + d*S(3)(S(80)ae*S(2) - S(10)bde + S(3)cd*S(2))atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(256)e(S(5)/2)) + dS(2)xsqrt(d + exS(2))(S(80)ae*S(2) - S(10)bde + S(3)cd*S(2))/(S(256)e*S(2)) + dx(d + exS(2))(S(3)/2)(S(80)aeS(2) - S(10)bde + S(3)cd*S(2))/(S(384)e*S(2)) - x(d + exS(2))(S(7)/2)(-S(10)be + S(3)cd)/(S(80)eS(2)) + x(d + exS(2))(S(5)/2)(S(80)ae*S(2) - S(10)bde + S(3)cd*S(2))/(S(480)e*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))(S(3)/2)(a + bx*S(2) + cx*S(4)), x), x, cx*S(3)(d + exS(2))(S(5)/2)/(S(8)e) + d*S(2)(S(48)ae*S(2) - S(8)bde + S(3)cd*S(2))atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(128)e*(S(5)/2)) + dxsqrt(d + ex*S(2))(S(48)ae*S(2) - S(8)bde + S(3)cd*S(2))/(S(128)e*S(2)) - x(d + exS(2))(S(5)/2)(-S(8)be + S(3)cd)/(S(48)eS(2)) + x(d + exS(2))(S(3)/2)(S(48)ae*S(2) - S(8)bde + S(3)cd*S(2))/(S(192)e*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + ex*S(2))(a + bxS(2) + cx*S(4)), x), x, cx*S(3)(d + exS(2))(S(3)/2)/(S(6)e) + d(S(8)aeS(2) - S(2)bde + cdS(2))atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(16)e*(S(5)/2)) - x(d + exS(2))(S(3)/2)(-S(2)be + cd)/(S(8)e*S(2)) + xsqrt(d + exS(2))(S(8)ae*S(2) - S(2)bde + cdS(2))/(S(16)e*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/sqrt(d + ex*S(2)), x), x, cx*S(3)sqrt(d + exS(2))/(S(4)e) - xsqrt(d + ex*S(2))(-S(4)be + S(3)cd)/(S(8)eS(2)) + (S(8)aeS(2) - S(4)bde + S(3)cd*S(2))atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(8)e*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(d + exS(2))(S(3)/2), x), x, cxsqrt(d + exS(2))/(S(2)e*S(2)) - (-S(2)be + S(3)cd)atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(2)e*(S(5)/2)) + x(aeS(2) - bde + cd*S(2))/(de*S(2)sqrt(d + exS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(d + exS(2))(S(5)/2), x), x, catanh(sqrt(e)x/sqrt(d + exS(2)))/e(S(5)/2) + x(aeS(2) - bde + cd*S(2))/(S(3)deS(2)(d + exS(2))(S(3)/2)) - x(S(4)cd*S(2) - e(S(2)ae + bd))/(S(3)d*S(2)e*S(2)sqrt(d + exS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(d + exS(2))(S(7)/2), x), x, x(ae*S(2) - bde + cd*S(2))/(S(5)deS(2)(d + exS(2))(S(5)/2)) - x(cdS(2) - S(5)cdexS(2) - e(S(4)ae + bd))/(S(15)d*S(2)e*S(2)(d + exS(2))(S(3)/2)) - x(S(2)cd*S(2) - S(2)e(S(4)ae + bd))/(S(15)dS(3)e*S(2)sqrt(d + exS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(d + exS(2))(S(9)/2), x), x, x(ae*S(2) - bde + cd*S(2))/(S(7)deS(2)(d + exS(2))(S(7)/2)) - x(S(8)cd*S(2) - e(S(6)ae + bd))/(S(35)d*S(2)e*S(2)(d + exS(2))(S(5)/2)) + x(S(3)cd*S(2) + S(4)e(S(6)ae + bd))/(S(105)dS(3)e*S(2)(d + exS(2))(S(3)/2)) + x(S(6)cd*S(2) + S(8)e(S(6)ae + bd))/(S(105)dS(4)e*S(2)sqrt(d + exS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(d + exS(2))(S(11)/2), x), x, x(ae*S(2) - bde + cd*S(2))/(S(9)deS(2)(d + exS(2))(S(9)/2)) - x(S(10)cd*S(2) - e(S(8)ae + bd))/(S(63)d*S(2)e*S(2)(d + exS(2))(S(7)/2)) + x(cdS(2) + S(2)e(S(8)ae + bd))/(S(105)dS(3)e*S(2)(d + exS(2))(S(5)/2)) + x(S(4)cd*S(2) + S(8)e(S(8)ae + bd))/(S(315)dS(4)e*S(2)(d + exS(2))(S(3)/2)) + x(S(8)cd*S(2) + S(16)e(S(8)ae + bd))/(S(315)dS(5)e*S(2)sqrt(d + exS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(3)/(a + cx(S(2)n)), x), x, S(3)de*S(2)x/c + e*S(3)x*(n + S(1))/(c(n + S(1))) - x(-ae*S(3) + sqrt(c)d(-S(3)aeS(2) + cd*S(2))/sqrt(-a) + S(3)cdS(2)e)hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(2)c*(S(3)/2)sqrt(-a)) + x(-S(3)asqrt(c)deS(2) + ae*S(3)sqrt(-a) + c*(S(3)/2)d*S(3) - S(3)cdS(2)esqrt(-a))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(S(2)ac(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(2)/(a + cx(S(2)n)), x), x, e*S(2)x/c + x(-ae*S(2) - S(2)sqrt(c)desqrt(-a) + cd*S(2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(S(2)ac) + x(-aeS(2) + S(2)sqrt(c)desqrt(-a) + cd*S(2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)xn/sqrt(-a))/(S(2)ac), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*n)/(a + cx*(S(2)n)), x), x, x(d - esqrt(-a)/sqrt(c))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(2)a) + x(d + esqrt(-a)/sqrt(c))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(2)a), expand=True, _diff=True, _numerical=True) # Apart assert rubi_test(rubi_integrate(S(1)/((a + cx(S(2)n))(d + exn)), x), x, eS(2)xhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(d(aeS(2) + cd*S(2))) + cx(d - esqrt(-a)/sqrt(c))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(2)a(ae*S(2) + cd*S(2))) + cx(d + esqrt(-a)/sqrt(c))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(2)a(ae*S(2) + cd*S(2))), expand=True, _diff=True, _numerical=True) # Apart assert rubi_test(rubi_integrate(S(1)/((a + cx*(S(2)n))(d + exn)S(2)), x), x, S(2)ce*S(2)xhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(ae*S(2) + cdS(2))S(2) + e*S(2)xhyper((S(2), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(dS(2)(ae*S(2) + cd*S(2))) + cx(-ae*S(2) - S(2)sqrt(c)desqrt(-a) + cd*S(2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)xn/sqrt(-a))/(S(2)a(ae*S(2) + cdS(2))S(2)) + cx(-aeS(2) + S(2)sqrt(c)desqrt(-a) + cd*S(2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(S(2)a(ae*S(2) + cdS(2))S(2)), expand=True, _diff=True, _numerical=True) # Apart assert rubi_test(rubi_integrate(S(1)/((a + cx(S(2)n))(d + exn)S(3)), x), x, c*(S(3)/2)x(-ae*S(3) - sqrt(c)d(-S(3)aeS(2) + cd*S(2))/sqrt(-a) + S(3)cdS(2)e)hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(2)sqrt(-a)(ae*S(2) + cdS(2))S(3)) + ceS(2)x(-ae*S(2) + S(3)cdS(2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(d(aeS(2) + cdS(2))S(3)) + S(2)ce*S(2)xhyper((S(2), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(d(aeS(2) + cdS(2))S(2)) + e*S(2)xhyper((S(3), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(dS(3)(ae*S(2) + cdS(2))) + c(S(3)/2)x(-S(3)asqrt(c)deS(2) + c(S(3)/2)dS(3) + S(3)cdS(2)esqrt(-a) + eS(3)(-a)*(S(3)/2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(S(2)a(ae*S(2) + cdS(2))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)/(a - cx*(S(2)n)), x), x, x(-sqrt(a)e/sqrt(c) + d)hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(a))/(S(2)a) + x(sqrt(a)e/sqrt(c) + d)hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(a))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(3)/(a + cx*(S(2)n))*S(2), x), x, -x(sqrt(c)(-S(2)n + S(1))(-S(3)ade*S(2) + cd*S(3))/sqrt(-a) - (-n + S(1))(-aeS(3) + S(3)cdS(2)e))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(4)c*(S(3)/2)n(-a)(S(3)/2)) - x(sqrt(c)(-S(2)n + S(1))(-S(3)ade*S(2) + cd*S(3))/sqrt(-a) + (-n + S(1))(-aeS(3) + S(3)cdS(2)e))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(4)c*(S(3)/2)n(-a)(S(3)/2)) + eS(2)x(S(3)d - esqrt(-a)/sqrt(c))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(S(2)ac) + eS(2)x(S(3)d + esqrt(-a)/sqrt(c))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)xn/sqrt(-a))/(S(2)ac) + x(d(-S(3)aeS(2) + cd*S(2)) + ex*n(-aeS(2) + S(3)cdS(2)))/(S(2)acn(a + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(2)/(a + cx*(S(2)n))S(2), x), x, eS(2)xhyper((S(1), S(1)/(S(2)n)), (S(1) + S(1)/(S(2)n),), -cx(S(2)n)/a)/(ac) + x(-aeS(2) + cd*S(2) + S(2)cdexn)/(S(2)acn(a + cx*(S(2)n))) - x(-ae*S(2)(-S(2)n + S(1)) - S(2)sqrt(c)desqrt(-a)(-n + S(1)) + cdS(2)(-S(2)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(S(4)a*S(2)cn) - x(-aeS(2)(-S(2)n + S(1)) + S(2)sqrt(c)desqrt(-a)(-n + S(1)) + cdS(2)(-S(2)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)xn/sqrt(-a))/(S(4)a*S(2)cn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*n)/(a + cx*(S(2)n))*S(2), x), x, x(d + exn)/(S(2)an(a + cx(S(2)n))) - x(sqrt(c)(-S(2)dn + d) + esqrt(-a)(-n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(4)a*S(2)sqrt(c)n) - x(sqrt(c)d(-S(2)n + S(1)) - esqrt(-a)(-n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(S(4)a*S(2)sqrt(c)n), expand=True, _diff=True, _numerical=True) # Apart assert rubi_test(rubi_integrate(S(1)/((a + cx*(S(2)n))*S(2)(d + exn)), x), x, eS(4)xhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(d(aeS(2) + cdS(2))S(2)) + ceS(2)x(d - esqrt(-a)/sqrt(c))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(2)a(ae*S(2) + cdS(2))S(2)) + ceS(2)x(d + esqrt(-a)/sqrt(c))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(2)a(ae*S(2) + cdS(2))S(2)) + cx(d - exn)/(S(2)an(a + cx(S(2)n))(ae*S(2) + cd*S(2))) - sqrt(c)x(sqrt(c)(-S(2)dn + d) + esqrt(-a)(-n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(4)a*S(2)n(ae*S(2) + cd*S(2))) - sqrt(c)x(sqrt(c)d(-S(2)n + S(1)) - esqrt(-a)(-n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(4)a*S(2)n(ae*S(2) + cd*S(2))), expand=True, _diff=True, _numerical=True) # apart assert rubi_test(rubi_integrate(S(1)/((a + cx*(S(2)n))*S(2)(d + exn)S(2)), x), x, S(4)ceS(4)xhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(ae*S(2) + cdS(2))S(3) + e*S(4)xhyper((S(2), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(dS(2)(ae*S(2) + cdS(2))S(2)) + ceS(2)x(-ae*S(2) - S(4)sqrt(c)desqrt(-a) + S(3)cdS(2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)xn/sqrt(-a))/(S(2)a(ae*S(2) + cdS(2))S(3)) + ceS(2)x(-ae*S(2) + S(4)sqrt(c)desqrt(-a) + S(3)cdS(2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(S(2)a(ae*S(2) + cdS(2))S(3)) + cx(-aeS(2) + cd*S(2) - S(2)cdexn)/(S(2)an(a + cx(S(2)n))(ae*S(2) + cdS(2))S(2)) - cx(-aeS(2)(-S(2)n + S(1)) - S(2)sqrt(c)desqrt(-a)(-n + S(1)) + cdS(2)(-S(2)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)xn/sqrt(-a))/(S(4)a*S(2)n(ae*S(2) + cdS(2))S(2)) - cx(-aeS(2)(-S(2)n + S(1)) + S(2)sqrt(c)desqrt(-a)(-n + S(1)) + cdS(2)(-S(2)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(S(4)a*S(2)n(ae*S(2) + cdS(2))S(2)), expand=True, _diff=True, _numerical=True) # apart assert rubi_test(rubi_integrate(S(1)/((a + cx(S(2)n))*S(2)(d + exn)S(3)), x), x, c(S(3)/2)e*S(2)x(-ae*S(3) - S(3)sqrt(c)d(-aeS(2) + cd*S(2))/sqrt(-a) + S(5)cdS(2)e)hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(sqrt(-a)(aeS(2) + cdS(2))S(4)) - c*(S(3)/2)x(sqrt(c)(-S(2)n + S(1))(-S(3)adeS(2) + cd*S(3))/sqrt(-a) - (-n + S(1))(-aeS(3) + S(3)cdS(2)e))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(4)n(-a)(S(3)/2)(aeS(2) + cdS(2))S(3)) - c*(S(3)/2)x(sqrt(c)(-S(2)n + S(1))(-S(3)adeS(2) + cd*S(3))/sqrt(-a) + (-n + S(1))(-aeS(3) + S(3)cdS(2)e))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(4)n(-a)(S(3)/2)(aeS(2) + cdS(2))S(3)) + S(2)ce*S(4)x(-ae*S(2) + S(5)cdS(2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(d(aeS(2) + cdS(2))S(4)) + S(4)ce*S(4)xhyper((S(2), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(d(aeS(2) + cdS(2))S(3)) + e*S(4)xhyper((S(3), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(dS(3)(ae*S(2) + cdS(2))S(2)) + c*(S(3)/2)e*S(2)x(-S(3)asqrt(c)deS(2) + S(3)c*(S(3)/2)d*S(3) + S(5)cdS(2)esqrt(-a) + eS(3)(-a)*(S(3)/2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(a(aeS(2) + cdS(2))S(4)) + c*S(2)x(d(-S(3)ae*S(2) + cd*S(2)) - ex*n(-aeS(2) + S(3)cdS(2)))/(S(2)an(a + cx(S(2)n))(ae*S(2) + cdS(2))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(3)/(a + cx*(S(2)n))*S(3), x), x, -x(sqrt(c)(-S(4)n + S(1))(-S(2)n + S(1))(-S(3)ade*S(2) + cd*S(3))/sqrt(-a) - (-S(3)n + S(1))(-n + S(1))(-aeS(3) + S(3)cdS(2)e))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(16)c*(S(3)/2)n*S(2)(-a)*(S(5)/2)) - x(sqrt(c)(-S(4)n + S(1))(-S(2)n + S(1))(-S(3)ade*S(2) + cd*S(3))/sqrt(-a) + (-S(3)n + S(1))(-n + S(1))(-aeS(3) + S(3)cdS(2)e))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(16)c*(S(3)/2)n*S(2)(-a)(S(5)/2)) + eS(2)x(S(3)d + ex*n)/(S(2)acn(a + cx*(S(2)n))) + x(d(-S(3)ae*S(2) + cd*S(2)) + ex*n(-aeS(2) + S(3)cdS(2)))/(S(4)acn(a + cx*(S(2)n))*S(2)) - x(d(-S(4)n + S(1))(-S(3)aeS(2) + cd*S(2)) + ex*n(-S(3)n + S(1))(-aeS(2) + S(3)cdS(2)))/(S(8)a*S(2)cnS(2)(a + cx(S(2)n))) - e*S(2)x(sqrt(c)d(-S(6)n + S(3)) - esqrt(-a)(-n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(4)a*S(2)c*(S(3)/2)n) - e*S(2)x(sqrt(c)d(-S(6)n + S(3)) + esqrt(-a)(-n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(4)a*S(2)c*(S(3)/2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(2)/(a + cx*(S(2)n))*S(3), x), x, x(-aeS(2) + cd*S(2) + S(2)cdexn)/(S(4)acn(a + cx*(S(2)n))S(2)) + eS(2)xhyper((S(2), S(1)/(S(2)n)), (S(1) + S(1)/(S(2)n),), -cx(S(2)n)/a)/(a*S(2)c) - x(S(2)cdexn(-S(3)n + S(1)) + (-S(4)n + S(1))(-ae*S(2) + cd*S(2)))/(S(8)a*S(2)cnS(2)(a + cx(S(2)n))) + x(-ae*S(2)(S(8)nS(2) - S(6)n + S(1)) + S(2)sqrt(c)desqrt(-a)(S(3)n*S(2) - S(4)n + S(1)) + cdS(2)(S(8)nS(2) - S(6)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(16)a*S(3)cnS(2)) - x(aeS(2)(S(8)nS(2) - S(6)n + S(1)) + S(2)sqrt(c)desqrt(-a)(S(3)n*S(2) - S(4)n + S(1)) - cdS(2)(S(8)nS(2) - S(6)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(16)a*S(3)cnS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*n)/(a + cx*(S(2)n))*S(3), x), x, x(d + exn)/(S(4)an(a + cx(S(2)n))*S(2)) - x(d(-S(4)n + S(1)) + exn(-S(3)n + S(1)))/(S(8)a*S(2)n*S(2)(a + cx(S(2)n))) - x(-sqrt(c)d(S(8)n*S(2) - S(6)n + S(1)) + esqrt(-a)(S(3)nS(2) - S(4)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(16)a*S(3)sqrt(c)nS(2)) + x(sqrt(c)d(S(8)nS(2) - S(6)n + S(1)) + esqrt(-a)(S(3)nS(2) - S(4)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(16)a*S(3)sqrt(c)nS(2)), expand=True, _diff=True, _numerical=True) # Apart assert rubi_test(rubi_integrate(S(1)/((a + cx*(S(2)n))*S(3)(d + exn)), x), x, eS(6)xhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(d(aeS(2) + cdS(2))S(3)) + ceS(4)x(d - esqrt(-a)/sqrt(c))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(2)a(ae*S(2) + cdS(2))S(3)) + ceS(4)x(d + esqrt(-a)/sqrt(c))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(2)a(ae*S(2) + cdS(2))S(3)) + ceS(2)x(d - ex*n)/(S(2)an(a + cx(S(2)n))(ae*S(2) + cdS(2))S(2)) + cx(d - exn)/(S(4)an(a + cx(S(2)n))*S(2)(aeS(2) + cd*S(2))) - sqrt(c)e*S(2)x(sqrt(c)(-S(2)dn + d) + esqrt(-a)(-n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(4)a*S(2)n(ae*S(2) + cdS(2))S(2)) - sqrt(c)eS(2)x(sqrt(c)d(-S(2)n + S(1)) - esqrt(-a)(-n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(4)a*S(2)n(ae*S(2) + cdS(2))S(2)) - cx(d(-S(4)n + S(1)) - exn(-S(3)n + S(1)))/(S(8)a*S(2)n*S(2)(a + cx(S(2)n))(ae*S(2) + cd*S(2))) - sqrt(c)x(-sqrt(c)d(S(8)n*S(2) - S(6)n + S(1)) + esqrt(-a)(S(3)nS(2) - S(4)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(16)a*S(3)n*S(2)(aeS(2) + cd*S(2))) + sqrt(c)x(sqrt(c)d(S(8)n*S(2) - S(6)n + S(1)) + esqrt(-a)(S(3)nS(2) - S(4)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(16)a*S(3)n*S(2)(aeS(2) + cd*S(2))), expand=True, _diff=True, _numerical=True) # Apart assert rubi_test(rubi_integrate(S(1)/((a + cx*(S(2)n))*S(3)(d + exn)S(2)), x), x, S(6)ceS(6)xhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(ae*S(2) + cdS(2))S(4) + e*S(6)xhyper((S(2), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(dS(2)(ae*S(2) + cdS(2))S(3)) + ceS(4)x(-ae*S(2) - S(6)sqrt(c)desqrt(-a) + S(5)cdS(2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)xn/sqrt(-a))/(S(2)a(ae*S(2) + cdS(2))S(4)) + ceS(4)x(-ae*S(2) + S(6)sqrt(c)desqrt(-a) + S(5)cdS(2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(S(2)a(ae*S(2) + cdS(2))S(4)) + ceS(2)x(-ae*S(2) + S(3)cdS(2) - S(4)cdexn)/(S(2)an(a + cx(S(2)n))(ae*S(2) + cdS(2))S(3)) + cx(-aeS(2) + cd*S(2) - S(2)cdexn)/(S(4)an(a + cx(S(2)n))*S(2)(aeS(2) + cdS(2))S(2)) - ceS(2)x(-ae*S(2)(-S(2)n + S(1)) - S(4)sqrt(c)desqrt(-a)(-n + S(1)) + S(3)cd*S(2)(-S(2)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)xn/sqrt(-a))/(S(4)a*S(2)n(ae*S(2) + cdS(2))S(3)) - ceS(2)x(-ae*S(2)(-S(2)n + S(1)) + S(4)sqrt(c)desqrt(-a)(-n + S(1)) + S(3)cd*S(2)(-S(2)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(S(4)a*S(2)n(ae*S(2) + cdS(2))S(3)) - cx(-S(2)cdex*n(-S(3)n + S(1)) + (-S(4)n + S(1))(-ae*S(2) + cd*S(2)))/(S(8)a*S(2)n*S(2)(a + cx(S(2)n))(ae*S(2) + cdS(2))S(2)) + cx(-aeS(2)(S(8)nS(2) - S(6)n + S(1)) + S(2)sqrt(c)desqrt(-a)(S(3)n*S(2) - S(4)n + S(1)) + cdS(2)(S(8)nS(2) - S(6)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(16)a*S(3)n*S(2)(aeS(2) + cdS(2))S(2)) - cx(aeS(2)(S(8)nS(2) - S(6)n + S(1)) + S(2)sqrt(c)desqrt(-a)(S(3)n*S(2) - S(4)n + S(1)) - cdS(2)(S(8)nS(2) - S(6)n + S(1)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(16)a*S(3)n*S(2)(aeS(2) + cdS(2))S(2)), expand=True, _diff=True, _numerical=True) # Apart assert rubi_test(rubi_integrate(S(1)/((a + cx(S(2)n))*S(3)(d + exn)S(3)), x), x, S(3)c*(S(3)/2)e*S(4)x(-ae*S(3) - sqrt(c)d(-S(3)aeS(2) + S(5)cdS(2))/sqrt(-a) + S(7)cdS(2)e)hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(2)sqrt(-a)(ae*S(2) + cdS(2))S(5)) - c*(S(3)/2)e*S(2)x(S(3)sqrt(c)(-S(2)n + S(1))(-adeS(2) + cd*S(3))/sqrt(-a) - (-n + S(1))(-aeS(3) + S(5)cdS(2)e))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(2)n(-a)(S(3)/2)(aeS(2) + cdS(2))S(4)) - c*(S(3)/2)e*S(2)x(S(3)sqrt(c)(-S(2)n + S(1))(-adeS(2) + cd*S(3))/sqrt(-a) + (-n + S(1))(-aeS(3) + S(5)cdS(2)e))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(2)n(-a)(S(3)/2)(aeS(2) + cdS(2))S(4)) - c*(S(3)/2)x(sqrt(c)(-S(4)n + S(1))(-S(2)n + S(1))(-S(3)adeS(2) + cd*S(3))/sqrt(-a) - (-S(3)n + S(1))(-n + S(1))(-aeS(3) + S(3)cdS(2)e))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), sqrt(c)x*n/sqrt(-a))/(S(16)n*S(2)(-a)*(S(5)/2)(aeS(2) + cdS(2))S(3)) - c*(S(3)/2)x(sqrt(c)(-S(4)n + S(1))(-S(2)n + S(1))(-S(3)adeS(2) + cd*S(3))/sqrt(-a) + (-S(3)n + S(1))(-n + S(1))(-aeS(3) + S(3)cdS(2)e))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)x*n/sqrt(-a))/(S(16)n*S(2)(-a)*(S(5)/2)(aeS(2) + cdS(2))S(3)) + S(3)ce*S(6)x(-ae*S(2) + S(7)cdS(2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(d(aeS(2) + cdS(2))S(5)) + S(6)ce*S(6)xhyper((S(2), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(d(aeS(2) + cdS(2))S(4)) + e*S(6)xhyper((S(3), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(dS(3)(ae*S(2) + cdS(2))S(3)) + S(3)c(S(3)/2)e*S(4)x(-S(3)asqrt(c)deS(2) + S(5)c*(S(3)/2)d*S(3) + S(7)cdS(2)esqrt(-a) + eS(3)(-a)*(S(3)/2))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -sqrt(c)xn/sqrt(-a))/(S(2)a(ae*S(2) + cdS(2))S(5)) + c*S(2)e*S(2)x(S(3)d(-ae*S(2) + cd*S(2)) - ex*n(-aeS(2) + S(5)cdS(2)))/(an(a + cx*(S(2)n))(ae*S(2) + cdS(2))S(4)) + c*S(2)x(d(-S(3)ae*S(2) + cd*S(2)) - ex*n(-aeS(2) + S(3)cdS(2)))/(S(4)an(a + cx(S(2)n))*S(2)(aeS(2) + cdS(2))S(3)) - c*S(2)x(d(-S(4)n + S(1))(-S(3)ae*S(2) + cd*S(2)) - ex*n(-S(3)n + S(1))(-aeS(2) + S(3)cdS(2)))/(S(8)a*S(2)n*S(2)(a + cx(S(2)n))(ae*S(2) + cdS(2))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cx(S(2)n))*p(d + exn)S(3), x), x, dS(3)x(S(1) + cx*(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper((S(1)/(S(2)n), -p), (S(1) + S(1)/(S(2)n),), -cx(S(2)n)/a) + S(3)dS(2)ex(n + S(1))(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper(((n + S(1))/(S(2)n), -p), (S(3)/2 + S(1)/(S(2)n),), -cx(S(2)n)/a)/(n + S(1)) + S(3)de*S(2)x*(S(2)n + S(1))(S(1) + cx*(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper((S(1) + S(1)/(S(2)n), -p), (S(2) + S(1)/(S(2)n),), -cx(S(2)n)/a)/(S(2)n + S(1)) + eS(3)x*(S(3)n + S(1))(S(1) + cx*(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper((S(3)/2 + S(1)/(S(2)n), -p), (S(5)/2 + S(1)/(S(2)n),), -cx(S(2)n)/a)/(S(3)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cx*(S(2)n))*p(d + exn)S(2), x), x, dS(2)x(S(1) + cx*(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper((S(1)/(S(2)n), -p), (S(1) + S(1)/(S(2)n),), -cx(S(2)n)/a) + S(2)dex(n + S(1))(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper(((n + S(1))/(S(2)n), -p), (S(3)/2 + S(1)/(S(2)n),), -cx(S(2)n)/a)/(n + S(1)) + e*S(2)x*(S(2)n + S(1))(S(1) + cx*(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper((S(1) + S(1)/(S(2)n), -p), (S(2) + S(1)/(S(2)n),), -cx(S(2)n)/a)/(S(2)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cx*(S(2)n))*p(d + exn), x), x, dx(S(1) + cx*(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper((S(1)/(S(2)n), -p), (S(1) + S(1)/(S(2)n),), -cx(S(2)n)/a) + ex(n + S(1))(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper(((n + S(1))/(S(2)n), -p), (S(3)/2 + S(1)/(S(2)n),), -cx(S(2)n)/a)/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)(a + bxn + cx*(S(2)n)), x), x, adx + cex*(S(3)n + S(1))/(S(3)n + S(1)) + x(n + S(1))(ae + bd)/(n + S(1)) + x*(S(2)n + S(1))(be + cd)/(S(2)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)(a + bxn + cx*(S(2)n))S(2), x), x, aS(2)dx + ax(n + S(1))(ae + S(2)bd)/(n + S(1)) + cS(2)ex(S(5)n + S(1))/(S(5)n + S(1)) + cx*(S(4)n + S(1))(S(2)be + cd)/(S(4)n + S(1)) + x(S(2)n + S(1))(S(2)abe + S(2)acd + bS(2)d)/(S(2)n + S(1)) + x(S(3)n + S(1))(S(2)ace + b*S(2)e + S(2)bcd)/(S(3)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)(a + bxn + cx*(S(2)n))S(3), x), x, aS(3)dx + a*S(2)x*(n + S(1))(ae + S(3)bd)/(n + S(1)) + S(3)ax(S(2)n + S(1))(abe + acd + bS(2)d)/(S(2)n + S(1)) + cS(3)ex(S(7)n + S(1))/(S(7)n + S(1)) + cS(2)x*(S(6)n + S(1))(S(3)be + cd)/(S(6)n + S(1)) + S(3)cx(S(5)n + S(1))(ace + bS(2)e + bcd)/(S(5)n + S(1)) + x(S(3)n + S(1))(S(3)a*S(2)ce + S(3)abS(2)e + S(6)abcd + b*S(3)d)/(S(3)n + S(1)) + x(S(4)n + S(1))(S(6)abce + S(3)acS(2)d + b*S(3)e + S(3)bS(2)cd)/(S(4)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(3)/(a + bx*n + cx*(S(2)n)), x), x, e*S(3)x*(n + S(1))/(c(n + S(1))) + e*S(2)x(-be + S(3)cd)/c*S(2) + x(-aceS(3) + bS(2)eS(3) - S(3)bcdeS(2) + S(3)c*S(2)d*S(2)e - (-be + S(2)cd)(b*S(2)eS(2) + cS(2)dS(2) - ce(S(3)ae + bd))/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(cS(2)(b + sqrt(-S(4)ac + b*S(2)))) + x(-aceS(3) + bS(2)eS(3) - S(3)bcdeS(2) + S(3)c*S(2)d*S(2)e + (-be + S(2)cd)(b*S(2)eS(2) + cS(2)dS(2) - ce(S(3)ae + bd))/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(cS(2)(b - sqrt(-S(4)ac + b*S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(2)/(a + bxn + cx*(S(2)n)), x), x, e*S(2)x/c + x(-be*S(2) + S(2)cde - (b*S(2)e*S(2) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd))/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(c(b + sqrt(-S(4)ac + b*S(2)))) + x(-beS(2) + S(2)cde + (b*S(2)e*S(2) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd))/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(c(b - sqrt(-S(4)ac + b*S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*n)/(a + bx*n + cx*(S(2)n)), x), x, x(e - (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(b + sqrt(-S(4)ac + bS(2))) + x(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(b - sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) # Apart assert rubi_test(rubi_integrate(S(1)/((d + ex*n)(a + bxn + cx*(S(2)n))), x), x, -cx(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cd*S(2))) - cx(e + (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/((b + sqrt(-S(4)ac + bS(2)))(aeS(2) - bde + cdS(2))) + eS(2)xhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(d(aeS(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) # Apart assert rubi_test(rubi_integrate(S(1)/((d + exn)S(2)(a + bx*n + cx*(S(2)n))), x), x, -cx(beS(2)(b - sqrt(-S(4)ac + b*S(2))) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd - dsqrt(-S(4)ac + bS(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(2)) - cx(beS(2)(b + sqrt(-S(4)ac + b*S(2))) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd + dsqrt(-S(4)ac + bS(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(2)) + e*S(2)x(-be + S(2)cd)hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(d(aeS(2) - bde + cdS(2))S(2)) + e*S(2)xhyper((S(2), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(dS(2)(ae*S(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) # Apart assert rubi_test(rubi_integrate(S(1)/((d + exn)S(3)(a + bx*n + cx*(S(2)n))), x), x, -cx(-b*S(2)e*S(3)(b - sqrt(-S(4)ac + b*S(2))) + S(2)c*S(3)d*S(3) - S(3)c*S(2)de(S(2)ae + bd - dsqrt(-S(4)ac + b*S(2))) + ce*S(2)(S(3)abe - aesqrt(-S(4)ac + bS(2)) + S(3)b*S(2)d - S(3)bdsqrt(-S(4)ac + bS(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(3)) - cx(-b*S(2)e*S(3)(b + sqrt(-S(4)ac + b*S(2))) + S(2)c*S(3)d*S(3) - S(3)c*S(2)de(S(2)ae + bd + dsqrt(-S(4)ac + b*S(2))) + ce*S(2)(aesqrt(-S(4)ac + b*S(2)) + S(3)b*S(2)d + S(3)b(ae + dsqrt(-S(4)ac + b*S(2)))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(3)) + e*S(2)x(bS(2)e*S(2) + S(3)c*S(2)d*S(2) - ce(ae + S(3)bd))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(d(aeS(2) - bde + cdS(2))S(3)) + e*S(2)x(-be + S(2)cd)hyper((S(2), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(dS(2)(ae*S(2) - bde + cdS(2))S(2)) + e*S(2)xhyper((S(3), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(dS(3)(ae*S(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(3)/(a + bxn + cx*(S(2)n))S(2), x), x, eS(2)x(e - (-S(3)be + S(6)cd)/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(c(b + sqrt(-S(4)ac + bS(2)))) + eS(2)x(e + (-S(3)be + S(6)cd)/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(c(b - sqrt(-S(4)ac + b*S(2)))) + x(-abe(ae*S(2) + S(3)cdS(2)) - S(2)acd(-S(3)aeS(2) + cdS(2)) + bS(2)cdS(3) - xn(ab*S(2)e*S(3) + S(2)ace(-ae*S(2) + S(3)cdS(2)) - bcd(S(3)ae*S(2) + cd*S(2))))/(acn(-S(4)ac + b*S(2))(a + bxn + cx*(S(2)n))) + x((-n + S(1))(abS(2)e*S(3) + S(2)ace(-ae*S(2) + S(3)cdS(2)) - bcd(S(3)ae*S(2) + cd*S(2))) - (-ab*S(3)e*S(3)(-S(3)n + S(1)) + S(2)abce(aeS(2)(-S(5)n + S(2)) + S(3)cdS(2)n) + S(4)ac*S(2)d(-S(2)n + S(1))(-S(3)aeS(2) + cdS(2)) + bS(2)cd(S(3)aeS(2)(-S(3)n + S(1)) - cd*S(2)(-n + S(1))))/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(acn(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + b*S(2))) + x((-n + S(1))(ab*S(2)e*S(3) + S(2)ace(-ae*S(2) + S(3)cdS(2)) - bcd(S(3)ae*S(2) + cd*S(2))) + (-ab*S(3)e*S(3)(-S(3)n + S(1)) + S(2)abce(aeS(2)(-S(5)n + S(2)) + S(3)cdS(2)n) + S(4)ac*S(2)d(-S(2)n + S(1))(-S(3)aeS(2) + cdS(2)) + bS(2)cd(S(3)aeS(2)(-S(3)n + S(1)) - cd*S(2)(-n + S(1))))/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(acn(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(2)/(a + bxn + cx*(S(2)n))*S(2), x), x, -S(2)e*S(2)xhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - S(2)e*S(2)xhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))) + x(-S(2)abde - S(2)a(-aeS(2) + cdS(2)) + bS(2)dS(2) + xn(abe*S(2) - S(4)acde + bcdS(2)))/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))) - x((-n + S(1))(abe*S(2) - S(4)acde + bcdS(2)) + (S(4)abcden + S(4)ac(-S(2)n + S(1))(-aeS(2) + cdS(2)) + bS(2)(ae*S(2)(-S(3)n + S(1)) - cd*S(2)(-n + S(1))))/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(an(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) - x((-n + S(1))(abeS(2) - S(4)acde + bcdS(2)) - (S(4)abcden + S(4)ac(-S(2)n + S(1))(-aeS(2) + cdS(2)) + bS(2)(ae*S(2)(-S(3)n + S(1)) - cd*S(2)(-n + S(1))))/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(an(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*n)/(a + bx*n + cx*(S(2)n))*S(2), x), x, -cx(S(2)a(cd(-S(4)n + S(2)) - e(-n + S(1))sqrt(-S(4)ac + bS(2))) - bS(2)d(-n + S(1)) + b(S(2)aen + d(-n + S(1))sqrt(-S(4)ac + b*S(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) - cx(S(2)a(S(2)cd(-S(2)n + S(1)) + e(-n + S(1))sqrt(-S(4)ac + bS(2))) - bS(2)(-dn + d) - b(-S(2)aen + d(-n + S(1))sqrt(-S(4)ac + bS(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) + x(-abe - S(2)acd + bS(2)d + cxn(-S(2)ae + bd))/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) #Apart assert rubi_test(rubi_integrate(S(1)/((d + exn)(a + bxn + cx*(S(2)n))*S(2)), x), x, -ce*S(2)x(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(2)) - ceS(2)x(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(2)) + e*S(4)xhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(d(aeS(2) - bde + cdS(2))S(2)) + cx(-S(2)ac(S(2)cd(-S(2)n + S(1)) + e(-n + S(1))sqrt(-S(4)ac + bS(2))) - bS(3)e(-n + S(1)) + bS(2)(-n + S(1))(cd + esqrt(-S(4)ac + bS(2))) + bc(S(2)ae(-S(3)n + S(2)) - d(-n + S(1))sqrt(-S(4)ac + bS(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cd*S(2))) - cx((-n + S(1))(S(2)ace - bS(2)e + bcd) + (S(2)abce(-S(3)n + S(2)) - S(4)ac*S(2)d(-S(2)n + S(1)) - b*S(3)e(-n + S(1)) + bS(2)cd(-n + S(1)))/sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(an(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))(aeS(2) - bde + cd*S(2))) + x(S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd + cx*n(S(2)ace - bS(2)e + bcd))/(an(-S(4)ac + b*S(2))(a + bxn + cx*(S(2)n))(ae*S(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) #Apart assert rubi_test(rubi_integrate(S(1)/((d + exn)S(2)(a + bx*n + cx*(S(2)n))*S(2)), x), x, -S(2)ceS(2)x(be*S(2)(b - sqrt(-S(4)ac + b*S(2))) + S(3)c*S(2)d*S(2) - ce(ae + S(3)bd - S(2)dsqrt(-S(4)ac + b*S(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(3)) - S(2)ce*S(2)x(be*S(2)(b + sqrt(-S(4)ac + b*S(2))) + S(3)c*S(2)d*S(2) - ce(ae + S(3)bd + S(2)dsqrt(-S(4)ac + b*S(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(3)) + S(2)eS(4)x(-be + S(2)cd)hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(d(aeS(2) - bde + cdS(2))S(3)) + e*S(4)xhyper((S(2), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(dS(2)(ae*S(2) - bde + cdS(2))S(2)) + cx(S(4)ac*S(2)(-cdS(2)(-S(2)n + S(1)) + e(ae(-S(2)n + S(1)) - d(-n + S(1))sqrt(-S(4)ac + bS(2)))) + bS(4)e*S(2)(-n + S(1)) - b*S(3)e(-n + S(1))(S(2)cd + esqrt(-S(4)ac + bS(2))) - bS(2)c(-cd*S(2)(-n + S(1)) + e(ae(-S(7)n + S(5)) - S(2)d(-n + S(1))sqrt(-S(4)ac + bS(2)))) + bc(S(3)aeS(2)(-n + S(1))sqrt(-S(4)ac + bS(2)) + cd(S(4)ae(-S(3)n + S(2)) - d(-n + S(1))sqrt(-S(4)ac + bS(2)))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(2)) + cx(S(4)ac*S(2)(-cdS(2)(-S(2)n + S(1)) + e(ae(-S(2)n + S(1)) + d(-n + S(1))sqrt(-S(4)ac + bS(2)))) + bS(4)e*S(2)(-n + S(1)) - b*S(3)e(-n + S(1))(S(2)cd - esqrt(-S(4)ac + bS(2))) - bS(2)c(-cd*S(2)(-n + S(1)) + e(ae(-S(7)n + S(5)) + S(2)d(-n + S(1))sqrt(-S(4)ac + bS(2)))) + bc(-S(3)aeS(2)(-n + S(1))sqrt(-S(4)ac + bS(2)) + cd(S(4)ae(-S(3)n + S(2)) + d(-n + S(1))sqrt(-S(4)ac + bS(2)))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(2)) - x(-S(6)abc*S(2)de + S(2)acS(2)(-aeS(2) + cdS(2)) - bS(4)eS(2) + S(2)b*S(3)cde - b*S(2)c(-S(4)aeS(2) + cd*S(2)) + cx*n(-S(4)ac*S(2)de - bS(3)e*S(2) + S(2)b*S(2)cde - bc(-S(3)ae*S(2) + cd*S(2))))/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))(ae*S(2) - bde + cdS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(3)/(a + bx*n + cx*(S(2)n))S(3), x), x, eS(2)x(-S(2)ac(S(6)cd(-S(2)n + S(1)) - e(-n + S(1))sqrt(-S(4)ac + bS(2))) - bS(3)e(-n + S(1)) + bS(2)(-n + S(1))(S(3)cd + esqrt(-S(4)ac + b*S(2))) + bc(S(2)ae(-S(5)n + S(2)) - S(3)d(-n + S(1))sqrt(-S(4)ac + b*S(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(acn(-S(4)ac + b*S(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + bS(2)))) + eS(2)x(-S(2)ac(S(6)cd(-S(2)n + S(1)) + e(-n + S(1))sqrt(-S(4)ac + bS(2))) - bS(3)e(-n + S(1)) + bS(2)(-n + S(1))(S(3)cd - esqrt(-S(4)ac + b*S(2))) + bc(S(2)ae(-S(5)n + S(2)) + S(3)d(-n + S(1))sqrt(-S(4)ac + b*S(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(acn(-S(4)ac + b*S(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) + x(-abe(ae*S(2) + S(3)cdS(2)) - S(2)acd(-S(3)aeS(2) + cdS(2)) + bS(2)cdS(3) - xn(ab*S(2)e*S(3) + S(2)ace(-ae*S(2) + S(3)cdS(2)) - bcd(S(3)ae*S(2) + cd*S(2))))/(S(2)acn(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))S(2)) + eS(2)x(abce - S(6)acS(2)d - b*S(3)e + S(3)bS(2)cd + cx*n(-S(2)ace - bS(2)e + S(3)bcd))/(ac*S(2)n(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))) + x((-n + S(1))(S(4)aS(2)c*S(2)e(-S(3)n + S(1))(-ae*S(2) + S(3)cdS(2)) - S(2)abS(4)e*S(3)n - abS(2)ce(-aeS(2)(S(2)n + S(1)) + S(3)cdS(2)) - S(2)abc*S(2)d(S(3)aeS(2)n + cdS(2)(-S(7)n + S(2))) + bS(3)cd(S(6)ae*S(2)n + cdS(2)(-S(2)n + S(1)))) + (-S(4)a*S(2)bcS(2)e(ae*S(2)(S(19)nS(2) - S(11)n + S(1)) + S(3)cd*S(2)(-S(3)nS(2) - n + S(1))) - S(8)a*S(2)c*S(3)d(-S(3)aeS(2) + cd*S(2))(S(8)nS(2) - S(6)n + S(1)) + S(2)ab*S(5)e*S(3)n(-n + S(1)) + ab*S(3)ce(aeS(2)(S(30)nS(2) - S(19)n + S(1)) + S(3)cd*S(2)(-n + S(1))) + S(6)ab*S(2)c*S(2)d(-ae*S(2)(S(15)nS(2) - S(10)n + S(1)) + cdS(2)(S(3)nS(2) - S(4)n + S(1))) - b*S(4)cd(-n + S(1))(S(6)aeS(2)n + cdS(2)(-S(2)n + S(1))))/sqrt(-S(4)ac + bS(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)cnS(2)(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + x((-n + S(1))(S(4)aS(2)c*S(2)e(-S(3)n + S(1))(-ae*S(2) + S(3)cdS(2)) - S(2)abS(4)e*S(3)n - abS(2)ce(-aeS(2)(S(2)n + S(1)) + S(3)cdS(2)) - S(2)abc*S(2)d(S(3)aeS(2)n + cdS(2)(-S(7)n + S(2))) + bS(3)cd(S(6)ae*S(2)n + cdS(2)(-S(2)n + S(1)))) - (-S(4)a*S(2)bcS(2)e(ae*S(2)(S(19)nS(2) - S(11)n + S(1)) + S(3)cd*S(2)(-S(3)nS(2) - n + S(1))) - S(8)a*S(2)c*S(3)d(-S(3)aeS(2) + cd*S(2))(S(8)nS(2) - S(6)n + S(1)) + S(2)ab*S(5)e*S(3)n(-n + S(1)) + ab*S(3)ce(aeS(2)(S(30)nS(2) - S(19)n + S(1)) + S(3)cd*S(2)(-n + S(1))) + S(6)ab*S(2)c*S(2)d(-ae*S(2)(S(15)nS(2) - S(10)n + S(1)) + cdS(2)(S(3)nS(2) - S(4)n + S(1))) - b*S(4)cd(-n + S(1))(S(6)aeS(2)n + cdS(2)(-S(2)n + S(1))))/sqrt(-S(4)ac + bS(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)cnS(2)(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) - x(S(2)a*S(2)bcS(2)e(-S(5)aeS(2)n + S(3)cd*S(2)(-S(3)n + S(2))) + S(4)a*S(2)c*S(3)d(-S(4)n + S(1))(-S(3)aeS(2) + cd*S(2)) - S(2)abS(5)e*S(3)n - S(3)ab*S(3)ce(-S(3)ae*S(2)n + cdS(2)) + ab*S(2)c*S(2)d(S(3)aeS(2)(-S(9)n + S(1)) - S(5)cdS(2)(-S(3)n + S(1))) + bS(4)cd(S(6)ae*S(2)n + cdS(2)(-S(2)n + S(1))) + cx*n(S(4)aS(2)c*S(2)e(-S(3)n + S(1))(-ae*S(2) + S(3)cdS(2)) - S(2)abS(4)e*S(3)n - abS(2)ce(-aeS(2)(S(2)n + S(1)) + S(3)cdS(2)) - S(2)abc*S(2)d(S(3)aeS(2)n + cdS(2)(-S(7)n + S(2))) + bS(3)cd(S(6)ae*S(2)n + cdS(2)(-S(2)n + S(1)))))/(S(2)a*S(2)c*S(2)n*S(2)(-S(4)ac + bS(2))S(2)(a + bx*n + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(2)/(a + bx*n + cx*(S(2)n))S(3), x), x, -eS(2)x(S(4)ac(-S(2)n + S(1)) - b*S(2)(-n + S(1)) + b(-n + S(1))sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + bS(2)))) - eS(2)x(S(4)ac(-S(2)n + S(1)) - b*S(2)(-n + S(1)) - b(-n + S(1))sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) + x(-S(2)abde - S(2)a(-aeS(2) + cdS(2)) + bS(2)dS(2) + xn(abe*S(2) - S(4)acde + bcdS(2)))/(S(2)an(-S(4)ac + b*S(2))(a + bxn + cx*(S(2)n))S(2)) + eS(2)x(-S(2)ac + b*S(2) + bcxn)/(acn(-S(4)ac + b*S(2))(a + bxn + cx*(S(2)n))) - x((-n + S(1))(-S(8)aS(2)c*S(2)de(-S(3)n + S(1)) + S(2)abS(2)cde + S(2)abc(aeS(2)n + cdS(2)(-S(7)n + S(2))) - bS(3)(S(2)ae*S(2)n + cdS(2)(-S(2)n + S(1)))) - (-S(8)a*S(2)bcS(2)de(-S(3)nS(2) - n + S(1)) - S(8)a*S(2)c*S(2)(-aeS(2) + cd*S(2))(S(8)nS(2) - S(6)n + S(1)) + S(2)ab*S(3)cde(-n + S(1)) + S(2)abS(2)c(-ae*S(2)(S(15)nS(2) - S(10)n + S(1)) + S(3)cd*S(2)(S(3)nS(2) - S(4)n + S(1))) - b*S(4)(-n + S(1))(S(2)aeS(2)n + cdS(2)(-S(2)n + S(1))))/sqrt(-S(4)ac + bS(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)n*S(2)(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) - x((-n + S(1))(-S(8)aS(2)c*S(2)de(-S(3)n + S(1)) + S(2)abS(2)cde + S(2)abc(aeS(2)n + cdS(2)(-S(7)n + S(2))) - bS(3)(S(2)ae*S(2)n + cdS(2)(-S(2)n + S(1)))) + (-S(8)a*S(2)bcS(2)de(-S(3)nS(2) - n + S(1)) - S(8)a*S(2)c*S(2)(-aeS(2) + cd*S(2))(S(8)nS(2) - S(6)n + S(1)) + S(2)ab*S(3)cde(-n + S(1)) + S(2)abS(2)c(-ae*S(2)(S(15)nS(2) - S(10)n + S(1)) + S(3)cd*S(2)(S(3)nS(2) - S(4)n + S(1))) - b*S(4)(-n + S(1))(S(2)aeS(2)n + cdS(2)(-S(2)n + S(1))))/sqrt(-S(4)ac + bS(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)n*S(2)(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + x(-S(4)a*S(2)bcS(2)de(-S(3)n + S(2)) - S(4)a*S(2)c*S(2)(-S(4)n + S(1))(-aeS(2) + cd*S(2)) + S(2)abS(3)cde - abS(2)c(ae*S(2)(-S(9)n + S(1)) - S(5)cdS(2)(-S(3)n + S(1))) - bS(4)(S(2)ae*S(2)n + cdS(2)(-S(2)n + S(1))) + cx*n(-S(8)aS(2)c*S(2)de(-S(3)n + S(1)) + S(2)abS(2)cde + S(2)abc(aeS(2)n + cdS(2)(-S(7)n + S(2))) - bS(3)(S(2)ae*S(2)n + cdS(2)(-S(2)n + S(1)))))/(S(2)a*S(2)cnS(2)(-S(4)ac + bS(2))S(2)(a + bx*n + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)/(a + bx*n + cx*(S(2)n))*S(3), x), x, x(-abe - S(2)acd + bS(2)d + cxn(-S(2)ae + bd))/(S(2)an(-S(4)ac + b*S(2))(a + bxn + cx*(S(2)n))*S(2)) - cx(-S(4)a*S(2)c(-S(2)cd(S(8)nS(2) - S(6)n + S(1)) + esqrt(-S(4)ac + bS(2))(S(3)nS(2) - S(4)n + S(1))) + abS(2)(-n + S(1))(-S(6)cd(-S(3)n + S(1)) + esqrt(-S(4)ac + b*S(2))) + S(2)abc(S(2)ae(-S(3)nS(2) - n + S(1)) + dsqrt(-S(4)ac + b*S(2))(S(7)nS(2) - S(9)n + S(2))) + b*S(4)d(S(2)n*S(2) - S(3)n + S(1)) - b*S(3)(-n + S(1))(ae + d(-S(2)n + S(1))sqrt(-S(4)ac + bS(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)n*S(2)(-S(4)ac + bS(2))S(2)(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2)))) + cx(-S(4)a*S(2)c(S(2)cd(S(8)nS(2) - S(6)n + S(1)) + esqrt(-S(4)ac + bS(2))(S(3)nS(2) - S(4)n + S(1))) + abS(2)(-n + S(1))(S(6)cd(-S(3)n + S(1)) + esqrt(-S(4)ac + b*S(2))) - S(2)abc(S(2)ae(-S(3)nS(2) - n + S(1)) - dsqrt(-S(4)ac + b*S(2))(S(7)nS(2) - S(9)n + S(2))) - b*S(4)d(S(2)n*S(2) - S(3)n + S(1)) + b*S(3)(-n + S(1))(ae - d(-S(2)n + S(1))sqrt(-S(4)ac + bS(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)n*S(2)(-S(4)ac + bS(2))S(2)(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2)))) + x(-S(2)aS(2)bce(-S(3)n + S(2)) - S(4)aS(2)c*S(2)d(-S(4)n + S(1)) + abS(3)e + S(5)ab*S(2)cd(-S(3)n + S(1)) - bS(4)d(-S(2)n + S(1)) + cxn(-S(4)aS(2)ce(-S(3)n + S(1)) + ab*S(2)e + S(2)abcd(-S(7)n + S(2)) - b*S(3)d(-S(2)n + S(1))))/(S(2)aS(2)n*S(2)(-S(4)ac + bS(2))S(2)(a + bx*n + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) #Apart# assert rubi_test(rubi_integrate(S(1)/((d + exn)(a + bxn + cx*(S(2)n))*S(3)), x), x, -ce*S(4)x(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(3)) - ceS(4)x(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(3)) + e*S(6)xhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(d(aeS(2) - bde + cdS(2))S(3)) + ceS(2)x(-S(2)ac(S(2)cd(-S(2)n + S(1)) + e(-n + S(1))sqrt(-S(4)ac + bS(2))) - bS(3)e(-n + S(1)) + b*S(2)(-n + S(1))(cd + esqrt(-S(4)ac + bS(2))) + bc(S(2)ae(-S(3)n + S(2)) - d(-n + S(1))sqrt(-S(4)ac + bS(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(2)) + ceS(2)x(-S(2)ac(S(2)cd(-S(2)n + S(1)) - e(-n + S(1))sqrt(-S(4)ac + bS(2))) - bS(3)e(-n + S(1)) + b*S(2)(-n + S(1))(cd - esqrt(-S(4)ac + bS(2))) + bc(S(2)ae(-S(3)n + S(2)) + d(-n + S(1))sqrt(-S(4)ac + bS(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(2)) + e*S(2)x(S(3)abce - S(2)acS(2)d - b*S(3)e + b*S(2)cd + cx*n(S(2)ace - bS(2)e + bcd))/(an(-S(4)ac + b*S(2))(a + bxn + cx*(S(2)n))(ae*S(2) - bde + cdS(2))S(2)) + x(S(3)abce - S(2)acS(2)d - b*S(3)e + b*S(2)cd + cx*n(S(2)ace - bS(2)e + bcd))/(S(2)an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))*S(2)(aeS(2) - bde + cd*S(2))) - cx(-S(4)a*S(2)c*S(2)(-S(2)cd(S(8)n*S(2) - S(6)n + S(1)) + esqrt(-S(4)ac + bS(2))(S(3)nS(2) - S(4)n + S(1))) + abS(2)c(-n + S(1))(-S(6)cd(-S(3)n + S(1)) + e(-S(14)n + S(5))sqrt(-S(4)ac + bS(2))) - S(2)abc*S(2)(S(2)ae(S(13)n*S(2) - S(13)n + S(3)) + dsqrt(-S(4)ac + bS(2))(S(7)nS(2) - S(9)n + S(2))) - b*S(5)e(S(2)n*S(2) - S(3)n + S(1)) + b*S(4)(cd - esqrt(-S(4)ac + b*S(2)))(S(2)nS(2) - S(3)n + S(1)) + b*S(3)c(-n + S(1))(ae(-S(18)n + S(7)) + d(-S(2)n + S(1))sqrt(-S(4)ac + b*S(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)n*S(2)(-S(4)ac + bS(2))S(2)(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cd*S(2))) + cx(-S(4)a*S(2)c*S(2)(S(2)cd(S(8)n*S(2) - S(6)n + S(1)) + esqrt(-S(4)ac + bS(2))(S(3)nS(2) - S(4)n + S(1))) + abS(2)c(-n + S(1))(S(6)cd(-S(3)n + S(1)) + e(-S(14)n + S(5))sqrt(-S(4)ac + bS(2))) - S(2)abc*S(2)(-S(2)ae(S(13)n*S(2) - S(13)n + S(3)) + dsqrt(-S(4)ac + bS(2))(S(7)nS(2) - S(9)n + S(2))) + b*S(5)e(S(2)n*S(2) - S(3)n + S(1)) - b*S(4)(cd + esqrt(-S(4)ac + b*S(2)))(S(2)nS(2) - S(3)n + S(1)) - b*S(3)c(-n + S(1))(ae(-S(18)n + S(7)) - d(-S(2)n + S(1))sqrt(-S(4)ac + b*S(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)n*S(2)(-S(4)ac + bS(2))S(2)(cd*S(2) - e(-ae + bd))(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2)))) + x(S(2)aS(2)bcS(2)e(-S(11)n + S(4)) - S(4)aS(2)c*S(3)d(-S(4)n + S(1)) - S(3)ab*S(3)ce(-S(5)n + S(2)) + S(5)abS(2)c*S(2)d(-S(3)n + S(1)) + b*S(5)(-S(2)en + e) - b*S(4)cd(-S(2)n + S(1)) - cx*n(-S(4)aS(2)c*S(2)e(-S(3)n + S(1)) + abS(2)ce(-S(14)n + S(5)) - S(2)abc*S(2)d(-S(7)n + S(2)) - b*S(4)e(-S(2)n + S(1)) + b*S(3)cd(-S(2)n + S(1))))/(S(2)a*S(2)n*S(2)(-S(4)ac + bS(2))S(2)(a + bx*n + cx*(S(2)n))(ae*S(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) #Apart# assert rubi_test(rubi_integrate(S(1)/((d + exn)S(2)(a + bx*n + cx*(S(2)n))*S(3)), x), x, -ce*S(4)x(S(3)beS(2)(b - sqrt(-S(4)ac + b*S(2))) + S(10)c*S(2)d*S(2) - S(2)ce(ae + S(5)bd - S(3)dsqrt(-S(4)ac + bS(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(4)) - ceS(4)x(S(3)beS(2)(b + sqrt(-S(4)ac + b*S(2))) + S(10)c*S(2)d*S(2) - S(2)ce(ae + S(5)bd + S(3)dsqrt(-S(4)ac + bS(2))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/((-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(4)) + S(3)eS(6)x(-be + S(2)cd)hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -ex*n/d)/(d(aeS(2) - bde + cdS(2))S(4)) + e*S(6)xhyper((S(2), S(1)/n), (S(1) + S(1)/n,), -exn/d)/(dS(2)(ae*S(2) - bde + cdS(2))S(3)) + ceS(2)x(S(4)acS(2)(-S(3)cd*S(2)(-S(2)n + S(1)) + e(ae(-S(2)n + S(1)) - S(2)d(-n + S(1))sqrt(-S(4)ac + b*S(2)))) + S(2)b*S(4)e*S(2)(-n + S(1)) - b*S(3)e(-n + S(1))(S(5)cd + S(2)esqrt(-S(4)ac + bS(2))) - bS(2)c(-S(3)cd*S(2)(-n + S(1)) + e(ae(-S(13)n + S(9)) - S(5)d(-n + S(1))sqrt(-S(4)ac + bS(2)))) + bc(S(5)aeS(2)(-n + S(1))sqrt(-S(4)ac + bS(2)) + cd(S(4)ae(-S(8)n + S(5)) - S(3)d(-n + S(1))sqrt(-S(4)ac + b*S(2)))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(3)) + ceS(2)x(S(4)acS(2)(-S(3)cd*S(2)(-S(2)n + S(1)) + e(ae(-S(2)n + S(1)) + S(2)d(-n + S(1))sqrt(-S(4)ac + b*S(2)))) + S(2)b*S(4)e*S(2)(-n + S(1)) - b*S(3)e(-n + S(1))(S(5)cd - S(2)esqrt(-S(4)ac + bS(2))) - bS(2)c(-S(3)cd*S(2)(-n + S(1)) + e(ae(-S(13)n + S(9)) + S(5)d(-n + S(1))sqrt(-S(4)ac + bS(2)))) + bc(-S(5)aeS(2)(-n + S(1))sqrt(-S(4)ac + bS(2)) + cd(S(4)ae(-S(8)n + S(5)) + S(3)d(-n + S(1))sqrt(-S(4)ac + b*S(2)))))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))S(3)) - e*S(2)x(-S(14)abc*S(2)de + S(2)acS(2)(-aeS(2) + S(3)cdS(2)) - S(2)b*S(4)e*S(2) + S(5)b*S(3)cde - b*S(2)c(-S(7)aeS(2) + S(3)cdS(2)) + cx*n(-S(8)ac*S(2)de - S(2)b*S(3)e*S(2) + S(5)b*S(2)cde - bc(-S(5)ae*S(2) + S(3)cdS(2))))/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))(ae*S(2) - bde + cdS(2))S(3)) - x(-S(6)abc*S(2)de + S(2)acS(2)(-aeS(2) + cdS(2)) - bS(4)eS(2) + S(2)b*S(3)cde - b*S(2)c(-S(4)aeS(2) + cd*S(2)) + cx*n(-S(4)ac*S(2)de - bS(3)e*S(2) + S(2)b*S(2)cde - bc(-S(3)ae*S(2) + cd*S(2))))/(S(2)an(-S(4)ac + b*S(2))(a + bxn + cx*(S(2)n))*S(2)(aeS(2) - bde + cdS(2))S(2)) + cx((-n + S(1))(-S(8)a*S(2)c*S(3)de(-S(3)n + S(1)) + S(2)abS(2)c*S(2)de(-S(14)n + S(5)) + S(2)abc*S(2)(aeS(2)(-S(13)n + S(4)) - cd*S(2)(-S(7)n + S(2))) + bS(5)e*S(2)(-S(2)n + S(1)) - S(2)b*S(4)cde(-S(2)n + S(1)) - b*S(3)c(S(2)aeS(2)(-S(8)n + S(3)) - cd*S(2)(-S(2)n + S(1)))) + (S(8)a*S(2)bcS(3)de(S(13)nS(2) - S(13)n + S(3)) - S(8)aS(2)c*S(3)(-aeS(2) + cd*S(2))(S(8)nS(2) - S(6)n + S(1)) - S(2)ab*S(3)c*S(2)de(S(18)nS(2) - S(25)n + S(7)) + S(2)ab*S(2)c*S(2)(-aeS(2)(S(35)nS(2) - S(38)n + S(9)) + S(3)cd*S(2)(S(3)nS(2) - S(4)n + S(1))) - b*S(6)e*S(2)(S(2)nS(2) - S(3)n + S(1)) + S(2)bS(5)cde(S(2)n*S(2) - S(3)n + S(1)) + b*S(4)c(-n + S(1))(S(4)ae*S(2)(-S(5)n + S(2)) - cd*S(2)(-S(2)n + S(1))))/sqrt(-S(4)ac + bS(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)n*S(2)(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)(ae*S(2) - bde + cdS(2))S(2)) + cx((-n + S(1))(-S(8)a*S(2)c*S(3)de(-S(3)n + S(1)) + S(2)abS(2)c*S(2)de(-S(14)n + S(5)) + S(2)abc*S(2)(aeS(2)(-S(13)n + S(4)) - cd*S(2)(-S(7)n + S(2))) + bS(5)e*S(2)(-S(2)n + S(1)) - S(2)b*S(4)cde(-S(2)n + S(1)) - b*S(3)c(S(2)aeS(2)(-S(8)n + S(3)) - cd*S(2)(-S(2)n + S(1)))) - (S(8)a*S(2)bcS(3)de(S(13)nS(2) - S(13)n + S(3)) - S(8)aS(2)c*S(3)(-aeS(2) + cd*S(2))(S(8)nS(2) - S(6)n + S(1)) - S(2)ab*S(3)c*S(2)de(S(18)nS(2) - S(25)n + S(7)) + S(2)ab*S(2)c*S(2)(-aeS(2)(S(35)nS(2) - S(38)n + S(9)) + S(3)cd*S(2)(S(3)nS(2) - S(4)n + S(1))) - b*S(6)e*S(2)(S(2)nS(2) - S(3)n + S(1)) + S(2)bS(5)cde(S(2)n*S(2) - S(3)n + S(1)) + b*S(4)c(-n + S(1))(S(4)ae*S(2)(-S(5)n + S(2)) - cd*S(2)(-S(2)n + S(1))))/sqrt(-S(4)ac + bS(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)n*S(2)(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)(ae*S(2) - bde + cdS(2))S(2)) - x(-S(4)a*S(2)bcS(3)de(-S(11)n + S(4)) + S(4)a*S(2)c*S(3)(-S(4)n + S(1))(-aeS(2) + cd*S(2)) + S(6)abS(3)c*S(2)de(-S(5)n + S(2)) + ab*S(2)c*S(2)(aeS(2)(-S(37)n + S(13)) - S(5)cdS(2)(-S(3)n + S(1))) + bS(6)e*S(2)(-S(2)n + S(1)) - S(2)b*S(5)cde(-S(2)n + S(1)) - b*S(4)c(ae*S(2)(-S(17)n + S(7)) - cd*S(2)(-S(2)n + S(1))) + cx*n(-S(8)aS(2)c*S(3)de(-S(3)n + S(1)) + S(2)abS(2)c*S(2)de(-S(14)n + S(5)) + S(2)abc*S(2)(aeS(2)(-S(13)n + S(4)) - cd*S(2)(-S(7)n + S(2))) + bS(5)e*S(2)(-S(2)n + S(1)) - S(2)b*S(4)cde(-S(2)n + S(1)) - b*S(3)c(S(2)aeS(2)(-S(8)n + S(3)) - cd*S(2)(-S(2)n + S(1)))))/(S(2)a*S(2)n*S(2)(-S(4)ac + bS(2))S(2)(a + bx*n + cx*(S(2)n))(ae*S(2) - bde + cdS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cx(S(2)n))*p/(d + ex*n), x), x, x(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1(S(1)/(S(2)n), -p, S(1), S(1) + S(1)/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/d*S(2))/d - ex*(n + S(1))(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1((n + S(1))/(S(2)n), -p, S(1), S(3)/2 + S(1)/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(2)(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cx*(S(2)n))*p/(d + exn)S(2), x), x, x(S(1) + cx*(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1(S(1)/(S(2)n), -p, S(2), S(1) + S(1)/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/dS(2) - S(2)ex*(n + S(1))(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1((n + S(1))/(S(2)n), -p, S(2), S(3)/2 + S(1)/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(3)(n + S(1))) + eS(2)x*(S(2)n + S(1))(S(1) + cx*(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1(S(1) + S(1)/(S(2)n), -p, S(2), S(2) + S(1)/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(4)(S(2)n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cx(S(2)n))*p/(d + exn)S(3), x), x, x(S(1) + cx*(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1(S(1)/(S(2)n), -p, S(3), S(1) + S(1)/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/dS(3) - S(3)ex*(n + S(1))(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1((n + S(1))/(S(2)n), -p, S(3), S(3)/2 + S(1)/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(4)(n + S(1))) + S(3)e*S(2)x*(S(2)n + S(1))(S(1) + cx*(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1(S(1) + S(1)/(S(2)n), -p, S(3), S(2) + S(1)/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(5)(S(2)n + S(1))) - e*S(3)x*(S(3)n + S(1))(S(1) + cx*(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1(S(3)/2 + S(1)/(S(2)n), -p, S(3), S(5)/2 + S(1)/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(6)(S(3)n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a + cx(S(2)n))(d + ex*n)), x), x, xsqrt(S(1) + cx(S(2)n)/a)AppellF1(S(1)/(S(2)n), S(1)/2, S(1), S(1) + S(1)/(S(2)n), -cx*(S(2)n)/a, e*S(2)x*(S(2)n)/d*S(2))/(dsqrt(a + cx(S(2)n))) - ex(n + S(1))sqrt(S(1) + cx(S(2)n)/a)AppellF1((n + S(1))/(S(2)n), S(1)/2, S(1), S(3)/2 + S(1)/(S(2)n), -cx*(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(2)sqrt(a + cx*(S(2)n))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*n)sqrt(a + bxn + cx*(S(2)n)), x), x, dxsqrt(a + bxn + cx*(S(2)n))AppellF1(S(1)/n, S(-1)/2, S(-1)/2, S(1) + S(1)/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(sqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))) + ex*(n + S(1))sqrt(a + bxn + cx*(S(2)n))AppellF1(S(1) + S(1)/n, S(-1)/2, S(-1)/2, S(2) + S(1)/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/((n + S(1))sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*n)(a + bxn + cx*(S(2)n))*(S(3)/2), x), x, adxsqrt(a + bxn + cx*(S(2)n))AppellF1(S(1)/n, S(-3)/2, S(-3)/2, S(1) + S(1)/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(sqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))) + aex(n + S(1))sqrt(a + bxn + cx*(S(2)n))AppellF1(S(1) + S(1)/n, S(-3)/2, S(-3)/2, S(2) + S(1)/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/((n + S(1))sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*n)/sqrt(a + bx*n + cx*(S(2)n)), x), x, dxsqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/n, S(1)/2, S(1)/2, S(1) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/sqrt(a + bx*n + cx*(S(2)n)) + ex(n + S(1))sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1) + S(1)/n, S(1)/2, S(1)/2, S(2) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/((n + S(1))sqrt(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)/(a + bx*n + cx*(S(2)n))*(S(3)/2), x), x, dxsqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/n, S(3)/2, S(3)/2, S(1) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(asqrt(a + bxn + cx*(S(2)n))) + ex(n + S(1))sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1) + S(1)/n, S(3)/2, S(3)/2, S(2) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(a(n + S(1))sqrt(a + bx*n + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)/(a + bx*n + cx*(S(2)n))*(S(5)/2), x), x, dxsqrt(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/n, S(5)/2, S(5)/2, S(1) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(aS(2)sqrt(a + bxn + cx*(S(2)n))) + ex(n + S(1))sqrt(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1) + S(1)/n, S(5)/2, S(5)/2, S(2) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(aS(2)(n + S(1))sqrt(a + bx*n + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(3)(a + bxn + cx*(S(2)n))p, x), x, dS(3)x(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1(S(1)/n, -p, -p, S(1) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2)))) + S(3)d*S(2)ex(n + S(1))(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1(S(1) + S(1)/n, -p, -p, S(2) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(n + S(1)) + S(3)deS(2)x*(S(2)n + S(1))(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1(S(2) + S(1)/n, -p, -p, S(3) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(2)n + S(1)) + e*S(3)x*(S(3)n + S(1))(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1(S(3) + S(1)/n, -p, -p, S(4) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(3)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)S(2)(a + bxn + cx*(S(2)n))p, x), x, dS(2)x(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1(S(1)/n, -p, -p, S(1) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2)))) + S(2)dex*(n + S(1))(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1(S(1) + S(1)/n, -p, -p, S(2) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(n + S(1)) + eS(2)x*(S(2)n + S(1))(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1(S(2) + S(1)/n, -p, -p, S(3) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(2)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)(a + bxn + cx*(S(2)n))*p, x), x, dx(S(2)cxn/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1(S(1)/n, -p, -p, S(1) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2)))) + ex*(n + S(1))(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1(S(1) + S(1)/n, -p, -p, S(2) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(n + S(1)), expand=True, _diff=True, _numerical=True) def test_3(): assert rubi_test(rubi_integrate(xS(3)(a + cxS(4))S(5)(d + exS(2)), x), x, aS(5)dxS(4)/S(4) + aS(5)exS(6)/S(6) + S(5)a*S(4)cdxS(8)/S(8) + aS(4)cexS(10)/S(2) + S(5)a*S(3)c*S(2)dxS(12)/S(6) + S(5)a*S(3)c*S(2)exS(14)/S(7) + S(5)a*S(2)c*S(3)dxS(16)/S(8) + S(5)a*S(2)c*S(3)exS(18)/S(9) + ac*S(4)dxS(20)/S(4) + S(5)acS(4)exS(22)/S(22) + cS(5)dxS(24)/S(24) + cS(5)exS(26)/S(26), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + cxS(4))S(5)(d + exS(2)), x), x, aS(5)dxS(3)/S(3) + aS(5)exS(5)/S(5) + S(5)a*S(4)cdx*S(7)/S(7) + S(5)a*S(4)cex*S(9)/S(9) + S(10)a*S(3)c*S(2)dxS(11)/S(11) + S(10)a*S(3)c*S(2)exS(13)/S(13) + S(2)a*S(2)c*S(3)dxS(15)/S(3) + S(10)a*S(2)c*S(3)exS(17)/S(17) + S(5)acS(4)dxS(19)/S(19) + S(5)acS(4)exS(21)/S(21) + cS(5)dxS(23)/S(23) + cS(5)exS(25)/S(25), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + cxS(4))S(5)(d + exS(2)), x), x, aS(5)dxS(2)/S(2) + S(5)a*S(4)cdxS(6)/S(6) + aS(3)cS(2)dxS(10) + S(5)a*S(2)c*S(3)dxS(14)/S(7) + S(5)acS(4)dxS(18)/S(18) + cS(5)dxS(22)/S(22) + e(a + cxS(4))S(6)/(S(24)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cxS(4))S(5)(d + exS(2)), x), x, aS(5)dx + aS(5)exS(3)/S(3) + aS(4)cdx*S(5) + S(5)a*S(4)cex*S(7)/S(7) + S(10)a*S(3)c*S(2)dxS(9)/S(9) + S(10)a*S(3)c*S(2)exS(11)/S(11) + S(10)a*S(2)c*S(3)dxS(13)/S(13) + S(2)a*S(2)c*S(3)exS(15)/S(3) + S(5)acS(4)dxS(17)/S(17) + S(5)acS(4)exS(19)/S(19) + cS(5)dxS(21)/S(21) + cS(5)exS(23)/S(23), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cxS(4))S(5)(d + exS(2))/x, x), x, aS(5)dlog(x) + a*S(5)exS(2)/S(2) + S(5)a*S(4)cdx*S(4)/S(4) + S(5)a*S(4)cex*S(6)/S(6) + S(5)a*S(3)c*S(2)dxS(8)/S(4) + aS(3)c*S(2)exS(10) + S(5)a*S(2)c*S(3)dxS(12)/S(6) + S(5)a*S(2)c*S(3)exS(14)/S(7) + S(5)acS(4)dxS(16)/S(16) + S(5)acS(4)exS(18)/S(18) + cS(5)dxS(20)/S(20) + cS(5)exS(22)/S(22), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cxS(4))S(5)(d + exS(2))/xS(2), x), x, -a*S(5)d/x + a*S(5)ex + S(5)a*S(4)cdxS(3)/S(3) + aS(4)cexS(5) + S(10)a*S(3)c*S(2)dxS(7)/S(7) + S(10)a*S(3)c*S(2)exS(9)/S(9) + S(10)a*S(2)c*S(3)dxS(11)/S(11) + S(10)a*S(2)c*S(3)exS(13)/S(13) + ac*S(4)dxS(15)/S(3) + S(5)acS(4)exS(17)/S(17) + cS(5)dxS(19)/S(19) + cS(5)exS(21)/S(21), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cxS(4))S(5)(d + exS(2))/xS(3), x), x, -a*S(5)d/(S(2)xS(2)) + aS(5)elog(x) + S(5)a*S(4)cdx*S(2)/S(2) + S(5)a*S(4)cex*S(4)/S(4) + S(5)a*S(3)c*S(2)dxS(6)/S(3) + S(5)a*S(3)c*S(2)exS(8)/S(4) + aS(2)c*S(3)dxS(10) + S(5)a*S(2)c*S(3)exS(12)/S(6) + S(5)acS(4)dxS(14)/S(14) + S(5)acS(4)exS(16)/S(16) + cS(5)dxS(18)/S(18) + cS(5)exS(20)/S(20), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(d + exS(2))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, d(fx)*(m + S(1))/(f(m + S(1))) + e(fx)(m + S(23))/(fS(23)(m + S(23))) + (fx)*(m + S(3))(S(10)d + e)/(fS(3)(m + S(3))) + (fx)(m + S(5))(S(45)d + S(10)e)/(f*S(5)(m + S(5))) + (fx)(m + S(7))(S(120)d + S(45)e)/(f*S(7)(m + S(7))) + (fx)(m + S(9))(S(210)d + S(120)e)/(f*S(9)(m + S(9))) + (fx)(m + S(11))(S(252)d + S(210)e)/(f*S(11)(m + S(11))) + (fx)(m + S(13))(S(210)d + S(252)e)/(f*S(13)(m + S(13))) + (fx)(m + S(15))(S(120)d + S(210)e)/(f*S(15)(m + S(15))) + (fx)(m + S(17))(S(45)d + S(120)e)/(f*S(17)(m + S(17))) + (fx)(m + S(19))(S(10)d + S(45)e)/(f*S(19)(m + S(19))) + (fx)(m + S(21))(d + S(10)e)/(fS(21)(m + S(21))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)(d + exS(2))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, e(xS(2) + S(1))S(14)/S(28) + (d/S(26) - S(3)e/S(26))(xS(2) + S(1))S(13) + (d/S(22) - e/S(22))(xS(2) + S(1))S(11) - (d/S(12) - e/S(8))(xS(2) + S(1))S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(d + exS(2))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, dxS(5)/S(5) + exS(27)/S(27) + xS(25)(d/S(25) + S(2)e/S(5)) + x*S(23)(S(10)d/S(23) + S(45)e/S(23)) + x*S(21)(S(15)d/S(7) + S(40)e/S(7)) + x*S(19)(S(120)d/S(19) + S(210)e/S(19)) + x*S(17)(S(210)d/S(17) + S(252)e/S(17)) + x*S(15)(S(84)d/S(5) + S(14)e) + x*S(13)(S(210)d/S(13) + S(120)e/S(13)) + x*S(11)(S(120)d/S(11) + S(45)e/S(11)) + x*S(9)(S(5)d + S(10)e/S(9)) + x*S(7)(S(10)d/S(7) + e/S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(d + exS(2))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, e(xS(2) + S(1))S(13)/S(26) + (-d/S(22) + e/S(22))(xS(2) + S(1))S(11) + (d/S(24) - e/S(12))(xS(2) + S(1))S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(d + exS(2))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, dxS(3)/S(3) + exS(25)/S(25) + xS(23)(d/S(23) + S(10)e/S(23)) + x*S(21)(S(10)d/S(21) + S(15)e/S(7)) + x*S(19)(S(45)d/S(19) + S(120)e/S(19)) + x*S(17)(S(120)d/S(17) + S(210)e/S(17)) + x*S(15)(S(14)d + S(84)e/S(5)) + x*S(13)(S(252)d/S(13) + S(210)e/S(13)) + x*S(11)(S(210)d/S(11) + S(120)e/S(11)) + x*S(9)(S(40)d/S(3) + S(5)e) + x*S(7)(S(45)d/S(7) + S(10)e/S(7)) + x*S(5)(S(2)d + e/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(d + exS(2))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, e(xS(2) + S(1))S(12)/S(24) + (d/S(22) - e/S(22))(xS(2) + S(1))S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, dx + exS(23)/S(23) + xS(21)(d/S(21) + S(10)e/S(21)) + x*S(19)(S(10)d/S(19) + S(45)e/S(19)) + x*S(17)(S(45)d/S(17) + S(120)e/S(17)) + x*S(15)(S(8)d + S(14)e) + x*S(13)(S(210)d/S(13) + S(252)e/S(13)) + x*S(11)(S(252)d/S(11) + S(210)e/S(11)) + x*S(9)(S(70)d/S(3) + S(40)e/S(3)) + x*S(7)(S(120)d/S(7) + S(45)e/S(7)) + x*S(5)(S(9)d + S(2)e) + x*S(3)(S(10)d/S(3) + e/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))(x*S(4) + S(2)xS(2) + S(1))S(5)/x, x), x, dxS(20)/S(20) + S(5)dxS(18)/S(9) + S(45)dxS(16)/S(16) + S(60)dxS(14)/S(7) + S(35)dxS(12)/S(2) + S(126)dxS(10)/S(5) + S(105)dxS(8)/S(4) + S(20)dxS(6) + S(45)dxS(4)/S(4) + S(5)dxS(2) + dlog(x) + e(xS(2) + S(1))S(11)/S(22), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))(x*S(4) + S(2)xS(2) + S(1))S(5)/x*S(2), x), x, -d/x + exS(21)/S(21) + xS(19)(d/S(19) + S(10)e/S(19)) + x*S(17)(S(10)d/S(17) + S(45)e/S(17)) + x*S(15)(S(3)d + S(8)e) + x*S(13)(S(120)d/S(13) + S(210)e/S(13)) + x*S(11)(S(210)d/S(11) + S(252)e/S(11)) + x*S(9)(S(28)d + S(70)e/S(3)) + x*S(7)(S(30)d + S(120)e/S(7)) + x*S(5)(S(24)d + S(9)e) + x*S(3)(S(15)d + S(10)e/S(3)) + x(S(10)d + e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))(x*S(4) + S(2)xS(2) + S(1))S(5)/x*S(3), x), x, -d/(S(2)x*S(2)) + exS(20)/S(20) + xS(18)(d/S(18) + S(5)e/S(9)) + x*S(16)(S(5)d/S(8) + S(45)e/S(16)) + x*S(14)(S(45)d/S(14) + S(60)e/S(7)) + x*S(12)(S(10)d + S(35)e/S(2)) + x*S(10)(S(21)d + S(126)e/S(5)) + x*S(8)(S(63)d/S(2) + S(105)e/S(4)) + x*S(6)(S(35)d + S(20)e) + x*S(4)(S(30)d + S(45)e/S(4)) + x*S(2)(S(45)d/S(2) + S(5)e) + (S(10)d + e)log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)m(x*S(2) + S(1))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, (fx)(m + S(1))/(f(m + S(1))) + S(11)(fx)(m + S(3))/(fS(3)(m + S(3))) + S(55)(fx)(m + S(5))/(fS(5)(m + S(5))) + S(165)(fx)(m + S(7))/(fS(7)(m + S(7))) + S(330)(fx)(m + S(9))/(fS(9)(m + S(9))) + S(462)(fx)(m + S(11))/(fS(11)(m + S(11))) + S(462)(fx)(m + S(13))/(fS(13)(m + S(13))) + S(330)(fx)(m + S(15))/(fS(15)(m + S(15))) + S(165)(fx)(m + S(17))/(fS(17)(m + S(17))) + S(55)(fx)(m + S(19))/(fS(19)(m + S(19))) + S(11)(fx)(m + S(21))/(fS(21)(m + S(21))) + (fx)(m + S(23))/(fS(23)(m + S(23))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)(x*S(2) + S(1))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, (xS(2) + S(1))S(14)/S(28) - (xS(2) + S(1))S(13)/S(13) + (xS(2) + S(1))S(12)/S(24), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)(x*S(2) + S(1))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, x*S(27)/S(27) + S(11)x*S(25)/S(25) + S(55)x*S(23)/S(23) + S(55)x*S(21)/S(7) + S(330)x*S(19)/S(19) + S(462)x*S(17)/S(17) + S(154)x*S(15)/S(5) + S(330)x*S(13)/S(13) + S(15)x*S(11) + S(55)x*S(9)/S(9) + S(11)xS(7)/S(7) + xS(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(x*S(2) + S(1))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, (xS(2) + S(1))S(13)/S(26) - (xS(2) + S(1))S(12)/S(24), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(x*S(2) + S(1))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, x*S(25)/S(25) + S(11)x*S(23)/S(23) + S(55)x*S(21)/S(21) + S(165)x*S(19)/S(19) + S(330)x*S(17)/S(17) + S(154)x*S(15)/S(5) + S(462)x*S(13)/S(13) + S(30)x*S(11) + S(55)x*S(9)/S(3) + S(55)x*S(7)/S(7) + S(11)xS(5)/S(5) + xS(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(xS(2) + S(1))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, (xS(2) + S(1))S(12)/S(24), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) + S(1))(x*S(4) + S(2)xS(2) + S(1))S(5), x), x, x*S(23)/S(23) + S(11)x*S(21)/S(21) + S(55)x*S(19)/S(19) + S(165)x*S(17)/S(17) + S(22)x*S(15) + S(462)x*S(13)/S(13) + S(42)x*S(11) + S(110)x*S(9)/S(3) + S(165)x*S(7)/S(7) + S(11)x*S(5) + S(11)xS(3)/S(3) + x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))(xS(4) + S(2)xS(2) + S(1))S(5)/x, x), x, x*S(22)/S(22) + S(11)x*S(20)/S(20) + S(55)x*S(18)/S(18) + S(165)x*S(16)/S(16) + S(165)x*S(14)/S(7) + S(77)x*S(12)/S(2) + S(231)x*S(10)/S(5) + S(165)x*S(8)/S(4) + S(55)x*S(6)/S(2) + S(55)x*S(4)/S(4) + S(11)xS(2)/S(2) + log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))(xS(4) + S(2)xS(2) + S(1))S(5)/xS(2), x), x, xS(21)/S(21) + S(11)xS(19)/S(19) + S(55)x*S(17)/S(17) + S(11)x*S(15) + S(330)x*S(13)/S(13) + S(42)x*S(11) + S(154)x*S(9)/S(3) + S(330)x*S(7)/S(7) + S(33)x*S(5) + S(55)x*S(3)/S(3) + S(11)x - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) + S(1))(x*S(4) + S(2)xS(2) + S(1))S(5)/xS(3), x), x, xS(20)/S(20) + S(11)xS(18)/S(18) + S(55)x*S(16)/S(16) + S(165)x*S(14)/S(14) + S(55)x*S(12)/S(2) + S(231)x*S(10)/S(5) + S(231)x*S(8)/S(4) + S(55)x*S(6) + S(165)x*S(4)/S(4) + S(55)x*S(2)/S(2) + S(11)log(x) - S(1)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(d + exS(2))/sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, e(fx)(m + S(1))(a + bxS(2))/(bf(m + S(1))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (fx)*(m + S(1))(a + bxS(2))(-ae + bd)hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -bx*S(2)/a)/(abf(m + S(1))sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(d + exS(2))/sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, -sqrt(a)(a + bxS(2))(-ae + bd)atan(sqrt(b)x/sqrt(a))/(b*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + ex*S(3)(a + bxS(2))/(S(3)bsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + x(a + bxS(2))(-ae + bd)/(b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(d + exS(2))/sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, esqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2)b*S(2)) + (a + bx*S(2))(-ae + bd)log(a + bx*S(2))/(S(2)b*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/sqrt(aS(2) + S(2)abxS(2) + bS(2)x*S(4)), x), x, ex(a + bx*S(2))/(bsqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (a + bx*S(2))(-ae + bd)atan(sqrt(b)x/sqrt(a))/(sqrt(a)b(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/(xsqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, d(a + bxS(2))log(x)/(asqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - (a + bx*S(2))(-ae + bd)log(a + bx*S(2))/(S(2)absqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/(xS(2)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, -d(a + bxS(2))/(axsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - (a + bx*S(2))(-ae + bd)atan(sqrt(b)x/sqrt(a))/(a*(S(3)/2)sqrt(b)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/(xS(3)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), x), x, -d(a + bxS(2))/(S(2)axS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - (a + bx*S(2))(-ae + bd)log(x)/(aS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (a + bx*S(2))(-ae + bd)log(a + bx*S(2))/(S(2)a*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(d + exS(2))/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, (fx)*(m + S(1))(-ae + bd)/(S(4)abf(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (fx)*(m + S(1))(a + bxS(2))(ae(m + S(1)) + bd(-m + S(3)))hyper((S(2), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -bx*S(2)/a)/(S(4)a*S(3)bf(m + S(1))sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(d + exS(2))/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, -x(-ae + bd)/(S(4)bS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + x(-S(5)ae + bd)/(S(8)abS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (a + bx*S(2))(S(3)ae + bd)atan(sqrt(b)x/sqrt(a))/(S(8)a*(S(3)/2)b*(S(5)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(d + exS(2))/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, -(a + bx*S(2))(d + exS(2))S(2)/((-S(4)ae + S(4)bd)(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2), x), x, x(-ae + bd)/(S(4)ab(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + x(ae + S(3)bd)/(S(8)a*S(2)bsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) + (a + bx*S(2))(ae + S(3)bd)atan(sqrt(b)x/sqrt(a))/(S(8)a*(S(5)/2)b*(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/(x(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), x), x, (-ae + bd)/(S(4)ab(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + d/(S(2)a*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + d(a + bxS(2))log(x)/(a*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - d(a + bxS(2))log(a + bxS(2))/(S(2)a*S(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/(xS(2)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), x), x, -x(-ae + bd)/(S(4)aS(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - d(a + bxS(2))/(aS(3)xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))) - x(-S(3)ae + S(7)bd)/(S(8)aS(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - (a + bx*S(2))(-S(3)ae + S(15)bd)atan(sqrt(b)x/sqrt(a))/(S(8)a(S(7)/2)sqrt(b)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))/(xS(3)(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(3)/2)), x), x, -(-ae + bd)/(S(4)a*S(2)(a + bxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - d(a + bxS(2))/(S(2)a*S(3)x*S(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - (-ae + S(2)bd)/(S(2)aS(3)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) - (a + bx*S(2))(-ae + S(3)bd)log(x)/(a*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))) + (a + bx*S(2))(-ae + S(3)bd)log(a + bxS(2))/(S(2)a*S(4)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxS(2))(a*S(2) + S(2)abxS(2) + bS(2)xS(4))p, x), x, (aS(2) + S(2)abxS(2) + bS(2)xS(4))(p + S(1))/(S(4)b(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(c + dxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, bx*S(3)(c + dxS(2))(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(6)d(a + bxS(2))) + cS(2)(-S(2)ad + bc)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))atanh(sqrt(d)x/sqrt(c + dx*S(2)))/(S(16)d*(S(5)/2)(a + bxS(2))) - cxsqrt(c + dx*S(2))(-S(2)ad + bc)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(16)d*S(2)(a + bxS(2))) - xS(3)sqrt(c + dxS(2))(-S(2)ad + bc)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(8)d(a + bx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(c + dxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, (c + dxS(2))(S(3)/2)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(5)d) - (c + dxS(2))(S(3)/2)(-S(2)ad + S(2)bc)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(15)d*S(2)(a + bxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(c + dx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4)), x), x, bx(c + dxS(2))(S(3)/2)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))/(S(4)d(a + bx*S(2))) - c(-S(4)ad + bc)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))atanh(sqrt(d)x/sqrt(c + dx*S(2)))/(S(8)d*(S(3)/2)(a + bxS(2))) - xsqrt(c + dxS(2))(-S(4)ad + bc)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(8)d(a + bx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(c + dx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/x, x), x, -asqrt(c)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))atanh(sqrt(c + dxS(2))/sqrt(c))/(a + bx*S(2)) + asqrt(c + dxS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(a + bx*S(2)) + b(c + dxS(2))(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(3)d(a + bx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(c + dx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/xS(2), x), x, -a(c + dxS(2))(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(cx(a + bx*S(2))) + (S(2)ad + bc)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))atanh(sqrt(d)x/sqrt(c + dx*S(2)))/(S(2)sqrt(d)(a + bx*S(2))) + xsqrt(c + dxS(2))(S(2)ad + bc)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2)c(a + bx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(c + dx*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/xS(3), x), x, -a(c + dxS(2))(S(3)/2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2)cxS(2)(a + bxS(2))) + sqrt(c + dx*S(2))(ad + S(2)bc)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))/(S(2)c(a + bx*S(2))) - (ad + S(2)bc)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))atanh(sqrt(c + dxS(2))/sqrt(c))/(S(2)sqrt(c)(a + bxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(A + Bx*S(2))(a + bxS(2) + cxS(4))S(3), x), x, AaS(3)x*S(4)/S(4) + Bc*S(3)xS(18)/S(18) + aS(2)xS(6)(S(3)Ab + Ba)/S(6) + S(3)axS(8)(A(ac + b*S(2)) + Bab)/S(8) + cS(2)x*S(16)(Ac + S(3)Bb)/S(16) + S(3)cxS(14)(Abc + Bac + BbS(2))/S(14) + xS(12)(Aac*S(2)/S(4) + Ab*S(2)c/S(4) + Babc/S(2) + BbS(3)/S(12)) + xS(10)(A(S(6)abc + bS(3))/S(10) + S(3)Ba(ac + bS(2))/S(10)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(A + BxS(2))(a + bxS(2) + cxS(4))S(3), x), x, AaS(3)x*S(3)/S(3) + Bc*S(3)xS(17)/S(17) + aS(2)xS(5)(S(3)Ab + Ba)/S(5) + S(3)axS(7)(A(ac + b*S(2)) + Bab)/S(7) + cS(2)x*S(15)(Ac + S(3)Bb)/S(15) + S(3)cxS(13)(Abc + Bac + BbS(2))/S(13) + xS(11)(S(3)AacS(2)/S(11) + S(3)AbS(2)c/S(11) + S(6)Babc/S(11) + BbS(3)/S(11)) + xS(9)(A(S(6)abc + b*S(3))/S(9) + Ba(ac + b*S(2))/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(A + BxS(2))(a + bxS(2) + cxS(4))S(3), x), x, AaS(3)x*S(2)/S(2) + Bc*S(3)xS(16)/S(16) + aS(2)xS(4)(S(3)Ab + Ba)/S(4) + ax*S(6)(A(ac + b*S(2)) + Bab)/S(2) + cS(2)x*S(14)(Ac + S(3)Bb)/S(14) + cx*S(12)(Abc + Bac + BbS(2))/S(4) + xS(10)(S(3)AacS(2)/S(10) + S(3)AbS(2)c/S(10) + S(3)Babc/S(5) + BbS(3)/S(10)) + xS(8)(A(S(6)abc + b*S(3))/S(8) + S(3)Ba(ac + bS(2))/S(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))(a + bxS(2) + cxS(4))S(3), x), x, AaS(3)x + BcS(3)xS(15)/S(15) + aS(2)xS(3)(S(3)Ab + Ba)/S(3) + S(3)axS(5)(A(ac + b*S(2)) + Bab)/S(5) + cS(2)x*S(13)(Ac + S(3)Bb)/S(13) + S(3)cxS(11)(Abc + Bac + BbS(2))/S(11) + xS(9)(Aac*S(2)/S(3) + Ab*S(2)c/S(3) + S(2)Babc/S(3) + BbS(3)/S(9)) + xS(7)(A(S(6)abc + b*S(3))/S(7) + S(3)Ba(ac + bS(2))/S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))(a + bxS(2) + cxS(4))S(3)/x, x), x, AaS(3)log(x) + BcS(3)xS(14)/S(14) + aS(2)xS(2)(S(3)Ab + Ba)/S(2) + S(3)axS(4)(A(ac + b*S(2)) + Bab)/S(4) + cS(2)x*S(12)(Ac + S(3)Bb)/S(12) + S(3)cxS(10)(Abc + Bac + BbS(2))/S(10) + xS(8)(S(3)AacS(2)/S(8) + S(3)AbS(2)c/S(8) + S(3)Babc/S(4) + BbS(3)/S(8)) + xS(6)(A(S(6)abc + b*S(3))/S(6) + Ba(ac + b*S(2))/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))(a + bxS(2) + cxS(4))S(3)/x*S(2), x), x, -Aa*S(3)/x + Bc*S(3)xS(13)/S(13) + aS(2)x(S(3)Ab + Ba) + ax*S(3)(A(ac + b*S(2)) + Bab) + cS(2)x*S(11)(Ac + S(3)Bb)/S(11) + cx*S(9)(Abc + Bac + BbS(2))/S(3) + xS(7)(S(3)AacS(2)/S(7) + S(3)AbS(2)c/S(7) + S(6)Babc/S(7) + BbS(3)/S(7)) + xS(5)(A(S(6)abc + b*S(3))/S(5) + S(3)Ba(ac + bS(2))/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))(a + bxS(2) + cxS(4))S(3)/x*S(3), x), x, -Aa*S(3)/(S(2)x*S(2)) + Bc*S(3)xS(12)/S(12) + aS(2)(S(3)Ab + Ba)log(x) + S(3)axS(2)(A(ac + b*S(2)) + Bab)/S(2) + cS(2)x*S(10)(Ac + S(3)Bb)/S(10) + S(3)cxS(8)(Abc + Bac + BbS(2))/S(8) + xS(6)(Aac*S(2)/S(2) + Ab*S(2)c/S(2) + Babc + BbS(3)/S(6)) + xS(4)(A(S(6)abc + bS(3))/S(4) + S(3)Ba(ac + bS(2))/S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(A + BxS(2))/(a + bx*S(2) + cx*S(4)), x), x, Bx*S(4)/(S(4)c) - x*S(2)(-Ac + Bb)/(S(2)cS(2)) + (-Abc - Bac + Bb*S(2))log(a + bxS(2) + cx*S(4))/(S(4)c*S(3)) + (S(2)Aac*S(2) - Ab*S(2)c - S(3)Babc + BbS(3))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(3)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(A + Bx*S(2))/(a + bx*S(2) + cx*S(4)), x), x, Bx*S(2)/(S(2)c) - (-Ac + Bb)log(a + bx*S(2) + cx*S(4))/(S(4)c*S(2)) - (-Abc - S(2)Bac + BbS(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(A + BxS(2))/(a + bx*S(2) + cx*S(4)), x), x, Blog(a + bxS(2) + cx*S(4))/(S(4)c) + (-S(2)Ac + Bb)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))/(x(a + bxS(2) + cx*S(4))), x), x, Alog(x)/a - Alog(a + bx*S(2) + cx*S(4))/(S(4)a) + (Ab - S(2)Ba)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)asqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(3)(a + bx*S(2) + cx*S(4))), x), x, -A/(S(2)axS(2)) - (Ab - Ba)log(x)/a*S(2) + (Ab - Ba)log(a + bxS(2) + cx*S(4))/(S(4)a*S(2)) - (-S(2)Aac + AbS(2) - Bab)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(A + Bx*S(2))/(a + bx*S(2) + cx*S(4)), x), x, Bx*S(3)/(S(3)c) - x(-Ac + Bb)/cS(2) + sqrt(S(2))(-Abc - Bac + BbS(2) + (S(2)Aac*S(2) - Ab*S(2)c - S(3)Babc + BbS(3))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) + sqrt(S(2))(-Abc - Bac + BbS(2) - (S(2)Aac*S(2) - Ab*S(2)c - S(3)Babc + BbS(3))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(A + Bx*S(2))/(a + bx*S(2) + cx*S(4)), x), x, Bx/c - sqrt(S(2))(-Ac + Bb + (-Abc - S(2)Bac + BbS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) - sqrt(S(2))(-Ac + Bb - (-Abc - S(2)Bac + Bb*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))/(a + bx*S(2) + cx*S(4)), x), x, sqrt(S(2))(B - (-S(2)Ac + Bb)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(B + (-S(2)Ac + Bb)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(2)(a + bx*S(2) + cx*S(4))), x), x, -A/(ax) - sqrt(S(2))sqrt(c)(A - (Ab - S(2)Ba)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(b + sqrt(-S(4)ac + bS(2)))) - sqrt(S(2))sqrt(c)(A + (Ab - S(2)Ba)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(4)(a + bx*S(2) + cx*S(4))), x), x, -A/(S(3)axS(3)) + sqrt(S(2))sqrt(c)(-A(-S(2)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2))) + Ba(b - sqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))) - sqrt(S(2))sqrt(c)(-A(-S(2)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))) + Ba(b + sqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))) + (Ab - Ba)/(aS(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(7)(A + BxS(2))/(a + bx*S(2) + cxS(4))S(2), x), x, -x*S(4)(a(-S(2)Ac + Bb) + x*S(2)(-Abc - S(2)Bac + Bb*S(2)))/(S(2)c(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))) + xS(2)(-Abc - S(6)Bac + S(2)Bb*S(2))/(S(2)c*S(2)(-S(4)ac + b*S(2))) - (-Ac + S(2)Bb)log(a + bx*S(2) + cx*S(4))/(S(4)c*S(3)) - (S(6)AabcS(2) - Ab*S(3)c + S(12)Ba*S(2)c*S(2) - S(12)Bab*S(2)c + S(2)Bb*S(4))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(3)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)(A + BxS(2))/(a + bx*S(2) + cxS(4))S(2), x), x, Blog(a + bx*S(2) + cx*S(4))/(S(4)cS(2)) - xS(2)(a(-S(2)Ac + Bb) + xS(2)(-Abc - S(2)Bac + Bb*S(2)))/(S(2)c(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))) + (S(4)Aac*S(2) - S(6)Babc + Bb*S(3))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(A + BxS(2))/(a + bx*S(2) + cxS(4))S(2), x), x, (A + BxS(2))(S(2)a + bx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) - (Ab - S(2)Ba)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(A + BxS(2))/(a + bx*S(2) + cxS(4))S(2), x), x, -(-S(2)Ac + Bb)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) - (Ab - S(2)Ba - x*S(2)(-S(2)Ac + Bb))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))/(x(a + bxS(2) + cxS(4))S(2)), x), x, Alog(x)/aS(2) - Alog(a + bxS(2) + cx*S(4))/(S(4)a*S(2)) - (-A(-S(2)ac + b*S(2)) + Bab - cx*S(2)(Ab - S(2)Ba))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))) + (A(-S(6)abc + bS(3)) + S(4)BaS(2)c)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(3)(a + bxS(2) + cxS(4))S(2)), x), x, -(-A(-S(2)ac + bS(2)) + Bab - cx*S(2)(Ab - S(2)Ba))/(S(2)axS(2)(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) - (-S(6)Aac + S(2)Ab*S(2) - Bab)/(S(2)a*S(2)x*S(2)(-S(4)ac + b*S(2))) - (S(2)Ab - Ba)log(x)/aS(3) + (S(2)Ab - Ba)log(a + bx*S(2) + cx*S(4))/(S(4)a*S(3)) + (-S(2)A(S(6)a*S(2)c*S(2) - S(6)abS(2)c + b*S(4)) + Bab(-S(6)ac + b*S(2)))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(3)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(6)(A + BxS(2))/(a + bx*S(2) + cxS(4))S(2), x), x, -x*S(5)(Ab - S(2)Ba - xS(2)(-S(2)Ac + Bb))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cxS(4))) - xS(3)(-S(2)Ac + Bb)/(S(2)c(-S(4)ac + b*S(2))) + x(-Abc - S(10)Bac + S(3)BbS(2))/(S(2)c*S(2)(-S(4)ac + b*S(2))) - sqrt(S(2))(S(6)AacS(2) - Ab*S(2)c - S(13)Babc + S(3)Bb*S(3) + (S(8)AabcS(2) - Ab*S(3)c + S(20)Ba*S(2)c*S(2) - S(19)Bab*S(2)c + S(3)Bb*S(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(5)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + b*S(2))) - sqrt(S(2))(S(6)AacS(2) - Ab*S(2)c - S(13)Babc + S(3)Bb*S(3) - (S(8)AabcS(2) - Ab*S(3)c + S(20)Ba*S(2)c*S(2) - S(19)Bab*S(2)c + S(3)Bb*S(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(5)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(A + Bx*S(2))/(a + bx*S(2) + cxS(4))S(2), x), x, -x*S(3)(Ab - S(2)Ba - xS(2)(-S(2)Ac + Bb))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) - x(-S(2)Ac + Bb)/(S(2)c(-S(4)ac + bS(2))) + sqrt(S(2))(Abc - S(6)Bac + Bb*S(2) + (S(4)Aac*S(2) + Ab*S(2)c - S(8)Babc + BbS(3))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))(Abc - S(6)Bac + Bb*S(2) - (S(4)Aac*S(2) + Ab*S(2)c - S(8)Babc + BbS(3))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(A + Bx*S(2))/(a + bx*S(2) + cxS(4))S(2), x), x, -x(Ab - S(2)Ba - x*S(2)(-S(2)Ac + Bb))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) + sqrt(S(2))(-S(2)Ac + Bb + (-S(4)Abc + S(4)Bac + Bb*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))(-S(2)Ac + Bb - (-S(4)Abc + S(4)Bac + Bb*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))/(a + bx*S(2) + cxS(4))S(2), x), x, sqrt(S(2))sqrt(c)(Ab - S(2)Ba - (-S(12)Aac + AbS(2) + S(4)Bab)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)(Ab - S(2)Ba + (A(-S(12)ac + bS(2)) + S(4)Bab)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) - x(-A(-S(2)ac + bS(2)) + Bab - cx*S(2)(Ab - S(2)Ba))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(2)(a + bx*S(2) + cxS(4))S(2)), x), x, -(-A(-S(2)ac + bS(2)) + Bab - cx*S(2)(Ab - S(2)Ba))/(S(2)ax(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) - sqrt(S(2))sqrt(c)(-S(10)Aac + S(3)Ab*S(2) - Bab + (-A(-S(16)abc + S(3)b*S(3)) + Ba(-S(12)ac + bS(2)))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)(-A(-S(16)abc - S(10)acsqrt(-S(4)ac + b*S(2)) + S(3)b*S(3) + S(3)b*S(2)sqrt(-S(4)ac + b*S(2))) + Ba(-S(12)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - (-S(10)Aac + S(3)AbS(2) - Bab)/(S(2)a*S(2)x(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(4)(a + bx*S(2) + cxS(4))S(2)), x), x, -(-A(-S(2)ac + bS(2)) + Bab - cx*S(2)(Ab - S(2)Ba))/(S(2)axS(3)(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) - (-S(14)Aac + S(5)Ab*S(2) - S(3)Bab)/(S(6)aS(2)x*S(3)(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)(-A(S(28)aS(2)c*S(2) - S(29)abS(2)c + S(19)abcsqrt(-S(4)ac + b*S(2)) + S(5)b*S(4) - S(5)b*S(3)sqrt(-S(4)ac + b*S(2))) + Ba(-S(16)abc + S(10)acsqrt(-S(4)ac + bS(2)) + S(3)b*S(3) - S(3)b*S(2)sqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(3)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - sqrt(S(2))sqrt(c)(-A(S(28)a*S(2)c*S(2) - S(29)abS(2)c - S(19)abcsqrt(-S(4)ac + b*S(2)) + S(5)b*S(4) + S(5)b*S(3)sqrt(-S(4)ac + b*S(2))) + Ba(-S(16)abc - S(10)acsqrt(-S(4)ac + bS(2)) + S(3)b*S(3) + S(3)b*S(2)sqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(3)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - (-A(-S(19)abc + S(5)bS(3)) + Ba(-S(10)ac + S(3)b*S(2)))/(S(2)a*S(3)x(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)(A + BxS(2))/(a + bx*S(2) + cxS(4))S(3), x), x, -x*S(8)(a(-S(2)Ac + Bb) + x*S(2)(-Abc - S(2)Bac + Bb*S(2)))/(S(4)c(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))S(2)) - x*S(4)(a(S(16)Aac*S(2) - Ab*S(2)c - S(18)Babc + S(3)BbS(3)) + xS(2)(S(10)AabcS(2) - Ab*S(3)c + S(20)Ba*S(2)c*S(2) - S(20)Bab*S(2)c + S(3)Bb*S(4)))/(S(4)c*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cxS(4))) - xS(2)(A(-S(7)abc + bS(3)) + S(3)B(-S(10)a*S(2)c + S(7)abS(2) - bS(4)/c))/(S(2)cS(2)(-S(4)ac + bS(2))S(2)) - (-Ac + S(3)Bb)log(a + bxS(2) + cx*S(4))/(S(4)c*S(4)) - (-S(30)AaS(2)bcS(3) + S(10)Aab*S(3)c*S(2) - Ab*S(5)c - S(60)Ba*S(3)c*S(3) + S(90)BaS(2)b*S(2)c*S(2) - S(30)Bab*S(4)c + S(3)Bb*S(6))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(4)(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(9)(A + BxS(2))/(a + bx*S(2) + cxS(4))S(3), x), x, Blog(a + bx*S(2) + cx*S(4))/(S(4)cS(3)) - xS(6)(a(-S(2)Ac + Bb) + xS(2)(-Abc - S(2)Bac + Bb*S(2)))/(S(4)c(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))S(2)) - x*S(2)(S(2)a(S(6)AacS(2) - S(7)Babc + BbS(3)) + xS(2)(S(6)AabcS(2) + S(16)BaS(2)c*S(2) - S(15)Bab*S(2)c + S(2)Bb*S(4)))/(S(4)c*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) + (-S(12)AaS(2)c*S(3) + S(30)BaS(2)bcS(2) - S(10)Bab*S(3)c + BbS(5))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(3)(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(7)(A + BxS(2))/(a + bx*S(2) + cxS(4))S(3), x), x, S(3)a(Ab - S(2)Ba)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) - xS(6)(Ab - S(2)Ba - xS(2)(-S(2)Ac + Bb))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + x*S(2)(S(2)a + bx*S(2))(S(3)Ab - S(6)Ba)/(S(4)(-S(4)ac + bS(2))S(2)(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(A + Bx*S(2))/(a + bx*S(2) + cxS(4))S(3), x), x, -x*S(4)(Ab - S(2)Ba - xS(2)(-S(2)Ac + Bb))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + (S(2)a + bx*S(2))(S(2)Ab - S(4)Ba + x*S(2)(-S(2)Ac + Bb))/(S(4)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) + (-A(S(2)ac + b*S(2)) + S(3)Bab)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(A + BxS(2))/(a + bx*S(2) + cxS(4))S(3), x), x, -(-S(3)Abc + S(2)Bac + BbS(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) + (b + S(2)cxS(2))(-S(3)Abc + S(2)Bac + BbS(2))/(S(4)c(-S(4)ac + bS(2))S(2)(a + bxS(2) + cx*S(4))) - (a(-S(2)Ac + Bb) + xS(2)(-Abc - S(2)Bac + Bb*S(2)))/(S(4)c(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(A + Bx*S(2))/(a + bx*S(2) + cxS(4))S(3), x), x, S(3)c(-S(2)Ac + Bb)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) - (b + S(2)cxS(2))(-S(6)Ac + S(3)Bb)/(S(4)(-S(4)ac + bS(2))S(2)(a + bxS(2) + cx*S(4))) - (Ab - S(2)Ba - x*S(2)(-S(2)Ac + Bb))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(x(a + bxS(2) + cxS(4))S(3)), x), x, Alog(x)/aS(3) - Alog(a + bxS(2) + cx*S(4))/(S(4)a*S(3)) - (-A(-S(2)ac + b*S(2)) + Bab - cx*S(2)(Ab - S(2)Ba))/(S(4)a(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))S(2)) + (A(S(16)a*S(2)c*S(2) - S(15)abS(2)c + S(2)bS(4)) + S(6)BaS(2)bc + S(2)cxS(2)(A(-S(7)abc + b*S(3)) + S(6)BaS(2)c))/(S(4)aS(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) - (-A(S(30)aS(2)bcS(2) - S(10)abS(3)c + b*S(5)) + S(12)BaS(3)c*S(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(3)(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(3)(a + bxS(2) + cxS(4))S(3)), x), x, -(-A(-S(2)ac + bS(2)) + Bab - cx*S(2)(Ab - S(2)Ba))/(S(4)axS(2)(-S(4)ac + b*S(2))(a + bxS(2) + cxS(4))S(2)) - (-A(S(20)a*S(2)c*S(2) - S(20)abS(2)c + S(3)bS(4)) + Bab(-S(10)ac + b*S(2)) + cx*S(2)(-S(3)A(-S(6)abc + bS(3)) + Ba(-S(16)ac + bS(2))))/(S(4)a*S(2)x*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) + (-S(3)A(S(10)a*S(2)c*S(2) - S(7)abS(2)c + b*S(4)) + Bab(-S(7)ac + b*S(2)))/(S(2)a*S(3)x*S(2)(-S(4)ac + bS(2))S(2)) - (S(3)Ab - Ba)log(x)/a*S(4) + (S(3)Ab - Ba)log(a + bx*S(2) + cx*S(4))/(S(4)a*S(4)) + (-S(3)A(-S(20)a*S(3)c*S(3) + S(30)a*S(2)b*S(2)c*S(2) - S(10)abS(4)c + b*S(6)) + Bab(S(30)aS(2)c*S(2) - S(10)abS(2)c + b*S(4)))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(4)(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(8)(A + BxS(2))/(a + bx*S(2) + cxS(4))S(3), x), x, -x*S(7)(Ab - S(2)Ba - xS(2)(-S(2)Ac + Bb))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) - x*S(5)(-S(4)Aac + S(7)AbS(2) - S(12)Bab + x*S(2)(S(12)Abc - S(28)Bac + BbS(2)))/(S(8)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cxS(4))) + xS(3)(S(12)Abc - S(28)Bac + Bb*S(2))/(S(8)c(-S(4)ac + bS(2))S(2)) - x(S(20)AacS(2) + Ab*S(2)c - S(24)Babc + S(3)Bb*S(3))/(S(8)c*S(2)(-S(4)ac + bS(2))S(2)) + sqrt(S(2))(-S(16)AabcS(2) + Ab*S(3)c + S(84)Ba*S(2)c*S(2) - S(27)Bab*S(2)c + S(3)Bb*S(4) + (-S(40)AaS(2)c*S(3) - S(18)Aab*S(2)c*S(2) + Ab*S(4)c + S(132)Ba*S(2)bcS(2) - S(33)Bab*S(3)c + S(3)Bb*S(5))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)c*(S(5)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + sqrt(S(2))(-S(16)AabcS(2) + Ab*S(3)c + S(84)Ba*S(2)c*S(2) - S(27)Bab*S(2)c + S(3)Bb*S(4) - (-S(40)AaS(2)c*S(3) - S(18)Aab*S(2)c*S(2) + Ab*S(4)c + S(132)Ba*S(2)bcS(2) - S(33)Bab*S(3)c + S(3)Bb*S(5))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)c*(S(5)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(6)(A + BxS(2))/(a + bx*S(2) + cxS(4))S(3), x), x, -x*S(5)(Ab - S(2)Ba - xS(2)(-S(2)Ac + Bb))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) - x*S(3)(S(4)Aac + S(5)AbS(2) - S(12)Bab - x*S(2)(-S(12)Abc + S(20)Bac + BbS(2)))/(S(8)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) - x(-S(12)Abc + S(20)Bac + BbS(2))/(S(8)c(-S(4)ac + bS(2))S(2)) + sqrt(S(2))(S(12)AacS(2) + S(3)AbS(2)c - S(16)Babc + BbS(3) + (S(36)AabcS(2) + S(3)AbS(3)c - S(40)Ba*S(2)c*S(2) - S(18)Bab*S(2)c + BbS(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + sqrt(S(2))(S(12)Aac*S(2) + S(3)AbS(2)c - S(16)Babc + BbS(3) - (S(36)AabcS(2) + S(3)AbS(3)c - S(40)Ba*S(2)c*S(2) - S(18)Bab*S(2)c + BbS(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)(A + BxS(2))/(a + bx*S(2) + cxS(4))S(3), x), x, -x*S(3)(Ab - S(2)Ba - xS(2)(-S(2)Ac + Bb))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) + S(3)x(-A(S(4)ac + bS(2)) + S(4)Bab + x*S(2)(-S(4)Abc + S(4)Bac + BbS(2)))/(S(8)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) + sqrt(S(2))(-S(12)Abc + S(12)Bac + S(3)Bb*S(2) + S(3)(-S(8)AacS(2) - S(6)AbS(2)c + S(12)Babc + BbS(3))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))S(2)) + sqrt(S(2))(-S(12)Abc + S(12)Bac + S(3)BbS(2) - S(3)(-S(8)AacS(2) - S(6)AbS(2)c + S(12)Babc + BbS(3))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(A + BxS(2))/(a + bx*S(2) + cxS(4))S(3), x), x, -x(Ab - S(2)Ba - x*S(2)(-S(2)Ac + Bb))/((-S(16)ac + S(4)b*S(2))(a + bxS(2) + cxS(4))S(2)) - sqrt(S(2))sqrt(c)(-A(S(20)ac + bS(2)) + S(12)Bab + (A(-S(52)abc + b*S(3)) + S(6)Ba(S(4)ac + S(3)bS(2)))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))S(2)) - sqrt(S(2))sqrt(c)(-A(S(20)ac + bS(2)) + S(12)Bab - (A(-S(52)abc + b*S(3)) + S(6)Ba(S(4)ac + S(3)bS(2)))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))S(2)) - x(-A(S(8)abc + bS(3)) + Ba(-S(4)ac + S(7)b*S(2)) + cx*S(2)(-A(S(20)ac + bS(2)) + S(12)Bab))/(S(8)a(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))/(a + bx*S(2) + cxS(4))S(3), x), x, -x(-A(-S(2)ac + b*S(2)) + Bab - cx*S(2)(Ab - S(2)Ba))/(S(4)a(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))S(2)) + sqrt(S(2))sqrt(c)(S(3)A(-S(8)abc + bS(3)) + Ba(S(20)ac + bS(2)) - (S(3)A(S(56)a*S(2)c*S(2) - S(10)abS(2)c + b*S(4)) + Bab(-S(52)ac + b*S(2)))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + sqrt(S(2))sqrt(c)(S(3)A(-S(8)abc + bS(3)) + Ba(S(20)ac + bS(2)) + (S(3)A(S(56)a*S(2)c*S(2) - S(10)abS(2)c + b*S(4)) + Bab(-S(52)ac + b*S(2)))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))S(2)) + x(A(S(28)aS(2)c*S(2) - S(25)abS(2)c + S(3)bS(4)) + Bab(S(8)ac + b*S(2)) + cx*S(2)(S(3)A(-S(8)abc + bS(3)) + Ba(S(20)ac + bS(2))))/(S(8)a*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(4)xS(2) + S(-7))/(xS(4) - S(5)xS(2) + S(4)), x), x, log(-xS(2) + S(1))/S(2) + S(3)log(-xS(2) + S(4))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x*S(3) - S(7)x)/(x*S(4) - S(5)xS(2) + S(4)), x), x, log(-xS(2) + S(1))/S(2) + S(3)log(-xS(2) + S(4))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(xS(2) + S(2))/(xS(4) + xS(2) + S(1)), x), x, log(xS(4) + x*S(2) + S(1))/S(4) + sqrt(S(3))atan(sqrt(S(3))(S(2)xS(2) + S(1))/S(3))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(2)x)/(xS(4) + xS(2) + S(1)), x), x, log(xS(4) + xS(2) + S(1))/S(4) + sqrt(S(3))atan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(3) + S(11)x)/(x*S(4) + S(2)xS(2) + S(3))S(2), x), x, (S(9)xS(2) + S(5))/(S(8)x*S(4) + S(16)x*S(2) + S(24)) + S(9)sqrt(S(2))atan(sqrt(S(2))(xS(2) + S(1))/S(2))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(A + Bx*S(2))(a + bxS(2) + cxS(4))(S(3)/2), x), x, -(a + bxS(2) + cxS(4))(S(5)/2)(-S(12)Ac + S(7)Bb - S(10)Bcx*S(2))/(S(120)c*S(2)) + (b + S(2)cxS(2))(a + bxS(2) + cxS(4))(S(3)/2)(-S(12)Abc - S(4)Bac + S(7)BbS(2))/(S(384)c*S(3)) - (b + S(2)cxS(2))(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))(-S(12)Abc - S(4)Bac + S(7)Bb*S(2))/(S(1024)c*S(4)) + (-S(4)ac + bS(2))S(2)(-S(12)Abc - S(4)Bac + S(7)Bb*S(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(2048)c*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(A + BxS(2))(a + bxS(2) + cxS(4))(S(3)/2), x), x, B(a + bx*S(2) + cxS(4))(S(5)/2)/(S(10)c) - (b + S(2)cxS(2))(-S(2)Ac + Bb)(a + bxS(2) + cxS(4))(S(3)/2)/(S(32)cS(2)) + (b + S(2)cxS(2))(-S(2)Ac + Bb)(-S(12)ac + S(3)bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(256)c*S(3)) - S(3)(-S(2)Ac + Bb)(-S(4)ac + bS(2))S(2)atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(512)c*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/x, x), x, -Aa(S(3)/2)atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/S(2) + (a + bx*S(2) + cxS(4))(S(3)/2)(S(8)Ac + S(3)Bb + S(6)Bcx*S(2))/(S(48)c) - sqrt(a + bxS(2) + cx*S(4))(-S(64)AacS(2) - S(8)AbS(2)c - S(12)Babc + S(3)Bb*S(3) + S(2)cxS(2)(-S(8)Abc - S(12)Bac + S(3)Bb*S(2)))/(S(128)c*S(2)) + (S(64)AabcS(2) + (-S(4)ac + bS(2))(-S(8)Abc - S(12)Bac + S(3)Bb*S(2)))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(256)c*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/x*S(3), x), x, -sqrt(a)(S(3)Ab + S(2)Ba)atanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/S(4) - (S(3)A - BxS(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(6)xS(2)) + sqrt(a + bx*S(2) + cx*S(4))(S(18)Abc + S(8)Bac + BbS(2) + S(2)cxS(2)(S(6)Ac + Bb))/(S(16)c) - (-S(24)AacS(2) - S(6)AbS(2)c - S(12)Babc + BbS(3))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)c*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/x*S(5), x), x, -sqrt(a + bx*S(2) + cx*S(4))(S(3)Ab + S(6)Ba - S(3)xS(2)(S(2)Ac + Bb))/(S(8)x*S(2)) - (A - Bx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(4)xS(4)) + (S(12)Abc + S(12)Bac + S(3)BbS(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)sqrt(c)) - (S(3)A(S(4)ac + b*S(2)) + S(12)Bab)atanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)sqrt(a)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))(a + bxS(2) + cxS(4))(S(3)/2), x), x, -a*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(144)Aabc*S(2) - S(18)AbS(3)c + S(84)Ba*S(2)c*S(2) - S(57)Bab*S(2)c + S(8)Bb*S(4))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(315)c*(S(11)/4)sqrt(a + bxS(2) + cxS(4))) + a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(144)Aabc*S(2) - S(18)AbS(3)c + S(84)Ba*S(2)c*S(2) - S(57)Bab*S(2)c + S(8)Bb*S(4) + sqrt(a)sqrt(c)(S(180)Aac*S(2) - S(9)AbS(2)c - S(24)Babc + S(4)Bb*S(3)))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(630)c*(S(11)/4)sqrt(a + bxS(2) + cx*S(4))) + x(a + bxS(2) + cxS(4))(S(3)/2)(S(9)Ac + S(3)Bb + S(7)Bcx*S(2))/(S(63)c) - xsqrt(a + bx*S(2) + cx*S(4))(-S(90)AacS(2) - S(9)AbS(2)c - S(9)Babc + S(4)Bb*S(3) + S(3)cxS(2)(-S(9)Abc - S(14)Bac + S(4)Bb*S(2)))/(S(315)c*S(2)) + xsqrt(a + bxS(2) + cx*S(4))(S(144)Aabc*S(2) - S(18)AbS(3)c + S(84)Ba*S(2)c*S(2) - S(57)Bab*S(2)c + S(8)Bb*S(4))/(S(315)c*(S(5)/2)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/xS(2), x), x, a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(84)AacS(2) - S(7)AbS(2)c - S(16)Babc + S(2)Bb*S(3))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(35)c*(S(7)/4)sqrt(a + bxS(2) + cxS(4))) - a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(84)AacS(2) - S(7)AbS(2)c - S(16)Babc + S(2)Bb*S(3) + sqrt(a)sqrt(c)(-S(56)Abc - S(20)Bac + Bb*S(2)))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(70)c*(S(7)/4)sqrt(a + bxS(2) + cx*S(4))) - (S(7)A - BxS(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(7)x) + xsqrt(a + bxS(2) + cx*S(4))(S(49)Abc + S(10)Bac + BbS(2) + S(3)cxS(2)(S(14)Ac + Bb))/(S(35)c) - xsqrt(a + bx*S(2) + cx*S(4))(-S(84)AacS(2) - S(7)AbS(2)c - S(16)Babc + S(2)Bb*S(3))/(S(35)c*(S(3)/2)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/xS(4), x), x, -a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(40)Abc + S(36)Bac + S(3)Bb*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(15)c*(S(3)/4)sqrt(a + bxS(2) + cxS(4))) + a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(40)Abc + S(36)Bac + S(3)Bb*S(2) + sqrt(c)(S(5)A(S(4)ac + S(3)bS(2)) + S(24)Bab)/sqrt(a))elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(30)c*(S(3)/4)sqrt(a + bxS(2) + cx*S(4))) - sqrt(a + bx*S(2) + cx*S(4))(S(15)Ab + S(18)Ba - x*S(2)(S(10)Ac + S(3)Bb))/(S(15)x) - (S(5)A - S(3)Bx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(15)xS(3)) + xsqrt(a + bxS(2) + cx*S(4))(S(40)Abc + S(36)Bac + S(3)Bb*S(2))/(S(15)sqrt(c)(sqrt(a) + sqrt(c)x*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/x*S(6), x), x, -sqrt(a + bx*S(2) + cx*S(4))(S(3)Ab + S(10)Ba - x*S(2)(S(18)Ac + S(15)Bb))/(S(15)xS(3)) - (S(3)A - S(5)Bx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(15)xS(5)) + sqrt(c)x(S(3)A(S(12)ac + bS(2)) + S(40)Bab)sqrt(a + bx*S(2) + cx*S(4))/(S(15)a(sqrt(a) + sqrt(c)x*S(2))) - (S(3)A(S(12)ac + bS(2)) + S(40)Bab)sqrt(a + bx*S(2) + cx*S(4))/(S(15)ax) - c(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(3)A(S(12)ac + b*S(2)) + S(40)Bab)elliptic_e(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(15)a*(S(3)/4)sqrt(a + bxS(2) + cx*S(4))) + sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(2)sqrt(a)sqrt(c) + b)(S(3)Absqrt(c) + S(10)Basqrt(c) + S(3)sqrt(a)(S(6)Ac + S(5)Bb))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(30)a*(S(3)/4)c*(S(1)/4)sqrt(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(A + Bx*S(2))/sqrt(a + bx*S(2) + cx*S(4)), x), x, Bx*S(4)sqrt(a + bxS(2) + cx*S(4))/(S(6)c) + sqrt(a + bxS(2) + cx*S(4))(-S(18)Abc - S(16)Bac + S(15)Bb*S(2) - S(2)cxS(2)(-S(6)Ac + S(5)Bb))/(S(48)cS(3)) - (S(8)Aac*S(2) - S(6)AbS(2)c - S(12)Babc + S(5)Bb*S(3))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)c(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(A + Bx*S(2))/sqrt(a + bx*S(2) + cx*S(4)), x), x, -sqrt(a + bx*S(2) + cx*S(4))(-S(4)Ac + S(3)Bb - S(2)BcxS(2))/(S(8)c*S(2)) + (-S(4)Abc - S(4)Bac + S(3)BbS(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)c*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(A + BxS(2))/sqrt(a + bx*S(2) + cx*S(4)), x), x, Bsqrt(a + bxS(2) + cx*S(4))/(S(2)c) - (-S(2)Ac + Bb)atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)c*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))/(xsqrt(a + bxS(2) + cx*S(4))), x), x, -Aatanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)sqrt(a)) + Batanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(3)sqrt(a + bxS(2) + cx*S(4))), x), x, -Asqrt(a + bxS(2) + cx*S(4))/(S(2)axS(2)) + (Ab - S(2)Ba)atanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)a*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(5)sqrt(a + bx*S(2) + cx*S(4))), x), x, -Asqrt(a + bxS(2) + cx*S(4))/(S(4)axS(4)) + (S(3)Ab - S(4)Ba)sqrt(a + bxS(2) + cx*S(4))/(S(8)a*S(2)x*S(2)) - (-S(4)Aac + S(3)Ab*S(2) - S(4)Bab)atanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)a*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(7)sqrt(a + bx*S(2) + cx*S(4))), x), x, -Asqrt(a + bxS(2) + cx*S(4))/(S(6)axS(6)) + (S(5)Ab - S(6)Ba)sqrt(a + bxS(2) + cx*S(4))/(S(24)a*S(2)x*S(4)) - sqrt(a + bx*S(2) + cx*S(4))(-S(16)Aac + S(15)AbS(2) - S(18)Bab)/(S(48)aS(3)x*S(2)) + (-S(12)Aabc + S(5)AbS(3) + S(8)BaS(2)c - S(6)BabS(2))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)a(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(A + Bx*S(2))/sqrt(a + bx*S(2) + cx*S(4)), x), x, Bx*S(3)sqrt(a + bxS(2) + cx*S(4))/(S(5)c) - a*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(10)Abc - S(9)Bac + S(8)Bb*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(15)c*(S(11)/4)sqrt(a + bxS(2) + cxS(4))) + a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(10)Abc - S(9)Bac + S(8)Bb*S(2) + sqrt(a)sqrt(c)(-S(5)Ac + S(4)Bb))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(30)c*(S(11)/4)sqrt(a + bxS(2) + cx*S(4))) - x(-S(5)Ac + S(4)Bb)sqrt(a + bx*S(2) + cx*S(4))/(S(15)c*S(2)) + xsqrt(a + bxS(2) + cx*S(4))(-S(10)Abc - S(9)Bac + S(8)Bb*S(2))/(S(15)c*(S(5)/2)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(A + BxS(2))/sqrt(a + bx*S(2) + cx*S(4)), x), x, Bxsqrt(a + bx*S(2) + cx*S(4))/(S(3)c) + a*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(3)Ac + S(2)Bb)elliptic_e(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(3)c*(S(7)/4)sqrt(a + bxS(2) + cxS(4))) - a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(3)Ac + Bsqrt(a)sqrt(c) + S(2)Bb)elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(6)c*(S(7)/4)sqrt(a + bxS(2) + cx*S(4))) - x(-S(3)Ac + S(2)Bb)sqrt(a + bx*S(2) + cx*S(4))/(S(3)c*(S(3)/2)(sqrt(a) + sqrt(c)xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))/sqrt(a + bx*S(2) + cx*S(4)), x), x, -Ba*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(c(S(3)/4)sqrt(a + bxS(2) + cx*S(4))) + Bxsqrt(a + bx*S(2) + cx*S(4))/(sqrt(c)(sqrt(a) + sqrt(c)xS(2))) + a(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(Asqrt(c)/sqrt(a) + B)elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)c*(S(3)/4)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(2)sqrt(a + bx*S(2) + cx*S(4))), x), x, Asqrt(c)xsqrt(a + bxS(2) + cx*S(4))/(a(sqrt(a) + sqrt(c)xS(2))) - Asqrt(a + bxS(2) + cx*S(4))/(ax) - Ac(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(a(S(3)/4)sqrt(a + bxS(2) + cx*S(4))) + sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(Asqrt(c) + Bsqrt(a))elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(3)/4)c*(S(1)/4)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(4)sqrt(a + bx*S(2) + cx*S(4))), x), x, -Asqrt(a + bxS(2) + cx*S(4))/(S(3)axS(3)) - sqrt(c)x(S(2)Ab - S(3)Ba)sqrt(a + bxS(2) + cx*S(4))/(S(3)a*S(2)(sqrt(a) + sqrt(c)xS(2))) + (S(2)Ab - S(3)Ba)sqrt(a + bxS(2) + cx*S(4))/(S(3)a*S(2)x) + c*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(2)Ab - S(3)Ba)elliptic_e(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(3)a*(S(7)/4)sqrt(a + bxS(2) + cxS(4))) - c(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(Asqrt(a)sqrt(c) + S(2)Ab - S(3)Ba)elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(6)a*(S(7)/4)sqrt(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(A + Bx*S(2))/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, -x*S(2)(a(-S(2)Ac + Bb) + x*S(2)(-Abc - S(2)Bac + Bb*S(2)))/(c(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) + sqrt(a + bx*S(2) + cx*S(4))(-S(2)Abc - S(8)Bac + S(3)Bb*S(2))/(S(2)c*S(2)(-S(4)ac + b*S(2))) - (-S(2)Ac + S(3)Bb)atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)c(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(A + Bx*S(2))/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, Batanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)c*(S(3)/2)) - (a(-S(2)Ac + Bb) + xS(2)(-Abc - S(2)Bac + Bb*S(2)))/(c(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(A + BxS(2))/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, -(Ab - S(2)Ba - xS(2)(-S(2)Ac + Bb))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))/(x(a + bxS(2) + cxS(4))(S(3)/2)), x), x, -Aatanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)a*(S(3)/2)) - (-A(-S(2)ac + b*S(2)) + Bab - cx*S(2)(Ab - S(2)Ba))/(a(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(3)(a + bx*S(2) + cxS(4))(S(3)/2)), x), x, -(-A(-S(2)ac + bS(2)) + Bab - cx*S(2)(Ab - S(2)Ba))/(ax*S(2)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) - sqrt(a + bx*S(2) + cx*S(4))(-S(8)Aac + S(3)AbS(2) - S(2)Bab)/(S(2)aS(2)x*S(2)(-S(4)ac + b*S(2))) + (S(3)Ab - S(2)Ba)atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(A + Bx*S(2))/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, -a*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-Abc - S(6)Bac + S(2)BbS(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(c(S(7)/4)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cxS(4))) + a(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-Ac - S(3)Bsqrt(a)sqrt(c) + S(2)Bb)elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(c(S(7)/4)(-S(4)sqrt(a)sqrt(c) + S(2)b)sqrt(a + bxS(2) + cxS(4))) - xS(3)(Ab - S(2)Ba - x*S(2)(-S(2)Ac + Bb))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))) - x(-S(2)Ac + Bb)sqrt(a + bxS(2) + cx*S(4))/(c(-S(4)ac + b*S(2))) + xsqrt(a + bxS(2) + cx*S(4))(-Abc - S(6)Bac + S(2)BbS(2))/(c(S(3)/2)(sqrt(a) + sqrt(c)xS(2))(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(A + Bx*S(2))/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, a*(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(2)Ac + Bb)elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(c(S(3)/4)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) - x(Ab - S(2)Ba - xS(2)(-S(2)Ac + Bb))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))) - x(-S(2)Ac + Bb)sqrt(a + bxS(2) + cx*S(4))/(sqrt(c)(sqrt(a) + sqrt(c)xS(2))(-S(4)ac + b*S(2))) - sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-Asqrt(c) + Bsqrt(a))elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(1)/4)c*(S(3)/4)(-S(2)sqrt(a)sqrt(c) + b)sqrt(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx*S(2))/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, -sqrt(c)x(Ab - S(2)Ba)sqrt(a + bxS(2) + cx*S(4))/(a(sqrt(a) + sqrt(c)xS(2))(-S(4)ac + b*S(2))) - x(-A(-S(2)ac + bS(2)) + Bab - cx*S(2)(Ab - S(2)Ba))/(a(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cxS(4))) + c(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(Ab - S(2)Ba)elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(a(S(3)/4)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) + sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-Asqrt(c) + Bsqrt(a))elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(3)/4)c*(S(1)/4)(-S(2)sqrt(a)sqrt(c) + b)sqrt(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + BxS(2))/(xS(2)(a + bx*S(2) + cxS(4))(S(3)/2)), x), x, -(-A(-S(2)ac + bS(2)) + Bab - cx*S(2)(Ab - S(2)Ba))/(ax(-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))) + sqrt(c)xsqrt(a + bx*S(2) + cx*S(4))(-S(6)Aac + S(2)AbS(2) - Bab)/(aS(2)(sqrt(a) + sqrt(c)xS(2))(-S(4)ac + b*S(2))) - sqrt(a + bx*S(2) + cx*S(4))(-S(6)Aac + S(2)AbS(2) - Bab)/(aS(2)x(-S(4)ac + bS(2))) - c(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(6)Aac + S(2)AbS(2) - Bab)elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(a(S(7)/4)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cxS(4))) + c(S(1)/4)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(3)Asqrt(a)sqrt(c) + S(2)Ab - Ba)elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(7)/4)(-S(2)sqrt(a)sqrt(c) + b)sqrt(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*(S(3)/2)(d + exS(2))sqrt(a + bxS(2) + cx*S(4)), x), x, S(2)d(fx)*(S(5)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(5)/4, S(-1)/2, S(-1)/2, S(9)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)fsqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))) + S(2)e(fx)*(S(9)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(9)/4, S(-1)/2, S(-1)/2, S(13)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(9)f*S(3)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(fx)(d + ex*S(2))sqrt(a + bxS(2) + cx*S(4)), x), x, S(2)d(fx)*(S(3)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(3)/4, S(-1)/2, S(-1)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)fsqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))) + S(2)e(fx)*(S(7)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(7)/4, S(-1)/2, S(-1)/2, S(11)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(7)f*S(3)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))sqrt(a + bxS(2) + cx*S(4))/sqrt(fx), x), x, S(2)dsqrt(fx)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(1)/4, S(-1)/2, S(-1)/2, S(5)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(fsqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))) + S(2)e(fx)*(S(5)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(5)/4, S(-1)/2, S(-1)/2, S(9)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)f*S(3)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))sqrt(a + bxS(2) + cx*S(4))/(fx)*(S(3)/2), x), x, -S(2)dsqrt(a + bx*S(2) + cx*S(4))AppellF1(S(-1)/4, S(-1)/2, S(-1)/2, S(3)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(fsqrt(fx)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))) + S(2)e(fx)*(S(3)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(3)/4, S(-1)/2, S(-1)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)f*S(3)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*(S(3)/2)(d + exS(2))(a + bxS(2) + cxS(4))(S(3)/2), x), x, S(2)ad(fx)*(S(5)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(5)/4, S(-3)/2, S(-3)/2, S(9)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)fsqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))) + S(2)ae(fx)(S(9)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(9)/4, S(-3)/2, S(-3)/2, S(13)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(9)f*S(3)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(fx)(d + ex*S(2))(a + bxS(2) + cxS(4))(S(3)/2), x), x, S(2)ad(fx)*(S(3)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(3)/4, S(-3)/2, S(-3)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)fsqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))) + S(2)ae(fx)(S(7)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(7)/4, S(-3)/2, S(-3)/2, S(11)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(7)f*S(3)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/sqrt(fx), x), x, S(2)adsqrt(fx)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(1)/4, S(-3)/2, S(-3)/2, S(5)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(fsqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))) + S(2)ae(fx)(S(5)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(5)/4, S(-3)/2, S(-3)/2, S(9)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)f*S(3)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/(fx)(S(3)/2), x), x, -S(2)adsqrt(a + bxS(2) + cx*S(4))AppellF1(S(-1)/4, S(-3)/2, S(-3)/2, S(3)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(fsqrt(fx)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))) + S(2)ae(fx)(S(3)/2)sqrt(a + bxS(2) + cx*S(4))AppellF1(S(3)/4, S(-3)/2, S(-3)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)f*S(3)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*(S(3)/2)(d + exS(2))/sqrt(a + bx*S(2) + cx*S(4)), x), x, S(2)d(fx)*(S(5)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(5)/4, S(1)/2, S(1)/2, S(9)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)fsqrt(a + bx*S(2) + cx*S(4))) + S(2)e(fx)*(S(9)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(9)/4, S(1)/2, S(1)/2, S(13)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(9)f*S(3)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(fx)(d + ex*S(2))/sqrt(a + bx*S(2) + cx*S(4)), x), x, S(2)d(fx)*(S(3)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(3)/4, S(1)/2, S(1)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)fsqrt(a + bx*S(2) + cx*S(4))) + S(2)e(fx)*(S(7)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(7)/4, S(1)/2, S(1)/2, S(11)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(7)f*S(3)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/(sqrt(fx)sqrt(a + bx*S(2) + cx*S(4))), x), x, S(2)dsqrt(fx)sqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/4, S(1)/2, S(1)/2, S(5)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(fsqrt(a + bxS(2) + cx*S(4))) + S(2)e(fx)*(S(5)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(5)/4, S(1)/2, S(1)/2, S(9)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)f*S(3)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/((fx)*(S(3)/2)sqrt(a + bxS(2) + cx*S(4))), x), x, -S(2)dsqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(-1)/4, S(1)/2, S(1)/2, S(3)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(fsqrt(fx)sqrt(a + bxS(2) + cx*S(4))) + S(2)e(fx)*(S(3)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(3)/4, S(1)/2, S(1)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)f*S(3)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*(S(3)/2)(d + exS(2))/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, S(2)d(fx)(S(5)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(5)/4, S(3)/2, S(3)/2, S(9)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)afsqrt(a + bxS(2) + cx*S(4))) + S(2)e(fx)*(S(9)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(9)/4, S(3)/2, S(3)/2, S(13)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(9)afS(3)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(fx)(d + ex*S(2))/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, S(2)d(fx)(S(3)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(3)/4, S(3)/2, S(3)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)afsqrt(a + bxS(2) + cx*S(4))) + S(2)e(fx)*(S(7)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(7)/4, S(3)/2, S(3)/2, S(11)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(7)afS(3)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/(sqrt(fx)(a + bx*S(2) + cxS(4))(S(3)/2)), x), x, S(2)dsqrt(fx)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(1)/4, S(3)/2, S(3)/2, S(5)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(afsqrt(a + bx*S(2) + cx*S(4))) + S(2)e(fx)*(S(5)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(5)/4, S(3)/2, S(3)/2, S(9)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(5)afS(3)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(2))/((fx)*(S(3)/2)(a + bxS(2) + cxS(4))(S(3)/2)), x), x, -S(2)dsqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(-1)/4, S(3)/2, S(3)/2, S(3)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(afsqrt(fx)sqrt(a + bx*S(2) + cx*S(4))) + S(2)e(fx)*(S(3)/2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(3)/4, S(3)/2, S(3)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)afS(3)sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(d + exS(2))/(a + bx*S(2) + cxS(4))S(2), x), x, -c(fx)*(m + S(1))(S(2)a(-S(2)cd(-m + S(3)) + e(-m + S(1))sqrt(-S(4)ac + bS(2))) + bS(2)d(-m + S(1)) + b(S(4)ae - d(-m + S(1))sqrt(-S(4)ac + b*S(2))))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)af(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))(S(3)/2)) + c(fx)(m + S(1))(-S(2)a(S(2)cd(-m + S(3)) + e(-m + S(1))sqrt(-S(4)ac + bS(2))) + bS(2)(-dm + d) + b(S(4)ae + d(-m + S(1))sqrt(-S(4)ac + b*S(2))))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))))/(S(2)af(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))(S(3)/2)) + (fx)*(m + S(1))(-abe - S(2)acd + bS(2)d + cxS(2)(-S(2)ae + bd))/(S(2)af(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(d + exS(2))(a + bxS(2) + cxS(4))(S(3)/2), x), x, ad(fx)(m + S(1))sqrt(a + bxS(2) + cx*S(4))AppellF1(m/S(2) + S(1)/2, S(-3)/2, S(-3)/2, m/S(2) + S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(f(m + S(1))sqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))) + ae(fx)*(m + S(3))sqrt(a + bxS(2) + cx*S(4))AppellF1(m/S(2) + S(3)/2, S(-3)/2, S(-3)/2, m/S(2) + S(5)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(fS(3)(m + S(3))sqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(d + exS(2))sqrt(a + bxS(2) + cx*S(4)), x), x, d(fx)(m + S(1))sqrt(a + bxS(2) + cx*S(4))AppellF1(m/S(2) + S(1)/2, S(-1)/2, S(-1)/2, m/S(2) + S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(f(m + S(1))sqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))) + e(fx)(m + S(3))sqrt(a + bxS(2) + cx*S(4))AppellF1(m/S(2) + S(3)/2, S(-1)/2, S(-1)/2, m/S(2) + S(5)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(fS(3)(m + S(3))sqrt(S(2)cxS(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(d + exS(2))/sqrt(a + bx*S(2) + cx*S(4)), x), x, d(fx)(m + S(1))sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(m/S(2) + S(1)/2, S(1)/2, S(1)/2, m/S(2) + S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(f(m + S(1))sqrt(a + bx*S(2) + cx*S(4))) + e(fx)(m + S(3))sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(m/S(2) + S(3)/2, S(1)/2, S(1)/2, m/S(2) + S(5)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(fS(3)(m + S(3))sqrt(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(d + exS(2))/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, d(fx)*(m + S(1))sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(m/S(2) + S(1)/2, S(3)/2, S(3)/2, m/S(2) + S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(af(m + S(1))sqrt(a + bxS(2) + cx*S(4))) + e(fx)(m + S(3))sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(m/S(2) + S(3)/2, S(3)/2, S(3)/2, m/S(2) + S(5)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(af*S(3)(m + S(3))sqrt(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(d + exS(2))(a + bxS(2) + cxS(4))S(3), x), x, a*S(3)d(fx)*(m + S(1))/(f(m + S(1))) + a*S(2)(fx)(m + S(3))(ae + S(3)bd)/(fS(3)(m + S(3))) + S(3)a(fx)(m + S(5))(abe + acd + b*S(2)d)/(f*S(5)(m + S(5))) + c*S(3)e(fx)(m + S(15))/(fS(15)(m + S(15))) + cS(2)(fx)(m + S(13))(S(3)be + cd)/(fS(13)(m + S(13))) + S(3)c(fx)(m + S(11))(ace + b*S(2)e + bcd)/(f*S(11)(m + S(11))) + (fx)(m + S(7))(S(3)aS(2)ce + S(3)abS(2)e + S(6)abcd + b*S(3)d)/(f*S(7)(m + S(7))) + (fx)(m + S(9))(S(6)abce + S(3)ac*S(2)d + b*S(3)e + S(3)bS(2)cd)/(fS(9)(m + S(9))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)m(d + exS(2))(a + bxS(2) + cxS(4))S(2), x), x, a*S(2)d(fx)*(m + S(1))/(f(m + S(1))) + a(fx)*(m + S(3))(ae + S(2)bd)/(fS(3)(m + S(3))) + c*S(2)e(fx)(m + S(11))/(fS(11)(m + S(11))) + c(fx)(m + S(9))(S(2)be + cd)/(fS(9)(m + S(9))) + (fx)(m + S(5))(S(2)abe + S(2)acd + b*S(2)d)/(f*S(5)(m + S(5))) + (fx)(m + S(7))(S(2)ace + bS(2)e + S(2)bcd)/(fS(7)(m + S(7))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)m(d + exS(2))(a + bxS(2) + cx*S(4)), x), x, ad(fx)*(m + S(1))/(f(m + S(1))) + ce(fx)(m + S(7))/(fS(7)(m + S(7))) + (fx)(m + S(3))(ae + bd)/(f*S(3)(m + S(3))) + (fx)(m + S(5))(be + cd)/(f*S(5)(m + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)m(d + exS(2))/(a + bx*S(2) + cx*S(4)), x), x, (fx)*(m + S(1))(e - (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))))/(f(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))) + (fx)(m + S(1))(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))))/(f(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(d + exS(2))S(2)(a + bxS(2) + cx*S(4)), x), x, ad*S(2)x*S(4)/S(4) + ce*S(2)x*S(12)/S(12) + dx*S(6)(S(2)ae + bd)/S(6) + ex*S(10)(be + S(2)cd)/S(10) + xS(8)(cdS(2)/S(8) + e(ae + S(2)bd)/S(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(d + exS(2))S(2)(a + bxS(2) + cx*S(4)), x), x, ad*S(2)x*S(3)/S(3) + ce*S(2)x*S(11)/S(11) + dx*S(5)(S(2)ae + bd)/S(5) + ex*S(9)(be + S(2)cd)/S(9) + xS(7)(cdS(2)/S(7) + e(ae + S(2)bd)/S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(d + exS(2))S(2)(a + bxS(2) + cx*S(4)), x), x, ad*S(2)x*S(2)/S(2) + ce*S(2)x*S(10)/S(10) + dx*S(4)(S(2)ae + bd)/S(4) + ex*S(8)(be + S(2)cd)/S(8) + xS(6)(cdS(2)/S(6) + e(ae + S(2)bd)/S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)(a + bx*S(2) + cx*S(4)), x), x, ad*S(2)x + ceS(2)x*S(9)/S(9) + dx*S(3)(S(2)ae + bd)/S(3) + ex*S(7)(be + S(2)cd)/S(7) + xS(5)(cdS(2)/S(5) + e(ae + S(2)bd)/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)(a + bx*S(2) + cx*S(4))/x, x), x, ad*S(2)log(x) + ceS(2)x*S(8)/S(8) + dx*S(2)(S(2)ae + bd)/S(2) + ex*S(6)(be + S(2)cd)/S(6) + xS(4)(cdS(2)/S(4) + e(ae + S(2)bd)/S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)(a + bx*S(2) + cxS(4))/xS(2), x), x, -adS(2)/x + ce*S(2)x*S(7)/S(7) + dx(S(2)ae + bd) + exS(5)(be + S(2)cd)/S(5) + xS(3)(cdS(2)/S(3) + e(ae + S(2)bd)/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))S(2)(a + bx*S(2) + cxS(4))/xS(3), x), x, -adS(2)/(S(2)x*S(2)) + ce*S(2)x*S(6)/S(6) + d(S(2)ae + bd)log(x) + exS(4)(be + S(2)cd)/S(4) + xS(2)(cdS(2)/S(2) + e(ae + S(2)bd)/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)(a + bxS(2) + cx*S(4))/(d + exS(2))S(2), x), x, cxS(7)/(S(7)eS(2)) + d(S(3)/2)(S(9)cdS(2) - e(-S(5)ae + S(7)bd))atan(sqrt(e)x/sqrt(d))/(S(2)e(S(11)/2)) - dS(2)x(ae*S(2) - bde + cd*S(2))/(S(2)e*S(5)(d + exS(2))) - dx(S(4)cdS(2) - e(-S(2)ae + S(3)bd))/eS(5) - xS(5)(-be + S(2)cd)/(S(5)eS(3)) + xS(3)(S(3)cd*S(2) - e(-ae + S(2)bd))/(S(3)eS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(a + bx*S(2) + cx*S(4))/(d + exS(2))S(2), x), x, cxS(5)/(S(5)e*S(2)) - sqrt(d)(S(7)cd*S(2) - e(-S(3)ae + S(5)bd))atan(sqrt(e)x/sqrt(d))/(S(2)e(S(9)/2)) + dx(ae*S(2) - bde + cd*S(2))/(S(2)e*S(4)(d + exS(2))) - xS(3)(-be + S(2)cd)/(S(3)e*S(3)) + x(S(3)cd*S(2) - e(-ae + S(2)bd))/eS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bxS(2) + cx*S(4))/(d + exS(2))S(2), x), x, cxS(3)/(S(3)e*S(2)) - x(-be + S(2)cd)/eS(3) - x(aeS(2) - bde + cd*S(2))/(S(2)e*S(3)(d + exS(2))) + (S(5)cdS(2) - e(-ae + S(3)bd))atan(sqrt(e)x/sqrt(d))/(S(2)sqrt(d)e(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(d + exS(2))S(2), x), x, cx/eS(2) + x(aeS(2) - bde + cd*S(2))/(S(2)deS(2)(d + exS(2))) - (S(3)cdS(2) - e(ae + bd))atan(sqrt(e)x/sqrt(d))/(S(2)d(S(3)/2)e*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(2)(d + exS(2))S(2)), x), x, -a/(d*S(2)x) - x(ae*S(2) - bde + cd*S(2))/(S(2)d*S(2)e(d + ex*S(2))) + (cd*S(2) + e(-S(3)ae + bd))atan(sqrt(e)x/sqrt(d))/(S(2)d*(S(5)/2)e*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(4)(d + exS(2))S(2)), x), x, -a/(S(3)dS(2)x*S(3)) + x(aeS(2) - bde + cd*S(2))/(S(2)d*S(3)(d + exS(2))) - (-S(2)ae + bd)/(d*S(3)x) + (cdS(2) - e(-S(5)ae + S(3)bd))atan(sqrt(e)x/sqrt(d))/(S(2)d(S(7)/2)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))/(xS(6)(d + exS(2))S(2)), x), x, -a/(S(5)dS(2)x*S(5)) - (-S(2)ae + bd)/(S(3)dS(3)x*S(3)) - ex(ae*S(2) - bde + cd*S(2))/(S(2)d*S(4)(d + exS(2))) - (cd*S(2) - e(-S(3)ae + S(2)bd))/(d*S(4)x) - sqrt(e)(S(3)cdS(2) - e(-S(7)ae + S(5)bd))atan(sqrt(e)x/sqrt(d))/(S(2)d(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(8)(d + exS(2))S(2)), x), x, -a/(S(7)dS(2)x*S(7)) - (-S(2)ae + bd)/(S(5)dS(3)x*S(5)) - (cd*S(2) - e(-S(3)ae + S(2)bd))/(S(3)dS(4)xS(3)) + eS(2)x(aeS(2) - bde + cd*S(2))/(S(2)d*S(5)(d + exS(2))) + e(S(2)cd*S(2) - e(-S(4)ae + S(3)bd))/(d*S(5)x) + e*(S(3)/2)(S(5)cd*S(2) - e(-S(9)ae + S(7)bd))atan(sqrt(e)x/sqrt(d))/(S(2)d(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)(a + bxS(2) + cx*S(4))/(d + exS(2))S(3), x), x, cxS(5)/(S(5)e*S(3)) - sqrt(d)(S(15)ae*S(2) - S(35)bde + S(63)cd*S(2))atan(sqrt(e)x/sqrt(d))/(S(8)e(S(11)/2)) - dS(2)x(aeS(2) - bde + cd*S(2))/(S(4)e*S(5)(d + exS(2))S(2)) + dx(S(17)cdS(2) - e(-S(9)ae + S(13)bd))/(S(8)eS(5)(d + exS(2))) - xS(3)(-be + S(3)cd)/(S(3)e*S(4)) + x(S(6)cd*S(2) - e(-ae + S(3)bd))/eS(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(a + bxS(2) + cx*S(4))/(d + exS(2))S(3), x), x, cxS(3)/(S(3)e*S(3)) + dx(ae*S(2) - bde + cd*S(2))/(S(4)e*S(4)(d + exS(2))S(2)) - x(-be + S(3)cd)/eS(4) - x(S(13)cd*S(2) - e(-S(5)ae + S(9)bd))/(S(8)eS(4)(d + exS(2))) + (S(35)cdS(2) - S(3)e(-ae + S(5)bd))atan(sqrt(e)x/sqrt(d))/(S(8)sqrt(d)e(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx*S(2) + cx*S(4))/(d + exS(2))S(3), x), x, cx/eS(3) - x(aeS(2) - bde + cd*S(2))/(S(4)e*S(3)(d + exS(2))S(2)) + x(S(9)cd*S(2) - e(-ae + S(5)bd))/(S(8)deS(3)(d + exS(2))) - (S(15)cdS(2) - e(ae + S(3)bd))atan(sqrt(e)x/sqrt(d))/(S(8)d*(S(3)/2)e*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(d + exS(2))S(3), x), x, x(ae*S(2) - bde + cd*S(2))/(S(4)deS(2)(d + exS(2))S(2)) - x(S(5)cd*S(2) - e(S(3)ae + bd))/(S(8)d*S(2)e*S(2)(d + exS(2))) + (S(3)cdS(2) + e(S(3)ae + bd))atan(sqrt(e)x/sqrt(d))/(S(8)d*(S(5)/2)e*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(2)(d + exS(2))S(3)), x), x, -a/(d*S(3)x) - x(ae*S(2) - bde + cd*S(2))/(S(4)d*S(2)e(d + exS(2))S(2)) + x(cd*S(2) + e(-S(7)ae + S(3)bd))/(S(8)dS(3)e(d + ex*S(2))) + (cd*S(2) + S(3)e(-S(5)ae + bd))atan(sqrt(e)x/sqrt(d))/(S(8)d(S(7)/2)e*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(4)(d + exS(2))S(3)), x), x, -a/(S(3)dS(3)x*S(3)) + x(aeS(2) - bde + cd*S(2))/(S(4)d*S(3)(d + exS(2))S(2)) + x(S(3)cd*S(2) - e(-S(11)ae + S(7)bd))/(S(8)dS(4)(d + exS(2))) - (-S(3)ae + bd)/(d*S(4)x) + (S(35)ae*S(2) - S(15)bde + S(3)cd*S(2))atan(sqrt(e)x/sqrt(d))/(S(8)d*(S(9)/2)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))/(xS(6)(d + exS(2))S(3)), x), x, -a/(S(5)dS(3)x*S(5)) - ex(ae*S(2) - bde + cd*S(2))/(S(4)d*S(4)(d + exS(2))S(2)) - (-S(3)ae + bd)/(S(3)dS(4)x*S(3)) - ex(S(7)cdS(2) - e(-S(15)ae + S(11)bd))/(S(8)dS(5)(d + exS(2))) - (S(6)aeS(2) - S(3)bde + cdS(2))/(dS(5)x) - sqrt(e)(S(63)aeS(2) - S(35)bde + S(15)cd*S(2))atan(sqrt(e)x/sqrt(d))/(S(8)d(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/((d + exS(2))(a + bxS(2) + cxS(4))), x), x, dS(4)log(d + ex*S(2))/(S(2)e*S(3)(aeS(2) - bde + cdS(2))) + xS(4)/(S(4)ce) - x*S(2)(be + cd)/(S(2)cS(2)eS(2)) - (aS(2)ce - abS(2)e - S(2)abcd + b*S(3)d)log(a + bx*S(2) + cx*S(4))/(S(4)c*S(3)(aeS(2) - bde + cd*S(2))) - (S(3)a*S(2)bce + S(2)aS(2)c*S(2)d - abS(3)e - S(4)ab*S(2)cd + bS(4)d)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(3)sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cdS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/((d + exS(2))(a + bxS(2) + cxS(4))), x), x, -dS(3)log(d + ex*S(2))/(S(2)e*S(2)(aeS(2) - bde + cdS(2))) + xS(2)/(S(2)ce) + (-abe - acd + b*S(2)d)log(a + bx*S(2) + cx*S(4))/(S(4)c*S(2)(aeS(2) - bde + cd*S(2))) + (S(2)a*S(2)ce - ab*S(2)e - S(3)abcd + b*S(3)d)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cdS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/((d + exS(2))(a + bxS(2) + cxS(4))), x), x, dS(2)log(d + ex*S(2))/(S(2)e(ae*S(2) - bde + cd*S(2))) - (-ae + bd)log(a + bxS(2) + cx*S(4))/(S(4)c(ae*S(2) - bde + cd*S(2))) - (-abe - S(2)acd + b*S(2)d)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-S(4)ac + bS(2))(aeS(2) - bde + cdS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/((d + exS(2))(a + bxS(2) + cx*S(4))), x), x, dlog(a + bxS(2) + cx*S(4))/(S(4)aeS(2) - S(4)bde + S(4)cd*S(2)) - dlog(d + exS(2))/(S(2)aeS(2) - S(2)bde + S(2)cd*S(2)) + (-S(2)ae + bd)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((d + ex*S(2))(a + bxS(2) + cx*S(4))), x), x, -elog(a + bxS(2) + cx*S(4))/(S(4)aeS(2) - S(4)bde + S(4)cd*S(2)) + elog(d + exS(2))/(S(2)aeS(2) - S(2)bde + S(2)cd*S(2)) - (-be + S(2)cd)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(d + exS(2))(a + bxS(2) + cxS(4))), x), x, -eS(2)log(d + ex*S(2))/(S(2)d(ae*S(2) - bde + cd*S(2))) - (-be + cd)log(a + bxS(2) + cx*S(4))/(S(4)a(ae*S(2) - bde + cd*S(2))) + (S(2)ace - b*S(2)e + bcd)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)asqrt(-S(4)ac + bS(2))(aeS(2) - bde + cd*S(2))) + log(x)/(ad), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)(d + exS(2))(a + bxS(2) + cxS(4))), x), x, eS(3)log(d + ex*S(2))/(S(2)d*S(2)(aeS(2) - bde + cd*S(2))) - S(1)/(S(2)adx*S(2)) + (ace - bS(2)e + bcd)log(a + bx*S(2) + cx*S(4))/(S(4)a*S(2)(aeS(2) - bde + cd*S(2))) - (S(3)abce - S(2)acS(2)d - b*S(3)e + b*S(2)cd)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cd*S(2))) - (ae + bd)log(x)/(a*S(2)dS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(d + ex*S(2))(a + bxS(2) + cxS(4))), x), x, -eS(4)log(d + ex*S(2))/(S(2)d*S(3)(aeS(2) - bde + cd*S(2))) - S(1)/(S(4)adx*S(4)) + (ae + bd)/(S(2)a*S(2)d*S(2)x*S(2)) - (S(2)abce - ac*S(2)d - b*S(3)e + b*S(2)cd)log(a + bxS(2) + cx*S(4))/(S(4)a*S(3)(aeS(2) - bde + cd*S(2))) + (-S(2)a*S(2)c*S(2)e + S(4)ab*S(2)ce - S(3)abc*S(2)d - b*S(4)e + b*S(3)cd)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(3)sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cd*S(2))) + (abde - a(-ae*S(2) + cdS(2)) + bS(2)dS(2))log(x)/(a*S(3)dS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/((d + exS(2))(a + bxS(2) + cxS(4))), x), x, d(S(7)/2)atan(sqrt(e)x/sqrt(d))/(e*(S(5)/2)(aeS(2) - bde + cdS(2))) + xS(3)/(S(3)ce) - x(be + cd)/(cS(2)e*S(2)) - sqrt(S(2))(a*S(2)ce - ab*S(2)e - S(2)abcd + b*S(3)d + (S(3)aS(2)bce + S(2)aS(2)c*S(2)d - abS(3)e - S(4)ab*S(2)cd + bS(4)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cd*S(2))) - sqrt(S(2))(a*S(2)ce - ab*S(2)e - S(2)abcd + b*S(3)d - (S(3)aS(2)bce + S(2)aS(2)c*S(2)d - abS(3)e - S(4)ab*S(2)cd + bS(4)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/((d + exS(2))(a + bxS(2) + cxS(4))), x), x, -d(S(5)/2)atan(sqrt(e)x/sqrt(d))/(e*(S(3)/2)(aeS(2) - bde + cd*S(2))) + x/(ce) + sqrt(S(2))(-abe - acd + bS(2)d + (S(2)aS(2)ce - ab*S(2)e - S(3)abcd + b*S(3)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cd*S(2))) + sqrt(S(2))(-abe - acd + b*S(2)d - (S(2)aS(2)ce - ab*S(2)e - S(3)abcd + b*S(3)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/((d + exS(2))(a + bxS(2) + cxS(4))), x), x, d(S(3)/2)atan(sqrt(e)x/sqrt(d))/(sqrt(e)(ae*S(2) - bde + cd*S(2))) - sqrt(S(2))(-ae + bd + (-abe - S(2)acd + bS(2)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(aeS(2) - bde + cd*S(2))) - sqrt(S(2))(-ae + bd - (-abe - S(2)acd + bS(2)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(aeS(2) - bde + cdS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((d + exS(2))(a + bxS(2) + cx*S(4))), x), x, sqrt(S(2))sqrt(c)(d + (-S(2)ae + bd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cd*S(2))) + sqrt(S(2))sqrt(c)(d - (-S(2)ae + bd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cd*S(2))) - sqrt(d)sqrt(e)atan(sqrt(e)x/sqrt(d))/(aeS(2) - bde + cd*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex*S(2))(a + bxS(2) + cx*S(4))), x), x, -sqrt(S(2))sqrt(c)(e + (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cd*S(2))) - sqrt(S(2))sqrt(c)(e - (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cdS(2))) + e(S(3)/2)atan(sqrt(e)x/sqrt(d))/(sqrt(d)(ae*S(2) - bde + cdS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(d + ex*S(2))(a + bxS(2) + cxS(4))), x), x, -e(S(5)/2)atan(sqrt(e)x/sqrt(d))/(d*(S(3)/2)(aeS(2) - bde + cd*S(2))) + sqrt(S(2))sqrt(c)(S(2)ace - b*S(2)e + bcd - sqrt(-S(4)ac + b*S(2))(-be + cd))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(2)asqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cd*S(2))) - sqrt(S(2))sqrt(c)(S(2)ace - b*S(2)e + bcd + sqrt(-S(4)ac + b*S(2))(-be + cd))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(2)asqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cd*S(2))) - S(1)/(adx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(d + exS(2))(a + bxS(2) + cxS(4))), x), x, e(S(7)/2)atan(sqrt(e)x/sqrt(d))/(d*(S(5)/2)(aeS(2) - bde + cd*S(2))) - S(1)/(S(3)adx*S(3)) + sqrt(S(2))sqrt(c)(ac(S(2)cd + esqrt(-S(4)ac + bS(2))) + bS(3)e - bS(2)(cd + esqrt(-S(4)ac + b*S(2))) + bc(-S(3)ae + dsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))(aeS(2) - bde + cd*S(2))) + sqrt(S(2))sqrt(c)(ace - bS(2)e + bcd + (S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(aeS(2) - bde + cd*S(2))) + (ae + bd)/(aS(2)d*S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(fx)(d + exS(2))(a + bxS(2) + cxS(4))), x), x, -S(2)(S(3)/4)c(S(3)/4)(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(fx)/(sqrt(f)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)))/(S(2)sqrt(f)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))(aeS(2) - bde + cdS(2))) - S(2)(S(3)/4)c(S(3)/4)(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(fx)/(sqrt(f)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)))/(S(2)sqrt(f)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))(aeS(2) - bde + cdS(2))) + S(2)(S(3)/4)c(S(3)/4)(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))atan(S(2)*(S(1)/4)c*(S(1)/4)sqrt(fx)/(sqrt(f)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)))/(S(2)sqrt(f)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))(aeS(2) - bde + cdS(2))) + S(2)(S(3)/4)c(S(3)/4)(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))atanh(S(2)*(S(1)/4)c*(S(1)/4)sqrt(fx)/(sqrt(f)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)))/(S(2)sqrt(f)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)sqrt(-S(4)ac + bS(2))(aeS(2) - bde + cd*S(2))) - sqrt(S(2))e*(S(7)/4)log(-sqrt(S(2))d(S(1)/4)e*(S(1)/4)sqrt(fx) + sqrt(d)sqrt(f) + sqrt(e)sqrt(f)x)/(S(4)d(S(3)/4)sqrt(f)(ae*S(2) - bde + cd*S(2))) + sqrt(S(2))e*(S(7)/4)log(sqrt(S(2))d(S(1)/4)e*(S(1)/4)sqrt(fx) + sqrt(d)sqrt(f) + sqrt(e)sqrt(f)x)/(S(4)d(S(3)/4)sqrt(f)(ae*S(2) - bde + cd*S(2))) - sqrt(S(2))e*(S(7)/4)atan(S(1) - sqrt(S(2))e(S(1)/4)sqrt(fx)/(d(S(1)/4)sqrt(f)))/(S(2)d(S(3)/4)sqrt(f)(ae*S(2) - bde + cd*S(2))) + sqrt(S(2))e*(S(7)/4)atan(S(1) + sqrt(S(2))e(S(1)/4)sqrt(fx)/(d(S(1)/4)sqrt(f)))/(S(2)d(S(3)/4)sqrt(f)(ae*S(2) - bde + cdS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)sqrt(a + bx*S(2) + cx*S(4))/(d + ex*S(2)), x), x, -b(b + S(2)cx*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(16)c*S(2)e) + b(-S(4)ac + bS(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)c*(S(5)/2)e) + d*S(2)sqrt(a + bxS(2) + cx*S(4))/(S(2)eS(3)) + dS(2)sqrt(ae*S(2) - bde + cd*S(2))atanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)e*S(4)) - d(b + S(2)cx*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(8)ceS(2)) + (a + bx*S(2) + cxS(4))(S(3)/2)/(S(6)ce) - d*S(2)(-be + S(2)cd)atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)sqrt(c)eS(4)) + d(-S(4)ac + b*S(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)c*(S(3)/2)eS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(a + bx*S(2) + cx*S(4))/(d + ex*S(2)), x), x, -dsqrt(aeS(2) - bde + cd*S(2))atanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)e*S(3)) - sqrt(a + bx*S(2) + cx*S(4))(-be + S(4)cd - S(2)cex*S(2))/(S(8)ceS(2)) + (-bS(2)e*S(2) + S(8)c*S(2)d*S(2) - S(4)ce(-ae + bd))atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)c*(S(3)/2)e*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(a + bxS(2) + cx*S(4))/(d + ex*S(2)), x), x, sqrt(a + bx*S(2) + cx*S(4))/(S(2)e) + sqrt(aeS(2) - bde + cd*S(2))atanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)e*S(2)) - (-be + S(2)cd)atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)sqrt(c)eS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cx*S(4))/(x(d + exS(2))), x), x, -sqrt(a)atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)d) + sqrt(c)atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)e) - sqrt(aeS(2) - bde + cd*S(2))atanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)de), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cxS(4))/(xS(3)(d + ex*S(2))), x), x, sqrt(a)eatanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)d*S(2)) - beatanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)sqrt(c)dS(2)) + sqrt(c)atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)d) - sqrt(a + bxS(2) + cx*S(4))/(S(2)dxS(2)) + sqrt(ae*S(2) - bde + cd*S(2))atanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)d*S(2)) - (-be + S(2)cd)atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)sqrt(c)dS(2)) - batanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)sqrt(a)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(2)x*S(2) + S(3)), x), x, x(S(3)xS(2) + S(1))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(30) - xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(4) + S(109)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(120)(sqrt(S(2))xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(-70) + S(263)sqrt(S(2)))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(120)(S(-2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(15)sqrt(S(2)) + S(45))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(32)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(109)sqrt(S(2))xS(2) + S(109))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(120)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(3)sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))))/S(16), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(4)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(2)x*S(2) + S(3)), x), x, x(S(3)xS(2) + S(1))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(30) - xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(4) + S(109)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(120)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(1) + sqrt(S(2)))(sqrt(S(2))xS(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(8)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-S(139)sqrt(S(2)) + S(139))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(480)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(15)sqrt(S(2)) + S(45))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(32)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(45)sqrt(S(2))xS(2) + S(45))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(16)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(109)sqrt(S(2))xS(2) + S(109))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(120)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(3)sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(2)x*S(2) + S(3)), x), x, xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(6) - S(7)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(12)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(-4) + S(17)sqrt(S(2)))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(12)(S(-2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(5)sqrt(S(2)) + S(15))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(16)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(7)sqrt(S(2))xS(2) + S(7))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(12)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) - sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)xS(2) + S(1))))/S(8), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(2)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(2)x*S(2) + S(3)), x), x, xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(6) - S(7)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(12)(sqrt(S(2))xS(2) + S(1))) + S(2)(S(3)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-S(3)sqrt(S(2)) + S(3))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(16)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(1) + sqrt(S(2)))(sqrt(S(2))xS(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(12)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(5)sqrt(S(2)) + S(15))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(16)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(7)sqrt(S(2))xS(2) + S(7))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(12)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(15)sqrt(S(2))xS(2) + S(15))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(8)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) - sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(2)x*S(2) + S(3)), x), x, sqrt(S(2))xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(2)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(2)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((S(-2) + S(3)sqrt(S(2)))sqrt(S(2)xS(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(5)sqrt(S(2)) + S(15))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(24)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))))/S(12), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(2)x*S(2) + S(3)), x), x, sqrt(S(2))xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(2)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(2)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-sqrt(S(2)) + S(1))(sqrt(S(2))xS(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(8)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(5)sqrt(S(2)) + S(15))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(24)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(5)sqrt(S(2))xS(2) + S(5))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(4)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))/(xS(2)(S(2)x*S(2) + S(3))), x), x, sqrt(S(2))xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(3)sqrt(S(2))xS(2) + S(3)) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((S(-6) + S(9)sqrt(S(2)))sqrt(S(2)xS(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(5)sqrt(S(2)) + S(15))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(36)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) - sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))))/S(18) - sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(3)x), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt(S(2)xS(4) + S(2)xS(2) + S(1))/(xS(2)(S(2)x*S(2) + S(3))), x), x, sqrt(S(2))xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(3)sqrt(S(2))xS(2) + S(3)) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(3))(sqrt(S(2))xS(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(12)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(5)sqrt(S(2)) + S(15))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(36)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(5)sqrt(S(2))xS(2) + S(5))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(6)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) - sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))))/S(18) - sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(2)xS(4) + S(2)xS(2) + S(1))/(xS(4)(S(2)xS(2) + S(3))), x), x, -S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(18)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) - S(5)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((-S(27)sqrt(S(2)) + S(18))sqrt(S(2)xS(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(5)sqrt(S(2)) + S(15))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(54)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))))/S(27) - sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(9)x*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))/(xS(6)(S(2)x*S(2) + S(3))), x), x, S(4)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(45)sqrt(S(2))xS(2) + S(45)) - S(4)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(45)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(10)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((-S(81)sqrt(S(2)) + S(54))sqrt(S(2)xS(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-S(2)sqrt(S(2)) + S(19))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(135)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(5)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(27)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) - S(5)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(3))(sqrt(S(2))xS(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((-S(243)sqrt(S(2)) + S(162))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) - S(2)sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))))/S(81) - S(4)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(45)x) + S(4)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(135)x*S(3)) - sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(15)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(a + bx*S(2) + cxS(4))(S(3)/2)/(d + exS(2)), x), x, -b(b + S(2)cx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(32)cS(2)e) + S(3)b(b + S(2)cx*S(2))(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(256)c*S(3)e) - S(3)b(-S(4)ac + bS(2))S(2)atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(512)c*(S(7)/2)e) + d*S(2)(a + bxS(2) + cxS(4))(S(3)/2)/(S(6)eS(3)) + dS(2)(aeS(2) - bde + cdS(2))(S(3)/2)atanh((-S(2)ae + bd + x*S(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)eS(6)) + dS(2)sqrt(a + bx*S(2) + cx*S(4))(b*S(2)e*S(2) + S(8)c*S(2)d*S(2) - S(2)cex*S(2)(-be + S(2)cd) - S(2)ce(-S(4)ae + S(5)bd))/(S(16)ce*S(5)) - d(b + S(2)cx*S(2))(a + bxS(2) + cxS(4))(S(3)/2)/(S(16)ce*S(2)) + (a + bx*S(2) + cxS(4))(S(5)/2)/(S(10)ce) + d(b + S(2)cxS(2))(-S(12)ac + S(3)bS(2))sqrt(a + bxS(2) + cx*S(4))/(S(128)c*S(2)eS(2)) - dS(2)(-be + S(2)cd)(-bS(2)e*S(2) + S(8)c*S(2)d*S(2) - S(4)ce(-S(3)ae + S(2)bd))atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)c*(S(3)/2)e*S(6)) - S(3)d(-S(4)ac + bS(2))S(2)atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(256)c*(S(5)/2)eS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a + bx*S(2) + cxS(4))(S(3)/2)/(d + exS(2)), x), x, -d(aeS(2) - bde + cdS(2))(S(3)/2)atanh((-S(2)ae + bd + x*S(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)e*S(5)) - (a + bx*S(2) + cxS(4))(S(3)/2)(-S(3)be + S(8)cd - S(6)cex*S(2))/(S(48)ceS(2)) - sqrt(a + bx*S(2) + cx*S(4))(S(3)bS(3)e*S(3) + S(4)bce*S(2)(-S(3)ae + S(2)bd) + S(64)cS(3)d*S(3) - S(16)c*S(2)de(-S(4)ae + S(5)bd) - S(2)cexS(2)(-S(3)bS(2)e*S(2) + S(16)c*S(2)d*S(2) - S(4)ce(-S(3)ae + S(2)bd)))/(S(128)cS(2)e*S(4)) + (S(3)b*S(4)e*S(4) + S(8)b*S(2)ceS(3)(-S(3)ae + bd) + S(128)c*S(4)d*S(4) - S(192)c*S(3)d*S(2)e(-ae + bd) + S(48)c*S(2)e*S(2)(-ae + bd)*S(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(256)c*(S(5)/2)e*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxS(2) + cxS(4))(S(3)/2)/(d + exS(2)), x), x, (a + bx*S(2) + cxS(4))(S(3)/2)/(S(6)e) + (ae*S(2) - bde + cdS(2))(S(3)/2)atanh((-S(2)ae + bd + x*S(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)e*S(4)) + sqrt(a + bx*S(2) + cx*S(4))(b*S(2)e*S(2) + S(8)c*S(2)d*S(2) - S(2)cex*S(2)(-be + S(2)cd) - S(2)ce(-S(4)ae + S(5)bd))/(S(16)ce*S(3)) - (-be + S(2)cd)(-bS(2)e*S(2) + S(8)c*S(2)d*S(2) - S(4)ce(-S(3)ae + S(2)bd))atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)c*(S(3)/2)e*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(3)/2)/(x(d + exS(2))), x), x, -a(S(3)/2)atanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)d) + abatanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)sqrt(c)d) + asqrt(a + bxS(2) + cx*S(4))/(S(2)d) - sqrt(a + bxS(2) + cx*S(4))(S(4)cd*S(2) - S(2)cdexS(2) - e(-S(4)ae + S(5)bd))/(S(8)de*S(2)) - (ae*S(2) - bde + cdS(2))(S(3)/2)atanh((-S(2)ae + bd + x*S(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)deS(3)) + (be*S(2)(-S(4)ae + S(3)bd) + S(8)cS(2)d*S(3) - S(12)cde(-ae + bd))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)sqrt(c)de*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))(S(3)/2)/(x*S(3)(d + exS(2))), x), x, a(S(3)/2)eatanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)d*S(2)) - S(3)sqrt(a)batanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)d) + be(-S(12)ac + b*S(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)c*(S(3)/2)d*S(2)) + (S(9)b + S(6)cx*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(8)d) - (a + bxS(2) + cxS(4))(S(3)/2)/(S(2)dx*S(2)) + (ae*S(2) - bde + cdS(2))(S(3)/2)atanh((-S(2)ae + bd + x*S(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)d*S(2)e*S(2)) - esqrt(a + bxS(2) + cx*S(4))(S(8)ac + b*S(2) + S(2)bcx*S(2))/(S(16)cdS(2)) + sqrt(a + bx*S(2) + cx*S(4))(b*S(2)e*S(2) + S(8)c*S(2)d*S(2) - S(2)cex*S(2)(-be + S(2)cd) - S(2)ce(-S(4)ae + S(5)bd))/(S(16)cd*S(2)e) + (S(12)ac + S(3)bS(2))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)sqrt(c)d) - (-be + S(2)cd)(-bS(2)e*S(2) + S(8)c*S(2)d*S(2) - S(4)ce(-S(3)ae + S(2)bd))atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(32)c*(S(3)/2)d*S(2)eS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(S(2)x*S(4) + S(2)xS(2) + S(1))(S(3)/2)/(-S(2)xS(2) + S(3)), x), x, -S(27)x*S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(70) - x(S(2)xS(4) + S(2)xS(2) + S(1))(S(3)/2)/S(14) - S(213)xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(140) - S(2211)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(280)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(1542) + S(8151)sqrt(S(2)))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(280)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-S(289)sqrt(S(2)) + S(867))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(S(1)/2 + S(11)sqrt(S(2))/S(24), S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(32)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(2211)sqrt(S(2))xS(2) + S(2211))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(280)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(17)sqrt(S(51))atanh(sqrt(S(51))x/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))))/S(16), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(2)(S(2)x*S(4) + S(2)xS(2) + S(1))(S(3)/2)/(-S(2)xS(2) + S(3)), x), x, -S(3)x(xS(2) + S(2))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(35) - x(S(3)xS(2) + S(1))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(10) - x(S(2)xS(4) + S(2)xS(2) + S(1))(S(3)/2)/S(14) - S(5)xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(4) - S(6)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(35)sqrt(S(2))xS(2) + S(35)) - S(309)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(40)(sqrt(S(2))xS(2) + S(1))) + S(6)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(35)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(51)sqrt(S(2)) + S(255))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(32)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(5) + S(5)sqrt(S(2)))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(8)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(6)sqrt(S(2)) + S(9))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(140)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-sqrt(S(2)) + S(1))(sqrt(S(2))xS(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(40)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-S(289)sqrt(S(2)) + S(867))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(S(1)/2 + S(11)sqrt(S(2))/S(24), S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(32)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(309)sqrt(S(2))xS(2) + S(309))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(40)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(867)sqrt(S(2))xS(2) + S(867))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(16)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(17)sqrt(S(51))atanh(sqrt(S(51))x/(S(3)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(4) + S(2)xS(2) + S(1))(S(3)/2)/(-S(2)xS(2) + S(3)), x), x, -xS(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(5) - S(9)xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/S(10) - S(103)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(20)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(66) + S(383)sqrt(S(2)))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(20)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-S(289)sqrt(S(2)) + S(867))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(S(1)/2 + S(11)sqrt(S(2))/S(24), S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(48)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(103)sqrt(S(2))xS(2) + S(103))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(20)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(17)sqrt(S(51))atanh(sqrt(S(51))x/(S(3)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))))/S(24), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((S(2)x*S(4) + S(2)xS(2) + S(1))(S(3)/2)/(-S(2)xS(2) + S(3)), x), x, -x(S(3)xS(2) + S(1))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(15) - S(5)xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/S(6) - S(103)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(20)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(17)sqrt(S(2)) + S(85))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(16)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(5) + S(5)sqrt(S(2)))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(12)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-sqrt(S(2)) + S(1))(sqrt(S(2))xS(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(60)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-S(289)sqrt(S(2)) + S(867))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(S(1)/2 + S(11)sqrt(S(2))/S(24), S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(48)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(103)sqrt(S(2))xS(2) + S(103))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(20)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(289)sqrt(S(2))xS(2) + S(289))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(8)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(17)sqrt(S(51))atanh(sqrt(S(51))x/(S(3)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))))/S(24), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(4) + S(2)xS(2) + S(1))(S(3)/2)/(x*S(2)(-S(2)xS(2) + S(3))), x), x, sqrt(S(2))xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(3)sqrt(S(2))xS(2) + S(3)) - S(17)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(6)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(17)sqrt(S(2)) + S(85))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(24)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(6)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-S(289)sqrt(S(2)) + S(867))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(S(1)/2 + S(11)sqrt(S(2))/S(24), S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(72)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(17)sqrt(S(2))xS(2) + S(17))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(6)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(289)sqrt(S(2))xS(2) + S(289))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(12)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(17)sqrt(S(51))atanh(sqrt(S(51))x/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))))/S(36) - (xS(2) + S(1))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(4) + S(2)xS(2) + S(1))(S(3)/2)/(x*S(4)(-S(2)xS(2) + S(3))), x), x, sqrt(S(2))xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(9)sqrt(S(2))xS(2) + S(9)) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(9)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(17)sqrt(S(2)) + S(85))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(36)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(5)sqrt(S(2)) + S(9))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(9)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-S(289)sqrt(S(2)) + S(867))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(S(1)/2 + S(11)sqrt(S(2))/S(24), S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(108)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(289)sqrt(S(2))xS(2) + S(289))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(18)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(17)sqrt(S(51))atanh(sqrt(S(51))x/(S(3)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))))/S(54) - S(2)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/x - (-S(8)x*S(2) + S(1))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(9)x*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(4) + S(2)xS(2) + S(1))(S(3)/2)/(x*S(6)(-S(2)xS(2) + S(3))), x), x, S(262)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(135)sqrt(S(2))xS(2) + S(135)) - S(262)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(135)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-S(17)sqrt(S(2)) + S(51))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(54)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(23)sqrt(S(2)) + S(37))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(135)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(289)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((S(54) + S(81)sqrt(S(2)))sqrt(S(2)xS(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-S(289)sqrt(S(2)) + S(867))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(S(1)/2 + S(11)sqrt(S(2))/S(24), S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(162)(S(2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(17)sqrt(S(51))atanh(sqrt(S(51))x/(S(3)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))))/S(81) - S(262)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(135)x) + S(74)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(135)x*S(3)) - (S(40)x*S(2) + S(3))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(45)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/((d + exS(2))sqrt(a + bxS(2) + cx*S(4))), x), x, -batanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)c*(S(3)/2)e) + d*S(2)atanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)e*S(2)sqrt(aeS(2) - bde + cd*S(2))) + sqrt(a + bx*S(2) + cx*S(4))/(S(2)ce) - datanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)sqrt(c)eS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/((d + ex*S(2))sqrt(a + bxS(2) + cx*S(4))), x), x, -datanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)esqrt(ae*S(2) - bde + cd*S(2))) + atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)sqrt(c)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((d + ex*S(2))sqrt(a + bxS(2) + cx*S(4))), x), x, atanh((-S(2)ae + bd + x*S(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)sqrt(aeS(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(d + exS(2))sqrt(a + bxS(2) + cx*S(4))), x), x, -eatanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)dsqrt(ae*S(2) - bde + cd*S(2))) - atanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)sqrt(a)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(d + exS(2))sqrt(a + bxS(2) + cxS(4))), x), x, eS(2)atanh((-S(2)ae + bd + x*S(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)d*S(2)sqrt(aeS(2) - bde + cd*S(2))) - sqrt(a + bx*S(2) + cx*S(4))/(S(2)adx*S(2)) + eatanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)sqrt(a)dS(2)) + batanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)a*(S(3)/2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)/((S(2)x*S(2) + S(3))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))), x), x, sqrt(S(2))xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(4)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(4)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(8)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(3)sqrt(S(2)) + S(9))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(16)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(3)sqrt(S(2))xS(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(16)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(9)sqrt(S(2))xS(2) + S(9))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(8)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(3)sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))))/S(40), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((S(2)xS(2) + S(3))sqrt(S(2)xS(4) + S(2)xS(2) + S(1))), x), x, S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(8)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(3))(sqrt(S(2))xS(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(8)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(3)sqrt(S(2))xS(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(4)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) - sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))))/S(20), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((S(2)x*S(2) + S(3))sqrt(S(2)xS(4) + S(2)xS(2) + S(1))), x), x, -S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(2)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(3))(sqrt(S(2))xS(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(12)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)xS(2) + S(1))))/S(30), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(S(2)x*S(2) + S(3))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))), x), x, sqrt(S(2))xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(3)sqrt(S(2))xS(2) + S(3)) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((-S(9)sqrt(S(2)) + S(6))sqrt(S(2)xS(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(6)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(3))(sqrt(S(2))xS(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(18)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) - sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))))/S(45) - sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(4)(S(2)xS(2) + S(3))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))), x), x, -S(2)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(3)sqrt(S(2))xS(2) + S(3)) + S(2)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(1) + S(2)sqrt(S(2)))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(18)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(9)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) - S(2)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((-S(27)sqrt(S(2)) + S(18))sqrt(S(2)xS(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(3))(sqrt(S(2))xS(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((-S(81)sqrt(S(2)) + S(54))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) + S(2)sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))))/S(135) + S(2)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(3)x) - sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(9)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/((d + exS(2))(a + bxS(2) + cxS(4))(S(3)/2)), x), x, -bsqrt(a + bx*S(2) + cx*S(4))/(ce(-S(4)ac + bS(2))) - dS(3)atanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)e(ae*S(2) - bde + cdS(2))(S(3)/2)) + d*S(3)(S(2)ace - bS(2)e + bcd + cxS(2)(-be + S(2)cd))/(eS(3)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))(aeS(2) - bde + cdS(2))) - dS(2)(b + S(2)cxS(2))/(eS(3)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) - d(S(2)a + bxS(2))/(eS(2)(-S(4)ac + bS(2))sqrt(a + bxS(2) + cxS(4))) + xS(2)(S(2)a + bxS(2))/(e(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) + atanh((b + S(2)cxS(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)c*(S(3)/2)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)/((d + ex*S(2))(a + bxS(2) + cxS(4))(S(3)/2)), x), x, d*S(2)atanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)(aeS(2) - bde + cdS(2))(S(3)/2)) - d*S(2)(S(2)ace - bS(2)e + bcd + cxS(2)(-be + S(2)cd))/(eS(2)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))(aeS(2) - bde + cd*S(2))) + d(b + S(2)cxS(2))/(eS(2)(-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))) + (S(2)a + bxS(2))/(e(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/((d + exS(2))(a + bxS(2) + cxS(4))(S(3)/2)), x), x, -deatanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)(aeS(2) - bde + cdS(2))(S(3)/2)) + (a(-be + S(2)cd) + cxS(2)(-S(2)ae + bd))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))(aeS(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((d + ex*S(2))(a + bxS(2) + cxS(4))(S(3)/2)), x), x, e*S(2)atanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)(aeS(2) - bde + cdS(2))(S(3)/2)) - (S(2)ace - bS(2)e + bcd + cxS(2)(-be + S(2)cd))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))(aeS(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(d + exS(2))(a + bxS(2) + cxS(4))(S(3)/2)), x), x, -e*S(3)atanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)d(ae*S(2) - bde + cdS(2))(S(3)/2)) + e(S(2)ace - b*S(2)e + bcd + cxS(2)(-be + S(2)cd))/(d(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))(aeS(2) - bde + cd*S(2))) + (-S(2)ac + bS(2) + bcxS(2))/(ad(-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))) - atanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)a*(S(3)/2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)(d + exS(2))(a + bxS(2) + cxS(4))(S(3)/2)), x), x, e*S(4)atanh((-S(2)ae + bd + xS(2)(-be + S(2)cd))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(2) - bde + cd*S(2))))/(S(2)d*S(2)(aeS(2) - bde + cdS(2))(S(3)/2)) - e*S(2)(S(2)ace - bS(2)e + bcd + cxS(2)(-be + S(2)cd))/(dS(2)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))(aeS(2) - bde + cd*S(2))) + (-S(2)ac + bS(2) + bcxS(2))/(adxS(2)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) - e(-S(2)ac + b*S(2) + bcxS(2))/(ad*S(2)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))) - (-S(8)ac + S(3)b*S(2))sqrt(a + bxS(2) + cx*S(4))/(S(2)a*S(2)dxS(2)(-S(4)ac + b*S(2))) + eatanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)a*(S(3)/2)d*S(2)) + S(3)batanh((S(2)a + bxS(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)a*(S(5)/2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(8)/((S(2)x*S(2) + S(3))(S(2)xS(4) + S(2)xS(2) + S(1))(S(3)/2)), x), x, x*S(3)(-S(2)xS(2) + S(1))/(S(20)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) + xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/S(20) + sqrt(S(2))xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(20)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(20)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(7))(sqrt(S(2))xS(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(80)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(27)sqrt(S(2)) + S(81))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(160)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(9)sqrt(S(2))xS(2) + S(9))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(80)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(27)sqrt(S(2))xS(2) + S(27))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(160)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(81)sqrt(S(2))xS(2) + S(81))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(80)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(27)sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))))/S(400), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/((S(2)xS(2) + S(3))(S(2)xS(4) + S(2)xS(2) + S(1))(S(3)/2)), x), x, x(-S(2)x*S(2) + S(1))/(S(20)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) + sqrt(S(2))xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(20)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(20)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-sqrt(S(2)) + S(1))(sqrt(S(2))xS(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(80)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(9)sqrt(S(2)) + S(27))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(80)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(9)sqrt(S(2))xS(2) + S(9))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(80)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(27)sqrt(S(2))xS(2) + S(27))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(40)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) - S(9)sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))))/S(200), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/((S(2)xS(2) + S(3))(S(2)xS(4) + S(2)xS(2) + S(1))(S(3)/2)), x), x, -x(xS(2) + S(2))/(S(10)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) + sqrt(S(2))xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(20)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(20)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-sqrt(S(2)) + S(1))(sqrt(S(2))xS(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(40)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(3)sqrt(S(2)) + S(9))(sqrt(S(2))x*S(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(40)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(9)sqrt(S(2))xS(2) + S(9))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(20)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(3)sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))))/S(100), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((S(2)xS(2) + S(3))(S(2)xS(4) + S(2)xS(2) + S(1))(S(3)/2)), x), x, x(S(4)x*S(2) + S(3))/(S(10)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) - sqrt(S(2))xsqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(5)sqrt(S(2))xS(2) + S(5)) + S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(5)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(1) + S(2)sqrt(S(2)))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(40)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(3))(sqrt(S(2))xS(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(20)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(3)sqrt(S(2))xS(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(10)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) - sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))))/S(50), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((S(2)x*S(2) + S(3))(S(2)xS(4) + S(2)xS(2) + S(1))(S(3)/2)), x), x, -x(S(3)x*S(2) + S(1))/(S(5)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) + S(3)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(10)(sqrt(S(2))xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(2))(sqrt(S(2))xS(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(4)(S(-2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(3))(sqrt(S(2))xS(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(30)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(3)sqrt(S(2))xS(2) + S(3))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(10)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))))/S(75), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/((S(2)x*S(2) + S(3))(S(2)xS(4) + S(2)xS(2) + S(1))(S(3)/2)), x), x, -x(S(3)x*S(2) + S(1))/(S(5)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) + S(3)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(10)(sqrt(S(2))xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((-S(15)sqrt(S(2)) + S(10))sqrt(S(2)xS(4) + S(2)xS(2) + S(1))) + S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(10)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(3))(sqrt(S(2))xS(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(30)(-S(3)sqrt(S(2)) + S(2))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(3)sqrt(S(2))xS(2) + S(3))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(10)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(3)sqrt(S(2))xS(2) + S(3))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(20)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)xS(4) + S(2)xS(2) + S(1))))/S(75), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(S(2)x*S(2) + S(3))(S(2)xS(4) + S(2)xS(2) + S(1))(S(3)/2)), x), x, S(2)x(S(3)xS(2) + S(1))/(S(15)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) - x/(S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) + S(2)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(15)sqrt(S(2))xS(2) + S(15)) - S(2)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(15)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) + S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(S(-7) + S(3)sqrt(S(2)))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(6)(S(-2) + S(3)sqrt(S(2)))sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(3))(sqrt(S(2))xS(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((-S(135)sqrt(S(2)) + S(90))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) - S(2)sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))))/S(225) - sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(3)x), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/(x*S(2)(S(2)xS(2) + S(3))(S(2)xS(4) + S(2)xS(2) + S(1))(S(3)/2)), x), x, S(2)x(S(3)xS(2) + S(1))/(S(15)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) - x/(S(3)sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) + S(2)sqrt(S(2))xsqrt(S(2)xS(4) + S(2)x*S(2) + S(1))/(S(15)sqrt(S(2))xS(2) + S(15)) - S(2)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_e(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(15)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(10)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(15)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))) + S(2)S(2)*(S(1)/4)sqrt((S(2)xS(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2))x*S(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((-S(45)sqrt(S(2)) + S(30))sqrt(S(2)xS(4) + S(2)xS(2) + S(1))) - S(2)(S(3)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(-sqrt(S(2)) + S(1))(sqrt(S(2))xS(2) + S(1))elliptic_f(S(2)atan(S(2)(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/(S(12)sqrt(S(2)x*S(4) + S(2)xS(2) + S(1))) - S(2)(S(1)/4)sqrt((S(2)x*S(4) + S(2)x*S(2) + S(1))/(sqrt(S(2))xS(2) + S(1))S(2))(sqrt(S(2)) + S(3))(sqrt(S(2))xS(2) + S(1))elliptic_pi(-S(11)sqrt(S(2))/S(24) + S(1)/2, S(2)atan(S(2)*(S(1)/4)x), -sqrt(S(2))/S(4) + S(1)/2)/((-S(135)sqrt(S(2)) + S(90))sqrt(S(2)xS(4) + S(2)x*S(2) + S(1))) - S(2)sqrt(S(15))atan(sqrt(S(15))x/(S(3)sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))))/S(225) - sqrt(S(2)x*S(4) + S(2)x*S(2) + S(1))/(S(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(7)sqrt(d + exS(2))/(a + bx*S(2) + cx*S(4)), x), x, (d + exS(2))(S(5)/2)/(S(5)ce*S(2)) - (d + exS(2))(S(3)/2)(be + cd)/(S(3)c*S(2)e*S(2)) + sqrt(d + ex*S(2))(-ac + bS(2))/cS(3) - sqrt(S(2))(S(2)abce - acS(2)d - b*S(3)e + b*S(2)cd + (-S(2)a*S(2)c*S(2)e + S(4)ab*S(2)ce - S(3)abc*S(2)d - b*S(4)e + b*S(3)cd)/sqrt(-S(4)ac + bS(2)))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)c*(S(7)/2)sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) - sqrt(S(2))(S(2)abce - acS(2)d - b*S(3)e + b*S(2)cd - (-S(2)a*S(2)c*S(2)e + S(4)ab*S(2)ce - S(3)abc*S(2)d - b*S(4)e + b*S(3)cd)/sqrt(-S(4)ac + bS(2)))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2)))))/(S(2)c*(S(7)/2)sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)sqrt(d + exS(2))/(a + bx*S(2) + cx*S(4)), x), x, -bsqrt(d + exS(2))/cS(2) + (d + exS(2))(S(3)/2)/(S(3)ce) + sqrt(S(2))(ace - bS(2)e + bcd + (S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)c*(S(5)/2)sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + sqrt(S(2))(ace - b*S(2)e + bcd - (S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2)))))/(S(2)c*(S(5)/2)sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(d + exS(2))/(a + bx*S(2) + cx*S(4)), x), x, sqrt(d + ex*S(2))/c - sqrt(S(2))(S(2)ace - bS(2)e + bcd + sqrt(-S(4)ac + b*S(2))(-be + cd))atanh(sqrt(S(2))sqrt(c)sqrt(d + ex*S(2))/sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)c*(S(3)/2)sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + sqrt(S(2))(S(2)ace - bS(2)e + bcd - sqrt(-S(4)ac + b*S(2))(-be + cd))atanh(sqrt(S(2))sqrt(c)sqrt(d + ex*S(2))/sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2)))))/(S(2)c*(S(3)/2)sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(d + exS(2))/(a + bx*S(2) + cx*S(4)), x), x, -sqrt(S(2))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2)))))/(S(2)sqrt(c)sqrt(-S(4)ac + bS(2))) + sqrt(S(2))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)sqrt(c)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + ex*S(2))/(x(a + bxS(2) + cx*S(4))), x), x, -sqrt(S(2))sqrt(c)(-S(2)ae + bd - dsqrt(-S(4)ac + bS(2)))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)asqrt(-S(4)ac + bS(2))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + sqrt(S(2))sqrt(c)(-S(2)ae + bd + dsqrt(-S(4)ac + bS(2)))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2)))))/(S(2)asqrt(-S(4)ac + bS(2))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) - sqrt(d)atanh(sqrt(d + exS(2))/sqrt(d))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + exS(2))/(xS(5)(a + bx*S(2) + cx*S(4))), x), x, -sqrt(d + ex*S(2))/(S(4)axS(4)) + S(3)esqrt(d + ex*S(2))/(S(8)adx*S(2)) - S(3)e*S(2)atanh(sqrt(d + exS(2))/sqrt(d))/(S(8)ad(S(3)/2)) + sqrt(d + ex*S(2))(-ae + bd)/(S(2)aS(2)dxS(2)) - e(-ae + bd)atanh(sqrt(d + ex*S(2))/sqrt(d))/(S(2)a*S(2)d*(S(3)/2)) - sqrt(S(2))sqrt(c)(-ab(S(3)cd - esqrt(-S(4)ac + b*S(2))) + ac(S(2)ae + dsqrt(-S(4)ac + bS(2))) + bS(3)d - bS(2)(ae + dsqrt(-S(4)ac + b*S(2))))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)a*S(3)sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + sqrt(S(2))sqrt(c)(-ab(S(3)cd + esqrt(-S(4)ac + b*S(2))) - ac(-S(2)ae + dsqrt(-S(4)ac + bS(2))) + bS(3)d + bS(2)(-ae + dsqrt(-S(4)ac + b*S(2))))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2)))))/(S(2)a*S(3)sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) - (-abe - acd + bS(2)d)atanh(sqrt(d + exS(2))/sqrt(d))/(aS(3)sqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)sqrt(d + exS(2))/(a + bx*S(2) + cx*S(4)), x), x, xsqrt(d + exS(2))/(S(2)c) - (ace - b*S(2)e + bcd + (S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(cS(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) - (ace - bS(2)e + bcd - (S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(cS(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) + (-S(2)be + cd)atanh(sqrt(e)x/sqrt(d + exS(2)))/(S(2)c*S(2)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)sqrt(d + exS(2))/(a + bx*S(2) + cx*S(4)), x), x, sqrt(e)atanh(sqrt(e)x/sqrt(d + ex*S(2)))/c + (-be + cd + (S(2)ace - b*S(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(csqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + (-be + cd - (S(2)ace - b*S(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(csqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + ex*S(2))/(a + bx*S(2) + cx*S(4)), x), x, -sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))) + sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + exS(2))/(xS(2)(a + bx*S(2) + cx*S(4))), x), x, -c(d - (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(asqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) - c(d + (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(asqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) - sqrt(d + ex*S(2))/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + exS(2))/(xS(4)(a + bxS(2) + cx*S(4))), x), x, -sqrt(d + ex*S(2))/(S(3)axS(3)) + S(2)esqrt(d + ex*S(2))/(S(3)adx) + c(-ae + bd - (-abe - S(2)acd + b*S(2)d)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(aS(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + c(-ae + bd + (-abe - S(2)acd + bS(2)d)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(aS(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) + sqrt(d + ex*S(2))(-ae + bd)/(a*S(2)dx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + exS(2))/(xS(6)(a + bx*S(2) + cx*S(4))), x), x, -sqrt(d + ex*S(2))/(S(5)axS(5)) + S(4)esqrt(d + ex*S(2))/(S(15)adx*S(3)) - S(8)e*S(2)sqrt(d + exS(2))/(S(15)adS(2)x) + sqrt(d + exS(2))(-ae + bd)/(S(3)aS(2)dxS(3)) - S(2)esqrt(d + ex*S(2))(-ae + bd)/(S(3)aS(2)d*S(2)x) - c(-abe - acd + bS(2)d - (S(2)aS(2)ce - ab*S(2)e - S(3)abcd + b*S(3)d)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(aS(3)sqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) - c(-abe - acd + b*S(2)d + (S(2)aS(2)ce - ab*S(2)e - S(3)abcd + b*S(3)d)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(aS(3)sqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) - sqrt(d + ex*S(2))(-abe - acd + b*S(2)d)/(a*S(3)dx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(d + exS(2))(S(3)/2)/(a + bx*S(2) + cx*S(4)), x), x, (d + exS(2))(S(3)/2)/(S(3)c) + sqrt(d + ex*S(2))(-be + cd)/c*S(2) - sqrt(S(2))(b*S(3)eS(2) - bS(2)e(S(2)cd - esqrt(-S(4)ac + bS(2))) + bc(cd*S(2) - e(S(3)ae + S(2)dsqrt(-S(4)ac + b*S(2)))) - c(aeS(2)sqrt(-S(4)ac + b*S(2)) - cd(S(4)ae + dsqrt(-S(4)ac + b*S(2)))))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)c*(S(5)/2)sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + sqrt(S(2))(b*S(3)eS(2) - bS(2)e(S(2)cd + esqrt(-S(4)ac + bS(2))) + bc(cd*S(2) + e(-S(3)ae + S(2)dsqrt(-S(4)ac + b*S(2)))) + c(aeS(2)sqrt(-S(4)ac + b*S(2)) - cd(-S(4)ae + dsqrt(-S(4)ac + b*S(2)))))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2)))))/(S(2)c*(S(5)/2)sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(d + exS(2))(S(3)/2)/(a + bx*S(2) + cx*S(4)), x), x, esqrt(d + exS(2))/c + sqrt(S(2))(beS(2)(b + sqrt(-S(4)ac + b*S(2))) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd + dsqrt(-S(4)ac + bS(2))))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)c*(S(3)/2)sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) - sqrt(S(2))(beS(2)(b - sqrt(-S(4)ac + b*S(2))) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd - dsqrt(-S(4)ac + bS(2))))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2)))))/(S(2)c*(S(3)/2)sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))(S(3)/2)/(x(a + bx*S(2) + cxS(4))), x), x, -d(S(3)/2)atanh(sqrt(d + ex*S(2))/sqrt(d))/a - sqrt(S(2))(aeS(2)sqrt(-S(4)ac + b*S(2)) + b(aeS(2) + cd*S(2)) - cd(S(4)ae + dsqrt(-S(4)ac + b*S(2))))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)asqrt(c)sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) - sqrt(S(2))(aeS(2)sqrt(-S(4)ac + b*S(2)) - b(aeS(2) + cd*S(2)) - cd(-S(4)ae + dsqrt(-S(4)ac + b*S(2))))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2)))))/(S(2)asqrt(c)sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))(S(3)/2)/(x*S(3)(a + bxS(2) + cx*S(4))), x), x, sqrt(d)eatanh(sqrt(d + ex*S(2))/sqrt(d))/(S(2)a) - dsqrt(d + ex*S(2))/(S(2)axS(2)) + sqrt(S(2))sqrt(c)(-S(2)a(cd*S(2) - e(ae + dsqrt(-S(4)ac + bS(2)))) + bS(2)dS(2) - bd(S(2)ae + dsqrt(-S(4)ac + b*S(2))))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)a*S(2)sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) - sqrt(S(2))sqrt(c)(-S(2)a(cd*S(2) + e(-ae + dsqrt(-S(4)ac + bS(2)))) + bS(2)dS(2) + bd(-S(2)ae + dsqrt(-S(4)ac + b*S(2))))atanh(sqrt(S(2))sqrt(c)sqrt(d + exS(2))/sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2)))))/(S(2)a*S(2)sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) + sqrt(d)(-S(2)ae + bd)atanh(sqrt(d + exS(2))/sqrt(d))/aS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(d + exS(2))(S(3)/2)/(a + bx*S(2) + cx*S(4)), x), x, x(d + exS(2))(S(3)/2)/(S(4)c) + d(-S(4)be + S(3)cd)atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(8)c*S(2)sqrt(e)) + xsqrt(d + ex*S(2))(-S(4)be + S(3)cd)/(S(8)cS(2)) - sqrt(e)(ace - b*S(2)e + bcd - (S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(2)c*S(3)) - sqrt(e)(ace - b*S(2)e + bcd + (S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(2)c*S(3)) - sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))(ace - b*S(2)e + bcd + (S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(S(2)c*S(3)sqrt(b + sqrt(-S(4)ac + b*S(2)))) - sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))(ace - b*S(2)e + bcd - (S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(S(2)c*S(3)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(d + exS(2))(S(3)/2)/(a + bxS(2) + cx*S(4)), x), x, dsqrt(e)atanh(sqrt(e)x/sqrt(d + exS(2)))/(S(2)c) + exsqrt(d + exS(2))/(S(2)c) + sqrt(e)(-be + cd - (S(2)ace - b*S(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(2)c*S(2)) + sqrt(e)(-be + cd + (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(2)c*S(2)) + sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))(-be + cd + (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(S(2)c*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) + sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))(-be + cd - (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(S(2)c*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))(S(3)/2)/(a + bxS(2) + cx*S(4)), x), x, sqrt(e)(S(3)cd - e(b - sqrt(-S(4)ac + bS(2))))atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(2)csqrt(-S(4)ac + bS(2))) - sqrt(e)(S(3)cd - e(b + sqrt(-S(4)ac + bS(2))))atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(2)csqrt(-S(4)ac + bS(2))) - (be*S(2)(b + sqrt(-S(4)ac + b*S(2))) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd + dsqrt(-S(4)ac + bS(2))))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(csqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + (be*S(2)(b - sqrt(-S(4)ac + b*S(2))) + S(2)c*S(2)d*S(2) - S(2)ce(ae + bd - dsqrt(-S(4)ac + bS(2))))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(csqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))(S(3)/2)/(x*S(2)(a + bxS(2) + cx*S(4))), x), x, (S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))(S(3)/2)atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/((b + sqrt(-S(4)ac + bS(2)))(S(3)/2)sqrt(-S(4)ac + b*S(2))) - (S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))(S(3)/2)atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/((b - sqrt(-S(4)ac + bS(2)))(S(3)/2)sqrt(-S(4)ac + b*S(2))) - dsqrt(d + exS(2))/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(2))(S(3)/2)/(xS(4)(a + bxS(2) + cx*S(4))), x), x, -(d + exS(2))(S(3)/2)/(S(3)ax*S(3)) - sqrt(e)(-ae + bd)atanh(sqrt(e)x/sqrt(d + exS(2)))/aS(2) + sqrt(e)(-ae + bd - (-abe - S(2)acd + bS(2)d)/sqrt(-S(4)ac + b*S(2)))atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(2)a*S(2)) + sqrt(e)(-ae + bd + (-abe - S(2)acd + bS(2)d)/sqrt(-S(4)ac + b*S(2)))atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(2)a*S(2)) + sqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))(-ae + bd - (-abe - S(2)acd + bS(2)d)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(S(2)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) + sqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))(-ae + bd + (-abe - S(2)acd + bS(2)d)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(S(2)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))) + sqrt(d + ex*S(2))(-ae + bd)/(a*S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)sqrt(-x*S(2) + S(1))/(a + bx*S(2) + cx*S(4)), x), x, -bsqrt(-xS(2) + S(1))/cS(2) - (-xS(2) + S(1))(S(3)/2)/(S(3)c) + sqrt(S(2))(-ac + bS(2) + bc + (-S(3)abc - S(2)acS(2) + bS(3) + bS(2)c)/sqrt(-S(4)ac + b*S(2)))atanh(sqrt(S(2))sqrt(c)sqrt(-x*S(2) + S(1))/sqrt(b + S(2)c + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b + S(2)c + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(-ac + bS(2) + bc - (-S(3)abc - S(2)acS(2) + bS(3) + bS(2)c)/sqrt(-S(4)ac + b*S(2)))atanh(sqrt(S(2))sqrt(c)sqrt(-x*S(2) + S(1))/sqrt(b + S(2)c - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b + S(2)c - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(-x*S(2) + S(1))/(a + bx*S(2) + cxS(4)), x), x, sqrt(-xS(2) + S(1))/c - sqrt(S(2))(b + c - (-S(2)ac + bS(2) + bc)/sqrt(-S(4)ac + b*S(2)))atanh(sqrt(S(2))sqrt(c)sqrt(-x*S(2) + S(1))/sqrt(b + S(2)c - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b + S(2)c - sqrt(-S(4)ac + bS(2)))) - sqrt(S(2))(b + c + (-S(2)ac + b*S(2) + bc)/sqrt(-S(4)ac + b*S(2)))atanh(sqrt(S(2))sqrt(c)sqrt(-x*S(2) + S(1))/sqrt(b + S(2)c + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b + S(2)c + sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(-x*S(2) + S(1))/(a + bx*S(2) + cx*S(4)), x), x, -sqrt(S(2))sqrt(b + S(2)c - sqrt(-S(4)ac + bS(2)))atanh(sqrt(S(2))sqrt(c)sqrt(-x*S(2) + S(1))/sqrt(b + S(2)c - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(-S(4)ac + bS(2))) + sqrt(S(2))sqrt(b + S(2)c + sqrt(-S(4)ac + bS(2)))atanh(sqrt(S(2))sqrt(c)sqrt(-x*S(2) + S(1))/sqrt(b + S(2)c + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-xS(2) + S(1))/(x(a + bxS(2) + cx*S(4))), x), x, -sqrt(S(2))sqrt(c)(S(2)a + b - sqrt(-S(4)ac + b*S(2)))atanh(sqrt(S(2))sqrt(c)sqrt(-x*S(2) + S(1))/sqrt(b + S(2)c + sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(-S(4)ac + bS(2))sqrt(b + S(2)c + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))sqrt(c)(S(2)a + b + sqrt(-S(4)ac + b*S(2)))atanh(sqrt(S(2))sqrt(c)sqrt(-x*S(2) + S(1))/sqrt(b + S(2)c - sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(-S(4)ac + bS(2))sqrt(b + S(2)c - sqrt(-S(4)ac + bS(2)))) - atanh(sqrt(-xS(2) + S(1)))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-xS(2) + S(1))/(xS(3)(a + bxS(2) + cx*S(4))), x), x, S(1)/(S(4)a(sqrt(-xS(2) + S(1)) + S(1))) - S(1)/(S(4)a(-sqrt(-xS(2) + S(1)) + S(1))) + sqrt(S(2))sqrt(c)(a(b - S(2)c - sqrt(-S(4)ac + bS(2))) + b(b - sqrt(-S(4)ac + b*S(2))))atanh(sqrt(S(2))sqrt(c)sqrt(-x*S(2) + S(1))/sqrt(b + S(2)c + sqrt(-S(4)ac + b*S(2))))/(S(2)a*S(2)sqrt(-S(4)ac + b*S(2))sqrt(b + S(2)c + sqrt(-S(4)ac + bS(2)))) - sqrt(S(2))sqrt(c)(a(b - S(2)c + sqrt(-S(4)ac + bS(2))) + b(b + sqrt(-S(4)ac + b*S(2))))atanh(sqrt(S(2))sqrt(c)sqrt(-x*S(2) + S(1))/sqrt(b + S(2)c - sqrt(-S(4)ac + b*S(2))))/(S(2)a*S(2)sqrt(-S(4)ac + b*S(2))sqrt(b + S(2)c - sqrt(-S(4)ac + bS(2)))) + (a + S(2)b)atanh(sqrt(-xS(2) + S(1)))/(S(2)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)sqrt(-xS(2) + S(1))/(a + bx*S(2) + cx*S(4)), x), x, xsqrt(-x*S(2) + S(1))/(S(2)c) + (S(2)b + c)asin(x)/(S(2)cS(2)) - (-ac + b*S(2) + bc + (-S(3)abc - S(2)acS(2) + bS(3) + bS(2)c)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(b + S(2)c + sqrt(-S(4)ac + b*S(2)))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-xS(2) + S(1))))/(cS(2)sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(b + S(2)c + sqrt(-S(4)ac + bS(2)))) - (-ac + b*S(2) + bc - (-S(3)abc - S(2)acS(2) + bS(3) + bS(2)c)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(b + S(2)c - sqrt(-S(4)ac + b*S(2)))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-xS(2) + S(1))))/(cS(2)sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(b + S(2)c - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(-x*S(2) + S(1))/(a + bx*S(2) + cx*S(4)), x), x, -asin(x)/c + (b + c + (-S(2)ac + bS(2) + bc)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(b + S(2)c + sqrt(-S(4)ac + b*S(2)))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-x*S(2) + S(1))))/(csqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(b + S(2)c + sqrt(-S(4)ac + bS(2)))) + (b + c - (-S(2)ac + bS(2) + bc)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(b + S(2)c - sqrt(-S(4)ac + b*S(2)))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-x*S(2) + S(1))))/(csqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(b + S(2)c - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-xS(2) + S(1))/(a + bx*S(2) + cx*S(4)), x), x, -sqrt(b + S(2)c + sqrt(-S(4)ac + b*S(2)))atan(xsqrt(b + S(2)c + sqrt(-S(4)ac + b*S(2)))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-x*S(2) + S(1))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))) + sqrt(b + S(2)c - sqrt(-S(4)ac + b*S(2)))atan(xsqrt(b + S(2)c - sqrt(-S(4)ac + b*S(2)))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-x*S(2) + S(1))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-xS(2) + S(1))/(x*S(2)(a + bxS(2) + cx*S(4))), x), x, -c(-(S(2)a + b)/sqrt(-S(4)ac + bS(2)) + S(1))atan(xsqrt(b + S(2)c + sqrt(-S(4)ac + b*S(2)))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-x*S(2) + S(1))))/(asqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(b + S(2)c + sqrt(-S(4)ac + bS(2)))) - c((S(2)a + b)/sqrt(-S(4)ac + bS(2)) + S(1))atan(xsqrt(b + S(2)c - sqrt(-S(4)ac + b*S(2)))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-x*S(2) + S(1))))/(asqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(b + S(2)c - sqrt(-S(4)ac + bS(2)))) - sqrt(-xS(2) + S(1))/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)sqrt(-xS(2) + S(1))/(xS(4) + x*S(2) + S(-1)), x), x, -asin(x) + sqrt(S(2)/5 + sqrt(S(5))/S(5))atan(xsqrt(S(1)/2 + sqrt(S(5))/S(2))/sqrt(-xS(2) + S(1))) - sqrt(S(-2)/5 + sqrt(S(5))/S(5))atanh(xsqrt(S(-1)/2 + sqrt(S(5))/S(2))/sqrt(-xS(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(sqrt(d + ex*S(2))(a + bxS(2) + cx*S(4))), x), x, bdatanh(sqrt(e)x/sqrt(d + exS(2)))/(S(2)c*S(2)e*(S(3)/2)) - bxsqrt(d + ex*S(2))/(S(2)c*S(2)e) + S(3)dS(2)atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(8)ce(S(5)/2)) - S(3)dxsqrt(d + exS(2))/(S(8)ceS(2)) + xS(3)sqrt(d + exS(2))/(S(4)ce) - (-S(2)abc + b*S(3) + (S(2)a*S(2)c*S(2) - S(4)abS(2)c + b*S(4))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(cS(3)sqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) - (-S(2)abc + b*S(3) - (S(2)a*S(2)c*S(2) - S(4)abS(2)c + b*S(4))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(cS(3)sqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) + (-ac + b*S(2))atanh(sqrt(e)x/sqrt(d + exS(2)))/(cS(3)sqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(sqrt(d + ex*S(2))(a + bxS(2) + cx*S(4))), x), x, -batanh(sqrt(e)x/sqrt(d + exS(2)))/(cS(2)sqrt(e)) - datanh(sqrt(e)x/sqrt(d + ex*S(2)))/(S(2)ce(S(3)/2)) + xsqrt(d + exS(2))/(S(2)ce) + (-ac + b*S(2) + b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(cS(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + (-ac + b*S(2) - b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(cS(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(sqrt(d + ex*S(2))(a + bxS(2) + cx*S(4))), x), x, -(b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(csqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) - (b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(csqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + atanh(sqrt(e)x/sqrt(d + exS(2)))/(csqrt(e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(sqrt(d + ex*S(2))(a + bxS(2) + cx*S(4))), x), x, -sqrt(b - sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(sqrt(-S(4)ac + bS(2))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) + sqrt(b + sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(sqrt(-S(4)ac + bS(2))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(d + ex*S(2))(a + bxS(2) + cx*S(4))), x), x, -S(2)catan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + S(2)catan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(d + exS(2))(a + bxS(2) + cx*S(4))), x), x, -c(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(asqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) - c(b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(asqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) - sqrt(d + ex*S(2))/(adx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)sqrt(d + exS(2))(a + bxS(2) + cx*S(4))), x), x, -sqrt(d + ex*S(2))/(S(3)adx*S(3)) + S(2)esqrt(d + ex*S(2))/(S(3)adS(2)x) + bsqrt(d + exS(2))/(aS(2)dx) + c(b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(aS(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + c(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(aS(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(6)sqrt(d + exS(2))(a + bxS(2) + cx*S(4))), x), x, -sqrt(d + ex*S(2))/(S(5)adx*S(5)) + S(4)esqrt(d + ex*S(2))/(S(15)adS(2)x*S(3)) - S(8)e*S(2)sqrt(d + exS(2))/(S(15)adS(3)x) + bsqrt(d + ex*S(2))/(S(3)a*S(2)dxS(3)) - S(2)besqrt(d + exS(2))/(S(3)a*S(2)d*S(2)x) - c(-ac + b*S(2) - b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(aS(3)sqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) - c(-ac + bS(2) + b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(aS(3)sqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) - sqrt(d + ex*S(2))(-ac + bS(2))/(aS(3)dx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/((d + exS(2))(S(3)/2)(a + bx*S(2) + cxS(4))), x), x, -dS(2)x/(esqrt(d + exS(2))(aeS(2) - bde + cd*S(2))) + (-S(2)ac + S(2)b*S(2) + S(2)b(-S(3)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(csqrt(b + sqrt(-S(4)ac + b*S(2)))(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))(S(3)/2)) + (-S(2)ac + S(2)b*S(2) - S(2)b(-S(3)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(csqrt(b - sqrt(-S(4)ac + b*S(2)))(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))(S(3)/2)) + atanh(sqrt(e)x/sqrt(d + exS(2)))/(ce(S(3)/2)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(6)/((d + exS(2))(S(3)/2)(a + bxS(2) + cxS(4))), x), x, -dS(2)x/(esqrt(d + exS(2))(aeS(2) - bde + cdS(2))) + dS(2)atanh(sqrt(e)x/sqrt(d + exS(2)))/(e(S(3)/2)(aeS(2) - bde + cd*S(2))) + (-abe - acd + bS(2)d + (S(2)aS(2)ce - ab*S(2)e - S(3)abcd + b*S(3)d)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(csqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))(aeS(2) - bde + cd*S(2))) + (-abe - acd + bS(2)d - (S(2)aS(2)ce - ab*S(2)e - S(3)abcd + b*S(3)d)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(csqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))(aeS(2) - bde + cd*S(2))) - (-ae + bd)atanh(sqrt(e)x/sqrt(d + ex*S(2)))/(csqrt(e)(ae*S(2) - bde + cdS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/((d + exS(2))(S(3)/2)(a + bxS(2) + cx*S(4))), x), x, dx/(sqrt(d + exS(2))(aeS(2) - bde + cd*S(2))) - (-ae + bd + (-abe - S(2)acd + b*S(2)d)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))(aeS(2) - bde + cd*S(2))) - (-ae + bd - (-abe - S(2)acd + b*S(2)d)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))(aeS(2) - bde + cdS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((d + exS(2))(S(3)/2)(a + bxS(2) + cx*S(4))), x), x, c(d + (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))(aeS(2) - bde + cd*S(2))) + c(d - (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))(aeS(2) - bde + cd*S(2))) - ex/(sqrt(d + exS(2))(aeS(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + exS(2))(S(3)/2)(a + bx*S(2) + cx*S(4))), x), x, -c(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))(aeS(2) - bde + cd*S(2))) - c(e - (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))(aeS(2) - bde + cdS(2))) + eS(2)x/(dsqrt(d + exS(2))(aeS(2) - bde + cdS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(d + exS(2))(S(3)/2)(a + bx*S(2) + cx*S(4))), x), x, -S(2)c*S(2)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(asqrt(b + sqrt(-S(4)ac + b*S(2)))(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))(S(3)/2)) - S(2)c*S(2)(b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(asqrt(b - sqrt(-S(4)ac + b*S(2)))(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))(S(3)/2)) + ex(-be + cd)/(adsqrt(d + ex*S(2))(cdS(2) + e(ae - bd))) + (-d - S(2)ex*S(2))/(ad*S(2)xsqrt(d + exS(2))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/(xS(2)(d + exS(2))(S(3)/2)(a + bx*S(2) + cxS(4))), x), x, -eS(2)/(dxsqrt(d + exS(2))(aeS(2) - bde + cd*S(2))) - S(2)e*S(3)x/(d*S(2)sqrt(d + exS(2))(aeS(2) - bde + cd*S(2))) - c(-be + cd - (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(asqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))(aeS(2) - bde + cd*S(2))) - c(-be + cd + (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(asqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))(aeS(2) - bde + cd*S(2))) - sqrt(d + ex*S(2))(-be + cd)/(adx(ae*S(2) - bde + cdS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(d + exS(2))(S(3)/2)(a + bx*S(2) + cx*S(4))), x), x, -S(1)/(S(3)adx*S(3)sqrt(d + exS(2))) + S(2)c*S(2)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(aS(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))(S(3)/2)) + S(2)c*S(2)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(aS(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))(S(3)/2)) - ex(ace - bS(2)e + bcd)/(a*S(2)dsqrt(d + ex*S(2))(cdS(2) + e(ae - bd))) + (S(4)ae + S(3)bd)/(S(3)aS(2)d*S(2)xsqrt(d + ex*S(2))) + S(2)ex(S(4)ae + S(3)bd)/(S(3)aS(2)d*S(3)sqrt(d + exS(2))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/(xS(4)(d + exS(2))(S(3)/2)(a + bxS(2) + cxS(4))), x), x, -eS(2)/(S(3)dx*S(3)sqrt(d + exS(2))(aeS(2) - bde + cd*S(2))) + S(4)e*S(3)/(S(3)d*S(2)xsqrt(d + ex*S(2))(aeS(2) - bde + cd*S(2))) + S(8)e*S(4)x/(S(3)dS(3)sqrt(d + exS(2))(aeS(2) - bde + cd*S(2))) - sqrt(d + ex*S(2))(-be + cd)/(S(3)adxS(3)(aeS(2) - bde + cd*S(2))) + S(2)esqrt(d + ex*S(2))(-be + cd)/(S(3)ad*S(2)x(ae*S(2) - bde + cd*S(2))) + c(ace - b*S(2)e + bcd - (S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(aS(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))(aeS(2) - bde + cd*S(2))) + c(ace - b*S(2)e + bcd + (S(3)abce - S(2)ac*S(2)d - b*S(3)e + b*S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(xsqrt(S(2)cd - e(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(d + exS(2))))/(aS(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))sqrt(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))(aeS(2) - bde + cd*S(2))) + sqrt(d + ex*S(2))(ace - b*S(2)e + bcd)/(a*S(2)dx(aeS(2) - bde + cd*S(2))), expand=True, _diff=True, _numerical=True) # '''Apart # assert rubi_test(rubi_integrate((fx)*m(d + exS(2))q/(a + bx*S(2) + cx*S(4)), x), x, -S(2)c(fx)*(m + S(1))(S(1) + exS(2)/d)(-q)(d + exS(2))qAppellF1(m/S(2) + S(1)/2, S(1), -q, m/S(2) + S(3)/2, -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))), -ex*S(2)/d)/(f(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))sqrt(-S(4)ac + bS(2))) + S(2)c(fx)*(m + S(1))(S(1) + exS(2)/d)(-q)(d + exS(2))qAppellF1(m/S(2) + S(1)/2, S(1), -q, m/S(2) + S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -ex*S(2)/d)/(f(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) # assert rubi_test(rubi_integrate(xS(7)(d + exS(2))q/(a + bx*S(2) + cx*S(4)), x), x, (d + exS(2))(q + S(1))(a - bS(2)/c - b(-S(3)ac + b*S(2))/(csqrt(-S(4)ac + b*S(2))))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exS(2))/(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)c(q + S(1))(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + (d + exS(2))(q + S(1))(a - bS(2)/c + b(-S(3)ac + b*S(2))/(csqrt(-S(4)ac + b*S(2))))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exS(2))/(-be + S(2)cd + esqrt(-S(4)ac + bS(2))))/(S(2)c(q + S(1))(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) + (d + exS(2))(q + S(2))/(S(2)ce*S(2)(q + S(2))) - (d + exS(2))(q + S(1))(be + cd)/(S(2)cS(2)e*S(2)(q + S(1))), expand=True, _diff=True, _numerical=True) # assert rubi_test(rubi_integrate(x*S(5)(d + exS(2))q/(a + bx*S(2) + cx*S(4)), x), x, (b - (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))(d + exS(2))(q + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exS(2))/(-be + S(2)cd + esqrt(-S(4)ac + bS(2))))/(S(2)c(q + S(1))(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) + (b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))(d + exS(2))(q + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exS(2))/(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)c(q + S(1))(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + (d + exS(2))(q + S(1))/(S(2)ce(q + S(1))), expand=True, _diff=True, _numerical=True) # assert rubi_test(rubi_integrate(xS(3)(d + exS(2))q/(a + bx*S(2) + cx*S(4)), x), x, -(d + exS(2))(q + S(1))(-b/sqrt(-S(4)ac + bS(2)) + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exS(2))/(-be + S(2)cd + esqrt(-S(4)ac + bS(2))))/((q + S(1))(S(4)cd - S(2)e(b - sqrt(-S(4)ac + b*S(2))))) - (d + exS(2))(q + S(1))(b/sqrt(-S(4)ac + bS(2)) + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exS(2))/(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/((q + S(1))(S(4)cd - S(2)e(b + sqrt(-S(4)ac + b*S(2))))), expand=True, _diff=True, _numerical=True) # NC assert rubi_test(rubi_integrate(x(d + exS(2))q/(a + bx*S(2) + cx*S(4)), x), x, c(d + exS(2))(q + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exS(2))/(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/((q + S(1))sqrt(-S(4)ac + b*S(2))(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) - c(d + exS(2))(q + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exS(2))/(-be + S(2)cd + esqrt(-S(4)ac + bS(2))))/((q + S(1))sqrt(-S(4)ac + b*S(2))(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) # NC assert rubi_test(rubi_integrate((d + exS(2))q/(x(a + bx*S(2) + cx*S(4))), x), x, c(d + exS(2))(q + S(1))(-b/sqrt(-S(4)ac + b*S(2)) + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exS(2))/(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)a(q + S(1))(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + c(d + exS(2))(q + S(1))(b/sqrt(-S(4)ac + b*S(2)) + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exS(2))/(-be + S(2)cd + esqrt(-S(4)ac + bS(2))))/(S(2)a(q + S(1))(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) - (d + exS(2))(q + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(1) + ex*S(2)/d)/(S(2)ad(q + S(1))), expand=True, _diff=True, _numerical=True) # NC assert rubi_test(rubi_integrate((d + exS(2))q/(xS(3)(a + bxS(2) + cx*S(4))), x), x, e(d + exS(2))(q + S(1))hyper((S(2), q + S(1)), (q + S(2),), S(1) + exS(2)/d)/(S(2)adS(2)(q + S(1))) + b(d + exS(2))(q + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(1) + ex*S(2)/d)/(S(2)a*S(2)d(q + S(1))) - c(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))(d + exS(2))(q + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exS(2))/(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(S(2)a*S(2)(q + S(1))(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2))))) - c(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))(d + exS(2))(q + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exS(2))/(-be + S(2)cd + esqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)(q + S(1))(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))), expand=True, _diff=True, _numerical=True) # NC assert rubi_test(rubi_integrate(xS(6)(d + exS(2))q/(a + bxS(2) + cx*S(4)), x), x, -bx(S(1) + exS(2)/d)(-q)(d + exS(2))qhyper((S(1)/2, -q), (S(3)/2,), -exS(2)/d)/cS(2) + x*S(3)(S(1) + exS(2)/d)(-q)(d + exS(2))qhyper((S(3)/2, -q), (S(5)/2,), -exS(2)/d)/(S(3)c) + x(S(1) + exS(2)/d)(-q)(d + exS(2))q(-ac + b*S(2) + b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))AppellF1(S(1)/2, S(1), -q, S(3)/2, -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))), -exS(2)/d)/(cS(2)(b + sqrt(-S(4)ac + bS(2)))) + x(S(1) + exS(2)/d)(-q)(d + exS(2))q(-ac + bS(2) - b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))AppellF1(S(1)/2, S(1), -q, S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -exS(2)/d)/(cS(2)(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) # NC assert rubi_test(rubi_integrate(xS(4)(d + exS(2))q/(a + bx*S(2) + cx*S(4)), x), x, -x(S(1) + exS(2)/d)(-q)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))(d + exS(2))qAppellF1(S(1)/2, S(1), -q, S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -ex*S(2)/d)/(c(b - sqrt(-S(4)ac + b*S(2)))) - x(S(1) + exS(2)/d)(-q)(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))(d + exS(2))qAppellF1(S(1)/2, S(1), -q, S(3)/2, -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))), -ex*S(2)/d)/(c(b + sqrt(-S(4)ac + b*S(2)))) + x(S(1) + exS(2)/d)(-q)(d + exS(2))qhyper((S(1)/2, -q), (S(3)/2,), -exS(2)/d)/c, expand=True, _diff=True, _numerical=True) # NC assert rubi_test(rubi_integrate(xS(2)(d + exS(2))q/(a + bx*S(2) + cx*S(4)), x), x, -x(S(1) + exS(2)/d)(-q)(d + exS(2))qAppellF1(S(1)/2, S(1), -q, S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -ex*S(2)/d)/sqrt(-S(4)ac + bS(2)) + x(S(1) + exS(2)/d)(-q)(d + exS(2))qAppellF1(S(1)/2, S(1), -q, S(3)/2, -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))), -ex*S(2)/d)/sqrt(-S(4)ac + bS(2)), expand=True, _diff=True, _numerical=True) # NC assert rubi_test(rubi_integrate((d + exS(2))q/(a + bxS(2) + cx*S(4)), x), x, -S(2)cx(S(1) + exS(2)/d)(-q)(d + exS(2))qAppellF1(S(1)/2, S(1), -q, S(3)/2, -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))), -ex*S(2)/d)/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - S(2)cx(S(1) + exS(2)/d)(-q)(d + exS(2))qAppellF1(S(1)/2, S(1), -q, S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -ex*S(2)/d)/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) # NC assert rubi_test(rubi_integrate((d + exS(2))q/(x*S(2)(a + bxS(2) + cx*S(4))), x), x, -cx(S(1) + exS(2)/d)(-q)(d + exS(2))q(-b/sqrt(-S(4)ac + bS(2)) + S(1))AppellF1(S(1)/2, S(1), -q, S(3)/2, -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))), -ex*S(2)/d)/(a(b + sqrt(-S(4)ac + b*S(2)))) - cx(S(1) + exS(2)/d)(-q)(d + exS(2))q(b/sqrt(-S(4)ac + bS(2)) + S(1))AppellF1(S(1)/2, S(1), -q, S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -ex*S(2)/d)/(a(b - sqrt(-S(4)ac + b*S(2)))) - (S(1) + exS(2)/d)(-q)(d + exS(2))qhyper((S(-1)/2, -q), (S(1)/2,), -ex*S(2)/d)/(ax), expand=True, _diff=True, _numerical=True) # NC assert rubi_test(rubi_integrate((d + exS(2))q/(xS(4)(a + bxS(2) + cx*S(4))), x), x, -(S(1) + exS(2)/d)(-q)(d + exS(2))qhyper((S(-3)/2, -q), (S(-1)/2,), -ex*S(2)/d)/(S(3)axS(3)) + b(S(1) + exS(2)/d)(-q)(d + exS(2))qhyper((S(-1)/2, -q), (S(1)/2,), -exS(2)/d)/(aS(2)x) + cx(S(1) + exS(2)/d)(-q)(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))(d + exS(2))qAppellF1(S(1)/2, S(1), -q, S(3)/2, -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))), -exS(2)/d)/(aS(2)(b + sqrt(-S(4)ac + bS(2)))) + cx(S(1) + exS(2)/d)(-q)(b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))(d + exS(2))qAppellF1(S(1)/2, S(1), -q, S(3)/2, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -exS(2)/d)/(aS(2)(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) # NC assert rubi_test(rubi_integrate((d + ex*S(3))/(a + cx*S(6)), x), x, -(sqrt(c)d - esqrt(-a))log(c*(S(1)/6)x + (-a)*(S(1)/6))/(S(6)c*(S(2)/3)(-a)*(S(5)/6)) + (sqrt(c)d - esqrt(-a))log(-c*(S(1)/6)x(-a)(S(1)/6) + c(S(1)/3)xS(2) + (-a)(S(1)/3))/(S(12)c(S(2)/3)(-a)*(S(5)/6)) + sqrt(S(3))(sqrt(c)d - esqrt(-a))atan(sqrt(S(3))(-S(2)c(S(1)/6)x/(-a)*(S(1)/6) + S(1))/S(3))/(S(6)c*(S(2)/3)(-a)*(S(5)/6)) + (sqrt(c)d + esqrt(-a))log(-c*(S(1)/6)x + (-a)*(S(1)/6))/(S(6)c*(S(2)/3)(-a)*(S(5)/6)) - (sqrt(c)d + esqrt(-a))log(c*(S(1)/6)x(-a)(S(1)/6) + c(S(1)/3)xS(2) + (-a)(S(1)/3))/(S(12)c(S(2)/3)(-a)*(S(5)/6)) - sqrt(S(3))(sqrt(c)d + esqrt(-a))atan(sqrt(S(3))(S(2)c(S(1)/6)x/(-a)*(S(1)/6) + S(1))/S(3))/(S(6)c*(S(2)/3)(-a)*(S(5)/6)), expand=True, _diff=True, _numerical=True) # NC assert rubi_test(rubi_integrate((d + ex*S(3))/(a - cx*S(6)), x), x, (-sqrt(a)e + sqrt(c)d)log(a(S(1)/6) + c(S(1)/6)x)/(S(6)a*(S(5)/6)c*(S(2)/3)) - (-sqrt(a)e + sqrt(c)d)log(-a*(S(1)/6)c*(S(1)/6)x + a(S(1)/3) + c(S(1)/3)xS(2))/(S(12)a*(S(5)/6)c*(S(2)/3)) - sqrt(S(3))(-sqrt(a)e + sqrt(c)d)atan(sqrt(S(3))(a*(S(1)/6) - S(2)c*(S(1)/6)x)/(S(3)a(S(1)/6)))/(S(6)a*(S(5)/6)c*(S(2)/3)) - (sqrt(a)e + sqrt(c)d)log(a(S(1)/6) - c(S(1)/6)x)/(S(6)a*(S(5)/6)c*(S(2)/3)) + (sqrt(a)e + sqrt(c)d)log(a*(S(1)/6)c*(S(1)/6)x + a(S(1)/3) + c(S(1)/3)xS(2))/(S(12)a*(S(5)/6)c*(S(2)/3)) + sqrt(S(3))(sqrt(a)e + sqrt(c)d)atan(sqrt(S(3))(a*(S(1)/6) + S(2)c*(S(1)/6)x)/(S(3)a(S(1)/6)))/(S(6)a*(S(5)/6)c*(S(2)/3)), expand=True, _diff=True, _numerical=True) # nc assert rubi_test(rubi_integrate((d + exS(3))S(5)(a + bx*S(3) + cx*S(6)), x), x, ad*S(5)x + ceS(5)xS(22)/S(22) + dS(4)xS(4)(S(5)ae + bd)/S(4) + dS(3)x*S(7)(cdS(2) + S(5)e(S(2)ae + bd))/S(7) + d*S(2)exS(10)(cdS(2) + S(2)e(ae + bd))/S(2) + S(5)deS(2)x*S(13)(S(2)cd*S(2) + e(ae + S(2)bd))/S(13) + eS(4)x*S(19)(be + S(5)cd)/S(19) + eS(3)x*S(16)(S(10)cd*S(2) + e(ae + S(5)bd))/S(16), expand=True, _diff=True, _numerical=True) # nc assert rubi_test(rubi_integrate((d + exS(3))S(4)(a + bx*S(3) + cx*S(6)), x), x, ad*S(4)x + ceS(4)xS(19)/S(19) + dS(3)xS(4)(S(4)ae + bd)/S(4) + dS(2)x*S(7)(S(6)ae*S(2) + S(4)bde + cdS(2))/S(7) + dexS(10)(S(2)cd*S(2) + e(S(2)ae + S(3)bd))/S(5) + e*S(3)x*S(16)(be + S(4)cd)/S(16) + eS(2)x*S(13)(S(6)cd*S(2) + e(ae + S(4)bd))/S(13), expand=True, _diff=True, _numerical=True) # nc assert rubi_test(rubi_integrate((d + exS(3))S(3)(a + bx*S(3) + cx*S(6)), x), x, ad*S(3)x + ceS(3)xS(16)/S(16) + dS(2)xS(4)(S(3)ae + bd)/S(4) + dx*S(7)(cdS(2) + S(3)e(ae + bd))/S(7) + eS(2)x*S(13)(be + S(3)cd)/S(13) + ex*S(10)(S(3)cd*S(2) + e(ae + S(3)bd))/S(10), expand=True, _diff=True, _numerical=True) # ncassert rubi_test(rubi_integrate((d + exS(3))S(2)(a + bx*S(3) + cx*S(6)), x), x, ad*S(2)x + ceS(2)x*S(13)/S(13) + dx*S(4)(S(2)ae + bd)/S(4) + ex*S(10)(be + S(2)cd)/S(10) + xS(7)(cdS(2) + e(ae + S(2)bd))/S(7), expand=True, _diff=True, _numerical=True) # nc assert rubi_test(rubi_integrate((d + ex*S(3))(a + bxS(3) + cx*S(6)), x), x, adx + cexS(10)/S(10) + xS(7)(be + cd)/S(7) + x*S(4)(ae + bd)/S(4), expand=True, _diff=True, _numerical=True) # nc assert rubi_test(rubi_integrate((a + bxS(3) + cx*S(6))/(d + ex*S(3)), x), x, cx*S(4)/(S(4)e) - x(-be + cd)/eS(2) + (ae*S(2) - bde + cd*S(2))log(d(S(1)/3) + e(S(1)/3)x)/(S(3)d*(S(2)/3)e*(S(7)/3)) - (ae*S(2) - bde + cd*S(2))log(d(S(2)/3) - d(S(1)/3)e(S(1)/3)x + e*(S(2)/3)x*S(2))/(S(6)d*(S(2)/3)e*(S(7)/3)) - sqrt(S(3))(aeS(2) - bde + cd*S(2))atan(sqrt(S(3))(d(S(1)/3) - S(2)e*(S(1)/3)x)/(S(3)d(S(1)/3)))/(S(3)d*(S(2)/3)e*(S(7)/3)), expand=True, _diff=True, _numerical=True) # nc assert rubi_test(rubi_integrate((a + bx*S(3) + cx*S(6))/(d + exS(3))S(2), x), x, cx/eS(2) + x(aeS(2) - bde + cd*S(2))/(S(3)deS(2)(d + exS(3))) - (S(4)cdS(2) - e(S(2)ae + bd))log(d(S(1)/3) + e(S(1)/3)x)/(S(9)d*(S(5)/3)e*(S(7)/3)) + (S(4)cdS(2) - e(S(2)ae + bd))log(d(S(2)/3) - d(S(1)/3)e(S(1)/3)x + e*(S(2)/3)x*S(2))/(S(18)d*(S(5)/3)e*(S(7)/3)) + sqrt(S(3))(S(4)cd*S(2) - e(S(2)ae + bd))atan(sqrt(S(3))(d(S(1)/3) - S(2)e*(S(1)/3)x)/(S(3)d(S(1)/3)))/(S(9)d*(S(5)/3)e*(S(7)/3)), expand=True, _diff=True, _numerical=True) # nc assert rubi_test(rubi_integrate((a + bx*S(3) + cx*S(6))/(d + exS(3))S(3), x), x, x(ae*S(2) - bde + cd*S(2))/(S(6)deS(2)(d + exS(3))S(2)) - x(S(7)cd*S(2) - e(S(5)ae + bd))/(S(18)d*S(2)e*S(2)(d + exS(3))) + (S(2)cdS(2) + e(S(5)ae + bd))log(d(S(1)/3) + e(S(1)/3)x)/(S(27)d*(S(8)/3)e*(S(7)/3)) - (S(2)cdS(2) + e(S(5)ae + bd))log(d(S(2)/3) - d(S(1)/3)e(S(1)/3)x + e*(S(2)/3)x*S(2))/(S(54)d*(S(8)/3)e*(S(7)/3)) - sqrt(S(3))(S(2)cd*S(2) + e(S(5)ae + bd))atan(sqrt(S(3))(d(S(1)/3) - S(2)e*(S(1)/3)x)/(S(3)d(S(1)/3)))/(S(27)d*(S(8)/3)e(S(7)/3)), expand=True, _diff=True, _numerical=True) # ''' assert rubi_test(rubi_integrate(xS(8)(d + ex*S(3))/(a + bx*S(3) + cx*S(6)), x), x, ex*S(6)/(S(6)c) + x*S(3)(-be + cd)/(S(3)cS(2)) - (ace - bS(2)e + bcd)log(a + bx*S(3) + cx*S(6))/(S(6)c*S(3)) - (S(3)abce - S(2)acS(2)d - b*S(3)e + b*S(2)cd)atanh((b + S(2)cx*S(3))/sqrt(-S(4)ac + bS(2)))/(S(3)c*S(3)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(d + ex*S(3))/(a + bx*S(3) + cx*S(6)), x), x, ex*S(3)/(S(3)c) + (-be + cd)log(a + bx*S(3) + cx*S(6))/(S(6)c*S(2)) + (S(2)ace - b*S(2)e + bcd)atanh((b + S(2)cxS(3))/sqrt(-S(4)ac + bS(2)))/(S(3)c*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(d + ex*S(3))/(a + bx*S(3) + cx*S(6)), x), x, elog(a + bxS(3) + cx*S(6))/(S(6)c) - (-be + S(2)cd)atanh((b + S(2)cx*S(3))/sqrt(-S(4)ac + bS(2)))/(S(3)csqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(3))/(x(a + bxS(3) + cx*S(6))), x), x, dlog(x)/a - dlog(a + bx*S(3) + cx*S(6))/(S(6)a) + (-S(2)ae + bd)atanh((b + S(2)cx*S(3))/sqrt(-S(4)ac + bS(2)))/(S(3)asqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(3))/(xS(4)(a + bx*S(3) + cx*S(6))), x), x, -d/(S(3)axS(3)) - (-ae + bd)log(x)/a*S(2) + (-ae + bd)log(a + bxS(3) + cx*S(6))/(S(6)a*S(2)) - (-abe - S(2)acd + b*S(2)d)atanh((b + S(2)cxS(3))/sqrt(-S(4)ac + bS(2)))/(S(3)a*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(d + ex*S(3))/(a + bx*S(3) + cx*S(6)), x), x, ex*S(2)/(S(2)c) - S(2)*(S(1)/3)(-be + cd + (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(5)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)*(S(1)/3)(-be + cd + (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(5)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)sqrt(S(3))(-be + cd + (S(2)ace - b*S(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(5)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)(-be + cd - (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(5)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)*(S(1)/3)(-be + cd - (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(5)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)sqrt(S(3))(-be + cd - (S(2)ace - b*S(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(5)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(d + exS(3))/(a + bx*S(3) + cx*S(6)), x), x, ex/c + S(2)*(S(2)/3)(-be + cd + (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(4)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)(-be + cd + (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(4)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)sqrt(S(3))(-be + cd + (S(2)ace - b*S(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(4)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)(-be + cd - (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(4)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)(-be + cd - (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(4)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)sqrt(S(3))(-be + cd - (S(2)ace - b*S(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(4)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(d + ex*S(3))/(a + bx*S(3) + cxS(6)), x), x, -S(2)(S(1)/3)(e - (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(2)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)*(S(1)/3)(e - (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(2)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)sqrt(S(3))(e - (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(2)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(2)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)*(S(1)/3)(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(2)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)sqrt(S(3))(e + (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(2)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(3))/(a + bx*S(3) + cxS(6)), x), x, S(2)(S(2)/3)(e - (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(1)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)(e - (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(1)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)sqrt(S(3))(e - (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(1)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(1)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(1)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)sqrt(S(3))(e + (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(1)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(3))/(xS(2)(a + bxS(3) + cxS(6))), x), x, S(2)(S(1)/3)c(S(1)/3)(d - (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)a(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)c*(S(1)/3)(d - (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)a(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)(S(1)/3)sqrt(S(3))c(S(1)/3)(d - (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)a(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)*(S(1)/3)c*(S(1)/3)(d + (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)a(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)c*(S(1)/3)(d + (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)a(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)(S(1)/3)sqrt(S(3))c(S(1)/3)(d + (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)a(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) - d/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(3))/(xS(3)(a + bx*S(3) + cxS(6))), x), x, -S(2)(S(2)/3)c(S(2)/3)(d - (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)a(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)c*(S(2)/3)(d - (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)a(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)(S(2)/3)sqrt(S(3))c(S(2)/3)(d - (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)a(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)c*(S(2)/3)(d + (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)a(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)c*(S(2)/3)(d + (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)a(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)(S(2)/3)sqrt(S(3))c(S(2)/3)(d + (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)a(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - d/(S(2)axS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)(-xS(3) + S(1))/(xS(6) - xS(3) + S(1)), x), x, -xS(6)/S(6) + log(xS(6) - xS(3) + S(1))/S(6) - sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(3) + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(-xS(3) + S(1))/(xS(6) - xS(3) + S(1)), x), x, -xS(3)/S(3) - S(2)sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(3) + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(-xS(3) + S(1))/(xS(6) - xS(3) + S(1)), x), x, -log(xS(6) - x*S(3) + S(1))/S(6) - sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(3) + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(3) + S(1))/(x(xS(6) - xS(3) + S(1))), x), x, log(x) - log(xS(6) - xS(3) + S(1))/S(6) + sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(3) + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(3) + S(1))/(x*S(4)(xS(6) - xS(3) + S(1))), x), x, S(2)sqrt(S(3))atan(sqrt(S(3))(-S(2)x*S(3) + S(1))/S(3))/S(9) - S(1)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)(-xS(3) + S(1))/(xS(6) - xS(3) + S(1)), x), x, -xS(4)/S(4) + S(2)(S(2)/3)(S(3) + sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(S(3) - sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(S(3) + sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) - sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(S(3) - sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) + sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) - sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(-sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) + sqrt(S(3))I)(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(-xS(3) + S(1))/(xS(6) - xS(3) + S(1)), x), x, -xS(2)/S(2) + sqrt(S(3))Ilog(-S(2)*(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(9)(S(1)/2 - sqrt(S(3))I/S(2))(S(1)/3)) - sqrt(S(3))Ilog(-S(2)(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(9)(S(1)/2 + sqrt(S(3))I/S(2))(S(1)/3)) - S(2)(S(1)/3)sqrt(S(3))Ilog(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)sqrt(S(3))Ilog(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(1)/3)) + Iatan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))(S(1)/3) + S(1))/S(3))/(S(3)(S(1)/2 - sqrt(S(3))I/S(2))(S(1)/3)) - Iatan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))(S(1)/3) + S(1))/S(3))/(S(3)(S(1)/2 + sqrt(S(3))I/S(2))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(-xS(3) + S(1))/(xS(6) - x*S(3) + S(1)), x), x, -x + sqrt(S(3))Ilog(-S(2)(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(9)(S(1)/2 - sqrt(S(3))I/S(2))(S(2)/3)) - sqrt(S(3))Ilog(-S(2)(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(9)(S(1)/2 + sqrt(S(3))I/S(2))(S(2)/3)) - S(2)(S(2)/3)sqrt(S(3))Ilog(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)sqrt(S(3))Ilog(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(2)/3)) - Iatan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))(S(1)/3) + S(1))/S(3))/(S(3)(S(1)/2 - sqrt(S(3))I/S(2))(S(2)/3)) + Iatan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))(S(1)/3) + S(1))/S(3))/(S(3)(S(1)/2 + sqrt(S(3))I/S(2))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-xS(3) + S(1))/(xS(6) - xS(3) + S(1)), x), x, -S(2)(S(1)/3)(S(3) - sqrt(S(3))I)log(-S(2)(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(1)/3)) - S(2)(S(1)/3)(S(3) + sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(S(3) - sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) - sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(S(3) + sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) + sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(-sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) - sqrt(S(3))I)(S(1)/3)) - S(2)(S(1)/3)(sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) + sqrt(S(3))I)(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(3) + S(1))/(xS(6) - xS(3) + S(1)), x), x, -S(2)(S(2)/3)(S(3) - sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(S(3) + sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(S(3) - sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) - sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(S(3) + sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) + sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(-sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) - sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) + sqrt(S(3))I)(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(3) + S(1))/(xS(2)(xS(6) - xS(3) + S(1))), x), x, -S(2)*(S(1)/3)(S(3) + sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(1)/3)) - S(2)(S(1)/3)(S(3) - sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(S(3) + sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) - sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(S(3) - sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) + sqrt(S(3))I)(S(1)/3)) - S(2)(S(1)/3)(sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) - sqrt(S(3))I)(S(1)/3)) + S(2)(S(1)/3)(-sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) + sqrt(S(3))I)(S(1)/3)) - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(3) + S(1))/(xS(3)(xS(6) - xS(3) + S(1))), x), x, -S(2)*(S(2)/3)(S(3) + sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) - sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) - sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(S(3) - sqrt(S(3))I)log(-S(2)*(S(1)/3)x + (S(1) + sqrt(S(3))I)(S(1)/3))/(S(18)(S(1) + sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(S(3) + sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) - S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) - sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) - sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(S(3) - sqrt(S(3))I)log(S(2)*(S(2)/3)x*S(2) + x(S(2) + S(2)sqrt(S(3))I)*(S(1)/3) + (S(1) + sqrt(S(3))I)*(S(2)/3))/(S(36)(S(1) + sqrt(S(3))I)(S(2)/3)) + S(2)(S(2)/3)(sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 - sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) - sqrt(S(3))I)(S(2)/3)) - S(2)(S(2)/3)(-sqrt(S(3)) + I)atan(sqrt(S(3))(S(2)x/(S(1)/2 + sqrt(S(3))I/S(2))*(S(1)/3) + S(1))/S(3))/(S(6)(S(1) + sqrt(S(3))I)(S(2)/3)) - S(1)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(xS(3) + S(-2))/(xS(6) - xS(3) + S(1)), x), x, log(xS(6) - xS(3) + S(1))/S(6) + sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(3) + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(1))/(x(xS(6) - xS(3) + S(1))), x), x, log(x) - log(xS(6) - xS(3) + S(1))/S(6) - sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(3) + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(1))/(xS(7) - xS(4) + x), x), x, log(x) - log(xS(6) - xS(3) + S(1))/S(6) - sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(3) + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(3))(S(5)/2)(a + bx*S(3) + cx*S(6)), x), x, S(2)cxS(4)(d + exS(3))(S(7)/2)/(S(29)e) + S(54)S(3)(S(3)/4)d*S(3)sqrt((d(S(2)/3) - d(S(1)/3)e(S(1)/3)x + e*(S(2)/3)xS(2))/(d(S(1)/3)(S(1) + sqrt(S(3))) + e(S(1)/3)x)*S(2))sqrt(sqrt(S(3)) + S(2))(d(S(1)/3) + e(S(1)/3)x)(S(667)aeS(2) - S(58)bde + S(16)cd*S(2))elliptic_f(asin((d*(S(1)/3)(-sqrt(S(3)) + S(1)) + e*(S(1)/3)x)/(d*(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)), S(-7) - S(4)sqrt(S(3)))/(S(124729)e*(S(7)/3)sqrt(d*(S(1)/3)(d(S(1)/3) + e(S(1)/3)x)/(d(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)*S(2))sqrt(d + exS(3))) + S(54)d*S(2)xsqrt(d + ex*S(3))(S(667)ae*S(2) - S(58)bde + S(16)cd*S(2))/(S(124729)e*S(2)) + S(30)dx(d + exS(3))(S(3)/2)(S(667)ae*S(2) - S(58)bde + S(16)cd*S(2))/(S(124729)e*S(2)) - x(d + exS(3))(S(7)/2)(-S(58)be + S(16)cd)/(S(667)eS(2)) + x(d + exS(3))(S(5)/2)(S(1334)ae*S(2) - S(116)bde + S(32)cd*S(2))/(S(11339)e*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(3))(S(3)/2)(a + bx*S(3) + cx*S(6)), x), x, S(2)cxS(4)(d + exS(3))(S(5)/2)/(S(23)e) + S(18)S(3)(S(3)/4)d*S(2)sqrt((d(S(2)/3) - d(S(1)/3)e(S(1)/3)x + e*(S(2)/3)xS(2))/(d(S(1)/3)(S(1) + sqrt(S(3))) + e(S(1)/3)x)*S(2))sqrt(sqrt(S(3)) + S(2))(d(S(1)/3) + e(S(1)/3)x)(S(391)aeS(2) - S(46)bde + S(16)cd*S(2))elliptic_f(asin((d*(S(1)/3)(-sqrt(S(3)) + S(1)) + e*(S(1)/3)x)/(d*(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)), S(-7) - S(4)sqrt(S(3)))/(S(21505)e*(S(7)/3)sqrt(d*(S(1)/3)(d(S(1)/3) + e(S(1)/3)x)/(d(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)*S(2))sqrt(d + exS(3))) + S(18)dxsqrt(d + exS(3))(S(391)ae*S(2) - S(46)bde + S(16)cd*S(2))/(S(21505)e*S(2)) - x(d + exS(3))(S(5)/2)(-S(46)be + S(16)cd)/(S(391)eS(2)) + x(d + exS(3))(S(3)/2)(S(782)ae*S(2) - S(92)bde + S(32)cd*S(2))/(S(4301)e*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + ex*S(3))(a + bxS(3) + cx*S(6)), x), x, S(2)cxS(4)(d + exS(3))(S(3)/2)/(S(17)e) + S(2)S(3)(S(3)/4)dsqrt((d(S(2)/3) - d(S(1)/3)e*(S(1)/3)x + e*(S(2)/3)xS(2))/(d(S(1)/3)(S(1) + sqrt(S(3))) + e(S(1)/3)x)*S(2))sqrt(sqrt(S(3)) + S(2))(d(S(1)/3) + e(S(1)/3)x)(S(187)aeS(2) - S(34)bde + S(16)cd*S(2))elliptic_f(asin((d*(S(1)/3)(-sqrt(S(3)) + S(1)) + e*(S(1)/3)x)/(d*(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)), S(-7) - S(4)sqrt(S(3)))/(S(935)e*(S(7)/3)sqrt(d*(S(1)/3)(d(S(1)/3) + e(S(1)/3)x)/(d(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)*S(2))sqrt(d + exS(3))) - x(d + exS(3))(S(3)/2)(-S(34)be + S(16)cd)/(S(187)eS(2)) + xsqrt(d + exS(3))(S(374)ae*S(2) - S(68)bde + S(32)cd*S(2))/(S(935)e*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cx*S(6))/sqrt(d + ex*S(3)), x), x, S(2)cxS(4)sqrt(d + exS(3))/(S(11)e) - xsqrt(d + ex*S(3))(-S(22)be + S(16)cd)/(S(55)eS(2)) + S(2)S(3)*(S(3)/4)sqrt((d(S(2)/3) - d(S(1)/3)e(S(1)/3)x + e*(S(2)/3)xS(2))/(d(S(1)/3)(S(1) + sqrt(S(3))) + e(S(1)/3)x)*S(2))sqrt(sqrt(S(3)) + S(2))(d(S(1)/3) + e(S(1)/3)x)(S(55)aeS(2) - S(22)bde + S(16)cd*S(2))elliptic_f(asin((d*(S(1)/3)(-sqrt(S(3)) + S(1)) + e*(S(1)/3)x)/(d*(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)), S(-7) - S(4)sqrt(S(3)))/(S(165)e*(S(7)/3)sqrt(d*(S(1)/3)(d(S(1)/3) + e(S(1)/3)x)/(d(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)*S(2))sqrt(d + exS(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cx*S(6))/(d + exS(3))(S(3)/2), x), x, S(2)cxsqrt(d + ex*S(3))/(S(5)e*S(2)) + x(S(2)ae*S(2) - S(2)bde + S(2)cd*S(2))/(S(3)deS(2)sqrt(d + exS(3))) - S(2)S(3)*(S(3)/4)sqrt((d(S(2)/3) - d(S(1)/3)e(S(1)/3)x + e*(S(2)/3)xS(2))/(d(S(1)/3)(S(1) + sqrt(S(3))) + e(S(1)/3)x)*S(2))sqrt(sqrt(S(3)) + S(2))(d(S(1)/3) + e(S(1)/3)x)(S(16)cdS(2) - S(5)e(ae + S(2)bd))elliptic_f(asin((d(S(1)/3)(-sqrt(S(3)) + S(1)) + e*(S(1)/3)x)/(d*(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)), S(-7) - S(4)sqrt(S(3)))/(S(45)de(S(7)/3)sqrt(d*(S(1)/3)(d(S(1)/3) + e(S(1)/3)x)/(d(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)*S(2))sqrt(d + exS(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cx*S(6))/(d + exS(3))(S(5)/2), x), x, x(S(2)aeS(2) - S(2)bde + S(2)cd*S(2))/(S(9)deS(2)(d + exS(3))(S(3)/2)) - x(-S(14)ae*S(2) - S(4)bde + S(22)cd*S(2))/(S(27)d*S(2)e*S(2)sqrt(d + exS(3))) + S(2)S(3)*(S(3)/4)sqrt((d(S(2)/3) - d(S(1)/3)e(S(1)/3)x + e*(S(2)/3)xS(2))/(d(S(1)/3)(S(1) + sqrt(S(3))) + e(S(1)/3)x)*S(2))sqrt(sqrt(S(3)) + S(2))(d(S(1)/3) + e(S(1)/3)x)(S(16)cdS(2) + e(S(7)ae + S(2)bd))elliptic_f(asin((d(S(1)/3)(-sqrt(S(3)) + S(1)) + e*(S(1)/3)x)/(d*(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)), S(-7) - S(4)sqrt(S(3)))/(S(81)d*S(2)e*(S(7)/3)sqrt(d*(S(1)/3)(d(S(1)/3) + e(S(1)/3)x)/(d(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)*S(2))sqrt(d + exS(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cx*S(6))/(d + exS(3))(S(7)/2), x), x, x(S(2)aeS(2) - S(2)bde + S(2)cd*S(2))/(S(15)deS(2)(d + exS(3))(S(5)/2)) - x(-S(26)ae*S(2) - S(4)bde + S(34)cd*S(2))/(S(135)d*S(2)e*S(2)(d + exS(3))(S(3)/2)) + x(S(182)ae*S(2) + S(28)bde + S(32)cd*S(2))/(S(405)d*S(3)e*S(2)sqrt(d + exS(3))) + S(2)S(3)*(S(3)/4)sqrt((d(S(2)/3) - d(S(1)/3)e(S(1)/3)x + e*(S(2)/3)xS(2))/(d(S(1)/3)(S(1) + sqrt(S(3))) + e(S(1)/3)x)*S(2))sqrt(sqrt(S(3)) + S(2))(d(S(1)/3) + e(S(1)/3)x)(S(91)aeS(2) + S(14)bde + S(16)cd*S(2))elliptic_f(asin((d*(S(1)/3)(-sqrt(S(3)) + S(1)) + e*(S(1)/3)x)/(d*(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)), S(-7) - S(4)sqrt(S(3)))/(S(1215)d*S(3)e*(S(7)/3)sqrt(d*(S(1)/3)(d(S(1)/3) + e(S(1)/3)x)/(d(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)*S(2))sqrt(d + exS(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(3) + cx*S(6))/(d + exS(3))(S(9)/2), x), x, x(S(2)aeS(2) - S(2)bde + S(2)cd*S(2))/(S(21)deS(2)(d + exS(3))(S(7)/2)) - x(-S(38)ae*S(2) - S(4)bde + S(46)cd*S(2))/(S(315)d*S(2)e*S(2)(d + exS(3))(S(5)/2)) + x(S(494)ae*S(2) + S(52)bde + S(32)cd*S(2))/(S(2835)d*S(3)e*S(2)(d + exS(3))(S(3)/2)) + x(S(494)ae*S(2) + S(52)bde + S(32)cd*S(2))/(S(1215)d*S(4)e*S(2)sqrt(d + exS(3))) + S(2)S(3)*(S(3)/4)sqrt((d(S(2)/3) - d(S(1)/3)e(S(1)/3)x + e*(S(2)/3)xS(2))/(d(S(1)/3)(S(1) + sqrt(S(3))) + e(S(1)/3)x)*S(2))sqrt(sqrt(S(3)) + S(2))(d(S(1)/3) + e(S(1)/3)x)(S(247)aeS(2) + S(26)bde + S(16)cd*S(2))elliptic_f(asin((d*(S(1)/3)(-sqrt(S(3)) + S(1)) + e*(S(1)/3)x)/(d*(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)), S(-7) - S(4)sqrt(S(3)))/(S(3645)d*S(4)e*(S(7)/3)sqrt(d*(S(1)/3)(d(S(1)/3) + e(S(1)/3)x)/(d(S(1)/3)(S(1) + sqrt(S(3))) + e*(S(1)/3)x)*S(2))sqrt(d + exS(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(4))/(a + cx*S(8)), x), x, sqrt(S(2))(sqrt(c)d - esqrt(-a))log(-sqrt(S(2))c*(S(1)/8)x(-a)(S(1)/8) + c(S(1)/4)xS(2) + (-a)(S(1)/4))/(S(16)c(S(5)/8)(-a)*(S(7)/8)) - sqrt(S(2))(sqrt(c)d - esqrt(-a))log(sqrt(S(2))c*(S(1)/8)x(-a)(S(1)/8) + c(S(1)/4)xS(2) + (-a)(S(1)/4))/(S(16)c(S(5)/8)(-a)*(S(7)/8)) - sqrt(S(2))(sqrt(c)d - esqrt(-a))atan(sqrt(S(2))c*(S(1)/8)x/(-a)*(S(1)/8) + S(-1))/(S(8)c*(S(5)/8)(-a)*(S(7)/8)) - sqrt(S(2))(sqrt(c)d - esqrt(-a))atan(sqrt(S(2))c*(S(1)/8)x/(-a)*(S(1)/8) + S(1))/(S(8)c*(S(5)/8)(-a)*(S(7)/8)) - (sqrt(c)d + esqrt(-a))atan(c*(S(1)/8)x/(-a)*(S(1)/8))/(S(4)c*(S(5)/8)(-a)*(S(7)/8)) - (sqrt(c)d + esqrt(-a))atanh(c*(S(1)/8)x/(-a)*(S(1)/8))/(S(4)c*(S(5)/8)(-a)*(S(7)/8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(4))/(a - cx*S(8)), x), x, -sqrt(S(2))(-sqrt(a)e + sqrt(c)d)log(-sqrt(S(2))a*(S(1)/8)c*(S(1)/8)x + a(S(1)/4) + c(S(1)/4)xS(2))/(S(16)a*(S(7)/8)c*(S(5)/8)) + sqrt(S(2))(-sqrt(a)e + sqrt(c)d)log(sqrt(S(2))a*(S(1)/8)c*(S(1)/8)x + a(S(1)/4) + c(S(1)/4)xS(2))/(S(16)a*(S(7)/8)c*(S(5)/8)) - sqrt(S(2))(-sqrt(a)e + sqrt(c)d)atan(S(1) - sqrt(S(2))c*(S(1)/8)x/a*(S(1)/8))/(S(8)a*(S(7)/8)c*(S(5)/8)) + sqrt(S(2))(-sqrt(a)e + sqrt(c)d)atan(S(1) + sqrt(S(2))c*(S(1)/8)x/a*(S(1)/8))/(S(8)a*(S(7)/8)c*(S(5)/8)) + (sqrt(a)e + sqrt(c)d)atan(c*(S(1)/8)x/a*(S(1)/8))/(S(4)a*(S(7)/8)c*(S(5)/8)) + (sqrt(a)e + sqrt(c)d)atanh(c*(S(1)/8)x/a*(S(1)/8))/(S(4)a*(S(7)/8)c(S(5)/8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(d + ex*S(4))/(a + bx*S(4) + cx*S(8)), x), x, ex/c - S(2)*(S(3)/4)(-be + cd - (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) - S(2)*(S(3)/4)(-be + cd - (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) - S(2)*(S(3)/4)(-be + cd + (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) - S(2)*(S(3)/4)(-be + cd + (S(2)ace - bS(2)e + bcd)/sqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(d + exS(4))/(a + bx*S(4) + cx*S(8)), x), x, elog(a + bxS(4) + cx*S(8))/(S(8)c) - (-be + S(2)cd)atanh((b + S(2)cx*S(4))/sqrt(-S(4)ac + bS(2)))/(S(4)csqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(d + exS(4))/(a + bx*S(4) + cxS(8)), x), x, S(2)(S(1)/4)(e + (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(3)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) - S(2)*(S(1)/4)(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(3)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) + S(2)*(S(1)/4)(e - (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(3)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)) - S(2)*(S(1)/4)(e - (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(3)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(d + ex*S(4))/(a + bx*S(4) + cx*S(8)), x), x, sqrt(S(2))(e - (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x*S(2)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x*S(2)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(4)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex*S(4))/(a + bx*S(4) + cxS(8)), x), x, -S(2)(S(3)/4)(e + (-be + S(2)cd)/sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(1)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) - S(2)*(S(3)/4)(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(1)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) - S(2)*(S(3)/4)(e - (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(1)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) - S(2)*(S(3)/4)(e - (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(1)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(4))/(x(a + bxS(4) + cx*S(8))), x), x, dlog(x)/a - dlog(a + bx*S(4) + cx*S(8))/(S(8)a) + (-S(2)ae + bd)atanh((b + S(2)cx*S(4))/sqrt(-S(4)ac + bS(2)))/(S(4)asqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(4))/(xS(2)(a + bx*S(4) + cxS(8))), x), x, -S(2)(S(1)/4)c(S(1)/4)(d + (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) + S(2)*(S(1)/4)c*(S(1)/4)(d + (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4)) - S(2)*(S(1)/4)c*(S(1)/4)(d - (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)) + S(2)*(S(1)/4)c*(S(1)/4)(d - (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4)) - d/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(4))/(xS(3)(a + bx*S(4) + cx*S(8))), x), x, -sqrt(S(2))sqrt(c)(d - (-S(2)ae + bd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x*S(2)/sqrt(b + sqrt(-S(4)ac + bS(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))) - sqrt(S(2))sqrt(c)(d + (-S(2)ae + bd)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x*S(2)/sqrt(b - sqrt(-S(4)ac + bS(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))) - d/(S(2)axS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exS(4))/(xS(4)(a + bx*S(4) + cxS(8))), x), x, S(2)(S(3)/4)c(S(3)/4)(d + (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)c*(S(3)/4)(d + (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)c*(S(3)/4)(d - (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)c*(S(3)/4)(d - (-S(2)ae + bd)/sqrt(-S(4)ac + bS(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)a(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) - d/(S(3)axS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(-xS(4) + S(1))/(xS(8) - xS(4) + S(1)), x), x, -x - sqrt(S(6))log(x*S(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(24) + sqrt(S(6))log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(24) - sqrt(S(6))log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(24) + sqrt(S(6))log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(24) - sqrt(S(6))atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(12) + sqrt(S(6))atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(12) - sqrt(S(6))atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(12) + sqrt(S(6))atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(-xS(4) + S(1))/(xS(8) - xS(4) + S(1)), x), x, -log(xS(8) - x*S(4) + S(1))/S(8) - sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(4) + S(1))/S(3))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(-xS(4) + S(1))/(xS(8) - xS(4) + S(1)), x), x, sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(x*S(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) + sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) - atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/(S(4)sqrt(S(3)sqrt(S(3)) + S(6))) + atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/(S(4)sqrt(S(3)sqrt(S(3)) + S(6))) + atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/(S(4)sqrt(-S(3)sqrt(S(3)) + S(6))) - atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/(S(4)sqrt(-S(3)sqrt(S(3)) + S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-xS(4) + S(1))/(xS(8) - xS(4) + S(1)), x), x, -sqrt(S(3))log(x*S(4) - sqrt(S(3))x*S(2) + S(1))/S(12) + sqrt(S(3))log(x*S(4) + sqrt(S(3))xS(2) + S(1))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(4) + S(1))/(xS(8) - xS(4) + S(1)), x), x, sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) + sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) + atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/(S(4)sqrt(S(3)sqrt(S(3)) + S(6))) - atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/(S(4)sqrt(S(3)sqrt(S(3)) + S(6))) - atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/(S(4)sqrt(-S(3)sqrt(S(3)) + S(6))) + atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/(S(4)sqrt(-S(3)sqrt(S(3)) + S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-x*S(4) + S(1))/(x(xS(8) - xS(4) + S(1))), x), x, log(x) - log(xS(8) - xS(4) + S(1))/S(8) + sqrt(S(3))atan(sqrt(S(3))(-S(2)xS(4) + S(1))/S(3))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(4) + S(1))/(xS(2)(xS(8) - xS(4) + S(1))), x), x, -sqrt(S(6))log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(24) + sqrt(S(6))log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(24) - sqrt(S(6))log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(24) + sqrt(S(6))log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(24) + sqrt(S(6))atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(12) - sqrt(S(6))atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(12) + sqrt(S(6))atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(12) - sqrt(S(6))atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(12) - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(4) + S(1))/(xS(3)(xS(8) - xS(4) + S(1))), x), x, -sqrt(S(3))log(x*S(4) - sqrt(S(3))x*S(2) + S(1))/S(24) + sqrt(S(3))log(x*S(4) + sqrt(S(3))x*S(2) + S(1))/S(24) - atan(S(2)x*S(2) - sqrt(S(3)))/S(4) - atan(S(2)x*S(2) + sqrt(S(3)))/S(4) - S(1)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(4) + S(1))/(x*S(4)(xS(8) - xS(4) + S(1))), x), x, sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(8) - sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) + sqrt(-sqrt(S(3))/S(3) + S(2)/3)log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/S(8) + sqrt(sqrt(S(3))/S(3) + S(2)/3)atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(4) - sqrt(sqrt(S(3))/S(3) + S(2)/3)atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(4) - sqrt(-sqrt(S(3))/S(3) + S(2)/3)atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(4) + sqrt(-sqrt(S(3))/S(3) + S(2)/3)atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(4) - S(1)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + S(1))/(xS(8) + xS(4) + S(1)), x), x, -log(xS(2) - x + S(1))/S(8) + log(xS(2) + x + S(1))/S(8) - sqrt(S(3))log(x*S(2) - sqrt(S(3))x + S(1))/S(24) + sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(24) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(12) + sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(12) + atan(S(2)x - sqrt(S(3)))/S(4) + atan(S(2)x + sqrt(S(3)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(4) + S(1))/(xS(8) + xS(4) + S(1)), x), x, log(xS(2) - x + S(1))/S(8) - log(x*S(2) + x + S(1))/S(8) - sqrt(S(3))log(x*S(2) - sqrt(S(3))x + S(1))/S(8) + sqrt(S(3))log(xS(2) + sqrt(S(3))x + S(1))/S(8) - sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(4) + sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(4) - atan(S(2)x - sqrt(S(3)))/S(4) - atan(S(2)x + sqrt(S(3)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + S(1))/(xS(8) - xS(4) + S(1)), x), x, -log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/(S(8)sqrt(-sqrt(S(3)) + S(2))) + log(x*S(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/(S(8)sqrt(-sqrt(S(3)) + S(2))) - log(xS(2) - xsqrt(sqrt(S(3)) + S(2)) + S(1))/(S(8)sqrt(sqrt(S(3)) + S(2))) + log(xS(2) + xsqrt(sqrt(S(3)) + S(2)) + S(1))/(S(8)sqrt(sqrt(S(3)) + S(2))) - sqrt(sqrt(S(3)) + S(2))atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(4) + sqrt(sqrt(S(3)) + S(2))atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(4) - sqrt(-sqrt(S(3)) + S(2))atan((-S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(4) + sqrt(-sqrt(S(3)) + S(2))atan((S(2)x + sqrt(-sqrt(S(3)) + S(2)))/sqrt(sqrt(S(3)) + S(2)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(4) + S(1))/(xS(8) - S(3)x*S(4) + S(1)), x), x, sqrt(S(5))(sqrt(S(5))/S(2) + S(3)/2)*(S(1)/4)atan(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(10) + sqrt(S(5))(-sqrt(S(5))/S(2) + S(3)/2)*(S(1)/4)atan(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))*(S(1)/4))/S(10) + sqrt(S(5))(sqrt(S(5))/S(2) + S(3)/2)*(S(1)/4)atanh(x(sqrt(S(5))/S(2) + S(3)/2)(S(1)/4))/S(10) + sqrt(S(5))(-sqrt(S(5))/S(2) + S(3)/2)*(S(1)/4)atanh(S(2)*(S(1)/4)x/(sqrt(S(5)) + S(3))*(S(1)/4))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(4) + S(-1) + sqrt(S(3)))/(xS(8) - x*S(4) + S(1)), x), x, -sqrt(S(2))log(x*S(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(4) + sqrt(S(2))log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(4) - sqrt(S(2))atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(2) + sqrt(S(2))atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(4)(S(1) + sqrt(S(3))) + S(1))/(xS(8) - xS(4) + S(1)), x), x, -sqrt(sqrt(S(3)) + S(2))log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(4) + sqrt(sqrt(S(3)) + S(2))log(xS(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(4) - sqrt(sqrt(S(3)) + S(2))atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(2) + sqrt(sqrt(S(3)) + S(2))atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(4)(S(-3) + sqrt(S(3))) - S(2)sqrt(S(3)) + S(3))/(xS(8) - xS(4) + S(1)), x), x, sqrt(-S(3)sqrt(S(3)) + S(6))log(xS(2) - xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(4) - sqrt(-S(3)sqrt(S(3)) + S(6))log(x*S(2) + xsqrt(-sqrt(S(3)) + S(2)) + S(1))/S(4) + sqrt(-S(3)sqrt(S(3)) + S(6))atan((-S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(2) - sqrt(-S(3)sqrt(S(3)) + S(6))atan((S(2)x + sqrt(sqrt(S(3)) + S(2)))/sqrt(-sqrt(S(3)) + S(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + e/x)/(a/x*S(2) + c), x), x, -sqrt(a)datan(sqrt(c)x/sqrt(a))/c*(S(3)/2) + dx/c + elog(a + cx*S(2))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + e/x)/(a/x*S(2) + b/x + c), x), x, dx/c - (bd - ce)log(a + bx + cxS(2))/(S(2)c*S(2)) - (-S(2)acd + b*S(2)d - bce)atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + e/xS(2))/(a/x*S(4) + c), x), x, dx/c + sqrt(S(2))(sqrt(a)d - sqrt(c)e)atan(S(1) - sqrt(S(2))c(S(1)/4)x/a*(S(1)/4))/(S(4)a*(S(1)/4)c*(S(5)/4)) - sqrt(S(2))(sqrt(a)d - sqrt(c)e)atan(S(1) + sqrt(S(2))c*(S(1)/4)x/a*(S(1)/4))/(S(4)a*(S(1)/4)c*(S(5)/4)) + sqrt(S(2))(sqrt(a)d + sqrt(c)e)log(-sqrt(S(2))a*(S(1)/4)c*(S(1)/4)x + sqrt(a) + sqrt(c)xS(2))/(S(8)a*(S(1)/4)c*(S(5)/4)) - sqrt(S(2))(sqrt(a)d + sqrt(c)e)log(sqrt(S(2))a*(S(1)/4)c*(S(1)/4)x + sqrt(a) + sqrt(c)xS(2))/(S(8)a*(S(1)/4)c(S(5)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + e/xS(2))/(a/xS(4) + b/xS(2) + c), x), x, dx/c - sqrt(S(2))(bd - ce + (-S(2)acd + bS(2)d - bce)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) - sqrt(S(2))(bd - ce - (-S(2)acd + bS(2)d - bce)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + e/xS(3))/(a/x*S(6) + c), x), x, dx/c - (-sqrt(c)e + dsqrt(-a))log(c(S(1)/6)x + (-a)*(S(1)/6))/(S(6)c*(S(7)/6)(-a)*(S(1)/3)) + (-sqrt(c)e + dsqrt(-a))log(-c*(S(1)/6)x(-a)(S(1)/6) + c(S(1)/3)xS(2) + (-a)(S(1)/3))/(S(12)c(S(7)/6)(-a)*(S(1)/3)) + sqrt(S(3))(-sqrt(c)e + dsqrt(-a))atan(sqrt(S(3))(-S(2)c(S(1)/6)x/(-a)*(S(1)/6) + S(1))/S(3))/(S(6)c*(S(7)/6)(-a)*(S(1)/3)) + (sqrt(c)e + dsqrt(-a))log(-c*(S(1)/6)x + (-a)*(S(1)/6))/(S(6)c*(S(7)/6)(-a)*(S(1)/3)) - (sqrt(c)e + dsqrt(-a))log(c*(S(1)/6)x(-a)(S(1)/6) + c(S(1)/3)xS(2) + (-a)(S(1)/3))/(S(12)c(S(7)/6)(-a)*(S(1)/3)) - sqrt(S(3))(sqrt(c)e + dsqrt(-a))atan(sqrt(S(3))(S(2)c(S(1)/6)x/(-a)*(S(1)/6) + S(1))/S(3))/(S(6)c*(S(7)/6)(-a)(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + e/xS(3))/(a/xS(6) + b/xS(3) + c), x), x, dx/c - S(2)(S(2)/3)(bd - ce + (-S(2)acd + bS(2)d - bce)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(4)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)(bd - ce + (-S(2)acd + bS(2)d - bce)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(4)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)sqrt(S(3))(bd - ce + (-S(2)acd + b*S(2)d - bce)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(4)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)(bd - ce - (-S(2)acd + bS(2)d - bce)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(4)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)(bd - ce - (-S(2)acd + bS(2)d - bce)/sqrt(-S(4)ac + b*S(2)))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(4)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)sqrt(S(3))(bd - ce - (-S(2)acd + b*S(2)d - bce)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(4)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + e/xS(4))/(a/xS(8) + c), x), x, dx/c + sqrt(S(2))(-sqrt(c)e + dsqrt(-a))log(-sqrt(S(2))c*(S(1)/8)x(-a)(S(1)/8) + c(S(1)/4)xS(2) + (-a)(S(1)/4))/(S(16)c(S(9)/8)(-a)*(S(3)/8)) - sqrt(S(2))(-sqrt(c)e + dsqrt(-a))log(sqrt(S(2))c*(S(1)/8)x(-a)(S(1)/8) + c(S(1)/4)xS(2) + (-a)(S(1)/4))/(S(16)c(S(9)/8)(-a)*(S(3)/8)) - sqrt(S(2))(-sqrt(c)e + dsqrt(-a))atan(sqrt(S(2))c*(S(1)/8)x/(-a)*(S(1)/8) + S(-1))/(S(8)c*(S(9)/8)(-a)*(S(3)/8)) - sqrt(S(2))(-sqrt(c)e + dsqrt(-a))atan(sqrt(S(2))c*(S(1)/8)x/(-a)*(S(1)/8) + S(1))/(S(8)c*(S(9)/8)(-a)*(S(3)/8)) - (sqrt(c)e + dsqrt(-a))atan(c*(S(1)/8)x/(-a)*(S(1)/8))/(S(4)c*(S(9)/8)(-a)*(S(3)/8)) - (sqrt(c)e + dsqrt(-a))atanh(c*(S(1)/8)x/(-a)*(S(1)/8))/(S(4)c*(S(9)/8)(-a)(S(3)/8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + e/xS(4))/(a/xS(8) + b/xS(4) + c), x), x, dx/c + S(2)(S(3)/4)(bd - ce - (-S(2)acd + bS(2)d - bce)/sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)(bd - ce - (-S(2)acd + bS(2)d - bce)/sqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b + sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b + sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)(bd - ce + (-S(2)acd + bS(2)d - bce)/sqrt(-S(4)ac + b*S(2)))atan(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)) + S(2)*(S(3)/4)(bd - ce + (-S(2)acd + bS(2)d - bce)/sqrt(-S(4)ac + b*S(2)))atanh(S(2)*(S(1)/4)c*(S(1)/4)x/(-b - sqrt(-S(4)ac + bS(2)))(S(1)/4))/(S(4)c(S(5)/4)(-b - sqrt(-S(4)ac + bS(2)))(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)m(a + cx(S(2)n))*p(d + exn)S(3), x), x, dS(3)(fx)(m + S(1))(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper(((m + S(1))/(S(2)n), -p), (S(1) + (m + S(1))/(S(2)n),), -cx(S(2)n)/a)/(f(m + S(1))) + S(3)d*S(2)ex(n + S(1))(fx)m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper(((m + n + S(1))/(S(2)n), -p), ((m + S(3)n + S(1))/(S(2)n),), -cx*(S(2)n)/a)/(m + n + S(1)) + S(3)de*S(2)x*(S(2)n + S(1))(fx)*m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper(((m + S(2)n + S(1))/(S(2)n), -p), ((m + S(4)n + S(1))/(S(2)n),), -cx(S(2)n)/a)/(m + S(2)n + S(1)) + eS(3)x*(S(3)n + S(1))(fx)*m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper(((m + S(3)n + S(1))/(S(2)n), -p), ((m + S(5)n + S(1))/(S(2)n),), -cx(S(2)n)/a)/(m + S(3)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(a + cx(S(2)n))*p(d + exn)S(2), x), x, dS(2)(fx)(m + S(1))(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper(((m + S(1))/(S(2)n), -p), (S(1) + (m + S(1))/(S(2)n),), -cx(S(2)n)/a)/(f(m + S(1))) + S(2)dex*(n + S(1))(fx)m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper(((m + n + S(1))/(S(2)n), -p), ((m + S(3)n + S(1))/(S(2)n),), -cx*(S(2)n)/a)/(m + n + S(1)) + e*S(2)x*(S(2)n + S(1))(fx)*m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper(((m + S(2)n + S(1))/(S(2)n), -p), ((m + S(4)n + S(1))/(S(2)n),), -cx(S(2)n)/a)/(m + S(2)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(a + cx(S(2)n))*p(d + exn), x), x, d(fx)(m + S(1))(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper(((m + S(1))/(S(2)n), -p), (S(1) + (m + S(1))/(S(2)n),), -cx(S(2)n)/a)/(f(m + S(1))) + ex*(n + S(1))(fx)m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*phyper(((m + n + S(1))/(S(2)n), -p), ((m + S(3)n + S(1))/(S(2)n),), -cx*(S(2)n)/a)/(m + n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(a + bx + cxS(2))S(13), x), x, (a + bx + cxS(2))S(14)/S(14), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(b + S(2)cx*S(2))(a + bxS(2) + cxS(4))S(13), x), x, (a + bxS(2) + cxS(4))S(14)/S(28), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(b + S(2)cx*S(3))(a + bxS(3) + cxS(6))S(13), x), x, (a + bxS(3) + cxS(6))S(14)/S(42), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(n + S(-1))(b + S(2)cx*n)(a + bxn + cx*(S(2)n))*S(13), x), x, (a + bx*n + cx*(S(2)n))*S(14)/(S(14)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(-a + bx + cxS(2))S(13), x), x, (-a + bx + cxS(2))S(14)/S(14), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(b + S(2)cx*S(2))(-a + bxS(2) + cxS(4))S(13), x), x, (a - bxS(2) - cxS(4))S(14)/S(28), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(b + S(2)cx*S(3))(-a + bxS(3) + cxS(6))S(13), x), x, (a - bxS(3) - cxS(6))S(14)/S(42), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(n + S(-1))(b + S(2)cx*n)(-a + bxn + cx*(S(2)n))*S(13), x), x, (a - bx*n - cx*(S(2)n))*S(14)/(S(14)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(bx + cxS(2))S(13), x), x, (bx + cxS(2))S(14)/S(14), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(b + S(2)cx*S(2))(bxS(2) + cxS(4))S(13), x), x, x*S(28)(b + cxS(2))S(14)/S(28), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(b + S(2)cx*S(3))(bxS(3) + cxS(6))S(13), x), x, x*S(42)(b + cxS(3))S(14)/S(42), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))(b + S(2)cx*n)(bxn + cx*(S(2)n))S(13), x), x, x(S(14)n)(b + cxn)S(14)/(S(14)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)/(a + bx + cx*S(2)), x), x, log(a + bx + cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(b + S(2)cx*S(2))/(a + bx*S(2) + cx*S(4)), x), x, log(a + bx*S(2) + cxS(4))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(b + S(2)cxS(3))/(a + bx*S(3) + cx*S(6)), x), x, log(a + bx*S(3) + cxS(6))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))(b + S(2)cxn)/(a + bx*n + cx*(S(2)n)), x), x, log(a + bxn + cx*(S(2)n))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)/(a + bx + cxS(2))S(8), x), x, -S(1)/(S(7)(a + bx + cxS(2))S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(b + S(2)cx*S(2))/(a + bx*S(2) + cxS(4))S(8), x), x, -S(1)/(S(14)(a + bx*S(2) + cxS(4))S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(b + S(2)cx*S(3))/(a + bx*S(3) + cxS(6))S(8), x), x, -S(1)/(S(21)(a + bx*S(3) + cxS(6))S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(n + S(-1))(b + S(2)cx*n)/(a + bx*n + cx*(S(2)n))*S(8), x), x, -S(1)/(S(7)n(a + bx*n + cx*(S(2)n))*S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)/(-a + bx + cxS(2)), x), x, log(a - bx - cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(b + S(2)cx*S(2))/(-a + bx*S(2) + cx*S(4)), x), x, log(a - bx*S(2) - cxS(4))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(b + S(2)cxS(3))/(-a + bx*S(3) + cx*S(6)), x), x, log(a - bx*S(3) - cxS(6))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))(b + S(2)cxn)/(-a + bx*n + cx*(S(2)n)), x), x, log(a - bxn - cx*(S(2)n))/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)/(-a + bx + cxS(2))S(8), x), x, -S(1)/(S(7)(-a + bx + cxS(2))S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(b + S(2)cx*S(2))/(-a + bx*S(2) + cxS(4))S(8), x), x, S(1)/(S(14)(a - bx*S(2) - cxS(4))S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(b + S(2)cx*S(3))/(-a + bx*S(3) + cxS(6))S(8), x), x, S(1)/(S(21)(a - bx*S(3) - cxS(6))S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(n + S(-1))(b + S(2)cx*n)/(-a + bx*n + cx*(S(2)n))*S(8), x), x, S(1)/(S(7)n(a - bx*n - cx*(S(2)n))*S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)/(bx + cxS(2)), x), x, log(bx + cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(b + S(2)cx*S(2))/(bx*S(2) + cx*S(4)), x), x, log(x) + log(b + cxS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(b + S(2)cxS(3))/(bx*S(3) + cx*S(6)), x), x, log(x) + log(b + cxS(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))(b + S(2)cxn)/(bx*n + cx*(S(2)n)), x), x, log(x) + log(b + cxn)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)/(bx + cxS(2))S(8), x), x, -S(1)/(S(7)(bx + cxS(2))S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(b + S(2)cxS(2))/(bx*S(2) + cxS(4))S(8), x), x, -S(1)/(S(14)xS(14)(b + cxS(2))S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(b + S(2)cx*S(3))/(bx*S(3) + cxS(6))S(8), x), x, -S(1)/(S(21)xS(21)(b + cxS(3))S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))(b + S(2)cx*n)/(bx*n + cx*(S(2)n))S(8), x), x, -x(-S(7)n)/(S(7)n(b + cxn)S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(a + bx + cxS(2))p, x), x, (a + bx + cxS(2))(p + S(1))/(p + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(b + S(2)cx*S(2))(a + bxS(2) + cxS(4))p, x), x, (a + bxS(2) + cxS(4))(p + S(1))/(S(2)p + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(b + S(2)cx*S(3))(a + bxS(3) + cxS(6))p, x), x, (a + bxS(3) + cxS(6))(p + S(1))/(S(3)p + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))(b + S(2)cx*n)(a + bxn + cx*(S(2)n))*p, x), x, (a + bx*n + cx*(S(2)n))*(p + S(1))/(n(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(-a + bx + cxS(2))p, x), x, (-a + bx + cxS(2))(p + S(1))/(p + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(b + S(2)cx*S(2))(-a + bxS(2) + cxS(4))p, x), x, (-a + bxS(2) + cxS(4))(p + S(1))/(S(2)p + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(b + S(2)cx*S(3))(-a + bxS(3) + cxS(6))p, x), x, (-a + bxS(3) + cxS(6))(p + S(1))/(S(3)p + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))(b + S(2)cx*n)(-a + bxn + cx*(S(2)n))*p, x), x, (-a + bx*n + cx*(S(2)n))*(p + S(1))/(n(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(bx + cxS(2))p, x), x, (bx + cxS(2))(p + S(1))/(p + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(b + S(2)cx*S(2))(bxS(2) + cxS(4))p, x), x, (bxS(2) + cxS(4))(p + S(1))/(S(2)p + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(b + S(2)cx*S(3))(bxS(3) + cxS(6))p, x), x, (bxS(3) + cxS(6))(p + S(1))/(S(3)p + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n + S(-1))(b + S(2)cx*n)(bxn + cx*(S(2)n))*p, x), x, (bx*n + cx*(S(2)n))*(p + S(1))/(n(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)m(d + exn)/(a + bx*n + cx*(S(2)n)), x), x, (fx)(m + S(1))(e - (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(f(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))) + (fx)(m + S(1))(e + (-be + S(2)cd)/sqrt(-S(4)ac + bS(2)))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(f(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)m(d + exn)/(a + bx*n + cx*(S(2)n))*S(2), x), x, -c(fx)(m + S(1))((-S(2)ae + bd)(m - n + S(1)) + (S(2)aben + S(4)acd(m - S(2)n + S(1)) - bS(2)d(m - n + S(1)))/sqrt(-S(4)ac + bS(2)))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(afn(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))) - c(fx)(m + S(1))((-S(2)ae + bd)(m - n + S(1)) - (S(2)aben + S(4)acd(m - S(2)n + S(1)) - bS(2)d(m - n + S(1)))/sqrt(-S(4)ac + bS(2)))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(afn(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))) + (fx)*(m + S(1))(-abe - S(2)acd + bS(2)d + cxn(-S(2)ae + bd))/(afn(-S(4)ac + b*S(2))(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) # large time assert rubi_test(rubi_integrate((d + exn)q/(x(a + bxn + cx*(S(2)n))), x), x, c(d + exn)(q + S(1))(-b/sqrt(-S(4)ac + bS(2)) + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exn)/(S(2)cd - e(b + sqrt(-S(4)ac + b*S(2)))))/(an(q + S(1))(S(2)cd - e(b + sqrt(-S(4)ac + bS(2))))) + c(d + exn)(q + S(1))(b/sqrt(-S(4)ac + b*S(2)) + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(2)c(d + exn)/(-be + S(2)cd + esqrt(-S(4)ac + bS(2))))/(an(q + S(1))(S(2)cd - e(b - sqrt(-S(4)ac + bS(2))))) - (d + exn)(q + S(1))hyper((S(1), q + S(1)), (q + S(2),), S(1) + ex*n/d)/(adn(q + S(1))), expand=True, _diff=True, _numerical=True) # Apart assert rubi_test(rubi_integrate((fx)m(d + exn)/(a + bx*n + cx*(S(2)n))*S(3), x), x, (fx)*(m + S(1))(-abe - S(2)acd + bS(2)d + cxn(-S(2)ae + bd))/(S(2)afn(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))*S(2)) - c(fx)(m + S(1))((m - n + S(1))(-S(4)a*S(2)ce(m - S(3)n + S(1)) + ab*S(2)e(m + S(1)) + S(2)abcd(S(2)m - S(7)n + S(2)) - b*S(3)d(m - S(2)n + S(1))) - (-S(4)aS(2)bce(mS(2) + m(-n + S(2)) - S(3)nS(2) - n + S(1)) - S(8)a*S(2)c*S(2)d(mS(2) + m(-S(6)n + S(2)) + S(8)n*S(2) - S(6)n + S(1)) + abS(3)e(m + S(1))(m - n + S(1)) + S(6)ab*S(2)cd(m*S(2) + m(-S(4)n + S(2)) + S(3)n*S(2) - S(4)n + S(1)) - b*S(4)d(mS(2) + m(-S(3)n + S(2)) + S(2)n*S(2) - S(3)n + S(1)))/sqrt(-S(4)ac + b*S(2)))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)fnS(2)(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))S(2)) - c(fx)(m + S(1))((m - n + S(1))(-S(4)a*S(2)ce(m - S(3)n + S(1)) + ab*S(2)e(m + S(1)) + S(2)abcd(S(2)m - S(7)n + S(2)) - b*S(3)d(m - S(2)n + S(1))) + (-S(4)aS(2)bce(mS(2) + m(-n + S(2)) - S(3)nS(2) - n + S(1)) - S(8)a*S(2)c*S(2)d(mS(2) + m(-S(6)n + S(2)) + S(8)n*S(2) - S(6)n + S(1)) + abS(3)e(m + S(1))(m - n + S(1)) + S(6)ab*S(2)cd(m*S(2) + m(-S(4)n + S(2)) + S(3)n*S(2) - S(4)n + S(1)) - b*S(4)d(mS(2) + m(-S(3)n + S(2)) + S(2)n*S(2) - S(3)n + S(1)))/sqrt(-S(4)ac + b*S(2)))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(S(2)a*S(2)fnS(2)(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))(-S(4)ac + bS(2))S(2)) + (fx)*(m + S(1))(abc(-S(2)ae + bd)(m - S(3)n + S(1)) + cxn(-S(4)aS(2)ce(m - S(3)n + S(1)) + ab*S(2)e(m + S(1)) + S(2)abcd(S(2)m - S(7)n + S(2)) - b*S(3)d(m - S(2)n + S(1))) + (-S(2)ac + b*S(2))(abe(m + S(1)) + S(2)acd(m - S(4)n + S(1)) - b*S(2)d(m - S(2)n + S(1))))/(S(2)aS(2)fnS(2)(-S(4)ac + bS(2))S(2)(a + bx*n + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c*(S(1)/3) - S(2)d*(S(1)/3)x(S(1)/3))/(-c(S(2)/3)d(S(2)/3)x + c*(S(1)/3)dx(S(4)/3) + cd*(S(1)/3)x*(S(2)/3)), x), x, -S(3)log(c(S(2)/3) - c(S(1)/3)d(S(1)/3)x(S(1)/3) + d(S(2)/3)x(S(2)/3))/(c(S(1)/3)d(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(a + bx*S(2) + cx*S(4))/(sqrt(d - ex)sqrt(d + ex)), x), x, -c(d - ex)*(S(9)/2)(d + ex)(S(9)/2)/(S(9)eS(10)) - dS(4)sqrt(d - ex)sqrt(d + ex)(ae*S(4) + bd*S(2)e*S(2) + cdS(4))/eS(10) + d*S(2)(d - ex)(S(3)/2)(d + ex)(S(3)/2)(S(2)ae*S(4) + S(3)bdS(2)e*S(2) + S(4)cdS(4))/(S(3)e*S(10)) + (d - ex)*(S(7)/2)(d + ex)(S(7)/2)(beS(2) + S(4)cdS(2))/(S(7)e*S(10)) - (d - ex)*(S(5)/2)(d + ex)(S(5)/2)(aeS(4) + S(3)bdS(2)e*S(2) + S(6)cdS(4))/(S(5)eS(10)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(5)(a + bx*S(2) + cx*S(4))/(sqrt(d - ex)sqrt(d + ex)), x), x, -c(dS(2) - eS(2)xS(2))S(5)/(S(9)eS(10)sqrt(d - ex)sqrt(d + ex)) - dS(4)(dS(2) - eS(2)xS(2))(aeS(4) + bd*S(2)e*S(2) + cdS(4))/(eS(10)sqrt(d - ex)sqrt(d + ex)) + d*S(2)(dS(2) - eS(2)xS(2))S(2)(S(2)ae*S(4) + S(3)bdS(2)e*S(2) + S(4)cdS(4))/(S(3)e*S(10)sqrt(d - ex)sqrt(d + ex)) + (dS(2) - eS(2)xS(2))S(4)(be*S(2) + S(4)cdS(2))/(S(7)e*S(10)sqrt(d - ex)sqrt(d + ex)) - (dS(2) - eS(2)xS(2))S(3)(ae*S(4) + S(3)bdS(2)e*S(2) + S(6)cdS(4))/(S(5)e*S(10)sqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a + bxS(2) + cx*S(4))/(sqrt(d - ex)sqrt(d + ex)), x), x, c(d - ex)*(S(7)/2)(d + ex)(S(7)/2)/(S(7)eS(8)) - dS(2)sqrt(d - ex)sqrt(d + ex)(ae*S(4) + bd*S(2)e*S(2) + cdS(4))/eS(8) - (d - ex)(S(5)/2)(d + ex)(S(5)/2)(beS(2) + S(3)cdS(2))/(S(5)e*S(8)) + (d - ex)*(S(3)/2)(d + ex)(S(3)/2)(aeS(4) + S(2)bdS(2)e*S(2) + S(3)cdS(4))/(S(3)eS(8)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(3)(a + bx*S(2) + cx*S(4))/(sqrt(d - ex)sqrt(d + ex)), x), x, c(dS(2) - eS(2)xS(2))S(4)/(S(7)eS(8)sqrt(d - ex)sqrt(d + ex)) - dS(2)(dS(2) - eS(2)xS(2))(aeS(4) + bd*S(2)e*S(2) + cdS(4))/(eS(8)sqrt(d - ex)sqrt(d + ex)) - (dS(2) - eS(2)xS(2))S(3)(beS(2) + S(3)cdS(2))/(S(5)e*S(8)sqrt(d - ex)sqrt(d + ex)) + (dS(2) - eS(2)xS(2))S(2)(ae*S(4) + S(2)bdS(2)e*S(2) + S(3)cdS(4))/(S(3)e*S(8)sqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxS(2) + cx*S(4))/(sqrt(d - ex)sqrt(d + ex)), x), x, -c(d - ex)*(S(5)/2)(d + ex)(S(5)/2)/(S(5)e*S(6)) + (d - ex)*(S(3)/2)(d + ex)(S(3)/2)(beS(2) + S(2)cdS(2))/(S(3)e*S(6)) - sqrt(d - ex)sqrt(d + ex)(ae*S(4) + bd*S(2)e*S(2) + cdS(4))/eS(6), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(x(a + bx*S(2) + cx*S(4))/(sqrt(d - ex)sqrt(d + ex)), x), x, -c(dS(2) - eS(2)xS(2))S(3)/(S(5)eS(6)sqrt(d - ex)sqrt(d + ex)) + (dS(2) - eS(2)xS(2))S(2)(be*S(2) + S(2)cdS(2))/(S(3)e*S(6)sqrt(d - ex)sqrt(d + ex)) - (dS(2) - eS(2)x*S(2))(aeS(4) + bd*S(2)e*S(2) + cdS(4))/(eS(6)sqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cx*S(4))/(xsqrt(d - ex)sqrt(d + ex)), x), x, -aatanh(sqrt(d - ex)sqrt(d + ex)/d)/d + c(d - ex)(S(3)/2)(d + ex)(S(3)/2)/(S(3)e*S(4)) - sqrt(d - ex)sqrt(d + ex)(be*S(2) + cdS(2))/eS(4), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + bxS(2) + cx*S(4))/(xsqrt(d - ex)sqrt(d + ex)), x), x, -asqrt(dS(2) - eS(2)xS(2))atanh(sqrt(dS(2) - eS(2)xS(2))/d)/(dsqrt(d - ex)sqrt(d + ex)) + c(dS(2) - eS(2)xS(2))S(2)/(S(3)e*S(4)sqrt(d - ex)sqrt(d + ex)) - (dS(2) - eS(2)x*S(2))(beS(2) + cdS(2))/(eS(4)sqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))/(xS(3)sqrt(d - ex)sqrt(d + ex)), x), x, -asqrt(d - ex)sqrt(d + ex)/(S(2)dS(2)x*S(2)) - csqrt(d - ex)sqrt(d + ex)/eS(2) - (ae*S(2) + S(2)bdS(2))atanh(sqrt(d - ex)sqrt(d + ex)/d)/(S(2)d*S(3)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(3)sqrt(d - ex)sqrt(d + ex)), x), x, -a(dS(2) - eS(2)x*S(2))/(S(2)d*S(2)x*S(2)sqrt(d - ex)sqrt(d + ex)) - c(dS(2) - eS(2)xS(2))/(eS(2)sqrt(d - ex)sqrt(d + ex)) - sqrt(dS(2) - eS(2)x*S(2))(aeS(2) + S(2)bdS(2))atanh(sqrt(dS(2) - eS(2)xS(2))/d)/(S(2)d*S(3)sqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(5)sqrt(d - ex)sqrt(d + ex)), x), x, -asqrt(d - ex)sqrt(d + ex)/(S(4)dS(2)x*S(4)) - sqrt(d - ex)sqrt(d + ex)(S(3)aeS(2) + S(4)bdS(2))/(S(8)d*S(4)x*S(2)) - (S(3)aeS(4) + S(4)bdS(2)e*S(2) + S(8)cdS(4))atanh(sqrt(d - ex)sqrt(d + ex)/d)/(S(8)d*S(5)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(5)sqrt(d - ex)sqrt(d + ex)), x), x, -a(dS(2) - eS(2)x*S(2))/(S(4)d*S(2)x*S(4)sqrt(d - ex)sqrt(d + ex)) - (dS(2) - eS(2)x*S(2))(S(3)ae*S(2) + S(4)bdS(2))/(S(8)d*S(4)x*S(2)sqrt(d - ex)sqrt(d + ex)) - sqrt(dS(2) - eS(2)x*S(2))(S(3)ae*S(4) + S(4)bdS(2)e*S(2) + S(8)cdS(4))atanh(sqrt(dS(2) - eS(2)xS(2))/d)/(S(8)d*S(5)sqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(7)sqrt(d - ex)sqrt(d + ex)), x), x, -asqrt(d - ex)sqrt(d + ex)/(S(6)dS(2)x*S(6)) - sqrt(d - ex)sqrt(d + ex)(S(5)aeS(2) + S(6)bdS(2))/(S(24)d*S(4)x*S(4)) - sqrt(d - ex)sqrt(d + ex)(S(5)aeS(4) + S(6)bdS(2)e*S(2) + S(8)cdS(4))/(S(16)d*S(6)xS(2)) - eS(2)sqrt(dS(2) - eS(2)x*S(2))(S(5)ae*S(4) + S(6)bdS(2)e*S(2) + S(8)cdS(4))atanh(sqrt(dS(2) - eS(2)xS(2))/d)/(S(16)d*S(7)sqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(7)sqrt(d - ex)sqrt(d + ex)), x), x, -a(dS(2) - eS(2)x*S(2))/(S(6)d*S(2)x*S(6)sqrt(d - ex)sqrt(d + ex)) - (dS(2) - eS(2)x*S(2))(S(5)ae*S(2) + S(6)bdS(2))/(S(24)d*S(4)x*S(4)sqrt(d - ex)sqrt(d + ex)) - (dS(2) - eS(2)x*S(2))(S(5)ae*S(4) + S(6)bdS(2)e*S(2) + S(8)cdS(4))/(S(16)d*S(6)x*S(2)sqrt(d - ex)sqrt(d + ex)) - eS(2)sqrt(dS(2) - eS(2)xS(2))(S(5)ae*S(4) + S(6)bdS(2)e*S(2) + S(8)cdS(4))atanh(sqrt(dS(2) - eS(2)xS(2))/d)/(S(16)d*S(7)sqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(a + bxS(2) + cx*S(4))/(sqrt(d - ex)sqrt(d + ex)), x), x, cxS(7)(-d + ex)sqrt(d + ex)/(S(8)e*S(2)sqrt(d - ex)) + dS(4)sqrt(dS(2) - eS(2)xS(2))(S(48)ae*S(4) + S(40)bdS(2)e*S(2) + S(35)cdS(4))atan(ex/sqrt(dS(2) - eS(2)x*S(2)))/(S(128)e*S(9)sqrt(d - ex)sqrt(d + ex)) - dS(2)xsqrt(d - ex)sqrt(d + ex)(S(48)aeS(4) + S(40)bdS(2)e*S(2) + S(35)cdS(4))/(S(128)eS(8)) - xS(5)sqrt(d - ex)sqrt(d + ex)(S(8)beS(2) + S(7)cdS(2))/(S(48)eS(4)) - xS(3)sqrt(d - ex)sqrt(d + ex)(S(48)aeS(4) + S(40)bdS(2)e*S(2) + S(35)cdS(4))/(S(192)eS(6)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(4)(a + bx*S(2) + cx*S(4))/(sqrt(d - ex)sqrt(d + ex)), x), x, -cxS(7)(dS(2) - eS(2)xS(2))/(S(8)e*S(2)sqrt(d - ex)sqrt(d + ex)) + dS(4)sqrt(dS(2) - eS(2)xS(2))(S(48)ae*S(4) + S(40)bdS(2)e*S(2) + S(35)cdS(4))atan(ex/sqrt(dS(2) - eS(2)x*S(2)))/(S(128)e*S(9)sqrt(d - ex)sqrt(d + ex)) - dS(2)x(dS(2) - eS(2)x*S(2))(S(48)ae*S(4) + S(40)bdS(2)e*S(2) + S(35)cdS(4))/(S(128)e*S(8)sqrt(d - ex)sqrt(d + ex)) - xS(5)(dS(2) - eS(2)xS(2))(S(8)be*S(2) + S(7)cdS(2))/(S(48)e*S(4)sqrt(d - ex)sqrt(d + ex)) - xS(3)(dS(2) - eS(2)xS(2))(S(48)ae*S(4) + S(40)bdS(2)e*S(2) + S(35)cdS(4))/(S(192)e*S(6)sqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bxS(2) + cx*S(4))/(sqrt(d - ex)sqrt(d + ex)), x), x, cxS(5)(-d + ex)sqrt(d + ex)/(S(6)e*S(2)sqrt(d - ex)) + dS(2)sqrt(dS(2) - eS(2)xS(2))(S(8)ae*S(4) + S(6)bdS(2)e*S(2) + S(5)cdS(4))atan(ex/sqrt(dS(2) - eS(2)x*S(2)))/(S(16)e*S(7)sqrt(d - ex)sqrt(d + ex)) - xS(3)sqrt(d - ex)sqrt(d + ex)(S(6)be*S(2) + S(5)cdS(2))/(S(24)e*S(4)) - xsqrt(d - ex)sqrt(d + ex)(S(8)ae*S(4) + S(6)bdS(2)e*S(2) + S(5)cdS(4))/(S(16)eS(6)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(2)(a + bx*S(2) + cx*S(4))/(sqrt(d - ex)sqrt(d + ex)), x), x, -cxS(5)(dS(2) - eS(2)xS(2))/(S(6)e*S(2)sqrt(d - ex)sqrt(d + ex)) + dS(2)sqrt(dS(2) - eS(2)xS(2))(S(8)ae*S(4) + S(6)bdS(2)e*S(2) + S(5)cdS(4))atan(ex/sqrt(dS(2) - eS(2)x*S(2)))/(S(16)e*S(7)sqrt(d - ex)sqrt(d + ex)) - xS(3)(dS(2) - eS(2)xS(2))(S(6)be*S(2) + S(5)cdS(2))/(S(24)e*S(4)sqrt(d - ex)sqrt(d + ex)) - x(dS(2) - eS(2)xS(2))(S(8)ae*S(4) + S(6)bdS(2)e*S(2) + S(5)cdS(4))/(S(16)e*S(6)sqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(sqrt(d - ex)sqrt(d + ex)), x), x, cxS(3)(-d + ex)sqrt(d + ex)/(S(4)e*S(2)sqrt(d - ex)) - xsqrt(d - ex)sqrt(d + ex)(S(4)be*S(2) + S(3)cdS(2))/(S(8)e*S(4)) - (S(8)aeS(4) + S(4)bdS(2)e*S(2) + S(3)cdS(4))atan(sqrt(d - ex)/sqrt(d + ex))/(S(4)eS(5)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))/(sqrt(d - ex)sqrt(d + ex)), x), x, -cxS(3)(dS(2) - eS(2)xS(2))/(S(4)e*S(2)sqrt(d - ex)sqrt(d + ex)) - x(dS(2) - eS(2)xS(2))(S(4)be*S(2) + S(3)cdS(2))/(S(8)e*S(4)sqrt(d - ex)sqrt(d + ex)) + sqrt(dS(2) - eS(2)x*S(2))(S(8)ae*S(4) + S(4)bdS(2)e*S(2) + S(3)cdS(4))atan(ex/sqrt(dS(2) - eS(2)x*S(2)))/(S(8)e*S(5)sqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(2)sqrt(d - ex)sqrt(d + ex)), x), x, -asqrt(d - ex)sqrt(d + ex)/(d*S(2)x) + cx(-d + ex)sqrt(d + ex)/(S(2)e*S(2)sqrt(d - ex)) - (S(2)beS(2) + cd*S(2))atan(sqrt(d - ex)/sqrt(d + ex))/e*S(3), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(2)sqrt(d - ex)sqrt(d + ex)), x), x, -a(dS(2) - eS(2)xS(2))/(dS(2)xsqrt(d - ex)sqrt(d + ex)) - cx(dS(2) - eS(2)x*S(2))/(S(2)e*S(2)sqrt(d - ex)sqrt(d + ex)) + sqrt(dS(2) - eS(2)x*S(2))(S(2)be*S(2) + cd*S(2))atan(ex/sqrt(dS(2) - eS(2)x*S(2)))/(S(2)e*S(3)sqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(4)sqrt(d - ex)sqrt(d + ex)), x), x, -asqrt(d - ex)sqrt(d + ex)/(S(3)dS(2)x*S(3)) - S(2)aeS(2)sqrt(d - ex)sqrt(d + ex)/(S(3)d*S(4)x) - bsqrt(d - ex)sqrt(d + ex)/(d*S(2)x) + csqrt(dS(2) - eS(2)x*S(2))atan(ex/sqrt(dS(2) - eS(2)x*S(2)))/(esqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(4)sqrt(d - ex)sqrt(d + ex)), x), x, -a(dS(2) - eS(2)x*S(2))/(S(3)d*S(2)x*S(3)sqrt(d - ex)sqrt(d + ex)) - S(2)aeS(2)(dS(2) - eS(2)xS(2))/(S(3)d*S(4)xsqrt(d - ex)sqrt(d + ex)) - b(dS(2) - eS(2)xS(2))/(dS(2)xsqrt(d - ex)sqrt(d + ex)) + csqrt(dS(2) - eS(2)xS(2))atan(ex/sqrt(dS(2) - eS(2)x*S(2)))/(esqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))/(xS(6)sqrt(d - ex)sqrt(d + ex)), x), x, -asqrt(d - ex)sqrt(d + ex)/(S(5)dS(2)x*S(5)) - c(-d + ex)sqrt(d + ex)/(S(2)e*S(2)x*S(3)sqrt(d - ex)) - sqrt(d - ex)sqrt(d + ex)(S(8)aeS(4) + S(10)bdS(2)e*S(2) + S(15)cdS(4))/(S(30)d*S(4)e*S(2)x*S(3)) - sqrt(d - ex)sqrt(d + ex)(S(8)aeS(4) + S(10)bdS(2)e*S(2) + S(15)cdS(4))/(S(15)d*S(6)x), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + bxS(2) + cxS(4))/(xS(6)sqrt(d - ex)sqrt(d + ex)), x), x, -a(dS(2) - eS(2)x*S(2))/(S(5)d*S(2)x*S(5)sqrt(d - ex)sqrt(d + ex)) + c(dS(2) - eS(2)xS(2))/(S(2)e*S(2)x*S(3)sqrt(d - ex)sqrt(d + ex)) - (dS(2) - eS(2)x*S(2))(S(8)ae*S(4) + S(10)bdS(2)e*S(2) + S(15)cdS(4))/(S(30)d*S(4)e*S(2)x*S(3)sqrt(d - ex)sqrt(d + ex)) - (dS(2) - eS(2)x*S(2))(S(8)ae*S(4) + S(10)bdS(2)e*S(2) + S(15)cdS(4))/(S(15)d*S(6)xsqrt(d - ex)sqrt(d + ex)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)m(a + cx(S(2)n))*p/(d + ex*n), x), x, x(fx)m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1((m + S(1))/(S(2)n), -p, S(1), S(1) + (m + S(1))/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/d*S(2))/(d(m + S(1))) - ex(n + S(1))(fx)m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1((m + n + S(1))/(S(2)n), -p, S(1), (m + S(3)n + S(1))/(S(2)n), -cx*(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(2)(m + n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(a + cx(S(2)n))*p/(d + exn)S(2), x), x, x(fx)*m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1((m + S(1))/(S(2)n), -p, S(2), S(1) + (m + S(1))/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(2)(m + S(1))) - S(2)ex(n + S(1))(fx)m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1((m + n + S(1))/(S(2)n), -p, S(2), (m + S(3)n + S(1))/(S(2)n), -cx*(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(3)(m + n + S(1))) + eS(2)x*(S(2)n + S(1))(fx)*m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1((m + S(2)n + S(1))/(S(2)n), -p, S(2), (m + S(4)n + S(1))/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(4)(m + S(2)n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)m(a + cx(S(2)n))*p/(d + exn)S(3), x), x, x(fx)*m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1((m + S(1))/(S(2)n), -p, S(3), S(1) + (m + S(1))/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(3)(m + S(1))) - S(3)ex(n + S(1))(fx)m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1((m + n + S(1))/(S(2)n), -p, S(3), (m + S(3)n + S(1))/(S(2)n), -cx*(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(4)(m + n + S(1))) + S(3)e*S(2)x*(S(2)n + S(1))(fx)*m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1((m + S(2)n + S(1))/(S(2)n), -p, S(3), (m + S(4)n + S(1))/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(5)(m + S(2)n + S(1))) - e*S(3)x*(S(3)n + S(1))(fx)*m(S(1) + cx(S(2)n)/a)*(-p)(a + cx(S(2)n))*pAppellF1((m + S(3)n + S(1))/(S(2)n), -p, S(3), (m + S(5)n + S(1))/(S(2)n), -cx(S(2)n)/a, e*S(2)x*(S(2)n)/dS(2))/(dS(6)(m + S(3)n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)m(d + exn)S(2)(a + bxn + cx*(S(2)n))p, x), x, dS(2)(fx)*(m + S(1))(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1((m + S(1))/n, -p, -p, (m + n + S(1))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(f(m + S(1))) + S(2)dex(n + S(1))(fx)m(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1((m + n + S(1))/n, -p, -p, (m + S(2)n + S(1))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(m + n + S(1)) + eS(2)x*(S(2)n + S(1))(fx)*m(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1((m + S(2)n + S(1))/n, -p, -p, (m + S(3)n + S(1))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(m + S(2)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)m(d + exn)(a + bxn + cx*(S(2)n))*p, x), x, d(fx)(m + S(1))(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1((m + S(1))/n, -p, -p, (m + n + S(1))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(f(m + S(1))) + ex(n + S(1))(fx)m(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1((m + n + S(1))/n, -p, -p, (m + S(2)n + S(1))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(m + n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)*m(a + bxn + cx*(S(2)n))*p, x), x, (fx)*(m + S(1))(S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))) + S(1))(-p)(S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))) + S(1))(-p)(a + bxn + cx*(S(2)n))*pAppellF1((m + S(1))/n, -p, -p, (m + n + S(1))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(f(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((fx)m(d + exn)q/(a + bx*n + cx*(S(2)n)), x), x, -S(2)c(fx)(m + S(1))(S(1) + exn/d)(-q)(d + exn)qAppellF1((m + S(1))/n, S(1), -q, (m + n + S(1))/n, -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))), -ex*n/d)/(f(b + sqrt(-S(4)ac + b*S(2)))(m + S(1))sqrt(-S(4)ac + bS(2))) + S(2)c(fx)*(m + S(1))(S(1) + exn/d)(-q)(d + exn)qAppellF1((m + S(1))/n, S(1), -q, (m + n + S(1))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -ex*n/d)/(f(b - sqrt(-S(4)ac + b*S(2)))(m + S(1))sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(d + exn)q/(a + bx*n + cx*(S(2)n)), x), x, -S(2)cx*S(3)(S(1) + exn/d)(-q)(d + exn)qAppellF1(S(3)/n, S(1), -q, (n + S(3))/n, -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))), -ex*n/d)/(-S(12)ac + S(3)b*S(2) + S(3)bsqrt(-S(4)ac + bS(2))) - S(2)cxS(3)(S(1) + exn/d)(-q)(d + exn)qAppellF1(S(3)/n, S(1), -q, (n + S(3))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -ex*n/d)/(-S(12)ac + S(3)b*S(2) - S(3)bsqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(d + exn)q/(a + bx*n + cx*(S(2)n)), x), x, -cxS(2)(S(1) + exn/d)(-q)(d + exn)qAppellF1(S(2)/n, S(1), -q, (n + S(2))/n, -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))), -ex*n/d)/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - cx*S(2)(S(1) + exn/d)(-q)(d + exn)qAppellF1(S(2)/n, S(1), -q, (n + S(2))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -ex*n/d)/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)q/(a + bxn + cx*(S(2)n)), x), x, -S(2)cx(S(1) + exn/d)(-q)(d + exn)qAppellF1(S(1)/n, S(1), -q, S(1) + S(1)/n, -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))), -ex*n/d)/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - S(2)cx(S(1) + exn/d)(-q)(d + exn)qAppellF1(S(1)/n, S(1), -q, S(1) + S(1)/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -ex*n/d)/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)q/(x*S(2)(a + bxn + cx*(S(2)n))), x), x, S(2)c(S(1) + exn/d)(-q)(d + exn)qAppellF1(-S(1)/n, S(1), -q, -(-n + S(1))/n, -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))), -ex*n/d)/(x(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) + S(2)c(S(1) + exn/d)(-q)(d + exn)qAppellF1(-S(1)/n, S(1), -q, -(-n + S(1))/n, -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))), -ex*n/d)/(x(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + exn)q/(x*S(3)(a + bxn + cx*(S(2)n))), x), x, c(S(1) + exn/d)(-q)(d + exn)qAppellF1(-S(2)/n, S(1), -q, -(-n + S(2))/n, -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))), -exn/d)/(xS(2)(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2)))) + c(S(1) + exn/d)(-q)(d + exn)qAppellF1(-S(2)/n, S(1), -q, -(-n + S(2))/n, -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))), -exn/d)/(xS(2)(-S(4)ac + bS(2) - bsqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) def test_4(): assert rubi_test(rubi_integrate((xS(3) + xS(2))/(xS(2) + x + S(-2)), x), x, x*S(2)/S(2) + S(2)log(-x + S(1))/S(3) + S(4)log(x + S(2))/S(3), expand=True, _diff=True, _numerical=True) # Large time assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4) + jx*S(5) + kx*S(6) + lx*S(7) + mx*S(8))/(a + bx + cxS(2)), x), x, mx*S(7)/(S(7)c) + x*S(6)(-bm + cl)/(S(6)cS(2)) + xS(5)(b*S(2)m + c*S(2)k - c(am + bl))/(S(5)cS(3)) + xS(4)(-bS(3)m + bc(S(2)am + bl) + cS(3)j - c*S(2)(al + bk))/(S(4)cS(4)) + xS(3)(b*S(4)m - b*S(2)c(S(3)am + bl) + c*S(4)h - c*S(3)(ak + bj) + c*S(2)(a*S(2)m + S(2)abl + bS(2)k))/(S(3)cS(5)) + xS(2)(-b*S(5)m + b*S(3)c(S(4)am + bl) - bcS(2)(S(3)aS(2)m + S(3)abl + bS(2)k) + c*S(5)g - c*S(4)(aj + bh) + c*S(3)(a*S(2)l + S(2)abk + bS(2)j))/(S(2)cS(6)) + x(b*S(6)m - b*S(4)c(S(5)am + bl) + b*S(2)c*S(2)(S(6)aS(2)m + S(4)abl + bS(2)k) + c*S(6)f - c*S(5)(ah + bg) + c*S(4)(a*S(2)k + S(2)abj + bS(2)h) - c*S(3)(a*S(3)m + S(3)aS(2)bl + S(3)abS(2)k + b*S(3)j))/cS(7) + (-bS(7)m + bS(5)c(S(6)am + bl) - b*S(3)c*S(2)(S(10)aS(2)m + S(5)abl + bS(2)k) + bcS(3)(S(4)aS(3)m + S(6)aS(2)bl + S(4)abS(2)k + b*S(3)j) + c*S(7)e - c*S(6)(ag + bf) + c*S(5)(a*S(2)j + S(2)abh + bS(2)g) - c*S(4)(a*S(3)l + S(3)aS(2)bk + S(3)abS(2)j + b*S(3)h))log(a + bx + cxS(2))/(S(2)cS(8)) - (bS(8)m - bS(6)c(S(8)am + bl) + b*S(4)c*S(2)(S(20)aS(2)m + S(7)abl + bS(2)k) - b*S(2)c*S(3)(S(16)aS(3)m + S(14)aS(2)bl + S(6)abS(2)k + b*S(3)j) + S(2)cS(8)d - c*S(7)(S(2)af + be) + cS(6)(S(2)aS(2)h + S(3)abg + bS(2)f) - c*S(5)(S(2)aS(3)k + S(5)aS(2)bj + S(4)abS(2)h + b*S(3)g) + c*S(4)(S(2)aS(4)m + S(7)aS(3)bl + S(9)a*S(2)b*S(2)k + S(5)ab*S(3)j + b*S(4)h))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(8)sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3))/(a + bx + cxS(2))S(2), x), x, glog(a + bx + cx*S(2))/(S(2)c*S(2)) - (-ab*S(2)g - S(2)ac(-ag + ce) + bc(af + cd) + x(-b*S(3)g + bc(S(3)ag + bf) + S(2)c*S(3)d - c*S(2)(S(2)af + be)))/(cS(2)(-S(4)ac + b*S(2))(a + bx + cx*S(2))) + (-S(6)abcg + bS(3)g + S(4)cS(3)d - c*S(2)(-S(4)af + S(2)be))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(2)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2) + gx*S(3) + hx*S(4) + ix*S(5))/(a + bx + cxS(2))S(3), x), x, -xS(3)(-S(5)bi + S(2)ch)/(S(2)cS(2)(a + bx + cxS(2))S(2)) + ilog(a + bx + cxS(2))/(S(2)cS(3)) - xS(2)(-S(4)aci - S(9)bS(2)i + S(2)bch + S(2)c*S(2)g)/(S(4)cS(3)(a + bx + cxS(2))S(2)) - (-S(30)aS(2)bcS(2)i + S(10)ab*S(3)ci - bS(5)i + S(12)cS(5)d - c*S(4)(-S(4)af + S(6)be) + S(2)cS(3)(S(6)aS(2)h - S(3)abg + bS(2)f))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(3)(-S(4)ac + bS(2))(S(5)/2)) + (b + S(2)cx)(-S(30)a*S(2)bcS(2)i + S(10)ab*S(3)ci - bS(5)i + S(12)cS(5)d - c*S(4)(-S(4)af + S(6)be) + S(2)cS(3)(S(6)aS(2)h - S(3)abg + bS(2)f))/(S(4)cS(4)(-S(4)ac + bS(2))S(2)(a + bx + cxS(2))) - (cS(3)(-abS(4)i/c*S(3) - S(4)a(-S(3)a*S(2)i/c + ag + ce) + S(2)b(a*S(2)h/c + af + cd)) + x(-S(8)abS(3)ci + S(2)abc*S(2)(S(23)ai + S(2)bh) - b*S(5)i + S(4)cS(5)d - S(2)cS(4)(S(2)af + be) + S(2)c*S(3)(-S(6)aS(2)h - abg + b*S(2)f)))/(S(4)cS(4)(-S(4)ac + b*S(2))(a + bx + cxS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2) + gx*S(3) + hx*S(4))/(a + bx + cxS(2))(S(5)/2), x), x, -xS(2)(-S(2)bh + cg)/(cS(2)(a + bx + cxS(2))(S(3)/2)) + (b + S(2)cx)(-S(4)b*S(4)h + b*S(2)c(S(28)ah + bg) + S(16)cS(4)d - c*S(3)(-S(8)af + S(8)be) + S(2)cS(2)(-S(16)aS(2)h - S(6)abg + bS(2)f))/(S(3)cS(3)(-S(4)ac + bS(2))S(2)sqrt(a + bx + cxS(2))) - (cS(2)(-S(4)ab*S(3)h/c*S(2) + ab*S(2)g/c - S(4)a(S(2)ag + ce) + S(2)b(S(9)a*S(2)h/c + af + cd)) + x(-S(4)b*S(4)h + b*S(2)c(S(16)ah + bg) + S(4)cS(4)d - S(2)cS(3)(S(2)af + be) + S(2)c*S(2)(S(2)aS(2)h - S(3)abg + bS(2)f)))/(S(3)cS(3)(-S(4)ac + b*S(2))(a + bx + cxS(2))(S(3)/2)) + hatanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cxS(2))))/c(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4))/(a + bx - cxS(2))(S(5)/2), x), x, xS(2)(S(2)bh + cg)/(cS(2)(a + bx - cxS(2))(S(3)/2)) - (b - S(2)cx)(-S(4)b*S(4)h - b*S(2)c(S(28)ah + bg) + S(16)cS(4)d + S(8)cS(3)(-af + be) + S(2)cS(2)(-S(16)aS(2)h - S(6)abg + bS(2)f))/(S(3)cS(3)(S(4)ac + bS(2))S(2)sqrt(a + bx - cxS(2))) - (cS(2)(S(4)ab*S(3)h/c*S(2) + ab*S(2)g/c - S(4)a(-S(2)ag + ce) + S(2)b(S(9)a*S(2)h/c - af + cd)) - x(-S(4)b*S(4)h - b*S(2)c(S(16)ah + bg) + S(4)cS(4)d + S(2)cS(3)(S(2)af + be) + S(2)c*S(2)(S(2)aS(2)h - S(3)abg + bS(2)f)))/(S(3)cS(3)(S(4)ac + b*S(2))(a + bx - cxS(2))(S(3)/2)) - hatan((b - S(2)cx)/(S(2)sqrt(c)sqrt(a + bx - cxS(2))))/c(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)(a + bx*S(2) + cx*S(4)), x), x, adx + aexS(2)/S(2) + bdxS(3)/S(3) + bexS(4)/S(4) + cdxS(5)/S(5) + cexS(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cx*S(4))(d + ex + fx*S(2)), x), x, adx + aexS(2)/S(2) + bexS(4)/S(4) + cexS(6)/S(6) + cfxS(7)/S(7) + xS(5)(bf/S(5) + cd/S(5)) + x*S(3)(af/S(3) + bd/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cx*S(4))(d + ex + fx*S(2) + gx*S(3)), x), x, adx + aexS(2)/S(2) + cfxS(7)/S(7) + cgxS(8)/S(8) + xS(6)(bg/S(6) + ce/S(6)) + x*S(5)(bf/S(5) + cd/S(5)) + x*S(4)(ag/S(4) + be/S(4)) + x*S(3)(af/S(3) + bd/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cx*S(4))(d + ex + fx*S(2) + gx*S(3) + hx*S(4)), x), x, adx + aexS(2)/S(2) + cgxS(8)/S(8) + chxS(9)/S(9) + xS(7)(bh/S(7) + cf/S(7)) + x*S(6)(bg/S(6) + ce/S(6)) + x*S(5)(ah/S(5) + bf/S(5) + cd/S(5)) + xS(4)(ag/S(4) + be/S(4)) + x*S(3)(af/S(3) + bd/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cx*S(4))(d + ex + fx*S(2) + gx*S(3) + hx*S(4) + ix*S(5)), x), x, adx + aexS(2)/S(2) + chxS(9)/S(9) + cixS(10)/S(10) + xS(8)(bi/S(8) + cg/S(8)) + x*S(7)(bh/S(7) + cf/S(7)) + x*S(6)(ai/S(6) + bg/S(6) + ce/S(6)) + xS(5)(ah/S(5) + bf/S(5) + cd/S(5)) + xS(4)(ag/S(4) + be/S(4)) + x*S(3)(af/S(3) + bd/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)(a + bxS(2) + cxS(4))S(2), x), x, a*S(2)dx + aS(2)exS(2)/S(2) + S(2)abdxS(3)/S(3) + abex*S(4)/S(2) + S(2)bcdxS(7)/S(7) + bcexS(8)/S(4) + cS(2)dxS(9)/S(9) + cS(2)ex*S(10)/S(10) + dx*S(5)(S(2)ac/S(5) + b*S(2)/S(5)) + ex*S(6)(ac/S(3) + bS(2)/S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2) + cxS(4))S(2)(d + ex + fxS(2)), x), x, aS(2)dx + aS(2)exS(2)/S(2) + abex*S(4)/S(2) + ax*S(3)(af + S(2)bd)/S(3) + bcexS(8)/S(4) + cS(2)exS(10)/S(10) + cS(2)fx*S(11)/S(11) + cx*S(9)(S(2)bf + cd)/S(9) + ex*S(6)(ac/S(3) + bS(2)/S(6)) + xS(7)(S(2)acf/S(7) + bS(2)f/S(7) + S(2)bcd/S(7)) + xS(5)(S(2)abf/S(5) + S(2)acd/S(5) + b*S(2)d/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))S(2)(d + ex + fxS(2) + gxS(3)), x), x, aS(2)dx + a*S(2)exS(2)/S(2) + ax*S(4)(ag + S(2)be)/S(4) + ax*S(3)(af + S(2)bd)/S(3) + cS(2)fxS(11)/S(11) + cS(2)gxS(12)/S(12) + cx*S(10)(S(2)bg + ce)/S(10) + cx*S(9)(S(2)bf + cd)/S(9) + xS(8)(acg/S(4) + b*S(2)g/S(8) + bce/S(4)) + x*S(7)(S(2)acf/S(7) + bS(2)f/S(7) + S(2)bcd/S(7)) + xS(6)(abg/S(3) + ace/S(3) + b*S(2)e/S(6)) + x*S(5)(S(2)abf/S(5) + S(2)acd/S(5) + b*S(2)d/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))S(2)(d + ex + fxS(2) + gx*S(3) + hxS(4)), x), x, aS(2)dx + a*S(2)exS(2)/S(2) + ax*S(4)(ag + S(2)be)/S(4) + ax*S(3)(af + S(2)bd)/S(3) + cS(2)gxS(12)/S(12) + cS(2)hxS(13)/S(13) + cx*S(11)(S(2)bh + cf)/S(11) + cx*S(10)(S(2)bg + ce)/S(10) + xS(9)(b*S(2)h/S(9) + c*S(2)d/S(9) + S(2)c(ah + bf)/S(9)) + x*S(8)(acg/S(4) + b*S(2)g/S(8) + bce/S(4)) + x*S(7)(S(2)acf/S(7) + bS(2)f/S(7) + S(2)b(ah + cd)/S(7)) + x*S(6)(abg/S(3) + ace/S(3) + b*S(2)e/S(6)) + x*S(5)(S(2)abf/S(5) + a(ah + S(2)cd)/S(5) + bS(2)d/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(xS(4) - S(5)x*S(2) + S(4)), x), x, -datanh(x/S(2))/S(6) + datanh(x)/S(3) - elog(-x*S(2) + S(1))/S(6) + elog(-x*S(2) + S(4))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2))/(xS(4) - S(5)x*S(2) + S(4)), x), x, -elog(-x*S(2) + S(1))/S(6) + elog(-x*S(2) + S(4))/S(6) + (-d/S(6) - S(2)f/S(3))atanh(x/S(2)) + (d/S(3) + f/S(3))atanh(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2) + gxS(3))/(xS(4) - S(5)xS(2) + S(4)), x), x, (-d/S(6) - S(2)f/S(3))atanh(x/S(2)) + (d/S(3) + f/S(3))atanh(x) - (e/S(6) + g/S(6))log(-xS(2) + S(1)) + (e/S(6) + S(2)g/S(3))log(-xS(2) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hxS(4))/(xS(4) - S(5)xS(2) + S(4)), x), x, hx - (e/S(6) + g/S(6))log(-xS(2) + S(1)) + (e/S(6) + S(2)g/S(3))log(-xS(2) + S(4)) - (d/S(6) + S(2)f/S(3) + S(8)h/S(3))atanh(x/S(2)) + (d/S(3) + f/S(3) + h/S(3))atanh(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4) + ixS(5))/(xS(4) - S(5)xS(2) + S(4)), x), x, hx + ixS(2)/S(2) - (d/S(6) + S(2)f/S(3) + S(8)h/S(3))atanh(x/S(2)) + (d/S(3) + f/S(3) + h/S(3))atanh(x) - (e/S(6) + g/S(6) + i/S(6))log(-x*S(2) + S(1)) + (e/S(6) + S(2)g/S(3) + S(8)i/S(3))log(-x*S(2) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(xS(4) + xS(2) + S(1)), x), x, -dlog(xS(2) - x + S(1))/S(4) + dlog(x*S(2) + x + S(1))/S(4) - sqrt(S(3))datan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) + sqrt(S(3))datan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(6) + sqrt(S(3))eatan(sqrt(S(3))(S(2)xS(2) + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2))/(xS(4) + xS(2) + S(1)), x), x, sqrt(S(3))eatan(sqrt(S(3))(S(2)xS(2) + S(1))/S(3))/S(3) - (d/S(4) - f/S(4))log(x*S(2) - x + S(1)) + (d/S(4) - f/S(4))log(x*S(2) + x + S(1)) - sqrt(S(3))(d + f)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) + sqrt(S(3))(d + f)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gxS(3))/(xS(4) + x*S(2) + S(1)), x), x, glog(xS(4) + xS(2) + S(1))/S(4) - (d/S(4) - f/S(4))log(xS(2) - x + S(1)) + (d/S(4) - f/S(4))log(x*S(2) + x + S(1)) - sqrt(S(3))(d + f)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) + sqrt(S(3))(d + f)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(6) + sqrt(S(3))(S(2)e - g)atan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hxS(4))/(xS(4) + x*S(2) + S(1)), x), x, glog(xS(4) + xS(2) + S(1))/S(4) + hx - (d/S(4) - f/S(4))log(x*S(2) - x + S(1)) + (d/S(4) - f/S(4))log(x*S(2) + x + S(1)) + sqrt(S(3))(S(2)e - g)atan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(6) - sqrt(S(3))(d + f - S(2)h)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) + sqrt(S(3))(d + f - S(2)h)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4) + ixS(5))/(xS(4) + x*S(2) + S(1)), x), x, hx + ixS(2)/S(2) - (d/S(4) - f/S(4))log(x*S(2) - x + S(1)) + (d/S(4) - f/S(4))log(x*S(2) + x + S(1)) + (g/S(4) - i/S(4))log(xS(4) + xS(2) + S(1)) - sqrt(S(3))(d + f - S(2)h)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(6) + sqrt(S(3))(d + f - S(2)h)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(6) + sqrt(S(3))(S(2)e - g - i)atan(sqrt(S(3))(S(2)xS(2) + S(1))/S(3))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(a + bxS(2) + cx*S(4)), x), x, -sqrt(S(2))sqrt(c)datan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)datan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))) - eatanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2))/(a + bx*S(2) + cx*S(4)), x), x, -eatanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)) + sqrt(S(2))(f - (-bf + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(f + (-bf + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3))/(a + bx*S(2) + cx*S(4)), x), x, glog(a + bxS(2) + cx*S(4))/(S(4)c) - (-bg + S(2)ce)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-S(4)ac + bS(2))) + sqrt(S(2))(f - (-bf + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(f + (-bf + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4))/(a + bx*S(2) + cx*S(4)), x), x, glog(a + bxS(2) + cx*S(4))/(S(4)c) + hx/c - (-bg + S(2)ce)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-S(4)ac + bS(2))) + sqrt(S(2))(-bh/c + f - (-S(2)ach + b*S(2)h - bcf + S(2)cS(2)d)/(csqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(-bh/c + f + (bS(2)h + S(2)cS(2)d - c(S(2)ah + bf))/(csqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4) + ix*S(5))/(a + bx*S(2) + cx*S(4)), x), x, hx/c + ixS(2)/(S(2)c) + (-bi + cg)log(a + bx*S(2) + cx*S(4))/(S(4)c*S(2)) - (-S(2)aci + b*S(2)i - bcg + S(2)cS(2)e)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)sqrt(-S(4)ac + b*S(2))) + sqrt(S(2))(-bh/c + f - (-S(2)ach + b*S(2)h - bcf + S(2)cS(2)d)/(csqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(-bh/c + f + (bS(2)h + S(2)cS(2)d - c(S(2)ah + bf))/(csqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) # failing assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4) + jx*S(5) + kx*S(6) + lx*S(7) + mx*S(8))/(a + bx*S(2) + cx*S(4)), x), x, lx*S(4)/(S(4)c) + mxS(5)/(S(5)c) + x*S(3)(-bm + ck)/(S(3)cS(2)) + xS(2)(-bl + cj)/(S(2)cS(2)) + x(b*S(2)m + c*S(2)h - c(am + bk))/cS(3) + (bS(2)l + c*S(2)g - c(al + bj))log(a + bxS(2) + cx*S(4))/(S(4)cS(3)) - (-bS(3)l + bc(S(3)al + bj) + S(2)cS(3)e - c*S(2)(S(2)aj + bg))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(3)sqrt(-S(4)ac + b*S(2))) + sqrt(S(2))(S(2)abm/cS(2) - ak/c - b*S(3)m/cS(3) + bS(2)k/cS(2) - bh/c + f - (b*S(4)m - b*S(2)c(S(4)am + bk) + S(2)cS(4)d - c*S(3)(S(2)ah + bf) + cS(2)(S(2)aS(2)m + S(3)abk + bS(2)h))/(c*S(3)sqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(S(2)abm/cS(2) - ak/c - b*S(3)m/cS(3) + bS(2)k/cS(2) - bh/c + f + (b*S(4)m - b*S(2)c(S(4)am + bk) + S(2)cS(4)d - c*S(3)(S(2)ah + bf) + cS(2)(S(2)aS(2)m + S(3)abk + bS(2)h))/(c*S(3)sqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(x*S(4) - S(5)xS(2) + S(4))S(2), x), x, S(19)datanh(x/S(2))/S(432) - datanh(x)/S(54) + elog(-x*S(2) + S(1))/S(27) - elog(-x*S(2) + S(4))/S(27) + x(-S(5)dx*S(2) + S(17)d - S(5)ex*S(3) + S(17)ex)/(S(72)x*S(4) - S(360)x*S(2) + S(288)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, elog(-xS(2) + S(1))/S(27) - elog(-x*S(2) + S(4))/S(27) + x(S(17)d - S(5)exS(3) + S(17)ex + S(20)f - x*S(2)(S(5)d + S(8)f))/(S(72)xS(4) - S(360)x*S(2) + S(288)) - (d/S(54) + S(7)f/S(54))atanh(x) + (S(19)d/S(432) + S(13)f/S(108))atanh(x/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2) + gxS(3))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, x(S(17)d + S(20)f - x*S(3)(S(5)e + S(8)g) - x*S(2)(S(5)d + S(8)f) + x(S(17)e + S(20)g))/(S(72)x*S(4) - S(360)x*S(2) + S(288)) - (d/S(54) + S(7)f/S(54))atanh(x) + (S(19)d/S(432) + S(13)f/S(108))atanh(x/S(2)) + (e/S(27) + S(5)g/S(54))log(-x*S(2) + S(1)) - (e/S(27) + S(5)g/S(54))log(-xS(2) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hxS(4))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, x(S(17)d + S(20)f + S(32)h - xS(3)(S(5)e + S(8)g) - x*S(2)(S(5)d + S(8)f + S(20)h) + x(S(17)e + S(20)g))/(S(72)xS(4) - S(360)x*S(2) + S(288)) + (e/S(27) + S(5)g/S(54))log(-xS(2) + S(1)) - (e/S(27) + S(5)g/S(54))log(-xS(2) + S(4)) - (d/S(54) + S(7)f/S(54) + S(13)h/S(54))atanh(x) + (S(19)d/S(432) + S(13)f/S(108) + S(7)h/S(27))atanh(x/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2) + gx*S(3) + hx*S(4) + ixS(5))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, x(S(17)d + S(20)f + S(32)h - xS(3)(S(5)e + S(8)g + S(20)i) - xS(2)(S(5)d + S(8)f + S(20)h) + x(S(17)e + S(20)g + S(32)i))/(S(72)x*S(4) - S(360)x*S(2) + S(288)) - (d/S(54) + S(7)f/S(54) + S(13)h/S(54))atanh(x) + (S(19)d/S(432) + S(13)f/S(108) + S(7)h/S(27))atanh(x/S(2)) + (e/S(27) + S(5)g/S(54) + S(4)i/S(27))log(-xS(2) + S(1)) - (e/S(27) + S(5)g/S(54) + S(4)i/S(27))log(-x*S(2) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(xS(4) + xS(2) + S(1))*S(2), x), x, -dlog(x*S(2) - x + S(1))/S(4) + dlog(x*S(2) + x + S(1))/S(4) - sqrt(S(3))datan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(9) + sqrt(S(3))datan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(9) + S(2)sqrt(S(3))eatan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(9) + x(-dxS(2) + d - ex*S(3) + ex)/(S(6)xS(4) + S(6)x*S(2) + S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2))/(xS(4) + xS(2) + S(1))S(2), x), x, S(2)sqrt(S(3))eatan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(9) + x(d - exS(3) + ex + f - x*S(2)(d - S(2)f))/(S(6)x*S(4) + S(6)x*S(2) + S(6)) - (d/S(4) - f/S(8))log(x*S(2) - x + S(1)) + (d/S(4) - f/S(8))log(x*S(2) + x + S(1)) - sqrt(S(3))(S(4)d + f)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(36) + sqrt(S(3))(S(4)d + f)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(36), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gxS(3))/(xS(4) + xS(2) + S(1))S(2), x), x, x(d + f - xS(3)(e - S(2)g) - xS(2)(d - S(2)f) + x(e + g))/(S(6)xS(4) + S(6)x*S(2) + S(6)) - (d/S(4) - f/S(8))log(x*S(2) - x + S(1)) + (d/S(4) - f/S(8))log(x*S(2) + x + S(1)) - sqrt(S(3))(S(4)d + f)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(36) + sqrt(S(3))(S(4)d + f)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(36) + sqrt(S(3))(S(2)e - g)atan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hxS(4))/(xS(4) + xS(2) + S(1))S(2), x), x, x(d + f - S(2)h - x*S(3)(e - S(2)g) - xS(2)(d - S(2)f + h) + x(e + g))/(S(6)xS(4) + S(6)x*S(2) + S(6)) + sqrt(S(3))(S(2)e - g)atan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(9) - (d/S(4) - f/S(8) + h/S(8))log(x*S(2) - x + S(1)) + (d/S(4) - f/S(8) + h/S(8))log(x*S(2) + x + S(1)) - sqrt(S(3))(S(4)d + f + h)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(36) + sqrt(S(3))(S(4)d + f + h)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(36), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4) + ixS(5))/(xS(4) + xS(2) + S(1))S(2), x), x, x(d + f - S(2)h - x*S(3)(e - S(2)g + i) - xS(2)(d - S(2)f + h) + x(e + g - S(2)i))/(S(6)x*S(4) + S(6)x*S(2) + S(6)) - (d/S(4) - f/S(8) + h/S(8))log(x*S(2) - x + S(1)) + (d/S(4) - f/S(8) + h/S(8))log(x*S(2) + x + S(1)) - sqrt(S(3))(S(4)d + f + h)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(36) + sqrt(S(3))(S(4)d + f + h)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(36) + sqrt(S(3))(S(2)e - g + S(2)i)atan(sqrt(S(3))(S(2)xS(2) + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(a + bxS(2) + cxS(4))S(2), x), x, S(2)ceatanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + sqrt(S(2))sqrt(c)d(b - (-S(12)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)d(b + (-S(12)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + x(bcdxS(2) + bcex*S(3) + d(-S(2)ac + b*S(2)) + ex(-S(2)ac + bS(2)))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2))/(a + bx*S(2) + cxS(4))S(2), x), x, S(2)ceatanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + sqrt(S(2))sqrt(c)(-S(2)af + bd - (S(4)abf - S(12)acd + b*S(2)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)(-S(2)af + bd + (S(4)abf - S(12)acd + b*S(2)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + x(-abf - S(2)acd + bS(2)d + bcexS(3) + cx*S(2)(-S(2)af + bd) + ex(-S(2)ac + bS(2)))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3))/(a + bx*S(2) + cxS(4))S(2), x), x, (-bg + S(2)ce)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + sqrt(S(2))sqrt(c)(-S(2)af + bd - (S(4)abf - S(12)acd + b*S(2)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)(-S(2)af + bd + (S(4)abf - S(12)acd + b*S(2)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + x(-abf - S(2)acd + bS(2)d + cxS(3)(-S(2)ag + be) + cx*S(2)(-S(2)af + bd) + x(-abg - S(2)ace + bS(2)e))/(S(2)a(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4))/(a + bx*S(2) + cxS(4))S(2), x), x, (-bg + S(2)ce)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + x(-abf - S(2)a(-ah + cd) + b*S(2)d + cxS(3)(-S(2)ag + be) + xS(2)(abh - S(2)acf + bcd) + x(-abg - S(2)ace + bS(2)e))/(S(2)a(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) - sqrt(S(2))(-S(2)ac(S(2)ah + S(6)cd - fsqrt(-S(4)ac + bS(2))) + bS(2)(-ah + cd) - b(-S(4)acf + ahsqrt(-S(4)ac + bS(2)) + cdsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(c)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))(-S(2)ac(S(2)ah + S(6)cd + fsqrt(-S(4)ac + bS(2))) + bS(2)(-ah + cd) + b(S(4)acf + ahsqrt(-S(4)ac + b*S(2)) + cdsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(c)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2) + gx*S(3) + hx*S(4) + ix*S(5))/(a + bx*S(2) + cxS(4))S(2), x), x, (S(2)ai - bg + S(2)ce)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + x(-abf - S(2)a(-ah + cd) + b*S(2)d + x*S(3)(abi - S(2)acg + bce) + xS(2)(abh - S(2)acf + bcd) + x(-abg - S(2)a(-ai + ce) + b*S(2)e))/(S(2)a(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) - sqrt(S(2))(-S(2)ac(S(2)ah + S(6)cd - fsqrt(-S(4)ac + bS(2))) + bS(2)(-ah + cd) - b(-S(4)acf + ahsqrt(-S(4)ac + bS(2)) + cdsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(c)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))(-S(2)ac(S(2)ah + S(6)cd + fsqrt(-S(4)ac + bS(2))) + bS(2)(-ah + cd) + b(S(4)acf + ahsqrt(-S(4)ac + b*S(2)) + cdsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(c)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2) + gx*S(3) + hx*S(4) + jx*S(5) + kx*S(6) + lx*S(7) + mx*S(8))/(a + bx*S(2) + cxS(4))S(2), x), x, llog(a + bx*S(2) + cx*S(4))/(S(4)c*S(2)) + mx/c*S(2) + (-S(6)abcl + bS(3)l + S(4)cS(3)e - c*S(2)(-S(4)aj + S(2)bg))atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)(-S(4)ac + bS(2))(S(3)/2)) - x(abc(ak + cf) + S(2)ac(aS(2)m - ach + c*S(2)d) - b*S(2)(a*S(2)m + c*S(2)d) - cxS(3)(-abS(2)l - S(2)ac(-al + cg) + bc(aj + ce)) - cx(-ab(al + cg) - S(2)ac(-aj + ce) + b*S(2)ce) + xS(2)(-abS(3)m + abS(2)ck + S(2)acS(2)(-ak + cf) - bc(-S(3)aS(2)m + ach + c*S(2)d)))/(S(2)ac*S(2)(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) - sqrt(S(2))(S(3)ab*S(4)m - abS(3)(ck - S(3)msqrt(-S(4)ac + bS(2))) - S(2)acS(2)(-S(10)aS(2)m + S(2)ach - S(3)aksqrt(-S(4)ac + b*S(2)) + S(6)c*S(2)d - cfsqrt(-S(4)ac + bS(2))) + bS(2)c(-ach - a(S(19)am + ksqrt(-S(4)ac + bS(2))) + cS(2)d) - bc(S(13)a*S(2)msqrt(-S(4)ac + bS(2)) + ac(-S(8)ak + hsqrt(-S(4)ac + bS(2))) + cS(2)(-S(4)af + dsqrt(-S(4)ac + b*S(2)))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)ac(S(5)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))(S(3)abS(4)m - abS(3)(ck + S(3)msqrt(-S(4)ac + bS(2))) - S(2)acS(2)(-S(10)aS(2)m + S(2)ach + S(3)aksqrt(-S(4)ac + b*S(2)) + S(6)c*S(2)d + cfsqrt(-S(4)ac + bS(2))) + bS(2)c(-ach + a(-S(19)am + ksqrt(-S(4)ac + bS(2))) + cS(2)d) + bc(S(13)a*S(2)msqrt(-S(4)ac + bS(2)) + ac(S(8)ak + hsqrt(-S(4)ac + bS(2))) + cS(2)(S(4)af + dsqrt(-S(4)ac + b*S(2)))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)ac(S(5)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(xS(4) - S(5)xS(2) + S(4))S(3), x), x, -S(313)datanh(x/S(2))/S(20736) + S(13)datanh(x)/S(648) - elog(-xS(2) + S(1))/S(81) + elog(-x*S(2) + S(4))/S(81) - x(-S(35)dx*S(2) + S(59)d - S(50)ex*S(3) + S(122)ex)/(S(3456)x*S(4) - S(17280)x*S(2) + S(13824)) + x(-S(5)dx*S(2) + S(17)d - S(5)ex*S(3) + S(17)ex)/(S(144)(x*S(4) - S(5)xS(2) + S(4))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2))/(xS(4) - S(5)xS(2) + S(4))S(3), x), x, -elog(-x*S(2) + S(1))/S(81) + elog(-x*S(2) + S(4))/S(81) - x(S(59)d - S(50)exS(3) + S(122)ex + S(380)f - x*S(2)(S(35)d + S(140)f))/(S(3456)xS(4) - S(17280)x*S(2) + S(13824)) + x(S(17)d - S(5)exS(3) + S(17)ex + S(20)f - x*S(2)(S(5)d + S(8)f))/(S(144)(xS(4) - S(5)xS(2) + S(4))S(2)) + (S(13)d/S(648) + S(25)f/S(648))atanh(x) - (S(313)d + S(820)f)atanh(x/S(2))/S(20736), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2) + gxS(3))/(xS(4) - S(5)xS(2) + S(4))S(3), x), x, -x(S(59)d + S(380)f - x*S(3)(S(50)e + S(152)g) - x*S(2)(S(35)d + S(140)f) + x(S(122)e + S(440)g))/(S(3456)x*S(4) - S(17280)x*S(2) + S(13824)) + x(S(17)d + S(20)f - x*S(3)(S(5)e + S(8)g) - x*S(2)(S(5)d + S(8)f) + x(S(17)e + S(20)g))/(S(144)(x*S(4) - S(5)xS(2) + S(4))S(2)) + (S(13)d/S(648) + S(25)f/S(648))atanh(x) - (S(313)d + S(820)f)atanh(x/S(2))/S(20736) - (e/S(81) + S(5)g/S(162))log(-x*S(2) + S(1)) + (e/S(81) + S(5)g/S(162))log(-xS(2) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hxS(4))/(xS(4) - S(5)xS(2) + S(4))S(3), x), x, -x(S(59)d + S(380)f + S(848)h - xS(3)(S(50)e + S(152)g) - x*S(2)(S(35)d + S(140)f + S(320)h) + x(S(122)e + S(440)g))/(S(3456)xS(4) - S(17280)x*S(2) + S(13824)) + x(S(17)d + S(20)f + S(32)h - xS(3)(S(5)e + S(8)g) - x*S(2)(S(5)d + S(8)f + S(20)h) + x(S(17)e + S(20)g))/(S(144)(xS(4) - S(5)xS(2) + S(4))S(2)) - (e/S(81) + S(5)g/S(162))log(-x*S(2) + S(1)) + (e/S(81) + S(5)g/S(162))log(-xS(2) + S(4)) + (S(13)d/S(648) + S(25)f/S(648) + S(61)h/S(648))atanh(x) - (S(313)d + S(820)f + S(1936)h)atanh(x/S(2))/S(20736), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4) + ixS(5))/(xS(4) - S(5)xS(2) + S(4))S(3), x), x, -x(S(59)d + S(380)f + S(848)h - xS(3)(S(50)e + S(152)g + S(320)i) - xS(2)(S(35)d + S(140)f + S(320)h) + x(S(122)e + S(440)g + S(896)i))/(S(3456)x*S(4) - S(17280)x*S(2) + S(13824)) + x(S(17)d + S(20)f + S(32)h - xS(3)(S(5)e + S(8)g + S(20)i) - xS(2)(S(5)d + S(8)f + S(20)h) + x(S(17)e + S(20)g + S(32)i))/(S(144)(x*S(4) - S(5)xS(2) + S(4))S(2)) + (S(13)d/S(648) + S(25)f/S(648) + S(61)h/S(648))atanh(x) - (S(313)d + S(820)f + S(1936)h)atanh(x/S(2))/S(20736) - (e/S(81) + S(5)g/S(162) + S(11)i/S(162))log(-xS(2) + S(1)) + (e/S(81) + S(5)g/S(162) + S(11)i/S(162))log(-x*S(2) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(xS(4) + xS(2) + S(1))*S(3), x), x, -S(9)dlog(xS(2) - x + S(1))/S(32) + S(9)dlog(xS(2) + x + S(1))/S(32) - S(13)sqrt(S(3))datan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(144) + S(13)sqrt(S(3))datan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(144) + S(2)sqrt(S(3))eatan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(9) + x(-S(7)dx*S(2) + S(2)d - S(6)ex*S(3) + S(2)ex)/(S(24)x*S(4) + S(24)x*S(2) + S(24)) + x(-dxS(2) + d - ex*S(3) + ex)/(S(12)(xS(4) + xS(2) + S(1))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2))/(xS(4) + xS(2) + S(1))S(3), x), x, S(2)sqrt(S(3))eatan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(9) + x(S(2)d - S(6)exS(3) + S(2)ex + S(3)f - x*S(2)(S(7)d - S(7)f))/(S(24)xS(4) + S(24)x*S(2) + S(24)) + x(d - exS(3) + ex + f - x*S(2)(d - S(2)f))/(S(12)(xS(4) + xS(2) + S(1))*S(2)) - (S(9)d/S(32) - f/S(8))log(xS(2) - x + S(1)) + (S(9)d/S(32) - f/S(8))log(xS(2) + x + S(1)) - sqrt(S(3))(S(13)d + S(2)f)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(144) + sqrt(S(3))(S(13)d + S(2)f)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(144), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gxS(3))/(xS(4) + xS(2) + S(1))S(3), x), x, x(S(2)d + S(3)f - xS(3)(S(6)e - S(6)g) - x*S(2)(S(7)d - S(7)f) + x(S(2)e + S(2)g))/(S(24)x*S(4) + S(24)x*S(2) + S(24)) + x(d + f - x*S(3)(e - S(2)g) - xS(2)(d - S(2)f) + x(e + g))/(S(12)(xS(4) + xS(2) + S(1))S(2)) - (S(9)d/S(32) - f/S(8))log(xS(2) - x + S(1)) + (S(9)d/S(32) - f/S(8))log(xS(2) + x + S(1)) - sqrt(S(3))(S(13)d + S(2)f)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(144) + sqrt(S(3))(S(13)d + S(2)f)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(144) + sqrt(S(3))(S(2)e - g)atan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hxS(4))/(xS(4) + xS(2) + S(1))S(3), x), x, x(S(2)d + S(3)f - h - xS(3)(S(6)e - S(6)g) - x*S(2)(S(7)d - S(7)f + S(4)h) + x(S(2)e + S(2)g))/(S(24)xS(4) + S(24)x*S(2) + S(24)) + x(d + f - S(2)h - xS(3)(e - S(2)g) - xS(2)(d - S(2)f + h) + x(e + g))/(S(12)(xS(4) + xS(2) + S(1))S(2)) + sqrt(S(3))(S(2)e - g)atan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(9) - (S(9)d/S(32) - f/S(8) + S(3)h/S(32))log(x*S(2) - x + S(1)) + (S(9)d/S(32) - f/S(8) + S(3)h/S(32))log(x*S(2) + x + S(1)) - sqrt(S(3))(S(13)d + S(2)f + h)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(144) + sqrt(S(3))(S(13)d + S(2)f + h)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(144), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4) + ixS(5))/(xS(4) + xS(2) + S(1))S(3), x), x, x(S(2)d + S(3)f - h - xS(3)(S(6)e - S(6)g + S(4)i) - xS(2)(S(7)d - S(7)f + S(4)h) + x(S(2)e + S(2)g))/(S(24)xS(4) + S(24)x*S(2) + S(24)) + x(d + f - S(2)h - xS(3)(e - S(2)g + i) - xS(2)(d - S(2)f + h) + x(e + g - S(2)i))/(S(12)(xS(4) + xS(2) + S(1))*S(2)) - (S(9)d/S(32) - f/S(8) + S(3)h/S(32))log(x*S(2) - x + S(1)) + (S(9)d/S(32) - f/S(8) + S(3)h/S(32))log(x*S(2) + x + S(1)) - sqrt(S(3))(S(13)d + S(2)f + h)atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(144) + sqrt(S(3))(S(13)d + S(2)f + h)atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(144) + sqrt(S(3))(S(2)e - g + i)atan(sqrt(S(3))(S(2)x*S(2) + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(a + bxS(2) + cxS(4))S(3), x), x, -S(6)cS(2)eatanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) + x(bcdxS(2) + bcex*S(3) + d(-S(2)ac + b*S(2)) + ex(-S(2)ac + bS(2)))/(S(4)a(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))S(2)) - S(3)sqrt(S(2))sqrt(c)d(S(56)aS(2)c*S(2) - S(10)abS(2)c + b*S(4) - b(-S(8)ac + b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + S(3)sqrt(S(2))sqrt(c)d(S(56)aS(2)c*S(2) - S(10)abS(2)c + b*S(4) + b(-S(8)ac + b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + x(S(3)bcdxS(2)(-S(8)ac + b*S(2)) + S(2)bcexS(3)(-S(10)ac + b*S(2)) + d(-S(7)ac + b*S(2))(-S(4)ac + S(3)bS(2)) + ex(S(24)a*S(2)c*S(2) - S(20)abS(2)c + S(2)bS(4)))/(S(8)a*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2))/(a + bx*S(2) + cxS(4))S(3), x), x, -S(6)cS(2)eatanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) + x(-abf - S(2)acd + bS(2)d + bcexS(3) + cx*S(2)(-S(2)af + bd) + ex(-S(2)ac + bS(2)))/(S(4)a(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))S(2)) - sqrt(S(2))sqrt(c)(S(4)aS(2)c(S(42)cd - S(5)fsqrt(-S(4)ac + bS(2))) - ab*S(2)(S(30)cd + fsqrt(-S(4)ac + bS(2))) + S(4)abc(-S(13)af + S(6)dsqrt(-S(4)ac + bS(2))) + S(3)b*S(4)d - b*S(3)(-af + S(3)dsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + sqrt(S(2))sqrt(c)(S(4)aS(2)c(S(42)cd + S(5)fsqrt(-S(4)ac + bS(2))) - ab*S(2)(S(30)cd - fsqrt(-S(4)ac + bS(2))) - S(4)abc(S(13)af + S(6)dsqrt(-S(4)ac + bS(2))) + S(3)b*S(4)d + b*S(3)(af + S(3)dsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + x(S(8)a*S(2)bcf + S(28)aS(2)c*S(2)d + abS(3)f - S(25)ab*S(2)cd + S(3)b*S(4)d + S(2)bcex*S(3)(-S(10)ac + b*S(2)) + cx*S(2)(S(20)aS(2)cf + ab*S(2)f - S(24)abcd + S(3)bS(3)d) + ex(S(24)aS(2)c*S(2) - S(20)abS(2)c + S(2)bS(4)))/(S(8)a*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3))/(a + bx*S(2) + cxS(4))S(3), x), x, -S(3)c(-bg + S(2)ce)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) + x(-abf - S(2)acd + bS(2)d + cxS(3)(-S(2)ag + be) + cx*S(2)(-S(2)af + bd) + x(-abg - S(2)ace + bS(2)e))/(S(4)a(-S(4)ac + b*S(2))(a + bxS(2) + cxS(4))S(2)) - sqrt(S(2))sqrt(c)(S(4)aS(2)c(S(42)cd - S(5)fsqrt(-S(4)ac + bS(2))) - ab*S(2)(S(30)cd + fsqrt(-S(4)ac + bS(2))) + S(4)abc(-S(13)af + S(6)dsqrt(-S(4)ac + bS(2))) + S(3)b*S(4)d - b*S(3)(-af + S(3)dsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + sqrt(S(2))sqrt(c)(S(4)aS(2)c(S(42)cd + S(5)fsqrt(-S(4)ac + bS(2))) - ab*S(2)(S(30)cd - fsqrt(-S(4)ac + bS(2))) - S(4)abc(S(13)af + S(6)dsqrt(-S(4)ac + bS(2))) + S(3)b*S(4)d + b*S(3)(af + S(3)dsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + x(S(8)a*S(2)bcf + S(28)aS(2)c*S(2)d + abS(3)f - S(25)ab*S(2)cd + S(3)b*S(4)d + S(2)cx*S(3)(S(8)aS(2)cg + ab*S(2)g - S(10)abce + b*S(3)e) + cxS(2)(S(20)aS(2)cf + ab*S(2)f - S(24)abcd + S(3)bS(3)d) + x(S(4)a*S(2)bcg + S(24)aS(2)c*S(2)e + S(2)ab*S(3)g - S(20)ab*S(2)ce + S(2)b*S(4)e))/(S(8)aS(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4))/(a + bx*S(2) + cxS(4))S(3), x), x, -S(3)c(-bg + S(2)ce)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) + x(-abf - S(2)a(-ah + cd) + b*S(2)d + cxS(3)(-S(2)ag + be) + xS(2)(abh - S(2)acf + bcd) + x(-abg - S(2)ace + bS(2)e))/(S(4)a(-S(4)ac + b*S(2))(a + bxS(2) + cxS(4))S(2)) - sqrt(S(2))sqrt(c)(S(4)aS(2)c(S(6)ah + S(42)cd - S(5)fsqrt(-S(4)ac + bS(2))) - ab*S(2)(-S(18)ah + S(30)cd + fsqrt(-S(4)ac + bS(2))) + S(4)ab(-S(13)acf + S(3)ahsqrt(-S(4)ac + b*S(2)) + S(6)cdsqrt(-S(4)ac + b*S(2))) + S(3)b*S(4)d - b*S(3)(-af + S(3)dsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + sqrt(S(2))sqrt(c)(S(4)aS(2)c(S(6)ah + S(42)cd + S(5)fsqrt(-S(4)ac + bS(2))) - ab*S(2)(-S(18)ah + S(30)cd - fsqrt(-S(4)ac + bS(2))) - S(4)ab(S(13)acf + S(3)ahsqrt(-S(4)ac + b*S(2)) + S(6)cdsqrt(-S(4)ac + b*S(2))) + S(3)b*S(4)d + b*S(3)(af + S(3)dsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + x(S(8)a*S(2)bcf + S(4)aS(2)c(ah + S(7)cd) + abS(3)f - abS(2)(S(7)ah + S(25)cd) + S(3)bS(4)d + S(2)cx*S(3)(S(8)aS(2)cg + ab*S(2)g - S(10)abce + b*S(3)e) + cxS(2)(S(20)aS(2)cf + ab*S(2)f - S(12)ab(ah + S(2)cd) + S(3)bS(3)d) + x(S(4)a*S(2)bcg + S(24)aS(2)c*S(2)e + S(2)ab*S(3)g - S(20)ab*S(2)ce + S(2)b*S(4)e))/(S(8)aS(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4) + ix*S(5))/(a + bx*S(2) + cxS(4))S(3), x), x, -(S(2)aci + bS(2)i - S(3)bcg + S(6)c*S(2)e)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) + x(-abf - S(2)a(-ah + cd) + b*S(2)d + x*S(3)(abi - S(2)acg + bce) + xS(2)(abh - S(2)acf + bcd) + x(-abg - S(2)a(-ai + ce) + b*S(2)e))/(S(4)a(-S(4)ac + b*S(2))(a + bxS(2) + cxS(4))S(2)) - sqrt(S(2))sqrt(c)(S(4)aS(2)c(S(6)ah + S(42)cd - S(5)fsqrt(-S(4)ac + bS(2))) - ab*S(2)(-S(18)ah + S(30)cd + fsqrt(-S(4)ac + bS(2))) + S(4)ab(-S(13)acf + S(3)ahsqrt(-S(4)ac + b*S(2)) + S(6)cdsqrt(-S(4)ac + b*S(2))) + S(3)b*S(4)d - b*S(3)(-af + S(3)dsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + sqrt(S(2))sqrt(c)(S(4)aS(2)c(S(6)ah + S(42)cd + S(5)fsqrt(-S(4)ac + bS(2))) - ab*S(2)(-S(18)ah + S(30)cd - fsqrt(-S(4)ac + bS(2))) - S(4)ab(S(13)acf + S(3)ahsqrt(-S(4)ac + b*S(2)) + S(6)cdsqrt(-S(4)ac + b*S(2))) + S(3)b*S(4)d + b*S(3)(af + S(3)dsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + x(S(8)a*S(2)bcf + S(4)aS(2)c(ah + S(7)cd) + abS(3)f - abS(2)(S(7)ah + S(25)cd) + S(3)bS(4)d + S(2)cx*S(3)(S(8)aS(2)cg + ab*S(2)g - S(2)ab(S(3)ai + S(5)ce) + bS(3)e) + cxS(2)(S(20)aS(2)cf + ab*S(2)f - S(12)ab(ah + S(2)cd) + S(3)bS(3)d) + x(S(4)a*S(2)bcg + S(8)aS(2)c(ai + S(3)ce) + S(2)ab*S(3)g - S(4)ab*S(2)(S(2)ai + S(5)ce) + S(2)bS(4)e))/(S(8)aS(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4) + jx*S(5) + kx*S(6) + lx*S(7) + mx*S(8))/(a + bx*S(2) + cxS(4))S(3), x), x, -(-S(3)abl + S(2)acj + b*S(2)j - S(3)bcg + S(6)c*S(2)e)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) - x(abc(ak + cf) + S(2)ac(a*S(2)m - ach + c*S(2)d) - b*S(2)(a*S(2)m + c*S(2)d) - cxS(3)(-abS(2)l - S(2)ac(-al + cg) + bc(aj + ce)) - cx(-ab(al + cg) - S(2)ac(-aj + ce) + b*S(2)ce) + xS(2)(-abS(3)m + abS(2)ck + S(2)acS(2)(-ak + cf) - bc(-S(3)aS(2)m + ach + c*S(2)d)))/(S(4)ac*S(2)(-S(4)ac + b*S(2))(a + bxS(2) + cxS(4))S(2)) + x(S(4)a*S(2)bcS(2)(ak + S(2)cf) + S(4)a*S(2)c*S(2)(-S(9)aS(2)m + ach + S(7)cS(2)d) + abS(3)c(S(2)ak + cf) - abS(2)c(-S(11)a*S(2)m + S(7)ach + S(25)c*S(2)d) + b*S(4)(-S(2)aS(2)m + S(3)cS(2)d) + S(2)cS(2)x*S(3)(S(8)aS(2)c(al + cg) + ab*S(2)(al + cg) - S(2)abc(S(3)aj + S(5)ce) + b*S(3)ce) + cx*S(2)(S(4)aS(2)c*S(2)(S(3)ak + S(5)cf) + abS(2)c(S(3)ak + cf) - S(4)abc(S(4)aS(2)m + S(3)ach + S(6)c*S(2)d) + b*S(3)(a*S(2)m + S(3)cS(2)d)) + S(2)cx(S(2)a*S(2)bc(al + cg) + S(4)aS(2)c*S(2)(aj + S(3)ce) + ab*S(3)(al + cg) - S(2)ab*S(2)c(S(2)aj + S(5)ce) + bS(4)ce))/(S(8)a*S(2)c*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) - sqrt(S(2))(S(4)aS(2)c*S(2)(S(10)aS(2)m + S(6)ach - S(3)aksqrt(-S(4)ac + b*S(2)) + S(42)c*S(2)d - S(5)cfsqrt(-S(4)ac + bS(2))) - ab*S(2)c(S(3)a(-S(6)am + ksqrt(-S(4)ac + b*S(2))) + S(30)c*S(2)d + c(-S(18)ah + fsqrt(-S(4)ac + b*S(2)))) + S(4)abc(S(4)a*S(2)msqrt(-S(4)ac + bS(2)) + S(3)ac(-S(3)ak + hsqrt(-S(4)ac + bS(2))) + cS(2)(-S(13)af + S(6)dsqrt(-S(4)ac + bS(2)))) + bS(4)(-aS(2)m + S(3)cS(2)d) - b*S(3)(S(3)aS(2)ck + aS(2)msqrt(-S(4)ac + bS(2)) + cS(2)(-af + S(3)dsqrt(-S(4)ac + bS(2)))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + sqrt(S(2))(S(4)a*S(2)c*S(2)(S(10)aS(2)m + S(6)ach + S(3)aksqrt(-S(4)ac + b*S(2)) + S(42)c*S(2)d + S(5)cfsqrt(-S(4)ac + bS(2))) - ab*S(2)c(-S(3)a(S(6)am + ksqrt(-S(4)ac + b*S(2))) + S(30)c*S(2)d - c(S(18)ah + fsqrt(-S(4)ac + b*S(2)))) - S(4)abc(S(4)a*S(2)msqrt(-S(4)ac + bS(2)) + S(3)ac(S(3)ak + hsqrt(-S(4)ac + bS(2))) + cS(2)(S(13)af + S(6)dsqrt(-S(4)ac + bS(2)))) + bS(4)(-aS(2)m + S(3)cS(2)d) + b*S(3)(-S(3)aS(2)ck + aS(2)msqrt(-S(4)ac + bS(2)) + cS(2)(af + S(3)dsqrt(-S(4)ac + bS(2)))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate((d + ex + fx*S(2) + gx*S(3) + hx*S(4) + ix*S(5) + jx*S(6) + kx*S(7))/(a + bx*S(2) + cxS(4))S(2), x), x, klog(a + bx*S(2) + cx*S(4))/(S(4)c*S(2)) + (-S(6)abck + bS(3)k + S(4)cS(3)e - c*S(2)(-S(4)ai + S(2)bg))atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)(-S(4)ac + bS(2))(S(3)/2)) + x(-ab(aj + cf) - S(2)ac(-ah + cd) + b*S(2)cd + xS(3)(-abS(2)k - S(2)ac(-ak + cg) + bc(ai + ce)) + xS(2)(-abS(2)j - S(2)ac(-aj + cf) + bc(ah + cd)) + x(-ab(ak + cg) - S(2)ac(-ai + ce) + bS(2)ce))/(S(2)ac(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) + sqrt(S(2))(abS(3)j + S(2)ac(S(2)ach - S(3)ajsqrt(-S(4)ac + bS(2)) + S(6)c*S(2)d - cfsqrt(-S(4)ac + bS(2))) - bS(2)(-ach - ajsqrt(-S(4)ac + bS(2)) + cS(2)d) + bc(-S(8)aS(2)j - S(4)acf + ahsqrt(-S(4)ac + bS(2)) + cdsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)ac(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - sqrt(S(2))(ab*S(3)j + S(2)ac(S(2)ach + S(3)ajsqrt(-S(4)ac + bS(2)) + S(6)c*S(2)d + cfsqrt(-S(4)ac + bS(2))) - bS(2)(-ach + ajsqrt(-S(4)ac + bS(2)) + cS(2)d) - bc(S(8)aS(2)j + S(4)acf + ahsqrt(-S(4)ac + bS(2)) + cdsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)ac(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate((d + ex + fx*S(2) + gx*S(3) + hx*S(4) + ix*S(5) + jx*S(8) + kx*S(11))/(a + bx*S(2) + cxS(4))S(3), x), x, klog(a + bx*S(2) + cx*S(4))/(S(4)c*S(3)) - (-S(30)a*S(2)bcS(2)k + S(10)ab*S(3)ck - bS(5)k + S(2)bS(2)c*S(3)i + S(12)cS(5)e - c*S(4)(-S(4)ai + S(6)bg))atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(3)(-S(4)ac + bS(2))(S(5)/2)) - x(cx*S(2)(-abS(3)j + S(2)ac*S(3)f - bc(-S(3)aS(2)j + ach + c*S(2)d)) + c(abcS(2)f + S(2)ac(aS(2)j - ach + c*S(2)d) - b*S(2)(a*S(2)j + c*S(2)d)) - x*S(3)(-S(2)aS(3)c*S(2)k + S(4)aS(2)b*S(2)ck - ab*S(4)k - S(2)ac*S(4)g + bcS(3)(ai + ce)) - x(-aS(2)b*S(3)k - S(2)ac*S(3)(-ai + ce) + b*S(2)c*S(3)e - b(-S(3)a*S(3)ck + ac*S(3)g)))/(S(4)ac*S(3)(-S(4)ac + b*S(2))(a + bxS(2) + cxS(4))S(2)) + x(cS(2)x*S(2)(S(20)aS(2)c*S(3)f + abS(2)c*S(2)f - S(4)abc(S(4)aS(2)j + S(3)ach + S(6)c*S(2)d) + b*S(3)(a*S(2)j + S(3)cS(2)d)) + S(2)cx*S(3)(-S(24)aS(4)c*S(2)k - S(3)aS(2)b*S(4)k + S(8)aS(2)c*S(4)g - S(2)abcS(3)(S(3)ai + S(5)ce) + b*S(3)c*S(3)e + b*S(2)(S(21)aS(3)ck + ac*S(3)g)) + c(S(8)a*S(2)bcS(3)f + S(4)aS(2)c*S(2)(-S(9)aS(2)j + ach + S(7)cS(2)d) + abS(3)c*S(2)f - abS(2)c(-S(11)a*S(2)j + S(7)ach + S(25)c*S(2)d) + b*S(4)(-S(2)aS(2)j + S(3)cS(2)d)) + x(S(2)a*S(2)b*S(5)k + S(8)aS(2)c*S(4)(ai + S(3)ce) - S(4)abS(2)c*S(3)(S(2)ai + S(5)ce) + S(2)bS(4)c*S(3)e + S(2)bS(3)(-S(9)aS(3)ck + ac*S(3)g) + S(4)b(S(13)aS(4)c*S(2)k + a*S(2)c*S(4)g)))/(S(8)aS(2)c*S(3)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cx*S(4))) + sqrt(S(2))(-S(4)aS(2)c*S(2)(S(10)aS(2)j + S(6)ach + S(42)c*S(2)d - S(5)cfsqrt(-S(4)ac + bS(2))) + ab*S(2)c(-S(18)a*S(2)j - S(18)ach + S(30)c*S(2)d + cfsqrt(-S(4)ac + b*S(2))) - S(4)abc(S(4)a*S(2)jsqrt(-S(4)ac + bS(2)) + S(3)achsqrt(-S(4)ac + bS(2)) + cS(2)(-S(13)af + S(6)dsqrt(-S(4)ac + bS(2)))) - bS(4)(-aS(2)j + S(3)cS(2)d) + b*S(3)(a*S(2)jsqrt(-S(4)ac + bS(2)) + cS(2)(-af + S(3)dsqrt(-S(4)ac + bS(2)))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) - sqrt(S(2))(-S(4)a*S(2)c*S(2)(S(10)aS(2)j + S(6)ach + S(42)c*S(2)d + S(5)cfsqrt(-S(4)ac + bS(2))) + ab*S(2)c(-S(18)a*S(2)j - S(18)ach + S(30)c*S(2)d - cfsqrt(-S(4)ac + b*S(2))) + S(4)abc(S(4)a*S(2)jsqrt(-S(4)ac + bS(2)) + S(3)achsqrt(-S(4)ac + bS(2)) + cS(2)(S(13)af + S(6)dsqrt(-S(4)ac + bS(2)))) - bS(4)(-aS(2)j + S(3)cS(2)d) - b*S(3)(a*S(2)jsqrt(-S(4)ac + bS(2)) + cS(2)(af + S(3)dsqrt(-S(4)ac + bS(2)))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))S(3)(ad + aex + bex*S(3) + cexS(5) + cfxS(6) + xS(4)(bf + cd) + x*S(2)(af + bd)), x), x, a*S(4)dx + aS(4)exS(2)/S(2) + aS(3)bexS(4) + aS(3)xS(3)(af + S(4)bd)/S(3) + aS(2)exS(6)(S(2)ac + S(3)bS(2))/S(3) + S(2)a*S(2)x*S(5)(S(2)abf + S(2)acd + S(3)bS(2)d)/S(5) + abexS(8)(S(3)ac + b*S(2))/S(2) + S(2)axS(7)(S(2)aS(2)cf + S(3)abS(2)f + S(6)abcd + S(2)bS(3)d)/S(7) + bcS(3)exS(16)/S(4) + bcex*S(12)(S(3)ac + bS(2))/S(3) + cS(4)exS(18)/S(18) + cS(4)fxS(19)/S(19) + cS(3)xS(17)(S(4)bf + cd)/S(17) + cS(2)exS(14)(S(2)ac + S(3)bS(2))/S(7) + S(2)c*S(2)x*S(15)(S(2)acf + S(3)b*S(2)f + S(2)bcd)/S(15) + S(2)cxS(13)(S(6)abcf + S(2)ac*S(2)d + S(2)bS(3)f + S(3)bS(2)cd)/S(13) + ex*S(10)(S(3)aS(2)c*S(2)/S(5) + S(6)abS(2)c/S(5) + bS(4)/S(10)) + xS(11)(S(6)a*S(2)c*S(2)f/S(11) + S(12)ab*S(2)cf/S(11) + S(12)abc*S(2)d/S(11) + b*S(4)f/S(11) + S(4)bS(3)cd/S(11)) + xS(9)(S(4)aS(2)bcf/S(3) + S(2)aS(2)c*S(2)d/S(3) + S(4)ab*S(3)f/S(9) + S(4)ab*S(2)cd/S(3) + bS(4)d/S(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cxS(4))S(2)(ad + aex + bex*S(3) + cexS(5) + cfxS(6) + xS(4)(bf + cd) + x*S(2)(af + bd)), x), x, a*S(3)dx + aS(3)exS(2)/S(2) + S(3)a*S(2)bexS(4)/S(4) + aS(2)xS(3)(af + S(3)bd)/S(3) + aexS(6)(ac + bS(2))/S(2) + S(3)axS(5)(abf + acd + b*S(2)d)/S(5) + bcS(2)exS(12)/S(4) + bexS(8)(S(6)ac + bS(2))/S(8) + cS(3)exS(14)/S(14) + cS(3)fxS(15)/S(15) + cS(2)xS(13)(S(3)bf + cd)/S(13) + S(3)cex*S(10)(ac + bS(2))/S(10) + S(3)cxS(11)(acf + b*S(2)f + bcd)/S(11) + x*S(9)(S(2)abcf/S(3) + acS(2)d/S(3) + b*S(3)f/S(9) + b*S(2)cd/S(3)) + xS(7)(S(3)aS(2)cf/S(7) + S(3)abS(2)f/S(7) + S(6)abcd/S(7) + b*S(3)d/S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2) + cx*S(4))(ad + aex + bexS(3) + cexS(5) + cfxS(6) + xS(4)(bf + cd) + x*S(2)(af + bd)), x), x, a*S(2)dx + aS(2)exS(2)/S(2) + abex*S(4)/S(2) + ax*S(3)(af + S(2)bd)/S(3) + bcexS(8)/S(4) + cS(2)exS(10)/S(10) + cS(2)fx*S(11)/S(11) + cx*S(9)(S(2)bf + cd)/S(9) + ex*S(6)(ac/S(3) + bS(2)/S(6)) + xS(7)(S(2)acf/S(7) + bS(2)f/S(7) + S(2)bcd/S(7)) + xS(5)(S(2)abf/S(5) + S(2)acd/S(5) + b*S(2)d/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ad + aex + bexS(3) + cexS(5) + cfxS(6) + xS(4)(bf + cd) + x*S(2)(af + bd))/(a + bxS(2) + cx*S(4)), x), x, dx + exS(2)/S(2) + fx*S(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ad + aex + bex*S(3) + cexS(5) + cfxS(6) + xS(4)(bf + cd) + x*S(2)(af + bd))/(a + bxS(2) + cxS(4))S(2), x), x, -eatanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)) + sqrt(S(2))(f - (-bf + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(f + (-bf + S(2)cd)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ad + aex + bex*S(3) + cexS(5) + cfxS(6) + xS(4)(bf + cd) + x*S(2)(af + bd))/(a + bxS(2) + cxS(4))S(3), x), x, S(2)ceatanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + sqrt(S(2))sqrt(c)(-S(2)af + bd - (S(4)abf - S(12)acd + b*S(2)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)(-S(2)af + bd + (S(4)abf - S(12)acd + b*S(2)d)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + x(-abf - S(2)acd + bS(2)d + bcexS(3) + cx*S(2)(-S(2)af + bd) + ex(-S(2)ac + bS(2)))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ad + aex + bex*S(3) + cexS(5) + cfxS(6) + xS(4)(bf + cd) + x*S(2)(af + bd))/(a + bxS(2) + cxS(4))S(4), x), x, -S(6)cS(2)eatanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(5)/2) + x(-abf - S(2)acd + bS(2)d + bcexS(3) + cx*S(2)(-S(2)af + bd) + ex(-S(2)ac + bS(2)))/(S(4)a(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))S(2)) - sqrt(S(2))sqrt(c)(S(4)aS(2)c(S(42)cd - S(5)fsqrt(-S(4)ac + bS(2))) - ab*S(2)(S(30)cd + fsqrt(-S(4)ac + bS(2))) + S(4)abc(-S(13)af + S(6)dsqrt(-S(4)ac + bS(2))) + S(3)b*S(4)d - b*S(3)(-af + S(3)dsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + sqrt(S(2))sqrt(c)(S(4)aS(2)c(S(42)cd + S(5)fsqrt(-S(4)ac + bS(2))) - ab*S(2)(S(30)cd - fsqrt(-S(4)ac + bS(2))) - S(4)abc(S(13)af + S(6)dsqrt(-S(4)ac + bS(2))) + S(3)b*S(4)d + b*S(3)(af + S(3)dsqrt(-S(4)ac + bS(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(16)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(5)/2)) + x(S(8)a*S(2)bcf + S(28)aS(2)c*S(2)d + abS(3)f - S(25)ab*S(2)cd + S(3)b*S(4)d + S(2)bcex*S(3)(-S(10)ac + b*S(2)) + cx*S(2)(S(20)aS(2)cf + ab*S(2)f - S(24)abcd + S(3)bS(3)d) + ex(S(24)aS(2)c*S(2) - S(20)abS(2)c + S(2)bS(4)))/(S(8)a*S(2)(-S(4)ac + bS(2))S(2)(a + bx*S(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) - S(2)xS(2) - x + S(2))/(xS(4) - S(5)x*S(2) + S(4)), x), x, log(x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)(xS(3) - S(2)xS(2) - x + S(2))/(xS(4) - S(5)xS(2) + S(4)), x), x, ex + (d - S(2)e)log(x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2))(x*S(3) - S(2)xS(2) - x + S(2))/(xS(4) - S(5)xS(2) + S(4)), x), x, f(x + S(2))*S(2)/S(2) + x(e - S(4)f) + (d - S(2)e + S(4)f)log(x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2) + gx*S(3))(x*S(3) - S(2)xS(2) - x + S(2))/(xS(4) - S(5)xS(2) + S(4)), x), x, g(x + S(2))*S(3)/S(3) + x(e - S(4)f + S(12)g) + (f/S(2) - S(3)g)(x + S(2))*S(2) + (d - S(2)e + S(4)f - S(8)g)log(x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) - S(2)x*S(2) - x + S(2))(d + ex + fx*S(2) + gx*S(3) + hxS(4))/(xS(4) - S(5)xS(2) + S(4)), x), x, hxS(4)/S(4) + xS(3)(g/S(3) - S(2)h/S(3)) + x*S(2)(f/S(2) - g + S(2)h) + x(e - S(2)f + S(4)g - S(8)h) + (d - S(2)e + S(4)f - S(8)g + S(16)h)log(x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(3) - S(2)x*S(2) - x + S(2))(d + ex + fx*S(2) + gx*S(3) + hx*S(4) + ixS(5))/(xS(4) - S(5)xS(2) + S(4)), x), x, ixS(5)/S(5) + xS(4)(h/S(4) - i/S(2)) + xS(3)(g/S(3) - S(2)h/S(3) + S(4)i/S(3)) + x*S(2)(f/S(2) - g + S(2)h - S(4)i) + x(e - S(2)f + S(4)g - S(8)h + S(16)i) + (d - S(2)e + S(4)f - S(8)g + S(16)h - S(32)i)log(x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) - S(3)x + S(2))/(x*S(4) - S(5)x*S(2) + S(4)), x), x, -S(2)atanh(S(2)x + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)(xS(2) - S(3)x + S(2))/(x*S(4) - S(5)x*S(2) + S(4)), x), x, -(d - S(2)e)log(x + S(2)) + (d - e)log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2))(x*S(2) - S(3)x + S(2))/(x*S(4) - S(5)x*S(2) + S(4)), x), x, fx - (d - S(2)e + S(4)f)log(x + S(2)) + (d - e + f)log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) - S(3)x + S(2))(d + ex + fxS(2) + gxS(3))/(xS(4) - S(5)xS(2) + S(4)), x), x, gx*S(2)/S(2) + x(f - S(3)g) - (d - S(2)e + S(4)f - S(8)g)log(x + S(2)) + (d - e + f - g)log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) - S(3)x + S(2))(d + ex + fxS(2) + gx*S(3) + hxS(4))/(xS(4) - S(5)xS(2) + S(4)), x), x, hxS(3)/S(3) + xS(2)(g/S(2) - S(3)h/S(2)) + x(f - S(3)g + S(7)h) - (d - S(2)e + S(4)f - S(8)g + S(16)h)log(x + S(2)) + (d - e + f - g + h)log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) - S(3)x + S(2))(d + ex + fxS(2) + gx*S(3) + hx*S(4) + ixS(5))/(xS(4) - S(5)xS(2) + S(4)), x), x, ixS(4)/S(4) + xS(3)(h/S(3) - i) + xS(2)(g/S(2) - S(3)h/S(2) + S(7)i/S(2)) + x(f - S(3)g + S(7)h - S(15)i) - (d - S(2)e + S(4)f - S(8)g + S(16)h - S(32)i)log(x + S(2)) + (d - e + f - g + h - i)log(x + S(1)), expand=True, _diff=True, _numerical=True) # wromg result (rule) assert rubi_test(rubi_integrate((x + S(2))/(xS(4) - S(5)x*S(2) + S(4)), x), x, -log(-x + S(1))/S(2) + log(-x + S(2))/S(3) + log(x + S(1))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)(x + S(2))/(xS(4) - S(5)x*S(2) + S(4)), x), x, (-d/S(2) - e/S(2))log(-x + S(1)) + (d/S(6) - e/S(6))log(x + S(1)) + (d/S(3) + S(2)e/S(3))log(-x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2))(d + ex + fxS(2))/(xS(4) - S(5)xS(2) + S(4)), x), x, (-d/S(2) - e/S(2) - f/S(2))log(-x + S(1)) + (d/S(6) - e/S(6) + f/S(6))log(x + S(1)) + (d/S(3) + S(2)e/S(3) + S(4)f/S(3))log(-x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2))(d + ex + fxS(2) + gxS(3))/(xS(4) - S(5)xS(2) + S(4)), x), x, gx + (d/S(6) - e/S(6) + f/S(6) - g/S(6))log(x + S(1)) + (d/S(3) + S(2)e/S(3) + S(4)f/S(3) + S(8)g/S(3))log(-x + S(2)) - (d/S(2) + e/S(2) + f/S(2) + g/S(2))log(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2))(d + ex + fxS(2) + gx*S(3) + hxS(4))/(xS(4) - S(5)xS(2) + S(4)), x), x, hx*S(2)/S(2) + x(g + S(2)h) + (d/S(6) - e/S(6) + f/S(6) - g/S(6) + h/S(6))log(x + S(1)) + (d/S(3) + S(2)e/S(3) + S(4)f/S(3) + S(8)g/S(3) + S(16)h/S(3))log(-x + S(2)) - (d/S(2) + e/S(2) + f/S(2) + g/S(2) + h/S(2))log(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2))(d + ex + fxS(2) + gx*S(3) + hx*S(4) + ixS(5))/(xS(4) - S(5)xS(2) + S(4)), x), x, ixS(3)/S(3) + xS(2)(h/S(2) + i) + x(g + S(2)h + S(5)i) + (d/S(6) - e/S(6) + f/S(6) - g/S(6) + h/S(6) - i/S(6))log(x + S(1)) + (d/S(3) + S(2)e/S(3) + S(4)f/S(3) + S(8)g/S(3) + S(16)h/S(3) + S(32)i/S(3))log(-x + S(2)) - (d/S(2) + e/S(2) + f/S(2) + g/S(2) + h/S(2) + i/S(2))log(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(3) - S(2)xS(2) - x + S(2))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, -log(-x + S(1))/S(18) + log(-x + S(2))/S(48) + log(x + S(1))/S(6) - S(19)log(x + S(2))/S(144) + S(1)/(S(12)x + S(24)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)(xS(3) - S(2)xS(2) - x + S(2))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(48) + e/S(24))log(-x + S(2)) - (d/S(18) + e/S(18))log(-x + S(1)) - (S(19)d/S(144) - S(13)e/S(72))log(x + S(2)) + (d/S(6) - e/S(6))log(x + S(1)) + (d - S(2)e)/(S(12)x + S(24)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2))(x*S(3) - S(2)xS(2) - x + S(2))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(48) + e/S(24) + f/S(12))log(-x + S(2)) - (d/S(18) + e/S(18) + f/S(18))log(-x + S(1)) - (S(19)d/S(144) - S(13)e/S(72) + S(7)f/S(36))log(x + S(2)) + (d/S(6) - e/S(6) + f/S(6))log(x + S(1)) + (d - S(2)e + S(4)f)/(S(12)x + S(24)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3))(x*S(3) - S(2)xS(2) - x + S(2))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(48) + e/S(24) + f/S(12) + g/S(6))log(-x + S(2)) - (d/S(18) + e/S(18) + f/S(18) + g/S(18))log(-x + S(1)) - (S(19)d/S(144) - S(13)e/S(72) + S(7)f/S(36) - g/S(18))log(x + S(2)) + (d/S(6) - e/S(6) + f/S(6) - g/S(6))log(x + S(1)) + (d - S(2)e + S(4)f - S(8)g)/(S(12)x + S(24)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(3) - S(2)x*S(2) - x + S(2))(d + ex + fx*S(2) + gx*S(3) + hxS(4))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(48) + e/S(24) + f/S(12) + g/S(6) + h/S(3))log(-x + S(2)) - (d/S(18) + e/S(18) + f/S(18) + g/S(18) + h/S(18))log(-x + S(1)) - (S(19)d/S(144) - S(13)e/S(72) + S(7)f/S(36) - g/S(18) - S(5)h/S(9))log(x + S(2)) + (d/S(6) - e/S(6) + f/S(6) - g/S(6) + h/S(6))log(x + S(1)) + (d - S(2)e + S(4)f - S(8)g + S(16)h)/(S(12)x + S(24)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(3) - S(2)x*S(2) - x + S(2))(d + ex + fx*S(2) + gx*S(3) + hx*S(4) + ixS(5))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, ix + (d/S(48) + e/S(24) + f/S(12) + g/S(6) + h/S(3) + S(2)i/S(3))log(-x + S(2)) - (d/S(18) + e/S(18) + f/S(18) + g/S(18) + h/S(18) + i/S(18))log(-x + S(1)) - (S(19)d/S(144) - S(13)e/S(72) + S(7)f/S(36) - g/S(18) - S(5)h/S(9) + S(22)i/S(9))log(x + S(2)) + (d/S(6) - e/S(6) + f/S(6) - g/S(6) + h/S(6) - i/S(6))log(x + S(1)) + (d - S(2)e + S(4)f - S(8)g + S(16)h - S(32)i)/(S(12)x + S(24)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) - S(3)x + S(2))/(x*S(4) - S(5)xS(2) + S(4))S(2), x), x, -(S(3)x + S(5))/(S(12)x*S(2) + S(36)x + S(24)) - log(-x + S(1))/S(36) + log(-x + S(2))/S(144) - S(7)log(x + S(1))/S(36) + S(31)log(x + S(2))/S(144), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)(x*S(2) - S(3)x + S(2))/(x*S(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(144) + e/S(72))log(-x + S(2)) - (d/S(36) + e/S(36))log(-x + S(1)) - (S(7)d/S(36) - S(13)e/S(36))log(x + S(1)) + (S(31)d/S(144) - S(25)e/S(72))log(x + S(2)) - (S(5)d - S(6)e + x(S(3)d - S(4)e))/(S(12)x*S(2) + S(36)x + S(24)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2))(x*S(2) - S(3)x + S(2))/(x*S(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(144) + e/S(72) + f/S(36))log(-x + S(2)) - (d/S(36) + e/S(36) + f/S(36))log(-x + S(1)) - (S(7)d/S(36) - S(13)e/S(36) + S(19)f/S(36))log(x + S(1)) + (S(31)d/S(144) - S(25)e/S(72) + S(19)f/S(36))log(x + S(2)) - (S(5)d - S(6)e + S(8)f + x(S(3)d - S(4)e + S(6)f))/(S(12)x*S(2) + S(36)x + S(24)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) - S(3)x + S(2))(d + ex + fxS(2) + gxS(3))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(144) + e/S(72) + f/S(36) + g/S(18))log(-x + S(2)) - (d/S(36) + e/S(36) + f/S(36) + g/S(36))log(-x + S(1)) - (S(7)d/S(36) - S(13)e/S(36) + S(19)f/S(36) - S(25)g/S(36))log(x + S(1)) + (S(31)d/S(144) - S(25)e/S(72) + S(19)f/S(36) - S(13)g/S(18))log(x + S(2)) - (d - S(2)e + S(4)f - S(8)g)/(S(12)x + S(24)) - (d - e + f - g)/(S(6)x + S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) - S(3)x + S(2))(d + ex + fxS(2) + gx*S(3) + hxS(4))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(144) + e/S(72) + f/S(36) + g/S(18) + h/S(9))log(-x + S(2)) - (d/S(36) + e/S(36) + f/S(36) + g/S(36) + h/S(36))log(-x + S(1)) - (S(7)d/S(36) - S(13)e/S(36) + S(19)f/S(36) - S(25)g/S(36) + S(31)h/S(36))log(x + S(1)) + (S(31)d/S(144) - S(25)e/S(72) + S(19)f/S(36) - S(13)g/S(18) + S(7)h/S(9))log(x + S(2)) - (d - S(2)e + S(4)f - S(8)g + S(16)h)/(S(12)x + S(24)) - (d - e + f - g + h)/(S(6)x + S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) - S(3)x + S(2))(d + ex + fxS(2) + gx*S(3) + hx*S(4) + ixS(5))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(144) + e/S(72) + f/S(36) + g/S(18) + h/S(9) + S(2)i/S(9))log(-x + S(2)) - (d/S(36) + e/S(36) + f/S(36) + g/S(36) + h/S(36) + i/S(36))log(-x + S(1)) - (S(7)d/S(36) - S(13)e/S(36) + S(19)f/S(36) - S(25)g/S(36) + S(31)h/S(36) - S(37)i/S(36))log(x + S(1)) + (S(31)d/S(144) - S(25)e/S(72) + S(19)f/S(36) - S(13)g/S(18) + S(7)h/S(9) - S(2)i/S(9))log(x + S(2)) - (d - S(2)e + S(4)f - S(8)g + S(16)h - S(32)i)/(S(12)x + S(24)) - (d - e + f - g + h - i)/(S(6)x + S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, log(-x + S(1))/S(18) - S(35)log(-x + S(2))/S(432) + log(x + S(1))/S(54) + log(x + S(2))/S(144) - S(1)/(S(36)x + S(36)) + S(1)/(-S(12)x + S(12)) + S(1)/(-S(36)x + S(72)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)(x + S(2))/(x*S(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(144) - e/S(72))log(x + S(2)) + (d/S(54) + e/S(108))log(x + S(1)) + (d/S(18) + S(5)e/S(36))log(-x + S(1)) - (S(35)d/S(432) + S(29)e/S(216))log(-x + S(2)) - (d - e)/(S(36)x + S(36)) + (d + e)/(-S(12)x + S(12)) + (d + S(2)e)/(-S(36)x + S(72)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2))(d + ex + fxS(2))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(144) - e/S(72) + f/S(36))log(x + S(2)) + (d/S(54) + e/S(108) - f/S(27))log(x + S(1)) + (d/S(18) + S(5)e/S(36) + S(2)f/S(9))log(-x + S(1)) - (S(35)d/S(432) + S(29)e/S(216) + S(23)f/S(108))log(-x + S(2)) - (d - e + f)/(S(36)x + S(36)) + (d + e + f)/(-S(12)x + S(12)) + (d + S(2)e + S(4)f)/(-S(36)x + S(72)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2))(d + ex + fx*S(2) + gxS(3))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(144) - e/S(72) + f/S(36) - g/S(18))log(x + S(2)) + (d/S(54) + e/S(108) - f/S(27) + S(7)g/S(108))log(x + S(1)) + (d/S(18) + S(5)e/S(36) + S(2)f/S(9) + S(11)g/S(36))log(-x + S(1)) - (S(35)d/S(432) + S(29)e/S(216) + S(23)f/S(108) + S(17)g/S(54))log(-x + S(2)) - (d - e + f - g)/(S(36)x + S(36)) + (d + e + f + g)/(-S(12)x + S(12)) + (d + S(2)e + S(4)f + S(8)g)/(-S(36)x + S(72)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2))(d + ex + fx*S(2) + gx*S(3) + hxS(4))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(144) - e/S(72) + f/S(36) - g/S(18) + h/S(9))log(x + S(2)) + (d/S(54) + e/S(108) - f/S(27) + S(7)g/S(108) - S(5)h/S(54))log(x + S(1)) + (d/S(18) + S(5)e/S(36) + S(2)f/S(9) + S(11)g/S(36) + S(7)h/S(18))log(-x + S(1)) - (S(35)d/S(432) + S(29)e/S(216) + S(23)f/S(108) + S(17)g/S(54) + S(11)h/S(27))log(-x + S(2)) - (d - e + f - g + h)/(S(36)x + S(36)) + (d + e + f + g + h)/(-S(12)x + S(12)) + (d + S(2)e + S(4)f + S(8)g + S(16)h)/(-S(36)x + S(72)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2))(d + ex + fx*S(2) + gx*S(3) + hx*S(4) + ixS(5))/(xS(4) - S(5)xS(2) + S(4))S(2), x), x, (d/S(144) - e/S(72) + f/S(36) - g/S(18) + h/S(9) - S(2)i/S(9))log(x + S(2)) + (d/S(54) + e/S(108) - f/S(27) + S(7)g/S(108) - S(5)h/S(54) + S(13)i/S(108))log(x + S(1)) + (d/S(18) + S(5)e/S(36) + S(2)f/S(9) + S(11)g/S(36) + S(7)h/S(18) + S(17)i/S(36))log(-x + S(1)) - (S(35)d/S(432) + S(29)e/S(216) + S(23)f/S(108) + S(17)g/S(54) + S(11)h/S(27) + S(10)i/S(27))log(-x + S(2)) - (d - e + f - g + h - i)/(S(36)x + S(36)) + (d + e + f + g + h + i)/(-S(12)x + S(12)) + (d + S(2)e + S(4)f + S(8)g + S(16)h + S(32)i)/(-S(36)x + S(72)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ag - cgxS(4))/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, gx/sqrt(a + bx*S(2) + cx*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ag - cgx*S(4) + ex)/(a + bxS(2) + cxS(4))(S(3)/2), x), x, -(be + S(2)cex*S(2) - gx(-S(4)ac + bS(2)))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ag - cgx*S(4) + fx*S(3))/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, (S(2)af + bfx*S(2) + gx(-S(4)ac + bS(2)))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ag - cgx*S(4) + ex + fxS(3))/(a + bx*S(2) + cxS(4))(S(3)/2), x), x, -(-S(2)af + be - gx(-S(4)ac + bS(2)) + xS(2)(-bf + S(2)ce))/((-S(4)ac + bS(2))sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) # large time assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3) + hx*S(4) + jx*S(5) + kx*S(6) + lx*S(7) + mx*S(8))/(a + bx*S(3) + cx*S(6)), x), x, kx/c + lxS(2)/(S(2)c) + mxS(3)/(S(3)c) + (-bm + cj)log(a + bx*S(3) + cx*S(6))/(S(6)c*S(2)) - (-S(2)acm + b*S(2)m - bcj + S(2)cS(2)f)atanh((b + S(2)cxS(3))/sqrt(-S(4)ac + bS(2)))/(S(3)c*S(2)sqrt(-S(4)ac + bS(2))) + S(2)(S(2)/3)(-bk/c + g - (-S(2)ack + bS(2)k - bcg + S(2)cS(2)d)/(csqrt(-S(4)ac + bS(2))))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(1)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)(-bk/c + g - (-S(2)ack + b*S(2)k - bcg + S(2)cS(2)d)/(csqrt(-S(4)ac + bS(2))))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(1)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)sqrt(S(3))(-bk/c + g - (-S(2)ack + bS(2)k - bcg + S(2)cS(2)d)/(csqrt(-S(4)ac + bS(2))))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(1)/3)(b + sqrt(-S(4)ac + bS(2)))(S(2)/3)) + S(2)*(S(2)/3)(-bk/c + g + (bS(2)k + S(2)cS(2)d - c(S(2)ak + bg))/(csqrt(-S(4)ac + bS(2))))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(1)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)(-bk/c + g + (bS(2)k + S(2)cS(2)d - c(S(2)ak + bg))/(csqrt(-S(4)ac + bS(2))))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(1)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(2)/3)sqrt(S(3))(-bk/c + g + (b*S(2)k + S(2)cS(2)d - c(S(2)ak + bg))/(csqrt(-S(4)ac + bS(2))))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(1)/3)(b - sqrt(-S(4)ac + bS(2)))(S(2)/3)) - S(2)*(S(1)/3)(-bl/c + h - (-S(2)acl + b*S(2)l - bch + S(2)cS(2)e)/(csqrt(-S(4)ac + bS(2))))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b + sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(2)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)*(S(1)/3)(-bl/c + h - (-S(2)acl + b*S(2)l - bch + S(2)cS(2)e)/(csqrt(-S(4)ac + bS(2))))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b + sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(2)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)sqrt(S(3))(-bl/c + h - (-S(2)acl + bS(2)l - bch + S(2)cS(2)e)/(csqrt(-S(4)ac + bS(2))))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b + sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(2)/3)(b + sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)(-bl/c + h + (bS(2)l + S(2)cS(2)e - c(S(2)al + bh))/(csqrt(-S(4)ac + bS(2))))log(S(2)*(S(1)/3)c*(S(1)/3)x + (b - sqrt(-S(4)ac + bS(2)))(S(1)/3))/(S(6)c(S(2)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) + S(2)*(S(1)/3)(-bl/c + h + (bS(2)l + S(2)cS(2)e - c(S(2)al + bh))/(csqrt(-S(4)ac + bS(2))))log(S(2)*(S(2)/3)c*(S(2)/3)xS(2) - S(2)(S(1)/3)c(S(1)/3)x(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + (b - sqrt(-S(4)ac + bS(2)))(S(2)/3))/(S(12)c*(S(2)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)) - S(2)*(S(1)/3)sqrt(S(3))(-bl/c + h + (b*S(2)l + S(2)cS(2)e - c(S(2)al + bh))/(csqrt(-S(4)ac + bS(2))))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)c*(S(1)/3)x/(b - sqrt(-S(4)ac + bS(2)))(S(1)/3) + S(1))/S(3))/(S(6)c(S(2)/3)(b - sqrt(-S(4)ac + bS(2)))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bxn + cx*(S(2)n)), x), x, -S(2)cxhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - S(2)cxhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(a + bxn + cx*(S(2)n)), x), x, -S(2)cdxhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - S(2)cdxhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))) - cexS(2)hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - cexS(2)hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2))/(a + bx*n + cx*(S(2)n)), x), x, -S(2)cdxhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - S(2)cdxhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))) - cexS(2)hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - cexS(2)hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))) - S(2)cfx*S(3)hyper((S(1), S(3)/n), ((n + S(3))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(-S(12)ac + S(3)b*S(2) + S(3)bsqrt(-S(4)ac + bS(2))) - S(2)cfx*S(3)hyper((S(1), S(3)/n), ((n + S(3))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(-S(12)ac + S(3)b*S(2) - S(3)bsqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fxS(2) + gx*S(3))/(a + bx*n + cx*(S(2)n)), x), x, -S(2)cdxhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - S(2)cdxhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))) - cexS(2)hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))) - cexS(2)hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(-S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))) - S(2)cfx*S(3)hyper((S(1), S(3)/n), ((n + S(3))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(-S(12)ac + S(3)b*S(2) + S(3)bsqrt(-S(4)ac + bS(2))) - S(2)cfx*S(3)hyper((S(1), S(3)/n), ((n + S(3))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(-S(12)ac + S(3)b*S(2) - S(3)bsqrt(-S(4)ac + bS(2))) - cgxS(4)hyper((S(1), S(4)/n), ((n + S(4))/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(-S(8)ac + S(2)b*S(2) + S(2)bsqrt(-S(4)ac + bS(2))) - cgxS(4)hyper((S(1), S(4)/n), ((n + S(4))/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(-S(8)ac + S(2)b*S(2) - S(2)bsqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*n + cx*(S(2)n))*(S(-2)), x), x, -cx(S(4)ac(-S(2)n + S(1)) - bS(2)(-n + S(1)) + b(-n + S(1))sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) - cx(S(4)ac(-S(2)n + S(1)) - bS(2)(-n + S(1)) - b(-n + S(1))sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) + x(-S(2)ac + b*S(2) + bcxn)/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)/(a + bx*n + cx*(S(2)n))*S(2), x), x, S(2)bcS(2)ex(n + S(2))(-n + S(2))hyper((S(1), (n + S(2))/n), (S(2) + S(2)/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(an(b + sqrt(-S(4)ac + bS(2)))(n + S(2))(-S(4)ac + bS(2))(S(3)/2)) - S(2)bcS(2)ex(n + S(2))(-n + S(2))hyper((S(1), (n + S(2))/n), (S(2) + S(2)/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(an(b - sqrt(-S(4)ac + bS(2)))(n + S(2))(-S(4)ac + bS(2))(S(3)/2)) - cdx(S(4)ac(-S(2)n + S(1)) - b*S(2)(-n + S(1)) + b(-n + S(1))sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) - cdx(S(4)ac(-S(2)n + S(1)) - b*S(2)(-n + S(1)) - b(-n + S(1))sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) - cexS(2)(S(4)ac(-n + S(1)) - bS(2)(-n + S(2)))hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) - cexS(2)(S(4)ac(-n + S(1)) - bS(2)(-n + S(2)))hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) + dx(-S(2)ac + bS(2) + bcxn)/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))) + exS(2)(-S(2)ac + b*S(2) + bcxn)/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2))/(a + bx*n + cx*(S(2)n))*S(2), x), x, S(2)bcS(2)ex(n + S(2))(-n + S(2))hyper((S(1), (n + S(2))/n), (S(2) + S(2)/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(an(b + sqrt(-S(4)ac + bS(2)))(n + S(2))(-S(4)ac + bS(2))(S(3)/2)) - S(2)bcS(2)ex(n + S(2))(-n + S(2))hyper((S(1), (n + S(2))/n), (S(2) + S(2)/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(an(b - sqrt(-S(4)ac + bS(2)))(n + S(2))(-S(4)ac + bS(2))(S(3)/2)) + S(2)bcS(2)fx(n + S(3))(-n + S(3))hyper((S(1), (n + S(3))/n), (S(2) + S(3)/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(an(b + sqrt(-S(4)ac + bS(2)))(n + S(3))(-S(4)ac + bS(2))(S(3)/2)) - S(2)bcS(2)fx(n + S(3))(-n + S(3))hyper((S(1), (n + S(3))/n), (S(2) + S(3)/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(an(b - sqrt(-S(4)ac + bS(2)))(n + S(3))(-S(4)ac + bS(2))(S(3)/2)) - cdx(S(4)ac(-S(2)n + S(1)) - b*S(2)(-n + S(1)) + b(-n + S(1))sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) - cdx(S(4)ac(-S(2)n + S(1)) - b*S(2)(-n + S(1)) - b(-n + S(1))sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) - cexS(2)(S(4)ac(-n + S(1)) - bS(2)(-n + S(2)))hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) - cexS(2)(S(4)ac(-n + S(1)) - bS(2)(-n + S(2)))hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) - S(2)cfx*S(3)(S(2)ac(-S(2)n + S(3)) - b*S(2)(-n + S(3)))hyper((S(1), S(3)/n), ((n + S(3))/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(3)an(-S(4)ac + b*S(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) - S(2)cfx*S(3)(S(2)ac(-S(2)n + S(3)) - b*S(2)(-n + S(3)))hyper((S(1), S(3)/n), ((n + S(3))/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(S(3)an(-S(4)ac + b*S(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) + dx(-S(2)ac + bS(2) + bcxn)/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))) + exS(2)(-S(2)ac + b*S(2) + bcxn)/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))) + fxS(3)(-S(2)ac + b*S(2) + bcxn)/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fx*S(2) + gx*S(3))/(a + bx*n + cx*(S(2)n))*S(2), x), x, S(2)bcS(2)ex(n + S(2))(-n + S(2))hyper((S(1), (n + S(2))/n), (S(2) + S(2)/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(an(b + sqrt(-S(4)ac + bS(2)))(n + S(2))(-S(4)ac + bS(2))(S(3)/2)) - S(2)bcS(2)ex(n + S(2))(-n + S(2))hyper((S(1), (n + S(2))/n), (S(2) + S(2)/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(an(b - sqrt(-S(4)ac + bS(2)))(n + S(2))(-S(4)ac + bS(2))(S(3)/2)) + S(2)bcS(2)fx(n + S(3))(-n + S(3))hyper((S(1), (n + S(3))/n), (S(2) + S(3)/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(an(b + sqrt(-S(4)ac + bS(2)))(n + S(3))(-S(4)ac + bS(2))(S(3)/2)) - S(2)bcS(2)fx(n + S(3))(-n + S(3))hyper((S(1), (n + S(3))/n), (S(2) + S(3)/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(an(b - sqrt(-S(4)ac + bS(2)))(n + S(3))(-S(4)ac + bS(2))(S(3)/2)) + S(2)bcS(2)gx(n + S(4))(-n + S(4))hyper((S(1), (n + S(4))/n), (S(2) + S(4)/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(an(b + sqrt(-S(4)ac + bS(2)))(n + S(4))(-S(4)ac + bS(2))(S(3)/2)) - S(2)bcS(2)gx(n + S(4))(-n + S(4))hyper((S(1), (n + S(4))/n), (S(2) + S(4)/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(an(b - sqrt(-S(4)ac + bS(2)))(n + S(4))(-S(4)ac + bS(2))(S(3)/2)) - cdx(S(4)ac(-S(2)n + S(1)) - b*S(2)(-n + S(1)) + b(-n + S(1))sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) - cdx(S(4)ac(-S(2)n + S(1)) - b*S(2)(-n + S(1)) - b(-n + S(1))sqrt(-S(4)ac + b*S(2)))hyper((S(1), S(1)/n), (S(1) + S(1)/n,), -S(2)cx*n/(b - sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) - cexS(2)(S(4)ac(-n + S(1)) - bS(2)(-n + S(2)))hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) - cexS(2)(S(4)ac(-n + S(1)) - bS(2)(-n + S(2)))hyper((S(1), S(2)/n), ((n + S(2))/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(an(-S(4)ac + bS(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) - S(2)cfx*S(3)(S(2)ac(-S(2)n + S(3)) - b*S(2)(-n + S(3)))hyper((S(1), S(3)/n), ((n + S(3))/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(3)an(-S(4)ac + b*S(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) - S(2)cfx*S(3)(S(2)ac(-S(2)n + S(3)) - b*S(2)(-n + S(3)))hyper((S(1), S(3)/n), ((n + S(3))/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(S(3)an(-S(4)ac + b*S(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) - cgxS(4)(S(4)ac(-n + S(2)) - bS(2)(-n + S(4)))hyper((S(1), S(4)/n), ((n + S(4))/n,), -S(2)cxn/(b + sqrt(-S(4)ac + bS(2))))/(S(2)an(-S(4)ac + b*S(2))(-S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))) - cgxS(4)(S(4)ac(-n + S(2)) - bS(2)(-n + S(4)))hyper((S(1), S(4)/n), ((n + S(4))/n,), -S(2)cxn/(b - sqrt(-S(4)ac + bS(2))))/(S(2)an(-S(4)ac + b*S(2))(-S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))) + dx(-S(2)ac + bS(2) + bcxn)/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))) + exS(2)(-S(2)ac + b*S(2) + bcxn)/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))) + fxS(3)(-S(2)ac + b*S(2) + bcxn)/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))) + gxS(4)(-S(2)ac + b*S(2) + bcxn)/(an(-S(4)ac + bS(2))(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-ahx*(n/S(2) + S(-1)) + cfx(n + S(-1)) + cgx(S(2)n + S(-1)) + chx*(S(5)n/S(2) + S(-1)))/(a + bxn + cx*(S(2)n))*(S(3)/2), x), x, -(S(2)cxn(-bg + S(2)cf) + S(2)c(-S(2)ag + bf) + S(2)hx*(n/S(2))(-S(4)ac + b*S(2)))/(n(-S(4)ac + b*S(2))sqrt(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxn + cx*(S(2)n))*p(a + bxn(np + n + S(1)) + cx*(S(2)n)(S(2)n(p + S(1)) + S(1))), x), x, x(a + bxn + cx*(S(2)n))(p + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(A + Bx + CxS(2))(a + bxS(2) + cx*S(4)), x), x, Aax(m + S(1))/(m + S(1)) + Bax(m + S(2))/(m + S(2)) + Bbx(m + S(4))/(m + S(4)) + Bcx(m + S(6))/(m + S(6)) + Ccx(m + S(7))/(m + S(7)) + x(m + S(3))(Ab + Ca)/(m + S(3)) + x*(m + S(5))(Ac + Cb)/(m + S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(A + Bx + Cx*S(2))(a + bxS(2) + cx*S(4)), x), x, AaxS(4)/S(4) + BaxS(5)/S(5) + BbxS(7)/S(7) + BcxS(9)/S(9) + CcxS(10)/S(10) + xS(8)(Ac/S(8) + Cb/S(8)) + x*S(6)(Ab/S(6) + Ca/S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(A + Bx + Cx*S(2))(a + bxS(2) + cx*S(4)), x), x, AaxS(3)/S(3) + BaxS(4)/S(4) + BbxS(6)/S(6) + BcxS(8)/S(8) + CcxS(9)/S(9) + xS(7)(Ac/S(7) + Cb/S(7)) + x*S(5)(Ab/S(5) + Ca/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(A + Bx + CxS(2))(a + bxS(2) + cx*S(4)), x), x, AaxS(2)/S(2) + BaxS(3)/S(3) + BbxS(5)/S(5) + BcxS(7)/S(7) + CcxS(8)/S(8) + xS(6)(Ac/S(6) + Cb/S(6)) + x*S(4)(Ab/S(4) + Ca/S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + Cx*S(2))(a + bxS(2) + cx*S(4)), x), x, Aax + BaxS(2)/S(2) + BbxS(4)/S(4) + BcxS(6)/S(6) + CcxS(7)/S(7) + xS(5)(Ac/S(5) + Cb/S(5)) + x*S(3)(Ab/S(3) + Ca/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + Cx*S(2))(a + bxS(2) + cx*S(4))/x, x), x, Aalog(x) + Bax + BbxS(3)/S(3) + BcxS(5)/S(5) + CcxS(6)/S(6) + xS(4)(Ac/S(4) + Cb/S(4)) + x*S(2)(Ab/S(2) + Ca/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + Cx*S(2))(a + bxS(2) + cxS(4))/xS(2), x), x, -Aa/x + Balog(x) + BbxS(2)/S(2) + BcxS(4)/S(4) + CcxS(5)/S(5) + xS(3)(Ac/S(3) + Cb/S(3)) + x(Ab + Ca), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))(a + bxS(2) + cxS(4))/xS(3), x), x, -Aa/(S(2)x*S(2)) - Ba/x + Bbx + Bcx*S(3)/S(3) + CcxS(4)/S(4) + xS(2)(Ac/S(2) + Cb/S(2)) + (Ab + Ca)log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))(a + bxS(2) + cxS(4))/xS(4), x), x, -Aa/(S(3)x*S(3)) - Ba/(S(2)xS(2)) + Bblog(x) + BcxS(2)/S(2) + CcxS(3)/S(3) + x(Ac + Cb) - (Ab + Ca)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + Cx*S(2))(a + bxS(2) + cxS(4))/xS(5), x), x, -Aa/(S(4)x*S(4)) - Ba/(S(3)xS(3)) - Bb/x + Bcx + Ccx*S(2)/S(2) + (Ac + Cb)log(x) - (Ab + Ca)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))(a + bxS(2) + cxS(4))/xS(6), x), x, -Aa/(S(5)x*S(5)) - Ba/(S(4)xS(4)) - Bb/(S(2)xS(2)) + Bclog(x) + Ccx - (Ac + Cb)/x - (Ab + Ca)/(S(3)x*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))(a + bxS(2) + cxS(4))/xS(7), x), x, -Aa/(S(6)x*S(6)) - Ba/(S(5)xS(5)) - Bb/(S(3)xS(3)) - Bc/x + Cclog(x) - (Ac + Cb)/(S(2)xS(2)) - (Ab + Ca)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(A + Bx + CxS(2))(a + bxS(2) + cxS(4))S(2), x), x, AaS(2)x*(m + S(1))/(m + S(1)) + Ba*S(2)x*(m + S(2))/(m + S(2)) + S(2)Babx(m + S(4))/(m + S(4)) + S(2)Bbcx(m + S(8))/(m + S(8)) + Bc*S(2)x*(m + S(10))/(m + S(10)) + Bx*(m + S(6))(S(2)ac + b*S(2))/(m + S(6)) + Cc*S(2)x*(m + S(11))/(m + S(11)) + ax*(m + S(3))(S(2)Ab + Ca)/(m + S(3)) + cx*(m + S(9))(Ac + S(2)Cb)/(m + S(9)) + x(m + S(5))(A(S(2)ac + bS(2)) + S(2)Cab)/(m + S(5)) + x*(m + S(7))(S(2)Abc + C(S(2)ac + bS(2)))/(m + S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(A + Bx + CxS(2))(a + bxS(2) + cxS(4))S(2), x), x, AaS(2)x*S(4)/S(4) + Ba*S(2)x*S(5)/S(5) + S(2)BabxS(7)/S(7) + S(2)BbcxS(11)/S(11) + Bc*S(2)x*S(13)/S(13) + Bx*S(9)(S(2)ac + b*S(2))/S(9) + Cc*S(2)x*S(14)/S(14) + ax*S(6)(S(2)Ab + Ca)/S(6) + cx*S(12)(Ac + S(2)Cb)/S(12) + xS(10)(Abc/S(5) + C(S(2)ac + bS(2))/S(10)) + xS(8)(A(S(2)ac + bS(2))/S(8) + Cab/S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(A + Bx + Cx*S(2))(a + bxS(2) + cxS(4))S(2), x), x, AaS(2)x*S(3)/S(3) + Ba*S(2)x*S(4)/S(4) + Babx*S(6)/S(3) + Bbcx*S(10)/S(5) + Bc*S(2)x*S(12)/S(12) + Bx*S(8)(S(2)ac + b*S(2))/S(8) + Cc*S(2)x*S(13)/S(13) + ax*S(5)(S(2)Ab + Ca)/S(5) + cx*S(11)(Ac + S(2)Cb)/S(11) + xS(9)(S(2)Abc/S(9) + C(S(2)ac + bS(2))/S(9)) + xS(7)(A(S(2)ac + b*S(2))/S(7) + S(2)Cab/S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(A + Bx + CxS(2))(a + bxS(2) + cxS(4))S(2), x), x, AaS(2)x*S(2)/S(2) + Ba*S(2)x*S(3)/S(3) + S(2)BabxS(5)/S(5) + S(2)BbcxS(9)/S(9) + Bc*S(2)x*S(11)/S(11) + Bx*S(7)(S(2)ac + b*S(2))/S(7) + Cc*S(2)x*S(12)/S(12) + ax*S(4)(S(2)Ab + Ca)/S(4) + cx*S(10)(Ac + S(2)Cb)/S(10) + xS(8)(Abc/S(4) + C(S(2)ac + bS(2))/S(8)) + xS(6)(A(S(2)ac + bS(2))/S(6) + Cab/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))(a + bxS(2) + cxS(4))S(2), x), x, AaS(2)x + BaS(2)x*S(2)/S(2) + Babx*S(4)/S(2) + Bbcx*S(8)/S(4) + Bc*S(2)x*S(10)/S(10) + Bx*S(6)(S(2)ac + b*S(2))/S(6) + Cc*S(2)x*S(11)/S(11) + ax*S(3)(S(2)Ab + Ca)/S(3) + cx*S(9)(Ac + S(2)Cb)/S(9) + xS(7)(S(2)Abc/S(7) + C(S(2)ac + bS(2))/S(7)) + xS(5)(A(S(2)ac + b*S(2))/S(5) + S(2)Cab/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + Cx*S(2))(a + bxS(2) + cxS(4))S(2)/x, x), x, AaS(2)log(x) + BaS(2)x + S(2)Babx*S(3)/S(3) + S(2)BbcxS(7)/S(7) + Bc*S(2)x*S(9)/S(9) + Bx*S(5)(S(2)ac + b*S(2))/S(5) + Cc*S(2)x*S(10)/S(10) + ax*S(2)(S(2)Ab + Ca)/S(2) + cx*S(8)(Ac + S(2)Cb)/S(8) + xS(6)(Abc/S(3) + C(S(2)ac + bS(2))/S(6)) + xS(4)(A(S(2)ac + bS(2))/S(4) + Cab/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))(a + bxS(2) + cxS(4))S(2)/x*S(2), x), x, -Aa*S(2)/x + Ba*S(2)log(x) + BabxS(2) + Bbcx*S(6)/S(3) + Bc*S(2)x*S(8)/S(8) + Bx*S(4)(S(2)ac + b*S(2))/S(4) + Cc*S(2)x*S(9)/S(9) + ax(S(2)Ab + Ca) + cxS(7)(Ac + S(2)Cb)/S(7) + xS(5)(S(2)Abc/S(5) + C(S(2)ac + bS(2))/S(5)) + xS(3)(A(S(2)ac + b*S(2))/S(3) + S(2)Cab/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + Cx*S(2))(a + bxS(2) + cxS(4))S(2)/x*S(3), x), x, -Aa*S(2)/(S(2)x*S(2)) - Ba*S(2)/x + S(2)Babx + S(2)BbcxS(5)/S(5) + Bc*S(2)x*S(7)/S(7) + Bx*S(3)(S(2)ac + b*S(2))/S(3) + Cc*S(2)x*S(8)/S(8) + a(S(2)Ab + Ca)log(x) + cxS(6)(Ac + S(2)Cb)/S(6) + xS(4)(Abc/S(2) + C(S(2)ac + bS(2))/S(4)) + xS(2)(A(S(2)ac + bS(2))/S(2) + Cab), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))(a + bxS(2) + cxS(4))S(2)/x*S(4), x), x, -Aa*S(2)/(S(3)x*S(3)) - Ba*S(2)/(S(2)x*S(2)) + S(2)Bablog(x) + Bbcx*S(4)/S(2) + Bc*S(2)x*S(6)/S(6) + Bx*S(2)(S(2)ac + b*S(2))/S(2) + Cc*S(2)x*S(7)/S(7) - a(S(2)Ab + Ca)/x + cx*S(5)(Ac + S(2)Cb)/S(5) + xS(3)(S(2)Abc/S(3) + C(S(2)ac + b*S(2))/S(3)) + x(A(S(2)ac + bS(2)) + S(2)Cab), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + Cx*S(2))(a + bxS(2) + cxS(4))S(2)/x*S(5), x), x, -Aa*S(2)/(S(4)x*S(4)) - Ba*S(2)/(S(3)x*S(3)) - S(2)Bab/x + S(2)Bbcx*S(3)/S(3) + Bc*S(2)x*S(5)/S(5) + Bx(S(2)ac + bS(2)) + Cc*S(2)x*S(6)/S(6) - a(S(2)Ab + Ca)/(S(2)x*S(2)) + cx*S(4)(Ac + S(2)Cb)/S(4) + xS(2)(Abc + C(S(2)ac + bS(2))/S(2)) + (A(S(2)ac + b*S(2)) + S(2)Cab)log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))(a + bxS(2) + cxS(4))S(2)/x*S(6), x), x, -Aa*S(2)/(S(5)x*S(5)) - Ba*S(2)/(S(4)x*S(4)) - Bab/xS(2) + Bbcx*S(2) + Bc*S(2)x*S(4)/S(4) + B(S(2)ac + b*S(2))log(x) + CcS(2)x*S(5)/S(5) - a(S(2)Ab + Ca)/(S(3)x*S(3)) + cx*S(3)(Ac + S(2)Cb)/S(3) + x(S(2)Abc + C(S(2)ac + b*S(2))) - (A(S(2)ac + b*S(2)) + S(2)Cab)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + Cx*S(2))(a + bxS(2) + cxS(4))S(2)/x*S(7), x), x, -Aa*S(2)/(S(6)x*S(6)) - Ba*S(2)/(S(5)x*S(5)) - S(2)Bab/(S(3)xS(3)) + S(2)Bbcx + Bc*S(2)x*S(3)/S(3) - B(S(2)ac + b*S(2))/x + Cc*S(2)x*S(4)/S(4) - a(S(2)Ab + Ca)/(S(4)x*S(4)) + cx*S(2)(Ac + S(2)Cb)/S(2) + (S(2)Abc + C(S(2)ac + bS(2)))log(x) - (A(S(2)ac + bS(2)) + S(2)Cab)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(A + Bx + Cx*S(2))/(a + bx*S(2) + cx*S(4)), x), x, -S(2)Bcx*(m + S(2))hyper((S(1), m/S(2) + S(1)), (m/S(2) + S(2),), -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))))/((b + sqrt(-S(4)ac + bS(2)))(m + S(2))sqrt(-S(4)ac + bS(2))) + S(2)Bcx*(m + S(2))hyper((S(1), m/S(2) + S(1)), (m/S(2) + S(2),), -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))))/((b - sqrt(-S(4)ac + bS(2)))(m + S(2))sqrt(-S(4)ac + bS(2))) + x(m + S(1))(C - (S(2)Ac - Cb)/sqrt(-S(4)ac + bS(2)))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))))/((b + sqrt(-S(4)ac + bS(2)))(m + S(1))) + x*(m + S(1))(C + (S(2)Ac - Cb)/sqrt(-S(4)ac + bS(2)))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))))/((b - sqrt(-S(4)ac + bS(2)))(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)(A + Bx + Cx*S(2))/(a + bx*S(2) + cx*S(4)), x), x, -Bblog(a + bx*S(2) + cx*S(4))/(S(4)c*S(2)) + Bx*S(2)/(S(2)c) - B(-S(2)ac + bS(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)sqrt(-S(4)ac + b*S(2))) + Cx*S(3)/(S(3)c) + x(Ac - Cb)/cS(2) - sqrt(S(2))(Abc + Cac - CbS(2) + (Ac(-S(2)ac + bS(2)) - Cb(-S(3)ac + bS(2)))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) - sqrt(S(2))(Abc + Cac - CbS(2) - (Ac(-S(2)ac + bS(2)) - Cb(-S(3)ac + bS(2)))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(A + Bx + CxS(2))/(a + bx*S(2) + cx*S(4)), x), x, Bx/c - sqrt(S(2))B(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))) - sqrt(S(2))B(b + (-S(2)ac + bS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) + Cx*S(2)/(S(2)c) + (Ac - Cb)log(a + bx*S(2) + cx*S(4))/(S(4)c*S(2)) + (Abc + S(2)Cac - CbS(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(A + Bx + CxS(2))/(a + bx*S(2) + cx*S(4)), x), x, Bbatanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-S(4)ac + bS(2))) + Blog(a + bxS(2) + cx*S(4))/(S(4)c) + Cx/c + sqrt(S(2))(Ac - Cb + (Abc + S(2)Cac - Cb*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) + sqrt(S(2))(Ac - Cb - (Abc - C(-S(2)ac + bS(2)))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(A + Bx + Cx*S(2))/(a + bx*S(2) + cx*S(4)), x), x, -sqrt(S(2))Bsqrt(b - sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(-S(4)ac + bS(2))) + sqrt(S(2))Bsqrt(b + sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(-S(4)ac + bS(2))) + Clog(a + bxS(2) + cx*S(4))/(S(4)c) - (S(2)Ac - Cb)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))/(a + bx*S(2) + cx*S(4)), x), x, -Batanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)) + sqrt(S(2))(C - (S(2)Ac - Cb)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(C + (S(2)Ac - Cb)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))/(x(a + bxS(2) + cx*S(4))), x), x, Alog(x)/a - Alog(a + bx*S(2) + cx*S(4))/(S(4)a) - sqrt(S(2))Bsqrt(c)atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + bS(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))) + sqrt(S(2))Bsqrt(c)atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))) + (Ab - S(2)Ca)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)asqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))/(xS(2)(a + bxS(2) + cx*S(4))), x), x, -A/(ax) + Bbatanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)asqrt(-S(4)ac + bS(2))) + Blog(x)/a - Blog(a + bx*S(2) + cx*S(4))/(S(4)a) - sqrt(S(2))sqrt(c)(A - (Ab - S(2)Ca)/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(b + sqrt(-S(4)ac + bS(2)))) - sqrt(S(2))sqrt(c)(A + (Ab - S(2)Ca)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))/(xS(3)(a + bxS(2) + cx*S(4))), x), x, -A/(S(2)axS(2)) - sqrt(S(2))Bsqrt(c)(-b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(b + sqrt(-S(4)ac + bS(2)))) - sqrt(S(2))Bsqrt(c)(b/sqrt(-S(4)ac + b*S(2)) + S(1))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(b - sqrt(-S(4)ac + bS(2)))) - B/(ax) - (Ab - Ca)log(x)/aS(2) + (Ab - Ca)log(a + bxS(2) + cx*S(4))/(S(4)a*S(2)) - (A(-S(2)ac + b*S(2)) - Cab)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(A + Bx + CxS(2))/(a + bx*S(2) + cxS(4))S(2), x), x, Bcx*(m + S(2))(S(4)ac(-m + S(2)) + bm(b - sqrt(-S(4)ac + bS(2))))hyper((S(1), m/S(2) + S(1)), (m/S(2) + S(2),), -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)a(b + sqrt(-S(4)ac + bS(2)))(m + S(2))(-S(4)ac + bS(2))(S(3)/2)) - Bcx(m + S(2))(S(4)ac(-m + S(2)) + bm(b + sqrt(-S(4)ac + bS(2))))hyper((S(1), m/S(2) + S(1)), (m/S(2) + S(2),), -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))))/(S(2)a(b - sqrt(-S(4)ac + bS(2)))(m + S(2))(-S(4)ac + bS(2))(S(3)/2)) + Bx*(m + S(2))(-S(2)ac + b*S(2) + bcxS(2))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))) - cx*(m + S(1))(A(-S(4)ac(-m + S(3)) + b*S(2)(-m + S(1)) - b(-m + S(1))sqrt(-S(4)ac + b*S(2))) + S(2)Ca(S(2)b + (-m + S(1))sqrt(-S(4)ac + b*S(2))))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(2)a(b + sqrt(-S(4)ac + bS(2)))(m + S(1))(-S(4)ac + bS(2))(S(3)/2)) + cx*(m + S(1))(A(-S(4)ac(-m + S(3)) + b*S(2)(-m + S(1)) + b(-m + S(1))sqrt(-S(4)ac + b*S(2))) + S(2)Ca(S(2)b - (-m + S(1))sqrt(-S(4)ac + b*S(2))))hyper((S(1), m/S(2) + S(1)/2), (m/S(2) + S(3)/2,), -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))))/(S(2)a(b - sqrt(-S(4)ac + bS(2)))(m + S(1))(-S(4)ac + bS(2))(S(3)/2)) + x(m + S(1))(A(-S(2)ac + bS(2)) - Cab + cx*S(2)(Ab - S(2)Ca))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(A + Bx + CxS(2))/(a + bx*S(2) + cxS(4))S(2), x), x, S(2)Baatanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + x(S(2)Bacx + BbcxS(3) + a(S(2)Ac - Cb) + xS(2)(Abc + S(2)Cac - Cb*S(2)))/(S(2)c(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))) + sqrt(S(2))(Ac(S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))) + C(-S(8)abc - S(6)acsqrt(-S(4)ac + bS(2)) + bS(3) + b*S(2)sqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - sqrt(S(2))(Ac(S(4)ac + bS(2) - bsqrt(-S(4)ac + b*S(2))) + C(-S(8)abc + S(6)acsqrt(-S(4)ac + bS(2)) + bS(3) - b*S(2)sqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(A + Bx + Cx*S(2))/(a + bx*S(2) + cxS(4))S(2), x), x, sqrt(S(2))B(S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) - sqrt(S(2))B(S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + x(S(2)Ba + BbxS(2) - xS(3)(S(2)Ac - Cb) - x(Ab - S(2)Ca))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) - (Ab - S(2)Ca)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(A + Bx + Cx*S(2))/(a + bx*S(2) + cxS(4))S(2), x), x, -Bbatanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) - x(Ab + Bbx + S(2)Bcx*S(3) - S(2)Ca + xS(2)(S(2)Ac - Cb))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) - sqrt(S(2))(S(2)Ac(S(2)b + sqrt(-S(4)ac + b*S(2))) - C(S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))(S(2)Ac(S(2)b - sqrt(-S(4)ac + bS(2))) - C(S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(A + Bx + CxS(2))/(a + bx*S(2) + cxS(4))S(2), x), x, -sqrt(S(2))Bsqrt(c)(S(2)b + sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))Bsqrt(c)(S(2)b - sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) + (S(2)Ac - Cb)atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) - x(Bab + S(2)Bacx*S(2) - cx*S(3)(Ab - S(2)Ca) - x(A(-S(2)ac + bS(2)) - Cab))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))/(a + bx*S(2) + cxS(4))S(2), x), x, S(2)Bcatanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + sqrt(S(2))sqrt(c)(Ab - S(2)Ca - (-S(12)Aac + Ab*S(2) + S(4)Cab)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)(Ab - S(2)Ca + (A(-S(12)ac + bS(2)) + S(4)Cab)/sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + x(A(-S(2)ac + bS(2)) + Bbcx*S(3) + Bx(-S(2)ac + bS(2)) - Cab + cx*S(2)(Ab - S(2)Ca))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))/(x(a + bxS(2) + cxS(4))S(2)), x), x, Alog(x)/aS(2) - Alog(a + bxS(2) + cx*S(4))/(S(4)a*S(2)) - sqrt(S(2))Bsqrt(c)(-S(12)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))Bsqrt(c)(-S(12)ac + bS(2) + bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + Bx(-S(2)ac + b*S(2) + bcxS(2))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))) + (A(-S(2)ac + b*S(2)) - Cab + cx*S(2)(Ab - S(2)Ca))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))) + (A(-S(6)abc + bS(3)) + S(4)CaS(2)c)atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))/(xS(2)(a + bx*S(2) + cxS(4))S(2)), x), x, B(-S(2)ac + bS(2) + bcxS(2))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))) + Bb(-S(6)ac + bS(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)(-S(4)ac + bS(2))(S(3)/2)) + Blog(x)/aS(2) - Blog(a + bxS(2) + cx*S(4))/(S(4)a*S(2)) + (A(-S(2)ac + b*S(2)) - Cab + cx*S(2)(Ab - S(2)Ca))/(S(2)ax(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) - sqrt(S(2))sqrt(c)(-S(10)Aac + S(3)Ab*S(2) - Cab - (A(-S(16)abc + S(3)b*S(3)) - Ca(-S(12)ac + bS(2)))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + b*S(2))) - sqrt(S(2))sqrt(c)(A(-S(16)abc - S(10)acsqrt(-S(4)ac + b*S(2)) + S(3)b*S(3) + S(3)b*S(2)sqrt(-S(4)ac + b*S(2))) - Ca(-S(12)ac + bS(2) + bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - (-S(10)Aac + S(3)AbS(2) - Cab)/(S(2)a*S(2)x(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2))/(xS(3)(a + bxS(2) + cxS(4))S(2)), x), x, B(-S(2)ac + bS(2) + bcxS(2))/(S(2)ax(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) + sqrt(S(2))Bsqrt(c)(-S(16)abc + S(3)b*S(3) - (-S(10)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - sqrt(S(2))Bsqrt(c)(-S(16)abc + S(3)bS(3) + (-S(10)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - B(-S(10)ac + S(3)b*S(2))/(S(2)a*S(2)x(-S(4)ac + bS(2))) + (A(-S(2)ac + b*S(2)) - Cab + cx*S(2)(Ab - S(2)Ca))/(S(2)axS(2)(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) - (-S(6)Aac + S(2)Ab*S(2) - Cab)/(S(2)a*S(2)x*S(2)(-S(4)ac + b*S(2))) - (S(2)Ab - Ca)log(x)/aS(3) + (S(2)Ab - Ca)log(a + bx*S(2) + cx*S(4))/(S(4)a*S(3)) - (S(2)A(S(6)a*S(2)c*S(2) - S(6)abS(2)c + b*S(4)) - Cab(-S(6)ac + b*S(2)))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(3)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(A + Bx + Cx*S(2))/(a + bx*S(2) + cxS(4))S(2), x), x, -Bbatanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) - x(Ab + Bbx + S(2)Bcx*S(3) - S(2)Ca + xS(2)(S(2)Ac - Cb))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) - sqrt(S(2))(S(2)Ac(S(2)b + sqrt(-S(4)ac + b*S(2))) - C(S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))(S(2)Ac(S(2)b - sqrt(-S(4)ac + bS(2))) - C(S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(Ax + BxS(2) + Cx*S(3))/(a + bx*S(2) + cxS(4))S(2), x), x, -Bbatanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) - x(Ab + Bbx + S(2)Bcx*S(3) - S(2)Ca + xS(2)(S(2)Ac - Cb))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) - sqrt(S(2))(S(2)Ac(S(2)b + sqrt(-S(4)ac + b*S(2))) - C(S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))(S(2)Ac(S(2)b - sqrt(-S(4)ac + bS(2))) - C(S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((AxS(2) + Bx*S(3) + Cx*S(4))/(a + bx*S(2) + cxS(4))S(2), x), x, -Bbatanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) - x(Ab + Bbx + S(2)Bcx*S(3) - S(2)Ca + xS(2)(S(2)Ac - Cb))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) - sqrt(S(2))(S(2)Ac(S(2)b + sqrt(-S(4)ac + b*S(2))) - C(S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))(S(2)Ac(S(2)b - sqrt(-S(4)ac + bS(2))) - C(S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((AxS(3) + Bx*S(4) + Cx*S(5))/(x(a + bxS(2) + cxS(4))S(2)), x), x, -Bbatanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) - x(Ab + Bbx + S(2)Bcx*S(3) - S(2)Ca + xS(2)(S(2)Ac - Cb))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) - sqrt(S(2))(S(2)Ac(S(2)b + sqrt(-S(4)ac + b*S(2))) - C(S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))(S(2)Ac(S(2)b - sqrt(-S(4)ac + bS(2))) - C(S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((AxS(4) + Bx*S(5) + CxS(6))/(xS(2)(a + bx*S(2) + cxS(4))S(2)), x), x, -Bbatanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) - x(Ab + Bbx + S(2)Bcx*S(3) - S(2)Ca + xS(2)(S(2)Ac - Cb))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) - sqrt(S(2))(S(2)Ac(S(2)b + sqrt(-S(4)ac + b*S(2))) - C(S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))(S(2)Ac(S(2)b - sqrt(-S(4)ac + bS(2))) - C(S(4)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2))))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(d + ex*S(2) + fx*S(4) + gx*S(6))/(a + bx*S(2) + cx*S(4)), x), x, gx*S(4)/(S(4)c) + x*S(2)(-bg + cf)/(S(2)cS(2)) + (bS(2)g + c*S(2)e - c(ag + bf))log(a + bxS(2) + cx*S(4))/(S(4)cS(3)) - (-bS(3)g + bc(S(3)ag + bf) + S(2)cS(3)d - c*S(2)(S(2)af + be))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(3)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx*S(2) + cxS(4))p(S(3)a + bxS(2)(S(2)p + S(5)) + cx*S(4)(S(4)p + S(7))), x), x, xS(3)(a + bxS(2) + cxS(4))(p + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(n/S(4) + S(-1))(-ah + cfx(n/S(4)) + cgx(S(3)n/S(4)) + chx*n)/(a + cxn)(S(3)/2), x), x, -(S(2)ag + S(4)ahx(n/S(4)) - S(2)cfx*(n/S(2)))/(ansqrt(a + cx*n)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(n/S(4) + S(-1))(-ah + cfx(n/S(4)) + cgx(S(3)n/S(4)) + chx*n)/(a + cxn)(S(3)/2), x), x, -S(2)x(-n/S(4) + S(1))(dx)(n/S(4) + S(-1))(ag + S(2)ahx*(n/S(4)) - cfx(n/S(2)))/(ansqrt(a + cxn)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(n/S(2) + S(-1))(-ah + cfx*(n/S(2)) + cgx(S(3)n/S(2)) + chx*(S(2)n))/(a + bxn + cx*(S(2)n))*(S(3)/2), x), x, -(S(2)cxn(-bg + S(2)cf) + S(2)c(-S(2)ag + bf) + S(2)hx*(n/S(2))(-S(4)ac + b*S(2)))/(n(-S(4)ac + b*S(2))sqrt(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(n/S(2) + S(-1))(-ah + cfx(n/S(2)) + cgx(S(3)n/S(2)) + chx*(S(2)n))/(a + bxn + cx*(S(2)n))*(S(3)/2), x), x, -S(2)x*(-n/S(2) + S(1))(dx)(n/S(2) + S(-1))(cxn(-bg + S(2)cf) + c(-S(2)ag + bf) + hx*(n/S(2))(-S(4)ac + b*S(2)))/(n(-S(4)ac + b*S(2))sqrt(a + bxn + cx*(S(2)n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((gx)m(a + bxn + cx*(S(2)n))*p(a(m + S(1)) + bx*n(m + np + n + S(1)) + cx*(S(2)n)(m + S(2)n(p + S(1)) + S(1))), x), x, (gx)*(m + S(1))(a + bxn + cx*(S(2)n))(p + S(1))/g, expand=True, _diff=True, _numerical=True) def test_5(): assert rubi_test(rubi_integrate(xS(2)(ax*S(2) + bx*S(3) + cx*S(4)), x), x, ax*S(5)/S(5) + bx*S(6)/S(6) + cx*S(7)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(axS(2) + bx*S(3) + cx*S(4)), x), x, ax*S(4)/S(4) + bx*S(5)/S(5) + cx*S(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(ax*S(2) + bx*S(3) + cx*S(4), x), x, ax*S(3)/S(3) + bx*S(4)/S(4) + cx*S(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cx*S(4))/x, x), x, ax*S(2)/S(2) + bx*S(3)/S(3) + cx*S(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))/xS(2), x), x, ax + bx*S(2)/S(2) + cxS(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(ax*S(2) + bx*S(3) + cxS(4))S(2), x), x, a*S(2)x*S(7)/S(7) + abxS(8)/S(4) + bcxS(10)/S(5) + cS(2)xS(11)/S(11) + xS(9)(S(2)ac + bS(2))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(axS(2) + bx*S(3) + cxS(4))S(2), x), x, a*S(2)x*S(6)/S(6) + S(2)abx*S(7)/S(7) + S(2)bcxS(9)/S(9) + cS(2)xS(10)/S(10) + xS(8)(S(2)ac + b*S(2))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))S(2), x), x, a*S(2)x*S(5)/S(5) + abxS(6)/S(3) + bcxS(8)/S(4) + cS(2)xS(9)/S(9) + xS(7)(S(2)ac + bS(2))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))S(2)/x, x), x, a*S(2)x*S(4)/S(4) + S(2)abx*S(5)/S(5) + S(2)bcxS(7)/S(7) + cS(2)xS(8)/S(8) + xS(6)(S(2)ac + b*S(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))S(2)/xS(2), x), x, aS(2)xS(3)/S(3) + abxS(4)/S(2) + bcxS(6)/S(3) + cS(2)xS(7)/S(7) + xS(5)(S(2)ac + bS(2))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(ax*S(2) + bx*S(3) + cx*S(4)), x), x, -bx/c*S(2) + b(-S(3)ac + b*S(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(3)sqrt(-S(4)ac + bS(2))) + xS(2)/(S(2)c) + (-ac + bS(2))log(a + bx + cx*S(2))/(S(2)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(axS(2) + bx*S(3) + cx*S(4)), x), x, -blog(a + bx + cx*S(2))/(S(2)c*S(2)) + x/c - (-S(2)ac + bS(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(ax*S(2) + bx*S(3) + cx*S(4)), x), x, batanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(csqrt(-S(4)ac + b*S(2))) + log(a + bx + cxS(2))/(S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(ax*S(2) + bx*S(3) + cx*S(4)), x), x, -S(2)atanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/sqrt(-S(4)ac + bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(ax*S(2) + bx*S(3) + cx*S(4)), x), x, batanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(asqrt(-S(4)ac + b*S(2))) + log(x)/a - log(a + bx + cxS(2))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(axS(2) + bx*S(3) + cx*S(4)), x), x, -S(1)/(ax) - blog(x)/aS(2) + blog(a + bx + cx*S(2))/(S(2)a*S(2)) - (-S(2)ac + bS(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(axS(2) + bx*S(3) + cx*S(4))), x), x, -S(1)/(S(2)axS(2)) + b/(aS(2)x) + b(-S(3)ac + bS(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(3)sqrt(-S(4)ac + bS(2))) + (-ac + b*S(2))log(x)/a*S(3) - (-ac + b*S(2))log(a + bx + cx*S(2))/(S(2)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(ax*S(2) + bx*S(3) + cx*S(4))), x), x, -S(1)/(S(3)axS(3)) + b/(S(2)a*S(2)x*S(2)) - (-ac + bS(2))/(aS(3)x) - b(-S(2)ac + b*S(2))log(x)/a*S(4) + b(-S(2)ac + b*S(2))log(a + bx + cx*S(2))/(S(2)a*S(4)) - (S(2)a*S(2)c*S(2) - S(4)abS(2)c + b*S(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(4)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(ax*S(2) + bx*S(3) + cxS(4))S(2), x), x, -bxS(2)/(c(-S(4)ac + b*S(2))) - blog(a + bx + cxS(2))/cS(3) + x*S(3)(S(2)a + bx)/((-S(4)ac + b*S(2))(a + bx + cx*S(2))) + x(-S(6)ac + S(2)bS(2))/(cS(2)(-S(4)ac + b*S(2))) - (S(12)a*S(2)c*S(2) - S(12)abS(2)c + S(2)bS(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(3)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(ax*S(2) + bx*S(3) + cxS(4))S(2), x), x, -bx/(c(-S(4)ac + b*S(2))) + b(-S(6)ac + b*S(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(cS(2)(-S(4)ac + bS(2))(S(3)/2)) + xS(2)(S(2)a + bx)/((-S(4)ac + b*S(2))(a + bx + cx*S(2))) + log(a + bx + cxS(2))/(S(2)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(axS(2) + bx*S(3) + cxS(4))S(2), x), x, S(4)aatanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(-S(4)ac + bS(2))(S(3)/2) + x(S(2)a + bx)/((-S(4)ac + b*S(2))(a + bx + cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(axS(2) + bx*S(3) + cxS(4))S(2), x), x, -S(2)batanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(-S(4)ac + bS(2))(S(3)/2) + (S(2)a + bx)/((-S(4)ac + bS(2))(a + bx + cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(axS(2) + bx*S(3) + cxS(4))S(2), x), x, S(4)catanh((b + S(2)cx)/sqrt(-S(4)ac + b*S(2)))/(-S(4)ac + bS(2))(S(3)/2) - (b + S(2)cx)/((-S(4)ac + bS(2))(a + bx + cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(axS(2) + bx*S(3) + cxS(4))S(2), x), x, (-S(2)ac + b*S(2) + bcx)/(a(-S(4)ac + b*S(2))(a + bx + cx*S(2))) + b(-S(6)ac + b*S(2))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(2)(-S(4)ac + bS(2))(S(3)/2)) + log(x)/aS(2) - log(a + bx + cxS(2))/(S(2)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(axS(2) + bx*S(3) + cxS(4))S(2), x), x, (-S(2)ac + b*S(2) + bcx)/(ax(-S(4)ac + bS(2))(a + bx + cx*S(2))) + (S(6)ac - S(2)bS(2))/(aS(2)x(-S(4)ac + b*S(2))) - S(2)blog(x)/aS(3) + blog(a + bx + cxS(2))/aS(3) - (S(12)aS(2)c*S(2) - S(12)abS(2)c + S(2)bS(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(3)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(ax*S(2) + bx*S(3) + cxS(4))S(2), x), x, (-S(2)ac + b*S(2) + bcx)/(ax*S(2)(-S(4)ac + b*S(2))(a + bx + cx*S(2))) - (-S(8)ac + S(3)b*S(2))/(S(2)a*S(2)x*S(2)(-S(4)ac + b*S(2))) + b(-S(11)ac + S(3)bS(2))/(aS(3)x(-S(4)ac + bS(2))) + b(S(30)aS(2)c*S(2) - S(20)abS(2)c + S(3)bS(4))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(4)(-S(4)ac + bS(2))(S(3)/2)) + (-S(2)ac + S(3)b*S(2))log(x)/a*S(4) - (-S(2)ac + S(3)b*S(2))log(a + bx + cx*S(2))/(S(2)a*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))(S(-2)), x), x, (-S(2)ac + b*S(2) + bcx)/(ax*S(3)(-S(4)ac + b*S(2))(a + bx + cx*S(2))) - (-S(10)ac + S(4)b*S(2))/(S(3)a*S(2)x*S(3)(-S(4)ac + b*S(2))) + b(-S(7)ac + S(2)bS(2))/(aS(3)x*S(2)(-S(4)ac + b*S(2))) - (S(10)a*S(2)c*S(2) - S(18)abS(2)c + S(4)bS(4))/(aS(4)x(-S(4)ac + bS(2))) - S(2)b(-S(3)ac + S(2)b*S(2))log(x)/a*S(5) + b(-S(3)ac + S(2)bS(2))log(a + bx + cxS(2))/aS(5) - (-S(20)aS(3)c*S(3) + S(60)a*S(2)b*S(2)c*S(2) - S(30)abS(4)c + S(4)bS(6))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(5)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(axS(2) + bx*S(3) + cxS(4))S(2)), x), x, (-S(2)ac + b*S(2) + bcx)/(ax*S(4)(-S(4)ac + b*S(2))(a + bx + cx*S(2))) - (-S(12)ac + S(5)b*S(2))/(S(4)a*S(2)x*S(4)(-S(4)ac + b*S(2))) + b(-S(17)ac + S(5)bS(2))/(S(3)a*S(3)x*S(3)(-S(4)ac + b*S(2))) - (S(12)a*S(2)c*S(2) - S(22)abS(2)c + S(5)bS(4))/(S(2)a*S(4)x*S(2)(-S(4)ac + b*S(2))) + b(S(29)aS(2)c*S(2) - S(27)abS(2)c + S(5)bS(4))/(aS(5)x(-S(4)ac + bS(2))) + b(-S(70)aS(3)c*S(3) + S(105)a*S(2)b*S(2)c*S(2) - S(42)abS(4)c + S(5)bS(6))atanh((b + S(2)cx)/sqrt(-S(4)ac + bS(2)))/(aS(6)(-S(4)ac + bS(2))(S(3)/2)) + (S(3)a*S(2)c*S(2) - S(12)abS(2)c + S(5)bS(4))log(x)/a*S(6) - (S(3)a*S(2)c*S(2) - S(12)abS(2)c + S(5)bS(4))log(a + bx + cx*S(2))/(S(2)aS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(ax*S(2) + bx*S(3) + cx*S(4)), x), x, b(-S(116)ac + S(35)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(960)c*S(3)) + bx(-S(12)ac + S(7)b*S(2))(-S(4)ac + b*S(2))sqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(256)c*(S(9)/2)sqrt(axS(2) + bx*S(3) + cxS(4))) + xS(2)(b + S(8)cx)sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(40)c) - x(-S(16)ac + S(7)b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(240)c*S(2)) - sqrt(ax*S(2) + bx*S(3) + cx*S(4))(S(256)aS(2)c*S(2) - S(460)abS(2)c + S(105)bS(4))/(S(1920)c*S(4)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(ax*S(2) + bx*S(3) + cx*S(4)), x), x, b(-S(52)ac + S(15)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(192)c*S(3)x) + x(b + S(6)cx)sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(24)c) - (-S(12)ac + S(5)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(96)c*S(2)) - x(-S(4)ac + b*S(2))(-S(4)ac + S(5)bS(2))sqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(128)c*(S(7)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax*S(2) + bx*S(3) + cx*S(4)), x), x, -b(b + S(2)cx)sqrt(ax*S(2) + bx*S(3) + cx*S(4))/(S(8)c*S(2)x) + b(-S(4)ac + bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(16)c*(S(5)/2)xsqrt(a + bx + cxS(2))) + (a + bx + cxS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(3)cx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax*S(2) + bx*S(3) + cx*S(4))/x, x), x, (b + S(2)cx)sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(4)cx) - x(-S(4)ac + b*S(2))sqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(8)c*(S(3)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax*S(2) + bx*S(3) + cxS(4))/xS(2), x), x, -sqrt(a)xsqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/sqrt(ax*S(2) + bx*S(3) + cx*S(4)) + bxsqrt(a + bx + cxS(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(2)sqrt(c)sqrt(ax*S(2) + bx*S(3) + cx*S(4))) + sqrt(ax*S(2) + bx*S(3) + cx*S(4))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax*S(2) + bx*S(3) + cxS(4))/xS(3), x), x, sqrt(c)xsqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/sqrt(ax*S(2) + bx*S(3) + cx*S(4)) - sqrt(ax*S(2) + bx*S(3) + cxS(4))/xS(2) - bxsqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(2)sqrt(a)sqrt(ax*S(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax*S(2) + bx*S(3) + cxS(4))/xS(4), x), x, -sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(2)x*S(3)) - bsqrt(axS(2) + bx*S(3) + cx*S(4))/(S(4)axS(2)) + x(-S(4)ac + b*S(2))sqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(8)a*(S(3)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax*S(2) + bx*S(3) + cxS(4))/xS(5), x), x, -sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(3)x*S(4)) - bsqrt(axS(2) + bx*S(3) + cx*S(4))/(S(12)axS(3)) + (-S(8)ac + S(3)b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(24)a*S(2)x*S(2)) - bx(-S(4)ac + bS(2))sqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(16)a*(S(5)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax*S(2) + bx*S(3) + cxS(4))/xS(6), x), x, -sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(4)x*S(5)) - bsqrt(axS(2) + bx*S(3) + cx*S(4))/(S(24)axS(4)) + (-S(12)ac + S(5)b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(96)a*S(2)x*S(3)) - b(-S(52)ac + S(15)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(192)a*S(3)x*S(2)) + x(-S(4)ac + b*S(2))(-S(4)ac + S(5)bS(2))sqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(128)a*(S(7)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(axS(2) + bx*S(3) + cxS(4))(S(3)/2), x), x, -bxsqrt(axS(2) + bx*S(3) + cx*S(4))(S(2416)aS(2)c*S(2) - S(1560)abS(2)c + S(231)bS(4))/(S(71680)c*S(4)) - bsqrt(axS(2) + bx*S(3) + cx*S(4))(-S(58816)aS(3)c*S(3) + S(81648)a*S(2)b*S(2)c*S(2) - S(30660)abS(4)c + S(3465)bS(6))/(S(573440)c*S(6)x) + x(S(3)b + S(14)cx)(ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/(S(112)c) - xS(3)(b(S(68)ac + S(11)b*S(2)) + S(10)cx(-S(28)ac + S(11)bS(2)))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(4480)cS(2)) + xS(2)sqrt(ax*S(2) + bx*S(3) + cx*S(4))(S(560)aS(2)c*S(2) - S(568)abS(2)c + S(99)bS(4))/(S(35840)c*S(3)) + sqrt(ax*S(2) + bx*S(3) + cx*S(4))(-S(6720)aS(3)c*S(3) + S(18896)a*S(2)b*S(2)c*S(2) - S(8988)abS(4)c + S(1155)bS(6))/(S(286720)c*S(5)) + S(3)x(-S(4)ac + bS(2))S(2)sqrt(a + bx + cx*S(2))(S(16)aS(2)c*S(2) - S(72)abS(2)c + S(33)bS(4))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(32768)c*(S(13)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))(S(3)/2), x), x, -bxS(2)(-S(44)ac + S(9)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(2240)c*S(2)) - bsqrt(axS(2) + bx*S(3) + cx*S(4))(S(1168)aS(2)c*S(2) - S(728)abS(2)c + S(105)bS(4))/(S(17920)c*S(4)) - S(3)bx(-S(4)ac + bS(2))S(2)(-S(4)ac + S(3)b*S(2))sqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(2048)c*(S(11)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))) + x(axS(2) + bx*S(3) + cxS(4))(S(3)/2)/S(7) + x*S(3)(S(24)ac + b*S(2) + S(10)bcx)sqrt(ax*S(2) + bx*S(3) + cx*S(4))/(S(280)c) + x(-S(32)ac + S(7)b*S(2))(-S(4)ac + S(3)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(4480)c*S(3)) + sqrt(ax*S(2) + bx*S(3) + cx*S(4))(-S(2048)aS(3)c*S(3) + S(5488)a*S(2)b*S(2)c*S(2) - S(2520)abS(4)c + S(315)bS(6))/(S(35840)c*S(5)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((axS(2) + bx*S(3) + cxS(4))(S(3)/2)/x, x), x, -bsqrt(ax*S(2) + bx*S(3) + cx*S(4))(S(1296)aS(2)c*S(2) - S(760)abS(2)c + S(105)bS(4))/(S(7680)c*S(4)x) + (S(3)b + S(10)cx)(axS(2) + bx*S(3) + cxS(4))(S(3)/2)/(S(60)cx) - x(b(S(12)ac + S(7)bS(2)) + S(6)cx(-S(20)ac + S(7)bS(2)))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(960)c*S(2)) + sqrt(ax*S(2) + bx*S(3) + cx*S(4))(S(240)aS(2)c*S(2) - S(216)abS(2)c + S(35)bS(4))/(S(3840)c*S(3)) + x(-S(4)ac + bS(2))S(2)(-S(4)ac + S(7)b*S(2))sqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(1024)c*(S(9)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/x*S(2), x), x, -b(b + S(2)cx)(ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/(S(16)cS(2)x*S(3)) + S(3)b(b + S(2)cx)(-S(4)ac + b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(128)c*S(3)x) - S(3)bx(-S(4)ac + bS(2))S(2)sqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(256)c*(S(7)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))) + (ax*S(2) + bx*S(3) + cxS(4))(S(5)/2)/(S(5)cx*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/x*S(3), x), x, (b + S(2)cx)(axS(2) + bx*S(3) + cxS(4))(S(3)/2)/(S(8)cx*S(3)) - (b + S(2)cx)(-S(12)ac + S(3)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(64)c*S(2)x) + S(3)x(-S(4)ac + bS(2))S(2)sqrt(a + bx + cxS(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(128)c*(S(5)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/xS(4), x), x, -a(S(3)/2)xsqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/sqrt(ax*S(2) + bx*S(3) + cx*S(4)) - bx(-S(12)ac + bS(2))sqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(16)c*(S(3)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))) + (ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/(S(3)xS(3)) + (S(8)ac + bS(2) + S(2)bcx)sqrt(ax*S(2) + bx*S(3) + cx*S(4))/(S(8)cx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/x*S(5), x), x, -S(3)sqrt(a)bxsqrt(a + bx + cxS(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(2)sqrt(axS(2) + bx*S(3) + cx*S(4))) + (S(9)b + S(6)cx)sqrt(ax*S(2) + bx*S(3) + cx*S(4))/(S(4)x) - (axS(2) + bx*S(3) + cxS(4))(S(3)/2)/x*S(4) + x(S(12)ac + S(3)bS(2))sqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(8)sqrt(c)sqrt(ax*S(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/x*S(6), x), x, S(3)bsqrt(c)xsqrt(a + bx + cxS(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(2)sqrt(axS(2) + bx*S(3) + cx*S(4))) - (S(3)b - S(6)cx)sqrt(ax*S(2) + bx*S(3) + cx*S(4))/(S(4)x*S(2)) - (ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/(S(2)xS(5)) - x(S(12)ac + S(3)bS(2))sqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(8)sqrt(a)sqrt(ax*S(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/xS(7), x), x, c(S(3)/2)xsqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/sqrt(ax*S(2) + bx*S(3) + cx*S(4)) - (ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/(S(3)xS(6)) - b(axS(2) + bx*S(3) + cxS(4))(S(3)/2)/(S(4)ax*S(5)) + (-S(8)ac + bS(2) + S(2)bcx)sqrt(ax*S(2) + bx*S(3) + cx*S(4))/(S(8)axS(2)) + bx(-S(12)ac + bS(2))sqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(16)a*(S(3)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/x*S(8), x), x, -(b + S(6)cx)sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(8)x*S(4)) - (ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/(S(4)xS(7)) - (-S(12)ac + bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(32)axS(3)) + b(-S(20)ac + S(3)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(64)a*S(2)x*S(2)) - S(3)x(-S(4)ac + bS(2))S(2)sqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(128)a*(S(5)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/x*S(9), x), x, -(S(3)b + S(12)cx)sqrt(ax*S(2) + bx*S(3) + cx*S(4))/(S(40)x*S(5)) - (ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)/(S(5)xS(8)) - (-S(8)ac + bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(80)axS(4)) + b(-S(28)ac + S(5)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(320)a*S(2)x*S(3)) - sqrt(ax*S(2) + bx*S(3) + cx*S(4))(S(128)aS(2)c*S(2) - S(100)abS(2)c + S(15)bS(4))/(S(640)a*S(3)x*S(2)) + S(3)bx(-S(4)ac + bS(2))S(2)sqrt(a + bx + cxS(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(256)a*(S(7)/2)sqrt(axS(2) + bx*S(3) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(axS(2) + bx*S(3) + cx*S(4)), x), x, -S(3)bsqrt(ax*S(2) + bx*S(3) + cx*S(4))/(S(4)c*S(2)x) + sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(2)c) + x(-S(4)ac + S(3)b*S(2))sqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(8)c*(S(5)/2)sqrt(axS(2) + bx*S(3) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(axS(2) + bx*S(3) + cx*S(4)), x), x, -bxsqrt(a + bx + cxS(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(2)c*(S(3)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))) + sqrt(ax*S(2) + bx*S(3) + cx*S(4))/(cx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(axS(2) + bx*S(3) + cx*S(4)), x), x, xsqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(sqrt(c)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(ax*S(2) + bx*S(3) + cx*S(4)), x), x, -xsqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(sqrt(a)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(axS(2) + bx*S(3) + cx*S(4))), x), x, -sqrt(ax*S(2) + bx*S(3) + cx*S(4))/(ax*S(2)) + bxsqrt(a + bx + cxS(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(2)a*(S(3)/2)sqrt(axS(2) + bx*S(3) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(ax*S(2) + bx*S(3) + cx*S(4))), x), x, -sqrt(ax*S(2) + bx*S(3) + cx*S(4))/(S(2)axS(3)) + S(3)bsqrt(ax*S(2) + bx*S(3) + cx*S(4))/(S(4)a*S(2)x*S(2)) - x(-S(4)ac + S(3)bS(2))sqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(8)a*(S(5)/2)sqrt(axS(2) + bx*S(3) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(axS(2) + bx*S(3) + cxS(4))(S(3)/2), x), x, -S(2)bxsqrt(ax*S(2) + bx*S(3) + cx*S(4))/(c(-S(4)ac + b*S(2))) - b(-S(52)ac + S(15)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(4)c*S(3)x(-S(4)ac + bS(2))) + S(2)x*S(4)(S(2)a + bx)/((-S(4)ac + b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))) + (-S(12)ac + S(5)b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(2)c*S(2)(-S(4)ac + b*S(2))) + x(-S(12)ac + S(15)bS(2))sqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(8)c*(S(7)/2)sqrt(axS(2) + bx*S(3) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(axS(2) + bx*S(3) + cxS(4))(S(3)/2), x), x, -S(2)bsqrt(axS(2) + bx*S(3) + cx*S(4))/(c(-S(4)ac + b*S(2))) - S(3)bxsqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cx*S(2))))/(S(2)c*(S(5)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))) + S(2)x*S(3)(S(2)a + bx)/((-S(4)ac + b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))) + (-S(8)ac + S(3)b*S(2))sqrt(axS(2) + bx*S(3) + cxS(4))/(cS(2)x(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(axS(2) + bx*S(3) + cxS(4))(S(3)/2), x), x, -S(2)bsqrt(axS(2) + bx*S(3) + cx*S(4))/(cx(-S(4)ac + bS(2))) + S(2)x*S(2)(S(2)a + bx)/((-S(4)ac + b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))) + xsqrt(a + bx + cx*S(2))atanh((b + S(2)cx)/(S(2)sqrt(c)sqrt(a + bx + cxS(2))))/(c(S(3)/2)sqrt(ax*S(2) + bx*S(3) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(axS(2) + bx*S(3) + cxS(4))(S(3)/2), x), x, S(2)x(S(2)a + bx)/((-S(4)ac + b*S(2))sqrt(axS(2) + bx*S(3) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(axS(2) + bx*S(3) + cxS(4))(S(3)/2), x), x, -S(2)x(b + S(2)cx)/((-S(4)ac + b*S(2))sqrt(axS(2) + bx*S(3) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(axS(2) + bx*S(3) + cxS(4))(S(3)/2), x), x, S(2)x(-S(2)ac + b*S(2) + bcx)/(a(-S(4)ac + b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))) - xsqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cxS(2))))/(a(S(3)/2)sqrt(ax*S(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(ax*S(2) + bx*S(3) + cxS(4))(S(3)/2), x), x, (-S(4)ac + S(2)bS(2) + S(2)bcx)/(a(-S(4)ac + bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))) - (-S(8)ac + S(3)b*S(2))sqrt(axS(2) + bx*S(3) + cxS(4))/(aS(2)xS(2)(-S(4)ac + b*S(2))) + S(3)bxsqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(2)a*(S(5)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax*S(2) + bx*S(3) + cxS(4))(S(-3)/2), x), x, (-S(4)ac + S(2)bS(2) + S(2)bcx)/(ax(-S(4)ac + b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))) - (-S(12)ac + S(5)b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(2)a*S(2)x*S(3)(-S(4)ac + b*S(2))) + b(-S(52)ac + S(15)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(4)a*S(3)x*S(2)(-S(4)ac + b*S(2))) - x(-S(12)ac + S(15)bS(2))sqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(8)a*(S(7)/2)sqrt(axS(2) + bx*S(3) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(axS(2) + bx*S(3) + cxS(4))(S(3)/2)), x), x, (-S(4)ac + S(2)bS(2) + S(2)bcx)/(axS(2)(-S(4)ac + b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))) - (-S(16)ac + S(7)b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(3)a*S(2)x*S(4)(-S(4)ac + b*S(2))) + b(-S(116)ac + S(35)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(12)a*S(3)x*S(3)(-S(4)ac + b*S(2))) - sqrt(ax*S(2) + bx*S(3) + cx*S(4))(S(256)aS(2)c*S(2) - S(460)abS(2)c + S(105)bS(4))/(S(24)a*S(4)x*S(2)(-S(4)ac + b*S(2))) + S(5)bx(-S(12)ac + S(7)bS(2))sqrt(a + bx + cx*S(2))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(16)a*(S(9)/2)sqrt(axS(2) + bx*S(3) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(ax*S(2) + bx*S(3) + cxS(4))(S(3)/2)), x), x, (-S(4)ac + S(2)bS(2) + S(2)bcx)/(axS(3)(-S(4)ac + b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))) - (-S(20)ac + S(9)b*S(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(4)a*S(2)x*S(5)(-S(4)ac + b*S(2))) + b(-S(68)ac + S(21)bS(2))sqrt(axS(2) + bx*S(3) + cx*S(4))/(S(8)a*S(3)x*S(4)(-S(4)ac + b*S(2))) - sqrt(ax*S(2) + bx*S(3) + cx*S(4))(S(240)aS(2)c*S(2) - S(448)abS(2)c + S(105)bS(4))/(S(32)a*S(4)x*S(3)(-S(4)ac + b*S(2))) + bsqrt(axS(2) + bx*S(3) + cx*S(4))(S(1808)aS(2)c*S(2) - S(1680)abS(2)c + S(315)bS(4))/(S(64)a*S(5)x*S(2)(-S(4)ac + b*S(2))) - xsqrt(a + bx + cx*S(2))(S(240)aS(2)c*S(2) - S(840)abS(2)c + S(315)bS(4))atanh((S(2)a + bx)/(S(2)sqrt(a)sqrt(a + bx + cx*S(2))))/(S(128)a*(S(11)/2)sqrt(axS(2) + bx*S(3) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(ax + bxS(3) + cx*S(5)), x), x, ax*(m + S(2))/(m + S(2)) + bx*(m + S(4))/(m + S(4)) + cx(m + S(6))/(m + S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(ax + bxS(3) + cx*S(5)), x), x, ax*S(4)/S(4) + bx*S(6)/S(6) + cx*S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(ax + bx*S(3) + cx*S(5)), x), x, ax*S(3)/S(3) + bx*S(5)/S(5) + cx*S(7)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(ax + bxS(3) + cx*S(5), x), x, ax*S(2)/S(2) + bx*S(4)/S(4) + cx*S(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax + bxS(3) + cx*S(5))/x, x), x, ax + bxS(3)/S(3) + cx*S(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax + bxS(3) + cxS(5))/xS(2), x), x, alog(x) + bx*S(2)/S(2) + cx*S(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax + bxS(3) + cxS(5))/xS(3), x), x, -a/x + bx + cxS(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(ax + bxS(3) + cxS(5))S(2), x), x, a*S(2)x*(m + S(3))/(m + S(3)) + S(2)abx*(m + S(5))/(m + S(5)) + S(2)bcx(m + S(9))/(m + S(9)) + cS(2)x(m + S(11))/(m + S(11)) + x(m + S(7))(S(2)ac + bS(2))/(m + S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(ax + bxS(3) + cxS(5))S(2), x), x, a*S(2)x*S(5)/S(5) + S(2)abx*S(7)/S(7) + S(2)bcxS(11)/S(11) + cS(2)xS(13)/S(13) + xS(9)(S(2)ac + b*S(2))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(ax + bx*S(3) + cxS(5))S(2), x), x, a*S(2)x*S(4)/S(4) + abxS(6)/S(3) + bcxS(10)/S(5) + cS(2)xS(12)/S(12) + xS(8)(S(2)ac + bS(2))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax + bxS(3) + cxS(5))S(2), x), x, a*S(2)x*S(3)/S(3) + S(2)abx*S(5)/S(5) + S(2)bcxS(9)/S(9) + cS(2)xS(11)/S(11) + xS(7)(S(2)ac + b*S(2))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax + bxS(3) + cxS(5))S(2)/x, x), x, a*S(2)x*S(2)/S(2) + abxS(4)/S(2) + bcxS(8)/S(4) + cS(2)xS(10)/S(10) + xS(6)(S(2)ac + bS(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax + bxS(3) + cxS(5))S(2)/xS(2), x), x, aS(2)x + S(2)abx*S(3)/S(3) + S(2)bcxS(7)/S(7) + cS(2)xS(9)/S(9) + xS(5)(S(2)ac + bS(2))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(ax + bx*S(3) + cx*S(5)), x), x, -bx*S(2)/(S(2)c*S(2)) + b(-S(3)ac + b*S(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(3)sqrt(-S(4)ac + bS(2))) + xS(4)/(S(4)c) + (-ac + b*S(2))log(a + bxS(2) + cx*S(4))/(S(4)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(ax + bx*S(3) + cx*S(5)), x), x, -bx/cS(2) + xS(3)/(S(3)c) + sqrt(S(2))(-ac + bS(2) + b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))) + sqrt(S(2))(-ac + bS(2) - b(-S(3)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(5)/2)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(ax + bx*S(3) + cx*S(5)), x), x, -blog(a + bxS(2) + cx*S(4))/(S(4)cS(2)) + xS(2)/(S(2)c) - (-S(2)ac + bS(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(ax + bx*S(3) + cx*S(5)), x), x, x/c - sqrt(S(2))(b - (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))) - sqrt(S(2))(b + (-S(2)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(ax + bx*S(3) + cx*S(5)), x), x, batanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)csqrt(-S(4)ac + bS(2))) + log(a + bx*S(2) + cx*S(4))/(S(4)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/(ax + bxS(3) + cx*S(5)), x), x, -sqrt(S(2))sqrt(b - sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(-S(4)ac + bS(2))) + sqrt(S(2))sqrt(b + sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(ax + bxS(3) + cx*S(5)), x), x, -atanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(ax + bxS(3) + cx*S(5)), x), x, -sqrt(S(2))sqrt(c)atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + bS(2))))/(sqrt(b + sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))) + sqrt(S(2))sqrt(c)atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + bS(2))))/(sqrt(b - sqrt(-S(4)ac + bS(2)))sqrt(-S(4)ac + b*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ax + bxS(3) + cx*S(5)), x), x, batanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)asqrt(-S(4)ac + bS(2))) + log(x)/a - log(a + bx*S(2) + cx*S(4))/(S(4)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(ax + bxS(3) + cx*S(5))), x), x, -sqrt(S(2))sqrt(c)(-b/sqrt(-S(4)ac + bS(2)) + S(1))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(b + sqrt(-S(4)ac + bS(2)))) - sqrt(S(2))sqrt(c)(b/sqrt(-S(4)ac + bS(2)) + S(1))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)asqrt(b - sqrt(-S(4)ac + bS(2)))) - S(1)/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(ax + bx*S(3) + cx*S(5))), x), x, -S(1)/(S(2)axS(2)) - blog(x)/a*S(2) + blog(a + bxS(2) + cx*S(4))/(S(4)a*S(2)) - (-S(2)ac + bS(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)sqrt(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(11)/(ax + bx*S(3) + cxS(5))S(2), x), x, -bxS(4)/(S(2)c(-S(4)ac + bS(2))) - blog(a + bxS(2) + cx*S(4))/(S(2)cS(3)) + xS(6)(S(2)a + bxS(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cxS(4))) + xS(2)(-S(3)ac + bS(2))/(cS(2)(-S(4)ac + b*S(2))) - (S(6)a*S(2)c*S(2) - S(6)abS(2)c + b*S(4))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(cS(3)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(10)/(ax + bxS(3) + cxS(5))S(2), x), x, -bxS(3)/(S(2)c(-S(4)ac + bS(2))) + xS(5)(S(2)a + bx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) + x(-S(10)ac + S(3)bS(2))/(S(2)c*S(2)(-S(4)ac + b*S(2))) - sqrt(S(2))(-S(13)abc + S(3)b*S(3) + (S(20)a*S(2)c*S(2) - S(19)abS(2)c + S(3)bS(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(5)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + b*S(2))) - sqrt(S(2))(-S(13)abc + S(3)b*S(3) - (S(20)a*S(2)c*S(2) - S(19)abS(2)c + S(3)bS(4))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(5)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)/(ax + bx*S(3) + cxS(5))S(2), x), x, -bxS(2)/(S(2)c(-S(4)ac + bS(2))) + b(-S(6)ac + b*S(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)c*S(2)(-S(4)ac + bS(2))(S(3)/2)) + x*S(4)(S(2)a + bx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) + log(a + bx*S(2) + cx*S(4))/(S(4)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)/(ax + bx*S(3) + cxS(5))S(2), x), x, -bx/(S(2)c(-S(4)ac + bS(2))) + xS(3)(S(2)a + bx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) + sqrt(S(2))(-S(6)ac + b*S(2) + b(-S(8)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(3)/2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))(-S(6)ac + b*S(2) - b(-S(8)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)c*(S(3)/2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)/(ax + bx*S(3) + cxS(5))S(2), x), x, S(2)aatanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + xS(2)(S(2)a + bx*S(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/(ax + bx*S(3) + cxS(5))S(2), x), x, x(S(2)a + bxS(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cx*S(4))) + sqrt(S(2))(b - (S(4)ac + b*S(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + b*S(2))) + sqrt(S(2))(S(4)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)/(ax + bxS(3) + cxS(5))S(2), x), x, -batanh((b + S(2)cxS(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) + (S(2)a + bxS(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(ax + bx*S(3) + cxS(5))S(2), x), x, -sqrt(S(2))sqrt(c)(S(2)b + sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))sqrt(c)(S(2)b - sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - x(b + S(2)cxS(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(ax + bx*S(3) + cxS(5))S(2), x), x, S(2)catanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(-S(4)ac + bS(2))(S(3)/2) - (b + S(2)cxS(2))/((-S(8)ac + S(2)b*S(2))(a + bxS(2) + cxS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(ax + bx*S(3) + cxS(5))S(2), x), x, -sqrt(S(2))sqrt(c)(-S(12)ac + b*S(2) - bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b + sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))sqrt(c)(-S(12)ac + b*S(2) + bsqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)asqrt(b - sqrt(-S(4)ac + bS(2)))(-S(4)ac + bS(2))(S(3)/2)) + x(-S(2)ac + bS(2) + bcxS(2))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(ax + bxS(3) + cxS(5))S(2), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(S(2)a(-S(4)ac + bS(2))(a + bxS(2) + cx*S(4))) + b(-S(6)ac + b*S(2))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(2)(-S(4)ac + bS(2))(S(3)/2)) + log(x)/a*S(2) - log(a + bx*S(2) + cx*S(4))/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax + bxS(3) + cxS(5))(S(-2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(S(2)ax(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) + sqrt(S(2))sqrt(c)(-S(16)abc + S(3)bS(3) - (-S(10)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(2)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - sqrt(S(2))sqrt(c)(-S(16)abc + S(3)b*S(3) + (-S(10)ac + S(3)b*S(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(2)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) - (-S(10)ac + S(3)bS(2))/(S(2)a*S(2)x(-S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(ax + bx*S(3) + cxS(5))S(2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(S(2)axS(2)(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) - (-S(3)ac + bS(2))/(aS(2)x*S(2)(-S(4)ac + b*S(2))) - S(2)blog(x)/aS(3) + blog(a + bxS(2) + cx*S(4))/(S(2)a*S(3)) - (S(6)a*S(2)c*S(2) - S(6)abS(2)c + b*S(4))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(aS(3)(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(ax + bx*S(3) + cxS(5))S(2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(S(2)axS(3)(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) - (-S(14)ac + S(5)b*S(2))/(S(6)a*S(2)x*S(3)(-S(4)ac + b*S(2))) + b(-S(19)ac + S(5)bS(2))/(S(2)a*S(3)x(-S(4)ac + bS(2))) - sqrt(S(2))sqrt(c)(S(28)a*S(2)c*S(2) - S(29)abS(2)c + S(5)bS(4) - b(-S(19)ac + S(5)bS(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(3)sqrt(b + sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)) + sqrt(S(2))sqrt(c)(S(28)aS(2)c*S(2) - S(29)abS(2)c + S(5)bS(4) + b(-S(19)ac + S(5)bS(2))sqrt(-S(4)ac + b*S(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(4)a*S(3)sqrt(b - sqrt(-S(4)ac + b*S(2)))(-S(4)ac + bS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)(ax + bx*S(3) + cxS(5))S(2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(S(2)axS(4)(-S(4)ac + b*S(2))(a + bxS(2) + cx*S(4))) - (-S(8)ac + S(3)b*S(2))/(S(4)a*S(2)x*S(4)(-S(4)ac + b*S(2))) + b(-S(11)ac + S(3)bS(2))/(S(2)a*S(3)x*S(2)(-S(4)ac + b*S(2))) + b(S(30)aS(2)c*S(2) - S(20)abS(2)c + S(3)bS(4))atanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/(S(2)a*S(4)(-S(4)ac + bS(2))(S(3)/2)) + (-S(2)ac + S(3)bS(2))log(x)/a*S(4) - (-S(2)ac + S(3)b*S(2))log(a + bxS(2) + cx*S(4))/(S(4)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)sqrt(ax + bxS(3) + cx*S(5)), x), x, S(2)a*(S(1)/4)sqrt(x)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(3)ac + b*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(15)c*(S(7)/4)sqrt(ax + bx*S(3) + cxS(5))) - a(S(1)/4)sqrt(x)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(sqrt(a)bsqrt(c) - S(6)ac + S(2)bS(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(30)c*(S(7)/4)sqrt(ax + bx*S(3) + cx*S(5))) + sqrt(x)(b + S(3)cx*S(2))sqrt(ax + bx*S(3) + cx*S(5))/(S(15)c) - x*(S(3)/2)(-S(6)ac + S(2)bS(2))(a + bxS(2) + cx*S(4))/(S(15)c*(S(3)/2)(sqrt(a) + sqrt(c)xS(2))sqrt(ax + bx*S(3) + cx*S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)sqrt(ax + bx*S(3) + cx*S(5)), x), x, (b + S(2)cxS(2))sqrt(ax + bx*S(3) + cx*S(5))/(S(8)csqrt(x)) - sqrt(x)(-S(4)ac + b*S(2))sqrt(a + bxS(2) + cx*S(4))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(16)c*(S(3)/2)sqrt(ax + bx*S(3) + cx*S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax + bxS(3) + cxS(5))/sqrt(x), x), x, -a(S(1)/4)bsqrt(x)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(3)c*(S(3)/4)sqrt(ax + bx*S(3) + cxS(5))) + a(S(1)/4)sqrt(x)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(2)sqrt(a)sqrt(c) + b)elliptic_f(S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(6)c*(S(3)/4)sqrt(ax + bx*S(3) + cx*S(5))) + bx*(S(3)/2)(a + bxS(2) + cx*S(4))/(S(3)sqrt(c)(sqrt(a) + sqrt(c)x*S(2))sqrt(ax + bx*S(3) + cx*S(5))) + sqrt(x)sqrt(ax + bx*S(3) + cx*S(5))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax + bxS(3) + cxS(5))/x(S(3)/2), x), x, -sqrt(a)sqrt(x)sqrt(a + bxS(2) + cx*S(4))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)sqrt(ax + bx*S(3) + cx*S(5))) + bsqrt(x)sqrt(a + bx*S(2) + cx*S(4))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)sqrt(c)sqrt(ax + bxS(3) + cx*S(5))) + sqrt(ax + bxS(3) + cx*S(5))/(S(2)sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(3)/2)(ax + bx*S(3) + cxS(5))(S(3)/2), x), x, -S(3)bsqrt(x)(-S(4)ac + bS(2))S(2)sqrt(a + bxS(2) + cx*S(4))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(512)c*(S(7)/2)sqrt(ax + bx*S(3) + cx*S(5))) + sqrt(x)(S(3)b + S(8)cxS(2))(ax + bx*S(3) + cxS(5))(S(3)/2)/(S(80)c) - x(S(3)/2)(b(-S(4)ac + S(5)b*S(2)) + S(4)cxS(2)(-S(16)ac + S(5)bS(2)))sqrt(ax + bx*S(3) + cx*S(5))/(S(640)c*S(2)) + sqrt(ax + bxS(3) + cx*S(5))(S(128)aS(2)c*S(2) - S(100)abS(2)c + S(15)bS(4))/(S(1280)c*S(3)sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)(ax + bxS(3) + cxS(5))(S(3)/2), x), x, -a*(S(1)/4)sqrt(x)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(84)aS(2)c*S(2) - S(57)abS(2)c + S(8)bS(4))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(315)c*(S(11)/4)sqrt(ax + bx*S(3) + cxS(5))) + a(S(1)/4)sqrt(x)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(S(4)sqrt(a)bsqrt(c)(-S(6)ac + b*S(2)) + S(84)a*S(2)c*S(2) - S(57)abS(2)c + S(8)bS(4))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(630)c*(S(11)/4)sqrt(ax + bx*S(3) + cx*S(5))) + (S(3)b + S(7)cx*S(2))(ax + bx*S(3) + cxS(5))(S(3)/2)/(S(63)csqrt(x)) - sqrt(x)(b(-S(9)ac + S(4)bS(2)) + S(6)cxS(2)(-S(7)ac + S(2)bS(2)))sqrt(ax + bx*S(3) + cx*S(5))/(S(315)cS(2)) + x(S(3)/2)(a + bx*S(2) + cx*S(4))(S(84)aS(2)c*S(2) - S(57)abS(2)c + S(8)bS(4))/(S(315)c*(S(5)/2)(sqrt(a) + sqrt(c)xS(2))sqrt(ax + bx*S(3) + cx*S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax + bxS(3) + cxS(5))(S(3)/2)/sqrt(x), x), x, (b + S(2)cx*S(2))(ax + bx*S(3) + cxS(5))(S(3)/2)/(S(16)cx*(S(3)/2)) - (b + S(2)cxS(2))(-S(12)ac + S(3)bS(2))sqrt(ax + bx*S(3) + cx*S(5))/(S(128)c*S(2)sqrt(x)) + S(3)sqrt(x)(-S(4)ac + bS(2))S(2)sqrt(a + bx*S(2) + cx*S(4))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(256)c*(S(5)/2)sqrt(ax + bx*S(3) + cx*S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax + bxS(3) + cxS(5))(S(3)/2)/x*(S(3)/2), x), x, S(2)a*(S(1)/4)bsqrt(x)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(8)ac + b*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(35)c*(S(7)/4)sqrt(ax + bx*S(3) + cxS(5))) - a(S(1)/4)sqrt(x)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(sqrt(a)sqrt(c)(-S(20)ac + b*S(2)) + S(2)b(-S(8)ac + bS(2)))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(70)c*(S(7)/4)sqrt(ax + bx*S(3) + cx*S(5))) - S(2)bx(S(3)/2)(-S(8)ac + b*S(2))(a + bxS(2) + cx*S(4))/(S(35)c*(S(3)/2)(sqrt(a) + sqrt(c)xS(2))sqrt(ax + bx*S(3) + cx*S(5))) + (ax + bxS(3) + cxS(5))(S(3)/2)/(S(7)sqrt(x)) + sqrt(x)(S(10)ac + b*S(2) + S(3)bcx*S(2))sqrt(ax + bx*S(3) + cx*S(5))/(S(35)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(3)/2)/sqrt(ax + bxS(3) + cx*S(5)), x), x, sqrt(x)sqrt(a + bxS(2) + cx*S(4))atanh((b + S(2)cx*S(2))/(S(2)sqrt(c)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)sqrt(c)sqrt(ax + bxS(3) + cx*S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/sqrt(ax + bxS(3) + cx*S(5)), x), x, sqrt(x)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(1)/4)c*(S(1)/4)sqrt(ax + bx*S(3) + cx*S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)sqrt(ax + bx*S(3) + cx*S(5))), x), x, -sqrt(x)sqrt(a + bxS(2) + cx*S(4))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)sqrt(a)sqrt(ax + bxS(3) + cxS(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(3)/2)sqrt(ax + bxS(3) + cx*S(5))), x), x, sqrt(c)x*(S(3)/2)(a + bxS(2) + cx*S(4))/(a(sqrt(a) + sqrt(c)xS(2))sqrt(ax + bx*S(3) + cx*S(5))) - sqrt(ax + bxS(3) + cx*S(5))/(ax(S(3)/2)) - c(S(1)/4)sqrt(x)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(a(S(3)/4)sqrt(ax + bx*S(3) + cxS(5))) + c(S(1)/4)sqrt(x)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(3)/4)sqrt(ax + bx*S(3) + cxS(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)/2)/(ax + bx*S(3) + cxS(5))(S(3)/2), x), x, -bsqrt(c)x*(S(3)/2)(a + bxS(2) + cx*S(4))/(a(sqrt(a) + sqrt(c)xS(2))(-S(4)ac + b*S(2))sqrt(ax + bx*S(3) + cxS(5))) + x(S(3)/2)(-S(2)ac + bS(2) + bcxS(2))/(a(-S(4)ac + b*S(2))sqrt(ax + bx*S(3) + cx*S(5))) + bc*(S(1)/4)sqrt(x)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(a(S(3)/4)(-S(4)ac + b*S(2))sqrt(ax + bx*S(3) + cxS(5))) - c(S(1)/4)sqrt(x)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(3)/4)(-S(2)sqrt(a)sqrt(c) + b)sqrt(ax + bxS(3) + cx*S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(ax + bxS(3) + cxS(5))(S(3)/2), x), x, sqrt(x)(-S(2)ac + bS(2) + bcxS(2))/(a(-S(4)ac + b*S(2))sqrt(ax + bx*S(3) + cx*S(5))) - sqrt(x)sqrt(a + bxS(2) + cx*S(4))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(2)a*(S(3)/2)sqrt(ax + bx*S(3) + cx*S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)(ax + bx*S(3) + cxS(5))(S(3)/2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(asqrt(x)(-S(4)ac + bS(2))sqrt(ax + bx*S(3) + cx*S(5))) + S(2)sqrt(c)x(S(3)/2)(-S(3)ac + b*S(2))(a + bxS(2) + cxS(4))/(aS(2)(sqrt(a) + sqrt(c)x*S(2))(-S(4)ac + b*S(2))sqrt(ax + bx*S(3) + cx*S(5))) - (-S(6)ac + S(2)b*S(2))sqrt(ax + bx*S(3) + cxS(5))/(aS(2)x(S(3)/2)(-S(4)ac + b*S(2))) - S(2)c*(S(1)/4)sqrt(x)sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-S(3)ac + b*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(a(S(7)/4)(-S(4)ac + b*S(2))sqrt(ax + bx*S(3) + cxS(5))) + c(S(1)/4)sqrt(x)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(sqrt(a)bsqrt(c) - S(6)ac + S(2)bS(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(7)/4)(-S(4)ac + b*S(2))sqrt(ax + bx*S(3) + cxS(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(3)/2)(ax + bxS(3) + cxS(5))(S(3)/2)), x), x, (-S(2)ac + b*S(2) + bcxS(2))/(ax*(S(3)/2)(-S(4)ac + b*S(2))sqrt(ax + bx*S(3) + cx*S(5))) - (-S(8)ac + S(3)b*S(2))sqrt(ax + bx*S(3) + cx*S(5))/(S(2)a*S(2)x*(S(5)/2)(-S(4)ac + b*S(2))) + S(3)bsqrt(x)sqrt(a + bxS(2) + cx*S(4))atanh((S(2)a + bx*S(2))/(S(2)sqrt(a)sqrt(a + bx*S(2) + cx*S(4))))/(S(4)a*(S(5)/2)sqrt(ax + bx*S(3) + cxS(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(3)n/S(2) + S(-3)/2)/(ax*(n + S(-1)) + bx*n + cx(n + S(1)))(S(3)/2), x), x, -S(2)x(n/S(2) + S(-1)/2)(b + S(2)cx)/((-S(4)ac + b*S(2))sqrt(ax(n + S(-1)) + bx*n + cx*(n + S(1)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(ax + bxS(3) + cx*S(5)), x), x, S(2)x*S(2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(3)/4, S(1)/2, S(1)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)sqrt(ax + bx*S(3) + cx*S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(d + exS(2))/sqrt(ax + bxS(3) + cx*S(5)), x), x, S(2)dxS(2)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(3)/4, S(1)/2, S(1)/2, S(7)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(3)sqrt(ax + bx*S(3) + cx*S(5))) + S(2)exS(4)sqrt(S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))) + S(1))sqrt(S(2)cx*S(2)/(b + sqrt(-S(4)ac + bS(2))) + S(1))AppellF1(S(7)/4, S(1)/2, S(1)/2, S(11)/4, -S(2)cx*S(2)/(b - sqrt(-S(4)ac + bS(2))), -S(2)cxS(2)/(b + sqrt(-S(4)ac + bS(2))))/(S(7)sqrt(ax + bx*S(3) + cx**S(5))), expand=True, _diff=True, _numerical=True)
19e48f8bf13adff8c7b8be60bc0da118449f1bbd70d73d39e63b1c51d19e7230	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.functions import log, sqrt, exp, cos, sin, tan, sec, csc, cot from sympy.functions.elementary.hyperbolic import atanh from sympy.functions.elementary.hyperbolic import asinh from sympy.functions.elementary.hyperbolic import acosh from sympy.functions.elementary.trigonometric import atan from sympy.functions.elementary.trigonometric import asin from sympy.functions.elementary.trigonometric import acos from sympy.integrals.rubi.utility_function import (EllipticE, EllipticF, hypergeom, rubi_test, AppellF1, EllipticPi, Log, Sqrt, ArcTan, ArcTanh, ArcSin, ArcCos, Hypergeometric2F1) from sympy.core.numbers import (I, pi as Pi) from sympy.core.singleton import S from sympy.core.symbol import symbols from sympy.functions.elementary.exponential import exp_polar from sympy.functions.special.elliptic_integrals import (elliptic_e, elliptic_f, elliptic_pi) from sympy.functions.special.hyper import hyper from sympy.simplify.simplify import simplify from sympy.testing.pytest import SKIP from sympy.functions.elementary.hyperbolic import acsch as arccsch from sympy.functions.elementary.trigonometric import acsc as arccsc a, b, c, d, e, f, m, n, x, u , k, p, r, s, t= symbols('a b c d e f m n x u k p r s t') A, B, C, D, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B C D a b c d e f g h y z m n p q u v w F',) def test_1(): # difference in apart assert rubi_test(rubi_integrate(S(1)/(S(2)sqrt(S(3))b*(S(3)/2) - S(9)bx + S(9)x*S(3)), x), x, -log(sqrt(b) - sqrt(S(3))x)/(S(27)b) + log(S(2)sqrt(b) + sqrt(S(3))x)/(S(27)b) + sqrt(S(3))/(S(9)sqrt(b)(sqrt(S(3))sqrt(b) - S(3)x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3))p, x), x, (a + bx)(aS(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3))p/(b(S(3)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3))S(3), x), x, (a + bx)*S(10)/(S(10)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3))S(2), x), x, (a + bx)*S(7)/(S(7)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a*S(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3), x), x, aS(3)x + S(3)aS(2)bxS(2)/S(2) + ab*S(2)xS(3) + bS(3)xS(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(aS(3) + S(3)a*S(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3)), x), x, -S(1)/(S(2)b(a + bx)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(3) + S(3)aS(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3))(S(-2)), x), x, -S(1)/(S(5)b(a + bx)S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(3) + S(3)aS(2)bx + S(3)abS(2)xS(2) + bS(3)xS(3))(S(-3)), x), x, -S(1)/(S(8)b(a + bx)*S(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)ab + S(3)b*S(2)x + S(3)bcxS(2) + cS(2)xS(3))S(3), x), x, -b*S(3)x(-S(3)ac + bS(2))S(3)/cS(3) + S(3)b*S(2)(b + cx)S(4)(-S(3)ac + bS(2))S(2)/(S(4)cS(4)) - S(3)b(b + cx)*S(7)(-S(3)ac + b*S(2))/(S(7)c*S(4)) + (b + cx)*S(10)/(S(10)c*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)ab + S(3)b*S(2)x + S(3)bcxS(2) + cS(2)xS(3))S(2), x), x, b*S(2)x(-S(3)ac + bS(2))S(2)/cS(2) - b(b + cx)S(4)(-S(3)ac + b*S(2))/(S(2)c*S(3)) + (b + cx)*S(7)/(S(7)c*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(3)ab + S(3)b*S(2)x + S(3)bcxS(2) + cS(2)x*S(3), x), x, S(3)abx + S(3)bS(2)x*S(2)/S(2) + bcxS(3) + cS(2)x*S(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(3)ab + S(3)b*S(2)x + S(3)bcxS(2) + cS(2)xS(3)), x), x, log(b(S(1)/3)(-S(3)ac + bS(2))(S(1)/3) - b - cx)/(S(3)b(S(2)/3)(-S(3)ac + bS(2))(S(2)/3)) - log(b*(S(2)/3)(-S(3)ac + bS(2))(S(2)/3) + b*(S(1)/3)(b + cx)(-S(3)ac + bS(2))(S(1)/3) + (b + cx)S(2))/(S(6)b*(S(2)/3)(-S(3)ac + bS(2))(S(2)/3)) - sqrt(S(3))atan(sqrt(S(3))(b*(S(1)/3) + (S(2)b + S(2)cx)/(-S(3)ac + bS(2))(S(1)/3))/(S(3)b(S(1)/3)))/(S(3)b*(S(2)/3)(-S(3)ac + bS(2))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)ab + S(3)bS(2)x + S(3)bcxS(2) + cS(2)xS(3))(S(-2)), x), x, c(b + cx)/(S(3)b(-S(3)ac + b*S(2))(b(-S(3)ac + bS(2)) - (b + cx)*S(3))) - S(2)clog(b(S(1)/3)(-S(3)ac + bS(2))(S(1)/3) - b - cx)/(S(9)b*(S(5)/3)(-S(3)ac + bS(2))(S(5)/3)) + clog(b(S(2)/3)(-S(3)ac + bS(2))(S(2)/3) + b*(S(1)/3)(b + cx)(-S(3)ac + bS(2))(S(1)/3) + (b + cx)S(2))/(S(9)b*(S(5)/3)(-S(3)ac + bS(2))(S(5)/3)) + S(2)sqrt(S(3))catan(sqrt(S(3))(b*(S(1)/3) + (S(2)b + S(2)cx)/(-S(3)ac + bS(2))(S(1)/3))/(S(3)b(S(1)/3)))/(S(9)b*(S(5)/3)(-S(3)ac + bS(2))(S(5)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)ab + S(3)bS(2)x + S(3)bcxS(2) + cS(2)xS(3))(S(-3)), x), x, -c*S(2)(b + cx)/(S(6)b(-S(3)ac + bS(2))(b(-S(3)ac + bS(2)) - (b + cx)S(3))S(2)) - S(5)cS(2)(b + cx)/(S(18)b*S(2)(-S(3)ac + bS(2))S(2)(b(-S(3)ac + b*S(2)) - (b + cx)*S(3))) + S(5)c*S(2)log(b*(S(1)/3)(-S(3)ac + bS(2))(S(1)/3) - b - cx)/(S(27)b*(S(8)/3)(-S(3)ac + bS(2))(S(8)/3)) - S(5)cS(2)log(b*(S(2)/3)(-S(3)ac + bS(2))(S(2)/3) + b*(S(1)/3)(b + cx)(-S(3)ac + bS(2))(S(1)/3) + (b + cx)S(2))/(S(54)b*(S(8)/3)(-S(3)ac + bS(2))(S(8)/3)) - S(5)sqrt(S(3))c*S(2)atan(sqrt(S(3))(b(S(1)/3) + (S(2)b + S(2)cx)/(-S(3)ac + bS(2))(S(1)/3))/(S(3)b(S(1)/3)))/(S(27)b*(S(8)/3)(-S(3)ac + bS(2))(S(8)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ace + bdfxS(3) + xS(2)(adf + bcf + bde) + x(acf + ade + bce))S(3), x), x, aS(3)c*S(3)e*S(3)x + S(3)aS(2)c*S(2)e*S(2)x*S(2)(acf + ade + bce)/S(2) + acexS(3)(a*S(2)(c*S(2)f*S(2) + S(3)cdef + dS(2)e*S(2)) + S(3)abce(cf + de) + b*S(2)c*S(2)eS(2)) + bS(3)dS(3)f*S(3)xS(10)/S(10) + bS(2)dS(2)f*S(2)x*S(9)(adf + bcf + bde)/S(3) + S(3)bdfx*S(8)(a*S(2)d*S(2)f*S(2) + S(3)abdf(cf + de) + b*S(2)(c*S(2)f*S(2) + S(3)cdef + dS(2)eS(2)))/S(8) + xS(7)(aS(3)d*S(3)f*S(3)/S(7) + S(9)a*S(2)bdS(2)f*S(2)(cf + de)/S(7) + S(9)ab*S(2)df(c*S(2)f*S(2) + S(3)cdef + dS(2)eS(2))/S(7) + bS(3)(cS(3)f*S(3) + S(9)c*S(2)def*S(2) + S(9)cdS(2)e*S(2)f + d*S(3)eS(3))/S(7)) + xS(6)(aS(3)d*S(2)f*S(2)(cf + de)/S(2) + S(3)aS(2)bdf(cS(2)f*S(2) + S(3)cdef + dS(2)e*S(2))/S(2) + ab*S(2)(c*S(3)f*S(3) + S(9)c*S(2)def*S(2) + S(9)cdS(2)e*S(2)f + d*S(3)eS(3))/S(2) + bS(3)ce(cS(2)f*S(2) + S(3)cdef + dS(2)eS(2))/S(2)) + xS(5)(S(3)a*S(3)df(c*S(2)f*S(2) + S(3)cdef + dS(2)e*S(2))/S(5) + S(3)a*S(2)b(cS(3)f*S(3) + S(9)c*S(2)def*S(2) + S(9)cdS(2)e*S(2)f + d*S(3)e*S(3))/S(5) + S(9)abS(2)ce(c*S(2)f*S(2) + S(3)cdef + dS(2)e*S(2))/S(5) + S(3)b*S(3)c*S(2)e*S(2)(cf + de)/S(5)) + x*S(4)(a*S(3)(c*S(3)f*S(3) + S(9)c*S(2)def*S(2) + S(9)cdS(2)e*S(2)f + d*S(3)e*S(3))/S(4) + S(9)a*S(2)bce(cS(2)f*S(2) + S(3)cdef + dS(2)e*S(2))/S(4) + S(9)abS(2)c*S(2)e*S(2)(cf + de)/S(4) + b*S(3)c*S(3)e*S(3)/S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ace + bdfxS(3) + xS(2)(adf + bcf + bde) + x(acf + ade + bce))S(2), x), x, aS(2)cS(2)e*S(2)x + acexS(2)(acf + ade + bce) + b*S(2)d*S(2)f*S(2)x*S(7)/S(7) + bdfx*S(6)(adf + bcf + bde)/S(3) + x*S(5)(a*S(2)d*S(2)f*S(2)/S(5) + S(4)abdf(cf + de)/S(5) + b*S(2)(c*S(2)f*S(2) + S(4)cdef + dS(2)eS(2))/S(5)) + xS(4)(aS(2)df(cf + de)/S(2) + ab(c*S(2)f*S(2) + S(4)cdef + dS(2)eS(2))/S(2) + bS(2)ce(cf + de)/S(2)) + xS(3)(a*S(2)(c*S(2)f*S(2) + S(4)cdef + dS(2)e*S(2))/S(3) + S(4)abce(cf + de)/S(3) + b*S(2)c*S(2)e*S(2)/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(ace + bdfxS(3) + xS(2)(adf + bcf + bde) + x(acf + ade + bce), x), x, acex + bdfxS(4)/S(4) + xS(3)(adf/S(3) + bcf/S(3) + bde/S(3)) + xS(2)(acf/S(2) + ade/S(2) + bce/S(2)), expand=True, _diff=True, _numerical=True) '''taking a long time assert rubi_test(rubi_integrate(S(1)/(ace + bdfxS(3) + xS(2)(adf + bcf + bde) + x(acf + ade + bce)), x), x, blog(a + bx)/((-ad + bc)(-af + be)) - dlog(c + dx)/((-ad + bc)(-cf + de)) + flog(e + fx)/((-af + be)(-cf + de)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ace + bdfxS(3) + xS(2)(adf + bcf + bde) + x(acf + ade + bce))(S(-2)), x), x, -S(2)b*S(3)(-S(2)adf + bcf + bde)log(a + bx)/((-ad + bc)S(3)(-af + be)S(3)) - bS(3)/((a + bx)(-ad + bc)*S(2)(-af + be)*S(2)) + S(2)d*S(3)(adf - S(2)bcf + bde)log(c + dx)/((-ad + bc)S(3)(-cf + de)S(3)) - dS(3)/((c + dx)(-ad + bc)*S(2)(-cf + de)*S(2)) + S(2)f*S(3)(-adf - bcf + S(2)bde)log(e + fx)/((-af + be)S(3)(-cf + de)S(3)) - fS(3)/((e + fx)(-af + be)*S(2)(-cf + de)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ace + bdfxS(3) + xS(2)(adf + bcf + bde) + x(acf + ade + bce))*(S(-3)), x), x, S(3)b*S(5)(S(7)aS(2)d*S(2)f*S(2) - S(7)abdf(cf + de) + b*S(2)(S(2)cS(2)f*S(2) + S(3)cdef + S(2)d*S(2)e*S(2)))log(a + bx)/((-ad + bc)S(5)(-af + be)*S(5)) + S(3)b*S(5)(-S(2)adf + bcf + bde)/((a + bx)(-ad + bc)S(4)(-af + be)S(4)) - bS(5)/(S(2)(a + bx)*S(2)(-ad + bc)*S(3)(-af + be)*S(3)) - S(3)d*S(5)(S(2)aS(2)d*S(2)f*S(2) + abdf(-S(7)cf + S(3)de) + bS(2)(S(7)cS(2)f*S(2) - S(7)cdef + S(2)d*S(2)e*S(2)))log(c + dx)/((-ad + bc)S(5)(-cf + de)*S(5)) + S(3)d*S(5)(adf - S(2)bcf + bde)/((c + dx)(-ad + bc)S(4)(-cf + de)S(4)) + dS(5)/(S(2)(c + dx)*S(2)(-ad + bc)*S(3)(-cf + de)*S(3)) + S(3)f*S(5)(S(2)aS(2)d*S(2)f*S(2) - abdf(-S(3)cf + S(7)de) + bS(2)(S(2)cS(2)f*S(2) - S(7)cdef + S(7)d*S(2)e*S(2)))log(e + fx)/((-af + be)S(5)(-cf + de)*S(5)) - S(3)f*S(5)(-adf - bcf + S(2)bde)/((e + fx)(-af + be)S(4)(-cf + de)S(4)) - fS(5)/(S(2)(e + fx)*S(2)(-af + be)*S(3)(-cf + de)*S(3)), expand=True, _diff=True, _numerical=True) ''' '''matchpy and mathematica difference assert rubi_test(rubi_integrate(S(1)/(S(16)x*S(3) - S(4)x*S(2) + S(4)x + S(-1)), x), x, log(-S(4)x + S(1))/S(5) - log(S(4)x*S(2) + S(1))/S(10) - atan(S(2)x)/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3) + xS(2) + x + S(1)), x), x, log(x + S(1))/S(2) - log(x*S(2) + S(1))/S(4) + atan(x)/S(2), expand=True, _diff=True, _numerical=True) ''' assert rubi_test(rubi_integrate(S(1)/(dx*S(3)), x), x, -S(1)/(S(2)dxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(cx*S(2) + dx*S(3)), x), x, -S(1)/(cx) - dlog(x)/cS(2) + dlog(c + dx)/cS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(bx + dxS(3)), x), x, log(x)/b - log(b + dx*S(2))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(bx + cx*S(2) + dx*S(3)), x), x, catanh((c + S(2)dx)/sqrt(-S(4)bd + c*S(2)))/(bsqrt(-S(4)bd + c*S(2))) + log(x)/b - log(b + cx + dxS(2))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + dxS(3)), x), x, log(a(S(1)/3) + d(S(1)/3)x)/(S(3)a(S(2)/3)d(S(1)/3)) - log(a(S(2)/3) - a*(S(1)/3)d*(S(1)/3)x + d*(S(2)/3)x*S(2))/(S(6)a*(S(2)/3)d*(S(1)/3)) - sqrt(S(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)d*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(3)a*(S(2)/3)d*(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dxS(3))n, x), x, x(dxS(3))n/(S(3)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cx*S(2) + dxS(3))n, x), x, x(S(1) + dx/c)*(-n)(cxS(2) + dxS(3))nhyper((-n, S(2)n + S(1)), (S(2)n + S(2),), -dx/c)/(S(2)n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx + dxS(3))n, x), x, x(b + dxS(2))(bx + dxS(3))nhyper((S(1), S(3)n/S(2) + S(3)/2), (n/S(2) + S(3)/2,), -dxS(2)/b)/(b(n + S(1))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((bx + dxS(3))n, x), x, x(S(1) + dxS(2)/b)(-n)(bx + dxS(3))nhyper((-n, n/S(2) + S(1)/2), (n/S(2) + S(3)/2,), -dxS(2)/b)/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx + cxS(2) + dxS(3))n, x), x, x(S(2)dx/(c - sqrt(-S(4)bd + cS(2))) + S(1))(-n)(S(2)dx/(c + sqrt(-S(4)bd + cS(2))) + S(1))(-n)(bx + cxS(2) + dxS(3))nAppellF1(n + S(1), -n, -n, n + S(2), -S(2)dx/(c - sqrt(-S(4)bd + cS(2))), -S(2)dx/(c + sqrt(-S(4)bd + cS(2))))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + dxS(3))n, x), x, x(a + dxS(3))(n + S(1))hyper((S(1), n + S(4)/3), (S(4)/3,), -dx*S(3)/a)/a, expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + dxS(3))n, x), x, x(S(1) + dxS(3)/a)(-n)(a + dxS(3))nhyper((S(1)/3, -n), (S(4)/3,), -dxS(3)/a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(5) + S(5)aS(4)bx + S(10)a*S(3)b*S(2)x*S(2) + S(10)a*S(2)b*S(3)x*S(3) + S(5)abS(4)xS(4) + bS(5)xS(5))S(3), x), x, (a + bx)*S(16)/(S(16)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a*S(5) + S(5)a*S(4)bx + S(10)a*S(3)b*S(2)x*S(2) + S(10)a*S(2)b*S(3)x*S(3) + S(5)abS(4)xS(4) + bS(5)xS(5))S(2), x), x, (a + bx)*S(11)/(S(11)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a*S(5) + S(5)a*S(4)bx + S(10)a*S(3)b*S(2)x*S(2) + S(10)a*S(2)b*S(3)x*S(3) + S(5)abS(4)xS(4) + bS(5)xS(5), x), x, (a + bx)*S(6)/(S(6)b), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(a*S(5) + S(5)a*S(4)bx + S(10)a*S(3)b*S(2)x*S(2) + S(10)a*S(2)b*S(3)x*S(3) + S(5)abS(4)xS(4) + bS(5)xS(5), x), x, aS(5)x + S(5)aS(4)bxS(2)/S(2) + S(10)a*S(3)b*S(2)x*S(3)/S(3) + S(5)a*S(2)b*S(3)x*S(4)/S(2) + ab*S(4)xS(5) + bS(5)xS(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(aS(5) + S(5)a*S(4)bx + S(10)a*S(3)b*S(2)x*S(2) + S(10)a*S(2)b*S(3)x*S(3) + S(5)abS(4)xS(4) + bS(5)xS(5)), x), x, -S(1)/(S(4)b(a + bx)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(5) + S(5)aS(4)bx + S(10)a*S(3)b*S(2)x*S(2) + S(10)a*S(2)b*S(3)x*S(3) + S(5)abS(4)xS(4) + bS(5)xS(5))(S(-2)), x), x, -S(1)/(S(9)b(a + bx)S(9)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(5) + S(5)aS(4)bx + S(10)a*S(3)b*S(2)x*S(2) + S(10)a*S(2)b*S(3)x*S(3) + S(5)abS(4)xS(4) + bS(5)xS(5))(S(-3)), x), x, -S(1)/(S(14)b(a + bx)S(14)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(c + (a + bx)S(2)), x), x, -S(3)ax/bS(3) - a(a*S(2) - S(3)c)atan((a + bx)/sqrt(c))/(b*S(4)sqrt(c)) + (a + bx)S(2)/(S(2)b*S(4)) + (S(3)a*S(2)/S(2) - c/S(2))log(c + (a + bx)S(2))/bS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(c + (a + bx)*S(2)), x), x, -alog(c + (a + bx)S(2))/bS(3) + x/bS(2) + (aS(2) - c)atan((a + bx)/sqrt(c))/(bS(3)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(c + (a + bx)S(2)), x), x, -aatan((a + bx)/sqrt(c))/(bS(2)sqrt(c)) + log(c + (a + bx)S(2))/(S(2)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(c + (a + bx)*S(2)), x), x, atan((a + bx)/sqrt(c))/(bsqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(c + (a + bx)S(2))), x), x, -aatan((a + bx)/sqrt(c))/(sqrt(c)(aS(2) + c)) + log(x)/(aS(2) + c) - log(c + (a + bx)S(2))/(S(2)(aS(2) + c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(c + (a + bx)*S(2))), x), x, -S(2)ablog(x)/(aS(2) + c)S(2) + ablog(c + (a + bx)S(2))/(aS(2) + c)S(2) + b(a*S(2) - c)atan((a + bx)/sqrt(c))/(sqrt(c)(aS(2) + c)S(2)) - S(1)/(x(aS(2) + c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(c + (a + bx)S(2))), x), x, -ab*S(2)(a*S(2) - S(3)c)atan((a + bx)/sqrt(c))/(sqrt(c)(aS(2) + c)S(3)) + S(2)ab/(x(aS(2) + c)S(2)) + b*S(2)(S(3)aS(2) - c)log(x)/(aS(2) + c)S(3) - b*S(2)(S(3)aS(2) - c)log(c + (a + bx)S(2))/(S(2)(aS(2) + c)S(3)) - S(1)/(S(2)xS(2)(a*S(2) + c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + b(c + dx)S(2)), x), x, atan(sqrt(b)(c + dx)/sqrt(a))/(sqrt(a)sqrt(b)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(c + dx)S(2))(S(-2)), x), x, (c/S(2) + dx/S(2))/(ad(a + b(c + dx)*S(2))) + atan(sqrt(b)(c + dx)/sqrt(a))/(S(2)a*(S(3)/2)sqrt(b)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(c + dx)S(2))(S(-3)), x), x, (c/S(4) + dx/S(4))/(ad(a + b(c + dx)S(2))S(2)) + (S(3)c/S(8) + S(3)dx/S(8))/(aS(2)d(a + b(c + dx)S(2))) + S(3)atan(sqrt(b)(c + dx)/sqrt(a))/(S(8)a(S(5)/2)sqrt(b)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(b(c + dx)S(2) + sqrt(-a)), x), x, atan(sqrt(b)(c + dx)/(-a)(S(1)/4))/(sqrt(b)d(-a)(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((c + dx)*S(2) + S(1)), x), x, atan(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(((c + dx)S(2) + S(1))(S(-2)), x), x, (c/S(2) + dx/S(2))/(d((c + dx)*S(2) + S(1))) + atan(c + dx)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(((c + dx)S(2) + S(1))(S(-3)), x), x, (c/S(4) + dx/S(4))/(d((c + dx)S(2) + S(1))S(2)) + (S(3)c/S(8) + S(3)dx/S(8))/(d((c + dx)*S(2) + S(1))) + S(3)atan(c + dx)/(S(8)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(-(c + dx)S(2) + S(1)), x), x, atanh(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-(c + dx)S(2) + S(1))(S(-2)), x), x, (c/S(2) + dx/S(2))/(d(-(c + dx)*S(2) + S(1))) + atanh(c + dx)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-(c + dx)S(2) + S(1))(S(-3)), x), x, (c/S(4) + dx/S(4))/(d(-(c + dx)S(2) + S(1))S(2)) + (S(3)c/S(8) + S(3)dx/S(8))/(d(-(c + dx)*S(2) + S(1))) + S(3)atanh(c + dx)/(S(8)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(-(x + S(1))S(2) + S(1)), x), x, atanh(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-(x + S(1))S(2) + S(1))(S(-2)), x), x, (x/S(2) + S(1)/2)/(-(x + S(1))S(2) + S(1)) + atanh(x + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-(x + S(1))S(2) + S(1))(S(-3)), x), x, (x/S(4) + S(1)/4)/(-(x + S(1))S(2) + S(1))S(2) + (S(3)x/S(8) + S(3)/8)/(-(x + S(1))S(2) + S(1)) + S(3)atanh(x + S(1))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(((a + bx)S(2) + S(1))S(2)/x, x), x, abx(a*S(2) + S(2)) + a(a + bx)S(3)/S(3) + (a + bx)*S(4)/S(4) + (a + bx)*S(2)(aS(2)/S(2) + S(1)) + (aS(2) + S(1))*S(2)log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((x + S(-1))S(2) + S(1)), x), x, x + log((x + S(-1))S(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(-(x + S(1))*S(2) + S(1)), x), x, -xsqrt(-(x + S(1))*S(2) + S(1))/S(2) + S(3)sqrt(-(x + S(1))*S(2) + S(1))/S(2) + S(3)asin(x + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/sqrt(-(a + bx)*S(2) + S(1)), x), x, S(3)asqrt(-(a + bx)*S(2) + S(1))/(S(2)b*S(3)) - xsqrt(-(a + bx)S(2) + S(1))/(S(2)bS(2)) + (aS(2) + S(1)/2)asin(a + bx)/bS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt((a + bx)S(2) + S(1)), x), x, -S(3)asqrt((a + bx)*S(2) + S(1))/(S(2)b*S(3)) + xsqrt((a + bx)S(2) + S(1))/(S(2)bS(2)) + (aS(2) + S(-1)/2)asinh(a + bx)/b*S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((A + Bx + CxS(2) + Dx*S(3))/(ax*S(4) + a + bx*S(3) + bx + cxS(2)), x), x, -(D(b - sqrt(S(8)aS(2) - S(4)ac + bS(2))) + S(2)a(A - C))log(S(2)ax*S(2) + S(2)a + x(b - sqrt(S(8)a*S(2) - S(4)ac + bS(2))))/(S(4)asqrt(S(8)a*S(2) - S(4)ac + bS(2))) + (D(b + sqrt(S(8)aS(2) - S(4)ac + bS(2))) + S(2)a(A - C))log(S(2)ax*S(2) + S(2)a + x(b + sqrt(S(8)a*S(2) - S(4)ac + bS(2))))/(S(4)asqrt(S(8)a*S(2) - S(4)ac + bS(2))) - sqrt(S(2))(S(4)Ba*S(2) + Db(b + sqrt(S(8)a*S(2) - S(4)ac + bS(2))) - a(A(b + sqrt(S(8)a*S(2) - S(4)ac + bS(2))) + Cb + Csqrt(S(8)a*S(2) - S(4)ac + bS(2)) + S(2)Dc))atan(sqrt(S(2))(S(4)ax + b + sqrt(S(8)a*S(2) - S(4)ac + bS(2)))/(S(2)sqrt(S(4)aS(2) + S(2)ac - b(b + sqrt(S(8)aS(2) - S(4)ac + bS(2))))))/(S(2)asqrt(S(4)a*S(2) + S(2)ac - b(b + sqrt(S(8)aS(2) - S(4)ac + bS(2))))sqrt(S(8)aS(2) - S(4)ac + bS(2))) + sqrt(S(2))(S(4)Ba*S(2) + Db(b - sqrt(S(8)a*S(2) - S(4)ac + bS(2))) - a(A(b - sqrt(S(8)a*S(2) - S(4)ac + bS(2))) + Cb - Csqrt(S(8)a*S(2) - S(4)ac + bS(2)) + S(2)Dc))atan(sqrt(S(2))(S(4)ax + b - sqrt(S(8)a*S(2) - S(4)ac + bS(2)))/(S(2)sqrt(S(4)aS(2) + S(2)ac - b(b - sqrt(S(8)aS(2) - S(4)ac + bS(2))))))/(S(2)asqrt(S(4)a*S(2) + S(2)ac - b(b - sqrt(S(8)aS(2) - S(4)ac + bS(2))))sqrt(S(8)aS(2) - S(4)ac + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(S(2)xS(3) + S(3)x*S(2) + x + S(5))/(S(2)xS(4) + xS(3) + S(3)xS(2) + x + S(2)), x), x, xS(4)/S(4) + xS(3)/S(3) - S(3)x*S(2)/S(4) + S(5)x/S(4) + log(x*S(2) + x + S(1))/S(3) - S(13)log(S(2)xS(2) - x + S(2))/S(48) + sqrt(S(15))atan(sqrt(S(15))(-S(4)x + S(1))/S(15))/S(72) - S(10)sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(S(2)xS(3) + S(3)x*S(2) + x + S(5))/(S(2)xS(4) + xS(3) + S(3)xS(2) + x + S(2)), x), x, xS(3)/S(3) + xS(2)/S(2) - S(3)x/S(2) + S(2)log(xS(2) + x + S(1))/S(3) - log(S(2)x*S(2) - x + S(2))/S(24) + S(5)sqrt(S(15))atan(sqrt(S(15))(-S(4)x + S(1))/S(15))/S(36) + S(8)sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(S(2)xS(3) + S(3)x*S(2) + x + S(5))/(S(2)xS(4) + xS(3) + S(3)xS(2) + x + S(2)), x), x, xS(2)/S(2) + x - log(xS(2) + x + S(1)) + log(S(2)x*S(2) - x + S(2))/S(4) + sqrt(S(15))atan(sqrt(S(15))(-S(4)x + S(1))/S(15))/S(18) + S(2)sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)x*S(3) + S(3)x*S(2) + x + S(5))/(S(2)xS(4) + xS(3) + S(3)xS(2) + x + S(2)), x), x, x + log(xS(2) + x + S(1))/S(3) + log(S(2)x*S(2) - x + S(2))/S(6) - sqrt(S(15))atan(sqrt(S(15))(-S(4)x + S(1))/S(15))/S(9) - S(10)sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(3) + S(3)x*S(2) + x + S(5))/(S(2)xS(4) + xS(3) + S(3)xS(2) + x + S(2)), x), x, S(2)log(x*S(2) + x + S(1))/S(3) - log(S(2)x*S(2) - x + S(2))/S(6) - sqrt(S(15))atan(sqrt(S(15))(-S(4)x + S(1))/S(15))/S(9) + S(8)sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(3) + S(3)x*S(2) + x + S(5))/(x(S(2)xS(4) + xS(3) + S(3)x*S(2) + x + S(2))), x), x, S(5)log(x)/S(2) - log(x*S(2) + x + S(1)) - log(S(2)x*S(2) - x + S(2))/S(4) + sqrt(S(15))atan(sqrt(S(15))(-S(4)x + S(1))/S(15))/S(18) + S(2)sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(3) + S(3)xS(2) + x + S(5))/(xS(2)(S(2)xS(4) + xS(3) + S(3)xS(2) + x + S(2))), x), x, -S(3)log(x)/S(4) + log(x*S(2) + x + S(1))/S(3) + log(S(2)x*S(2) - x + S(2))/S(24) + S(5)sqrt(S(15))atan(sqrt(S(15))(-S(4)x + S(1))/S(15))/S(36) - S(10)sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(9) - S(5)/(S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(3) + S(3)xS(2) + x + S(5))/(xS(3)(S(2)xS(4) + xS(3) + S(3)xS(2) + x + S(2))), x), x, -S(15)log(x)/S(8) + S(2)log(xS(2) + x + S(1))/S(3) + S(13)log(S(2)xS(2) - x + S(2))/S(48) + sqrt(S(15))atan(sqrt(S(15))(-S(4)x + S(1))/S(15))/S(72) + S(8)sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(9) + S(3)/(S(4)x) - S(5)/(S(4)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(S(2)x*S(3) + S(3)x*S(2) + x + S(5))/(S(2)xS(4) + xS(3) + S(5)xS(2) + x + S(2)), x), x, xS(3)(S(7) - S(5)sqrt(S(7))I)/S(42) + x*S(3)(S(7) + S(5)sqrt(S(7))I)/S(42) + x*S(2)(S(7) - S(5)sqrt(S(7))I)/S(28) + x*S(2)(S(7) + S(5)sqrt(S(7))I)/S(28) - x(S(35) + S(9)sqrt(S(7))I)/S(28) - x(S(35) - S(9)sqrt(S(7))I)/S(28) + S(3)(S(7) - S(11)sqrt(S(7))I)log(S(4)xS(2) + x(S(1) - sqrt(S(7))I) + S(4))/S(112) + S(3)(S(7) + S(11)sqrt(S(7))I)log(S(4)x*S(2) + x(S(1) + sqrt(S(7))I) + S(4))/S(112) - S(11)(-S(5)sqrt(S(7)) + S(9)I)atan((S(8)x + S(1) + sqrt(S(7))I)/sqrt(S(70) - S(2)sqrt(S(7))I))/(S(4)sqrt(S(490) - S(14)sqrt(S(7))I)) + S(11)(S(5)sqrt(S(7)) + S(9)I)atan((S(8)x + S(1) - sqrt(S(7))I)/sqrt(S(70) + S(2)sqrt(S(7))I))/(S(4)sqrt(S(490) + S(14)sqrt(S(7))I)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(S(2)xS(3) + S(3)x*S(2) + x + S(5))/(S(2)xS(4) + xS(3) + S(5)xS(2) + x + S(2)), x), x, xS(2)(S(7) - S(5)sqrt(S(7))I)/S(28) + x*S(2)(S(7) + S(5)sqrt(S(7))I)/S(28) + x(S(7) - S(5)sqrt(S(7))I)/S(14) + x(S(7) + S(5)sqrt(S(7))I)/S(14) - (S(35) + S(9)sqrt(S(7))I)log(S(4)x*S(2) + x(S(1) - sqrt(S(7))I) + S(4))/S(56) - (S(35) - S(9)sqrt(S(7))I)log(S(4)xS(2) + x(S(1) + sqrt(S(7))I) + S(4))/S(56) + (-sqrt(S(7)) + S(53)I)atan((S(8)x + S(1) + sqrt(S(7))I)/sqrt(S(70) - S(2)sqrt(S(7))I))/(S(2)sqrt(S(490) - S(14)sqrt(S(7))I)) - (sqrt(S(7)) + S(53)I)atan((S(8)x + S(1) - sqrt(S(7))I)/sqrt(S(70) + S(2)sqrt(S(7))I))/(S(2)sqrt(S(490) + S(14)sqrt(S(7))I)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)xS(3) + S(3)x*S(2) + x + S(5))/(S(2)xS(4) + xS(3) + S(5)xS(2) + x + S(2)), x), x, x(S(7) - S(5)sqrt(S(7))I)/S(14) + x(S(7) + S(5)sqrt(S(7))I)/S(14) + (S(7) + S(5)sqrt(S(7))I)log(S(4)xS(2) + x(S(1) - sqrt(S(7))I) + S(4))/S(28) + (S(7) - S(5)sqrt(S(7))I)log(S(4)xS(2) + x(S(1) + sqrt(S(7))I) + S(4))/S(28) + (-S(7)sqrt(S(7)) + S(19)I)atan((S(8)x + S(1) + sqrt(S(7))I)/sqrt(S(70) - S(2)sqrt(S(7))I))/sqrt(S(490) - S(14)sqrt(S(7))I) - (S(7)sqrt(S(7)) + S(19)I)atan((S(8)x + S(1) - sqrt(S(7))I)/sqrt(S(70) + S(2)sqrt(S(7))I))/sqrt(S(490) + S(14)sqrt(S(7))I), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(3) + S(3)x*S(2) + x + S(5))/(S(2)xS(4) + xS(3) + S(5)xS(2) + x + S(2)), x), x, (S(7) + S(5)sqrt(S(7))I)log(S(4)xS(2) + x(S(1) - sqrt(S(7))I) + S(4))/S(28) + (S(7) - S(5)sqrt(S(7))I)log(S(4)xS(2) + x(S(1) + sqrt(S(7))I) + S(4))/S(28) - (-S(7)sqrt(S(7)) + S(19)I)atan((S(8)x + S(1) + sqrt(S(7))I)/sqrt(S(70) - S(2)sqrt(S(7))I))/sqrt(S(490) - S(14)sqrt(S(7))I) + (S(7)sqrt(S(7)) + S(19)I)atan((S(8)x + S(1) - sqrt(S(7))I)/sqrt(S(70) + S(2)sqrt(S(7))I))/sqrt(S(490) + S(14)sqrt(S(7))I), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(3) + S(3)x*S(2) + x + S(5))/(x(S(2)xS(4) + xS(3) + S(5)x*S(2) + x + S(2))), x), x, (S(35) - S(9)sqrt(S(7))I)log(x)/S(28) + (S(35) + S(9)sqrt(S(7))I)log(x)/S(28) - (S(35) + S(9)sqrt(S(7))I)log(S(4)xS(2) + x(S(1) - sqrt(S(7))I) + S(4))/S(56) - (S(35) - S(9)sqrt(S(7))I)log(S(4)xS(2) + x(S(1) + sqrt(S(7))I) + S(4))/S(56) - (-sqrt(S(7)) + S(53)I)atan((S(8)x + S(1) + sqrt(S(7))I)/sqrt(S(70) - S(2)sqrt(S(7))I))/(S(2)sqrt(S(490) - S(14)sqrt(S(7))I)) + (sqrt(S(7)) + S(53)I)atan((S(8)x + S(1) - sqrt(S(7))I)/sqrt(S(70) + S(2)sqrt(S(7))I))/(S(2)sqrt(S(490) + S(14)sqrt(S(7))I)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(3) + S(3)xS(2) + x + S(5))/(xS(2)(S(2)xS(4) + xS(3) + S(5)xS(2) + x + S(2))), x), x, -S(3)(S(7) + S(11)sqrt(S(7))I)log(x)/S(56) - S(3)(S(7) - S(11)sqrt(S(7))I)log(x)/S(56) + S(3)(S(7) + S(11)sqrt(S(7))I)log(S(4)IxS(2) + x(-sqrt(S(7)) + I) + S(4)I)/S(112) + S(3)(S(7) - S(11)sqrt(S(7))I)log(S(4)IxS(2) + x(sqrt(S(7)) + I) + S(4)I)/S(112) + S(11)(S(9) + S(5)sqrt(S(7))I)atanh((S(8)Ix - sqrt(S(7)) + I)/sqrt(S(70) - S(2)sqrt(S(7))I))/(S(4)sqrt(S(490) - S(14)sqrt(S(7))I)) - S(11)(S(9) - S(5)sqrt(S(7))I)atanh((S(8)Ix + sqrt(S(7)) + I)/sqrt(S(70) + S(2)sqrt(S(7))I))/(S(4)sqrt(S(490) + S(14)sqrt(S(7))I)) + (S(-5)/4 - S(9)sqrt(S(7))I/S(28))/x + (S(-5)/4 + S(9)sqrt(S(7))I/S(28))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(3) + S(3)xS(2) + x + S(5))/(xS(3)(S(2)xS(4) + xS(3) + S(5)xS(2) + x + S(2))), x), x, -(S(35) + S(9)sqrt(S(7))I)log(x)/S(16) - (S(35) - S(9)sqrt(S(7))I)log(x)/S(16) + (S(35) - S(9)sqrt(S(7))I)log(S(4)Ix*S(2) + x(-sqrt(S(7)) + I) + S(4)I)/S(32) + (S(35) + S(9)sqrt(S(7))I)log(S(4)Ix*S(2) + x(sqrt(S(7)) + I) + S(4)I)/S(32) + (S(355) - S(73)sqrt(S(7))I)atanh((S(8)Ix - sqrt(S(7)) + I)/sqrt(S(70) - S(2)sqrt(S(7))I))/(S(8)sqrt(S(490) - S(14)sqrt(S(7))I)) - (S(355) + S(73)sqrt(S(7))I)atanh((S(8)Ix + sqrt(S(7)) + I)/sqrt(S(70) + S(2)sqrt(S(7))I))/(S(8)sqrt(S(490) + S(14)sqrt(S(7))I)) + (S(3)/8 - S(33)sqrt(S(7))I/S(56))/x + (S(3)/8 + S(33)sqrt(S(7))I/S(56))/x + (S(-5)/8 - S(9)sqrt(S(7))I/S(56))/xS(2) + (S(-5)/8 + S(9)sqrt(S(7))I/S(56))/xS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(3)xS(2) + x + S(9))/((xS(2) + S(1))(xS(2) + S(3))), x), x, log(xS(2) + S(3))/S(2) + S(3)atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + xS(2) + x + S(3))/((x*S(2) + S(1))(xS(2) + S(3))), x), x, log(xS(2) + S(3))/S(2) + atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)xS(3) - xS(2) + S(6)x + S(-4))/((x*S(2) + S(1))(x*S(2) + S(2))), x), x, S(3)log(x*S(2) + S(1))/S(2) - S(3)atan(x) + sqrt(S(2))atan(sqrt(S(2))x/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(3)xS(4) + S(1))/((x + S(-2))(xS(2) + S(1))S(2)), x), x, (S(2)x/S(5) + S(-1)/5)/(xS(2) + S(1)) - S(47)log(-x + S(2))/S(25) - S(14)log(xS(2) + S(1))/S(25) - S(46)atan(x)/S(25), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(2) - S(9)x + S(-9))/(x*S(3) - S(9)x), x), x, log(x) - log(-x + S(3)) + S(2)log(x + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(5) + S(2)xS(2) + S(1))/(xS(3) - x), x), x, x*S(3)/S(3) + x - log(x) + S(2)log(-x + S(1)) + log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(2) + S(3))/(x(x + S(-1))*S(2)), x), x, S(3)log(x) - log(-x + S(1)) + S(5)/(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(2) + S(-1))/((S(4)x + S(-1))(xS(2) + S(1))), x), x, -S(7)log(-S(4)x + S(1))/S(34) + S(6)log(x*S(2) + S(1))/S(17) + S(3)atan(x)/S(17), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(3) - S(3)x*S(2) + S(2)x + S(-3))/(xS(2) + S(1)), x), x, xS(2)/S(2) - S(3)x + log(xS(2) + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + S(6)x*S(3) + S(10)xS(2) + x)/(xS(2) + S(6)x + S(10)), x), x, xS(3)/S(3) + log(xS(2) + S(6)x + S(10))/S(2) - S(3)atan(x + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4) - S(3)x*S(3) - S(7)x*S(2) + S(27)x + S(-18)), x), x, log(-x + S(1))/S(8) - log(-x + S(2))/S(5) + log(-x + S(3))/S(12) - log(x + S(3))/S(120), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(1))/(x + S(-2)), x), x, xS(3)/S(3) + x*S(2) + S(4)x + S(9)log(-x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x*S(3) - S(4)x*S(2) + S(3)x)/(x*S(2) + S(1)), x), x, S(3)x*S(2)/S(2) - S(4)x + S(4)atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x + S(5))/(xS(3) - xS(2) - x + S(1)), x), x, atanh(x) + S(4)/(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) - xS(3) - x + S(-1))/(xS(3) - xS(2)), x), x, x*S(2)/S(2) + S(2)log(x) - S(2)log(-x + S(1)) - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + xS(2) + x + S(2))/(xS(4) + S(3)xS(2) + S(2)), x), x, log(xS(2) + S(2))/S(2) + atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(5) - xS(4) + S(4)xS(3) - S(4)x*S(2) + S(8)x + S(-4))/(xS(2) + S(2))S(3), x), x, log(x*S(2) + S(2))/S(2) - sqrt(S(2))atan(sqrt(S(2))x/S(2))/S(2) - S(1)/(xS(2) + S(2))S(2), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((xS(5) - xS(4) + S(4)x*S(3) - S(4)x*S(2) + S(8)x + S(-4))/(xS(2) + S(2))S(3), x), x, x*S(2)/(S(4)(xS(2) + S(2))) + xS(2)/(S(2)(xS(2) + S(2))S(2)) + log(xS(2) + S(2))/S(2) - sqrt(S(2))atan(sqrt(S(2))x/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) - S(3)x + S(-1))/(xS(3) + xS(2) - S(2)x), x), x, log(x)/S(2) - log(-x + S(1)) + S(3)log(x + S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(4) - S(2)x*S(3) + S(3)xS(2) - x + S(3))/(xS(3) - S(2)xS(2) + S(3)x), x), x, xS(2)/S(2) + log(x) - log(xS(2) - S(2)x + S(3))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + x + S(-1))/(xS(2) + S(1))S(2), x), x, -x/(S(2)(xS(2) + S(1))) + log(xS(2) + S(1))/S(2) - atan(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(4) + S(8)xS(3) - xS(2) + S(2)x + S(1))/((xS(2) + x)(x*S(3) + S(1))), x), x, log(x) - S(2)log(x + S(1)) + log(x*S(2) - x + S(1)) - S(2)sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(3) - S(3)/(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + xS(2) - S(5)x + S(15))/((x*S(2) + S(5))(x*S(2) + S(2)x + S(3))), x), x, log(x*S(2) + S(2)x + S(3))/S(2) + S(5)sqrt(S(2))atan(sqrt(S(2))(x + S(1))/S(2))/S(2) - sqrt(S(5))atan(sqrt(S(5))x/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(6) + S(7)x*S(5) + S(15)x*S(4) + S(32)x*S(3) + S(23)x*S(2) + S(25)x + S(-3))/((xS(2) + S(1))S(2)(xS(2) + x + S(2))S(2)), x), x, log(xS(2) + S(1)) - log(xS(2) + x + S(2)) + S(1)/(xS(2) + x + S(2)) - S(3)/(xS(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((xS(2) + S(1))(x*S(2) + S(4))), x), x, -atan(x/S(2))/S(6) + atan(x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(3))/(xS(2) + S(1)), x), x, aatan(x) + bx*S(2)/S(2) - blog(xS(2) + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + x)/((x + S(4))(xS(2) + S(-4))), x), x, log(x + S(4)) - atanh(x/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(4))/((xS(2) + S(1))(x*S(2) + S(2))), x), x, S(3)atan(x) - sqrt(S(2))atan(sqrt(S(2))x/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(4) + S(3)x*S(2) - S(4)x + S(5))/((x + S(-1))*S(2)(x*S(2) + S(1))), x), x, x + log(-x + S(1))/S(2) + S(3)log(x*S(2) + S(1))/S(4) + S(2)atan(x) + S(5)/(S(2)(-x + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + S(1))/(xS(2) + S(2)), x), x, xS(3)/S(3) - S(2)x + S(5)sqrt(S(2))atan(sqrt(S(2))x/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + S(2)x + S(2))/(xS(5) + xS(4)), x), x, log(x + S(1)) - S(2)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(2) - S(5)x + S(-1))/(x*S(3) - S(2)x*S(2) - x + S(2)), x), x, S(2)log(-x + S(1)) - log(-x + S(2)) + log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + x + S(2))/(xS(4) + S(2)xS(2) + S(1)), x), x, x/(xS(2) + S(1)) + log(xS(2) + S(1))/S(2) + atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + xS(2) + S(2)x + S(1))/(x*S(4) + S(2)xS(2) + S(1)), x), x, log(xS(2) + S(1))/S(2) + atan(x) - S(1)/(S(2)(xS(2) + S(1))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((xS(3) + xS(2) + S(2)x + S(1))/(x*S(4) + S(2)xS(2) + S(1)), x), x, xS(2)/(S(2)(xS(2) + S(1))) + log(xS(2) + S(1))/S(2) + atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x + S(3))/((x*S(2) + S(1))(x*S(2) + S(2))), x), x, S(2)log(x*S(2) + S(1)) - S(2)log(x*S(2) + S(2)) + S(3)atan(x) - S(3)sqrt(S(2))atan(sqrt(S(2))x/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2))/((xS(2) + S(1))(xS(2) + S(4))), x), x, log(xS(2) + S(1))/S(6) - log(x*S(2) + S(4))/S(6) - atan(x/S(2))/S(3) + S(2)atan(x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) - x + S(2))/(xS(2) - S(6)x + S(-7)), x), x, xS(2)/S(2) + S(6)x + S(169)log(-x + S(7))/S(4) - log(x + S(1))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(5) + S(-1))/(xS(2) + S(-1)), x), x, xS(4)/S(4) + xS(2)/S(2) + log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) - xS(2) + S(2)x + S(5))/(xS(2) + x + S(1)), x), x, xS(2)/S(2) - S(2)x + S(3)log(x*S(2) + x + S(1))/S(2) + S(11)sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) - S(2)x*S(3) + x + S(-3))/(S(2)x*S(2) - S(8)x + S(10)), x), x, xS(3)/S(6) + xS(2)/S(2) + S(3)x/S(2) + S(3)log(x*S(2) - S(4)x + S(5))/S(4) - S(6)atan(x + S(-2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(3)x*S(2) + S(2)x + S(1))/((x + S(-3))(x + S(-2))(x + S(-1))), x), x, x + S(7)log(-x + S(1))/S(2) - S(25)log(-x + S(2)) + S(61)log(-x + S(3))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) - xS(3) + xS(2) - S(7)x + S(2))/(xS(3) + xS(2) - S(14)x + S(-24)), x), x, xS(2)/S(2) - S(2)x + S(13)log(-x + S(4))/S(3) - S(22)log(x + S(2))/S(3) + S(20)log(x + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(2))/(x(x + S(-1))*S(2)(x + S(1))), x), x, S(2)log(x) - S(5)log(-x + S(1))/S(4) - S(3)log(x + S(1))/S(4) + S(3)/(S(2)(-x + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + xS(2) + S(3))/(xS(2) + S(2))S(2), x), x, (x/S(4) + S(1))/(xS(2) + S(2)) + log(xS(2) + S(2))/S(2) + S(5)sqrt(S(2))atan(sqrt(S(2))x/S(2))/S(8), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((xS(3) + xS(2) + S(3))/(xS(2) + S(2))S(2), x), x, x(-x/S(2) + S(1)/4)/(xS(2) + S(2)) + log(xS(2) + S(2))/S(2) + S(5)sqrt(S(2))atan(sqrt(S(2))x/S(2))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(3) - S(4)x*S(2) + S(70)x + S(-35))/((x*S(2) - S(10)x + S(26))(xS(2) - S(2)x + S(17))), x), x, S(1003)log(xS(2) - S(10)x + S(26))/S(1025) + S(22)log(xS(2) - S(2)x + S(17))/S(1025) - S(4607)atan(x/S(4) + S(-1)/4)/S(4100) + S(15033)atan(x + S(-5))/S(1025), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) + S(2))/((x + S(-5))(x + S(-3))(x + S(4))), x), x, -S(11)log(-x + S(3))/S(14) + S(3)log(-x + S(5))/S(2) + S(2)log(x + S(4))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)/((x + S(-1))(xS(2) + S(2))), x), x, xS(2)/S(2) + x + log(-x + S(1))/S(3) - S(2)log(xS(2) + S(2))/S(3) - S(2)sqrt(S(2))atan(sqrt(S(2))x/S(2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(2) + S(7)x + S(-1))/(xS(3) + xS(2) - x + S(-1)), x), x, S(2)log(-x + S(1)) - S(3)/(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x + S(1))/(x*S(3) - S(3)x*S(2) + S(3)x + S(-1)), x), x, -(S(2)x + S(1))S(2)/(S(6)(-x + S(1))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(7)xS(2) - S(5)x + S(5))/((x + S(-1))*S(2)(x + S(1))S(3)), x), x, -S(2)/(x + S(1))S(2) + S(1)/(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)xS(2) + S(3)x + S(1))/(x*S(3) + S(2)x*S(2) + S(2)x + S(1)), x), x, log(x + S(1)) + log(x*S(2) + x + S(1)) - S(2)sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(2)x + S(-1))/(S(2)xS(3) + S(3)x*S(2) - S(2)x), x), x, log(x)/S(2) + log(-S(2)x + S(1))/S(10) - log(x + S(2))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) - S(2)x*S(2) + S(4)x + S(1))/(xS(3) - xS(2) - x + S(1)), x), x, x*S(2)/S(2) + x - S(2)atanh(x) + S(2)/(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(2) - x + S(4))/(xS(3) + S(4)x), x), x, log(x) + log(xS(2) + S(4))/S(2) - atan(x/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + x*S(2) + S(1))/(x(x + S(-1))(xS(2) + S(1))S(3)(x*S(2) + x + S(1))), x), x, S(3)x/(S(16)(xS(2) + S(1))) - (-S(3)x/S(8) + S(3)/8)/(xS(2) + S(1)) + (x/S(8) + S(1)/8)/(xS(2) + S(1))*S(2) - log(x) + log(-x + S(1))/S(8) + S(15)log(xS(2) + S(1))/S(16) - log(xS(2) + x + S(1))/S(2) + S(7)atan(x)/S(16) - sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-x*S(3) + S(2)x*S(2) - S(3)x + S(1))/(xS(2) + S(1))S(2), x), x, (-x/S(2) + S(1))/(xS(2) + S(1)) - log(xS(2) + S(1))/S(2) + S(3)atan(x)/S(2), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((-xS(3) + S(2)x*S(2) - S(3)x + S(1))/(xS(2) + S(1))S(2), x), x, -x(S(2)x + S(1))/(S(2)(xS(2) + S(1))) - log(xS(2) + S(1))/S(2) + S(3)atan(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-x*S(3) + S(2)x*S(2) - S(3)x + S(1))/(x(xS(2) + S(1))S(2)), x), x, (-x + S(-1)/2)/(xS(2) + S(1)) + log(x) - log(xS(2) + S(1))/S(2) - S(2)atan(x), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((-x*S(3) + S(2)x*S(2) - S(3)x + S(1))/(x(xS(2) + S(1))S(2)), x), x, x(x/S(2) + S(-1))/(xS(2) + S(1)) + log(x) - log(xS(2) + S(1))/S(2) - S(2)atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + xS(3) - xS(2) - x + S(1))/(xS(3) - x), x), x, xS(2)/S(2) + x - log(x) + log(-xS(2) + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) - S(4)xS(2) + S(2))/((xS(2) + S(1))(xS(2) + S(2))), x), x, -log(xS(2) + S(1))/S(2) + log(xS(2) + S(2)) + S(6)atan(x) - S(5)sqrt(S(2))atan(sqrt(S(2))x/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + xS(2) + S(1))/((xS(2) + S(1))(xS(2) + S(4))S(2)), x), x, -S(13)x/(S(24)(x*S(2) + S(4))) + S(25)atan(x/S(2))/S(144) + atan(x)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + xS(2) + S(1))/(xS(4) + xS(3) + S(2)xS(2)), x), x, -log(x)/S(4) + S(5)log(x*S(2) + x + S(2))/S(8) + sqrt(S(7))atan(sqrt(S(7))(S(2)x + S(1))/S(7))/S(28) - S(1)/(S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + xS(2) - S(12)x + S(1))/(xS(2) + x + S(-12)), x), x, xS(2)/S(2) - S(2)atanh(S(2)x/S(7) + S(1)/7)/S(7), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((xS(3) + xS(2) - S(12)x + S(1))/(xS(2) + x + S(-12)), x), x, xS(2)/S(2) + log(-x + S(3))/S(7) - log(x + S(4))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(6)x*S(2) + S(5)x + S(-3))/(x*S(3) + S(2)x*S(2) - S(3)x), x), x, log(x) + S(2)log(-x + S(1)) + S(3)log(x + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)xS(2) + S(3)x + S(-2))/(x*S(3) + S(2)x*S(2)), x), x, S(2)log(x) + S(3)log(x + S(2)) + S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(4)x*S(2) - S(2)x + S(18))/(x*S(3) + S(4)x*S(2) + x + S(-6)), x), x, log(-x + S(1)) - S(2)log(x + S(2)) - S(3)log(x + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) - S(2)xS(2) + x + S(1))/(xS(4) + S(5)xS(2) + S(4)), x), x, log(xS(2) + S(4))/S(2) - S(3)atan(x/S(2))/S(2) + atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)xS(3) - S(27)x*S(2) + S(5)x + S(-32))/(S(30)xS(5) - S(13)x*S(4) + S(50)x*S(3) - S(286)x*S(2) - S(299)x + S(-70)), x), x, -S(3146)log(-S(3)x + S(7))/S(80155) - S(334)log(S(2)x + S(1))/S(323) + S(4822)log(S(5)x + S(2))/S(4879) + S(11049)log(xS(2) + x + S(5))/S(260015) + S(3988)sqrt(S(19))atan(sqrt(S(19))(S(2)x + S(1))/S(19))/S(260015), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(12)x*S(5) - S(7)x*S(3) - S(13)x*S(2) + S(8))/(S(100)x*S(6) - S(80)x*S(5) + S(116)x*S(4) - S(80)x*S(3) + S(41)x*S(2) - S(20)x + S(4)), x), x, (-S(251)x/S(726) + S(-313)/1452)/(S(2)x*S(2) + S(1)) - S(59096)log(-S(5)x + S(2))/S(99825) + S(2843)log(S(2)xS(2) + S(1))/S(7986) + S(503)sqrt(S(2))atan(sqrt(S(2))x)/S(15972) + S(5828)/(S(9075)(-S(5)x + S(2))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((S(12)xS(5) - S(7)x*S(3) - S(13)x*S(2) + S(8))/(S(100)x*S(6) - S(80)x*S(5) + S(116)x*S(4) - S(80)x*S(3) + S(41)x*S(2) - S(20)x + S(4)), x), x, (-S(251)x/S(726) + S(-313)/1452)/(S(2)x*S(2) + S(1)) - S(59096)log(-S(5)x + S(2))/S(99825) + S(2843)log(S(2)xS(2) + S(1))/S(7986) + S(503)sqrt(S(2))atan(sqrt(S(2))x)/S(15972) + S(5828)/(S(9075)(-S(5)x + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + S(9))/(xS(2)(xS(2) + S(9))), x), x, x - S(10)atan(x/S(3))/S(3) - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(4) + S(2)x)/(xS(2) + S(1)), x), x, xS(3)/S(3) - x + log(xS(2) + S(1)) + atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) - x)/((x + S(-1))*S(2)(x*S(2) + S(1))), x), x, log(-x + S(1)) + atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(3) + S(3)x*S(2) + S(5)x + S(2))/(xS(2) + x + S(1)), x), x, xS(2) + x + log(x*S(2) + x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x*S(3) - S(5)x*S(2) - S(4)x + S(3))/(x*S(3)(x*S(2) + x + S(-1))), x), x, S(3)log(x) - (sqrt(S(5)) + S(15))log(S(2)x + S(1) + sqrt(S(5)))/S(10) - (-sqrt(S(5)) + S(15))log(S(2)x - sqrt(S(5)) + S(1))/S(10) - S(1)/x + S(3)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(3) + S(5)x*S(2) + S(8)x + S(4))/(x*S(2) + S(2)x + S(2))S(2), x), x, log(xS(2) + S(2)x + S(2)) - atan(x + S(1)) - S(1)/(xS(2) + S(2)x + S(2)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((S(2)xS(3) + S(5)x*S(2) + S(8)x + S(4))/(x*S(2) + S(2)x + S(2))*S(2), x), x, x(x + S(2))/(S(2)(xS(2) + S(2)x + S(2))) + log(x*S(2) + S(2)x + S(2)) - atan(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)(x + S(-1))S(4)/(xS(2) + S(1)), x), x, x*S(7)/S(7) - S(2)xS(6)/S(3) + xS(5) - S(4)xS(3)/S(3) + S(4)x - S(4)atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x*S(2) - S(20)x)/(x*S(4) - S(10)x*S(2) + S(9)), x), x, log(-x + S(1)) - log(-x + S(3))/S(2) + S(3)log(x + S(1))/S(2) - S(2)log(x + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)xS(3) + x + S(-1))/(xS(2)(x + S(-1))(x*S(2) + S(1))), x), x, S(2)log(-x + S(1)) - log(xS(2) + S(1)) + atan(x) - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) - S(4)xS(3) + S(2)x*S(2) - S(3)x + S(1))/(xS(2) + S(1))S(3), x), x, atan(x) - (S(4)xS(2) + S(3))S(2)/(S(4)(xS(2) + S(1))S(2)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((x*S(4) - S(4)x*S(3) + S(2)x*S(2) - S(3)x + S(1))/(xS(2) + S(1))S(3), x), x, x*S(2)/(S(4)(xS(2) + S(1))S(2)) + atan(x) + S(7)/(S(4)(xS(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) - S(4)x*S(3) + S(2)x*S(2) - S(3)x + S(1))/(x*S(6) + S(3)x*S(4) + S(3)xS(2) + S(1)), x), x, atan(x) + S(2)/(xS(2) + S(1)) - S(1)/(S(4)(xS(2) + S(1))S(2)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((xS(4) - S(4)x*S(3) + S(2)x*S(2) - S(3)x + S(1))/(x*S(6) + S(3)x*S(4) + S(3)xS(2) + S(1)), x), x, xS(2)/(S(4)(xS(2) + S(1))S(2)) + atan(x) + S(7)/(S(4)(x*S(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x*S(3) + S(2)xS(2) + x + S(1))/(xS(4) + xS(3) + xS(2)), x), x, log(xS(2) + x + S(1)) - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((xS(2) - S(4)x + S(4))(x*S(2) - S(4)x + S(5))), x), x, -atan(x + S(-2)) + S(1)/(-x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + x + S(-3))/(xS(2)(x + S(-3))), x), x, log(-x + S(3)) - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x*S(2) + x + S(1))/(S(4)x*S(3) + x), x), x, log(x) + atan(S(2)x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)xS(2) - x + S(1))/(xS(3) - xS(2)), x), x, S(3)log(-x + S(1)) + S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) + S(3)x + S(4))/(x*S(2) + x), x), x, x + S(4)log(x) - S(2)log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)xS(2) + x + S(4))/(xS(3) + x), x), x, S(4)log(x) - log(xS(2) + S(1))/S(2) + atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)x*S(2) - S(4)x + S(7))/((S(4)x + S(1))(x*S(2) + S(1))), x), x, S(2)log(S(4)x + S(1)) - atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((x + S(-1))(x*S(2) + S(2)x + S(1))), x), x, log(-x + S(1))/S(4) + S(3)log(x + S(1))/S(4) + S(1)/(S(2)(x + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) + S(3)x + S(-4))/((S(2)x + S(-1))S(2)(S(2)x + S(3))), x), x, S(41)log(-S(2)x + S(1))/S(128) - S(25)log(S(2)x + S(3))/S(128) - S(9)/(S(32)(-S(2)x + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x*S(2) - S(4)x + S(5))/((x + S(-1))(xS(2) + S(1))), x), x, S(2)log(-x + S(1)) + log(x*S(2) + S(1))/S(2) - S(3)atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) - S(2)x + S(-1))/((x + S(-1))*S(2)(xS(2) + S(1))), x), x, log(-x + S(1)) - log(xS(2) + S(1))/S(2) + atan(x) + S(1)/(x + S(-1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(5))/((xS(2) - S(6)x + S(10))(x*S(2) - x + S(1)/2)), x), x, S(56)log(x*S(2) - S(6)x + S(10))/S(221) + S(109)log(S(2)x*S(2) - S(2)x + S(1))/S(442) + S(1026)atan(x + S(-3))/S(221) + S(261)atan(S(2)x + S(-1))/S(221), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(3)x + S(4))/((x + S(-3))(x + S(-2))(x + S(-1))), x), x, S(4)log(-x + S(1)) - S(14)log(-x + S(2)) + S(11)log(-x + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(16)x + S(1))/((x + S(5))*S(2)(S(2)x + S(-3))(x*S(2) + x + S(1))), x), x, S(200)log(-S(2)x + S(3))/S(3211) + S(2731)log(x + S(5))/S(24843) - S(481)log(xS(2) + x + S(1))/S(5586) + S(451)sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(8379) - S(79)/(S(273)(x + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(-1))/(xS(2) + x + S(1)), x), x, xS(2)/S(2) - x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(-3))/(x*S(2) - S(6)x + S(-7)), x), x, x*S(2)/S(2) + S(6)x + S(85)log(-x + S(7))/S(2) + log(x + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(1))/(xS(2) + S(4)x + S(13))*S(2), x), x, (S(47)x/S(18) + S(67)/18)/(x*S(2) + S(4)x + S(13)) + log(x*S(2) + S(4)x + S(13))/S(2) - S(61)atan(x/S(3) + S(2)/3)/S(54), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x*S(5) - S(10)x*S(4) + S(21)x*S(3) - S(42)x*S(2) + S(36)x + S(-32))/(x(xS(2) + S(1))(xS(2) + S(4))S(2)), x), x, -S(2)log(x) + log(xS(2) + S(4)) + atan(x/S(2))/S(2) + S(2)atan(x) + S(1)/(xS(2) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(9) + S(7)xS(5) + xS(4) + S(-1))/(xS(8) + S(6)xS(4) + S(-7)), x), x, xS(2)/S(2) - sqrt(S(2))S(7)(S(1)/4)log(x*S(2) - sqrt(S(2))S(7)*(S(1)/4)x + sqrt(S(7)))/S(56) + sqrt(S(2))S(7)(S(1)/4)log(x*S(2) + sqrt(S(2))S(7)*(S(1)/4)x + sqrt(S(7)))/S(56) + sqrt(S(2))S(7)(S(1)/4)atan(sqrt(S(2))S(7)(S(3)/4)x/S(7) + S(-1))/S(28) + sqrt(S(2))S(7)(S(1)/4)atan(sqrt(S(2))S(7)(S(3)/4)x/S(7) + S(1))/S(28) - atanh(xS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(6) + xS(3) + S(1))/(xS(5) + x), x), x, xS(2)/S(2) + log(x) - log(xS(4) + S(1))/S(4) + sqrt(S(2))log(xS(2) - sqrt(S(2))x + S(1))/S(8) - sqrt(S(2))log(xS(2) + sqrt(S(2))x + S(1))/S(8) - atan(x*S(2))/S(2) + sqrt(S(2))atan(sqrt(S(2))x + S(-1))/S(4) + sqrt(S(2))atan(sqrt(S(2))x + S(1))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(xS(2) - x), x), x, x - log(x) + S(2)log(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(1))/(xS(3) - x), x), x, x - log(x) + log(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(1))/(xS(3) - x*S(2)), x), x, x - log(x) + S(2)log(-x + S(1)) + S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(5) + S(-1))/(xS(3) - x), x), x, xS(3)/S(3) + x + log(x) - log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + S(1))/(xS(5) + xS(3)), x), x, -log(x) + log(x*S(2) + S(1)) - S(1)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(x*S(3) + S(2)xS(2) + x), x), x, log(x) + S(2)/(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(5) + S(1))/(x*S(3) - S(3)x*S(2) - S(10)x), x), x, x*S(3)/S(3) + S(3)x*S(2)/S(2) + S(19)x - log(x)/S(10) + S(3126)log(-x + S(5))/S(35) - S(31)log(x + S(2))/S(14), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + xS(2) - S(5)x + S(15))/((xS(2) + S(5))(x*S(2) + S(2)x + S(3))), x), x, log(x*S(2) + S(2)x + S(3))/S(2) + S(5)sqrt(S(2))atan(sqrt(S(2))(x + S(1))/S(2))/S(2) - sqrt(S(5))atan(sqrt(S(5))x/S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((xS(2) + S(1))(S(10)x/(xS(2) + S(1)) + S(3))), x), x, -log(x + S(3))/S(8) + log(S(3)x + S(1))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/(S(15)x + S(13) + S(2)/x), x), x, x*S(3)/S(45) - S(13)x*S(2)/S(450) + S(139)x/S(3375) - S(16)log(S(3)x + S(2))/S(567) + log(S(5)x + S(1))/S(4375), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(S(15)x + S(13) + S(2)/x), x), x, x*S(2)/S(30) - S(13)x/S(225) + S(8)log(S(3)x + S(2))/S(189) - log(S(5)x + S(1))/S(875), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(S(15)x + S(13) + S(2)/x), x), x, x/S(15) - S(4)log(S(3)x + S(2))/S(63) + log(S(5)x + S(1))/S(175), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(15)x + S(13) + S(2)/x), x), x, S(2)log(S(3)x + S(2))/S(21) - log(S(5)x + S(1))/S(35), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(15)x + S(13) + S(2)/x)), x), x, -log(S(3)x + S(2))/S(7) + log(S(5)x + S(1))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(S(15)x + S(13) + S(2)/x)), x), x, log(x)/S(2) + S(3)log(S(3)x + S(2))/S(14) - S(5)log(S(5)x + S(1))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(S(15)x + S(13) + S(2)/x)), x), x, -S(13)log(x)/S(4) - S(9)log(S(3)x + S(2))/S(28) + S(25)log(S(5)x + S(1))/S(7) - S(1)/(S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)(S(15)x + S(13) + S(2)/x)), x), x, S(139)log(x)/S(8) + S(27)log(S(3)x + S(2))/S(56) - S(125)log(S(5)x + S(1))/S(7) + S(13)/(S(4)x) - S(1)/(S(4)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(5)(S(15)x + S(13) + S(2)/x)), x), x, -S(1417)log(x)/S(16) - S(81)log(S(3)x + S(2))/S(112) + S(625)log(S(5)x + S(1))/S(7) - S(139)/(S(8)x) + S(13)/(S(8)xS(2)) - S(1)/(S(6)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(a + b(-xS(2) + S(1))S(4)), x), x, -atanh(b(S(1)/8)x/sqrt(b*(S(1)/4) + I(-a)*(S(1)/4)))/(S(4)b*(S(3)/8)sqrt(-a)sqrt(b(S(1)/4) + I(-a)(S(1)/4))) + atanh(b(S(1)/8)x/sqrt(b(S(1)/4) + (-a)(S(1)/4)))/(S(4)b*(S(3)/8)sqrt(-a)sqrt(b(S(1)/4) + (-a)(S(1)/4))) + atan(b(S(1)/8)x/sqrt(-b*(S(1)/4) + I(-a)*(S(1)/4)))/(S(4)b*(S(3)/8)sqrt(-a)sqrt(-b(S(1)/4) + I(-a)(S(1)/4))) - atan(b(S(1)/8)x/sqrt(-b(S(1)/4) + (-a)(S(1)/4)))/(S(4)b*(S(3)/8)sqrt(-a)sqrt(-b(S(1)/4) + (-a)(S(1)/4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/(a + b(xS(2) + S(-1))S(4)), x), x, -atanh(b*(S(1)/8)x/sqrt(b*(S(1)/4) + I(-a)*(S(1)/4)))/(S(4)b*(S(3)/8)sqrt(-a)sqrt(b(S(1)/4) + I(-a)(S(1)/4))) + atanh(b(S(1)/8)x/sqrt(b(S(1)/4) + (-a)(S(1)/4)))/(S(4)b*(S(3)/8)sqrt(-a)sqrt(b(S(1)/4) + (-a)(S(1)/4))) + atan(b(S(1)/8)x/sqrt(-b*(S(1)/4) + I(-a)*(S(1)/4)))/(S(4)b*(S(3)/8)sqrt(-a)sqrt(-b(S(1)/4) + I(-a)(S(1)/4))) - atan(b(S(1)/8)x/sqrt(-b(S(1)/4) + (-a)(S(1)/4)))/(S(4)b*(S(3)/8)sqrt(-a)sqrt(-b(S(1)/4) + (-a)(S(1)/4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)xS(5) + S(-1))/(xS(5) + x + S(1))S(2), x), x, -x/(xS(5) + x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-ad - S(2)aex - S(3)afxS(2) + bc - bex*S(2) - S(2)bfx*S(3))/(c + dx + exS(2) + fxS(3))S(2), x), x, a/(c + dx + ex*S(2) + fx*S(3)) + bx/(c + dx + ex*S(2) + fx*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(39)x*S(8) + S(26)x*S(6) + S(24)x*S(5) + S(174)x*S(4) - S(18)x*S(2) - S(40)x + S(9))/(x*S(4) + S(2)xS(2) + S(3))S(3), x), x, S(13)x/(xS(4) + S(2)x*S(2) + S(3)) + (-S(26)x*S(3) - S(4)x*S(2) - S(36)x + S(2))/(x*S(4) + S(2)xS(2) + S(3))S(2), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((-S(39)xS(8) + S(26)x*S(6) + S(24)x*S(5) + S(174)x*S(4) - S(18)x*S(2) - S(40)x + S(9))/(x*S(4) + S(2)xS(2) + S(3))S(3), x), x, x(-S(2)x*S(3) - S(4)x + S(117))/(S(9)(xS(4) + S(2)x*S(2) + S(3))) - S(2)x(xS(3) + S(39)x*S(2) + S(8)x + S(54))/(S(3)(xS(4) + S(2)xS(2) + S(3))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(30)xS(9) - S(8)x*S(7) - S(15)x*S(6) - S(140)x*S(5) + S(34)x*S(4) - S(12)x*S(3) - S(5)x*S(2) + S(36)x + S(-15))/(xS(4) + x + S(3))S(4), x), x, -S(5)xS(6)/(xS(4) + x + S(3))S(3) + xS(4)/(xS(4) + x + S(3))S(3) + S(5)xS(2)/(xS(4) + x + S(3))*S(3) - S(3)x/(xS(4) + x + S(3))S(3) + S(2)/(xS(4) + x + S(3))S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(30)x/(xS(4) + x + S(3))S(2) + (-S(8)x*S(3) - S(75)x*S(2) - S(320)x + S(42))/(xS(4) + x + S(3))S(3) + (S(57)xS(3) + S(360)x*S(2) + S(684)x + S(-141))/(xS(4) + x + S(3))S(4), x), x, (-S(5)xS(6) + xS(4) + S(5)x*S(2) - S(3)x + S(2))/(xS(4) + x + S(3))S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-(S(12)xS(3) + S(3))(-S(5)xS(6) + xS(4) + S(5)x*S(2) - S(3)x + S(2))/(xS(4) + x + S(3))S(4) + (-S(30)xS(5) + S(4)x*S(3) + S(10)x + S(-3))/(xS(4) + x + S(3))S(3), x), x, (-S(5)xS(6) + xS(4) + S(5)x*S(2) - S(3)x + S(2))/(xS(4) + x + S(3))S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(-1))/(xS(2) - x + S(1)), x), x, log(xS(2) - x + S(1))/S(2) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(-1))/(xS(3) + S(1)), x), x, log(xS(2) - x + S(1))/S(2) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x + S(-4))/(xS(2) - S(2)x + S(4)), x), x, S(3)log(xS(2) - S(2)x + S(4))/S(2) + sqrt(S(3))atan(sqrt(S(3))(-x + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)xS(2) + S(2)x + S(-8))/(x*S(3) + S(8)), x), x, S(3)log(x*S(2) - S(2)x + S(4))/S(2) + sqrt(S(3))atan(sqrt(S(3))(-x + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x + S(4))/(xS(2)(x*S(2) + S(1))), x), x, S(4)log(x) - S(2)log(xS(2) + S(1)) - S(4)atan(x) - S(4)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)x + S(24))/(x(x*S(2) + S(-4))), x), x, -S(6)log(x) + S(5)log(-x + S(2)) + log(x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(-1))/(xS(3) - S(2)x), x), x, log(x)/S(2) + log(-xS(2) + S(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(x*S(3) + S(3)x), x), x, log(x)/S(3) + log(x*S(2) + S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + S(3)bxS(2))/(ax + bxS(3)), x), x, log(x) + log(a + bx*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x + S(-2))/(x*S(3) - x), x), x, S(2)log(x) + log(-x + S(1)) - S(3)log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(4))/(xS(3) + S(4)x), x), x, log(x) - log(x*S(2) + S(4))/S(2) + atan(x/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(3) - x)/(xS(4) - xS(2) + S(1)), x), x, log(xS(4) - xS(2) + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(-3))/(xS(3) + S(3)xS(2) + S(2)x), x), x, -S(3)log(x)/S(2) + S(4)log(x + S(1)) - S(5)log(x + S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x + S(2))/(x*S(4) + S(2)xS(3) + xS(2)), x), x, -S(2)/(x(x + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(1))/(xS(3) + xS(2) - S(6)x), x), x, -log(x)/S(6) + S(3)log(-x + S(2))/S(10) - S(2)log(x + S(3))/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(3) + S(4)xS(2))/(xS(3) + x), x), x, x + S(2)log(xS(2) + S(1)) - atan(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(3) + x)/(xS(4) + xS(2))S(3), x), x, -S(1)/(S(4)xS(4)(xS(2) + S(1))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((axS(2) + bx*S(3))/(cx*S(2) + dx*S(3)), x), x, bx/d - (-ad + bc)log(c + dx)/dS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + x)/(xS(3) - xS(2) - S(2)x), x), x, log(-x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(5)xS(2) + S(1))/(xS(3)(xS(2) + S(1))), x), x, -S(6)log(x) + S(3)log(xS(2) + S(1)) - S(1)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)x/((x + S(-1))(xS(2) + S(5))), x), x, log(-x + S(1))/S(3) - log(xS(2) + S(5))/S(6) + sqrt(S(5))atan(sqrt(S(5))x/S(5))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(2))/(x + S(2)), x), x, xS(2)/S(2) - S(2)x + S(6)log(x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x + S(-3))(xS(2) + S(4))), x), x, log(-x + S(3))/S(13) - log(xS(2) + S(4))/S(26) - S(3)atan(x/S(2))/S(26), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x*S(6) + S(-2))/(x(S(2)xS(6) + S(5))), x), x, -S(2)log(x)/S(5) + S(19)log(S(2)x*S(6) + S(5))/S(60), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x + S(3))/((x + S(-2))(x + S(5))), x), x, log(-x + S(2)) + log(x + S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(xS(4) + S(5)x*S(2) + S(4)), x), x, x - S(8)atan(x/S(2))/S(3) + atan(x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x + S(1))(x + S(2))S(2)(x + S(3))*S(3)), x), x, log(x + S(1))/S(8) + S(2)log(x + S(2)) - S(17)log(x + S(3))/S(8) + S(5)/(S(4)(x + S(3))) + S(1)/(S(4)(x + S(3))S(2)) + S(1)/(x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(2) + S(-1)), x), x, log(-xS(2) + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(-1))(S(-2)), x), x, x/(S(2)(-xS(2) + S(1))) + atanh(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(xS(2) + S(1))S(2), x), x, -x/(S(2)(xS(2) + S(1))) + atan(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(3)x + S(2)), x), x, log(S(3)x + S(2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(aS(2) + xS(2)), x), x, atan(x/a)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bx*S(2)), x), x, atan(sqrt(b)x/sqrt(a))/(sqrt(a)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2) - x + S(2)), x), x, -S(2)sqrt(S(7))atan(sqrt(S(7))(-S(2)x + S(1))/S(7))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(-xS(2) + S(4))S(2), x), x, x*S(7)/S(7) - S(8)x*S(5)/S(5) + S(16)x*S(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-xS(3) + S(1))S(2), x), x, x*S(8)/S(8) - S(2)xS(5)/S(5) + xS(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(3) + S(5)xS(2) + S(-4))/xS(2), x), x, x*S(2)/S(2) + S(5)x + S(4)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(-1))/(S(3)xS(2) - S(4)x + S(3)), x), x, log(S(3)xS(2) - S(4)x + S(3))/S(6) + sqrt(S(5))atan(sqrt(S(5))(-S(3)x + S(2))/S(5))/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(2))S(2), x), x, xS(7)/S(7) + xS(4) + S(4)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(-4))/(x + S(2)), x), x, xS(2)/S(2) - S(2)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x + S(2))(xS(2) + S(1))), x), x, log(x + S(2))/S(5) - log(xS(2) + S(1))/S(10) + S(2)atan(x)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x + S(1))(xS(2) + S(1))), x), x, log(x + S(1))/S(2) - log(xS(2) + S(1))/S(4) + atan(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((x + S(1))(xS(2) + S(1))), x), x, -log(x + S(1))/S(2) + log(xS(2) + S(1))/S(4) + atan(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(2)x)/(x + S(1))S(2), x), x, (x + S(2))S(2)/(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) + S(-10))/(S(2)x*S(4) + S(9)x*S(2) + S(4)), x), x, atan(x/S(2)) - S(3)sqrt(S(2))atan(sqrt(S(2))x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)x + S(31))/(S(3)x*S(2) - S(4)x + S(11)), x), x, S(5)log(S(3)x*S(2) - S(4)x + S(11))/S(6) - S(103)sqrt(S(29))atan(sqrt(S(29))(-S(3)x + S(2))/S(29))/S(87), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + xS(2) + S(-2))/x*S(4), x), x, log(x) - S(1)/x + S(2)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + x + S(1))/xS(2), x), x, xS(2)/S(2) + log(x) - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) + S(-2))/(x(xS(2) + S(2))), x), x, -log(x) + log(xS(2) + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(-3))(S(4)xS(2) + S(-7)), x), x, xS(4) - S(4)xS(3) - S(7)x*S(2)/S(2) + S(21)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(7)x + S(-2))S(3), x), x, (-S(7)x + S(2))*S(4)/S(28), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x*S(2) + S(-7))/(S(2)x + S(3)), x), x, x*S(2) - S(3)x + log(S(2)x + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(1))/(xS(2)(x + S(-1))), x), x, -S(2)log(x) + S(2)log(-x + S(1)) + S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(4) + S(4)x*S(3) + S(4)x*S(2)), x), x, atanh(x + S(1))/S(2) - S(1)/(S(4)(x + S(2))) - S(1)/(S(4)x), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/(xS(4) + S(4)x*S(3) + S(4)x*S(2)), x), x, -log(x)/S(4) + log(x + S(2))/S(4) - S(1)/(S(4)(x + S(2))) - S(1)/(S(4)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(1))/(x + S(1)), x), x, xS(2)/S(2) - x + S(2)log(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(3) - S(3)x*S(2) + S(3)x + S(-1))/xS(2), x), x, xS(2)/S(2) - S(3)x + S(3)log(x) + S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(3)/2 + sqrt(S(37))/S(2))(x - sqrt(S(37))/S(2) + S(3)/2), x), x, xS(3)/S(3) + S(3)x*S(2)/S(2) - S(7)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(3) + S(3)xS(2) + S(4))/(x + S(1))S(4), x), x, S(2)log(x + S(1)) + S(3)/(x + S(1)) - S(5)/(S(3)(x + S(1))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((x + S(1))S(2)(xS(2) + S(1))), x), x, atan(x)/S(2) + S(1)/(S(2)(x + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) - xS(3) + S(3)xS(2) - S(2)x + S(7))/(x + S(2)), x), x, xS(4)/S(4) - xS(3) + S(9)xS(2)/S(2) - S(20)x + S(47)log(x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(-1))/(x + S(-1)), x), x, xS(3)/S(3) + xS(2)/S(2) + x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x + S(2))/((x + S(-1))*S(3)(xS(2) + S(1))), x), x, atan(x) + S(1)/(x + S(-1)) - S(1)/(-x + S(1))S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(bx + c(d + ex)S(2)), x), x, -S(2)atanh((b + S(2)cde + S(2)ceS(2)x)/(sqrt(b)sqrt(b + S(4)cde)))/(sqrt(b)sqrt(b + S(4)cde)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bx + c(d + ex)S(2)), x), x, -S(2)atanh((b + S(2)cde + S(2)ceS(2)x)/sqrt(-S(4)aceS(2) + bS(2) + S(4)bcde))/sqrt(-S(4)aceS(2) + bS(2) + S(4)bcde), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((xS(2) + S(-1))S(2) + S(1)), x), x, log(xS(2) - xsqrt(S(2) + S(2)sqrt(S(2))) + sqrt(S(2)))/(S(4)sqrt(S(2) + S(2)sqrt(S(2)))) - log(x*S(2) + xsqrt(S(2) + S(2)sqrt(S(2))) + sqrt(S(2)))/(S(4)sqrt(S(2) + S(2)sqrt(S(2)))) - sqrt(S(1)/2 + sqrt(S(2))/S(2))atan((-S(2)x + sqrt(S(2) + S(2)sqrt(S(2))))/sqrt(S(-2) + S(2)sqrt(S(2))))/S(2) + sqrt(S(1)/2 + sqrt(S(2))/S(2))atan((S(2)x + sqrt(S(2) + S(2)sqrt(S(2))))/sqrt(S(-2) + S(2)sqrt(S(2))))/S(2), expand=True, _diff=True, _numerical=True) def test_2(): assert rubi_test(rubi_integrate(xS(5)(a + bxS(2))/(sqrt(-c + dx)sqrt(c + dx)), x), x, bxS(6)sqrt(-c + dx)sqrt(c + dx)/(S(7)d*S(2)) + S(8)c*S(4)sqrt(-c + dx)sqrt(c + dx)(S(7)ad*S(2) + S(6)bcS(2))/(S(105)d*S(8)) + S(4)c*S(2)x*S(2)sqrt(-c + dx)sqrt(c + dx)(S(7)ad*S(2) + S(6)bcS(2))/(S(105)dS(6)) + xS(4)sqrt(-c + dx)sqrt(c + dx)(S(7)adS(2) + S(6)bcS(2))/(S(35)dS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(a + bx*S(2))/(sqrt(-c + dx)sqrt(c + dx)), x), x, bxS(5)sqrt(-c + dx)sqrt(c + dx)/(S(6)dS(2)) + cS(4)(S(6)adS(2) + S(5)bcS(2))atanh(sqrt(-c + dx)/sqrt(c + dx))/(S(8)dS(7)) + cS(2)xsqrt(-c + dx)sqrt(c + dx)(S(6)adS(2) + S(5)bcS(2))/(S(16)dS(6)) + xS(3)sqrt(-c + dx)sqrt(c + dx)(S(6)adS(2) + S(5)bcS(2))/(S(24)dS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a + bx*S(2))/(sqrt(-c + dx)sqrt(c + dx)), x), x, bxS(4)sqrt(-c + dx)sqrt(c + dx)/(S(5)d*S(2)) + S(2)c*S(2)sqrt(-c + dx)sqrt(c + dx)(S(5)ad*S(2) + S(4)bcS(2))/(S(15)dS(6)) + xS(2)sqrt(-c + dx)sqrt(c + dx)(S(5)adS(2) + S(4)bcS(2))/(S(15)dS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx*S(2))/(sqrt(-c + dx)sqrt(c + dx)), x), x, bxS(3)sqrt(-c + dx)sqrt(c + dx)/(S(4)dS(2)) + cS(2)(S(4)adS(2) + S(3)bcS(2))atanh(sqrt(-c + dx)/sqrt(c + dx))/(S(4)dS(5)) + xsqrt(-c + dx)sqrt(c + dx)(S(4)ad*S(2) + S(3)bcS(2))/(S(8)d*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxS(2))/(sqrt(-c + dx)sqrt(c + dx)), x), x, bxS(2)sqrt(-c + dx)sqrt(c + dx)/(S(3)d*S(2)) + sqrt(-c + dx)sqrt(c + dx)(S(3)adS(2) + S(2)bcS(2))/(S(3)d*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2))/(sqrt(-c + dx)sqrt(c + dx)), x), x, bxsqrt(-c + dx)sqrt(c + dx)/(S(2)d*S(2)) + (S(2)adS(2) + bc*S(2))atanh(sqrt(-c + dx)/sqrt(c + dx))/d*S(3), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + bx*S(2))/(sqrt(-c + dx)sqrt(c + dx)), x), x, S(2)aatanh(sqrt(-c + dx)/sqrt(c + dx))/d + bcS(2)atanh(sqrt(-c + dx)/sqrt(c + dx))/d*S(3) + bxsqrt(-c + dx)sqrt(c + dx)/(S(2)dS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2))/(xsqrt(-c + dx)sqrt(c + dx)), x), x, aatan(sqrt(-c + dx)sqrt(c + dx)/c)/c + bsqrt(-c + dx)sqrt(c + dx)/dS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(2)sqrt(-c + dx)sqrt(c + dx)), x), x, asqrt(-c + dx)sqrt(c + dx)/(c*S(2)x) + S(2)batanh(sqrt(-c + dx)/sqrt(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(3)sqrt(-c + dx)sqrt(c + dx)), x), x, asqrt(-c + dx)sqrt(c + dx)/(S(2)c*S(2)x*S(2)) + (ad*S(2) + S(2)bcS(2))atan(sqrt(-c + dx)sqrt(c + dx)/c)/(S(2)c*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(4)sqrt(-c + dx)sqrt(c + dx)), x), x, asqrt(-c + dx)sqrt(c + dx)/(S(3)cS(2)x*S(3)) + sqrt(-c + dx)sqrt(c + dx)(S(2)adS(2) + S(3)bcS(2))/(S(3)c*S(4)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(5)sqrt(-c + dx)sqrt(c + dx)), x), x, asqrt(-c + dx)sqrt(c + dx)/(S(4)c*S(2)x*S(4)) + sqrt(-c + dx)sqrt(c + dx)(S(3)adS(2) + S(4)bcS(2))/(S(8)c*S(4)xS(2)) + dS(2)(S(3)adS(2) + S(4)bcS(2))atan(sqrt(-c + dx)sqrt(c + dx)/c)/(S(8)c*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(6)sqrt(-c + dx)sqrt(c + dx)), x), x, asqrt(-c + dx)sqrt(c + dx)/(S(5)cS(2)x*S(5)) + sqrt(-c + dx)sqrt(c + dx)(S(4)adS(2) + S(5)bcS(2))/(S(15)c*S(4)x*S(3)) + S(2)d*S(2)sqrt(-c + dx)sqrt(c + dx)(S(4)ad*S(2) + S(5)bcS(2))/(S(15)c*S(6)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)(a + bxS(2))/((-c + dx)*(S(3)/2)(c + dx)(S(3)/2)), x), x, bx*S(6)/(S(5)d*S(2)sqrt(-c + dx)sqrt(c + dx)) + S(8)c*S(2)sqrt(-c + dx)sqrt(c + dx)(S(5)ad*S(2) + S(6)bcS(2))/(S(15)dS(8)) - xS(4)(S(5)adS(2) + S(6)bcS(2))/(S(5)d*S(4)sqrt(-c + dx)sqrt(c + dx)) + xS(2)sqrt(-c + dx)sqrt(c + dx)(S(20)ad*S(2) + S(24)bcS(2))/(S(15)dS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(a + bx*S(2))/((-c + dx)*(S(3)/2)(c + dx)(S(3)/2)), x), x, bx*S(5)/(S(4)d*S(2)sqrt(-c + dx)sqrt(c + dx)) + S(3)c*S(2)(S(4)ad*S(2) + S(5)bcS(2))atanh(sqrt(-c + dx)/sqrt(c + dx))/(S(4)dS(7)) - xS(3)(S(4)ad*S(2) + S(5)bcS(2))/(S(4)d*S(4)sqrt(-c + dx)sqrt(c + dx)) + xsqrt(-c + dx)sqrt(c + dx)(S(12)ad*S(2) + S(15)bcS(2))/(S(8)dS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a + bx*S(2))/((-c + dx)*(S(3)/2)(c + dx)(S(3)/2)), x), x, bx*S(4)/(S(3)d*S(2)sqrt(-c + dx)sqrt(c + dx)) - xS(2)(S(3)ad*S(2) + S(4)bcS(2))/(S(3)d*S(4)sqrt(-c + dx)sqrt(c + dx)) + sqrt(-c + dx)sqrt(c + dx)(S(6)adS(2) + S(8)bcS(2))/(S(3)dS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx*S(2))/((-c + dx)*(S(3)/2)(c + dx)(S(3)/2)), x), x, bx*S(3)/(S(2)d*S(2)sqrt(-c + dx)sqrt(c + dx)) - c(S(2)ad*S(2) + S(3)bcS(2))/(S(2)d*S(5)sqrt(-c + dx)sqrt(c + dx)) - sqrt(-c + dx)(S(2)adS(2) + S(3)bcS(2))/(S(2)d*S(5)sqrt(c + dx)) + (S(2)adS(2) + S(3)bcS(2))atanh(sqrt(-c + dx)/sqrt(c + dx))/d*S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxS(2))/((-c + dx)*(S(3)/2)(c + dx)(S(3)/2)), x), x, -xS(2)(a/cS(2) + b/dS(2))/(sqrt(-c + dx)sqrt(c + dx)) + sqrt(-c + dx)sqrt(c + dx)(ad*S(2) + S(2)bcS(2))/(cS(2)d*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2))/((-c + dx)*(S(3)/2)(c + dx)(S(3)/2)), x), x, -ax/(c*S(2)sqrt(-c + dx)sqrt(c + dx)) - bc/(d*S(3)sqrt(-c + dx)sqrt(c + dx)) - bsqrt(-c + dx)/(dS(3)sqrt(c + dx)) + S(2)batanh(sqrt(-c + dx)/sqrt(c + dx))/dS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*S(2))/(x(-c + dx)(S(3)/2)(c + dx)(S(3)/2)), x), x, -aatan(sqrt(-c + dx)sqrt(c + dx)/c)/cS(3) - (a/cS(2) + b/dS(2))/(sqrt(-c + dx)sqrt(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(2)(-c + dx)(S(3)/2)(c + dx)(S(3)/2)), x), x, a/(cS(2)xsqrt(-c + dx)sqrt(c + dx)) - x(S(2)adS(2) + bcS(2))/(cS(4)sqrt(-c + dx)sqrt(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(3)(-c + dx)(S(3)/2)(c + dx)(S(3)/2)), x), x, a/(S(2)c*S(2)x*S(2)sqrt(-c + dx)sqrt(c + dx)) - (S(3)adS(2) + S(2)bcS(2))/(S(2)c*S(4)sqrt(-c + dx)sqrt(c + dx)) - (S(3)adS(2) + S(2)bcS(2))atan(sqrt(-c + dx)sqrt(c + dx)/c)/(S(2)c*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(4)(-c + dx)*(S(3)/2)(c + dx)(S(3)/2)), x), x, a/(S(3)c*S(2)x*S(3)sqrt(-c + dx)sqrt(c + dx)) + (S(4)adS(2) + S(3)bcS(2))/(S(3)c*S(4)xsqrt(-c + dx)sqrt(c + dx)) - S(2)dS(2)x(S(4)adS(2) + S(3)bcS(2))/(S(3)c*S(6)sqrt(-c + dx)sqrt(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(5)(-c + dx)*(S(3)/2)(c + dx)(S(3)/2)), x), x, a/(S(4)c*S(2)x*S(4)sqrt(-c + dx)sqrt(c + dx)) + (S(5)adS(2) + S(4)bcS(2))/(S(8)c*S(4)x*S(2)sqrt(-c + dx)sqrt(c + dx)) - S(3)d*S(2)(S(5)ad*S(2) + S(4)bcS(2))/(S(8)c*S(6)sqrt(-c + dx)sqrt(c + dx)) - S(3)d*S(2)(S(5)ad*S(2) + S(4)bcS(2))atan(sqrt(-c + dx)sqrt(c + dx)/c)/(S(8)c*S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))/(xS(6)(-c + dx)*(S(3)/2)(c + dx)(S(3)/2)), x), x, a/(S(5)c*S(2)x*S(5)sqrt(-c + dx)sqrt(c + dx)) + (S(6)adS(2) + S(5)bcS(2))/(S(15)c*S(4)x*S(3)sqrt(-c + dx)sqrt(c + dx)) + S(4)d*S(2)(S(6)ad*S(2) + S(5)bcS(2))/(S(15)c*S(6)xsqrt(-c + dx)sqrt(c + dx)) - S(8)dS(4)x(S(6)adS(2) + S(5)bcS(2))/(S(15)c*S(8)sqrt(-c + dx)sqrt(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cS(2)x*S(2) + S(1))/(xsqrt(cx + S(-1))sqrt(cx + S(1))), x), x, sqrt(cx + S(-1))sqrt(cx + S(1)) + atan(sqrt(cx + S(-1))sqrt(cx + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx)n(c + dxS(3)), x), x, S(10)a*S(2)d(a + bx)(n + S(4))/(bS(6)(n + S(4))) + aS(2)(a + bx)(n + S(1))(-a*S(3)d + b*S(3)c)/(b*S(6)(n + S(1))) - S(5)ad(a + bx)(n + S(5))/(bS(6)(n + S(5))) - a(a + bx)(n + S(2))(-S(5)aS(3)d + S(2)bS(3)c)/(b*S(6)(n + S(2))) + d(a + bx)(n + S(6))/(bS(6)(n + S(6))) + (a + bx)*(n + S(3))(-S(10)aS(3)d + b*S(3)c)/(b*S(6)(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)*n(c + dxS(3)), x), x, S(6)a*S(2)d(a + bx)(n + S(3))/(bS(5)(n + S(3))) - S(4)ad(a + bx)(n + S(4))/(bS(5)(n + S(4))) - a(a + bx)*(n + S(1))(-a*S(3)d + b*S(3)c)/(b*S(5)(n + S(1))) + d(a + bx)(n + S(5))/(bS(5)(n + S(5))) + (a + bx)*(n + S(2))(-S(4)aS(3)d + b*S(3)c)/(b*S(5)(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)n(c + dxS(3)), x), x, S(3)a*S(2)d(a + bx)(n + S(2))/(bS(4)(n + S(2))) - S(3)ad(a + bx)(n + S(3))/(bS(4)(n + S(3))) + d(a + bx)(n + S(4))/(bS(4)(n + S(4))) + (a + bx)*(n + S(1))(-a*S(3)d + b*S(3)c)/(b*S(4)(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)n(c + dxS(3))/x, x), x, aS(2)d(a + bx)(n + S(1))/(bS(3)(n + S(1))) - S(2)ad(a + bx)(n + S(2))/(bS(3)(n + S(2))) + d(a + bx)(n + S(3))/(bS(3)(n + S(3))) - c(a + bx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), S(1) + bx/a)/(a(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(a + bx)n(c + dxS(3))S(2), x), x, S(28)a*S(2)d*S(2)(a + bx)(n + S(7))/(bS(9)(n + S(7))) + S(4)aS(2)d(a + bx)*(n + S(4))(-S(14)aS(3)d + S(5)bS(3)c)/(b*S(9)(n + S(4))) + a*S(2)(a + bx)(n + S(1))(-a*S(3)d + b*S(3)c)S(2)/(bS(9)(n + S(1))) - S(8)adS(2)(a + bx)(n + S(8))/(bS(9)(n + S(8))) - S(10)ad(a + bx)*(n + S(5))(-S(7)aS(3)d + b*S(3)c)/(b*S(9)(n + S(5))) - S(2)a(a + bx)(n + S(2))(-S(4)aS(3)d + b*S(3)c)(-aS(3)d + b*S(3)c)/(b*S(9)(n + S(2))) + d*S(2)(a + bx)(n + S(9))/(bS(9)(n + S(9))) + S(2)d(a + bx)(n + S(6))(-S(28)aS(3)d + b*S(3)c)/(b*S(9)(n + S(6))) + (a + bx)(n + S(3))(S(28)aS(6)d*S(2) - S(20)a*S(3)b*S(3)cd + bS(6)cS(2))/(bS(9)(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)n(c + dxS(3))S(2), x), x, S(21)a*S(2)d*S(2)(a + bx)(n + S(6))/(bS(8)(n + S(6))) + S(3)aS(2)d(a + bx)*(n + S(3))(-S(7)aS(3)d + S(4)bS(3)c)/(b*S(8)(n + S(3))) - S(7)ad*S(2)(a + bx)(n + S(7))/(bS(8)(n + S(7))) - ad(a + bx)(n + S(4))(-S(35)aS(3)d + S(8)bS(3)c)/(b*S(8)(n + S(4))) - a(a + bx)*(n + S(1))(-a*S(3)d + b*S(3)c)S(2)/(bS(8)(n + S(1))) + dS(2)(a + bx)(n + S(8))/(bS(8)(n + S(8))) + d(a + bx)*(n + S(5))(-S(35)aS(3)d + S(2)bS(3)c)/(b*S(8)(n + S(5))) + (a + bx)(n + S(2))(-S(7)aS(3)d + b*S(3)c)(-aS(3)d + b*S(3)c)/(b*S(8)(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)n(c + dxS(3))S(2), x), x, S(15)a*S(2)d*S(2)(a + bx)(n + S(5))/(bS(7)(n + S(5))) + S(6)aS(2)d(a + bx)*(n + S(2))(-a*S(3)d + b*S(3)c)/(b*S(7)(n + S(2))) - S(6)ad*S(2)(a + bx)(n + S(6))/(bS(7)(n + S(6))) - S(3)ad(a + bx)*(n + S(3))(-S(5)aS(3)d + S(2)bS(3)c)/(b*S(7)(n + S(3))) + d*S(2)(a + bx)(n + S(7))/(bS(7)(n + S(7))) + S(2)d(a + bx)(n + S(4))(-S(10)aS(3)d + b*S(3)c)/(b*S(7)(n + S(4))) + (a + bx)(n + S(1))(-a*S(3)d + b*S(3)c)S(2)/(bS(7)(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*n(c + dxS(3))S(2)/x, x), x, S(10)a*S(2)d*S(2)(a + bx)(n + S(4))/(bS(6)(n + S(4))) + a*S(2)d(a + bx)*(n + S(1))(-a*S(3)d + S(2)bS(3)c)/(b*S(6)(n + S(1))) - S(5)ad*S(2)(a + bx)(n + S(5))/(bS(6)(n + S(5))) - ad(a + bx)(n + S(2))(-S(5)aS(3)d + S(4)bS(3)c)/(b*S(6)(n + S(2))) + d*S(2)(a + bx)(n + S(6))/(bS(6)(n + S(6))) + S(2)d(a + bx)(n + S(3))(-S(5)aS(3)d + b*S(3)c)/(b*S(6)(n + S(3))) - c*S(2)(a + bx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), S(1) + bx/a)/(a(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(a + bx)n(c + dxS(3))S(3), x), x, S(55)a*S(2)d*S(3)(a + bx)(n + S(10))/(bS(12)(n + S(10))) + S(42)aS(2)d*S(2)(a + bx)(n + S(7))(-S(11)aS(3)d + S(2)bS(3)c)/(b*S(12)(n + S(7))) + S(3)aS(2)d(a + bx)*(n + S(4))(S(55)aS(6)d*S(2) - S(56)a*S(3)b*S(3)cd + S(10)b*S(6)cS(2))/(bS(12)(n + S(4))) + aS(2)(a + bx)(n + S(1))(-a*S(3)d + b*S(3)c)S(3)/(bS(12)(n + S(1))) - S(11)adS(3)(a + bx)(n + S(11))/(bS(12)(n + S(11))) - S(6)ad*S(2)(a + bx)(n + S(8))(-S(55)aS(3)d + S(4)bS(3)c)/(b*S(12)(n + S(8))) - S(15)ad(a + bx)*(n + S(5))(S(22)aS(6)d*S(2) - S(14)a*S(3)b*S(3)cd + bS(6)cS(2))/(bS(12)(n + S(5))) - a(a + bx)(n + S(2))(-S(11)aS(3)d + S(2)bS(3)c)(-aS(3)d + b*S(3)c)S(2)/(bS(12)(n + S(2))) + dS(3)(a + bx)(n + S(12))/(bS(12)(n + S(12))) + S(3)dS(2)(a + bx)(n + S(9))(-S(55)aS(3)d + b*S(3)c)/(b*S(12)(n + S(9))) + S(3)d(a + bx)(n + S(6))(S(154)aS(6)d*S(2) - S(56)a*S(3)b*S(3)cd + bS(6)cS(2))/(bS(12)(n + S(6))) + (a + bx)*(n + S(3))(-a*S(3)d + b*S(3)c)(S(55)a*S(6)d*S(2) - S(29)a*S(3)b*S(3)cd + bS(6)cS(2))/(bS(12)(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)n(c + dxS(3))S(3), x), x, S(45)a*S(2)d*S(3)(a + bx)(n + S(9))/(bS(11)(n + S(9))) + S(63)aS(2)d*S(2)(a + bx)(n + S(6))(-S(4)aS(3)d + b*S(3)c)/(b*S(11)(n + S(6))) + S(9)aS(2)d(a + bx)*(n + S(3))(-S(5)aS(3)d + S(2)bS(3)c)(-aS(3)d + b*S(3)c)/(b*S(11)(n + S(3))) - S(10)ad*S(3)(a + bx)(n + S(10))/(bS(11)(n + S(10))) - S(21)ad*S(2)(a + bx)(n + S(7))(-S(10)aS(3)d + b*S(3)c)/(b*S(11)(n + S(7))) - S(3)ad(a + bx)*(n + S(4))(S(40)aS(6)d*S(2) - S(35)a*S(3)b*S(3)cd + S(4)b*S(6)cS(2))/(bS(11)(n + S(4))) - a(a + bx)(n + S(1))(-a*S(3)d + b*S(3)c)S(3)/(bS(11)(n + S(1))) + dS(3)(a + bx)(n + S(11))/(bS(11)(n + S(11))) + S(3)dS(2)(a + bx)(n + S(8))(-S(40)aS(3)d + b*S(3)c)/(b*S(11)(n + S(8))) + S(3)d(a + bx)(n + S(5))(S(70)aS(6)d*S(2) - S(35)a*S(3)b*S(3)cd + bS(6)cS(2))/(bS(11)(n + S(5))) + (a + bx)*(n + S(2))(-S(10)aS(3)d + b*S(3)c)(-aS(3)d + b*S(3)c)S(2)/(bS(11)(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*n(c + dxS(3))S(3), x), x, S(36)a*S(2)d*S(3)(a + bx)(n + S(8))/(bS(10)(n + S(8))) + S(9)aS(2)d*S(2)(a + bx)(n + S(5))(-S(14)aS(3)d + S(5)bS(3)c)/(b*S(10)(n + S(5))) + S(9)aS(2)d(a + bx)*(n + S(2))(-a*S(3)d + b*S(3)c)S(2)/(bS(10)(n + S(2))) - S(9)adS(3)(a + bx)(n + S(9))/(bS(10)(n + S(9))) - S(18)ad*S(2)(a + bx)(n + S(6))(-S(7)aS(3)d + b*S(3)c)/(b*S(10)(n + S(6))) - S(9)ad(a + bx)*(n + S(3))(-S(4)aS(3)d + b*S(3)c)(-aS(3)d + b*S(3)c)/(b*S(10)(n + S(3))) + d*S(3)(a + bx)(n + S(10))/(bS(10)(n + S(10))) + S(3)dS(2)(a + bx)(n + S(7))(-S(28)aS(3)d + b*S(3)c)/(b*S(10)(n + S(7))) + S(3)d(a + bx)(n + S(4))(S(28)aS(6)d*S(2) - S(20)a*S(3)b*S(3)cd + bS(6)cS(2))/(bS(10)(n + S(4))) + (a + bx)*(n + S(1))(-a*S(3)d + b*S(3)c)S(3)/(bS(10)(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*n(c + dxS(3))S(3)/x, x), x, S(28)a*S(2)d*S(3)(a + bx)(n + S(7))/(bS(9)(n + S(7))) + S(2)aS(2)d*S(2)(a + bx)(n + S(4))(-S(28)aS(3)d + S(15)bS(3)c)/(b*S(9)(n + S(4))) + a*S(2)d(a + bx)*(n + S(1))(a*S(6)d*S(2) - S(3)a*S(3)b*S(3)cd + S(3)b*S(6)cS(2))/(bS(9)(n + S(1))) - S(8)adS(3)(a + bx)(n + S(8))/(bS(9)(n + S(8))) - S(5)ad*S(2)(a + bx)(n + S(5))(-S(14)aS(3)d + S(3)bS(3)c)/(b*S(9)(n + S(5))) - ad(a + bx)(n + S(2))(S(8)aS(6)d*S(2) - S(15)a*S(3)b*S(3)cd + S(6)b*S(6)cS(2))/(bS(9)(n + S(2))) + dS(3)(a + bx)(n + S(9))/(bS(9)(n + S(9))) + d*S(2)(a + bx)(n + S(6))(-S(56)aS(3)d + S(3)bS(3)c)/(b*S(9)(n + S(6))) + d(a + bx)*(n + S(3))(S(28)aS(6)d*S(2) - S(30)a*S(3)b*S(3)cd + S(3)b*S(6)cS(2))/(bS(9)(n + S(3))) - cS(3)(a + bx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), S(1) + bx/a)/(a(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*m(e + fx)n/(a + bxS(3)), x), x, x(m + S(1))(S(1) + fx/e)*(-n)(e + fx)nAppellF1(m + S(1), -n, S(1), m + S(2), -fx/e, -b(S(1)/3)x/a*(S(1)/3))/(S(3)a(m + S(1))) + x(m + S(1))(S(1) + fx/e)(-n)(e + fx)nAppellF1(m + S(1), -n, S(1), m + S(2), -fx/e, (S(-1))(S(1)/3)b*(S(1)/3)x/a*(S(1)/3))/(S(3)a(m + S(1))) + x(m + S(1))(S(1) + fx/e)(-n)(e + fx)nAppellF1(m + S(1), -n, S(1), m + S(2), -fx/e, -(S(-1))(S(2)/3)b*(S(1)/3)x/a*(S(1)/3))/(S(3)a(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)(e + fx)n/(a + bx*S(3)), x), x, a(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/(-(S(-1))(S(2)/3)a*(S(1)/3)f + b*(S(1)/3)e))/(S(3)b(S(5)/3)(n + S(1))(-(S(-1))(S(2)/3)a*(S(1)/3)f + b*(S(1)/3)e)) + a(e + fx)*(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/((S(-1))(S(1)/3)a*(S(1)/3)f + b*(S(1)/3)e))/(S(3)b(S(5)/3)(n + S(1))((S(-1))(S(1)/3)a*(S(1)/3)f + b*(S(1)/3)e)) + a(e + fx)*(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + b*(S(1)/3)e))/(S(3)b(S(5)/3)(n + S(1))(-a(S(1)/3)f + b*(S(1)/3)e)) + e*S(2)(e + fx)(n + S(1))/(bf*S(3)(n + S(1))) - S(2)e(e + fx)(n + S(2))/(bf*S(3)(n + S(2))) + (e + fx)(n + S(3))/(bf*S(3)(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)(e + fx)n/(a + bxS(3)), x), x, (S(-1))(S(2)/3)a(S(2)/3)(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), (S(-1))*(S(1)/3)b*(S(1)/3)(e + fx)/(a(S(1)/3)f + (S(-1))*(S(1)/3)b*(S(1)/3)e))/(S(3)b(S(4)/3)(n + S(1))(a(S(1)/3)f + (S(-1))*(S(1)/3)b*(S(1)/3)e)) + (S(-1))*(S(1)/3)a*(S(2)/3)(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), (S(-1))*(S(2)/3)b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + (S(-1))*(S(2)/3)b*(S(1)/3)e))/(S(3)b(S(4)/3)(n + S(1))(-a(S(1)/3)f + (S(-1))*(S(2)/3)b*(S(1)/3)e)) - a*(S(2)/3)(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + b*(S(1)/3)e))/(S(3)b(S(4)/3)(n + S(1))(-a(S(1)/3)f + b*(S(1)/3)e)) - e(e + fx)*(n + S(1))/(bf*S(2)(n + S(1))) + (e + fx)(n + S(2))/(bf*S(2)(n + S(2))), expand=True, _diff=True, _numerical=True) # difference in simplify assert rubi_test(rubi_integrate(x*S(3)(e + fx)n/(a + bxS(3)), x), x, -a(S(1)/3)(e + fx)*(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), (S(-1))*(S(1)/3)b*(S(1)/3)(e + fx)/(a(S(1)/3)f + (S(-1))*(S(1)/3)b*(S(1)/3)e))/(S(3)b(n + S(1))(a(S(1)/3)f + (S(-1))*(S(1)/3)b*(S(1)/3)e)) + a*(S(1)/3)(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), (S(-1))*(S(2)/3)b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + (S(-1))*(S(2)/3)b*(S(1)/3)e))/(S(3)b(n + S(1))(-a(S(1)/3)f + (S(-1))*(S(2)/3)b*(S(1)/3)e)) + a*(S(1)/3)(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + b*(S(1)/3)e))/(S(3)b(n + S(1))(-a(S(1)/3)f + b*(S(1)/3)e)) + (e + fx)(n + S(1))/(bf(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(e + fx)n/(a + bx*S(3)), x), x, -(e + fx)*(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/(-(S(-1))(S(2)/3)a*(S(1)/3)f + b*(S(1)/3)e))/(S(3)b(S(2)/3)(n + S(1))(-(S(-1))(S(2)/3)a*(S(1)/3)f + b*(S(1)/3)e)) - (e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/((S(-1))(S(1)/3)a*(S(1)/3)f + b*(S(1)/3)e))/(S(3)b(S(2)/3)(n + S(1))((S(-1))(S(1)/3)a*(S(1)/3)f + b*(S(1)/3)e)) - (e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + b*(S(1)/3)e))/(S(3)b(S(2)/3)(n + S(1))(-a(S(1)/3)f + b*(S(1)/3)e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(e + fx)*n/(a + bxS(3)), x), x, -(S(-1))(S(2)/3)(e + fx)*(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), (S(-1))*(S(1)/3)b*(S(1)/3)(e + fx)/(a(S(1)/3)f + (S(-1))*(S(1)/3)b*(S(1)/3)e))/(S(3)a(S(1)/3)b*(S(1)/3)(n + S(1))(a(S(1)/3)f + (S(-1))*(S(1)/3)b*(S(1)/3)e)) - (S(-1))*(S(1)/3)(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), (S(-1))*(S(2)/3)b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + (S(-1))*(S(2)/3)b*(S(1)/3)e))/(S(3)a(S(1)/3)b*(S(1)/3)(n + S(1))(-a(S(1)/3)f + (S(-1))*(S(2)/3)b*(S(1)/3)e)) + (e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + b*(S(1)/3)e))/(S(3)a(S(1)/3)b*(S(1)/3)(n + S(1))(-a(S(1)/3)f + b*(S(1)/3)e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((e + fx)n/(a + bx*S(3)), x), x, (e + fx)*(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), (S(-1))*(S(1)/3)b*(S(1)/3)(e + fx)/(a(S(1)/3)f + (S(-1))*(S(1)/3)b*(S(1)/3)e))/(S(3)a(S(2)/3)(n + S(1))(a(S(1)/3)f + (S(-1))*(S(1)/3)b*(S(1)/3)e)) - (e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), (S(-1))*(S(2)/3)b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + (S(-1))*(S(2)/3)b*(S(1)/3)e))/(S(3)a(S(2)/3)(n + S(1))(-a(S(1)/3)f + (S(-1))*(S(2)/3)b*(S(1)/3)e)) - (e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + b*(S(1)/3)e))/(S(3)a(S(2)/3)(n + S(1))(-a(S(1)/3)f + b*(S(1)/3)e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((e + fx)n/(x(a + bxS(3))), x), x, b(S(1)/3)(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/(-(S(-1))(S(2)/3)a*(S(1)/3)f + b*(S(1)/3)e))/(S(3)a(n + S(1))(-(S(-1))(S(2)/3)a*(S(1)/3)f + b*(S(1)/3)e)) + b*(S(1)/3)(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/((S(-1))(S(1)/3)a*(S(1)/3)f + b*(S(1)/3)e))/(S(3)a(n + S(1))((S(-1))(S(1)/3)a*(S(1)/3)f + b*(S(1)/3)e)) + b*(S(1)/3)(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + b*(S(1)/3)e))/(S(3)a(n + S(1))(-a(S(1)/3)f + b*(S(1)/3)e)) - (e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), S(1) + fx/e)/(ae(n + S(1))), expand=True, _diff=True, _numerical=True) # large time in rubi_test assert rubi_test(rubi_integrate((e + fx)n/(xS(2)(a + bx*S(3))), x), x, f(e + fx)(n + S(1))hyper((S(2), n + S(1)), (n + S(2),), S(1) + fx/e)/(ae*S(2)(n + S(1))) + (S(-1))*(S(2)/3)b*(S(2)/3)(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), (S(-1))*(S(1)/3)b*(S(1)/3)(e + fx)/(a(S(1)/3)f + (S(-1))*(S(1)/3)b*(S(1)/3)e))/(S(3)a(S(4)/3)(n + S(1))(a(S(1)/3)f + (S(-1))*(S(1)/3)b*(S(1)/3)e)) + (S(-1))*(S(1)/3)b*(S(2)/3)(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), (S(-1))*(S(2)/3)b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + (S(-1))*(S(2)/3)b*(S(1)/3)e))/(S(3)a(S(4)/3)(n + S(1))(-a(S(1)/3)f + (S(-1))*(S(2)/3)b*(S(1)/3)e)) - b*(S(2)/3)(e + fx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/3)(e + fx)/(-a(S(1)/3)f + b*(S(1)/3)e))/(S(3)a(S(4)/3)(n + S(1))(-a(S(1)/3)f + b*(S(1)/3)e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(c + dx)(n + S(1))/(a + bx*S(3)), x), x, -(c + dx)*(n + S(2))hyper((S(1), n + S(2)), (n + S(3),), b*(S(1)/3)(c + dx)/(-(S(-1))(S(2)/3)a*(S(1)/3)d + b*(S(1)/3)c))/(S(3)b(S(2)/3)(n + S(2))(-(S(-1))(S(2)/3)a*(S(1)/3)d + b*(S(1)/3)c)) - (c + dx)(n + S(2))hyper((S(1), n + S(2)), (n + S(3),), b*(S(1)/3)(c + dx)/((S(-1))(S(1)/3)a*(S(1)/3)d + b*(S(1)/3)c))/(S(3)b(S(2)/3)(n + S(2))((S(-1))(S(1)/3)a*(S(1)/3)d + b*(S(1)/3)c)) - (c + dx)(n + S(2))hyper((S(1), n + S(2)), (n + S(3),), b*(S(1)/3)(c + dx)/(-a(S(1)/3)d + b*(S(1)/3)c))/(S(3)b(S(2)/3)(n + S(2))(-a(S(1)/3)d + b*(S(1)/3)c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(c + dx)n/(a + bx*S(4)), x), x, -(c + dx)*(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/4)(c + dx)/(b(S(1)/4)c + Id(-a)*(S(1)/4)))/(S(4)b*(S(3)/4)(n + S(1))(b(S(1)/4)c + Id(-a)*(S(1)/4))) - (c + dx)*(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/4)(c + dx)/(b(S(1)/4)c - Id(-a)*(S(1)/4)))/(S(4)b*(S(3)/4)(n + S(1))(b(S(1)/4)c - Id(-a)*(S(1)/4))) - (c + dx)*(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/4)(c + dx)/(b(S(1)/4)c + d(-a)(S(1)/4)))/(S(4)b*(S(3)/4)(n + S(1))(b(S(1)/4)c + d(-a)(S(1)/4))) - (c + dx)*(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), b*(S(1)/4)(c + dx)/(b(S(1)/4)c - d(-a)(S(1)/4)))/(S(4)b*(S(3)/4)(n + S(1))(b(S(1)/4)c - d(-a)(S(1)/4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(c + dx)(n + S(1))/(a + bx*S(4)), x), x, -(c + dx)*(n + S(2))hyper((S(1), n + S(2)), (n + S(3),), b*(S(1)/4)(c + dx)/(b(S(1)/4)c + Id(-a)*(S(1)/4)))/(S(4)b*(S(3)/4)(n + S(2))(b(S(1)/4)c + Id(-a)*(S(1)/4))) - (c + dx)*(n + S(2))hyper((S(1), n + S(2)), (n + S(3),), b*(S(1)/4)(c + dx)/(b(S(1)/4)c - Id(-a)*(S(1)/4)))/(S(4)b*(S(3)/4)(n + S(2))(b(S(1)/4)c - Id(-a)*(S(1)/4))) - (c + dx)*(n + S(2))hyper((S(1), n + S(2)), (n + S(3),), b*(S(1)/4)(c + dx)/(b(S(1)/4)c + d(-a)(S(1)/4)))/(S(4)b*(S(3)/4)(n + S(2))(b(S(1)/4)c + d(-a)(S(1)/4))) - (c + dx)*(n + S(2))hyper((S(1), n + S(2)), (n + S(3),), b*(S(1)/4)(c + dx)/(b(S(1)/4)c - d(-a)(S(1)/4)))/(S(4)b*(S(3)/4)(n + S(2))(b(S(1)/4)c - d(-a)(S(1)/4))), expand=True, _diff=True, _numerical=True) # large time in rubi_test assert rubi_test(rubi_integrate(S(1)/(sqrt(a + bx*S(4))(c + dx + ex*S(2))), x), x, sqrt(S(2))e*S(2)atanh(sqrt(S(2))(S(4)aeS(2) + bx*S(2)(d + sqrt(-S(4)ce + dS(2)))S(2))/(S(4)sqrt(a + bx*S(4))sqrt(S(2)ae*S(4) + S(2)bcS(2)e*S(2) - S(4)bcd*S(2)e + bdS(4) + bdsqrt(-S(4)ce + dS(2))(-S(2)ce + d*S(2)))))/(S(2)sqrt(-S(4)ce + d*S(2))sqrt(S(2)ae*S(4) + S(2)bcS(2)e*S(2) - S(4)bcd*S(2)e + bdS(4) + bdsqrt(-S(4)ce + dS(2))(-S(2)ce + d*S(2)))) - sqrt(S(2))e*S(2)atanh(sqrt(S(2))(S(4)aeS(2) + bx*S(2)(d - sqrt(-S(4)ce + dS(2)))S(2))/(S(4)sqrt(a + bx*S(4))sqrt(S(2)ae*S(4) + S(2)bcS(2)e*S(2) - S(4)bcd*S(2)e + bdS(4) - bdsqrt(-S(4)ce + dS(2))(-S(2)ce + d*S(2)))))/(S(2)sqrt(-S(4)ce + d*S(2))sqrt(S(2)ae*S(4) + S(2)bcS(2)e*S(2) - S(4)bcd*S(2)e + bdS(4) - bdsqrt(-S(4)ce + dS(2))(-S(2)ce + d*S(2)))) - S(2)eatan(xsqrt(-(S(16)ae*S(4) + b(d + sqrt(-S(4)ce + dS(2)))S(4))/(e*S(2)(d + sqrt(-S(4)ce + dS(2)))S(2)))/(S(2)sqrt(a + bx*S(4))))/(sqrt(-(S(16)aeS(4) + b(d + sqrt(-S(4)ce + dS(2)))S(4))/(e*S(2)(d + sqrt(-S(4)ce + dS(2)))S(2)))(d + sqrt(-S(4)ce + dS(2)))sqrt(-S(4)ce + d*S(2))) + S(2)eatan(xsqrt(-(S(16)ae*S(4) + b(d - sqrt(-S(4)ce + dS(2)))S(4))/(e*S(2)(d - sqrt(-S(4)ce + dS(2)))S(2)))/(S(2)sqrt(a + bx*S(4))))/(sqrt(-(S(16)aeS(4) + b(d - sqrt(-S(4)ce + dS(2)))S(4))/(e*S(2)(d - sqrt(-S(4)ce + dS(2)))S(2)))(d - sqrt(-S(4)ce + dS(2)))sqrt(-S(4)ce + d*S(2))) - esqrt((a + bxS(4))/(sqrt(a) + sqrt(b)xS(2))S(2))(sqrt(a) + sqrt(b)x*S(2))(S(4)eS(2) - sqrt(b)(d + sqrt(-S(4)ce + dS(2)))S(2)/sqrt(a))elliptic_pi(sqrt(a)(S(4)eS(2) + sqrt(b)(d + sqrt(-S(4)ce + dS(2)))S(2)/sqrt(a))*S(2)/(S(16)sqrt(b)eS(2)(d + sqrt(-S(4)ce + dS(2)))S(2)), S(2)atan(b(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(2)a*(S(1)/4)b*(S(1)/4)sqrt(a + bxS(4))(d + sqrt(-S(4)ce + d*S(2)))(S(4)eS(2) + sqrt(b)(d + sqrt(-S(4)ce + dS(2)))S(2)/sqrt(a))sqrt(-S(4)ce + dS(2))) + esqrt((a + bxS(4))/(sqrt(a) + sqrt(b)xS(2))S(2))(sqrt(a) + sqrt(b)x*S(2))(S(4)eS(2) - sqrt(b)(d - sqrt(-S(4)ce + dS(2)))S(2)/sqrt(a))elliptic_pi(sqrt(a)(S(4)eS(2) + sqrt(b)(d - sqrt(-S(4)ce + dS(2)))S(2)/sqrt(a))*S(2)/(S(16)sqrt(b)eS(2)(d - sqrt(-S(4)ce + dS(2)))S(2)), S(2)atan(b(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(2)a*(S(1)/4)b*(S(1)/4)sqrt(a + bxS(4))(d - sqrt(-S(4)ce + d*S(2)))(S(4)eS(2) + sqrt(b)(d - sqrt(-S(4)ce + dS(2)))S(2)/sqrt(a))sqrt(-S(4)ce + dS(2))) + b(S(1)/4)esqrt((a + bx*S(4))/(sqrt(a) + sqrt(b)xS(2))S(2))(sqrt(a) + sqrt(b)x*S(2))(d - sqrt(-S(4)ce + d*S(2)))elliptic_f(S(2)atan(b(S(1)/4)x/a(S(1)/4)), S(1)/2)/(a(S(3)/4)sqrt(a + bx*S(4))(S(4)eS(2) + sqrt(b)(d - sqrt(-S(4)ce + dS(2)))S(2)/sqrt(a))sqrt(-S(4)ce + dS(2))) - b(S(1)/4)esqrt((a + bx*S(4))/(sqrt(a) + sqrt(b)xS(2))S(2))(sqrt(a) + sqrt(b)x*S(2))(d + sqrt(-S(4)ce + d*S(2)))elliptic_f(S(2)atan(b(S(1)/4)x/a(S(1)/4)), S(1)/2)/(a(S(3)/4)sqrt(a + bx*S(4))(S(4)eS(2) + sqrt(b)(d + sqrt(-S(4)ce + dS(2)))S(2)/sqrt(a))sqrt(-S(4)ce + dS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(1)/n))p, x), x, x(cxn)(-S(1)/n)(a + b(cxn)(S(1)/n))*(p + S(1))/(b(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(1)/n))*S(3), x), x, x(cxn)(-S(1)/n)(a + b(cxn)(S(1)/n))*S(4)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(1)/n))*S(2), x), x, x(cxn)(-S(1)/n)(a + b(cxn)(S(1)/n))*S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a + b(cxn)(S(1)/n), x), x, ax + bx(cxn)(S(1)/n)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + b(cxn)(S(1)/n)), x), x, x(cxn)(-S(1)/n)log(a + b(cxn)(S(1)/n))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(1)/n))(S(-2)), x), x, x/(aS(2) + ab(cxn)(S(1)/n)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + b(cxn)(S(1)/n))*(S(-2)), x), x, -x(cxn)(-S(1)/n)/(b(a + b(cxn)(S(1)/n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(1)/n))*(S(-3)), x), x, -x(cxn)(-S(1)/n)/(S(2)b(a + b(cxn)(S(1)/n))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(2)/n))S(3), x), x, aS(3)x + a*S(2)bx(cxn)(S(2)/n) + S(3)abS(2)x(cxn)(S(4)/n)/S(5) + b*S(3)x(cxn)(S(6)/n)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(2)/n))S(2), x), x, aS(2)x + S(2)abx(cxn)(S(2)/n)/S(3) + b*S(2)x(cxn)(S(4)/n)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a + b(cxn)(S(2)/n), x), x, ax + bx(cxn)(S(2)/n)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + b(cxn)(S(2)/n)), x), x, x(cxn)(-S(1)/n)atan(sqrt(b)(cxn)(S(1)/n)/sqrt(a))/(sqrt(a)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(2)/n))*(S(-2)), x), x, x/(S(2)a(a + b(cxn)(S(2)/n))) + x(cxn)(-S(1)/n)atan(sqrt(b)(cxn)(S(1)/n)/sqrt(a))/(S(2)a(S(3)/2)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(2)/n))*(S(-3)), x), x, x/(S(4)a(a + b(cxn)(S(2)/n))S(2)) + S(3)x/(S(8)aS(2)(a + b(cxn)(S(2)/n))) + S(3)x(cxn)(-S(1)/n)atan(sqrt(b)(cxn)(S(1)/n)/sqrt(a))/(S(8)a(S(5)/2)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(3)/n))S(3), x), x, aS(3)x + S(3)a*S(2)bx(cxn)(S(3)/n)/S(4) + S(3)abS(2)x(cxn)(S(6)/n)/S(7) + b*S(3)x(cxn)(S(9)/n)/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(3)/n))S(2), x), x, aS(2)x + abx(cxn)(S(3)/n)/S(2) + bS(2)x(cxn)(S(6)/n)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a + b(cxn)(S(3)/n), x), x, ax + bx(cxn)(S(3)/n)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + b(cxn)(S(3)/n)), x), x, x(cxn)(-S(1)/n)log(a(S(1)/3) + b(S(1)/3)(cxn)(S(1)/n))/(S(3)a*(S(2)/3)b*(S(1)/3)) - x(cxn)(-S(1)/n)log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)(cxn)(S(1)/n) + b(S(2)/3)(cxn)(S(2)/n))/(S(6)a*(S(2)/3)b*(S(1)/3)) - sqrt(S(3))x(cxn)(-S(1)/n)atan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)(cxn)(S(1)/n))/(S(3)a*(S(1)/3)))/(S(3)a*(S(2)/3)b*(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(3)/n))(S(-2)), x), x, x/(S(3)a(a + b(cxn)(S(3)/n))) + S(2)x(cxn)(-S(1)/n)log(a(S(1)/3) + b(S(1)/3)(cxn)(S(1)/n))/(S(9)a*(S(5)/3)b*(S(1)/3)) - x(cxn)(-S(1)/n)log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)(cxn)(S(1)/n) + b(S(2)/3)(cxn)(S(2)/n))/(S(9)a*(S(5)/3)b*(S(1)/3)) - S(2)sqrt(S(3))x(cxn)(-S(1)/n)atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)(cxn)(S(1)/n))/(S(3)a*(S(1)/3)))/(S(9)a*(S(5)/3)b*(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(3)/n))(S(-3)), x), x, x/(S(6)a(a + b(cxn)(S(3)/n))S(2)) + S(5)x/(S(18)aS(2)(a + b(cxn)(S(3)/n))) + S(5)x(cxn)(-S(1)/n)log(a(S(1)/3) + b(S(1)/3)(cxn)(S(1)/n))/(S(27)a(S(8)/3)b*(S(1)/3)) - S(5)x(cxn)(-S(1)/n)log(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)(cxn)(S(1)/n) + b(S(2)/3)(cxn)(S(2)/n))/(S(54)a*(S(8)/3)b*(S(1)/3)) - S(5)sqrt(S(3))x(cxn)(-S(1)/n)atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)(cxn)(S(1)/n))/(S(3)a*(S(1)/3)))/(S(27)a*(S(8)/3)b(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((xS(3))*(S(2)/3) + S(1)), x), x, xatan((xS(3))(S(1)/3))/(xS(3))(S(1)/3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((xS(2))(S(3)/2) + S(1)), x), x, xlog(sqrt(xS(2)) + S(1))/(S(3)sqrt(x*S(2))) - xlog(xS(2) - sqrt(xS(2)) + S(1))/(S(6)sqrt(xS(2))) - sqrt(S(3))xatan(sqrt(S(3))(-S(2)sqrt(xS(2)) + S(1))/S(3))/(S(3)sqrt(x*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(4)sqrt(x*S(4)) + S(1)), x), x, xatan(S(2)(xS(4))(S(1)/4))/(S(2)(xS(4))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(-S(4)sqrt(xS(4)) + S(1)), x), x, xatanh(S(2)(xS(4))(S(1)/4))/(S(2)(xS(4))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(4)(xS(6))(S(1)/3) + S(1)), x), x, xatan(S(2)(xS(6))(S(1)/6))/(S(2)(xS(6))(S(1)/6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(-S(4)(xS(6))(S(1)/3) + S(1)), x), x, xatanh(S(2)(xS(6))(S(1)/6))/(S(2)(xS(6))(S(1)/6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(4)(x(S(2)n))*(S(1)/n) + S(1)), x), x, x(x*(S(2)n))*(-S(1)/(S(2)n))atan(S(2)(x*(S(2)n))*(S(1)/(S(2)n)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(-S(4)(x(S(2)n))*(S(1)/n) + S(1)), x), x, x(x*(S(2)n))*(-S(1)/(S(2)n))atanh(S(2)(x*(S(2)n))*(S(1)/(S(2)n)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/(a + b(cxn)(S(1)/n)), x), x, -aS(3)x*S(4)(cxn)(-S(4)/n)log(a + b(cxn)(S(1)/n))/bS(4) + aS(2)xS(4)(cxn)(-S(3)/n)/bS(3) - ax*S(4)(cxn)(-S(2)/n)/(S(2)bS(2)) + xS(4)(cxn)(-S(1)/n)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + b(cxn)(S(1)/n)), x), x, aS(2)x*S(3)(cxn)(-S(3)/n)log(a + b(cxn)(S(1)/n))/b*S(3) - ax*S(3)(cxn)(-S(2)/n)/bS(2) + xS(3)(cxn)(-S(1)/n)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + b(cxn)(S(1)/n)), x), x, -axS(2)(cxn)(-S(2)/n)log(a + b(cxn)(S(1)/n))/bS(2) + xS(2)(cxn)(-S(1)/n)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + b(cxn)(S(1)/n)), x), x, x(cxn)(-S(1)/n)log(a + b(cxn)(S(1)/n))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + b(cxn)(S(1)/n))), x), x, log(x)/a - log(a + b(cxn)(S(1)/n))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(a + b(cxn)(S(1)/n))), x), x, -S(1)/(ax) - b(cxn)(S(1)/n)log(x)/(a*S(2)x) + b(cxn)(S(1)/n)log(a + b(cxn)(S(1)/n))/(aS(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)(a + b(cxn)(S(1)/n))), x), x, -S(1)/(S(2)ax*S(2)) + b(cxn)(S(1)/n)/(aS(2)xS(2)) + bS(2)(cxn)(S(2)/n)log(x)/(aS(3)xS(2)) - bS(2)(cxn)(S(2)/n)log(a + b(cxn)(S(1)/n))/(aS(3)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + b(cxn)(S(1)/n))S(2), x), x, aS(3)xS(4)(cxn)(-S(4)/n)/(bS(4)(a + b(cxn)(S(1)/n))) + S(3)aS(2)x*S(4)(cxn)(-S(4)/n)log(a + b(cxn)(S(1)/n))/b*S(4) - S(2)axS(4)(cxn)(-S(3)/n)/bS(3) + xS(4)(cxn)(-S(2)/n)/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + b(cxn)(S(1)/n))S(2), x), x, -aS(2)xS(3)(cxn)(-S(3)/n)/(bS(3)(a + b(cxn)(S(1)/n))) - S(2)ax*S(3)(cxn)(-S(3)/n)log(a + b(cxn)(S(1)/n))/bS(3) + xS(3)(cxn)(-S(2)/n)/b*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + b(cxn)(S(1)/n))S(2), x), x, ax*S(2)(cxn)(-S(2)/n)/(bS(2)(a + b(cxn)(S(1)/n))) + x*S(2)(cxn)(-S(2)/n)log(a + b(cxn)(S(1)/n))/b*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(1)/n))(S(-2)), x), x, -x(cxn)(-S(1)/n)/(b(a + b(cxn)(S(1)/n))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + b(cxn)(S(1)/n))S(2)), x), x, S(1)/(a(a + b(cxn)(S(1)/n))) + log(x)/a*S(2) - log(a + b(cxn)(S(1)/n))/aS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + b(cxn)(S(1)/n))*S(2)), x), x, -b(cxn)(S(1)/n)/(aS(2)x(a + b(cxn)(S(1)/n))) - S(1)/(aS(2)x) - S(2)b(cxn)(S(1)/n)log(x)/(a*S(3)x) + S(2)b(cxn)(S(1)/n)log(a + b(cxn)(S(1)/n))/(a*S(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)(a + b(cxn)(S(1)/n))*S(2)), x), x, -S(1)/(S(2)a*S(2)xS(2)) + bS(2)(cxn)(S(2)/n)/(a*S(3)x*S(2)(a + b(cxn)(S(1)/n))) + S(2)b(cxn)(S(1)/n)/(aS(3)x*S(2)) + S(3)b*S(2)(cxn)(S(2)/n)log(x)/(a*S(4)x*S(2)) - S(3)b*S(2)(cxn)(S(2)/n)log(a + b(cxn)(S(1)/n))/(a*S(4)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a + b(cxn)(S(1)/n))p, x), x, -aS(3)x*S(4)(cxn)(-S(4)/n)(a + b(cxn)(S(1)/n))(p + S(1))/(bS(4)(p + S(1))) + S(3)a*S(2)x*S(4)(cxn)(-S(4)/n)(a + b(cxn)(S(1)/n))(p + S(2))/(bS(4)(p + S(2))) - S(3)axS(4)(cxn)(-S(4)/n)(a + b(cxn)(S(1)/n))(p + S(3))/(bS(4)(p + S(3))) + xS(4)(cxn)(-S(4)/n)(a + b(cxn)(S(1)/n))(p + S(4))/(bS(4)(p + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + b(cxn)(S(1)/n))p, x), x, aS(2)xS(3)(cxn)(-S(3)/n)(a + b(cxn)(S(1)/n))(p + S(1))/(bS(3)(p + S(1))) - S(2)axS(3)(cxn)(-S(3)/n)(a + b(cxn)(S(1)/n))(p + S(2))/(bS(3)(p + S(2))) + xS(3)(cxn)(-S(3)/n)(a + b(cxn)(S(1)/n))(p + S(3))/(bS(3)(p + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + b(cxn)(S(1)/n))*p, x), x, -ax*S(2)(cxn)(-S(2)/n)(a + b(cxn)(S(1)/n))(p + S(1))/(bS(2)(p + S(1))) + xS(2)(cxn)(-S(2)/n)(a + b(cxn)(S(1)/n))(p + S(2))/(bS(2)(p + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(1)/n))p, x), x, x(cxn)(-S(1)/n)(a + b(cxn)(S(1)/n))*(p + S(1))/(b(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cxn)(S(1)/n))*p/x, x), x, -(a + b(cxn)(S(1)/n))(p + S(1))hyper((S(1), p + S(1)), (p + S(2),), S(1) + b(cxn)(S(1)/n)/a)/(a(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((xn)(S(1)/n) + S(1))S(2), x), x, xS(2)(xn)(-S(2)/n)log((xn)(S(1)/n) + S(1)) + xS(2)(xn)(-S(2)/n)/((xn)(S(1)/n) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*m(a(bxn)p)q, x), x, x(m + S(1))(a(bxn)p)q/(m + npq + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a(bxn)p)q, x), x, xS(3)(a(bxn)p)q/(npq + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a(bxn)p)q, x), x, xS(2)(a(bxn)p)q/(npq + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a(bxn)p)q, x), x, x(a(bxn)p)*q/(npq + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a(bxn)p)q/x, x), x, (a(bxn)p)q/(npq), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a(bxn)p)q/xS(2), x), x, -(a(bxn)p)q/(x(-npq + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a(bxn)p)q/xS(3), x), x, -(a(bxn)p)q/(xS(2)(-npq + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a(bxm)n)*(-S(1)/(mn)), x), x, x*S(3)(a(bxm)n)*(-S(1)/(mn))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a(bxm)n)(-S(1)/(mn)), x), x, x*S(2)(a(bxm)n)*(-S(1)/(mn)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a(bxm)n)*(-S(1)/(mn)), x), x, x(a(bxm)n)(-S(1)/(mn))log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a(bxm)n)(-S(1)/(mn))/x, x), x, -(a(bxm)n)*(-S(1)/(mn)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a(bxm)n)*(-S(1)/(mn))/x*S(2), x), x, -(a(bxm)n)(-S(1)/(mn))/(S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-npq + S(2))(a(bxn)p)q, x), x, x(-npq + S(3))(a(bxn)p)q/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-npq + S(1))(a(bxn)p)q, x), x, x(-npq + S(2))(a(bxn)p)q/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-npq)(a(bxn)p)q, x), x, x(-npq + S(1))(a(bxn)p)q, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-npq + S(-1))(a(bxn)p)q, x), x, x(-npq)(a(bxn)p)qlog(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(-npq + S(-2))(a(bxn)p)q, x), x, -x(-npq + S(-1))(a(bxn)p)q, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(cxS(2))(a + bx)n, x), x, -aS(3)sqrt(cxS(2))(a + bx)(n + S(1))/(bS(4)x(n + S(1))) + S(3)a*S(2)sqrt(cxS(2))(a + bx)(n + S(2))/(bS(4)x(n + S(2))) - S(3)asqrt(cx*S(2))(a + bx)(n + S(3))/(bS(4)x(n + S(3))) + sqrt(cx*S(2))(a + bx)(n + S(4))/(bS(4)x(n + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(cxS(2))(a + bx)n, x), x, aS(2)sqrt(cxS(2))(a + bx)(n + S(1))/(bS(3)x(n + S(1))) - S(2)asqrt(cx*S(2))(a + bx)(n + S(2))/(bS(3)x(n + S(2))) + sqrt(cx*S(2))(a + bx)(n + S(3))/(bS(3)x(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)n, x), x, -asqrt(cxS(2))(a + bx)(n + S(1))/(bS(2)x(n + S(1))) + sqrt(cx*S(2))(a + bx)(n + S(2))/(bS(2)x(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)n/x, x), x, sqrt(cx*S(2))(a + bx)(n + S(1))/(bx(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)n/xS(2), x), x, -sqrt(cx*S(2))(a + bx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), S(1) + bx/a)/(ax(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)n/xS(3), x), x, bsqrt(cxS(2))(a + bx)(n + S(1))hyper((S(2), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(2)x(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)n/xS(4), x), x, -bS(2)sqrt(cxS(2))(a + bx)(n + S(1))hyper((S(3), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(3)x(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(cxS(2))(S(3)/2)(a + bx)n, x), x, aS(4)csqrt(cx*S(2))(a + bx)(n + S(1))/(bS(5)x(n + S(1))) - S(4)a*S(3)csqrt(cx*S(2))(a + bx)(n + S(2))/(bS(5)x(n + S(2))) + S(6)a*S(2)csqrt(cx*S(2))(a + bx)(n + S(3))/(bS(5)x(n + S(3))) - S(4)acsqrt(cxS(2))(a + bx)(n + S(4))/(bS(5)x(n + S(4))) + csqrt(cxS(2))(a + bx)(n + S(5))/(bS(5)x(n + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)n, x), x, -aS(3)csqrt(cxS(2))(a + bx)(n + S(1))/(bS(4)x(n + S(1))) + S(3)a*S(2)csqrt(cx*S(2))(a + bx)(n + S(2))/(bS(4)x(n + S(2))) - S(3)acsqrt(cxS(2))(a + bx)(n + S(3))/(bS(4)x(n + S(3))) + csqrt(cxS(2))(a + bx)(n + S(4))/(bS(4)x(n + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)n/x, x), x, aS(2)csqrt(cxS(2))(a + bx)(n + S(1))/(bS(3)x(n + S(1))) - S(2)acsqrt(cxS(2))(a + bx)(n + S(2))/(bS(3)x(n + S(2))) + csqrt(cxS(2))(a + bx)(n + S(3))/(bS(3)x(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)n/xS(2), x), x, -acsqrt(cxS(2))(a + bx)(n + S(1))/(bS(2)x(n + S(1))) + csqrt(cxS(2))(a + bx)(n + S(2))/(bS(2)x(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)n/xS(3), x), x, csqrt(cx*S(2))(a + bx)(n + S(1))/(bx(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)n/xS(4), x), x, -csqrt(cx*S(2))(a + bx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), S(1) + bx/a)/(ax(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)n/xS(5), x), x, bcsqrt(cxS(2))(a + bx)(n + S(1))hyper((S(2), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(2)x(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)n/xS(6), x), x, -b*S(2)csqrt(cx*S(2))(a + bx)(n + S(1))hyper((S(3), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(3)x(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)n, x), x, -aS(5)cS(2)sqrt(cxS(2))(a + bx)(n + S(1))/(bS(6)x(n + S(1))) + S(5)a*S(4)c*S(2)sqrt(cxS(2))(a + bx)(n + S(2))/(bS(6)x(n + S(2))) - S(10)a*S(3)c*S(2)sqrt(cxS(2))(a + bx)(n + S(3))/(bS(6)x(n + S(3))) + S(10)a*S(2)c*S(2)sqrt(cxS(2))(a + bx)(n + S(4))/(bS(6)x(n + S(4))) - S(5)acS(2)sqrt(cxS(2))(a + bx)(n + S(5))/(bS(6)x(n + S(5))) + cS(2)sqrt(cxS(2))(a + bx)(n + S(6))/(bS(6)x(n + S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)n/x, x), x, aS(4)cS(2)sqrt(cxS(2))(a + bx)(n + S(1))/(bS(5)x(n + S(1))) - S(4)a*S(3)c*S(2)sqrt(cxS(2))(a + bx)(n + S(2))/(bS(5)x(n + S(2))) + S(6)a*S(2)c*S(2)sqrt(cxS(2))(a + bx)(n + S(3))/(bS(5)x(n + S(3))) - S(4)acS(2)sqrt(cxS(2))(a + bx)(n + S(4))/(bS(5)x(n + S(4))) + cS(2)sqrt(cxS(2))(a + bx)(n + S(5))/(bS(5)x(n + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)n/xS(2), x), x, -a*S(3)c*S(2)sqrt(cxS(2))(a + bx)(n + S(1))/(bS(4)x(n + S(1))) + S(3)a*S(2)c*S(2)sqrt(cxS(2))(a + bx)(n + S(2))/(bS(4)x(n + S(2))) - S(3)acS(2)sqrt(cxS(2))(a + bx)(n + S(3))/(bS(4)x(n + S(3))) + cS(2)sqrt(cxS(2))(a + bx)(n + S(4))/(bS(4)x(n + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)n/xS(3), x), x, a*S(2)c*S(2)sqrt(cxS(2))(a + bx)(n + S(1))/(bS(3)x(n + S(1))) - S(2)acS(2)sqrt(cxS(2))(a + bx)(n + S(2))/(bS(3)x(n + S(2))) + cS(2)sqrt(cxS(2))(a + bx)(n + S(3))/(bS(3)x(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)n/xS(4), x), x, -acS(2)sqrt(cxS(2))(a + bx)(n + S(1))/(bS(2)x(n + S(1))) + cS(2)sqrt(cxS(2))(a + bx)(n + S(2))/(bS(2)x(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)n/xS(5), x), x, c*S(2)sqrt(cxS(2))(a + bx)(n + S(1))/(bx(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)n/xS(6), x), x, -c*S(2)sqrt(cxS(2))(a + bx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), S(1) + bx/a)/(ax(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)n/xS(7), x), x, bcS(2)sqrt(cxS(2))(a + bx)(n + S(1))hyper((S(2), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(2)x(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)(a + bx)n/sqrt(cxS(2)), x), x, -aS(3)x(a + bx)(n + S(1))/(bS(4)sqrt(cxS(2))(n + S(1))) + S(3)aS(2)x(a + bx)(n + S(2))/(bS(4)sqrt(cx*S(2))(n + S(2))) - S(3)ax(a + bx)(n + S(3))/(bS(4)sqrt(cx*S(2))(n + S(3))) + x(a + bx)(n + S(4))/(bS(4)sqrt(cx*S(2))(n + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a + bx)n/sqrt(cxS(2)), x), x, aS(2)x(a + bx)(n + S(1))/(bS(3)sqrt(cxS(2))(n + S(1))) - S(2)ax(a + bx)(n + S(2))/(bS(3)sqrt(cx*S(2))(n + S(2))) + x(a + bx)(n + S(3))/(bS(3)sqrt(cx*S(2))(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(a + bx)n/sqrt(cx*S(2)), x), x, -asqrt(cxS(2))(a + bx)(n + S(1))/(bS(2)cx(n + S(1))) + sqrt(cxS(2))(a + bx)(n + S(2))/(bS(2)cx(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)*n/sqrt(cx*S(2)), x), x, x(a + bx)(n + S(1))/(bsqrt(cxS(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)n/sqrt(cx*S(2)), x), x, -x(a + bx)(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), S(1) + bx/a)/(asqrt(cxS(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)n/(xsqrt(cxS(2))), x), x, bx(a + bx)*(n + S(1))hyper((S(2), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(2)sqrt(cxS(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)n/(xS(2)sqrt(cxS(2))), x), x, -bS(2)x(a + bx)*(n + S(1))hyper((S(3), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(3)sqrt(cxS(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)n/(xS(3)sqrt(cxS(2))), x), x, bS(3)x(a + bx)*(n + S(1))hyper((S(4), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(4)sqrt(cxS(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(6)(a + bx)n/(cxS(2))(S(3)/2), x), x, -a*S(3)x(a + bx)(n + S(1))/(bS(4)csqrt(cxS(2))(n + S(1))) + S(3)aS(2)x(a + bx)(n + S(2))/(bS(4)csqrt(cxS(2))(n + S(2))) - S(3)ax(a + bx)(n + S(3))/(bS(4)csqrt(cxS(2))(n + S(3))) + x(a + bx)(n + S(4))/(bS(4)csqrt(cxS(2))(n + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)(a + bx)n/(cxS(2))(S(3)/2), x), x, a*S(2)x(a + bx)(n + S(1))/(bS(3)csqrt(cxS(2))(n + S(1))) - S(2)ax(a + bx)(n + S(2))/(bS(3)csqrt(cxS(2))(n + S(2))) + x(a + bx)(n + S(3))/(bS(3)csqrt(cxS(2))(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)(a + bx)n/(cxS(2))(S(3)/2), x), x, -ax(a + bx)(n + S(1))/(bS(2)csqrt(cx*S(2))(n + S(1))) + x(a + bx)(n + S(2))/(bS(2)csqrt(cxS(2))(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a + bx)n/(cxS(2))(S(3)/2), x), x, x(a + bx)*(n + S(1))/(bcsqrt(cx*S(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(a + bx)n/(cxS(2))(S(3)/2), x), x, -x(a + bx)*(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), S(1) + bx/a)/(acsqrt(cx*S(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)*n/(cxS(2))(S(3)/2), x), x, bx(a + bx)(n + S(1))hyper((S(2), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(2)csqrt(cx*S(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)n/(cxS(2))(S(3)/2), x), x, -b*S(2)x(a + bx)*(n + S(1))hyper((S(3), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(3)csqrt(cx*S(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)n/(x(cxS(2))(S(3)/2)), x), x, bS(3)x(a + bx)*(n + S(1))hyper((S(4), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(4)csqrt(cx*S(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(8)(a + bx)n/(cxS(2))(S(5)/2), x), x, -a*S(3)x(a + bx)(n + S(1))/(bS(4)cS(2)sqrt(cxS(2))(n + S(1))) + S(3)aS(2)x(a + bx)(n + S(2))/(bS(4)cS(2)sqrt(cxS(2))(n + S(2))) - S(3)ax(a + bx)(n + S(3))/(bS(4)cS(2)sqrt(cxS(2))(n + S(3))) + x(a + bx)(n + S(4))/(bS(4)cS(2)sqrt(cxS(2))(n + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(7)(a + bx)n/(cxS(2))(S(5)/2), x), x, a*S(2)x(a + bx)(n + S(1))/(bS(3)cS(2)sqrt(cxS(2))(n + S(1))) - S(2)ax(a + bx)(n + S(2))/(bS(3)cS(2)sqrt(cxS(2))(n + S(2))) + x(a + bx)(n + S(3))/(bS(3)cS(2)sqrt(cxS(2))(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(6)(a + bx)n/(cxS(2))(S(5)/2), x), x, -ax(a + bx)(n + S(1))/(bS(2)c*S(2)sqrt(cxS(2))(n + S(1))) + x(a + bx)(n + S(2))/(bS(2)cS(2)sqrt(cxS(2))(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)(a + bx)n/(cxS(2))(S(5)/2), x), x, x(a + bx)*(n + S(1))/(bc*S(2)sqrt(cxS(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)(a + bx)n/(cxS(2))(S(5)/2), x), x, -x(a + bx)*(n + S(1))hyper((S(1), n + S(1)), (n + S(2),), S(1) + bx/a)/(ac*S(2)sqrt(cxS(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a + bx)n/(cxS(2))(S(5)/2), x), x, bx(a + bx)(n + S(1))hyper((S(2), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(2)c*S(2)sqrt(cxS(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(a + bx)n/(cxS(2))(S(5)/2), x), x, -b*S(2)x(a + bx)*(n + S(1))hyper((S(3), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(3)c*S(2)sqrt(cxS(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)*n/(cxS(2))(S(5)/2), x), x, b*S(3)x(a + bx)*(n + S(1))hyper((S(4), n + S(1)), (n + S(2),), S(1) + bx/a)/(aS(4)c*S(2)sqrt(cxS(2))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*msqrt(cxS(2))(a + bx), x), x, ax*(m + S(1))sqrt(cxS(2))/(m + S(2)) + bx*(m + S(2))sqrt(cxS(2))/(m + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(cxS(2))(a + bx), x), x, ax*S(4)sqrt(cxS(2))/S(5) + bx*S(5)sqrt(cxS(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(cxS(2))(a + bx), x), x, ax*S(3)sqrt(cxS(2))/S(4) + bx*S(4)sqrt(cxS(2))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(cxS(2))(a + bx), x), x, ax*S(2)sqrt(cxS(2))/S(3) + bx*S(3)sqrt(cxS(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx), x), x, axsqrt(cx*S(2))/S(2) + bx*S(2)sqrt(cxS(2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)/x, x), x, asqrt(cxS(2)) + bxsqrt(cx*S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)/xS(2), x), x, asqrt(cxS(2))log(x)/x + bsqrt(cx*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)/xS(3), x), x, -asqrt(cxS(2))/xS(2) + bsqrt(cxS(2))log(x)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cxS(2))(a + bx)/xS(4), x), x, -asqrt(cxS(2))/(S(2)x*S(3)) - bsqrt(cxS(2))/xS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(cxS(2))(S(3)/2)(a + bx), x), x, acx(m + S(3))sqrt(cxS(2))/(m + S(4)) + bcx(m + S(4))sqrt(cxS(2))/(m + S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(cxS(2))(S(3)/2)(a + bx), x), x, acxS(6)sqrt(cxS(2))/S(7) + bcxS(7)sqrt(cxS(2))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(cxS(2))(S(3)/2)(a + bx), x), x, acxS(5)sqrt(cxS(2))/S(6) + bcxS(6)sqrt(cxS(2))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(cxS(2))(S(3)/2)(a + bx), x), x, acxS(4)sqrt(cxS(2))/S(5) + bcxS(5)sqrt(cxS(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx), x), x, acx*S(3)sqrt(cxS(2))/S(4) + bcxS(4)sqrt(cxS(2))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)/x, x), x, acx*S(2)sqrt(cxS(2))/S(3) + bcxS(3)sqrt(cxS(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)/x*S(2), x), x, acxsqrt(cxS(2))/S(2) + bcxS(2)sqrt(cxS(2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)/x*S(3), x), x, acsqrt(cx*S(2)) + bcxsqrt(cxS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)/x*S(4), x), x, acsqrt(cx*S(2))log(x)/x + bcsqrt(cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(cxS(2))(S(5)/2)(a + bx), x), x, ac*S(2)x*(m + S(5))sqrt(cxS(2))/(m + S(6)) + bc*S(2)x*(m + S(6))sqrt(cxS(2))/(m + S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(cxS(2))(S(5)/2)(a + bx), x), x, ac*S(2)x*S(8)sqrt(cxS(2))/S(9) + bc*S(2)x*S(9)sqrt(cxS(2))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(cxS(2))(S(5)/2)(a + bx), x), x, ac*S(2)x*S(7)sqrt(cxS(2))/S(8) + bc*S(2)x*S(8)sqrt(cxS(2))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(cxS(2))(S(5)/2)(a + bx), x), x, ac*S(2)x*S(6)sqrt(cxS(2))/S(7) + bc*S(2)x*S(7)sqrt(cxS(2))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx), x), x, acS(2)x*S(5)sqrt(cxS(2))/S(6) + bc*S(2)x*S(6)sqrt(cxS(2))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)/x, x), x, acS(2)x*S(4)sqrt(cxS(2))/S(5) + bc*S(2)x*S(5)sqrt(cxS(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)/x*S(2), x), x, ac*S(2)x*S(3)sqrt(cxS(2))/S(4) + bc*S(2)x*S(4)sqrt(cxS(2))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)/x*S(3), x), x, ac*S(2)x*S(2)sqrt(cxS(2))/S(3) + bc*S(2)x*S(3)sqrt(cxS(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)/x*S(4), x), x, ac*S(2)xsqrt(cx*S(2))/S(2) + bc*S(2)x*S(2)sqrt(cxS(2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(a + bx)/sqrt(cx*S(2)), x), x, ax*(m + S(1))/(msqrt(cxS(2))) + bx*(m + S(2))/(sqrt(cx*S(2))(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a + bx)/sqrt(cx*S(2)), x), x, ax*S(4)/(S(3)sqrt(cxS(2))) + bx*S(5)/(S(4)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx)/sqrt(cx*S(2)), x), x, axsqrt(cx*S(2))/(S(2)c) + bxS(2)sqrt(cxS(2))/(S(3)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)/sqrt(cxS(2)), x), x, ax*S(2)/sqrt(cx*S(2)) + bx*S(3)/(S(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/sqrt(cxS(2)), x), x, axlog(x)/sqrt(cx*S(2)) + bx*S(2)/sqrt(cx*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(xsqrt(cx*S(2))), x), x, -a/sqrt(cx*S(2)) + bxlog(x)/sqrt(cx*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(x*S(2)sqrt(cxS(2))), x), x, -a/(S(2)xsqrt(cx*S(2))) - b/sqrt(cx*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(x*S(3)sqrt(cxS(2))), x), x, -a/(S(3)x*S(2)sqrt(cxS(2))) - b/(S(2)xsqrt(cx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(x*S(4)sqrt(cxS(2))), x), x, -a/(S(4)x*S(3)sqrt(cxS(2))) - b/(S(3)x*S(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(a + bx)/(cxS(2))(S(3)/2), x), x, -ax(m + S(-1))/(csqrt(cxS(2))(-m + S(2))) - bxm/(csqrt(cxS(2))(-m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a + bx)/(cxS(2))(S(3)/2), x), x, axS(2)/(csqrt(cxS(2))) + bx*S(3)/(S(2)csqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx)/(cxS(2))(S(3)/2), x), x, axlog(x)/(csqrt(cxS(2))) + bx*S(2)/(csqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)/(cxS(2))(S(3)/2), x), x, -a/(csqrt(cx*S(2))) + bxlog(x)/(csqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(cxS(2))(S(3)/2), x), x, -a/(S(2)cxsqrt(cxS(2))) - b/(csqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(x(cxS(2))(S(3)/2)), x), x, -a/(S(3)cx*S(2)sqrt(cxS(2))) - b/(S(2)cxsqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(x*S(2)(cxS(2))(S(3)/2)), x), x, -a/(S(4)cxS(3)sqrt(cxS(2))) - b/(S(3)cxS(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(x*S(3)(cxS(2))(S(3)/2)), x), x, -a/(S(5)cxS(4)sqrt(cxS(2))) - b/(S(4)cxS(3)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(x*S(4)(cxS(2))(S(3)/2)), x), x, -a/(S(6)cxS(5)sqrt(cxS(2))) - b/(S(5)cxS(4)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(a + bx)/(cxS(2))(S(5)/2), x), x, -ax(m + S(-3))/(cS(2)sqrt(cxS(2))(-m + S(4))) - bx(m + S(-2))/(cS(2)sqrt(cxS(2))(-m + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a + bx)/(cxS(2))(S(5)/2), x), x, -a/(c*S(2)sqrt(cxS(2))) + bxlog(x)/(cS(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx)/(cxS(2))(S(5)/2), x), x, -a/(S(2)cS(2)xsqrt(cxS(2))) - b/(cS(2)sqrt(cx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)/(cxS(2))(S(5)/2), x), x, -a/(S(3)cS(2)x*S(2)sqrt(cxS(2))) - b/(S(2)c*S(2)xsqrt(cx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(cxS(2))(S(5)/2), x), x, -a/(S(4)c*S(2)x*S(3)sqrt(cxS(2))) - b/(S(3)c*S(2)x*S(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(x(cxS(2))(S(5)/2)), x), x, -a/(S(5)cS(2)x*S(4)sqrt(cxS(2))) - b/(S(4)c*S(2)x*S(3)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(x*S(2)(cxS(2))(S(5)/2)), x), x, -a/(S(6)c*S(2)x*S(5)sqrt(cxS(2))) - b/(S(5)c*S(2)x*S(4)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(x*S(3)(cxS(2))(S(5)/2)), x), x, -a/(S(7)c*S(2)x*S(6)sqrt(cxS(2))) - b/(S(6)c*S(2)x*S(5)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)/(x*S(4)(cxS(2))(S(5)/2)), x), x, -a/(S(8)c*S(2)x*S(7)sqrt(cxS(2))) - b/(S(7)c*S(2)x*S(6)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmsqrt(cxS(2))(a + bx)S(2), x), x, aS(2)x*(m + S(1))sqrt(cxS(2))/(m + S(2)) + S(2)abx*(m + S(2))sqrt(cxS(2))/(m + S(3)) + bS(2)x*(m + S(3))sqrt(cxS(2))/(m + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(cxS(2))(a + bx)S(2), x), x, aS(2)x*S(4)sqrt(cxS(2))/S(5) + abxS(5)sqrt(cxS(2))/S(3) + bS(2)x*S(6)sqrt(cxS(2))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(cxS(2))(a + bx)S(2), x), x, aS(2)x*S(3)sqrt(cxS(2))/S(4) + S(2)abx*S(4)sqrt(cxS(2))/S(5) + bS(2)x*S(5)sqrt(cxS(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(cxS(2))(a + bx)S(2), x), x, aS(2)x*S(2)sqrt(cxS(2))/S(3) + abxS(3)sqrt(cxS(2))/S(2) + bS(2)x*S(4)sqrt(cxS(2))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)S(2), x), x, aS(2)xsqrt(cx*S(2))/S(2) + S(2)abx*S(2)sqrt(cxS(2))/S(3) + bS(2)x*S(3)sqrt(cxS(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)S(2)/x, x), x, sqrt(cx*S(2))(a + bx)S(3)/(S(3)bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)S(2)/xS(2), x), x, aS(2)sqrt(cxS(2))log(x)/x + S(2)absqrt(cxS(2)) + bS(2)xsqrt(cxS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)S(2)/xS(3), x), x, -aS(2)sqrt(cxS(2))/xS(2) + S(2)absqrt(cxS(2))log(x)/x + b*S(2)sqrt(cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))(a + bx)S(2)/xS(4), x), x, -aS(2)sqrt(cxS(2))/(S(2)x*S(3)) - S(2)absqrt(cxS(2))/xS(2) + bS(2)sqrt(cxS(2))log(x)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*m(cxS(2))(S(3)/2)(a + bx)S(2), x), x, aS(2)cx(m + S(3))sqrt(cxS(2))/(m + S(4)) + S(2)abcx(m + S(4))sqrt(cxS(2))/(m + S(5)) + bS(2)cx(m + S(5))sqrt(cxS(2))/(m + S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(cxS(2))(S(3)/2)(a + bx)S(2), x), x, aS(2)cxS(6)sqrt(cxS(2))/S(7) + abcx*S(7)sqrt(cxS(2))/S(4) + bS(2)cxS(8)sqrt(cxS(2))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(cxS(2))(S(3)/2)(a + bx)S(2), x), x, aS(2)cxS(5)sqrt(cxS(2))/S(6) + S(2)abcxS(6)sqrt(cxS(2))/S(7) + bS(2)cxS(7)sqrt(cxS(2))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(cxS(2))(S(3)/2)(a + bx)S(2), x), x, aS(2)cxS(4)sqrt(cxS(2))/S(5) + abcx*S(5)sqrt(cxS(2))/S(3) + bS(2)cxS(6)sqrt(cxS(2))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)S(2), x), x, aS(2)cx*S(3)sqrt(cxS(2))/S(4) + S(2)abcxS(4)sqrt(cxS(2))/S(5) + bS(2)cxS(5)sqrt(cxS(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)S(2)/x, x), x, aS(2)cx*S(2)sqrt(cxS(2))/S(3) + abcx*S(3)sqrt(cxS(2))/S(2) + bS(2)cxS(4)sqrt(cxS(2))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)S(2)/xS(2), x), x, a*S(2)cxsqrt(cxS(2))/S(2) + S(2)abcxS(2)sqrt(cxS(2))/S(3) + bS(2)cxS(3)sqrt(cxS(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)S(2)/xS(3), x), x, csqrt(cx*S(2))(a + bx)S(3)/(S(3)bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)(a + bx)S(2)/xS(4), x), x, a*S(2)csqrt(cx*S(2))log(x)/x + S(2)abcsqrt(cxS(2)) + bS(2)cxsqrt(cxS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(cxS(2))(S(5)/2)(a + bx)S(2), x), x, aS(2)c*S(2)x*(m + S(5))sqrt(cxS(2))/(m + S(6)) + S(2)abc*S(2)x*(m + S(6))sqrt(cxS(2))/(m + S(7)) + bS(2)c*S(2)x*(m + S(7))sqrt(cxS(2))/(m + S(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(cxS(2))(S(5)/2)(a + bx)S(2), x), x, aS(2)c*S(2)x*S(8)sqrt(cxS(2))/S(9) + abcS(2)x*S(9)sqrt(cxS(2))/S(5) + bS(2)c*S(2)x*S(10)sqrt(cxS(2))/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(cxS(2))(S(5)/2)(a + bx)S(2), x), x, aS(2)c*S(2)x*S(7)sqrt(cxS(2))/S(8) + S(2)abc*S(2)x*S(8)sqrt(cxS(2))/S(9) + bS(2)c*S(2)x*S(9)sqrt(cxS(2))/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(cxS(2))(S(5)/2)(a + bx)S(2), x), x, aS(2)c*S(2)x*S(6)sqrt(cxS(2))/S(7) + abcS(2)x*S(7)sqrt(cxS(2))/S(4) + bS(2)c*S(2)x*S(8)sqrt(cxS(2))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)S(2), x), x, aS(2)cS(2)x*S(5)sqrt(cxS(2))/S(6) + S(2)abc*S(2)x*S(6)sqrt(cxS(2))/S(7) + bS(2)c*S(2)x*S(7)sqrt(cxS(2))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)S(2)/x, x), x, aS(2)cS(2)x*S(4)sqrt(cxS(2))/S(5) + abcS(2)x*S(5)sqrt(cxS(2))/S(3) + bS(2)c*S(2)x*S(6)sqrt(cxS(2))/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)S(2)/xS(2), x), x, a*S(2)c*S(2)x*S(3)sqrt(cxS(2))/S(4) + S(2)abc*S(2)x*S(4)sqrt(cxS(2))/S(5) + bS(2)c*S(2)x*S(5)sqrt(cxS(2))/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)S(2)/xS(3), x), x, a*S(2)c*S(2)x*S(2)sqrt(cxS(2))/S(3) + abcS(2)x*S(3)sqrt(cxS(2))/S(2) + bS(2)c*S(2)x*S(4)sqrt(cxS(2))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)(a + bx)S(2)/xS(4), x), x, a*S(2)c*S(2)xsqrt(cx*S(2))/S(2) + S(2)abc*S(2)x*S(2)sqrt(cxS(2))/S(3) + bS(2)c*S(2)x*S(3)sqrt(cxS(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(a + bx)S(2)/sqrt(cxS(2)), x), x, aS(2)x(m + S(1))/(msqrt(cxS(2))) + S(2)abx*(m + S(2))/(sqrt(cx*S(2))(m + S(1))) + b*S(2)x*(m + S(3))/(sqrt(cx*S(2))(m + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a + bx)S(2)/sqrt(cxS(2)), x), x, aS(2)xS(4)/(S(3)sqrt(cxS(2))) + abxS(5)/(S(2)sqrt(cxS(2))) + bS(2)x*S(6)/(S(5)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx)S(2)/sqrt(cxS(2)), x), x, aS(2)xsqrt(cxS(2))/(S(2)c) + S(2)abxS(2)sqrt(cxS(2))/(S(3)c) + b*S(2)x*S(3)sqrt(cxS(2))/(S(4)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)*S(2)/sqrt(cx*S(2)), x), x, x(a + bx)S(3)/(S(3)bsqrt(cx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(2)/sqrt(cxS(2)), x), x, aS(2)xlog(x)/sqrt(cxS(2)) + S(2)abx*S(2)/sqrt(cxS(2)) + bS(2)xS(3)/(S(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(2)/(xsqrt(cxS(2))), x), x, -aS(2)/sqrt(cx*S(2)) + S(2)abxlog(x)/sqrt(cxS(2)) + bS(2)xS(2)/sqrt(cx*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)/(xS(2)sqrt(cxS(2))), x), x, -aS(2)/(S(2)xsqrt(cxS(2))) - S(2)ab/sqrt(cxS(2)) + bS(2)xlog(x)/sqrt(cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)/(xS(3)sqrt(cx*S(2))), x), x, -(a + bx)*S(3)/(S(3)axS(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)/(xS(4)sqrt(cxS(2))), x), x, -aS(2)/(S(4)xS(3)sqrt(cxS(2))) - S(2)ab/(S(3)x*S(2)sqrt(cxS(2))) - bS(2)/(S(2)xsqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(a + bx)*S(2)/(cxS(2))(S(3)/2), x), x, -a*S(2)x*(m + S(-1))/(csqrt(cxS(2))(-m + S(2))) - S(2)abxm/(csqrt(cxS(2))(-m + S(1))) + b*S(2)x*(m + S(1))/(cmsqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a + bx)*S(2)/(cxS(2))(S(3)/2), x), x, x(a + bx)*S(3)/(S(3)bcsqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx)S(2)/(cxS(2))(S(3)/2), x), x, a*S(2)xlog(x)/(csqrt(cxS(2))) + S(2)abx*S(2)/(csqrt(cxS(2))) + bS(2)x*S(3)/(S(2)csqrt(cx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)S(2)/(cxS(2))(S(3)/2), x), x, -a*S(2)/(csqrt(cxS(2))) + S(2)abxlog(x)/(csqrt(cxS(2))) + bS(2)x*S(2)/(csqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(2)/(cxS(2))(S(3)/2), x), x, -a*S(2)/(S(2)cxsqrt(cxS(2))) - S(2)ab/(csqrt(cxS(2))) + bS(2)xlog(x)/(csqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(2)/(x(cxS(2))(S(3)/2)), x), x, -(a + bx)*S(3)/(S(3)acx*S(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)/(xS(2)(cxS(2))(S(3)/2)), x), x, -a*S(2)/(S(4)cxS(3)sqrt(cxS(2))) - S(2)ab/(S(3)cxS(2)sqrt(cxS(2))) - bS(2)/(S(2)cxsqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)/(xS(3)(cxS(2))(S(3)/2)), x), x, -a*S(2)/(S(5)cxS(4)sqrt(cxS(2))) - ab/(S(2)cx*S(3)sqrt(cxS(2))) - bS(2)/(S(3)cxS(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)/(xS(4)(cxS(2))(S(3)/2)), x), x, -a*S(2)/(S(6)cxS(5)sqrt(cxS(2))) - S(2)ab/(S(5)cxS(4)sqrt(cxS(2))) - bS(2)/(S(4)cxS(3)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(a + bx)S(2)/(cxS(2))(S(5)/2), x), x, -a*S(2)x(m + S(-3))/(cS(2)sqrt(cx*S(2))(-m + S(4))) - S(2)abx(m + S(-2))/(cS(2)sqrt(cxS(2))(-m + S(3))) - b*S(2)x(m + S(-1))/(cS(2)sqrt(cx*S(2))(-m + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a + bx)S(2)/(cxS(2))(S(5)/2), x), x, -aS(2)/(cS(2)sqrt(cx*S(2))) + S(2)abxlog(x)/(cS(2)sqrt(cxS(2))) + bS(2)xS(2)/(cS(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bx)*S(2)/(cxS(2))(S(5)/2), x), x, -a*S(2)/(S(2)c*S(2)xsqrt(cx*S(2))) - S(2)ab/(cS(2)sqrt(cxS(2))) + bS(2)xlog(x)/(cS(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bx)S(2)/(cxS(2))(S(5)/2), x), x, -(a + bx)S(3)/(S(3)acS(2)x*S(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(2)/(cxS(2))(S(5)/2), x), x, -a*S(2)/(S(4)c*S(2)x*S(3)sqrt(cxS(2))) - S(2)ab/(S(3)c*S(2)x*S(2)sqrt(cxS(2))) - bS(2)/(S(2)c*S(2)xsqrt(cx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)*S(2)/(x(cxS(2))(S(5)/2)), x), x, -aS(2)/(S(5)c*S(2)x*S(4)sqrt(cxS(2))) - ab/(S(2)cS(2)x*S(3)sqrt(cxS(2))) - bS(2)/(S(3)c*S(2)x*S(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)/(xS(2)(cxS(2))(S(5)/2)), x), x, -a*S(2)/(S(6)c*S(2)x*S(5)sqrt(cxS(2))) - S(2)ab/(S(5)c*S(2)x*S(4)sqrt(cxS(2))) - bS(2)/(S(4)c*S(2)x*S(3)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)/(xS(3)(cxS(2))(S(5)/2)), x), x, -a*S(2)/(S(7)c*S(2)x*S(6)sqrt(cxS(2))) - ab/(S(3)cS(2)x*S(5)sqrt(cxS(2))) - bS(2)/(S(5)c*S(2)x*S(4)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)S(2)/(xS(4)(cxS(2))(S(5)/2)), x), x, -a*S(2)/(S(8)c*S(2)x*S(7)sqrt(cxS(2))) - S(2)ab/(S(7)c*S(2)x*S(6)sqrt(cxS(2))) - bS(2)/(S(6)c*S(2)x*S(5)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(cxS(2))/(a + bx), x), x, a*S(4)sqrt(cxS(2))log(a + bx)/(bS(5)x) - a*S(3)sqrt(cxS(2))/bS(4) + aS(2)xsqrt(cx*S(2))/(S(2)b*S(3)) - ax*S(2)sqrt(cxS(2))/(S(3)bS(2)) + xS(3)sqrt(cx*S(2))/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)sqrt(cxS(2))/(a + bx), x), x, -a*S(3)sqrt(cxS(2))log(a + bx)/(bS(4)x) + a*S(2)sqrt(cxS(2))/bS(3) - axsqrt(cx*S(2))/(S(2)bS(2)) + xS(2)sqrt(cx*S(2))/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(cx*S(2))/(a + bx), x), x, a*S(2)sqrt(cxS(2))log(a + bx)/(bS(3)x) - asqrt(cxS(2))/bS(2) + xsqrt(cx*S(2))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cxS(2))/(a + bx), x), x, -asqrt(cx*S(2))log(a + bx)/(bS(2)x) + sqrt(cxS(2))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))/(x(a + bx)), x), x, sqrt(cx*S(2))log(a + bx)/(bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cxS(2))/(xS(2)(a + bx)), x), x, sqrt(cx*S(2))log(x)/(ax) - sqrt(cx*S(2))log(a + bx)/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cxS(2))/(xS(3)(a + bx)), x), x, -sqrt(cx*S(2))/(ax*S(2)) - bsqrt(cxS(2))log(x)/(a*S(2)x) + bsqrt(cx*S(2))log(a + bx)/(aS(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cxS(2))/(xS(4)(a + bx)), x), x, -sqrt(cx*S(2))/(S(2)axS(3)) + bsqrt(cxS(2))/(aS(2)xS(2)) + bS(2)sqrt(cx*S(2))log(x)/(a*S(3)x) - b*S(2)sqrt(cxS(2))log(a + bx)/(aS(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(cxS(2))(S(3)/2)/(a + bx), x), x, aS(4)csqrt(cx*S(2))log(a + bx)/(bS(5)x) - a*S(3)csqrt(cxS(2))/bS(4) + a*S(2)cxsqrt(cxS(2))/(S(2)b*S(3)) - acxS(2)sqrt(cxS(2))/(S(3)b*S(2)) + cx*S(3)sqrt(cxS(2))/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(a + bx), x), x, -a*S(3)csqrt(cx*S(2))log(a + bx)/(bS(4)x) + a*S(2)csqrt(cxS(2))/bS(3) - acxsqrt(cx*S(2))/(S(2)b*S(2)) + cx*S(2)sqrt(cxS(2))/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(x(a + bx)), x), x, aS(2)csqrt(cx*S(2))log(a + bx)/(bS(3)x) - acsqrt(cxS(2))/bS(2) + cxsqrt(cx*S(2))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(xS(2)(a + bx)), x), x, -acsqrt(cx*S(2))log(a + bx)/(bS(2)x) + csqrt(cx*S(2))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(x*S(3)(a + bx)), x), x, csqrt(cxS(2))log(a + bx)/(bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(xS(4)(a + bx)), x), x, csqrt(cxS(2))log(x)/(ax) - csqrt(cxS(2))log(a + bx)/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(xS(5)(a + bx)), x), x, -csqrt(cxS(2))/(ax*S(2)) - bcsqrt(cx*S(2))log(x)/(a*S(2)x) + bcsqrt(cxS(2))log(a + bx)/(aS(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(xS(6)(a + bx)), x), x, -csqrt(cxS(2))/(S(2)axS(3)) + bcsqrt(cxS(2))/(aS(2)xS(2)) + bS(2)csqrt(cx*S(2))log(x)/(a*S(3)x) - b*S(2)csqrt(cx*S(2))log(a + bx)/(aS(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(xS(7)(a + bx)), x), x, -csqrt(cxS(2))/(S(3)axS(4)) + bcsqrt(cx*S(2))/(S(2)a*S(2)xS(3)) - bS(2)csqrt(cxS(2))/(aS(3)xS(2)) - bS(3)csqrt(cxS(2))log(x)/(a*S(4)x) + b*S(3)csqrt(cx*S(2))log(a + bx)/(aS(4)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)/(a + bx), x), x, -a*S(5)c*S(2)sqrt(cxS(2))log(a + bx)/(bS(6)x) + a*S(4)c*S(2)sqrt(cxS(2))/bS(5) - aS(3)c*S(2)xsqrt(cx*S(2))/(S(2)bS(4)) + aS(2)cS(2)x*S(2)sqrt(cxS(2))/(S(3)b*S(3)) - ac*S(2)x*S(3)sqrt(cxS(2))/(S(4)bS(2)) + cS(2)xS(4)sqrt(cxS(2))/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)/(x(a + bx)), x), x, aS(4)c*S(2)sqrt(cxS(2))log(a + bx)/(bS(5)x) - a*S(3)c*S(2)sqrt(cxS(2))/bS(4) + aS(2)c*S(2)xsqrt(cx*S(2))/(S(2)b*S(3)) - ac*S(2)x*S(2)sqrt(cxS(2))/(S(3)bS(2)) + cS(2)xS(3)sqrt(cxS(2))/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)/(xS(2)(a + bx)), x), x, -aS(3)c*S(2)sqrt(cxS(2))log(a + bx)/(bS(4)x) + a*S(2)c*S(2)sqrt(cxS(2))/bS(3) - ac*S(2)xsqrt(cx*S(2))/(S(2)bS(2)) + cS(2)xS(2)sqrt(cxS(2))/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)/(xS(3)(a + bx)), x), x, aS(2)c*S(2)sqrt(cxS(2))log(a + bx)/(bS(3)x) - acS(2)sqrt(cxS(2))/bS(2) + cS(2)xsqrt(cx*S(2))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)/(xS(4)(a + bx)), x), x, -ac*S(2)sqrt(cxS(2))log(a + bx)/(bS(2)x) + c*S(2)sqrt(cxS(2))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)/(x*S(5)(a + bx)), x), x, cS(2)sqrt(cxS(2))log(a + bx)/(bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)/(xS(6)(a + bx)), x), x, cS(2)sqrt(cxS(2))log(x)/(ax) - cS(2)sqrt(cxS(2))log(a + bx)/(ax), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(5)/2)/(xS(7)(a + bx)), x), x, -cS(2)sqrt(cxS(2))/(ax*S(2)) - bc*S(2)sqrt(cxS(2))log(x)/(a*S(2)x) + bcS(2)sqrt(cxS(2))log(a + bx)/(aS(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)/(sqrt(cx*S(2))(a + bx)), x), x, -aS(3)xlog(a + bx)/(b*S(4)sqrt(cxS(2))) + aS(2)xS(2)/(bS(3)sqrt(cx*S(2))) - ax*S(3)/(S(2)b*S(2)sqrt(cxS(2))) + xS(4)/(S(3)bsqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(sqrt(cxS(2))(a + bx)), x), x, aS(2)xlog(a + bx)/(b*S(3)sqrt(cxS(2))) - axS(2)/(bS(2)sqrt(cxS(2))) + xS(3)/(S(2)bsqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(sqrt(cx*S(2))(a + bx)), x), x, -asqrt(cxS(2))log(a + bx)/(bS(2)cx) + sqrt(cx*S(2))/(bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt(cxS(2))(a + bx)), x), x, xlog(a + bx)/(bsqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(cx*S(2))(a + bx)), x), x, xlog(x)/(asqrt(cx*S(2))) - xlog(a + bx)/(asqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(cxS(2))(a + bx)), x), x, -S(1)/(asqrt(cxS(2))) - bxlog(x)/(aS(2)sqrt(cxS(2))) + bxlog(a + bx)/(a*S(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(cxS(2))(a + bx)), x), x, -S(1)/(S(2)axsqrt(cxS(2))) + b/(aS(2)sqrt(cxS(2))) + bS(2)xlog(x)/(aS(3)sqrt(cxS(2))) - bS(2)xlog(a + bx)/(a*S(3)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)sqrt(cxS(2))(a + bx)), x), x, -S(1)/(S(3)axS(2)sqrt(cxS(2))) + b/(S(2)a*S(2)xsqrt(cxS(2))) - bS(2)/(a*S(3)sqrt(cxS(2))) - bS(3)xlog(x)/(aS(4)sqrt(cxS(2))) + bS(3)xlog(a + bx)/(a*S(4)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/((cxS(2))(S(3)/2)(a + bx)), x), x, -a*S(3)xlog(a + bx)/(b*S(4)csqrt(cxS(2))) + aS(2)xS(2)/(bS(3)csqrt(cx*S(2))) - ax*S(3)/(S(2)b*S(2)csqrt(cxS(2))) + xS(4)/(S(3)bcsqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/((cxS(2))(S(3)/2)(a + bx)), x), x, aS(2)xlog(a + bx)/(b*S(3)csqrt(cx*S(2))) - axS(2)/(bS(2)csqrt(cxS(2))) + xS(3)/(S(2)bcsqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/((cxS(2))(S(3)/2)(a + bx)), x), x, -axlog(a + bx)/(bS(2)csqrt(cxS(2))) + xS(2)/(bcsqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/((cxS(2))(S(3)/2)(a + bx)), x), x, xlog(a + bx)/(bcsqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((cxS(2))(S(3)/2)(a + bx)), x), x, xlog(x)/(acsqrt(cx*S(2))) - xlog(a + bx)/(acsqrt(cx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((cxS(2))(S(3)/2)(a + bx)), x), x, -S(1)/(acsqrt(cxS(2))) - bxlog(x)/(aS(2)csqrt(cx*S(2))) + bxlog(a + bx)/(a*S(2)csqrt(cx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((cxS(2))(S(3)/2)(a + bx)), x), x, -S(1)/(S(2)acxsqrt(cxS(2))) + b/(aS(2)csqrt(cxS(2))) + bS(2)xlog(x)/(a*S(3)csqrt(cxS(2))) - bS(2)xlog(a + bx)/(aS(3)csqrt(cx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(cxS(2))(S(3)/2)(a + bx)), x), x, -S(1)/(S(3)acx*S(2)sqrt(cxS(2))) + b/(S(2)a*S(2)cxsqrt(cxS(2))) - bS(2)/(aS(3)csqrt(cxS(2))) - bS(3)xlog(x)/(a*S(4)csqrt(cxS(2))) + bS(3)xlog(a + bx)/(aS(4)csqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(cx*S(2))/(a + bx)S(2), x), x, -aS(4)sqrt(cxS(2))/(bS(5)x(a + bx)) - S(4)a*S(3)sqrt(cxS(2))log(a + bx)/(bS(5)x) + S(3)aS(2)sqrt(cxS(2))/bS(4) - axsqrt(cxS(2))/bS(3) + x*S(2)sqrt(cxS(2))/(S(3)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(cx*S(2))/(a + bx)S(2), x), x, aS(3)sqrt(cxS(2))/(bS(4)x(a + bx)) + S(3)a*S(2)sqrt(cxS(2))log(a + bx)/(bS(4)x) - S(2)asqrt(cxS(2))/bS(3) + xsqrt(cxS(2))/(S(2)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(cxS(2))/(a + bx)S(2), x), x, -aS(2)sqrt(cxS(2))/(bS(3)x(a + bx)) - S(2)asqrt(cx*S(2))log(a + bx)/(bS(3)x) + sqrt(cxS(2))/bS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cx*S(2))/(a + bx)*S(2), x), x, asqrt(cxS(2))/(bS(2)x(a + bx)) + sqrt(cxS(2))log(a + bx)/(bS(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cxS(2))/(x(a + bx)S(2)), x), x, -sqrt(cx*S(2))/(bx(a + bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cxS(2))/(xS(2)(a + bx)S(2)), x), x, sqrt(cx*S(2))/(ax(a + bx)) + sqrt(cxS(2))log(x)/(a*S(2)x) - sqrt(cxS(2))log(a + bx)/(aS(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cxS(2))/(xS(3)(a + bx)S(2)), x), x, -bsqrt(cxS(2))/(aS(2)x(a + bx)) - sqrt(cxS(2))/(aS(2)x*S(2)) - S(2)bsqrt(cx*S(2))log(x)/(a*S(3)x) + S(2)bsqrt(cxS(2))log(a + bx)/(aS(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cxS(2))/(xS(4)(a + bx)S(2)), x), x, -sqrt(cx*S(2))/(S(2)a*S(2)xS(3)) + bS(2)sqrt(cxS(2))/(aS(3)x(a + bx)) + S(2)bsqrt(cxS(2))/(aS(3)xS(2)) + S(3)b*S(2)sqrt(cxS(2))log(x)/(a*S(4)x) - S(3)bS(2)sqrt(cxS(2))log(a + bx)/(aS(4)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(cxS(2))(S(3)/2)/(a + bx)S(2), x), x, -aS(4)csqrt(cxS(2))/(bS(5)x(a + bx)) - S(4)a*S(3)csqrt(cx*S(2))log(a + bx)/(bS(5)x) + S(3)aS(2)csqrt(cxS(2))/bS(4) - acxsqrt(cxS(2))/bS(3) + cxS(2)sqrt(cxS(2))/(S(3)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(a + bx)S(2), x), x, aS(3)csqrt(cxS(2))/(bS(4)x(a + bx)) + S(3)a*S(2)csqrt(cx*S(2))log(a + bx)/(bS(4)x) - S(2)acsqrt(cxS(2))/bS(3) + cxsqrt(cxS(2))/(S(2)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(x(a + bx)S(2)), x), x, -aS(2)csqrt(cxS(2))/(bS(3)x(a + bx)) - S(2)acsqrt(cx*S(2))log(a + bx)/(bS(3)x) + csqrt(cxS(2))/bS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(xS(2)(a + bx)S(2)), x), x, acsqrt(cxS(2))/(bS(2)x(a + bx)) + csqrt(cxS(2))log(a + bx)/(bS(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(xS(3)(a + bx)S(2)), x), x, -csqrt(cxS(2))/(bx(a + bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(xS(4)(a + bx)S(2)), x), x, csqrt(cxS(2))/(ax(a + bx)) + csqrt(cx*S(2))log(x)/(a*S(2)x) - csqrt(cx*S(2))log(a + bx)/(aS(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(xS(5)(a + bx)S(2)), x), x, -bcsqrt(cxS(2))/(aS(2)x(a + bx)) - csqrt(cxS(2))/(aS(2)x*S(2)) - S(2)bcsqrt(cxS(2))log(x)/(a*S(3)x) + S(2)bcsqrt(cx*S(2))log(a + bx)/(aS(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))(S(3)/2)/(xS(6)(a + bx)S(2)), x), x, -csqrt(cxS(2))/(S(2)a*S(2)xS(3)) + bS(2)csqrt(cxS(2))/(aS(3)x(a + bx)) + S(2)bcsqrt(cxS(2))/(aS(3)xS(2)) + S(3)b*S(2)csqrt(cx*S(2))log(x)/(a*S(4)x) - S(3)bS(2)csqrt(cx*S(2))log(a + bx)/(aS(4)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(5)/(sqrt(cx*S(2))(a + bx)S(2)), x), x, -aS(4)x/(b*S(5)sqrt(cxS(2))(a + bx)) - S(4)a*S(3)xlog(a + bx)/(b*S(5)sqrt(cxS(2))) + S(3)a*S(2)xS(2)/(bS(4)sqrt(cx*S(2))) - axS(3)/(bS(3)sqrt(cxS(2))) + xS(4)/(S(3)bS(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(sqrt(cx*S(2))(a + bx)S(2)), x), x, aS(3)x/(b*S(4)sqrt(cxS(2))(a + bx)) + S(3)a*S(2)xlog(a + bx)/(b*S(4)sqrt(cxS(2))) - S(2)axS(2)/(bS(3)sqrt(cxS(2))) + xS(3)/(S(2)b*S(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(sqrt(cx*S(2))(a + bx)S(2)), x), x, -aS(2)x/(b*S(3)sqrt(cxS(2))(a + bx)) - S(2)axlog(a + bx)/(bS(3)sqrt(cxS(2))) + xS(2)/(bS(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(sqrt(cx*S(2))(a + bx)S(2)), x), x, asqrt(cxS(2))/(bS(2)cx(a + bx)) + sqrt(cx*S(2))log(a + bx)/(bS(2)cx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt(cx*S(2))(a + bx)S(2)), x), x, -x/(bsqrt(cxS(2))(a + bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(cx*S(2))(a + bx)S(2)), x), x, x/(asqrt(cxS(2))(a + bx)) + xlog(x)/(a*S(2)sqrt(cxS(2))) - xlog(a + bx)/(aS(2)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(cxS(2))(a + bx)S(2)), x), x, -bx/(a*S(2)sqrt(cxS(2))(a + bx)) - S(1)/(aS(2)sqrt(cxS(2))) - S(2)bxlog(x)/(a*S(3)sqrt(cxS(2))) + S(2)bxlog(a + bx)/(aS(3)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(cxS(2))(a + bx)S(2)), x), x, -S(1)/(S(2)a*S(2)xsqrt(cxS(2))) + bS(2)x/(aS(3)sqrt(cxS(2))(a + bx)) + S(2)b/(a*S(3)sqrt(cxS(2))) + S(3)b*S(2)xlog(x)/(aS(4)sqrt(cxS(2))) - S(3)b*S(2)xlog(a + bx)/(a*S(4)sqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/((cxS(2))(S(3)/2)(a + bx)S(2)), x), x, -aS(2)x/(bS(3)csqrt(cx*S(2))(a + bx)) - S(2)axlog(a + bx)/(bS(3)csqrt(cxS(2))) + xS(2)/(b*S(2)csqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/((cxS(2))(S(3)/2)(a + bx)S(2)), x), x, ax/(b*S(2)csqrt(cx*S(2))(a + bx)) + xlog(a + bx)/(bS(2)csqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/((cxS(2))(S(3)/2)(a + bx)S(2)), x), x, -x/(bcsqrt(cx*S(2))(a + bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((cxS(2))(S(3)/2)(a + bx)*S(2)), x), x, x/(acsqrt(cx*S(2))(a + bx)) + xlog(x)/(a*S(2)csqrt(cx*S(2))) - xlog(a + bx)/(aS(2)csqrt(cx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((cxS(2))(S(3)/2)(a + bx)*S(2)), x), x, -bx/(a*S(2)csqrt(cx*S(2))(a + bx)) - S(1)/(aS(2)csqrt(cx*S(2))) - S(2)bxlog(x)/(a*S(3)csqrt(cx*S(2))) + S(2)bxlog(a + bx)/(aS(3)csqrt(cx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((cxS(2))(S(3)/2)(a + bx)*S(2)), x), x, -S(1)/(S(2)a*S(2)cxsqrt(cxS(2))) + bS(2)x/(a*S(3)csqrt(cx*S(2))(a + bx)) + S(2)b/(a*S(3)csqrt(cx*S(2))) + S(3)b*S(2)xlog(x)/(aS(4)csqrt(cx*S(2))) - S(3)b*S(2)xlog(a + bx)/(a*S(4)csqrt(cxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(cxS(2))p(a + bx)*(-m - S(2)p + S(-2)), x), x, x*(m + S(1))(cxS(2))p(a + bx)(-m - S(2)p + S(-1))/(a(m + S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(cxS(2))p(a + bx)(-S(2)p + S(-5)), x), x, x*S(4)(cxS(2))p(a + bx)(-S(2)p + S(-4))/(S(2)a(p + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(cxS(2))p(a + bx)(-S(2)p + S(-4)), x), x, x*S(3)(cxS(2))p(a + bx)(-S(2)p + S(-3))/(a(S(2)p + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(cxS(2))p(a + bx)*(-S(2)p + S(-3)), x), x, x*S(2)(cxS(2))p(a + bx)(-S(2)p + S(-2))/(S(2)a(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))p(a + bx)(-S(2)p + S(-2)), x), x, x(cxS(2))p(a + bx)*(-S(2)p + S(-1))/(a(S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))p(a + bx)(-S(2)p + S(-1))/x, x), x, (cxS(2))p(a + bx)(-S(2)p)/(S(2)ap), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))p(a + bx)(-S(2)p)/x*S(2), x), x, -(cxS(2))p(a + bx)*(-S(2)p + S(1))/(ax(-S(2)p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))p(a + bx)*(-S(2)p + S(1))/x*S(3), x), x, -(cxS(2))p(a + bx)*(-S(2)p + S(2))/(S(2)ax*S(2)(-p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxS(2))p(a + bx)(-S(2)p + S(2))/x*S(4), x), x, -(cxS(2))p(a + bx)*(-S(2)p + S(3))/(axS(3)(-S(2)p + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axS(23))/sqrt(xS(5) + S(1)), x), x, sqrt(axS(23))sqrt(x*S(5) + S(1))/(S(10)x*S(4)) - S(3)sqrt(axS(23))sqrt(x*S(5) + S(1))/(S(20)x*S(9)) + S(3)sqrt(axS(23))asinh(x*(S(5)/2))/(S(20)x*(S(23)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axS(13))/sqrt(xS(5) + S(1)), x), x, sqrt(axS(13))sqrt(x*S(5) + S(1))/(S(5)x*S(4)) - sqrt(ax*S(13))asinh(x*(S(5)/2))/(S(5)x*(S(13)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axS(3))/sqrt(xS(5) + S(1)), x), x, S(2)sqrt(ax*S(3))asinh(x*(S(5)/2))/(S(5)x(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a/xS(7))/sqrt(x*S(5) + S(1)), x), x, -S(2)xsqrt(a/xS(7))sqrt(xS(5) + S(1))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a/xS(17))/sqrt(x*S(5) + S(1)), x), x, S(4)x*S(6)sqrt(a/x*S(17))sqrt(x*S(5) + S(1))/S(15) - S(2)xsqrt(a/xS(17))sqrt(x*S(5) + S(1))/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax*S(6))/(x(-x*S(4) + S(1))), x), x, -sqrt(ax*S(6))atan(x)/(S(2)xS(3)) + sqrt(ax*S(6))atanh(x)/(S(2)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axS(6))/(-xS(5) + x), x), x, -sqrt(axS(6))atan(x)/(S(2)xS(3)) + sqrt(ax*S(6))atanh(x)/(S(2)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((axS(6))(S(3)/2)/(x(-xS(4) + S(1))), x), x, -ax*S(2)sqrt(axS(6))/S(5) - asqrt(axS(6))/xS(2) + asqrt(axS(6))atan(x)/(S(2)xS(3)) + asqrt(axS(6))atanh(x)/(S(2)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(-xS(4) + S(1)) - sqrt(ax*S(6))/(x(-x*S(4) + S(1))), x), x, atan(x)/S(2) + atanh(x)/S(2) + sqrt(ax*S(6))atan(x)/(S(2)xS(3)) - sqrt(ax*S(6))atanh(x)/(S(2)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-sqrt(axS(6))/(-xS(5) + x) + S(1)/(-x*S(4) + S(1)), x), x, atan(x)/S(2) + atanh(x)/S(2) + sqrt(ax*S(6))atan(x)/(S(2)xS(3)) - sqrt(ax*S(6))atanh(x)/(S(2)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axS(3))/(-xS(3) + x), x), x, -sqrt(axS(3))atan(sqrt(x))/x*(S(3)/2) + sqrt(ax*S(3))atanh(sqrt(x))/x*(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axS(4))/sqrt(xS(2) + S(1)), x), x, sqrt(axS(4))sqrt(x*S(2) + S(1))/(S(2)x) - sqrt(axS(4))asinh(x)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axS(3))/sqrt(xS(2) + S(1)), x), x, S(2)sqrt(ax*S(3))sqrt(x*S(2) + S(1))/(S(3)x) - sqrt(axS(3))sqrt((xS(2) + S(1))/(x + S(1))S(2))(x + S(1))elliptic_f(S(2)atan(sqrt(x)), S(1)/2)/(S(3)x*(S(3)/2)sqrt(x*S(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axS(2))/sqrt(xS(2) + S(1)), x), x, sqrt(axS(2))sqrt(x*S(2) + S(1))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax)/sqrt(x*S(2) + S(1)), x), x, -S(2)sqrt(a)sqrt((xS(2) + S(1))/(x + S(1))S(2))(x + S(1))elliptic_e(S(2)atan(sqrt(ax)/sqrt(a)), S(1)/2)/sqrt(xS(2) + S(1)) + sqrt(a)sqrt((xS(2) + S(1))/(x + S(1))S(2))(x + S(1))elliptic_f(S(2)atan(sqrt(ax)/sqrt(a)), S(1)/2)/sqrt(x*S(2) + S(1)) + S(2)sqrt(ax)sqrt(xS(2) + S(1))/(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a/x)/sqrt(xS(2) + S(1)), x), x, sqrt(x)sqrt(a/x)sqrt((xS(2) + S(1))/(x + S(1))S(2))(x + S(1))elliptic_f(S(2)atan(sqrt(x)), S(1)/2)/sqrt(xS(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a/xS(2))/sqrt(xS(2) + S(1)), x), x, -xsqrt(a/x*S(2))atanh(sqrt(xS(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a/xS(3))/sqrt(x*S(2) + S(1)), x), x, -S(2)x*(S(3)/2)sqrt(a/x*S(3))sqrt((xS(2) + S(1))/(x + S(1))S(2))(x + S(1))elliptic_e(S(2)atan(sqrt(x)), S(1)/2)/sqrt(xS(2) + S(1)) + x(S(3)/2)sqrt(a/x*S(3))sqrt((xS(2) + S(1))/(x + S(1))S(2))(x + S(1))elliptic_f(S(2)atan(sqrt(x)), S(1)/2)/sqrt(xS(2) + S(1)) + S(2)x*S(2)sqrt(a/x*S(3))sqrt(x*S(2) + S(1))/(x + S(1)) - S(2)xsqrt(a/xS(3))sqrt(xS(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a/xS(4))/sqrt(x*S(2) + S(1)), x), x, -xsqrt(a/x*S(4))sqrt(x*S(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axS(4))/sqrt(xS(3) + S(1)), x), x, S(2)sqrt(ax*S(4))sqrt(x*S(3) + S(1))/(S(3)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axS(3))/sqrt(xS(3) + S(1)), x), x, -S(3)*(S(1)/4)sqrt(axS(3))sqrt((x*S(2) - x + S(1))/(x(S(1) + sqrt(S(3))) + S(1))*S(2))(x + S(1))elliptic_e(acos((x(-sqrt(S(3)) + S(1)) + S(1))/(x(S(1) + sqrt(S(3))) + S(1))), sqrt(S(3))/S(4) + S(1)/2)/(xsqrt(x(x + S(1))/(x(S(1) + sqrt(S(3))) + S(1))*S(2))sqrt(xS(3) + S(1))) - S(3)(S(3)/4)sqrt(ax*S(3))sqrt((x*S(2) - x + S(1))/(x(S(1) + sqrt(S(3))) + S(1))*S(2))(-sqrt(S(3)) + S(1))(x + S(1))elliptic_f(acos((x(-sqrt(S(3)) + S(1)) + S(1))/(x(S(1) + sqrt(S(3))) + S(1))), sqrt(S(3))/S(4) + S(1)/2)/(S(6)xsqrt(x(x + S(1))/(x(S(1) + sqrt(S(3))) + S(1))*S(2))sqrt(x*S(3) + S(1))) + sqrt(ax*S(3))(S(1) + sqrt(S(3)))sqrt(xS(3) + S(1))/(x(x(S(1) + sqrt(S(3))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axS(2))/sqrt(xS(3) + S(1)), x), x, S(2)sqrt(ax*S(2))sqrt(x*S(3) + S(1))/(x(x + S(1) + sqrt(S(3)))) - S(3)*(S(1)/4)sqrt(axS(2))sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))sqrt(-sqrt(S(3)) + S(2))(x + S(1))elliptic_e(asin((x - sqrt(S(3)) + S(1))/(x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(xsqrt((x + S(1))/(x + S(1) + sqrt(S(3)))S(2))sqrt(x*S(3) + S(1))) + S(2)sqrt(S(2))S(3)(S(3)/4)sqrt(axS(2))sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))(x + S(1))elliptic_f(asin((x - sqrt(S(3)) + S(1))/(x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(S(3)xsqrt((x + S(1))/(x + S(1) + sqrt(S(3)))S(2))sqrt(x*S(3) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax)/sqrt(x*S(3) + S(1)), x), x, S(2)sqrt(a)asinh((ax)(S(3)/2)/a(S(3)/2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a/x)/sqrt(xS(3) + S(1)), x), x, S(3)(S(3)/4)xsqrt(a/x)sqrt((xS(2) - x + S(1))/(x(S(1) + sqrt(S(3))) + S(1))*S(2))(x + S(1))elliptic_f(acos((x(-sqrt(S(3)) + S(1)) + S(1))/(x(S(1) + sqrt(S(3))) + S(1))), sqrt(S(3))/S(4) + S(1)/2)/(S(3)sqrt(x(x + S(1))/(x(S(1) + sqrt(S(3))) + S(1))*S(2))sqrt(xS(3) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a/xS(2))/sqrt(x*S(3) + S(1)), x), x, -S(2)xsqrt(a/xS(2))atanh(sqrt(xS(3) + S(1)))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a/xS(3))/sqrt(x*S(3) + S(1)), x), x, -S(2)S(3)*(S(1)/4)x*S(2)sqrt(a/x*S(3))sqrt((x*S(2) - x + S(1))/(x(S(1) + sqrt(S(3))) + S(1))*S(2))(x + S(1))elliptic_e(acos((x(-sqrt(S(3)) + S(1)) + S(1))/(x(S(1) + sqrt(S(3))) + S(1))), sqrt(S(3))/S(4) + S(1)/2)/(sqrt(x(x + S(1))/(x(S(1) + sqrt(S(3))) + S(1))S(2))sqrt(xS(3) + S(1))) - S(3)(S(3)/4)xS(2)sqrt(a/x*S(3))sqrt((x*S(2) - x + S(1))/(x(S(1) + sqrt(S(3))) + S(1))*S(2))(-sqrt(S(3)) + S(1))(x + S(1))elliptic_f(acos((x(-sqrt(S(3)) + S(1)) + S(1))/(x(S(1) + sqrt(S(3))) + S(1))), sqrt(S(3))/S(4) + S(1)/2)/(S(3)sqrt(x(x + S(1))/(x(S(1) + sqrt(S(3))) + S(1))S(2))sqrt(xS(3) + S(1))) + xS(2)sqrt(a/xS(3))(S(2) + S(2)sqrt(S(3)))sqrt(x*S(3) + S(1))/(x(S(1) + sqrt(S(3))) + S(1)) - S(2)xsqrt(a/x*S(3))sqrt(xS(3) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a/xS(4))/sqrt(xS(3) + S(1)), x), x, xS(2)sqrt(a/xS(4))sqrt(xS(3) + S(1))/(x + S(1) + sqrt(S(3))) - S(3)(S(1)/4)xS(2)sqrt(a/x*S(4))sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))sqrt(-sqrt(S(3)) + S(2))(x + S(1))elliptic_e(asin((x - sqrt(S(3)) + S(1))/(x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(S(2)sqrt((x + S(1))/(x + S(1) + sqrt(S(3)))S(2))sqrt(x*S(3) + S(1))) + sqrt(S(2))S(3)*(S(3)/4)x*S(2)sqrt(a/x*S(4))sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))(x + S(1))elliptic_f(asin((x - sqrt(S(3)) + S(1))/(x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(S(3)sqrt((x + S(1))/(x + S(1) + sqrt(S(3)))*S(2))sqrt(x*S(3) + S(1))) - xsqrt(a/x*S(4))sqrt(x*S(3) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax*(S(2)n))/sqrt(x*n + S(1)), x), x, xsqrt(ax(S(2)n))hyper((S(1)/2, S(1) + S(1)/n), (S(2) + S(1)/n,), -xn)/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axn)/sqrt(xn + S(1)), x), x, S(2)xsqrt(axn)hyper((S(1)/2, S(1)/2 + S(1)/n), (S(3)/2 + S(1)/n,), -x*n)/(n + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax(n/S(2)))/sqrt(xn + S(1)), x), x, S(4)xsqrt(ax(n/S(2)))hyper((S(1)/2, S(1)/4 + S(1)/n), (S(5)/4 + S(1)/n,), -x*n)/(n + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax*(S(2)n))/sqrt(x*n + S(1)) + S(2)x*(-n)sqrt(ax(S(2)n))/((n + S(2))sqrt(xn + S(1))), x), x, S(2)x*(-n + S(1))sqrt(ax(S(2)n))sqrt(xn + S(1))/(n + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax)/(sqrt(d + ex)sqrt(e + fx)), x), x, S(2)sqrt(ax)sqrt(e(e + fx)/(-df + eS(2)))sqrt(df - eS(2))elliptic_e(asin(sqrt(f)sqrt(d + ex)/sqrt(df - eS(2))), S(1) - eS(2)/(df))/(esqrt(f)sqrt(-ex/d)sqrt(e + fx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((axm)r, x), x, x(axm)r/(mr + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((axm)r(bxn)s, x), x, x(axm)r(bxn)s/(mr + ns + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((axm)r(bxn)s(cxp)t, x), x, x(axm)r(bxn)s(cxp)t/(mr + ns + pt + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxn)(S(1)/n)(a + b(cxn)(S(1)/n))*p, x), x, -ax(cxn)(-S(1)/n)(a + b(cxn)(S(1)/n))(p + S(1))/(bS(2)(p + S(1))) + x(cxn)(-S(1)/n)(a + b(cxn)(S(1)/n))(p + S(2))/(bS(2)(p + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxn)(S(1)/n)(a + b(cxn)(S(1)/n))*S(3), x), x, -ax(cxn)(-S(1)/n)(a + b(cxn)(S(1)/n))S(4)/(S(4)b*S(2)) + x(cxn)(-S(1)/n)(a + b(cxn)(S(1)/n))*S(5)/(S(5)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxn)(S(1)/n)(a + b(cxn)(S(1)/n))S(2), x), x, aS(2)x(cxn)(S(1)/n)/S(2) + S(2)abx(cxn)(S(2)/n)/S(3) + bS(2)x(cxn)(S(3)/n)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxn)(S(1)/n)(a + b(cxn)(S(1)/n)), x), x, ax(cxn)(S(1)/n)/S(2) + bx(cxn)(S(2)/n)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxn)(S(1)/n)/(a + b(cxn)(S(1)/n)), x), x, -ax(cxn)(-S(1)/n)log(a + b(cxn)(S(1)/n))/bS(2) + x/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxn)(S(1)/n)/(a + b(cxn)(S(1)/n))*S(2), x), x, ax(cxn)(-S(1)/n)/(b*S(2)(a + b(cxn)(S(1)/n))) + x(cxn)(-S(1)/n)log(a + b(cxn)(S(1)/n))/bS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxn)(S(1)/n)/(a + b(cxn)(S(1)/n))*S(3), x), x, x(cxn)(S(1)/n)/(S(2)a(a + b(cxn)(S(1)/n))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxn)(S(1)/n)/(a + b(cxn)(S(1)/n))*S(4), x), x, ax(cxn)(-S(1)/n)/(S(3)bS(2)(a + b(cxn)(S(1)/n))*S(3)) - x(cxn)(-S(1)/n)/(S(2)b*S(2)(a + b(cxn)(S(1)/n))*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((cxn)(S(1)/n)/(a + b(cxn)(S(1)/n))*S(5), x), x, ax(cxn)(-S(1)/n)/(S(4)bS(2)(a + b(cxn)(S(1)/n))*S(4)) - x(cxn)(-S(1)/n)/(S(3)b*S(2)(a + b(cxn)(S(1)/n))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(sqrt(a + bx) + sqrt(bx + c)), x), x, S(2)aS(2)(a + bx)(S(3)/2)/(S(3)b*S(3)(a - c)) - S(4)a(a + bx)(S(5)/2)/(S(5)b*S(3)(a - c)) - S(2)cS(2)(bx + c)(S(3)/2)/(S(3)b*S(3)(a - c)) + S(4)c(bx + c)(S(5)/2)/(S(5)b*S(3)(a - c)) + S(2)(a + bx)*(S(7)/2)/(S(7)b*S(3)(a - c)) - S(2)(bx + c)*(S(7)/2)/(S(7)b*S(3)(a - c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt(a + bx) + sqrt(bx + c)), x), x, -S(2)a(a + bx)(S(3)/2)/(S(3)b*S(2)(a - c)) + S(2)c(bx + c)(S(3)/2)/(S(3)b*S(2)(a - c)) + S(2)(a + bx)*(S(5)/2)/(S(5)b*S(2)(a - c)) - S(2)(bx + c)*(S(5)/2)/(S(5)b*S(2)(a - c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a + bx) + sqrt(bx + c)), x), x, S(2)(a + bx)*(S(3)/2)/(S(3)b(a - c)) - S(2)(bx + c)(S(3)/2)/(S(3)b(a - c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(sqrt(a + bx) + sqrt(bx + c))), x), x, -S(2)sqrt(a)atanh(sqrt(a + bx)/sqrt(a))/(a - c) + S(2)sqrt(c)atanh(sqrt(bx + c)/sqrt(c))/(a - c) + S(2)sqrt(a + bx)/(a - c) - S(2)sqrt(bx + c)/(a - c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(sqrt(a + bx) + sqrt(bx + c))), x), x, batanh(sqrt(bx + c)/sqrt(c))/(sqrt(c)(a - c)) - sqrt(a + bx)/(x(a - c)) + sqrt(bx + c)/(x(a - c)) - batanh(sqrt(a + bx)/sqrt(a))/(sqrt(a)(a - c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(sqrt(a + bx) + sqrt(bx + c))S(2), x), x, bx*S(4)/(S(2)(a - c)S(2)) + xS(3)(a + c)/(S(3)(a - c)*S(2)) - x(a + bx)(S(3)/2)(bx + c)(S(3)/2)/(S(2)b*S(2)(a - c)*S(2)) - (S(4)ac - S(5)(a + c)*S(2))atanh(sqrt(a + bx)/sqrt(bx + c))/(S(32)bS(3)) - sqrt(a + bx)(S(4)ac - S(5)(a + c)*S(2))sqrt(bx + c)/(S(32)b*S(3)(a - c)) + (a + bx)(S(3)/2)(S(5)a + S(5)c)(bx + c)*(S(3)/2)/(S(12)b*S(3)(a - c)*S(2)) + (a + bx)*(S(3)/2)(S(4)ac - S(5)(a + c)S(2))sqrt(bx + c)/(S(16)b*S(3)(a - c)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt(a + bx) + sqrt(bx + c))S(2), x), x, S(2)bxS(3)/(S(3)(a - c)S(2)) + xS(2)(a + c)/(S(2)(a - c)*S(2)) - (a + c)atanh(sqrt(a + bx)/sqrt(bx + c))/(S(4)bS(2)) - (a + c)sqrt(a + bx)sqrt(bx + c)/(S(4)b*S(2)(a - c)) + (a + c)(a + bx)*(S(3)/2)sqrt(bx + c)/(S(2)b*S(2)(a - c)*S(2)) - S(2)(a + bx)(S(3)/2)(bx + c)(S(3)/2)/(S(3)b*S(2)(a - c)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(a + bx) + sqrt(bx + c))(S(-2)), x), x, (a - c)S(2)/(S(8)b(sqrt(a + bx) + sqrt(bx + c))S(4)) + atanh(sqrt(a + bx)/sqrt(bx + c))/(S(2)b), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((sqrt(a + bx) + sqrt(bx + c))*(S(-2)), x), x, bxS(2)/(a - c)S(2) + x(a + c)/(a - c)S(2) + atanh(sqrt(a + bx)/sqrt(bx + c))/(S(2)b) + sqrt(a + bx)sqrt(bx + c)/(S(2)b(a - c)) - (a + bx)*(S(3)/2)sqrt(bx + c)/(b(a - c)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(sqrt(a + bx) + sqrt(bx + c))*S(2)), x), x, S(4)sqrt(a)sqrt(c)atanh(sqrt(c)sqrt(a + bx)/(sqrt(a)sqrt(bx + c)))/(a - c)*S(2) + S(2)bx/(a - c)S(2) + (a + c)log(x)/(a - c)*S(2) - S(2)sqrt(a + bx)sqrt(bx + c)/(a - c)S(2) - (S(2)a + S(2)c)atanh(sqrt(a + bx)/sqrt(bx + c))/(a - c)S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(sqrt(a + bx) + sqrt(bx + c))S(2)), x), x, S(2)blog(x)/(a - c)S(2) - S(4)batanh(sqrt(a + bx)/sqrt(bx + c))/(a - c)S(2) - (a + c)/(x(a - c)*S(2)) + S(2)sqrt(a + bx)sqrt(bx + c)/(x(a - c)*S(2)) + S(2)b(a + c)atanh(sqrt(c)sqrt(a + bx)/(sqrt(a)sqrt(bx + c)))/(sqrt(a)sqrt(c)(a - c)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(sqrt(a + bx) + sqrt(bx + c))*S(3), x), x, -S(8)a*S(3)(a + bx)(S(3)/2)/(S(3)b*S(3)(a - c)*S(3)) + S(2)a*S(2)(a + S(3)c)(a + bx)(S(3)/2)/(S(3)b*S(3)(a - c)*S(3)) + S(24)a*S(2)(a + bx)(S(5)/2)/(S(5)b*S(3)(a - c)*S(3)) - S(4)a(a + S(3)c)(a + bx)*(S(5)/2)/(S(5)b*S(3)(a - c)*S(3)) - S(24)a(a + bx)*(S(7)/2)/(S(7)b*S(3)(a - c)*S(3)) + S(8)c*S(3)(bx + c)(S(3)/2)/(S(3)b*S(3)(a - c)*S(3)) - S(2)c*S(2)(S(3)a + c)(bx + c)(S(3)/2)/(S(3)b*S(3)(a - c)*S(3)) - S(24)c*S(2)(bx + c)(S(5)/2)/(S(5)b*S(3)(a - c)*S(3)) + S(4)c(S(3)a + c)(bx + c)*(S(5)/2)/(S(5)b*S(3)(a - c)*S(3)) + S(24)c(bx + c)*(S(7)/2)/(S(7)b*S(3)(a - c)*S(3)) + S(8)(a + bx)(S(9)/2)/(S(9)b*S(3)(a - c)*S(3)) + (a + bx)*(S(7)/2)(S(2)a + S(6)c)/(S(7)bS(3)(a - c)*S(3)) - (S(6)a + S(2)c)(bx + c)(S(7)/2)/(S(7)b*S(3)(a - c)*S(3)) - S(8)(bx + c)(S(9)/2)/(S(9)b*S(3)(a - c)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt(a + bx) + sqrt(bx + c))S(3), x), x, S(8)a*S(2)(a + bx)(S(3)/2)/(S(3)b*S(2)(a - c)*S(3)) - S(2)a(a + S(3)c)(a + bx)*(S(3)/2)/(S(3)b*S(2)(a - c)*S(3)) - S(16)a(a + bx)*(S(5)/2)/(S(5)b*S(2)(a - c)*S(3)) - S(8)c*S(2)(bx + c)(S(3)/2)/(S(3)b*S(2)(a - c)*S(3)) + S(2)c(S(3)a + c)(bx + c)*(S(3)/2)/(S(3)b*S(2)(a - c)*S(3)) + S(16)c(bx + c)*(S(5)/2)/(S(5)b*S(2)(a - c)*S(3)) + S(8)(a + bx)(S(7)/2)/(S(7)b*S(2)(a - c)*S(3)) + (a + bx)*(S(5)/2)(S(2)a + S(6)c)/(S(5)bS(2)(a - c)*S(3)) - (S(6)a + S(2)c)(bx + c)(S(5)/2)/(S(5)b*S(2)(a - c)*S(3)) - S(8)(bx + c)(S(7)/2)/(S(7)b*S(2)(a - c)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(a + bx) + sqrt(bx + c))(S(-3)), x), x, (a - c)S(2)/(S(10)b(sqrt(a + bx) + sqrt(bx + c))S(5)) - S(1)/(S(2)b(sqrt(a + bx) + sqrt(bx + c))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((sqrt(a + bx) + sqrt(bx + c))(S(-3)), x), x, -S(8)a(a + bx)*(S(3)/2)/(S(3)b(a - c)S(3)) + S(8)c(bx + c)*(S(3)/2)/(S(3)b(a - c)S(3)) + S(8)(a + bx)(S(5)/2)/(S(5)b(a - c)S(3)) + (a + bx)*(S(3)/2)(S(2)a + S(6)c)/(S(3)b(a - c)*S(3)) - (S(6)a + S(2)c)(bx + c)(S(3)/2)/(S(3)b(a - c)S(3)) - S(8)(bx + c)(S(5)/2)/(S(5)b(a - c)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(sqrt(a + bx) + sqrt(bx + c))*S(3)), x), x, -S(2)sqrt(a)(a + S(3)c)atanh(sqrt(a + bx)/sqrt(a))/(a - c)*S(3) + S(2)sqrt(c)(S(3)a + c)atanh(sqrt(bx + c)/sqrt(c))/(a - c)*S(3) + S(8)(a + bx)(S(3)/2)/(S(3)(a - c)*S(3)) + sqrt(a + bx)(S(2)a + S(6)c)/(a - c)S(3) - (S(6)a + S(2)c)sqrt(bx + c)/(a - c)S(3) - S(8)(bx + c)(S(3)/2)/(S(3)(a - c)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(sqrt(a + bx) + sqrt(bx + c))S(3)), x), x, S(8)bsqrt(a + bx)/(a - c)*S(3) - S(8)bsqrt(bx + c)/(a - c)*S(3) - S(3)b(a + S(3)c)atanh(sqrt(bx + c)/sqrt(c))/(sqrt(c)(-a + c)S(3)) - (a + S(3)c)sqrt(a + bx)/(x(a - c)S(3)) + (S(3)a + c)sqrt(bx + c)/(x(a - c)S(3)) - S(3)b(S(3)a + c)atanh(sqrt(a + bx)/sqrt(a))/(sqrt(a)(a - c)S(3)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/(xS(2)(sqrt(a + bx) + sqrt(bx + c))*S(3)), x), x, -S(8)sqrt(a)batanh(sqrt(a + bx)/sqrt(a))/(a - c)S(3) + S(8)bsqrt(c)atanh(sqrt(bx + c)/sqrt(c))/(a - c)S(3) + S(8)bsqrt(a + bx)/(a - c)*S(3) - S(8)bsqrt(bx + c)/(a - c)*S(3) + b(S(3)a + c)atanh(sqrt(bx + c)/sqrt(c))/(sqrt(c)(a - c)*S(3)) - (a + S(3)c)sqrt(a + bx)/(x(a - c)S(3)) + (S(3)a + c)sqrt(bx + c)/(x(a - c)S(3)) - b(a + S(3)c)atanh(sqrt(a + bx)/sqrt(a))/(sqrt(a)(a - c)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x) + sqrt(x + S(1))), x), x, -S(2)x*(S(3)/2)/S(3) + S(2)(x + S(1))*(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x) + sqrt(x + S(-1))), x), x, S(2)x*(S(3)/2)/S(3) - S(2)(x + S(-1))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x + S(-1)) + sqrt(x + S(1))), x), x, -(x + S(-1))(S(3)/2)/S(3) + (x + S(1))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(sqrt(-x + S(1)) + sqrt(x + S(1)))S(2), x), x, xS(4)/S(2) + S(2)(-xS(2) + S(1))(S(5)/2)/S(5) - S(2)(-xS(2) + S(1))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(sqrt(-x + S(1)) + sqrt(x + S(1)))S(2), x), x, xS(3)sqrt(-xS(2) + S(1))/S(2) + S(2)x*S(3)/S(3) - xsqrt(-x*S(2) + S(1))/S(4) + asin(x)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(sqrt(-x + S(1)) + sqrt(x + S(1)))S(2), x), x, xS(2) - S(2)(-xS(2) + S(1))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(-x + S(1)) + sqrt(x + S(1)))S(2), x), x, xsqrt(-x*S(2) + S(1)) + S(2)x + asin(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(-x + S(1)) + sqrt(x + S(1)))*S(2)/x, x), x, S(2)sqrt(-x*S(2) + S(1)) + S(2)log(x) - S(2)atanh(sqrt(-xS(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(-x + S(1)) + sqrt(x + S(1)))S(2)/xS(2), x), x, -S(2)asin(x) - S(2)sqrt(-xS(2) + S(1))/x - S(2)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(-x + S(1)) + sqrt(x + S(1)))S(2)/xS(3), x), x, atanh(sqrt(-xS(2) + S(1))) - sqrt(-xS(2) + S(1))/xS(2) - S(1)/xS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(sqrt(a + bx) + sqrt(a + cx)), x), x, -S(2)a*S(2)(a + cx)(S(3)/2)/(cS(3)(S(3)b - S(3)c)) + S(2)aS(2)(a + bx)(S(3)/2)/(S(3)b*S(3)(b - c)) + S(4)a(a + cx)(S(5)/2)/(cS(3)(S(5)b - S(5)c)) - S(4)a(a + bx)(S(5)/2)/(S(5)b*S(3)(b - c)) - S(2)(a + cx)(S(7)/2)/(cS(3)(S(7)b - S(7)c)) + S(2)(a + bx)(S(7)/2)/(S(7)b*S(3)(b - c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(sqrt(a + bx) + sqrt(a + cx)), x), x, S(2)a(a + cx)(S(3)/2)/(cS(2)(S(3)b - S(3)c)) - S(2)a(a + bx)*(S(3)/2)/(S(3)b*S(2)(b - c)) - S(2)(a + cx)(S(5)/2)/(cS(2)(S(5)b - S(5)c)) + S(2)(a + bx)(S(5)/2)/(S(5)b*S(2)(b - c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt(a + bx) + sqrt(a + cx)), x), x, -S(2)(a + cx)*(S(3)/2)/(c(S(3)b - S(3)c)) + S(2)(a + bx)*(S(3)/2)/(S(3)b(b - c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a + bx) + sqrt(a + cx)), x), x, -S(2)sqrt(a)atanh(sqrt(a + bx)/sqrt(a))/(b - c) + S(2)sqrt(a)atanh(sqrt(a + cx)/sqrt(a))/(b - c) + S(2)sqrt(a + bx)/(b - c) - S(2)sqrt(a + cx)/(b - c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(sqrt(a + bx) + sqrt(a + cx))), x), x, -sqrt(a + bx)/(x(b - c)) + sqrt(a + cx)/(x(b - c)) - batanh(sqrt(a + bx)/sqrt(a))/(sqrt(a)(b - c)) + catanh(sqrt(a + cx)/sqrt(a))/(sqrt(a)(b - c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(sqrt(a + bx) + sqrt(a + cx))), x), x, -sqrt(a + bx)/(xS(2)(S(2)b - S(2)c)) + sqrt(a + cx)/(xS(2)(S(2)b - S(2)c)) - bsqrt(a + bx)/(S(4)ax(b - c)) + csqrt(a + cx)/(S(4)ax(b - c)) + b*S(2)atanh(sqrt(a + bx)/sqrt(a))/(S(4)a*(S(3)/2)(b - c)) - c*S(2)atanh(sqrt(a + cx)/sqrt(a))/(S(4)a*(S(3)/2)(b - c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/(sqrt(a + bx) + sqrt(a + cx))S(2), x), x, -aS(3)(b + c)atanh(sqrt(c)sqrt(a + bx)/(sqrt(b)sqrt(a + cx)))/(S(4)b*(S(5)/2)c(S(5)/2)) + aS(2)sqrt(a + bx)sqrt(a + cx)(b + c)/(S(4)b*S(2)c*S(2)(b - c)) + axS(2)/(b - c)S(2) + a(a + bx)(S(3)/2)sqrt(a + cx)(b + c)/(S(2)bS(2)c(b - c)S(2)) + xS(3)(b + c)/(S(3)(b - c)S(2)) - S(2)(a + bx)(S(3)/2)(a + cx)(S(3)/2)/(S(3)bc(b - c)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(sqrt(a + bx) + sqrt(a + cx))S(2), x), x, aS(2)atanh(sqrt(c)sqrt(a + bx)/(sqrt(b)sqrt(a + cx)))/(S(2)b*(S(3)/2)c*(S(3)/2)) + S(2)ax/(b - c)S(2) - asqrt(a + bx)sqrt(a + cx)/(S(2)bc(b - c)) + x*S(2)(b + c)/(S(2)(b - c)S(2)) - (a + bx)*(S(3)/2)sqrt(a + cx)/(b(b - c)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt(a + bx) + sqrt(a + cx))S(2), x), x, S(2)alog(x)/(b - c)S(2) + S(4)aatanh(sqrt(a + bx)/sqrt(a + cx))/(b - c)S(2) - S(2)a(b + c)atanh(sqrt(c)sqrt(a + bx)/(sqrt(b)sqrt(a + cx)))/(sqrt(b)sqrt(c)(b - c)*S(2)) + x(b + c)/(b - c)*S(2) - S(2)sqrt(a + bx)sqrt(a + cx)/(b - c)S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(a + bx) + sqrt(a + cx))(S(-2)), x), x, -S(2)a/(x(b - c)S(2)) - S(4)sqrt(b)sqrt(c)atanh(sqrt(c)sqrt(a + bx)/(sqrt(b)sqrt(a + cx)))/(b - c)*S(2) + (b + c)log(x)/(b - c)*S(2) + (S(2)b + S(2)c)atanh(sqrt(a + bx)/sqrt(a + cx))/(b - c)*S(2) + S(2)sqrt(a + bx)sqrt(a + cx)/(x(b - c)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(sqrt(a + bx) + sqrt(a + cx))S(2)), x), x, -a/(xS(2)(b - c)S(2)) - (b + c)/(x(b - c)*S(2)) - atanh(sqrt(a + bx)/sqrt(a + cx))/(S(2)a) + sqrt(a + bx)sqrt(a + cx)/(S(2)ax(b - c)) + sqrt(a + bx)(a + cx)(S(3)/2)/(ax*S(2)(b - c)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(sqrt(a + bx) + sqrt(a + cx))S(2)), x), x, -S(2)a/(S(3)xS(3)(b - c)*S(2)) - (b + c)/(S(2)x*S(2)(b - c)*S(2)) + (b + c)atanh(sqrt(a + bx)/sqrt(a + cx))/(S(4)aS(2)) - sqrt(a + bx)sqrt(a + cx)(b + c)/(S(4)a*S(2)x(b - c)) - sqrt(a + bx)(a + cx)*(S(3)/2)(b + c)/(S(2)aS(2)x*S(2)(b - c)*S(2)) + S(2)(a + bx)(S(3)/2)(a + cx)(S(3)/2)/(S(3)a*S(2)x*S(3)(b - c)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/(sqrt(a + bx) + sqrt(a + cx))*S(3), x), x, S(8)a*S(2)(a + cx)(S(3)/2)/(S(3)c*S(2)(b - c)*S(3)) - S(2)a*S(2)(a + cx)(S(3)/2)(S(3)b + c)/(S(3)c*S(3)(b - c)*S(3)) - S(8)a*S(2)(a + bx)(S(3)/2)/(S(3)b*S(2)(b - c)*S(3)) + S(2)a*S(2)(a + bx)(S(3)/2)(b + S(3)c)/(S(3)b*S(3)(b - c)*S(3)) - S(8)a(a + cx)*(S(5)/2)/(S(5)c*S(2)(b - c)*S(3)) + S(4)a(a + cx)*(S(5)/2)(S(3)b + c)/(S(5)c*S(3)(b - c)*S(3)) + S(8)a(a + bx)*(S(5)/2)/(S(5)b*S(2)(b - c)*S(3)) - S(4)a(a + bx)*(S(5)/2)(b + S(3)c)/(S(5)b*S(3)(b - c)*S(3)) - (a + cx)*(S(7)/2)(S(6)b + S(2)c)/(S(7)cS(3)(b - c)*S(3)) + (a + bx)*(S(7)/2)(S(2)b + S(6)c)/(S(7)bS(3)(b - c)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(sqrt(a + bx) + sqrt(a + cx))*S(3), x), x, -S(8)a(a + cx)*(S(3)/2)/(S(3)c(b - c)S(3)) + S(2)a(a + cx)*(S(3)/2)(S(3)b + c)/(S(3)c*S(2)(b - c)*S(3)) + S(8)a(a + bx)*(S(3)/2)/(S(3)b(b - c)S(3)) - S(2)a(a + bx)*(S(3)/2)(b + S(3)c)/(S(3)b*S(2)(b - c)*S(3)) - (a + cx)*(S(5)/2)(S(6)b + S(2)c)/(S(5)cS(2)(b - c)*S(3)) + (a + bx)*(S(5)/2)(S(2)b + S(6)c)/(S(5)bS(2)(b - c)S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(sqrt(a + bx) + sqrt(a + cx))*S(3), x), x, -S(8)a*(S(3)/2)atanh(sqrt(a + bx)/sqrt(a))/(b - c)S(3) + S(8)a*(S(3)/2)atanh(sqrt(a + cx)/sqrt(a))/(b - c)S(3) + S(8)asqrt(a + bx)/(b - c)*S(3) - S(8)asqrt(a + cx)/(b - c)*S(3) - (a + cx)*(S(3)/2)(S(6)b + S(2)c)/(S(3)c(b - c)*S(3)) + (a + bx)*(S(3)/2)(S(2)b + S(6)c)/(S(3)b(b - c)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt(a + bx) + sqrt(a + cx))S(3), x), x, -S(6)sqrt(a)(b + c)atanh(sqrt(a + bx)/sqrt(a))/(b - c)S(3) + S(6)sqrt(a)(b + c)atanh(sqrt(a + cx)/sqrt(a))/(b - c)S(3) - S(4)asqrt(a + bx)/(x(b - c)S(3)) + S(4)asqrt(a + cx)/(x(b - c)S(3)) + sqrt(a + bx)(S(2)b + S(6)c)/(b - c)S(3) - sqrt(a + cx)(S(6)b + S(2)c)/(b - c)S(3), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(x/(sqrt(a + bx) + sqrt(a + cx))S(3), x), x, -S(4)sqrt(a)batanh(sqrt(a + bx)/sqrt(a))/(b - c)S(3) + S(4)sqrt(a)catanh(sqrt(a + cx)/sqrt(a))/(b - c)S(3) - S(2)sqrt(a)(b + S(3)c)atanh(sqrt(a + bx)/sqrt(a))/(b - c)*S(3) + S(2)sqrt(a)(S(3)b + c)atanh(sqrt(a + cx)/sqrt(a))/(b - c)*S(3) - S(4)asqrt(a + bx)/(x(b - c)S(3)) + S(4)asqrt(a + cx)/(x(b - c)S(3)) + sqrt(a + bx)(S(2)b + S(6)c)/(b - c)S(3) - sqrt(a + cx)(S(6)b + S(2)c)/(b - c)S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(a + bx) + sqrt(a + cx))(S(-3)), x), x, -S(2)asqrt(a + bx)/(x*S(2)(b - c)*S(3)) + S(2)asqrt(a + cx)/(x*S(2)(b - c)*S(3)) - sqrt(a + bx)(S(2)b + S(3)c)/(x(b - c)*S(3)) + sqrt(a + cx)(S(3)b + S(2)c)/(x(b - c)*S(3)) - S(3)bcatanh(sqrt(a + bx)/sqrt(a))/(sqrt(a)(b - c)*S(3)) + S(3)bcatanh(sqrt(a + cx)/sqrt(a))/(sqrt(a)(b - c)*S(3)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((sqrt(a + bx) + sqrt(a + cx))(S(-3)), x), x, -S(2)asqrt(a + bx)/(x*S(2)(b - c)*S(3)) + S(2)asqrt(a + cx)/(x*S(2)(b - c)*S(3)) - bsqrt(a + bx)/(x(b - c)*S(3)) + csqrt(a + cx)/(x(b - c)*S(3)) - sqrt(a + bx)(b + S(3)c)/(x(b - c)S(3)) + sqrt(a + cx)(S(3)b + c)/(x(b - c)S(3)) + bS(2)atanh(sqrt(a + bx)/sqrt(a))/(sqrt(a)(b - c)*S(3)) - b(b + S(3)c)atanh(sqrt(a + bx)/sqrt(a))/(sqrt(a)(b - c)S(3)) - cS(2)atanh(sqrt(a + cx)/sqrt(a))/(sqrt(a)(b - c)S(3)) + c(S(3)b + c)atanh(sqrt(a + cx)/sqrt(a))/(sqrt(a)(b - c)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-x + S(1))(sqrt(-x + S(1)) + sqrt(x + S(1))), x), x, -x*S(2)/S(2) + xsqrt(-xS(2) + S(1))/S(2) + x + asin(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(-sqrt(-x + S(1)) - sqrt(x + S(1)))(sqrt(-x + S(1)) + sqrt(x + S(1))), x), x, -x*S(4)/S(2) - S(2)(-xS(2) + S(1))(S(5)/2)/S(5) + S(2)(-xS(2) + S(1))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(-sqrt(-x + S(1)) - sqrt(x + S(1)))(sqrt(-x + S(1)) + sqrt(x + S(1))), x), x, -xS(3)sqrt(-x*S(2) + S(1))/S(2) - S(2)x*S(3)/S(3) + xsqrt(-x*S(2) + S(1))/S(4) - asin(x)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-sqrt(-x + S(1)) - sqrt(x + S(1)))(sqrt(-x + S(1)) + sqrt(x + S(1))), x), x, -xS(2) + S(2)(-xS(2) + S(1))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-sqrt(-x + S(1)) - sqrt(x + S(1)))(sqrt(-x + S(1)) + sqrt(x + S(1))), x), x, -xsqrt(-x*S(2) + S(1)) - S(2)x - asin(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-sqrt(-x + S(1)) - sqrt(x + S(1)))(sqrt(-x + S(1)) + sqrt(x + S(1)))/x, x), x, -S(2)sqrt(-x*S(2) + S(1)) - S(2)log(x) + S(2)atanh(sqrt(-xS(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-sqrt(-x + S(1)) - sqrt(x + S(1)))(sqrt(-x + S(1)) + sqrt(x + S(1)))/x*S(2), x), x, S(2)asin(x) + S(2)sqrt(-xS(2) + S(1))/x + S(2)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-sqrt(-x + S(1)) - sqrt(x + S(1)))(sqrt(-x + S(1)) + sqrt(x + S(1)))/xS(3), x), x, -atanh(sqrt(-xS(2) + S(1))) + sqrt(-xS(2) + S(1))/xS(2) + x(S(-2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(-x + S(1)) + sqrt(x + S(1)))/(-sqrt(-x + S(1)) + sqrt(x + S(1))), x), x, sqrt(-xS(2) + S(1)) + log(x) - atanh(sqrt(-xS(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-sqrt(x + S(-1)) + sqrt(x + S(1)))/(sqrt(x + S(-1)) + sqrt(x + S(1))), x), x, xS(2)/S(2) - xsqrt(x + S(-1))sqrt(x + S(1))/S(2) + acosh(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*n, x), x, af*S(2)(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*(n + S(1))hyper((S(2), n + S(1)), (n + S(2),), (d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/d)/(S(2)dS(2)e(n + S(1))) + (d + ex + fsqrt(a + eS(2)xS(2)/fS(2)))*(n + S(1))/(S(2)e(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + eS(2)xS(2)/fS(2)))*S(3), x), x, -ad*S(3)f*S(2)/(S(2)e(ex + fsqrt(a + eS(2)xS(2)/fS(2)))) + S(3)ad*S(2)f*S(2)log(ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(S(2)e) + adfS(2)(ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/e + afS(2)(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*S(2)/(S(4)e) + (d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*S(4)/(S(8)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*S(2), x), x, -ad*S(2)f*S(2)/(S(2)e(ex + fsqrt(a + eS(2)xS(2)/fS(2)))) + adf*S(2)log(ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/e + afS(2)(ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(S(2)e) + (d + ex + fsqrt(a + eS(2)xS(2)/fS(2)))*S(3)/(S(6)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)), x), x, afS(2)atanh(ex/(fsqrt(a + e*S(2)xS(2)/fS(2))))/(S(2)e) + dx + exS(2)/S(2) + fxsqrt(a + eS(2)xS(2)/fS(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2))), x), x, -afS(2)/(S(2)de(ex + fsqrt(a + e*S(2)xS(2)/fS(2)))) - afS(2)log(ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(S(2)dS(2)e) + (afS(2)/dS(2) + S(1))log(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(S(2)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + eS(2)xS(2)/fS(2)))*(S(-2)), x), x, -af*S(2)/(S(2)d*S(2)e(ex + fsqrt(a + eS(2)xS(2)/fS(2)))) - afS(2)log(ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(d*S(3)e) + afS(2)log(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(d*S(3)e) - (afS(2)/dS(2) + S(1))/(S(2)e(d + ex + fsqrt(a + eS(2)xS(2)/fS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*(S(-3)), x), x, -afS(2)/(dS(3)e(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))) - afS(2)/(S(2)d*S(3)e(ex + fsqrt(a + eS(2)xS(2)/fS(2)))) - S(3)af*S(2)log(ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(S(2)dS(4)e) + S(3)af*S(2)log(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(S(2)dS(4)e) - (afS(2)/dS(2) + S(1))/(S(4)e(d + ex + fsqrt(a + eS(2)xS(2)/fS(2)))*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + eS(2)xS(2)/fS(2)))*(S(5)/2), x), x, -S(5)ad(S(3)/2)f*S(2)atanh(sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/sqrt(d))/(S(2)e) - ad*S(2)f*S(2)sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(S(2)e(ex + fsqrt(a + e*S(2)xS(2)/fS(2)))) + S(2)adfS(2)sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/e + afS(2)(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*(S(3)/2)/(S(3)e) + (d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*(S(7)/2)/(S(7)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*(S(3)/2), x), x, -S(3)asqrt(d)f*S(2)atanh(sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/sqrt(d))/(S(2)e) - adfS(2)sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(S(2)e(ex + fsqrt(a + e*S(2)xS(2)/fS(2)))) + afS(2)sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/e + (d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*(S(5)/2)/(S(5)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2))), x), x, -afS(2)sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(S(2)e(ex + fsqrt(a + e*S(2)xS(2)/fS(2)))) - afS(2)atanh(sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/sqrt(d))/(S(2)sqrt(d)e) + (d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*(S(3)/2)/(S(3)e), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2))), x), x, -afS(2)sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(S(2)de(ex + fsqrt(a + eS(2)xS(2)/fS(2)))) + afS(2)atanh(sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/sqrt(d))/(S(2)d(S(3)/2)e) + sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/e, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*(S(-3)/2), x), x, -af*S(2)sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(S(2)dS(2)e(ex + fsqrt(a + eS(2)xS(2)/fS(2)))) + S(3)af*S(2)atanh(sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/sqrt(d))/(S(2)d(S(5)/2)e) - (afS(2)/dS(2) + S(1))/(esqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))*(S(-5)/2), x), x, -S(2)afS(2)/(dS(3)esqrt(d + ex + fsqrt(a + eS(2)xS(2)/fS(2)))) - afS(2)sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/(S(2)dS(3)e(ex + fsqrt(a + eS(2)xS(2)/fS(2)))) + S(5)af*S(2)atanh(sqrt(d + ex + fsqrt(a + e*S(2)xS(2)/fS(2)))/sqrt(d))/(S(2)d(S(7)/2)e) - (afS(2)/dS(2) + S(1))/(S(3)e(d + ex + fsqrt(a + eS(2)xS(2)/fS(2)))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x - sqrt(xS(2) + S(-4))), x), x, (x - sqrt(xS(2) + S(-4)))(S(3)/2)/S(3) + S(4)/sqrt(x - sqrt(x*S(2) + S(-4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(ax + bsqrt(aS(2)xS(2)/bS(2) + c)), x), x, -b*S(2)c/(asqrt(ax + bsqrt(aS(2)xS(2)/bS(2) + c))) + (ax + bsqrt(a*S(2)xS(2)/bS(2) + c))*(S(3)/2)/(S(3)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sqrt(-x*S(2) + S(1)) + S(1)), x), x, -S(2)x*S(3)/(S(3)(sqrt(-xS(2) + S(1)) + S(1))(S(3)/2)) + S(2)x/sqrt(sqrt(-xS(2) + S(1)) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sqrt(xS(2) + S(1)) + S(1)), x), x, S(2)x*S(3)/(S(3)(sqrt(xS(2) + S(1)) + S(1))(S(3)/2)) + S(2)x/sqrt(sqrt(xS(2) + S(1)) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sqrt(xS(2) + S(25)) + S(5)), x), x, S(2)x*S(3)/(S(3)(sqrt(xS(2) + S(25)) + S(5))(S(3)/2)) + S(10)x/sqrt(sqrt(xS(2) + S(25)) + S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(aS(2)/bS(2) + cxS(2))), x), x, S(2)ax/sqrt(a + bsqrt(aS(2)/bS(2) + cxS(2))) + S(2)b*S(2)cxS(3)/(S(3)(a + bsqrt(aS(2)/bS(2) + cxS(2)))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))n, x), x, fS(2)(S(4)aeS(2) - bS(2)f*S(2))(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))*(n + S(1))hyper((S(2), n + S(1)), (n + S(2),), S(2)e(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/(-bfS(2) + S(2)de))/(S(2)e(n + S(1))(-bfS(2) + S(2)de)S(2)) + (d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))*(n + S(1))/(S(2)e(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))*S(3), x), x, (d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))*S(4)/(S(8)e) + f*S(2)(S(4)aeS(2) - bS(2)fS(2))(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))*S(2)/(S(16)eS(3)) + fS(2)(S(4)aeS(2) - bS(2)f*S(2))(-bfS(2) + S(2)de)(ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/(S(8)eS(4)) + fS(2)(S(4)aeS(2) - bS(2)fS(2))(-bfS(2) + S(2)de)S(3)/(S(32)e*S(5)(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))) + S(3)fS(2)(S(4)aeS(2) - bS(2)fS(2))(-bfS(2) + S(2)de)S(2)log(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))/(S(32)eS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))*S(2), x), x, (d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))*S(3)/(S(6)e) + f*S(2)(S(4)aeS(2) - bS(2)fS(2))(ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/(S(8)eS(3)) + fS(2)(S(4)aeS(2) - bS(2)fS(2))(-bfS(2) + S(2)de)S(2)/(S(16)e*S(4)(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))) + f*S(2)(S(4)aeS(2) - bS(2)fS(2))(-bfS(2) + S(2)de)log(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))/(S(8)eS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)), x), x, dx + ex*S(2)/S(2) + f(bfS(2) + S(2)e*S(2)x)sqrt(a + bx + e*S(2)xS(2)/fS(2))/(S(4)eS(2)) + fS(2)(S(4)aeS(2) - bS(2)fS(2))atanh((bfS(2) + S(2)e*S(2)x)/(S(2)efsqrt(a + bx + e*S(2)xS(2)/fS(2))))/(S(8)eS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))), x), x, (S(2)aefS(2) - S(2)bdf*S(2) + S(2)d*S(2)e)log(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))/(-bfS(2) + S(2)de)S(2) + fS(2)(S(4)aeS(2) - bS(2)fS(2))/(S(2)e(-bf*S(2) + S(2)de)(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))) - f*S(2)(S(4)aeS(2) - bS(2)fS(2))log(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))/(S(2)e(-bfS(2) + S(2)de)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))(S(-2)), x), x, fS(2)(S(4)aeS(2) - bS(2)f*S(2))/((-bf*S(2) + S(2)de)S(2)(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))) + S(2)fS(2)(S(4)aeS(2) - bS(2)fS(2))log(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/(-bfS(2) + S(2)de)S(3) - S(2)f*S(2)(S(4)aeS(2) - bS(2)fS(2))log(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))/(-bfS(2) + S(2)de)S(3) - (S(2)aef*S(2) - S(2)bdf*S(2) + S(2)d*S(2)e)/((-bfS(2) + S(2)de)S(2)(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))*(S(-3)), x), x, S(2)efS(2)(S(4)aeS(2) - bS(2)fS(2))/((-bf*S(2) + S(2)de)S(3)(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))) + S(6)ef*S(2)(S(4)aeS(2) - bS(2)fS(2))log(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/(-bfS(2) + S(2)de)S(4) - S(6)efS(2)(S(4)aeS(2) - bS(2)fS(2))log(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))/(-bfS(2) + S(2)de)S(4) - S(2)f*S(2)(S(4)aeS(2) - bS(2)fS(2))/((-bf*S(2) + S(2)de)S(3)(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))) - (aef*S(2) - bdfS(2) + dS(2)e)/((-bfS(2) + S(2)de)S(2)(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))*(S(5)/2), x), x, (d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))*(S(7)/2)/(S(7)e) + f*S(2)(S(4)aeS(2) - bS(2)fS(2))(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))*(S(3)/2)/(S(12)eS(3)) + fS(2)(S(4)aeS(2) - bS(2)f*S(2))(-bfS(2) + S(2)de)S(2)sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/(S(16)eS(4)(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))) + f*S(2)(S(4)aeS(2) - bS(2)fS(2))(-bfS(2) + S(2)de)sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/(S(4)eS(4)) - S(5)sqrt(S(2))fS(2)(S(4)aeS(2) - bS(2)fS(2))(-bfS(2) + S(2)de)(S(3)/2)atanh(sqrt(S(2))sqrt(e)sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/sqrt(-bfS(2) + S(2)de))/(S(32)e*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))*(S(3)/2), x), x, (d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))*(S(5)/2)/(S(5)e) + f*S(2)(S(4)aeS(2) - bS(2)fS(2))(-bfS(2) + S(2)de)sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/(S(8)eS(3)(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))) + f*S(2)(S(4)aeS(2) - bS(2)fS(2))sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/(S(4)eS(3)) - S(3)sqrt(S(2))fS(2)(S(4)aeS(2) - bS(2)fS(2))sqrt(-bfS(2) + S(2)de)atanh(sqrt(S(2))sqrt(e)sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/sqrt(-bfS(2) + S(2)de))/(S(16)e*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))), x), x, f*S(2)(S(4)a - bS(2)fS(2)/eS(2))sqrt(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))/(-S(4)bf*S(2) + S(8)de - S(8)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))) + (d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))*(S(3)/2)/(S(3)e) - sqrt(S(2))fS(2)(S(4)aeS(2) - bS(2)fS(2))atanh(sqrt(S(2))sqrt(e)sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/sqrt(-bfS(2) + S(2)de))/(S(8)e*(S(5)/2)sqrt(-bfS(2) + S(2)de)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))), x), x, f*S(2)(S(4)aeS(2) - bS(2)fS(2))sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/(S(2)e(-bfS(2) + S(2)de)(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))) + sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/e + sqrt(S(2))fS(2)(S(4)aeS(2) - bS(2)fS(2))atanh(sqrt(S(2))sqrt(e)sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/sqrt(-bfS(2) + S(2)de))/(S(4)e*(S(3)/2)(-bfS(2) + S(2)de)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))(S(-3)/2), x), x, fS(2)(S(4)aeS(2) - bS(2)f*S(2))sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/((-bfS(2) + S(2)de)S(2)(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))) - (S(4)aefS(2) - S(4)bdf*S(2) + S(4)d*S(2)e)/((-bfS(2) + S(2)de)S(2)sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))) + S(3)sqrt(S(2))f*S(2)(S(4)aeS(2) - bS(2)fS(2))atanh(sqrt(S(2))sqrt(e)sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/sqrt(-bfS(2) + S(2)de))/(S(2)sqrt(e)(-bf*S(2) + S(2)de)(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2)))*(S(-5)/2), x), x, S(5)sqrt(S(2))sqrt(e)f*S(2)(S(4)aeS(2) - bS(2)fS(2))atanh(sqrt(S(2))sqrt(e)sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/sqrt(-bfS(2) + S(2)de))/(-bf*S(2) + S(2)de)(S(7)/2) + S(2)efS(2)(S(4)aeS(2) - bS(2)fS(2))sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))/((-bfS(2) + S(2)de)S(3)(-bfS(2) + S(2)de - S(2)e(d + ex + fsqrt(a + bx + e*S(2)xS(2)/fS(2))))) - S(4)fS(2)(S(4)aeS(2) - bS(2)fS(2))/((-bf*S(2) + S(2)de)S(3)sqrt(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))) - (S(4)aefS(2) - S(4)bdf*S(2) + S(4)d*S(2)e)/(S(3)(-bf*S(2) + S(2)de)S(2)(d + ex + fsqrt(a + bx + eS(2)xS(2)/fS(2)))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + xS(2))*S(2)(x + sqrt(a + xS(2)))n, x), x, -a*S(5)(x + sqrt(a + xS(2)))(n + S(-5))/(-S(32)n + S(160)) - S(5)a*S(4)(x + sqrt(a + xS(2)))(n + S(-3))/(-S(32)n + S(96)) - S(5)a*S(3)(x + sqrt(a + xS(2)))(n + S(-1))/(-S(16)n + S(16)) + S(5)a*S(2)(x + sqrt(a + xS(2)))(n + S(1))/(S(16)n + S(16)) + S(5)a(x + sqrt(a + xS(2)))(n + S(3))/(S(32)n + S(96)) + (x + sqrt(a + xS(2)))(n + S(5))/(S(32)n + S(160)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + xS(2))(x + sqrt(a + xS(2)))n, x), x, -a*S(3)(x + sqrt(a + xS(2)))(n + S(-3))/(-S(8)n + S(24)) - S(3)a*S(2)(x + sqrt(a + xS(2)))(n + S(-1))/(-S(8)n + S(8)) + S(3)a(x + sqrt(a + xS(2)))(n + S(1))/(S(8)n + S(8)) + (x + sqrt(a + xS(2)))(n + S(3))/(S(8)n + S(24)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(a + xS(2)))n, x), x, -a(x + sqrt(a + xS(2)))(n + S(-1))/(-S(2)n + S(2)) + (x + sqrt(a + xS(2)))(n + S(1))/(S(2)n + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(a + xS(2)))n/(a + x*S(2)), x), x, S(2)(x + sqrt(a + xS(2)))(n + S(1))hyper((S(1), n/S(2) + S(1)/2), (n/S(2) + S(3)/2,), -(x + sqrt(a + xS(2)))S(2)/a)/(a(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(a + xS(2)))n/(a + xS(2))S(2), x), x, S(8)(x + sqrt(a + xS(2)))(n + S(3))hyper((S(3), n/S(2) + S(3)/2), (n/S(2) + S(5)/2,), -(x + sqrt(a + xS(2)))S(2)/a)/(a*S(3)(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + xS(2))S(2)(x - sqrt(a + xS(2)))n, x), x, -aS(5)(x - sqrt(a + xS(2)))(n + S(-5))/(-S(32)n + S(160)) - S(5)a*S(4)(x - sqrt(a + xS(2)))(n + S(-3))/(-S(32)n + S(96)) - S(5)a*S(3)(x - sqrt(a + xS(2)))(n + S(-1))/(-S(16)n + S(16)) + S(5)a*S(2)(x - sqrt(a + xS(2)))(n + S(1))/(S(16)n + S(16)) + S(5)a(x - sqrt(a + xS(2)))(n + S(3))/(S(32)n + S(96)) + (x - sqrt(a + xS(2)))(n + S(5))/(S(32)n + S(160)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + xS(2))(x - sqrt(a + xS(2)))n, x), x, -a*S(3)(x - sqrt(a + xS(2)))(n + S(-3))/(-S(8)n + S(24)) - S(3)a*S(2)(x - sqrt(a + xS(2)))(n + S(-1))/(-S(8)n + S(8)) + S(3)a(x - sqrt(a + xS(2)))(n + S(1))/(S(8)n + S(8)) + (x - sqrt(a + xS(2)))(n + S(3))/(S(8)n + S(24)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x - sqrt(a + xS(2)))n, x), x, -a(x - sqrt(a + xS(2)))(n + S(-1))/(-S(2)n + S(2)) + (x - sqrt(a + xS(2)))(n + S(1))/(S(2)n + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x - sqrt(a + xS(2)))n/(a + x*S(2)), x), x, S(2)(x - sqrt(a + xS(2)))(n + S(1))hyper((S(1), n/S(2) + S(1)/2), (n/S(2) + S(3)/2,), -(x - sqrt(a + xS(2)))S(2)/a)/(a(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x - sqrt(a + xS(2)))n/(a + xS(2))S(2), x), x, S(8)(x - sqrt(a + xS(2)))(n + S(3))hyper((S(3), n/S(2) + S(3)/2), (n/S(2) + S(5)/2,), -(x - sqrt(a + xS(2)))S(2)/a)/(a*S(3)(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + xS(2))(S(5)/2)(x + sqrt(a + xS(2)))n, x), x, -aS(6)(x + sqrt(a + xS(2)))(n + S(-6))/(-S(64)n + S(384)) - S(3)a*S(5)(x + sqrt(a + xS(2)))(n + S(-4))/(-S(32)n + S(128)) - S(15)a*S(4)(x + sqrt(a + xS(2)))(n + S(-2))/(-S(64)n + S(128)) + S(5)a*S(3)(x + sqrt(a + xS(2)))n/(S(16)n) + S(15)a*S(2)(x + sqrt(a + xS(2)))(n + S(2))/(S(64)n + S(128)) + S(3)a(x + sqrt(a + xS(2)))(n + S(4))/(S(32)n + S(128)) + (x + sqrt(a + xS(2)))(n + S(6))/(S(64)n + S(384)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + xS(2))(S(3)/2)(x + sqrt(a + xS(2)))n, x), x, -a*S(4)(x + sqrt(a + xS(2)))(n + S(-4))/(-S(16)n + S(64)) - aS(3)(x + sqrt(a + xS(2)))(n + S(-2))/(-S(4)n + S(8)) + S(3)a*S(2)(x + sqrt(a + xS(2)))n/(S(8)n) + a(x + sqrt(a + xS(2)))(n + S(2))/(S(4)n + S(8)) + (x + sqrt(a + xS(2)))(n + S(4))/(S(16)n + S(64)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + x*S(2))(x + sqrt(a + xS(2)))n, x), x, -a*S(2)(x + sqrt(a + xS(2)))(n + S(-2))/(-S(4)n + S(8)) + a(x + sqrt(a + xS(2)))n/(S(2)n) + (x + sqrt(a + xS(2)))(n + S(2))/(S(4)n + S(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(a + xS(2)))n/sqrt(a + xS(2)), x), x, (x + sqrt(a + xS(2)))n/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(a + xS(2)))n/(a + xS(2))*(S(3)/2), x), x, S(4)(x + sqrt(a + xS(2)))(n + S(2))hyper((S(2), n/S(2) + S(1)), (n/S(2) + S(2),), -(x + sqrt(a + xS(2)))S(2)/a)/(aS(2)(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(a + xS(2)))n/(a + xS(2))(S(5)/2), x), x, S(16)(x + sqrt(a + xS(2)))(n + S(4))hyper((S(4), n/S(2) + S(2)), (n/S(2) + S(3),), -(x + sqrt(a + xS(2)))S(2)/a)/(a*S(4)(n + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + xS(2))(S(5)/2)(x - sqrt(a + xS(2)))n, x), x, aS(6)(x - sqrt(a + xS(2)))(n + S(-6))/(-S(64)n + S(384)) + S(3)a*S(5)(x - sqrt(a + xS(2)))(n + S(-4))/(-S(32)n + S(128)) + S(15)a*S(4)(x - sqrt(a + xS(2)))(n + S(-2))/(-S(64)n + S(128)) - S(5)a*S(3)(x - sqrt(a + xS(2)))n/(S(16)n) - S(15)a*S(2)(x - sqrt(a + xS(2)))(n + S(2))/(S(64)n + S(128)) - S(3)a(x - sqrt(a + xS(2)))(n + S(4))/(S(32)n + S(128)) - (x - sqrt(a + xS(2)))(n + S(6))/(S(64)n + S(384)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + xS(2))(S(3)/2)(x - sqrt(a + xS(2)))n, x), x, a*S(4)(x - sqrt(a + xS(2)))(n + S(-4))/(-S(16)n + S(64)) + aS(3)(x - sqrt(a + xS(2)))(n + S(-2))/(-S(4)n + S(8)) - S(3)a*S(2)(x - sqrt(a + xS(2)))n/(S(8)n) - a(x - sqrt(a + xS(2)))(n + S(2))/(S(4)n + S(8)) - (x - sqrt(a + xS(2)))(n + S(4))/(S(16)n + S(64)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + x*S(2))(x - sqrt(a + xS(2)))n, x), x, a*S(2)(x - sqrt(a + xS(2)))(n + S(-2))/(-S(4)n + S(8)) - a(x - sqrt(a + xS(2)))n/(S(2)n) - (x - sqrt(a + xS(2)))(n + S(2))/(S(4)n + S(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x - sqrt(a + xS(2)))n/sqrt(a + xS(2)), x), x, -(x - sqrt(a + xS(2)))n/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x - sqrt(a + xS(2)))n/(a + xS(2))*(S(3)/2), x), x, -S(4)(x - sqrt(a + xS(2)))(n + S(2))hyper((S(2), n/S(2) + S(1)), (n/S(2) + S(2),), -(x - sqrt(a + xS(2)))S(2)/a)/(aS(2)(n + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x - sqrt(a + xS(2)))n/(a + xS(2))(S(5)/2), x), x, -S(16)(x - sqrt(a + xS(2)))(n + S(4))hyper((S(4), n/S(2) + S(2)), (n/S(2) + S(3),), -(x - sqrt(a + xS(2)))S(2)/a)/(a*S(4)(n + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2))*S(2)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*n, x), x, (d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))(n + S(5))/(S(32)efS(4)(n + S(5))) - (-S(5)af*S(2) + S(5)d*S(2))(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(3))/(S(32)efS(4)(n + S(3))) + S(5)(-afS(2) + dS(2))*S(2)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(1))/(S(16)efS(4)(n + S(1))) + (-afS(2) + dS(2))S(5)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(-5))/(S(32)efS(4)(-n + S(5))) - S(5)(-afS(2) + dS(2))*S(4)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(-3))/(S(32)efS(4)(-n + S(3))) + S(5)(-afS(2) + dS(2))*S(3)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(-1))/(S(16)efS(4)(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2))(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))n, x), x, (d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))(n + S(3))/(S(8)efS(2)(n + S(3))) - (-S(3)af*S(2) + S(3)d*S(2))(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(1))/(S(8)efS(2)(n + S(1))) + (-afS(2) + dS(2))S(3)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(-3))/(S(8)efS(2)(-n + S(3))) - S(3)(-afS(2) + dS(2))*S(2)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(-1))/(S(8)efS(2)(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*n, x), x, (d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))(n + S(1))/(S(2)e(n + S(1))) + (-afS(2) + dS(2))(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))(n + S(-1))/(S(2)e(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))n/(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)), x), x, -S(2)f*S(2)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(1))hyper((S(1), n/S(2) + S(1)/2), (n/S(2) + S(3)/2,), (d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*S(2)/(-afS(2) + dS(2)))/(e(n + S(1))(-afS(2) + dS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))n/(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2))S(2), x), x, -S(8)f*S(4)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(3))hyper((S(3), n/S(2) + S(3)/2), (n/S(2) + S(5)/2,), (d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*S(2)/(-afS(2) + dS(2)))/(e(n + S(3))(-afS(2) + dS(2))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt((af*S(2) + ex(S(2)d + ex))/fS(2)))n, x), x, (d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))(n + S(1))/(S(2)e(n + S(1))) + (-afS(2) + dS(2))(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))(n + S(-1))/(S(2)e(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt((af*S(2) + ex(S(2)d + ex))/fS(2)))n/(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)), x), x, -S(2)f*S(2)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(1))hyper((S(1), n/S(2) + S(1)/2), (n/S(2) + S(3)/2,), (d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*S(2)/(-afS(2) + dS(2)))/(e(n + S(1))(-afS(2) + dS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2))(S(3)/2)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*n, x), x, (d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))(n + S(4))/(S(16)efS(3)(n + S(4))) - (-afS(2) + dS(2))(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(2))/(S(4)efS(3)(n + S(2))) - (-afS(2) + dS(2))S(4)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(-4))/(S(16)efS(3)(-n + S(4))) + (-afS(2) + dS(2))S(3)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(-2))/(S(4)efS(3)(-n + S(2))) + S(3)(-afS(2) + dS(2))*S(2)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*n/(S(8)efS(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2))(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))n, x), x, (d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))(n + S(2))/(S(4)ef(n + S(2))) - (-afS(2) + dS(2))S(2)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(-2))/(S(4)ef(-n + S(2))) - (-afS(2) + dS(2))(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*n/(S(2)efn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*n/sqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)), x), x, f(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*n/(en), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*n/(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2))(S(3)/2), x), x, S(4)f*S(3)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(2))hyper((S(2), n/S(2) + S(1)), (n/S(2) + S(2),), (d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*S(2)/(-afS(2) + dS(2)))/(e(n + S(2))(-afS(2) + dS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt((af*S(2) + ex(S(2)d + ex))/fS(2)))n/sqrt((af*S(2) + ex(S(2)d + ex))/fS(2)), x), x, f(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*n/(en), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*nsqrt(ag + S(2)degx/fS(2) + eS(2)gxS(2)/fS(2)), x), x, (d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))(n + S(2))sqrt(ag + S(2)degx/fS(2) + eS(2)gxS(2)/fS(2))/(S(4)ef(n + S(2))sqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2))) - (-afS(2) + dS(2))*S(2)(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*(n + S(-2))sqrt(ag + S(2)degx/fS(2) + eS(2)gxS(2)/fS(2))/(S(4)ef(-n + S(2))sqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2))) - (-afS(2) + dS(2))(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))nsqrt(ag + S(2)degx/fS(2) + eS(2)gxS(2)/fS(2))/(S(2)efnsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))n/sqrt(ag + S(2)degx/fS(2) + eS(2)gxS(2)/fS(2)), x), x, fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2))(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*n/(ensqrt(ag + S(2)degx/fS(2) + eS(2)gxS(2)/fS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*n/(ag + S(2)degx/fS(2) + eS(2)gxS(2)/fS(2))*(S(3)/2), x), x, S(4)f*S(3)sqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2))(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))(n + S(2))hyper((S(2), n/S(2) + S(1)), (n/S(2) + S(2),), (d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))*S(2)/(-afS(2) + dS(2)))/(eg(n + S(2))(-afS(2) + dS(2))*S(2)sqrt(ag + S(2)degx/fS(2) + eS(2)gxS(2)/fS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex + fsqrt((af*S(2) + ex(S(2)d + ex))/fS(2)))n/sqrt((af*S(2)g + egx(S(2)d + ex))/fS(2)), x), x, fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2))(d + ex + fsqrt(a + S(2)dex/fS(2) + eS(2)xS(2)/fS(2)))n/(ensqrt(ag + S(2)degx/fS(2) + eS(2)gxS(2)/fS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a + cxS(4))(d + ex)), x), x, -eatanh((aeS(2) + cd*S(2)x*S(2))/(sqrt(a + cx*S(4))sqrt(aeS(4) + cd*S(4))))/(S(2)sqrt(aeS(4) + cd*S(4))) + atan(xsqrt(-(aeS(4) + cdS(4))/(dS(2)eS(2)))/sqrt(a + cx*S(4)))/(S(2)dsqrt(-(ae*S(4) + cdS(4))/(dS(2)eS(2)))) + c(S(1)/4)dsqrt((a + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(2)a*(S(1)/4)sqrt(a + cxS(4))(sqrt(a)eS(2) + sqrt(c)d*S(2))) - sqrt((a + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-sqrt(a)eS(2) + sqrt(c)d*S(2))elliptic_pi((sqrt(a)eS(2) + sqrt(c)dS(2))S(2)/(S(4)sqrt(a)sqrt(c)dS(2)e*S(2)), S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(4)a*(S(1)/4)c*(S(1)/4)dsqrt(a + cx*S(4))(sqrt(a)eS(2) + sqrt(c)d*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a + cx*S(4))(d + ex)S(2)), x), x, -a(S(1)/4)c*(S(1)/4)e*S(2)sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(sqrt(a + cx*S(4))(aeS(4) + cdS(4))) - a(S(1)/4)c(S(1)/4)sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-e*S(2) + sqrt(c)d*S(2)/sqrt(a))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(sqrt(a + cx*S(4))(S(2)ae*S(4) + S(2)cdS(4))) + sqrt(c)e*S(2)xsqrt(a + cx*S(4))/((sqrt(a) + sqrt(c)x*S(2))(aeS(4) + cd*S(4))) - cd*S(3)eatanh((ae*S(2) + cd*S(2)x*S(2))/(sqrt(a + cx*S(4))sqrt(aeS(4) + cd*S(4))))/(ae*S(4) + cdS(4))(S(3)/2) - catan(xsqrt(-(aeS(4) + cdS(4))/(dS(2)eS(2)))/sqrt(a + cxS(4)))/(eS(2)(-(ae*S(4) + cdS(4))/(dS(2)eS(2)))(S(3)/2)) - eS(3)sqrt(a + cxS(4))/((d + ex)(ae*S(4) + cdS(4))) + c(S(5)/4)dS(4)sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a(S(1)/4)), S(1)/2)/(a(S(1)/4)sqrt(a + cx*S(4))(sqrt(a)eS(2) + sqrt(c)d*S(2))(aeS(4) + cdS(4))) - c(S(3)/4)dS(2)sqrt((a + cxS(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-sqrt(a)eS(2) + sqrt(c)d*S(2))elliptic_pi((sqrt(a)eS(2) + sqrt(c)dS(2))S(2)/(S(4)sqrt(a)sqrt(c)dS(2)e*S(2)), S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2)/(S(2)a*(S(1)/4)sqrt(a + cxS(4))(sqrt(a)eS(2) + sqrt(c)d*S(2))(aeS(4) + cd*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)sqrt(a + bx*S(2) + cx*S(4))), x), x, -eatanh((S(2)ae*S(2) + bdS(2) + xS(2)(be*S(2) + S(2)cdS(2)))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(4) + bd*S(2)e*S(2) + cd*S(4))))/(S(2)sqrt(aeS(4) + bd*S(2)e*S(2) + cd*S(4))) + atan(xsqrt(-b - (aeS(4) + cdS(4))/(dS(2)eS(2)))/sqrt(a + bx*S(2) + cx*S(4)))/(S(2)dsqrt(-b - (ae*S(4) + cdS(4))/(dS(2)eS(2)))) + c(S(1)/4)dsqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(1)/4)(sqrt(a)eS(2) + sqrt(c)d*S(2))sqrt(a + bxS(2) + cx*S(4))) - sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-sqrt(a)eS(2) + sqrt(c)d*S(2))elliptic_pi((sqrt(a)eS(2) + sqrt(c)dS(2))S(2)/(S(4)sqrt(a)sqrt(c)dS(2)e*S(2)), S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(4)a*(S(1)/4)c*(S(1)/4)d(sqrt(a)e*S(2) + sqrt(c)d*S(2))sqrt(a + bxS(2) + cx*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((d + ex)*S(2)sqrt(a + bxS(2) + cxS(4))), x), x, -a(S(1)/4)c(S(1)/4)e*S(2)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))elliptic_e(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(sqrt(a + bx*S(2) + cx*S(4))(aeS(4) + bd*S(2)e*S(2) + cdS(4))) - a(S(1)/4)c(S(1)/4)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-e*S(2) + sqrt(c)d*S(2)/sqrt(a))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(sqrt(a + bx*S(2) + cx*S(4))(S(2)ae*S(4) + S(2)bdS(2)e*S(2) + S(2)cdS(4))) + sqrt(c)e*S(2)xsqrt(a + bx*S(2) + cx*S(4))/((sqrt(a) + sqrt(c)x*S(2))(aeS(4) + bd*S(2)e*S(2) + cd*S(4))) - de(be*S(2) + S(2)cdS(2))atanh((S(2)ae*S(2) + bdS(2) + xS(2)(be*S(2) + S(2)cdS(2)))/(S(2)sqrt(a + bxS(2) + cx*S(4))sqrt(aeS(4) + bd*S(2)e*S(2) + cd*S(4))))/(S(2)(aeS(4) + bd*S(2)e*S(2) + cdS(4))(S(3)/2)) - e*S(3)sqrt(a + bxS(2) + cx*S(4))/((d + ex)(ae*S(4) + bd*S(2)e*S(2) + cd*S(4))) - (be*S(2) + S(2)cdS(2))atan(xsqrt(-b - (ae*S(4) + cdS(4))/(dS(2)eS(2)))/sqrt(a + bx*S(2) + cx*S(4)))/(S(2)d*S(2)e*S(2)(-b - (aeS(4) + cdS(4))/(dS(2)eS(2)))(S(3)/2)) + c(S(1)/4)d*S(2)sqrt((a + bxS(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(beS(2) + S(2)cdS(2))elliptic_f(S(2)atan(c(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(2)a*(S(1)/4)(sqrt(a)eS(2) + sqrt(c)d*S(2))sqrt(a + bxS(2) + cx*S(4))(aeS(4) + bd*S(2)e*S(2) + cd*S(4))) - sqrt((a + bx*S(2) + cx*S(4))/(sqrt(a) + sqrt(c)xS(2))S(2))(sqrt(a) + sqrt(c)x*S(2))(-sqrt(a)eS(2) + sqrt(c)d*S(2))(beS(2) + S(2)cdS(2))elliptic_pi((sqrt(a)eS(2) + sqrt(c)dS(2))S(2)/(S(4)sqrt(a)sqrt(c)dS(2)e*S(2)), S(2)atan(c*(S(1)/4)x/a*(S(1)/4)), S(1)/2 - b/(S(4)sqrt(a)sqrt(c)))/(S(4)a*(S(1)/4)c*(S(1)/4)(sqrt(a)eS(2) + sqrt(c)d*S(2))sqrt(a + bxS(2) + cx*S(4))(aeS(4) + bd*S(2)e*S(2) + cd*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bx*S(2) + cx*S(4))/(ad - cdx*S(4)), x), x, -sqrt(-S(2)sqrt(a)sqrt(c) + b)atanh(xsqrt(-S(2)sqrt(a)sqrt(c) + b)/sqrt(a + bx*S(2) + cx*S(4)))/(S(4)sqrt(a)sqrt(c)d) + sqrt(S(2)sqrt(a)sqrt(c) + b)atanh(xsqrt(S(2)sqrt(a)sqrt(c) + b)/sqrt(a + bxS(2) + cx*S(4)))/(S(4)sqrt(a)sqrt(c)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bxS(2) - cx*S(4))/(ad + cdx*S(4)), x), x, sqrt(S(2))sqrt(-b + sqrt(S(4)ac + b*S(2)))atanh(sqrt(S(2))xsqrt(-b + sqrt(S(4)ac + b*S(2)))(b - S(2)cx*S(2) + sqrt(S(4)ac + bS(2)))/(S(4)sqrt(a)sqrt(c)sqrt(a + bxS(2) - cx*S(4))))/(S(4)sqrt(a)sqrt(c)d) - sqrt(S(2))sqrt(b + sqrt(S(4)ac + bS(2)))atan(sqrt(S(2))xsqrt(b + sqrt(S(4)ac + b*S(2)))(b - S(2)cx*S(2) - sqrt(S(4)ac + bS(2)))/(S(4)sqrt(a)sqrt(c)sqrt(a + bxS(2) - cx*S(4))))/(S(4)sqrt(a)sqrt(c)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))sqrt(c + dxS(2) + ex), x), x, bxS(2)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(c + dxS(2) + ex)*(S(3)/2)/(S(5)d(a + bxS(2))) - sqrt(aS(2) + S(2)abxS(2) + bS(2)x*S(4))(c + dxS(2) + ex)*(S(3)/2)(-S(80)ad*S(2) + S(32)bcd + S(42)bdex - S(35)be*S(2))/(S(240)d*S(3)(a + bxS(2))) + e(S(2)dx + e)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))sqrt(c + dxS(2) + ex)(-S(16)adS(2) + S(12)bcd - S(7)be*S(2))/(S(128)d*S(4)(a + bxS(2))) + e(S(4)cd - e*S(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(-S(16)ad*S(2) + S(12)bcd - S(7)be*S(2))atanh((S(2)dx + e)/(S(2)sqrt(d)sqrt(c + dxS(2) + ex)))/(S(256)d(S(9)/2)(a + bxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))sqrt(c + dxS(2) + ex), x), x, -b(-S(6)dx + S(5)e)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))(c + dxS(2) + ex)*(S(3)/2)/(S(24)d*S(2)(a + bxS(2))) - (S(2)dx + e)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))sqrt(c + dxS(2) + ex)(-S(16)adS(2) + S(4)bcd - S(5)be*S(2))/(S(64)d*S(3)(a + bxS(2))) - (S(4)cd - eS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(-S(16)ad*S(2) + S(4)bcd - S(5)be*S(2))atanh((S(2)dx + e)/(S(2)sqrt(d)sqrt(c + dxS(2) + ex)))/(S(128)d(S(7)/2)(a + bxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))sqrt(c + dxS(2) + ex)/x, x), x, -asqrt(c)sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))atanh((S(2)c + ex)/(S(2)sqrt(c)sqrt(c + dxS(2) + ex)))/(a + bxS(2)) + bsqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(c + dxS(2) + ex)*(S(3)/2)/(S(3)d(a + bxS(2))) + sqrt(aS(2) + S(2)abxS(2) + bS(2)x*S(4))sqrt(c + dxS(2) + ex)(S(8)adS(2) - S(2)bdex - be*S(2))/(S(8)d*S(2)(a + bxS(2))) + e(S(8)ad*S(2) - b(S(4)cd - e*S(2)))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))atanh((S(2)dx + e)/(S(2)sqrt(d)sqrt(c + dxS(2) + ex)))/(S(16)d(S(5)/2)(a + bxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))sqrt(c + dxS(2) + ex)/x*S(2), x), x, -asqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(c + dxS(2) + ex)*(S(3)/2)/(cx(a + bx*S(2))) - aesqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))atanh((S(2)c + ex)/(S(2)sqrt(c)sqrt(c + dxS(2) + ex)))/(S(2)sqrt(c)(a + bxS(2))) + sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))(S(8)ad*S(2) + S(4)bcd - beS(2))atanh((S(2)dx + e)/(S(2)sqrt(d)sqrt(c + dxS(2) + ex)))/(S(8)d(S(3)/2)(a + bxS(2))) + (S(2)dx(S(2)ad + bc) + e(S(4)ad + bc))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))sqrt(c + dxS(2) + ex)/(S(4)cd(a + bxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)x*S(4))sqrt(c + dxS(2) + ex)/x*S(3), x), x, -asqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(c + dxS(2) + ex)*(S(3)/2)/(S(2)cxS(2)(a + bxS(2))) + besqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))atanh((S(2)dx + e)/(S(2)sqrt(d)sqrt(c + dxS(2) + ex)))/(S(2)sqrt(d)(a + bxS(2))) + (ae + x(S(2)ad + S(4)bc))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))sqrt(c + dxS(2) + ex)/(S(4)cx(a + bxS(2))) - sqrt(aS(2) + S(2)abxS(2) + bS(2)x*S(4))(S(4)acd - ae*S(2) + S(8)bcS(2))atanh((S(2)c + ex)/(S(2)sqrt(c)sqrt(c + dxS(2) + ex)))/(S(8)c(S(3)/2)(a + bxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))sqrt(c + dxS(2) + ex)/x*S(4), x), x, -asqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))(c + dxS(2) + ex)*(S(3)/2)/(S(3)cxS(3)(a + bxS(2))) + bsqrt(d)sqrt(aS(2) + S(2)abxS(2) + bS(2)xS(4))atanh((S(2)dx + e)/(S(2)sqrt(d)sqrt(c + dxS(2) + ex)))/(a + bxS(2)) + (S(2)ace - x(-ae*S(2) + S(8)bcS(2)))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))sqrt(c + dxS(2) + ex)/(S(8)cS(2)x*S(2)(a + bxS(2))) - e(-a(S(4)cd - eS(2)) + S(8)bcS(2))sqrt(a*S(2) + S(2)abxS(2) + bS(2)xS(4))atanh((S(2)c + ex)/(S(2)sqrt(c)sqrt(c + dxS(2) + ex)))/(S(16)c(S(5)/2)(a + bxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x + S(3))sqrt(x*S(3) + S(1))), x), x, sqrt(S(26))sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))(x + S(1))atan(sqrt(S(2))S(3)(S(3)/4)sqrt(S(1) - (x - sqrt(S(3)) + S(1))S(2)/(x + S(1) + sqrt(S(3)))S(2))sqrt(-S(13)sqrt(S(3)) + S(26))/(S(6)sqrt(-S(4)sqrt(S(3)) + S(7) + (x - sqrt(S(3)) + S(1))S(2)/(x + S(1) + sqrt(S(3)))S(2))))/(S(26)sqrt((x + S(1))/(x + S(1) + sqrt(S(3)))S(2))sqrt(x*S(3) + S(1))) + S(2)S(3)*(S(3)/4)sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))sqrt(S(15)sqrt(S(3)) + S(26))(x + S(1))elliptic_f(asin((x - sqrt(S(3)) + S(1))/(x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(S(3)sqrt((x + S(1))/(x + S(1) + sqrt(S(3)))*S(2))sqrt(x*S(3) + S(1))) - S(4)S(3)*(S(1)/4)sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))(x + S(1))elliptic_pi(-S(56)sqrt(S(3)) + S(97), asin((x - sqrt(S(3)) + S(1))/(x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(sqrt((x + S(1))/(x + S(1) + sqrt(S(3)))*S(2))sqrt(-sqrt(S(3)) + S(2))sqrt(xS(3) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x + S(3))sqrt(-x*S(3) + S(1))), x), x, -sqrt(S(7))sqrt((xS(2) + x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))(-x + S(1))atanh(S(3)*(S(3)/4)sqrt(S(1) - (-x - sqrt(S(3)) + S(1))S(2)/(-x + S(1) + sqrt(S(3)))S(2))sqrt(-S(7)sqrt(S(3)) + S(14))/(S(6)sqrt(-S(4)sqrt(S(3)) + S(7) + (-x - sqrt(S(3)) + S(1))S(2)/(-x + S(1) + sqrt(S(3)))S(2))))/(S(14)sqrt((-x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))sqrt(-x*S(3) + S(1))) - S(2)S(3)*(S(3)/4)sqrt((xS(2) + x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))sqrt(sqrt(S(3)) + S(2))(-x + S(1))elliptic_f(asin((-x - sqrt(S(3)) + S(1))/(-x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(S(3)sqrt((-x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))(sqrt(S(3)) + S(4))sqrt(-xS(3) + S(1))) - S(4)S(3)*(S(1)/4)sqrt((xS(2) + x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))sqrt(sqrt(S(3)) + S(2))(-x + S(1))elliptic_pi(S(304)sqrt(S(3))/S(169) + S(553)/169, asin((-x - sqrt(S(3)) + S(1))/(-x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(S(13)sqrt((-x + S(1))/(-x + S(1) + sqrt(S(3)))*S(2))sqrt(-x*S(3) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x + S(3))sqrt(x*S(3) + S(-1))), x), x, -sqrt(S(7))sqrt((xS(2) + x + S(1))/(-x - sqrt(S(3)) + S(1))S(2))(-x + S(1))atan(S(3)*(S(3)/4)sqrt(S(7)sqrt(S(3)) + S(14))sqrt(-(-x + S(1) + sqrt(S(3)))S(2)/(-x - sqrt(S(3)) + S(1))S(2) + S(1))/(S(6)sqrt((-x + S(1) + sqrt(S(3)))S(2)/(-x - sqrt(S(3)) + S(1))S(2) + S(4)sqrt(S(3)) + S(7))))/(S(14)sqrt(-(-x + S(1))/(-x - sqrt(S(3)) + S(1))S(2))sqrt(x*S(3) + S(-1))) - S(2)S(3)*(S(3)/4)sqrt((xS(2) + x + S(1))/(-x - sqrt(S(3)) + S(1))S(2))sqrt(-S(3)sqrt(S(3)) + S(14))(-x + S(1))elliptic_f(asin((-x + S(1) + sqrt(S(3)))/(-x - sqrt(S(3)) + S(1))), S(-7) + S(4)sqrt(S(3)))/(S(39)sqrt(-(-x + S(1))/(-x - sqrt(S(3)) + S(1))*S(2))sqrt(x*S(3) + S(-1))) + S(4)S(3)*(S(1)/4)sqrt((xS(2) + x + S(1))/(-x - sqrt(S(3)) + S(1))S(2))sqrt(-sqrt(S(3)) + S(2))(-x + S(1))elliptic_pi(-S(304)sqrt(S(3))/S(169) + S(553)/169, asin((-x + S(1) + sqrt(S(3)))/(-x - sqrt(S(3)) + S(1))), S(-7) + S(4)sqrt(S(3)))/(S(13)sqrt(-(-x + S(1))/(-x - sqrt(S(3)) + S(1))*S(2))sqrt(x*S(3) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x + S(3))sqrt(-x*S(3) + S(-1))), x), x, sqrt(S(26))sqrt((xS(2) - x + S(1))/(x - sqrt(S(3)) + S(1))S(2))(x + S(1))atanh(sqrt(S(2))S(3)(S(3)/4)sqrt(S(13)sqrt(S(3)) + S(26))sqrt(-(x + S(1) + sqrt(S(3)))S(2)/(x - sqrt(S(3)) + S(1))S(2) + S(1))/(S(6)sqrt((x + S(1) + sqrt(S(3)))S(2)/(x - sqrt(S(3)) + S(1))S(2) + S(4)sqrt(S(3)) + S(7))))/(S(26)sqrt(-(x + S(1))/(x - sqrt(S(3)) + S(1))S(2))sqrt(-x*S(3) + S(-1))) + S(2)S(3)*(S(3)/4)sqrt((xS(2) - x + S(1))/(x - sqrt(S(3)) + S(1))S(2))sqrt(-S(15)sqrt(S(3)) + S(26))(x + S(1))elliptic_f(asin((x + S(1) + sqrt(S(3)))/(x - sqrt(S(3)) + S(1))), S(-7) + S(4)sqrt(S(3)))/(S(3)sqrt(-(x + S(1))/(x - sqrt(S(3)) + S(1))*S(2))sqrt(-x*S(3) + S(-1))) + S(4)S(3)*(S(1)/4)sqrt((xS(2) - x + S(1))/(x - sqrt(S(3)) + S(1))S(2))(x + S(1))elliptic_pi(S(56)sqrt(S(3)) + S(97), asin((x + S(1) + sqrt(S(3)))/(x - sqrt(S(3)) + S(1))), S(-7) + S(4)sqrt(S(3)))/(sqrt(-(x + S(1))/(x - sqrt(S(3)) + S(1))*S(2))sqrt(sqrt(S(3)) + S(2))sqrt(-xS(3) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((x + S(3))sqrt(x*S(3) + S(1))), x), x, -S(2)S(3)*(S(3)/4)sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))sqrt(S(112)sqrt(S(3)) + S(194))(x + S(1))elliptic_f(asin((x - sqrt(S(3)) + S(1))/(x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(S(3)sqrt((x + S(1))/(x + S(1) + sqrt(S(3)))*S(2))sqrt(x*S(3) + S(1))) + S(12)S(3)*(S(1)/4)sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))(x + S(1))elliptic_pi(-S(56)sqrt(S(3)) + S(97), asin((x - sqrt(S(3)) + S(1))/(x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(sqrt((x + S(1))/(x + S(1) + sqrt(S(3)))*S(2))sqrt(-sqrt(S(3)) + S(2))sqrt(xS(3) + S(1))) - sqrt(S(26))sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))(S(3)x + S(3))atan(sqrt(S(2))S(3)*(S(3)/4)sqrt(S(1) - (x - sqrt(S(3)) + S(1))S(2)/(x + S(1) + sqrt(S(3)))S(2))sqrt(-S(13)sqrt(S(3)) + S(26))/(S(6)sqrt(-S(4)sqrt(S(3)) + S(7) + (x - sqrt(S(3)) + S(1))S(2)/(x + S(1) + sqrt(S(3)))S(2))))/(S(26)sqrt((x + S(1))/(x + S(1) + sqrt(S(3)))S(2))sqrt(x*S(3) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((x + S(3))sqrt(-x*S(3) + S(1))), x), x, sqrt(S(7))sqrt((xS(2) + x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))(-S(3)x + S(3))atanh(S(3)(S(3)/4)sqrt(S(1) - (-x - sqrt(S(3)) + S(1))S(2)/(-x + S(1) + sqrt(S(3)))S(2))sqrt(-S(7)sqrt(S(3)) + S(14))/(S(6)sqrt(-S(4)sqrt(S(3)) + S(7) + (-x - sqrt(S(3)) + S(1))S(2)/(-x + S(1) + sqrt(S(3)))S(2))))/(S(14)sqrt((-x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))sqrt(-x*S(3) + S(1))) - S(2)S(3)*(S(3)/4)sqrt((xS(2) + x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))sqrt(S(40)sqrt(S(3)) + S(74))(-x + S(1))elliptic_f(asin((-x - sqrt(S(3)) + S(1))/(-x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(S(39)sqrt((-x + S(1))/(-x + S(1) + sqrt(S(3)))*S(2))sqrt(-x*S(3) + S(1))) + S(12)S(3)*(S(1)/4)sqrt((xS(2) + x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))sqrt(sqrt(S(3)) + S(2))(-x + S(1))elliptic_pi(S(304)sqrt(S(3))/S(169) + S(553)/169, asin((-x - sqrt(S(3)) + S(1))/(-x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(S(13)sqrt((-x + S(1))/(-x + S(1) + sqrt(S(3)))*S(2))sqrt(-x*S(3) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((x + S(3))sqrt(x*S(3) + S(-1))), x), x, sqrt(S(7))sqrt((xS(2) + x + S(1))/(-x - sqrt(S(3)) + S(1))S(2))(-S(3)x + S(3))atan(S(3)(S(3)/4)sqrt(S(7)sqrt(S(3)) + S(14))sqrt(-(-x + S(1) + sqrt(S(3)))S(2)/(-x - sqrt(S(3)) + S(1))S(2) + S(1))/(S(6)sqrt((-x + S(1) + sqrt(S(3)))S(2)/(-x - sqrt(S(3)) + S(1))S(2) + S(4)sqrt(S(3)) + S(7))))/(S(14)sqrt(-(-x + S(1))/(-x - sqrt(S(3)) + S(1))S(2))sqrt(x*S(3) + S(-1))) + S(2)S(3)*(S(3)/4)sqrt((xS(2) + x + S(1))/(-x - sqrt(S(3)) + S(1))S(2))sqrt(-S(40)sqrt(S(3)) + S(74))(-x + S(1))elliptic_f(asin((-x + S(1) + sqrt(S(3)))/(-x - sqrt(S(3)) + S(1))), S(-7) + S(4)sqrt(S(3)))/(S(39)sqrt(-(-x + S(1))/(-x - sqrt(S(3)) + S(1))*S(2))sqrt(x*S(3) + S(-1))) - S(12)S(3)*(S(1)/4)sqrt((xS(2) + x + S(1))/(-x - sqrt(S(3)) + S(1))S(2))sqrt(-sqrt(S(3)) + S(2))(-x + S(1))elliptic_pi(-S(304)sqrt(S(3))/S(169) + S(553)/169, asin((-x + S(1) + sqrt(S(3)))/(-x - sqrt(S(3)) + S(1))), S(-7) + S(4)sqrt(S(3)))/(S(13)sqrt(-(-x + S(1))/(-x - sqrt(S(3)) + S(1))*S(2))sqrt(x*S(3) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((x + S(3))sqrt(-x*S(3) + S(-1))), x), x, S(2)S(3)*(S(3)/4)sqrt((xS(2) - x + S(1))/(x - sqrt(S(3)) + S(1))S(2))sqrt(-S(112)sqrt(S(3)) + S(194))(x + S(1))elliptic_f(asin((x + S(1) + sqrt(S(3)))/(x - sqrt(S(3)) + S(1))), S(-7) + S(4)sqrt(S(3)))/(S(3)sqrt(-(x + S(1))/(x - sqrt(S(3)) + S(1))*S(2))sqrt(-x*S(3) + S(-1))) - S(12)S(3)*(S(1)/4)sqrt((xS(2) - x + S(1))/(x - sqrt(S(3)) + S(1))S(2))(x + S(1))elliptic_pi(S(56)sqrt(S(3)) + S(97), asin((x + S(1) + sqrt(S(3)))/(x - sqrt(S(3)) + S(1))), S(-7) + S(4)sqrt(S(3)))/(sqrt(-(x + S(1))/(x - sqrt(S(3)) + S(1))*S(2))sqrt(sqrt(S(3)) + S(2))sqrt(-xS(3) + S(-1))) - sqrt(S(26))sqrt((xS(2) - x + S(1))/(x - sqrt(S(3)) + S(1))S(2))(S(3)x + S(3))atanh(sqrt(S(2))S(3)*(S(3)/4)sqrt(S(13)sqrt(S(3)) + S(26))sqrt(-(x + S(1) + sqrt(S(3)))S(2)/(x - sqrt(S(3)) + S(1))S(2) + S(1))/(S(6)sqrt((x + S(1) + sqrt(S(3)))S(2)/(x - sqrt(S(3)) + S(1))S(2) + S(4)sqrt(S(3)) + S(7))))/(S(26)sqrt(-(x + S(1))/(x - sqrt(S(3)) + S(1))S(2))sqrt(-x*S(3) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(3)x + S(2))/((x + S(3))sqrt(xS(3) + S(1))), x), x, S(2)S(3)*(S(3)/4)sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))sqrt(S(1560)sqrt(S(3)) + S(2702))(x + S(1))elliptic_f(asin((x - sqrt(S(3)) + S(1))/(x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(S(3)sqrt((x + S(1))/(x + S(1) + sqrt(S(3)))*S(2))sqrt(x*S(3) + S(1))) - S(44)S(3)*(S(1)/4)sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))(x + S(1))elliptic_pi(-S(56)sqrt(S(3)) + S(97), asin((x - sqrt(S(3)) + S(1))/(x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(sqrt((x + S(1))/(x + S(1) + sqrt(S(3)))*S(2))sqrt(-sqrt(S(3)) + S(2))sqrt(xS(3) + S(1))) + sqrt(S(26))sqrt((xS(2) - x + S(1))/(x + S(1) + sqrt(S(3)))S(2))(S(11)x + S(11))atan(sqrt(S(2))S(3)*(S(3)/4)sqrt(S(1) - (x - sqrt(S(3)) + S(1))S(2)/(x + S(1) + sqrt(S(3)))S(2))sqrt(-S(13)sqrt(S(3)) + S(26))/(S(6)sqrt(-S(4)sqrt(S(3)) + S(7) + (x - sqrt(S(3)) + S(1))S(2)/(x + S(1) + sqrt(S(3)))S(2))))/(S(26)sqrt((x + S(1))/(x + S(1) + sqrt(S(3)))S(2))sqrt(x*S(3) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(3)x + S(2))/((x + S(3))sqrt(-xS(3) + S(1))), x), x, -sqrt(S(7))sqrt((xS(2) + x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))(-S(11)x + S(11))atanh(S(3)(S(3)/4)sqrt(S(1) - (-x - sqrt(S(3)) + S(1))S(2)/(-x + S(1) + sqrt(S(3)))S(2))sqrt(-S(7)sqrt(S(3)) + S(14))/(S(6)sqrt(-S(4)sqrt(S(3)) + S(7) + (-x - sqrt(S(3)) + S(1))S(2)/(-x + S(1) + sqrt(S(3)))S(2))))/(S(14)sqrt((-x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))sqrt(-x*S(3) + S(1))) + S(2)S(3)*(S(3)/4)sqrt((xS(2) + x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))sqrt(S(168)sqrt(S(3)) + S(446))(-x + S(1))elliptic_f(asin((-x - sqrt(S(3)) + S(1))/(-x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(S(39)sqrt((-x + S(1))/(-x + S(1) + sqrt(S(3)))*S(2))sqrt(-x*S(3) + S(1))) - S(44)S(3)*(S(1)/4)sqrt((xS(2) + x + S(1))/(-x + S(1) + sqrt(S(3)))S(2))sqrt(sqrt(S(3)) + S(2))(-x + S(1))elliptic_pi(S(304)sqrt(S(3))/S(169) + S(553)/169, asin((-x - sqrt(S(3)) + S(1))/(-x + S(1) + sqrt(S(3)))), S(-7) - S(4)sqrt(S(3)))/(S(13)sqrt((-x + S(1))/(-x + S(1) + sqrt(S(3)))*S(2))sqrt(-x*S(3) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(3)x + S(2))/((x + S(3))sqrt(xS(3) + S(-1))), x), x, -sqrt(S(7))sqrt((xS(2) + x + S(1))/(-x - sqrt(S(3)) + S(1))S(2))(-S(11)x + S(11))atan(S(3)(S(3)/4)sqrt(S(7)sqrt(S(3)) + S(14))sqrt(-(-x + S(1) + sqrt(S(3)))S(2)/(-x - sqrt(S(3)) + S(1))S(2) + S(1))/(S(6)sqrt((-x + S(1) + sqrt(S(3)))S(2)/(-x - sqrt(S(3)) + S(1))S(2) + S(4)sqrt(S(3)) + S(7))))/(S(14)sqrt(-(-x + S(1))/(-x - sqrt(S(3)) + S(1))S(2))sqrt(x*S(3) + S(-1))) - S(2)S(3)*(S(3)/4)sqrt((xS(2) + x + S(1))/(-x - sqrt(S(3)) + S(1))S(2))sqrt(-S(168)sqrt(S(3)) + S(446))(-x + S(1))elliptic_f(asin((-x + S(1) + sqrt(S(3)))/(-x - sqrt(S(3)) + S(1))), S(-7) + S(4)sqrt(S(3)))/(S(39)sqrt(-(-x + S(1))/(-x - sqrt(S(3)) + S(1))*S(2))sqrt(x*S(3) + S(-1))) + S(44)S(3)*(S(1)/4)sqrt((xS(2) + x + S(1))/(-x - sqrt(S(3)) + S(1))S(2))sqrt(-sqrt(S(3)) + S(2))(-x + S(1))elliptic_pi(-S(304)sqrt(S(3))/S(169) + S(553)/169, asin((-x + S(1) + sqrt(S(3)))/(-x - sqrt(S(3)) + S(1))), S(-7) + S(4)sqrt(S(3)))/(S(13)sqrt(-(-x + S(1))/(-x - sqrt(S(3)) + S(1))*S(2))sqrt(x*S(3) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(3)x + S(2))/((x + S(3))sqrt(-xS(3) + S(-1))), x), x, -S(2)S(3)*(S(3)/4)sqrt((xS(2) - x + S(1))/(x - sqrt(S(3)) + S(1))S(2))sqrt(-S(1560)sqrt(S(3)) + S(2702))(x + S(1))elliptic_f(asin((x + S(1) + sqrt(S(3)))/(x - sqrt(S(3)) + S(1))), S(-7) + S(4)sqrt(S(3)))/(S(3)sqrt(-(x + S(1))/(x - sqrt(S(3)) + S(1))*S(2))sqrt(-x*S(3) + S(-1))) + S(44)S(3)*(S(1)/4)sqrt((xS(2) - x + S(1))/(x - sqrt(S(3)) + S(1))S(2))(x + S(1))elliptic_pi(S(56)sqrt(S(3)) + S(97), asin((x + S(1) + sqrt(S(3)))/(x - sqrt(S(3)) + S(1))), S(-7) + S(4)sqrt(S(3)))/(sqrt(-(x + S(1))/(x - sqrt(S(3)) + S(1))*S(2))sqrt(sqrt(S(3)) + S(2))sqrt(-xS(3) + S(-1))) + sqrt(S(26))sqrt((xS(2) - x + S(1))/(x - sqrt(S(3)) + S(1))S(2))(S(11)x + S(11))atanh(sqrt(S(2))S(3)*(S(3)/4)sqrt(S(13)sqrt(S(3)) + S(26))sqrt(-(x + S(1) + sqrt(S(3)))S(2)/(x - sqrt(S(3)) + S(1))S(2) + S(1))/(S(6)sqrt((x + S(1) + sqrt(S(3)))S(2)/(x - sqrt(S(3)) + S(1))S(2) + S(4)sqrt(S(3)) + S(7))))/(S(26)sqrt(-(x + S(1))/(x - sqrt(S(3)) + S(1))S(2))sqrt(-xS(3) + S(-1))), expand=True, _diff=True, _numerical=True) # sympy and mathematica assert rubi_test(rubi_integrate((dS(3) + e*S(3)xS(3))p/(d + ex), x), x, (S(1) + (S(2)d + S(2)ex)/(d(S(-3) + sqrt(S(3))I)))*(-p)(S(1) - (S(2)d + S(2)ex)/(d(S(3) + sqrt(S(3))I)))(-p)(dS(3) + eS(3)xS(3))pAppellF1(p, -p, -p, p + S(1), -(S(2)d + S(2)ex)/(d(S(-3) + sqrt(S(3))I)), (S(2)d + S(2)ex)/(d(S(3) + sqrt(S(3))I)))/(ep), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bx)sqrt(c + dx*S(2))sqrt(e + fxS(2))), x), x, asqrt(e)sqrt(f)sqrt(c + dxS(2))elliptic_f(atan(sqrt(f)x/sqrt(e)), S(1) - de/(cf))/(csqrt(e(c + dx*S(2))/(c(e + fxS(2))))sqrt(e + fxS(2))(a*S(2)f + b*S(2)e)) - batanh(sqrt(c + dx*S(2))sqrt(a*S(2)f + b*S(2)e)/(sqrt(e + fxS(2))sqrt(a*S(2)d + b*S(2)c)))/(sqrt(a*S(2)d + b*S(2)c)sqrt(aS(2)f + b*S(2)e)) + b*S(2)e*(S(3)/2)sqrt(c + dxS(2))elliptic_pi(S(1) + b*S(2)e/(a*S(2)f), atan(sqrt(f)x/sqrt(e)), S(1) - de/(cf))/(acsqrt(f)sqrt(e(c + dx*S(2))/(c(e + fxS(2))))sqrt(e + fxS(2))(a*S(2)f + b*S(2)e)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((e - S(2)fx*S(2))/(S(4)dfxS(2) + eS(2) + S(4)efxS(2) + S(4)f*S(2)x*S(4)), x), x, -log(e - S(2)sqrt(f)xsqrt(-d) + S(2)fx*S(2))/(S(4)sqrt(f)sqrt(-d)) + log(e + S(2)sqrt(f)xsqrt(-d) + S(2)fx*S(2))/(S(4)sqrt(f)sqrt(-d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((e - S(2)fxS(2))/(-S(4)dfxS(2) + eS(2) + S(4)efxS(2) + S(4)f*S(2)x*S(4)), x), x, -log(-S(2)sqrt(d)sqrt(f)x + e + S(2)fx*S(2))/(S(4)sqrt(d)sqrt(f)) + log(S(2)sqrt(d)sqrt(f)x + e + S(2)fx*S(2))/(S(4)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((e - S(4)fxS(3))/(S(4)dfxS(2) + eS(2) + S(4)efxS(3) + S(4)f*S(2)x*S(6)), x), x, atan(S(2)sqrt(d)sqrt(f)x/(e + S(2)fx*S(3)))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((e - S(4)fxS(3))/(-S(4)dfxS(2) + eS(2) + S(4)efxS(3) + S(4)f*S(2)x*S(6)), x), x, atanh(S(2)sqrt(d)sqrt(f)x/(e + S(2)fx*S(3)))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((e - S(2)fxn(n + S(-1)))/(S(4)dfxS(2) + eS(2) + S(4)efx*n + S(4)f*S(2)x*(S(2)n)), x), x, atan(S(2)sqrt(d)sqrt(f)x/(e + S(2)fxn))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((e - S(2)fxn(n + S(-1)))/(-S(4)dfxS(2) + eS(2) + S(4)efx*n + S(4)f*S(2)x*(S(2)n)), x), x, atanh(S(2)sqrt(d)sqrt(f)x/(e + S(2)fxn))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(S(4)dfxS(4) + eS(2) + S(4)efxS(2) + S(4)f*S(2)x*S(4)), x), x, atan(sqrt(f)(e + x*S(2)(S(2)d + S(2)f))/(sqrt(d)e))/(S(4)sqrt(d)esqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(-S(4)dfxS(4) + eS(2) + S(4)efx*S(2) + S(4)f*S(2)x*S(4)), x), x, -atanh(sqrt(f)(e - x*S(2)(S(2)d - S(2)f))/(sqrt(d)e))/(S(4)sqrt(d)esqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(S(3)e + S(2)fxS(2))/(S(4)dfxS(6) + eS(2) + S(4)efxS(2) + S(4)f*S(2)x*S(4)), x), x, atan(S(2)sqrt(d)sqrt(f)x*S(3)/(e + S(2)fxS(2)))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(S(3)e + S(2)fxS(2))/(-S(4)dfxS(6) + eS(2) + S(4)efxS(2) + S(4)f*S(2)x*S(4)), x), x, atanh(S(2)sqrt(d)sqrt(f)x*S(3)/(e + S(2)fxS(2)))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(e(m + S(1)) + S(2)fxS(2)(m + S(-1)))/(S(4)dfx(S(2)m + S(2)) + e*S(2) + S(4)efx*S(2) + S(4)f*S(2)x*S(4)), x), x, atan(S(2)sqrt(d)sqrt(f)x*(m + S(1))/(e + S(2)fxS(2)))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xm(e(m + S(1)) + S(2)fxS(2)(m + S(-1)))/(S(4)dfx(S(2)m + S(2)) + e*S(2) + S(4)efx*S(2) + S(4)f*S(2)x*S(4)), x), x, atan(S(2)sqrt(d)sqrt(f)x*(m + S(1))(-m*S(2) + S(1))/((e + S(2)fxS(2))(-m + S(1))(m + S(1))))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(e(m + S(1)) + S(2)fxS(2)(m + S(-1)))/(-S(4)dfx(S(2)m + S(2)) + e*S(2) + S(4)efx*S(2) + S(4)f*S(2)x*S(4)), x), x, atanh(S(2)sqrt(d)sqrt(f)x*(m + S(1))/(e + S(2)fxS(2)))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xm(e(m + S(1)) + S(2)fxS(2)(m + S(-1)))/(-S(4)dfx(S(2)m + S(2)) + e*S(2) + S(4)efx*S(2) + S(4)f*S(2)x*S(4)), x), x, atanh(S(2)sqrt(d)sqrt(f)x*(m + S(1))(-m*S(2) + S(1))/((e + S(2)fxS(2))(-m + S(1))(m + S(1))))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)e - S(2)fxS(3))/(S(4)dfxS(4) + eS(2) + S(4)efxS(3) + S(4)f*S(2)x*S(6)), x), x, atan(S(2)sqrt(d)sqrt(f)x*S(2)/(e + S(2)fxS(3)))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)e - S(2)fxS(3))/(-S(4)dfxS(4) + eS(2) + S(4)efxS(3) + S(4)f*S(2)x*S(6)), x), x, atanh(S(2)sqrt(d)sqrt(f)x*S(2)/(e + S(2)fxS(3)))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(S(4)dfxS(6) + eS(2) + S(4)efxS(3) + S(4)f*S(2)x*S(6)), x), x, atan(sqrt(f)(e + x*S(3)(S(2)d + S(2)f))/(sqrt(d)e))/(S(6)sqrt(d)esqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(-S(4)dfxS(6) + eS(2) + S(4)efxS(3) + S(4)f*S(2)x*S(6)), x), x, -atanh(sqrt(f)(e - x*S(3)(S(2)d - S(2)f))/(sqrt(d)e))/(S(6)sqrt(d)esqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*m(e(m + S(1)) + S(2)fxS(3)(m + S(-2)))/(S(4)dfx(S(2)m + S(2)) + e*S(2) + S(4)efx*S(3) + S(4)f*S(2)x*S(6)), x), x, atan(S(2)sqrt(d)sqrt(f)x*(m + S(1))/(e + S(2)fxS(3)))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(e(m + S(1)) + S(2)fxS(3)(m + S(-2)))/(-S(4)dfx(S(2)m + S(2)) + e*S(2) + S(4)efx*S(3) + S(4)f*S(2)x*S(6)), x), x, atanh(S(2)sqrt(d)sqrt(f)x*(m + S(1))/(e + S(2)fxS(3)))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(e(m + S(1)) + S(2)fxn(m - n + S(1)))/(S(4)dfx(S(2)m + S(2)) + e*S(2) + S(4)efx*n + S(4)f*S(2)x*(S(2)n)), x), x, atan(S(2)sqrt(d)sqrt(f)x(m + S(1))/(e + S(2)fxn))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm(e(m + S(1)) + S(2)fxn(m - n + S(1)))/(-S(4)dfx(S(2)m + S(2)) + e*S(2) + S(4)efx*n + S(4)f*S(2)x*(S(2)n)), x), x, atanh(S(2)sqrt(d)sqrt(f)x(m + S(1))/(e + S(2)fxn))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(ac + bcx*S(2) + dsqrt(a + bxS(2))), x), x, -xS(2)(S(2)acS(2) - dS(2))/(S(2)bS(2)c*S(3)) + (a + bxS(2))S(2)/(S(4)bS(3)c) - d(a + bxS(2))(S(3)/2)/(S(3)bS(3)c*S(2)) + dsqrt(a + bxS(2))(S(2)acS(2) - dS(2))/(b*S(3)c*S(4)) + (acS(2) - dS(2))*S(2)log(csqrt(a + bxS(2)) + d)/(bS(3)cS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(ac + bcx*S(2) + dsqrt(a + bxS(2))), x), x, xS(2)/(S(2)bc) - dsqrt(a + bxS(2))/(bS(2)c*S(2)) - (acS(2) - dS(2))log(csqrt(a + bxS(2)) + d)/(bS(2)c*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(ac + bcx*S(2) + dsqrt(a + bxS(2))), x), x, log(csqrt(a + bxS(2)) + d)/(bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(ac + bcx*S(2) + dsqrt(a + bxS(2)))), x), x, clog(x)/(acS(2) - dS(2)) - clog(csqrt(a + bx*S(2)) + d)/(acS(2) - dS(2)) + datanh(sqrt(a + bx*S(2))/sqrt(a))/(sqrt(a)(acS(2) - dS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(ac + bcxS(2) + dsqrt(a + bxS(2)))), x), x, -bc*S(3)log(x)/(acS(2) - dS(2))S(2) + bc*S(3)log(csqrt(a + bx*S(2)) + d)/(acS(2) - dS(2))*S(2) - (ac - dsqrt(a + bx*S(2)))/(S(2)axS(2)(acS(2) - dS(2))) - bd(S(3)acS(2) - dS(2))atanh(sqrt(a + bxS(2))/sqrt(a))/(S(2)a*(S(3)/2)(acS(2) - dS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(ac + bcx*S(2) + dsqrt(a + bxS(2))), x), x, x/(bc) - datanh(sqrt(b)x/sqrt(a + bxS(2)))/(b(S(3)/2)c*S(2)) - sqrt(acS(2) - dS(2))atan(sqrt(b)cx/sqrt(acS(2) - dS(2)))/(b*(S(3)/2)c*S(2)) + sqrt(acS(2) - dS(2))atan(sqrt(b)dx/(sqrt(a + bx*S(2))sqrt(acS(2) - dS(2))))/(b(S(3)/2)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ac + bcx*S(2) + dsqrt(a + bxS(2))), x), x, atan(sqrt(b)cx/sqrt(acS(2) - dS(2)))/(sqrt(b)sqrt(acS(2) - dS(2))) - atan(sqrt(b)dx/(sqrt(a + bxS(2))sqrt(acS(2) - dS(2))))/(sqrt(b)sqrt(acS(2) - dS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(ac + bcxS(2) + dsqrt(a + bxS(2)))), x), x, -sqrt(b)c*S(2)atan(sqrt(b)cx/sqrt(acS(2) - dS(2)))/(acS(2) - dS(2))*(S(3)/2) + sqrt(b)c*S(2)atan(sqrt(b)dx/(sqrt(a + bxS(2))sqrt(acS(2) - dS(2))))/(acS(2) - dS(2))*(S(3)/2) - c/(x(acS(2) - dS(2))) + dsqrt(a + bxS(2))/(ax(acS(2) - dS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(8)/(ac + bcx*S(3) + dsqrt(a + bxS(3))), x), x, -xS(3)(S(2)acS(2) - dS(2))/(S(3)bS(2)c*S(3)) + (a + bxS(3))S(2)/(S(6)bS(3)c) - S(2)d(a + bxS(3))(S(3)/2)/(S(9)b*S(3)c*S(2)) + S(2)dsqrt(a + bx*S(3))(S(2)acS(2) - dS(2))/(S(3)bS(3)c*S(4)) + S(2)(acS(2) - dS(2))S(2)log(csqrt(a + bx*S(3)) + d)/(S(3)b*S(3)cS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/(ac + bcxS(3) + dsqrt(a + bxS(3))), x), x, xS(3)/(S(3)bc) - S(2)dsqrt(a + bx*S(3))/(S(3)b*S(2)c*S(2)) - (S(2)acS(2) - S(2)d*S(2))log(csqrt(a + bx*S(3)) + d)/(S(3)b*S(2)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(ac + bcxS(3) + dsqrt(a + bxS(3))), x), x, S(2)log(csqrt(a + bx*S(3)) + d)/(S(3)bc), expand=True, _diff=True, _numerical=True) # taking a long time assert rubi_test(rubi_integrate(S(1)/(x(ac + bcxS(3) + dsqrt(a + bxS(3)))), x), x, -S(2)clog(csqrt(a + bxS(3)) + d)/(S(3)acS(2) - S(3)d*S(2)) + clog(x)/(acS(2) - dS(2)) + S(2)datanh(sqrt(a + bx*S(3))/sqrt(a))/(S(3)sqrt(a)(acS(2) - dS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(4)(ac + bcxS(3) + dsqrt(a + bxS(3)))), x), x, -bc*S(3)log(x)/(acS(2) - dS(2))S(2) + S(2)bcS(3)log(csqrt(a + bx*S(3)) + d)/(S(3)(acS(2) - dS(2))S(2)) - (ac - dsqrt(a + bx*S(3)))/(S(3)axS(3)(acS(2) - dS(2))) - bd(S(3)acS(2) - dS(2))atanh(sqrt(a + bxS(3))/sqrt(a))/(S(3)a*(S(3)/2)(acS(2) - dS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(ac + bcx*S(3) + dsqrt(a + bxS(3))), x), x, -dx*S(4)sqrt(S(1) + bxS(3)/a)AppellF1(S(4)/3, S(1)/2, S(1), S(7)/3, -bxS(3)/a, -bc*S(2)x*S(3)/(acS(2) - dS(2)))/(sqrt(a + bxS(3))(S(4)ac*S(2) - S(4)d*S(2))) + x/(bc) - (acS(2) - dS(2))(S(1)/3)log(b*(S(1)/3)c*(S(2)/3)x + (acS(2) - dS(2))(S(1)/3))/(S(3)b*(S(4)/3)c*(S(5)/3)) + (acS(2) - dS(2))*(S(1)/3)log(b*(S(2)/3)c*(S(4)/3)xS(2) - b(S(1)/3)c(S(2)/3)x(acS(2) - dS(2))*(S(1)/3) + (acS(2) - dS(2))*(S(2)/3))/(S(6)b*(S(4)/3)c*(S(5)/3)) + sqrt(S(3))(acS(2) - dS(2))(S(1)/3)atan(sqrt(S(3))(-S(2)b*(S(1)/3)c*(S(2)/3)x/(acS(2) - dS(2))(S(1)/3) + S(1))/S(3))/(S(3)b*(S(4)/3)c*(S(5)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(ac + bcx*S(3) + dsqrt(a + bxS(3))), x), x, -dx*S(2)sqrt(S(1) + bxS(3)/a)AppellF1(S(2)/3, S(1)/2, S(1), S(5)/3, -bxS(3)/a, -bc*S(2)x*S(3)/(acS(2) - dS(2)))/(sqrt(a + bxS(3))(S(2)ac*S(2) - S(2)dS(2))) - log(b(S(1)/3)c(S(2)/3)x + (acS(2) - dS(2))(S(1)/3))/(S(3)b*(S(2)/3)c*(S(1)/3)(acS(2) - dS(2))(S(1)/3)) + log(b(S(2)/3)c*(S(4)/3)xS(2) - b(S(1)/3)c(S(2)/3)x(acS(2) - dS(2))*(S(1)/3) + (acS(2) - dS(2))*(S(2)/3))/(S(6)b*(S(2)/3)c*(S(1)/3)(acS(2) - dS(2))(S(1)/3)) - sqrt(S(3))atan(sqrt(S(3))(-S(2)b*(S(1)/3)c*(S(2)/3)x/(acS(2) - dS(2))(S(1)/3) + S(1))/S(3))/(S(3)b*(S(2)/3)c*(S(1)/3)(acS(2) - dS(2))(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ac + bcx*S(3) + dsqrt(a + bxS(3))), x), x, -dxsqrt(S(1) + bx*S(3)/a)AppellF1(S(1)/3, S(1)/2, S(1), S(4)/3, -bxS(3)/a, -bc*S(2)x*S(3)/(acS(2) - dS(2)))/(sqrt(a + bxS(3))(acS(2) - dS(2))) + c(S(1)/3)log(b*(S(1)/3)c*(S(2)/3)x + (acS(2) - dS(2))(S(1)/3))/(S(3)b*(S(1)/3)(acS(2) - dS(2))(S(2)/3)) - c(S(1)/3)log(b*(S(2)/3)c*(S(4)/3)xS(2) - b(S(1)/3)c(S(2)/3)x(acS(2) - dS(2))*(S(1)/3) + (acS(2) - dS(2))*(S(2)/3))/(S(6)b*(S(1)/3)(acS(2) - dS(2))(S(2)/3)) - sqrt(S(3))c*(S(1)/3)atan(sqrt(S(3))(-S(2)b*(S(1)/3)c*(S(2)/3)x/(acS(2) - dS(2))(S(1)/3) + S(1))/S(3))/(S(3)b*(S(1)/3)(acS(2) - dS(2))(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(ac + bcxS(3) + dsqrt(a + bxS(3)))), x), x, b(S(1)/3)c*(S(5)/3)log(b*(S(1)/3)c*(S(2)/3)x + (acS(2) - dS(2))(S(1)/3))/(S(3)(acS(2) - dS(2))(S(4)/3)) - b(S(1)/3)c*(S(5)/3)log(b*(S(2)/3)c*(S(4)/3)xS(2) - b(S(1)/3)c(S(2)/3)x(acS(2) - dS(2))*(S(1)/3) + (acS(2) - dS(2))*(S(2)/3))/(S(6)(acS(2) - dS(2))(S(4)/3)) + sqrt(S(3))b*(S(1)/3)c*(S(5)/3)atan(sqrt(S(3))(-S(2)b*(S(1)/3)c*(S(2)/3)x/(acS(2) - dS(2))(S(1)/3) + S(1))/S(3))/(S(3)(acS(2) - dS(2))(S(4)/3)) - c/(x(acS(2) - dS(2))) + dsqrt(S(1) + bxS(3)/a)AppellF1(S(-1)/3, S(1)/2, S(1), S(2)/3, -bxS(3)/a, -bc*S(2)x*S(3)/(acS(2) - dS(2)))/(xsqrt(a + bx*S(3))(acS(2) - dS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(ac + bcxS(3) + dsqrt(a + bxS(3)))), x), x, -b(S(2)/3)c*(S(7)/3)log(b*(S(1)/3)c*(S(2)/3)x + (acS(2) - dS(2))(S(1)/3))/(S(3)(acS(2) - dS(2))(S(5)/3)) + b(S(2)/3)c*(S(7)/3)log(b*(S(2)/3)c*(S(4)/3)xS(2) - b(S(1)/3)c(S(2)/3)x(acS(2) - dS(2))*(S(1)/3) + (acS(2) - dS(2))*(S(2)/3))/(S(6)(acS(2) - dS(2))(S(5)/3)) + sqrt(S(3))b*(S(2)/3)c*(S(7)/3)atan(sqrt(S(3))(-S(2)b*(S(1)/3)c*(S(2)/3)x/(acS(2) - dS(2))(S(1)/3) + S(1))/S(3))/(S(3)(acS(2) - dS(2))(S(5)/3)) - c/(xS(2)(S(2)ac*S(2) - S(2)d*S(2))) + dsqrt(S(1) + bxS(3)/a)AppellF1(S(-2)/3, S(1)/2, S(1), S(1)/3, -bxS(3)/a, -bc*S(2)x*S(3)/(acS(2) - dS(2)))/(x*S(2)sqrt(a + bxS(3))(S(2)ac*S(2) - S(2)d*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(ac + bcx*n + dsqrt(a + bxn)), x), x, cxhyper((S(1), S(1)/n), (S(1) + S(1)/n,), -bc*S(2)x*n/(acS(2) - dS(2)))/(acS(2) - dS(2)) - dxsqrt(S(1) + bx*n/a)AppellF1(S(1)/n, S(1)/2, S(1), S(1) + S(1)/n, -bxn/a, -bc*S(2)x*n/(acS(2) - dS(2)))/(sqrt(a + bxn)(acS(2) - dS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/(ac + bcx*n + dsqrt(a + bxn)), x), x, cx*(m + S(1))hyper((S(1), (m + S(1))/n), ((m + n + S(1))/n,), -bcS(2)x*n/(acS(2) - dS(2)))/((m + S(1))(acS(2) - dS(2))) - dx(m + S(1))sqrt(S(1) + bxn/a)AppellF1((m + S(1))/n, S(1)/2, S(1), (m + n + S(1))/n, -bxn/a, -bc*S(2)x*n/(acS(2) - dS(2)))/(sqrt(a + bxn)(m + S(1))(acS(2) - dS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(n + S(-1))/(ac + bcx*n + dsqrt(a + bxn)), x), x, S(2)log(csqrt(a + bx*n) + d)/(bcn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(4)x*(S(3)/2) + sqrt(x)), x), x, atan(S(2)sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(-x(S(5)/2) + sqrt(x)), x), x, atan(sqrt(x)) + atanh(sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(-x(S(1)/4) + sqrt(x)), x), x, S(4)x(S(1)/4) + S(2)sqrt(x) + S(4)log(-x(S(1)/4) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(1)/3) + sqrt(x)), x), x, S(6)x*(S(1)/6) - S(3)x*(S(1)/3) + S(2)sqrt(x) - S(6)log(x(S(1)/6) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(1)/4) + sqrt(x)), x), x, -S(4)x*(S(1)/4) + S(2)sqrt(x) + S(4)log(x(S(1)/4) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(2)/3) - x(S(1)/3)), x), x, S(3)x*(S(1)/3) + S(3)log(-x(S(1)/3) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x) + x(S(-1)/4)), x), x, S(2)sqrt(x) + S(4)log(x*(S(1)/4) + S(1))/S(3) - S(2)log(-x*(S(1)/4) + sqrt(x) + S(1))/S(3) + S(4)sqrt(S(3))atan(sqrt(S(3))(-S(2)x(S(1)/4) + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(1)/4) + x(S(1)/3)), x), x, -S(12)x*(S(7)/12)/S(7) - S(12)x*(S(5)/12)/S(5) - S(12)x*(S(1)/12) + S(6)x*(S(1)/6) - S(4)x*(S(1)/4) + S(3)x*(S(2)/3)/S(2) + S(3)x*(S(1)/3) + S(2)sqrt(x) + S(12)log(x(S(1)/12) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(S(-1)/3) + x(S(-1)/4)), x), x, S(12)x*(S(13)/12)/S(13) + S(12)x*(S(11)/12)/S(11) + S(12)x*(S(7)/12)/S(7) + S(12)x*(S(5)/12)/S(5) + S(12)x*(S(1)/12) - S(6)x*(S(7)/6)/S(7) - S(6)x*(S(5)/6)/S(5) - S(6)x*(S(1)/6) + S(4)x*(S(5)/4)/S(5) + S(4)x*(S(3)/4)/S(3) + S(4)x*(S(1)/4) - S(3)x*(S(2)/3)/S(2) - S(3)x*(S(1)/3) - S(2)sqrt(x) - x - S(12)log(x(S(1)/12) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x) - S(1)/x(S(1)/3)), x), x, S(2)sqrt(x) + S(6)log(-x(S(1)/6) + S(1))/S(5) - (-S(3)sqrt(S(5))/S(10) + S(3)/10)log(x(S(1)/6) + sqrt(S(5))x*(S(1)/6) + S(2)x*(S(1)/3) + S(2)) - (S(3)/10 + S(3)sqrt(S(5))/S(10))log(-sqrt(S(5))x(S(1)/6) + x(S(1)/6) + S(2)x(S(1)/3) + S(2)) - S(3)sqrt(S(2)sqrt(S(5)) + S(10))atan(sqrt(sqrt(S(5))/S(10) + S(1)/2)(S(4)x*(S(1)/6) + S(1) + sqrt(S(5)))/S(2))/S(5) + S(3)sqrt(-S(2)sqrt(S(5)) + S(10))atan((S(4)x(S(1)/6) - sqrt(S(5)) + S(1))/sqrt(S(2)sqrt(S(5)) + S(10)))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(x*S(2) + x), x), x, S(2)atan(sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(S(4)sqrt(x) + x), x), x, -S(8)sqrt(x) + x + S(32)log(sqrt(x) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(x(S(1)/3) + x), x), x, S(2)sqrt(x) - S(3)sqrt(S(2))log(-sqrt(S(2))x(S(1)/6) + x(S(1)/3) + S(1))/S(4) + S(3)sqrt(S(2))log(sqrt(S(2))x(S(1)/6) + x(S(1)/3) + S(1))/S(4) - S(3)sqrt(S(2))atan(sqrt(S(2))x(S(1)/6) + S(-1))/S(2) - S(3)sqrt(S(2))atan(sqrt(S(2))x(S(1)/6) + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(1)/3)/(x*(S(1)/4) + sqrt(x)), x), x, -S(12)x*(S(7)/12)/S(7) - S(12)x*(S(1)/12) + S(6)x*(S(5)/6)/S(5) + S(3)x*(S(1)/3) + S(6)log(x*(S(1)/12) + S(1)) - S(2)log(x*(S(1)/4) + S(1)) - S(4)sqrt(S(3))atan(sqrt(S(3))(-S(2)x(S(1)/12) + S(1))/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(x(S(1)/4) + x(S(1)/3)), x), x, -S(12)x*(S(13)/12)/S(13) - S(12)x*(S(11)/12)/S(11) - S(12)x*(S(7)/12)/S(7) - S(12)x*(S(5)/12)/S(5) - S(12)x*(S(1)/12) + S(6)x*(S(7)/6)/S(7) + S(6)x*(S(5)/6)/S(5) + S(6)x*(S(1)/6) - S(4)x*(S(3)/4)/S(3) - S(4)x*(S(1)/4) + S(3)x*(S(2)/3)/S(2) + S(3)x*(S(1)/3) + S(2)sqrt(x) + x + S(12)log(x(S(1)/12) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(sqrt(x) - S(1)/x(S(1)/3)), x), x, S(6)x*(S(1)/6) + x + S(6)log(-x*(S(1)/6) + S(1))/S(5) - (S(3)/10 + S(3)sqrt(S(5))/S(10))log(x(S(1)/6) + sqrt(S(5))x*(S(1)/6) + S(2)x*(S(1)/3) + S(2)) - (-S(3)sqrt(S(5))/S(10) + S(3)/10)log(-sqrt(S(5))x(S(1)/6) + x(S(1)/6) + S(2)x(S(1)/3) + S(2)) - S(3)sqrt(-S(2)sqrt(S(5)) + S(10))atan(sqrt(sqrt(S(5))/S(10) + S(1)/2)(S(4)x*(S(1)/6) + S(1) + sqrt(S(5)))/S(2))/S(5) - S(3)sqrt(S(2)sqrt(S(5)) + S(10))atan((S(4)x(S(1)/6) - sqrt(S(5)) + S(1))/sqrt(S(2)sqrt(S(5)) + S(10)))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bxS(2) + sqrt(a + bS(2)xS(4)))/sqrt(a + bS(2)xS(4)), x), x, sqrt(S(2))atanh(sqrt(S(2))sqrt(b)x/sqrt(bxS(2) + sqrt(a + bS(2)x*S(4))))/(S(2)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-bxS(2) + sqrt(a + bS(2)xS(4)))/sqrt(a + bS(2)xS(4)), x), x, sqrt(S(2))atan(sqrt(S(2))sqrt(b)x/sqrt(-bxS(2) + sqrt(a + bS(2)x*S(4))))/(S(2)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(2)xS(2) + sqrt(S(4)x*S(4) + S(3)))/((c + dx)sqrt(S(4)x*S(4) + S(3))), x), x, -(S(1)/2 + I/S(2))atanh((-S(2)Icx + sqrt(S(3))d)/(sqrt(S(2)Ic*S(2) + sqrt(S(3))d*S(2))sqrt(S(2)Ix*S(2) + sqrt(S(3)))))/sqrt(S(2)IcS(2) + sqrt(S(3))d*S(2)) + (S(1)/2 - I/S(2))atan((S(2)Icx + sqrt(S(3))d)/(sqrt(S(2)Ic*S(2) - sqrt(S(3))d*S(2))sqrt(-S(2)Ix*S(2) + sqrt(S(3)))))/sqrt(S(2)IcS(2) - sqrt(S(3))d*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(2)x*S(2) + sqrt(S(4)x*S(4) + S(3)))/((c + dx)*S(2)sqrt(S(4)xS(4) + S(3))), x), x, c(S(1) - I)atanh((-S(2)Icx + sqrt(S(3))d)/(sqrt(S(2)IcS(2) + sqrt(S(3))d*S(2))sqrt(S(2)Ix*S(2) + sqrt(S(3)))))/(S(2)IcS(2) + sqrt(S(3))dS(2))(S(3)/2) + c(S(1) + I)atan((S(2)Icx + sqrt(S(3))d)/(sqrt(S(2)Ic*S(2) - sqrt(S(3))d*S(2))sqrt(-S(2)Ix*S(2) + sqrt(S(3)))))/(S(2)IcS(2) - sqrt(S(3))dS(2))(S(3)/2) - d(S(1)/2 + I/S(2))sqrt(S(2)Ix*S(2) + sqrt(S(3)))/((c + dx)(S(2)IcS(2) + sqrt(S(3))d*S(2))) + d(S(1)/2 - I/S(2))sqrt(-S(2)IxS(2) + sqrt(S(3)))/((c + dx)(S(2)IcS(2) - sqrt(S(3))d*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(-4))/(sqrt(x)(x*(S(1)/3) + S(1))), x), x, S(6)x*(S(7)/6)/S(7) - S(6)x*(S(5)/6)/S(5) - S(30)x*(S(1)/6) + S(2)sqrt(x) + S(30)atan(x(S(1)/6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(x) + S(1))/(x(S(7)/6) + x(S(5)/6)), x), x, S(3)x*(S(1)/3) - S(3)log(x*(S(1)/3) + S(1)) + S(6)atan(x*(S(1)/6)), expand=True, _diff=True, _numerical=True) # difference in simplify assert rubi_test(rubi_integrate((sqrt(x) + S(1))/(sqrt(x)(x*(S(1)/3) + S(1))), x), x, S(6)x*(S(1)/6) + S(3)x*(S(2)/3)/S(2) - S(3)x*(S(1)/3) + S(3)log(x*(S(1)/3) + S(1)) - S(6)atan(x(S(1)/6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(b/xS(2) + S(2))/(b + S(2)xS(2)), x), x, -acsch(sqrt(S(2))x/sqrt(b))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-b/x*S(2) + S(2))/(-b + S(2)x*S(2)), x), x, -acsc(sqrt(S(2))x/sqrt(b))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + c/x*S(2))/(d + ex), x), x, sqrt(a)atanh(sqrt(a + c/xS(2))/sqrt(a))/e - sqrt(c)atanh(sqrt(c)/(xsqrt(a + c/xS(2))))/d - sqrt(ad*S(2) + ce*S(2))atanh((ad - ce/x)/(sqrt(a + c/x*S(2))sqrt(adS(2) + ce*S(2))))/(de), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + b/x + c/x*S(2))/(d + ex), x), x, sqrt(a)atanh((S(2)a + b/x)/(S(2)sqrt(a)sqrt(a + b/x + c/x*S(2))))/e - sqrt(c)atanh((b + S(2)c/x)/(S(2)sqrt(c)sqrt(a + b/x + c/xS(2))))/d - sqrt(ad*S(2) - e(bd - ce))atanh((S(2)ad - be + (bd - S(2)ce)/x)/(S(2)sqrt(adS(2) - e(bd - ce))sqrt(a + b/x + c/xS(2))))/(de), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x(S(1)/6) + (xS(3))*(S(1)/5))/sqrt(x), x), x, S(3)x*(S(2)/3)/S(2) + S(10)sqrt(x)(xS(3))(S(1)/5)/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2))/sqrt(-xS(2) + S(4)x), x), x, -sqrt(-x*S(2) + S(4)x) + S(4)asin(x/S(2) + S(-1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(3))/(xS(2) + S(6)x)*(S(1)/3), x), x, S(3)(x*S(2) + S(6)x)(S(2)/3)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(4))/(-xS(2) + S(6)x)(S(3)/2), x), x, -(-S(7)x + S(12))/(S(9)sqrt(-xS(2) + S(6)x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x + S(1))sqrt(xS(2) + S(2)x)), x), x, atan(sqrt(x*S(2) + S(2)x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((S(2)x + S(1))sqrt(x*S(2) + x)), x), x, atan(S(2)sqrt(xS(2) + x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(-1))/sqrt(-xS(2) + S(2)x), x), x, -sqrt(-xS(2) + S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-xS(2) + x)/(x + S(1)), x), x, sqrt(-xS(2) + x) + S(3)asin(S(2)x + S(-1))/S(2) + sqrt(S(2))atan(sqrt(S(2))(-S(3)x + S(1))/(S(4)sqrt(-xS(2) + x))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x(S(1)/4) + x), x), x, x*(S(1)/4)sqrt(x*(S(1)/4) + x)/S(3) + S(2)xsqrt(x(S(1)/4) + x)/S(3) - atanh(sqrt(x)/sqrt(x(S(1)/4) + x))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x(S(3)/2) + x), x), x, -S(16)(x(S(3)/2) + x)(S(3)/2)/(S(35)x) + S(4)(x(S(3)/2) + x)(S(3)/2)/(S(7)sqrt(x)) + S(32)(x(S(3)/2) + x)(S(3)/2)/(S(105)x(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(x*(S(3)/2) + x), x), x, S(4)sqrt(x)(x(S(3)/2) + x)(S(3)/2)/S(11) - S(32)(x(S(3)/2) + x)(S(3)/2)/S(99) - S(256)(x(S(3)/2) + x)(S(3)/2)/(S(1155)x) + S(64)(x(S(3)/2) + x)(S(3)/2)/(S(231)sqrt(x)) + S(512)(x(S(3)/2) + x)(S(3)/2)/(S(3465)x(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))sqrt(S(1)/(-xS(2) + S(2))), x), x, x/(S(2)sqrt(S(1)/(-xS(2) + S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-xS(4) + xS(3) + xS(2)), x), x, -(-S(2)x + S(1))sqrt(-xS(4) + xS(3) + x*S(2))/(S(8)x) - (-x*S(2) + x + S(1))sqrt(-xS(4) + xS(3) + x*S(2))/(S(3)x) - S(5)sqrt(-xS(4) + xS(3) + xS(2))asin(sqrt(S(5))(-S(2)x + S(1))/S(5))/(S(16)xsqrt(-xS(2) + x + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt((aS(2) + xS(2))S(3)), x), x, x(aS(2) + xS(2))/(aS(2)sqrt((aS(2) + xS(2))*S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(sqrt(x) + x + S(1)), x), x, S(2)sqrt(x) - log(sqrt(x) + x + S(1)) - S(2)sqrt(S(3))atan(sqrt(S(3))(S(2)sqrt(x) + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt(x) + x + S(1)), x), x, -S(2)sqrt(x) + x + S(4)sqrt(S(3))atan(sqrt(S(3))(S(2)sqrt(x) + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)(sqrt(x) + x + S(1))*(S(7)/2)), x), x, (S(8)sqrt(x) + S(4))/(S(15)(sqrt(x) + x + S(1))(S(5)/2)) + (S(128)sqrt(x) + S(64))/(S(135)(sqrt(x) + x + S(1))(S(3)/2)) + (S(1024)sqrt(x) + S(512))/(S(405)sqrt(sqrt(x) + x + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(-1))/(sqrt(xS(2) + S(1)) + S(1)), x), x, sqrt(xS(2) + S(1)) - log(sqrt(xS(2) + S(1)) + S(1)) - asinh(x) + sqrt(xS(2) + S(1))/x - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x + S(1))(S(2)/3)(xS(2) + S(-1))(S(2)/3)), x), x, S(3)(xS(2) + S(-1))(S(1)/3)/(S(2)(x + S(1))*(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(-xS(2) + S(1))/(x + S(1)), x), x, -sqrt(-xS(2) + S(1))/S(2) - asin(x)/S(2) - (-xS(2) + S(1))(S(3)/2)/(S(2)x + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(6) + S(1))(S(2)/3) + (-xS(6) + S(1))(S(2)/3)/xS(6), x), x, x(-xS(6) + S(1))(S(2)/3)/S(5) - (-xS(6) + S(1))(S(2)/3)/(S(5)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(m + S(-1))(S(2)am + bxn(S(2)m - n))/(S(2)(a + bxn)(S(3)/2)), x), x, xm/sqrt(a + bx*n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(2)x*S(3) + x)/sqrt(S(3)x + S(2)), x), x, -S(4)(S(3)x + S(2))*(S(7)/2)/S(567) + S(8)(S(3)x + S(2))(S(5)/2)/S(135) - S(10)(S(3)x + S(2))(S(3)/2)/S(81) - S(4)sqrt(S(3)x + S(2))/S(81), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x + S(1))(S(1)/4) + sqrt(x + S(1))), x), x, -S(4)(x + S(1))*(S(1)/4) + S(2)sqrt(x + S(1)) + S(4)log((x + S(1))(S(1)/4) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x + S(1))/sqrt(x*S(2) + x), x), x, S(2)sqrt(x*S(2) + x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(2)sqrt(x)(x + S(1))), x), x, atan(sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(-x*S(2) + S(6)x)), x), x, -sqrt(-x*S(2) + S(6)x)/(S(3)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)(sqrt(x) + S(1)), x), x, S(2)x(S(3)/2)/S(3) + xS(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-sqrt(x) + S(1))/x(S(1)/3), x), x, -S(6)x*(S(7)/6)/S(7) + S(3)x(S(2)/3)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(x(S(1)/3) + S(1)), x), x, S(6)x(S(7)/6)/S(7) - S(6)x*(S(5)/6)/S(5) - S(6)x*(S(1)/6) + S(2)sqrt(x) + S(6)atan(x(S(1)/6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(x) + S(1))(S(1)/3)/x, x), x, S(6)(sqrt(x) + S(1))*(S(1)/3) - log(x)/S(2) + S(3)log(-(sqrt(x) + S(1))*(S(1)/3) + S(1)) - S(2)sqrt(S(3))atan(sqrt(S(3))(S(2)(sqrt(x) + S(1))(S(1)/3) + S(1))/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-sqrt(x) + S(1), x), x, -S(2)x(S(3)/2)/S(3) + x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-x(S(1)/4) + S(1), x), x, -S(4)x(S(5)/4)/S(5) + x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-sqrt(x) + S(1))/(x(S(1)/4) + S(1)), x), x, -S(4)x*(S(5)/4)/S(5) + x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt((a + bx)(c + dx)), x), x, atanh((ad + bc + S(2)bdx)/(S(2)sqrt(b)sqrt(d)sqrt(ac + bdxS(2) + x(ad + bc))))/(sqrt(b)sqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt((a + bx)(c - dx)), x), x, -atan((-ad + bc - S(2)bdx)/(S(2)sqrt(b)sqrt(d)sqrt(ac - bdxS(2) + x(-ad + bc))))/(sqrt(b)sqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)(-xS(2) + S(1))), x), x, atan(sqrt(x)) + atanh(sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(-xS(3) + x), x), x, atan(sqrt(x)) + atanh(sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(x*S(2) + x(S(1) + sqrt(S(3))) - sqrt(S(3)) + S(2)), x), x, log(x*S(2) + x(S(1) + sqrt(S(3))) - sqrt(S(3)) + S(2))/S(2) + sqrt(S(13)/23 + S(8)sqrt(S(3))/S(23))atanh((S(2)x + S(1) + sqrt(S(3)))/sqrt(S(-4) + S(6)sqrt(S(3)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xS(3) + xS(2)), x), x, S(2)(xS(3) + xS(2))(S(3)/2)/(S(5)x*S(2)) - S(4)(xS(3) + xS(2))*(S(3)/2)/(S(15)x*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x + S(1))sqrt(x*S(2) + S(2)x)), x), x, atan(sqrt(x*S(2) + S(2)x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)sqrt(-sqrt(x) - x + S(1)), x), x, -sqrt(x)(-sqrt(x) - x + S(1))*(S(3)/2)/S(2) + (S(9)sqrt(x)/S(16) + S(9)/32)sqrt(-sqrt(x) - x + S(1)) + S(5)(-sqrt(x) - x + S(1))*(S(3)/2)/S(12) + S(45)asin(sqrt(S(5))(S(2)sqrt(x) + S(1))/S(5))/S(64), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(x + S(-3)) + S(1))*(S(1)/3), x), x, S(6)(sqrt(x + S(-3)) + S(1))*(S(7)/3)/S(7) - S(3)(sqrt(x + S(-3)) + S(1))*(S(4)/3)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(sqrt(S(2)x + S(-1)) + S(3)), x), x, S(2)(sqrt(S(2)x + S(-1)) + S(3))*(S(3)/2)/S(3) - S(6)sqrt(sqrt(S(2)x + S(-1)) + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-x + S(1))/(sqrt(x) + S(1)), x), x, -sqrt(-x + S(1)) - asin(sqrt(x)) - (-x + S(1))(S(3)/2)/(sqrt(x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-x + S(1))/(-sqrt(x) + S(1)), x), x, -sqrt(-x + S(1)) + asin(sqrt(x)) - (-x + S(1))(S(3)/2)/(-sqrt(x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(x - sqrt(xS(2) + S(1))), x), x, -xS(3)/S(3) - (xS(2) + S(1))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(x - sqrt(-xS(2) + S(1))), x), x, x/S(2) + sqrt(-xS(2) + S(1))/S(2) - sqrt(S(2))atanh(sqrt(S(2))x)/S(4) - sqrt(S(2))atanh(sqrt(S(2))sqrt(-xS(2) + S(1)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(x - sqrt(S(2)x*S(2) + S(1))), x), x, -x - sqrt(S(2)x*S(2) + S(1)) + atan(x) + atan(sqrt(S(2)x*S(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)sqrt(sqrt(x) + x), x), x, sqrt(x)(sqrt(x) + x)(S(3)/2)/S(2) + (S(5)sqrt(x)/S(16) + S(5)/32)sqrt(sqrt(x) + x) - S(5)(sqrt(x) + x)*(S(3)/2)/S(12) - S(5)atanh(sqrt(x)/sqrt(sqrt(x) + x))/S(32), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*(S(1)/3) + S(1))/(sqrt(x) + S(1)), x), x, S(6)x*(S(5)/6)/S(5) - S(3)x*(S(1)/3) + S(2)sqrt(x) - S(4)log(x(S(1)/6) + S(1)) - log(-x(S(1)/6) + x(S(1)/3) + S(1)) - S(2)sqrt(S(3))atan(sqrt(S(3))(-S(2)x(S(1)/6) + S(1))/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x(S(1)/3) + S(1))/(x(S(1)/4) + S(1)), x), x, S(12)x*(S(13)/12)/S(13) + S(12)x*(S(7)/12)/S(7) + S(12)x*(S(1)/12) - S(6)x*(S(5)/6)/S(5) + S(4)x*(S(3)/4)/S(3) + S(4)x*(S(1)/4) - S(3)x*(S(1)/3) - S(2)sqrt(x) - S(8)log(x(S(1)/12) + S(1)) - S(2)log(-x(S(1)/12) + x(S(1)/6) + S(1)) + S(4)sqrt(S(3))atan(sqrt(S(3))(-S(2)x(S(1)/12) + S(1))/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(xS(2) + sqrt(-xS(2) + S(1)) + S(-1)), x), x, x + asin(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((x + S(1))/x), x), x, xsqrt(S(1) + S(1)/x) + atanh(sqrt(S(1) + S(1)/x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((-x + S(1))/x), x), x, xsqrt(S(-1) + S(1)/x) - atan(sqrt(S(-1) + S(1)/x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((x + S(-1))/x), x), x, sqrt(x)sqrt(x + S(-1)) - asinh(sqrt(x + S(-1))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt((x + S(-1))/x), x), x, xsqrt(S(1) - S(1)/x) - atanh(sqrt(S(1) - S(1)/x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((x + S(1))/x)/x, x), x, -S(2)sqrt(S(1) + S(1)/x) + S(2)atanh(sqrt(S(1) + S(1)/x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x/(x + S(1))), x), x, sqrt(x)sqrt(x + S(1)) - asinh(sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt((-x + S(-1))/x), x), x, -xsqrt(S(-1) - S(1)/x) + atan(sqrt(S(-1) - S(1)/x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x(-x + S(4))), x), x, (x/S(2) + S(-1))sqrt(-x*S(2) + S(4)x) + S(2)asin(x/S(2) + S(-1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(x(-x + S(1))), x), x, asin(S(2)x + S(-1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(x(x + S(2)))(S(3)/2), x), x, x/sqrt(xS(2) + S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(1) + S(1)/x)/(-xS(2) + S(1)), x), x, sqrt(S(2))atanh(sqrt(S(2))sqrt(S(1) + S(1)/x)/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(-xS(2) + sqrt(S(5))x*S(2) + S(1) + sqrt(S(5))), x), x, atan(xsqrt(-sqrt(S(5))/S(2) + S(3)/2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((-a + x)(b - x)), x), x, -(a - b)S(2)atan((a + b - S(2)x)/(S(2)sqrt(-ab - xS(2) + x(a + b))))/S(8) + (-a/S(4) - b/S(4) + x/S(2))sqrt(-ab - x*S(2) + x(a + b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt((-a + x)(b - x)), x), x, -atan((a + b - S(2)x)/(S(2)sqrt(-ab - x*S(2) + x(a + b)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((-x*S(2) + S(1))(x*S(2) + S(3))), x), x, xsqrt(-x*S(4) - S(2)x*S(2) + S(3))/S(3) - S(2)sqrt(S(3))elliptic_e(asin(x), S(-1)/3)/S(3) + S(4)sqrt(S(3))elliptic_f(asin(x), S(-1)/3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt((-xS(2) + S(1))(x*S(2) + S(3))), x), x, sqrt(S(3))elliptic_f(asin(x), S(-1)/3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(ax + bx*S(2)), x), x, S(2)atanh(sqrt(b)x/sqrt(ax + bxS(2)))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(x(a + bx)), x), x, S(2)atanh(sqrt(b)x/sqrt(ax + bxS(2)))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(xS(2)(a/x + b)), x), x, S(2)atanh(sqrt(b)x/sqrt(ax + bxS(2)))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(xS(3)(a/xS(2) + b/x)), x), x, S(2)atanh(sqrt(b)x/sqrt(ax + bxS(2)))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt((ax*S(2) + bx*S(3))/x), x), x, S(2)atanh(sqrt(b)x/sqrt(ax + bxS(2)))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt((ax*S(3) + bxS(4))/xS(2)), x), x, S(2)atanh(sqrt(b)x/sqrt(ax + bx*S(2)))/sqrt(b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(acx + bcxS(2)), x), x, S(2)atanh(sqrt(b)sqrt(c)x/sqrt(acx + bcx*S(2)))/(sqrt(b)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(c(ax + bxS(2))), x), x, S(2)atanh(sqrt(b)sqrt(c)x/sqrt(acx + bcx*S(2)))/(sqrt(b)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(cx(a + bx)), x), x, S(2)atanh(sqrt(b)sqrt(c)x/sqrt(acx + bcx*S(2)))/(sqrt(b)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(cxS(2)(a/x + b)), x), x, S(2)atanh(sqrt(b)sqrt(c)x/sqrt(acx + bcxS(2)))/(sqrt(b)sqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-x*S(2) + xsqrt(x*S(2) + S(-1)) + S(1)), x), x, (S(3)x/S(4) + sqrt(x*S(2) + S(-1))/S(4))sqrt(-x*S(2) + xsqrt(x*S(2) + S(-1)) + S(1)) + S(3)sqrt(S(2))asin(x - sqrt(xS(2) + S(-1)))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sqrt(x)sqrt(x + S(1)) - x)/sqrt(x + S(1)), x), x, (sqrt(x)/S(2) + S(3)sqrt(x + S(1))/S(2))sqrt(sqrt(x)sqrt(x + S(1)) - x) - S(3)sqrt(S(2))asin(sqrt(x) - sqrt(x + S(1)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-(x + S(2)sqrt(xS(2) + S(1)))/(xS(3) + x + sqrt(x*S(2) + S(1))), x), x, -sqrt(S(2) + S(2)sqrt(S(5)))atan(sqrt(S(-2) + sqrt(S(5)))(x + sqrt(x*S(2) + S(1)))) + sqrt(S(-2) + S(2)sqrt(S(5)))atanh(sqrt(S(2) + sqrt(S(5)))(x + sqrt(x*S(2) + S(1)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x + S(1))/((x*S(2) + S(1))sqrt(x*S(2) + S(2)x + S(2))), x), x, -sqrt(S(1)/2 + sqrt(S(5))/S(2))atan((-x(sqrt(S(5)) + S(5)) + S(2)sqrt(S(5)))/(sqrt(S(10) + S(10)sqrt(S(5)))sqrt(xS(2) + S(2)x + S(2)))) - sqrt(S(-1)/2 + sqrt(S(5))/S(2))atanh((x(-sqrt(S(5)) + S(5)) + S(2)sqrt(S(5)))/(sqrt(S(-10) + S(10)sqrt(S(5)))sqrt(xS(2) + S(2)x + S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(-xS(2) + sqrt(xS(4) + S(1)))(xS(4) + S(1))), x), x, atan(x/sqrt(-xS(2) + sqrt(xS(4) + S(1)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bx*S(4))sqrt(cxS(2) + dsqrt(a + bxS(4)))), x), x, atanh(sqrt(c)x/sqrt(cxS(2) + dsqrt(a + bxS(4))))/(asqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bxS(4))sqrt(-cxS(2) + dsqrt(a + bxS(4)))), x), x, atan(sqrt(c)x/sqrt(-cxS(2) + dsqrt(a + bxS(4))))/(asqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + bcS(4) + S(4)bcS(3)dx + S(6)bcS(2)d*S(2)x*S(2) + S(4)bcd*S(3)x*S(3) + bd*S(4)x*S(4)), x), x, atanh(sqrt(b)d*S(2)(c/d + x)*S(2)/sqrt(a + bd*S(4)(c/d + x)*S(4)))/(S(2)sqrt(b)dS(2)) - csqrt((a + bdS(4)(c/d + x)*S(4))/(sqrt(a) + sqrt(b)d*S(2)(c/d + x)S(2))S(2))(sqrt(a) + sqrt(b)d*S(2)(c/d + x)*S(2))elliptic_f(S(2)atan(b(S(1)/4)d(c/d + x)/a(S(1)/4)), S(1)/2)/(S(2)a*(S(1)/4)b*(S(1)/4)d*S(2)sqrt(a + bdS(4)(c/d + x)*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + bc*S(4) + S(4)bcS(3)dx + S(6)bcS(2)d*S(2)x*S(2) + S(4)bcd*S(3)x*S(3) + bd*S(4)x*S(4)), x), x, sqrt((a + bd*S(4)(c/d + x)*S(4))/(sqrt(a) + sqrt(b)d*S(2)(c/d + x)S(2))S(2))(sqrt(a) + sqrt(b)d*S(2)(c/d + x)*S(2))elliptic_f(S(2)atan(b(S(1)/4)d(c/d + x)/a(S(1)/4)), S(1)/2)/(S(2)a*(S(1)/4)b*(S(1)/4)dsqrt(a + bd*S(4)(c/d + x)*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a - cx*S(4))/(sqrt(a + bx*S(2) + cx*S(4))(ad + aexS(2) + cdxS(4))), x), x, atanh(xsqrt(-ae + bd)/(sqrt(d)sqrt(a + bx*S(2) + cx*S(4))))/(sqrt(d)sqrt(-ae + bd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a - cxS(4))/(sqrt(a - bx*S(2) + cx*S(4))(ad + aexS(2) + cdxS(4))), x), x, atan(xsqrt(ae + bd)/(sqrt(d)sqrt(a - bx*S(2) + cx*S(4))))/(sqrt(d)sqrt(ae + bd)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((x*S(3) + S(8))sqrt(x*S(2) - S(2)x + S(5))), x), x, -sqrt(S(3))atan(sqrt(S(3))(-x + S(1))/(S(3)sqrt(xS(2) - S(2)x + S(5))))/S(12) - sqrt(S(13))atanh(sqrt(S(13))(-S(3)x + S(7))/(S(13)sqrt(x*S(2) - S(2)x + S(5))))/S(156) + atanh(sqrt(x*S(2) - S(2)x + S(5)))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xS(2)/(xS(2) + S(1))), x), x, sqrt(x*S(2) + S(1))sqrt(xS(2))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xn/(x*n + S(1))), x), x, S(2)xsqrt(xn)hyper((S(1)/2, S(1)/2 + S(1)/n), (S(3)/2 + S(1)/n,), -x*n)/(n + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-efxS(2) + ef)/((adx*S(2) + ad + bdx)sqrt(ax*S(4) + a + bx*S(3) + bx + cxS(2))), x), x, efatan((abxS(2) + ab + x(S(4)a*S(2) - S(2)ac + bS(2)))/(S(2)asqrt(S(2)a - c)sqrt(ax*S(4) + a + bx*S(3) + bx + cxS(2))))/(adsqrt(S(2)a - c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-efx*S(2) + ef)/((-adx*S(2) - ad + bdx)sqrt(-ax*S(4) - a + bx*S(3) + bx + cxS(2))), x), x, efatanh((abxS(2) + ab - x(S(4)a*S(2) + S(2)ac + bS(2)))/(S(2)asqrt(S(2)a + c)sqrt(-ax*S(4) - a + bx*S(3) + bx + cxS(2))))/(adsqrt(S(2)a + c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(axS(2) + bxsqrt(aS(2)xS(2)/bS(2) - a/b*S(2)))/(xsqrt(a*S(2)xS(2)/bS(2) - a/b*S(2))), x), x, sqrt(S(2))basinh((ax + bsqrt(aS(2)xS(2)/bS(2) - a/b*S(2)))/sqrt(a))/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-ax*S(2) + bxsqrt(aS(2)xS(2)/bS(2) + a/b*S(2)))/(xsqrt(a*S(2)xS(2)/bS(2) + a/b*S(2))), x), x, sqrt(S(2))basin((ax - bsqrt(aS(2)xS(2)/bS(2) + a/b*S(2)))/sqrt(a))/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x(ax + bsqrt(a*S(2)xS(2)/bS(2) - a/b*S(2))))/(xsqrt(a*S(2)xS(2)/bS(2) - a/b*S(2))), x), x, sqrt(S(2))basinh((ax + bsqrt(aS(2)xS(2)/bS(2) - a/b*S(2)))/sqrt(a))/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x(-ax + bsqrt(a*S(2)xS(2)/bS(2) + a/b*S(2))))/(xsqrt(a*S(2)xS(2)/bS(2) + a/b*S(2))), x), x, sqrt(S(2))basin((ax - bsqrt(aS(2)xS(2)/bS(2) + a/b*S(2)))/sqrt(a))/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xsqrt(x + S(-4)) + xsqrt(x + S(-1)) - sqrt(x + S(-4)) - S(4)sqrt(x + S(-1)))/((x*S(2) - S(5)x + S(4))(sqrt(x + S(-4)) + sqrt(x + S(-1)) + S(1))), x), x, S(2)log(sqrt(x + S(-4)) + sqrt(x + S(-1)) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(xS(2) + S(3)x + S(3))(xS(3) + S(3)x*S(2) + S(3)x + S(3))(S(1)/3)), x), x, S(3)(S(2)/3)log(-S(3)(S(1)/3)(x + S(1))/((x + S(1))S(3) + S(2))(S(1)/3) + S(1))/S(9) - S(3)*(S(2)/3)log(S(3)*(S(2)/3)(x + S(1))S(2)/((x + S(1))S(3) + S(2))(S(2)/3) + S(3)(S(1)/3)(x + S(1))/((x + S(1))S(3) + S(2))(S(1)/3) + S(1))/S(18) - S(3)(S(1)/6)atan(sqrt(S(3))(S(2)S(3)*(S(1)/3)(x + S(1))/((x + S(1))S(3) + S(2))(S(1)/3) + S(1))/S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(1))/((-xS(3) + S(1))*(S(2)/3)(x*S(2) - x + S(1))), x), x, S(3)S(2)*(S(1)/3)log(S(2)*(S(1)/3)(-x + S(1)) + (-xS(3) + S(1))(S(1)/3))/S(4) - S(2)*(S(1)/3)log(-x*S(3) + S(2)(-x + S(1))S(3) + S(1))/S(4) + S(2)(S(1)/3)sqrt(S(3))atan(sqrt(S(3))(-S(2)S(2)*(S(1)/3)(-x + S(1))/(-xS(3) + S(1))(S(1)/3) + S(1))/S(3))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(sqrt(xS(4) + S(-1))(xS(4) + S(1))), x), x, -atan((xS(2) + S(1))/(xsqrt(xS(4) + S(-1))))/S(4) - atanh((-xS(2) + S(1))/(xsqrt(xS(4) + S(-1))))/S(4), expand=True, _diff=True, _numerical=True) def test_3(): assert rubi_test(rubi_integrate(sqrt(xS(2) + S(-1))/sqrt(xS(4) + S(-1)), x), x, sqrt(xS(2) + S(-1))sqrt(x*S(2) + S(1))asinh(x)/sqrt(xS(4) + S(-1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xS(2) + S(1))/sqrt(xS(4) + S(-1)), x), x, -sqrt(xS(4) + S(-1))asin(x)/sqrt(-xS(4) + S(1)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt(xS(2) + S(1))/sqrt(xS(4) + S(-1)), x), x, sqrt(xS(2) + S(-1))sqrt(x*S(2) + S(1))atanh(x/sqrt(xS(2) + S(-1)))/sqrt(xS(4) + S(-1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-sqrt(xS(2) + S(-1)) + sqrt(xS(2) + S(1)))/sqrt(xS(4) + S(-1)), x), x, sqrt(xS(2) + S(-1))sqrt(xS(4) + S(-1))asinh(x)/((-x*S(2) + S(1))sqrt(xS(2) + S(1))) - sqrt(xS(4) + S(-1))asin(x)/(sqrt(-xS(2) + S(1))sqrt(xS(2) + S(1))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((-sqrt(xS(2) + S(-1)) + sqrt(xS(2) + S(1)))/sqrt(xS(4) + S(-1)), x), x, -sqrt(x*S(2) + S(-1))sqrt(x*S(2) + S(1))asinh(x)/sqrt(xS(4) + S(-1)) + sqrt(xS(2) + S(-1))sqrt(xS(2) + S(1))atanh(x/sqrt(xS(2) + S(-1)))/sqrt(xS(4) + S(-1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(-xS(2) + S(1))S(5), x), x, S(1)/(S(8)(-xS(2) + S(1))S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-S(5)/(S(256)(x + S(1))*S(2)) - S(5)/(S(128)(x + S(1))*S(3)) - S(3)/(S(64)(x + S(1))*S(4)) - S(1)/(S(32)(x + S(1))*S(5)) + S(5)/(S(256)(x + S(-1))*S(2)) - S(5)/(S(128)(x + S(-1))*S(3)) + S(3)/(S(64)(x + S(-1))*S(4)) - S(1)/(S(32)(x + S(-1))*S(5)), x), x, S(1)/(S(8)(-xS(2) + S(1))S(4)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(-S(5)/(S(256)(x + S(1))S(2)) - S(5)/(S(128)(x + S(1))*S(3)) - S(3)/(S(64)(x + S(1))*S(4)) - S(1)/(S(32)(x + S(1))*S(5)) + S(5)/(S(256)(x + S(-1))*S(2)) - S(5)/(S(128)(x + S(-1))*S(3)) + S(3)/(S(64)(x + S(-1))*S(4)) - S(1)/(S(32)(x + S(-1))*S(5)), x), x, S(5)/(S(256)(x + S(1))) + S(5)/(S(256)(x + S(1))S(2)) + S(1)/(S(64)(x + S(1))*S(3)) + S(1)/(S(128)(x + S(1))*S(4)) + S(5)/(S(256)(-x + S(1))) + S(5)/(S(256)(-x + S(1))S(2)) + S(1)/(S(64)(-x + S(1))*S(3)) + S(1)/(S(128)(-x + S(1))S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2))/(xS(2) + S(2)x + S(-1)), x), x, (-sqrt(S(2)) + S(2))log(x + S(1) + sqrt(S(2)))/S(4) + (sqrt(S(2)) + S(2))log(x - sqrt(S(2)) + S(1))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(-4))/(xS(3) - S(5)x + S(2)), x), x, (-sqrt(S(2)) + S(2))log(x + S(1) + sqrt(S(2)))/S(4) + (sqrt(S(2)) + S(2))log(x - sqrt(S(2)) + S(1))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)xS(8) + S(1))/(x(xS(8) + S(1))(S(3)/2)), x), x, -atanh(sqrt(x*S(8) + S(1)))/S(4) - S(1)/(S(4)sqrt(xS(8) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xS(8) + S(1))(S(2)xS(8) + S(1))/(xS(17) + S(2)xS(9) + x), x), x, -atanh(sqrt(xS(8) + S(1)))/S(4) - S(1)/(S(4)sqrt(x*S(8) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-S(9)x*S(2) + x/sqrt(-S(9)x*S(2) + S(1)) + S(1), x), x, -S(3)x*S(3) + x - sqrt(-S(9)x*S(2) + S(1))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + (-S(9)xS(2) + S(1))(S(3)/2))/sqrt(-S(9)xS(2) + S(1)), x), x, -S(3)x*S(3) + x - sqrt(-S(9)x*S(2) + S(1))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(3)sqrt(x) + x)*(S(2)/3)(S(2)sqrt(x) + S(-3))/sqrt(x), x), x, S(6)(-S(3)sqrt(x) + x)(S(5)/3)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(9)sqrt(x) + S(2)x + S(9))/(-S(3)sqrt(x) + x)*(S(1)/3), x), x, S(6)(-S(3)sqrt(x) + x)(S(5)/3)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(2)/(S(4)x*S(2) + S(-1)), x), x, -atanh(S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-S(1)/(S(2)x + S(1)) + S(1)/(S(2)x + S(-1)), x), x, log(-S(2)x + S(1))/S(2) - log(S(2)x + S(1))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-S(9)xS(2) + S(4)), x), x, asin(S(3)x/S(2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(-S(3)x + S(2))sqrt(S(3)x + S(2))), x), x, asin(S(3)x/S(2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt((-S(3)x + S(2))(S(3)x + S(2))), x), x, asin(S(3)x/S(2))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-x*S(2) - S(2)x + S(15)), x), x, asin(x/S(4) + S(1)/4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(-x + S(3))sqrt(x + S(5))), x), x, asin(x/S(4) + S(1)/4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt((-x + S(3))(x + S(5))), x), x, asin(x/S(4) + S(1)/4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-x*S(2) - S(8)x + S(-15)), x), x, asin(x + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(-x + S(-3))sqrt(x + S(5))), x), x, asin(x + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt((-x + S(-3))(x + S(5))), x), x, asin(x + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-sqrt(x) + S(1), x), x, -S(2)x(S(3)/2)/S(3) + x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-x + S(1))/(sqrt(x) + S(1)), x), x, -S(2)x(S(3)/2)/S(3) + x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(1)/(-xS(2) + S(1))), x), x, sqrt(-x*S(2) + S(1))sqrt(S(1)/(-x*S(2) + S(1)))asin(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((xS(2) + S(1))/(-xS(4) + S(1))), x), x, sqrt(-x*S(2) + S(1))sqrt(S(1)/(-x*S(2) + S(1)))asin(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(1)/(xS(2) + S(-1))), x), x, sqrt(xS(2) + S(-1))sqrt(S(1)/(xS(2) + S(-1)))atanh(x/sqrt(xS(2) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((xS(2) + S(1))/(xS(4) + S(-1))), x), x, sqrt(xS(2) + S(-1))sqrt(S(1)/(xS(2) + S(-1)))atanh(x/sqrt(xS(2) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(6) + S(1))/(xS(6) + S(-1)), x), x, x + log(xS(2) - x + S(1))/S(6) - log(x*S(2) + x + S(1))/S(6) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(3) - sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(3) - S(2)atanh(x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + x(S(-3)))/(xS(3) - S(1)/xS(3)), x), x, x + log(xS(2) - x + S(1))/S(6) - log(xS(2) + x + S(1))/S(6) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(3) - sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(3) - S(2)atanh(x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-x + S(1)), x), x, -S(2)sqrt(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x + S(1))/sqrt(-x*S(2) + S(1)), x), x, -S(2)sqrt(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(x + S(1)), x), x, S(2)sqrt(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-x + S(1))/sqrt(-xS(2) + S(1)), x), x, S(2)sqrt(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-x + S(1)), x), x, -S(2)(-x + S(1))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-xS(2) + S(1))/sqrt(x + S(1)), x), x, -S(2)(-x + S(1))*(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x + S(1)), x), x, S(2)(x + S(1))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-xS(2) + S(1))/sqrt(-x + S(1)), x), x, S(2)(x + S(1))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(3)x + S(2))/sqrt(x + S(1)), x), x, sqrt(x + S(1))sqrt(S(3)x + S(2)) - sqrt(S(3))asinh(sqrt(S(3)x + S(2)))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-x + S(1))sqrt(S(3)x + S(2))/sqrt(-x*S(2) + S(1)), x), x, sqrt(x + S(1))sqrt(S(3)x + S(2)) - sqrt(S(3))asinh(sqrt(S(3)x + S(2)))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(1))(S(3)/2)/(x(-x + S(1))*(S(3)/2)), x), x, -asin(x) - atanh(sqrt(-x + S(1))sqrt(x + S(1))) + S(4)sqrt(x + S(1))/sqrt(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(1))S(3)/(x(-xS(2) + S(1))(S(3)/2)), x), x, -asin(x) - atanh(sqrt(-x*S(2) + S(1))) + S(4)sqrt(-x*S(2) + S(1))/(-x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax + S(1))*(S(3)/2)/(x(-ax + S(1))(S(3)/2)), x), x, -asin(ax) - atanh(sqrt(-ax + S(1))sqrt(ax + S(1))) + S(4)sqrt(ax + S(1))/sqrt(-ax + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((ax + S(1))S(3)/(x(-a*S(2)xS(2) + S(1))(S(3)/2)), x), x, -asin(ax) - atanh(sqrt(-aS(2)x*S(2) + S(1))) + S(4)sqrt(-a*S(2)x*S(2) + S(1))/(-ax + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-xS(2) + S(1)), x), x, asin(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xS(2) + S(1))/sqrt(-xS(4) + S(1)), x), x, asin(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(xS(2) + S(1)), x), x, asinh(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-xS(2) + S(1))/sqrt(-xS(4) + S(1)), x), x, asinh(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-x*S(2) + S(1)), x), x, xsqrt(-xS(2) + S(1))/S(2) + asin(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-xS(4) + S(1))/sqrt(x*S(2) + S(1)), x), x, xsqrt(-xS(2) + S(1))/S(2) + asin(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xS(2) + S(1)), x), x, xsqrt(xS(2) + S(1))/S(2) + asinh(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-xS(4) + S(1))/sqrt(-xS(2) + S(1)), x), x, xsqrt(xS(2) + S(1))/S(2) + asinh(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2)c + aS(2)dx + S(2)abcxS(2) + S(2)abdxS(3) + bS(2)cxS(4) + bS(2)dxS(5))/(c + dx), x), x, a*S(2)x + S(2)abxS(3)/S(3) + bS(2)xS(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2)c + aS(2)dx + S(2)abcxS(2) + S(2)abdxS(3) + bS(2)cxS(4) + bS(2)dxS(5))/(c + dx)S(2), x), x, -bS(2)cx*S(3)/(S(3)dS(2)) + bS(2)xS(4)/(S(4)d) - bcx(S(2)adS(2) + bcS(2))/dS(4) + bxS(2)(S(2)ad*S(2) + bc*S(2))/(S(2)d*S(3)) + (ad*S(2) + bcS(2))S(2)log(c + dx)/dS(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2)c + aS(2)dx + S(2)abcxS(2) + S(2)abdxS(3) + bS(2)cxS(4) + bS(2)dxS(5))/(a + bx*S(2)), x), x, acx + adxS(2)/S(2) + bcxS(3)/S(3) + bdxS(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2)c + a*S(2)dx + S(2)abcxS(2) + S(2)abdxS(3) + bS(2)cxS(4) + bS(2)dxS(5))/(a + bxS(2))S(2), x), x, cx + dxS(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((aS(2)c + aS(2)dx + S(2)abcxS(2) + S(2)abdxS(3) + bS(2)cxS(4) + bS(2)dxS(5))/(a + bxS(2))S(3), x), x, dlog(a + bx*S(2))/(S(2)b) + catan(sqrt(b)x/sqrt(a))/(sqrt(a)sqrt(b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(((a + b + cxS(2))/d)m, x), x, dx(cxS(2)/d + (a + b)/d)(m + S(1))hyper((S(1), m + S(3)/2), (S(3)/2,), -cxS(2)/(a + b))/(a + b), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(((a + b + cxS(2))/d)m, x), x, x(cxS(2)/d + (a + b)/d)m(cxS(2)/(a + b) + S(1))(-m)hyper((S(1)/2, -m), (S(3)/2,), -cxS(2)/(a + b)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x - sqrt(xS(2) + S(1))), x), x, -x*S(2)/S(2) - xsqrt(xS(2) + S(1))/S(2) - asinh(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x - sqrt(-xS(2) + S(1))), x), x, log(-S(2)xS(2) + S(1))/S(4) - asin(x)/S(2) - atanh(x/sqrt(-xS(2) + S(1)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x - sqrt(S(2)xS(2) + S(1))), x), x, -log(xS(2) + S(1))/S(2) - sqrt(S(2))asinh(sqrt(S(2))x) + atanh(x/sqrt(S(2)xS(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(3) + xS(2)sqrt(-x*S(2) + S(2)) + S(2)x)/(S(2)xS(2) + S(-2)), x), x, -xS(2)/S(4) + xsqrt(-xS(2) + S(2))/S(4) + log(-xS(2) + S(1))/S(4) - atanh(x/sqrt(-x*S(2) + S(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(-xS(2) + S(2))/(x - sqrt(-xS(2) + S(2))), x), x, -x*S(2)/S(4) + xsqrt(-xS(2) + S(2))/S(4) + log(-x + S(1))/S(4) + log(x + S(1))/S(4) - atanh(x/sqrt(-xS(2) + S(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(-x + sqrt(-x*S(2) + S(2)x)), x), x, -x/S(2) - sqrt(-x*S(2) + S(2)x)/S(2) - log(-x + S(1))/S(2) + atanh(sqrt(-x*S(2) + S(2)x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(-x*S(2) + S(2)x))/(-S(2)x + S(2)), x), x, -x/S(2) - sqrt(-xS(2) + S(2)x)/S(2) - log(-x + S(1))/S(2) + atanh(sqrt(-x*S(2) + S(2)x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(x)sqrt(-x + S(2)) + x)/(-S(2)x + S(2)), x), x, -x/S(2) - sqrt(-x*S(2) + S(2)x)/S(2) - log(-x + S(1))/S(2) + atanh(sqrt(-x*S(2) + S(2)x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/(-sqrt(x) + sqrt(-x + S(2))), x), x, -sqrt(x)sqrt(-x + S(2))/S(2) - x/S(2) - log(-x + S(1))/S(2) + atanh(sqrt(x)sqrt(-x + S(2)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)sqrt(-x + S(3)) + S(3)/sqrt(x + S(1)))S(2)/x, x), x, -S(4)x + S(21)log(x) - S(9)log(x + S(1)) - S(12)asin(x/S(2) + S(-1)/2) - S(24)sqrt(S(3))atanh(sqrt(S(3))sqrt(x + S(1))/sqrt(-x + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + x + S(-1))/(sqrt(xS(2) + S(1)) + S(1)), x), x, xsqrt(xS(2) + S(1))/S(2) - x + sqrt(xS(2) + S(1)) - log(sqrt(xS(2) + S(1)) + S(1)) - asinh(x)/S(2) + sqrt(xS(2) + S(1))/x - S(1)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + x + S(-1))/(x + sqrt(xS(2) + S(1)) + S(1)), x), x, xS(3)/S(6) + xS(2)/S(2) + sqrt(xS(2) + S(1))(-S(2)xS(2) - S(3)x + S(4))/S(12) - log(sqrt(xS(2) + S(1)) + S(1))/S(2) - asinh(x)/S(4), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((xS(2) + x + S(-1))/(x + sqrt(xS(2) + S(1)) + S(1)), x), x, xS(3)/S(6) + x*S(2)/S(2) - xsqrt(xS(2) + S(1))/S(4) + x/S(2) - (xS(2) + S(1))(S(3)/2)/S(6) + log(x + sqrt(xS(2) + S(1)))/S(2) - log(x + sqrt(x*S(2) + S(1)) + S(1)) - asinh(x)/S(4) + S(1)/(S(2)(x + sqrt(x*S(2) + S(1)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(2)sqrt(x + S(-1)))/(xsqrt(x + S(-1))), x), x, S(2)sqrt(x + S(-1)) + S(2)log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*(S(2)/3) + csqrt(x))S(2), x), x, aS(2)x + S(6)abx*(S(5)/3)/S(5) + S(4)acx*(S(3)/2)/S(3) + S(3)b*S(2)x*(S(7)/3)/S(7) + S(12)bcx(S(13)/6)/S(13) + cS(2)xS(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx*(S(2)/3) + csqrt(x))S(3), x), x, aS(3)x + S(9)a*S(2)bx(S(5)/3)/S(5) + S(2)a*S(2)cx(S(3)/2) + S(9)abS(2)x*(S(7)/3)/S(7) + S(36)abcx(S(13)/6)/S(13) + S(3)acS(2)xS(2)/S(2) + bS(3)xS(3)/S(3) + S(18)b*S(2)cx(S(17)/6)/S(17) + S(9)bcS(2)x*(S(8)/3)/S(8) + S(2)c*S(3)x(S(5)/2)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(-1))/(x*S(3)sqrt(a - b + b/xS(2))), x), x, atanh(sqrt(a - b + b/xS(2))/sqrt(a - b))/sqrt(a - b) + sqrt(a - b + b/xS(2))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(-1))/(x*S(3)sqrt(a + b(S(-1) + x(S(-2))))), x), x, atanh(sqrt(a - b + b/xS(2))/sqrt(a - b))/sqrt(a - b) + sqrt(a - b + b/xS(2))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(c + dx)S(2)/(a + bxS(3)), x), x, -a(S(1)/3)d(-a*(S(1)/3)d + S(2)b(S(1)/3)c)log(a(S(1)/3) + b(S(1)/3)x)/(S(3)b(S(5)/3)) + a(S(1)/3)d(-a(S(1)/3)d + S(2)b(S(1)/3)c)log(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(6)b*(S(5)/3)) + sqrt(S(3))a*(S(1)/3)d(a(S(1)/3)d + S(2)b(S(1)/3)c)atan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(3)b(S(5)/3)) + cS(2)log(a + bx*S(3))/(S(3)b) + S(2)cdx/b + dS(2)x*S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(1))/((x*S(2) + S(4))sqrt(x*S(2) + S(9))), x), x, sqrt(S(5))atan(sqrt(S(5))x/(S(2)sqrt(x*S(2) + S(9))))/S(10) - sqrt(S(5))atanh(sqrt(S(5))sqrt(xS(2) + S(9))/S(5))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(sqrt(-xS(2) + S(1)) + S(1)), x), x, xS(2)/S(2) - (-xS(2) + S(1))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(sqrt(-x + S(1))sqrt(x + S(1)) + S(1)), x), x, xS(2)/S(2) - (-xS(2) + S(1))*(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(1) + S(1)/(sqrt(x + S(2))sqrt(x + S(3)))), x), x, xS(2)/S(2) + sqrt(x + S(2))sqrt(x + S(3)) - S(5)asinh(sqrt(x + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x - sqrt(xS(6)))/(x(-xS(4) + S(1))), x), x, atan(x)/S(2) + atanh(x)/S(2) + sqrt(xS(6))atan(x)/(S(2)xS(3)) - sqrt(xS(6))atanh(x)/(S(2)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(1) - sqrt(xS(6))/x)/(-xS(4) + S(1)), x), x, atan(x)/S(2) + atanh(x)/S(2) + sqrt(xS(6))atan(x)/(S(2)xS(3)) - sqrt(xS(6))atanh(x)/(S(2)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x - sqrt(xS(6)))/(-xS(5) + x), x), x, atan(x)/S(2) + atanh(x)/S(2) + sqrt(xS(6))atan(x)/(S(2)xS(3)) - sqrt(xS(6))atanh(x)/(S(2)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(x + sqrt(xS(6))), x), x, atan(x)/S(2) + atanh(x)/S(2) + sqrt(x*S(6))atan(x)/(S(2)xS(3)) - sqrt(xS(6))atanh(x)/(S(2)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(x) - sqrt(xS(3)))/(-xS(3) + x), x), x, atan(sqrt(x)) + atanh(sqrt(x)) + sqrt(xS(3))atan(sqrt(x))/x(S(3)/2) - sqrt(xS(3))atanh(sqrt(x))/x(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x) + sqrt(xS(3))), x), x, atan(sqrt(x)) + atanh(sqrt(x)) + sqrt(xS(3))atan(sqrt(x))/x(S(3)/2) - sqrt(xS(3))atanh(sqrt(x))/x(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x + S(-1)) + sqrt((x + S(-1))S(3))), x), x, atan(sqrt(x + S(-1))) + atanh(sqrt(x + S(-1))) + sqrt((x + S(-1))S(3))atan(sqrt(x + S(-1)))/(x + S(-1))(S(3)/2) - sqrt((x + S(-1))S(3))atanh(sqrt(x + S(-1)))/(x + S(-1))(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(4)x + S(-5))/((S(5)x + S(4))S(2)sqrt(-x*S(2) + S(1))) - S(3)/(S(5)x + S(4))S(2), x), x, sqrt(-xS(2) + S(1))/(S(5)x + S(4)) + S(3)/(S(5)(S(5)x + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(4)x - S(3)sqrt(-xS(2) + S(1)) + S(-5))/((S(5)x + S(4))*S(2)sqrt(-xS(2) + S(1))), x), x, sqrt(-xS(2) + S(1))/(S(5)x + S(4)) + S(3)/(S(5)(S(5)x + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(-S(3)x*S(2) + (-S(4)x + S(-5))sqrt(-xS(2) + S(1)) + S(3)), x), x, sqrt(-xS(2) + S(1))/(S(5)x + S(4)) + S(3)/(S(5)(S(5)x + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(-S(3)xS(2) - S(4)xsqrt(-xS(2) + S(1)) - S(5)sqrt(-xS(2) + S(1)) + S(3)), x), x, sqrt(-xS(2) + S(1))/(S(5)x + S(4)) + S(3)/(S(5)(S(5)x + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(-xS(2) + S(1)) + S(-1))/(sqrt(-xS(2) + S(1))(x - S(2)sqrt(-xS(2) + S(1)) + S(2))S(2)), x), x, sqrt(-xS(2) + S(1))/(S(5)x + S(4)) + S(3)/(S(5)(S(5)x + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx(n + S(-1)))/(cx + dxn), x), x, blog(x)/d - (-ad + bc)log(cx*(-n + S(1)) + d)/(cd(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x*S(5) + S(2)xS(3) - x)/(xS(4) + S(2)xS(2) + S(3))S(2), x), x, (-S(7)xS(2)/S(8) + S(5)/8)/(xS(4) + S(2)xS(2) + S(3)) + S(9)sqrt(S(2))atan(sqrt(S(2))(xS(2) + S(1))/S(2))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(5) + x)/(S(2)xS(4) + S(2)xS(2) + S(1))S(3), x), x, (x*S(2)/S(4) + S(3)/16)/(S(2)x*S(4) + S(2)xS(2) + S(1))S(2) + (x*S(2) + S(1)/2)/(S(2)x*S(4) + S(2)x*S(2) + S(1)) + atan(S(2)x*S(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx + cxS(2))/(d + ex*S(2) + fx*S(4)), x), x, -batanh((e + S(2)fx*S(2))/sqrt(-S(4)df + eS(2)))/sqrt(-S(4)df + eS(2)) + sqrt(S(2))(c + (-S(2)af + ce)/sqrt(-S(4)df + eS(2)))atan(sqrt(S(2))sqrt(f)x/sqrt(e + sqrt(-S(4)df + e*S(2))))/(S(2)sqrt(f)sqrt(e + sqrt(-S(4)df + eS(2)))) + sqrt(S(2))(c + (S(2)af - ce)/sqrt(-S(4)df + eS(2)))atan(sqrt(S(2))sqrt(f)x/sqrt(e - sqrt(-S(4)df + e*S(2))))/(S(2)sqrt(f)sqrt(e - sqrt(-S(4)df + eS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((d + ex)*S(2)/(a + bx*S(2) + cx*S(4)), x), x, -S(2)deatanh((b + S(2)cx*S(2))/sqrt(-S(4)ac + bS(2)))/sqrt(-S(4)ac + bS(2)) + sqrt(S(2))(e*S(2) + (be*S(2) - S(2)cdS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b + sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b + sqrt(-S(4)ac + bS(2)))) + sqrt(S(2))(e*S(2) + (-be*S(2) + S(2)cdS(2))/sqrt(-S(4)ac + bS(2)))atan(sqrt(S(2))sqrt(c)x/sqrt(b - sqrt(-S(4)ac + b*S(2))))/(S(2)sqrt(c)sqrt(b - sqrt(-S(4)ac + bS(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(2)x*S(2) + S(1))/(sqrt(S(2)x*S(2) + S(1)) + S(1)), x), x, x - sqrt(S(2))asinh(sqrt(S(2))x)/S(2) + sqrt(S(2)x*S(2) + S(1))/(S(2)x) - S(1)/(S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(4)x*S(2) + S(-1))/(x + sqrt(S(4)x*S(2) + S(-1))), x), x, S(4)x/S(3) - sqrt(S(4)xS(2) + S(-1))/S(3) - sqrt(S(3))atanh(sqrt(S(3))x)/S(9) + sqrt(S(3))atanh(sqrt(S(3))sqrt(S(4)xS(2) + S(-1)))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((a + bx)(c + dx)), x), x, aS(2)log(a + bx)/(bS(2)(-ad + bc)) - c*S(2)log(c + dx)/(dS(2)(-ad + bc)) + x/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((a + bx*S(2))(c + dx)), x), x, -sqrt(a)catan(sqrt(b)x/sqrt(a))/(sqrt(b)(ad*S(2) + bc*S(2))) + adlog(a + bx*S(2))/(S(2)b(ad*S(2) + bcS(2))) + cS(2)log(c + dx)/(d(ad*S(2) + bcS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((a + bxS(3))(c + dx)), x), x, a(S(1)/3)d(a(S(1)/3)d + b*(S(1)/3)c)log(a(S(1)/3) + b(S(1)/3)x)/(S(3)b(S(2)/3)(-adS(3) + bcS(3))) - a(S(1)/3)d(a*(S(1)/3)d + b*(S(1)/3)c)log(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)x + b*(S(2)/3)x*S(2))/(S(6)b*(S(2)/3)(-adS(3) + bc*S(3))) - sqrt(S(3))a*(S(1)/3)datan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)x)/(S(3)a(S(1)/3)))/(S(3)b*(S(2)/3)(a*(S(2)/3)dS(2) + a(S(1)/3)b(S(1)/3)cd + b(S(2)/3)cS(2))) + cS(2)log(a + bx*S(3))/(S(3)(-adS(3) + bcS(3))) - cS(2)log(c + dx)/(-adS(3) + bcS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((a + bxS(4))(c + dx)), x), x, sqrt(a)d*S(3)atan(sqrt(b)xS(2)/sqrt(a))/(S(2)sqrt(b)(ad*S(4) + bcS(4))) - cS(2)dlog(a + bxS(4))/(S(4)(adS(4) + bcS(4))) + cS(2)dlog(c + dx)/(ad*S(4) + bc*S(4)) - sqrt(S(2))c(-sqrt(a)d*S(2) + sqrt(b)c*S(2))atan(S(1) - sqrt(S(2))b(S(1)/4)x/a*(S(1)/4))/(S(4)a*(S(1)/4)b*(S(1)/4)(adS(4) + bc*S(4))) + sqrt(S(2))c(-sqrt(a)d*S(2) + sqrt(b)c*S(2))atan(S(1) + sqrt(S(2))b(S(1)/4)x/a*(S(1)/4))/(S(4)a*(S(1)/4)b*(S(1)/4)(adS(4) + bc*S(4))) + sqrt(S(2))c(sqrt(a)d*S(2) + sqrt(b)c*S(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)x + sqrt(a) + sqrt(b)xS(2))/(S(8)a*(S(1)/4)b*(S(1)/4)(adS(4) + bc*S(4))) - sqrt(S(2))c(sqrt(a)d*S(2) + sqrt(b)c*S(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)x + sqrt(a) + sqrt(b)xS(2))/(S(8)a*(S(1)/4)b*(S(1)/4)(adS(4) + bc*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/((-x + S(1))(x + S(1))*S(2)), x), x, atanh(x)/S(2) + S(1)/(S(2)(x + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/((-xS(2) + S(1))(xS(2) + S(1))S(2)), x), x, -x/(S(4)(xS(2) + S(1))) + atanh(x)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/((-x*S(3) + S(1))(xS(3) + S(1))S(2)), x), x, -x/(S(6)(xS(3) + S(1))) - log(-x + S(1))/S(12) - log(x + S(1))/S(36) + log(xS(2) - x + S(1))/S(72) + log(xS(2) + x + S(1))/S(24) + sqrt(S(3))atan(sqrt(S(3))(-S(2)x + S(1))/S(3))/S(36) + sqrt(S(3))atan(sqrt(S(3))(S(2)x + S(1))/S(3))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx + cxS(2))/((d + ex)*S(3)sqrt(x*S(2) + S(-1))), x), x, (a(S(2)dS(2) + eS(2))/S(2) - S(3)bde/S(2) + c(dS(2) + S(2)e*S(2))/S(2))atanh((dx + e)/(sqrt(dS(2) - eS(2))sqrt(xS(2) + S(-1))))/(dS(2) - eS(2))(S(5)/2) + sqrt(x*S(2) + S(-1))(c(dS(3) - S(4)deS(2))/S(2) - e(S(3)ade - b(d*S(2) + S(2)e*S(2)))/S(2))/(e(d + ex)(dS(2) - eS(2))S(2)) - sqrt(xS(2) + S(-1))(ae*S(2)/S(2) - bde/S(2) + cd*S(2)/S(2))/(e(d + ex)S(2)(dS(2) - eS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx + cx*S(2))/((d + ex)*S(3)sqrt(x + S(-1))sqrt(x + S(1))), x), x, (-S(3)bde + d*S(2)(S(2)a + c) + eS(2)(a + S(2)c))atanh(sqrt(d + e)sqrt(x + S(1))/(sqrt(d - e)sqrt(x + S(-1))))/((d - e)*(S(5)/2)(d + e)*(S(5)/2)) + sqrt(x + S(-1))sqrt(x + S(1))(bd*S(2)e/S(2) + beS(3) + cd*S(3)/S(2) - de*S(2)(S(3)a + S(4)c)/S(2))/(e(d + ex)(dS(2) - eS(2))S(2)) - sqrt(x + S(-1))sqrt(x + S(1))(ae*S(2)/S(2) - bde/S(2) + cd*S(2)/S(2))/(e(d + ex)S(2)(dS(2) - eS(2))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + bx + cx*S(2))/((d + ex)*S(3)sqrt(x + S(-1))sqrt(x + S(1))), x), x, S(2)catanh(sqrt(d + e)sqrt(x + S(1))/(sqrt(d - e)sqrt(x + S(-1))))/(eS(2)sqrt(d - e)sqrt(d + e)) - S(3)dsqrt(x + S(-1))sqrt(x + S(1))(ae*S(2) - bde + cd*S(2))/(S(2)e(d + ex)(dS(2) - eS(2))S(2)) - S(2)d(-be + S(2)cd)atanh(sqrt(d + e)sqrt(x + S(1))/(sqrt(d - e)sqrt(x + S(-1))))/(eS(2)(d - e)*(S(3)/2)(d + e)*(S(3)/2)) + sqrt(x + S(-1))sqrt(x + S(1))(-be + S(2)cd)/(e(d + ex)(dS(2) - eS(2))) - sqrt(x + S(-1))sqrt(x + S(1))(ae*S(2)/S(2) - bde/S(2) + cd*S(2)/S(2))/(e(d + ex)S(2)(dS(2) - eS(2))) + (S(2)dS(2) + eS(2))(aeS(2) - bde + cd*S(2))atanh(sqrt(d + e)sqrt(x + S(1))/(sqrt(d - e)sqrt(x + S(-1))))/(e*S(2)(d - e)*(S(5)/2)(d + e)*(S(5)/2)), expand=True, _diff=True, _numerical=True) def test_4(): assert rubi_test(rubi_integrate((b + S(2)cx + S(3)dxS(2))(a + bx + cx*S(2) + dxS(3))n, x), x, (a + bx + cx*S(2) + dxS(3))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx + S(3)dx*S(2))(bx + cx*S(2) + dxS(3))n, x), x, (bx + cx*S(2) + dxS(3))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*n(b + cx + dxS(2))n(b + S(2)cx + S(3)dxS(2)), x), x, x(n + S(1))(b + cx + dxS(2))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(3)dx*S(2))(a + bx + dxS(3))n, x), x, (a + bx + dxS(3))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(3)dx*S(2))(bx + dxS(3))n, x), x, (bx + dxS(3))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*n(b + dxS(2))n(b + S(3)dxS(2)), x), x, x(n + S(1))(b + dxS(2))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)cx + S(3)dx*S(2))(a + cxS(2) + dxS(3))n, x), x, (a + cxS(2) + dxS(3))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)cx + S(3)dx*S(2))(cxS(2) + dxS(3))n, x), x, (cxS(2) + dxS(3))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*n(cx + dxS(2))n(S(2)cx + S(3)dxS(2)), x), x, x(n + S(1))(cx + dxS(2))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(S(2)n)(c + dx)*n(S(2)cx + S(3)dxS(2)), x), x, x(S(2)n + S(2))(c + dx)(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)c + S(3)dx)(a + cxS(2) + dxS(3))n, x), x, (a + cxS(2) + dxS(3))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)c + S(3)dx)(cx*S(2) + dxS(3))n, x), x, (cxS(2) + dxS(3))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx + S(3)dx*S(2))(a + bx + cx*S(2) + dxS(3))S(7), x), x, (a + bx + cx*S(2) + dxS(3))S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx + S(3)dx*S(2))(bx + cx*S(2) + dxS(3))S(7), x), x, x*S(8)(b + cx + dxS(2))S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(7)(b + cx + dxS(2))S(7)(b + S(2)cx + S(3)dxS(2)), x), x, xS(8)(b + cx + dxS(2))S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(3)dx*S(2))(a + bx + dxS(3))S(7), x), x, (a + bx + dxS(3))S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(7)(b + dxS(2))S(7)(b + S(3)dxS(2)), x), x, xS(8)(b + dxS(2))S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(3)dx*S(2))(bx + dxS(3))S(7), x), x, x*S(8)(b + dxS(2))S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)cx + S(3)dxS(2))(a + cxS(2) + dxS(3))S(7), x), x, (a + cxS(2) + dxS(3))S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)cx + S(3)dx*S(2))(cxS(2) + dxS(3))S(7), x), x, x*S(16)(c + dx)S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(7)(cx + dxS(2))S(7)(S(2)cx + S(3)dxS(2)), x), x, xS(16)(c + dx)S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(14)(c + dx)S(7)(S(2)cx + S(3)dxS(2)), x), x, xS(16)(c + dx)*S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)c + S(3)dx)(a + cxS(2) + dxS(3))S(7), x), x, (a + cxS(2) + dxS(3))S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)c + S(3)dx)(cx*S(2) + dxS(3))S(7), x), x, x*S(16)(c + dx)S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(8)(S(2)c + S(3)dx)(cx + dxS(2))S(7), x), x, x*S(16)(c + dx)S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(15)(c + dx)S(7)(S(2)c + S(3)dx), x), x, xS(16)(c + dx)S(8)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)((ax + bxS(2)/S(2))S(4) + S(1)), x), x, ax + bxS(2)/S(2) + (ax + bxS(2)/S(2))S(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)((ax + bxS(2)/S(2) + c)S(4) + S(1)), x), x, ax + bxS(2)/S(2) + (ax + bxS(2)/S(2) + c)S(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)((ax + bxS(2)/S(2))n + S(1)), x), x, ax + bxS(2)/S(2) + (ax + bxS(2)/S(2))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bx)((ax + bxS(2)/S(2) + c)n + S(1)), x), x, ax + bxS(2)/S(2) + (ax + bxS(2)/S(2) + c)(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cx*S(2))((ax + cxS(3)/S(3))S(5) + S(1)), x), x, ax + cx*S(3)/S(3) + (ax + cxS(3)/S(3))S(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cx*S(2))((ax + cxS(3)/S(3) + d)S(5) + S(1)), x), x, ax + cx*S(3)/S(3) + (ax + cxS(3)/S(3) + d)S(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx + cxS(2))((bxS(2)/S(2) + cxS(3)/S(3))S(5) + S(1)), x), x, bxS(2)/S(2) + cx*S(3)/S(3) + (bx*S(2)/S(2) + cxS(3)/S(3))S(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx + cx*S(2))((bxS(2)/S(2) + cxS(3)/S(3) + d)S(5) + S(1)), x), x, bxS(2)/S(2) + cx*S(3)/S(3) + (bx*S(2)/S(2) + cxS(3)/S(3) + d)S(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(((ax + bx*S(2)/S(2) + cxS(3)/S(3))S(5) + S(1))(a + bx + cxS(2)), x), x, ax + bxS(2)/S(2) + cx*S(3)/S(3) + (ax + bxS(2)/S(2) + cxS(3)/S(3))S(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(((ax + bx*S(2)/S(2) + cxS(3)/S(3) + d)S(5) + S(1))(a + bx + cxS(2)), x), x, ax + bxS(2)/S(2) + cx*S(3)/S(3) + (ax + bxS(2)/S(2) + cxS(3)/S(3) + d)S(6)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + cxS(2))((ax + cxS(3)/S(3))n + S(1)), x), x, ax + cx*S(3)/S(3) + (ax + cxS(3)/S(3))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx + cxS(2))((bxS(2)/S(2) + cxS(3)/S(3))n + S(1)), x), x, bxS(2)/S(2) + cx*S(3)/S(3) + (bx*S(2)/S(2) + cxS(3)/S(3))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(((ax + bx*S(2)/S(2) + cxS(3)/S(3))n + S(1))(a + bx + cxS(2)), x), x, ax + bxS(2)/S(2) + cx*S(3)/S(3) + (ax + bxS(2)/S(2) + cxS(3)/S(3))(n + S(1))/(n + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(bx + cxS(2))S(13), x), x, (bx + cxS(2))S(14)/S(14), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(14)(b + S(2)cx*S(2))(bx + cxS(3))S(13), x), x, x*S(28)(b + cxS(2))S(14)/S(28), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(28)(b + S(2)cx*S(3))(bx + cxS(4))S(13), x), x, x*S(42)(b + cxS(3))S(14)/S(42), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(14)n + S(-14))(b + S(2)cxn)(bx + cx(n + S(1)))S(13), x), x, x*(S(14)n)(b + cxn)S(14)/(S(14)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)/(bx + cxS(2)), x), x, log(bx + cxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cxS(2))/(bx + cxS(3)), x), x, log(x) + log(b + cx*S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cxS(3))/(bx + cxS(4)), x), x, log(x) + log(b + cx*S(3))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cxn)/(bx + cx(n + S(1))), x), x, log(x) + log(b + cx*n)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)/(bx + cxS(2))S(8), x), x, -S(1)/(S(7)(bx + cxS(2))S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cxS(2))/(xS(7)(bx + cxS(3))S(8)), x), x, -S(1)/(S(14)x*S(14)(b + cxS(2))S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cxS(3))/(xS(14)(bx + cxS(4))S(8)), x), x, -S(1)/(S(21)xS(21)(b + cxS(3))S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-S(7)n + S(7))(b + S(2)cxn)/(bx + cx(n + S(1)))S(8), x), x, -x(-S(7)n)/(S(7)n(b + cxn)S(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((b + S(2)cx)(bx + cxS(2))p, x), x, (bx + cxS(2))(p + S(1))/(p + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(p + S(1))(b + S(2)cx*S(2))(bx + cxS(3))p, x), x, x*(p + S(1))(bx + cxS(3))(p + S(1))/(S(2)p + S(2)), expand=True, _diff=True, _numerical=True) # fails in mathematica too assert rubi_test(rubi_integrate(bx*(p + S(1))(bx + cxS(3))p + S(2)cx*(p + S(3))(bx + cxS(3))p, x), x, x*(p + S(1))(bx + cxS(3))(p + S(1))/(S(2)p + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(S(2)p + S(2))(b + S(2)cxS(3))(bx + cxS(4))p, x), x, x*(S(2)p + S(2))(bx + cxS(4))(p + S(1))/(S(3)p + S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*((n + S(-1))(p + S(1)))(b + S(2)cxn)(bx + cx(n + S(1)))p, x), x, x*(-(-n + S(1))(p + S(1)))(bx + cx(n + S(1)))(p + S(1))/(n(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(2) + S(4)x + S(-4))(xS(3) + S(6)x*S(2) - S(12)x + S(5)), x), x, (x*S(3) + S(6)x*S(2) - S(12)x + S(5))S(2)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(2)x)(x*S(4) + S(4)xS(2) + S(1)), x), x, (xS(4) + S(4)xS(2) + S(1))S(2)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x + S(1))(xS(2) + x)S(3)(S(7)(xS(2) + x)S(3) + S(-18))S(2), x), x, S(49)x*S(10)(x + S(1))*S(10)/S(10) - S(36)x*S(7)(x + S(1))*S(7) + S(81)x*S(4)(x + S(1))S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(x + S(1))S(3)(S(2)x + S(1))(S(7)xS(3)(x + S(1))S(3) + S(-18))S(2), x), x, S(49)xS(10)(x + S(1))*S(10)/S(10) - S(36)x*S(7)(x + S(1))*S(7) + S(81)x*S(4)(x + S(1))S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(2))/(x*S(3) - S(6)x + S(1))*S(5), x), x, S(1)/(S(12)(x*S(3) - S(6)x + S(1))S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(2)x)/(xS(3) + S(3)xS(2) + S(4)), x), x, log(xS(3) + S(3)xS(2) + S(4))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + x + S(1))/(xS(4) + S(2)x*S(2) + S(4)x), x), x, log(x(xS(3) + S(2)x + S(4)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) + S(-1))/(xS(4) - S(4)x)(S(2)/3), x), x, S(3)(x*S(4) - S(4)x)(S(1)/3)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(2) + S(2))(-xS(3) + S(6)x)*(S(1)/4), x), x, S(4)(-x*S(3) + S(6)x)(S(5)/4)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(4) + S(1))sqrt(xS(5) + S(5)x), x), x, S(2)(xS(5) + S(5)x)*(S(3)/2)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(5)x*S(4) + S(2))sqrt(x*S(5) + S(2)x), x), x, S(2)(xS(5) + S(2)x)*(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x*S(2) + x)/sqrt(S(2)xS(3) + xS(2)), x), x, sqrt(S(2)xS(3) + xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(((-S(5)x + S(1))*(S(1)/3) + S(2))/((-S(5)x + S(1))*(S(1)/3) + S(3)), x), x, x + S(3)(-S(5)x + S(1))(S(2)/3)/S(10) - S(9)(-S(5)x + S(1))(S(1)/3)/S(5) + S(27)log((-S(5)x + S(1))(S(1)/3) + S(3))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(x) + S(1))/(sqrt(x) + S(-1)), x), x, S(4)sqrt(x) + x + S(4)log(-sqrt(x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-sqrt(S(3)x + S(2)) + S(1))/(sqrt(S(3)x + S(2)) + S(1)), x), x, -x + S(4)sqrt(S(3)x + S(2))/S(3) - S(4)log(sqrt(S(3)x + S(2)) + S(1))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sqrt(a + bx) + S(-1))/(sqrt(a + bx) + S(1)), x), x, x - S(4)sqrt(a + bx)/b + S(4)log(sqrt(a + bx) + S(1))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bnx(n + S(-1)))/(ax + bxn), x), x, log(ax + bxn), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a + bnx(n + S(-1)))/(ax + bxn), x), x, nlog(x) + log(ax(-n + S(1)) + b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(-n)(a + bnx*(n + S(-1)))/(ax*(-n + S(1)) + b), x), x, nlog(x) + log(ax(-n + S(1)) + b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + b(c + dx)S(3)), x), x, -clog(a + b(c + dx)*S(3))/(bd*S(4)) + x/(bd*S(3)) + sqrt(S(3))(-S(3)a(S(1)/3)b*(S(2)/3)c*S(2) + a + bc*S(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)(c + dx))/(S(3)a*(S(1)/3)))/(S(3)a*(S(2)/3)b*(S(4)/3)d*S(4)) - (S(3)a*(S(1)/3)b*(S(2)/3)c*S(2) + a + bc*S(3))log(a(S(1)/3) + b(S(1)/3)(c + dx))/(S(3)a(S(2)/3)b*(S(4)/3)d*S(4)) + (S(3)a*(S(1)/3)b*(S(2)/3)c*S(2) + a + bc*S(3))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)(c + dx) + b(S(2)/3)(c + dx)S(2))/(S(6)a*(S(2)/3)b*(S(4)/3)dS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + b(c + dx)*S(3)), x), x, log(a + b(c + dx)S(3))/(S(3)bdS(3)) + sqrt(S(3))c(S(2)a(S(1)/3) - b(S(1)/3)c)atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)(c + dx))/(S(3)a*(S(1)/3)))/(S(3)a*(S(2)/3)b*(S(2)/3)d*S(3)) + c(S(2)a(S(1)/3) + b(S(1)/3)c)log(a(S(1)/3) + b(S(1)/3)(c + dx))/(S(3)a*(S(2)/3)b*(S(2)/3)d*S(3)) - c(S(2)a(S(1)/3) + b(S(1)/3)c)log(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)(c + dx) + b(S(2)/3)(c + dx)S(2))/(S(6)a*(S(2)/3)b*(S(2)/3)d*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + b(c + dx)S(3)), x), x, -sqrt(S(3))(a(S(1)/3) - b(S(1)/3)c)atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)(c + dx))/(S(3)a*(S(1)/3)))/(S(3)a*(S(2)/3)b*(S(2)/3)dS(2)) - (a(S(1)/3) + b*(S(1)/3)c)log(a(S(1)/3) + b(S(1)/3)(c + dx))/(S(3)a*(S(2)/3)b*(S(2)/3)dS(2)) + (a(S(1)/3) + b*(S(1)/3)c)log(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)(c + dx) + b(S(2)/3)(c + dx)S(2))/(S(6)a*(S(2)/3)b*(S(2)/3)d*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + b(c + dx)S(3)), x), x, log(a(S(1)/3) + b(S(1)/3)(c + dx))/(S(3)a*(S(2)/3)b*(S(1)/3)d) - log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)(c + dx) + b(S(2)/3)(c + dx)S(2))/(S(6)a*(S(2)/3)b*(S(1)/3)d) - sqrt(S(3))atan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)(c + dx))/(S(3)a*(S(1)/3)))/(S(3)a*(S(2)/3)b*(S(1)/3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + b(c + dx)S(3))), x), x, log(x)/(a + bc*S(3)) + sqrt(S(3))b*(S(1)/3)catan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)(c + dx))/(S(3)a*(S(1)/3)))/(S(3)a*(S(2)/3)(a(S(2)/3) - a(S(1)/3)b(S(1)/3)c + b*(S(2)/3)c*S(2))) - (S(2)a(S(1)/3) - b(S(1)/3)c)log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)(c + dx) + b(S(2)/3)(c + dx)S(2))/(S(6)a*(S(2)/3)(a(S(2)/3) - a(S(1)/3)b(S(1)/3)c + b*(S(2)/3)cS(2))) - log(a(S(1)/3) + b*(S(1)/3)(c + dx))/(S(3)a*(S(2)/3)(a(S(1)/3) + b(S(1)/3)c)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/(x(a + b(c + dx)*S(3))), x), x, -log(a + b(c + dx)S(3))/(S(3)a + S(3)bc*S(3)) + log(x)/(a + bcS(3)) + b(S(1)/3)c(a(S(1)/3) - b(S(1)/3)c)log(a(S(1)/3) + b(S(1)/3)(c + dx))/(S(3)a(S(2)/3)(a + bcS(3))) - b(S(1)/3)c(a(S(1)/3) - b(S(1)/3)c)log(a(S(2)/3) - a(S(1)/3)b*(S(1)/3)(c + dx) + b(S(2)/3)(c + dx)S(2))/(S(6)a*(S(2)/3)(a + bcS(3))) + sqrt(S(3))b*(S(1)/3)c(a(S(1)/3) + b(S(1)/3)c)atan(sqrt(S(3))(a*(S(1)/3) - S(2)b*(S(1)/3)(c + dx))/(S(3)a*(S(1)/3)))/(S(3)a*(S(2)/3)(a + bcS(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + b(c + dx)*S(3))), x), x, -S(3)bcS(2)dlog(x)/(a + bcS(3))S(2) + bcS(2)dlog(a + b(c + dx)S(3))/(a + bcS(3))S(2) - S(1)/(x(a + bc*S(3))) + sqrt(S(3))b*(S(1)/3)d(a(S(1)/3) - b(S(1)/3)c)(a(S(1)/3) + b(S(1)/3)c)*S(3)atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)(c + dx))/(S(3)a*(S(1)/3)))/(S(3)a*(S(2)/3)(a + bcS(3))S(2)) + b(S(1)/3)d(a(S(1)/3)(a - S(2)bcS(3)) - b(S(1)/3)(S(2)ac - bc*S(4)))log(a(S(1)/3) + b(S(1)/3)(c + dx))/(S(3)a(S(2)/3)(a + bcS(3))S(2)) - b(S(1)/3)d(a(S(1)/3)(a - S(2)bcS(3)) - b(S(1)/3)(S(2)ac - bc*S(4)))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)(c + dx) + b(S(2)/3)(c + dx)S(2))/(S(6)a*(S(2)/3)(a + bcS(3))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a + b(c + dx)*S(3))), x), x, S(3)bcS(2)d/(x(a + bcS(3))S(2)) - S(3)bcdS(2)(a - S(2)bc*S(3))log(x)/(a + bcS(3))S(3) + bcdS(2)(a - S(2)bc*S(3))log(a + b(c + dx)*S(3))/(a + bcS(3))S(3) - S(1)/(x*S(2)(S(2)a + S(2)bcS(3))) + sqrt(S(3))b*(S(2)/3)d*S(2)(a(S(1)/3) + b(S(1)/3)c)S(3)(-S(3)a(S(2)/3)b*(S(1)/3)c + a + bcS(3))atan(sqrt(S(3))(a(S(1)/3) - S(2)b*(S(1)/3)(c + dx))/(S(3)a*(S(1)/3)))/(S(3)a*(S(2)/3)(a + bcS(3))S(3)) - b(S(2)/3)d*S(2)(S(6)a(S(4)/3)b*(S(2)/3)c*S(2) - S(3)a*(S(1)/3)b*(S(5)/3)cS(5) + aS(2) - S(7)abcS(3) + bS(2)c*S(6))log(a(S(1)/3) + b(S(1)/3)(c + dx))/(S(3)a(S(2)/3)(a + bcS(3))S(3)) + b(S(2)/3)d*S(2)(S(6)a(S(4)/3)b*(S(2)/3)c*S(2) - S(3)a*(S(1)/3)b*(S(5)/3)cS(5) + aS(2) - S(7)abcS(3) + bS(2)c*S(6))log(a(S(2)/3) - a(S(1)/3)b(S(1)/3)(c + dx) + b(S(2)/3)(c + dx)S(2))/(S(6)a*(S(2)/3)(a + bcS(3))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + b(c + dx)S(4)), x), x, log(a + b(c + dx)S(4))/(S(4)bdS(4)) + S(3)c*S(2)atan(sqrt(b)(c + dx)*S(2)/sqrt(a))/(S(2)sqrt(a)sqrt(b)d*S(4)) - sqrt(S(2))c(S(3)sqrt(a) - sqrt(b)cS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)(c + dx) + sqrt(a) + sqrt(b)(c + dx)S(2))/(S(8)a*(S(3)/4)b*(S(3)/4)d*S(4)) + sqrt(S(2))c(S(3)sqrt(a) - sqrt(b)cS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)(c + dx) + sqrt(a) + sqrt(b)(c + dx)S(2))/(S(8)a*(S(3)/4)b*(S(3)/4)d*S(4)) + sqrt(S(2))c(S(3)sqrt(a) + sqrt(b)cS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)(c + dx)/a(S(1)/4))/(S(4)a*(S(3)/4)b*(S(3)/4)d*S(4)) - sqrt(S(2))c(S(3)sqrt(a) + sqrt(b)cS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)(c + dx)/a(S(1)/4))/(S(4)a*(S(3)/4)b*(S(3)/4)dS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + b(c + dx)*S(4)), x), x, -catan(sqrt(b)(c + dx)*S(2)/sqrt(a))/(sqrt(a)sqrt(b)dS(3)) + sqrt(S(2))(sqrt(a) - sqrt(b)cS(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)(c + dx) + sqrt(a) + sqrt(b)(c + dx)S(2))/(S(8)a*(S(3)/4)b*(S(3)/4)d*S(3)) - sqrt(S(2))(sqrt(a) - sqrt(b)cS(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)(c + dx) + sqrt(a) + sqrt(b)(c + dx)S(2))/(S(8)a*(S(3)/4)b*(S(3)/4)d*S(3)) - sqrt(S(2))(sqrt(a) + sqrt(b)cS(2))atan(S(1) - sqrt(S(2))b(S(1)/4)(c + dx)/a(S(1)/4))/(S(4)a*(S(3)/4)b*(S(3)/4)d*S(3)) + sqrt(S(2))(sqrt(a) + sqrt(b)cS(2))atan(S(1) + sqrt(S(2))b(S(1)/4)(c + dx)/a(S(1)/4))/(S(4)a*(S(3)/4)b*(S(3)/4)d*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + b(c + dx)S(4)), x), x, atan(sqrt(b)(c + dx)S(2)/sqrt(a))/(S(2)sqrt(a)sqrt(b)d*S(2)) + sqrt(S(2))clog(-sqrt(S(2))a*(S(1)/4)b*(S(1)/4)(c + dx) + sqrt(a) + sqrt(b)(c + dx)S(2))/(S(8)a*(S(3)/4)b*(S(1)/4)d*S(2)) - sqrt(S(2))clog(sqrt(S(2))a*(S(1)/4)b*(S(1)/4)(c + dx) + sqrt(a) + sqrt(b)(c + dx)S(2))/(S(8)a*(S(3)/4)b*(S(1)/4)d*S(2)) + sqrt(S(2))catan(S(1) - sqrt(S(2))b*(S(1)/4)(c + dx)/a(S(1)/4))/(S(4)a*(S(3)/4)b*(S(1)/4)d*S(2)) - sqrt(S(2))catan(S(1) + sqrt(S(2))b*(S(1)/4)(c + dx)/a(S(1)/4))/(S(4)a*(S(3)/4)b*(S(1)/4)d*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + b(c + dx)S(4)), x), x, -sqrt(S(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)(c + dx) + sqrt(a) + sqrt(b)(c + dx)S(2))/(S(8)a*(S(3)/4)b*(S(1)/4)d) + sqrt(S(2))log(sqrt(S(2))a*(S(1)/4)b*(S(1)/4)(c + dx) + sqrt(a) + sqrt(b)(c + dx)S(2))/(S(8)a*(S(3)/4)b*(S(1)/4)d) - sqrt(S(2))atan(S(1) - sqrt(S(2))b*(S(1)/4)(c + dx)/a(S(1)/4))/(S(4)a*(S(3)/4)b*(S(1)/4)d) + sqrt(S(2))atan(S(1) + sqrt(S(2))b*(S(1)/4)(c + dx)/a(S(1)/4))/(S(4)a*(S(3)/4)b*(S(1)/4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + b(c + dx)S(4))), x), x, -log(a + b(c + dx)S(4))/(S(4)a + S(4)bc*S(4)) + log(x)/(a + bc*S(4)) - sqrt(b)c*S(2)atan(sqrt(b)(c + dx)*S(2)/sqrt(a))/(S(2)sqrt(a)(a + bc*S(4))) - sqrt(S(2))b*(S(1)/4)c(sqrt(a) - sqrt(b)c*S(2))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)(c + dx) + sqrt(a) + sqrt(b)(c + dx)S(2))/(S(8)a*(S(3)/4)(a + bcS(4))) + sqrt(S(2))b*(S(1)/4)c(sqrt(a) - sqrt(b)c*S(2))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)(c + dx) + sqrt(a) + sqrt(b)(c + dx)S(2))/(S(8)a*(S(3)/4)(a + bcS(4))) + sqrt(S(2))b*(S(1)/4)c(sqrt(a) + sqrt(b)c*S(2))atan(S(1) - sqrt(S(2))b(S(1)/4)(c + dx)/a(S(1)/4))/(S(4)a*(S(3)/4)(a + bcS(4))) - sqrt(S(2))b*(S(1)/4)c(sqrt(a) + sqrt(b)c*S(2))atan(S(1) + sqrt(S(2))b(S(1)/4)(c + dx)/a(S(1)/4))/(S(4)a*(S(3)/4)(a + bcS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + b(c + dx)*S(4))), x), x, -S(4)bcS(3)dlog(x)/(a + bcS(4))S(2) + bcS(3)dlog(a + b(c + dx)S(4))/(a + bcS(4))S(2) - S(1)/(x(a + bc*S(4))) - sqrt(b)cd(a - bcS(4))atan(sqrt(b)(c + dx)*S(2)/sqrt(a))/(sqrt(a)(a + bcS(4))S(2)) + sqrt(S(2))b*(S(1)/4)d(sqrt(a)(a - S(3)bc*S(4)) + sqrt(b)c*S(2)(S(3)a - bc*S(4)))atan(S(1) - sqrt(S(2))b(S(1)/4)(c + dx)/a(S(1)/4))/(S(4)a*(S(3)/4)(a + bcS(4))S(2)) - sqrt(S(2))b*(S(1)/4)d(sqrt(a)(a - S(3)bc*S(4)) + sqrt(b)c*S(2)(S(3)a - bc*S(4)))atan(S(1) + sqrt(S(2))b(S(1)/4)(c + dx)/a(S(1)/4))/(S(4)a*(S(3)/4)(a + bcS(4))S(2)) - sqrt(S(2))b*(S(1)/4)d(a(S(3)/2) - S(3)sqrt(a)bc*S(4) - S(3)asqrt(b)cS(2) + b(S(3)/2)cS(6))log(-sqrt(S(2))a(S(1)/4)b*(S(1)/4)(c + dx) + sqrt(a) + sqrt(b)(c + dx)S(2))/(S(8)a*(S(3)/4)(a + bcS(4))S(2)) + sqrt(S(2))b*(S(1)/4)d(a(S(3)/2) - S(3)sqrt(a)bc*S(4) - S(3)asqrt(b)cS(2) + b(S(3)/2)cS(6))log(sqrt(S(2))a(S(1)/4)b*(S(1)/4)(c + dx) + sqrt(a) + sqrt(b)(c + dx)S(2))/(S(8)a*(S(3)/4)(a + bcS(4))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a + bsqrt(c + dx))S(2), x), x, -aS(2)cS(3)x/d*S(3) - S(4)abc*S(3)(c + dx)(S(3)/2)/(S(3)d*S(4)) + S(12)abc*S(2)(c + dx)(S(5)/2)/(S(5)d*S(4)) - S(12)abc(c + dx)*(S(7)/2)/(S(7)d*S(4)) + S(4)ab(c + dx)(S(9)/2)/(S(9)dS(4)) + bS(2)(c + dx)*S(5)/(S(5)dS(4)) + cS(2)(S(3)aS(2) - bS(2)c)(c + dx)S(2)/(S(2)d*S(4)) - c(aS(2) - bS(2)c)(c + dx)S(3)/dS(4) + (aS(2) - S(3)b*S(2)c)(c + dx)*S(4)/(S(4)dS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + bsqrt(c + dx))S(2), x), x, aS(2)c*S(2)x/d*S(2) + S(4)abc*S(2)(c + dx)(S(3)/2)/(S(3)d*S(3)) - S(8)abc(c + dx)*(S(5)/2)/(S(5)d*S(3)) + S(4)ab(c + dx)(S(7)/2)/(S(7)dS(3)) + bS(2)(c + dx)*S(4)/(S(4)d*S(3)) - c(S(2)aS(2) - bS(2)c)(c + dx)*S(2)/(S(2)dS(3)) + (aS(2) - S(2)bS(2)c)(c + dx)*S(3)/(S(3)d*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bsqrt(c + dx))S(2), x), x, -aS(2)cx/d - S(4)abc(c + dx)(S(3)/2)/(S(3)d*S(2)) + S(4)ab(c + dx)(S(5)/2)/(S(5)dS(2)) + bS(2)(c + dx)*S(3)/(S(3)dS(2)) + (aS(2) - b*S(2)c)(c + dx)*S(2)/(S(2)d*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsqrt(c + dx))S(2), x), x, aS(2)x + S(4)ab(c + dx)*(S(3)/2)/(S(3)d) + b*S(2)(c + dx)S(2)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsqrt(c + dx))*S(2)/x, x), x, -S(4)absqrt(c)atanh(sqrt(c + dx)/sqrt(c)) + S(4)absqrt(c + dx) + b*S(2)dx + (aS(2) + bS(2)c)log(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsqrt(c + dx))S(2)/xS(2), x), x, -S(2)abdatanh(sqrt(c + dx)/sqrt(c))/sqrt(c) + b*S(2)dlog(x) - (a + bsqrt(c + dx))S(2)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsqrt(c + dx))S(2)/xS(3), x), x, abdS(2)atanh(sqrt(c + dx)/sqrt(c))/(S(2)c*(S(3)/2)) - bd(asqrt(c + dx) + bc)/(S(2)cx) - (a + bsqrt(c + dx))*S(2)/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(a + bsqrt(c + dx)), x), x, -S(28)a(a + bsqrt(c + dx))(S(15)/2)/(S(15)b*S(8)d*S(4)) - S(20)a(a + bsqrt(c + dx))(S(11)/2)(S(7)aS(2) - S(3)b*S(2)c)/(S(11)bS(8)d*S(4)) - S(12)a(a + bsqrt(c + dx))(S(7)/2)(aS(2) - bS(2)c)(S(7)aS(2) - S(3)b*S(2)c)/(S(7)bS(8)d*S(4)) - S(4)a(a + bsqrt(c + dx))(S(3)/2)(aS(2) - bS(2)c)S(3)/(S(3)b*S(8)d*S(4)) + S(4)(a + bsqrt(c + dx))*(S(17)/2)/(S(17)b*S(8)d*S(4)) + (a + bsqrt(c + dx))(S(13)/2)(S(84)aS(2) - S(12)b*S(2)c)/(S(13)bS(8)d*S(4)) + (a + bsqrt(c + dx))(S(9)/2)(S(140)aS(4) - S(120)a*S(2)b*S(2)c + S(12)bS(4)c*S(2))/(S(9)b*S(8)d*S(4)) + S(4)(a + bsqrt(c + dx))*(S(5)/2)(aS(2) - bS(2)c)S(2)(S(7)aS(2) - bS(2)c)/(S(5)bS(8)dS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(a + bsqrt(c + dx)), x), x, -S(20)a(a + bsqrt(c + dx))(S(11)/2)/(S(11)b*S(6)d*S(3)) - S(8)a(a + bsqrt(c + dx))(S(7)/2)(S(5)aS(2) - S(3)b*S(2)c)/(S(7)bS(6)d*S(3)) - S(4)a(a + bsqrt(c + dx))(S(3)/2)(aS(2) - bS(2)c)S(2)/(S(3)b*S(6)d*S(3)) + S(4)(a + bsqrt(c + dx))*(S(13)/2)/(S(13)b*S(6)d*S(3)) + (a + bsqrt(c + dx))(S(9)/2)(S(40)aS(2) - S(8)b*S(2)c)/(S(9)bS(6)d*S(3)) + (a + bsqrt(c + dx))(S(5)/2)(S(20)aS(4) - S(24)a*S(2)b*S(2)c + S(4)bS(4)c*S(2))/(S(5)b*S(6)d*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(a + bsqrt(c + dx)), x), x, -S(12)a(a + bsqrt(c + dx))*(S(7)/2)/(S(7)b*S(4)d*S(2)) - S(4)a(a + bsqrt(c + dx))(S(3)/2)(aS(2) - bS(2)c)/(S(3)b*S(4)d*S(2)) + S(4)(a + bsqrt(c + dx))*(S(9)/2)/(S(9)b*S(4)d*S(2)) + (a + bsqrt(c + dx))(S(5)/2)(S(12)aS(2) - S(4)b*S(2)c)/(S(5)bS(4)d*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(c + dx)), x), x, -S(4)a(a + bsqrt(c + dx))(S(3)/2)/(S(3)b*S(2)d) + S(4)(a + bsqrt(c + dx))(S(5)/2)/(S(5)b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(c + dx))/x, x), x, -S(2)sqrt(a - bsqrt(c))atanh(sqrt(a + bsqrt(c + dx))/sqrt(a - bsqrt(c))) - S(2)sqrt(a + bsqrt(c))atanh(sqrt(a + bsqrt(c + dx))/sqrt(a + bsqrt(c))) + S(4)sqrt(a + bsqrt(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(c + dx))/xS(2), x), x, -bdatanh(sqrt(a + bsqrt(c + dx))/sqrt(a + bsqrt(c)))/(S(2)sqrt(c)sqrt(a + bsqrt(c))) + bdatanh(sqrt(a + bsqrt(c + dx))/sqrt(a - bsqrt(c)))/(S(2)sqrt(c)sqrt(a - bsqrt(c))) - sqrt(a + bsqrt(c + dx))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(c + dx))/xS(3), x), x, bdsqrt(a + bsqrt(c + dx))(-asqrt(c + dx) + bc)/(S(8)cx(aS(2) - bS(2)c)) + bd*S(2)(S(2)a + S(3)bsqrt(c))atanh(sqrt(a + bsqrt(c + dx))/sqrt(a + bsqrt(c)))/(S(16)c*(S(3)/2)(a + bsqrt(c))(S(3)/2)) - bd*S(2)(S(2)a - S(3)bsqrt(c))atanh(sqrt(a + bsqrt(c + dx))/sqrt(a - bsqrt(c)))/(S(16)c*(S(3)/2)(a - bsqrt(c))(S(3)/2)) - sqrt(a + bsqrt(c + dx))/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/(a + bsqrt(c + dx)), x), x, -a(c + dx)*S(3)/(S(3)b*S(2)d*S(4)) - a(a*S(2) - S(3)b*S(2)c)(c + dx)*S(2)/(S(2)b*S(4)d*S(4)) - ax(aS(4) - S(3)a*S(2)b*S(2)c + S(3)bS(4)cS(2))/(bS(6)dS(3)) - S(2)a(aS(2) - bS(2)c)*S(3)log(a + bsqrt(c + dx))/(b*S(8)d*S(4)) + S(2)(c + dx)(S(7)/2)/(S(7)bdS(4)) + (S(2)a*S(2) - S(6)b*S(2)c)(c + dx)*(S(5)/2)/(S(5)b*S(3)d*S(4)) + (c + dx)*(S(3)/2)(S(2)aS(4) - S(6)a*S(2)b*S(2)c + S(6)bS(4)c*S(2))/(S(3)b*S(5)d*S(4)) + S(2)(aS(2) - bS(2)c)S(3)sqrt(c + dx)/(bS(7)dS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bsqrt(c + dx)), x), x, -a(c + dx)*S(2)/(S(2)b*S(2)d*S(3)) - ax(aS(2) - S(2)b*S(2)c)/(b*S(4)d*S(2)) - S(2)a(aS(2) - bS(2)c)*S(2)log(a + bsqrt(c + dx))/(b*S(6)d*S(3)) + S(2)(c + dx)(S(5)/2)/(S(5)bdS(3)) + (S(2)a*S(2) - S(4)b*S(2)c)(c + dx)*(S(3)/2)/(S(3)b*S(3)d*S(3)) + S(2)(aS(2) - bS(2)c)S(2)sqrt(c + dx)/(bS(5)d*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bsqrt(c + dx)), x), x, -ax/(b*S(2)d) - S(2)a(aS(2) - bS(2)c)log(a + bsqrt(c + dx))/(b*S(4)d*S(2)) + S(2)(c + dx)(S(3)/2)/(S(3)bdS(2)) + (S(2)a*S(2) - S(2)b*S(2)c)sqrt(c + dx)/(b*S(3)d*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + bsqrt(c + dx)), x), x, -S(2)alog(a + bsqrt(c + dx))/(bS(2)d) + S(2)sqrt(c + dx)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bsqrt(c + dx))), x), x, alog(x)/(aS(2) - bS(2)c) - S(2)alog(a + bsqrt(c + dx))/(aS(2) - bS(2)c) + S(2)bsqrt(c)atanh(sqrt(c + dx)/sqrt(c))/(aS(2) - bS(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)(a + bsqrt(c + dx))), x), x, abS(2)dlog(x)/(aS(2) - bS(2)c)*S(2) - S(2)abS(2)dlog(a + bsqrt(c + dx))/(aS(2) - bS(2)c)*S(2) + bd(aS(2) + bS(2)c)atanh(sqrt(c + dx)/sqrt(c))/(sqrt(c)(aS(2) - bS(2)c)*S(2)) - (a - bsqrt(c + dx))/(x(aS(2) - bS(2)c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)(a + bsqrt(c + dx))), x), x, abS(4)d*S(2)log(x)/(aS(2) - bS(2)c)S(3) - S(2)abS(4)d*S(2)log(a + bsqrt(c + dx))/(aS(2) - bS(2)c)S(3) - bd(S(4)abc - (a*S(2) + S(3)b*S(2)c)sqrt(c + dx))/(S(4)cx(aS(2) - bS(2)c)*S(2)) - bd*S(2)(a*S(4) - S(6)a*S(2)b*S(2)c - S(3)bS(4)c*S(2))atanh(sqrt(c + dx)/sqrt(c))/(S(4)c*(S(3)/2)(aS(2) - bS(2)c)S(3)) - (a - bsqrt(c + dx))/(xS(2)(S(2)aS(2) - S(2)b*S(2)c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/(a + bsqrt(c + dx))S(2), x), x, -S(4)a(c + dx)*(S(5)/2)/(S(5)b*S(3)d*S(4)) - S(4)a(S(2)a*S(2) - S(3)b*S(2)c)(c + dx)*(S(3)/2)/(S(3)b*S(5)d*S(4)) - S(12)a(aS(2) - bS(2)c)*S(2)sqrt(c + dx)/(bS(7)d*S(4)) + S(2)a(aS(2) - bS(2)c)S(3)/(bS(8)dS(4)(a + bsqrt(c + dx))) + (c + dx)S(3)/(S(3)b*S(2)d*S(4)) + (S(3)a*S(2) - S(3)b*S(2)c)(c + dx)*S(2)/(S(2)b*S(4)d*S(4)) + x(S(5)aS(4) - S(9)a*S(2)b*S(2)c + S(3)bS(4)cS(2))/(bS(6)dS(3)) + S(2)(aS(2) - bS(2)c)S(2)(S(7)aS(2) - bS(2)c)log(a + bsqrt(c + dx))/(bS(8)dS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + bsqrt(c + dx))*S(2), x), x, -S(4)a(c + dx)*(S(3)/2)/(S(3)b*S(3)d*S(3)) - S(8)a(aS(2) - bS(2)c)sqrt(c + dx)/(b*S(5)d*S(3)) + S(2)a(aS(2) - bS(2)c)S(2)/(bS(6)dS(3)(a + bsqrt(c + dx))) + (c + dx)S(2)/(S(2)b*S(2)d*S(3)) + x(S(3)aS(2) - S(2)b*S(2)c)/(b*S(4)d*S(2)) + (S(10)a*S(4) - S(12)a*S(2)b*S(2)c + S(2)bS(4)c*S(2))log(a + bsqrt(c + dx))/(b*S(6)d*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bsqrt(c + dx))S(2), x), x, -S(4)asqrt(c + dx)/(b*S(3)d*S(2)) + S(2)a(aS(2) - bS(2)c)/(b*S(4)d*S(2)(a + bsqrt(c + dx))) + x/(b*S(2)d) + (S(6)aS(2) - S(2)b*S(2)c)log(a + bsqrt(c + dx))/(bS(4)d*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsqrt(c + dx))(S(-2)), x), x, S(2)a/(b*S(2)d(a + bsqrt(c + dx))) + S(2)log(a + bsqrt(c + dx))/(b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + bsqrt(c + dx))S(2)), x), x, S(4)absqrt(c)atanh(sqrt(c + dx)/sqrt(c))/(aS(2) - bS(2)c)S(2) + S(2)a/((a + bsqrt(c + dx))(aS(2) - bS(2)c)) + (aS(2) + bS(2)c)log(x)/(aS(2) - bS(2)c)S(2) - (S(2)a*S(2) + S(2)b*S(2)c)log(a + bsqrt(c + dx))/(aS(2) - bS(2)c)S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + bsqrt(c + dx))S(2)), x), x, S(4)abS(2)d/((a + bsqrt(c + dx))(aS(2) - bS(2)c)*S(2)) + S(2)abd(aS(2) + S(3)b*S(2)c)atanh(sqrt(c + dx)/sqrt(c))/(sqrt(c)(aS(2) - bS(2)c)S(3)) + bS(2)d(S(3)aS(2) + bS(2)c)log(x)/(aS(2) - bS(2)c)*S(3) - S(2)b*S(2)d(S(3)aS(2) + bS(2)c)log(a + bsqrt(c + dx))/(aS(2) - bS(2)c)S(3) - (a - bsqrt(c + dx))/(x(a + bsqrt(c + dx))(aS(2) - bS(2)c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)(a + bsqrt(c + dx))*S(2)), x), x, ab*S(2)d*S(2)(a*S(2) + S(11)b*S(2)c)/(S(2)c(a + bsqrt(c + dx))(aS(2) - bS(2)c)*S(3)) - abdS(2)(a*S(4) - S(10)a*S(2)b*S(2)c - S(15)bS(4)c*S(2))atanh(sqrt(c + dx)/sqrt(c))/(S(2)c*(S(3)/2)(aS(2) - bS(2)c)S(4)) + bS(4)d*S(2)(S(5)aS(2) + bS(2)c)log(x)/(aS(2) - bS(2)c)*S(4) - S(2)b*S(4)d*S(2)(S(5)aS(2) + bS(2)c)log(a + bsqrt(c + dx))/(aS(2) - bS(2)c)*S(4) - bd(S(3)abc - (a*S(2) + S(2)b*S(2)c)sqrt(c + dx))/(S(2)cx(a + bsqrt(c + dx))(aS(2) - bS(2)c)S(2)) - (a - bsqrt(c + dx))/(xS(2)(a + bsqrt(c + dx))(S(2)a*S(2) - S(2)b*S(2)c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)/sqrt(a + bsqrt(c + dx)), x), x, -S(28)a(a + bsqrt(c + dx))(S(13)/2)/(S(13)b*S(8)d*S(4)) - S(20)a(a + bsqrt(c + dx))(S(9)/2)(S(7)aS(2) - S(3)b*S(2)c)/(S(9)bS(8)d*S(4)) - S(12)a(a + bsqrt(c + dx))(S(5)/2)(aS(2) - bS(2)c)(S(7)aS(2) - S(3)b*S(2)c)/(S(5)bS(8)d*S(4)) - S(4)asqrt(a + bsqrt(c + dx))(aS(2) - bS(2)c)S(3)/(bS(8)d*S(4)) + S(4)(a + bsqrt(c + dx))*(S(15)/2)/(S(15)b*S(8)d*S(4)) + (a + bsqrt(c + dx))(S(11)/2)(S(84)aS(2) - S(12)b*S(2)c)/(S(11)bS(8)d*S(4)) + (a + bsqrt(c + dx))(S(7)/2)(S(140)aS(4) - S(120)a*S(2)b*S(2)c + S(12)bS(4)c*S(2))/(S(7)b*S(8)d*S(4)) + S(4)(a + bsqrt(c + dx))*(S(3)/2)(aS(2) - bS(2)c)S(2)(S(7)aS(2) - bS(2)c)/(S(3)bS(8)dS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(a + bsqrt(c + dx)), x), x, -S(20)a(a + bsqrt(c + dx))*(S(9)/2)/(S(9)b*S(6)d*S(3)) - S(8)a(a + bsqrt(c + dx))(S(5)/2)(S(5)aS(2) - S(3)b*S(2)c)/(S(5)bS(6)d*S(3)) - S(4)asqrt(a + bsqrt(c + dx))(aS(2) - bS(2)c)S(2)/(bS(6)d*S(3)) + S(4)(a + bsqrt(c + dx))*(S(11)/2)/(S(11)b*S(6)d*S(3)) + (a + bsqrt(c + dx))(S(7)/2)(S(40)aS(2) - S(8)b*S(2)c)/(S(7)bS(6)d*S(3)) + (a + bsqrt(c + dx))(S(3)/2)(S(20)aS(4) - S(24)a*S(2)b*S(2)c + S(4)bS(4)c*S(2))/(S(3)b*S(6)d*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + bsqrt(c + dx)), x), x, -S(12)a(a + bsqrt(c + dx))(S(5)/2)/(S(5)b*S(4)d*S(2)) - S(4)asqrt(a + bsqrt(c + dx))(aS(2) - bS(2)c)/(bS(4)d*S(2)) + S(4)(a + bsqrt(c + dx))*(S(7)/2)/(S(7)b*S(4)d*S(2)) + (a + bsqrt(c + dx))(S(3)/2)(S(12)aS(2) - S(4)b*S(2)c)/(S(3)bS(4)d*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + bsqrt(c + dx)), x), x, -S(4)asqrt(a + bsqrt(c + dx))/(bS(2)d) + S(4)(a + bsqrt(c + dx))(S(3)/2)/(S(3)b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + bsqrt(c + dx))), x), x, -S(2)atanh(sqrt(a + bsqrt(c + dx))/sqrt(a + bsqrt(c)))/sqrt(a + bsqrt(c)) - S(2)atanh(sqrt(a + bsqrt(c + dx))/sqrt(a - bsqrt(c)))/sqrt(a - bsqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(a + bsqrt(c + dx))), x), x, bdatanh(sqrt(a + bsqrt(c + dx))/sqrt(a + bsqrt(c)))/(S(2)sqrt(c)(a + bsqrt(c))*(S(3)/2)) - bdatanh(sqrt(a + bsqrt(c + dx))/sqrt(a - bsqrt(c)))/(S(2)sqrt(c)(a - bsqrt(c))(S(3)/2)) - (a - bsqrt(c + dx))sqrt(a + bsqrt(c + dx))/(x(aS(2) - bS(2)c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)sqrt(a + bsqrt(c + dx))), x), x, -bdsqrt(a + bsqrt(c + dx))(S(6)abc - (a*S(2) + S(5)b*S(2)c)sqrt(c + dx))/(S(8)cx(aS(2) - bS(2)c)*S(2)) - bd*S(2)(S(2)a + S(5)bsqrt(c))atanh(sqrt(a + bsqrt(c + dx))/sqrt(a + bsqrt(c)))/(S(16)c*(S(3)/2)(a + bsqrt(c))(S(5)/2)) + bd*S(2)(S(2)a - S(5)bsqrt(c))atanh(sqrt(a + bsqrt(c + dx))/sqrt(a - bsqrt(c)))/(S(16)c*(S(3)/2)(a - bsqrt(c))(S(5)/2)) - (a - bsqrt(c + dx))sqrt(a + bsqrt(c + dx))/(x*S(2)(S(2)aS(2) - S(2)b*S(2)c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)(a + bsqrt(c + dx))*p, x), x, -S(2)a(a + bsqrt(c + dx))(p + S(1))(aS(2) - bS(2)c)S(3)/(bS(8)d*S(4)(p + S(1))) - S(6)a(a + bsqrt(c + dx))*(p + S(3))(aS(2) - bS(2)c)(S(7)aS(2) - S(3)b*S(2)c)/(b*S(8)d*S(4)(p + S(3))) - S(10)a(a + bsqrt(c + dx))*(p + S(5))(S(7)aS(2) - S(3)b*S(2)c)/(b*S(8)d*S(4)(p + S(5))) - S(14)a(a + bsqrt(c + dx))(p + S(7))/(bS(8)dS(4)(p + S(7))) + S(2)(a + bsqrt(c + dx))(p + S(2))(aS(2) - bS(2)c)S(2)(S(7)aS(2) - bS(2)c)/(b*S(8)d*S(4)(p + S(2))) + (a + bsqrt(c + dx))*(p + S(4))(S(70)aS(4) - S(60)a*S(2)b*S(2)c + S(6)bS(4)cS(2))/(bS(8)dS(4)(p + S(4))) + (a + bsqrt(c + dx))*(p + S(6))(S(42)aS(2) - S(6)b*S(2)c)/(b*S(8)d*S(4)(p + S(6))) + S(2)(a + bsqrt(c + dx))(p + S(8))/(bS(8)d*S(4)(p + S(8))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(a + bsqrt(c + dx))*p, x), x, -S(2)a(a + bsqrt(c + dx))(p + S(1))(aS(2) - bS(2)c)S(2)/(bS(6)d*S(3)(p + S(1))) - S(4)a(a + bsqrt(c + dx))*(p + S(3))(S(5)aS(2) - S(3)b*S(2)c)/(b*S(6)d*S(3)(p + S(3))) - S(10)a(a + bsqrt(c + dx))(p + S(5))/(bS(6)dS(3)(p + S(5))) + (a + bsqrt(c + dx))*(p + S(2))(S(10)aS(4) - S(12)a*S(2)b*S(2)c + S(2)bS(4)cS(2))/(bS(6)dS(3)(p + S(2))) + (a + bsqrt(c + dx))*(p + S(4))(S(20)aS(2) - S(4)b*S(2)c)/(b*S(6)d*S(3)(p + S(4))) + S(2)(a + bsqrt(c + dx))(p + S(6))/(bS(6)d*S(3)(p + S(6))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bsqrt(c + dx))p, x), x, -S(2)a(a + bsqrt(c + dx))(p + S(1))(aS(2) - bS(2)c)/(bS(4)d*S(2)(p + S(1))) - S(6)a(a + bsqrt(c + dx))(p + S(3))/(bS(4)dS(2)(p + S(3))) + (a + bsqrt(c + dx))*(p + S(2))(S(6)aS(2) - S(2)b*S(2)c)/(b*S(4)d*S(2)(p + S(2))) + S(2)(a + bsqrt(c + dx))(p + S(4))/(bS(4)d*S(2)(p + S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsqrt(c + dx))*p, x), x, -S(2)a(a + bsqrt(c + dx))(p + S(1))/(bS(2)d(p + S(1))) + S(2)(a + bsqrt(c + dx))(p + S(2))/(bS(2)d(p + S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsqrt(c + dx))*p/x, x), x, -(a + bsqrt(c + dx))(p + S(1))hyper((S(1), p + S(1)), (p + S(2),), (a + bsqrt(c + dx))/(a + bsqrt(c)))/((a + bsqrt(c))(p + S(1))) - (a + bsqrt(c + dx))(p + S(1))hyper((S(1), p + S(1)), (p + S(2),), (a + bsqrt(c + dx))/(a - bsqrt(c)))/((a - bsqrt(c))(p + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cx)n)(S(5)/2)/x, x), x, -S(2)a*(S(5)/2)atanh(sqrt(a + b(cx)*n)/sqrt(a))/n + S(2)a*S(2)sqrt(a + b(cx)*n)/n + S(2)a(a + b(cx)n)(S(3)/2)/(S(3)n) + S(2)(a + b(cx)n)(S(5)/2)/(S(5)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b(cx)n)(S(3)/2)/x, x), x, -S(2)a(S(3)/2)atanh(sqrt(a + b(cx)*n)/sqrt(a))/n + S(2)asqrt(a + b(cx)n)/n + S(2)(a + b(cx)n)(S(3)/2)/(S(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + b(cx)n)/x, x), x, -S(2)sqrt(a)atanh(sqrt(a + b(cx)n)/sqrt(a))/n + S(2)sqrt(a + b(cx)*n)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + b(cx)*n)), x), x, -S(2)atanh(sqrt(a + b(cx)*n)/sqrt(a))/(sqrt(a)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + b(cx)n)(S(3)/2)), x), x, S(2)/(ansqrt(a + b(cx)n)) - S(2)atanh(sqrt(a + b(cx)n)/sqrt(a))/(a(S(3)/2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + b(cx)n)(S(5)/2)), x), x, S(2)/(S(3)an(a + b(cx)n)(S(3)/2)) + S(2)/(aS(2)nsqrt(a + b(cx)n)) - S(2)atanh(sqrt(a + b(cx)n)/sqrt(a))/(a(S(5)/2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-a + b(cx)n)(S(5)/2)/x, x), x, -S(2)a*(S(5)/2)atan(sqrt(-a + b(cx)*n)/sqrt(a))/n + S(2)a*S(2)sqrt(-a + b(cx)*n)/n - S(2)a(-a + b(cx)n)(S(3)/2)/(S(3)n) + S(2)(-a + b(cx)n)(S(5)/2)/(S(5)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-a + b(cx)n)(S(3)/2)/x, x), x, S(2)a(S(3)/2)atan(sqrt(-a + b(cx)*n)/sqrt(a))/n - S(2)asqrt(-a + b(cx)n)/n + S(2)(-a + b(cx)n)(S(3)/2)/(S(3)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-a + b(cx)n)/x, x), x, -S(2)sqrt(a)atan(sqrt(-a + b(cx)n)/sqrt(a))/n + S(2)sqrt(-a + b(cx)*n)/n, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(-a + b(cx)*n)), x), x, S(2)atan(sqrt(-a + b(cx)*n)/sqrt(a))/(sqrt(a)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(-a + b(cx)n)(S(3)/2)), x), x, -S(2)/(ansqrt(-a + b(cx)n)) - S(2)atan(sqrt(-a + b(cx)n)/sqrt(a))/(a(S(3)/2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(-a + b(cx)n)(S(5)/2)), x), x, -S(2)/(S(3)an(-a + b(cx)n)(S(3)/2)) + S(2)/(aS(2)nsqrt(-a + b(cx)n)) + S(2)atan(sqrt(-a + b(cx)n)/sqrt(a))/(a(S(5)/2)n), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + bx)), x), x, -S(2)atanh(sqrt(a + bx)/sqrt(a))/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + b(cx)*m)), x), x, -S(2)atanh(sqrt(a + b(cx)*m)/sqrt(a))/(sqrt(a)m), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + b(c(dx)m)n)), x), x, -S(2)atanh(sqrt(a + b(c(dx)m)n)/sqrt(a))/(sqrt(a)mn), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + b(c(d(ex)m)n)p)), x), x, -S(2)atanh(sqrt(a + b(c(d(ex)m)n)*p)/sqrt(a))/(sqrt(a)mnp), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + b(c(d(e(fx)m)n)p)q)), x), x, -S(2)atanh(sqrt(a + b(c(d(e(fx)m)n)p)q)/sqrt(a))/(sqrt(a)mnpq), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(-1) + x*(S(-2)))(xS(2) + S(-1))S(3)/x, x), x, -x*S(6)(S(-1) + x(S(-2)))(S(7)/2)/S(6) - S(7)xS(4)(S(-1) + x(S(-2)))(S(5)/2)/S(24) - S(35)xS(2)(S(-1) + x(S(-2)))(S(3)/2)/S(48) + S(35)sqrt(S(-1) + x(S(-2)))/S(16) - S(35)atan(sqrt(S(-1) + x(S(-2))))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(-1) + x(S(-2)))(xS(2) + S(-1))S(2)/x, x), x, xS(4)(S(-1) + x(S(-2)))(S(5)/2)/S(4) + S(5)xS(2)(S(-1) + x(S(-2)))(S(3)/2)/S(8) - S(15)sqrt(S(-1) + x(S(-2)))/S(8) + S(15)atan(sqrt(S(-1) + x(S(-2))))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(-1) + x(S(-2)))(xS(2) + S(-1))/x, x), x, -xS(2)(S(-1) + x(S(-2)))(S(3)/2)/S(2) + S(3)sqrt(S(-1) + x(S(-2)))/S(2) - S(3)atan(sqrt(S(-1) + x(S(-2))))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(-1) + x(S(-2)))/(x(xS(2) + S(-1))), x), x, sqrt(S(-1) + x(S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(-1) + x(S(-2)))/(x(xS(2) + S(-1))S(2)), x), x, -sqrt(S(-1) + x(S(-2))) + S(1)/sqrt(S(-1) + x(S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(-1) + x*(S(-2)))/(x(xS(2) + S(-1))S(3)), x), x, sqrt(S(-1) + x(S(-2))) - S(2)/sqrt(S(-1) + x(S(-2))) - S(1)/(S(3)(S(-1) + x(S(-2)))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(S(1) + x(S(-2)))/(xS(2) + S(1))S(2), x), x, S(1)/sqrt(S(1) + x(S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(S(1) + x(S(-2)))(xS(2) + S(1))), x), x, S(1)/sqrt(S(1) + x(S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + bxS(2) + sqrt(a + bx*S(2))), x), x, log(sqrt(a + bxS(2)) + S(1))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(xS(2) - (xS(2))(S(1)/3)), x), x, S(3)log(-(xS(2))(S(2)/3) + S(1))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(xS(2) + S(1))S(3)sqrt(xS(4) + S(2)xS(2) + S(2)), x), x, (xS(2) + S(1))*S(2)(x*S(4) + S(2)xS(2) + S(2))(S(3)/2)/S(10) - (x*S(4) + S(2)xS(2) + S(2))(S(3)/2)/S(15), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt((-xS(2) + S(1))/(xS(2) + S(1))), x), x, sqrt((-xS(2) + S(1))/(xS(2) + S(1)))(xS(2) + S(1))/S(2) - atan(sqrt((-xS(2) + S(1))/(x*S(2) + S(1)))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xsqrt((-xS(2) + S(1))/(xS(2) + S(1))), x), x, sqrt((-xS(2) + S(1))/(xS(2) + S(1)))/((-xS(2) + S(1))/(xS(2) + S(1)) + S(1)) - atan(sqrt((-xS(2) + S(1))/(xS(2) + S(1)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt((-S(7)x*S(2) + S(5))/(S(5)x*S(2) + S(7))), x), x, sqrt((-S(7)x*S(2) + S(5))/(S(5)x*S(2) + S(7)))(S(5)xS(2) + S(7))/S(10) - S(37)sqrt(S(35))atan(sqrt(S(35))sqrt((-S(7)xS(2) + S(5))/(S(5)x*S(2) + S(7)))/S(7))/S(175), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xsqrt((-S(7)xS(2) + S(5))/(S(5)x*S(2) + S(7))), x), x, S(37)sqrt((-S(7)xS(2) + S(5))/(S(5)x*S(2) + S(7)))/(S(5)(-S(35)xS(2) + S(25))/(S(5)x*S(2) + S(7)) + S(35)) - S(37)sqrt(S(35))atan(sqrt(S(35))sqrt((-S(7)xS(2) + S(5))/(S(5)xS(2) + S(7)))/S(7))/S(175), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt((-xS(3) + S(1))/(xS(3) + S(1))), x), x, sqrt((-xS(3) + S(1))/(xS(3) + S(1)))(x*S(3) + S(1))/S(3) - S(2)atan(sqrt((-xS(3) + S(1))/(xS(3) + S(1))))/S(3), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(x*S(2)sqrt((-xS(3) + S(1))/(xS(3) + S(1))), x), x, S(2)sqrt((-xS(3) + S(1))/(xS(3) + S(1)))/(S(3)(-xS(3) + S(1))/(xS(3) + S(1)) + S(3)) - S(2)atan(sqrt((-xS(3) + S(1))/(xS(3) + S(1))))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)sqrt(-x*S(3) + S(1))(xS(9) + S(1))S(2), x), x, S(2)(-xS(3) + S(1))(S(17)/2)/S(51) - S(14)(-xS(3) + S(1))(S(15)/2)/S(45) + S(14)(-xS(3) + S(1))(S(13)/2)/S(13) - S(74)(-xS(3) + S(1))(S(11)/2)/S(33) + S(86)(-xS(3) + S(1))(S(9)/2)/S(27) - S(22)(-xS(3) + S(1))(S(7)/2)/S(7) + S(32)(-xS(3) + S(1))(S(5)/2)/S(15) - S(8)(-xS(3) + S(1))(S(3)/2)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(8)sqrt((-xS(3) + S(1))/(xS(3) + S(1))), x), x, -S(8)((-xS(3) + S(1))/(xS(3) + S(1)))(S(3)/2)/(S(9)((-xS(3) + S(1))/(xS(3) + S(1)) + S(1))S(3)) + sqrt((-xS(3) + S(1))/(xS(3) + S(1)))/((-xS(3) + S(1))/(x*S(3) + S(1)) + S(1)) - S(2)sqrt((-xS(3) + S(1))/(xS(3) + S(1)))/(S(3)((-xS(3) + S(1))/(xS(3) + S(1)) + S(1))S(2)) - atan(sqrt((-xS(3) + S(1))/(xS(3) + S(1))))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(9)sqrt((-S(7)xS(5) + S(5))/(S(5)x*S(5) + S(7))), x), x, -S(999)sqrt((-S(7)xS(5) + S(5))/(S(5)x*S(5) + S(7)))/(S(175)(-S(35)xS(5) + S(25))/(S(5)x*S(5) + S(7)) + S(1225)) + S(2738)sqrt((-S(7)xS(5) + S(5))/(S(5)x*S(5) + S(7)))/(S(125)((-S(35)xS(5) + S(25))/(S(5)xS(5) + S(7)) + S(7))S(2)) + S(2257)sqrt(S(35))atan(sqrt(S(35))sqrt((-S(7)x*S(5) + S(5))/(S(5)x*S(5) + S(7)))/S(7))/S(30625), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt(a + bx*S(2))(x*S(2) + S(1))) + x/(a + bxS(2))(S(3)/2), x), x, -atanh(sqrt(a + bxS(2))/sqrt(a - b))/sqrt(a - b) - S(1)/(bsqrt(a + bxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + bxS(2) + xS(2) + S(1))/((a + bxS(2))(S(3)/2)(xS(2) + S(1))), x), x, -atanh(sqrt(a + bx*S(2))/sqrt(a - b))/sqrt(a - b) - S(1)/(bsqrt(a + bxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt(a + bx*S(2))(x*S(2) + S(1))) + x/(a + bxS(2))(S(3)/2) + x/(a + bxS(2))(S(5)/2), x), x, -atanh(sqrt(a + bx*S(2))/sqrt(a - b))/sqrt(a - b) - S(1)/(bsqrt(a + bxS(2))) - S(1)/(S(3)b(a + bxS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(aS(2) + S(2)abx*S(2) + axS(2) + a + bS(2)xS(4) + bx*S(4) + bxS(2) + xS(2) + S(1))/((a + bxS(2))(S(5)/2)(x*S(2) + S(1))), x), x, -atanh(sqrt(a + bx*S(2))/sqrt(a - b))/sqrt(a - b) - S(1)/(bsqrt(a + bxS(2))) - S(1)/(S(3)b(a + bxS(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(sqrt(x) + x), x), x, S(2)sqrt(sqrt(x) + x) - S(2)atanh(sqrt(x)/sqrt(sqrt(x) + x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sqrt(x) + x), x), x, sqrt(x)sqrt(sqrt(x) + x)/S(6) + S(2)xsqrt(sqrt(x) + x)/S(3) - sqrt(sqrt(x) + x)/S(4) + atanh(sqrt(x)/sqrt(sqrt(x) + x))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-x)(x + sqrt(-x)), x), x, -x*S(2)/S(2) + S(2)(-x)(S(5)/2)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x(S(1)/4) + S(5))/(x + S(-6)), x), x, S(4)x(S(1)/4) + S(5)log(-x + S(6)) - S(2)S(6)(S(1)/4)atan(S(6)*(S(3)/4)x*(S(1)/4)/S(6)) - S(2)S(6)*(S(1)/4)atanh(S(6)*(S(3)/4)x*(S(1)/4)/S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(-x + sqrt(-x + S(4)) + S(4)), x), x, -S(2)log(sqrt(-x + S(4)) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x - sqrt(x + S(2)) + S(1)), x), x, (sqrt(S(5))/S(5) + S(1))log(-S(2)sqrt(x + S(2)) + S(1) + sqrt(S(5))) + (-sqrt(S(5))/S(5) + S(1))log(-S(2)sqrt(x + S(2)) - sqrt(S(5)) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x + sqrt(x + S(1)) + S(4)), x), x, log(x + sqrt(x + S(1)) + S(4)) - S(2)sqrt(S(11))atan(sqrt(S(11))(S(2)sqrt(x + S(1)) + S(1))/S(11))/S(11), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x - sqrt(x + S(1))), x), x, (sqrt(S(5))/S(5) + S(1))log(-S(2)sqrt(x + S(1)) + S(1) + sqrt(S(5))) + (-sqrt(S(5))/S(5) + S(1))log(-S(2)sqrt(x + S(1)) - sqrt(S(5)) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x - sqrt(x + S(2))), x), x, S(4)log(-sqrt(x + S(2)) + S(2))/S(3) + S(2)log(sqrt(x + S(2)) + S(1))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x - sqrt(-x + S(1))), x), x, (sqrt(S(5))/S(5) + S(1))log(S(2)sqrt(-x + S(1)) + S(1) + sqrt(S(5))) + (-sqrt(S(5))/S(5) + S(1))log(S(2)sqrt(-x + S(1)) - sqrt(S(5)) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sqrt(x) + x + S(1)), x), x, (-sqrt(x)/S(2) + S(-1)/4)sqrt(sqrt(x) + x + S(1)) + S(2)(sqrt(x) + x + S(1))*(S(3)/2)/S(3) - S(3)asinh(sqrt(S(3))(S(2)sqrt(x) + S(1))/S(3))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x + sqrt(x + S(1)) + S(1)), x), x, -(S(2)sqrt(x + S(1)) + S(1))sqrt(x + sqrt(x + S(1)) + S(1))/S(4) + S(2)(x + sqrt(x + S(1)) + S(1))(S(3)/2)/S(3) + atanh(sqrt(x + S(1))/sqrt(x + sqrt(x + S(1)) + S(1)))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x + sqrt(x + S(-1))), x), x, S(2)(x + sqrt(x + S(-1)))*(S(3)/2)/S(3) + sqrt(x + sqrt(x + S(-1)))(-sqrt(x + S(-1))/S(2) + S(-1)/4) - S(3)asinh(sqrt(S(3))(S(2)sqrt(x + S(-1)) + S(1))/S(3))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(2)x + sqrt(S(2)x + S(-1))), x), x, (S(2)x + sqrt(S(2)x + S(-1)))(S(3)/2)/S(3) - sqrt(S(2)x + sqrt(S(2)x + S(-1)))(S(2)sqrt(S(2)x + S(-1)) + S(1))/S(8) - S(3)asinh(sqrt(S(3))(S(2)sqrt(S(2)x + S(-1)) + S(1))/S(3))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(3)x + sqrt(S(8)x + S(-7))), x), x, sqrt(S(2))(S(24)x + S(8)sqrt(S(8)x + S(-7)))*(S(3)/2)/S(144) - sqrt(S(2))sqrt(S(24)x + S(8)sqrt(S(8)x + S(-7)))(S(3)sqrt(S(8)x + S(-7)) + S(4))/S(72) - S(47)sqrt(S(6))asinh(sqrt(S(47))(S(3)sqrt(S(8)x + S(-7)) + S(4))/S(47))/S(216), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(x + sqrt(x + S(1))), x), x, S(2)sqrt(x + sqrt(x + S(1))) - atanh((S(2)sqrt(x + S(1)) + S(1))/(S(2)sqrt(x + sqrt(x + S(1))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + S(1))/(x + sqrt(S(6)x + S(-9)) + S(4)), x), x, x - S(2)sqrt(S(3))sqrt(S(2)x + S(-3)) + S(3)log(x + sqrt(S(3))sqrt(S(2)x + S(-3)) + S(4)) + S(4)sqrt(S(6))atan(sqrt(S(6))(sqrt(S(6)x + S(-9)) + S(3))/S(12)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-x + S(12))/(x + sqrt(S(6)x + S(-9)) + S(4)), x), x, -x + S(2)sqrt(S(3))sqrt(S(2)x + S(-3)) + S(10)log(x + sqrt(S(3))sqrt(S(2)x + S(-3)) + S(4)) - S(21)sqrt(S(6))atan(sqrt(S(6))(sqrt(S(6)x + S(-9)) + S(3))/S(12))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(3) + S(-1))/(sqrt(x)(x*S(2) + S(1))), x), x, S(2)x*(S(3)/2)/S(3) - sqrt(S(2))atan(sqrt(S(2))sqrt(x) + S(-1)) - sqrt(S(2))atan(sqrt(S(2))sqrt(x) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(2)sqrt(x + S(-1))sqrt(x - sqrt(x + S(-1)))), x), x, -asinh(sqrt(S(3))(-S(2)sqrt(x + S(-1)) + S(1))/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x(S(7)/2) + S(1))/(-xS(2) + S(1)), x), x, -S(2)x*(S(5)/2)/S(5) - S(2)sqrt(x) - log(-sqrt(x) + S(1)) + log(x + S(1))/S(2) + atan(sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x + S(4))/((S(2)x + S(-1))*(S(1)/3) + sqrt(S(2)x + S(-1))), x), x, -x + S(3)(S(2)x + S(-1))*(S(7)/6)/S(7) + S(3)(S(2)x + S(-1))(S(5)/6)/S(5) + S(18)(S(2)x + S(-1))(S(1)/6) - S(3)(S(2)x + S(-1))(S(4)/3)/S(8) - S(3)(S(2)x + S(-1))(S(2)/3)/S(4) - S(9)(S(2)x + S(-1))(S(1)/3) + (S(2)x + S(-1))*(S(3)/2)/S(3) + S(6)sqrt(S(2)x + S(-1)) - S(18)log((S(2)x + S(-1))(S(1)/6) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(sqrt(sqrt(x) + S(1)) + S(2)), x), x, S(8)(sqrt(sqrt(x) + S(1)) + S(2))*(S(7)/2)/S(7) - S(48)(sqrt(sqrt(x) + S(1)) + S(2))*(S(5)/2)/S(5) + S(88)(sqrt(sqrt(x) + S(1)) + S(2))*(S(3)/2)/S(3) - S(48)sqrt(sqrt(sqrt(x) + S(1)) + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sqrt(sqrt(x) + S(4)) + S(2)), x), x, S(8)(sqrt(sqrt(x) + S(4)) + S(2))(S(9)/2)/S(9) - S(48)(sqrt(sqrt(x) + S(4)) + S(2))*(S(7)/2)/S(7) + S(64)(sqrt(sqrt(x) + S(4)) + S(2))*(S(5)/2)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-sqrt(sqrt(S(5)x + S(-9)) + S(4)) + S(2)), x), x, S(8)(-sqrt(sqrt(S(5)x + S(-9)) + S(4)) + S(2))*(S(9)/2)/S(45) - S(48)(-sqrt(sqrt(S(5)x + S(-9)) + S(4)) + S(2))(S(7)/2)/S(35) + S(64)(-sqrt(sqrt(S(5)x + S(-9)) + S(4)) + S(2))(S(5)/2)/S(25), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(sqrt(sqrt(x) + S(1)) + S(2)), x), x, S(8)(sqrt(sqrt(x) + S(1)) + S(2))*(S(7)/2)/S(7) - S(48)(sqrt(sqrt(x) + S(1)) + S(2))*(S(5)/2)/S(5) + S(88)(sqrt(sqrt(x) + S(1)) + S(2))*(S(3)/2)/S(3) - S(48)sqrt(sqrt(sqrt(x) + S(1)) + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sqrt(sqrt(sqrt(x) + S(1)) + S(1)) + S(1)), x), x, S(16)(sqrt(sqrt(sqrt(x) + S(1)) + S(1)) + S(1))(S(17)/2)/S(17) - S(112)(sqrt(sqrt(sqrt(x) + S(1)) + S(1)) + S(1))*(S(15)/2)/S(15) + S(288)(sqrt(sqrt(sqrt(x) + S(1)) + S(1)) + S(1))*(S(13)/2)/S(13) - S(320)(sqrt(sqrt(sqrt(x) + S(1)) + S(1)) + S(1))*(S(11)/2)/S(11) + S(112)(sqrt(sqrt(sqrt(x) + S(1)) + S(1)) + S(1))*(S(9)/2)/S(9) + S(48)(sqrt(sqrt(sqrt(x) + S(1)) + S(1)) + S(1))*(S(7)/2)/S(7) - S(32)(sqrt(sqrt(sqrt(x) + S(1)) + S(1)) + S(1))*(S(5)/2)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sqrt(sqrt(S(2)sqrt(x) + S(-1)) + S(3)) + S(2)), x), x, S(4)(sqrt(sqrt(S(2)sqrt(x) + S(-1)) + S(3)) + S(2))*(S(17)/2)/S(17) - S(56)(sqrt(sqrt(S(2)sqrt(x) + S(-1)) + S(3)) + S(2))(S(15)/2)/S(15) + S(300)(sqrt(sqrt(S(2)sqrt(x) + S(-1)) + S(3)) + S(2))(S(13)/2)/S(13) - S(760)(sqrt(sqrt(S(2)sqrt(x) + S(-1)) + S(3)) + S(2))(S(11)/2)/S(11) + S(304)(sqrt(sqrt(S(2)sqrt(x) + S(-1)) + S(3)) + S(2))(S(9)/2)/S(3) - S(480)(sqrt(sqrt(S(2)sqrt(x) + S(-1)) + S(3)) + S(2))(S(7)/2)/S(7) + S(136)(sqrt(sqrt(S(2)sqrt(x) + S(-1)) + S(3)) + S(2))(S(5)/2)/S(5) - S(16)(sqrt(sqrt(S(2)sqrt(x) + S(-1)) + S(3)) + S(2))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(sqrt(sqrt(x + S(-1)) + S(1)) + S(1)), x), x, S(8)(sqrt(sqrt(x + S(-1)) + S(1)) + S(1))(S(17)/2)/S(17) - S(56)(sqrt(sqrt(x + S(-1)) + S(1)) + S(1))*(S(15)/2)/S(15) + S(144)(sqrt(sqrt(x + S(-1)) + S(1)) + S(1))*(S(13)/2)/S(13) - S(160)(sqrt(sqrt(x + S(-1)) + S(1)) + S(1))*(S(11)/2)/S(11) + S(8)(sqrt(sqrt(x + S(-1)) + S(1)) + S(1))*(S(9)/2) - S(24)(sqrt(sqrt(x + S(-1)) + S(1)) + S(1))*(S(7)/2)/S(7) + S(16)(sqrt(sqrt(x + S(-1)) + S(1)) + S(1))*(S(5)/2)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x + S(-1))sqrt(x - sqrt(x + S(-1)))), x), x, -S(2)asinh(sqrt(S(3))(-S(2)sqrt(x + S(-1)) + S(1))/S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(x + sqrt(S(2)x + S(-1)) + S(1)), x), x, S(2)sqrt(x + sqrt(S(2)x + S(-1)) + S(1)) - sqrt(S(2))asinh(sqrt(S(2))(sqrt(S(2)x + S(-1)) + S(1))/S(2)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(S(1)/sqrt(x + sqrt(S(2)x + S(-1)) + S(1)), x), x, sqrt(S(2))sqrt(S(2)x + S(2)sqrt(S(2)x + S(-1)) + S(2)) - sqrt(S(2))asinh(sqrt(S(2))(sqrt(S(2)x + S(-1)) + S(1))/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((px + q)/((f + sqrt(ax + b))sqrt(ax + b)), x), x, px/a - S(2)fpsqrt(ax + b)/a*S(2) - (-S(2)aq + S(2)bp - S(2)f*S(2)p)log(f + sqrt(ax + b))/a*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-sqrt(x) - x + S(1)), x), x, (-sqrt(x)/S(2) + S(-1)/4)sqrt(-sqrt(x) - x + S(1)) - S(2)(-sqrt(x) - x + S(1))(S(3)/2)/S(3) - S(5)asin(sqrt(S(5))(S(2)sqrt(x) + S(1))/S(5))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(6)sqrt(x) + x + S(9))/(S(4)sqrt(x) + x), x), x, S(4)sqrt(x) + x + S(2)log(sqrt(x) + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-S(8)x(S(7)/2) + S(6))/(-S(9)sqrt(x) + S(5)), x), x, S(80)x(S(7)/2)/S(567) + S(400)x*(S(5)/2)/S(6561) + S(50000)x*(S(3)/2)/S(1594323) - S(56145628)sqrt(x)/S(43046721) + S(2)xS(4)/S(9) + S(200)x*S(3)/S(2187) + S(2500)x*S(2)/S(59049) + S(125000)x/S(4782969) - S(280728140)log(-S(9)sqrt(x) + S(5))/S(387420489), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x + S(1))(xS(3) + S(1))/(xS(2) + S(1)), x), x, S(2)(x + S(1))*(S(5)/2)/S(5) - S(2)(x + S(1))*(S(3)/2)/S(3) - S(2)sqrt(x + S(1)) + (S(1) - I)*(S(3)/2)atanh(sqrt(x + S(1))/sqrt(S(1) - I)) + (S(1) + I)*(S(3)/2)atanh(sqrt(x + S(1))/sqrt(S(1) + I)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt(x + S(1))(xS(3) + S(1))/(xS(2) + S(1)), x), x, S(2)(x + S(1))*(S(5)/2)/S(5) - S(2)(x + S(1))*(S(3)/2)/S(3) - S(2)sqrt(x + S(1)) - log(x - sqrt(S(2) + S(2)sqrt(S(2)))sqrt(x + S(1)) + S(1) + sqrt(S(2)))/(S(2)sqrt(S(1) + sqrt(S(2)))) + log(x + sqrt(S(2) + S(2)sqrt(S(2)))sqrt(x + S(1)) + S(1) + sqrt(S(2)))/(S(2)sqrt(S(1) + sqrt(S(2)))) - sqrt(S(1) + sqrt(S(2)))atan((-S(2)sqrt(x + S(1)) + sqrt(S(2) + S(2)sqrt(S(2))))/sqrt(S(-2) + S(2)sqrt(S(2)))) + sqrt(S(1) + sqrt(S(2)))atan((S(2)sqrt(x + S(1)) + sqrt(S(2) + S(2)sqrt(S(2))))/sqrt(S(-2) + S(2)sqrt(S(2)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-sqrt(x) + x + S(-1))/(sqrt(x)(x + S(-1))), x), x, atan((-sqrt(x) + S(3))/(S(2)sqrt(-sqrt(x) + x + S(-1)))) - S(2)atanh((-S(2)sqrt(x) + S(1))/(S(2)sqrt(-sqrt(x) + x + S(-1)))) - atanh((S(3)sqrt(x) + S(1))/(S(2)sqrt(-sqrt(x) + x + S(-1)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)sqrt(x + S(1)) + S(1))/(xsqrt(x + S(1))sqrt(x + sqrt(x + S(1)))), x), x, -atan((sqrt(x + S(1)) + S(3))/(S(2)sqrt(x + sqrt(x + S(1))))) + S(3)atanh((-S(3)sqrt(x + S(1)) + S(1))/(S(2)sqrt(x + sqrt(x + S(1))))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(x)sqrt(x + S(1))), x), x, S(2)asinh(sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x/(x + S(1)))/x, x), x, S(2)asinh(sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x)/sqrt(x + S(1)), x), x, sqrt(x)sqrt(x + S(1)) - asinh(sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x/(x + S(1))), x), x, sqrt(x)sqrt(x + S(1)) - asinh(sqrt(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x + S(-1))/(xS(2)sqrt(x + S(1))), x), x, atan(sqrt(x + S(-1))sqrt(x + S(1))) - sqrt(x + S(-1))sqrt(x + S(1))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((x + S(-1))/(x + S(1)))/x*S(2), x), x, atan(sqrt(x + S(-1))sqrt(x + S(1))) - sqrt(x + S(-1))sqrt(x + S(1))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(x + S(-1))/sqrt(x + S(1)), x), x, x*S(2)(x + S(-1))*(S(3)/2)sqrt(x + S(1))/S(4) + (-x/S(12) + S(7)/24)(x + S(-1))(S(3)/2)sqrt(x + S(1)) - S(3)sqrt(x + S(-1))sqrt(x + S(1))/S(8) + S(3)acosh(x)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt((x + S(-1))/(x + S(1))), x), x, x*S(2)(x + S(-1))*(S(3)/2)sqrt(x + S(1))/S(4) + (-x/S(12) + S(7)/24)(x + S(-1))(S(3)/2)sqrt(x + S(1)) - S(3)sqrt(x + S(-1))sqrt(x + S(1))/S(8) + S(3)acosh(x)/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-x/(x + S(1)))/x, x), x, S(2)atan(sqrt(-x/(x + S(1)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((-x + S(1))/(x + S(1)))/(x + S(-1)), x), x, S(2)atan(sqrt((-x + S(1))/(x + S(1)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((a + bx)/(-bx + c))/(a + bx), x), x, S(2)atan(sqrt((a + bx)/(-bx + c)))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((a + bx)/(c + dx))/(a + bx), x), x, S(2)atanh(sqrt(d)sqrt((a + bx)/(c + dx))/sqrt(b))/(sqrt(b)sqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-x/(x + S(1))), x), x, sqrt(-x/(x + S(1)))(x + S(1)) - atan(sqrt(-x/(x + S(1)))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt(-x/(x + S(1))), x), x, sqrt(-x/(x + S(1)))/(-x/(x + S(1)) + S(1)) - atan(sqrt(-x/(x + S(1)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((-x + S(1))/(x + S(1))), x), x, sqrt((-x + S(1))/(x + S(1)))(x + S(1)) - S(2)atan(sqrt((-x + S(1))/(x + S(1)))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt((-x + S(1))/(x + S(1))), x), x, S(2)sqrt((-x + S(1))/(x + S(1)))/((-x + S(1))/(x + S(1)) + S(1)) - S(2)atan(sqrt((-x + S(1))/(x + S(1)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((a + x)/(a - x)), x), x, S(2)aatan(sqrt((a + x)/(a - x))) - sqrt((a + x)/(a - x))(a - x), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt((a + x)/(a - x)), x), x, -S(2)asqrt((a + x)/(a - x))/(S(1) + (a + x)/(a - x)) + S(2)aatan(sqrt((a + x)/(a - x))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((-a + x)/(a + x)), x), x, -S(2)aatanh(sqrt(-(a - x)/(a + x))) + sqrt(-(a - x)/(a + x))(a + x), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt((-a + x)/(a + x)), x), x, S(2)asqrt(-(a - x)/(a + x))/((a - x)/(a + x) + S(1)) - S(2)aatanh(sqrt(-(a - x)/(a + x))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((a + bx)/(c + dx)), x), x, sqrt((a + bx)/(c + dx))(c + dx)/d - (-ad + bc)atanh(sqrt(d)sqrt((a + bx)/(c + dx))/sqrt(b))/(sqrt(b)d(S(3)/2)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt((a + bx)/(c + dx)), x), x, sqrt((a + bx)/(c + dx))(-ad + bc)/(d(b - d(a + bx)/(c + dx))) - (-ad + bc)atanh(sqrt(d)sqrt((a + bx)/(c + dx))/sqrt(b))/(sqrt(b)d(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((x + S(-1))/(S(3)x + S(5))), x), x, sqrt(x + S(-1))sqrt(S(3)x + S(5))/S(3) - S(8)sqrt(S(3))asinh(sqrt(S(6))sqrt(x + S(-1))/S(4))/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((S(5)x + S(-1))/(S(7)x + S(1)))/xS(2), x), x, -S(12)atan(sqrt(S(7)x + S(1))/sqrt(S(5)x + S(-1))) - sqrt(S(5)x + S(-1))sqrt(S(7)x + S(1))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt((-x + S(1))/(x + S(1)))(x + S(1))), x), x, -(-x + S(1))/sqrt((-x + S(1))/(x + S(1))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(x/(sqrt((-x + S(1))/(x + S(1)))(x + S(1))), x), x, -S(2)sqrt((-x + S(1))/(x + S(1)))/((-x + S(1))/(x + S(1)) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt(S(-1) + S(2)/(x + S(1)))(x + S(1))), x), x, -sqrt(S(-1) + S(2)/(x + S(1)))(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(sqrt((x + S(2))/(x + S(3)))(x + S(1))), x), x, sqrt(x + S(2))sqrt(x + S(3)) - asinh(sqrt(x + S(2))) + S(2)sqrt(S(2))atanh(sqrt(S(2))sqrt(x + S(2))/sqrt(x + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(1) + S(1)/x)/(x + S(1))S(2), x), x, S(2)/sqrt(S(1) + S(1)/x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(1) + S(1)/x)/sqrt(-xS(2) + S(1)), x), x, sqrt(x)sqrt(S(1) + S(1)/x)asin(S(2)x + S(-1))/sqrt(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*msqrt(a + bsqrt(c/x)), x), x, S(4)x*(m + S(1))(a + bsqrt(c/x))(S(3)/2)hyper((S(1), -S(2)m + S(-1)/2), (S(5)/2,), (a + bsqrt(c/x))/a)/(S(3)a), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xmsqrt(a + bsqrt(c/x)), x), x, S(4)b*S(2)cxm(-bsqrt(c/x)/a)(S(2)m)(a + bsqrt(c/x))*(S(3)/2)hyper((S(3)/2, S(2)m + S(3)), (S(5)/2,), S(1) + bsqrt(c/x)/a)/(S(3)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(a + bsqrt(c/x)), x), x, xS(2)sqrt(a + bsqrt(c/x))/S(2) + bc*S(2)sqrt(a + bsqrt(c/x))/(S(12)a(c/x)(S(3)/2)) - S(5)b*S(2)cxsqrt(a + bsqrt(c/x))/(S(48)a*S(2)) + S(5)b*S(3)c*S(2)sqrt(a + bsqrt(c/x))/(S(32)a*S(3)sqrt(c/x)) - S(5)bS(4)c*S(2)atanh(sqrt(a + bsqrt(c/x))/sqrt(a))/(S(32)a*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(c/x)), x), x, xsqrt(a + bsqrt(c/x)) + bcsqrt(a + bsqrt(c/x))/(S(2)asqrt(c/x)) - bS(2)catanh(sqrt(a + bsqrt(c/x))/sqrt(a))/(S(2)a(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(c/x))/x, x), x, S(4)sqrt(a)atanh(sqrt(a + bsqrt(c/x))/sqrt(a)) - S(4)sqrt(a + bsqrt(c/x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(c/x))/x*S(2), x), x, S(4)a(a + bsqrt(c/x))*(S(3)/2)/(S(3)b*S(2)c) - S(4)(a + bsqrt(c/x))*(S(5)/2)/(S(5)b*S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(c/x))/xS(3), x), x, S(4)a*S(3)(a + bsqrt(c/x))(S(3)/2)/(S(3)b*S(4)c*S(2)) - S(12)a*S(2)(a + bsqrt(c/x))(S(5)/2)/(S(5)b*S(4)c*S(2)) + S(12)a(a + bsqrt(c/x))*(S(7)/2)/(S(7)b*S(4)c*S(2)) - S(4)(a + bsqrt(c/x))(S(9)/2)/(S(9)b*S(4)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(c/x))/x*S(4), x), x, S(4)a*S(5)(a + bsqrt(c/x))(S(3)/2)/(S(3)b*S(6)c*S(3)) - S(4)a*S(4)(a + bsqrt(c/x))(S(5)/2)/(bS(6)c*S(3)) + S(40)a*S(3)(a + bsqrt(c/x))(S(7)/2)/(S(7)b*S(6)c*S(3)) - S(40)a*S(2)(a + bsqrt(c/x))(S(9)/2)/(S(9)b*S(6)c*S(3)) + S(20)a(a + bsqrt(c/x))*(S(11)/2)/(S(11)b*S(6)c*S(3)) - S(4)(a + bsqrt(c/x))(S(13)/2)/(S(13)b*S(6)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/sqrt(a + bsqrt(c/x)), x), x, S(4)x*(m + S(1))sqrt(a + bsqrt(c/x))hyper((S(1), -S(2)m + S(-3)/2), (S(3)/2,), (a + bsqrt(c/x))/a)/a, expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(x*m/sqrt(a + bsqrt(c/x)), x), x, S(4)bS(2)cxm(-bsqrt(c/x)/a)(S(2)m)sqrt(a + bsqrt(c/x))hyper((S(1)/2, S(2)m + S(3)), (S(3)/2,), S(1) + bsqrt(c/x)/a)/aS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + bsqrt(c/x)), x), x, x*S(2)sqrt(a + bsqrt(c/x))/(S(2)a) - S(7)bc*S(2)sqrt(a + bsqrt(c/x))/(S(12)a*S(2)(c/x)*(S(3)/2)) + S(35)b*S(2)cxsqrt(a + bsqrt(c/x))/(S(48)a*S(3)) - S(35)b*S(3)c*S(2)sqrt(a + bsqrt(c/x))/(S(32)a*S(4)sqrt(c/x)) + S(35)bS(4)c*S(2)atanh(sqrt(a + bsqrt(c/x))/sqrt(a))/(S(32)a*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + bsqrt(c/x)), x), x, xsqrt(a + bsqrt(c/x))/a - S(3)bcsqrt(a + bsqrt(c/x))/(S(2)aS(2)sqrt(c/x)) + S(3)bS(2)catanh(sqrt(a + bsqrt(c/x))/sqrt(a))/(S(2)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + bsqrt(c/x))), x), x, S(4)atanh(sqrt(a + bsqrt(c/x))/sqrt(a))/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(a + bsqrt(c/x))), x), x, S(4)asqrt(a + bsqrt(c/x))/(b*S(2)c) - S(4)(a + bsqrt(c/x))*(S(3)/2)/(S(3)b*S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(3)sqrt(a + bsqrt(c/x))), x), x, S(4)a*S(3)sqrt(a + bsqrt(c/x))/(bS(4)c*S(2)) - S(4)a*S(2)(a + bsqrt(c/x))(S(3)/2)/(bS(4)c*S(2)) + S(12)a(a + bsqrt(c/x))*(S(5)/2)/(S(5)b*S(4)c*S(2)) - S(4)(a + bsqrt(c/x))(S(7)/2)/(S(7)b*S(4)cS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)sqrt(a + bsqrt(c/x))), x), x, S(4)aS(5)sqrt(a + bsqrt(c/x))/(bS(6)c*S(3)) - S(20)a*S(4)(a + bsqrt(c/x))(S(3)/2)/(S(3)b*S(6)c*S(3)) + S(8)a*S(3)(a + bsqrt(c/x))(S(5)/2)/(bS(6)c*S(3)) - S(40)a*S(2)(a + bsqrt(c/x))(S(7)/2)/(S(7)b*S(6)c*S(3)) + S(20)a(a + bsqrt(c/x))*(S(9)/2)/(S(9)b*S(6)c*S(3)) - S(4)(a + bsqrt(c/x))(S(11)/2)/(S(11)b*S(6)c*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(sqrt(S(1)/x) + S(1)), x), x, xsqrt(sqrt(S(1)/x) + S(1)) - S(3)sqrt(sqrt(S(1)/x) + S(1))/(S(2)sqrt(S(1)/x)) + S(3)atanh(sqrt(sqrt(S(1)/x) + S(1)))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmsqrt(a + bsqrt(d/x) + c/x), x), x, x(m + S(1))sqrt(a + bsqrt(d/x) + c/x)AppellF1(-S(2)m + S(-2), S(-1)/2, S(-1)/2, -S(2)m + S(-1), -S(2)csqrt(d/x)/(sqrt(d)(bsqrt(d) - sqrt(-S(4)ac + b*S(2)d))), -S(2)csqrt(d/x)/(sqrt(d)(bsqrt(d) + sqrt(-S(4)ac + b*S(2)d))))/((m + S(1))sqrt(S(2)csqrt(d/x)/(sqrt(d)(bsqrt(d) - sqrt(-S(4)ac + bS(2)d))) + S(1))sqrt(S(2)csqrt(d/x)/(sqrt(d)(bsqrt(d) + sqrt(-S(4)ac + bS(2)d))) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)sqrt(a + bsqrt(d/x) + c/x), x), x, xS(3)(a + bsqrt(d/x) + c/x)(S(3)/2)/(S(3)a) - S(3)bd*S(3)(a + bsqrt(d/x) + c/x)(S(3)/2)/(S(10)a*S(2)(d/x)(S(5)/2)) - xS(2)(S(20)ac - S(21)b*S(2)d)(a + bsqrt(d/x) + c/x)*(S(3)/2)/(S(80)a*S(3)) + S(7)bdS(2)(S(28)ac - S(15)bS(2)d)(a + bsqrt(d/x) + c/x)*(S(3)/2)/(S(480)a*S(4)(d/x)*(S(3)/2)) + x(S(2)a + bsqrt(d/x))sqrt(a + bsqrt(d/x) + c/x)(S(16)a*S(2)c*S(2) - S(56)abS(2)cd + S(21)b*S(4)d*S(2))/(S(256)a*S(5)) + (S(4)ac - bS(2)d)(S(16)a*S(2)c*S(2) - S(56)abS(2)cd + S(21)b*S(4)d*S(2))atanh((S(2)a + bsqrt(d/x))/(S(2)sqrt(a)sqrt(a + bsqrt(d/x) + c/x)))/(S(512)a*(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(a + bsqrt(d/x) + c/x), x), x, xS(2)(a + bsqrt(d/x) + c/x)(S(3)/2)/(S(2)a) - S(5)bd*S(2)(a + bsqrt(d/x) + c/x)(S(3)/2)/(S(12)a*S(2)(d/x)*(S(3)/2)) - x(S(2)a + bsqrt(d/x))(S(4)ac - S(5)b*S(2)d)sqrt(a + bsqrt(d/x) + c/x)/(S(32)aS(3)) - (S(4)ac - S(5)b*S(2)d)(S(4)ac - bS(2)d)atanh((S(2)a + bsqrt(d/x))/(S(2)sqrt(a)sqrt(a + bsqrt(d/x) + c/x)))/(S(64)a(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(d/x) + c/x), x), x, x(S(2)a + bsqrt(d/x))sqrt(a + bsqrt(d/x) + c/x)/(S(2)a) + (S(4)ac - b*S(2)d)atanh((S(2)a + bsqrt(d/x))/(S(2)sqrt(a)sqrt(a + bsqrt(d/x) + c/x)))/(S(4)a(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(d/x) + c/x)/x, x), x, S(2)sqrt(a)atanh((S(2)a + bsqrt(d/x))/(S(2)sqrt(a)sqrt(a + bsqrt(d/x) + c/x))) - bsqrt(d)atanh((bd + S(2)csqrt(d/x))/(S(2)sqrt(c)sqrt(d)sqrt(a + bsqrt(d/x) + c/x)))/sqrt(c) - S(2)sqrt(a + bsqrt(d/x) + c/x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(d/x) + c/x)/xS(2), x), x, b(bd + S(2)csqrt(d/x))sqrt(a + bsqrt(d/x) + c/x)/(S(4)c*S(2)) + bsqrt(d)(S(4)ac - bS(2)d)atanh((bd + S(2)csqrt(d/x))/(S(2)sqrt(c)sqrt(d)sqrt(a + bsqrt(d/x) + c/x)))/(S(8)c(S(5)/2)) - S(2)(a + bsqrt(d/x) + c/x)(S(3)/2)/(S(3)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(d/x) + c/x)/xS(3), x), x, -b(S(12)ac - S(7)bS(2)d)(bd + S(2)csqrt(d/x))sqrt(a + bsqrt(d/x) + c/x)/(S(64)cS(4)) - bsqrt(d)(S(4)ac - bS(2)d)(S(12)ac - S(7)b*S(2)d)atanh((bd + S(2)csqrt(d/x))/(S(2)sqrt(c)sqrt(d)sqrt(a + bsqrt(d/x) + c/x)))/(S(128)c(S(9)/2)) - S(2)(a + bsqrt(d/x) + c/x)(S(3)/2)/(S(5)cx) + (a + bsqrt(d/x) + c/x)*(S(3)/2)(S(32)ac - S(35)bS(2)d + S(42)bcsqrt(d/x))/(S(120)c*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bsqrt(d/x) + c/x)/x*S(4), x), x, S(11)b(d/x)(S(3)/2)(a + bsqrt(d/x) + c/x)(S(3)/2)/(S(42)c*S(2)d) + b(bd + S(2)csqrt(d/x))sqrt(a + bsqrt(d/x) + c/x)(S(80)a*S(2)c*S(2) - S(120)abS(2)cd + S(33)b*S(4)d*S(2))/(S(512)c*S(6)) + bsqrt(d)(S(4)ac - bS(2)d)(S(80)a*S(2)c*S(2) - S(120)abS(2)cd + S(33)b*S(4)d*S(2))atanh((bd + S(2)csqrt(d/x))/(S(2)sqrt(c)sqrt(d)sqrt(a + bsqrt(d/x) + c/x)))/(S(1024)c*(S(13)/2)) - S(2)(a + bsqrt(d/x) + c/x)(S(3)/2)/(S(7)cxS(2)) + (S(32)ac - S(33)b*S(2)d)(a + bsqrt(d/x) + c/x)*(S(3)/2)/(S(140)c*S(3)x) - (a + bsqrt(d/x) + c/x)(S(3)/2)(S(1024)aS(2)c*S(2) - S(3276)abS(2)cd + S(1155)b*S(4)d*S(2) + S(18)bcsqrt(d/x)(S(148)ac - S(77)b*S(2)d))/(S(6720)cS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/sqrt(a + bsqrt(d/x) + c/x), x), x, x*(m + S(1))sqrt(S(2)csqrt(d/x)/(sqrt(d)(bsqrt(d) - sqrt(-S(4)ac + b*S(2)d))) + S(1))sqrt(S(2)csqrt(d/x)/(sqrt(d)(bsqrt(d) + sqrt(-S(4)ac + bS(2)d))) + S(1))AppellF1(-S(2)m + S(-2), S(1)/2, S(1)/2, -S(2)m + S(-1), -S(2)csqrt(d/x)/(sqrt(d)(bsqrt(d) - sqrt(-S(4)ac + bS(2)d))), -S(2)csqrt(d/x)/(sqrt(d)(bsqrt(d) + sqrt(-S(4)ac + b*S(2)d))))/((m + S(1))sqrt(a + bsqrt(d/x) + c/x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/sqrt(a + bsqrt(d/x) + c/x), x), x, x*S(3)sqrt(a + bsqrt(d/x) + c/x)/(S(3)a) - S(11)bd*S(3)sqrt(a + bsqrt(d/x) + c/x)/(S(30)a*S(2)(d/x)(S(5)/2)) - xS(2)(S(100)ac - S(99)b*S(2)d)sqrt(a + bsqrt(d/x) + c/x)/(S(240)aS(3)) + bd*S(2)(S(156)ac - S(77)bS(2)d)sqrt(a + bsqrt(d/x) + c/x)/(S(160)aS(4)(d/x)*(S(3)/2)) + xsqrt(a + bsqrt(d/x) + c/x)(S(400)aS(2)c*S(2) - S(1176)abS(2)cd + S(385)b*S(4)d*S(2))/(S(640)a*S(5)) - S(7)bdsqrt(a + bsqrt(d/x) + c/x)(S(528)aS(2)c*S(2) - S(680)abS(2)cd + S(165)b*S(4)d*S(2))/(S(1280)a*S(6)sqrt(d/x)) - (S(320)aS(3)c*S(3) - S(1680)a*S(2)b*S(2)c*S(2)d + S(1260)ab*S(4)cdS(2) - S(231)b*S(6)d*S(3))atanh((S(2)a + bsqrt(d/x))/(S(2)sqrt(a)sqrt(a + bsqrt(d/x) + c/x)))/(S(512)a*(S(13)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + bsqrt(d/x) + c/x), x), x, x*S(2)sqrt(a + bsqrt(d/x) + c/x)/(S(2)a) - S(7)bd*S(2)sqrt(a + bsqrt(d/x) + c/x)/(S(12)a*S(2)(d/x)*(S(3)/2)) - x(S(36)ac - S(35)bS(2)d)sqrt(a + bsqrt(d/x) + c/x)/(S(48)aS(3)) + S(5)bd(S(44)ac - S(21)bS(2)d)sqrt(a + bsqrt(d/x) + c/x)/(S(96)aS(4)sqrt(d/x)) + (S(48)aS(2)c*S(2) - S(120)abS(2)cd + S(35)b*S(4)d*S(2))atanh((S(2)a + bsqrt(d/x))/(S(2)sqrt(a)sqrt(a + bsqrt(d/x) + c/x)))/(S(64)a*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + bsqrt(d/x) + c/x), x), x, xsqrt(a + bsqrt(d/x) + c/x)/a - S(3)bdsqrt(a + bsqrt(d/x) + c/x)/(S(2)aS(2)sqrt(d/x)) - (S(4)ac - S(3)bS(2)d)atanh((S(2)a + bsqrt(d/x))/(S(2)sqrt(a)sqrt(a + bsqrt(d/x) + c/x)))/(S(4)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + bsqrt(d/x) + c/x)), x), x, S(2)atanh((S(2)a + bsqrt(d/x))/(S(2)sqrt(a)sqrt(a + bsqrt(d/x) + c/x)))/sqrt(a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(a + bsqrt(d/x) + c/x)), x), x, bsqrt(d)atanh((bd + S(2)csqrt(d/x))/(S(2)sqrt(c)sqrt(d)sqrt(a + bsqrt(d/x) + c/x)))/c*(S(3)/2) - S(2)sqrt(a + bsqrt(d/x) + c/x)/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(3)sqrt(a + bsqrt(d/x) + c/x)), x), x, -bsqrt(d)(S(12)ac - S(5)b*S(2)d)atanh((bd + S(2)csqrt(d/x))/(S(2)sqrt(c)sqrt(d)sqrt(a + bsqrt(d/x) + c/x)))/(S(8)c(S(7)/2)) - S(2)sqrt(a + bsqrt(d/x) + c/x)/(S(3)cx) + sqrt(a + bsqrt(d/x) + c/x)(S(16)ac - S(15)b*S(2)d + S(10)bcsqrt(d/x))/(S(12)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(4)sqrt(a + bsqrt(d/x) + c/x)), x), x, S(9)b(d/x)*(S(3)/2)sqrt(a + bsqrt(d/x) + c/x)/(S(20)c*S(2)d) + bsqrt(d)(S(240)aS(2)c*S(2) - S(280)abS(2)cd + S(63)b*S(4)d*S(2))atanh((bd + S(2)csqrt(d/x))/(S(2)sqrt(c)sqrt(d)sqrt(a + bsqrt(d/x) + c/x)))/(S(128)c*(S(11)/2)) - S(2)sqrt(a + bsqrt(d/x) + c/x)/(S(5)cxS(2)) + (S(64)ac - S(63)b*S(2)d)sqrt(a + bsqrt(d/x) + c/x)/(S(120)cS(3)x) - sqrt(a + bsqrt(d/x) + c/x)(S(1024)aS(2)c*S(2) - S(2940)abS(2)cd + S(945)b*S(4)d*S(2) + S(14)bcsqrt(d/x)(S(92)ac - S(45)b*S(2)d))/(S(960)cS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sqrt(S(1)/x) + S(1)/x), x), x, S(4)(sqrt(S(1)/x) + S(1)/x)*(S(3)/2)/(S(3)(S(1)/x)*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sqrt(S(1)/x) + S(2) + S(1)/x), x), x, x(sqrt(S(1)/x)/S(4) + S(1))sqrt(sqrt(S(1)/x) + S(2) + S(1)/x) + S(7)sqrt(S(2))atanh(sqrt(S(2))(sqrt(S(1)/x) + S(4))/(S(4)sqrt(sqrt(S(1)/x) + S(2) + S(1)/x)))/S(16), expand=True, _diff=True, _numerical=True) # difference in simplify assert rubi_test(rubi_integrate(S(1)/(x + sqrt(-xS(2) - S(2)x + S(3))), x), x, -log(-(-x - sqrt(S(3))sqrt(-xS(2) - S(2)x + S(3)) + S(3))/x*S(2))/S(2) + (-sqrt(S(7))/S(14) + S(1)/2)log(S(1) + sqrt(S(3)) + sqrt(S(7)) - sqrt(S(3))(-sqrt(-xS(2) - S(2)x + S(3)) + sqrt(S(3)))/x) + (sqrt(S(7))/S(14) + S(1)/2)log(-sqrt(S(7)) + S(1) + sqrt(S(3)) - sqrt(S(3))(-sqrt(-x*S(2) - S(2)x + S(3)) + sqrt(S(3)))/x) + atan((-sqrt(-x*S(2) - S(2)x + S(3)) + sqrt(S(3)))/x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(-x*S(2) - S(2)x + S(3)))*(S(-2)), x), x, (-S(2)sqrt(S(3)) + S(8) + S(2)(-S(3)sqrt(-x*S(2) - S(2)x + S(3)) + S(3)sqrt(S(3)))/x)/(-S(7)sqrt(S(3)) + S(14) - S(7)(S(2) + S(2)sqrt(S(3)))(-sqrt(-xS(2) - S(2)x + S(3)) + sqrt(S(3)))/x + S(7)sqrt(S(3))(-sqrt(-x*S(2) - S(2)x + S(3)) + sqrt(S(3)))S(2)/xS(2)) - S(8)sqrt(S(7))atanh(sqrt(S(7))(S(1) + sqrt(S(3)) - sqrt(S(3))(-sqrt(-x*S(2) - S(2)x + S(3)) + sqrt(S(3)))/x)/S(7))/S(49), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(-x*S(2) - S(2)x + S(3)))*(S(-3)), x), x, S(4)sqrt(S(3))(S(1) + sqrt(S(3)) - sqrt(S(3))(-sqrt(-x*S(2) - S(2)x + S(3)) + sqrt(S(3)))/x)/(-S(49)sqrt(S(3)) + S(98) - S(49)(S(2) + S(2)sqrt(S(3)))(-sqrt(-x*S(2) - S(2)x + S(3)) + sqrt(S(3)))/x + S(49)sqrt(S(3))(-sqrt(-x*S(2) - S(2)x + S(3)) + sqrt(S(3)))S(2)/xS(2)) - sqrt(S(3))(-S(2)sqrt(S(3)) + S(8) + S(2)(-S(7)sqrt(S(3)) + S(10))(-sqrt(-xS(2) - S(2)x + S(3)) + sqrt(S(3)))/x)/(S(21)(-sqrt(S(3)) + S(2) - (S(2) + S(2)sqrt(S(3)))(-sqrt(-xS(2) - S(2)x + S(3)) + sqrt(S(3)))/x + sqrt(S(3))(-sqrt(-xS(2) - S(2)x + S(3)) + sqrt(S(3)))S(2)/xS(2))*S(2)) - S(12)sqrt(S(7))atanh(sqrt(S(7))(S(1) + sqrt(S(3)) - sqrt(S(3))(-sqrt(-xS(2) - S(2)x + S(3)) + sqrt(S(3)))/x)/S(7))/S(343) + (S(6) + S(4)sqrt(S(3)))(-sqrt(-x*S(2) - S(2)x + S(3)) + sqrt(S(3)))*S(2)/(S(3)x*S(2)(-sqrt(S(3)) + S(2) - (S(2) + S(2)sqrt(S(3)))(-sqrt(-x*S(2) - S(2)x + S(3)) + sqrt(S(3)))/x + sqrt(S(3))(-sqrt(-xS(2) - S(2)x + S(3)) + sqrt(S(3)))S(2)/xS(2))*S(2)) - S(2)(-sqrt(-x*S(2) - S(2)x + S(3)) + sqrt(S(3)))S(3)/(xS(3)(-sqrt(S(3)) + S(2) - (S(2) + S(2)sqrt(S(3)))(-sqrt(-xS(2) - S(2)x + S(3)) + sqrt(S(3)))/x + sqrt(S(3))(-sqrt(-xS(2) - S(2)x + S(3)) + sqrt(S(3)))S(2)/xS(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x + sqrt(xS(2) - S(2)x + S(-3))), x), x, -S(3)log(x + sqrt(x*S(2) - S(2)x + S(-3)))/S(2) + S(2)log(-x - sqrt(xS(2) - S(2)x + S(-3)) + S(1)) - S(2)/(-x - sqrt(x*S(2) - S(2)x + S(-3)) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(x*S(2) - S(2)x + S(-3)))*(S(-2)), x), x, -S(4)log(x + sqrt(x*S(2) - S(2)x + S(-3))) + S(4)log(-x - sqrt(xS(2) - S(2)x + S(-3)) + S(1)) - S(2)/(-x - sqrt(x*S(2) - S(2)x + S(-3)) + S(1)) + S(3)/(S(2)x + S(2)sqrt(x*S(2) - S(2)x + S(-3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(x*S(2) - S(2)x + S(-3)))*(S(-3)), x), x, -S(6)log(x + sqrt(x*S(2) - S(2)x + S(-3))) + S(6)log(-x - sqrt(xS(2) - S(2)x + S(-3)) + S(1)) - S(2)/(-x - sqrt(x*S(2) - S(2)x + S(-3)) + S(1)) + S(4)/(x + sqrt(x*S(2) - S(2)x + S(-3))) + S(3)/(S(4)(x + sqrt(xS(2) - S(2)x + S(-3)))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x + sqrt(-xS(2) - S(4)x + S(-3))), x), x, log((xsqrt(-x + S(-1)) + xsqrt(x + S(3)) + S(3)sqrt(-x + S(-1)))/(x + S(3))*(S(3)/2))/S(2) - log(S(1)/(x + S(3)))/S(2) - sqrt(S(2))atan(sqrt(S(2))(-S(3)sqrt(-x + S(-1))/sqrt(x + S(3)) + S(1))/S(2)) - atan(sqrt(-x + S(-1))/sqrt(x + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(-x*S(2) - S(4)x + S(-3)))*(S(-2)), x), x, (-sqrt(-x + S(-1))/sqrt(x + S(3)) + S(1))/(-S(2)sqrt(-x + S(-1))/sqrt(x + S(3)) + S(1) - (S(3)x + S(3))/(x + S(3))) + sqrt(S(2))atan(sqrt(S(2))(-S(3)sqrt(-x + S(-1))/sqrt(x + S(3)) + S(1))/S(2))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x + sqrt(-x*S(2) - S(4)x + S(-3)))*(S(-3)), x), x, -(-S(9)sqrt(-x + S(-1))/sqrt(x + S(3)) + S(5))/(-S(18)sqrt(-x + S(-1))/sqrt(x + S(3)) + S(9) - S(9)(S(3)x + S(3))/(x + S(3))) - (-S(3)sqrt(-x + S(-1))/sqrt(x + S(3)) + S(1))/(-S(12)sqrt(-x + S(-1))/sqrt(x + S(3)) + S(6) - S(6)(S(3)x + S(3))/(x + S(3))) - (-S(2)sqrt(-x + S(-1))/sqrt(x + S(3)) + S(4))/(S(9)(-S(2)sqrt(-x + S(-1))/sqrt(x + S(3)) + S(1) - (S(3)x + S(3))/(x + S(3)))S(2)) - S(3)sqrt(S(2))atan(sqrt(S(2))(-S(3)sqrt(-x + S(-1))/sqrt(x + S(3)) + S(1))/S(2))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(x + S(1))*S(3)(S(2)x + S(1))sqrt(-x*S(4) - S(2)xS(3) - xS(2) + S(1)), x), x, -(-x*S(4) - S(2)xS(3) - xS(2) + S(1))*(S(3)/2)(S(3)xS(4) + S(6)x*S(3) + S(3)xS(2) + S(2))/S(15), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(3)(x + S(1))S(3)(S(2)x + S(1))sqrt(-x*S(4) - S(2)xS(3) - xS(2) + S(1)), x), x, -(-S(4)(x + S(1)/2)S(2) + S(1))S(2)(-S(16)(x + S(1)/2)S(4) + S(8)(x + S(1)/2)S(2) + S(15))(S(3)/2)/S(5120) - (-S(16)(x + S(1)/2)S(4) + S(8)(x + S(1)/2)S(2) + S(15))(S(3)/2)/S(480), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x + S(1))(xS(2) + x)S(3)sqrt(-(xS(2) + x)S(2) + S(1)), x), x, -(-xS(4) - S(2)xS(3) - xS(2) + S(1))*(S(3)/2)(S(3)xS(4) + S(6)x*S(3) + S(3)x*S(2) + S(2))/S(15), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((S(2)x + S(1))(xS(2) + x)S(3)sqrt(-(xS(2) + x)S(2) + S(1)), x), x, -(-S(4)(x + S(1)/2)S(2) + S(1))S(2)(-S(16)(x + S(1)/2)S(4) + S(8)(x + S(1)/2)S(2) + S(15))(S(3)/2)/S(5120) - (-S(16)(x + S(1)/2)S(4) + S(8)(x + S(1)/2)S(2) + S(15))(S(3)/2)/S(480), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x*S(3) + S(3)x*S(2) + S(3)x)/(x*S(4) + S(4)x*S(3) + S(6)x*S(2) + S(4)x + S(1)), x), x, log(x + S(1)) + S(1)/(S(3)(x + S(1))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(3) - S(3)x*S(2) + S(3)x + S(-1))/(x*S(4) + S(4)x*S(3) + S(6)x*S(2) + S(4)x + S(1)), x), x, log(x + S(1)) + S(6)/(x + S(1)) - S(6)/(x + S(1))*S(2) + S(8)/(S(3)(x + S(1))S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(4) + S(4)xS(3) - S(8)x*S(2) + S(8)x)*(S(3)/2), x), x, (x + S(-1))(-S(6)(x + S(-1))S(2)/S(35) + S(26)/35)sqrt(-(x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3)) + (x + S(-1))(-(x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))(S(3)/2)/S(7) - S(16)sqrt(S(3))elliptic_e(asin(x + S(-1)), S(-1)/3)/S(5) + S(176)sqrt(S(3))elliptic_f(asin(x + S(-1)), S(-1)/3)/S(35), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x), x), x, (x + S(-1))sqrt(-(x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))/S(3) - S(2)sqrt(S(3))elliptic_e(asin(x + S(-1)), S(-1)/3)/S(3) + S(4)sqrt(S(3))elliptic_f(asin(x + S(-1)), S(-1)/3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(-xS(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x), x), x, sqrt(S(3))elliptic_f(asin(x + S(-1)), S(-1)/3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)*(S(-3)/2), x), x, (x + S(-1))((x + S(-1))*S(2) + S(5))/(S(24)sqrt(-(x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3))) - sqrt(S(3))elliptic_e(asin(x + S(-1)), S(-1)/3)/S(24) + sqrt(S(3))elliptic_f(asin(x + S(-1)), S(-1)/3)/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((-xS(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)*(S(-5)/2), x), x, (x + S(-1))((x + S(-1))*S(2) + S(5))/(S(72)(-(x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))(S(3)/2)) + (x + S(-1))(S(7)(x + S(-1))*S(2) + S(26))/(S(432)sqrt(-(x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3))) - S(7)sqrt(S(3))elliptic_e(asin(x + S(-1)), S(-1)/3)/S(432) + S(11)sqrt(S(3))elliptic_f(asin(x + S(-1)), S(-1)/3)/S(432), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x(-x + S(2))(xS(2) - S(2)x + S(4)))*(S(3)/2), x), x, (x + S(-1))(-S(6)(x + S(-1))S(2)/S(35) + S(26)/35)sqrt(-(x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3)) + (x + S(-1))(-(x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))(S(3)/2)/S(7) - S(16)sqrt(S(3))elliptic_e(asin(x + S(-1)), S(-1)/3)/S(5) + S(176)sqrt(S(3))elliptic_f(asin(x + S(-1)), S(-1)/3)/S(35), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(x(-x + S(2))(x*S(2) - S(2)x + S(4))), x), x, (x + S(-1))sqrt(-(x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))/S(3) - S(2)sqrt(S(3))elliptic_e(asin(x + S(-1)), S(-1)/3)/S(3) + S(4)sqrt(S(3))elliptic_f(asin(x + S(-1)), S(-1)/3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(x(-x + S(2))(xS(2) - S(2)x + S(4))), x), x, sqrt(S(3))elliptic_f(asin(x + S(-1)), S(-1)/3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x(-x + S(2))(xS(2) - S(2)x + S(4)))*(S(-3)/2), x), x, (x + S(-1))((x + S(-1))*S(2) + S(5))/(S(24)sqrt(-(x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3))) - sqrt(S(3))elliptic_e(asin(x + S(-1)), S(-1)/3)/S(24) + sqrt(S(3))elliptic_f(asin(x + S(-1)), S(-1)/3)/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((x(-x + S(2))(xS(2) - S(2)x + S(4)))*(S(-5)/2), x), x, (x + S(-1))((x + S(-1))*S(2) + S(5))/(S(72)(-(x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))(S(3)/2)) + (x + S(-1))(S(7)(x + S(-1))*S(2) + S(26))/(S(432)sqrt(-(x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3))) - S(7)sqrt(S(3))elliptic_e(asin(x + S(-1)), S(-1)/3)/S(432) + S(11)sqrt(S(3))elliptic_f(asin(x + S(-1)), S(-1)/3)/S(432), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)ac + S(4)c*S(2)x*S(2) + S(4)cdxS(3) + dS(2)xS(4))S(4), x), x, -S(8)c*S(5)(S(4)adS(2) + cS(3))*S(3)(c/d + x)*S(3)/(S(3)d*S(6)) - S(8)c*S(4)(S(4)adS(2) + cS(3))(S(12)adS(2) + S(7)c*S(3))(c/d + x)*S(7)/(S(7)dS(2)) + cS(4)x(S(4)adS(2) + cS(3))S(4)/dS(8) - S(8)cS(3)d*S(2)(S(12)ad*S(2) + S(7)c*S(3))(c/d + x)*S(11)/S(11) + S(4)c*S(3)(S(4)adS(2) + cS(3))*S(2)(S(4)ad*S(2) + S(7)c*S(3))(c/d + x)*S(5)/(S(5)d*S(4)) - S(8)c*S(2)d*S(6)(c/d + x)*S(15)/S(15) + S(2)c*S(2)(c/d + x)*S(9)(S(48)aS(2)d*S(4) + S(120)acS(3)d*S(2) + S(35)c*S(6))/S(9) + S(4)cdS(4)(S(4)ad*S(2) + S(7)c*S(3))(c/d + x)S(13)/S(13) + dS(8)(c/d + x)S(17)/S(17), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)ac + S(4)c*S(2)x*S(2) + S(4)cdxS(3) + dS(2)xS(4))S(3), x), x, S(64)a*S(3)c*S(3)x + S(64)aS(2)c*S(4)x*S(3) + S(48)a*S(2)c*S(3)dxS(4) + S(64)acS(4)dxS(6) + S(48)acS(2)x*S(5)(adS(2) + S(4)c*S(3))/S(5) + S(16)c*S(3)d*S(3)x*S(10) + S(32)c*S(3)x*S(7)(S(9)ad*S(2) + S(2)c*S(3))/S(7) + S(60)c*S(2)d*S(4)x*S(11)/S(11) + S(12)c*S(2)dxS(8)(adS(2) + S(2)c*S(3)) + cd*S(5)x*S(12) + S(4)cdS(2)x*S(9)(adS(2) + S(20)cS(3))/S(3) + dS(6)xS(13)/S(13), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)ac + S(4)c*S(2)x*S(2) + S(4)cdxS(3) + dS(2)xS(4))S(2), x), x, S(16)a*S(2)c*S(2)x + S(32)ac*S(3)x*S(3)/S(3) + S(8)acS(2)dxS(4) + S(16)c*S(3)dxS(6)/S(3) + S(24)c*S(2)d*S(2)x*S(7)/S(7) + cd*S(3)x*S(8) + S(8)cxS(5)(adS(2) + S(2)cS(3))/S(5) + dS(4)xS(9)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(4)ac + S(4)c*S(2)x*S(2) + S(4)cdxS(3) + dS(2)xS(4), x), x, S(4)acx + S(4)cS(2)x*S(3)/S(3) + cdxS(4) + dS(2)x*S(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(4)ac + S(4)c*S(2)x*S(2) + S(4)cdxS(3) + dS(2)xS(4)), x), x, -atanh(d(c/d + x)/(c*(S(1)/4)sqrt(c*(S(3)/2) + S(2)dsqrt(-a))))/(S(4)c*(S(3)/4)sqrt(-a)sqrt(c(S(3)/2) + S(2)dsqrt(-a))) + atanh(d(c/d + x)/(c*(S(1)/4)sqrt(c*(S(3)/2) - S(2)dsqrt(-a))))/(S(4)c*(S(3)/4)sqrt(-a)sqrt(c(S(3)/2) - S(2)dsqrt(-a))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)ac + S(4)c*S(2)x*S(2) + S(4)cdxS(3) + dS(2)xS(4))(S(-2)), x), x, (S(6)adS(2) + c(S(3)/2)dsqrt(-a) + cS(3))atanh(d(c/d + x)/(c(S(1)/4)sqrt(c*(S(3)/2) + S(2)dsqrt(-a))))/(S(32)c*(S(7)/4)(-a)*(S(3)/2)sqrt(c*(S(3)/2) + S(2)dsqrt(-a))(S(4)adS(2) + cS(3))) - (S(6)adS(2) - c(S(3)/2)dsqrt(-a) + c*S(3))atanh(d(c/d + x)/(c(S(1)/4)sqrt(c*(S(3)/2) - S(2)dsqrt(-a))))/(S(32)c*(S(7)/4)(-a)*(S(3)/2)sqrt(c*(S(3)/2) - S(2)dsqrt(-a))(S(4)adS(2) + cS(3))) - (c/d + x)(-S(4)adS(2) + cS(3) - cd*S(2)(c/d + x)*S(2))/(S(16)ac(S(4)adS(2) + cS(3))(-S(2)c*S(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)ac + S(4)c*S(2)x*S(2) + S(4)cdxS(3) + dS(2)xS(4))(S(3)/2), x), x, S(16)c*(S(13)/4)sqrt((-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))/((S(4)a + cS(3)/dS(2))(sqrt(c) + dS(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))S(2)))(sqrt(c) + d*S(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))(S(4)adS(2) + cS(3))*(S(3)/4)(S(8)adS(2) + cS(3))elliptic_e(S(2)atan(d(c/d + x)/(c(S(1)/4)(S(4)adS(2) + cS(3))(S(1)/4))), c(S(3)/2)/(S(2)sqrt(S(4)adS(2) + cS(3))) + S(1)/2)/(S(35)d*S(5)sqrt(-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))) + S(8)c*(S(7)/4)sqrt((-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))/((S(4)a + cS(3)/dS(2))(sqrt(c) + dS(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))S(2)))(sqrt(c) + d*S(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))(S(4)adS(2) + cS(3))*(S(3)/4)(-c*(S(3)/2)(S(8)adS(2) + cS(3)) + sqrt(S(4)adS(2) + cS(3))(S(5)adS(2) + cS(3)))elliptic_f(S(2)atan(d(c/d + x)/(c*(S(1)/4)(S(4)adS(2) + cS(3))(S(1)/4))), c(S(3)/2)/(S(2)sqrt(S(4)adS(2) + cS(3))) + S(1)/2)/(S(35)d*S(5)sqrt(-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))) - S(16)c*S(3)(S(8)adS(2) + cS(3))(c/d + x)sqrt(-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))/(S(35)d*S(2)(sqrt(c) + d*S(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))sqrt(S(4)adS(2) + cS(3))) + S(2)c(c/d + x)(S(20)adS(2) + S(7)c*S(3) - S(3)cdS(2)(c/d + x)*S(2))sqrt(-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))/(S(35)d*S(2)) + (c/(S(7)d) + x/S(7))(-S(2)c*S(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)S(4))(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(4)ac + S(4)cS(2)x*S(2) + S(4)cdxS(3) + dS(2)xS(4)), x), x, S(2)c*(S(9)/4)sqrt((-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))/((S(4)a + cS(3)/dS(2))(sqrt(c) + dS(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))S(2)))(sqrt(c) + d*S(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))(S(4)adS(2) + cS(3))*(S(3)/4)elliptic_e(S(2)atan(d(c/d + x)/(c*(S(1)/4)(S(4)adS(2) + cS(3))(S(1)/4))), c(S(3)/2)/(S(2)sqrt(S(4)adS(2) + cS(3))) + S(1)/2)/(S(3)d*S(3)sqrt(-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)S(4))) + c(S(3)/4)sqrt((-S(2)c*S(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))/((S(4)a + cS(3)/dS(2))(sqrt(c) + dS(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))S(2)))(sqrt(c) + d*S(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))(S(4)adS(2) + cS(3))*(S(1)/4)(S(4)adS(2) - c(S(3)/2)sqrt(S(4)adS(2) + cS(3)) + cS(3))elliptic_f(S(2)atan(d(c/d + x)/(c*(S(1)/4)(S(4)adS(2) + cS(3))(S(1)/4))), c(S(3)/2)/(S(2)sqrt(S(4)adS(2) + cS(3))) + S(1)/2)/(S(3)d*S(3)sqrt(-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))) - S(2)c*S(2)(c/d + x)sqrt(-S(2)c*S(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))/(S(3)(sqrt(c) + d*S(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))sqrt(S(4)adS(2) + cS(3))) + (c/(S(3)d) + x/S(3))sqrt(-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(4)ac + S(4)c*S(2)x*S(2) + S(4)cdxS(3) + dS(2)xS(4)), x), x, sqrt((-S(2)c*S(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))/((S(4)a + cS(3)/dS(2))(sqrt(c) + dS(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))S(2)))(sqrt(c) + d*S(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))(S(4)adS(2) + cS(3))*(S(1)/4)elliptic_f(S(2)atan(d(c/d + x)/(c*(S(1)/4)(S(4)adS(2) + cS(3))(S(1)/4))), c(S(3)/2)/(S(2)sqrt(S(4)adS(2) + cS(3))) + S(1)/2)/(S(2)c*(S(1)/4)dsqrt(-S(2)c*S(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)ac + S(4)c*S(2)x*S(2) + S(4)cdxS(3) + dS(2)xS(4))(S(-3)/2), x), x, c(S(1)/4)sqrt((-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))/((S(4)a + cS(3)/dS(2))(sqrt(c) + dS(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))S(2)))(sqrt(c) + d*S(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))elliptic_e(S(2)atan(d(c/d + x)/(c*(S(1)/4)(S(4)adS(2) + cS(3))(S(1)/4))), c(S(3)/2)/(S(2)sqrt(S(4)adS(2) + cS(3))) + S(1)/2)/(S(8)ad(S(4)adS(2) + cS(3))*(S(1)/4)sqrt(-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)S(4))) - dS(2)(c/d + x)sqrt(-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))/(S(8)a(sqrt(c) + dS(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))(S(4)adS(2) + cS(3))*(S(3)/2)) - (c/d + x)(-S(4)adS(2) + cS(3) - cdS(2)(c/d + x)*S(2))/(S(8)ac(S(4)adS(2) + cS(3))sqrt(-S(2)c*S(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))) + sqrt((-S(2)c*S(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))/((S(4)a + cS(3)/dS(2))(sqrt(c) + dS(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))S(2)))(sqrt(c) + d*S(2)(c/d + x)*S(2)/sqrt(S(4)adS(2) + cS(3)))(S(4)adS(2) - c(S(3)/2)sqrt(S(4)adS(2) + cS(3)) + cS(3))elliptic_f(S(2)atan(d(c/d + x)/(c*(S(1)/4)(S(4)adS(2) + cS(3))(S(1)/4))), c(S(3)/2)/(S(2)sqrt(S(4)adS(2) + cS(3))) + S(1)/2)/(S(16)ac(S(5)/4)d(S(4)adS(2) + cS(3))(S(3)/4)sqrt(-S(2)cS(2)(c/d + x)*S(2) + c(S(4)a + cS(3)/dS(2)) + dS(2)(c/d + x)*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)aeS(2) - dS(3)x + S(8)de*S(2)x*S(3) + S(8)e*S(3)xS(4))S(4), x), x, -S(2048)dS(2)e*S(10)(d/(S(4)e) + x)S(15)/S(5) - S(72)d*S(2)e*S(6)(S(256)ae*S(3) + S(17)d*S(4))(d/(S(4)e) + x)S(11)/S(11) - S(9)d*S(2)e*S(2)(d/(S(4)e) + x)S(7)(S(65536)aS(2)e*S(6) + S(5632)adS(4)e*S(3) + S(85)dS(8))/S(224) - dS(2)(S(256)aeS(3) + S(5)dS(4))S(3)(d/(S(4)e) + x)*S(3)/(S(8192)e*S(2)) + S(4096)e*S(12)(d/(S(4)e) + x)S(17)/S(17) + S(64)e*S(8)(S(256)ae*S(3) + S(59)d*S(4))(d/(S(4)e) + x)S(13)/S(13) + eS(4)(d/(S(4)e) + x)S(9)(S(65536)aS(2)e*S(6) + S(20992)adS(4)e*S(3) + S(601)d*S(8))/S(24) + (S(256)aeS(3) + S(5)dS(4))S(2)(S(256)aeS(3) + S(59)d*S(4))(d/(S(4)e) + x)S(5)/S(5120) + x(S(256)ae*S(3) + S(5)dS(4))S(4)/(S(1048576)eS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)aeS(2) - dS(3)x + S(8)de*S(2)x*S(3) + S(8)e*S(3)xS(4))S(3), x), x, S(512)aS(3)e*S(6)x - S(96)aS(2)d*S(3)e*S(4)x*S(2) + S(8)adS(6)e*S(2)x*S(3) - S(384)aeS(4)x*S(5)(-S(4)aeS(3) + dS(4))/S(5) + S(32)dS(3)e*S(6)x*S(10) + S(4)d*S(3)e*S(2)x*S(6)(-S(16)aeS(3) + dS(4)) + S(1536)dS(2)e*S(7)x*S(11)/S(11) + S(24)d*S(2)e*S(3)x*S(7)(S(64)aeS(3) + dS(4))/S(7) + S(128)de*S(8)x*S(12) - S(24)deS(4)x*S(8)(-S(16)aeS(3) + dS(4)) - dxS(4)(-S(1536)aS(2)eS(6) + dS(8))/S(4) + S(512)eS(9)x*S(13)/S(13) - S(128)e*S(5)x*S(9)(-S(4)aeS(3) + dS(4))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)aeS(2) - dS(3)x + S(8)deS(2)x*S(3) + S(8)e*S(3)xS(4))S(2), x), x, S(64)aS(2)e*S(4)x - S(8)ad*S(3)e*S(2)x*S(2) + S(32)ade*S(4)xS(4) + dS(6)xS(3)/S(3) - S(8)d*S(3)e*S(3)x*S(6)/S(3) + S(64)d*S(2)e*S(4)x*S(7)/S(7) + S(16)deS(5)x*S(8) + S(64)e*S(6)x*S(9)/S(9) - S(16)e*S(2)x*S(5)(-S(8)aeS(3) + dS(4))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(8)aeS(2) - dS(3)x + S(8)deS(2)x*S(3) + S(8)e*S(3)x*S(4), x), x, S(8)aeS(2)x - d*S(3)x*S(2)/S(2) + S(2)deS(2)x*S(4) + S(8)e*S(3)x*S(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(8)aeS(2) - dS(3)x + S(8)de*S(2)x*S(3) + S(8)e*S(3)x*S(4)), x), x, -S(2)atanh(S(4)e(d/(S(4)e) + x)/sqrt(S(3)d*S(2) + S(2)sqrt(-S(64)aeS(3) + dS(4))))/(sqrt(S(3)dS(2) + S(2)sqrt(-S(64)aeS(3) + dS(4)))sqrt(-S(64)aeS(3) + dS(4))) + S(2)atanh(S(4)e(d/(S(4)e) + x)/sqrt(S(3)d*S(2) - S(2)sqrt(-S(64)aeS(3) + dS(4))))/(sqrt(S(3)dS(2) - S(2)sqrt(-S(64)aeS(3) + dS(4)))sqrt(-S(64)aeS(3) + dS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)aeS(2) - dS(3)x + S(8)de*S(2)x*S(3) + S(8)e*S(3)xS(4))(S(-2)), x), x, S(64)e(d/(S(4)e) + x)(-S(256)ae*S(3) + S(13)d*S(4) - S(48)d*S(2)e*S(2)(d/(S(4)e) + x)S(2))/((-S(16384)a*S(2)e*S(6) - S(64)adS(4)e*S(3) + S(5)d*S(8))(S(256)ae*S(2) + S(5)d*S(4)/e - S(96)d*S(2)e(d/(S(4)e) + x)*S(2) + S(256)e*S(3)(d/(S(4)e) + x)S(4))) + S(24)e(S(128)aeS(3) + dS(4) + dS(2)sqrt(-S(64)aeS(3) + dS(4)))atanh(S(4)e(d/(S(4)e) + x)/sqrt(S(3)dS(2) + S(2)sqrt(-S(64)aeS(3) + dS(4))))/(sqrt(S(3)dS(2) + S(2)sqrt(-S(64)aeS(3) + dS(4)))(-S(64)aeS(3) + dS(4))(S(3)/2)(S(256)ae*S(3) + S(5)d*S(4))) - S(24)e(S(128)aeS(3) + dS(4) - dS(2)sqrt(-S(64)aeS(3) + dS(4)))atanh(S(4)e(d/(S(4)e) + x)/sqrt(S(3)dS(2) - S(2)sqrt(-S(64)aeS(3) + dS(4))))/(sqrt(S(3)dS(2) - S(2)sqrt(-S(64)aeS(3) + dS(4)))(-S(64)aeS(3) + dS(4))(S(3)/2)(S(256)ae*S(3) + S(5)d*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(8)aeS(2) - dS(3)x + S(8)de*S(2)x*S(3) + S(8)e*S(3)x*S(4)), x), x, -sqrt(S(2))d*S(2)(d/(S(4)e) + x)sqrt(S(256)ae*S(2) + S(5)d*S(4)/e - S(96)d*S(2)e(d/(S(4)e) + x)*S(2) + S(256)e*S(3)(d/(S(4)e) + x)S(4))/(S(4)sqrt(S(256)ae*S(3) + S(5)d*S(4))(S(16)eS(2)(d/(S(4)e) + x)S(2)/sqrt(S(256)aeS(3) + S(5)d*S(4)) + S(1))) + sqrt(S(2))d*S(2)sqrt((S(256)ae*S(3) + S(5)d*S(4) - S(96)d*S(2)e*S(2)(d/(S(4)e) + x)S(2) + S(256)e*S(4)(d/(S(4)e) + x)S(4))/((S(256)aeS(3) + S(5)d*S(4))(S(16)eS(2)(d/(S(4)e) + x)S(2)/sqrt(S(256)aeS(3) + S(5)dS(4)) + S(1))S(2)))(S(256)aeS(3) + S(5)dS(4))(S(3)/4)(S(16)e*S(2)(d/(S(4)e) + x)S(2)/sqrt(S(256)aeS(3) + S(5)d*S(4)) + S(1))elliptic_e(S(2)atan(S(4)e(d/(S(4)e) + x)/(S(256)ae*S(3) + S(5)dS(4))(S(1)/4)), S(3)dS(2)/(S(2)sqrt(S(256)ae*S(3) + S(5)d*S(4))) + S(1)/2)/(S(16)e*S(2)sqrt(S(256)ae*S(2) + S(5)d*S(4)/e - S(96)d*S(2)e(d/(S(4)e) + x)*S(2) + S(256)e*S(3)(d/(S(4)e) + x)S(4))) + sqrt(S(2))(d/(S(4)e) + x)sqrt(S(256)ae*S(2) + S(5)d*S(4)/e - S(96)d*S(2)e(d/(S(4)e) + x)*S(2) + S(256)e*S(3)(d/(S(4)e) + x)S(4))/S(24) + sqrt(S(2))sqrt((S(256)ae*S(3) + S(5)d*S(4) - S(96)d*S(2)e*S(2)(d/(S(4)e) + x)S(2) + S(256)e*S(4)(d/(S(4)e) + x)S(4))/((S(256)aeS(3) + S(5)d*S(4))(S(16)eS(2)(d/(S(4)e) + x)S(2)/sqrt(S(256)aeS(3) + S(5)dS(4)) + S(1))S(2)))(S(256)aeS(3) + S(5)dS(4))(S(1)/4)(S(16)e*S(2)(d/(S(4)e) + x)S(2)/sqrt(S(256)aeS(3) + S(5)d*S(4)) + S(1))(S(256)ae*S(3) + S(5)d*S(4) - S(3)d*S(2)sqrt(S(256)ae*S(3) + S(5)d*S(4)))elliptic_f(S(2)atan(S(4)e(d/(S(4)e) + x)/(S(256)ae*S(3) + S(5)dS(4))(S(1)/4)), S(3)dS(2)/(S(2)sqrt(S(256)ae*S(3) + S(5)d*S(4))) + S(1)/2)/(S(96)e*S(2)sqrt(S(256)ae*S(2) + S(5)d*S(4)/e - S(96)d*S(2)e(d/(S(4)e) + x)*S(2) + S(256)e*S(3)(d/(S(4)e) + x)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(8)aeS(2) - dS(3)x + S(8)de*S(2)x*S(3) + S(8)e*S(3)x*S(4)), x), x, sqrt(S(2))sqrt((S(256)ae*S(3) + S(5)d*S(4) - S(96)d*S(2)e*S(2)(d/(S(4)e) + x)S(2) + S(256)e*S(4)(d/(S(4)e) + x)S(4))/((S(256)aeS(3) + S(5)d*S(4))(S(16)eS(2)(d/(S(4)e) + x)S(2)/sqrt(S(256)aeS(3) + S(5)dS(4)) + S(1))S(2)))(S(256)aeS(3) + S(5)dS(4))(S(1)/4)(S(16)e*S(2)(d/(S(4)e) + x)S(2)/sqrt(S(256)aeS(3) + S(5)d*S(4)) + S(1))elliptic_f(S(2)atan(S(4)e(d/(S(4)e) + x)/(S(256)ae*S(3) + S(5)dS(4))(S(1)/4)), S(3)dS(2)/(S(2)sqrt(S(256)ae*S(3) + S(5)d*S(4))) + S(1)/2)/(S(2)esqrt(S(256)aeS(2) + S(5)d*S(4)/e - S(96)d*S(2)e(d/(S(4)e) + x)*S(2) + S(256)e*S(3)(d/(S(4)e) + x)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a - xS(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)*S(4), x), x, x(a + S(3))*S(4) + (-S(4)a/S(5) + S(12)/5)(a + S(3))S(2)(x + S(-1))*S(5) + (-S(4)a/S(13) + S(12)/13)(x + S(-1))S(13) - S(8)(a + S(3))*S(3)(x + S(-1))*S(3)/S(3) + (S(8)a/S(7) + S(24)/7)(S(3)a + S(5))(x + S(-1))S(7) - (S(24)a/S(11) + S(40)/11)(x + S(-1))S(11) + (x + S(-1))S(17)/S(17) + S(8)(x + S(-1))S(15)/S(15) - (x + S(-1))S(9)(-S(2)a*S(2)/S(3) + S(4)a/S(3) + S(74)/9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)S(3), x), x, aS(3)x + S(12)a*S(2)x*S(2) + ax*S(3)(-S(8)a + S(64)) - xS(13)/S(13) + xS(12) - S(72)x*S(11)/S(11) + S(28)xS(10) - xS(9)(-a/S(3) + S(256)/3) + xS(8)(-S(3)a + S(192)) - xS(7)(-S(96)a/S(7) + S(320)) + xS(6)(-S(40)a + S(384)) - xS(5)(S(3)aS(2)/S(5) - S(384)a/S(5) + S(1536)/5) + x*S(4)(S(3)aS(2) - S(96)a + S(128)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)S(2), x), x, aS(2)x + S(8)axS(2) + xS(9)/S(9) - xS(8) + S(32)x*S(7)/S(7) - S(40)xS(6)/S(3) + xS(5)(-S(2)a/S(5) + S(128)/5) - x*S(4)(-S(2)a + S(32)) + xS(3)(-S(16)a/S(3) + S(64)/3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a - xS(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x, x), x, ax - xS(5)/S(5) + xS(4) - S(8)x*S(3)/S(3) + S(4)xS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a - xS(4) + S(4)xS(3) - S(8)x*S(2) + S(8)x), x), x, atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1)))/(S(2)sqrt(a + S(4))sqrt(sqrt(a + S(4)) + S(1))) - atan((x + S(-1))/sqrt(-sqrt(a + S(4)) + S(1)))/(S(2)sqrt(a + S(4))sqrt(-sqrt(a + S(4)) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)*(S(-2)), x), x, (x + S(-1))(a + (x + S(-1))*S(2) + S(5))/((S(4)a*S(2) + S(28)a + S(48))(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))) + (S(3)a - sqrt(a + S(4)) + S(10))atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1)))/((a + S(4))(S(3)/2)(S(8)a + S(24))sqrt(sqrt(a + S(4)) + S(1))) - (S(3)a + sqrt(a + S(4)) + S(10))atan((x + S(-1))/sqrt(-sqrt(a + S(4)) + S(1)))/((a + S(4))*(S(3)/2)(S(8)a + S(24))sqrt(-sqrt(a + S(4)) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a - xS(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)S(4), x), x, aS(4)xS(2)/S(2) + S(32)a*S(3)xS(3)/S(3) + aS(2)xS(4)(-S(8)a + S(96)) + S(16)axS(5)(a*S(2) - S(48)a + S(128))/S(5) + x*S(18)/S(18) - S(16)x*S(17)/S(17) + S(8)x*S(16) - S(224)xS(15)/S(5) + xS(14)(-S(2)a/S(7) + S(1280)/7) - x*S(13)(-S(48)a/S(13) + S(7424)/13) + xS(12)(-S(24)a + S(4192)/3) - xS(11)(-S(1120)a/S(11) + S(29696)/11) + xS(10)(S(3)aS(2)/S(5) - S(1536)a/S(5) + S(4096)) - x*S(9)(S(16)aS(2)/S(3) - S(2048)a/S(3) + S(14336)/3) + x*S(8)(-S(24)a + S(1024))(-a + S(4)) - x*S(7)(S(480)aS(2)/S(7) - S(9216)a/S(7) + S(16384)/7) + x*S(6)(-S(2)aS(3)/S(3) + S(128)a*S(2) - S(1024)a + S(2048)/3), expand=True, _diff=True, _numerical=True) # long time in rubi_int assert rubi_test(rubi_integrate(x(a - xS(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)S(3), x), x, aS(3)xS(2)/S(2) + S(8)a*S(2)x*S(3) + ax*S(4)(-S(6)a + S(48)) - xS(14)/S(14) + S(12)x*S(13)/S(13) - S(6)x*S(12) + S(280)xS(11)/S(11) - xS(10)(-S(3)a/S(10) + S(384)/5) + x*S(9)(-S(8)a/S(3) + S(512)/3) - xS(8)(-S(12)a + S(280)) + xS(7)(-S(240)a/S(7) + S(2304)/7) - xS(6)(a*S(2)/S(2) - S(64)a + S(256)) + x*S(5)(S(12)aS(2)/S(5) - S(384)a/S(5) + S(512)/5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a - xS(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)S(2), x), x, aS(2)xS(2)/S(2) + S(16)axS(3)/S(3) + xS(10)/S(10) - S(8)x*S(9)/S(9) + S(4)x*S(8) - S(80)xS(7)/S(7) + xS(6)(-a/S(3) + S(64)/3) - xS(5)(-S(8)a/S(5) + S(128)/5) + xS(4)(-S(4)a + S(16)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x), x), x, axS(2)/S(2) - xS(6)/S(6) + S(4)x*S(5)/S(5) - S(2)x*S(4) + S(8)xS(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a - xS(4) + S(4)xS(3) - S(8)x*S(2) + S(8)x), x), x, atanh(((x + S(-1))*S(2) + S(1))/sqrt(a + S(4)))/(S(2)sqrt(a + S(4))) + atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1)))/(S(2)sqrt(a + S(4))sqrt(sqrt(a + S(4)) + S(1))) - atan((x + S(-1))/sqrt(-sqrt(a + S(4)) + S(1)))/(S(2)sqrt(a + S(4))sqrt(-sqrt(a + S(4)) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)*S(2), x), x, (x + S(-1))(a + (a + S(5))(x + S(-1)) + (x + S(-1))S(3) + (x + S(-1))S(2) + S(5))/((S(4)a*S(2) + S(28)a + S(48))(a - (x + S(-1))S(4) - S(2)(x + S(-1))S(2) + S(3))) + atanh(((x + S(-1))S(2) + S(1))/sqrt(a + S(4)))/(S(4)(a + S(4))(S(3)/2)) + (S(3)a - sqrt(a + S(4)) + S(10))atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1)))/((a + S(4))(S(3)/2)(S(8)a + S(24))sqrt(sqrt(a + S(4)) + S(1))) - (S(3)a + sqrt(a + S(4)) + S(10))atan((x + S(-1))/sqrt(-sqrt(a + S(4)) + S(1)))/((a + S(4))*(S(3)/2)(S(8)a + S(24))sqrt(-sqrt(a + S(4)) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)S(4), x), x, aS(4)xS(3)/S(3) + S(8)a*S(3)xS(4) + aS(2)xS(5)(-S(32)a/S(5) + S(384)/5) + S(8)axS(6)(a*S(2) - S(48)a + S(128))/S(3) + x*S(19)/S(19) - S(8)x*S(18)/S(9) + S(128)x*S(17)/S(17) - S(42)xS(16) + xS(15)(-S(4)a/S(15) + S(512)/3) - x*S(14)(-S(24)a/S(7) + S(3712)/7) + xS(13)(-S(288)a/S(13) + S(16768)/13) - xS(12)(-S(280)a/S(3) + S(7424)/3) + xS(11)(S(6)aS(2)/S(11) - S(3072)a/S(11) + S(40960)/11) - x*S(10)(S(24)aS(2)/S(5) - S(3072)a/S(5) + S(21504)/5) + x*S(9)(-S(64)a/S(3) + S(8192)/9)(-a + S(4)) - x*S(8)(S(60)aS(2) - S(1152)a + S(2048)) + x*S(7)(-S(4)aS(3)/S(7) + S(768)a*S(2)/S(7) - S(6144)a/S(7) + S(4096)/7), expand=True, _diff=True, _numerical=True) # long time in rubi_int assert rubi_test(rubi_integrate(x*S(2)(a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)S(3), x), x, aS(3)xS(3)/S(3) + S(6)a*S(2)x*S(4) + ax*S(5)(-S(24)a/S(5) + S(192)/5) - xS(15)/S(15) + S(6)x*S(14)/S(7) - S(72)x*S(13)/S(13) + S(70)xS(12)/S(3) - xS(11)(-S(3)a/S(11) + S(768)/11) + x*S(10)(-S(12)a/S(5) + S(768)/5) - xS(9)(-S(32)a/S(3) + S(2240)/9) + xS(8)(-S(30)a + S(288)) - xS(7)(S(3)aS(2)/S(7) - S(384)a/S(7) + S(1536)/7) + x*S(6)(S(2)aS(2) - S(64)a + S(256)/3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)S(2), x), x, aS(2)xS(3)/S(3) + S(4)axS(4) + xS(11)/S(11) - S(4)x*S(10)/S(5) + S(32)x*S(9)/S(9) - S(10)xS(8) + xS(7)(-S(2)a/S(7) + S(128)/7) - x*S(6)(-S(4)a/S(3) + S(64)/3) + xS(5)(-S(16)a/S(5) + S(64)/5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x), x), x, axS(3)/S(3) - xS(7)/S(7) + S(2)x*S(6)/S(3) - S(8)x*S(5)/S(5) + S(2)xS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x), x), x, -atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1)))/(S(2)sqrt(sqrt(a + S(4)) + S(1))) - atan((x + S(-1))/sqrt(-sqrt(a + S(4)) + S(1)))/(S(2)sqrt(-sqrt(a + S(4)) + S(1))) + atanh(((x + S(-1))S(2) + S(1))/sqrt(a + S(4)))/sqrt(a + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)*S(2), x), x, (x + S(-1))(S(2)a + (a + S(4))(x + S(-1))*S(2) + (S(2)a + S(10))(x + S(-1)) + S(2)(x + S(-1))*S(3) + S(8))/((S(4)a*S(2) + S(28)a + S(48))(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))) + (-sqrt(a + S(4)) + S(1))atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1)))/(sqrt(a + S(4))(S(8)a + S(24))sqrt(sqrt(a + S(4)) + S(1))) - (sqrt(a + S(4)) + S(1))atan((x + S(-1))/sqrt(-sqrt(a + S(4)) + S(1)))/(sqrt(a + S(4))(S(8)a + S(24))sqrt(-sqrt(a + S(4)) + S(1))) + atanh(((x + S(-1))S(2) + S(1))/sqrt(a + S(4)))/(S(2)(a + S(4))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a - xS(4) + S(4)xS(3) - S(8)x*S(2) + S(8)x)*(S(3)/2), x), x, -(S(32)a + S(112))(x + S(-1))((x + S(-1))*S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-sqrt(a + S(4)) + S(1))/(S(35)sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))) + (x + S(-1))(S(2)a/S(7) - S(6)(x + S(-1))*S(2)/S(35) + S(26)/35)sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3)) + (x + S(-1))(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))(S(3)/2)/S(7) + (S(4)a + S(12))(S(5)a + S(16))((x + S(-1))*S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_f(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(S(35)sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3))) + (S(32)a + S(112))((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-sqrt(a + S(4)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_e(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(S(35)sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a - xS(4) + S(4)xS(3) - S(8)x*S(2) + S(8)x), x), x, -(x + S(-1))((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-S(2)sqrt(a + S(4)) + S(2))/(S(3)sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3))) + (x + S(-1))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3))/S(3) + (S(2)a + S(6))((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_f(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(S(3)sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))) + ((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-S(2)sqrt(a + S(4)) + S(2))sqrt(sqrt(a + S(4)) + S(1))elliptic_e(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(S(3)sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a - xS(4) + S(4)xS(3) - S(8)x*S(2) + S(8)x), x), x, ((x + S(-1))*S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_f(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)*(S(3)/2), x), x, (S(3)a/S(16) + S(3)/4)((x + S(-1))S(2) + S(1))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3)) + S(3)(a + S(4))*S(2)atan(((x + S(-1))S(2) + S(1))/sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))S(2) + S(3)))/S(16) - (S(32)a + S(112))(x + S(-1))((x + S(-1))*S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-sqrt(a + S(4)) + S(1))/(S(35)sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))) + (x + S(-1))(S(2)a/S(7) - S(6)(x + S(-1))*S(2)/S(35) + S(26)/35)sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3)) + (x + S(-1))(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))(S(3)/2)/S(7) + ((x + S(-1))*S(2)/S(8) + S(1)/8)(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))(S(3)/2) + (S(4)a + S(12))(S(5)a + S(16))((x + S(-1))*S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_f(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(S(35)sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3))) + (S(32)a + S(112))((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-sqrt(a + S(4)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_e(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(S(35)sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x), x), x, (a/S(4) + S(1))atan(((x + S(-1))S(2) + S(1))/sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))) - (x + S(-1))((x + S(-1))*S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-S(2)sqrt(a + S(4)) + S(2))/(S(3)sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3))) + (x + S(-1))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))/S(3) + ((x + S(-1))S(2)/S(4) + S(1)/4)sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3)) + (S(2)a + S(6))((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_f(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(S(3)sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))) + ((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-S(2)sqrt(a + S(4)) + S(2))sqrt(sqrt(a + S(4)) + S(1))elliptic_e(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(S(3)sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a - xS(4) + S(4)xS(3) - S(8)x*S(2) + S(8)x), x), x, atan(((x + S(-1))S(2) + S(1))/sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))S(2) + S(3)))/S(2) + ((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_f(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a - xS(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)*(S(3)/2), x), x, (S(3)a/S(8) + S(3)/2)((x + S(-1))S(2) + S(1))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))*S(2) + S(3)) + S(3)(a + S(4))*S(2)atan(((x + S(-1))S(2) + S(1))/sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))S(2) + S(3)))/S(8) + (x + S(-1))((x + S(-1))*S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-sqrt(a + S(4)) + S(1))(S(84)a*S(2) + S(444)a + S(560))/(S(315)sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))) + (x + S(-1))((x + S(-1))*S(2)/S(9) + S(5)/21)(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))(S(3)/2) + (x + S(-1))(S(12)a/S(35) + S(2)(S(21)a + S(60))(x + S(-1))S(2)/S(315) + S(64)/63)sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3)) + ((x + S(-1))S(2)/S(4) + S(1)/4)(a - (x + S(-1))S(4) - S(2)(x + S(-1))S(2) + S(3))(S(3)/2) + (S(4)a + S(12))(S(33)a + S(100))((x + S(-1))*S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_f(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(S(315)sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))) - ((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-sqrt(a + S(4)) + S(1))sqrt(sqrt(a + S(4)) + S(1))(S(84)a*S(2) + S(444)a + S(560))elliptic_e(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(S(315)sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(a - xS(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x), x), x, (a/S(2) + S(2))atan(((x + S(-1))S(2) + S(1))/sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))) + (S(6)a + S(16))(x + S(-1))((x + S(-1))*S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-sqrt(a + S(4)) + S(1))/(S(15)sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))) + (x + S(-1))((x + S(-1))*S(2)/S(5) + S(7)/15)sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3)) + ((x + S(-1))S(2)/S(2) + S(1)/2)sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3)) - (S(6)a + S(16))((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-sqrt(a + S(4)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_e(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(S(15)sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))) + (S(8)a + S(24))((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_f(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(S(15)sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x), x), x, (x + S(-1))((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-sqrt(a + S(4)) + S(1))/sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3)) + atan(((x + S(-1))S(2) + S(1))/sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))) - ((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-sqrt(a + S(4)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_e(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))S(2) + S(3))) + ((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_f(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a - x*S(4) + S(4)x*S(3) - S(8)x*S(2) + S(8)x)*(S(3)/2), x), x, (x + S(-1))(S(2)a + (a + S(4))(x + S(-1))*S(2) + (S(2)a + S(10))(x + S(-1)) + S(2)(x + S(-1))*S(3) + S(8))/((S(2)a*S(2) + S(14)a + S(24))sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))S(2) + S(3))) + sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))S(2) + S(3))/(aS(2) + S(7)a + S(12)) - (x + S(-1))((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-sqrt(a + S(4)) + S(1))/((S(2)a + S(6))sqrt(a - (x + S(-1))*S(4) - S(2)(x + S(-1))S(2) + S(3))) + ((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))(-sqrt(a + S(4)) + S(1))sqrt(sqrt(a + S(4)) + S(1))elliptic_e(atan((x + S(-1))/sqrt(sqrt(a + S(4)) + S(1))), -S(2)sqrt(a + S(4))/(-sqrt(a + S(4)) + S(1)))/(sqrt(((x + S(-1))S(2)/(-sqrt(a + S(4)) + S(1)) + S(1))/((x + S(-1))S(2)/(sqrt(a + S(4)) + S(1)) + S(1)))(S(2)a + S(6))sqrt(a - (x + S(-1))S(4) - S(2)(x + S(-1))*S(2) + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)xS(4) - xS(3) + S(8)x + S(8))S(4), x), x, S(4096)x*S(17)/S(17) - S(128)x*S(16) + S(128)x*S(15)/S(5) + S(1168)x*S(14) + S(10241)x*S(13)/S(13) - S(448)x*S(12) + S(25312)x*S(11)/S(11) + S(21488)x*S(10)/S(5) + S(1408)x*S(9) + S(1376)x*S(8) + S(6784)x*S(7) + S(7168)x*S(6) + S(14336)x*S(5)/S(5) + S(3584)x*S(4) + S(8192)x*S(3) + S(8192)x*S(2) + S(4096)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)xS(4) - xS(3) + S(8)x + S(8))*S(3), x), x, S(512)x*S(13)/S(13) - S(16)x*S(12) + S(24)x*S(11)/S(11) + S(307)x*S(10)/S(2) + S(128)x*S(9) - S(45)x*S(8) + S(1560)x*S(7)/S(7) + S(480)x*S(6) + S(1152)x*S(5)/S(5) + S(80)x*S(4) + S(512)x*S(3) + S(768)x*S(2) + S(512)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)xS(4) - xS(3) + S(8)x + S(8))*S(2), x), x, S(64)x*S(9)/S(9) - S(2)xS(8) + xS(7)/S(7) + S(64)xS(6)/S(3) + S(112)x*S(5)/S(5) - S(4)x*S(4) + S(64)x*S(3)/S(3) + S(64)x*S(2) + S(64)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(8)xS(4) - xS(3) + S(8)x + S(8), x), x, S(8)xS(5)/S(5) - xS(4)/S(4) + S(4)x*S(2) + S(8)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(8)xS(4) - xS(3) + S(8)x + S(8)), x), x, -sqrt(S(-109)/1218 + S(67)sqrt(S(29))/S(1218))log((S(1) + S(4)/x)*S(2) - (S(1) + S(4)/x)sqrt(S(6) + S(6)sqrt(S(29))) + S(3)sqrt(S(29)))/S(24) + sqrt(S(-109)/1218 + S(67)sqrt(S(29))/S(1218))log((S(1) + S(4)/x)*S(2) + (S(1) + S(4)/x)sqrt(S(6) + S(6)sqrt(S(29))) + S(3)sqrt(S(29)))/S(24) - sqrt(S(7))atan(sqrt(S(7))(-(S(1) + S(4)/x)*S(2) + S(3))/S(42))/S(84) + sqrt(S(109)/1218 + S(67)sqrt(S(29))/S(1218))atan((S(-2) + sqrt(S(6) + S(6)sqrt(S(29))) - S(8)/x)/sqrt(S(-6) + S(6)sqrt(S(29))))/S(12) - sqrt(S(109)/1218 + S(67)sqrt(S(29))/S(1218))atan((S(2) + sqrt(S(6) + S(6)sqrt(S(29))) + S(8)/x)/sqrt(S(-6) + S(6)sqrt(S(29))))/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)xS(4) - xS(3) + S(8)x + S(8))(S(-2)), x), x, (S(1) + S(4)/x)(S(207)(S(1) + S(4)/x)S(3) + S(995)(S(1) + S(4)/x)*S(2) + S(16974) - S(35244)/x)/(S(87696)(S(1) + S(4)/x)*S(4) - S(526176)(S(1) + S(4)/x)*S(2) + S(22888656)) - sqrt(S(-180983329)/1218 + S(1583563)sqrt(S(29))/S(42))log((S(1) + S(4)/x)S(2) - (S(1) + S(4)/x)sqrt(S(6) + S(6)sqrt(S(29))) + S(3)sqrt(S(29)))/S(175392) + sqrt(S(-180983329)/1218 + S(1583563)sqrt(S(29))/S(42))log((S(1) + S(4)/x)*S(2) + (S(1) + S(4)/x)sqrt(S(6) + S(6)sqrt(S(29))) + S(3)sqrt(S(29)))/S(175392) - S(17)sqrt(S(7))atan(sqrt(S(7))(-(S(1) + S(4)/x)S(2) + S(3))/S(42))/S(7056) + sqrt(S(180983329)/1218 + S(1583563)sqrt(S(29))/S(42))atan((S(-2) + sqrt(S(6) + S(6)sqrt(S(29))) - S(8)/x)/sqrt(S(-6) + S(6)sqrt(S(29))))/S(87696) - sqrt(S(180983329)/1218 + S(1583563)sqrt(S(29))/S(42))atan((S(2) + sqrt(S(6) + S(6)sqrt(S(29))) + S(8)/x)/sqrt(S(-6) + S(6)sqrt(S(29))))/S(87696), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x*S(4) + S(4)x*S(2) + S(4)x + S(1))*S(4), x), x, S(256)x*S(17)/S(17) + S(1024)x*S(15)/S(15) + S(512)x*S(14)/S(7) + S(1792)x*S(13)/S(13) + S(256)x*S(12) + S(3328)x*S(11)/S(11) + S(384)x*S(10) + S(4192)x*S(9)/S(9) + S(448)x*S(8) + S(2752)x*S(7)/S(7) + S(992)x*S(6)/S(3) + S(1136)x*S(5)/S(5) + S(112)x*S(4) + S(112)x*S(3)/S(3) + S(8)x*S(2) + x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x*S(4) + S(4)x*S(2) + S(4)x + S(1))*S(3), x), x, S(64)x*S(13)/S(13) + S(192)x*S(11)/S(11) + S(96)x*S(10)/S(5) + S(80)x*S(9)/S(3) + S(48)x*S(8) + S(352)x*S(7)/S(7) + S(48)x*S(6) + S(252)x*S(5)/S(5) + S(40)x*S(4) + S(20)x*S(3) + S(6)x*S(2) + x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x*S(4) + S(4)x*S(2) + S(4)x + S(1))*S(2), x), x, S(16)x*S(9)/S(9) + S(32)x*S(7)/S(7) + S(16)x*S(6)/S(3) + S(24)x*S(5)/S(5) + S(8)x*S(4) + S(8)x*S(3) + S(4)x*S(2) + x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(4)x*S(4) + S(4)x*S(2) + S(4)x + S(1), x), x, S(4)xS(5)/S(5) + S(4)x*S(3)/S(3) + S(2)x*S(2) + x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(4)x*S(4) + S(4)x*S(2) + S(4)x + S(1)), x), x, -sqrt(S(-2)/5 + sqrt(S(5))/S(5))log((S(1) + S(1)/x)S(2) - (S(1) + S(1)/x)sqrt(S(2) + S(2)sqrt(S(5))) + sqrt(S(5)))/S(4) + sqrt(S(-2)/5 + sqrt(S(5))/S(5))log((S(1) + S(1)/x)*S(2) + (S(1) + S(1)/x)sqrt(S(2) + S(2)sqrt(S(5))) + sqrt(S(5)))/S(4) + sqrt(S(2)/5 + sqrt(S(5))/S(5))atan((S(-2) + sqrt(S(2) + S(2)sqrt(S(5))) - S(2)/x)/sqrt(S(-2) + S(2)sqrt(S(5))))/S(2) - sqrt(S(2)/5 + sqrt(S(5))/S(5))atan((S(2) + sqrt(S(2) + S(2)sqrt(S(5))) + S(2)/x)/sqrt(S(-2) + S(2)sqrt(S(5))))/S(2) + atan((S(1) + S(1)/x)S(2)/S(2) + S(-1)/2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x*S(4) + S(4)x*S(2) + S(4)x + S(1))*(S(-2)), x), x, (S(1) + S(1)/x)(S(17)(S(1) + S(1)/x)S(3) - S(17)(S(1) + S(1)/x)*S(2) + S(30) - S(29)/x)/(S(10)(S(1) + S(1)/x)*S(4) - S(20)(S(1) + S(1)/x)*S(2) + S(50)) + sqrt(S(-5959)/10 + S(533)sqrt(S(5))/S(2))log((S(1) + S(1)/x)S(2) - (S(1) + S(1)/x)sqrt(S(2) + S(2)sqrt(S(5))) + sqrt(S(5)))/S(40) - sqrt(S(-5959)/10 + S(533)sqrt(S(5))/S(2))log((S(1) + S(1)/x)S(2) + (S(1) + S(1)/x)sqrt(S(2) + S(2)sqrt(S(5))) + sqrt(S(5)))/S(40) + sqrt(S(5959)/10 + S(533)sqrt(S(5))/S(2))atan((S(-2) + sqrt(S(2) + S(2)sqrt(S(5))) - S(2)/x)/sqrt(S(-2) + S(2)sqrt(S(5))))/S(20) - sqrt(S(5959)/10 + S(533)sqrt(S(5))/S(2))atan((S(2) + sqrt(S(2) + S(2)sqrt(S(5))) + S(2)/x)/sqrt(S(-2) + S(2)sqrt(S(5))))/S(20) + S(7)atan((S(1) + S(1)/x)*S(2)/S(2) + S(-1)/2)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)x*S(4) - S(15)x*S(3) + S(8)x*S(2) + S(24)x + S(8))*S(4), x), x, S(4096)x*S(17)/S(17) - S(1920)x*S(16) + S(102784)x*S(15)/S(15) - S(75504)x*S(14)/S(7) - S(12095)x*S(13)/S(13) + S(31128)x*S(12) - S(331040)x*S(11)/S(11) - S(169584)x*S(10)/S(5) + S(641152)x*S(9)/S(9) + S(36384)x*S(8) - S(566912)x*S(7)/S(7) - S(30720)x*S(6) + S(538624)x*S(5)/S(5) + S(139776)x*S(4) + S(237568)x*S(3)/S(3) + S(24576)x*S(2) + S(4096)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)xS(4) - S(15)x*S(3) + S(8)x*S(2) + S(24)x + S(8))*S(3), x), x, S(512)x*S(13)/S(13) - S(240)x*S(12) + S(6936)x*S(11)/S(11) - S(4527)x*S(10)/S(10) - S(2936)x*S(9)/S(3) + S(2097)x*S(8) + S(5528)x*S(7)/S(7) - S(2976)x*S(6) - S(384)x*S(5)/S(5) + S(5040)x*S(4) + S(5120)x*S(3) + S(2304)x*S(2) + S(512)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)xS(4) - S(15)x*S(3) + S(8)x*S(2) + S(24)x + S(8))*S(2), x), x, S(64)x*S(9)/S(9) - S(30)x*S(8) + S(353)x*S(7)/S(7) + S(24)x*S(6) - S(528)x*S(5)/S(5) + S(36)x*S(4) + S(704)x*S(3)/S(3) + S(192)x*S(2) + S(64)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(8)xS(4) - S(15)x*S(3) + S(8)x*S(2) + S(24)x + S(8), x), x, S(8)xS(5)/S(5) - S(15)x*S(4)/S(4) + S(8)x*S(3)/S(3) + S(12)x*S(2) + S(8)x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(S(8)xS(4) - S(15)x*S(3) + S(8)x*S(2) + S(24)x + S(8)), x), x, -sqrt(S(-5167)/40326 + S(5)sqrt(S(517))/S(858))log((S(3) + S(4)/x)*S(2) - (S(3) + S(4)/x)sqrt(S(38) + S(2)sqrt(S(517))) + sqrt(S(517)))/S(8) + sqrt(S(-5167)/40326 + S(5)sqrt(S(517))/S(858))log((S(3) + S(4)/x)S(2) + (S(3) + S(4)/x)sqrt(S(38) + S(2)sqrt(S(517))) + sqrt(S(517)))/S(8) - sqrt(S(39))atan(sqrt(S(39))(-(S(3) + S(4)/x)S(2) + S(19))/S(78))/S(52) + sqrt(S(5167)/40326 + S(5)sqrt(S(517))/S(858))atan((S(-6) + sqrt(S(38) + S(2)sqrt(S(517))) - S(8)/x)/sqrt(S(-38) + S(2)sqrt(S(517))))/S(4) - sqrt(S(5167)/40326 + S(5)sqrt(S(517))/S(858))atan((S(6) + sqrt(S(38) + S(2)sqrt(S(517))) + S(8)/x)/sqrt(S(-38) + S(2)sqrt(S(517))))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)x*S(4) - S(15)x*S(3) + S(8)x*S(2) + S(24)x + S(8))*(S(-2)), x), x, (S(3) + S(4)/x)(S(30231)(S(3) + S(4)/x)S(3) - S(129631)(S(3) + S(4)/x)*S(2) + S(1375210) - S(2603628)/x)/(S(322608)(S(3) + S(4)/x)*S(4) - S(12259104)(S(3) + S(4)/x)*S(2) + S(166788336)) - sqrt(S(-59644114671451)/40326 + S(5073830635)sqrt(S(517))/S(78))log((S(3) + S(4)/x)S(2) - (S(3) + S(4)/x)sqrt(S(38) + S(2)sqrt(S(517))) + sqrt(S(517)))/S(645216) + sqrt(S(-59644114671451)/40326 + S(5073830635)sqrt(S(517))/S(78))log((S(3) + S(4)/x)S(2) + (S(3) + S(4)/x)sqrt(S(38) + S(2)sqrt(S(517))) + sqrt(S(517)))/S(645216) - S(73)sqrt(S(39))atan(sqrt(S(39))(-(S(3) + S(4)/x)*S(2) + S(19))/S(78))/S(2704) + sqrt(S(19)/40326 + sqrt(S(517))/S(40326))(S(1678181) + S(74897)sqrt(S(517)))atan((S(-6) + sqrt(S(38) + S(2)sqrt(S(517))) - S(8)/x)/sqrt(S(-38) + S(2)sqrt(S(517))))/S(645216) - sqrt(S(19)/40326 + sqrt(S(517))/S(40326))(S(1678181) + S(74897)sqrt(S(517)))atan((S(6) + sqrt(S(38) + S(2)sqrt(S(517))) + S(8)/x)/sqrt(S(-38) + S(2)sqrt(S(517))))/S(645216), expand=True, _diff=True, _numerical=True) '''Takes a long time in rubi test, final results contain subs with Integral assert rubi_test(rubi_integrate(S(1)/sqrt(S(8)xS(4) - xS(3) + S(8)x + S(8)), x), x, -S(29)(S(3)/4)sqrt(S(6))xS(2)sqrt(((S(1) + S(4)/x)*S(4) - S(6)(S(1) + S(4)/x)*S(2) + S(261))/(sqrt(S(29))(S(1) + S(4)/x)S(2) + S(87))S(2))(sqrt(S(29))(S(1) + S(4)/x)*S(2) + S(87))elliptic_f(S(2)atan(S(29)(S(3)/4)sqrt(S(3))(S(1) + S(4)/x)/S(87)), sqrt(S(29))/S(58) + S(1)/2)/(S(174)sqrt(x*S(4)((S(1) + S(4)/x)*S(4) - S(6)(S(1) + S(4)/x)*S(2) + S(261)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)xS(4) - xS(3) + S(8)x + S(8))(S(-3)/2), x), x, S(29)(S(1)/4)sqrt(S(6))xS(2)sqrt(((S(1) + S(4)/x)*S(4) - S(6)(S(1) + S(4)/x)*S(2) + S(261))/(sqrt(S(29))(S(1) + S(4)/x)S(2) + S(87))S(2))(-S(5)sqrt(S(29)) + S(14))(sqrt(S(29))(S(1) + S(4)/x)*S(2) + S(87))elliptic_f(S(2)atan(S(29)(S(3)/4)sqrt(S(3))(S(1) + S(4)/x)/S(87)), sqrt(S(29))/S(58) + S(1)/2)/(S(12528)sqrt(x*S(4)((S(1) + S(4)/x)*S(4) - S(6)(S(1) + S(4)/x)S(2) + S(261)))) - S(29)(S(1)/4)sqrt(S(6))x*S(2)sqrt(((S(1) + S(4)/x)*S(4) - S(6)(S(1) + S(4)/x)*S(2) + S(261))/(sqrt(S(29))(S(1) + S(4)/x)S(2) + S(87))S(2))(S(7)sqrt(S(29))(S(1) + S(4)/x)S(2) + S(609))elliptic_e(S(2)atan(S(29)(S(3)/4)sqrt(S(3))(S(1) + S(4)/x)/S(87)), sqrt(S(29))/S(58) + S(1)/2)/(S(3132)sqrt(x*S(4)((S(1) + S(4)/x)*S(4) - S(6)(S(1) + S(4)/x)*S(2) + S(261)))) + sqrt(S(2))x*S(2)(S(1) + S(4)/x)(S(22)(S(1) + S(4)/x)*S(3) - S(49)(S(1) + S(4)/x)*S(2) + S(1467) - S(180)/x)/(S(21924)sqrt(x*S(4)((S(1) + S(4)/x)*S(4) - S(6)(S(1) + S(4)/x)*S(2) + S(261)))) + sqrt(S(58))x*S(2)(S(1) + S(4)/x)(S(7)(S(1) + S(4)/x)*S(4) - S(42)(S(1) + S(4)/x)*S(2) + S(1827))/(S(3132)sqrt(x*S(4)((S(1) + S(4)/x)*S(4) - S(6)(S(1) + S(4)/x)*S(2) + S(261)))(sqrt(S(29))(S(1) + S(4)/x)S(2) + S(87))) - sqrt(S(2))x*S(2)(S(11)(S(1) + S(4)/x)S(4) - S(66)(S(1) + S(4)/x)*S(2) + S(2871))/(S(10962)sqrt(x*S(4)((S(1) + S(4)/x)*S(4) - S(6)(S(1) + S(4)/x)*S(2) + S(261)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(4)x*S(4) + S(4)x*S(2) + S(4)x + S(1)), x), x, -S(5)*(S(3)/4)x*S(2)sqrt(((S(1) + S(1)/x)*S(4) - S(2)(S(1) + S(1)/x)S(2) + S(5))/((S(1) + S(1)/x)S(2) + sqrt(S(5)))*S(2))((S(1) + S(1)/x)*S(2) + sqrt(S(5)))elliptic_f(S(2)atan(S(5)(S(3)/4)(S(1) + S(1)/x)/S(5)), sqrt(S(5))/S(10) + S(1)/2)/(S(10)sqrt(xS(4)((S(1) + S(1)/x)*S(4) - S(2)(S(1) + S(1)/x)*S(2) + S(5)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(4)x*S(4) + S(4)x*S(2) + S(4)x + S(1))(S(-3)/2), x), x, S(5)(S(1)/4)xS(2)sqrt(((S(1) + S(1)/x)*S(4) - S(2)(S(1) + S(1)/x)S(2) + S(5))/((S(1) + S(1)/x)S(2) + sqrt(S(5)))*S(2))(-S(3)sqrt(S(5)) + S(9))((S(1) + S(1)/x)*S(2) + sqrt(S(5)))elliptic_f(S(2)atan(S(5)(S(3)/4)(S(1) + S(1)/x)/S(5)), sqrt(S(5))/S(10) + S(1)/2)/(S(20)sqrt(xS(4)((S(1) + S(1)/x)*S(4) - S(2)(S(1) + S(1)/x)S(2) + S(5)))) - S(5)(S(1)/4)xS(2)sqrt(((S(1) + S(1)/x)*S(4) - S(2)(S(1) + S(1)/x)S(2) + S(5))/((S(1) + S(1)/x)S(2) + sqrt(S(5)))*S(2))(S(9)(S(1) + S(1)/x)S(2) + S(9)sqrt(S(5)))elliptic_e(S(2)atan(S(5)*(S(3)/4)(S(1) + S(1)/x)/S(5)), sqrt(S(5))/S(10) + S(1)/2)/(S(10)sqrt(xS(4)((S(1) + S(1)/x)*S(4) - S(2)(S(1) + S(1)/x)S(2) + S(5)))) + xS(2)(S(1) + S(1)/x)(S(6)(S(1) + S(1)/x)S(3) - S(9)(S(1) + S(1)/x)*S(2) + S(11) - S(2)/x)/(S(10)sqrt(x*S(4)((S(1) + S(1)/x)*S(4) - S(2)(S(1) + S(1)/x)S(2) + S(5)))) + xS(2)(S(1) + S(1)/x)(S(9)(S(1) + S(1)/x)S(4) - S(18)(S(1) + S(1)/x)S(2) + S(45))/(sqrt(xS(4)((S(1) + S(1)/x)S(4) - S(2)(S(1) + S(1)/x)*S(2) + S(5)))(S(10)(S(1) + S(1)/x)S(2) + S(10)sqrt(S(5)))) - x*S(2)(S(3)(S(1) + S(1)/x)S(4) - S(6)(S(1) + S(1)/x)*S(2) + S(15))/(S(5)sqrt(x*S(4)((S(1) + S(1)/x)*S(4) - S(2)(S(1) + S(1)/x)*S(2) + S(5)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(8)x*S(4) - S(15)x*S(3) + S(8)x*S(2) + S(24)x + S(8)), x), x, -sqrt(S(2))S(517)(S(3)/4)x*S(2)sqrt(((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)S(2) + S(517))/((S(3) + S(4)/x)S(2) + sqrt(S(517)))*S(2))((S(3) + S(4)/x)*S(2) + sqrt(S(517)))elliptic_f(S(2)atan(S(517)(S(3)/4)(S(3) + S(4)/x)/S(517)), S(19)sqrt(S(517))/S(1034) + S(1)/2)/(S(1034)sqrt(x*S(4)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)x*S(4) - S(15)x*S(3) + S(8)x*S(2) + S(24)x + S(8))*(S(-3)/2), x), x, sqrt(S(2))S(517)*(S(1)/4)x*S(2)sqrt(((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)S(2) + S(517))/((S(3) + S(4)/x)S(2) + sqrt(S(517)))*S(2))(-S(203)sqrt(S(517)) + S(4910))((S(3) + S(4)/x)*S(2) + sqrt(S(517)))elliptic_f(S(2)atan(S(517)(S(3)/4)(S(3) + S(4)/x)/S(517)), S(19)sqrt(S(517))/S(1034) + S(1)/2)/(S(322608)sqrt(x*S(4)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517)))) - sqrt(S(2))S(517)*(S(1)/4)x*S(2)sqrt(((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)S(2) + S(517))/((S(3) + S(4)/x)S(2) + sqrt(S(517)))*S(2))(S(2455)(S(3) + S(4)/x)S(2) + S(2455)sqrt(S(517)))elliptic_e(S(2)atan(S(517)*(S(3)/4)(S(3) + S(4)/x)/S(517)), S(19)sqrt(S(517))/S(1034) + S(1)/2)/(S(80652)sqrt(x*S(4)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517)))) + sqrt(S(2))x*S(2)(S(3) + S(4)/x)(S(516)(S(3) + S(4)/x)*S(3) - S(2455)(S(3) + S(4)/x)*S(2) + S(24643) - S(35004)/x)/(S(80652)sqrt(x*S(4)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517)))) + sqrt(S(2))x*S(2)(S(3) + S(4)/x)(S(2455)(S(3) + S(4)/x)*S(4) - S(93290)(S(3) + S(4)/x)*S(2) + S(1269235))/(S(80652)sqrt(x*S(4)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517)))((S(3) + S(4)/x)*S(2) + sqrt(S(517)))) - S(43)sqrt(S(2))xS(2)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517))/(S(6721)sqrt(x*S(4)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(8)x*S(4) - S(15)x*S(3) + S(8)x*S(2) + S(24)x + S(8))*(S(-5)/2), x), x, sqrt(S(2))S(517)*(S(1)/4)x*S(2)sqrt(((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)S(2) + S(517))/((S(3) + S(4)/x)S(2) + sqrt(S(517)))*S(2))(-S(175318963)sqrt(S(517)) + S(4346103976))((S(3) + S(4)/x)*S(2) + sqrt(S(517)))elliptic_f(S(2)atan(S(517)(S(3)/4)(S(3) + S(4)/x)/S(517)), S(19)sqrt(S(517))/S(1034) + S(1)/2)/(S(156113882496)sqrt(x*S(4)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517)))) - sqrt(S(2))S(517)*(S(1)/4)x*S(2)sqrt(((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)S(2) + S(517))/((S(3) + S(4)/x)S(2) + sqrt(S(517)))*S(2))(S(543262997)(S(3) + S(4)/x)S(2) + S(543262997)sqrt(S(517)))elliptic_e(S(2)atan(S(517)*(S(3)/4)(S(3) + S(4)/x)/S(517)), S(19)sqrt(S(517))/S(1034) + S(1)/2)/(S(9757117656)sqrt(x*S(4)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517)))) + sqrt(S(2))x*S(2)(S(3) + S(4)/x)(S(223148517)(S(3) + S(4)/x)*S(3) - S(1086525994)(S(3) + S(4)/x)*S(2) + S(8668521901) - S(13685866440)/x)/(S(19514235312)sqrt(x*S(4)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517)))) + sqrt(S(2))x*S(2)(S(3) + S(4)/x)(S(193467)(S(3) + S(4)/x)*S(3) - S(718994)(S(3) + S(4)/x)*S(2) + S(8297705) - S(20727588)/x)/(S(241956)sqrt(x*S(4)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517)))((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517))) + sqrt(S(2))x*S(2)(S(3) + S(4)/x)(S(543262997)(S(3) + S(4)/x)*S(4) - S(20643993886)(S(3) + S(4)/x)*S(2) + S(280866969449))/(S(9757117656)sqrt(x*S(4)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517)))((S(3) + S(4)/x)*S(2) + sqrt(S(517)))) - sqrt(S(2))x*S(2)(S(74382839)(S(3) + S(4)/x)S(4) - S(2826547882)(S(3) + S(4)/x)*S(2) + S(38455927763))/(S(6504745104)sqrt(x*S(4)((S(3) + S(4)/x)*S(4) - S(38)(S(3) + S(4)/x)*S(2) + S(517)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(3)x*S(4) + S(15)x*S(3) - S(44)x*S(2) - S(6)x + S(9)), x), x, S(613)*(S(3)/4)x*S(2)sqrt(((S(-1) + S(6)/x)*S(4) - S(182)(S(1) - S(6)/x)S(2) + S(613))/((S(-1) + S(6)/x)S(2) + sqrt(S(613)))*S(2))((S(-1) + S(6)/x)*S(2) + sqrt(S(613)))elliptic_f(S(2)atan(S(613)(S(3)/4)(S(1) - S(6)/x)/S(613)), S(1)/2 + S(91)sqrt(S(613))/S(1226))/(S(613)sqrt(x*S(4)((S(-1) + S(6)/x)*S(4) - S(182)(S(1) - S(6)/x)*S(2) + S(613)))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(3)x*S(4) + S(15)x*S(3) - S(44)x*S(2) - S(6)x + S(9))(S(-3)/2), x), x, S(613)(S(1)/4)xS(2)sqrt(((S(-1) + S(6)/x)*S(4) - S(182)(S(1) - S(6)/x)S(2) + S(613))/((S(-1) + S(6)/x)S(2) + sqrt(S(613)))*S(2))(-S(145)sqrt(S(613)) + S(7444))((S(-1) + S(6)/x)*S(2) + sqrt(S(613)))elliptic_f(S(2)atan(S(613)(S(3)/4)(S(1) - S(6)/x)/S(613)), S(1)/2 + S(91)sqrt(S(613))/S(1226))/(S(10576089)sqrt(x*S(4)((S(-1) + S(6)/x)*S(4) - S(182)(S(1) - S(6)/x)S(2) + S(613)))) - S(613)(S(1)/4)xS(2)sqrt(((S(-1) + S(6)/x)*S(4) - S(182)(S(1) - S(6)/x)S(2) + S(613))/((S(-1) + S(6)/x)S(2) + sqrt(S(613)))*S(2))(S(14888)(S(-1) + S(6)/x)S(2) + S(14888)sqrt(S(613)))elliptic_e(S(2)atan(S(613)*(S(3)/4)(S(1) - S(6)/x)/S(613)), S(1)/2 + S(91)sqrt(S(613))/S(1226))/(S(10576089)sqrt(x*S(4)((S(-1) + S(6)/x)*S(4) - S(182)(S(1) - S(6)/x)S(2) + S(613)))) + xS(2)(S(1) - S(6)/x)(S(704)(S(1) - S(6)/x)S(3) - S(14888)(S(1) - S(6)/x)*S(2) + S(109872) + S(430392)/x)/(S(10576089)sqrt(x*S(4)((S(-1) + S(6)/x)*S(4) - S(182)(S(1) - S(6)/x)S(2) + S(613)))) + xS(2)(S(1) - S(6)/x)(S(14888)(S(-1) + S(6)/x)S(4) - S(2709616)(S(1) - S(6)/x)S(2) + S(9126344))/(sqrt(xS(4)((S(-1) + S(6)/x)S(4) - S(182)(S(1) - S(6)/x)*S(2) + S(613)))(S(10576089)(S(-1) + S(6)/x)S(2) + S(10576089)sqrt(S(613)))) - x*S(2)(S(704)(S(-1) + S(6)/x)S(4) - S(128128)(S(1) - S(6)/x)*S(2) + S(431552))/(S(10576089)sqrt(x*S(4)((S(-1) + S(6)/x)*S(4) - S(182)(S(1) - S(6)/x)S(2) + S(613)))), expand=True, _diff=True, _numerical=True) ''' def test_5(): assert rubi_test(rubi_integrate(xmsqrt(-a/x + b)/sqrt(a - bx), x), x, S(2)x(m + S(1))sqrt(-a/x + b)/(sqrt(a - bx)(S(2)m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(-a/x + b)/sqrt(a - bx), x), x, S(2)x*S(3)sqrt(-a/x + b)/(S(5)sqrt(a - bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(-a/x + b)/sqrt(a - bx), x), x, S(2)xS(2)sqrt(-a/x + b)/(S(3)sqrt(a - bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-a/x + b)/sqrt(a - bx), x), x, S(2)xsqrt(-a/x + b)/sqrt(a - bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-a/x + b)/(xsqrt(a - bx)), x), x, -S(2)sqrt(-a/x + b)/sqrt(a - bx), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-a/x + b)/(x*S(2)sqrt(a - bx)), x), x, -S(2)sqrt(-a/x + b)/(S(3)xsqrt(a - bx)), expand=True, _diff=True, _numerical=True) # appellf1 assert rubi_test(rubi_integrate((a + b/x)m(c + dx)n, x), x, x(S(1) + dx/c)(-n)(a + b/x)*m(c + dx)n(ax/b + S(1))(-m)AppellF1(-m + S(1), -m, -n, -m + S(2), -ax/b, -dx/c)/(-m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b/x)*m(c + dx)S(2), x), x, dS(2)x*S(3)(a + b/x)*(m + S(1))/(S(3)a) + dxS(2)(a + b/x)*(m + S(1))(S(6)ac - bd(-m + S(2)))/(S(6)aS(2)) - b(a + b/x)*(m + S(1))(S(6)aS(2)c*S(2) - S(6)abcd(-m + S(1)) + b*S(2)d*S(2)(m*S(2) - S(3)m + S(2)))hyper((S(2), m + S(1)), (m + S(2),), S(1) + b/(ax))/(S(6)aS(4)(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b/x)*m(c + dx), x), x, dx*S(2)(a + b/x)*(m + S(1))/(S(2)a) - b(a + b/x)(m + S(1))(S(2)ac - bd(-m + S(1)))hyper((S(2), m + S(1)), (m + S(2),), S(1) + b/(ax))/(S(2)aS(3)(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b/x)*m, x), x, -b(a + b/x)*(m + S(1))hyper((S(2), m + S(1)), (m + S(2),), S(1) + b/(ax))/(aS(2)(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b/x)*m/(c + dx), x), x, -c(a + b/x)(m + S(1))hyper((S(1), m + S(1)), (m + S(2),), c(a + b/x)/(ac - bd))/(d(m + S(1))(ac - bd)) + (a + b/x)(m + S(1))hyper((S(1), m + S(1)), (m + S(2),), S(1) + b/(ax))/(ad(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b/x)m/(c + dx)*S(2), x), x, -b(a + b/x)*(m + S(1))hyper((S(2), m + S(1)), (m + S(2),), c(a + b/x)/(ac - bd))/((m + S(1))(ac - bd)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b/x)m/(c + dx)S(3), x), x, -b(a + b/x)*(m + S(1))(S(2)ac - bd(m + S(1)))hyper((S(2), m + S(1)), (m + S(2),), c(a + b/x)/(ac - bd))/(S(2)c(m + S(1))(ac - bd)S(3)) - d(a + b/x)*(m + S(1))/(S(2)c(ac - bd)(c/x + d)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + b/x)m/(c + dx)S(4), x), x, -b(a + b/x)*(m + S(1))(S(6)aS(2)c*S(2) - S(6)abcd(m + S(1)) + b*S(2)d*S(2)(m*S(2) + S(3)m + S(2)))hyper((S(2), m + S(1)), (m + S(2),), c(a + b/x)/(ac - bd))/(S(6)cS(2)(m + S(1))(ac - bd)S(4)) + dS(2)(a + b/x)*(m + S(1))/(S(3)c*S(2)(ac - bd)(c/x + d)S(3)) - d(a + b/x)*(m + S(1))(S(6)ac - bd(m + S(4)))/(S(6)cS(2)(ac - bd)*S(2)(c/x + d)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmsqrt(-a/xS(2) + b)/sqrt(a - bxS(2)), x), x, x(m + S(1))sqrt(-a/xS(2) + b)/(msqrt(a - bxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(-a/x*S(2) + b)/sqrt(a - bxS(2)), x), x, xS(3)sqrt(-a/xS(2) + b)/(S(2)sqrt(a - bxS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(-a/x*S(2) + b)/sqrt(a - bxS(2)), x), x, xS(2)sqrt(-a/xS(2) + b)/sqrt(a - bxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-a/xS(2) + b)/sqrt(a - bxS(2)), x), x, xsqrt(-a/x*S(2) + b)log(x)/sqrt(a - bxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-a/xS(2) + b)/(xsqrt(a - bxS(2))), x), x, -sqrt(-a/xS(2) + b)/sqrt(a - bxS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(-a/xS(2) + b)/(x*S(2)sqrt(a - bxS(2))), x), x, -sqrt(-a/xS(2) + b)/(S(2)xsqrt(a - bx*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)(S(3)/2)/sqrt(a + b/xS(2)), x), x, -2sqrt(b)csqrt(a(c + dx)/(ac - sqrt(b)dsqrt(-a)))(ac*2 + bd*2)sqrt(ax2/b + 1)EllipticF(asin(sqrt(2)sqrt(1 - xsqrt(-a)/sqrt(b))/2), -2sqrt(b)dsqrt(-a)/(ac - sqrt(b)dsqrt(-a)))/(5dx(-a)(3/2)sqrt(a + b/x*2)sqrt(c + dx)) + 2sqrt(b)sqrt(c + dx)(ac*2 - 3bd2)sqrt(ax2/b + 1)EllipticE(asin(sqrt(2)sqrt(1 - xsqrt(-a)/sqrt(b))/2), -2sqrt(b)dsqrt(-a)/(ac - sqrt(b)dsqrt(-a)))/(5dx(-a)(3/2)sqrt(a(c + dx)/(ac - sqrt(b)dsqrt(-a)))sqrt(a + b/x*2)) + 2csqrt(c + dx)(ax*2 + b)/(5axsqrt(a + b/x*2)) + 2(c + dx)(3/2)(ax2 + b)/(5axsqrt(a + b/x*2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))(S(5)/2)/sqrt(aS(2) - bS(2)xS(4)), x), x, S(19)a*S(2)sqrt(a - bxS(2))sqrt(a + bxS(2))atan(sqrt(b)x/sqrt(a - bx*S(2)))/(S(8)sqrt(b)sqrt(aS(2) - bS(2)x*S(4))) - S(9)ax(a - bxS(2))sqrt(a + bxS(2))/(S(8)sqrt(aS(2) - bS(2)xS(4))) - x(a - bxS(2))(a + bxS(2))(S(3)/2)/(S(4)sqrt(aS(2) - bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bxS(2))(S(3)/2)/sqrt(aS(2) - bS(2)xS(4)), x), x, S(3)asqrt(a - bx*S(2))sqrt(a + bxS(2))atan(sqrt(b)x/sqrt(a - bx*S(2)))/(S(2)sqrt(b)sqrt(aS(2) - bS(2)x*S(4))) - x(a - bxS(2))sqrt(a + bxS(2))/(S(2)sqrt(aS(2) - bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + bxS(2))/sqrt(aS(2) - b*S(2)x*S(4)), x), x, sqrt(a - bx*S(2))sqrt(a + bxS(2))atan(sqrt(b)x/sqrt(a - bx*S(2)))/(sqrt(b)sqrt(aS(2) - bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a + bx*S(2))sqrt(aS(2) - bS(2)xS(4))), x), x, sqrt(S(2))sqrt(a - bxS(2))sqrt(a + bxS(2))atan(sqrt(S(2))sqrt(b)x/sqrt(a - bxS(2)))/(S(2)asqrt(b)sqrt(aS(2) - bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bxS(2))(S(3)/2)sqrt(aS(2) - bS(2)x*S(4))), x), x, x(a - bxS(2))/(S(4)a*S(2)sqrt(a + bxS(2))sqrt(aS(2) - bS(2)xS(4))) + S(3)sqrt(S(2))sqrt(a - bx*S(2))sqrt(a + bxS(2))atan(sqrt(S(2))sqrt(b)x/sqrt(a - bxS(2)))/(S(8)a*S(2)sqrt(b)sqrt(aS(2) - bS(2)x*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bxS(2))(S(5)/2)sqrt(aS(2) - bS(2)x*S(4))), x), x, x(a - bxS(2))/(S(8)a*S(2)(a + bxS(2))(S(3)/2)sqrt(aS(2) - bS(2)xS(4))) + S(9)x(a - bx*S(2))/(S(32)a*S(3)sqrt(a + bxS(2))sqrt(aS(2) - bS(2)xS(4))) + S(19)sqrt(S(2))sqrt(a - bx*S(2))sqrt(a + bxS(2))atan(sqrt(S(2))sqrt(b)x/sqrt(a - bxS(2)))/(S(64)a*S(3)sqrt(b)sqrt(aS(2) - bS(2)x*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a - bxS(2))(S(5)/2)/sqrt(aS(2) - bS(2)xS(4)), x), x, S(19)a*S(2)sqrt(a - bxS(2))sqrt(a + bxS(2))atanh(sqrt(b)x/sqrt(a + bx*S(2)))/(S(8)sqrt(b)sqrt(aS(2) - bS(2)x*S(4))) - S(9)axsqrt(a - bxS(2))(a + bxS(2))/(S(8)sqrt(aS(2) - bS(2)xS(4))) - x(a - bxS(2))(S(3)/2)(a + bxS(2))/(S(4)sqrt(aS(2) - bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a - bxS(2))(S(3)/2)/sqrt(aS(2) - bS(2)xS(4)), x), x, S(3)asqrt(a - bx*S(2))sqrt(a + bxS(2))atanh(sqrt(b)x/sqrt(a + bx*S(2)))/(S(2)sqrt(b)sqrt(aS(2) - bS(2)x*S(4))) - xsqrt(a - bxS(2))(a + bxS(2))/(S(2)sqrt(aS(2) - bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a - bxS(2))/sqrt(aS(2) - b*S(2)x*S(4)), x), x, sqrt(a - bx*S(2))sqrt(a + bxS(2))atanh(sqrt(b)x/sqrt(a + bx*S(2)))/(sqrt(b)sqrt(aS(2) - bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a - bx*S(2))sqrt(aS(2) - bS(2)xS(4))), x), x, sqrt(S(2))sqrt(a - bxS(2))sqrt(a + bxS(2))atanh(sqrt(S(2))sqrt(b)x/sqrt(a + bxS(2)))/(S(2)asqrt(b)sqrt(aS(2) - bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a - bxS(2))(S(3)/2)sqrt(aS(2) - bS(2)x*S(4))), x), x, x(a + bxS(2))/(S(4)a*S(2)sqrt(a - bxS(2))sqrt(aS(2) - bS(2)xS(4))) + S(3)sqrt(S(2))sqrt(a - bx*S(2))sqrt(a + bxS(2))atanh(sqrt(S(2))sqrt(b)x/sqrt(a + bxS(2)))/(S(8)a*S(2)sqrt(b)sqrt(aS(2) - bS(2)x*S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a - bxS(2))(S(5)/2)sqrt(aS(2) - bS(2)x*S(4))), x), x, x(a + bxS(2))/(S(8)a*S(2)(a - bxS(2))(S(3)/2)sqrt(aS(2) - bS(2)xS(4))) + S(9)x(a + bx*S(2))/(S(32)a*S(3)sqrt(a - bxS(2))sqrt(aS(2) - bS(2)xS(4))) + S(19)sqrt(S(2))sqrt(a - bx*S(2))sqrt(a + bxS(2))atanh(sqrt(S(2))sqrt(b)x/sqrt(a + bxS(2)))/(S(64)a*S(3)sqrt(b)sqrt(aS(2) - bS(2)xS(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xS(2)/(xS(2) + S(-1)))/(xS(2) + S(1)), x), x, sqrt(S(2))sqrt(-xS(2)/(-xS(2) + S(1)))sqrt(x*S(2) + S(-1))atan(sqrt(S(2))sqrt(xS(2) + S(-1))/S(2))/(S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xS(2)/(a + xS(2)(a + S(1)) + S(-1)))/(xS(2) + S(1)), x), x, sqrt(S(2))sqrt(-xS(2)/(-a - xS(2)(a + S(1)) + S(1)))sqrt(a + x*S(2)(a + S(1)) + S(-1))atan(sqrt(S(2))sqrt(a + x*S(2)(a + S(1)) + S(-1))/S(2))/(S(2)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(((x + S(1))(xS(2) + S(-1)))(S(-2)/3), x), x, (S(3)xS(2)/S(2) + S(-3)/2)/((-x + S(-1))(-xS(2) + S(1)))(S(2)/3), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(((x + S(1))(xS(2) + S(-1)))(S(-2)/3), x), x, (x + S(1))(S(3)x/S(2) + S(-3)/2)/(xS(3) + xS(2) - x + S(-1))(S(2)/3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(-1))/(sqrt(x(x*S(2) + S(1)))(x*S(2) + S(1))), x), x, -S(2)x/sqrt(x(xS(2) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((xS(2) + S(-1))/((xS(2) + S(1))sqrt(x*S(3) + x)), x), x, -S(2)x/sqrt(xS(3) + x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((xS(2) + S(-1))*S(2)/(x(xS(2) + S(1))))/(xS(2) + S(1)), x), x, S(2)xsqrt((-xS(2) + S(1))S(2)/(x(xS(2) + S(1))))/(-xS(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((xS(2) + S(-1))S(2)/(xS(3) + x))/(xS(2) + S(1)), x), x, S(2)xsqrt((-xS(2) + S(1))S(2)/(xS(3) + x))/(-xS(2) + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(a + b/xS(2))sqrt(c + dxS(2))), x), x, sqrt(ax*S(2) + b)atanh(sqrt(d)sqrt(ax*S(2) + b)/(sqrt(a)sqrt(c + dxS(2))))/(sqrt(a)sqrt(d)xsqrt(a + b/xS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(xS(4) - S(2)xS(2))/((xS(2) + S(-1))(x*S(2) + S(2))), x), x, S(2)sqrt(x*S(4) - S(2)x*S(2))atan(sqrt(x*S(2) + S(-2))/S(2))/(S(3)xsqrt(xS(2) + S(-2))) - sqrt(xS(4) - S(2)x*S(2))atan(sqrt(x*S(2) + S(-2)))/(S(3)xsqrt(xS(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(1) - S(1)/(xS(2) + S(-1))S(2))/(-xS(2) + S(2)), x), x, sqrt(S(1) - S(1)/(-xS(2) + S(1))S(2))(-x*S(2) + S(1))atan(sqrt(x*S(2) + S(-2)))/(xsqrt(xS(2) + S(-2))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(sqrt(S(1) - S(1)/(xS(2) + S(-1))S(2))/(-xS(2) + S(2)), x), x, sqrt(S(1) - S(1)/(-xS(2) + S(1))S(2))(-xS(2) + S(1))sqrt(x*S(4) - S(2)x*S(2))atan(sqrt(x*S(2) + S(-2)))/(xsqrt(x*S(2) + S(-2))sqrt((xS(2) + S(-1))S(2) + S(-1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt((x*S(4) - S(2)xS(2))/(xS(2) + S(-1))S(2))/(xS(2) + S(2)), x), x, sqrt((x*S(4) - S(2)xS(2))/(-xS(2) + S(1))*S(2))(-x*S(2)/S(3) + S(1)/3)atan(sqrt(x*S(2) + S(-2)))/(xsqrt(xS(2) + S(-2))) + sqrt((xS(4) - S(2)xS(2))/(-xS(2) + S(1))S(2))(S(2)xS(2)/S(3) + S(-2)/3)atan(sqrt(x*S(2) + S(-2))/S(2))/(xsqrt(x*S(2) + S(-2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x/(xS(2) + S(1)) + S(1))(S(5)/2), x), x, -(-x/S(3) + S(1)/3)(x + S(1))S(3)sqrt(S(2)x/(xS(2) + S(1)) + S(1))/(xS(2) + S(1)) + (x + S(1))(S(8)x/S(3) + S(-4)/3)sqrt(S(2)x/(xS(2) + S(1)) + S(1)) - (S(3)x + S(4))(xS(2) + S(1))sqrt(S(2)x/(xS(2) + S(1)) + S(1))/(x + S(1)) + S(5)sqrt(x*S(2) + S(1))sqrt(S(2)x/(xS(2) + S(1)) + S(1))asinh(x)/(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x/(xS(2) + S(1)) + S(1))(S(3)/2), x), x, -x(x*S(2) + S(1))sqrt(S(2)x/(xS(2) + S(1)) + S(1))/(x + S(1)) + (x + S(-1))(x + S(1))sqrt(S(2)x/(x*S(2) + S(1)) + S(1)) + S(3)sqrt(x*S(2) + S(1))sqrt(S(2)x/(xS(2) + S(1)) + S(1))asinh(x)/(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(2)x/(xS(2) + S(1)) + S(1)), x), x, sqrt(xS(2) + S(1))sqrt(S(2)x/(xS(2) + S(1)) + S(1))asinh(x)/(x + S(1)) + (x*S(2) + S(1))sqrt(S(2)x/(xS(2) + S(1)) + S(1))/(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(S(2)x/(x*S(2) + S(1)) + S(1)), x), x, (x + S(1))/sqrt(S(2)x/(x*S(2) + S(1)) + S(1)) - (x + S(1))asinh(x)/(sqrt(x*S(2) + S(1))sqrt(S(2)x/(xS(2) + S(1)) + S(1))) - sqrt(S(2))(x + S(1))atanh(sqrt(S(2))(-x + S(1))/(S(2)sqrt(xS(2) + S(1))))/(sqrt(xS(2) + S(1))sqrt(S(2)x/(xS(2) + S(1)) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((S(2)x/(xS(2) + S(1)) + S(1))(S(-3)/2), x), x, (S(3)x/S(2) + S(3))/sqrt(S(2)x/(x*S(2) + S(1)) + S(1)) - (S(3)x + S(3))asinh(x)/(sqrt(xS(2) + S(1))sqrt(S(2)x/(xS(2) + S(1)) + S(1))) - sqrt(S(2))(S(9)x/S(2) + S(9)/2)atanh(sqrt(S(2))(-x + S(1))/(S(2)sqrt(x*S(2) + S(1))))/(S(2)sqrt(x*S(2) + S(1))sqrt(S(2)x/(xS(2) + S(1)) + S(1))) + (-xS(2)/S(2) + S(-1)/2)/((x + S(1))sqrt(S(2)x/(xS(2) + S(1)) + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(S(2)x/(xS(2) + S(1)) + S(1))/(xS(2) + S(1)), x), x, (x + S(-1))sqrt(S(2)x/(xS(2) + S(1)) + S(1))/(x + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(c/(a + bxS(2)))(S(3)/2), x), x, -cxsqrt(c/(a + bxS(2)))/b + csqrt(c/(a + bxS(2)))sqrt(a + bxS(2))atanh(sqrt(b)x/sqrt(a + bxS(2)))/b(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(c/(a + bxS(2)))(S(3)/2), x), x, -csqrt(c/(a + bx*S(2)))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c/(a + bxS(2)))(S(3)/2), x), x, cxsqrt(c/(a + bxS(2)))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c/(a + bxS(2)))(S(3)/2)/x, x), x, csqrt(c/(a + bx*S(2)))/a - csqrt(c/(a + bxS(2)))sqrt(a + bxS(2))atanh(sqrt(a + bxS(2))/sqrt(a))/a(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c/(a + bxS(2)))(S(3)/2)/x*S(2), x), x, -csqrt(c/(a + bxS(2)))/(ax) - S(2)bcxsqrt(c/(a + bxS(2)))/aS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c/(a + bxS(2)))(S(3)/2)/x*S(3), x), x, csqrt(c/(a + bxS(2)))/(ax*S(2)) - S(3)csqrt(c/(a + bx*S(2)))(a + bxS(2))/(S(2)a*S(2)x*S(2)) + S(3)bcsqrt(c/(a + bxS(2)))sqrt(a + bxS(2))atanh(sqrt(a + bxS(2))/sqrt(a))/(S(2)a(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(c(a + bxS(2))S(3))(S(3)/2), x), x, -S(21)a*S(6)csqrt(c(a + bxS(2))S(3))atanh(sqrt(b)x/sqrt(a + bx*S(2)))/(S(1024)b*(S(3)/2)(a + bxS(2))(S(3)/2)) + S(21)a*S(5)cxsqrt(c(a + bxS(2))S(3))/(S(1024)b(a + bxS(2))) + S(21)a*S(4)cxS(3)sqrt(c(a + bxS(2))S(3))/(S(512)(a + bx*S(2))) + S(7)a*S(3)cxS(3)sqrt(c(a + bxS(2))S(3))/S(128) + S(21)aS(2)cxS(3)sqrt(c(a + bxS(2))S(3))(a + bx*S(2))/S(320) + S(3)acx*S(3)sqrt(c(a + bxS(2))S(3))(a + bxS(2))S(2)/S(40) + cxS(3)sqrt(c(a + bxS(2))S(3))(a + bxS(2))S(3)/S(12), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(c(a + bxS(2))S(3))(S(3)/2), x), x, csqrt(c(a + bxS(2))S(3))(a + bxS(2))S(4)/(S(11)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c(a + bxS(2))S(3))(S(3)/2), x), x, S(63)a*S(5)csqrt(c(a + bxS(2))S(3))atanh(sqrt(b)x/sqrt(a + bx*S(2)))/(S(256)sqrt(b)(a + bxS(2))(S(3)/2)) + S(63)aS(4)cxsqrt(c(a + bxS(2))S(3))/(S(256)(a + bx*S(2))) + S(21)a*S(3)cxsqrt(c(a + bxS(2))S(3))/S(128) + S(21)aS(2)cxsqrt(c(a + bxS(2))S(3))(a + bx*S(2))/S(160) + S(9)acxsqrt(c(a + bxS(2))S(3))(a + bxS(2))S(2)/S(80) + cxsqrt(c(a + bxS(2))S(3))(a + bxS(2))S(3)/S(10), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c(a + bxS(2))S(3))(S(3)/2)/x, x), x, -a(S(9)/2)csqrt(c(a + bxS(2))S(3))atanh(sqrt(a + bxS(2))/sqrt(a))/(a + bxS(2))(S(3)/2) + a*S(4)csqrt(c(a + bxS(2))S(3))/(a + bxS(2)) + aS(3)csqrt(c(a + bxS(2))S(3))/S(3) + a*S(2)csqrt(c(a + bxS(2))S(3))(a + bxS(2))/S(5) + acsqrt(c(a + bxS(2))S(3))(a + bxS(2))S(2)/S(7) + csqrt(c(a + bxS(2))S(3))(a + bxS(2))S(3)/S(9), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c(a + bxS(2))S(3))(S(3)/2)/xS(2), x), x, S(315)aS(4)sqrt(b)csqrt(c(a + bxS(2))S(3))atanh(sqrt(b)x/sqrt(a + bxS(2)))/(S(128)(a + bxS(2))(S(3)/2)) + S(315)a*S(3)bcxsqrt(c(a + bxS(2))S(3))/(S(128)(a + bxS(2))) + S(105)a*S(2)bcxsqrt(c(a + bxS(2))S(3))/S(64) + S(21)abcxsqrt(c(a + bxS(2))S(3))(a + bx*S(2))/S(16) + S(9)bcxsqrt(c(a + bxS(2))S(3))(a + bxS(2))S(2)/S(8) - csqrt(c(a + bxS(2))S(3))(a + bxS(2))S(3)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c(a + bxS(2))S(3))(S(3)/2)/xS(3), x), x, -S(9)a(S(7)/2)bcsqrt(c(a + bxS(2))S(3))atanh(sqrt(a + bx*S(2))/sqrt(a))/(S(2)(a + bxS(2))(S(3)/2)) + S(9)a*S(3)bcsqrt(c(a + bxS(2))S(3))/(S(2)(a + bx*S(2))) + S(3)a*S(2)bcsqrt(c(a + bxS(2))S(3))/S(2) + S(9)abcsqrt(c(a + bxS(2))S(3))(a + bx*S(2))/S(10) + S(9)bcsqrt(c(a + bxS(2))S(3))(a + bxS(2))S(2)/S(14) - csqrt(c(a + bxS(2))S(3))(a + bxS(2))S(3)/(S(2)x**S(2)), expand=True, _diff=True, _numerical=True)
7ec6296ee65adc30d6bf67204c47ca05b2de818da92be6df09b7585d63d8768b	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.utility_function import ( sympy_op_factory, Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ, ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ, PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ, NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest, SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient, Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan, ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec, ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less, Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ, InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ, EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, ExpandLinearProduct, GCD, ContentFactor, NumericFactor, NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, ExpandExpression, Apart, SmartApart, MatchQ, PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, RationalFunctionFactors, NonrationalFunctionFactors, Reverse, RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, GeneralizedBinomialParts, GeneralizedTrinomialDegree, GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, MinimumMonomialExponent, MonomialExponent, LinearMatchQ, PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, AlgebraicFunctionFactors, NonalgebraicFunctionFactors, QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, CombineExponents, FactorInteger, FactorAbsurdNumber, SubstForInverseFunction, SubstForFractionalPower, SubstForFractionalPowerOfQuotientOfLinears, FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, NormalizeIntegrandAux, NormalizeIntegrandFactor, NormalizeIntegrandFactorBase, NormalizeTogether, NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand, SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, KnownTangentIntegrandQ, KnownCotangentIntegrandQ, KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, FunctionOfLog, PowerVariableExpn, PowerVariableDegree, PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, ConstantFactor, SameQ, ReplacePart, CommonFactors, MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, _SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify, _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor, PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral, CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial, Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ ) from sympy.core.add import Add from sympy.core.mod import Mod from sympy.core.mul import Mul from sympy.core.numbers import (Float, I, Integer) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.functions.elementary.complexes import Abs from sympy.functions.elementary.miscellaneous import sqrt from sympy.integrals.integrals import Integral as Integrate from sympy.logic.boolalg import (And, Or) from sympy.simplify.simplify import simplify from sympy.integrals.rubi.symbol import WC from sympy.core.symbol import symbols, Symbol from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf, exp, log) from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch) from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec) from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.core.numbers import pi as Pi a, b, c, d, e, f, m, n, x, u , k, p, r, s, t, i, j= symbols('a b c d e f m n x u k p r s t i j') A, B, C, D, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B C D a b c d e f g h y z m n p q u v w F', ) def test_1(): assert rubi_test(rubi_integrate(x*S(4)asinh(ax), x), x, xS(5)asinh(ax)/S(5) - (aS(2)xS(2) + S(1))(S(5)/2)/(S(25)aS(5)) + S(2)(a*S(2)xS(2) + S(1))(S(3)/2)/(S(15)aS(5)) - sqrt(aS(2)x*S(2) + S(1))/(S(5)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)asinh(ax), x), x, x*S(4)asinh(ax)/S(4) - xS(3)sqrt(a*S(2)x*S(2) + S(1))/(S(16)a) + S(3)xsqrt(a*S(2)x*S(2) + S(1))/(S(32)a*S(3)) - S(3)asinh(ax)/(S(32)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asinh(ax), x), x, x*S(3)asinh(ax)/S(3) - (aS(2)xS(2) + S(1))(S(3)/2)/(S(9)aS(3)) + sqrt(aS(2)x*S(2) + S(1))/(S(3)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasinh(ax), x), x, xS(2)asinh(ax)/S(2) - xsqrt(a*S(2)x*S(2) + S(1))/(S(4)a) + asinh(ax)/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax), x), x, xasinh(ax) - sqrt(a*S(2)x*S(2) + S(1))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)/x, x), x, PolyLog(S(2), exp(S(2)asinh(ax)))/S(2) + log(-exp(S(2)asinh(ax)) + S(1))asinh(ax) - asinh(ax)S(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)/x*S(2), x), x, -aatanh(sqrt(a*S(2)x*S(2) + S(1))) - asinh(ax)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)/xS(3), x), x, -asqrt(a*S(2)x*S(2) + S(1))/(S(2)x) - asinh(ax)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)/xS(4), x), x, aS(3)atanh(sqrt(aS(2)x*S(2) + S(1)))/S(6) - asqrt(a*S(2)x*S(2) + S(1))/(S(6)x*S(2)) - asinh(ax)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)/xS(5), x), x, aS(3)sqrt(aS(2)x*S(2) + S(1))/(S(6)x) - asqrt(aS(2)x*S(2) + S(1))/(S(12)x*S(3)) - asinh(ax)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)/x*S(6), x), x, -S(3)a*S(5)atanh(sqrt(a*S(2)x*S(2) + S(1)))/S(40) + S(3)a*S(3)sqrt(a*S(2)x*S(2) + S(1))/(S(40)x*S(2)) - asqrt(a*S(2)x*S(2) + S(1))/(S(20)x*S(4)) - asinh(ax)/(S(5)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)asinh(ax)S(2), x), x, xS(5)asinh(ax)S(2)/S(5) + S(2)x*S(5)/S(125) - S(2)x*S(4)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(25)a) - S(8)xS(3)/(S(225)a*S(2)) + S(8)x*S(2)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(75)a*S(3)) + S(16)x/(S(75)aS(4)) - S(16)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(75)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)asinh(ax)S(2), x), x, xS(4)asinh(ax)S(2)/S(4) + xS(4)/S(32) - x*S(3)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(8)a) - S(3)xS(2)/(S(32)a*S(2)) + S(3)xsqrt(aS(2)x*S(2) + S(1))asinh(ax)/(S(16)a*S(3)) - S(3)asinh(ax)S(2)/(S(32)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asinh(ax)S(2), x), x, xS(3)asinh(ax)*S(2)/S(3) + S(2)x*S(3)/S(27) - S(2)x*S(2)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(9)a) - S(4)x/(S(9)a*S(2)) + S(4)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(9)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasinh(ax)S(2), x), x, xS(2)asinh(ax)S(2)/S(2) + xS(2)/S(4) - xsqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(2)a) + asinh(ax)S(2)/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*S(2), x), x, xasinh(ax)S(2) + S(2)x - S(2)sqrt(aS(2)x*S(2) + S(1))asinh(ax)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*S(2)/x, x), x, PolyLog(S(2), exp(S(2)asinh(ax)))asinh(ax) - PolyLog(S(3), exp(S(2)asinh(ax)))/S(2) + log(-exp(S(2)asinh(ax)) + S(1))asinh(ax)S(2) - asinh(ax)*S(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)S(2)/xS(2), x), x, -S(2)aPolyLog(S(2), -exp(asinh(ax))) + S(2)aPolyLog(S(2), exp(asinh(ax))) - S(4)aasinh(ax)atanh(exp(asinh(ax))) - asinh(ax)*S(2)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)S(2)/xS(3), x), x, a*S(2)log(x) - asqrt(aS(2)x*S(2) + S(1))asinh(ax)/x - asinh(ax)*S(2)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)S(2)/xS(4), x), x, a*S(3)PolyLog(S(2), -exp(asinh(ax)))/S(3) - aS(3)PolyLog(S(2), exp(asinh(ax)))/S(3) + S(2)a*S(3)asinh(ax)atanh(exp(asinh(ax)))/S(3) - aS(2)/(S(3)x) - asqrt(aS(2)x*S(2) + S(1))asinh(ax)/(S(3)x*S(2)) - asinh(ax)*S(2)/(S(3)x*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)S(2)/xS(5), x), x, -a*S(4)log(x)/S(3) + a*S(3)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(3)x) - a*S(2)/(S(12)x*S(2)) - asqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(6)x*S(3)) - asinh(ax)*S(2)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)asinh(ax)S(3), x), x, xS(5)asinh(ax)*S(3)/S(5) + S(6)x*S(5)asinh(ax)/S(125) - S(3)x*S(4)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(2)/(S(25)a) - S(8)xS(3)asinh(ax)/(S(75)a*S(2)) + S(4)x*S(2)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(2)/(S(25)a*S(3)) + S(16)xasinh(ax)/(S(25)aS(4)) - S(6)(a*S(2)xS(2) + S(1))(S(5)/2)/(S(625)aS(5)) + S(76)(a*S(2)xS(2) + S(1))(S(3)/2)/(S(1125)aS(5)) - S(8)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(2)/(S(25)a*S(5)) - S(298)sqrt(a*S(2)x*S(2) + S(1))/(S(375)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)asinh(ax)S(3), x), x, xS(4)asinh(ax)*S(3)/S(4) + S(3)x*S(4)asinh(ax)/S(32) - S(3)x*S(3)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(2)/(S(16)a) - S(3)xS(3)sqrt(a*S(2)x*S(2) + S(1))/(S(128)a) - S(9)xS(2)asinh(ax)/(S(32)a*S(2)) + S(9)xsqrt(aS(2)x*S(2) + S(1))asinh(ax)S(2)/(S(32)a*S(3)) + S(45)xsqrt(aS(2)x*S(2) + S(1))/(S(256)a*S(3)) - S(3)asinh(ax)S(3)/(S(32)a*S(4)) - S(45)asinh(ax)/(S(256)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asinh(ax)S(3), x), x, xS(3)asinh(ax)*S(3)/S(3) + S(2)x*S(3)asinh(ax)/S(9) - xS(2)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(2)/(S(3)a) - S(4)xasinh(ax)/(S(3)a*S(2)) - S(2)(a*S(2)xS(2) + S(1))(S(3)/2)/(S(27)aS(3)) + S(2)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(2)/(S(3)a*S(3)) + S(14)sqrt(a*S(2)x*S(2) + S(1))/(S(9)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasinh(ax)S(3), x), x, xS(2)asinh(ax)S(3)/S(2) + S(3)x*S(2)asinh(ax)/S(4) - S(3)xsqrt(aS(2)x*S(2) + S(1))asinh(ax)S(2)/(S(4)a) - S(3)xsqrt(a*S(2)x*S(2) + S(1))/(S(8)a) + asinh(ax)S(3)/(S(4)a*S(2)) + S(3)asinh(ax)/(S(8)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*S(3), x), x, xasinh(ax)S(3) + S(6)xasinh(ax) - S(3)sqrt(aS(2)x*S(2) + S(1))asinh(ax)S(2)/a - S(6)sqrt(a*S(2)x*S(2) + S(1))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*S(3)/x, x), x, S(3)PolyLog(S(2), exp(S(2)asinh(ax)))asinh(ax)*S(2)/S(2) - S(3)PolyLog(S(3), exp(S(2)asinh(ax)))asinh(ax)/S(2) + S(3)PolyLog(S(4), exp(S(2)asinh(ax)))/S(4) + log(-exp(S(2)asinh(ax)) + S(1))asinh(ax)S(3) - asinh(ax)*S(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)S(3)/xS(2), x), x, -S(6)aPolyLog(S(2), -exp(asinh(ax)))asinh(ax) + S(6)aPolyLog(S(2), exp(asinh(ax)))asinh(ax) + S(6)aPolyLog(S(3), -exp(asinh(ax))) - S(6)aPolyLog(S(3), exp(asinh(ax))) - S(6)aasinh(ax)S(2)atanh(exp(asinh(ax))) - asinh(ax)*S(3)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)S(3)/xS(3), x), x, S(3)aS(2)PolyLog(S(2), exp(S(2)asinh(ax)))/S(2) + S(3)aS(2)log(-exp(S(2)asinh(ax)) + S(1))asinh(ax) - S(3)aS(2)asinh(ax)S(2)/S(2) - S(3)asqrt(aS(2)x*S(2) + S(1))asinh(ax)S(2)/(S(2)x) - asinh(ax)S(3)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)S(3)/xS(4), x), x, a*S(3)PolyLog(S(2), -exp(asinh(ax)))asinh(ax) - aS(3)PolyLog(S(2), exp(asinh(ax)))asinh(ax) - aS(3)PolyLog(S(3), -exp(asinh(ax))) + aS(3)PolyLog(S(3), exp(asinh(ax))) + aS(3)asinh(ax)S(2)atanh(exp(asinh(ax))) - aS(3)atanh(sqrt(a*S(2)xS(2) + S(1))) - aS(2)asinh(ax)/x - asqrt(aS(2)x*S(2) + S(1))asinh(ax)S(2)/(S(2)x*S(2)) - asinh(ax)*S(3)/(S(3)x*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)S(3)/xS(5), x), x, -a*S(4)PolyLog(S(2), exp(S(2)asinh(ax)))/S(2) - a*S(4)log(-exp(S(2)asinh(ax)) + S(1))asinh(ax) + a*S(4)asinh(ax)S(2)/S(2) + aS(3)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(2)/(S(2)x) - a*S(3)sqrt(a*S(2)x*S(2) + S(1))/(S(4)x) - a*S(2)asinh(ax)/(S(4)x*S(2)) - asqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(2)/(S(4)x*S(3)) - asinh(ax)*S(3)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)asinh(ax)S(4), x), x, xS(6)asinh(ax)S(4)/S(6) + xS(6)asinh(ax)S(2)/S(18) + xS(6)/S(324) - x*S(5)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(3)/(S(9)a) - x*S(5)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(54)a) - S(5)xS(4)asinh(ax)S(2)/(S(48)a*S(2)) - S(65)x*S(4)/(S(3456)a*S(2)) + S(5)x*S(3)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(3)/(S(36)a*S(3)) + S(65)x*S(3)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(864)a*S(3)) + S(5)x*S(2)asinh(ax)S(2)/(S(16)a*S(4)) + S(245)x*S(2)/(S(1152)a*S(4)) - S(5)xsqrt(aS(2)x*S(2) + S(1))asinh(ax)S(3)/(S(24)a*S(5)) - S(245)xsqrt(aS(2)x*S(2) + S(1))asinh(ax)/(S(576)a*S(5)) + S(5)asinh(ax)S(4)/(S(96)a*S(6)) + S(245)asinh(ax)S(2)/(S(1152)aS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)asinh(ax)S(4), x), x, xS(5)asinh(ax)*S(4)/S(5) + S(12)x*S(5)asinh(ax)S(2)/S(125) + S(24)x*S(5)/S(3125) - S(4)x*S(4)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(3)/(S(25)a) - S(24)xS(4)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(625)a) - S(16)xS(3)asinh(ax)S(2)/(S(75)a*S(2)) - S(1088)x*S(3)/(S(16875)a*S(2)) + S(16)x*S(2)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(3)/(S(75)a*S(3)) + S(1088)x*S(2)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(5625)a*S(3)) + S(32)xasinh(ax)*S(2)/(S(25)a*S(4)) + S(16576)x/(S(5625)aS(4)) - S(32)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(3)/(S(75)a*S(5)) - S(16576)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(5625)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)asinh(ax)S(4), x), x, xS(4)asinh(ax)*S(4)/S(4) + S(3)x*S(4)asinh(ax)S(2)/S(16) + S(3)xS(4)/S(128) - xS(3)sqrt(aS(2)x*S(2) + S(1))asinh(ax)S(3)/(S(4)a) - S(3)xS(3)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(32)a) - S(9)xS(2)asinh(ax)S(2)/(S(16)a*S(2)) - S(45)x*S(2)/(S(128)a*S(2)) + S(3)xsqrt(aS(2)x*S(2) + S(1))asinh(ax)S(3)/(S(8)a*S(3)) + S(45)xsqrt(aS(2)x*S(2) + S(1))asinh(ax)/(S(64)a*S(3)) - S(3)asinh(ax)S(4)/(S(32)a*S(4)) - S(45)asinh(ax)S(2)/(S(128)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asinh(ax)S(4), x), x, xS(3)asinh(ax)*S(4)/S(3) + S(4)x*S(3)asinh(ax)S(2)/S(9) + S(8)x*S(3)/S(81) - S(4)x*S(2)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(3)/(S(9)a) - S(8)xS(2)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(27)a) - S(8)xasinh(ax)S(2)/(S(3)a*S(2)) - S(160)x/(S(27)aS(2)) + S(8)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(3)/(S(9)a*S(3)) + S(160)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/(S(27)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasinh(ax)S(4), x), x, xS(2)asinh(ax)S(4)/S(2) + S(3)x*S(2)asinh(ax)S(2)/S(2) + S(3)x*S(2)/S(4) - xsqrt(a*S(2)x*S(2) + S(1))asinh(ax)S(3)/a - S(3)xsqrt(aS(2)x*S(2) + S(1))asinh(ax)/(S(2)a) + asinh(ax)S(4)/(S(4)a*S(2)) + S(3)asinh(ax)S(2)/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*S(4), x), x, xasinh(ax)S(4) + S(12)xasinh(ax)*S(2) + S(24)x - S(4)sqrt(aS(2)x*S(2) + S(1))asinh(ax)S(3)/a - S(24)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*S(4)/x, x), x, S(2)PolyLog(S(2), exp(S(2)asinh(ax)))asinh(ax)*S(3) - S(3)PolyLog(S(3), exp(S(2)asinh(ax)))asinh(ax)*S(2) + S(3)PolyLog(S(4), exp(S(2)asinh(ax)))asinh(ax) - S(3)PolyLog(S(5), exp(S(2)asinh(ax)))/S(2) + log(-exp(S(2)asinh(ax)) + S(1))asinh(ax)S(4) - asinh(ax)*S(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)S(4)/xS(2), x), x, -S(12)aPolyLog(S(2), -exp(asinh(ax)))asinh(ax)S(2) + S(12)aPolyLog(S(2), exp(asinh(ax)))asinh(ax)*S(2) + S(24)aPolyLog(S(3), -exp(asinh(ax)))asinh(ax) - S(24)aPolyLog(S(3), exp(asinh(ax)))asinh(ax) - S(24)aPolyLog(S(4), -exp(asinh(ax))) + S(24)aPolyLog(S(4), exp(asinh(ax))) - S(8)aasinh(ax)*S(3)atanh(exp(asinh(ax))) - asinh(ax)*S(4)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)S(4)/xS(3), x), x, S(6)aS(2)PolyLog(S(2), exp(S(2)asinh(ax)))asinh(ax) - S(3)aS(2)PolyLog(S(3), exp(S(2)asinh(ax))) + S(6)aS(2)log(-exp(S(2)asinh(ax)) + S(1))asinh(ax)*S(2) - S(2)a*S(2)asinh(ax)S(3) - S(2)asqrt(aS(2)x*S(2) + S(1))asinh(ax)S(3)/x - asinh(ax)*S(4)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)S(4)/xS(4), x), x, S(2)aS(3)PolyLog(S(2), -exp(asinh(ax)))asinh(ax)S(2) - S(4)a*S(3)PolyLog(S(2), -exp(asinh(ax))) - S(2)a*S(3)PolyLog(S(2), exp(asinh(ax)))asinh(ax)S(2) + S(4)a*S(3)PolyLog(S(2), exp(asinh(ax))) - S(4)a*S(3)PolyLog(S(3), -exp(asinh(ax)))asinh(ax) + S(4)a*S(3)PolyLog(S(3), exp(asinh(ax)))asinh(ax) + S(4)a*S(3)PolyLog(S(4), -exp(asinh(ax))) - S(4)a*S(3)PolyLog(S(4), exp(asinh(ax))) + S(4)a*S(3)asinh(ax)S(3)atanh(exp(asinh(ax)))/S(3) - S(8)a*S(3)asinh(ax)atanh(exp(asinh(ax))) - S(2)a*S(2)asinh(ax)S(2)/x - S(2)asqrt(aS(2)x*S(2) + S(1))asinh(ax)S(3)/(S(3)x*S(2)) - asinh(ax)*S(4)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/asinh(ax), x), x, -S(5)CoshIntegral(asinh(ax))/(S(64)a*S(7)) + S(9)CoshIntegral(S(3)asinh(ax))/(S(64)aS(7)) - S(5)CoshIntegral(S(5)asinh(ax))/(S(64)aS(7)) + CoshIntegral(S(7)asinh(ax))/(S(64)aS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/asinh(ax), x), x, S(5)SinhIntegral(S(2)asinh(ax))/(S(32)aS(6)) - SinhIntegral(S(4)asinh(ax))/(S(8)a*S(6)) + SinhIntegral(S(6)asinh(ax))/(S(32)aS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asinh(ax), x), x, CoshIntegral(asinh(ax))/(S(8)aS(5)) - S(3)CoshIntegral(S(3)asinh(ax))/(S(16)aS(5)) + CoshIntegral(S(5)asinh(ax))/(S(16)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asinh(ax), x), x, -SinhIntegral(S(2)asinh(ax))/(S(4)a*S(4)) + SinhIntegral(S(4)asinh(ax))/(S(8)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/asinh(ax), x), x, -CoshIntegral(asinh(ax))/(S(4)aS(3)) + CoshIntegral(S(3)asinh(ax))/(S(4)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asinh(ax), x), x, SinhIntegral(S(2)asinh(ax))/(S(2)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/asinh(ax), x), x, CoshIntegral(asinh(ax))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasinh(ax)), x), x, Integrate(S(1)/(xasinh(ax)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)asinh(ax)), x), x, Integrate(S(1)/(xS(2)asinh(ax)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/asinh(ax)S(2), x), x, -xS(6)sqrt(aS(2)x*S(2) + S(1))/(aasinh(ax)) - S(5)SinhIntegral(asinh(ax))/(S(64)a*S(7)) + S(27)SinhIntegral(S(3)asinh(ax))/(S(64)aS(7)) - S(25)SinhIntegral(S(5)asinh(ax))/(S(64)aS(7)) + S(7)SinhIntegral(S(7)asinh(ax))/(S(64)aS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/asinh(ax)S(2), x), x, -xS(5)sqrt(aS(2)x*S(2) + S(1))/(aasinh(ax)) + S(5)CoshIntegral(S(2)asinh(ax))/(S(16)aS(6)) - CoshIntegral(S(4)asinh(ax))/(S(2)a*S(6)) + S(3)CoshIntegral(S(6)asinh(ax))/(S(16)aS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asinh(ax)S(2), x), x, -xS(4)sqrt(aS(2)x*S(2) + S(1))/(aasinh(ax)) + SinhIntegral(asinh(ax))/(S(8)aS(5)) - S(9)SinhIntegral(S(3)asinh(ax))/(S(16)aS(5)) + S(5)SinhIntegral(S(5)asinh(ax))/(S(16)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asinh(ax)S(2), x), x, -xS(3)sqrt(aS(2)x*S(2) + S(1))/(aasinh(ax)) - CoshIntegral(S(2)asinh(ax))/(S(2)a*S(4)) + CoshIntegral(S(4)asinh(ax))/(S(2)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/asinh(ax)S(2), x), x, -xS(2)sqrt(a*S(2)x*S(2) + S(1))/(aasinh(ax)) - SinhIntegral(asinh(ax))/(S(4)aS(3)) + S(3)SinhIntegral(S(3)asinh(ax))/(S(4)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asinh(ax)*S(2), x), x, -xsqrt(a*S(2)x*S(2) + S(1))/(aasinh(ax)) + CoshIntegral(S(2)asinh(ax))/aS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)(S(-2)), x), x, -sqrt(aS(2)xS(2) + S(1))/(aasinh(ax)) + SinhIntegral(asinh(ax))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasinh(ax)*S(2)), x), x, Integrate(S(1)/(xasinh(ax)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)asinh(ax)S(2)), x), x, Integrate(S(1)/(xS(2)asinh(ax)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asinh(ax)*S(3), x), x, -S(5)x*S(5)/(S(2)asinh(ax)) - xS(4)sqrt(a*S(2)x*S(2) + S(1))/(S(2)aasinh(ax)*S(2)) - S(2)xS(3)/(aS(2)asinh(ax)) + CoshIntegral(asinh(ax))/(S(16)a*S(5)) - S(27)CoshIntegral(S(3)asinh(ax))/(S(32)aS(5)) + S(25)CoshIntegral(S(5)asinh(ax))/(S(32)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asinh(ax)*S(3), x), x, -S(2)x*S(4)/asinh(ax) - x*S(3)sqrt(a*S(2)x*S(2) + S(1))/(S(2)aasinh(ax)*S(2)) - S(3)x*S(2)/(S(2)a*S(2)asinh(ax)) - SinhIntegral(S(2)asinh(ax))/(S(2)a*S(4)) + SinhIntegral(S(4)asinh(ax))/aS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/asinh(ax)*S(3), x), x, -S(3)x*S(3)/(S(2)asinh(ax)) - xS(2)sqrt(a*S(2)x*S(2) + S(1))/(S(2)aasinh(ax)S(2)) - x/(aS(2)asinh(ax)) - CoshIntegral(asinh(ax))/(S(8)a*S(3)) + S(9)CoshIntegral(S(3)asinh(ax))/(S(8)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asinh(ax)S(3), x), x, -xS(2)/asinh(ax) - xsqrt(a*S(2)x*S(2) + S(1))/(S(2)aasinh(ax)*S(2)) + SinhIntegral(S(2)asinh(ax))/aS(2) - S(1)/(S(2)a*S(2)asinh(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*(S(-3)), x), x, -x/(S(2)asinh(ax)) - sqrt(aS(2)x*S(2) + S(1))/(S(2)aasinh(ax)*S(2)) + CoshIntegral(asinh(ax))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasinh(ax)S(3)), x), x, Integrate(S(1)/(xasinh(ax)S(3)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)asinh(ax)S(3)), x), x, Integrate(S(1)/(xS(2)asinh(ax)S(3)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asinh(ax)*S(4), x), x, -S(5)x*S(5)/(S(6)asinh(ax)S(2)) - S(25)x*S(4)sqrt(a*S(2)x*S(2) + S(1))/(S(6)aasinh(ax)) - x*S(4)sqrt(a*S(2)x*S(2) + S(1))/(S(3)aasinh(ax)*S(3)) - S(2)x*S(3)/(S(3)a*S(2)asinh(ax)S(2)) - S(2)x*S(2)sqrt(a*S(2)xS(2) + S(1))/(aS(3)asinh(ax)) + SinhIntegral(asinh(ax))/(S(48)a*S(5)) - S(27)SinhIntegral(S(3)asinh(ax))/(S(32)aS(5)) + S(125)SinhIntegral(S(5)asinh(ax))/(S(96)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asinh(ax)*S(4), x), x, -S(2)x*S(4)/(S(3)asinh(ax)S(2)) - S(8)x*S(3)sqrt(a*S(2)x*S(2) + S(1))/(S(3)aasinh(ax)) - x*S(3)sqrt(a*S(2)x*S(2) + S(1))/(S(3)aasinh(ax)S(3)) - xS(2)/(S(2)aS(2)asinh(ax)S(2)) - xsqrt(a*S(2)xS(2) + S(1))/(aS(3)asinh(ax)) - CoshIntegral(S(2)asinh(ax))/(S(3)aS(4)) + S(4)CoshIntegral(S(4)asinh(ax))/(S(3)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/asinh(ax)S(4), x), x, -xS(3)/(S(2)asinh(ax)*S(2)) - S(3)x*S(2)sqrt(a*S(2)x*S(2) + S(1))/(S(2)aasinh(ax)) - x*S(2)sqrt(a*S(2)x*S(2) + S(1))/(S(3)aasinh(ax)*S(3)) - x/(S(3)a*S(2)asinh(ax)S(2)) - sqrt(aS(2)x*S(2) + S(1))/(S(3)a*S(3)asinh(ax)) - SinhIntegral(asinh(ax))/(S(24)aS(3)) + S(9)SinhIntegral(S(3)asinh(ax))/(S(8)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asinh(ax)S(4), x), x, -xS(2)/(S(3)asinh(ax)*S(2)) - S(2)xsqrt(aS(2)x*S(2) + S(1))/(S(3)aasinh(ax)) - xsqrt(aS(2)x*S(2) + S(1))/(S(3)aasinh(ax)*S(3)) + S(2)CoshIntegral(S(2)asinh(ax))/(S(3)aS(2)) - S(1)/(S(6)a*S(2)asinh(ax)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*(S(-4)), x), x, -x/(S(6)asinh(ax)S(2)) - sqrt(aS(2)x*S(2) + S(1))/(S(6)aasinh(ax)) - sqrt(a*S(2)x*S(2) + S(1))/(S(3)aasinh(ax)*S(3)) + SinhIntegral(asinh(ax))/(S(6)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasinh(ax)S(4)), x), x, Integrate(S(1)/(xasinh(ax)S(4)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)asinh(ax)S(4)), x), x, Integrate(S(1)/(xS(2)asinh(ax)S(4)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)sqrt(asinh(ax)), x), x, -sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(192)a*S(5)) + sqrt(S(5))sqrt(Pi)Erf(sqrt(S(5))sqrt(asinh(ax)))/(S(1600)a*S(5)) + sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(32)a*S(5)) + sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(192)a*S(5)) - sqrt(S(5))sqrt(Pi)Erfi(sqrt(S(5))sqrt(asinh(ax)))/(S(1600)a*S(5)) - sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(32)aS(5)) + xS(5)sqrt(asinh(ax))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(3)sqrt(asinh(ax)), x), x, sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(64)a*S(4)) - sqrt(Pi)Erf(S(2)sqrt(asinh(ax)))/(S(256)aS(4)) + sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(64)a*S(4)) - sqrt(Pi)Erfi(S(2)sqrt(asinh(ax)))/(S(256)aS(4)) + xS(4)sqrt(asinh(ax))/S(4) - S(3)sqrt(asinh(ax))/(S(32)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(asinh(ax)), x), x, sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(144)aS(3)) - sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(16)a*S(3)) - sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(144)a*S(3)) + sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(16)aS(3)) + xS(3)sqrt(asinh(ax))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(asinh(ax)), x), x, -sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(32)aS(2)) - sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(32)aS(2)) + xS(2)sqrt(asinh(ax))/S(2) + sqrt(asinh(ax))/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(asinh(ax)), x), x, sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(4)a) - sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(4)a) + xsqrt(asinh(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(asinh(ax))/x, x), x, Integrate(sqrt(asinh(ax))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)asinh(ax)(S(3)/2), x), x, -sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(384)a*S(5)) + S(3)sqrt(S(5))sqrt(Pi)Erf(sqrt(S(5))sqrt(asinh(ax)))/(S(16000)aS(5)) + S(3)sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(64)aS(5)) - sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(384)a*S(5)) + S(3)sqrt(S(5))sqrt(Pi)Erfi(sqrt(S(5))sqrt(asinh(ax)))/(S(16000)aS(5)) + S(3)sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(64)aS(5)) + xS(5)asinh(ax)(S(3)/2)/S(5) - S(3)x*S(4)sqrt(a*S(2)x*S(2) + S(1))sqrt(asinh(ax))/(S(50)a) + S(2)xS(2)sqrt(a*S(2)x*S(2) + S(1))sqrt(asinh(ax))/(S(25)a*S(3)) - S(4)sqrt(a*S(2)x*S(2) + S(1))sqrt(asinh(ax))/(S(25)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)asinh(ax)*(S(3)/2), x), x, S(3)sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(256)aS(4)) - S(3)sqrt(Pi)Erf(S(2)sqrt(asinh(ax)))/(S(2048)a*S(4)) - S(3)sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(256)aS(4)) + S(3)sqrt(Pi)Erfi(S(2)sqrt(asinh(ax)))/(S(2048)aS(4)) + xS(4)asinh(ax)*(S(3)/2)/S(4) - S(3)x*S(3)sqrt(a*S(2)x*S(2) + S(1))sqrt(asinh(ax))/(S(32)a) + S(9)xsqrt(a*S(2)x*S(2) + S(1))sqrt(asinh(ax))/(S(64)a*S(3)) - S(3)asinh(ax)(S(3)/2)/(S(32)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asinh(ax)*(S(3)/2), x), x, sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(288)a*S(3)) - S(3)sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(32)aS(3)) + sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(288)a*S(3)) - S(3)sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(32)aS(3)) + xS(3)asinh(ax)(S(3)/2)/S(3) - xS(2)sqrt(a*S(2)x*S(2) + S(1))sqrt(asinh(ax))/(S(6)a) + sqrt(a*S(2)x*S(2) + S(1))sqrt(asinh(ax))/(S(3)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasinh(ax)(S(3)/2), x), x, -S(3)sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(128)aS(2)) + S(3)sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(128)aS(2)) + xS(2)asinh(ax)(S(3)/2)/S(2) - S(3)xsqrt(aS(2)x*S(2) + S(1))sqrt(asinh(ax))/(S(8)a) + asinh(ax)(S(3)/2)/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*(S(3)/2), x), x, S(3)sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(8)a) + S(3)sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(8)a) + xasinh(ax)(S(3)/2) - S(3)sqrt(a*S(2)x*S(2) + S(1))sqrt(asinh(ax))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)(S(3)/2)/x, x), x, Integrate(asinh(ax)(S(3)/2)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)asinh(ax)*(S(5)/2), x), x, -S(5)sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(2304)aS(5)) + S(3)sqrt(S(5))sqrt(Pi)Erf(sqrt(S(5))sqrt(asinh(ax)))/(S(32000)aS(5)) + S(15)sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(128)aS(5)) + S(5)sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(2304)aS(5)) - S(3)sqrt(S(5))sqrt(Pi)Erfi(sqrt(S(5))sqrt(asinh(ax)))/(S(32000)aS(5)) - S(15)sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(128)aS(5)) + xS(5)asinh(ax)(S(5)/2)/S(5) + S(3)x*S(5)sqrt(asinh(ax))/S(100) - xS(4)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)(S(3)/2)/(S(10)a) - x*S(3)sqrt(asinh(ax))/(S(15)a*S(2)) + S(2)x*S(2)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)(S(3)/2)/(S(15)a*S(3)) + S(2)xsqrt(asinh(ax))/(S(5)aS(4)) - S(4)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)(S(3)/2)/(S(15)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)asinh(ax)*(S(5)/2), x), x, S(15)sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(1024)aS(4)) - S(15)sqrt(Pi)Erf(S(2)sqrt(asinh(ax)))/(S(16384)a*S(4)) + S(15)sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(1024)aS(4)) - S(15)sqrt(Pi)Erfi(S(2)sqrt(asinh(ax)))/(S(16384)aS(4)) + xS(4)asinh(ax)*(S(5)/2)/S(4) + S(15)x*S(4)sqrt(asinh(ax))/S(256) - S(5)x*S(3)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)(S(3)/2)/(S(32)a) - S(45)xS(2)sqrt(asinh(ax))/(S(256)a*S(2)) + S(15)xsqrt(aS(2)x*S(2) + S(1))asinh(ax)(S(3)/2)/(S(64)a*S(3)) - S(3)asinh(ax)(S(5)/2)/(S(32)a*S(4)) - S(225)sqrt(asinh(ax))/(S(2048)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asinh(ax)*(S(5)/2), x), x, S(5)sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(1728)aS(3)) - S(15)sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(64)aS(3)) - S(5)sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(1728)aS(3)) + S(15)sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(64)aS(3)) + xS(3)asinh(ax)(S(5)/2)/S(3) + S(5)x*S(3)sqrt(asinh(ax))/S(36) - S(5)x*S(2)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)(S(3)/2)/(S(18)a) - S(5)xsqrt(asinh(ax))/(S(6)a*S(2)) + S(5)sqrt(a*S(2)x*S(2) + S(1))asinh(ax)(S(3)/2)/(S(9)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasinh(ax)(S(5)/2), x), x, -S(15)sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(512)aS(2)) - S(15)sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(512)aS(2)) + xS(2)asinh(ax)(S(5)/2)/S(2) + S(15)x*S(2)sqrt(asinh(ax))/S(32) - S(5)xsqrt(aS(2)x*S(2) + S(1))asinh(ax)(S(3)/2)/(S(8)a) + asinh(ax)(S(5)/2)/(S(4)a*S(2)) + S(15)sqrt(asinh(ax))/(S(64)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*(S(5)/2), x), x, S(15)sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(16)a) - S(15)sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(16)a) + xasinh(ax)(S(5)/2) + S(15)xsqrt(asinh(ax))/S(4) - S(5)sqrt(aS(2)x*S(2) + S(1))asinh(ax)(S(3)/2)/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)(S(5)/2)/x, x), x, Integrate(asinh(ax)(S(5)/2)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/sqrt(asinh(ax)), x), x, -sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(32)a*S(5)) + sqrt(S(5))sqrt(Pi)Erf(sqrt(S(5))sqrt(asinh(ax)))/(S(160)a*S(5)) + sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(16)a*S(5)) - sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(32)a*S(5)) + sqrt(S(5))sqrt(Pi)Erfi(sqrt(S(5))sqrt(asinh(ax)))/(S(160)a*S(5)) + sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(16)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(asinh(ax)), x), x, sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(16)a*S(4)) - sqrt(Pi)Erf(S(2)sqrt(asinh(ax)))/(S(32)aS(4)) - sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(16)a*S(4)) + sqrt(Pi)Erfi(S(2)sqrt(asinh(ax)))/(S(32)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(asinh(ax)), x), x, sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(24)aS(3)) - sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(8)a*S(3)) + sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(24)a*S(3)) - sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(8)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(asinh(ax)), x), x, -sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(8)aS(2)) + sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(8)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(asinh(ax)), x), x, sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(2)a) + sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(asinh(ax))), x), x, Integrate(S(1)/(xsqrt(asinh(ax))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x*S(2)sqrt(asinh(ax))), x), x, Integrate(S(1)/(xS(2)sqrt(asinh(ax))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asinh(ax)*(S(3)/2), x), x, S(3)sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(16)aS(5)) - sqrt(S(5))sqrt(Pi)Erf(sqrt(S(5))sqrt(asinh(ax)))/(S(16)a*S(5)) - sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(8)a*S(5)) - S(3)sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(16)aS(5)) + sqrt(S(5))sqrt(Pi)Erfi(sqrt(S(5))sqrt(asinh(ax)))/(S(16)a*S(5)) + sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(8)a*S(5)) - S(2)x*S(4)sqrt(a*S(2)x*S(2) + S(1))/(asqrt(asinh(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asinh(ax)*(S(3)/2), x), x, -sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(4)a*S(4)) + sqrt(Pi)Erf(S(2)sqrt(asinh(ax)))/(S(4)aS(4)) - sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(4)a*S(4)) + sqrt(Pi)Erfi(S(2)sqrt(asinh(ax)))/(S(4)aS(4)) - S(2)x*S(3)sqrt(a*S(2)x*S(2) + S(1))/(asqrt(asinh(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/asinh(ax)*(S(3)/2), x), x, -sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(4)a*S(3)) + sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(4)a*S(3)) + sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(4)a*S(3)) - sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(4)a*S(3)) - S(2)x*S(2)sqrt(a*S(2)x*S(2) + S(1))/(asqrt(asinh(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asinh(ax)*(S(3)/2), x), x, sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(2)a*S(2)) + sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(2)a*S(2)) - S(2)xsqrt(aS(2)x*S(2) + S(1))/(asqrt(asinh(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*(S(-3)/2), x), x, -sqrt(Pi)Erf(sqrt(asinh(ax)))/a + sqrt(Pi)Erfi(sqrt(asinh(ax)))/a - S(2)sqrt(a*S(2)x*S(2) + S(1))/(asqrt(asinh(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasinh(ax)(S(3)/2)), x), x, Integrate(S(1)/(xasinh(ax)(S(3)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asinh(ax)*(S(5)/2), x), x, -S(3)sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(8)aS(5)) + S(5)sqrt(S(5))sqrt(Pi)Erf(sqrt(S(5))sqrt(asinh(ax)))/(S(24)aS(5)) + sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(12)a*S(5)) - S(3)sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(8)aS(5)) + S(5)sqrt(S(5))sqrt(Pi)Erfi(sqrt(S(5))sqrt(asinh(ax)))/(S(24)aS(5)) + sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(12)a*S(5)) - S(20)x*S(5)/(S(3)sqrt(asinh(ax))) - S(2)x*S(4)sqrt(a*S(2)x*S(2) + S(1))/(S(3)aasinh(ax)*(S(3)/2)) - S(16)x*S(3)/(S(3)a*S(2)sqrt(asinh(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asinh(ax)*(S(5)/2), x), x, sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(3)a*S(4)) - S(2)sqrt(Pi)Erf(S(2)sqrt(asinh(ax)))/(S(3)a*S(4)) - sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(3)a*S(4)) + S(2)sqrt(Pi)Erfi(S(2)sqrt(asinh(ax)))/(S(3)a*S(4)) - S(16)x*S(4)/(S(3)sqrt(asinh(ax))) - S(2)x*S(3)sqrt(a*S(2)x*S(2) + S(1))/(S(3)aasinh(ax)*(S(3)/2)) - S(4)xS(2)/(aS(2)sqrt(asinh(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/asinh(ax)*(S(5)/2), x), x, sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(2)a*S(3)) - sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(6)a*S(3)) + sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(2)a*S(3)) - sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(6)a*S(3)) - S(4)x*S(3)/sqrt(asinh(ax)) - S(2)xS(2)sqrt(a*S(2)x*S(2) + S(1))/(S(3)aasinh(ax)*(S(3)/2)) - S(8)x/(S(3)aS(2)sqrt(asinh(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asinh(ax)*(S(5)/2), x), x, -S(2)sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(3)aS(2)) + S(2)sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(3)aS(2)) - S(8)x*S(2)/(S(3)sqrt(asinh(ax))) - S(2)xsqrt(aS(2)x*S(2) + S(1))/(S(3)aasinh(ax)*(S(3)/2)) - S(4)/(S(3)a*S(2)sqrt(asinh(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*(S(-5)/2), x), x, S(2)sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(3)a) + S(2)sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(3)a) - S(4)x/(S(3)sqrt(asinh(ax))) - S(2)sqrt(aS(2)x*S(2) + S(1))/(S(3)aasinh(ax)*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasinh(ax)(S(5)/2)), x), x, Integrate(S(1)/(xasinh(ax)(S(5)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asinh(ax)*(S(7)/2), x), x, S(9)sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(20)aS(5)) - S(5)sqrt(S(5))sqrt(Pi)Erf(sqrt(S(5))sqrt(asinh(ax)))/(S(12)aS(5)) - sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(30)a*S(5)) - S(9)sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(20)aS(5)) + S(5)sqrt(S(5))sqrt(Pi)Erfi(sqrt(S(5))sqrt(asinh(ax)))/(S(12)aS(5)) + sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(30)a*S(5)) - S(4)x*S(5)/(S(3)asinh(ax)(S(3)/2)) - S(40)x*S(4)sqrt(a*S(2)x*S(2) + S(1))/(S(3)asqrt(asinh(ax))) - S(2)xS(4)sqrt(a*S(2)x*S(2) + S(1))/(S(5)aasinh(ax)*(S(5)/2)) - S(16)x*S(3)/(S(15)a*S(2)asinh(ax)(S(3)/2)) - S(32)x*S(2)sqrt(a*S(2)x*S(2) + S(1))/(S(5)a*S(3)sqrt(asinh(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asinh(ax)*(S(7)/2), x), x, -S(4)sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(15)aS(4)) + S(16)sqrt(Pi)Erf(S(2)sqrt(asinh(ax)))/(S(15)a*S(4)) - S(4)sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(15)aS(4)) + S(16)sqrt(Pi)Erfi(S(2)sqrt(asinh(ax)))/(S(15)a*S(4)) - S(16)x*S(4)/(S(15)asinh(ax)(S(3)/2)) - S(128)x*S(3)sqrt(a*S(2)x*S(2) + S(1))/(S(15)asqrt(asinh(ax))) - S(2)xS(3)sqrt(a*S(2)x*S(2) + S(1))/(S(5)aasinh(ax)*(S(5)/2)) - S(4)x*S(2)/(S(5)a*S(2)asinh(ax)(S(3)/2)) - S(16)xsqrt(aS(2)x*S(2) + S(1))/(S(5)a*S(3)sqrt(asinh(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/asinh(ax)*(S(7)/2), x), x, -S(3)sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(asinh(ax)))/(S(5)aS(3)) + sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(15)a*S(3)) + S(3)sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(asinh(ax)))/(S(5)aS(3)) - sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(15)a*S(3)) - S(4)x*S(3)/(S(5)asinh(ax)(S(3)/2)) - S(24)x*S(2)sqrt(a*S(2)x*S(2) + S(1))/(S(5)asqrt(asinh(ax))) - S(2)xS(2)sqrt(a*S(2)x*S(2) + S(1))/(S(5)aasinh(ax)*(S(5)/2)) - S(8)x/(S(15)aS(2)asinh(ax)(S(3)/2)) - S(16)sqrt(a*S(2)x*S(2) + S(1))/(S(15)a*S(3)sqrt(asinh(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asinh(ax)*(S(7)/2), x), x, S(8)sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(asinh(ax)))/(S(15)aS(2)) + S(8)sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(asinh(ax)))/(S(15)aS(2)) - S(8)x*S(2)/(S(15)asinh(ax)(S(3)/2)) - S(32)xsqrt(aS(2)x*S(2) + S(1))/(S(15)asqrt(asinh(ax))) - S(2)xsqrt(a*S(2)x*S(2) + S(1))/(S(5)aasinh(ax)*(S(5)/2)) - S(4)/(S(15)a*S(2)asinh(ax)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)*(S(-7)/2), x), x, -S(4)sqrt(Pi)Erf(sqrt(asinh(ax)))/(S(15)a) + S(4)sqrt(Pi)Erfi(sqrt(asinh(ax)))/(S(15)a) - S(4)x/(S(15)asinh(ax)*(S(3)/2)) - S(8)sqrt(a*S(2)x*S(2) + S(1))/(S(15)asqrt(asinh(ax))) - S(2)sqrt(aS(2)x*S(2) + S(1))/(S(5)aasinh(ax)*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasinh(ax)(S(7)/2)), x), x, Integrate(S(1)/(xasinh(ax)(S(7)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmasinh(ax)S(4), x), x, -S(4)aIntegrate(x(m + S(1))asinh(ax)S(3)/sqrt(aS(2)xS(2) + S(1)), x)/(m + S(1)) + x(m + S(1))asinh(ax)S(4)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmasinh(ax)*S(3), x), x, -S(3)aIntegrate(x(m + S(1))asinh(ax)S(2)/sqrt(aS(2)xS(2) + S(1)), x)/(m + S(1)) + x(m + S(1))asinh(ax)S(3)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmasinh(ax)*S(2), x), x, S(2)a*S(2)x*(m + S(3))HypergeometricPFQ(List(S(1), m/S(2) + S(3)/2, m/S(2) + S(3)/2), List(m/S(2) + S(2), m/S(2) + S(5)/2), -a*S(2)xS(2))/(mS(3) + S(6)mS(2) + S(11)m + S(6)) - S(2)ax*(m + S(2))Hypergeometric2F1(S(1)/2, m/S(2) + S(1), m/S(2) + S(2), -a*S(2)x*S(2))asinh(ax)/(mS(2) + S(3)m + S(2)) + x*(m + S(1))asinh(ax)S(2)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmasinh(ax), x), x, -ax*(m + S(2))Hypergeometric2F1(S(1)/2, m/S(2) + S(1), m/S(2) + S(2), -a*S(2)xS(2))/(mS(2) + S(3)m + S(2)) + x(m + S(1))asinh(ax)/(m + S(1)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/asinh(ax), x), x, Integrate(x*m/asinh(ax), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*m/asinh(ax)S(2), x), x, Integrate(xm/asinh(ax)S(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmasinh(ax)(S(5)/2), x), x, Integrate(xmasinh(ax)(S(5)/2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmasinh(ax)(S(3)/2), x), x, Integrate(xmasinh(ax)(S(3)/2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmsqrt(asinh(ax)), x), x, Integrate(xmsqrt(asinh(ax)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xm/sqrt(asinh(ax)), x), x, Integrate(x*m/sqrt(asinh(ax)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*m/asinh(ax)(S(3)/2), x), x, Integrate(xm/asinh(ax)(S(3)/2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)*masinh(ax)n, x), x, Integrate((bx)*masinh(ax)n, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)asinh(ax)n, x), x, -S(5)(-n + S(-1))Gamma(n + S(1), S(5)asinh(ax))/(S(32)aS(5)) + S(5)(-n + S(-1))(-asinh(ax))(-n)Gamma(n + S(1), -S(5)asinh(ax))asinh(ax)*n/(S(32)a*S(5)) - Gamma(n + S(1), asinh(ax))/(S(16)aS(5)) + (-asinh(ax))*(-n)Gamma(n + S(1), -asinh(ax))asinh(ax)n/(S(16)aS(5)) + S(3)(-n)Gamma(n + S(1), S(3)asinh(ax))/(S(32)aS(5)) - S(3)(-n)(-asinh(ax))*(-n)Gamma(n + S(1), -S(3)asinh(ax))asinh(ax)*n/(S(32)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asinh(ax)n, x), x, -S(3)(-n + S(-1))Gamma(n + S(1), S(3)asinh(ax))/(S(8)aS(3)) + S(3)(-n + S(-1))(-asinh(ax))*(-n)Gamma(n + S(1), -S(3)asinh(ax))asinh(ax)*n/(S(8)a*S(3)) + Gamma(n + S(1), asinh(ax))/(S(8)aS(3)) - (-asinh(ax))*(-n)Gamma(n + S(1), -asinh(ax))asinh(ax)n/(S(8)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasinh(ax)n, x), x, S(2)(-n + S(-3))Gamma(n + S(1), S(2)asinh(ax))/aS(2) + S(2)(-n + S(-3))(-asinh(ax))*(-n)Gamma(n + S(1), -S(2)asinh(ax))asinh(ax)n/aS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)n, x), x, -Gamma(n + S(1), asinh(ax))/(S(2)a) + (-asinh(ax))*(-n)Gamma(n + S(1), -asinh(ax))asinh(ax)n/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)n/x, x), x, Integrate(asinh(ax)*n/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asinh(ax)n/xS(2), x), x, Integrate(asinh(ax)n/xS(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(a + basinh(cx)), x), x, -sqrt(Pi)sqrt(b)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(16)c*S(3)) + sqrt(S(3))sqrt(Pi)sqrt(b)Erf(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(S(3)a/b)/(S(144)c*S(3)) + sqrt(Pi)sqrt(b)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(16)cS(3)) - sqrt(S(3))sqrt(Pi)sqrt(b)Erfi(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(-S(3)a/b)/(S(144)cS(3)) + xS(3)sqrt(a + basinh(cx))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(a + basinh(cx)), x), x, -sqrt(S(2))sqrt(Pi)sqrt(b)Erf(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(S(2)a/b)/(S(32)cS(2)) - sqrt(S(2))sqrt(Pi)sqrt(b)Erfi(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(-S(2)a/b)/(S(32)cS(2)) + xS(2)sqrt(a + basinh(cx))/S(2) + sqrt(a + basinh(cx))/(S(4)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + basinh(cx)), x), x, sqrt(Pi)sqrt(b)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(4)c) - sqrt(Pi)sqrt(b)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(4)c) + xsqrt(a + basinh(cx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(a + basinh(cx))*(S(3)/2), x), x, -S(3)sqrt(Pi)b(S(3)/2)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(32)c*S(3)) + sqrt(S(3))sqrt(Pi)b(S(3)/2)Erf(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(S(3)a/b)/(S(288)c*S(3)) - S(3)sqrt(Pi)b(S(3)/2)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(32)c*S(3)) + sqrt(S(3))sqrt(Pi)b(S(3)/2)Erfi(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(-S(3)a/b)/(S(288)c*S(3)) - bx*S(2)sqrt(a + basinh(cx))sqrt(cS(2)x*S(2) + S(1))/(S(6)c) + bsqrt(a + basinh(cx))sqrt(c*S(2)x*S(2) + S(1))/(S(3)cS(3)) + xS(3)(a + basinh(cx))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + basinh(cx))*(S(3)/2), x), x, -S(3)sqrt(S(2))sqrt(Pi)b*(S(3)/2)Erf(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(S(2)a/b)/(S(128)c*S(2)) + S(3)sqrt(S(2))sqrt(Pi)b*(S(3)/2)Erfi(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(-S(2)a/b)/(S(128)c*S(2)) - S(3)bxsqrt(a + basinh(cx))sqrt(cS(2)x*S(2) + S(1))/(S(8)c) + x*S(2)(a + basinh(cx))*(S(3)/2)/S(2) + (a + basinh(cx))(S(3)/2)/(S(4)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basinh(cx))(S(3)/2), x), x, S(3)sqrt(Pi)b(S(3)/2)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(8)c) + S(3)sqrt(Pi)b*(S(3)/2)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(8)c) - S(3)bsqrt(a + basinh(cx))sqrt(cS(2)x*S(2) + S(1))/(S(2)c) + x(a + basinh(cx))(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + basinh(cx))*(S(5)/2), x), x, -S(15)sqrt(Pi)b(S(5)/2)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(64)c*S(3)) + S(5)sqrt(S(3))sqrt(Pi)b*(S(5)/2)Erf(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(S(3)a/b)/(S(1728)c*S(3)) + S(15)sqrt(Pi)b(S(5)/2)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(64)c*S(3)) - S(5)sqrt(S(3))sqrt(Pi)b*(S(5)/2)Erfi(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(-S(3)a/b)/(S(1728)c*S(3)) + S(5)b*S(2)x*S(3)sqrt(a + basinh(cx))/S(36) - S(5)bS(2)xsqrt(a + basinh(cx))/(S(6)c*S(2)) - S(5)bxS(2)(a + basinh(cx))*(S(3)/2)sqrt(c*S(2)x*S(2) + S(1))/(S(18)c) + S(5)b(a + basinh(cx))*(S(3)/2)sqrt(c*S(2)x*S(2) + S(1))/(S(9)cS(3)) + xS(3)(a + basinh(cx))(S(5)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + basinh(cx))*(S(5)/2), x), x, -S(15)sqrt(S(2))sqrt(Pi)b*(S(5)/2)Erf(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(S(2)a/b)/(S(512)c*S(2)) - S(15)sqrt(S(2))sqrt(Pi)b*(S(5)/2)Erfi(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(-S(2)a/b)/(S(512)c*S(2)) + S(15)b*S(2)x*S(2)sqrt(a + basinh(cx))/S(32) + S(15)bS(2)sqrt(a + basinh(cx))/(S(64)cS(2)) - S(5)bx(a + basinh(cx))*(S(3)/2)sqrt(c*S(2)x*S(2) + S(1))/(S(8)c) + x*S(2)(a + basinh(cx))*(S(5)/2)/S(2) + (a + basinh(cx))(S(5)/2)/(S(4)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basinh(cx))(S(5)/2), x), x, S(15)sqrt(Pi)b(S(5)/2)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(16)c) - S(15)sqrt(Pi)b*(S(5)/2)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(16)c) + S(15)bS(2)xsqrt(a + basinh(cx))/S(4) - S(5)b(a + basinh(cx))(S(3)/2)sqrt(c*S(2)x*S(2) + S(1))/(S(2)c) + x(a + basinh(cx))(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(a + basinh(cx)), x), x, -sqrt(Pi)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(8)sqrt(b)cS(3)) + sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(S(3)a/b)/(S(24)sqrt(b)c*S(3)) - sqrt(Pi)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(8)sqrt(b)cS(3)) + sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(-S(3)a/b)/(S(24)sqrt(b)c*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + basinh(cx)), x), x, -sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(S(2)a/b)/(S(8)sqrt(b)c*S(2)) + sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(-S(2)a/b)/(S(8)sqrt(b)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + basinh(cx)), x), x, sqrt(Pi)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(2)sqrt(b)c) + sqrt(Pi)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(2)sqrt(b)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + basinh(cx))(S(3)/2), x), x, sqrt(Pi)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(4)b*(S(3)/2)c*S(3)) - sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(S(3)a/b)/(S(4)b(S(3)/2)c*S(3)) - sqrt(Pi)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(4)b*(S(3)/2)c*S(3)) + sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(-S(3)a/b)/(S(4)b(S(3)/2)c*S(3)) - S(2)x*S(2)sqrt(c*S(2)x*S(2) + S(1))/(bcsqrt(a + basinh(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + basinh(cx))(S(3)/2), x), x, sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(S(2)a/b)/(S(2)b(S(3)/2)c*S(2)) + sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(-S(2)a/b)/(S(2)b(S(3)/2)c*S(2)) - S(2)xsqrt(cS(2)x*S(2) + S(1))/(bcsqrt(a + basinh(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basinh(cx))(S(-3)/2), x), x, -sqrt(Pi)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(b(S(3)/2)c) + sqrt(Pi)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(b*(S(3)/2)c) - S(2)sqrt(cS(2)x*S(2) + S(1))/(bcsqrt(a + basinh(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + basinh(cx))(S(5)/2), x), x, -sqrt(Pi)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(6)b*(S(5)/2)c*S(3)) + sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(S(3)a/b)/(S(2)b(S(5)/2)c*S(3)) - sqrt(Pi)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(6)b*(S(5)/2)c*S(3)) + sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(-S(3)a/b)/(S(2)b(S(5)/2)c*S(3)) - S(2)x*S(2)sqrt(c*S(2)x*S(2) + S(1))/(S(3)bc(a + basinh(cx))*(S(3)/2)) - S(4)xS(3)/(bS(2)sqrt(a + basinh(cx))) - S(8)x/(S(3)bS(2)c*S(2)sqrt(a + basinh(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + basinh(cx))*(S(5)/2), x), x, -S(2)sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(S(2)a/b)/(S(3)b*(S(5)/2)c*S(2)) + S(2)sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(-S(2)a/b)/(S(3)b*(S(5)/2)c*S(2)) - S(2)xsqrt(cS(2)x*S(2) + S(1))/(S(3)bc(a + basinh(cx))*(S(3)/2)) - S(8)x*S(2)/(S(3)b*S(2)sqrt(a + basinh(cx))) - S(4)/(S(3)bS(2)c*S(2)sqrt(a + basinh(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basinh(cx))*(S(-5)/2), x), x, S(2)sqrt(Pi)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(3)b(S(5)/2)c) + S(2)sqrt(Pi)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(3)b*(S(5)/2)c) - S(2)sqrt(cS(2)x*S(2) + S(1))/(S(3)bc(a + basinh(cx))*(S(3)/2)) - S(4)x/(S(3)bS(2)sqrt(a + basinh(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(a + basinh(cx))(S(7)/2), x), x, sqrt(Pi)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(15)b*(S(7)/2)c*S(3)) - S(3)sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(S(3)a/b)/(S(5)b*(S(7)/2)c*S(3)) - sqrt(Pi)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(15)b*(S(7)/2)c*S(3)) + S(3)sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(a + basinh(cx))/sqrt(b))exp(-S(3)a/b)/(S(5)b*(S(7)/2)c*S(3)) - S(2)x*S(2)sqrt(c*S(2)x*S(2) + S(1))/(S(5)bc(a + basinh(cx))*(S(5)/2)) - S(4)x*S(3)/(S(5)b*S(2)(a + basinh(cx))*(S(3)/2)) - S(8)x/(S(15)bS(2)c*S(2)(a + basinh(cx))*(S(3)/2)) - S(24)x*S(2)sqrt(c*S(2)x*S(2) + S(1))/(S(5)b*S(3)csqrt(a + basinh(cx))) - S(16)sqrt(c*S(2)x*S(2) + S(1))/(S(15)b*S(3)c*S(3)sqrt(a + basinh(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + basinh(cx))*(S(7)/2), x), x, S(8)sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(S(2)a/b)/(S(15)b*(S(7)/2)c*S(2)) + S(8)sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(a + basinh(cx))/sqrt(b))exp(-S(2)a/b)/(S(15)b*(S(7)/2)c*S(2)) - S(2)xsqrt(cS(2)x*S(2) + S(1))/(S(5)bc(a + basinh(cx))*(S(5)/2)) - S(8)x*S(2)/(S(15)b*S(2)(a + basinh(cx))*(S(3)/2)) - S(4)/(S(15)b*S(2)c*S(2)(a + basinh(cx))*(S(3)/2)) - S(32)xsqrt(cS(2)x*S(2) + S(1))/(S(15)b*S(3)csqrt(a + basinh(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basinh(cx))(S(-7)/2), x), x, -S(4)sqrt(Pi)Erf(sqrt(a + basinh(cx))/sqrt(b))exp(a/b)/(S(15)b(S(7)/2)c) + S(4)sqrt(Pi)Erfi(sqrt(a + basinh(cx))/sqrt(b))exp(-a/b)/(S(15)b*(S(7)/2)c) - S(2)sqrt(cS(2)x*S(2) + S(1))/(S(5)bc(a + basinh(cx))*(S(5)/2)) - S(4)x/(S(15)bS(2)(a + basinh(cx))*(S(3)/2)) - S(8)sqrt(c*S(2)x*S(2) + S(1))/(S(15)b*S(3)csqrt(a + basinh(c*x))), expand=True, _diff=True, _numerical=True)
075f332a1671e701781092ca748017bd9df4919f99fc66e422c1344da75c1329	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.functions import log, sqrt, exp, cos, sin, tan, sec, csc, cot from sympy.functions.elementary.hyperbolic import atanh as arctanh from sympy.functions.elementary.hyperbolic import asinh as arcsinh from sympy.functions.elementary.hyperbolic import acosh as arccosh from sympy.functions.elementary.trigonometric import atan as arctan from sympy.functions.elementary.trigonometric import asin as arcsin from sympy.functions.elementary.trigonometric import acos as arccos from sympy.integrals.rubi.utility_function import EllipticE, EllipticF, EllipticPi, hypergeom, rubi_test, AppellF1 from sympy.core.numbers import (I, pi as Pi) from sympy.core.singleton import S from sympy.core.symbol import symbols from sympy.functions.elementary.exponential import exp_polar from sympy.functions.special.hyper import hyper from sympy.integrals.integrals import Integral from sympy.simplify.simplify import simplify from sympy.testing.pytest import XFAIL A, B, C, D, a, b, c, d, e, f, g, h, i, m, n, p, x, u = symbols('A B C D a b c d e f g h i m n p x u') def test_1(): ''' Tests for Rubi Algebraic 1.2 rules. Parsed from Maple syntax All tests: http://www.apmaths.uwo.ca/~arich/IntegrationProblems/MapleSyntaxFiles/MapleSyntaxFiles.html Note: Some tests are commented since they depend rules other than Algebraic1.2. ''' test = [ [(a + bx)S(2)(e + fx)sqrt(c + dx)/x, x, S(5), S(2)/S(7)f(a + bx)*S(2)(c + dx)(S(3)/S(2))/d + S(2)/S(105)(c + dx)(S(3)/S(2))(S(2)(S(10)a*S(2)d*S(2)f - b*S(2)c(S(7)de - S(4)cf) + S(7)abd(S(5)de - S(2)cf)) + S(3)bd(S(7)bde - S(4)bcf + S(4)adf)x)/d*S(3) - S(2)a*S(2)earctanh(sqrt(c + dx)/sqrt(c))sqrt(c) + S(2)a*S(2)esqrt(c + dx)], [(a + bx)(e + fx)sqrt(c + dx)/x, x, S(4), - S(2)/S(15)(c + dx)(S(3)/S(2))(S(2)bcf - S(5)d(be + af) - S(3)bdfx)/dS(2) - S(2)aearctanh(sqrt(c + dx)/sqrt(c))sqrt(c) + S(2)aesqrt(c + dx)], [(c + dx)S(2)(e + fx)sqrt(a + bx)/x, x, S(5), S(2)/S(7)f(a + bx)*(S(3)/S(2))(c + dx)S(2)/b + S(2)/S(105)(a + bx)(S(3)/S(2))(S(2)(S(4)a*S(2)d*S(2)f - S(7)abd(de + S(2)cf) + S(5)b*S(2)c(S(7)de + S(2)cf)) + S(3)bd(S(7)bde + S(4)bcf - S(4)adf)x)/b*S(3) - S(2)c*S(2)earctanh(sqrt(a + bx)/sqrt(a))sqrt(a) + S(2)c*S(2)esqrt(a + bx)], [(c + dx)(e + fx)sqrt(a + bx)/x, x, S(4), - S(2)/S(15)(a + bx)(S(3)/S(2))(S(2)adf - S(5)b(de + cf) - S(3)bdfx)/bS(2) - S(2)cearctanh(sqrt(a + bx)/sqrt(a))sqrt(a) + S(2)cesqrt(a + bx)], [x*S(4)(e + fx)n/((a + bx)(c + dx)), x, S(8), e*S(2)(e + fx)(S(1) + n)/(bdfS(3)(S(1) + n)) + (bc + ad)e(e + fx)(S(1) + n)/(bS(2)d*S(2)f*S(2)(S(1) + n)) + (b*S(2)c*S(2) + abcd + a*S(2)d*S(2))(e + fx)(S(1) + n)/(bS(3)d*S(3)f(S(1) + n)) - S(2)e(e + fx)*(S(2) + n)/(bdfS(3)(S(2) + n)) - (bc + ad)(e + fx)(S(2) + n)/(bS(2)dS(2)f*S(2)(S(2) + n)) + (e + fx)(S(3) + n)/(bdfS(3)(S(3) + n)) - a*S(4)(e + fx)(S(1) + n)hypergeom([S(1), S(1) + n], [S(2) + n], b(e + fx)/(be - af))/(b*S(3)(bc - ad)(be - af)(S(1) + n)) + c*S(4)(e + fx)(S(1) + n)hypergeom([S(1), S(1) + n], [S(2) + n], d(e + fx)/(de - cf))/(d*S(3)(bc - ad)(de - cf)(S(1) + n))], [(a + bx)(c + dx)(e + fx)(g + hx), x, S(2), acegx + S(1)/S(2)(bceg + a(deg + cfg + ceh))xS(2) + S(1)/S(3)(b(deg + cfg + ceh) + a(dfg + deh + cfh))xS(3) + S(1)/S(4)(adfh + b(dfg + deh + cfh))xS(4) + S(1)/S(5)bdfhx*S(5)], [(a + bx)(c + dx)(e + fx)/(g + hx), x, S(2), (b(dg - ch)(fg - eh) - ah(dfg - deh - cfh))x/h*S(3) + S(1)/S(2)(adfh - b(dfg - deh - cfh))xS(2)/hS(2) + S(1)/S(3)bdfxS(3)/h - (bg - ah)(dg - ch)(fg - eh)log(g + hx)/hS(4)], [(a + bx)*m(c + dx)(e + fx)(g + hx), x, S(2), (bc - ad)(be - af)(bg - ah)(a + bx)(S(1) + m)/(bS(4)(S(1) + m)) + (S(3)aS(2)dfh + b*S(2)(deg + cfg + ceh) - S(2)ab(dfg + deh + cfh))(a + bx)(S(2) + m)/(bS(4)(S(2) + m)) - (S(3)adfh - b(dfg + deh + cfh))(a + bx)(S(3) + m)/(bS(4)(S(3) + m)) + dfh(a + bx)(S(4) + m)/(bS(4)(S(4) + m))], [(c + dx)*( - S(4) - m)(e + fx)m(g + hx), x, S(3), - (dg - ch)(c + dx)( - S(3) - m)(e + fx)(S(1) + m)/(d(de - cf)(S(3) + m)) + (cfh(S(1) + m) + d(S(2)fg - eh(S(3) + m)))(c + dx)( - S(2) - m)(e + fx)(S(1) + m)/(d(de - cf)*S(2)(S(2) + m)(S(3) + m)) - f(cfh(S(1) + m) + d(S(2)fg - eh(S(3) + m)))(c + dx)*( - S(1) - m)(e + fx)(S(1) + m)/(d(de - cf)*S(3)(S(1) + m)(S(2) + m)(S(3) + m))], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True) else: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) @XFAIL def test_2(): test = [ [x*m(e + fx)n/((a + bx)(c + dx)), x, S(6), bx(S(1) + m)(e + fx)nAppellF1(S(1) + m, - n, S(1), S(2) + m, - fx/e, - bx/a)/(a(bc - ad)(S(1) + m)(S(1) + fx/e)*n) - dx*(S(1) + m)(e + fx)nAppellF1(S(1) + m, - n, S(1), S(2) + m, - fx/e, - dx/c)/(c(bc - ad)(S(1) + m)(S(1) + fx/e)*n)], [(a + bx)*m(c + dx)n(e + fx)(g + hx), x, S(3), - (a + bx)*(S(1) + m)(c + dx)(S(1) + n)(bcfh(S(2) + m) + adfh(S(2) + n) - bd(fg + eh)(S(3) + m + n) - bdfh(S(2) + m + n)x)/(b*S(2)d*S(2)(S(2) + m + n)(S(3) + m + n)) + (aS(2)d*S(2)fh(S(1) + n)(S(2) + n) + abd(S(1) + n)(S(2)cfh(S(1) + m) - d(fg + eh)(S(3) + m + n)) + bS(2)(c*S(2)fh(S(1) + m)(S(2) + m) - cd(fg + eh)(S(1) + m)(S(3) + m + n) + dS(2)eg(S(2) + m + n)(S(3) + m + n)))(a + bx)(S(1) + m)(c + dx)nhypergeom([S(1) + m, - n], [S(2) + m], - d(a + bx)/(bc - ad))/(b*S(3)d*S(2)(S(1) + m)(S(2) + m + n)(S(3) + m + n)(b(c + dx)/(bc - ad))n)], [(a + bx)*m(A + Bx)(c + dx)n(e + fx)p, x, S(7), (Ab - aB)(a + bx)(S(1) + m)(c + dx)n(e + fx)pAppellF1(S(1) + m, - n, - p, S(2) + m, - d(a + bx)/(bc - ad), - f(a + bx)/(be - af))/(b*S(2)(S(1) + m)(b(c + dx)/(bc - ad))n(b(e + fx)/(be - af))*p) + B(a + bx)(S(2) + m)(c + dx)n(e + fx)pAppellF1(S(2) + m, - n, - p, S(3) + m, - d(a + bx)/(bc - ad), - f(a + bx)/(be - af))/(b*S(2)(S(2) + m)(b(c + dx)/(bc - ad))n(b(e + fx)/(be - af))*p)], [(A + Bx)(c + dx)*n(e + fx)p/(a + bx), x, S(5), - (Ab - aB)(c + dx)*(S(1) + n)(e + fx)pAppellF1(S(1) + n, S(1), - p, S(2) + n, b(c + dx)/(bc - ad), - f(c + dx)/(de - cf))/(b(bc - ad)(S(1) + n)(d(e + fx)/(de - cf))p) - B(c + dx)(S(1) + n)(e + fx)(S(1) + p)hypergeom([S(1), S(2) + n + p], [S(2) + p], d(e + fx)/(de - cf))/(b(de - cf)(S(1) + p)), - (Ab - aB)(c + dx)*(S(1) + n)(e + fx)pAppellF1(S(1) + n, - p, S(1), S(2) + n, - f(c + dx)/(de - cf), b(c + dx)/(bc - ad))/(b(bc - ad)(S(1) + n)(d(e + fx)/(de - cf))p) + B(c + dx)(S(1) + n)(e + fx)phypergeom([S(1) + n, - p], [S(2) + n], - f(c + dx)/(de - cf))/(bd(S(1) + n)(d(e + fx)/(de - cf))p)], [(ci + dix)/(sqrt(c + dx)sqrt(e + fx)sqrt(g + hx)), x, S(3), S(2)iEllipticE(sqrt(h)sqrt(e + fx)/sqrt( - fg + eh), sqrt( - d(fg - eh)/((de - cf)h)))sqrt( - fg + eh)sqrt(c + dx)sqrt(f(g + hx)/(fg - eh))/(fsqrt(h)sqrt( - f(c + dx)/(de - cf))sqrt(g + hx))], [(a + bx)*m(c + dx)n(e + fx)p, x, S(3), (a + bx)*(S(1) + m)(c + dx)n(e + fx)pAppellF1(S(1) + m, - n, - p, S(2) + m, - d(a + bx)/(bc - ad), - f(a + bx)/(be - af))/(b(S(1) + m)(b(c + dx)/(bc - ad))*n(b(e + fx)/(be - af))*p)], [(a + bx)*m(c + dx)n(e + fx)p/(g + hx), x, S(0), Integral((a + bx)m(c + dx)n(e + fx)p/(g + hx), x)], [x*S(3)(S(1) + ax)/(sqrt(ax)sqrt(S(1) - ax)), x, S(8), - S(75)/S(128)arcsin(S(1) - S(2)ax)/aS(4) - S(25)/S(32)(ax)(S(3)/S(2))sqrt(S(1) - ax)/aS(4) - S(5)/S(8)(ax)(S(5)/S(2))sqrt(S(1) - ax)/aS(4) - S(1)/S(4)(ax)(S(7)/S(2))sqrt(S(1) - ax)/aS(4) - S(75)/S(64)sqrt(ax)sqrt(S(1) - ax)/aS(4)], ] for index in test: r = rubi_integrate(index[0], index[1]) if len(index) == 5: assert rubi_test(r, index[1], index[3], expand=True, _diff=True) or rubi_test(r, index[1], index[4], expand=True, _diff=True) else: assert rubi_test(r, index[1], index[3], expand=True, _diff=True) @XFAIL def test_numerical(): test = [ #[S(1)/((a + bx)sqrt(c + dx)sqrt(e + fx)sqrt(g + hx)), x, S(1), - S(2)EllipticPi(sqrt( - f/(de - cf))sqrt(c + dx), - b(de - cf)/((bc - ad)f), sqrt((de - cf)h/(f(dg - ch))))sqrt(d(e + fx)/(de - cf))sqrt(d(g + hx)/(dg - ch))/((bc - ad)sqrt( - f/(de - cf))sqrt(e + fx)sqrt(g + hx))], #[S(1)/(sqrt(a + bx)sqrt(c + dx)sqrt(e + fx)sqrt(g + hx)), x, S(2), - S(2)(a + bx)sqrt(cos(arctan(sqrt(be - af)sqrt(g + hx)/(sqrt(fg - eh)sqrt(a + bx))))*S(2))/cos(arctan(sqrt(be - af)sqrt(g + hx)/(sqrt(fg - eh)sqrt(a + bx))))EllipticF(sin(arctan(sqrt(be - af)sqrt(g + hx)/(sqrt(fg - eh)sqrt(a + bx)))), sqrt((de - cf)(bg - ah)/((be - af)(dg - ch))))sqrt(fg - eh)sqrt((bg - ah)(c + dx)/((dg - ch)(a + bx)))sqrt((bg - ah)(e + fx)/((fg - eh)(a + bx)))sqrt(S(1) + (bc - ad)(g + hx)/((dg - ch)(a + bx)))/((bg - ah)sqrt(be - af)sqrt(c + dx)sqrt(e + fx)sqrt((S(1) + (bc - ad)(g + hx)/((dg - ch)(a + bx)))/(S(1) + (be - af)(g + hx)/((fg - eh)(a + bx))))sqrt(S(1) + (be - af)(g + hx)/((fg - eh)(a + b*x))))], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True, _diff=True, _numerical=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True, _numerical=True) else: assert rubi_test(r, i[1], i[3], expand=True, _diff=True, _numerical=True)
cdd6834e34d61e897fd40e65f6fddc81801d38938a8b67887b94423927dccd54	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.utility_function import ( sympy_op_factory, Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ, ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ, PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ, NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest, SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient, Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan, ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec, ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less, Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ, InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ, EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, ExpandLinearProduct, GCD, ContentFactor, NumericFactor, NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, ExpandExpression, Apart, SmartApart, MatchQ, PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, RationalFunctionFactors, NonrationalFunctionFactors, Reverse, RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, GeneralizedBinomialParts, GeneralizedTrinomialDegree, GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, MinimumMonomialExponent, MonomialExponent, LinearMatchQ, PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, AlgebraicFunctionFactors, NonalgebraicFunctionFactors, QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, CombineExponents, FactorInteger, FactorAbsurdNumber, SubstForInverseFunction, SubstForFractionalPower, SubstForFractionalPowerOfQuotientOfLinears, FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, NormalizeIntegrandAux, NormalizeIntegrandFactor, NormalizeIntegrandFactorBase, NormalizeTogether, NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand, SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, KnownTangentIntegrandQ, KnownCotangentIntegrandQ, KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, FunctionOfLog, PowerVariableExpn, PowerVariableDegree, PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, ConstantFactor, SameQ, ReplacePart, CommonFactors, MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, _SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify, _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor, PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral, CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial, Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ ) from sympy.core.add import Add from sympy.core.mod import Mod from sympy.core.mul import Mul from sympy.core.numbers import (Float, I, Integer) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.functions.elementary.complexes import Abs from sympy.functions.elementary.miscellaneous import sqrt from sympy.integrals.integrals import Integral from sympy.logic.boolalg import (And, Or) from sympy.simplify.simplify import simplify from sympy.integrals.rubi.symbol import WC from sympy.core.symbol import symbols, Symbol from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf, exp, log) from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch) from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec) from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.core.numbers import pi as Pi a, b, c, d, e, f, m, n, x, u , k, p, r, s, t, i, j= symbols('a b c d e f m n x u k p r s t i j') A, B, C, D, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B C D a b c d e f g h y z m n p q u v w F', ) def test_1(): assert rubi_test(rubi_integrate(sec(a + bx), x), x, atanh(sin(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(2), x), x, tan(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(3), x), x, tan(a + bx)sec(a + bx)/(S(2)b) + atanh(sin(a + bx))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(4), x), x, tan(a + bx)*S(3)/(S(3)b) + tan(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(5), x), x, tan(a + bx)sec(a + bx)*S(3)/(S(4)b) + S(3)tan(a + bx)sec(a + bx)/(S(8)b) + S(3)atanh(sin(a + bx))/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(6), x), x, tan(a + bx)*S(5)/(S(5)b) + S(2)tan(a + bx)*S(3)/(S(3)b) + tan(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(7), x), x, tan(a + bx)sec(a + bx)*S(5)/(S(6)b) + S(5)tan(a + bx)sec(a + bx)*S(3)/(S(24)b) + S(5)tan(a + bx)sec(a + bx)/(S(16)b) + S(5)atanh(sin(a + bx))/(S(16)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(8), x), x, tan(a + bx)*S(7)/(S(7)b) + S(3)tan(a + bx)*S(5)/(S(5)b) + tan(a + bx)S(3)/b + tan(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)(S(7)/2), x), x, -S(6)EllipticE(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))sqrt(sec(a + bx))/(S(5)b) + S(2)sin(a + bx)sec(a + bx)*(S(5)/2)/(S(5)b) + S(6)sin(a + bx)sqrt(sec(a + bx))/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*(S(5)/2), x), x, S(2)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))sqrt(sec(a + bx))/(S(3)b) + S(2)sin(a + bx)sec(a + bx)*(S(3)/2)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)(S(3)/2), x), x, -S(2)EllipticE(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))sqrt(sec(a + bx))/b + S(2)sin(a + bx)sqrt(sec(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sec(a + bx)), x), x, S(2)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))sqrt(sec(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(sec(a + bx)), x), x, S(2)EllipticE(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))sqrt(sec(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*(S(-3)/2), x), x, S(2)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))sqrt(sec(a + bx))/(S(3)b) + S(2)sin(a + bx)/(S(3)bsqrt(sec(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*(S(-5)/2), x), x, S(6)EllipticE(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))sqrt(sec(a + bx))/(S(5)b) + S(2)sin(a + bx)/(S(5)bsec(a + bx)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*(S(-7)/2), x), x, S(10)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))sqrt(sec(a + bx))/(S(21)b) + S(10)sin(a + bx)/(S(21)bsqrt(sec(a + bx))) + S(2)sin(a + bx)/(S(7)bsec(a + bx)*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(7)/2), x), x, -S(6)c*S(4)EllipticE(a/S(2) + bx/S(2), S(2))/(S(5)bsqrt(csec(a + bx))sqrt(cos(a + bx))) + S(6)c*S(3)sqrt(csec(a + bx))sin(a + bx)/(S(5)b) + S(2)c(csec(a + bx))(S(5)/2)sin(a + bx)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(5)/2), x), x, S(2)c*S(2)sqrt(csec(a + bx))EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(3)b) + S(2)c(csec(a + bx))(S(3)/2)sin(a + bx)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(3)/2), x), x, -S(2)c*S(2)EllipticE(a/S(2) + bx/S(2), S(2))/(bsqrt(csec(a + bx))sqrt(cos(a + bx))) + S(2)csqrt(csec(a + bx))sin(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csec(a + bx)), x), x, S(2)sqrt(csec(a + bx))EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(csec(a + bx)), x), x, S(2)EllipticE(a/S(2) + bx/S(2), S(2))/(bsqrt(csec(a + bx))sqrt(cos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(-3)/2), x), x, S(2)sin(a + bx)/(S(3)bcsqrt(csec(a + bx))) + S(2)sqrt(csec(a + bx))EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(3)bcS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(-5)/2), x), x, S(2)sin(a + bx)/(S(5)bc(csec(a + bx))*(S(3)/2)) + S(6)EllipticE(a/S(2) + bx/S(2), S(2))/(S(5)bcS(2)sqrt(csec(a + bx))sqrt(cos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(-7)/2), x), x, S(2)sin(a + bx)/(S(7)bc(csec(a + bx))*(S(5)/2)) + S(10)sin(a + bx)/(S(21)bcS(3)sqrt(csec(a + bx))) + S(10)sqrt(csec(a + bx))EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(21)bcS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*(S(4)/3), x), x, S(3)Hypergeometric2F1(S(-1)/6, S(1)/2, S(5)/6, cos(a + bx)S(2))sin(a + bx)sec(a + bx)(S(1)/3)/(bsqrt(sin(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*(S(2)/3), x), x, -S(3)Hypergeometric2F1(S(1)/6, S(1)/2, S(7)/6, cos(a + bx)S(2))sin(a + bx)/(bsqrt(sin(a + bx)S(2))sec(a + bx)(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*(S(1)/3), x), x, -S(3)Hypergeometric2F1(S(1)/3, S(1)/2, S(4)/3, cos(a + bx)S(2))sin(a + bx)/(S(2)bsqrt(sin(a + bx)*S(2))sec(a + bx)(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*(S(-1)/3), x), x, -S(3)Hypergeometric2F1(S(1)/2, S(2)/3, S(5)/3, cos(a + bx)S(2))sin(a + bx)/(S(4)bsqrt(sin(a + bx)*S(2))sec(a + bx)(S(4)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*(S(-2)/3), x), x, -S(3)Hypergeometric2F1(S(1)/2, S(5)/6, S(11)/6, cos(a + bx)S(2))sin(a + bx)/(S(5)bsqrt(sin(a + bx)*S(2))sec(a + bx)(S(5)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*(S(-4)/3), x), x, -S(3)Hypergeometric2F1(S(1)/2, S(7)/6, S(13)/6, cos(a + bx)S(2))sin(a + bx)/(S(7)bsqrt(sin(a + bx)*S(2))sec(a + bx)(S(7)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(4)/3), x), x, S(3)c(csec(a + bx))(S(1)/3)Hypergeometric2F1(S(-1)/6, S(1)/2, S(5)/6, cos(a + bx)S(2))sin(a + bx)/(bsqrt(sin(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(2)/3), x), x, -S(3)cHypergeometric2F1(S(1)/6, S(1)/2, S(7)/6, cos(a + bx)*S(2))sin(a + bx)/(b(csec(a + bx))*(S(1)/3)sqrt(sin(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(1)/3), x), x, -S(3)cHypergeometric2F1(S(1)/3, S(1)/2, S(4)/3, cos(a + bx)*S(2))sin(a + bx)/(S(2)b(csec(a + bx))(S(2)/3)sqrt(sin(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(-1)/3), x), x, -S(3)cHypergeometric2F1(S(1)/2, S(2)/3, S(5)/3, cos(a + bx)*S(2))sin(a + bx)/(S(4)b(csec(a + bx))(S(4)/3)sqrt(sin(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(-2)/3), x), x, -S(3)cHypergeometric2F1(S(1)/2, S(5)/6, S(11)/6, cos(a + bx)*S(2))sin(a + bx)/(S(5)b(csec(a + bx))(S(5)/3)sqrt(sin(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(-4)/3), x), x, -S(3)cHypergeometric2F1(S(1)/2, S(7)/6, S(13)/6, cos(a + bx)*S(2))sin(a + bx)/(S(7)b(csec(a + bx))(S(7)/3)sqrt(sin(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*n, x), x, -Hypergeometric2F1(S(1)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(a + bx)*S(2))sin(a + bx)sec(a + bx)(n + S(-1))/(b(-n + S(1))sqrt(sin(a + bx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))n, x), x, -c(csec(a + bx))*(n + S(-1))Hypergeometric2F1(S(1)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(a + bx)S(2))sin(a + bx)/(b(-n + S(1))sqrt(sin(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sec(x)S(2))(S(7)/2), x), x, (sec(x)S(2))*(S(5)/2)tan(x)/S(6) + S(5)(sec(x)S(2))(S(3)/2)tan(x)/S(24) + S(5)sqrt(sec(x)S(2))tan(x)/S(16) + S(5)asinh(tan(x))/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sec(x)S(2))(S(5)/2), x), x, (sec(x)S(2))(S(3)/2)tan(x)/S(4) + S(3)sqrt(sec(x)S(2))tan(x)/S(8) + S(3)asinh(tan(x))/S(8), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sec(x)S(2))(S(3)/2), x), x, sqrt(sec(x)S(2))tan(x)/S(2) + asinh(tan(x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sec(x)S(2)), x), x, asinh(tan(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(sec(x)S(2)), x), x, tan(x)/sqrt(sec(x)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sec(x)S(2))*(S(-3)/2), x), x, S(2)tan(x)/(S(3)sqrt(sec(x)S(2))) + tan(x)/(S(3)(sec(x)S(2))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sec(x)S(2))(S(-5)/2), x), x, S(8)tan(x)/(S(15)sqrt(sec(x)*S(2))) + S(4)tan(x)/(S(15)(sec(x)S(2))(S(3)/2)) + tan(x)/(S(5)(sec(x)S(2))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((sec(x)S(2))(S(-7)/2), x), x, S(16)tan(x)/(S(35)sqrt(sec(x)*S(2))) + S(8)tan(x)/(S(35)(sec(x)S(2))(S(3)/2)) + S(6)tan(x)/(S(35)(sec(x)S(2))(S(5)/2)) + tan(x)/(S(7)(sec(x)S(2))(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(2))(S(7)/2), x), x, S(5)a*(S(7)/2)atanh(sqrt(a)tan(x)/sqrt(asec(x)*S(2)))/S(16) + S(5)a*S(3)sqrt(asec(x)S(2))tan(x)/S(16) + S(5)aS(2)(asec(x)S(2))(S(3)/2)tan(x)/S(24) + a(asec(x)S(2))(S(5)/2)tan(x)/S(6), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(2))(S(5)/2), x), x, S(3)a(S(5)/2)atanh(sqrt(a)tan(x)/sqrt(asec(x)*S(2)))/S(8) + S(3)a*S(2)sqrt(asec(x)S(2))tan(x)/S(8) + a(asec(x)S(2))(S(3)/2)tan(x)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(2))(S(3)/2), x), x, a*(S(3)/2)atanh(sqrt(a)tan(x)/sqrt(asec(x)*S(2)))/S(2) + asqrt(asec(x)S(2))tan(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(asec(x)S(2)), x), x, sqrt(a)atanh(sqrt(a)tan(x)/sqrt(asec(x)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(asec(x)*S(2)), x), x, tan(x)/sqrt(asec(x)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(2))(S(-3)/2), x), x, tan(x)/(S(3)(asec(x)S(2))(S(3)/2)) + S(2)tan(x)/(S(3)asqrt(asec(x)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(2))(S(-5)/2), x), x, tan(x)/(S(5)(asec(x)S(2))(S(5)/2)) + S(4)tan(x)/(S(15)a(asec(x)S(2))(S(3)/2)) + S(8)tan(x)/(S(15)a*S(2)sqrt(asec(x)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(2))(S(-7)/2), x), x, tan(x)/(S(7)(asec(x)S(2))(S(7)/2)) + S(6)tan(x)/(S(35)a(asec(x)S(2))(S(5)/2)) + S(8)tan(x)/(S(35)a*S(2)(asec(x)S(2))(S(3)/2)) + S(16)tan(x)/(S(35)aS(3)sqrt(asec(x)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(3))(S(5)/2), x), x, -S(154)aS(2)sqrt(asec(x)S(3))EllipticE(x/S(2), S(2))cos(x)(S(3)/2)/S(195) + S(154)a*S(2)sqrt(asec(x)S(3))sin(x)cos(x)/S(195) + S(2)a*S(2)sqrt(asec(x)S(3))tan(x)sec(x)S(4)/S(13) + S(22)a*S(2)sqrt(asec(x)S(3))tan(x)sec(x)S(2)/S(117) + S(154)a*S(2)sqrt(asec(x)S(3))tan(x)/S(585), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(3))(S(3)/2), x), x, S(10)asqrt(asec(x)*S(3))EllipticF(x/S(2), S(2))cos(x)(S(3)/2)/S(21) + S(10)asqrt(asec(x)*S(3))sin(x)/S(21) + S(2)asqrt(asec(x)S(3))tan(x)sec(x)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(asec(x)*S(3)), x), x, -S(2)sqrt(asec(x)S(3))EllipticE(x/S(2), S(2))cos(x)(S(3)/2) + S(2)sqrt(asec(x)S(3))sin(x)cos(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(asec(x)*S(3)), x), x, S(2)EllipticF(x/S(2), S(2))/(S(3)sqrt(asec(x)*S(3))cos(x)*(S(3)/2)) + S(2)tan(x)/(S(3)sqrt(asec(x)*S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(3))(S(-3)/2), x), x, S(14)EllipticE(x/S(2), S(2))/(S(15)asqrt(asec(x)*S(3))cos(x)*(S(3)/2)) + S(2)sin(x)cos(x)S(2)/(S(9)asqrt(asec(x)*S(3))) + S(14)sin(x)/(S(45)asqrt(asec(x)S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(3))(S(-5)/2), x), x, S(26)EllipticF(x/S(2), S(2))/(S(77)a*S(2)sqrt(asec(x)S(3))cos(x)*(S(3)/2)) + S(2)sin(x)cos(x)S(5)/(S(15)a*S(2)sqrt(asec(x)S(3))) + S(26)sin(x)cos(x)S(3)/(S(165)a*S(2)sqrt(asec(x)S(3))) + S(78)sin(x)cos(x)/(S(385)a*S(2)sqrt(asec(x)S(3))) + S(26)tan(x)/(S(77)aS(2)sqrt(asec(x)S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(4))(S(7)/2), x), x, a*S(3)sqrt(asec(x)S(4))sin(x)*S(2)tan(x)*S(11)/S(13) + S(6)a*S(3)sqrt(asec(x)S(4))sin(x)*S(2)tan(x)*S(9)/S(11) + S(5)a*S(3)sqrt(asec(x)S(4))sin(x)*S(2)tan(x)*S(7)/S(3) + S(20)a*S(3)sqrt(asec(x)S(4))sin(x)*S(2)tan(x)*S(5)/S(7) + S(3)a*S(3)sqrt(asec(x)S(4))sin(x)*S(2)tan(x)*S(3) + S(2)a*S(3)sqrt(asec(x)S(4))sin(x)*S(2)tan(x) + a*S(3)sqrt(asec(x)S(4))sin(x)cos(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(4))(S(5)/2), x), x, a*S(2)sqrt(asec(x)S(4))sin(x)*S(2)tan(x)*S(7)/S(9) + S(4)a*S(2)sqrt(asec(x)S(4))sin(x)*S(2)tan(x)*S(5)/S(7) + S(6)a*S(2)sqrt(asec(x)S(4))sin(x)*S(2)tan(x)*S(3)/S(5) + S(4)a*S(2)sqrt(asec(x)S(4))sin(x)*S(2)tan(x)/S(3) + a*S(2)sqrt(asec(x)S(4))sin(x)cos(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(4))(S(3)/2), x), x, asqrt(asec(x)*S(4))sin(x)*S(2)tan(x)*S(3)/S(5) + S(2)asqrt(asec(x)*S(4))sin(x)*S(2)tan(x)/S(3) + asqrt(asec(x)*S(4))sin(x)cos(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(asec(x)*S(4)), x), x, sqrt(asec(x)*S(4))sin(x)cos(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(asec(x)*S(4)), x), x, xsec(x)*S(2)/(S(2)sqrt(asec(x)S(4))) + tan(x)/(S(2)sqrt(asec(x)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(4))(S(-3)/2), x), x, S(5)xsec(x)*S(2)/(S(16)asqrt(asec(x)*S(4))) + sin(x)cos(x)*S(3)/(S(6)asqrt(asec(x)*S(4))) + S(5)sin(x)cos(x)/(S(24)asqrt(asec(x)*S(4))) + S(5)tan(x)/(S(16)asqrt(asec(x)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(x)S(4))(S(-5)/2), x), x, S(63)xsec(x)*S(2)/(S(256)a*S(2)sqrt(asec(x)S(4))) + sin(x)cos(x)*S(7)/(S(10)a*S(2)sqrt(asec(x)S(4))) + S(9)sin(x)cos(x)S(5)/(S(80)a*S(2)sqrt(asec(x)S(4))) + S(21)sin(x)cos(x)S(3)/(S(160)a*S(2)sqrt(asec(x)S(4))) + S(21)sin(x)cos(x)/(S(128)a*S(2)sqrt(asec(x)S(4))) + S(63)tan(x)/(S(256)aS(2)sqrt(asec(x)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(((bsec(c + dx))p)n, x), x, -((bsec(c + dx))p)nHypergeometric2F1(S(1)/2, -np/S(2) + S(1)/2, -np/S(2) + S(3)/2, cos(c + dx)S(2))sin(c + dx)cos(c + dx)/(d(-np + S(1))sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a(bsec(c + dx))p)n, x), x, -(a(bsec(c + dx))p)nHypergeometric2F1(S(1)/2, -np/S(2) + S(1)/2, -np/S(2) + S(3)/2, cos(c + dx)S(2))sin(c + dx)cos(c + dx)/(d(-np + S(1))sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))sec(c + dx)S(4), x), x, S(10)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)d) + S(10)(bsec(c + dx))*(S(3)/2)sin(c + dx)/(S(21)bd) + S(2)(bsec(c + dx))*(S(7)/2)sin(c + dx)/(S(7)b*S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))sec(c + dx)*S(3), x), x, -S(6)bEllipticE(c/S(2) + dx/S(2), S(2))/(S(5)dsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(6)sqrt(bsec(c + dx))sin(c + dx)/(S(5)d) + S(2)(bsec(c + dx))(S(5)/2)sin(c + dx)/(S(5)b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))sec(c + dx)*S(2), x), x, S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(3)d) + S(2)(bsec(c + dx))*(S(3)/2)sin(c + dx)/(S(3)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))sec(c + dx), x), x, -S(2)bEllipticE(c/S(2) + dx/S(2), S(2))/(dsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(2)sqrt(bsec(c + dx))sin(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx)), x), x, S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))cos(c + dx), x), x, S(2)bEllipticE(c/S(2) + dx/S(2), S(2))/(dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))cos(c + dx)S(2), x), x, S(2)bsin(c + dx)/(S(3)dsqrt(bsec(c + dx))) + S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))cos(c + dx)*S(3), x), x, S(2)b*S(2)sin(c + dx)/(S(5)d(bsec(c + dx))(S(3)/2)) + S(6)bEllipticE(c/S(2) + dx/S(2), S(2))/(S(5)dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))cos(c + dx)*S(4), x), x, S(2)b*S(3)sin(c + dx)/(S(7)d(bsec(c + dx))(S(5)/2)) + S(10)bsin(c + dx)/(S(21)dsqrt(bsec(c + dx))) + S(10)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))cos(c + dx)*S(5), x), x, S(2)b*S(4)sin(c + dx)/(S(9)d(bsec(c + dx))(S(7)/2)) + S(14)b*S(2)sin(c + dx)/(S(45)d(bsec(c + dx))(S(3)/2)) + S(14)bEllipticE(c/S(2) + dx/S(2), S(2))/(S(15)dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(3)/2)sec(c + dx)S(3), x), x, S(10)bsqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)d) + S(10)(bsec(c + dx))(S(3)/2)sin(c + dx)/(S(21)d) + S(2)(bsec(c + dx))(S(7)/2)sin(c + dx)/(S(7)b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(3)/2)sec(c + dx)S(2), x), x, -S(6)b*S(2)EllipticE(c/S(2) + dx/S(2), S(2))/(S(5)dsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(6)bsqrt(bsec(c + dx))sin(c + dx)/(S(5)d) + S(2)(bsec(c + dx))(S(5)/2)sin(c + dx)/(S(5)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(3)/2)sec(c + dx), x), x, S(2)bsqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(3)d) + S(2)(bsec(c + dx))(S(3)/2)sin(c + dx)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(3)/2), x), x, -S(2)b*S(2)EllipticE(c/S(2) + dx/S(2), S(2))/(dsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(2)bsqrt(bsec(c + dx))sin(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(3)/2)cos(c + dx), x), x, S(2)bsqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(3)/2)cos(c + dx)S(2), x), x, S(2)b*S(2)EllipticE(c/S(2) + dx/S(2), S(2))/(dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(3)/2)cos(c + dx)S(3), x), x, S(2)b*S(2)sin(c + dx)/(S(3)dsqrt(bsec(c + dx))) + S(2)bsqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(3)/2)cos(c + dx)S(4), x), x, S(2)b*S(3)sin(c + dx)/(S(5)d(bsec(c + dx))(S(3)/2)) + S(6)b*S(2)EllipticE(c/S(2) + dx/S(2), S(2))/(S(5)dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(3)/2)cos(c + dx)S(5), x), x, S(2)b*S(4)sin(c + dx)/(S(7)d(bsec(c + dx))(S(5)/2)) + S(10)b*S(2)sin(c + dx)/(S(21)dsqrt(bsec(c + dx))) + S(10)bsqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(3)/2)cos(c + dx)S(6), x), x, S(2)b*S(5)sin(c + dx)/(S(9)d(bsec(c + dx))(S(7)/2)) + S(14)b*S(3)sin(c + dx)/(S(45)d(bsec(c + dx))(S(3)/2)) + S(14)b*S(2)EllipticE(c/S(2) + dx/S(2), S(2))/(S(15)dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(5)/2)sec(c + dx)S(2), x), x, S(10)b*S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)d) + S(10)b(bsec(c + dx))(S(3)/2)sin(c + dx)/(S(21)d) + S(2)(bsec(c + dx))(S(7)/2)sin(c + dx)/(S(7)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(5)/2)sec(c + dx), x), x, -S(6)b*S(3)EllipticE(c/S(2) + dx/S(2), S(2))/(S(5)dsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(6)b*S(2)sqrt(bsec(c + dx))sin(c + dx)/(S(5)d) + S(2)(bsec(c + dx))*(S(5)/2)sin(c + dx)/(S(5)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(5)/2), x), x, S(2)b*S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(3)d) + S(2)b(bsec(c + dx))(S(3)/2)sin(c + dx)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(5)/2)cos(c + dx), x), x, -S(2)b*S(3)EllipticE(c/S(2) + dx/S(2), S(2))/(dsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(2)bS(2)sqrt(bsec(c + dx))sin(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(5)/2)cos(c + dx)S(2), x), x, S(2)b*S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(5)/2)cos(c + dx)S(3), x), x, S(2)b*S(3)EllipticE(c/S(2) + dx/S(2), S(2))/(dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(5)/2)cos(c + dx)S(4), x), x, S(2)b*S(3)sin(c + dx)/(S(3)dsqrt(bsec(c + dx))) + S(2)b*S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(5)/2)cos(c + dx)S(5), x), x, S(2)b*S(4)sin(c + dx)/(S(5)d(bsec(c + dx))(S(3)/2)) + S(6)b*S(3)EllipticE(c/S(2) + dx/S(2), S(2))/(S(5)dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(5)/2)cos(c + dx)S(6), x), x, S(2)b*S(5)sin(c + dx)/(S(7)d(bsec(c + dx))(S(5)/2)) + S(10)b*S(3)sin(c + dx)/(S(21)dsqrt(bsec(c + dx))) + S(10)b*S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(5)/2)cos(c + dx)S(7), x), x, S(2)b*S(6)sin(c + dx)/(S(9)d(bsec(c + dx))(S(7)/2)) + S(14)b*S(4)sin(c + dx)/(S(45)d(bsec(c + dx))(S(3)/2)) + S(14)b*S(3)EllipticE(c/S(2) + dx/S(2), S(2))/(S(15)dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(7)/2), x), x, -S(6)b*S(4)EllipticE(c/S(2) + dx/S(2), S(2))/(S(5)dsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(6)b*S(3)sqrt(bsec(c + dx))sin(c + dx)/(S(5)d) + S(2)b(bsec(c + dx))(S(5)/2)sin(c + dx)/(S(5)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(5)/sqrt(bsec(c + dx)), x), x, S(10)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)bd) + S(10)(bsec(c + dx))(S(3)/2)sin(c + dx)/(S(21)b*S(2)d) + S(2)(bsec(c + dx))(S(7)/2)sin(c + dx)/(S(7)b*S(4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(4)/sqrt(bsec(c + dx)), x), x, -S(6)EllipticE(c/S(2) + dx/S(2), S(2))/(S(5)dsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(6)sqrt(bsec(c + dx))sin(c + dx)/(S(5)bd) + S(2)(bsec(c + dx))(S(5)/2)sin(c + dx)/(S(5)b*S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(3)/sqrt(bsec(c + dx)), x), x, S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(3)bd) + S(2)(bsec(c + dx))(S(3)/2)sin(c + dx)/(S(3)b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(2)/sqrt(bsec(c + dx)), x), x, -S(2)EllipticE(c/S(2) + dx/S(2), S(2))/(dsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(2)sqrt(bsec(c + dx))sin(c + dx)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)/sqrt(bsec(c + dx)), x), x, S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(bsec(c + dx)), x), x, S(2)EllipticE(c/S(2) + dx/S(2), S(2))/(dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)/sqrt(bsec(c + dx)), x), x, S(2)sin(c + dx)/(S(3)dsqrt(bsec(c + dx))) + S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(3)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)S(2)/sqrt(bsec(c + dx)), x), x, S(2)bsin(c + dx)/(S(5)d(bsec(c + dx))*(S(3)/2)) + S(6)EllipticE(c/S(2) + dx/S(2), S(2))/(S(5)dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)*S(3)/sqrt(bsec(c + dx)), x), x, S(2)b*S(2)sin(c + dx)/(S(7)d(bsec(c + dx))(S(5)/2)) + S(10)sin(c + dx)/(S(21)dsqrt(bsec(c + dx))) + S(10)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)S(4)/sqrt(bsec(c + dx)), x), x, S(2)b*S(3)sin(c + dx)/(S(9)d(bsec(c + dx))(S(7)/2)) + S(14)bsin(c + dx)/(S(45)d(bsec(c + dx))*(S(3)/2)) + S(14)EllipticE(c/S(2) + dx/S(2), S(2))/(S(15)dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)*S(6)/(bsec(c + dx))(S(3)/2), x), x, S(10)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)bS(2)d) + S(10)(bsec(c + dx))(S(3)/2)sin(c + dx)/(S(21)b*S(3)d) + S(2)(bsec(c + dx))(S(7)/2)sin(c + dx)/(S(7)b*S(5)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(5)/(bsec(c + dx))(S(3)/2), x), x, -S(6)EllipticE(c/S(2) + dx/S(2), S(2))/(S(5)bdsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(6)sqrt(bsec(c + dx))sin(c + dx)/(S(5)b*S(2)d) + S(2)(bsec(c + dx))(S(5)/2)sin(c + dx)/(S(5)b*S(4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(4)/(bsec(c + dx))(S(3)/2), x), x, S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(3)bS(2)d) + S(2)(bsec(c + dx))(S(3)/2)sin(c + dx)/(S(3)b*S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(3)/(bsec(c + dx))(S(3)/2), x), x, -S(2)EllipticE(c/S(2) + dx/S(2), S(2))/(bdsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(2)sqrt(bsec(c + dx))sin(c + dx)/(b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(2)/(bsec(c + dx))(S(3)/2), x), x, S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)/(bsec(c + dx))(S(3)/2), x), x, S(2)EllipticE(c/S(2) + dx/S(2), S(2))/(bdsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(-3)/2), x), x, S(2)sin(c + dx)/(S(3)bdsqrt(bsec(c + dx))) + S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(3)b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)/(bsec(c + dx))(S(3)/2), x), x, S(2)sin(c + dx)/(S(5)d(bsec(c + dx))(S(3)/2)) + S(6)EllipticE(c/S(2) + dx/S(2), S(2))/(S(5)bdsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)S(2)/(bsec(c + dx))(S(3)/2), x), x, S(2)bsin(c + dx)/(S(7)d(bsec(c + dx))*(S(5)/2)) + S(10)sin(c + dx)/(S(21)bdsqrt(bsec(c + dx))) + S(10)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)b*S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)S(3)/(bsec(c + dx))(S(3)/2), x), x, S(2)b*S(2)sin(c + dx)/(S(9)d(bsec(c + dx))(S(7)/2)) + S(14)sin(c + dx)/(S(45)d(bsec(c + dx))(S(3)/2)) + S(14)EllipticE(c/S(2) + dx/S(2), S(2))/(S(15)bdsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(7)/(bsec(c + dx))(S(5)/2), x), x, S(10)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)bS(3)d) + S(10)(bsec(c + dx))(S(3)/2)sin(c + dx)/(S(21)b*S(4)d) + S(2)(bsec(c + dx))(S(7)/2)sin(c + dx)/(S(7)b*S(6)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(6)/(bsec(c + dx))(S(5)/2), x), x, -S(6)EllipticE(c/S(2) + dx/S(2), S(2))/(S(5)b*S(2)dsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(6)sqrt(bsec(c + dx))sin(c + dx)/(S(5)bS(3)d) + S(2)(bsec(c + dx))(S(5)/2)sin(c + dx)/(S(5)b*S(5)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(5)/(bsec(c + dx))(S(5)/2), x), x, S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(3)bS(3)d) + S(2)(bsec(c + dx))(S(3)/2)sin(c + dx)/(S(3)b*S(4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(4)/(bsec(c + dx))(S(5)/2), x), x, -S(2)EllipticE(c/S(2) + dx/S(2), S(2))/(bS(2)dsqrt(bsec(c + dx))sqrt(cos(c + dx))) + S(2)sqrt(bsec(c + dx))sin(c + dx)/(b*S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(3)/(bsec(c + dx))(S(5)/2), x), x, S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(b*S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)S(2)/(bsec(c + dx))(S(5)/2), x), x, S(2)EllipticE(c/S(2) + dx/S(2), S(2))/(bS(2)dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)/(bsec(c + dx))*(S(5)/2), x), x, S(2)sin(c + dx)/(S(3)b*S(2)dsqrt(bsec(c + dx))) + S(2)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(3)bS(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(-5)/2), x), x, S(2)sin(c + dx)/(S(5)bd(bsec(c + dx))*(S(3)/2)) + S(6)EllipticE(c/S(2) + dx/S(2), S(2))/(S(5)b*S(2)dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)/(bsec(c + dx))*(S(5)/2), x), x, S(2)sin(c + dx)/(S(7)d(bsec(c + dx))(S(5)/2)) + S(10)sin(c + dx)/(S(21)b*S(2)dsqrt(bsec(c + dx))) + S(10)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)bS(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)S(2)/(bsec(c + dx))(S(5)/2), x), x, S(2)bsin(c + dx)/(S(9)d(bsec(c + dx))*(S(7)/2)) + S(14)sin(c + dx)/(S(45)bd(bsec(c + dx))*(S(3)/2)) + S(14)EllipticE(c/S(2) + dx/S(2), S(2))/(S(15)b*S(2)dsqrt(bsec(c + dx))sqrt(cos(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(-7)/2), x), x, S(2)sin(c + dx)/(S(7)bd(bsec(c + dx))*(S(5)/2)) + S(10)sin(c + dx)/(S(21)b*S(3)dsqrt(bsec(c + dx))) + S(10)sqrt(bsec(c + dx))EllipticF(c/S(2) + dx/S(2), S(2))sqrt(cos(c + dx))/(S(21)bS(4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))sec(c + dx)*(S(9)/2), x), x, sqrt(bsec(c + dx))sin(c + dx)sec(c + dx)(S(7)/2)/(S(4)d) + S(3)sqrt(bsec(c + dx))sin(c + dx)sec(c + dx)(S(3)/2)/(S(8)d) + S(3)sqrt(bsec(c + dx))atanh(sin(c + dx))/(S(8)dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))sec(c + dx)*(S(7)/2), x), x, sqrt(bsec(c + dx))sin(c + dx)S(3)sec(c + dx)(S(5)/2)/(S(3)d) + sqrt(bsec(c + dx))sin(c + dx)sqrt(sec(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))sec(c + dx)*(S(5)/2), x), x, sqrt(bsec(c + dx))sin(c + dx)sec(c + dx)(S(3)/2)/(S(2)d) + sqrt(bsec(c + dx))atanh(sin(c + dx))/(S(2)dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))sec(c + dx)(S(3)/2), x), x, sqrt(bsec(c + dx))sin(c + dx)sqrt(sec(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))sqrt(sec(c + dx)), x), x, sqrt(bsec(c + dx))atanh(sin(c + dx))/(dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))/sqrt(sec(c + dx)), x), x, xsqrt(bsec(c + dx))/sqrt(sec(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))/sec(c + dx)(S(3)/2), x), x, sqrt(bsec(c + dx))sin(c + dx)/(dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))/sec(c + dx)*(S(5)/2), x), x, xsqrt(bsec(c + dx))/(S(2)sqrt(sec(c + dx))) + sqrt(bsec(c + dx))sin(c + dx)/(S(2)dsec(c + dx)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))/sec(c + dx)*(S(7)/2), x), x, -sqrt(bsec(c + dx))sin(c + dx)S(3)/(S(3)dsqrt(sec(c + dx))) + sqrt(bsec(c + dx))sin(c + dx)/(dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(c + dx))/sec(c + dx)(S(9)/2), x), x, S(3)xsqrt(bsec(c + dx))/(S(8)sqrt(sec(c + dx))) + S(3)sqrt(bsec(c + dx))sin(c + dx)/(S(8)dsec(c + dx)(S(3)/2)) + sqrt(bsec(c + dx))sin(c + dx)/(S(4)dsec(c + dx)*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(3)/2)sec(c + dx)(S(7)/2), x), x, bsqrt(bsec(c + dx))sin(c + dx)sec(c + dx)*(S(7)/2)/(S(4)d) + S(3)bsqrt(bsec(c + dx))sin(c + dx)sec(c + dx)*(S(3)/2)/(S(8)d) + S(3)bsqrt(bsec(c + dx))atanh(sin(c + dx))/(S(8)dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(3)/2)sec(c + dx)(S(5)/2), x), x, bsqrt(bsec(c + dx))sin(c + dx)*S(3)sec(c + dx)(S(5)/2)/(S(3)d) + bsqrt(bsec(c + dx))sin(c + dx)sqrt(sec(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(3)/2)sec(c + dx)(S(3)/2), x), x, bsqrt(bsec(c + dx))sin(c + dx)sec(c + dx)*(S(3)/2)/(S(2)d) + bsqrt(bsec(c + dx))atanh(sin(c + dx))/(S(2)dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(3)/2)sqrt(sec(c + dx)), x), x, bsqrt(bsec(c + dx))sin(c + dx)sqrt(sec(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(3)/2)/sqrt(sec(c + dx)), x), x, bsqrt(bsec(c + dx))atanh(sin(c + dx))/(dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(3)/2)/sec(c + dx)*(S(3)/2), x), x, bxsqrt(bsec(c + dx))/sqrt(sec(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(3)/2)/sec(c + dx)*(S(5)/2), x), x, bsqrt(bsec(c + dx))sin(c + dx)/(dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(3)/2)/sec(c + dx)*(S(7)/2), x), x, bxsqrt(bsec(c + dx))/(S(2)sqrt(sec(c + dx))) + bsqrt(bsec(c + dx))sin(c + dx)/(S(2)dsec(c + dx)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(3)/2)/sec(c + dx)*(S(9)/2), x), x, -bsqrt(bsec(c + dx))sin(c + dx)*S(3)/(S(3)dsqrt(sec(c + dx))) + bsqrt(bsec(c + dx))sin(c + dx)/(dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(3)/2)/sec(c + dx)*(S(11)/2), x), x, S(3)bxsqrt(bsec(c + dx))/(S(8)sqrt(sec(c + dx))) + S(3)bsqrt(bsec(c + dx))sin(c + dx)/(S(8)dsec(c + dx)(S(3)/2)) + bsqrt(bsec(c + dx))sin(c + dx)/(S(4)dsec(c + dx)(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(5)/2)sec(c + dx)(S(7)/2), x), x, bS(2)sqrt(bsec(c + dx))sin(c + dx)*S(5)sec(c + dx)(S(9)/2)/(S(5)d) + S(2)bS(2)sqrt(bsec(c + dx))sin(c + dx)*S(3)sec(c + dx)(S(5)/2)/(S(3)d) + b*S(2)sqrt(bsec(c + dx))sin(c + dx)sqrt(sec(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(5)/2)sec(c + dx)(S(3)/2), x), x, bS(2)sqrt(bsec(c + dx))sin(c + dx)*S(3)sec(c + dx)(S(5)/2)/(S(3)d) + b*S(2)sqrt(bsec(c + dx))sin(c + dx)sqrt(sec(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(5)/2)sqrt(sec(c + dx)), x), x, bS(2)sqrt(bsec(c + dx))sin(c + dx)sec(c + dx)*(S(3)/2)/(S(2)d) + b*S(2)sqrt(bsec(c + dx))atanh(sin(c + dx))/(S(2)dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(5)/2)/sqrt(sec(c + dx)), x), x, b*S(2)sqrt(bsec(c + dx))sin(c + dx)sqrt(sec(c + dx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))*(S(5)/2)/sec(c + dx)(S(3)/2), x), x, bS(2)sqrt(bsec(c + dx))atanh(sin(c + dx))/(dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(5)/2)/sec(c + dx)(S(5)/2), x), x, bS(2)xsqrt(bsec(c + dx))/sqrt(sec(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(5)/2)/sec(c + dx)(S(7)/2), x), x, bS(2)sqrt(bsec(c + dx))sin(c + dx)/(dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(5)/2)/sec(c + dx)(S(9)/2), x), x, bS(2)xsqrt(bsec(c + dx))/(S(2)sqrt(sec(c + dx))) + b*S(2)sqrt(bsec(c + dx))sin(c + dx)/(S(2)dsec(c + dx)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(5)/2)/sec(c + dx)(S(11)/2), x), x, -bS(2)sqrt(bsec(c + dx))sin(c + dx)S(3)/(S(3)dsqrt(sec(c + dx))) + b*S(2)sqrt(bsec(c + dx))sin(c + dx)/(dsqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)(S(7)/2)/sqrt(bsec(c + dx)), x), x, sin(c + dx)sec(c + dx)*(S(5)/2)/(S(2)dsqrt(bsec(c + dx))) + atanh(sin(c + dx))sqrt(sec(c + dx))/(S(2)dsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)(S(5)/2)/sqrt(bsec(c + dx)), x), x, sin(c + dx)sec(c + dx)*(S(3)/2)/(dsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)(S(3)/2)/sqrt(bsec(c + dx)), x), x, atanh(sin(c + dx))sqrt(sec(c + dx))/(dsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sec(c + dx))/sqrt(bsec(c + dx)), x), x, xsqrt(sec(c + dx))/sqrt(bsec(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(bsec(c + dx))sqrt(sec(c + dx))), x), x, sin(c + dx)sqrt(sec(c + dx))/(dsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(bsec(c + dx))sec(c + dx)*(S(3)/2)), x), x, xsqrt(sec(c + dx))/(S(2)sqrt(bsec(c + dx))) + sin(c + dx)/(S(2)dsqrt(bsec(c + dx))sqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(bsec(c + dx))sec(c + dx)(S(5)/2)), x), x, -sin(c + dx)*S(3)sqrt(sec(c + dx))/(S(3)dsqrt(bsec(c + dx))) + sin(c + dx)sqrt(sec(c + dx))/(dsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)*(S(9)/2)/(bsec(c + dx))(S(3)/2), x), x, sin(c + dx)sec(c + dx)*(S(5)/2)/(S(2)bdsqrt(bsec(c + dx))) + atanh(sin(c + dx))sqrt(sec(c + dx))/(S(2)bdsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)(S(7)/2)/(bsec(c + dx))(S(3)/2), x), x, sin(c + dx)sec(c + dx)*(S(3)/2)/(bdsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)*(S(5)/2)/(bsec(c + dx))(S(3)/2), x), x, atanh(sin(c + dx))sqrt(sec(c + dx))/(bdsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)(S(3)/2)/(bsec(c + dx))(S(3)/2), x), x, xsqrt(sec(c + dx))/(bsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sec(c + dx))/(bsec(c + dx))(S(3)/2), x), x, sin(c + dx)sqrt(sec(c + dx))/(bdsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((bsec(c + dx))*(S(3)/2)sqrt(sec(c + dx))), x), x, xsqrt(sec(c + dx))/(S(2)bsqrt(bsec(c + dx))) + sin(c + dx)/(S(2)bdsqrt(bsec(c + dx))sqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((bsec(c + dx))(S(3)/2)sec(c + dx)(S(3)/2)), x), x, -sin(c + dx)*S(3)sqrt(sec(c + dx))/(S(3)bdsqrt(bsec(c + dx))) + sin(c + dx)sqrt(sec(c + dx))/(bdsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((bsec(c + dx))(S(3)/2)sec(c + dx)(S(5)/2)), x), x, S(3)xsqrt(sec(c + dx))/(S(8)bsqrt(bsec(c + dx))) + S(3)sin(c + dx)/(S(8)bdsqrt(bsec(c + dx))sqrt(sec(c + dx))) + sin(c + dx)/(S(4)bdsqrt(bsec(c + dx))sec(c + dx)(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)*(S(11)/2)/(bsec(c + dx))(S(5)/2), x), x, sin(c + dx)sec(c + dx)*(S(5)/2)/(S(2)b*S(2)dsqrt(bsec(c + dx))) + atanh(sin(c + dx))sqrt(sec(c + dx))/(S(2)bS(2)dsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)*(S(9)/2)/(bsec(c + dx))(S(5)/2), x), x, sin(c + dx)sec(c + dx)(S(3)/2)/(bS(2)dsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)(S(7)/2)/(bsec(c + dx))(S(5)/2), x), x, atanh(sin(c + dx))sqrt(sec(c + dx))/(b*S(2)dsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)*(S(5)/2)/(bsec(c + dx))(S(5)/2), x), x, xsqrt(sec(c + dx))/(bS(2)sqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)(S(3)/2)/(bsec(c + dx))(S(5)/2), x), x, sin(c + dx)sqrt(sec(c + dx))/(b*S(2)dsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sec(c + dx))/(bsec(c + dx))*(S(5)/2), x), x, xsqrt(sec(c + dx))/(S(2)b*S(2)sqrt(bsec(c + dx))) + sin(c + dx)/(S(2)b*S(2)dsqrt(bsec(c + dx))sqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((bsec(c + dx))(S(5)/2)sqrt(sec(c + dx))), x), x, -sin(c + dx)*S(3)sqrt(sec(c + dx))/(S(3)b*S(2)dsqrt(bsec(c + dx))) + sin(c + dx)sqrt(sec(c + dx))/(b*S(2)dsqrt(bsec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((bsec(c + dx))(S(5)/2)sec(c + dx)(S(3)/2)), x), x, S(3)xsqrt(sec(c + dx))/(S(8)bS(2)sqrt(bsec(c + dx))) + S(3)sin(c + dx)/(S(8)bS(2)dsqrt(bsec(c + dx))sqrt(sec(c + dx))) + sin(c + dx)/(S(4)bS(2)dsqrt(bsec(c + dx))sec(c + dx)(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(1)/3)sec(c + dx)S(2), x), x, S(3)(bsec(c + dx))*(S(4)/3)Hypergeometric2F1(S(-2)/3, S(1)/2, S(1)/3, cos(c + dx)S(2))sin(c + dx)/(S(4)bdsqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(1)/3)sec(c + dx), x), x, S(3)(bsec(c + dx))*(S(1)/3)Hypergeometric2F1(S(-1)/6, S(1)/2, S(5)/6, cos(c + dx)S(2))sin(c + dx)/(dsqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(1)/3), x), x, -S(3)bHypergeometric2F1(S(1)/3, S(1)/2, S(4)/3, cos(c + dx)*S(2))sin(c + dx)/(S(2)d(bsec(c + dx))(S(2)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(1)/3)cos(c + dx), x), x, -S(3)b*S(2)Hypergeometric2F1(S(1)/2, S(5)/6, S(11)/6, cos(c + dx)S(2))sin(c + dx)/(S(5)d(bsec(c + dx))(S(5)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(1)/3)cos(c + dx)S(2), x), x, -S(3)b*S(3)Hypergeometric2F1(S(1)/2, S(4)/3, S(7)/3, cos(c + dx)S(2))sin(c + dx)/(S(8)d(bsec(c + dx))(S(8)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(4)/3)sec(c + dx)S(2), x), x, S(3)(bsec(c + dx))*(S(7)/3)Hypergeometric2F1(S(-7)/6, S(1)/2, S(-1)/6, cos(c + dx)S(2))sin(c + dx)/(S(7)bdsqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(4)/3)sec(c + dx), x), x, S(3)(bsec(c + dx))*(S(4)/3)Hypergeometric2F1(S(-2)/3, S(1)/2, S(1)/3, cos(c + dx)S(2))sin(c + dx)/(S(4)dsqrt(sin(c + dx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(4)/3), x), x, S(3)b(bsec(c + dx))(S(1)/3)Hypergeometric2F1(S(-1)/6, S(1)/2, S(5)/6, cos(c + dx)S(2))sin(c + dx)/(dsqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(4)/3)cos(c + dx), x), x, -S(3)b*S(2)Hypergeometric2F1(S(1)/3, S(1)/2, S(4)/3, cos(c + dx)S(2))sin(c + dx)/(S(2)d(bsec(c + dx))(S(2)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(4)/3)cos(c + dx)S(2), x), x, -S(3)b*S(3)Hypergeometric2F1(S(1)/2, S(5)/6, S(11)/6, cos(c + dx)S(2))sin(c + dx)/(S(5)d(bsec(c + dx))(S(5)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)*S(2)/(bsec(c + dx))(S(1)/3), x), x, S(3)(bsec(c + dx))*(S(2)/3)Hypergeometric2F1(S(-1)/3, S(1)/2, S(2)/3, cos(c + dx)S(2))sin(c + dx)/(S(2)bdsqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)/(bsec(c + dx))*(S(1)/3), x), x, -S(3)Hypergeometric2F1(S(1)/6, S(1)/2, S(7)/6, cos(c + dx)S(2))sin(c + dx)/(d(bsec(c + dx))*(S(1)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(-1)/3), x), x, -S(3)bHypergeometric2F1(S(1)/2, S(2)/3, S(5)/3, cos(c + dx)*S(2))sin(c + dx)/(S(4)d(bsec(c + dx))(S(4)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)/(bsec(c + dx))*(S(1)/3), x), x, -S(3)b*S(2)Hypergeometric2F1(S(1)/2, S(7)/6, S(13)/6, cos(c + dx)S(2))sin(c + dx)/(S(7)d(bsec(c + dx))(S(7)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)*S(2)/(bsec(c + dx))(S(1)/3), x), x, -S(3)b*S(3)Hypergeometric2F1(S(1)/2, S(5)/3, S(8)/3, cos(c + dx)S(2))sin(c + dx)/(S(10)d(bsec(c + dx))(S(10)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)*S(2)/(bsec(c + dx))(S(4)/3), x), x, -S(3)Hypergeometric2F1(S(1)/6, S(1)/2, S(7)/6, cos(c + dx)S(2))sin(c + dx)/(bd(bsec(c + dx))(S(1)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)/(bsec(c + dx))*(S(4)/3), x), x, -S(3)Hypergeometric2F1(S(1)/2, S(2)/3, S(5)/3, cos(c + dx)S(2))sin(c + dx)/(S(4)d(bsec(c + dx))(S(4)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(-4)/3), x), x, -S(3)bHypergeometric2F1(S(1)/2, S(7)/6, S(13)/6, cos(c + dx)*S(2))sin(c + dx)/(S(7)d(bsec(c + dx))(S(7)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)/(bsec(c + dx))*(S(4)/3), x), x, -S(3)b*S(2)Hypergeometric2F1(S(1)/2, S(5)/3, S(8)/3, cos(c + dx)S(2))sin(c + dx)/(S(10)d(bsec(c + dx))(S(10)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(c + dx)*S(2)/(bsec(c + dx))(S(4)/3), x), x, -S(3)b*S(3)Hypergeometric2F1(S(1)/2, S(13)/6, S(19)/6, cos(c + dx)S(2))sin(c + dx)/(S(13)d(bsec(c + dx))(S(13)/3)sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(4)/3)sec(c + dx)m, x), x, S(3)b(bsec(c + dx))(S(1)/3)Hypergeometric2F1(S(1)/2, -m/S(2) + S(-1)/6, -m/S(2) + S(5)/6, cos(c + dx)S(2))sin(c + dx)sec(c + dx)m/(d(S(3)m + S(1))sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(2)/3)sec(c + dx)m, x), x, -S(3)(bsec(c + dx))*(S(2)/3)Hypergeometric2F1(S(1)/2, -m/S(2) + S(1)/6, -m/S(2) + S(7)/6, cos(c + dx)S(2))sin(c + dx)sec(c + dx)(m + S(-1))/(d(-S(3)m + S(1))sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))(S(1)/3)sec(c + dx)m, x), x, -S(3)(bsec(c + dx))*(S(1)/3)Hypergeometric2F1(S(1)/2, -m/S(2) + S(1)/3, -m/S(2) + S(4)/3, cos(c + dx)S(2))sin(c + dx)sec(c + dx)(m + S(-1))/(d(-S(3)m + S(2))sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)*m/(bsec(c + dx))(S(1)/3), x), x, -S(3)Hypergeometric2F1(S(1)/2, -m/S(2) + S(2)/3, -m/S(2) + S(5)/3, cos(c + dx)S(2))sin(c + dx)sec(c + dx)(m + S(-1))/(d(bsec(c + dx))*(S(1)/3)(-S(3)m + S(4))sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)*m/(bsec(c + dx))(S(2)/3), x), x, -S(3)Hypergeometric2F1(S(1)/2, -m/S(2) + S(5)/6, -m/S(2) + S(11)/6, cos(c + dx)S(2))sin(c + dx)sec(c + dx)(m + S(-1))/(d(bsec(c + dx))*(S(2)/3)(-S(3)m + S(5))sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(c + dx)*m/(bsec(c + dx))(S(4)/3), x), x, -S(3)Hypergeometric2F1(S(1)/2, -m/S(2) + S(7)/6, -m/S(2) + S(13)/6, cos(c + dx)S(2))sin(c + dx)sec(c + dx)(m + S(-2))/(bd(bsec(c + dx))(S(1)/3)(-S(3)m + S(7))sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))nsec(c + dx)m, x), x, -(bsec(c + dx))nHypergeometric2F1(S(1)/2, -m/S(2) - n/S(2) + S(1)/2, -m/S(2) - n/S(2) + S(3)/2, cos(c + dx)S(2))sin(c + dx)sec(c + dx)(m + S(-1))/(d(-m - n + S(1))sqrt(sin(c + dx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))nsec(c + dx)S(2), x), x, (bsec(c + dx))(n + S(1))Hypergeometric2F1(S(1)/2, -n/S(2) + S(-1)/2, -n/S(2) + S(1)/2, cos(c + dx)S(2))sin(c + dx)/(bd(n + S(1))sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))nsec(c + dx), x), x, (bsec(c + dx))nHypergeometric2F1(S(1)/2, -n/S(2), -n/S(2) + S(1), cos(c + dx)S(2))sin(c + dx)/(dnsqrt(sin(c + dx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))n, x), x, -b(bsec(c + dx))*(n + S(-1))Hypergeometric2F1(S(1)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(c + dx)S(2))sin(c + dx)/(d(-n + S(1))sqrt(sin(c + dx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))ncos(c + dx), x), x, -bS(2)(bsec(c + dx))*(n + S(-2))Hypergeometric2F1(S(1)/2, -n/S(2) + S(1), -n/S(2) + S(2), cos(c + dx)S(2))sin(c + dx)/(d(-n + S(2))sqrt(sin(c + dx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))ncos(c + dx)S(2), x), x, -bS(3)(bsec(c + dx))*(n + S(-3))Hypergeometric2F1(S(1)/2, -n/S(2) + S(3)/2, -n/S(2) + S(5)/2, cos(c + dx)S(2))sin(c + dx)/(d(-n + S(3))sqrt(sin(c + dx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))ncos(c + dx)S(3), x), x, -bS(4)(bsec(c + dx))*(n + S(-4))Hypergeometric2F1(S(1)/2, -n/S(2) + S(2), -n/S(2) + S(3), cos(c + dx)S(2))sin(c + dx)/(d(-n + S(4))sqrt(sin(c + dx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))nsec(c + dx)(S(5)/2), x), x, S(2)(bsec(c + dx))*nHypergeometric2F1(S(1)/2, -n/S(2) + S(-3)/4, -n/S(2) + S(1)/4, cos(c + dx)S(2))sin(c + dx)sec(c + dx)(S(3)/2)/(d(S(2)n + S(3))sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))nsec(c + dx)(S(3)/2), x), x, S(2)(bsec(c + dx))*nHypergeometric2F1(S(1)/2, -n/S(2) + S(-1)/4, -n/S(2) + S(3)/4, cos(c + dx)S(2))sin(c + dx)sqrt(sec(c + dx))/(d(S(2)n + S(1))sqrt(sin(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))nsqrt(sec(c + dx)), x), x, -S(2)(bsec(c + dx))*nHypergeometric2F1(S(1)/2, -n/S(2) + S(1)/4, -n/S(2) + S(5)/4, cos(c + dx)S(2))sin(c + dx)/(d(-S(2)n + S(1))sqrt(sin(c + dx)S(2))sqrt(sec(c + dx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))n/sqrt(sec(c + dx)), x), x, -S(2)(bsec(c + dx))nHypergeometric2F1(S(1)/2, -n/S(2) + S(3)/4, -n/S(2) + S(7)/4, cos(c + dx)S(2))sin(c + dx)/(d(-S(2)n + S(3))sqrt(sin(c + dx)S(2))sec(c + dx)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))n/sec(c + dx)*(S(3)/2), x), x, -S(2)(bsec(c + dx))*nHypergeometric2F1(S(1)/2, -n/S(2) + S(5)/4, -n/S(2) + S(9)/4, cos(c + dx)S(2))sin(c + dx)/(d(-S(2)n + S(5))sqrt(sin(c + dx)S(2))sec(c + dx)(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(c + dx))n/sec(c + dx)*(S(5)/2), x), x, -S(2)(bsec(c + dx))*nHypergeometric2F1(S(1)/2, -n/S(2) + S(7)/4, -n/S(2) + S(11)/4, cos(c + dx)S(2))sin(c + dx)/(d(-S(2)n + S(7))sqrt(sin(c + dx)S(2))sec(c + dx)(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(a + bx))(S(7)/2)sin(a + bx), x), x, S(2)d(dsec(a + bx))(S(5)/2)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(a + bx))*(S(5)/2)sin(a + bx), x), x, S(2)d(dsec(a + bx))(S(3)/2)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(a + bx))*(S(3)/2)sin(a + bx), x), x, S(2)dsqrt(dsec(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dsec(a + bx))sin(a + bx), x), x, -S(2)d/(bsqrt(dsec(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)/sqrt(dsec(a + bx)), x), x, -S(2)d/(S(3)b(dsec(a + bx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(a + bx))(S(5)/2)sin(a + bx)S(3), x), x, S(2)d*S(3)/(bsqrt(dsec(a + bx))) + S(2)d(dsec(a + bx))*(S(3)/2)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dsec(a + bx))*(S(9)/2)sin(a + bx)S(3), x), x, -S(2)d*S(3)(dsec(a + bx))*(S(3)/2)/(S(3)b) + S(2)d(dsec(a + bx))*(S(7)/2)/(S(7)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csec(a + bx))(dcsc(a + bx))(S(9)/2), x), x, -S(4)cdS(3)(dcsc(a + bx))*(S(3)/2)/(S(7)bsqrt(csec(a + bx))) - S(2)cd(dcsc(a + bx))*(S(7)/2)/(S(7)bsqrt(csec(a + bx))) + S(4)d*S(4)sqrt(csec(a + bx))sqrt(dcsc(a + bx))EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(7)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csec(a + bx))(dcsc(a + bx))(S(7)/2), x), x, -S(8)cdS(3)sqrt(dcsc(a + bx))/(S(5)bsqrt(csec(a + bx))) - S(2)cd(dcsc(a + bx))(S(5)/2)/(S(5)bsqrt(csec(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csec(a + bx))(dcsc(a + bx))*(S(5)/2), x), x, -S(2)cd(dcsc(a + bx))*(S(3)/2)/(S(3)bsqrt(csec(a + bx))) + S(2)d*S(2)sqrt(csec(a + bx))sqrt(dcsc(a + bx))EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csec(a + bx))(dcsc(a + bx))(S(3)/2), x), x, -S(2)cdsqrt(dcsc(a + bx))/(bsqrt(csec(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csec(a + bx))sqrt(dcsc(a + bx)), x), x, sqrt(csec(a + bx))sqrt(dcsc(a + bx))EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csec(a + bx))/sqrt(dcsc(a + bx)), x), x, -sqrt(S(2))sqrt(csec(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))/(S(2)bsqrt(dcsc(a + bx))sqrt(tan(a + bx))) + sqrt(S(2))sqrt(csec(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))/(S(2)bsqrt(dcsc(a + bx))sqrt(tan(a + bx))) + sqrt(S(2))sqrt(csec(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))/(S(4)bsqrt(dcsc(a + bx))sqrt(tan(a + bx))) - sqrt(S(2))sqrt(csec(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))/(S(4)bsqrt(dcsc(a + bx))sqrt(tan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csec(a + bx))/(dcsc(a + bx))(S(3)/2), x), x, -c/(bdsqrt(csec(a + bx))sqrt(dcsc(a + bx))) + sqrt(csec(a + bx))sqrt(dcsc(a + bx))EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(2)bdS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csec(a + bx))/(dcsc(a + bx))(S(5)/2), x), x, -c/(S(2)bdsqrt(csec(a + bx))(dcsc(a + bx))(S(3)/2)) - S(3)sqrt(S(2))sqrt(csec(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))/(S(8)bd*S(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) + S(3)sqrt(S(2))sqrt(csec(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))/(S(8)bdS(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) + S(3)sqrt(S(2))sqrt(csec(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))/(S(16)bd*S(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) - S(3)sqrt(S(2))sqrt(csec(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))/(S(16)bd*S(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(3)/2)(dcsc(a + bx))*(S(9)/2), x), x, S(64)cdS(5)sqrt(csec(a + bx))/(S(21)bsqrt(dcsc(a + bx))) - S(16)cd*S(3)sqrt(csec(a + bx))(dcsc(a + bx))(S(3)/2)/(S(21)b) - S(2)cdsqrt(csec(a + bx))(dcsc(a + bx))*(S(7)/2)/(S(7)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(3)/2)(dcsc(a + bx))*(S(7)/2), x), x, -S(24)c*S(2)d*S(4)EllipticE(-Pi/S(4) + a + bx, S(2))/(S(5)bsqrt(csec(a + bx))sqrt(dcsc(a + bx))sqrt(sin(S(2)a + S(2)bx))) + S(24)cd*S(5)sqrt(csec(a + bx))/(S(5)b(dcsc(a + bx))*(S(3)/2)) - S(12)cdS(3)sqrt(csec(a + bx))sqrt(dcsc(a + bx))/(S(5)b) - S(2)cdsqrt(csec(a + bx))(dcsc(a + bx))*(S(5)/2)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(3)/2)(dcsc(a + bx))*(S(5)/2), x), x, S(8)cdS(3)sqrt(csec(a + bx))/(S(3)bsqrt(dcsc(a + bx))) - S(2)cdsqrt(csec(a + bx))(dcsc(a + bx))*(S(3)/2)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(3)/2)(dcsc(a + bx))*(S(3)/2), x), x, -S(4)c*S(2)d*S(2)EllipticE(-Pi/S(4) + a + bx, S(2))/(bsqrt(csec(a + bx))sqrt(dcsc(a + bx))sqrt(sin(S(2)a + S(2)bx))) + S(4)cdS(3)sqrt(csec(a + bx))/(b(dcsc(a + bx))(S(3)/2)) - S(2)cdsqrt(csec(a + bx))sqrt(dcsc(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(3)/2)sqrt(dcsc(a + bx)), x), x, S(2)cdsqrt(csec(a + bx))/(bsqrt(dcsc(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(3)/2)/sqrt(dcsc(a + bx)), x), x, -S(2)c*S(2)EllipticE(-Pi/S(4) + a + bx, S(2))/(bsqrt(csec(a + bx))sqrt(dcsc(a + bx))sqrt(sin(S(2)a + S(2)bx))) + S(2)cdsqrt(csec(a + bx))/(b(dcsc(a + bx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(3)/2)/(dcsc(a + bx))(S(3)/2), x), x, sqrt(S(2))c*S(2)sqrt(dcsc(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(2)bdS(2)sqrt(csec(a + bx))) - sqrt(S(2))cS(2)sqrt(dcsc(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(2)bdS(2)sqrt(csec(a + bx))) + sqrt(S(2))cS(2)sqrt(dcsc(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(4)bd*S(2)sqrt(csec(a + bx))) - sqrt(S(2))cS(2)sqrt(dcsc(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(4)bd*S(2)sqrt(csec(a + bx))) + S(2)csqrt(csec(a + bx))/(bdsqrt(dcsc(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(3)/2)/(dcsc(a + bx))(S(5)/2), x), x, -S(3)c*S(2)EllipticE(-Pi/S(4) + a + bx, S(2))/(bd*S(2)sqrt(csec(a + bx))sqrt(dcsc(a + bx))sqrt(sin(S(2)a + S(2)bx))) + S(2)csqrt(csec(a + bx))/(bd(dcsc(a + bx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(5)/2)(dcsc(a + bx))*(S(9)/2), x), x, S(40)c*S(2)d*S(4)sqrt(csec(a + bx))sqrt(dcsc(a + bx))EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(21)b) + S(40)cd*S(5)(csec(a + bx))*(S(3)/2)/(S(21)bsqrt(dcsc(a + bx))) - S(20)cdS(3)(csec(a + bx))*(S(3)/2)(dcsc(a + bx))*(S(3)/2)/(S(21)b) - S(2)cd(csec(a + bx))(S(3)/2)(dcsc(a + bx))*(S(7)/2)/(S(7)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(5)/2)(dcsc(a + bx))*(S(7)/2), x), x, S(64)cdS(5)(csec(a + bx))*(S(3)/2)/(S(15)b(dcsc(a + bx))(S(3)/2)) - S(16)cdS(3)(csec(a + bx))*(S(3)/2)sqrt(dcsc(a + bx))/(S(5)b) - S(2)cd(csec(a + bx))*(S(3)/2)(dcsc(a + bx))*(S(5)/2)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(5)/2)(dcsc(a + bx))*(S(5)/2), x), x, S(4)c*S(2)d*S(2)sqrt(csec(a + bx))sqrt(dcsc(a + bx))EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(3)b) + S(4)cd*S(3)(csec(a + bx))*(S(3)/2)/(S(3)bsqrt(dcsc(a + bx))) - S(2)cd(csec(a + bx))*(S(3)/2)(dcsc(a + bx))*(S(3)/2)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(5)/2)(dcsc(a + bx))*(S(3)/2), x), x, S(8)cdS(3)(csec(a + bx))*(S(3)/2)/(S(3)b(dcsc(a + bx))(S(3)/2)) - S(2)cd(csec(a + bx))*(S(3)/2)sqrt(dcsc(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(5)/2)sqrt(dcsc(a + bx)), x), x, S(2)cS(2)sqrt(csec(a + bx))sqrt(dcsc(a + bx))EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(3)b) + S(2)cd(csec(a + bx))(S(3)/2)/(S(3)bsqrt(dcsc(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(5)/2)/sqrt(dcsc(a + bx)), x), x, S(2)cd(csec(a + bx))*(S(3)/2)/(S(3)b(dcsc(a + bx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))(S(5)/2)/(dcsc(a + bx))(S(3)/2), x), x, -cS(2)sqrt(csec(a + bx))sqrt(dcsc(a + bx))EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(3)bdS(2)) + S(2)c(csec(a + bx))(S(3)/2)/(S(3)bdsqrt(dcsc(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csec(a + bx))*(S(5)/2)/(dcsc(a + bx))(S(5)/2), x), x, sqrt(S(2))c*S(2)sqrt(csec(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))/(S(2)bdS(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) - sqrt(S(2))cS(2)sqrt(csec(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))/(S(2)bdS(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) - sqrt(S(2))cS(2)sqrt(csec(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))/(S(4)bd*S(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) + sqrt(S(2))cS(2)sqrt(csec(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))/(S(4)bd*S(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) + S(2)c(csec(a + bx))*(S(3)/2)/(S(3)bd(dcsc(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))(S(9)/2)/sqrt(csec(a + bx)), x), x, -S(8)cdS(3)(dcsc(a + bx))*(S(3)/2)/(S(21)b(csec(a + bx))(S(3)/2)) - S(2)cd(dcsc(a + bx))*(S(7)/2)/(S(7)b(csec(a + bx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))(S(7)/2)/sqrt(csec(a + bx)), x), x, -S(4)cdS(3)sqrt(dcsc(a + bx))/(S(5)b(csec(a + bx))*(S(3)/2)) - S(2)cd(dcsc(a + bx))*(S(5)/2)/(S(5)b(csec(a + bx))(S(3)/2)) - S(4)d*S(4)EllipticE(-Pi/S(4) + a + bx, S(2))/(S(5)bsqrt(csec(a + bx))sqrt(dcsc(a + bx))sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))*(S(5)/2)/sqrt(csec(a + bx)), x), x, -S(2)cd(dcsc(a + bx))*(S(3)/2)/(S(3)b(csec(a + bx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))(S(3)/2)/sqrt(csec(a + bx)), x), x, -S(2)cdsqrt(dcsc(a + bx))/(b(csec(a + bx))(S(3)/2)) - S(2)d*S(2)EllipticE(-Pi/S(4) + a + bx, S(2))/(bsqrt(csec(a + bx))sqrt(dcsc(a + bx))sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcsc(a + bx))/sqrt(csec(a + bx)), x), x, -sqrt(S(2))sqrt(dcsc(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(2)bsqrt(csec(a + bx))) + sqrt(S(2))sqrt(dcsc(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(2)bsqrt(csec(a + bx))) - sqrt(S(2))sqrt(dcsc(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(4)bsqrt(csec(a + bx))) + sqrt(S(2))sqrt(dcsc(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(4)bsqrt(csec(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(csec(a + bx))sqrt(dcsc(a + bx))), x), x, EllipticE(-Pi/S(4) + a + bx, S(2))/(bsqrt(csec(a + bx))sqrt(dcsc(a + bx))sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(csec(a + bx))(dcsc(a + bx))*(S(3)/2)), x), x, -c/(S(2)bd(csec(a + bx))*(S(3)/2)sqrt(dcsc(a + bx))) - sqrt(S(2))sqrt(dcsc(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(8)bd*S(2)sqrt(csec(a + bx))) + sqrt(S(2))sqrt(dcsc(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(8)bd*S(2)sqrt(csec(a + bx))) - sqrt(S(2))sqrt(dcsc(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(16)bdS(2)sqrt(csec(a + bx))) + sqrt(S(2))sqrt(dcsc(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(16)bdS(2)sqrt(csec(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(csec(a + bx))(dcsc(a + bx))(S(5)/2)), x), x, -c/(S(3)bd(csec(a + bx))*(S(3)/2)(dcsc(a + bx))*(S(3)/2)) + EllipticE(-Pi/S(4) + a + bx, S(2))/(S(2)bd*S(2)sqrt(csec(a + bx))sqrt(dcsc(a + bx))sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))(S(11)/2)/(csec(a + bx))(S(3)/2), x), x, S(8)d*S(5)sqrt(dcsc(a + bx))/(S(45)bcsqrt(csec(a + bx))) + S(2)d*S(3)(dcsc(a + bx))*(S(5)/2)/(S(45)bcsqrt(csec(a + bx))) - S(2)d(dcsc(a + bx))*(S(9)/2)/(S(9)bcsqrt(csec(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))*(S(9)/2)/(csec(a + bx))(S(3)/2), x), x, S(2)d*S(3)(dcsc(a + bx))*(S(3)/2)/(S(21)bcsqrt(csec(a + bx))) - S(2)d(dcsc(a + bx))*(S(7)/2)/(S(7)bcsqrt(csec(a + bx))) - S(2)dS(4)sqrt(csec(a + bx))sqrt(dcsc(a + bx))EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(21)bcS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))(S(7)/2)/(csec(a + bx))(S(3)/2), x), x, -S(2)cd(dcsc(a + bx))*(S(5)/2)/(S(5)b(csec(a + bx))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))(S(5)/2)/(csec(a + bx))(S(3)/2), x), x, -S(2)d(dcsc(a + bx))(S(3)/2)/(S(3)bcsqrt(csec(a + bx))) - d*S(2)sqrt(csec(a + bx))sqrt(dcsc(a + bx))EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(3)bcS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))(S(3)/2)/(csec(a + bx))(S(3)/2), x), x, -S(2)dsqrt(dcsc(a + bx))/(bcsqrt(csec(a + bx))) + sqrt(S(2))d*S(2)sqrt(csec(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))/(S(2)bcS(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) - sqrt(S(2))dS(2)sqrt(csec(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))/(S(2)bcS(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) - sqrt(S(2))dS(2)sqrt(csec(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))/(S(4)bc*S(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) + sqrt(S(2))dS(2)sqrt(csec(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))/(S(4)bc*S(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcsc(a + bx))/(csec(a + bx))*(S(3)/2), x), x, d/(bcsqrt(csec(a + bx))sqrt(dcsc(a + bx))) + sqrt(csec(a + bx))sqrt(dcsc(a + bx))EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(2)bcS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((csec(a + bx))(S(3)/2)sqrt(dcsc(a + bx))), x), x, d/(S(2)bcsqrt(csec(a + bx))(dcsc(a + bx))*(S(3)/2)) - sqrt(S(2))sqrt(csec(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))/(S(8)bcS(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) + sqrt(S(2))sqrt(csec(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))/(S(8)bc*S(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) + sqrt(S(2))sqrt(csec(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))/(S(16)bcS(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) - sqrt(S(2))sqrt(csec(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))/(S(16)bcS(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((csec(a + bx))*(S(3)/2)(dcsc(a + bx))*(S(3)/2)), x), x, -c/(S(3)bd(csec(a + bx))*(S(5)/2)sqrt(dcsc(a + bx))) + S(1)/(S(6)bcdsqrt(csec(a + bx))sqrt(dcsc(a + bx))) + sqrt(csec(a + bx))sqrt(dcsc(a + bx))EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(12)bc*S(2)d*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((csec(a + bx))(S(3)/2)(dcsc(a + bx))*(S(5)/2)), x), x, -c/(S(4)bd(csec(a + bx))*(S(5)/2)(dcsc(a + bx))*(S(3)/2)) + S(3)/(S(16)bcdsqrt(csec(a + bx))(dcsc(a + bx))*(S(3)/2)) - S(3)sqrt(S(2))sqrt(csec(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))/(S(64)bc*S(2)d*S(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) + S(3)sqrt(S(2))sqrt(csec(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))/(S(64)bcS(2)d*S(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) + S(3)sqrt(S(2))sqrt(csec(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))/(S(128)bc*S(2)d*S(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))) - S(3)sqrt(S(2))sqrt(csec(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))/(S(128)bc*S(2)d*S(2)sqrt(dcsc(a + bx))sqrt(tan(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))*(S(9)/2)/(csec(a + bx))(S(5)/2), x), x, -S(2)cd(dcsc(a + bx))*(S(7)/2)/(S(7)b(csec(a + bx))(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))(S(7)/2)/(csec(a + bx))(S(5)/2), x), x, S(6)d*S(3)sqrt(dcsc(a + bx))/(S(5)bc(csec(a + bx))(S(3)/2)) - S(2)d(dcsc(a + bx))(S(5)/2)/(S(5)bc(csec(a + bx))*(S(3)/2)) + S(6)d*S(4)EllipticE(-Pi/S(4) + a + bx, S(2))/(S(5)bcS(2)sqrt(csec(a + bx))sqrt(dcsc(a + bx))sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))(S(5)/2)/(csec(a + bx))(S(5)/2), x), x, -S(2)d(dcsc(a + bx))(S(3)/2)/(S(3)bc(csec(a + bx))*(S(3)/2)) + sqrt(S(2))d*S(2)sqrt(dcsc(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(2)bcS(2)sqrt(csec(a + bx))) - sqrt(S(2))dS(2)sqrt(dcsc(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(2)bcS(2)sqrt(csec(a + bx))) + sqrt(S(2))dS(2)sqrt(dcsc(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(4)bc*S(2)sqrt(csec(a + bx))) - sqrt(S(2))dS(2)sqrt(dcsc(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(4)bc*S(2)sqrt(csec(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(a + bx))*(S(3)/2)/(csec(a + bx))(S(5)/2), x), x, -S(2)dsqrt(dcsc(a + bx))/(bc(csec(a + bx))(S(3)/2)) - S(3)d*S(2)EllipticE(-Pi/S(4) + a + bx, S(2))/(bc*S(2)sqrt(csec(a + bx))sqrt(dcsc(a + bx))sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcsc(a + bx))/(csec(a + bx))(S(5)/2), x), x, d/(S(2)bc(csec(a + bx))*(S(3)/2)sqrt(dcsc(a + bx))) - S(3)sqrt(S(2))sqrt(dcsc(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(8)bcS(2)sqrt(csec(a + bx))) + S(3)sqrt(S(2))sqrt(dcsc(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(8)bcS(2)sqrt(csec(a + bx))) - S(3)sqrt(S(2))sqrt(dcsc(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(16)bc*S(2)sqrt(csec(a + bx))) + S(3)sqrt(S(2))sqrt(dcsc(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(16)bc*S(2)sqrt(csec(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((csec(a + bx))*(S(5)/2)sqrt(dcsc(a + bx))), x), x, d/(S(3)bc(csec(a + bx))(S(3)/2)(dcsc(a + bx))*(S(3)/2)) + EllipticE(-Pi/S(4) + a + bx, S(2))/(S(2)bc*S(2)sqrt(csec(a + bx))sqrt(dcsc(a + bx))sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((csec(a + bx))(S(5)/2)(dcsc(a + bx))*(S(3)/2)), x), x, -c/(S(4)bd(csec(a + bx))*(S(7)/2)sqrt(dcsc(a + bx))) + S(1)/(S(16)bcd(csec(a + bx))*(S(3)/2)sqrt(dcsc(a + bx))) - S(3)sqrt(S(2))sqrt(dcsc(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(64)bcS(2)d*S(2)sqrt(csec(a + bx))) + S(3)sqrt(S(2))sqrt(dcsc(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(64)bcS(2)d*S(2)sqrt(csec(a + bx))) - S(3)sqrt(S(2))sqrt(dcsc(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(128)bc*S(2)d*S(2)sqrt(csec(a + bx))) + S(3)sqrt(S(2))sqrt(dcsc(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(128)bc*S(2)d*S(2)sqrt(csec(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((csec(a + bx))*(S(5)/2)(dcsc(a + bx))*(S(5)/2)), x), x, -c/(S(5)bd(csec(a + bx))*(S(7)/2)(dcsc(a + bx))*(S(3)/2)) + S(1)/(S(10)bcd(csec(a + bx))(S(3)/2)(dcsc(a + bx))*(S(3)/2)) + S(3)EllipticE(-Pi/S(4) + a + bx, S(2))/(S(20)bcS(2)d*S(2)sqrt(csec(a + bx))sqrt(dcsc(a + bx))sqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((csec(a + bx))(S(5)/2)(dcsc(a + bx))*(S(7)/2)), x), x, -c/(S(6)bd(csec(a + bx))*(S(7)/2)(dcsc(a + bx))*(S(5)/2)) - S(5)c/(S(48)bd*S(3)(csec(a + bx))*(S(7)/2)sqrt(dcsc(a + bx))) + S(5)/(S(192)bcdS(3)(csec(a + bx))*(S(3)/2)sqrt(dcsc(a + bx))) - S(5)sqrt(S(2))sqrt(dcsc(a + bx))ArcTan(-sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(256)bcS(2)d*S(4)sqrt(csec(a + bx))) + S(5)sqrt(S(2))sqrt(dcsc(a + bx))ArcTan(sqrt(S(2))sqrt(tan(a + bx)) + S(1))sqrt(tan(a + bx))/(S(256)bcS(2)d*S(4)sqrt(csec(a + bx))) - S(5)sqrt(S(2))sqrt(dcsc(a + bx))log(-sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(512)bc*S(2)d*S(4)sqrt(csec(a + bx))) + S(5)sqrt(S(2))sqrt(dcsc(a + bx))log(sqrt(S(2))sqrt(tan(a + bx)) + tan(a + bx) + S(1))sqrt(tan(a + bx))/(S(512)bc*S(2)d*S(4)sqrt(csec(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)nsec(e + fx)m, x), x, (cos(e + fx)S(2))(m/S(2) + S(1)/2)Hypergeometric2F1(m/S(2) + S(1)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, sin(e + fx)*S(2))csc(e + fx)(n + S(-1))sec(e + fx)(m + S(1))/(f(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asec(e + fx))*mcsc(e + fx)n, x), x, -(asec(e + fx))m(sin(e + fx)S(2))(n/S(2) + S(1)/2)Hypergeometric2F1(-m/S(2) + S(1)/2, n/S(2) + S(1)/2, -m/S(2) + S(3)/2, cos(e + fx)S(2))cos(e + fx)csc(e + fx)(n + S(1))/(f(-m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bcsc(e + fx))*nsec(e + fx)m, x), x, (bcsc(e + fx))n(cos(e + fx)S(2))(m/S(2) + S(1)/2)Hypergeometric2F1(m/S(2) + S(1)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, sin(e + fx)S(2))sin(e + fx)sec(e + fx)(m + S(1))/(f(-n + S(1))), expand=True, _diff=True, _numerical=True)
8ffed12ae3d960494eee372ccde5a0bbee4150a550ebcaa1d7b3dae1e628021f	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.functions import log, sqrt, exp, cos, sin, tan, sec, csc, cot from sympy.functions.elementary.hyperbolic import atanh as arctanh from sympy.functions.elementary.hyperbolic import asinh as arcsinh from sympy.functions.elementary.hyperbolic import acosh as arccosh from sympy.functions.elementary.trigonometric import atan as arctan from sympy.functions.elementary.trigonometric import asin as arcsin from sympy.functions.elementary.trigonometric import acos as arccos from sympy.integrals.rubi.utility_function import EllipticE, EllipticF, hypergeom, rubi_test from sympy.core.numbers import (I, pi as Pi) from sympy.core.singleton import S from sympy.core.symbol import symbols from sympy.functions.elementary.exponential import exp_polar from sympy.functions.special.hyper import hyper from sympy.simplify.simplify import simplify from sympy.testing.pytest import slow, skip, ON_TRAVIS A, B, C, D, a, b, c, d, e, f, m, n, p, x, u = symbols('A B C D a b c d e f m n p x u', real=True, imaginary=False) @slow def test_1(): if ON_TRAVIS: skip('Too slow for travis.') test = [ [x*S(2)(a + bx)(ac - bcx)S(3), x, S(2), S(1)/S(3)a*S(4)c*S(3)x*S(3) - S(1)/S(2)a*S(3)bcS(3)x*S(4) + S(1)/S(3)abS(3)c*S(3)x*S(6) - S(1)/S(7)b*S(4)c*S(3)x*S(7)], [x(a + bx)(ac - bcx)S(3), x, S(2), S(1)/S(2)a*S(4)c*S(3)x*S(2) - S(2)/S(3)a*S(3)bcS(3)x*S(3) + S(2)/S(5)abS(3)c*S(3)x*S(5) - S(1)/S(6)b*S(4)c*S(3)xS(6)], [xS(3)(a + bx)(A + Bx), x, S(2), S(1)/S(4)aAxS(4) + S(1)/S(5)(Ab + aB)xS(5) + S(1)/S(6)bBxS(6)], [xS(4)(A + Bx)/(a + bx), x, S(2), - aS(3)(Ab - aB)x/bS(5) + S(1)/S(2)a*S(2)(Ab - aB)xS(2)/bS(4) - S(1)/S(3)a(Ab - aB)xS(3)/bS(3) + S(1)/S(4)(Ab - aB)xS(4)/bS(2) + S(1)/S(5)BxS(5)/b + aS(4)(Ab - aB)log(a + bx)/bS(6)], [xS(2)(c + dx)/(a + bx), x, S(2), - a(bc - ad)x/b*S(3) + S(1)/S(2)(bc - ad)xS(2)/bS(2) + S(1)/S(3)dxS(3)/b + aS(2)(bc - ad)log(a + bx)/bS(4)], [xS(3)(c + dx)*S(2)/(a + bx)*S(2), x, S(2), - S(2)a(bc - S(2)ad)(bc - ad)x/b*S(5) + S(1)/S(2)(bc - S(3)ad)(bc - ad)xS(2)/bS(4) + S(2)/S(3)d(bc - ad)xS(3)/bS(3) + S(1)/S(4)dS(2)xS(4)/bS(2) + a*S(3)(bc - ad)S(2)/(bS(6)(a + bx)) + a*S(2)(S(3)bc - S(5)ad)(bc - ad)log(a + bx)/bS(6)], [xS(2)(c + dx)S(3)/(a + bx)*S(3), x, S(2), S(3)d(bc - S(2)ad)(bc - ad)x/b*S(5) + S(3)/S(2)d*S(2)(bc - ad)xS(2)/bS(4) + S(1)/S(3)d*S(3)xS(3)/bS(3) - S(1)/S(2)aS(2)(bc - ad)S(3)/(bS(6)(a + bx)*S(2)) + a(S(2)bc - S(5)ad)(bc - ad)S(2)/(bS(6)(a + bx)) + (bc - ad)(b*S(2)c*S(2) - S(8)abcd + S(10)a*S(2)d*S(2))log(a + bx)/bS(6)], [x(S(5)/S(2))(A + Bx)/(a + bx), x, S(6), - S(2)/S(3)a(Ab - aB)x(S(3)/S(2))/bS(3) + S(2)/S(5)(Ab - aB)x(S(5)/S(2))/bS(2) + S(2)/S(7)Bx(S(7)/S(2))/b - S(2)a*(S(5)/S(2))(Ab - aB)arctan(sqrt(b)sqrt(x)/sqrt(a))/b*(S(9)/S(2)) + S(2)a*S(2)(Ab - aB)sqrt(x)/bS(4)], [xm(a + bx)S(3)(A + Bx), x, S(2), aS(3)Ax(S(1) + m)/(S(1) + m) + aS(2)(S(3)Ab + aB)x*(S(2) + m)/(S(2) + m) + S(3)ab(Ab + aB)x(S(3) + m)/(S(3) + m) + bS(2)(Ab + S(3)aB)x(S(4) + m)/(S(4) + m) + bS(3)Bx(S(5) + m)/(S(5) + m)], [xm(c + dx)*S(3)/(a + bx), x, S(7), d(S(3)b*S(2)c*S(2) - S(3)abcd + aS(2)d*S(2))x(S(1) + m)/(bS(3)(S(1) + m)) + dS(2)(S(3)bc - ad)x(S(2) + m)/(bS(2)(S(2) + m)) + dS(3)x*(S(3) + m)/(b(S(3) + m)) + (bc - ad)*S(3)x*(S(1) + m)hypergeom([S(1), S(1)], [S(1) - m], a/(a + bx))/(bS(3)m(a + bx)), c*S(2)dx(S(1) + m)/(b(S(1) + m)) + cd(bc - ad)x(S(1) + m)/(bS(2)(S(1) + m)) + d(bc - ad)S(2)x(S(1) + m)/(bS(3)(S(1) + m)) + S(2)cdS(2)x*(S(2) + m)/(b(S(2) + m)) + d*S(2)(bc - ad)x(S(2) + m)/(bS(2)(S(2) + m)) + d*S(3)x*(S(3) + m)/(b(S(3) + m)) + (bc - ad)*S(3)x*(S(1) + m)hypergeom([S(1), S(1) + m], [S(2) + m], - bx/a)/(ab*S(3)(S(1) + m))], [x*m(c + dx)S(2)/(a + bx), x, S(5), cdx*(S(1) + m)/(b(S(1) + m)) + d(bc - ad)x(S(1) + m)/(bS(2)(S(1) + m)) + dS(2)x*(S(2) + m)/(b(S(2) + m)) + (bc - ad)*S(2)x*(S(1) + m)hypergeom([S(1), S(1) + m], [S(2) + m], - bx/a)/(ab*S(2)(S(1) + m))], [b*S(2)x*m/(b + axS(2))S(2), x, S(2), x*(S(1) + m)hypergeom([S(2), S(1)/S(2)(S(1) + m)], [S(1)/S(2)(S(3) + m)], - axS(2)/b)/(S(1) + m)], [xm/((S(1) - xsqrt(a)/sqrt( - b))*S(2)(S(1) + xsqrt(a)/sqrt( - b))S(2)), x, S(2), x(S(1) + m)hypergeom([S(2), S(1)/S(2)(S(1) + m)], [S(1)/S(2)(S(3) + m)], - axS(2)/b)/(S(1) + m)], [xS(3)(A + Bx)sqrt(a + bx), x, S(2), - S(2)/S(3)a*S(3)(Ab - aB)(a + bx)(S(3)/S(2))/bS(5) + S(2)/S(5)aS(2)(S(3)Ab - S(4)aB)(a + bx)(S(5)/S(2))/bS(5) - S(6)/S(7)a(Ab - S(2)aB)(a + bx)(S(7)/S(2))/bS(5) + S(2)/S(9)(Ab - S(4)aB)(a + bx)(S(9)/S(2))/bS(5) + S(2)/S(11)B(a + bx)(S(11)/S(2))/bS(5)], [x*S(3)(A + Bx)/sqrt(a + bx), x, S(2), S(2)/S(3)aS(2)(S(3)Ab - S(4)aB)(a + bx)(S(3)/S(2))/bS(5) - S(6)/S(5)a(Ab - S(2)aB)(a + bx)(S(5)/S(2))/bS(5) + S(2)/S(7)(Ab - S(4)aB)(a + bx)(S(7)/S(2))/bS(5) + S(2)/S(9)B(a + bx)(S(9)/S(2))/bS(5) - S(2)aS(3)(Ab - aB)sqrt(a + bx)/bS(5)], [x(S(5)/S(2))(A + Bx)sqrt(a + bx), x, S(7), S(1)/S(5)Bx*(S(7)/S(2))(a + bx)(S(3)/S(2))/b - S(1)/S(128)a*S(4)(S(10)Ab - S(7)aB)arctanh(sqrt(b)sqrt(x)/sqrt(a + bx))/b(S(9)/S(2)) - S(1)/S(192)a*S(2)(S(10)Ab - S(7)aB)x(S(3)/S(2))sqrt(a + bx)/bS(3) + S(1)/S(240)a(S(10)Ab - S(7)aB)x*(S(5)/S(2))sqrt(a + bx)/bS(2) + S(1)/S(40)(S(10)Ab - S(7)aB)x(S(7)/S(2))sqrt(a + bx)/b + S(1)/S(128)a*S(3)(S(10)Ab - S(7)aB)sqrt(x)sqrt(a + bx)/bS(4)], [x(S(3)/S(2))(A + Bx)sqrt(a + bx), x, S(6), S(1)/S(4)Bx(S(5)/S(2))(a + bx)(S(3)/S(2))/b + S(1)/S(64)a*S(3)(S(8)Ab - S(5)aB)arctanh(sqrt(b)sqrt(x)/sqrt(a + bx))/b(S(7)/S(2)) + S(1)/S(96)a(S(8)Ab - S(5)aB)x*(S(3)/S(2))sqrt(a + bx)/bS(2) + S(1)/S(24)(S(8)Ab - S(5)aB)x(S(5)/S(2))sqrt(a + bx)/b - S(1)/S(64)a*S(2)(S(8)Ab - S(5)aB)sqrt(x)sqrt(a + bx)/bS(3)], [x(S(7)/S(2))(A + Bx)/sqrt(a + bx), x, S(7), S(7)/S(128)aS(4)(S(10)Ab - S(9)aB)arctanh(sqrt(b)sqrt(x)/sqrt(a + bx))/b(S(11)/S(2)) + S(7)/S(192)a*S(2)(S(10)Ab - S(9)aB)x(S(3)/S(2))sqrt(a + bx)/bS(4) - S(7)/S(240)a(S(10)Ab - S(9)aB)x*(S(5)/S(2))sqrt(a + bx)/bS(3) + S(1)/S(40)(S(10)Ab - S(9)aB)x(S(7)/S(2))sqrt(a + bx)/bS(2) + S(1)/S(5)Bx(S(9)/S(2))sqrt(a + bx)/b - S(7)/S(128)a*S(3)(S(10)Ab - S(9)aB)sqrt(x)sqrt(a + bx)/bS(5)], [x(S(5)/S(2))(A + Bx)/sqrt(a + bx), x, S(6), - S(5)/S(64)aS(3)(S(8)Ab - S(7)aB)arctanh(sqrt(b)sqrt(x)/sqrt(a + bx))/b(S(9)/S(2)) - S(5)/S(96)a(S(8)Ab - S(7)aB)x*(S(3)/S(2))sqrt(a + bx)/bS(3) + S(1)/S(24)(S(8)Ab - S(7)aB)x(S(5)/S(2))sqrt(a + bx)/bS(2) + S(1)/S(4)Bx(S(7)/S(2))sqrt(a + bx)/b + S(5)/S(64)a*S(2)(S(8)Ab - S(7)aB)sqrt(x)sqrt(a + bx)/bS(4)], [xS(3)sqrt(a + bx)sqrt(c + dx), x, S(6), S(1)/S(5)x*S(2)(a + bx)(S(3)/S(2))(c + dx)(S(3)/S(2))/(bd) + S(1)/S(240)(a + bx)*(S(3)/S(2))(c + dx)(S(3)/S(2))(S(35)bS(2)c*S(2) + S(38)abcd + S(35)a*S(2)d*S(2) - S(42)bd(bc + ad)x)/(bS(3)d*S(3)) + S(1)/S(128)(bc - ad)*S(2)(bc + ad)(S(7)b*S(2)c*S(2) + S(2)abcd + S(7)a*S(2)d*S(2))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(9)/S(2))d*(S(9)/S(2))) - S(1)/S(64)(bc + ad)(S(7)b*S(2)c*S(2) + S(2)abcd + S(7)a*S(2)d*S(2))(a + bx)(S(3)/S(2))sqrt(c + dx)/(bS(4)d*S(3)) - S(1)/S(128)(S(7)bS(4)c*S(4) + S(2)abS(3)c*S(3)d - S(2)aS(3)bcd*S(3) - S(7)a*S(4)d*S(4))sqrt(a + bx)sqrt(c + dx)/(bS(4)dS(4))], [xS(2)sqrt(a + bx)sqrt(c + dx), x, S(6), - S(5)/S(24)(bc + ad)(a + bx)(S(3)/S(2))(c + dx)(S(3)/S(2))/(bS(2)d*S(2)) + S(1)/S(4)x(a + bx)*(S(3)/S(2))(c + dx)(S(3)/S(2))/(bd) + S(1)/S(64)(bc - ad)S(2)(S(4)abcd - S(5)(bc + ad)S(2))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(7)/S(2))d*(S(7)/S(2))) - S(1)/S(32)(S(4)abcd - S(5)(bc + ad)S(2))(a + bx)(S(3)/S(2))sqrt(c + dx)/(bS(3)d*S(2)) - S(1)/S(64)(bc - ad)(S(4)abcd - S(5)(bc + ad)*S(2))sqrt(a + bx)sqrt(c + dx)/(bS(3)dS(3))], [xS(3)sqrt(a + bx)/sqrt(c + dx), x, S(5), S(1)/S(64)(bc - ad)(S(35)b*S(3)c*S(3) + S(15)abS(2)c*S(2)d + S(9)aS(2)bcd*S(2) + S(5)a*S(3)d*S(3))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(7)/S(2))d*(S(9)/S(2))) + S(1)/S(4)x*S(2)(a + bx)(S(3)/S(2))sqrt(c + dx)/(bd) + S(1)/S(96)(a + bx)*(S(3)/S(2))(S(35)bS(2)c*S(2) + S(22)abcd + S(15)a*S(2)d*S(2) - S(4)bd(S(7)bc + S(5)ad)x)sqrt(c + dx)/(bS(3)d*S(3)) - S(1)/S(64)(S(35)bS(3)c*S(3) + S(15)abS(2)c*S(2)d + S(9)aS(2)bcd*S(2) + S(5)a*S(3)d*S(3))sqrt(a + bx)sqrt(c + dx)/(bS(3)dS(4))], [xS(2)sqrt(a + bx)/sqrt(c + dx), x, S(5), - S(1)/S(8)(bc - ad)(S(5)b*S(2)c*S(2) + S(2)abcd + aS(2)d*S(2))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(5)/S(2))d*(S(7)/S(2))) - S(1)/S(12)(S(5)bc + S(3)ad)(a + bx)*(S(3)/S(2))sqrt(c + dx)/(bS(2)d*S(2)) + S(1)/S(3)x(a + bx)*(S(3)/S(2))sqrt(c + dx)/(bd) + S(1)/S(8)(S(5)b*S(2)c*S(2) + S(2)abcd + aS(2)d*S(2))sqrt(a + bx)sqrt(c + dx)/(bS(2)dS(3))], [xS(2)(a + bx)*(S(3)/S(2))sqrt(c + dx), x, S(7), - S(1)/S(40)(S(7)bc + S(5)ad)(a + bx)*(S(5)/S(2))(c + dx)(S(3)/S(2))/(bS(2)d*S(2)) + S(1)/S(5)x(a + bx)*(S(5)/S(2))(c + dx)(S(3)/S(2))/(bd) + S(1)/S(128)(bc - ad)S(3)(S(7)bS(2)c*S(2) + S(6)abcd + S(3)a*S(2)d*S(2))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(7)/S(2))d*(S(9)/S(2))) + S(1)/S(192)(bc - ad)(S(7)b*S(2)c*S(2) + S(6)abcd + S(3)a*S(2)d*S(2))(a + bx)(S(3)/S(2))sqrt(c + dx)/(bS(3)d*S(3)) + S(1)/S(48)(S(7)bS(2)c*S(2) + S(6)abcd + S(3)a*S(2)d*S(2))(a + bx)(S(5)/S(2))sqrt(c + dx)/(bS(3)d*S(2)) - S(1)/S(128)(bc - ad)*S(2)(S(7)bS(2)c*S(2) + S(6)abcd + S(3)a*S(2)d*S(2))sqrt(a + bx)sqrt(c + dx)/(bS(3)d*S(4))], [x(a + bx)(S(3)/S(2))sqrt(c + dx), x, S(6), S(1)/S(4)(a + bx)(S(5)/S(2))(c + dx)(S(3)/S(2))/(bd) - S(1)/S(64)(bc - ad)S(3)(S(5)bc + S(3)ad)arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b(S(5)/S(2))d*(S(7)/S(2))) - S(1)/S(96)(bc - ad)(S(5)bc + S(3)ad)(a + bx)(S(3)/S(2))sqrt(c + dx)/(bS(2)d*S(2)) - S(1)/S(24)(S(5)bc + S(3)ad)(a + bx)*(S(5)/S(2))sqrt(c + dx)/(bS(2)d) + S(1)/S(64)(bc - ad)S(2)(S(5)bc + S(3)ad)sqrt(a + bx)sqrt(c + dx)/(b*S(2)dS(3))], [xS(2)(a + bx)*(S(3)/S(2))/sqrt(c + dx), x, S(6), S(1)/S(64)(bc - ad)S(2)(S(35)bS(2)c*S(2) + S(10)abcd + S(3)a*S(2)d*S(2))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(5)/S(2))d*(S(9)/S(2))) + S(1)/S(96)(S(35)bS(2)c*S(2) + S(10)abcd + S(3)a*S(2)d*S(2))(a + bx)(S(3)/S(2))sqrt(c + dx)/(bS(2)d*S(3)) - S(1)/S(24)(S(7)bc + S(3)ad)(a + bx)*(S(5)/S(2))sqrt(c + dx)/(bS(2)d*S(2)) + S(1)/S(4)x(a + bx)*(S(5)/S(2))sqrt(c + dx)/(bd) - S(1)/S(64)(bc - ad)(S(35)bS(2)c*S(2) + S(10)abcd + S(3)a*S(2)d*S(2))sqrt(a + bx)sqrt(c + dx)/(bS(2)d*S(4))], [x(a + bx)(S(3)/S(2))/sqrt(c + dx), x, S(5), - S(1)/S(8)(bc - ad)S(2)(S(5)bc + ad)arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(3)/S(2))d*(S(7)/S(2))) - S(1)/S(12)(S(5)bc + ad)(a + bx)(S(3)/S(2))sqrt(c + dx)/(bd*S(2)) + S(1)/S(3)(a + bx)(S(5)/S(2))sqrt(c + dx)/(bd) + S(1)/S(8)(bc - ad)(S(5)bc + ad)sqrt(a + bx)sqrt(c + dx)/(bdS(3))], [xS(2)(a + bx)*(S(5)/S(2))sqrt(c + dx), x, S(8), - S(1)/S(60)(S(9)bc + S(5)ad)(a + bx)*(S(7)/S(2))(c + dx)(S(3)/S(2))/(bS(2)d*S(2)) + S(1)/S(6)x(a + bx)*(S(7)/S(2))(c + dx)(S(3)/S(2))/(bd) - S(1)/S(512)(bc - ad)S(4)(S(21)bS(2)c*S(2) + S(14)abcd + S(5)a*S(2)d*S(2))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(7)/S(2))d*(S(11)/S(2))) - S(1)/S(768)(bc - ad)*S(2)(S(21)bS(2)c*S(2) + S(14)abcd + S(5)a*S(2)d*S(2))(a + bx)(S(3)/S(2))sqrt(c + dx)/(bS(3)d*S(4)) + S(1)/S(960)(bc - ad)(S(21)b*S(2)c*S(2) + S(14)abcd + S(5)a*S(2)d*S(2))(a + bx)(S(5)/S(2))sqrt(c + dx)/(bS(3)d*S(3)) + S(1)/S(160)(S(21)bS(2)c*S(2) + S(14)abcd + S(5)a*S(2)d*S(2))(a + bx)(S(7)/S(2))sqrt(c + dx)/(bS(3)d*S(2)) + S(1)/S(512)(bc - ad)*S(3)(S(21)bS(2)c*S(2) + S(14)abcd + S(5)a*S(2)d*S(2))sqrt(a + bx)sqrt(c + dx)/(bS(3)d*S(5))], [x(a + bx)(S(5)/S(2))sqrt(c + dx), x, S(7), S(1)/S(5)(a + bx)(S(7)/S(2))(c + dx)(S(3)/S(2))/(bd) + S(1)/S(128)(bc - ad)S(4)(S(7)bc + S(3)ad)arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b(S(5)/S(2))d*(S(9)/S(2))) + S(1)/S(192)(bc - ad)*S(2)(S(7)bc + S(3)ad)(a + bx)*(S(3)/S(2))sqrt(c + dx)/(bS(2)d*S(3)) - S(1)/S(240)(bc - ad)(S(7)bc + S(3)ad)(a + bx)(S(5)/S(2))sqrt(c + dx)/(bS(2)d*S(2)) - S(1)/S(40)(S(7)bc + S(3)ad)(a + bx)*(S(7)/S(2))sqrt(c + dx)/(bS(2)d) - S(1)/S(128)(bc - ad)S(3)(S(7)bc + S(3)ad)sqrt(a + bx)sqrt(c + dx)/(b*S(2)dS(4))], [xS(2)sqrt(c + dx)/sqrt(a + bx), x, S(5), S(1)/S(8)(bc - ad)(bS(2)c*S(2) + S(2)abcd + S(5)a*S(2)d*S(2))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(7)/S(2))d*(S(5)/S(2))) - S(1)/S(12)(S(3)bc + S(5)ad)(c + dx)*(S(3)/S(2))sqrt(a + bx)/(bS(2)d*S(2)) + S(1)/S(3)x(c + dx)*(S(3)/S(2))sqrt(a + bx)/(bd) + S(1)/S(8)(bS(2)c*S(2) + S(2)abcd + S(5)a*S(2)d*S(2))sqrt(a + bx)sqrt(c + dx)/(bS(3)d*S(2))], [xsqrt(c + dx)/sqrt(a + bx), x, S(4), - S(1)/S(4)(bc - ad)(bc + S(3)ad)arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(5)/S(2))d*(S(3)/S(2))) + S(1)/S(2)(c + dx)(S(3)/S(2))sqrt(a + bx)/(bd) - S(1)/S(4)(bc + S(3)ad)sqrt(a + bx)sqrt(c + dx)/(b*S(2)d)], [x*S(3)/(sqrt(a + bx)sqrt(c + dx)), x, S(4), - S(1)/S(8)(bc + ad)(S(5)bS(2)c*S(2) - S(2)abcd + S(5)a*S(2)d*S(2))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(7)/S(2))d*(S(7)/S(2))) + S(1)/S(3)x*S(2)sqrt(a + bx)sqrt(c + dx)/(bd) + S(1)/S(24)(S(15)b*S(2)c*S(2) + S(14)abcd + S(15)a*S(2)d*S(2) - S(10)bd(bc + ad)x)sqrt(a + bx)sqrt(c + dx)/(bS(3)dS(3))], [xS(2)/(sqrt(a + bx)sqrt(c + dx)), x, S(4), - S(1)/S(4)(S(4)abcd - S(3)(bc + ad)S(2))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(5)/S(2))d*(S(5)/S(2))) - S(3)/S(4)(bc + ad)sqrt(a + bx)sqrt(c + dx)/(b*S(2)d*S(2)) + S(1)/S(2)xsqrt(a + bx)sqrt(c + dx)/(bd)], [xS(4)/((a + bx)*(S(3)/S(2))(c + dx)(S(3)/S(2))), x, S(5), S(3)/S(4)(S(5)bS(2)c*S(2) + S(6)abcd + S(5)a*S(2)d*S(2))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(7)/S(2))d*(S(7)/S(2))) + S(2)axS(3)/(b(bc - ad)sqrt(a + bx)sqrt(c + dx)) - S(2)c(bc + ad)xS(2)sqrt(a + bx)/(bd(bc - ad)S(2)sqrt(c + dx)) - S(1)/S(4)((bc + ad)(S(15)b*S(2)c*S(2) - S(22)abcd + S(15)a*S(2)d*S(2)) - S(2)bd(S(5)bS(2)c*S(2) - S(2)abcd + S(5)a*S(2)d*S(2))x)sqrt(a + bx)sqrt(c + dx)/(b*S(3)d*S(3)(bc - ad)S(2))], [xS(3)/((a + bx)(S(3)/S(2))(c + dx)(S(3)/S(2))), x, S(4), - S(3)(bc + ad)arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b(S(5)/S(2))d*(S(5)/S(2))) + S(2)axS(2)/(b(bc - ad)sqrt(a + bx)sqrt(c + dx)) + (c(S(3)b*S(2)c*S(2) - S(2)abcd + S(3)a*S(2)d*S(2)) + d(bc - S(3)ad)(bc - ad)x)sqrt(a + bx)/(bS(2)d*S(2)(bc - ad)*S(2)sqrt(c + dx))], [xS(3)(a + bx)(S(1)/S(4))/(c + dx)*(S(1)/S(4)), x, S(7), - S(1)/S(512)(S(195)bS(3)c*S(3) + S(135)abS(2)c*S(2)d + S(105)aS(2)bcd*S(2) + S(77)a*S(3)d*S(3))(a + bx)(S(1)/S(4))(c + dx)(S(3)/S(4))/(bS(3)d*S(4)) + S(1)/S(4)x*S(2)(a + bx)(S(5)/S(4))(c + dx)(S(3)/S(4))/(bd) + S(1)/S(384)(a + bx)*(S(5)/S(4))(c + dx)(S(3)/S(4))(S(117)bS(2)c*S(2) + S(94)abcd + S(77)a*S(2)d*S(2) - S(8)bd(S(13)bc + S(11)ad)x)/(bS(3)d*S(3)) + S(1)/S(1024)(bc - ad)(S(195)b*S(3)c*S(3) + S(135)abS(2)c*S(2)d + S(105)aS(2)bcd*S(2) + S(77)a*S(3)d*S(3))arctan(d*(S(1)/S(4))(a + bx)(S(1)/S(4))/(b(S(1)/S(4))(c + dx)(S(1)/S(4))))/(b(S(15)/S(4))d*(S(17)/S(4))) + S(1)/S(1024)(bc - ad)(S(195)b*S(3)c*S(3) + S(135)abS(2)c*S(2)d + S(105)aS(2)bcd*S(2) + S(77)a*S(3)d*S(3))arctanh(d*(S(1)/S(4))(a + bx)(S(1)/S(4))/(b(S(1)/S(4))(c + dx)(S(1)/S(4))))/(b(S(15)/S(4))d(S(17)/S(4)))], [xS(2)(a + bx)*(S(1)/S(4))/(c + dx)*(S(1)/S(4)), x, S(7), S(1)/S(32)(S(15)bS(2)c*S(2) + S(10)abcd + S(7)a*S(2)d*S(2))(a + bx)(S(1)/S(4))(c + dx)(S(3)/S(4))/(bS(2)d*S(3)) - S(1)/S(24)(S(9)bc + S(7)ad)(a + bx)*(S(5)/S(4))(c + dx)(S(3)/S(4))/(bS(2)d*S(2)) + S(1)/S(3)x(a + bx)*(S(5)/S(4))(c + dx)(S(3)/S(4))/(bd) - S(1)/S(64)(bc - ad)(S(15)bS(2)c*S(2) + S(10)abcd + S(7)a*S(2)d*S(2))arctan(d*(S(1)/S(4))(a + bx)(S(1)/S(4))/(b(S(1)/S(4))(c + dx)(S(1)/S(4))))/(b(S(11)/S(4))d*(S(13)/S(4))) - S(1)/S(64)(bc - ad)(S(15)b*S(2)c*S(2) + S(10)abcd + S(7)a*S(2)d*S(2))arctanh(d*(S(1)/S(4))(a + bx)(S(1)/S(4))/(b(S(1)/S(4))(c + dx)(S(1)/S(4))))/(b(S(11)/S(4))d*(S(13)/S(4)))], [x(a + bx)n(c + dx), x, S(2), - a(bc - ad)(a + bx)(S(1) + n)/(bS(3)(S(1) + n)) + (bc - S(2)ad)(a + bx)(S(2) + n)/(bS(3)(S(2) + n)) + d(a + bx)(S(3) + n)/(bS(3)(S(3) + n))], [x*S(2)(a + bx)n/(c + dx), x, S(3), - (bc + ad)(a + bx)(S(1) + n)/(bS(2)dS(2)(S(1) + n)) + (a + bx)(S(2) + n)/(bS(2)d(S(2) + n)) + cS(2)(a + bx)(S(1) + n)hypergeom([S(1), S(1) + n], [S(2) + n], - d(a + bx)/(bc - ad))/(d*S(2)(bc - ad)(S(1) + n))], [x(a + bx)n/(c + dx), x, S(2), (a + bx)(S(1) + n)/(bd(S(1) + n)) - c(a + bx)(S(1) + n)hypergeom([S(1), S(1) + n], [S(2) + n], - d(a + bx)/(bc - ad))/(d(bc - ad)(S(1) + n))], [x*m(S(3) - S(2)ax)*(S(2) + n)(S(6) + S(4)ax)n, x, S(8), S(2)nS(9)(S(1) + n)x*(S(1) + m)hypergeom([S(1)/S(2)(S(1) + m), - n], [S(1)/S(2)(S(3) + m)], S(4)/S(9)aS(2)xS(2))/(S(1) + m) - S(2)(S(2) + n)S(3)(S(1) + S(2)n)ax*(S(2) + m)hypergeom([S(1)/S(2)(S(2) + m), - n], [S(1)/S(2)(S(4) + m)], S(4)/S(9)aS(2)xS(2))/(S(2) + m) + S(2)(S(2) + n)S(9)na*S(2)x*(S(3) + m)hypergeom([S(1)/S(2)(S(3) + m), - n], [S(1)/S(2)(S(5) + m)], S(4)/S(9)aS(2)xS(2))/(S(3) + m)], [xm(S(3) - S(2)ax)(S(1) + n)(S(6) + S(4)ax)n, x, S(5), S(2)nS(3)(S(1) + S(2)n)x(S(1) + m)hypergeom([S(1)/S(2)(S(1) + m), - n], [S(1)/S(2)(S(3) + m)], S(4)/S(9)aS(2)xS(2))/(S(1) + m) - S(2)(S(1) + n)S(9)nax(S(2) + m)hypergeom([S(1)/S(2)(S(2) + m), - n], [S(1)/S(2)(S(4) + m)], S(4)/S(9)aS(2)x*S(2))/(S(2) + m)], [(a + bx)(A + Bx)(d + ex)*m, x, S(2), (bd - ae)(Bd - Ae)(d + ex)(S(1) + m)/(eS(3)(S(1) + m)) - (S(2)bBd - Abe - aBe)(d + ex)(S(2) + m)/(eS(3)(S(2) + m)) + bB(d + ex)(S(3) + m)/(eS(3)(S(3) + m))], [(A + Bx)(d + ex)*S(5)/(a + bx), x, S(2), (Ab - aB)e(bd - ae)*S(4)x/b*S(6) + S(1)/S(2)(Ab - aB)(bd - ae)S(3)(d + ex)S(2)/bS(5) + S(1)/S(3)(Ab - aB)(bd - ae)S(2)(d + ex)S(3)/bS(4) + S(1)/S(4)(Ab - aB)(bd - ae)(d + ex)S(4)/bS(3) + S(1)/S(5)(Ab - aB)(d + ex)S(5)/bS(2) + S(1)/S(6)B(d + ex)S(6)/(be) + (Ab - aB)(bd - ae)S(5)log(a + bx)/bS(7)], [(S(1) - S(2)x)(S(2) + S(3)x)*m(S(3) + S(5)x), x, S(2), - S(7)/S(27)(S(2) + S(3)x)(S(1) + m)/(S(1) + m) + S(37)/S(27)(S(2) + S(3)x)(S(2) + m)/(S(2) + m) - S(10)/S(27)(S(2) + S(3)x)(S(3) + m)/(S(3) + m)], [(S(1) - S(2)x)(S(2) + S(3)x)*S(8)(S(3) + S(5)x), x, S(2), - S(7)/S(243)(S(2) + S(3)x)S(9) + S(37)/S(270)(S(2) + S(3)x)S(10) - S(10)/S(297)(S(2) + S(3)x)S(11)], [(S(1) - S(2)x)(S(2) + S(3)x)*m/(S(3) + S(5)x), x, S(2), - S(2)/S(15)(S(2) + S(3)x)*(S(1) + m)/(S(1) + m) - S(11)/S(5)(S(2) + S(3)x)(S(1) + m)hypergeom([S(1), S(1) + m], [S(2) + m], S(5)(S(2) + S(3)x))/(S(1) + m)], [(S(1) - S(2)x)(S(2) + S(3)x)S(6)/(S(3) + S(5)x), x, S(2), S(1666663)/S(78125)x + S(1777779)/S(31250)x*S(2) + S(152469)/S(3125)x*S(3) - S(152469)/S(2500)x*S(4) - S(106677)/S(625)x*S(5) - S(7047)/S(50)x*S(6) - S(1458)/S(35)x*S(7) + S(11)/S(390625)log(S(3) + S(5)x)], [(S(1) - S(2)x)*S(2)(S(2) + S(3)x)S(8)(S(3) + S(5)x), x, S(2), - S(49)/S(729)(S(2) + S(3)x)S(9) + S(91)/S(270)(S(2) + S(3)x)S(10) - S(16)/S(99)(S(2) + S(3)x)S(11) + S(5)/S(243)(S(2) + S(3)x)S(12)], [(S(1) - S(2)x)*S(2)(S(2) + S(3)x)S(7)(S(3) + S(5)x), x, S(2), - S(49)/S(648)(S(2) + S(3)x)S(8) + S(91)/S(243)(S(2) + S(3)x)S(9) - S(8)/S(45)(S(2) + S(3)x)S(10) + S(20)/S(891)(S(2) + S(3)x)S(11)], [(S(1) - S(2)x)*S(2)(S(2) + S(3)x)S(7)/(S(3) + S(5)x), x, S(2), S(83333293)/S(1953125)x + S(80555569)/S(781250)x*S(2) + S(1327159)/S(78125)x*S(3) - S(20577159)/S(62500)x*S(4) - S(7315947)/S(15625)x*S(5) + S(130383)/S(1250)x*S(6) + S(672867)/S(875)x*S(7) + S(16767)/S(25)x*S(8) + S(972)/S(5)x*S(9) + S(121)/S(9765625)log(S(3) + S(5)x)], [(S(1) - S(2)x)*S(2)(S(2) + S(3)x)S(6)/(S(3) + S(5)x), x, S(2), S(8333293)/S(390625)x + S(5555569)/S(156250)x*S(2) - S(422841)/S(15625)x*S(3) - S(1677159)/S(12500)x*S(4) - S(228447)/S(3125)x*S(5) + S(35883)/S(250)x*S(6) + S(34992)/S(175)x*S(7) + S(729)/S(10)x*S(8) + S(121)/S(1953125)log(S(3) + S(5)x)], [(S(1) - S(2)x)*S(3)(S(2) + S(3)x)S(8)(S(3) + S(5)x), x, S(2), - S(343)/S(2187)(S(2) + S(3)x)S(9) + S(2009)/S(2430)(S(2) + S(3)x)S(10) - S(518)/S(891)(S(2) + S(3)x)S(11) + S(107)/S(729)(S(2) + S(3)x)S(12) - S(40)/S(3159)(S(2) + S(3)x)S(13)], [(S(1) - S(2)x)*S(3)(S(2) + S(3)x)S(7)(S(3) + S(5)x), x, S(2), S(384)x + S(1184)xS(2) + S(480)x*S(3) - S(5148)x*S(4) - S(48968)/S(5)x*S(5) + S(3514)x*S(6) + S(29106)x*S(7) + S(208035)/S(8)x*S(8) - S(15507)x*S(9) - S(217971)/S(5)x*S(10) - S(329508)/S(11)x*S(11) - S(7290)x*S(12)], [(S(1) - S(2)x)*S(3)(S(2) + S(3)x)S(6)/(S(3) + S(5)x), x, S(2), S(41666223)/S(1953125)x + S(11111259)/S(781250)x*S(2) - S(17453753)/S(234375)x*S(3) - S(5848749)/S(62500)x*S(4) + S(2212083)/S(15625)x*S(5) + S(331713)/S(1250)x*S(6) - S(40338)/S(875)x*S(7) - S(13851)/S(50)x*S(8) - S(648)/S(5)x*S(9) + S(1331)/S(9765625)log(S(3) + S(5)x)], [(S(1) - S(2)x)*S(3)(S(2) + S(3)x)S(5)/(S(3) + S(5)x), x, S(2), S(4166223)/S(390625)x - S(138741)/S(156250)x*S(2) - S(1703753)/S(46875)x*S(3) - S(73749)/S(12500)x*S(4) + S(243333)/S(3125)x*S(5) + S(4419)/S(125)x*S(6) - S(11988)/S(175)x*S(7) - S(243)/S(5)x*S(8) + S(1331)/S(1953125)log(S(3) + S(5)x)], [(S(2) + S(3)x)*m(S(3) + S(5)x)/(S(1) - S(2)x), x, S(2), - S(5)/S(6)(S(2) + S(3)x)*(S(1) + m)/(S(1) + m) + S(11)/S(14)(S(2) + S(3)x)(S(1) + m)hypergeom([S(1), S(1) + m], [S(2) + m], S(2)/S(7)(S(2) + S(3)x))/(S(1) + m)], [(S(2) + S(3)x)S(8)(S(3) + S(5)x)/(S(1) - S(2)x), x, S(2), - S(63019595)/S(512)x - S(60332619)/S(512)x*S(2) - S(17391129)/S(128)x*S(3) - S(37722699)/S(256)x*S(4) - S(21272139)/S(160)x*S(5) - S(2929689)/S(32)x*S(6) - S(353565)/S(8)x*S(7) - S(422091)/S(32)x*S(8) - S(3645)/S(2)x*S(9) - S(63412811)/S(1024)log(S(1) - S(2)x)], [(S(2) + S(3)x)*m/((S(1) - S(2)x)(S(3) + S(5)x)), x, S(3), S(2)/S(77)(S(2) + S(3)x)*(S(1) + m)hypergeom([S(1), S(1) + m], [S(2) + m], S(2)/S(7)(S(2) + S(3)x))/(S(1) + m) - S(5)/S(11)(S(2) + S(3)x)*(S(1) + m)hypergeom([S(1), S(1) + m], [S(2) + m], S(5)(S(2) + S(3)x))/(S(1) + m)], [(S(2) + S(3)x)S(8)(S(3) + S(5)x)/(S(1) - S(2)x)*S(2), x, S(2), S(63412811)/S(1024)/(S(1) - S(2)x) + S(91609881)/S(256)x + S(122887143)/S(512)x*S(2) + S(5892813)/S(32)x*S(3) + S(32991057)/S(256)x*S(4) + S(5859459)/S(80)x*S(5) + S(976617)/S(32)x*S(6) + S(56862)/S(7)x*S(7) + S(32805)/S(32)x*S(8) + S(246239357)/S(1024)log(S(1) - S(2)x)], [(S(2) + S(3)x)*S(7)(S(3) + S(5)x)/(S(1) - S(2)x)*S(2), x, S(2), S(9058973)/S(512)/(S(1) - S(2)x) + S(22333965)/S(256)x + S(873207)/S(16)x*S(2) + S(2399985)/S(64)x*S(3) + S(1423899)/S(64)x*S(4) + S(793881)/S(80)x*S(5) + S(11421)/S(4)x*S(6) + S(10935)/S(28)x*S(7) + S(15647317)/S(256)log(S(1) - S(2)x)], [(a + bx)*m/(e + fx)*S(2), x, S(1), b(a + bx)(S(1) + m)hypergeom([S(2), S(1) + m], [S(2) + m], - f(a + bx)/(be - af))/((be - af)*S(2)(S(1) + m))], [(a + bx)m/((c + dx)(e + fx)*S(2)), x, S(4), - f(a + bx)(S(1) + m)/((be - af)(de - cf)(e + fx)) + d*S(2)(a + bx)(S(1) + m)hypergeom([S(1), S(1) + m], [S(2) + m], - d(a + bx)/(bc - ad))/((bc - ad)(de - cf)S(2)(S(1) + m)) + f(adf - b(de(S(1) - m) + cfm))(a + bx)*(S(1) + m)hypergeom([S(1), S(1) + m], [S(2) + m], - f(a + bx)/(be - af))/((be - af)*S(2)(de - cf)*S(2)(S(1) + m))], [(S(2) + S(3)x)S(7)(S(3) + S(5)x)/(S(1) - S(2)x)*S(3), x, S(2), S(9058973)/S(1024)/(S(1) - S(2)x)*S(2) + ( - S(15647317)/S(256))/(S(1) - S(2)x) - S(24960933)/S(256)x - S(10989621)/S(256)x*S(2) - S(631611)/S(32)x*S(3) - S(235467)/S(32)x*S(4) - S(147987)/S(80)x*S(5) - S(3645)/S(16)x*S(6) - S(23647449)/S(256)log(S(1) - S(2)x)], [(S(2) + S(3)x)*S(8)/((S(1) - S(2)x)*S(3)(S(3) + S(5)x)), x, S(2), S(5764801)/S(5632)/(S(1) - S(2)x)*S(2) + ( - S(188591347)/S(30976))/(S(1) - S(2)x) - S(2941619571)/S(400000)x - S(110180817)/S(40000)x*S(2) - S(124416)/S(125)x*S(3) - S(408969)/S(1600)x*S(4) - S(6561)/S(200)x*S(5) - S(2644396573)/S(340736)log(S(1) - S(2)x) + S(1)/S(20796875)log(S(3) + S(5)x)], [(S(2) + S(3)x)*S(7)/((S(1) - S(2)x)*S(3)(S(3) + S(5)x)), x, S(2), S(823543)/S(2816)/(S(1) - S(2)x)*S(2) + ( - S(5764801)/S(3872))/(S(1) - S(2)x) - S(26161299)/S(20000)x - S(792423)/S(2000)x*S(2) - S(40581)/S(400)x*S(3) - S(2187)/S(160)x*S(4) - S(269063263)/S(170368)log(S(1) - S(2)x) + S(1)/S(4159375)log(S(3) + S(5)x)], [(S(2) + S(3)x)*S(6)(S(3) + S(5)x)sqrt(S(1) - S(2)x), x, S(2), - S(1294139)/S(384)(S(1) - S(2)x)(S(3)/S(2)) + S(3916031)/S(640)(S(1) - S(2)x)(S(5)/S(2)) - S(725445)/S(128)(S(1) - S(2)x)(S(7)/S(2)) + S(406455)/S(128)(S(1) - S(2)x)(S(9)/S(2)) - S(1580985)/S(1408)(S(1) - S(2)x)(S(11)/S(2)) + S(409941)/S(1664)(S(1) - S(2)x)(S(13)/S(2)) - S(19683)/S(640)(S(1) - S(2)x)(S(15)/S(2)) + S(3645)/S(2176)(S(1) - S(2)x)(S(17)/S(2))], [(S(2) + S(3)x)*S(5)(S(3) + S(5)x)sqrt(S(1) - S(2)x), x, S(2), - S(184877)/S(192)(S(1) - S(2)x)(S(3)/S(2)) + S(12005)/S(8)(S(1) - S(2)x)(S(5)/S(2)) - S(74235)/S(64)(S(1) - S(2)x)(S(7)/S(2)) + S(4165)/S(8)(S(1) - S(2)x)(S(9)/S(2)) - S(97335)/S(704)(S(1) - S(2)x)(S(11)/S(2)) + S(81)/S(4)(S(1) - S(2)x)(S(13)/S(2)) - S(81)/S(64)(S(1) - S(2)x)(S(15)/S(2))], [(S(2) + S(3)x)*S(4)sqrt(S(1) - S(2)x)/(S(3) + S(5)x), x, S(5), - S(45473)/S(5000)(S(1) - S(2)x)*(S(3)/S(2)) + S(34371)/S(5000)(S(1) - S(2)x)(S(5)/S(2)) - S(2889)/S(1400)(S(1) - S(2)x)(S(7)/S(2)) + S(9)/S(40)(S(1) - S(2)x)(S(9)/S(2)) - S(2)/S(3125)arctanh(sqrt(S(5)/S(11))sqrt(S(1) - S(2)x))sqrt(S(11)/S(5)) + S(2)/S(3125)sqrt(S(1) - S(2)x)], [(S(2) + S(3)x)*S(3)sqrt(S(1) - S(2)x)/(S(3) + S(5)x), x, S(5), - S(1299)/S(500)(S(1) - S(2)x)*(S(3)/S(2)) + S(162)/S(125)(S(1) - S(2)x)(S(5)/S(2)) - S(27)/S(140)(S(1) - S(2)x)(S(7)/S(2)) - S(2)/S(625)arctanh(sqrt(S(5)/S(11))sqrt(S(1) - S(2)x))sqrt(S(11)/S(5)) + S(2)/S(625)sqrt(S(1) - S(2)x)], [(S(1) - S(2)x)*(S(3)/S(2))(S(2) + S(3)x)S(6)(S(3) + S(5)x), x, S(2), - S(1294139)/S(640)(S(1) - S(2)x)(S(5)/S(2)) + S(559433)/S(128)(S(1) - S(2)x)(S(7)/S(2)) - S(564235)/S(128)(S(1) - S(2)x)(S(9)/S(2)) + S(3658095)/S(1408)(S(1) - S(2)x)(S(11)/S(2)) - S(1580985)/S(1664)(S(1) - S(2)x)(S(13)/S(2)) + S(136647)/S(640)(S(1) - S(2)x)(S(15)/S(2)) - S(59049)/S(2176)(S(1) - S(2)x)(S(17)/S(2)) + S(3645)/S(2432)(S(1) - S(2)x)(S(19)/S(2))], [(S(1) - S(2)x)*(S(3)/S(2))(S(2) + S(3)x)S(5)(S(3) + S(5)x), x, S(2), - S(184877)/S(320)(S(1) - S(2)x)(S(5)/S(2)) + S(8575)/S(8)(S(1) - S(2)x)(S(7)/S(2)) - S(173215)/S(192)(S(1) - S(2)x)(S(9)/S(2)) + S(37485)/S(88)(S(1) - S(2)x)(S(11)/S(2)) - S(97335)/S(832)(S(1) - S(2)x)(S(13)/S(2)) + S(351)/S(20)(S(1) - S(2)x)(S(15)/S(2)) - S(1215)/S(1088)(S(1) - S(2)x)(S(17)/S(2))], [(S(1) - S(2)x)*(S(3)/S(2))(S(2) + S(3)x)S(6)/(S(3) + S(5)x), x, S(6), S(2)/S(234375)(S(1) - S(2)x)*(S(3)/S(2)) - S(167115051)/S(2500000)(S(1) - S(2)x)(S(5)/S(2)) + S(70752609)/S(700000)(S(1) - S(2)x)(S(7)/S(2)) - S(665817)/S(10000)(S(1) - S(2)x)(S(9)/S(2)) + S(507627)/S(22000)(S(1) - S(2)x)(S(11)/S(2)) - S(43011)/S(10400)(S(1) - S(2)x)(S(13)/S(2)) + S(243)/S(800)(S(1) - S(2)x)(S(15)/S(2)) - S(22)/S(390625)arctanh(sqrt(S(5)/S(11))sqrt(S(1) - S(2)x))sqrt(S(11)/S(5)) + S(22)/S(390625)sqrt(S(1) - S(2)x)], [(S(1) - S(2)x)*(S(3)/S(2))(S(2) + S(3)x)S(5)/(S(3) + S(5)x), x, S(6), S(2)/S(46875)(S(1) - S(2)x)*(S(3)/S(2)) - S(4774713)/S(250000)(S(1) - S(2)x)(S(5)/S(2)) + S(806121)/S(35000)(S(1) - S(2)x)(S(7)/S(2)) - S(5673)/S(500)(S(1) - S(2)x)(S(9)/S(2)) + S(5751)/S(2200)(S(1) - S(2)x)(S(11)/S(2)) - S(243)/S(1040)(S(1) - S(2)x)(S(13)/S(2)) - S(22)/S(78125)arctanh(sqrt(S(5)/S(11))sqrt(S(1) - S(2)x))sqrt(S(11)/S(5)) + S(22)/S(78125)sqrt(S(1) - S(2)x)], [(S(1) - S(2)x)*(S(5)/S(2))(S(2) + S(3)x)S(6)(S(3) + S(5)x), x, S(2), - S(184877)/S(128)(S(1) - S(2)x)(S(7)/S(2)) + S(3916031)/S(1152)(S(1) - S(2)x)(S(9)/S(2)) - S(5078115)/S(1408)(S(1) - S(2)x)(S(11)/S(2)) + S(3658095)/S(1664)(S(1) - S(2)x)(S(13)/S(2)) - S(105399)/S(128)(S(1) - S(2)x)(S(15)/S(2)) + S(409941)/S(2176)(S(1) - S(2)x)(S(17)/S(2)) - S(59049)/S(2432)(S(1) - S(2)x)(S(19)/S(2)) + S(1215)/S(896)(S(1) - S(2)x)(S(21)/S(2))], [(S(1) - S(2)x)*(S(5)/S(2))(S(2) + S(3)x)S(5)(S(3) + S(5)x), x, S(2), - S(26411)/S(64)(S(1) - S(2)x)(S(7)/S(2)) + S(60025)/S(72)(S(1) - S(2)x)(S(9)/S(2)) - S(519645)/S(704)(S(1) - S(2)x)(S(11)/S(2)) + S(37485)/S(104)(S(1) - S(2)x)(S(13)/S(2)) - S(6489)/S(64)(S(1) - S(2)x)(S(15)/S(2)) + S(1053)/S(68)(S(1) - S(2)x)(S(17)/S(2)) - S(1215)/S(1216)(S(1) - S(2)x)(S(19)/S(2))], [(S(1) - S(2)x)*(S(5)/S(2))(S(2) + S(3)x)S(4)/(S(3) + S(5)x), x, S(7), S(22)/S(46875)(S(1) - S(2)x)*(S(3)/S(2)) + S(2)/S(15625)(S(1) - S(2)x)(S(5)/S(2)) - S(136419)/S(35000)(S(1) - S(2)x)(S(7)/S(2)) + S(3819)/S(1000)(S(1) - S(2)x)(S(9)/S(2)) - S(2889)/S(2200)(S(1) - S(2)x)(S(11)/S(2)) + S(81)/S(520)(S(1) - S(2)x)(S(13)/S(2)) - S(242)/S(78125)arctanh(sqrt(S(5)/S(11))sqrt(S(1) - S(2)x))sqrt(S(11)/S(5)) + S(242)/S(78125)sqrt(S(1) - S(2)x)], [(S(1) - S(2)x)*(S(5)/S(2))(S(2) + S(3)x)S(3)/(S(3) + S(5)x), x, S(7), S(22)/S(9375)(S(1) - S(2)x)*(S(3)/S(2)) + S(2)/S(3125)(S(1) - S(2)x)(S(5)/S(2)) - S(3897)/S(3500)(S(1) - S(2)x)(S(7)/S(2)) + S(18)/S(25)(S(1) - S(2)x)(S(9)/S(2)) - S(27)/S(220)(S(1) - S(2)x)(S(11)/S(2)) - S(242)/S(15625)arctanh(sqrt(S(5)/S(11))sqrt(S(1) - S(2)x))sqrt(S(11)/S(5)) + S(242)/S(15625)sqrt(S(1) - S(2)x)], [(S(2) + S(3)x)*S(5)(S(3) + S(5)x)/sqrt(S(1) - S(2)x), x, S(2), S(60025)/S(24)(S(1) - S(2)x)*(S(3)/S(2)) - S(103929)/S(64)(S(1) - S(2)x)(S(5)/S(2)) + S(5355)/S(8)(S(1) - S(2)x)(S(7)/S(2)) - S(10815)/S(64)(S(1) - S(2)x)(S(9)/S(2)) + S(1053)/S(44)(S(1) - S(2)x)(S(11)/S(2)) - S(1215)/S(832)(S(1) - S(2)x)(S(13)/S(2)) - S(184877)/S(64)sqrt(S(1) - S(2)x)], [(S(2) + S(3)x)*S(4)(S(3) + S(5)x)/sqrt(S(1) - S(2)x), x, S(2), S(57281)/S(96)(S(1) - S(2)x)*(S(3)/S(2)) - S(24843)/S(80)(S(1) - S(2)x)(S(5)/S(2)) + S(1539)/S(16)(S(1) - S(2)x)(S(7)/S(2)) - S(519)/S(32)(S(1) - S(2)x)(S(9)/S(2)) + S(405)/S(352)(S(1) - S(2)x)(S(11)/S(2)) - S(26411)/S(32)sqrt(S(1) - S(2)x)], [(S(2) + S(3)x)*S(5)/((S(3) + S(5)x)sqrt(S(1) - S(2)x)), x, S(4), S(268707)/S(5000)(S(1) - S(2)x)*(S(3)/S(2)) - S(51057)/S(2500)(S(1) - S(2)x)(S(5)/S(2)) + S(5751)/S(1400)(S(1) - S(2)x)(S(7)/S(2)) - S(27)/S(80)(S(1) - S(2)x)(S(9)/S(2)) - S(2)/S(3125)arctanh(sqrt(S(5)/S(11))sqrt(S(1) - S(2)x))/sqrt(S(55)) - S(4774713)/S(50000)sqrt(S(1) - S(2)x)], [(S(2) + S(3)x)S(7)(S(3) + S(5)x)/(S(1) - S(2)x)*(S(3)/S(2)), x, S(2), - S(7882483)/S(128)(S(1) - S(2)x)(S(3)/S(2)) + S(4084101)/S(128)(S(1) - S(2)x)(S(5)/S(2)) - S(787185)/S(64)(S(1) - S(2)x)(S(7)/S(2)) + S(422919)/S(128)(S(1) - S(2)x)(S(9)/S(2)) - S(821583)/S(1408)(S(1) - S(2)x)(S(11)/S(2)) + S(101331)/S(1664)(S(1) - S(2)x)(S(13)/S(2)) - S(729)/S(256)(S(1) - S(2)x)(S(15)/S(2)) + S(9058973)/S(256)/sqrt(S(1) - S(2)x) + S(15647317)/S(128)sqrt(S(1) - S(2)x)], [(S(2) + S(3)x)S(6)(S(3) + S(5)x)/(S(1) - S(2)x)*(S(3)/S(2)), x, S(2), - S(1692705)/S(128)(S(1) - S(2)x)(S(3)/S(2)) + S(731619)/S(128)(S(1) - S(2)x)(S(5)/S(2)) - S(225855)/S(128)(S(1) - S(2)x)(S(7)/S(2)) + S(45549)/S(128)(S(1) - S(2)x)(S(9)/S(2)) - S(59049)/S(1408)(S(1) - S(2)x)(S(11)/S(2)) + S(3645)/S(1664)(S(1) - S(2)x)(S(13)/S(2)) + S(1294139)/S(128)/sqrt(S(1) - S(2)x) + S(3916031)/S(128)sqrt(S(1) - S(2)x)], [(S(2) + S(3)x)S(5)(S(3) + S(5)x)/(S(1) - S(2)x)*(S(5)/S(2)), x, S(2), S(184877)/S(192)/(S(1) - S(2)x)*(S(3)/S(2)) + S(12495)/S(8)(S(1) - S(2)x)(S(3)/S(2)) - S(19467)/S(64)(S(1) - S(2)x)(S(5)/S(2)) + S(1053)/S(28)(S(1) - S(2)x)(S(7)/S(2)) - S(135)/S(64)(S(1) - S(2)x)(S(9)/S(2)) + ( - S(60025)/S(8))/sqrt(S(1) - S(2)x) - S(519645)/S(64)sqrt(S(1) - S(2)x)], [(S(2) + S(3)x)S(4)(S(3) + S(5)x)/(S(1) - S(2)x)*(S(5)/S(2)), x, S(2), S(26411)/S(96)/(S(1) - S(2)x)*(S(3)/S(2)) + S(3591)/S(16)(S(1) - S(2)x)(S(3)/S(2)) - S(4671)/S(160)(S(1) - S(2)x)(S(5)/S(2)) + S(405)/S(224)(S(1) - S(2)x)(S(7)/S(2)) + ( - S(57281)/S(32))/sqrt(S(1) - S(2)x) - S(24843)/S(16)sqrt(S(1) - S(2)x)], [(A + Bx)(d + ex)(S(5)/S(2))sqrt(a + bx), x, S(7), - S(1)/S(48)(bd - ae)(S(3)bBd - S(10)Abe + S(7)aBe)(a + bx)*(S(3)/S(2))(d + ex)(S(3)/S(2))/(bS(3)e) - S(1)/S(40)(S(3)bBd - S(10)Abe + S(7)aBe)(a + bx)*(S(3)/S(2))(d + ex)(S(5)/S(2))/(bS(2)e) + S(1)/S(5)B(a + bx)(S(3)/S(2))(d + ex)(S(7)/S(2))/(be) + S(1)/S(128)(bd - ae)S(4)(S(3)bBd - S(10)Abe + S(7)aBe)arctanh(sqrt(e)sqrt(a + bx)/(sqrt(b)sqrt(d + ex)))/(b*(S(9)/S(2))e*(S(5)/S(2))) - S(1)/S(64)(bd - ae)*S(2)(S(3)bBd - S(10)Abe + S(7)aBe)(a + bx)(S(3)/S(2))sqrt(d + ex)/(bS(4)e) - S(1)/S(128)(bd - ae)S(3)(S(3)bBd - S(10)Abe + S(7)aBe)sqrt(a + bx)sqrt(d + ex)/(bS(4)e*S(2))], [(A + Bx)(d + ex)*(S(3)/S(2))sqrt(a + bx), x, S(6), - S(1)/S(24)(S(3)bBd - S(8)Abe + S(5)aBe)(a + bx)(S(3)/S(2))(d + ex)(S(3)/S(2))/(bS(2)e) + S(1)/S(4)B(a + bx)(S(3)/S(2))(d + ex)(S(5)/S(2))/(be) + S(1)/S(64)(bd - ae)S(3)(S(3)bBd - S(8)Abe + S(5)aBe)arctanh(sqrt(e)sqrt(a + bx)/(sqrt(b)sqrt(d + ex)))/(b*(S(7)/S(2))e*(S(5)/S(2))) - S(1)/S(32)(bd - ae)(S(3)bBd - S(8)Abe + S(5)aBe)(a + bx)*(S(3)/S(2))sqrt(d + ex)/(bS(3)e) - S(1)/S(64)(bd - ae)S(2)(S(3)bBd - S(8)Abe + S(5)aBe)sqrt(a + bx)sqrt(d + ex)/(bS(3)e*S(2))], [(A + Bx)(d + ex)*(S(5)/S(2))/sqrt(a + bx), x, S(6), - S(5)/S(64)(bd - ae)S(3)(bBd - S(8)Abe + S(7)aBe)arctanh(sqrt(e)sqrt(a + bx)/(sqrt(b)sqrt(d + ex)))/(b(S(9)/S(2))e*(S(3)/S(2))) - S(5)/S(96)(bd - ae)(bBd - S(8)Abe + S(7)aBe)(d + ex)(S(3)/S(2))sqrt(a + bx)/(bS(3)e) - S(1)/S(24)(bBd - S(8)Abe + S(7)aBe)(d + ex)(S(5)/S(2))sqrt(a + bx)/(bS(2)e) + S(1)/S(4)B(d + ex)(S(7)/S(2))sqrt(a + bx)/(be) - S(5)/S(64)(bd - ae)S(2)(bBd - S(8)Abe + S(7)aBe)sqrt(a + bx)sqrt(d + ex)/(b*S(4)e)], [(A + Bx)(d + ex)(S(3)/S(2))/sqrt(a + bx), x, S(5), - S(1)/S(8)(bd - ae)S(2)(bBd - S(6)Abe + S(5)aBe)arctanh(sqrt(e)sqrt(a + bx)/(sqrt(b)sqrt(d + ex)))/(b(S(7)/S(2))e*(S(3)/S(2))) - S(1)/S(12)(bBd - S(6)Abe + S(5)aBe)(d + ex)*(S(3)/S(2))sqrt(a + bx)/(bS(2)e) + S(1)/S(3)B(d + ex)(S(5)/S(2))sqrt(a + bx)/(be) - S(1)/S(8)(bd - ae)(bBd - S(6)Abe + S(5)aBe)sqrt(a + bx)sqrt(d + ex)/(b*S(3)e)], [(S(2) + S(3)x)S(4)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x), x, S(7), - S(333)/S(2000)(S(1) - S(2)x)(S(3)/S(2))(S(2) + S(3)x)S(2)(S(3) + S(5)x)(S(3)/S(2)) - S(1)/S(20)(S(1) - S(2)x)(S(3)/S(2))(S(2) + S(3)x)S(3)(S(3) + S(5)x)(S(3)/S(2)) - S(7)/S(640000)(S(1) - S(2)x)(S(3)/S(2))(S(3) + S(5)x)(S(3)/S(2))(S(231223) + S(140652)x) + S(4122385421)/S(51200000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) - S(34069301)/S(5120000)(S(1) - S(2)x)*(S(3)/S(2))sqrt(S(3) + S(5)x) + S(374762311)/S(51200000)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(3)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x), x, S(6), - S(3)/S(50)(S(1) - S(2)x)(S(3)/S(2))(S(2) + S(3)x)S(2)(S(3) + S(5)x)(S(3)/S(2)) - S(21)/S(16000)(S(1) - S(2)x)(S(3)/S(2))(S(3) + S(5)x)(S(3)/S(2))(S(731) + S(444)x) + S(39142411)/S(1280000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) - S(323491)/S(128000)(S(1) - S(2)x)*(S(3)/S(2))sqrt(S(3) + S(5)x) + S(3558401)/S(1280000)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(3)sqrt(S(1) - S(2)x)/sqrt(S(3) + S(5)x), x, S(5), S(525371)/S(64000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) - S(3)/S(40)(S(1) - S(2)x)(S(3)/S(2))(S(2) + S(3)x)S(2)sqrt(S(3) + S(5)x) - S(21)/S(6400)(S(1) - S(2)x)(S(3)/S(2))(S(335) + S(216)x)sqrt(S(3) + S(5)x) + S(47761)/S(64000)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(2)sqrt(S(1) - S(2)x)/sqrt(S(3) + S(5)x), x, S(5), S(3047)/S(800)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) - S(23)/S(80)(S(1) - S(2)x)(S(3)/S(2))sqrt(S(3) + S(5)x) - S(1)/S(10)(S(1) - S(2)x)(S(3)/S(2))(S(2) + S(3)x)sqrt(S(3) + S(5)x) + S(277)/S(800)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(1) - S(2)x)*(S(3)/S(2))(S(2) + S(3)x)S(3)sqrt(S(3) + S(5)x), x, S(7), - S(1)/S(20)(S(1) - S(2)x)(S(5)/S(2))(S(2) + S(3)x)S(2)(S(3) + S(5)x)(S(3)/S(2)) - S(1)/S(160000)(S(1) - S(2)x)(S(5)/S(2))(S(3) + S(5)x)(S(3)/S(2))(S(88987) + S(63120)x) + S(452517373)/S(25600000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(3739813)/S(7680000)(S(1) - S(2)x)*(S(3)/S(2))sqrt(S(3) + S(5)x) - S(339983)/S(384000)(S(1) - S(2)x)(S(5)/S(2))sqrt(S(3) + S(5)x) + S(41137943)/S(25600000)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(1) - S(2)x)*(S(3)/S(2))(S(2) + S(3)x)S(2)sqrt(S(3) + S(5)x), x, S(7), - S(567)/S(4000)(S(1) - S(2)x)(S(5)/S(2))(S(3) + S(5)x)(S(3)/S(2)) - S(3)/S(50)(S(1) - S(2)x)(S(5)/S(2))(S(2) + S(3)x)(S(3) + S(5)x)(S(3)/S(2)) + S(5487713)/S(640000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(45353)/S(192000)(S(1) - S(2)x)*(S(3)/S(2))sqrt(S(3) + S(5)x) - S(4123)/S(9600)(S(1) - S(2)x)(S(5)/S(2))sqrt(S(3) + S(5)x) + S(498883)/S(640000)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(1) - S(2)x)*(S(3)/S(2))(S(2) + S(3)x)S(3)/sqrt(S(3) + S(5)x), x, S(6), S(18648399)/S(3200000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(51373)/S(320000)(S(1) - S(2)x)(S(3)/S(2))sqrt(S(3) + S(5)x) - S(3)/S(50)(S(1) - S(2)x)(S(5)/S(2))(S(2) + S(3)x)S(2)sqrt(S(3) + S(5)x) - S(3)/S(80000)(S(1) - S(2)x)(S(5)/S(2))(S(14629) + S(11580)x)sqrt(S(3) + S(5)x) + S(1695309)/S(3200000)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(1) - S(2)x)*(S(3)/S(2))(S(2) + S(3)x)S(2)/sqrt(S(3) + S(5)x), x, S(6), S(109263)/S(32000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(301)/S(3200)(S(1) - S(2)x)(S(3)/S(2))sqrt(S(3) + S(5)x) - S(119)/S(800)(S(1) - S(2)x)(S(5)/S(2))sqrt(S(3) + S(5)x) - S(3)/S(40)(S(1) - S(2)x)(S(5)/S(2))(S(2) + S(3)x)sqrt(S(3) + S(5)x) + S(9933)/S(32000)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(1) - S(2)x)*(S(5)/S(2))(S(2) + S(3)x)S(3)sqrt(S(3) + S(5)x), x, S(8), - S(3)/S(70)(S(1) - S(2)x)(S(7)/S(2))(S(2) + S(3)x)S(2)(S(3) + S(5)x)(S(3)/S(2)) - S(3)/S(280000)(S(1) - S(2)x)(S(7)/S(2))(S(3) + S(5)x)(S(3)/S(2))(S(33857) + S(26700)x) + S(3735929329)/S(256000000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(30875449)/S(76800000)(S(1) - S(2)x)*(S(3)/S(2))sqrt(S(3) + S(5)x) + S(2806859)/S(19200000)(S(1) - S(2)x)(S(5)/S(2))sqrt(S(3) + S(5)x) - S(255169)/S(640000)(S(1) - S(2)x)(S(7)/S(2))sqrt(S(3) + S(5)x) + S(339629939)/S(256000000)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(1) - S(2)x)*(S(5)/S(2))(S(2) + S(3)x)S(2)sqrt(S(3) + S(5)x), x, S(8), - S(193)/S(2000)(S(1) - S(2)x)(S(7)/S(2))(S(3) + S(5)x)(S(3)/S(2)) - S(1)/S(20)(S(1) - S(2)x)(S(7)/S(2))(S(2) + S(3)x)(S(3) + S(5)x)(S(3)/S(2)) + S(105254149)/S(12800000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(869869)/S(3840000)(S(1) - S(2)x)*(S(3)/S(2))sqrt(S(3) + S(5)x) + S(79079)/S(960000)(S(1) - S(2)x)(S(5)/S(2))sqrt(S(3) + S(5)x) - S(7189)/S(32000)(S(1) - S(2)x)(S(7)/S(2))sqrt(S(3) + S(5)x) + S(9568559)/S(12800000)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(1) - S(2)x)*(S(5)/S(2))(S(2) + S(3)x)S(4)/sqrt(S(3) + S(5)x), x, S(8), S(12679836719)/S(1280000000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(104792039)/S(384000000)(S(1) - S(2)x)(S(3)/S(2))sqrt(S(3) + S(5)x) + S(9526549)/S(96000000)(S(1) - S(2)x)(S(5)/S(2))sqrt(S(3) + S(5)x) - S(271)/S(2800)(S(1) - S(2)x)(S(7)/S(2))(S(2) + S(3)x)S(2)sqrt(S(3) + S(5)x) - S(3)/S(70)(S(1) - S(2)x)(S(7)/S(2))(S(2) + S(3)x)S(3)sqrt(S(3) + S(5)x) - S(1)/S(22400000)(S(1) - S(2)x)(S(7)/S(2))(S(12923401) + S(11603280)x)sqrt(S(3) + S(5)x) + S(1152712429)/S(1280000000)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(1) - S(2)x)*(S(5)/S(2))(S(2) + S(3)x)S(3)/sqrt(S(3) + S(5)x), x, S(7), S(368012183)/S(64000000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(3041423)/S(19200000)(S(1) - S(2)x)(S(3)/S(2))sqrt(S(3) + S(5)x) + S(276493)/S(4800000)(S(1) - S(2)x)(S(5)/S(2))sqrt(S(3) + S(5)x) - S(1)/S(20)(S(1) - S(2)x)(S(7)/S(2))(S(2) + S(3)x)S(2)sqrt(S(3) + S(5)x) - S(1)/S(160000)(S(1) - S(2)x)(S(7)/S(2))(S(52951) + S(47280)x)sqrt(S(3) + S(5)x) + S(33455653)/S(64000000)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(4)sqrt(S(3) + S(5)x)/sqrt(S(1) - S(2)x), x, S(6), S(1067352517)/S(2560000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) - S(987)/S(4000)(S(2) + S(3)x)S(2)(S(3) + S(5)x)(S(3)/S(2))sqrt(S(1) - S(2)x) - S(3)/S(50)(S(2) + S(3)x)S(3)(S(3) + S(5)x)(S(3)/S(2))sqrt(S(1) - S(2)x) - S(21)/S(640000)(S(3) + S(5)x)(S(3)/S(2))(S(194923) + S(92040)x)sqrt(S(1) - S(2)x) - S(97032047)/S(2560000)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(3)sqrt(S(3) + S(5)x)/sqrt(S(1) - S(2)x), x, S(5), S(677017)/S(5120)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) - S(3)/S(40)(S(2) + S(3)x)S(2)(S(3) + S(5)x)(S(3)/S(2))sqrt(S(1) - S(2)x) - S(3)/S(1280)(S(3) + S(5)x)(S(3)/S(2))(S(865) + S(408)x)sqrt(S(1) - S(2)x) - S(61547)/S(5120)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(4)/(sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)), x, S(5), S(10866247)/S(128000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) - S(259)/S(800)(S(2) + S(3)x)S(2)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) - S(3)/S(40)(S(2) + S(3)x)S(3)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) - S(7)/S(128000)(S(187559) + S(77820)x)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(3)/(sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)), x, S(4), S(44437)/S(1600)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) - S(1)/S(10)(S(2) + S(3)x)S(2)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) - S(1)/S(1600)(S(5363) + S(2220)x)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(5)sqrt(S(3) + S(5)x)/(S(1) - S(2)x)*(S(3)/S(2)), x, S(7), - S(35439958001)/S(5120000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + (S(2) + S(3)x)S(5)sqrt(S(3) + S(5)x)/sqrt(S(1) - S(2)x) + S(847637)/S(32000)(S(2) + S(3)x)*S(2)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) + S(10389)/S(1600)(S(2) + S(3)x)S(3)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) + S(33)/S(20)(S(2) + S(3)x)S(4)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) + S(49)/S(5120000)(S(87394471) + S(36265980)x)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(4)sqrt(S(3) + S(5)x)/(S(1) - S(2)x)*(S(3)/S(2)), x, S(6), - S(92108287)/S(51200)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + (S(2) + S(3)x)S(4)sqrt(S(3) + S(5)x)/sqrt(S(1) - S(2)x) + S(2203)/S(320)(S(2) + S(3)x)*S(2)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) + S(27)/S(16)(S(2) + S(3)x)S(3)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) + S(1)/S(51200)(S(11129753) + S(4618500)x)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(5)/((S(1) - S(2)x)*(S(3)/S(2))sqrt(S(3) + S(5)x)), x, S(6), - S(291096141)/S(256000)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(7)/S(11)(S(2) + S(3)x)*S(4)sqrt(S(3) + S(5)x)/sqrt(S(1) - S(2)x) + S(76587)/S(17600)(S(2) + S(3)x)*S(2)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) + S(939)/S(880)(S(2) + S(3)x)S(3)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) + S(21)/S(2816000)(S(18424549) + S(7645620)x)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(4)/((S(1) - S(2)x)*(S(3)/S(2))sqrt(S(3) + S(5)x)), x, S(5), - S(184641)/S(640)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(7)/S(11)(S(2) + S(3)x)*S(3)sqrt(S(3) + S(5)x)/sqrt(S(1) - S(2)x) + S(243)/S(220)(S(2) + S(3)x)*S(2)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) + S(9)/S(7040)(S(27269) + S(11316)x)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(4)sqrt(S(3) + S(5)x)/(S(1) - S(2)x)*(S(5)/S(2)), x, S(6), S(13246251)/S(6400)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(1)/S(3)(S(2) + S(3)x)*S(4)sqrt(S(3) + S(5)x)/(S(1) - S(2)x)*(S(3)/S(2)) - S(299)/S(66)(S(2) + S(3)x)S(3)sqrt(S(3) + S(5)x)/sqrt(S(1) - S(2)x) - S(697)/S(88)(S(2) + S(3)x)*S(2)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) - S(1)/S(70400)(S(17606479) + S(7306140)x)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(3)sqrt(S(3) + S(5)x)/(S(1) - S(2)x)*(S(5)/S(2)), x, S(5), S(126513)/S(320)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(1)/S(3)(S(2) + S(3)x)*S(3)sqrt(S(3) + S(5)x)/(S(1) - S(2)x)*(S(3)/S(2)) - S(233)/S(66)(S(2) + S(3)x)S(2)sqrt(S(3) + S(5)x)/sqrt(S(1) - S(2)x) - S(1)/S(3520)(S(168157) + S(69780)x)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)S(5)/((S(1) - S(2)x)*(S(5)/S(2))sqrt(S(3) + S(5)x)), x, S(6), S(8261577)/S(6400)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(7)/S(33)(S(2) + S(3)x)*S(4)sqrt(S(3) + S(5)x)/(S(1) - S(2)x)*(S(3)/S(2)) - S(2051)/S(726)(S(2) + S(3)x)S(3)sqrt(S(3) + S(5)x)/sqrt(S(1) - S(2)x) - S(23909)/S(4840)(S(2) + S(3)x)*S(2)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x) - S(1)/S(774400)(S(120791143) + S(50124540)x)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(S(2) + S(3)x)*S(4)/((S(1) - S(2)x)*(S(5)/S(2))sqrt(S(3) + S(5)x)), x, S(5), S(392283)/S(1600)arcsin(sqrt(S(2)/S(11))sqrt(S(3) + S(5)x))/sqrt(S(10)) + S(7)/S(33)(S(2) + S(3)x)*S(3)sqrt(S(3) + S(5)x)/(S(1) - S(2)x)*(S(3)/S(2)) - S(1589)/S(726)(S(2) + S(3)x)S(2)sqrt(S(3) + S(5)x)/sqrt(S(1) - S(2)x) - S(1)/S(193600)(S(5735477) + S(2380020)x)sqrt(S(1) - S(2)x)sqrt(S(3) + S(5)x)], [(c + dx)(S(1)/S(2))/(xS(2)(a + bx)S(2)), x, S(7), (S(4)bc - ad)arctanh(sqrt(c + dx)/sqrt(c))/(a*S(3)sqrt(c)) - (S(4)bc - S(3)ad)arctanh(sqrt(b)sqrt(c + dx)/sqrt(bc - ad))sqrt(b)/(a*S(3)sqrt(bc - ad)) - S(2)bsqrt(c + dx)/(aS(2)(a + bx)) - sqrt(c + dx)/(ax(a + bx))], [S(1)/(xS(2)(a + bx)S(2)(c + dx)(S(1)/S(2))), x, S(7), (S(4)bc + ad)arctanh(sqrt(c + dx)/sqrt(c))/(a*S(3)c(S(3)/S(2))) - b(S(3)/S(2))(S(4)bc - S(5)ad)arctanh(sqrt(b)sqrt(c + dx)/sqrt(bc - ad))/(a*S(3)(bc - ad)*(S(3)/S(2))) - b(S(2)bc - ad)sqrt(c + dx)/(aS(2)c(bc - ad)(a + bx)) - sqrt(c + dx)/(acx(a + bx))], [S(1)/(x*S(2)(a + bx)S(2)(c + dx)(S(3)/S(2))), x, S(8), (S(4)bc + S(3)ad)arctanh(sqrt(c + dx)/sqrt(c))/(aS(3)c(S(5)/S(2))) - b(S(5)/S(2))(S(4)bc - S(7)ad)arctanh(sqrt(b)sqrt(c + dx)/sqrt(bc - ad))/(a*S(3)(bc - ad)*(S(5)/S(2))) - d(S(2)bS(2)c*S(2) - S(2)abcd + S(3)a*S(2)dS(2))/(aS(2)cS(2)(bc - ad)*S(2)sqrt(c + dx)) - b(S(2)bc - ad)/(aS(2)c(bc - ad)(a + bx)sqrt(c + dx)) + ( - S(1))/(acx(a + bx)sqrt(c + dx))], [xS(3)(c + dx)(S(3)/S(2))/(a + bx)*(S(3)/S(2)), x, S(6), S(3)/S(64)(bc - ad)(bS(3)c*S(3) + S(5)abS(2)c*S(2)d + S(35)aS(2)bcd*S(2) - S(105)a*S(3)d*S(3))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(11)/S(2))d*(S(5)/S(2))) - S(2)x*S(3)(c + dx)(S(3)/S(2))/(bsqrt(a + bx)) + S(9)/S(4)x*S(2)(c + dx)(S(3)/S(2))sqrt(a + bx)/bS(2) - S(1)/S(32)(c + dx)(S(3)/S(2))(S(3)bS(2)c*S(2) + S(14)abcd - S(105)a*S(2)d*S(2) - S(4)bd(bc - S(21)ad)x)sqrt(a + bx)/(b*S(4)d*S(2)) + S(3)/S(64)(b*S(3)c*S(3) + S(5)abS(2)c*S(2)d + S(35)aS(2)bcd*S(2) - S(105)a*S(3)d*S(3))sqrt(a + bx)sqrt(c + dx)/(bS(5)dS(2))], [xS(2)(c + dx)*(S(3)/S(2))/(a + bx)*(S(3)/S(2)), x, S(6), - S(1)/S(8)(bc - ad)(bS(2)c*S(2) + S(10)abcd - S(35)a*S(2)d*S(2))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(9)/S(2))d*(S(3)/S(2))) - S(2)a*S(2)(c + dx)(S(5)/S(2))/(bS(2)(bc - ad)sqrt(a + bx)) - S(1)/S(12)(S(10)ac + bc*S(2)/d - S(35)a*S(2)d/b)(c + dx)*(S(3)/S(2))sqrt(a + bx)/(bS(2)(bc - ad)) + S(1)/S(3)(c + dx)*(S(5)/S(2))sqrt(a + bx)/(bS(2)d) - S(1)/S(8)(bS(2)c*S(2) + S(10)abcd - S(35)a*S(2)d*S(2))sqrt(a + bx)sqrt(c + dx)/(bS(4)d)], [x*S(3)(c + dx)(S(5)/S(2))/(a + bx)*(S(5)/S(2)), x, S(7), - S(2)/S(3)x*S(3)(c + dx)(S(5)/S(2))/(b(a + bx)(S(3)/S(2))) - S(5)/S(64)(bc - ad)(bS(3)c*S(3) + S(21)abS(2)c*S(2)d - S(189)aS(2)bcd*S(2) + S(231)a*S(3)d*S(3))arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(13)/S(2))d*(S(3)/S(2))) - S(2)/S(3)(S(6)bc - S(11)ad)xS(2)(c + dx)(S(5)/S(2))/(bS(2)(bc - ad)sqrt(a + bx)) - S(5)/S(96)(bS(3)c*S(3) + S(21)abS(2)c*S(2)d - S(189)aS(2)bcd*S(2) + S(231)a*S(3)d*S(3))(c + dx)(S(3)/S(2))sqrt(a + bx)/(bS(5)d(bc - ad)) + S(1)/S(24)(c + dx)(S(5)/S(2))(S(5)bS(2)c*S(2) - S(156)abcd + S(231)a*S(2)d*S(2) + S(2)bd(S(59)bc - S(99)ad)x)sqrt(a + bx)/(bS(4)d(bc - ad)) - S(5)/S(64)(b*S(3)c*S(3) + S(21)abS(2)c*S(2)d - S(189)aS(2)bcd*S(2) + S(231)a*S(3)d*S(3))sqrt(a + bx)sqrt(c + dx)/(bS(6)d)], [x*S(2)/((a + bx)*(S(5)/S(2))(c + dx)(S(1)/S(2))), x, S(4), S(2)arctanh(sqrt(d)sqrt(a + bx)/(sqrt(b)sqrt(c + dx)))/(b*(S(5)/S(2))sqrt(d)) - S(2)/S(3)aS(2)sqrt(c + dx)/(bS(2)(bc - ad)(a + bx)*(S(3)/S(2))) + S(4)/S(3)a(S(3)bc - S(2)ad)sqrt(c + dx)/(bS(2)(bc - ad)*S(2)sqrt(a + bx))], [xsqrt(a + bx)/sqrt( - a - bx), x, S(2), S(1)/S(2)xS(2)sqrt(a + bx)/sqrt( - a - bx)], [(c + dx)(S(3)/S(2))/(x(a + bx)S(2)), x, S(6), - S(2)c*(S(3)/S(2))arctanh(sqrt(c + dx)/sqrt(c))/aS(2) + (S(2)bc + ad)arctanh(sqrt(b)sqrt(c + dx)/sqrt(bc - ad))sqrt(bc - ad)/(a*S(2)b*(S(3)/S(2))) + (bc - ad)sqrt(c + dx)/(ab(a + bx))], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True, _diff=True, _numerical=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True, _numerical=True) else: assert rubi_test(r, i[1], i[3], expand=True, _diff=True, _numerical=True) def test_simplify(): test = [ [x*S(3)(a + bx)S(2)(c + dx)S(16), x, S(2), - S(1)/S(17)c*S(3)(bc - ad)*S(2)(c + dx)S(17)/dS(6) + S(1)/S(18)c*S(2)(S(5)bc - S(3)ad)(bc - ad)(c + dx)S(18)/dS(6) - S(1)/S(19)c(S(10)b*S(2)c*S(2) - S(12)abcd + S(3)a*S(2)d*S(2))(c + dx)S(19)/dS(6) + S(1)/S(20)(S(10)bS(2)c*S(2) - S(8)abcd + aS(2)d*S(2))(c + dx)S(20)/dS(6) - S(1)/S(21)b(S(5)bc - S(2)ad)(c + dx)S(21)/dS(6) + S(1)/S(22)b*S(2)(c + dx)S(22)/dS(6)], [xS(5)/((a + bx)*S(2)(c + dx)S(2)), x, S(2), - S(2)(bc + ad)x/(bS(3)d*S(3)) + S(1)/S(2)xS(2)/(bS(2)dS(2)) + aS(5)/(bS(4)(bc - ad)*S(2)(a + bx)) + cS(5)/(dS(4)(bc - ad)*S(2)(c + dx)) + aS(4)(S(5)bc - S(3)ad)log(a + bx)/(b*S(4)(bc - ad)S(3)) + cS(4)(S(3)bc - S(5)ad)log(c + dx)/(dS(4)(bc - ad)S(3))], [xS(5)/((a + bx)S(2)(c + dx)S(2)), x, S(2), - S(2)(bc + ad)x/(bS(3)d*S(3)) + S(1)/S(2)xS(2)/(bS(2)dS(2)) + aS(5)/(bS(4)(bc - ad)*S(2)(a + bx)) + cS(5)/(dS(4)(bc - ad)*S(2)(c + dx)) + aS(4)(S(5)bc - S(3)ad)log(a + bx)/(b*S(4)(bc - ad)S(3)) + cS(4)(S(3)bc - S(5)ad)log(c + dx)/(dS(4)(bc - ad)S(3))], [xS(4)/((a + bx)(c + dx)), x, S(2), (bS(2)c*S(2) + abcd + a*S(2)d*S(2))x/(b*S(3)d*S(3)) - S(1)/S(2)(bc + ad)xS(2)/(bS(2)d*S(2)) + S(1)/S(3)x*S(3)/(bd) + a*S(4)log(a + bx)/(bS(4)(bc - ad)) - c*S(4)log(c + dx)/(dS(4)(bc - ad))], [(a + bx)(A + Bx)(d + ex)S(4), x, S(2), S(1)/S(5)(bd - ae)(Bd - Ae)(d + ex)S(5)/eS(3) - S(1)/S(6)(S(2)bBd - Abe - aBe)(d + ex)S(6)/eS(3) + S(1)/S(7)bB(d + ex)S(7)/eS(3)], [(a + bx)*S(3)(c + dx)S(3)(e + fx)S(3), x, S(2), S(1)/S(4)(bc - ad)*S(3)(be - af)*S(3)(a + bx)S(4)/bS(7) + S(3)/S(5)(bc - ad)*S(2)(be - af)*S(2)(bde + bcf - S(2)adf)(a + bx)S(5)/bS(7) + S(1)/S(2)(bc - ad)(be - af)(S(5)aS(2)d*S(2)f*S(2) - S(5)abdf(de + cf) + b*S(2)(d*S(2)e*S(2) + S(3)cdef + cS(2)f*S(2)))(a + bx)S(6)/bS(7) + S(1)/S(7)(bde + bcf - S(2)adf)(S(10)aS(2)d*S(2)f*S(2) - S(10)abdf(de + cf) + b*S(2)(d*S(2)e*S(2) + S(8)cdef + cS(2)f*S(2)))(a + bx)S(7)/bS(7) + S(3)/S(8)df(S(5)aS(2)d*S(2)f*S(2) - S(5)abdf(de + cf) + b*S(2)(d*S(2)e*S(2) + S(3)cdef + cS(2)f*S(2)))(a + bx)S(8)/bS(7) + S(1)/S(3)d*S(2)f*S(2)(bde + bcf - S(2)adf)(a + bx)S(9)/bS(7) + S(1)/S(10)d*S(3)f*S(3)(a + bx)S(10)/bS(7)], [(a + bx)(A + Bx)(d + ex)*(S(5)/S(2)), x, S(2), S(2)/S(7)(bd - ae)(Bd - Ae)(d + ex)(S(7)/S(2))/eS(3) - S(2)/S(9)(S(2)bBd - Abe - aBe)(d + ex)(S(9)/S(2))/eS(3) + S(2)/S(11)bB(d + ex)(S(11)/S(2))/eS(3)], [(S(5) - S(4)x)*S(4)(S(2) + S(3)x)m/(S(1) + S(2)x)*m, x, S(4), - S(1)/S(45)(S(88) - m)(S(5) - S(4)x)*S(2)(S(1) + S(2)x)(S(1) - m)(S(2) + S(3)x)(S(1) + m) - S(2)/S(15)(S(5) - S(4)x)S(3)(S(1) + S(2)x)(S(1) - m)(S(2) + S(3)x)(S(1) + m) - S(1)/S(1215)(S(1) + S(2)x)(S(1) - m)(S(2) + S(3)x)(S(1) + m)(S(386850) - S(25441)m + S(426)m*S(2) - S(2)m*S(3) - S(24)(S(4359) - S(154)m + mS(2))x) + S(1)/S(1215)S(2)( - S(1) - m)(S(3528363) - S(639760)m + S(29050)m*S(2) - S(440)m*S(3) + S(2)m*S(4))(S(1) + S(2)x)(S(1) - m)hypergeom([S(1) - m, - m], [S(2) - m], - S(3)(S(1) + S(2)x))/(S(1) - m)], [(S(5) - S(4)x)S(3)(S(1) + S(2)x)( - S(1) - m)(S(2) + S(3)x)m, x, S(3), - S(2)/S(9)(S(5) - S(4)x)S(2)(S(2) + S(3)x)(S(1) + m)/(S(1) + S(2)x)*m - S(1)/S(27)(S(2) + S(3)x)(S(1) + m)(S(9261) - S(512)m + S(4)m*S(2) - S(4)(S(109) - S(2)m)mx)/(m(S(1) + S(2)x)m) + S(1)/S(27)S(2)*( - S(1) - m)(S(27783) - S(8324)m + S(390)m*S(2) - S(4)m*S(3))(S(1) + S(2)x)(S(1) - m)hypergeom([S(1) - m, - m], [S(2) - m], - S(3)(S(1) + S(2)x))/((S(1) - m)m)], [(a + bx)*m(c + dx)n((bcf + adf + adfm + bcfn)/(bd(S(2) + m + n)) + fx)( - S(3) - m - n), x, S(1), bd(S(2) + m + n)(a + bx)(S(1) + m)(c + dx)(S(1) + n)(f(ad(S(1) + m) + bc(S(1) + n))/(bd(S(2) + m + n)) + fx)*( - S(2) - m - n)/((bc - ad)S(2)f(S(1) + m)(S(1) + n))], [x*S(3)(c + dx)S(3)/(a + bx)*S(3), x, S(2), (bc - ad)(b*S(2)c*S(2) - S(8)abcd + S(10)a*S(2)d*S(2))x/b*S(6) + S(3)/S(2)d(bc - S(2)ad)(bc - ad)xS(2)/bS(5) + d*S(2)(bc - ad)xS(3)/bS(4) + S(1)/S(4)d*S(3)xS(4)/bS(3) + S(1)/S(2)aS(3)(bc - ad)S(3)/(bS(7)(a + bx)*S(2)) - S(3)a*S(2)(bc - S(2)ad)(bc - ad)S(2)/(bS(7)(a + bx)) - S(3)a(bc - ad)(bS(2)c*S(2) - S(5)abcd + S(5)a*S(2)d*S(2))log(a + bx)/bS(7)], [(S(2) + S(3)x)*S(8)(S(3) + S(5)x)/(S(1) - S(2)x)*S(3), x, S(2), S(63412811)/S(2048)/(S(1) - S(2)x)*S(2) + ( - S(246239357)/S(1024))/(S(1) - S(2)x) - S(120864213)/S(256)x - S(118841283)/S(512)x*S(2) - S(16042509)/S(128)x*S(3) - S(7568235)/S(128)x*S(4) - S(213597)/S(10)x*S(5) - S(162567)/S(32)x*S(6) - S(32805)/S(56)x*S(7) - S(106237047)/S(256)log(S(1) - S(2)x)], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True) or rubi_test(r, i[1], i[4], expand=True) else: assert rubi_test(r, i[1], i[3], expand=True) def test_diff(): test = [ [(a + bx)(e + fx)*(S(3)/S(2))/(c + dx), x, S(5), - S(2)/S(3)(bc - ad)(e + fx)(S(3)/S(2))/dS(2) + S(2)/S(5)b(e + fx)*(S(5)/S(2))/(df) + S(2)(bc - ad)(de - cf)*(S(3)/S(2))arctanh(sqrt(d)sqrt(e + fx)/sqrt(de - cf))/d*(S(7)/S(2)) - S(2)(bc - ad)(de - cf)sqrt(e + fx)/dS(3)], [x(S(5)/S(2))(A + Bx)/(a + bx), x, S(6), - S(2)/S(3)a(Ab - aB)x(S(3)/S(2))/bS(3) + S(2)/S(5)(Ab - aB)x(S(5)/S(2))/bS(2) + S(2)/S(7)Bx(S(7)/S(2))/b - S(2)a*(S(5)/S(2))(Ab - aB)arctan(sqrt(b)sqrt(x)/sqrt(a))/b*(S(9)/S(2)) + S(2)a*S(2)(Ab - aB)sqrt(x)/bS(4)], [(a + bx)*S(2)/((c + dx)*S(2)sqrt(e + fx)), x, S(4), (bc - ad)(S(4)bde - S(3)bcf - adf)arctanh(sqrt(d)sqrt(e + fx)/sqrt(de - cf))/(d(S(5)/S(2))(de - cf)*(S(3)/S(2))) + S(2)b*S(2)sqrt(e + fx)/(dS(2)f) - (bc - ad)*S(2)sqrt(e + fx)/(dS(2)(de - cf)(c + dx))], ] for i in test: r = rubi_integrate(i[0], i[1]) if len(i) == 5: assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True) else: assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
956d57f8cb810af076c65678a028d919fba770fa13ee8f4b467b15b8d742c979	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: disabled = True if sys.version_info[:2] < (3, 6): disabled = True if matchpy: from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match from sympy.integrals.rubi.utility_function import ( sympy_op_factory, Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ, ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ, PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ, NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest, SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient, Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan, ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec, ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less, Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ, InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ, EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, ExpandLinearProduct, GCD, ContentFactor, NumericFactor, NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, ExpandExpression, Apart, SmartApart, MatchQ, PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, RationalFunctionFactors, NonrationalFunctionFactors, Reverse, RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, GeneralizedBinomialParts, GeneralizedTrinomialDegree, GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, MinimumMonomialExponent, MonomialExponent, LinearMatchQ, PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, AlgebraicFunctionFactors, NonalgebraicFunctionFactors, QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, CombineExponents, FactorInteger, FactorAbsurdNumber, SubstForInverseFunction, SubstForFractionalPower, SubstForFractionalPowerOfQuotientOfLinears, FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, NormalizeIntegrandAux, NormalizeIntegrandFactor, NormalizeIntegrandFactorBase, NormalizeTogether, NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand, SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, KnownTangentIntegrandQ, KnownCotangentIntegrandQ, KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, FunctionOfLog, PowerVariableExpn, PowerVariableDegree, PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, ConstantFactor, SameQ, ReplacePart, CommonFactors, MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, _SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify, _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor, PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral, CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial, Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, exp, log, Discriminant ) from sympy.core.add import Add from sympy.core.mod import Mod from sympy.core.mul import Mul from sympy.core.numbers import (Float, I, Integer) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.functions.elementary.complexes import Abs from sympy.functions.elementary.miscellaneous import sqrt from sympy.integrals.integrals import Integral from sympy.logic.boolalg import (And, Or) from sympy.simplify.simplify import simplify from sympy.integrals.rubi.symbol import WC from sympy.core.symbol import symbols, Symbol from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf) from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch) from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2) from sympy.core.numbers import pi as Pi from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.functions.elementary.exponential import (exp, log) from sympy.integrals.integrals import Integral as Integrate a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z = symbols('a b c d e f g h i j k l m n o p q r s t u v w x y z') A, B, C, F, G, H, J, K, L, M, N, O, P, Q, R, T, U, V, W, X, Y, Z = symbols('A B C F G H J K L M N O P Q R T U V W X Y Z') def test_error_functions(): assert rubi_test(rubi_integrate(x*S(5)Erf(bx)S(2), x), x, xS(6)Erf(bx)S(2)/S(6) - S(5)Erf(bx)S(2)/(S(16)bS(6)) + xS(4)exp(-S(2)b*S(2)x*S(2))/(S(6)PibS(2)) + S(7)x*S(2)exp(-S(2)bS(2)x*S(2))/(S(12)PibS(4)) + S(11)exp(-S(2)bS(2)x*S(2))/(S(12)PibS(6)) + xS(5)Erf(bx)exp(-b*S(2)x*S(2))/(S(3)sqrt(Pi)b) + S(5)x*S(3)Erf(bx)exp(-b*S(2)x*S(2))/(S(6)sqrt(Pi)bS(3)) + S(5)xErf(bx)exp(-bS(2)x*S(2))/(S(4)sqrt(Pi)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)Erf(bx)S(2), x), x, xS(5)Erf(bx)S(2)/S(5) + xS(3)exp(-S(2)bS(2)x*S(2))/(S(5)PibS(2)) + S(11)xexp(-S(2)b*S(2)x*S(2))/(S(20)PibS(4)) + S(2)x*S(4)Erf(bx)exp(-b*S(2)x*S(2))/(S(5)sqrt(Pi)b) + S(4)x*S(2)Erf(bx)exp(-b*S(2)x*S(2))/(S(5)sqrt(Pi)bS(3)) + S(4)Erf(bx)exp(-b*S(2)x*S(2))/(S(5)sqrt(Pi)bS(5)) - S(43)sqrt(S(2))Erf(sqrt(S(2))bx)/(S(80)sqrt(Pi)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)Erf(bx)S(2), x), x, xS(4)Erf(bx)S(2)/S(4) - S(3)Erf(bx)S(2)/(S(16)bS(4)) + xS(2)exp(-S(2)b*S(2)x*S(2))/(S(4)PibS(2)) + exp(-S(2)b*S(2)x*S(2))/(S(2)PibS(4)) + xS(3)Erf(bx)exp(-b*S(2)x*S(2))/(S(2)sqrt(Pi)b) + S(3)xErf(bx)exp(-bS(2)x*S(2))/(S(4)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)Erf(bx)S(2), x), x, xS(3)Erf(bx)S(2)/S(3) + xexp(-S(2)bS(2)x*S(2))/(S(3)PibS(2)) + S(2)x*S(2)Erf(bx)exp(-b*S(2)x*S(2))/(S(3)sqrt(Pi)b) + S(2)Erf(bx)exp(-b*S(2)x*S(2))/(S(3)sqrt(Pi)bS(3)) - S(5)sqrt(S(2))Erf(sqrt(S(2))bx)/(S(12)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xErf(bx)S(2), x), x, xS(2)Erf(bx)S(2)/S(2) - Erf(bx)*S(2)/(S(4)b*S(2)) + exp(-S(2)b*S(2)x*S(2))/(S(2)PibS(2)) + xErf(bx)exp(-b*S(2)x*S(2))/(sqrt(Pi)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)S(2), x), x, xErf(bx)S(2) - sqrt(S(2))sqrt(S(1)/Pi)Erf(sqrt(S(2))bx)/b + S(2)Erf(bx)exp(-b*S(2)x*S(2))/(sqrt(Pi)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)S(2)/x, x), x, Integrate(Erf(bx)*S(2)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)S(2)/xS(2), x), x, Integrate(Erf(bx)S(2)/xS(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)S(2)/xS(3), x), x, -b*S(2)Erf(bx)S(2) - Erf(bx)*S(2)/(S(2)x*S(2)) + S(2)b*S(2)ExpIntegralEi(-S(2)bS(2)x*S(2))/Pi - S(2)bErf(bx)exp(-bS(2)x*S(2))/(sqrt(Pi)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)S(2)/xS(4), x), x, Integrate(Erf(bx)S(2)/xS(4), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)S(2)/xS(5), x), x, bS(4)Erf(bx)S(2)/S(3) - Erf(bx)*S(2)/(S(4)x*S(4)) - S(4)b*S(4)ExpIntegralEi(-S(2)bS(2)x*S(2))/(S(3)Pi) - b*S(2)exp(-S(2)bS(2)x*S(2))/(S(3)PixS(2)) + S(2)b*S(3)Erf(bx)exp(-b*S(2)x*S(2))/(S(3)sqrt(Pi)x) - bErf(bx)exp(-b*S(2)x*S(2))/(S(3)sqrt(Pi)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)S(2)/xS(6), x), x, Integrate(Erf(bx)S(2)/xS(6), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)S(2)/xS(7), x), x, -S(4)bS(6)Erf(bx)S(2)/S(45) - Erf(bx)*S(2)/(S(6)x*S(6)) + S(28)b*S(6)ExpIntegralEi(-S(2)bS(2)x*S(2))/(S(45)Pi) + S(2)bS(4)exp(-S(2)bS(2)x*S(2))/(S(9)PixS(2)) - bS(2)exp(-S(2)bS(2)x*S(2))/(S(15)PixS(4)) - S(8)b*S(5)Erf(bx)exp(-b*S(2)x*S(2))/(S(45)sqrt(Pi)x) + S(4)b*S(3)Erf(bx)exp(-b*S(2)x*S(2))/(S(45)sqrt(Pi)xS(3)) - S(2)bErf(bx)exp(-bS(2)x*S(2))/(S(15)sqrt(Pi)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)S(2)/xS(8), x), x, Integrate(Erf(bx)S(2)/xS(8), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)Erf(a + bx), x), x, -aS(4)Erf(a + bx)/(S(4)b*S(4)) - S(3)a*S(2)Erf(a + bx)/(S(4)bS(4)) + xS(4)Erf(a + bx)/S(4) - S(3)Erf(a + bx)/(S(16)bS(4)) - aS(3)exp(-(a + bx)S(2))/(sqrt(Pi)b*S(4)) + S(3)a*S(2)(a + bx)exp(-(a + bx)S(2))/(S(2)sqrt(Pi)bS(4)) - a(a + bx)S(2)exp(-(a + bx)S(2))/(sqrt(Pi)b*S(4)) - aexp(-(a + bx)S(2))/(sqrt(Pi)b*S(4)) + (a + bx)*S(3)exp(-(a + bx)S(2))/(S(4)sqrt(Pi)bS(4)) + (S(3)a + S(3)bx)exp(-(a + bx)*S(2))/(S(8)sqrt(Pi)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)Erf(a + bx), x), x, aS(3)Erf(a + bx)/(S(3)b*S(3)) + aErf(a + bx)/(S(2)bS(3)) + xS(3)Erf(a + bx)/S(3) + a*S(2)exp(-(a + bx)S(2))/(sqrt(Pi)b*S(3)) - a(a + bx)exp(-(a + bx)S(2))/(sqrt(Pi)b*S(3)) + (a + bx)*S(2)exp(-(a + bx)S(2))/(S(3)sqrt(Pi)bS(3)) + exp(-(a + bx)*S(2))/(S(3)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xErf(a + bx), x), x, -aS(2)Erf(a + bx)/(S(2)bS(2)) + xS(2)Erf(a + bx)/S(2) - Erf(a + bx)/(S(4)b*S(2)) - aexp(-(a + bx)S(2))/(sqrt(Pi)b*S(2)) + (a + bx)exp(-(a + bx)*S(2))/(S(2)sqrt(Pi)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(a + bx), x), x, (a + bx)Erf(a + bx)/b + exp(-(a + bx)*S(2))/(sqrt(Pi)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(a + bx)/x, x), x, Integrate(Erf(a + bx)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(a + bx)/xS(2), x), x, -Erf(a + bx)/x + S(2)bIntegrate(exp(-(a + bx)S(2))/x, x)/sqrt(Pi), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)Erf(a + bx)S(2), x), x, aS(2)(a + bx)Erf(a + bx)S(2)/bS(3) - sqrt(S(2))a*S(2)sqrt(S(1)/Pi)Erf(sqrt(S(2))(a + bx))/bS(3) - a(a + bx)S(2)Erf(a + bx)S(2)/bS(3) + aErf(a + bx)S(2)/(S(2)b*S(3)) + (a + bx)*S(3)Erf(a + bx)S(2)/(S(3)b*S(3)) - aexp(-S(2)(a + bx)*S(2))/(Pib*S(3)) + (a + bx)exp(-S(2)(a + bx)S(2))/(S(3)PibS(3)) + S(2)a*S(2)Erf(a + bx)exp(-(a + bx)S(2))/(sqrt(Pi)b*S(3)) - S(2)a(a + bx)Erf(a + bx)exp(-(a + bx)*S(2))/(sqrt(Pi)b*S(3)) + S(2)(a + bx)S(2)Erf(a + bx)exp(-(a + bx)S(2))/(S(3)sqrt(Pi)bS(3)) - S(5)sqrt(S(2))Erf(sqrt(S(2))(a + bx))/(S(12)sqrt(Pi)bS(3)) + S(2)Erf(a + bx)exp(-(a + bx)S(2))/(S(3)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xErf(a + bx)S(2), x), x, -a(a + bx)Erf(a + bx)S(2)/bS(2) + sqrt(S(2))asqrt(S(1)/Pi)Erf(sqrt(S(2))(a + bx))/b*S(2) + (a + bx)*S(2)Erf(a + bx)S(2)/(S(2)b*S(2)) - Erf(a + bx)*S(2)/(S(4)b*S(2)) + exp(-S(2)(a + bx)S(2))/(S(2)PibS(2)) - S(2)aErf(a + bx)exp(-(a + bx)*S(2))/(sqrt(Pi)b*S(2)) + (a + bx)Erf(a + bx)exp(-(a + bx)*S(2))/(sqrt(Pi)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(a + bx)*S(2), x), x, (a + bx)Erf(a + bx)*S(2)/b - sqrt(S(2))sqrt(S(1)/Pi)Erf(sqrt(S(2))(a + bx))/b + S(2)Erf(a + bx)exp(-(a + bx)S(2))/(sqrt(Pi)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(a + bx)S(2)/x, x), x, Integrate(Erf(a + bx)*S(2)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(a + bx)S(2)/xS(2), x), x, Integrate(Erf(a + bx)S(2)/xS(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)Erf(bx)exp(-b*S(2)x*S(2)), x), x, S(15)sqrt(Pi)Erf(bx)*S(2)/(S(32)bS(7)) - xS(5)Erf(bx)exp(-bS(2)x*S(2))/(S(2)b*S(2)) - S(5)x*S(3)Erf(bx)exp(-b*S(2)x*S(2))/(S(4)b*S(4)) - S(15)xErf(bx)exp(-bS(2)x*S(2))/(S(8)bS(6)) - xS(4)exp(-S(2)b*S(2)x*S(2))/(S(4)sqrt(Pi)bS(3)) - S(7)x*S(2)exp(-S(2)bS(2)x*S(2))/(S(8)sqrt(Pi)bS(5)) - S(11)exp(-S(2)bS(2)x*S(2))/(S(8)sqrt(Pi)bS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)Erf(bx)exp(-b*S(2)xS(2)), x), x, -xS(4)Erf(bx)exp(-bS(2)x*S(2))/(S(2)bS(2)) - xS(2)Erf(bx)exp(-bS(2)xS(2))/bS(4) - Erf(bx)exp(-b*S(2)xS(2))/bS(6) + S(43)sqrt(S(2))Erf(sqrt(S(2))bx)/(S(64)bS(6)) - xS(3)exp(-S(2)bS(2)x*S(2))/(S(4)sqrt(Pi)bS(3)) - S(11)xexp(-S(2)b*S(2)x*S(2))/(S(16)sqrt(Pi)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)Erf(bx)exp(-b*S(2)x*S(2)), x), x, S(3)sqrt(Pi)Erf(bx)*S(2)/(S(16)bS(5)) - xS(3)Erf(bx)exp(-bS(2)x*S(2))/(S(2)b*S(2)) - S(3)xErf(bx)exp(-bS(2)x*S(2))/(S(4)bS(4)) - xS(2)exp(-S(2)b*S(2)x*S(2))/(S(4)sqrt(Pi)bS(3)) - exp(-S(2)b*S(2)x*S(2))/(S(2)sqrt(Pi)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)Erf(bx)exp(-b*S(2)xS(2)), x), x, -xS(2)Erf(bx)exp(-bS(2)x*S(2))/(S(2)b*S(2)) - Erf(bx)exp(-bS(2)x*S(2))/(S(2)b*S(4)) + S(5)sqrt(S(2))Erf(sqrt(S(2))bx)/(S(16)b*S(4)) - xexp(-S(2)bS(2)x*S(2))/(S(4)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)Erf(bx)exp(-b*S(2)x*S(2)), x), x, sqrt(Pi)Erf(bx)S(2)/(S(8)b*S(3)) - xErf(bx)exp(-b*S(2)x*S(2))/(S(2)b*S(2)) - exp(-S(2)b*S(2)x*S(2))/(S(4)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xErf(bx)exp(-b*S(2)x*S(2)), x), x, -Erf(bx)exp(-bS(2)x*S(2))/(S(2)b*S(2)) + sqrt(S(2))Erf(sqrt(S(2))bx)/(S(4)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)exp(-bS(2)x*S(2)), x), x, sqrt(Pi)Erf(bx)S(2)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)exp(-b*S(2)x*S(2))/x, x), x, Integrate(Erf(bx)exp(-bS(2)x*S(2))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)exp(-bS(2)xS(2))/xS(2), x), x, -sqrt(Pi)bErf(bx)S(2)/S(2) - Erf(bx)exp(-bS(2)x*S(2))/x + bExpIntegralEi(-S(2)bS(2)x*S(2))/sqrt(Pi), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)exp(-bS(2)xS(2))/xS(3), x), x, -sqrt(S(2))bS(2)Erf(sqrt(S(2))bx) - b*S(2)Integrate(Erf(bx)exp(-b*S(2)x*S(2))/x, x) - Erf(bx)exp(-bS(2)x*S(2))/(S(2)x*S(2)) - bexp(-S(2)bS(2)x*S(2))/(sqrt(Pi)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)exp(-b*S(2)xS(2))/xS(4), x), x, sqrt(Pi)bS(3)Erf(bx)S(2)/S(3) + S(2)b*S(2)Erf(bx)exp(-b*S(2)x*S(2))/(S(3)x) - Erf(bx)exp(-b*S(2)x*S(2))/(S(3)x*S(3)) - S(4)b*S(3)ExpIntegralEi(-S(2)bS(2)x*S(2))/(S(3)sqrt(Pi)) - bexp(-S(2)b*S(2)x*S(2))/(S(3)sqrt(Pi)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)exp(-bS(2)xS(2))/xS(5), x), x, S(7)sqrt(S(2))b*S(4)Erf(sqrt(S(2))bx)/S(6) + b*S(4)Integrate(Erf(bx)exp(-b*S(2)xS(2))/x, x)/S(2) + bS(2)Erf(bx)exp(-bS(2)x*S(2))/(S(4)x*S(2)) - Erf(bx)exp(-bS(2)x*S(2))/(S(4)x*S(4)) + S(7)b*S(3)exp(-S(2)bS(2)x*S(2))/(S(6)sqrt(Pi)x) - bexp(-S(2)bS(2)x*S(2))/(S(6)sqrt(Pi)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erf(bx)exp(-bS(2)xS(2))/xS(6), x), x, -S(2)sqrt(Pi)b*S(5)Erf(bx)S(2)/S(15) - S(4)b*S(4)Erf(bx)exp(-b*S(2)x*S(2))/(S(15)x) + S(2)bS(2)Erf(bx)exp(-b*S(2)x*S(2))/(S(15)x*S(3)) - Erf(bx)exp(-bS(2)x*S(2))/(S(5)x*S(5)) + S(14)b*S(5)ExpIntegralEi(-S(2)bS(2)x*S(2))/(S(15)sqrt(Pi)) + b*S(3)exp(-S(2)bS(2)x*S(2))/(S(3)sqrt(Pi)xS(2)) - bexp(-S(2)bS(2)x*S(2))/(S(10)sqrt(Pi)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(bS(2)Erf(bx)exp(-b*S(2)x*S(2))/x + Erf(bx)exp(-bS(2)xS(2))/xS(3), x), x, -sqrt(S(2))bS(2)Erf(sqrt(S(2))bx) - Erf(bx)exp(-b*S(2)x*S(2))/(S(2)x*S(2)) - bexp(-S(2)bS(2)x*S(2))/(sqrt(Pi)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)S(2)/xS(8), x), x, Integrate(Erfc(bx)S(2)/xS(8), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)S(2)/xS(7), x), x, -S(4)b*S(6)Erfc(bx)S(2)/S(45) - Erfc(bx)*S(2)/(S(6)x*S(6)) + S(28)b*S(6)ExpIntegralEi(-S(2)bS(2)x*S(2))/(S(45)Pi) + S(2)bS(4)exp(-S(2)bS(2)x*S(2))/(S(9)PixS(2)) - bS(2)exp(-S(2)bS(2)x*S(2))/(S(15)PixS(4)) + S(8)b*S(5)Erfc(bx)exp(-b*S(2)x*S(2))/(S(45)sqrt(Pi)x) - S(4)b*S(3)Erfc(bx)exp(-b*S(2)x*S(2))/(S(45)sqrt(Pi)xS(3)) + S(2)bErfc(bx)exp(-bS(2)x*S(2))/(S(15)sqrt(Pi)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)S(2)/xS(6), x), x, Integrate(Erfc(bx)S(2)/xS(6), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)S(2)/xS(5), x), x, b*S(4)Erfc(bx)S(2)/S(3) - Erfc(bx)*S(2)/(S(4)x*S(4)) - S(4)b*S(4)ExpIntegralEi(-S(2)bS(2)x*S(2))/(S(3)Pi) - b*S(2)exp(-S(2)bS(2)x*S(2))/(S(3)PixS(2)) - S(2)b*S(3)Erfc(bx)exp(-b*S(2)x*S(2))/(S(3)sqrt(Pi)x) + bErfc(bx)exp(-b*S(2)x*S(2))/(S(3)sqrt(Pi)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)S(2)/xS(4), x), x, Integrate(Erfc(bx)S(2)/xS(4), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)S(2)/xS(3), x), x, -b*S(2)Erfc(bx)S(2) - Erfc(bx)*S(2)/(S(2)x*S(2)) + S(2)b*S(2)ExpIntegralEi(-S(2)bS(2)x*S(2))/Pi + S(2)bErfc(bx)exp(-bS(2)x*S(2))/(sqrt(Pi)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)S(2)/xS(2), x), x, Integrate(Erfc(bx)S(2)/xS(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)S(2)/x, x), x, Integrate(Erfc(bx)*S(2)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)*S(2), x), x, xErfc(bx)S(2) - sqrt(S(2))sqrt(S(1)/Pi)Erf(sqrt(S(2))bx)/b - S(2)Erfc(bx)exp(-b*S(2)x*S(2))/(sqrt(Pi)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xErfc(bx)S(2), x), x, xS(2)Erfc(bx)*S(2)/S(2) - Erfc(bx)*S(2)/(S(4)b*S(2)) + exp(-S(2)b*S(2)x*S(2))/(S(2)PibS(2)) - xErfc(bx)exp(-b*S(2)x*S(2))/(sqrt(Pi)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)Erfc(bx)S(2), x), x, xS(3)Erfc(bx)S(2)/S(3) + xexp(-S(2)bS(2)x*S(2))/(S(3)PibS(2)) - S(2)x*S(2)Erfc(bx)exp(-b*S(2)x*S(2))/(S(3)sqrt(Pi)b) - S(5)sqrt(S(2))Erf(sqrt(S(2))bx)/(S(12)sqrt(Pi)bS(3)) - S(2)Erfc(bx)exp(-b*S(2)x*S(2))/(S(3)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)Erfc(bx)S(2), x), x, xS(4)Erfc(bx)S(2)/S(4) - S(3)Erfc(bx)S(2)/(S(16)bS(4)) + xS(2)exp(-S(2)b*S(2)x*S(2))/(S(4)PibS(2)) + exp(-S(2)b*S(2)x*S(2))/(S(2)PibS(4)) - xS(3)Erfc(bx)exp(-b*S(2)x*S(2))/(S(2)sqrt(Pi)b) - S(3)xErfc(bx)exp(-bS(2)x*S(2))/(S(4)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)Erfc(bx)S(2), x), x, xS(5)Erfc(bx)S(2)/S(5) + xS(3)exp(-S(2)bS(2)x*S(2))/(S(5)PibS(2)) + S(11)xexp(-S(2)b*S(2)x*S(2))/(S(20)PibS(4)) - S(2)x*S(4)Erfc(bx)exp(-b*S(2)x*S(2))/(S(5)sqrt(Pi)b) - S(4)x*S(2)Erfc(bx)exp(-b*S(2)x*S(2))/(S(5)sqrt(Pi)bS(3)) - S(43)sqrt(S(2))Erf(sqrt(S(2))bx)/(S(80)sqrt(Pi)bS(5)) - S(4)Erfc(bx)exp(-b*S(2)x*S(2))/(S(5)sqrt(Pi)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)Erfc(bx)S(2), x), x, xS(6)Erfc(bx)S(2)/S(6) - S(5)Erfc(bx)S(2)/(S(16)bS(6)) + xS(4)exp(-S(2)b*S(2)x*S(2))/(S(6)PibS(2)) + S(7)x*S(2)exp(-S(2)bS(2)x*S(2))/(S(12)PibS(4)) + S(11)exp(-S(2)bS(2)x*S(2))/(S(12)PibS(6)) - xS(5)Erfc(bx)exp(-b*S(2)x*S(2))/(S(3)sqrt(Pi)b) - S(5)x*S(3)Erfc(bx)exp(-b*S(2)x*S(2))/(S(6)sqrt(Pi)bS(3)) - S(5)xErfc(bx)exp(-bS(2)x*S(2))/(S(4)sqrt(Pi)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(a + bx)/x, x), x, Integrate(Erfc(a + bx)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(a + bx), x), x, (a + bx)Erfc(a + bx)/b - exp(-(a + bx)*S(2))/(sqrt(Pi)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xErfc(a + bx), x), x, a*S(2)Erf(a + bx)/(S(2)bS(2)) + xS(2)Erfc(a + bx)/S(2) + Erf(a + bx)/(S(4)b*S(2)) + aexp(-(a + bx)S(2))/(sqrt(Pi)b*S(2)) - (a + bx)exp(-(a + bx)*S(2))/(S(2)sqrt(Pi)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)Erfc(a + bx), x), x, -aS(3)Erf(a + bx)/(S(3)b*S(3)) - aErf(a + bx)/(S(2)bS(3)) + xS(3)Erfc(a + bx)/S(3) - a*S(2)exp(-(a + bx)S(2))/(sqrt(Pi)b*S(3)) + a(a + bx)exp(-(a + bx)S(2))/(sqrt(Pi)b*S(3)) - (a + bx)*S(2)exp(-(a + bx)S(2))/(S(3)sqrt(Pi)bS(3)) - exp(-(a + bx)*S(2))/(S(3)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)Erfc(a + bx), x), x, aS(4)Erf(a + bx)/(S(4)b*S(4)) + S(3)a*S(2)Erf(a + bx)/(S(4)bS(4)) + xS(4)Erfc(a + bx)/S(4) + S(3)Erf(a + bx)/(S(16)bS(4)) + aS(3)exp(-(a + bx)S(2))/(sqrt(Pi)b*S(4)) - S(3)a*S(2)(a + bx)exp(-(a + bx)S(2))/(S(2)sqrt(Pi)bS(4)) + a(a + bx)S(2)exp(-(a + bx)S(2))/(sqrt(Pi)b*S(4)) + aexp(-(a + bx)S(2))/(sqrt(Pi)b*S(4)) - (a + bx)*S(3)exp(-(a + bx)S(2))/(S(4)sqrt(Pi)bS(4)) - (S(3)a + S(3)bx)exp(-(a + bx)*S(2))/(S(8)sqrt(Pi)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(a + bx)*S(2)/x, x), x, Integrate(Erfc(a + bx)*S(2)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(a + bx)*S(2), x), x, (a + bx)Erfc(a + bx)*S(2)/b - sqrt(S(2))sqrt(S(1)/Pi)Erf(sqrt(S(2))(a + bx))/b - S(2)Erfc(a + bx)exp(-(a + bx)S(2))/(sqrt(Pi)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xErfc(a + bx)*S(2), x), x, -a(a + bx)Erfc(a + bx)S(2)/bS(2) + sqrt(S(2))asqrt(S(1)/Pi)Erf(sqrt(S(2))(a + bx))/b*S(2) + (a + bx)*S(2)Erfc(a + bx)S(2)/(S(2)b*S(2)) - Erfc(a + bx)*S(2)/(S(4)b*S(2)) + exp(-S(2)(a + bx)S(2))/(S(2)PibS(2)) + S(2)aErfc(a + bx)exp(-(a + bx)*S(2))/(sqrt(Pi)b*S(2)) - (a + bx)Erfc(a + bx)exp(-(a + bx)*S(2))/(sqrt(Pi)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)Erfc(a + bx)S(2), x), x, aS(2)(a + bx)Erfc(a + bx)S(2)/bS(3) - sqrt(S(2))aS(2)sqrt(S(1)/Pi)Erf(sqrt(S(2))(a + bx))/bS(3) - a(a + bx)S(2)Erfc(a + bx)S(2)/bS(3) + aErfc(a + bx)S(2)/(S(2)b*S(3)) + (a + bx)*S(3)Erfc(a + bx)S(2)/(S(3)b*S(3)) - aexp(-S(2)(a + bx)*S(2))/(Pib*S(3)) + (a + bx)exp(-S(2)(a + bx)S(2))/(S(3)PibS(3)) - S(2)a*S(2)Erfc(a + bx)exp(-(a + bx)S(2))/(sqrt(Pi)b*S(3)) + S(2)a(a + bx)Erfc(a + bx)exp(-(a + bx)*S(2))/(sqrt(Pi)b*S(3)) - S(2)(a + bx)S(2)Erfc(a + bx)exp(-(a + bx)S(2))/(S(3)sqrt(Pi)bS(3)) - S(5)sqrt(S(2))Erf(sqrt(S(2))(a + bx))/(S(12)sqrt(Pi)bS(3)) - S(2)Erfc(a + bx)exp(-(a + bx)S(2))/(S(3)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)exp(-bS(2)xS(2))/xS(8), x), x, -S(4)sqrt(Pi)b*S(7)Erfc(bx)S(2)/S(105) + S(8)b*S(6)Erfc(bx)exp(-b*S(2)x*S(2))/(S(105)x) - S(4)bS(4)Erfc(bx)exp(-b*S(2)x*S(2))/(S(105)x*S(3)) + S(2)b*S(2)Erfc(bx)exp(-b*S(2)x*S(2))/(S(35)x*S(5)) - Erfc(bx)exp(-bS(2)x*S(2))/(S(7)x*S(7)) + S(16)b*S(7)ExpIntegralEi(-S(2)bS(2)x*S(2))/(S(35)sqrt(Pi)) + S(4)bS(5)exp(-S(2)bS(2)x*S(2))/(S(21)sqrt(Pi)xS(2)) - S(8)b*S(3)exp(-S(2)bS(2)x*S(2))/(S(105)sqrt(Pi)xS(4)) + bexp(-S(2)bS(2)x*S(2))/(S(21)sqrt(Pi)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)exp(-bS(2)xS(2))/xS(7), x), x, S(67)sqrt(S(2))b*S(6)Erf(sqrt(S(2))bx)/S(90) - b*S(6)Integrate(Erfc(bx)exp(-b*S(2)xS(2))/x, x)/S(6) - bS(4)Erfc(bx)exp(-bS(2)x*S(2))/(S(12)xS(2)) + bS(2)Erfc(bx)exp(-bS(2)x*S(2))/(S(12)x*S(4)) - Erfc(bx)exp(-bS(2)x*S(2))/(S(6)x*S(6)) + S(67)b*S(5)exp(-S(2)bS(2)x*S(2))/(S(90)sqrt(Pi)x) - S(13)b*S(3)exp(-S(2)bS(2)x*S(2))/(S(90)sqrt(Pi)xS(3)) + bexp(-S(2)bS(2)x*S(2))/(S(15)sqrt(Pi)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)exp(-bS(2)xS(2))/xS(6), x), x, S(2)sqrt(Pi)b*S(5)Erfc(bx)S(2)/S(15) - S(4)b*S(4)Erfc(bx)exp(-b*S(2)x*S(2))/(S(15)x) + S(2)bS(2)Erfc(bx)exp(-b*S(2)x*S(2))/(S(15)x*S(3)) - Erfc(bx)exp(-bS(2)x*S(2))/(S(5)x*S(5)) - S(14)b*S(5)ExpIntegralEi(-S(2)bS(2)x*S(2))/(S(15)sqrt(Pi)) - b*S(3)exp(-S(2)bS(2)x*S(2))/(S(3)sqrt(Pi)xS(2)) + bexp(-S(2)bS(2)x*S(2))/(S(10)sqrt(Pi)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)exp(-bS(2)xS(2))/xS(5), x), x, -S(7)sqrt(S(2))b*S(4)Erf(sqrt(S(2))bx)/S(6) + b*S(4)Integrate(Erfc(bx)exp(-b*S(2)xS(2))/x, x)/S(2) + bS(2)Erfc(bx)exp(-bS(2)x*S(2))/(S(4)x*S(2)) - Erfc(bx)exp(-bS(2)x*S(2))/(S(4)x*S(4)) - S(7)b*S(3)exp(-S(2)bS(2)x*S(2))/(S(6)sqrt(Pi)x) + bexp(-S(2)bS(2)x*S(2))/(S(6)sqrt(Pi)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)exp(-bS(2)xS(2))/xS(4), x), x, -sqrt(Pi)bS(3)Erfc(bx)S(2)/S(3) + S(2)b*S(2)Erfc(bx)exp(-b*S(2)x*S(2))/(S(3)x) - Erfc(bx)exp(-b*S(2)x*S(2))/(S(3)x*S(3)) + S(4)b*S(3)ExpIntegralEi(-S(2)bS(2)x*S(2))/(S(3)sqrt(Pi)) + bexp(-S(2)b*S(2)x*S(2))/(S(3)sqrt(Pi)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)exp(-bS(2)xS(2))/xS(3), x), x, sqrt(S(2))bS(2)Erf(sqrt(S(2))bx) - b*S(2)Integrate(Erfc(bx)exp(-b*S(2)x*S(2))/x, x) - Erfc(bx)exp(-bS(2)x*S(2))/(S(2)x*S(2)) + bexp(-S(2)bS(2)x*S(2))/(sqrt(Pi)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)exp(-b*S(2)xS(2))/xS(2), x), x, sqrt(Pi)bErfc(bx)S(2)/S(2) - Erfc(bx)exp(-bS(2)x*S(2))/x - bExpIntegralEi(-S(2)bS(2)x*S(2))/sqrt(Pi), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)exp(-bS(2)x*S(2))/x, x), x, Integrate(Erfc(bx)exp(-bS(2)x*S(2))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfc(bx)exp(-bS(2)x*S(2)), x), x, -sqrt(Pi)Erfc(bx)S(2)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xErfc(bx)exp(-bS(2)x*S(2)), x), x, -sqrt(S(2))Erf(sqrt(S(2))bx)/(S(4)bS(2)) - Erfc(bx)exp(-bS(2)x*S(2))/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)Erfc(bx)exp(-bS(2)x*S(2)), x), x, -sqrt(Pi)Erfc(bx)S(2)/(S(8)b*S(3)) - xErfc(bx)exp(-b*S(2)x*S(2))/(S(2)b*S(2)) + exp(-S(2)b*S(2)x*S(2))/(S(4)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)Erfc(bx)exp(-b*S(2)xS(2)), x), x, -xS(2)Erfc(bx)exp(-bS(2)x*S(2))/(S(2)b*S(2)) - S(5)sqrt(S(2))Erf(sqrt(S(2))bx)/(S(16)b*S(4)) - Erfc(bx)exp(-bS(2)x*S(2))/(S(2)b*S(4)) + xexp(-S(2)bS(2)x*S(2))/(S(4)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)Erfc(bx)exp(-b*S(2)x*S(2)), x), x, -S(3)sqrt(Pi)Erfc(bx)*S(2)/(S(16)bS(5)) - xS(3)Erfc(bx)exp(-bS(2)x*S(2))/(S(2)b*S(2)) - S(3)xErfc(bx)exp(-bS(2)x*S(2))/(S(4)bS(4)) + xS(2)exp(-S(2)b*S(2)x*S(2))/(S(4)sqrt(Pi)bS(3)) + exp(-S(2)b*S(2)x*S(2))/(S(2)sqrt(Pi)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)Erfc(bx)exp(-b*S(2)xS(2)), x), x, -xS(4)Erfc(bx)exp(-bS(2)x*S(2))/(S(2)bS(2)) - xS(2)Erfc(bx)exp(-bS(2)xS(2))/bS(4) - S(43)sqrt(S(2))Erf(sqrt(S(2))bx)/(S(64)bS(6)) - Erfc(bx)exp(-bS(2)xS(2))/bS(6) + x*S(3)exp(-S(2)bS(2)x*S(2))/(S(4)sqrt(Pi)bS(3)) + S(11)xexp(-S(2)b*S(2)x*S(2))/(S(16)sqrt(Pi)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)Erfc(bx)exp(-b*S(2)x*S(2)), x), x, -S(15)sqrt(Pi)Erfc(bx)*S(2)/(S(32)bS(7)) - xS(5)Erfc(bx)exp(-bS(2)x*S(2))/(S(2)b*S(2)) - S(5)x*S(3)Erfc(bx)exp(-b*S(2)x*S(2))/(S(4)b*S(4)) - S(15)xErfc(bx)exp(-bS(2)x*S(2))/(S(8)bS(6)) + xS(4)exp(-S(2)b*S(2)x*S(2))/(S(4)sqrt(Pi)bS(3)) + S(7)x*S(2)exp(-S(2)bS(2)x*S(2))/(S(8)sqrt(Pi)bS(5)) + S(11)exp(-S(2)bS(2)x*S(2))/(S(8)sqrt(Pi)bS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)S(2)/xS(8), x), x, Integrate(Erfi(bx)S(2)/xS(8), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)S(2)/xS(7), x), x, S(4)bS(6)Erfi(bx)S(2)/S(45) - Erfi(bx)*S(2)/(S(6)x*S(6)) + S(28)b*S(6)ExpIntegralEi(S(2)bS(2)x*S(2))/(S(45)Pi) - S(2)bS(4)exp(S(2)bS(2)x*S(2))/(S(9)PixS(2)) - bS(2)exp(S(2)bS(2)x*S(2))/(S(15)PixS(4)) - S(8)b*S(5)Erfi(bx)exp(b*S(2)x*S(2))/(S(45)sqrt(Pi)x) - S(4)b*S(3)Erfi(bx)exp(b*S(2)x*S(2))/(S(45)sqrt(Pi)xS(3)) - S(2)bErfi(bx)exp(bS(2)x*S(2))/(S(15)sqrt(Pi)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)S(2)/xS(6), x), x, Integrate(Erfi(bx)S(2)/xS(6), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)S(2)/xS(5), x), x, b*S(4)Erfi(bx)S(2)/S(3) - Erfi(bx)*S(2)/(S(4)x*S(4)) + S(4)b*S(4)ExpIntegralEi(S(2)bS(2)x*S(2))/(S(3)Pi) - b*S(2)exp(S(2)bS(2)x*S(2))/(S(3)PixS(2)) - S(2)b*S(3)Erfi(bx)exp(b*S(2)x*S(2))/(S(3)sqrt(Pi)x) - bErfi(bx)exp(b*S(2)x*S(2))/(S(3)sqrt(Pi)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)S(2)/xS(4), x), x, Integrate(Erfi(bx)S(2)/xS(4), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)S(2)/xS(3), x), x, b*S(2)Erfi(bx)S(2) - Erfi(bx)*S(2)/(S(2)x*S(2)) + S(2)b*S(2)ExpIntegralEi(S(2)bS(2)x*S(2))/Pi - S(2)bErfi(bx)exp(bS(2)x*S(2))/(sqrt(Pi)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)S(2)/xS(2), x), x, Integrate(Erfi(bx)S(2)/xS(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)S(2)/x, x), x, Integrate(Erfi(bx)*S(2)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)*S(2), x), x, xErfi(bx)S(2) + sqrt(S(2))sqrt(S(1)/Pi)Erfi(sqrt(S(2))bx)/b - S(2)Erfi(bx)exp(b*S(2)x*S(2))/(sqrt(Pi)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xErfi(bx)S(2), x), x, xS(2)Erfi(bx)*S(2)/S(2) + Erfi(bx)*S(2)/(S(4)b*S(2)) + exp(S(2)b*S(2)x*S(2))/(S(2)PibS(2)) - xErfi(bx)exp(b*S(2)x*S(2))/(sqrt(Pi)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)Erfi(bx)S(2), x), x, xS(3)Erfi(bx)S(2)/S(3) + xexp(S(2)bS(2)x*S(2))/(S(3)PibS(2)) - S(2)x*S(2)Erfi(bx)exp(b*S(2)x*S(2))/(S(3)sqrt(Pi)b) + S(2)Erfi(bx)exp(b*S(2)x*S(2))/(S(3)sqrt(Pi)bS(3)) - S(5)sqrt(S(2))Erfi(sqrt(S(2))bx)/(S(12)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)Erfi(bx)S(2), x), x, xS(4)Erfi(bx)S(2)/S(4) - S(3)Erfi(bx)S(2)/(S(16)bS(4)) + xS(2)exp(S(2)b*S(2)x*S(2))/(S(4)PibS(2)) - exp(S(2)b*S(2)x*S(2))/(S(2)PibS(4)) - xS(3)Erfi(bx)exp(b*S(2)x*S(2))/(S(2)sqrt(Pi)b) + S(3)xErfi(bx)exp(bS(2)x*S(2))/(S(4)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)Erfi(bx)S(2), x), x, xS(5)Erfi(bx)S(2)/S(5) + xS(3)exp(S(2)bS(2)x*S(2))/(S(5)PibS(2)) - S(11)xexp(S(2)b*S(2)x*S(2))/(S(20)PibS(4)) - S(2)x*S(4)Erfi(bx)exp(b*S(2)x*S(2))/(S(5)sqrt(Pi)b) + S(4)x*S(2)Erfi(bx)exp(b*S(2)x*S(2))/(S(5)sqrt(Pi)bS(3)) - S(4)Erfi(bx)exp(b*S(2)x*S(2))/(S(5)sqrt(Pi)bS(5)) + S(43)sqrt(S(2))Erfi(sqrt(S(2))bx)/(S(80)sqrt(Pi)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)Erfi(bx)S(2), x), x, xS(6)Erfi(bx)S(2)/S(6) + S(5)Erfi(bx)S(2)/(S(16)bS(6)) + xS(4)exp(S(2)b*S(2)x*S(2))/(S(6)PibS(2)) - S(7)x*S(2)exp(S(2)bS(2)x*S(2))/(S(12)PibS(4)) + S(11)exp(S(2)bS(2)x*S(2))/(S(12)PibS(6)) - xS(5)Erfi(bx)exp(b*S(2)x*S(2))/(S(3)sqrt(Pi)b) + S(5)x*S(3)Erfi(bx)exp(b*S(2)x*S(2))/(S(6)sqrt(Pi)bS(3)) - S(5)xErfi(bx)exp(bS(2)x*S(2))/(S(4)sqrt(Pi)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(a + bx)/x, x), x, Integrate(Erfi(a + bx)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(a + bx), x), x, (a + bx)Erfi(a + bx)/b - exp((a + bx)*S(2))/(sqrt(Pi)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xErfi(a + bx), x), x, -a*S(2)Erfi(a + bx)/(S(2)bS(2)) + xS(2)Erfi(a + bx)/S(2) + Erfi(a + bx)/(S(4)b*S(2)) + aexp((a + bx)S(2))/(sqrt(Pi)b*S(2)) - (a + bx)exp((a + bx)*S(2))/(S(2)sqrt(Pi)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)Erfi(a + bx), x), x, aS(3)Erfi(a + bx)/(S(3)b*S(3)) - aErfi(a + bx)/(S(2)bS(3)) + xS(3)Erfi(a + bx)/S(3) - a*S(2)exp((a + bx)S(2))/(sqrt(Pi)b*S(3)) + a(a + bx)exp((a + bx)S(2))/(sqrt(Pi)b*S(3)) - (a + bx)*S(2)exp((a + bx)S(2))/(S(3)sqrt(Pi)bS(3)) + exp((a + bx)*S(2))/(S(3)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)Erfi(a + bx), x), x, -aS(4)Erfi(a + bx)/(S(4)b*S(4)) + S(3)a*S(2)Erfi(a + bx)/(S(4)bS(4)) + xS(4)Erfi(a + bx)/S(4) - S(3)Erfi(a + bx)/(S(16)bS(4)) + aS(3)exp((a + bx)S(2))/(sqrt(Pi)b*S(4)) - S(3)a*S(2)(a + bx)exp((a + bx)S(2))/(S(2)sqrt(Pi)bS(4)) + a(a + bx)S(2)exp((a + bx)S(2))/(sqrt(Pi)b*S(4)) - aexp((a + bx)S(2))/(sqrt(Pi)b*S(4)) - (a + bx)*S(3)exp((a + bx)S(2))/(S(4)sqrt(Pi)bS(4)) + S(3)(a + bx)exp((a + bx)S(2))/(S(8)sqrt(Pi)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(a + bx)*S(2)/x, x), x, Integrate(Erfi(a + bx)*S(2)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(a + bx)*S(2), x), x, (a + bx)Erfi(a + bx)*S(2)/b + sqrt(S(2))sqrt(S(1)/Pi)Erfi(sqrt(S(2))(a + bx))/b - S(2)Erfi(a + bx)exp((a + bx)S(2))/(sqrt(Pi)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xErfi(a + bx)*S(2), x), x, -a(a + bx)Erfi(a + bx)S(2)/bS(2) - sqrt(S(2))asqrt(S(1)/Pi)Erfi(sqrt(S(2))(a + bx))/b*S(2) + (a + bx)*S(2)Erfi(a + bx)S(2)/(S(2)b*S(2)) + Erfi(a + bx)*S(2)/(S(4)b*S(2)) + exp(S(2)(a + bx)S(2))/(S(2)PibS(2)) + S(2)aErfi(a + bx)exp((a + bx)*S(2))/(sqrt(Pi)b*S(2)) - (a + bx)Erfi(a + bx)exp((a + bx)*S(2))/(sqrt(Pi)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)Erfi(a + bx)S(2), x), x, aS(2)(a + bx)Erfi(a + bx)S(2)/bS(3) + sqrt(S(2))aS(2)sqrt(S(1)/Pi)Erfi(sqrt(S(2))(a + bx))/bS(3) - a(a + bx)S(2)Erfi(a + bx)S(2)/bS(3) - aErfi(a + bx)S(2)/(S(2)b*S(3)) + (a + bx)*S(3)Erfi(a + bx)S(2)/(S(3)b*S(3)) - aexp(S(2)(a + bx)*S(2))/(Pib*S(3)) + (a + bx)exp(S(2)(a + bx)S(2))/(S(3)PibS(3)) - S(2)a*S(2)Erfi(a + bx)exp((a + bx)S(2))/(sqrt(Pi)b*S(3)) + S(2)a(a + bx)Erfi(a + bx)exp((a + bx)*S(2))/(sqrt(Pi)b*S(3)) - S(2)(a + bx)S(2)Erfi(a + bx)exp((a + bx)S(2))/(S(3)sqrt(Pi)bS(3)) - S(5)sqrt(S(2))Erfi(sqrt(S(2))(a + bx))/(S(12)sqrt(Pi)bS(3)) + S(2)Erfi(a + bx)exp((a + bx)S(2))/(S(3)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)exp(bS(2)xS(2))/xS(8), x), x, S(4)sqrt(Pi)b*S(7)Erfi(bx)S(2)/S(105) - S(8)b*S(6)Erfi(bx)exp(b*S(2)x*S(2))/(S(105)x) - S(4)bS(4)Erfi(bx)exp(b*S(2)x*S(2))/(S(105)x*S(3)) - S(2)b*S(2)Erfi(bx)exp(b*S(2)x*S(2))/(S(35)x*S(5)) - Erfi(bx)exp(bS(2)x*S(2))/(S(7)x*S(7)) + S(16)b*S(7)ExpIntegralEi(S(2)bS(2)x*S(2))/(S(35)sqrt(Pi)) - S(4)bS(5)exp(S(2)bS(2)x*S(2))/(S(21)sqrt(Pi)xS(2)) - S(8)b*S(3)exp(S(2)bS(2)x*S(2))/(S(105)sqrt(Pi)xS(4)) - bexp(S(2)bS(2)x*S(2))/(S(21)sqrt(Pi)xS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)exp(bS(2)xS(2))/xS(7), x), x, S(67)sqrt(S(2))b*S(6)Erfi(sqrt(S(2))bx)/S(90) + b*S(6)Integrate(Erfi(bx)exp(b*S(2)xS(2))/x, x)/S(6) - bS(4)Erfi(bx)exp(bS(2)x*S(2))/(S(12)xS(2)) - bS(2)Erfi(bx)exp(bS(2)x*S(2))/(S(12)x*S(4)) - Erfi(bx)exp(bS(2)x*S(2))/(S(6)x*S(6)) - S(67)b*S(5)exp(S(2)bS(2)x*S(2))/(S(90)sqrt(Pi)x) - S(13)b*S(3)exp(S(2)bS(2)x*S(2))/(S(90)sqrt(Pi)xS(3)) - bexp(S(2)bS(2)x*S(2))/(S(15)sqrt(Pi)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)exp(bS(2)xS(2))/xS(6), x), x, S(2)sqrt(Pi)b*S(5)Erfi(bx)S(2)/S(15) - S(4)b*S(4)Erfi(bx)exp(b*S(2)x*S(2))/(S(15)x) - S(2)bS(2)Erfi(bx)exp(b*S(2)x*S(2))/(S(15)x*S(3)) - Erfi(bx)exp(bS(2)x*S(2))/(S(5)x*S(5)) + S(14)b*S(5)ExpIntegralEi(S(2)bS(2)x*S(2))/(S(15)sqrt(Pi)) - b*S(3)exp(S(2)bS(2)x*S(2))/(S(3)sqrt(Pi)xS(2)) - bexp(S(2)bS(2)x*S(2))/(S(10)sqrt(Pi)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)exp(bS(2)xS(2))/xS(5), x), x, S(7)sqrt(S(2))b*S(4)Erfi(sqrt(S(2))bx)/S(6) + b*S(4)Integrate(Erfi(bx)exp(b*S(2)xS(2))/x, x)/S(2) - bS(2)Erfi(bx)exp(bS(2)x*S(2))/(S(4)x*S(2)) - Erfi(bx)exp(bS(2)x*S(2))/(S(4)x*S(4)) - S(7)b*S(3)exp(S(2)bS(2)x*S(2))/(S(6)sqrt(Pi)x) - bexp(S(2)bS(2)x*S(2))/(S(6)sqrt(Pi)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)exp(bS(2)xS(2))/xS(4), x), x, sqrt(Pi)bS(3)Erfi(bx)S(2)/S(3) - S(2)b*S(2)Erfi(bx)exp(b*S(2)x*S(2))/(S(3)x) - Erfi(bx)exp(b*S(2)x*S(2))/(S(3)x*S(3)) + S(4)b*S(3)ExpIntegralEi(S(2)bS(2)x*S(2))/(S(3)sqrt(Pi)) - bexp(S(2)b*S(2)x*S(2))/(S(3)sqrt(Pi)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)exp(bS(2)xS(2))/xS(3), x), x, sqrt(S(2))bS(2)Erfi(sqrt(S(2))bx) + b*S(2)Integrate(Erfi(bx)exp(b*S(2)x*S(2))/x, x) - Erfi(bx)exp(bS(2)x*S(2))/(S(2)x*S(2)) - bexp(S(2)bS(2)x*S(2))/(sqrt(Pi)x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)exp(b*S(2)xS(2))/xS(2), x), x, sqrt(Pi)bErfi(bx)S(2)/S(2) - Erfi(bx)exp(bS(2)x*S(2))/x + bExpIntegralEi(S(2)bS(2)x*S(2))/sqrt(Pi), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)exp(bS(2)x*S(2))/x, x), x, Integrate(Erfi(bx)exp(bS(2)x*S(2))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(Erfi(bx)exp(bS(2)x*S(2)), x), x, sqrt(Pi)Erfi(bx)S(2)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xErfi(bx)exp(bS(2)x*S(2)), x), x, Erfi(bx)exp(bS(2)x*S(2))/(S(2)b*S(2)) - sqrt(S(2))Erfi(sqrt(S(2))bx)/(S(4)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)Erfi(bx)exp(b*S(2)x*S(2)), x), x, -sqrt(Pi)Erfi(bx)S(2)/(S(8)b*S(3)) + xErfi(bx)exp(b*S(2)x*S(2))/(S(2)b*S(2)) - exp(S(2)b*S(2)x*S(2))/(S(4)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)Erfi(bx)exp(b*S(2)xS(2)), x), x, xS(2)Erfi(bx)exp(bS(2)x*S(2))/(S(2)b*S(2)) - Erfi(bx)exp(bS(2)x*S(2))/(S(2)b*S(4)) + S(5)sqrt(S(2))Erfi(sqrt(S(2))bx)/(S(16)b*S(4)) - xexp(S(2)bS(2)x*S(2))/(S(4)sqrt(Pi)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)Erfi(bx)exp(b*S(2)x*S(2)), x), x, S(3)sqrt(Pi)Erfi(bx)*S(2)/(S(16)bS(5)) + xS(3)Erfi(bx)exp(bS(2)x*S(2))/(S(2)b*S(2)) - S(3)xErfi(bx)exp(bS(2)x*S(2))/(S(4)bS(4)) - xS(2)exp(S(2)b*S(2)x*S(2))/(S(4)sqrt(Pi)bS(3)) + exp(S(2)b*S(2)x*S(2))/(S(2)sqrt(Pi)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)Erfi(bx)exp(b*S(2)xS(2)), x), x, xS(4)Erfi(bx)exp(bS(2)x*S(2))/(S(2)bS(2)) - xS(2)Erfi(bx)exp(bS(2)xS(2))/bS(4) + Erfi(bx)exp(b*S(2)xS(2))/bS(6) - S(43)sqrt(S(2))Erfi(sqrt(S(2))bx)/(S(64)bS(6)) - xS(3)exp(S(2)bS(2)x*S(2))/(S(4)sqrt(Pi)bS(3)) + S(11)xexp(S(2)b*S(2)x*S(2))/(S(16)sqrt(Pi)bS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)Erfi(bx)exp(b*S(2)x*S(2)), x), x, -S(15)sqrt(Pi)Erfi(bx)*S(2)/(S(32)bS(7)) + xS(5)Erfi(bx)exp(bS(2)x*S(2))/(S(2)b*S(2)) - S(5)x*S(3)Erfi(bx)exp(b*S(2)x*S(2))/(S(4)b*S(4)) + S(15)xErfi(bx)exp(bS(2)x*S(2))/(S(8)bS(6)) - xS(4)exp(S(2)b*S(2)x*S(2))/(S(4)sqrt(Pi)bS(3)) + S(7)x*S(2)exp(S(2)bS(2)x*S(2))/(S(8)sqrt(Pi)bS(5)) - S(11)exp(S(2)bS(2)x*S(2))/(S(8)sqrt(Pi)b*S(7)), expand=True, _diff=True, _numerical=True)
da062ff073a690245ab07b0a25f3be83b94869f9b46f98ebb127cccc45fb8542	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.utility_function import ( sympy_op_factory, Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ, ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ, PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ, NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest, SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient, Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan, ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec, ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less, Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ, InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ, EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, ExpandLinearProduct, GCD, ContentFactor, NumericFactor, NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, ExpandExpression, Apart, SmartApart, MatchQ, PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, RationalFunctionFactors, NonrationalFunctionFactors, Reverse, RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, GeneralizedBinomialParts, GeneralizedTrinomialDegree, GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, MinimumMonomialExponent, MonomialExponent, LinearMatchQ, PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, AlgebraicFunctionFactors, NonalgebraicFunctionFactors, QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, CombineExponents, FactorInteger, FactorAbsurdNumber, SubstForInverseFunction, SubstForFractionalPower, SubstForFractionalPowerOfQuotientOfLinears, FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, NormalizeIntegrandAux, NormalizeIntegrandFactor, NormalizeIntegrandFactorBase, NormalizeTogether, NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand, SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, KnownTangentIntegrandQ, KnownCotangentIntegrandQ, KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, FunctionOfLog, PowerVariableExpn, PowerVariableDegree, PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, ConstantFactor, SameQ, ReplacePart, CommonFactors, MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, _SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify, _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor, PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral, CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial, Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ ) from sympy.core.add import Add from sympy.core.mod import Mod from sympy.core.mul import Mul from sympy.core.numbers import (Float, I, Integer) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.functions.elementary.complexes import Abs from sympy.functions.elementary.miscellaneous import sqrt from sympy.integrals.integrals import Integral as Integrate from sympy.logic.boolalg import (And, Or) from sympy.simplify.simplify import simplify from sympy.integrals.rubi.symbol import WC from sympy.core.symbol import symbols, Symbol from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf, exp, log) from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch) from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec) from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.core.numbers import pi as Pi a, b, c, d, e, f, m, n, x, u , k, p, r, s, t, i, j= symbols('a b c d e f m n x u k p r s t i j') A, B, C, D, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B C D a b c d e f g h y z m n p q u v w F', ) def test_1(): assert rubi_test(rubi_integrate(x*S(4)asin(ax), x), x, xS(5)asin(ax)/S(5) + (-aS(2)xS(2) + S(1))(S(5)/2)/(S(25)aS(5)) - S(2)(-a*S(2)xS(2) + S(1))(S(3)/2)/(S(15)aS(5)) + sqrt(-aS(2)x*S(2) + S(1))/(S(5)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)asin(ax), x), x, x*S(4)asin(ax)/S(4) + xS(3)sqrt(-a*S(2)x*S(2) + S(1))/(S(16)a) + S(3)xsqrt(-a*S(2)x*S(2) + S(1))/(S(32)a*S(3)) - S(3)asin(ax)/(S(32)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asin(ax), x), x, x*S(3)asin(ax)/S(3) - (-aS(2)xS(2) + S(1))(S(3)/2)/(S(9)aS(3)) + sqrt(-aS(2)x*S(2) + S(1))/(S(3)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasin(ax), x), x, xS(2)asin(ax)/S(2) + xsqrt(-a*S(2)x*S(2) + S(1))/(S(4)a) - asin(ax)/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax), x), x, xasin(ax) + sqrt(-a*S(2)x*S(2) + S(1))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)/x, x), x, -IPolyLog(S(2), exp(S(2)Iasin(ax)))/S(2) + log(-exp(S(2)Iasin(ax)) + S(1))asin(ax) - Iasin(ax)S(2)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)/x*S(2), x), x, -aatanh(sqrt(-a*S(2)x*S(2) + S(1))) - asin(ax)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)/xS(3), x), x, -asqrt(-a*S(2)x*S(2) + S(1))/(S(2)x) - asin(ax)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)/xS(4), x), x, -aS(3)atanh(sqrt(-aS(2)x*S(2) + S(1)))/S(6) - asqrt(-a*S(2)x*S(2) + S(1))/(S(6)x*S(2)) - asin(ax)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)/xS(5), x), x, -aS(3)sqrt(-aS(2)x*S(2) + S(1))/(S(6)x) - asqrt(-aS(2)x*S(2) + S(1))/(S(12)x*S(3)) - asin(ax)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)/x*S(6), x), x, -S(3)a*S(5)atanh(sqrt(-a*S(2)x*S(2) + S(1)))/S(40) - S(3)a*S(3)sqrt(-a*S(2)x*S(2) + S(1))/(S(40)x*S(2)) - asqrt(-a*S(2)x*S(2) + S(1))/(S(20)x*S(4)) - asin(ax)/(S(5)xS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)asin(ax)S(2), x), x, xS(5)asin(ax)S(2)/S(5) - S(2)x*S(5)/S(125) + S(2)x*S(4)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(25)a) - S(8)xS(3)/(S(225)a*S(2)) + S(8)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(75)a*S(3)) - S(16)x/(S(75)aS(4)) + S(16)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(75)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)asin(ax)S(2), x), x, xS(4)asin(ax)S(2)/S(4) - xS(4)/S(32) + x*S(3)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(8)a) - S(3)xS(2)/(S(32)a*S(2)) + S(3)xsqrt(-aS(2)x*S(2) + S(1))asin(ax)/(S(16)a*S(3)) - S(3)asin(ax)S(2)/(S(32)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asin(ax)S(2), x), x, xS(3)asin(ax)*S(2)/S(3) - S(2)x*S(3)/S(27) + S(2)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(9)a) - S(4)x/(S(9)a*S(2)) + S(4)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(9)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasin(ax)S(2), x), x, xS(2)asin(ax)S(2)/S(2) - xS(2)/S(4) + xsqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(2)a) - asin(ax)S(2)/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*S(2), x), x, xasin(ax)S(2) - S(2)x + S(2)sqrt(-aS(2)x*S(2) + S(1))asin(ax)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*S(2)/x, x), x, -IPolyLog(S(2), exp(S(2)Iasin(ax)))asin(ax) + PolyLog(S(3), exp(S(2)Iasin(ax)))/S(2) + log(-exp(S(2)Iasin(ax)) + S(1))asin(ax)S(2) - Iasin(ax)S(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)S(2)/xS(2), x), x, S(2)IaPolyLog(S(2), -exp(Iasin(ax))) - S(2)IaPolyLog(S(2), exp(Iasin(ax))) - S(4)aasin(ax)atanh(exp(Iasin(ax))) - asin(ax)S(2)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)S(2)/xS(3), x), x, a*S(2)log(x) - asqrt(-aS(2)x*S(2) + S(1))asin(ax)/x - asin(ax)*S(2)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) # sympy and mathematica assert rubi_test(rubi_integrate(asin(ax)S(2)/xS(4), x), x, IaS(3)PolyLog(S(2), -exp(Iasin(ax)))/S(3) - IaS(3)PolyLog(S(2), exp(Iasin(ax)))/S(3) - S(2)aS(3)asin(ax)atanh(exp(Iasin(ax)))/S(3) - a*S(2)/(S(3)x) - asqrt(-aS(2)x*S(2) + S(1))asin(ax)/(S(3)x*S(2)) - asin(ax)*S(2)/(S(3)x*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)S(2)/xS(5), x), x, a*S(4)log(x)/S(3) - a*S(3)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(3)x) - a*S(2)/(S(12)x*S(2)) - asqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(6)x*S(3)) - asin(ax)*S(2)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)asin(ax)S(3), x), x, xS(5)asin(ax)*S(3)/S(5) - S(6)x*S(5)asin(ax)/S(125) + S(3)x*S(4)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(2)/(S(25)a) - S(8)xS(3)asin(ax)/(S(75)a*S(2)) + S(4)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(2)/(S(25)a*S(3)) - S(16)xasin(ax)/(S(25)aS(4)) - S(6)(-a*S(2)xS(2) + S(1))(S(5)/2)/(S(625)aS(5)) + S(76)(-a*S(2)xS(2) + S(1))(S(3)/2)/(S(1125)aS(5)) + S(8)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(2)/(S(25)a*S(5)) - S(298)sqrt(-a*S(2)x*S(2) + S(1))/(S(375)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)asin(ax)S(3), x), x, xS(4)asin(ax)*S(3)/S(4) - S(3)x*S(4)asin(ax)/S(32) + S(3)x*S(3)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(2)/(S(16)a) - S(3)xS(3)sqrt(-a*S(2)x*S(2) + S(1))/(S(128)a) - S(9)xS(2)asin(ax)/(S(32)a*S(2)) + S(9)xsqrt(-aS(2)x*S(2) + S(1))asin(ax)S(2)/(S(32)a*S(3)) - S(45)xsqrt(-aS(2)x*S(2) + S(1))/(S(256)a*S(3)) - S(3)asin(ax)S(3)/(S(32)a*S(4)) + S(45)asin(ax)/(S(256)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asin(ax)S(3), x), x, xS(3)asin(ax)*S(3)/S(3) - S(2)x*S(3)asin(ax)/S(9) + xS(2)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(2)/(S(3)a) - S(4)xasin(ax)/(S(3)a*S(2)) + S(2)(-a*S(2)xS(2) + S(1))(S(3)/2)/(S(27)aS(3)) + S(2)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(2)/(S(3)a*S(3)) - S(14)sqrt(-a*S(2)x*S(2) + S(1))/(S(9)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasin(ax)S(3), x), x, xS(2)asin(ax)S(3)/S(2) - S(3)x*S(2)asin(ax)/S(4) + S(3)xsqrt(-aS(2)x*S(2) + S(1))asin(ax)S(2)/(S(4)a) - S(3)xsqrt(-a*S(2)x*S(2) + S(1))/(S(8)a) - asin(ax)S(3)/(S(4)a*S(2)) + S(3)asin(ax)/(S(8)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*S(3), x), x, xasin(ax)S(3) - S(6)xasin(ax) + S(3)sqrt(-aS(2)x*S(2) + S(1))asin(ax)S(2)/a - S(6)sqrt(-a*S(2)x*S(2) + S(1))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*S(3)/x, x), x, -S(3)IPolyLog(S(2), exp(S(2)Iasin(ax)))asin(ax)*S(2)/S(2) + S(3)PolyLog(S(3), exp(S(2)Iasin(ax)))asin(ax)/S(2) + S(3)IPolyLog(S(4), exp(S(2)Iasin(ax)))/S(4) + log(-exp(S(2)Iasin(ax)) + S(1))asin(ax)S(3) - Iasin(ax)S(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)S(3)/xS(2), x), x, S(6)IaPolyLog(S(2), -exp(Iasin(ax)))asin(ax) - S(6)IaPolyLog(S(2), exp(Iasin(ax)))asin(ax) - S(6)aPolyLog(S(3), -exp(Iasin(ax))) + S(6)aPolyLog(S(3), exp(Iasin(ax))) - S(6)aasin(ax)S(2)atanh(exp(Iasin(ax))) - asin(ax)S(3)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)S(3)/xS(3), x), x, -S(3)Ia*S(2)PolyLog(S(2), exp(S(2)Iasin(ax)))/S(2) + S(3)a*S(2)log(-exp(S(2)Iasin(ax)) + S(1))asin(ax) - S(3)IaS(2)asin(ax)S(2)/S(2) - S(3)asqrt(-aS(2)x*S(2) + S(1))asin(ax)S(2)/(S(2)x) - asin(ax)S(3)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) # sympy and mathematica assert rubi_test(rubi_integrate(asin(ax)S(3)/xS(4), x), x, IaS(3)PolyLog(S(2), -exp(Iasin(ax)))asin(ax) - IaS(3)PolyLog(S(2), exp(Iasin(ax)))asin(ax) - a*S(3)PolyLog(S(3), -exp(Iasin(ax))) + a*S(3)PolyLog(S(3), exp(Iasin(ax))) - a*S(3)asin(ax)S(2)atanh(exp(Iasin(ax))) - a*S(3)atanh(sqrt(-a*S(2)xS(2) + S(1))) - aS(2)asin(ax)/x - asqrt(-aS(2)x*S(2) + S(1))asin(ax)S(2)/(S(2)x*S(2)) - asin(ax)*S(3)/(S(3)x*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)S(3)/xS(5), x), x, -IaS(4)PolyLog(S(2), exp(S(2)Iasin(ax)))/S(2) + aS(4)log(-exp(S(2)Iasin(ax)) + S(1))asin(ax) - Ia*S(4)asin(ax)S(2)/S(2) - aS(3)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(2)/(S(2)x) - a*S(3)sqrt(-a*S(2)x*S(2) + S(1))/(S(4)x) - a*S(2)asin(ax)/(S(4)x*S(2)) - asqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(2)/(S(4)x*S(3)) - asin(ax)*S(3)/(S(4)xS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)asin(ax)S(4), x), x, xS(6)asin(ax)S(4)/S(6) - xS(6)asin(ax)S(2)/S(18) + xS(6)/S(324) + x*S(5)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(3)/(S(9)a) - x*S(5)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(54)a) - S(5)xS(4)asin(ax)S(2)/(S(48)a*S(2)) + S(65)x*S(4)/(S(3456)a*S(2)) + S(5)x*S(3)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(3)/(S(36)a*S(3)) - S(65)x*S(3)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(864)a*S(3)) - S(5)x*S(2)asin(ax)S(2)/(S(16)a*S(4)) + S(245)x*S(2)/(S(1152)a*S(4)) + S(5)xsqrt(-aS(2)x*S(2) + S(1))asin(ax)S(3)/(S(24)a*S(5)) - S(245)xsqrt(-aS(2)x*S(2) + S(1))asin(ax)/(S(576)a*S(5)) - S(5)asin(ax)S(4)/(S(96)a*S(6)) + S(245)asin(ax)S(2)/(S(1152)aS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)asin(ax)S(4), x), x, xS(5)asin(ax)*S(4)/S(5) - S(12)x*S(5)asin(ax)S(2)/S(125) + S(24)x*S(5)/S(3125) + S(4)x*S(4)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(3)/(S(25)a) - S(24)xS(4)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(625)a) - S(16)xS(3)asin(ax)S(2)/(S(75)a*S(2)) + S(1088)x*S(3)/(S(16875)a*S(2)) + S(16)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(3)/(S(75)a*S(3)) - S(1088)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(5625)a*S(3)) - S(32)xasin(ax)*S(2)/(S(25)a*S(4)) + S(16576)x/(S(5625)aS(4)) + S(32)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(3)/(S(75)a*S(5)) - S(16576)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(5625)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)asin(ax)S(4), x), x, xS(4)asin(ax)*S(4)/S(4) - S(3)x*S(4)asin(ax)S(2)/S(16) + S(3)xS(4)/S(128) + xS(3)sqrt(-aS(2)x*S(2) + S(1))asin(ax)S(3)/(S(4)a) - S(3)xS(3)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(32)a) - S(9)xS(2)asin(ax)S(2)/(S(16)a*S(2)) + S(45)x*S(2)/(S(128)a*S(2)) + S(3)xsqrt(-aS(2)x*S(2) + S(1))asin(ax)S(3)/(S(8)a*S(3)) - S(45)xsqrt(-aS(2)x*S(2) + S(1))asin(ax)/(S(64)a*S(3)) - S(3)asin(ax)S(4)/(S(32)a*S(4)) + S(45)asin(ax)S(2)/(S(128)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asin(ax)S(4), x), x, xS(3)asin(ax)*S(4)/S(3) - S(4)x*S(3)asin(ax)S(2)/S(9) + S(8)x*S(3)/S(81) + S(4)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(3)/(S(9)a) - S(8)xS(2)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(27)a) - S(8)xasin(ax)S(2)/(S(3)a*S(2)) + S(160)x/(S(27)aS(2)) + S(8)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(3)/(S(9)a*S(3)) - S(160)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/(S(27)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasin(ax)S(4), x), x, xS(2)asin(ax)S(4)/S(2) - S(3)x*S(2)asin(ax)S(2)/S(2) + S(3)x*S(2)/S(4) + xsqrt(-a*S(2)x*S(2) + S(1))asin(ax)S(3)/a - S(3)xsqrt(-aS(2)x*S(2) + S(1))asin(ax)/(S(2)a) - asin(ax)S(4)/(S(4)a*S(2)) + S(3)asin(ax)S(2)/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*S(4), x), x, xasin(ax)S(4) - S(12)xasin(ax)*S(2) + S(24)x + S(4)sqrt(-aS(2)x*S(2) + S(1))asin(ax)S(3)/a - S(24)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*S(4)/x, x), x, -S(2)IPolyLog(S(2), exp(S(2)Iasin(ax)))asin(ax)*S(3) + S(3)PolyLog(S(3), exp(S(2)Iasin(ax)))asin(ax)S(2) + S(3)IPolyLog(S(4), exp(S(2)Iasin(ax)))asin(ax) - S(3)PolyLog(S(5), exp(S(2)Iasin(ax)))/S(2) + log(-exp(S(2)Iasin(ax)) + S(1))asin(ax)S(4) - Iasin(ax)S(5)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)S(4)/xS(2), x), x, S(12)IaPolyLog(S(2), -exp(Iasin(ax)))asin(ax)S(2) - S(12)IaPolyLog(S(2), exp(Iasin(ax)))asin(ax)*S(2) - S(24)aPolyLog(S(3), -exp(Iasin(ax)))asin(ax) + S(24)aPolyLog(S(3), exp(Iasin(ax)))asin(ax) - S(24)IaPolyLog(S(4), -exp(Iasin(ax))) + S(24)IaPolyLog(S(4), exp(Iasin(ax))) - S(8)aasin(ax)*S(3)atanh(exp(Iasin(ax))) - asin(ax)S(4)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)S(4)/xS(3), x), x, -S(6)Ia*S(2)PolyLog(S(2), exp(S(2)Iasin(ax)))asin(ax) + S(3)a*S(2)PolyLog(S(3), exp(S(2)Iasin(ax))) + S(6)a*S(2)log(-exp(S(2)Iasin(ax)) + S(1))asin(ax)S(2) - S(2)IaS(2)asin(ax)S(3) - S(2)asqrt(-aS(2)x*S(2) + S(1))asin(ax)S(3)/x - asin(ax)*S(4)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)S(4)/xS(4), x), x, S(2)Ia*S(3)PolyLog(S(2), -exp(Iasin(ax)))asin(ax)*S(2) + S(4)IaS(3)PolyLog(S(2), -exp(Iasin(ax))) - S(2)Ia*S(3)PolyLog(S(2), exp(Iasin(ax)))asin(ax)*S(2) - S(4)IaS(3)PolyLog(S(2), exp(Iasin(ax))) - S(4)aS(3)PolyLog(S(3), -exp(Iasin(ax)))asin(ax) + S(4)aS(3)PolyLog(S(3), exp(Iasin(ax)))asin(ax) - S(4)Ia*S(3)PolyLog(S(4), -exp(Iasin(ax))) + S(4)Ia*S(3)PolyLog(S(4), exp(Iasin(ax))) - S(4)aS(3)asin(ax)S(3)atanh(exp(Iasin(ax)))/S(3) - S(8)aS(3)asin(ax)atanh(exp(Iasin(ax))) - S(2)aS(2)asin(ax)S(2)/x - S(2)asqrt(-aS(2)x*S(2) + S(1))asin(ax)S(3)/(S(3)x*S(2)) - asin(ax)*S(4)/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/asin(ax), x), x, S(5)CosIntegral(asin(ax))/(S(64)a*S(7)) - S(9)CosIntegral(S(3)asin(ax))/(S(64)aS(7)) + S(5)CosIntegral(S(5)asin(ax))/(S(64)aS(7)) - CosIntegral(S(7)asin(ax))/(S(64)aS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/asin(ax), x), x, S(5)SinIntegral(S(2)asin(ax))/(S(32)aS(6)) - SinIntegral(S(4)asin(ax))/(S(8)a*S(6)) + SinIntegral(S(6)asin(ax))/(S(32)aS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asin(ax), x), x, CosIntegral(asin(ax))/(S(8)aS(5)) - S(3)CosIntegral(S(3)asin(ax))/(S(16)aS(5)) + CosIntegral(S(5)asin(ax))/(S(16)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asin(ax), x), x, SinIntegral(S(2)asin(ax))/(S(4)a*S(4)) - SinIntegral(S(4)asin(ax))/(S(8)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/asin(ax), x), x, CosIntegral(asin(ax))/(S(4)aS(3)) - CosIntegral(S(3)asin(ax))/(S(4)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asin(ax), x), x, SinIntegral(S(2)asin(ax))/(S(2)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/asin(ax), x), x, CosIntegral(asin(ax))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasin(ax)), x), x, Integrate(S(1)/(xasin(ax)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)asin(ax)), x), x, Integrate(S(1)/(xS(2)asin(ax)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/asin(ax)S(2), x), x, -xS(6)sqrt(-aS(2)x*S(2) + S(1))/(aasin(ax)) - S(5)SinIntegral(asin(ax))/(S(64)a*S(7)) + S(27)SinIntegral(S(3)asin(ax))/(S(64)aS(7)) - S(25)SinIntegral(S(5)asin(ax))/(S(64)aS(7)) + S(7)SinIntegral(S(7)asin(ax))/(S(64)aS(7)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/asin(ax)S(2), x), x, -xS(5)sqrt(-aS(2)x*S(2) + S(1))/(aasin(ax)) + S(5)CosIntegral(S(2)asin(ax))/(S(16)aS(6)) - CosIntegral(S(4)asin(ax))/(S(2)a*S(6)) + S(3)CosIntegral(S(6)asin(ax))/(S(16)aS(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asin(ax)S(2), x), x, -xS(4)sqrt(-aS(2)x*S(2) + S(1))/(aasin(ax)) - SinIntegral(asin(ax))/(S(8)aS(5)) + S(9)SinIntegral(S(3)asin(ax))/(S(16)aS(5)) - S(5)SinIntegral(S(5)asin(ax))/(S(16)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asin(ax)S(2), x), x, -xS(3)sqrt(-aS(2)x*S(2) + S(1))/(aasin(ax)) + CosIntegral(S(2)asin(ax))/(S(2)a*S(4)) - CosIntegral(S(4)asin(ax))/(S(2)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/asin(ax)S(2), x), x, -xS(2)sqrt(-a*S(2)x*S(2) + S(1))/(aasin(ax)) - SinIntegral(asin(ax))/(S(4)aS(3)) + S(3)SinIntegral(S(3)asin(ax))/(S(4)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asin(ax)*S(2), x), x, -xsqrt(-a*S(2)x*S(2) + S(1))/(aasin(ax)) + CosIntegral(S(2)asin(ax))/aS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)(S(-2)), x), x, -sqrt(-aS(2)xS(2) + S(1))/(aasin(ax)) - SinIntegral(asin(ax))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasin(ax)*S(2)), x), x, Integrate(S(1)/(xasin(ax)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)asin(ax)S(2)), x), x, Integrate(S(1)/(xS(2)asin(ax)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asin(ax)*S(3), x), x, S(5)x*S(5)/(S(2)asin(ax)) - xS(4)sqrt(-a*S(2)x*S(2) + S(1))/(S(2)aasin(ax)*S(2)) - S(2)xS(3)/(aS(2)asin(ax)) - CosIntegral(asin(ax))/(S(16)a*S(5)) + S(27)CosIntegral(S(3)asin(ax))/(S(32)aS(5)) - S(25)CosIntegral(S(5)asin(ax))/(S(32)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asin(ax)*S(3), x), x, S(2)x*S(4)/asin(ax) - x*S(3)sqrt(-a*S(2)x*S(2) + S(1))/(S(2)aasin(ax)*S(2)) - S(3)x*S(2)/(S(2)a*S(2)asin(ax)) - SinIntegral(S(2)asin(ax))/(S(2)a*S(4)) + SinIntegral(S(4)asin(ax))/aS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/asin(ax)*S(3), x), x, S(3)x*S(3)/(S(2)asin(ax)) - xS(2)sqrt(-a*S(2)x*S(2) + S(1))/(S(2)aasin(ax)S(2)) - x/(aS(2)asin(ax)) - CosIntegral(asin(ax))/(S(8)a*S(3)) + S(9)CosIntegral(S(3)asin(ax))/(S(8)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asin(ax)S(3), x), x, xS(2)/asin(ax) - xsqrt(-a*S(2)x*S(2) + S(1))/(S(2)aasin(ax)*S(2)) - SinIntegral(S(2)asin(ax))/aS(2) - S(1)/(S(2)a*S(2)asin(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*(S(-3)), x), x, x/(S(2)asin(ax)) - sqrt(-aS(2)x*S(2) + S(1))/(S(2)aasin(ax)*S(2)) - CosIntegral(asin(ax))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasin(ax)S(3)), x), x, Integrate(S(1)/(xasin(ax)S(3)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)asin(ax)S(3)), x), x, Integrate(S(1)/(xS(2)asin(ax)S(3)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asin(ax)*S(4), x), x, S(5)x*S(5)/(S(6)asin(ax)S(2)) + S(25)x*S(4)sqrt(-a*S(2)x*S(2) + S(1))/(S(6)aasin(ax)) - x*S(4)sqrt(-a*S(2)x*S(2) + S(1))/(S(3)aasin(ax)*S(3)) - S(2)x*S(3)/(S(3)a*S(2)asin(ax)S(2)) - S(2)x*S(2)sqrt(-a*S(2)xS(2) + S(1))/(aS(3)asin(ax)) + SinIntegral(asin(ax))/(S(48)a*S(5)) - S(27)SinIntegral(S(3)asin(ax))/(S(32)aS(5)) + S(125)SinIntegral(S(5)asin(ax))/(S(96)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asin(ax)*S(4), x), x, S(2)x*S(4)/(S(3)asin(ax)S(2)) + S(8)x*S(3)sqrt(-a*S(2)x*S(2) + S(1))/(S(3)aasin(ax)) - x*S(3)sqrt(-a*S(2)x*S(2) + S(1))/(S(3)aasin(ax)S(3)) - xS(2)/(S(2)aS(2)asin(ax)S(2)) - xsqrt(-a*S(2)xS(2) + S(1))/(aS(3)asin(ax)) - CosIntegral(S(2)asin(ax))/(S(3)aS(4)) + S(4)CosIntegral(S(4)asin(ax))/(S(3)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/asin(ax)S(4), x), x, xS(3)/(S(2)asin(ax)*S(2)) + S(3)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))/(S(2)aasin(ax)) - x*S(2)sqrt(-a*S(2)x*S(2) + S(1))/(S(3)aasin(ax)*S(3)) - x/(S(3)a*S(2)asin(ax)S(2)) - sqrt(-aS(2)x*S(2) + S(1))/(S(3)a*S(3)asin(ax)) + SinIntegral(asin(ax))/(S(24)aS(3)) - S(9)SinIntegral(S(3)asin(ax))/(S(8)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asin(ax)S(4), x), x, xS(2)/(S(3)asin(ax)*S(2)) + S(2)xsqrt(-aS(2)x*S(2) + S(1))/(S(3)aasin(ax)) - xsqrt(-aS(2)x*S(2) + S(1))/(S(3)aasin(ax)*S(3)) - S(2)CosIntegral(S(2)asin(ax))/(S(3)aS(2)) - S(1)/(S(6)a*S(2)asin(ax)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*(S(-4)), x), x, x/(S(6)asin(ax)S(2)) + sqrt(-aS(2)x*S(2) + S(1))/(S(6)aasin(ax)) - sqrt(-a*S(2)x*S(2) + S(1))/(S(3)aasin(ax)*S(3)) + SinIntegral(asin(ax))/(S(6)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasin(ax)S(4)), x), x, Integrate(S(1)/(xasin(ax)S(4)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)asin(ax)S(4)), x), x, Integrate(S(1)/(xS(2)asin(ax)S(4)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)sqrt(asin(ax)), x), x, -sqrt(S(10))sqrt(Pi)FresnelS(sqrt(S(10))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(800)aS(5)) - sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(16)aS(5)) + sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(96)aS(5)) + xS(5)sqrt(asin(ax))/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)sqrt(asin(ax)), x), x, sqrt(Pi)FresnelC(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(16)aS(4)) - sqrt(S(2))sqrt(Pi)FresnelC(S(2)sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(128)aS(4)) + xS(4)sqrt(asin(ax))/S(4) - S(3)sqrt(asin(ax))/(S(32)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(asin(ax)), x), x, -sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(8)aS(3)) + sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(72)aS(3)) + xS(3)sqrt(asin(ax))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(asin(ax)), x), x, sqrt(Pi)FresnelC(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(8)aS(2)) + xS(2)sqrt(asin(ax))/S(2) - sqrt(asin(ax))/(S(4)aS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(asin(ax)), x), x, -sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(2)a) + xsqrt(asin(ax)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(asin(ax))/x, x), x, Integrate(sqrt(asin(ax))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(4)asin(ax)(S(3)/2), x), x, -S(3)sqrt(S(10))sqrt(Pi)FresnelC(sqrt(S(10))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(8000)a*S(5)) - S(3)sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(32)a*S(5)) + sqrt(S(6))sqrt(Pi)FresnelC(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(192)aS(5)) + xS(5)asin(ax)(S(3)/2)/S(5) + S(3)x*S(4)sqrt(-a*S(2)x*S(2) + S(1))sqrt(asin(ax))/(S(50)a) + S(2)xS(2)sqrt(-a*S(2)x*S(2) + S(1))sqrt(asin(ax))/(S(25)a*S(3)) + S(4)sqrt(-a*S(2)x*S(2) + S(1))sqrt(asin(ax))/(S(25)aS(5)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(4)asin(ax)*(S(3)/2), x), x, -S(3)sqrt(S(10))sqrt(Pi)FresnelC(sqrt(S(10))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(8000)a*S(5)) - S(3)sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(32)a*S(5)) + sqrt(S(6))sqrt(Pi)FresnelC(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(192)aS(5)) + xS(5)asin(ax)(S(3)/2)/S(5) + S(3)x*S(4)sqrt(-a*S(2)x*S(2) + S(1))sqrt(asin(ax))/(S(50)a) + S(2)xS(2)sqrt(-a*S(2)x*S(2) + S(1))sqrt(asin(ax))/(S(25)a*S(3)) + S(4)sqrt(-a*S(2)x*S(2) + S(1))sqrt(asin(ax))/(S(25)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)asin(ax)*(S(3)/2), x), x, -S(3)sqrt(Pi)FresnelS(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(64)a*S(4)) + S(3)sqrt(S(2))sqrt(Pi)FresnelS(S(2)sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(1024)aS(4)) + xS(4)asin(ax)(S(3)/2)/S(4) + S(3)x*S(3)sqrt(-a*S(2)x*S(2) + S(1))sqrt(asin(ax))/(S(32)a) + S(9)xsqrt(-a*S(2)x*S(2) + S(1))sqrt(asin(ax))/(S(64)a*S(3)) - S(3)asin(ax)(S(3)/2)/(S(32)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asin(ax)*(S(3)/2), x), x, -S(3)sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(16)a*S(3)) + sqrt(S(6))sqrt(Pi)FresnelC(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(144)aS(3)) + xS(3)asin(ax)(S(3)/2)/S(3) + xS(2)sqrt(-a*S(2)x*S(2) + S(1))sqrt(asin(ax))/(S(6)a) + sqrt(-a*S(2)x*S(2) + S(1))sqrt(asin(ax))/(S(3)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasin(ax)(S(3)/2), x), x, -S(3)sqrt(Pi)FresnelS(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(32)aS(2)) + xS(2)asin(ax)*(S(3)/2)/S(2) + S(3)xsqrt(-aS(2)x*S(2) + S(1))sqrt(asin(ax))/(S(8)a) - asin(ax)(S(3)/2)/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*(S(3)/2), x), x, -S(3)sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(4)a) + xasin(ax)*(S(3)/2) + S(3)sqrt(-a*S(2)x*S(2) + S(1))sqrt(asin(ax))/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)(S(3)/2)/x, x), x, Integrate(asin(ax)(S(3)/2)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)asin(ax)*(S(5)/2), x), x, S(3)sqrt(S(10))sqrt(Pi)FresnelS(sqrt(S(10))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(16000)a*S(5)) + S(15)sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(64)a*S(5)) - S(5)sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(1152)aS(5)) + xS(5)asin(ax)*(S(5)/2)/S(5) - S(3)x*S(5)sqrt(asin(ax))/S(100) + xS(4)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)(S(3)/2)/(S(10)a) - x*S(3)sqrt(asin(ax))/(S(15)a*S(2)) + S(2)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)(S(3)/2)/(S(15)a*S(3)) - S(2)xsqrt(asin(ax))/(S(5)aS(4)) + S(4)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)(S(3)/2)/(S(15)aS(5)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(4)asin(ax)*(S(5)/2), x), x, S(3)sqrt(S(10))sqrt(Pi)FresnelS(sqrt(S(10))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(16000)a*S(5)) + S(15)sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(64)a*S(5)) - S(5)sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(1152)aS(5)) + xS(5)asin(ax)*(S(5)/2)/S(5) - S(3)x*S(5)sqrt(asin(ax))/S(100) + xS(4)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)(S(3)/2)/(S(10)a) - x*S(3)sqrt(asin(ax))/(S(15)a*S(2)) + S(2)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)(S(3)/2)/(S(15)a*S(3)) - S(2)xsqrt(asin(ax))/(S(5)aS(4)) + S(4)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)(S(3)/2)/(S(15)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)asin(ax)*(S(5)/2), x), x, -S(15)sqrt(Pi)FresnelC(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(256)a*S(4)) + S(15)sqrt(S(2))sqrt(Pi)FresnelC(S(2)sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(8192)aS(4)) + xS(4)asin(ax)(S(5)/2)/S(4) - S(15)x*S(4)sqrt(asin(ax))/S(256) + S(5)x*S(3)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)(S(3)/2)/(S(32)a) - S(45)xS(2)sqrt(asin(ax))/(S(256)a*S(2)) + S(15)xsqrt(-aS(2)x*S(2) + S(1))asin(ax)(S(3)/2)/(S(64)a*S(3)) - S(3)asin(ax)(S(5)/2)/(S(32)a*S(4)) + S(225)sqrt(asin(ax))/(S(2048)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asin(ax)*(S(5)/2), x), x, S(15)sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(32)a*S(3)) - S(5)sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(864)aS(3)) + xS(3)asin(ax)*(S(5)/2)/S(3) - S(5)x*S(3)sqrt(asin(ax))/S(36) + S(5)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)(S(3)/2)/(S(18)a) - S(5)xsqrt(asin(ax))/(S(6)a*S(2)) + S(5)sqrt(-a*S(2)x*S(2) + S(1))asin(ax)(S(3)/2)/(S(9)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasin(ax)(S(5)/2), x), x, -S(15)sqrt(Pi)FresnelC(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(128)aS(2)) + xS(2)asin(ax)*(S(5)/2)/S(2) - S(15)x*S(2)sqrt(asin(ax))/S(32) + S(5)xsqrt(-aS(2)x*S(2) + S(1))asin(ax)(S(3)/2)/(S(8)a) - asin(ax)(S(5)/2)/(S(4)a*S(2)) + S(15)sqrt(asin(ax))/(S(64)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*(S(5)/2), x), x, S(15)sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(8)a) + xasin(ax)*(S(5)/2) - S(15)xsqrt(asin(ax))/S(4) + S(5)sqrt(-aS(2)x*S(2) + S(1))asin(ax)(S(3)/2)/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)(S(5)/2)/x, x), x, Integrate(asin(ax)(S(5)/2)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/sqrt(asin(ax)), x), x, sqrt(S(10))sqrt(Pi)FresnelC(sqrt(S(10))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(80)aS(5)) + sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(8)aS(5)) - sqrt(S(6))sqrt(Pi)FresnelC(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(16)aS(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/sqrt(asin(ax)), x), x, sqrt(Pi)FresnelS(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(4)a*S(4)) - sqrt(S(2))sqrt(Pi)FresnelS(S(2)sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(16)aS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(asin(ax)), x), x, sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(4)aS(3)) - sqrt(S(6))sqrt(Pi)FresnelC(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(12)aS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(asin(ax)), x), x, sqrt(Pi)FresnelS(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(2)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(asin(ax)), x), x, sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/a, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(asin(ax))), x), x, Integrate(S(1)/(xsqrt(asin(ax))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(asin(ax))), x), x, Integrate(S(1)/(xS(2)sqrt(asin(ax))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(6)/asin(ax)*(S(3)/2), x), x, -S(5)sqrt(S(10))sqrt(Pi)FresnelS(sqrt(S(10))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(32)a*S(7)) + sqrt(S(14))sqrt(Pi)FresnelS(sqrt(S(14))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(32)aS(7)) - S(5)sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(32)a*S(7)) + S(9)sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(32)a*S(7)) - S(2)x*S(6)sqrt(-a*S(2)x*S(2) + S(1))/(asqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(5)/asin(ax)*(S(3)/2), x), x, S(5)sqrt(Pi)FresnelC(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(8)a*S(6)) - sqrt(S(2))sqrt(Pi)FresnelC(S(2)sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(2)a*S(6)) + sqrt(S(3))sqrt(Pi)FresnelC(S(2)sqrt(S(3))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(8)a*S(6)) - S(2)x*S(5)sqrt(-a*S(2)x*S(2) + S(1))/(asqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asin(ax)*(S(3)/2), x), x, -sqrt(S(10))sqrt(Pi)FresnelS(sqrt(S(10))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(8)aS(5)) - sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(4)aS(5)) + S(3)sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(8)a*S(5)) - S(2)x*S(4)sqrt(-a*S(2)x*S(2) + S(1))/(asqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asin(ax)*(S(3)/2), x), x, sqrt(Pi)FresnelC(S(2)sqrt(asin(ax))/sqrt(Pi))/a*S(4) - sqrt(S(2))sqrt(Pi)FresnelC(S(2)sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(2)a*S(4)) - S(2)x*S(3)sqrt(-a*S(2)x*S(2) + S(1))/(asqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/asin(ax)*(S(3)/2), x), x, -sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(2)aS(3)) + sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(2)aS(3)) - S(2)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))/(asqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asin(ax)*(S(3)/2), x), x, S(2)sqrt(Pi)FresnelC(S(2)sqrt(asin(ax))/sqrt(Pi))/aS(2) - S(2)xsqrt(-aS(2)x*S(2) + S(1))/(asqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*(S(-3)/2), x), x, -S(2)sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/a - S(2)sqrt(-a*S(2)x*S(2) + S(1))/(asqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasin(ax)(S(3)/2)), x), x, Integrate(S(1)/(xasin(ax)(S(3)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asin(ax)*(S(5)/2), x), x, -S(5)sqrt(S(10))sqrt(Pi)FresnelC(sqrt(S(10))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(12)a*S(5)) - sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(6)aS(5)) + S(3)sqrt(S(6))sqrt(Pi)FresnelC(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(4)a*S(5)) + S(20)x*S(5)/(S(3)sqrt(asin(ax))) - S(2)x*S(4)sqrt(-a*S(2)x*S(2) + S(1))/(S(3)aasin(ax)*(S(3)/2)) - S(16)x*S(3)/(S(3)a*S(2)sqrt(asin(ax))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(xS(4)/asin(ax)*(S(5)/2), x), x, -S(5)sqrt(S(10))sqrt(Pi)FresnelC(sqrt(S(10))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(12)a*S(5)) - sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(6)aS(5)) + S(3)sqrt(S(6))sqrt(Pi)FresnelC(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(4)a*S(5)) + S(20)x*S(5)/(S(3)sqrt(asin(ax))) - S(2)x*S(4)sqrt(-a*S(2)x*S(2) + S(1))/(S(3)aasin(ax)*(S(3)/2)) - S(16)x*S(3)/(S(3)a*S(2)sqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asin(ax)*(S(5)/2), x), x, -S(4)sqrt(Pi)FresnelS(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(3)a*S(4)) + S(4)sqrt(S(2))sqrt(Pi)FresnelS(S(2)sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(3)aS(4)) + S(16)x*S(4)/(S(3)sqrt(asin(ax))) - S(2)x*S(3)sqrt(-a*S(2)x*S(2) + S(1))/(S(3)aasin(ax)*(S(3)/2)) - S(4)xS(2)/(aS(2)sqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/asin(ax)*(S(5)/2), x), x, -sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(3)aS(3)) + sqrt(S(6))sqrt(Pi)FresnelC(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/a*S(3) + S(4)x*S(3)/sqrt(asin(ax)) - S(2)xS(2)sqrt(-a*S(2)x*S(2) + S(1))/(S(3)aasin(ax)*(S(3)/2)) - S(8)x/(S(3)aS(2)sqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asin(ax)*(S(5)/2), x), x, -S(8)sqrt(Pi)FresnelS(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(3)a*S(2)) + S(8)x*S(2)/(S(3)sqrt(asin(ax))) - S(2)xsqrt(-aS(2)x*S(2) + S(1))/(S(3)aasin(ax)*(S(3)/2)) - S(4)/(S(3)a*S(2)sqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*(S(-5)/2), x), x, -S(4)sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(3)a) + S(4)x/(S(3)sqrt(asin(ax))) - S(2)sqrt(-a*S(2)x*S(2) + S(1))/(S(3)aasin(ax)*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasin(ax)(S(5)/2)), x), x, Integrate(S(1)/(xasin(ax)(S(5)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(4)/asin(ax)*(S(7)/2), x), x, S(5)sqrt(S(10))sqrt(Pi)FresnelS(sqrt(S(10))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(6)a*S(5)) + sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(15)aS(5)) - S(9)sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(10)a*S(5)) + S(4)x*S(5)/(S(3)asin(ax)(S(3)/2)) + S(40)x*S(4)sqrt(-a*S(2)x*S(2) + S(1))/(S(3)asqrt(asin(ax))) - S(2)xS(4)sqrt(-a*S(2)x*S(2) + S(1))/(S(5)aasin(ax)*(S(5)/2)) - S(16)x*S(3)/(S(15)a*S(2)asin(ax)(S(3)/2)) - S(32)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))/(S(5)a*S(3)sqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)/asin(ax)*(S(7)/2), x), x, -S(16)sqrt(Pi)FresnelC(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(15)a*S(4)) + S(32)sqrt(S(2))sqrt(Pi)FresnelC(S(2)sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(15)aS(4)) + S(16)x*S(4)/(S(15)asin(ax)(S(3)/2)) + S(128)x*S(3)sqrt(-a*S(2)x*S(2) + S(1))/(S(15)asqrt(asin(ax))) - S(2)xS(3)sqrt(-a*S(2)x*S(2) + S(1))/(S(5)aasin(ax)*(S(5)/2)) - S(4)x*S(2)/(S(5)a*S(2)asin(ax)(S(3)/2)) - S(16)xsqrt(-aS(2)x*S(2) + S(1))/(S(5)a*S(3)sqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/asin(ax)*(S(7)/2), x), x, S(2)sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(15)a*S(3)) - S(6)sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(5)a*S(3)) + S(4)x*S(3)/(S(5)asin(ax)(S(3)/2)) + S(24)x*S(2)sqrt(-a*S(2)x*S(2) + S(1))/(S(5)asqrt(asin(ax))) - S(2)xS(2)sqrt(-a*S(2)x*S(2) + S(1))/(S(5)aasin(ax)*(S(5)/2)) - S(8)x/(S(15)aS(2)asin(ax)(S(3)/2)) - S(16)sqrt(-a*S(2)x*S(2) + S(1))/(S(15)a*S(3)sqrt(asin(ax))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/asin(ax)*(S(7)/2), x), x, -S(32)sqrt(Pi)FresnelC(S(2)sqrt(asin(ax))/sqrt(Pi))/(S(15)a*S(2)) + S(8)x*S(2)/(S(15)asin(ax)(S(3)/2)) + S(32)xsqrt(-aS(2)x*S(2) + S(1))/(S(15)asqrt(asin(ax))) - S(2)xsqrt(-a*S(2)x*S(2) + S(1))/(S(5)aasin(ax)*(S(5)/2)) - S(4)/(S(15)a*S(2)asin(ax)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*(S(-7)/2), x), x, S(8)sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(S(1)/Pi)sqrt(asin(ax)))/(S(15)a) + S(4)x/(S(15)asin(ax)(S(3)/2)) + S(8)sqrt(-a*S(2)x*S(2) + S(1))/(S(15)asqrt(asin(ax))) - S(2)sqrt(-aS(2)x*S(2) + S(1))/(S(5)aasin(ax)*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xasin(ax)(S(7)/2)), x), x, Integrate(S(1)/(xasin(ax)(S(7)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)*masin(ax)S(4), x), x, -S(4)aIntegrate((bx)*(m + S(1))asin(ax)S(3)/sqrt(-aS(2)x*S(2) + S(1)), x)/(b(m + S(1))) + (bx)(m + S(1))asin(ax)S(4)/(b(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)masin(ax)S(3), x), x, -S(3)aIntegrate((bx)*(m + S(1))asin(ax)S(2)/sqrt(-aS(2)x*S(2) + S(1)), x)/(b(m + S(1))) + (bx)(m + S(1))asin(ax)S(3)/(b(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)masin(ax)S(2), x), x, S(2)a*S(2)(bx)(m + S(3))HypergeometricPFQ(List(S(1), m/S(2) + S(3)/2, m/S(2) + S(3)/2), List(m/S(2) + S(2), m/S(2) + S(5)/2), a*S(2)xS(2))/(bS(3)(m + S(1))(m + S(2))(m + S(3))) - S(2)a(bx)*(m + S(2))Hypergeometric2F1(S(1)/2, m/S(2) + S(1), m/S(2) + S(2), a*S(2)x*S(2))asin(ax)/(bS(2)(m + S(1))(m + S(2))) + (bx)*(m + S(1))asin(ax)S(2)/(b(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)masin(ax), x), x, -a(bx)(m + S(2))Hypergeometric2F1(S(1)/2, m/S(2) + S(1), m/S(2) + S(2), a*S(2)xS(2))/(bS(2)(m + S(1))(m + S(2))) + (bx)(m + S(1))asin(ax)/(b(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)m/asin(ax), x), x, Integrate((bx)m/asin(ax), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)m/asin(ax)*S(2), x), x, Integrate((bx)*m/asin(ax)*S(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)*masin(ax)(S(3)/2), x), x, Integrate((bx)*masin(ax)(S(3)/2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)*msqrt(asin(ax)), x), x, Integrate((bx)*msqrt(asin(ax)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)*m/sqrt(asin(ax)), x), x, Integrate((bx)m/sqrt(asin(ax)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)m/asin(ax)*(S(3)/2), x), x, Integrate((bx)*m/asin(ax)*(S(3)/2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)*masin(ax)n, x), x, Integrate((bx)*masin(ax)n, x), expand=True, _diff=True, _numerical=True) # sympy and mathematicA assert rubi_test(rubi_integrate(xS(3)asin(ax)n, x), x, S(2)(-S(2)n + S(-6))(-Iasin(ax))(-n)Gamma(n + S(1), -S(4)Iasin(ax))asin(ax)n/aS(4) + S(2)(-S(2)n + S(-6))(Iasin(ax))(-n)Gamma(n + S(1), S(4)Iasin(ax))asin(ax)n/aS(4) - S(2)(-n + S(-4))(-Iasin(ax))*(-n)Gamma(n + S(1), -S(2)Iasin(ax))asin(ax)n/aS(4) - S(2)(-n + S(-4))(Iasin(ax))*(-n)Gamma(n + S(1), S(2)Iasin(ax))asin(ax)n/aS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)asin(ax)n, x), x, S(3)(-n + S(-1))I(-Iasin(ax))(-n)Gamma(n + S(1), -S(3)Iasin(ax))asin(ax)n/(S(8)aS(3)) - S(3)(-n + S(-1))I(Iasin(ax))*(-n)Gamma(n + S(1), S(3)Iasin(ax))asin(ax)n/(S(8)a*S(3)) - I(-Iasin(ax))*(-n)Gamma(n + S(1), -Iasin(ax))asin(ax)*n/(S(8)a*S(3)) + I(Iasin(ax))*(-n)Gamma(n + S(1), Iasin(ax))asin(ax)*n/(S(8)a*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xasin(ax)n, x), x, -S(2)(-n + S(-3))(-Iasin(ax))*(-n)Gamma(n + S(1), -S(2)Iasin(ax))asin(ax)n/aS(2) - S(2)(-n + S(-3))(Iasin(ax))*(-n)Gamma(n + S(1), S(2)Iasin(ax))asin(ax)n/aS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*n, x), x, -I(-Iasin(ax))*(-n)Gamma(n + S(1), -Iasin(ax))asin(ax)*n/(S(2)a) + I(Iasin(ax))(-n)Gamma(n + S(1), Iasin(ax))asin(ax)*n/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)n/x, x), x, Integrate(asin(ax)*n/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)n/xS(2), x), x, Integrate(asin(ax)n/xS(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bx)*(S(3)/2)asin(ax)n, x), x, Integrate((bx)*(S(3)/2)asin(ax)n, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bx)asin(ax)*n, x), x, Integrate(sqrt(bx)asin(ax)*n, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)*n/sqrt(bx), x), x, Integrate(asin(ax)n/sqrt(bx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(asin(ax)n/(bx)*(S(3)/2), x), x, Integrate(asin(ax)*n/(bx)(S(3)/2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(3)(a + basin(cx)), x), x, bx*S(3)sqrt(-c*S(2)x*S(2) + S(1))/(S(16)c) + S(3)bxsqrt(-cS(2)x*S(2) + S(1))/(S(32)c*S(3)) - S(3)basin(cx)/(S(32)cS(4)) + xS(4)(a + basin(cx))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)(a + basin(cx)), x), x, -b(-cS(2)xS(2) + S(1))(S(3)/2)/(S(9)cS(3)) + bsqrt(-c*S(2)x*S(2) + S(1))/(S(3)cS(3)) + xS(3)(a + basin(cx))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + basin(cx)), x), x, bxsqrt(-c*S(2)x*S(2) + S(1))/(S(4)c) - basin(cx)/(S(4)cS(2)) + xS(2)(a + basin(cx))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(a + basin(cx), x), x, ax + bxasin(cx) + bsqrt(-cS(2)x*S(2) + S(1))/c, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))/x, x), x, -IbPolyLog(S(2), exp(S(2)Iasin(cx)))/S(2) + (a + basin(cx))log(-exp(S(2)Iasin(cx)) + S(1)) - I(a + basin(cx))S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))/x*S(2), x), x, -bcatanh(sqrt(-cS(2)x*S(2) + S(1))) - (a + basin(cx))/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))/xS(3), x), x, -bcsqrt(-cS(2)x*S(2) + S(1))/(S(2)x) - (a + basin(cx))/(S(2)xS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))/xS(4), x), x, -bc*S(3)atanh(sqrt(-c*S(2)x*S(2) + S(1)))/S(6) - bcsqrt(-cS(2)x*S(2) + S(1))/(S(6)x*S(2)) - (a + basin(cx))/(S(3)xS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + basin(cx))S(2), x), x, -S(2)b*S(2)x*S(3)/S(27) - S(4)b*S(2)x/(S(9)cS(2)) + S(2)bxS(2)(a + basin(cx))sqrt(-cS(2)x*S(2) + S(1))/(S(9)c) + S(4)b(a + basin(cx))sqrt(-cS(2)x*S(2) + S(1))/(S(9)cS(3)) + xS(3)(a + basin(cx))S(2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + basin(cx))S(2), x), x, -bS(2)xS(2)/S(4) + bx(a + basin(cx))sqrt(-c*S(2)x*S(2) + S(1))/(S(2)c) + x*S(2)(a + basin(cx))*S(2)/S(2) - (a + basin(cx))S(2)/(S(4)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))S(2), x), x, -S(2)b*S(2)x + S(2)b(a + basin(cx))sqrt(-cS(2)x*S(2) + S(1))/c + x(a + basin(cx))*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))S(2)/x, x), x, bS(2)PolyLog(S(3), exp(S(2)Iasin(cx)))/S(2) - Ib(a + basin(cx))PolyLog(S(2), exp(S(2)Iasin(cx))) + (a + basin(cx))S(2)log(-exp(S(2)Iasin(cx)) + S(1)) - I(a + basin(cx))*S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))S(2)/xS(2), x), x, S(2)Ib*S(2)cPolyLog(S(2), -exp(Iasin(cx))) - S(2)IbS(2)cPolyLog(S(2), exp(Iasin(cx))) - S(4)bc(a + basin(cx))atanh(exp(Iasin(cx))) - (a + basin(cx))S(2)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + basin(cx))*S(3), x), x, -S(4)abS(2)x/(S(3)cS(2)) - S(4)b*S(3)xasin(cx)/(S(3)cS(2)) + S(2)b*S(3)(-c*S(2)xS(2) + S(1))(S(3)/2)/(S(27)cS(3)) - S(14)b*S(3)sqrt(-c*S(2)x*S(2) + S(1))/(S(9)c*S(3)) - S(2)b*S(2)x*S(3)(a + basin(cx))/S(9) + bxS(2)(a + basin(cx))*S(2)sqrt(-c*S(2)x*S(2) + S(1))/(S(3)c) + S(2)b(a + basin(cx))*S(2)sqrt(-c*S(2)x*S(2) + S(1))/(S(3)cS(3)) + xS(3)(a + basin(cx))S(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + basin(cx))*S(3), x), x, -S(3)b*S(3)xsqrt(-cS(2)x*S(2) + S(1))/(S(8)c) + S(3)bS(3)asin(cx)/(S(8)c*S(2)) - S(3)b*S(2)x*S(2)(a + basin(cx))/S(4) + S(3)bx(a + basin(cx))S(2)sqrt(-c*S(2)x*S(2) + S(1))/(S(4)c) + x*S(2)(a + basin(cx))*S(3)/S(2) - (a + basin(cx))S(3)/(S(4)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))S(3), x), x, -S(6)abS(2)x - S(6)bS(3)xasin(cx) - S(6)bS(3)sqrt(-c*S(2)x*S(2) + S(1))/c + S(3)b(a + basin(cx))S(2)sqrt(-c*S(2)x*S(2) + S(1))/c + x(a + basin(cx))*S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))S(3)/x, x), x, S(3)IbS(3)PolyLog(S(4), exp(S(2)Iasin(cx)))/S(4) + S(3)b*S(2)(a + basin(cx))PolyLog(S(3), exp(S(2)Iasin(cx)))/S(2) - S(3)Ib(a + basin(cx))S(2)PolyLog(S(2), exp(S(2)Iasin(cx)))/S(2) + (a + basin(cx))S(3)log(-exp(S(2)Iasin(cx)) + S(1)) - I(a + basin(cx))*S(4)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))S(3)/xS(2), x), x, -S(6)bS(3)cPolyLog(S(3), -exp(Iasin(cx))) + S(6)b*S(3)cPolyLog(S(3), exp(Iasin(cx))) + S(6)IbS(2)c(a + basin(cx))PolyLog(S(2), -exp(Iasin(cx))) - S(6)Ib*S(2)c(a + basin(cx))PolyLog(S(2), exp(Iasin(cx))) - S(6)bc(a + basin(cx))S(2)atanh(exp(Iasin(cx))) - (a + basin(cx))S(3)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + basin(cx)), x), x, CosIntegral(a/b + asin(cx))cos(a/b)/(S(4)bc*S(3)) - CosIntegral(S(3)a/b + S(3)asin(cx))cos(S(3)a/b)/(S(4)bc*S(3)) + SinIntegral(a/b + asin(cx))sin(a/b)/(S(4)bcS(3)) - SinIntegral(S(3)a/b + S(3)asin(cx))sin(S(3)a/b)/(S(4)bc*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + basin(cx)), x), x, -CosIntegral(S(2)a/b + S(2)asin(cx))sin(S(2)a/b)/(S(2)bc*S(2)) + SinIntegral(S(2)a/b + S(2)asin(cx))cos(S(2)a/b)/(S(2)bc*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(a + basin(cx)), x), x, CosIntegral((a + basin(cx))/b)cos(a/b)/(bc) + SinIntegral((a + basin(cx))/b)sin(a/b)/(bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + basin(cx))), x), x, Integrate(S(1)/(x(a + basin(cx))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + basin(cx))), x), x, Integrate(S(1)/(x*S(2)(a + basin(cx))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(a + basin(cx))S(2), x), x, -xS(2)sqrt(-c*S(2)x*S(2) + S(1))/(bc(a + basin(cx))) + CosIntegral(a/b + asin(cx))sin(a/b)/(S(4)b*S(2)c*S(3)) - S(3)CosIntegral(S(3)a/b + S(3)asin(cx))sin(S(3)a/b)/(S(4)b*S(2)c*S(3)) - SinIntegral(a/b + asin(cx))cos(a/b)/(S(4)b*S(2)c*S(3)) + S(3)SinIntegral(S(3)a/b + S(3)asin(cx))cos(S(3)a/b)/(S(4)b*S(2)c*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + basin(cx))S(2), x), x, -xsqrt(-c*S(2)x*S(2) + S(1))/(bc(a + basin(cx))) + CosIntegral(S(2)a/b + S(2)asin(cx))cos(S(2)a/b)/(b*S(2)c*S(2)) + SinIntegral(S(2)a/b + S(2)asin(cx))sin(S(2)a/b)/(b*S(2)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))(S(-2)), x), x, -sqrt(-cS(2)x*S(2) + S(1))/(bc(a + basin(cx))) + CosIntegral(a/b + asin(cx))sin(a/b)/(bS(2)c) - SinIntegral(a/b + asin(cx))cos(a/b)/(b*S(2)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + basin(cx))S(2)), x), x, Integrate(S(1)/(x(a + basin(cx))S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + basin(cx))S(2)), x), x, Integrate(S(1)/(xS(2)(a + basin(cx))S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + basin(cx))S(3), x), x, -xS(2)sqrt(-cS(2)x*S(2) + S(1))/(S(2)bc(a + basin(cx))*S(2)) + S(3)x*S(3)/(S(2)b*S(2)(a + basin(cx))) - x/(b*S(2)c*S(2)(a + basin(cx))) + CosIntegral((a + basin(cx))/b)cos(a/b)/(bS(3)c*S(3)) - S(9)CosIntegral(a/b + asin(cx))cos(a/b)/(S(8)bS(3)c*S(3)) + S(9)CosIntegral(S(3)a/b + S(3)asin(cx))cos(S(3)a/b)/(S(8)b*S(3)c*S(3)) + SinIntegral((a + basin(cx))/b)sin(a/b)/(b*S(3)c*S(3)) - S(9)SinIntegral(a/b + asin(cx))sin(a/b)/(S(8)bS(3)c*S(3)) + S(9)SinIntegral(S(3)a/b + S(3)asin(cx))sin(S(3)a/b)/(S(8)b*S(3)c*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + basin(cx))S(3), x), x, -xsqrt(-c*S(2)x*S(2) + S(1))/(S(2)bc(a + basin(cx))S(2)) + xS(2)/(b*S(2)(a + basin(cx))) - S(1)/(S(2)bS(2)c*S(2)(a + basin(cx))) + CosIntegral(S(2)a/b + S(2)asin(cx))sin(S(2)a/b)/(bS(3)c*S(2)) - SinIntegral(S(2)a/b + S(2)asin(cx))cos(S(2)a/b)/(b*S(3)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))(S(-3)), x), x, -sqrt(-cS(2)x*S(2) + S(1))/(S(2)bc(a + basin(cx))*S(2)) + x/(S(2)b*S(2)(a + basin(cx))) - CosIntegral((a + basin(cx))/b)cos(a/b)/(S(2)b*S(3)c) - SinIntegral((a + basin(cx))/b)sin(a/b)/(S(2)b*S(3)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + basin(cx))S(3)), x), x, Integrate(S(1)/(x(a + basin(cx))S(3)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + basin(cx))S(3)), x), x, Integrate(S(1)/(xS(2)(a + basin(cx))S(3)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(a + basin(cx)), x), x, sqrt(S(2))sqrt(Pi)sqrt(b)FresnelC(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(a/b)/(S(8)c*S(3)) - sqrt(S(6))sqrt(Pi)sqrt(b)FresnelC(sqrt(S(6))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(S(3)a/b)/(S(72)cS(3)) - sqrt(S(2))sqrt(Pi)sqrt(b)FresnelS(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(a/b)/(S(8)c*S(3)) + sqrt(S(6))sqrt(Pi)sqrt(b)FresnelS(sqrt(S(6))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(S(3)a/b)/(S(72)cS(3)) + xS(3)sqrt(a + basin(cx))/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(a + basin(cx)), x), x, sqrt(Pi)sqrt(b)FresnelC(S(2)sqrt(a + basin(cx))/(sqrt(Pi)sqrt(b)))cos(S(2)a/b)/(S(8)c*S(2)) + sqrt(Pi)sqrt(b)FresnelS(S(2)sqrt(a + basin(cx))/(sqrt(Pi)sqrt(b)))sin(S(2)a/b)/(S(8)cS(2)) + xS(2)sqrt(a + basin(cx))/S(2) - sqrt(a + basin(cx))/(S(4)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + basin(cx)), x), x, sqrt(S(2))sqrt(Pi)sqrt(b)FresnelC(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(a/b)/(S(2)c) - sqrt(S(2))sqrt(Pi)sqrt(b)FresnelS(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(a/b)/(S(2)c) + xsqrt(a + basin(cx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + basin(cx))/x, x), x, Integrate(sqrt(a + basin(cx))/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(a + basin(cx))/x*S(2), x), x, Integrate(sqrt(a + basin(cx))/xS(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + basin(cx))*(S(3)/2), x), x, -S(3)sqrt(S(2))sqrt(Pi)b*(S(3)/2)FresnelC(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(a/b)/(S(16)c*S(3)) + sqrt(S(6))sqrt(Pi)b(S(3)/2)FresnelC(sqrt(S(6))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(S(3)a/b)/(S(144)cS(3)) - S(3)sqrt(S(2))sqrt(Pi)b*(S(3)/2)FresnelS(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(a/b)/(S(16)c*S(3)) + sqrt(S(6))sqrt(Pi)b(S(3)/2)FresnelS(sqrt(S(6))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(S(3)a/b)/(S(144)cS(3)) + bx*S(2)sqrt(a + basin(cx))sqrt(-cS(2)x*S(2) + S(1))/(S(6)c) + bsqrt(a + basin(cx))sqrt(-c*S(2)x*S(2) + S(1))/(S(3)cS(3)) + xS(3)(a + basin(cx))(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + basin(cx))*(S(3)/2), x), x, S(3)sqrt(Pi)b(S(3)/2)FresnelC(S(2)sqrt(a + basin(cx))/(sqrt(Pi)sqrt(b)))sin(S(2)a/b)/(S(32)cS(2)) - S(3)sqrt(Pi)b(S(3)/2)FresnelS(S(2)sqrt(a + basin(cx))/(sqrt(Pi)sqrt(b)))cos(S(2)a/b)/(S(32)cS(2)) + S(3)bxsqrt(a + basin(cx))sqrt(-cS(2)x*S(2) + S(1))/(S(8)c) + x*S(2)(a + basin(cx))*(S(3)/2)/S(2) - (a + basin(cx))(S(3)/2)/(S(4)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))(S(3)/2), x), x, -S(3)sqrt(S(2))sqrt(Pi)b*(S(3)/2)FresnelC(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(a/b)/(S(4)c) - S(3)sqrt(S(2))sqrt(Pi)b(S(3)/2)FresnelS(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(a/b)/(S(4)c) + S(3)bsqrt(a + basin(cx))sqrt(-cS(2)x*S(2) + S(1))/(S(2)c) + x(a + basin(cx))(S(3)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))(S(3)/2)/x, x), x, Integrate((a + basin(cx))(S(3)/2)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))(S(3)/2)/xS(2), x), x, Integrate((a + basin(cx))(S(3)/2)/xS(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)(a + basin(cx))*(S(5)/2), x), x, -S(15)sqrt(S(2))sqrt(Pi)b*(S(5)/2)FresnelC(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(a/b)/(S(32)c*S(3)) + S(5)sqrt(S(6))sqrt(Pi)b*(S(5)/2)FresnelC(sqrt(S(6))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(S(3)a/b)/(S(864)cS(3)) + S(15)sqrt(S(2))sqrt(Pi)b*(S(5)/2)FresnelS(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(a/b)/(S(32)c*S(3)) - S(5)sqrt(S(6))sqrt(Pi)b*(S(5)/2)FresnelS(sqrt(S(6))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(S(3)a/b)/(S(864)cS(3)) - S(5)b*S(2)x*S(3)sqrt(a + basin(cx))/S(36) - S(5)bS(2)xsqrt(a + basin(cx))/(S(6)c*S(2)) + S(5)bxS(2)(a + basin(cx))*(S(3)/2)sqrt(-c*S(2)x*S(2) + S(1))/(S(18)c) + S(5)b(a + basin(cx))*(S(3)/2)sqrt(-c*S(2)x*S(2) + S(1))/(S(9)cS(3)) + xS(3)(a + basin(cx))(S(5)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(a + basin(cx))*(S(5)/2), x), x, -S(15)sqrt(Pi)b(S(5)/2)FresnelC(S(2)sqrt(a + basin(cx))/(sqrt(Pi)sqrt(b)))cos(S(2)a/b)/(S(128)cS(2)) - S(15)sqrt(Pi)b(S(5)/2)FresnelS(S(2)sqrt(a + basin(cx))/(sqrt(Pi)sqrt(b)))sin(S(2)a/b)/(S(128)cS(2)) - S(15)b*S(2)x*S(2)sqrt(a + basin(cx))/S(32) + S(15)bS(2)sqrt(a + basin(cx))/(S(64)cS(2)) + S(5)bx(a + basin(cx))*(S(3)/2)sqrt(-c*S(2)x*S(2) + S(1))/(S(8)c) + x*S(2)(a + basin(cx))*(S(5)/2)/S(2) - (a + basin(cx))(S(5)/2)/(S(4)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))(S(5)/2), x), x, -S(15)sqrt(S(2))sqrt(Pi)b*(S(5)/2)FresnelC(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(a/b)/(S(8)c) + S(15)sqrt(S(2))sqrt(Pi)b(S(5)/2)FresnelS(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(a/b)/(S(8)c) - S(15)bS(2)xsqrt(a + basin(cx))/S(4) + S(5)b(a + basin(cx))(S(3)/2)sqrt(-c*S(2)x*S(2) + S(1))/(S(2)c) + x(a + basin(cx))(S(5)/2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))(S(5)/2)/x, x), x, Integrate((a + basin(cx))(S(5)/2)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))(S(5)/2)/xS(2), x), x, Integrate((a + basin(cx))(S(5)/2)/xS(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/sqrt(a + basin(cx)), x), x, sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(a/b)/(S(4)sqrt(b)c*S(3)) - sqrt(S(6))sqrt(Pi)FresnelC(sqrt(S(6))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(S(3)a/b)/(S(12)sqrt(b)cS(3)) + sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(a/b)/(S(4)sqrt(b)c*S(3)) - sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(S(3)a/b)/(S(12)sqrt(b)cS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/sqrt(a + basin(cx)), x), x, -sqrt(Pi)FresnelC(S(2)sqrt(a + basin(cx))/(sqrt(Pi)sqrt(b)))sin(S(2)a/b)/(S(2)sqrt(b)c*S(2)) + sqrt(Pi)FresnelS(S(2)sqrt(a + basin(cx))/(sqrt(Pi)sqrt(b)))cos(S(2)a/b)/(S(2)sqrt(b)c*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(a + basin(cx)), x), x, sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(a/b)/(sqrt(b)c) + sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(a/b)/(sqrt(b)c), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(a + basin(cx))), x), x, Integrate(S(1)/(xsqrt(a + basin(cx))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(a + basin(cx))), x), x, Integrate(S(1)/(x*S(2)sqrt(a + basin(cx))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*S(2)/(a + basin(cx))(S(3)/2), x), x, sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(a/b)/(S(2)b(S(3)/2)c*S(3)) - sqrt(S(6))sqrt(Pi)FresnelC(sqrt(S(6))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(S(3)a/b)/(S(2)b*(S(3)/2)c*S(3)) - sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(a/b)/(S(2)b(S(3)/2)c*S(3)) + sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(S(3)a/b)/(S(2)b*(S(3)/2)c*S(3)) - S(2)x*S(2)sqrt(-c*S(2)x*S(2) + S(1))/(bcsqrt(a + basin(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + basin(cx))(S(3)/2), x), x, S(2)sqrt(Pi)FresnelC(S(2)sqrt(a + basin(cx))/(sqrt(Pi)sqrt(b)))cos(S(2)a/b)/(b(S(3)/2)c*S(2)) + S(2)sqrt(Pi)FresnelS(S(2)sqrt(a + basin(cx))/(sqrt(Pi)sqrt(b)))sin(S(2)a/b)/(b(S(3)/2)c*S(2)) - S(2)xsqrt(-cS(2)x*S(2) + S(1))/(bcsqrt(a + basin(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))(S(-3)/2), x), x, S(2)sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(a/b)/(b(S(3)/2)c) - S(2)sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(a/b)/(b*(S(3)/2)c) - S(2)sqrt(-cS(2)x*S(2) + S(1))/(bcsqrt(a + basin(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + basin(cx))*(S(3)/2)), x), x, Integrate(S(1)/(x(a + basin(cx))(S(3)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + basin(cx))(S(3)/2)), x), x, Integrate(S(1)/(xS(2)(a + basin(cx))(S(3)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)/(a + basin(cx))*(S(5)/2), x), x, -sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(a/b)/(S(3)b(S(5)/2)c*S(3)) + sqrt(S(6))sqrt(Pi)FresnelC(sqrt(S(6))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(S(3)a/b)/(b(S(5)/2)c*S(3)) - sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(a/b)/(S(3)b(S(5)/2)c*S(3)) + sqrt(S(6))sqrt(Pi)FresnelS(sqrt(S(6))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(S(3)a/b)/(b(S(5)/2)c*S(3)) - S(2)x*S(2)sqrt(-c*S(2)x*S(2) + S(1))/(S(3)bc(a + basin(cx))*(S(3)/2)) + S(4)xS(3)/(bS(2)sqrt(a + basin(cx))) - S(8)x/(S(3)bS(2)c*S(2)sqrt(a + basin(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(a + basin(cx))*(S(5)/2), x), x, S(8)sqrt(Pi)FresnelC(S(2)sqrt(a + basin(cx))/(sqrt(Pi)sqrt(b)))sin(S(2)a/b)/(S(3)b*(S(5)/2)c*S(2)) - S(8)sqrt(Pi)FresnelS(S(2)sqrt(a + basin(cx))/(sqrt(Pi)sqrt(b)))cos(S(2)a/b)/(S(3)b*(S(5)/2)c*S(2)) - S(2)xsqrt(-cS(2)x*S(2) + S(1))/(S(3)bc(a + basin(cx))*(S(3)/2)) + S(8)x*S(2)/(S(3)b*S(2)sqrt(a + basin(cx))) - S(4)/(S(3)bS(2)c*S(2)sqrt(a + basin(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))*(S(-5)/2), x), x, -S(4)sqrt(S(2))sqrt(Pi)FresnelC(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))cos(a/b)/(S(3)b*(S(5)/2)c) - S(4)sqrt(S(2))sqrt(Pi)FresnelS(sqrt(S(2))sqrt(a + basin(cx))sqrt(S(1)/Pi)/sqrt(b))sin(a/b)/(S(3)b(S(5)/2)c) - S(2)sqrt(-cS(2)x*S(2) + S(1))/(S(3)bc(a + basin(cx))*(S(3)/2)) + S(4)x/(S(3)bS(2)sqrt(a + basin(cx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(a + basin(cx))(S(5)/2)), x), x, Integrate(S(1)/(x(a + basin(cx))(S(5)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(a + basin(cx))(S(5)/2)), x), x, Integrate(S(1)/(xS(2)(a + basin(cx))*(S(5)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(5)/2)(a + basin(cx)), x), x, S(4)b(dx)(S(5)/2)sqrt(-c*S(2)x*S(2) + S(1))/(S(49)c) + S(20)bd*S(2)sqrt(dx)sqrt(-c*S(2)x*S(2) + S(1))/(S(147)c*S(3)) - S(20)bd(S(5)/2)EllipticF(asin(sqrt(c)sqrt(dx)/sqrt(d)), S(-1))/(S(147)c(S(7)/2)) + S(2)(dx)(S(7)/2)(a + basin(cx))/(S(7)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(3)/2)(a + basin(cx)), x), x, S(4)b(dx)(S(3)/2)sqrt(-c*S(2)x*S(2) + S(1))/(S(25)c) - S(12)bd*(S(3)/2)EllipticE(asin(sqrt(c)sqrt(dx)/sqrt(d)), S(-1))/(S(25)c(S(5)/2)) + S(12)bd(S(3)/2)EllipticF(asin(sqrt(c)sqrt(dx)/sqrt(d)), S(-1))/(S(25)c(S(5)/2)) + S(2)(dx)(S(5)/2)(a + basin(cx))/(S(5)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)(a + basin(cx)), x), x, S(4)bsqrt(dx)sqrt(-cS(2)x*S(2) + S(1))/(S(9)c) - S(4)bsqrt(d)EllipticF(asin(sqrt(c)sqrt(dx)/sqrt(d)), S(-1))/(S(9)c*(S(3)/2)) + S(2)(dx)(S(3)/2)(a + basin(cx))/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))/sqrt(dx), x), x, -S(4)bEllipticE(asin(sqrt(c)sqrt(dx)/sqrt(d)), S(-1))/(sqrt(c)sqrt(d)) + S(4)bEllipticF(asin(sqrt(c)sqrt(dx)/sqrt(d)), S(-1))/(sqrt(c)sqrt(d)) + S(2)sqrt(dx)(a + basin(cx))/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))/(dx)*(S(3)/2), x), x, S(4)bsqrt(c)EllipticF(asin(sqrt(c)sqrt(dx)/sqrt(d)), S(-1))/d*(S(3)/2) - (S(2)a + S(2)basin(cx))/(dsqrt(dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))/(dx)*(S(5)/2), x), x, -S(4)bc(S(3)/2)EllipticE(asin(sqrt(c)sqrt(dx)/sqrt(d)), S(-1))/(S(3)d(S(5)/2)) + S(4)bc(S(3)/2)EllipticF(asin(sqrt(c)sqrt(dx)/sqrt(d)), S(-1))/(S(3)d(S(5)/2)) - S(4)bcsqrt(-c*S(2)x*S(2) + S(1))/(S(3)d*S(2)sqrt(dx)) - (S(2)a + S(2)basin(cx))/(S(3)d(dx)*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(5)/2)(a + basin(cx))*S(2), x), x, S(16)b*S(2)c*S(2)(dx)(S(11)/2)HypergeometricPFQ(List(S(1), S(11)/4, S(11)/4), List(S(13)/4, S(15)/4), c*S(2)x*S(2))/(S(693)d*S(3)) - S(8)bc(dx)(S(9)/2)(a + basin(cx))Hypergeometric2F1(S(1)/2, S(9)/4, S(13)/4, cS(2)x*S(2))/(S(63)d*S(2)) + S(2)(dx)(S(7)/2)(a + basin(cx))*S(2)/(S(7)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(3)/2)(a + basin(cx))*S(2), x), x, S(16)b*S(2)c*S(2)(dx)(S(9)/2)HypergeometricPFQ(List(S(1), S(9)/4, S(9)/4), List(S(11)/4, S(13)/4), c*S(2)x*S(2))/(S(315)d*S(3)) - S(8)bc(dx)(S(7)/2)(a + basin(cx))Hypergeometric2F1(S(1)/2, S(7)/4, S(11)/4, cS(2)x*S(2))/(S(35)d*S(2)) + S(2)(dx)(S(5)/2)(a + basin(cx))*S(2)/(S(5)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)(a + basin(cx))*S(2), x), x, S(16)b*S(2)c*S(2)(dx)(S(7)/2)HypergeometricPFQ(List(S(1), S(7)/4, S(7)/4), List(S(9)/4, S(11)/4), c*S(2)x*S(2))/(S(105)d*S(3)) - S(8)bc(dx)(S(5)/2)(a + basin(cx))Hypergeometric2F1(S(1)/2, S(5)/4, S(9)/4, cS(2)x*S(2))/(S(15)d*S(2)) + S(2)(dx)(S(3)/2)(a + basin(cx))*S(2)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))*S(2)/sqrt(dx), x), x, S(16)bS(2)c*S(2)(dx)(S(5)/2)HypergeometricPFQ(List(S(1), S(5)/4, S(5)/4), List(S(7)/4, S(9)/4), c*S(2)x*S(2))/(S(15)d*S(3)) - S(8)bc(dx)(S(3)/2)(a + basin(cx))Hypergeometric2F1(S(1)/2, S(3)/4, S(7)/4, cS(2)x*S(2))/(S(3)d*S(2)) + S(2)sqrt(dx)(a + basin(cx))*S(2)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))S(2)/(dx)*(S(3)/2), x), x, -S(16)b*S(2)c*S(2)(dx)(S(3)/2)HypergeometricPFQ(List(S(3)/4, S(3)/4, S(1)), List(S(5)/4, S(7)/4), c*S(2)x*S(2))/(S(3)d*S(3)) + S(8)bcsqrt(dx)(a + basin(cx))Hypergeometric2F1(S(1)/4, S(1)/2, S(5)/4, cS(2)xS(2))/dS(2) - S(2)(a + basin(cx))S(2)/(dsqrt(dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))S(2)/(dx)*(S(5)/2), x), x, S(16)b*S(2)c*S(2)sqrt(dx)HypergeometricPFQ(List(S(1)/4, S(1)/4, S(1)), List(S(3)/4, S(5)/4), c*S(2)x*S(2))/(S(3)d*S(3)) - S(8)bc(a + basin(cx))Hypergeometric2F1(S(-1)/4, S(1)/2, S(3)/4, cS(2)x*S(2))/(S(3)d*S(2)sqrt(dx)) - S(2)(a + basin(cx))*S(2)/(S(3)d(dx)*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(3)/2)(a + basin(cx))*S(3), x), x, -S(6)bcIntegrate((dx)(S(5)/2)(a + basin(cx))S(2)/sqrt(-cS(2)xS(2) + S(1)), x)/(S(5)d) + S(2)(dx)*(S(5)/2)(a + basin(cx))*S(3)/(S(5)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)(a + basin(cx))*S(3), x), x, -S(2)bcIntegrate((dx)(S(3)/2)(a + basin(cx))S(2)/sqrt(-cS(2)xS(2) + S(1)), x)/d + S(2)(dx)(S(3)/2)(a + basin(cx))*S(3)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))*S(3)/sqrt(dx), x), x, -S(6)bcIntegrate(sqrt(dx)(a + basin(cx))S(2)/sqrt(-cS(2)x*S(2) + S(1)), x)/d + S(2)sqrt(dx)(a + basin(cx))*S(3)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))S(3)/(dx)*(S(3)/2), x), x, S(6)bcIntegrate((a + basin(cx))*S(2)/(sqrt(dx)sqrt(-cS(2)x*S(2) + S(1))), x)/d - S(2)(a + basin(cx))*S(3)/(dsqrt(dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + basin(cx))S(3)/(dx)*(S(5)/2), x), x, S(2)bcIntegrate((a + basin(cx))*S(2)/((dx)*(S(3)/2)sqrt(-c*S(2)x*S(2) + S(1))), x)/d - S(2)(a + basin(cx))*S(3)/(S(3)d(dx)*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(3)/2)/(a + basin(cx)), x), x, Integrate((dx)*(S(3)/2)/(a + basin(cx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)/(a + basin(cx)), x), x, Integrate(sqrt(dx)/(a + basin(cx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(dx)(a + basin(cx))), x), x, Integrate(S(1)/(sqrt(dx)(a + basin(cx))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(3)/2)(a + basin(cx))), x), x, Integrate(S(1)/((dx)(S(3)/2)(a + basin(cx))), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)(S(3)/2)/(a + basin(cx))S(2), x), x, Integrate((dx)*(S(3)/2)/(a + basin(cx))S(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)/(a + basin(cx))*S(2), x), x, Integrate(sqrt(dx)/(a + basin(cx))*S(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(dx)(a + basin(cx))S(2)), x), x, Integrate(S(1)/(sqrt(dx)(a + basin(cx))S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((dx)*(S(3)/2)(a + basin(cx))*S(2)), x), x, Integrate(S(1)/((dx)*(S(3)/2)(a + basin(cx))**S(2)), x), expand=True, _diff=True, _numerical=True)
cbc4f9e81003123f13126be47639bb79f609062a604a468e149919731f3a2468	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.utility_function import ( sympy_op_factory, Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ, ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ, PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ, NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest, SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient, Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan, ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec, ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less, Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ, InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ, EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, ExpandLinearProduct, GCD, ContentFactor, NumericFactor, NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, ExpandExpression, Apart, SmartApart, MatchQ, PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, RationalFunctionFactors, NonrationalFunctionFactors, Reverse, RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, GeneralizedBinomialParts, GeneralizedTrinomialDegree, GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, MinimumMonomialExponent, MonomialExponent, LinearMatchQ, PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, AlgebraicFunctionFactors, NonalgebraicFunctionFactors, QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, CombineExponents, FactorInteger, FactorAbsurdNumber, SubstForInverseFunction, SubstForFractionalPower, SubstForFractionalPowerOfQuotientOfLinears, FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, NormalizeIntegrandAux, NormalizeIntegrandFactor, NormalizeIntegrandFactorBase, NormalizeTogether, NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand, SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, KnownTangentIntegrandQ, KnownCotangentIntegrandQ, KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, FunctionOfLog, PowerVariableExpn, PowerVariableDegree, PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, ConstantFactor, SameQ, ReplacePart, CommonFactors, MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, _SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify, _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor, PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral, CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial, Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ ) from sympy.core.add import Add from sympy.core.mod import Mod from sympy.core.mul import Mul from sympy.core.numbers import (Float, I, Integer) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.functions.elementary.complexes import Abs from sympy.functions.elementary.miscellaneous import sqrt from sympy.integrals.integrals import Integral as Integrate from sympy.logic.boolalg import (And, Or) from sympy.simplify.simplify import simplify from sympy.integrals.rubi.symbol import WC from sympy.core.symbol import symbols, Symbol from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf, exp, log) from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch) from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec) from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.core.numbers import pi as Pi a, b, c, d, e, f, m, n, x, u , k, p, r, s, t, i, j= symbols('a b c d e f m n x u k p r s t i j') A, B, C, D, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B C D a b c d e f g h y z m n p q u v w F', ) def test_1(): assert rubi_test(rubi_integrate((c + dx)S(4)sinh(a + bx), x), x, (c + dx)*S(4)cosh(a + bx)/b - S(4)d(c + dx)*S(3)sinh(a + bx)/bS(2) + S(12)d*S(2)(c + dx)S(2)cosh(a + bx)/bS(3) - S(24)d*S(3)(c + dx)sinh(a + bx)/bS(4) + S(24)d*S(4)cosh(a + bx)/bS(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)sinh(a + bx), x), x, (c + dx)*S(3)cosh(a + bx)/b - S(3)d(c + dx)*S(2)sinh(a + bx)/bS(2) + S(6)d*S(2)(c + dx)cosh(a + bx)/bS(3) - S(6)d*S(3)sinh(a + bx)/bS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(2)sinh(a + bx), x), x, (c + dx)*S(2)cosh(a + bx)/b - S(2)d(c + dx)sinh(a + bx)/b*S(2) + S(2)d*S(2)cosh(a + bx)/bS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)sinh(a + bx), x), x, (c + dx)cosh(a + bx)/b - dsinh(a + bx)/bS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)/(c + dx), x), x, CoshIntegral(bc/d + bx)sinh(a - bc/d)/d + SinhIntegral(bc/d + bx)cosh(a - bc/d)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)/(c + dx)S(2), x), x, bCoshIntegral(bc/d + bx)cosh(a - bc/d)/d*S(2) + bSinhIntegral(bc/d + bx)sinh(a - bc/d)/d*S(2) - sinh(a + bx)/(d(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)/(c + dx)S(3), x), x, bS(2)CoshIntegral(bc/d + bx)sinh(a - bc/d)/(S(2)dS(3)) + bS(2)SinhIntegral(bc/d + bx)cosh(a - bc/d)/(S(2)d*S(3)) - bcosh(a + bx)/(S(2)d*S(2)(c + dx)) - sinh(a + bx)/(S(2)d(c + dx)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(4)sinh(a + bx)S(2), x), x, -(c + dx)*S(5)/(S(10)d) + (c + dx)S(4)sinh(a + bx)cosh(a + bx)/(S(2)b) - d(c + dx)*S(3)sinh(a + bx)S(2)/bS(2) - d(c + dx)S(3)/(S(2)b*S(2)) + S(3)d*S(2)(c + dx)S(2)sinh(a + bx)cosh(a + bx)/(S(2)b*S(3)) - S(3)d*S(4)x/(S(4)bS(4)) - S(3)d*S(3)(c + dx)sinh(a + bx)S(2)/(S(2)b*S(4)) + S(3)d*S(4)sinh(a + bx)cosh(a + bx)/(S(4)b*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)sinh(a + bx)S(2), x), x, -(c + dx)*S(4)/(S(8)d) + (c + dx)S(3)sinh(a + bx)cosh(a + bx)/(S(2)b) - S(3)cd*S(2)x/(S(4)bS(2)) - S(3)d*S(3)x*S(2)/(S(8)b*S(2)) - S(3)d(c + dx)*S(2)sinh(a + bx)S(2)/(S(4)b*S(2)) + S(3)d*S(2)(c + dx)sinh(a + bx)cosh(a + bx)/(S(4)b*S(3)) - S(3)d*S(3)sinh(a + bx)S(2)/(S(8)b*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(2)sinh(a + bx)S(2), x), x, -(c + dx)*S(3)/(S(6)d) + (c + dx)S(2)sinh(a + bx)cosh(a + bx)/(S(2)b) - d*S(2)x/(S(4)bS(2)) - d(c + dx)sinh(a + bx)S(2)/(S(2)bS(2)) + dS(2)sinh(a + bx)cosh(a + bx)/(S(4)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)sinh(a + bx)*S(2), x), x, -cx/S(2) - dxS(2)/S(4) + (c + dx)sinh(a + bx)cosh(a + bx)/(S(2)b) - dsinh(a + bx)S(2)/(S(4)b*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)*S(2)/(c + dx), x), x, CoshIntegral(S(2)bc/d + S(2)bx)cosh(S(2)a - S(2)bc/d)/(S(2)d) + SinhIntegral(S(2)bc/d + S(2)bx)sinh(S(2)a - S(2)bc/d)/(S(2)d) - log(c + dx)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)S(2)/(c + dx)*S(2), x), x, bCoshIntegral(S(2)bc/d + S(2)bx)sinh(S(2)a - S(2)bc/d)/d*S(2) + bSinhIntegral(S(2)bc/d + S(2)bx)cosh(S(2)a - S(2)bc/d)/d*S(2) - sinh(a + bx)*S(2)/(d(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)*S(2)/(c + dx)S(3), x), x, bS(2)CoshIntegral(S(2)bc/d + S(2)bx)cosh(S(2)a - S(2)bc/d)/dS(3) + bS(2)SinhIntegral(S(2)bc/d + S(2)bx)sinh(S(2)a - S(2)bc/d)/d*S(3) - bsinh(a + bx)cosh(a + bx)/(dS(2)(c + dx)) - sinh(a + bx)*S(2)/(S(2)d(c + dx)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)*S(2)/(c + dx)*S(4), x), x, S(2)b*S(3)CoshIntegral(S(2)bc/d + S(2)bx)sinh(S(2)a - S(2)bc/d)/(S(3)dS(4)) + S(2)b*S(3)SinhIntegral(S(2)bc/d + S(2)bx)cosh(S(2)a - S(2)bc/d)/(S(3)dS(4)) - S(2)b*S(2)sinh(a + bx)S(2)/(S(3)d*S(3)(c + dx)) - bS(2)/(S(3)d*S(3)(c + dx)) - bsinh(a + bx)cosh(a + bx)/(S(3)d*S(2)(c + dx)S(2)) - sinh(a + bx)*S(2)/(S(3)d(c + dx)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(4)sinh(a + bx)S(3), x), x, (c + dx)*S(4)sinh(a + bx)S(2)cosh(a + bx)/(S(3)b) - S(2)(c + dx)*S(4)cosh(a + bx)/(S(3)b) - S(4)d(c + dx)S(3)sinh(a + bx)S(3)/(S(9)b*S(2)) + S(8)d(c + dx)*S(3)sinh(a + bx)/(S(3)b*S(2)) + S(4)d*S(2)(c + dx)S(2)sinh(a + bx)S(2)cosh(a + bx)/(S(9)b*S(3)) - S(80)d*S(2)(c + dx)S(2)cosh(a + bx)/(S(9)b*S(3)) - S(8)d*S(3)(c + dx)sinh(a + bx)S(3)/(S(27)b*S(4)) + S(160)d*S(3)(c + dx)sinh(a + bx)/(S(9)b*S(4)) + S(8)d*S(4)cosh(a + bx)S(3)/(S(81)b*S(5)) - S(488)d*S(4)cosh(a + bx)/(S(27)b*S(5)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)sinh(a + bx)S(3), x), x, (c + dx)*S(3)sinh(a + bx)S(2)cosh(a + bx)/(S(3)b) - S(2)(c + dx)*S(3)cosh(a + bx)/(S(3)b) - d(c + dx)*S(2)sinh(a + bx)S(3)/(S(3)b*S(2)) + S(2)d(c + dx)*S(2)sinh(a + bx)/bS(2) + S(2)d*S(2)(c + dx)sinh(a + bx)S(2)cosh(a + bx)/(S(9)b*S(3)) - S(40)d*S(2)(c + dx)cosh(a + bx)/(S(9)b*S(3)) - S(2)d*S(3)sinh(a + bx)S(3)/(S(27)b*S(4)) + S(40)d*S(3)sinh(a + bx)/(S(9)b*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(2)sinh(a + bx)S(3), x), x, (c + dx)*S(2)sinh(a + bx)S(2)cosh(a + bx)/(S(3)b) - S(2)(c + dx)*S(2)cosh(a + bx)/(S(3)b) - S(2)d(c + dx)sinh(a + bx)S(3)/(S(9)b*S(2)) + S(4)d(c + dx)sinh(a + bx)/(S(3)bS(2)) + S(2)d*S(2)cosh(a + bx)S(3)/(S(27)b*S(3)) - S(14)d*S(2)cosh(a + bx)/(S(9)b*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)sinh(a + bx)*S(3), x), x, (c + dx)sinh(a + bx)*S(2)cosh(a + bx)/(S(3)b) - S(2)(c + dx)cosh(a + bx)/(S(3)b) - dsinh(a + bx)S(3)/(S(9)b*S(2)) + S(2)dsinh(a + bx)/(S(3)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)*S(3)/(c + dx), x), x, -S(3)CoshIntegral(bc/d + bx)sinh(a - bc/d)/(S(4)d) + CoshIntegral(S(3)bc/d + S(3)bx)sinh(S(3)a - S(3)bc/d)/(S(4)d) - S(3)SinhIntegral(bc/d + bx)cosh(a - bc/d)/(S(4)d) + SinhIntegral(S(3)bc/d + S(3)bx)cosh(S(3)a - S(3)bc/d)/(S(4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)S(3)/(c + dx)*S(2), x), x, -S(3)bCoshIntegral(bc/d + bx)cosh(a - bc/d)/(S(4)d*S(2)) + S(3)bCoshIntegral(S(3)bc/d + S(3)bx)cosh(S(3)a - S(3)bc/d)/(S(4)d*S(2)) - S(3)bSinhIntegral(bc/d + bx)sinh(a - bc/d)/(S(4)d*S(2)) + S(3)bSinhIntegral(S(3)bc/d + S(3)bx)sinh(S(3)a - S(3)bc/d)/(S(4)d*S(2)) - sinh(a + bx)*S(3)/(d(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)*S(3)/(c + dx)*S(3), x), x, -S(3)b*S(2)CoshIntegral(bc/d + bx)sinh(a - bc/d)/(S(8)dS(3)) + S(9)b*S(2)CoshIntegral(S(3)bc/d + S(3)bx)sinh(S(3)a - S(3)bc/d)/(S(8)dS(3)) - S(3)b*S(2)SinhIntegral(bc/d + bx)cosh(a - bc/d)/(S(8)dS(3)) + S(9)b*S(2)SinhIntegral(S(3)bc/d + S(3)bx)cosh(S(3)a - S(3)bc/d)/(S(8)dS(3)) - S(3)bsinh(a + bx)*S(2)cosh(a + bx)/(S(2)d*S(2)(c + dx)) - sinh(a + bx)*S(3)/(S(2)d(c + dx)*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)/sinh(a + bx), x), x, -S(2)(c + dx)*S(3)atanh(exp(a + bx))/b - S(3)d(c + dx)*S(2)PolyLog(S(2), -exp(a + bx))/bS(2) + S(3)d(c + dx)*S(2)PolyLog(S(2), exp(a + bx))/bS(2) + S(6)d*S(2)(c + dx)PolyLog(S(3), -exp(a + bx))/bS(3) - S(6)d*S(2)(c + dx)PolyLog(S(3), exp(a + bx))/bS(3) - S(6)d*S(3)PolyLog(S(4), -exp(a + bx))/bS(4) + S(6)d*S(3)PolyLog(S(4), exp(a + bx))/bS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(2)/sinh(a + bx), x), x, -S(2)(c + dx)*S(2)atanh(exp(a + bx))/b - S(2)d(c + dx)PolyLog(S(2), -exp(a + bx))/b*S(2) + S(2)d(c + dx)PolyLog(S(2), exp(a + bx))/b*S(2) + S(2)d*S(2)PolyLog(S(3), -exp(a + bx))/bS(3) - S(2)d*S(2)PolyLog(S(3), exp(a + bx))/bS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)/sinh(a + bx), x), x, -S(2)(c + dx)atanh(exp(a + bx))/b - dPolyLog(S(2), -exp(a + bx))/bS(2) + dPolyLog(S(2), exp(a + bx))/bS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((c + dx)sinh(a + bx)), x), x, Integrate(S(1)/((c + dx)sinh(a + bx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((c + dx)*S(2)sinh(a + bx)), x), x, Integrate(S(1)/((c + dx)*S(2)sinh(a + bx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)/sinh(a + bx)*S(2), x), x, -(c + dx)*S(3)/b - (c + dx)*S(3)/(btanh(a + bx)) + S(3)d(c + dx)*S(2)log(-exp(S(2)a + S(2)bx) + S(1))/bS(2) + S(3)d*S(2)(c + dx)PolyLog(S(2), exp(S(2)a + S(2)bx))/bS(3) - S(3)d*S(3)PolyLog(S(3), exp(S(2)a + S(2)bx))/(S(2)b*S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(2)/sinh(a + bx)*S(2), x), x, -(c + dx)*S(2)/b - (c + dx)*S(2)/(btanh(a + bx)) + S(2)d(c + dx)log(-exp(S(2)a + S(2)bx) + S(1))/bS(2) + dS(2)PolyLog(S(2), exp(S(2)a + S(2)bx))/b*S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)/sinh(a + bx)S(2), x), x, -(c + dx)/(btanh(a + bx)) + dlog(sinh(a + bx))/b*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((c + dx)sinh(a + bx)*S(2)), x), x, Integrate(S(1)/((c + dx)sinh(a + bx)*S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((c + dx)*S(2)sinh(a + bx)S(2)), x), x, Integrate(S(1)/((c + dx)*S(2)sinh(a + bx)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)/sinh(a + bx)*S(3), x), x, (c + dx)*S(3)atanh(exp(a + bx))/b - (c + dx)*S(3)/(S(2)bsinh(a + bx)tanh(a + bx)) + S(3)d(c + dx)S(2)PolyLog(S(2), -exp(a + bx))/(S(2)b*S(2)) - S(3)d(c + dx)*S(2)PolyLog(S(2), exp(a + bx))/(S(2)b*S(2)) - S(3)d(c + dx)*S(2)/(S(2)b*S(2)sinh(a + bx)) - S(3)d*S(2)(c + dx)PolyLog(S(3), -exp(a + bx))/bS(3) + S(3)d*S(2)(c + dx)PolyLog(S(3), exp(a + bx))/bS(3) - S(6)d*S(2)(c + dx)atanh(exp(a + bx))/bS(3) - S(3)d*S(3)PolyLog(S(2), -exp(a + bx))/bS(4) + S(3)d*S(3)PolyLog(S(2), exp(a + bx))/bS(4) + S(3)d*S(3)PolyLog(S(4), -exp(a + bx))/bS(4) - S(3)d*S(3)PolyLog(S(4), exp(a + bx))/bS(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(2)/sinh(a + bx)*S(3), x), x, (c + dx)*S(2)atanh(exp(a + bx))/b - (c + dx)*S(2)/(S(2)bsinh(a + bx)tanh(a + bx)) + d(c + dx)PolyLog(S(2), -exp(a + bx))/b*S(2) - d(c + dx)PolyLog(S(2), exp(a + bx))/bS(2) - d(c + dx)/(bS(2)sinh(a + bx)) - dS(2)PolyLog(S(3), -exp(a + bx))/bS(3) + dS(2)PolyLog(S(3), exp(a + bx))/bS(3) - dS(2)atanh(cosh(a + bx))/bS(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)/sinh(a + bx)S(3), x), x, (c + dx)atanh(exp(a + bx))/b - (c + dx)/(S(2)bsinh(a + bx)tanh(a + bx)) + dPolyLog(S(2), -exp(a + bx))/(S(2)bS(2)) - dPolyLog(S(2), exp(a + bx))/(S(2)b*S(2)) - d/(S(2)b*S(2)sinh(a + bx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((c + dx)sinh(a + bx)*S(3)), x), x, Integrate(S(1)/((c + dx)sinh(a + bx)*S(3)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((c + dx)*S(2)sinh(a + bx)S(3)), x), x, Integrate(S(1)/((c + dx)*S(2)sinh(a + bx)S(3)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*(S(5)/2)sinh(a + bx), x), x, -S(15)sqrt(Pi)d(S(5)/2)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(16)b(S(7)/2)) - S(15)sqrt(Pi)d(S(5)/2)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(16)b(S(7)/2)) + (c + dx)*(S(5)/2)cosh(a + bx)/b - S(5)d(c + dx)*(S(3)/2)sinh(a + bx)/(S(2)b*S(2)) + S(15)d*S(2)sqrt(c + dx)cosh(a + bx)/(S(4)b*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*(S(3)/2)sinh(a + bx), x), x, -S(3)sqrt(Pi)d(S(3)/2)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(8)b(S(5)/2)) + S(3)sqrt(Pi)d(S(3)/2)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(8)b(S(5)/2)) + (c + dx)*(S(3)/2)cosh(a + bx)/b - S(3)dsqrt(c + dx)sinh(a + bx)/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(c + dx)sinh(a + bx), x), x, -sqrt(Pi)sqrt(d)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(4)b(S(3)/2)) - sqrt(Pi)sqrt(d)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(4)b*(S(3)/2)) + sqrt(c + dx)cosh(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)/sqrt(c + dx), x), x, -sqrt(Pi)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(2)sqrt(b)sqrt(d)) + sqrt(Pi)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(2)sqrt(b)sqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)/(c + dx)*(S(3)/2), x), x, sqrt(Pi)sqrt(b)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/d(S(3)/2) + sqrt(Pi)sqrt(b)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/d(S(3)/2) - S(2)sinh(a + bx)/(dsqrt(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)/(c + dx)(S(5)/2), x), x, -S(2)sqrt(Pi)b(S(3)/2)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(3)d(S(5)/2)) + S(2)sqrt(Pi)b(S(3)/2)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(3)d(S(5)/2)) - S(4)bcosh(a + bx)/(S(3)dS(2)sqrt(c + dx)) - S(2)sinh(a + bx)/(S(3)d(c + dx)*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)/(c + dx)(S(7)/2), x), x, S(4)sqrt(Pi)b(S(5)/2)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(15)d(S(7)/2)) + S(4)sqrt(Pi)b(S(5)/2)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(15)d(S(7)/2)) - S(8)b*S(2)sinh(a + bx)/(S(15)d*S(3)sqrt(c + dx)) - S(4)bcosh(a + bx)/(S(15)dS(2)(c + dx)(S(3)/2)) - S(2)sinh(a + bx)/(S(5)d(c + dx)*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*(S(5)/2)sinh(a + bx)S(2), x), x, S(15)sqrt(S(2))sqrt(Pi)d*(S(5)/2)Erf(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(2)a + S(2)bc/d)/(S(512)b*(S(7)/2)) - S(15)sqrt(S(2))sqrt(Pi)d*(S(5)/2)Erfi(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(2)a - S(2)bc/d)/(S(512)b*(S(7)/2)) - (c + dx)*(S(7)/2)/(S(7)d) + (c + dx)(S(5)/2)sinh(a + bx)cosh(a + bx)/(S(2)b) - S(5)d(c + dx)(S(3)/2)sinh(a + bx)S(2)/(S(8)b*S(2)) - S(5)d(c + dx)*(S(3)/2)/(S(16)b*S(2)) + S(15)d*S(2)sqrt(c + dx)sinh(S(2)a + S(2)bx)/(S(64)b*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*(S(3)/2)sinh(a + bx)S(2), x), x, S(3)sqrt(S(2))sqrt(Pi)d*(S(3)/2)Erf(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(2)a + S(2)bc/d)/(S(128)b*(S(5)/2)) + S(3)sqrt(S(2))sqrt(Pi)d*(S(3)/2)Erfi(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(2)a - S(2)bc/d)/(S(128)b*(S(5)/2)) - (c + dx)*(S(5)/2)/(S(5)d) + (c + dx)(S(3)/2)sinh(a + bx)cosh(a + bx)/(S(2)b) - S(3)dsqrt(c + dx)sinh(a + bx)S(2)/(S(8)b*S(2)) - S(3)dsqrt(c + dx)/(S(16)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(c + dx)sinh(a + bx)*S(2), x), x, sqrt(S(2))sqrt(Pi)sqrt(d)Erf(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(2)a + S(2)bc/d)/(S(32)b*(S(3)/2)) - sqrt(S(2))sqrt(Pi)sqrt(d)Erfi(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(2)a - S(2)bc/d)/(S(32)b*(S(3)/2)) - (c + dx)*(S(3)/2)/(S(3)d) + sqrt(c + dx)sinh(S(2)a + S(2)bx)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)S(2)/sqrt(c + dx), x), x, sqrt(S(2))sqrt(Pi)Erf(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(2)a + S(2)bc/d)/(S(8)sqrt(b)sqrt(d)) + sqrt(S(2))sqrt(Pi)Erfi(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(2)a - S(2)bc/d)/(S(8)sqrt(b)sqrt(d)) - sqrt(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)*S(2)/(c + dx)*(S(3)/2), x), x, -sqrt(S(2))sqrt(Pi)sqrt(b)Erf(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(2)a + S(2)bc/d)/(S(2)d*(S(3)/2)) + sqrt(S(2))sqrt(Pi)sqrt(b)Erfi(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(2)a - S(2)bc/d)/(S(2)d*(S(3)/2)) - S(2)sinh(a + bx)S(2)/(dsqrt(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)*S(2)/(c + dx)*(S(5)/2), x), x, S(2)sqrt(S(2))sqrt(Pi)b*(S(3)/2)Erf(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(2)a + S(2)bc/d)/(S(3)d*(S(5)/2)) + S(2)sqrt(S(2))sqrt(Pi)b*(S(3)/2)Erfi(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(2)a - S(2)bc/d)/(S(3)d*(S(5)/2)) - S(8)bsinh(a + bx)cosh(a + bx)/(S(3)dS(2)sqrt(c + dx)) - S(2)sinh(a + bx)S(2)/(S(3)d(c + dx)*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)*S(2)/(c + dx)*(S(7)/2), x), x, -S(8)sqrt(S(2))sqrt(Pi)b*(S(5)/2)Erf(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(2)a + S(2)bc/d)/(S(15)d*(S(7)/2)) + S(8)sqrt(S(2))sqrt(Pi)b*(S(5)/2)Erfi(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(2)a - S(2)bc/d)/(S(15)d*(S(7)/2)) - S(32)b*S(2)sinh(a + bx)S(2)/(S(15)d*S(3)sqrt(c + dx)) - S(16)b*S(2)/(S(15)d*S(3)sqrt(c + dx)) - S(8)bsinh(a + bx)cosh(a + bx)/(S(15)dS(2)(c + dx)(S(3)/2)) - S(2)sinh(a + bx)S(2)/(S(5)d(c + dx)*(S(5)/2)), expand=True, _diff=True, _numerical=True) # taking long time assert rubi_test(rubi_integrate(sinh(a + bx)*S(2)/(c + dx)*(S(9)/2), x), x, S(32)sqrt(S(2))sqrt(Pi)b*(S(7)/2)Erf(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(2)a + S(2)bc/d)/(S(105)d*(S(9)/2)) + S(32)sqrt(S(2))sqrt(Pi)b*(S(7)/2)Erfi(sqrt(S(2))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(2)a - S(2)bc/d)/(S(105)d*(S(9)/2)) - S(128)b*S(3)sinh(a + bx)cosh(a + bx)/(S(105)d*S(4)sqrt(c + dx)) - S(32)b*S(2)sinh(a + bx)S(2)/(S(105)d*S(3)(c + dx)(S(3)/2)) - S(16)b*S(2)/(S(105)d*S(3)(c + dx)(S(3)/2)) - S(8)bsinh(a + bx)cosh(a + bx)/(S(35)dS(2)(c + dx)(S(5)/2)) - S(2)sinh(a + bx)S(2)/(S(7)d(c + dx)*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*(S(5)/2)sinh(a + bx)S(3), x), x, S(45)sqrt(Pi)d(S(5)/2)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(64)b(S(7)/2)) - S(5)sqrt(S(3))sqrt(Pi)d*(S(5)/2)Erf(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(3)a + S(3)bc/d)/(S(1728)b*(S(7)/2)) + S(45)sqrt(Pi)d(S(5)/2)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(64)b(S(7)/2)) - S(5)sqrt(S(3))sqrt(Pi)d*(S(5)/2)Erfi(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(3)a - S(3)bc/d)/(S(1728)b*(S(7)/2)) + (c + dx)*(S(5)/2)sinh(a + bx)S(2)cosh(a + bx)/(S(3)b) - S(2)(c + dx)*(S(5)/2)cosh(a + bx)/(S(3)b) - S(5)d(c + dx)(S(3)/2)sinh(a + bx)S(3)/(S(18)b*S(2)) + S(5)d(c + dx)*(S(3)/2)sinh(a + bx)/(S(3)b*S(2)) - S(45)d*S(2)sqrt(c + dx)cosh(a + bx)/(S(16)b*S(3)) + S(5)d*S(2)sqrt(c + dx)cosh(S(3)a + S(3)bx)/(S(144)b*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*(S(3)/2)sinh(a + bx)S(3), x), x, S(9)sqrt(Pi)d(S(3)/2)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(32)b(S(5)/2)) - sqrt(S(3))sqrt(Pi)d(S(3)/2)Erf(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(3)a + S(3)bc/d)/(S(288)b*(S(5)/2)) - S(9)sqrt(Pi)d(S(3)/2)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(32)b(S(5)/2)) + sqrt(S(3))sqrt(Pi)d(S(3)/2)Erfi(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(3)a - S(3)bc/d)/(S(288)b*(S(5)/2)) + (c + dx)*(S(3)/2)sinh(a + bx)S(2)cosh(a + bx)/(S(3)b) - S(2)(c + dx)*(S(3)/2)cosh(a + bx)/(S(3)b) - dsqrt(c + dx)sinh(a + bx)*S(3)/(S(6)b*S(2)) + dsqrt(c + dx)sinh(a + bx)/bS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(c + dx)sinh(a + bx)*S(3), x), x, S(3)sqrt(Pi)sqrt(d)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(16)b(S(3)/2)) - sqrt(S(3))sqrt(Pi)sqrt(d)Erf(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(3)a + S(3)bc/d)/(S(144)b*(S(3)/2)) + S(3)sqrt(Pi)sqrt(d)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(16)b(S(3)/2)) - sqrt(S(3))sqrt(Pi)sqrt(d)Erfi(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(3)a - S(3)bc/d)/(S(144)b*(S(3)/2)) - S(3)sqrt(c + dx)cosh(a + bx)/(S(4)b) + sqrt(c + dx)cosh(S(3)a + S(3)bx)/(S(12)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)S(3)/sqrt(c + dx), x), x, S(3)sqrt(Pi)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(8)sqrt(b)sqrt(d)) - sqrt(S(3))sqrt(Pi)Erf(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(3)a + S(3)bc/d)/(S(24)sqrt(b)sqrt(d)) - S(3)sqrt(Pi)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(8)sqrt(b)sqrt(d)) + sqrt(S(3))sqrt(Pi)Erfi(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(3)a - S(3)bc/d)/(S(24)sqrt(b)sqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)S(3)/(c + dx)*(S(3)/2), x), x, -S(3)sqrt(Pi)sqrt(b)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(4)d(S(3)/2)) + sqrt(S(3))sqrt(Pi)sqrt(b)Erf(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(3)a + S(3)bc/d)/(S(4)d*(S(3)/2)) - S(3)sqrt(Pi)sqrt(b)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(4)d(S(3)/2)) + sqrt(S(3))sqrt(Pi)sqrt(b)Erfi(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(3)a - S(3)bc/d)/(S(4)d*(S(3)/2)) - S(2)sinh(a + bx)S(3)/(dsqrt(c + dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(a + bx)*S(3)/(c + dx)*(S(5)/2), x), x, sqrt(Pi)b*(S(3)/2)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(2)d(S(5)/2)) - sqrt(S(3))sqrt(Pi)b(S(3)/2)Erf(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(3)a + S(3)bc/d)/(S(2)d*(S(5)/2)) - sqrt(Pi)b*(S(3)/2)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(2)d(S(5)/2)) + sqrt(S(3))sqrt(Pi)b(S(3)/2)Erfi(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(3)a - S(3)bc/d)/(S(2)d*(S(5)/2)) - S(4)bsinh(a + bx)*S(2)cosh(a + bx)/(dS(2)sqrt(c + dx)) - S(2)sinh(a + bx)S(3)/(S(3)d(c + dx)*(S(3)/2)), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate(sinh(a + bx)*S(3)/(c + dx)*(S(7)/2), x), x, -sqrt(Pi)b*(S(5)/2)Erf(sqrt(b)sqrt(c + dx)/sqrt(d))exp(-a + bc/d)/(S(5)d(S(7)/2)) + S(3)sqrt(S(3))sqrt(Pi)b*(S(5)/2)Erf(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(-S(3)a + S(3)bc/d)/(S(5)d*(S(7)/2)) - sqrt(Pi)b*(S(5)/2)Erfi(sqrt(b)sqrt(c + dx)/sqrt(d))exp(a - bc/d)/(S(5)d(S(7)/2)) + S(3)sqrt(S(3))sqrt(Pi)b*(S(5)/2)Erfi(sqrt(S(3))sqrt(b)sqrt(c + dx)/sqrt(d))exp(S(3)a - S(3)bc/d)/(S(5)d*(S(7)/2)) - S(24)b*S(2)sinh(a + bx)S(3)/(S(5)d*S(3)sqrt(c + dx)) - S(16)b*S(2)sinh(a + bx)/(S(5)d*S(3)sqrt(c + dx)) - S(4)bsinh(a + bx)*S(2)cosh(a + bx)/(S(5)d*S(2)(c + dx)(S(3)/2)) - S(2)sinh(a + bx)S(3)/(S(5)d(c + dx)*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dx)*(S(3)/2)sinh(fx), x), x, -S(3)sqrt(Pi)d(S(3)/2)Erf(sqrt(f)sqrt(dx)/sqrt(d))/(S(8)f(S(5)/2)) + S(3)sqrt(Pi)d(S(3)/2)Erfi(sqrt(f)sqrt(dx)/sqrt(d))/(S(8)f(S(5)/2)) - S(3)dsqrt(dx)sinh(fx)/(S(2)fS(2)) + (dx)*(S(3)/2)cosh(fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dx)sinh(fx), x), x, -sqrt(Pi)sqrt(d)Erf(sqrt(f)sqrt(dx)/sqrt(d))/(S(4)f(S(3)/2)) - sqrt(Pi)sqrt(d)Erfi(sqrt(f)sqrt(dx)/sqrt(d))/(S(4)f*(S(3)/2)) + sqrt(dx)cosh(fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(fx)/sqrt(dx), x), x, -sqrt(Pi)Erf(sqrt(f)sqrt(dx)/sqrt(d))/(S(2)sqrt(d)sqrt(f)) + sqrt(Pi)Erfi(sqrt(f)sqrt(dx)/sqrt(d))/(S(2)sqrt(d)sqrt(f)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(fx)/(dx)*(S(3)/2), x), x, sqrt(Pi)sqrt(f)Erf(sqrt(f)sqrt(dx)/sqrt(d))/d(S(3)/2) + sqrt(Pi)sqrt(f)Erfi(sqrt(f)sqrt(dx)/sqrt(d))/d(S(3)/2) - S(2)sinh(fx)/(dsqrt(dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(fx)/(dx)(S(5)/2), x), x, -S(2)sqrt(Pi)f(S(3)/2)Erf(sqrt(f)sqrt(dx)/sqrt(d))/(S(3)d(S(5)/2)) + S(2)sqrt(Pi)f(S(3)/2)Erfi(sqrt(f)sqrt(dx)/sqrt(d))/(S(3)d(S(5)/2)) - S(2)sinh(fx)/(S(3)d(dx)*(S(3)/2)) - S(4)fcosh(fx)/(S(3)dS(2)sqrt(dx)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(c + dx)/sinh(a + bx), x), x, Integrate(sqrt(c + dx)/sinh(a + bx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(c + dx)sinh(a + bx)), x), x, Integrate(S(1)/(sqrt(c + dx)sinh(a + bx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sinh(x)(S(3)/2)/xS(3), x), x, S(3)Integrate(S(1)/(xsqrt(sinh(x))), x)/S(8) + S(9)Integrate(sinh(x)*(S(3)/2)/x, x)/S(8) - S(3)sqrt(sinh(x))cosh(x)/(S(4)x) - sinh(x)*(S(3)/2)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-xsqrt(sinh(x)) + x/sinh(x)*(S(3)/2), x), x, -S(2)xcosh(x)/sqrt(sinh(x)) + S(4)sqrt(sinh(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x/(S(3)sqrt(sinh(x))) + x/sinh(x)(S(5)/2), x), x, -S(2)xcosh(x)/(S(3)sinh(x)*(S(3)/2)) - S(4)/(S(3)sqrt(sinh(x))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(3)xsqrt(sinh(x))/S(5) + x/sinh(x)*(S(7)/2), x), x, S(6)xcosh(x)/(S(5)sqrt(sinh(x))) - S(2)xcosh(x)/(S(5)sinh(x)(S(5)/2)) - S(12)sqrt(sinh(x))/S(5) - S(4)/(S(15)sinh(x)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-xS(2)sqrt(sinh(x)) + xS(2)/sinh(x)(S(3)/2), x), x, -S(2)xS(2)cosh(x)/sqrt(sinh(x)) + S(8)xsqrt(sinh(x)) - S(16)IEllipticE(Pi/S(4) - Ix/S(2), S(2))sqrt(sinh(x))/sqrt(Isinh(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsinh(e + fx))n(c + dx)m, x), x, Integrate((bsinh(e + fx))n(c + dx)m, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*msinh(a + bx)S(3), x), x, S(3)(-m + S(-1))(-b(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), -S(3)b(c + dx)/d)exp(S(3)a - S(3)bc/d)/(S(8)b) + S(3)*(-m + S(-1))(b(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), S(3)b(c + dx)/d)exp(-S(3)a + S(3)bc/d)/(S(8)b) - S(3)(-b(c + dx)/d)(-m)(c + dx)mGamma(m + S(1), -b(c + dx)/d)exp(a - bc/d)/(S(8)b) - S(3)(b(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), b(c + dx)/d)exp(-a + bc/d)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*msinh(a + bx)S(2), x), x, S(2)(-m + S(-3))(-b(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), -S(2)b(c + dx)/d)exp(S(2)a - S(2)bc/d)/b - S(2)(-m + S(-3))(b(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), S(2)b(c + dx)/d)exp(-S(2)a + S(2)bc/d)/b - (c + dx)*(m + S(1))/(S(2)d(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*msinh(a + bx), x), x, (-b(c + dx)/d)(-m)(c + dx)mGamma(m + S(1), -b(c + dx)/d)exp(a - bc/d)/(S(2)b) + (b(c + dx)/d)(-m)(c + dx)mGamma(m + S(1), b(c + dx)/d)exp(-a + bc/d)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*m/sinh(a + bx), x), x, Integrate((c + dx)m/sinh(a + bx), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)m/sinh(a + bx)*S(2), x), x, Integrate((c + dx)*m/sinh(a + bx)S(2), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(m + S(3))sinh(a + bx), x), x, -x*m(-bx)(-m)Gamma(m + S(4), -bx)exp(a)/(S(2)bS(4)) + xm(bx)(-m)Gamma(m + S(4), bx)exp(-a)/(S(2)bS(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(m + S(2))sinh(a + bx), x), x, xm(-bx)(-m)Gamma(m + S(3), -bx)exp(a)/(S(2)bS(3)) + xm(bx)(-m)Gamma(m + S(3), bx)exp(-a)/(S(2)bS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(m + S(1))sinh(a + bx), x), x, -xm(-bx)(-m)Gamma(m + S(2), -bx)exp(a)/(S(2)bS(2)) + xm(bx)(-m)Gamma(m + S(2), bx)exp(-a)/(S(2)bS(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmsinh(a + bx), x), x, xm(-bx)(-m)Gamma(m + S(1), -bx)exp(a)/(S(2)b) + xm(bx)(-m)Gamma(m + S(1), bx)exp(-a)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(m + S(-1))sinh(a + bx), x), x, -xm(-bx)(-m)Gamma(m, -bx)exp(a)/S(2) + x*m(bx)(-m)Gamma(m, bx)exp(-a)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(m + S(-2))sinh(a + bx), x), x, bx*m(-bx)(-m)Gamma(m + S(-1), -bx)exp(a)/S(2) + bxm(bx)(-m)Gamma(m + S(-1), bx)exp(-a)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(m + S(-3))sinh(a + bx), x), x, -bS(2)x*m(-bx)(-m)Gamma(m + S(-2), -bx)exp(a)/S(2) + b*S(2)x*m(bx)(-m)Gamma(m + S(-2), bx)exp(-a)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(m + S(3))sinh(a + bx)S(2), x), x, -S(2)(-m + S(-6))x*m(-bx)(-m)Gamma(m + S(4), -S(2)bx)exp(S(2)a)/bS(4) - S(2)(-m + S(-6))xm(bx)(-m)Gamma(m + S(4), S(2)bx)exp(-S(2)a)/bS(4) - x(m + S(4))/(S(2)m + S(8)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(m + S(2))sinh(a + bx)S(2), x), x, S(2)(-m + S(-5))x*m(-bx)(-m)Gamma(m + S(3), -S(2)bx)exp(S(2)a)/bS(3) - S(2)(-m + S(-5))xm(bx)(-m)Gamma(m + S(3), S(2)bx)exp(-S(2)a)/bS(3) - x(m + S(3))/(S(2)m + S(6)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x(m + S(1))sinh(a + bx)S(2), x), x, -S(2)(-m + S(-4))x*m(-bx)(-m)Gamma(m + S(2), -S(2)bx)exp(S(2)a)/bS(2) - S(2)(-m + S(-4))xm(bx)(-m)Gamma(m + S(2), S(2)bx)exp(-S(2)a)/bS(2) - x(m + S(2))/(S(2)m + S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xmsinh(a + bx)S(2), x), x, S(2)(-m + S(-3))x*m(-bx)(-m)Gamma(m + S(1), -S(2)bx)exp(S(2)a)/b - S(2)*(-m + S(-3))x*m(bx)(-m)Gamma(m + S(1), S(2)bx)exp(-S(2)a)/b - x*(m + S(1))/(S(2)m + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(m + S(-1))sinh(a + bx)S(2), x), x, -S(2)(-m + S(-2))x*m(-bx)(-m)Gamma(m, -S(2)bx)exp(S(2)a) - S(2)*(-m + S(-2))x*m(bx)(-m)Gamma(m, S(2)bx)exp(-S(2)a) - x*m/(S(2)m), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(m + S(-2))sinh(a + bx)S(2), x), x, S(2)(-m + S(-1))bxm(-bx)(-m)Gamma(m + S(-1), -S(2)bx)exp(S(2)a) - S(2)*(-m + S(-1))bxm(bx)(-m)Gamma(m + S(-1), S(2)bx)exp(-S(2)a) + x*(m + S(-1))/(-S(2)m + S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(x*(m + S(-3))sinh(a + bx)S(2), x), x, x(m + S(-2))/(-S(2)m + S(4)) - S(2)*(-m)b*S(2)x*m(-bx)(-m)Gamma(m + S(-2), -S(2)bx)exp(S(2)a) - S(2)*(-m)b*S(2)x*m(bx)(-m)Gamma(m + S(-2), S(2)bx)exp(-S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xsqrt(S(1)/sinh(x))/S(3) + x/(S(1)/sinh(x))(S(3)/2), x), x, S(2)xcosh(x)/(S(3)sqrt(S(1)/sinh(x))) - S(4)/(S(9)(S(1)/sinh(x))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(3)x/(S(5)sqrt(S(1)/sinh(x))) + x/(S(1)/sinh(x))(S(5)/2), x), x, S(2)xcosh(x)/(S(5)(S(1)/sinh(x))*(S(3)/2)) - S(4)/(S(25)(S(1)/sinh(x))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(-S(5)xsqrt(S(1)/sinh(x))/S(21) + x/(S(1)/sinh(x))(S(7)/2), x), x, -S(10)xcosh(x)/(S(21)sqrt(S(1)/sinh(x))) + S(2)xcosh(x)/(S(7)(S(1)/sinh(x))(S(5)/2)) + S(20)/(S(63)(S(1)/sinh(x))*(S(3)/2)) - S(4)/(S(49)(S(1)/sinh(x))(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(xS(2)sqrt(S(1)/sinh(x))/S(3) + xS(2)/(S(1)/sinh(x))(S(3)/2), x), x, S(2)x*S(2)cosh(x)/(S(3)sqrt(S(1)/sinh(x))) - S(8)x/(S(9)(S(1)/sinh(x))(S(3)/2)) - S(16)Isqrt(Isinh(x))sqrt(S(1)/sinh(x))EllipticF(Pi/S(4) - Ix/S(2), S(2))/S(27) + S(16)cosh(x)/(S(27)sqrt(S(1)/sinh(x))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)(Iasinh(e + fx) + a), x), x, -S(6)Iad*S(3)sinh(e + fx)/fS(4) + S(6)Iad*S(2)(c + dx)cosh(e + fx)/fS(3) - S(3)Iad(c + dx)*S(2)sinh(e + fx)/fS(2) + Ia(c + dx)*S(3)cosh(e + fx)/f + a(c + dx)S(4)/(S(4)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)S(2)(Iasinh(e + fx) + a), x), x, S(2)Iad*S(2)cosh(e + fx)/fS(3) - S(2)Iad(c + dx)sinh(e + fx)/f*S(2) + Ia(c + dx)*S(2)cosh(e + fx)/f + a(c + dx)S(3)/(S(3)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)(Iasinh(e + fx) + a), x), x, -Iadsinh(e + fx)/fS(2) + Ia(c + dx)cosh(e + fx)/f + a(c + dx)*S(2)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((Iasinh(e + fx) + a)/(c + dx), x), x, IaCoshIntegral(cf/d + fx)sinh(-cf/d + e)/d + IaSinhIntegral(cf/d + fx)cosh(-cf/d + e)/d + alog(c + dx)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((Iasinh(e + fx) + a)/(c + dx)*S(2), x), x, -Iasinh(e + fx)/(d(c + dx)) - a/(d(c + dx)) + IafCoshIntegral(cf/d + fx)cosh(-cf/d + e)/dS(2) + IafSinhIntegral(cf/d + fx)sinh(-cf/d + e)/d*S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((Iasinh(e + fx) + a)/(c + dx)S(3), x), x, -Iasinh(e + fx)/(S(2)d(c + dx)S(2)) - a/(S(2)d(c + dx)*S(2)) - Iafcosh(e + fx)/(S(2)d*S(2)(c + dx)) + IafS(2)CoshIntegral(cf/d + fx)sinh(-cf/d + e)/(S(2)dS(3)) + IafS(2)SinhIntegral(cf/d + fx)cosh(-cf/d + e)/(S(2)dS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)(Iasinh(e + fx) + a)S(2), x), x, S(3)a*S(2)cdS(2)x/(S(4)fS(2)) + S(3)a*S(2)d*S(3)x*S(2)/(S(8)f*S(2)) + S(3)a*S(2)d*S(3)sinh(e + fx)S(2)/(S(8)f*S(4)) - S(12)IaS(2)d*S(3)sinh(e + fx)/fS(4) - S(3)a*S(2)d*S(2)(c + dx)sinh(e + fx)cosh(e + fx)/(S(4)f*S(3)) + S(12)IaS(2)d*S(2)(c + dx)cosh(e + fx)/fS(3) + S(3)a*S(2)d(c + dx)*S(2)sinh(e + fx)S(2)/(S(4)f*S(2)) - S(6)IaS(2)d(c + dx)*S(2)sinh(e + fx)/fS(2) - aS(2)(c + dx)S(3)sinh(e + fx)cosh(e + fx)/(S(2)f) + S(2)Ia*S(2)(c + dx)S(3)cosh(e + fx)/f + S(3)a*S(2)(c + dx)S(4)/(S(8)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)S(2)(Iasinh(e + fx) + a)S(2), x), x, aS(2)d*S(2)x/(S(4)fS(2)) - aS(2)d*S(2)sinh(e + fx)cosh(e + fx)/(S(4)f*S(3)) + S(4)IaS(2)d*S(2)cosh(e + fx)/fS(3) + aS(2)d(c + dx)sinh(e + fx)*S(2)/(S(2)f*S(2)) - S(4)IaS(2)d(c + dx)sinh(e + fx)/fS(2) - aS(2)(c + dx)*S(2)sinh(e + fx)cosh(e + fx)/(S(2)f) + S(2)Ia*S(2)(c + dx)S(2)cosh(e + fx)/f + aS(2)(c + dx)S(3)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)(Iasinh(e + fx) + a)S(2), x), x, aS(2)cx/S(2) + aS(2)dxS(2)/S(4) + aS(2)dsinh(e + fx)*S(2)/(S(4)f*S(2)) - S(2)IaS(2)dsinh(e + fx)/fS(2) - aS(2)(c + dx)sinh(e + fx)cosh(e + fx)/(S(2)f) + S(2)IaS(2)(c + dx)cosh(e + fx)/f + aS(2)(c + dx)S(2)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((Iasinh(e + fx) + a)S(2)/(c + dx), x), x, S(2)Ia*S(2)CoshIntegral(cf/d + fx)sinh(-cf/d + e)/d - a*S(2)CoshIntegral(S(2)cf/d + S(2)fx)cosh(-S(2)cf/d + S(2)e)/(S(2)d) + S(2)IaS(2)SinhIntegral(cf/d + fx)cosh(-cf/d + e)/d - a*S(2)SinhIntegral(S(2)cf/d + S(2)fx)sinh(-S(2)cf/d + S(2)e)/(S(2)d) + S(3)a*S(2)log(c + dx)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((Iasinh(e + fx) + a)S(2)/(c + dx)*S(2), x), x, -S(4)a*S(2)cosh(IPi/S(4) + e/S(2) + fx/S(2))*S(4)/(d(c + dx)) + S(2)IaS(2)fCoshIntegral(cf/d + fx)cosh(-cf/d + e)/dS(2) - aS(2)fCoshIntegral(S(2)cf/d + S(2)fx)sinh(-S(2)cf/d + S(2)e)/dS(2) + S(2)IaS(2)fSinhIntegral(cf/d + fx)sinh(-cf/d + e)/dS(2) - aS(2)fSinhIntegral(S(2)cf/d + S(2)fx)cosh(-S(2)cf/d + S(2)e)/dS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((Iasinh(e + fx) + a)*S(2)/(c + dx)*S(3), x), x, -S(2)a*S(2)cosh(IPi/S(4) + e/S(2) + fx/S(2))*S(4)/(d(c + dx)S(2)) - S(4)a*S(2)fsinh(IPi/S(4) + e/S(2) + fx/S(2))cosh(IPi/S(4) + e/S(2) + fx/S(2))S(3)/(dS(2)(c + dx)) + IaS(2)f*S(2)CoshIntegral(cf/d + fx)sinh(-cf/d + e)/dS(3) - aS(2)fS(2)CoshIntegral(S(2)cf/d + S(2)fx)cosh(-S(2)cf/d + S(2)e)/d*S(3) + Ia*S(2)f*S(2)SinhIntegral(cf/d + fx)cosh(-cf/d + e)/dS(3) - aS(2)fS(2)SinhIntegral(S(2)cf/d + S(2)fx)sinh(-S(2)cf/d + S(2)e)/d*S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)/(Iasinh(e + fx) + a), x), x, S(12)dS(3)PolyLog(S(3), -exp(IPi/S(2) + e + fx))/(afS(4)) - S(12)d*S(2)(c + dx)PolyLog(S(2), -exp(IPi/S(2) + e + fx))/(afS(3)) - S(6)d(c + dx)*S(2)log(exp(IPi/S(2) + e + fx) + S(1))/(afS(2)) + (c + dx)*S(3)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(af) + (c + dx)*S(3)/(af), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)S(2)/(Iasinh(e + fx) + a), x), x, -S(4)dS(2)PolyLog(S(2), -exp(IPi/S(2) + e + fx))/(afS(3)) - S(4)d(c + dx)log(exp(IPi/S(2) + e + fx) + S(1))/(af*S(2)) + (c + dx)*S(2)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(af) + (c + dx)*S(2)/(af), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)/(Iasinh(e + fx) + a), x), x, -S(2)dlog(cosh(IPi/S(4) + e/S(2) + fx/S(2)))/(afS(2)) + (c + dx)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(af), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((c + dx)(Iasinh(e + fx) + a)), x), x, Integrate(S(1)/((c + dx)cosh(IPi/S(4) + e/S(2) + fx/S(2))S(2)), x)/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((c + dx)S(2)(Iasinh(e + fx) + a)), x), x, Integrate(S(1)/((c + dx)*S(2)cosh(IPi/S(4) + e/S(2) + fx/S(2))*S(2)), x)/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)S(3)/(Iasinh(e + fx) + a)*S(2), x), x, S(4)d*S(3)PolyLog(S(3), -exp(IPi/S(2) + e + fx))/(a*S(2)f*S(4)) + S(4)d*S(3)log(cosh(IPi/S(4) + e/S(2) + fx/S(2)))/(a*S(2)f*S(4)) - S(4)d*S(2)(c + dx)PolyLog(S(2), -exp(IPi/S(2) + e + fx))/(a*S(2)f*S(3)) - S(2)d*S(2)(c + dx)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(a*S(2)f*S(3)) - S(2)d(c + dx)*S(2)log(exp(IPi/S(2) + e + fx) + S(1))/(a*S(2)f*S(2)) + d(c + dx)S(2)/(S(2)a*S(2)f*S(2)cosh(IPi/S(4) + e/S(2) + fx/S(2))*S(2)) + (c + dx)*S(3)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(S(3)aS(2)f) + (c + dx)S(3)/(S(3)a*S(2)f) + (c + dx)S(3)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(S(6)aS(2)fcosh(IPi/S(4) + e/S(2) + fx/S(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(2)/(Iasinh(e + fx) + a)*S(2), x), x, -S(4)d*S(2)PolyLog(S(2), -exp(IPi/S(2) + e + fx))/(S(3)aS(2)f*S(3)) - S(2)d*S(2)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(S(3)aS(2)f*S(3)) - S(4)d(c + dx)log(exp(IPi/S(2) + e + fx) + S(1))/(S(3)a*S(2)f*S(2)) + d(c + dx)/(S(3)a*S(2)f*S(2)cosh(IPi/S(4) + e/S(2) + fx/S(2))*S(2)) + (c + dx)*S(2)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(S(3)aS(2)f) + (c + dx)S(2)/(S(3)a*S(2)f) + (c + dx)S(2)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(S(6)aS(2)fcosh(IPi/S(4) + e/S(2) + fx/S(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)/(Iasinh(e + fx) + a)S(2), x), x, -S(2)dlog(cosh(IPi/S(4) + e/S(2) + fx/S(2)))/(S(3)a*S(2)f*S(2)) + d/(S(6)a*S(2)f*S(2)cosh(IPi/S(4) + e/S(2) + fx/S(2))*S(2)) + (c + dx)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(S(3)a*S(2)f) + (c + dx)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(S(6)aS(2)fcosh(IPi/S(4) + e/S(2) + fx/S(2))S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((c + dx)(Iasinh(e + fx) + a)*S(2)), x), x, Integrate(S(1)/((c + dx)cosh(IPi/S(4) + e/S(2) + fx/S(2))S(4)), x)/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((c + dx)*S(2)(Iasinh(e + fx) + a)S(2)), x), x, Integrate(S(1)/((c + dx)*S(2)cosh(IPi/S(4) + e/S(2) + fx/S(2))*S(4)), x)/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(Iasinh(e + fx) + a)/x, x), x, sqrt(Iasinh(e + fx) + a)CoshIntegral(fx/S(2))cosh(IPi/S(4) + e/S(2))/cosh(IPi/S(4) + e/S(2) + fx/S(2)) + sqrt(Iasinh(e + fx) + a)SinhIntegral(fx/S(2))sinh(IPi/S(4) + e/S(2))/cosh(IPi/S(4) + e/S(2) + fx/S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(Iasinh(e + fx) + a)/xS(2), x), x, fsqrt(Iasinh(e + fx) + a)CoshIntegral(fx/S(2))sinh(IPi/S(4) + e/S(2))/(S(2)cosh(IPi/S(4) + e/S(2) + fx/S(2))) + fsqrt(Iasinh(e + fx) + a)SinhIntegral(fx/S(2))cosh(IPi/S(4) + e/S(2))/(S(2)cosh(IPi/S(4) + e/S(2) + fx/S(2))) - sqrt(Iasinh(e + fx) + a)/x, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(Iasinh(e + fx) + a)/xS(3), x), x, fS(2)sqrt(Iasinh(e + fx) + a)CoshIntegral(fx/S(2))cosh(IPi/S(4) + e/S(2))/(S(8)cosh(IPi/S(4) + e/S(2) + fx/S(2))) + f*S(2)sqrt(Iasinh(e + fx) + a)SinhIntegral(fx/S(2))sinh(IPi/S(4) + e/S(2))/(S(8)cosh(IPi/S(4) + e/S(2) + fx/S(2))) - fsqrt(Iasinh(e + fx) + a)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(S(4)x) - sqrt(Iasinh(e + fx) + a)/(S(2)x*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xsqrt(Iasinh(e + fx) + a)), x), x, Integrate(S(1)/(xsqrt(Iasinh(e + fx) + a)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)sqrt(Iasinh(e + fx) + a)), x), x, Integrate(S(1)/(xS(2)sqrt(Iasinh(e + fx) + a)), x), expand=True, _diff=True, _numerical=True) ''' long time # assert rubi_test(rubi_integrate(xS(3)/(Iasinh(e + fx) + a)(S(3)/2), x), x, xS(3)ArcTan(exp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(afsqrt(Iasinh(e + fx) + a)) + xS(3)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(S(2)afsqrt(Iasinh(e + fx) + a)) - S(3)Ix*S(2)PolyLog(S(2), -Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(afS(2)sqrt(Iasinh(e + fx) + a)) + S(3)IxS(2)PolyLog(S(2), Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(afS(2)sqrt(Iasinh(e + fx) + a)) + S(3)x*S(2)/(af*S(2)sqrt(Iasinh(e + fx) + a)) - S(24)xArcTan(exp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(afS(3)sqrt(Iasinh(e + fx) + a)) + S(12)IxPolyLog(S(3), -Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(afS(3)sqrt(Iasinh(e + fx) + a)) - S(12)IxPolyLog(S(3), Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(afS(3)sqrt(Iasinh(e + fx) + a)) + S(24)IPolyLog(S(2), -Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(af*S(4)sqrt(Iasinh(e + fx) + a)) - S(24)IPolyLog(S(2), Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(af*S(4)sqrt(Iasinh(e + fx) + a)) - S(24)IPolyLog(S(4), -Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(af*S(4)sqrt(Iasinh(e + fx) + a)) + S(24)IPolyLog(S(4), Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(af*S(4)sqrt(Iasinh(e + fx) + a)), expand=True, _diff=True, _numerical=True) # assert rubi_test(rubi_integrate(xS(2)/(Iasinh(e + fx) + a)(S(3)/2), x), x, xS(2)ArcTan(exp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(afsqrt(Iasinh(e + fx) + a)) + xS(2)tanh(IPi/S(4) + e/S(2) + fx/S(2))/(S(2)afsqrt(Iasinh(e + fx) + a)) - S(2)IxPolyLog(S(2), -Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(af*S(2)sqrt(Iasinh(e + fx) + a)) + S(2)IxPolyLog(S(2), Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(afS(2)sqrt(Iasinh(e + fx) + a)) + S(2)x/(afS(2)sqrt(Iasinh(e + fx) + a)) - S(4)ArcTan(sinh(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(af*S(3)sqrt(Iasinh(e + fx) + a)) + S(4)IPolyLog(S(3), -Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(af*S(3)sqrt(Iasinh(e + fx) + a)) - S(4)IPolyLog(S(3), Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(af*S(3)sqrt(Iasinh(e + fx) + a)), expand=True, _diff=True, _numerical=True) # assert rubi_test(rubi_integrate(x/(Iasinh(e + fx) + a)*(S(3)/2), x), x, xArcTan(exp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(afsqrt(Iasinh(e + fx) + a)) + xtanh(IPi/S(4) + e/S(2) + fx/S(2))/(S(2)afsqrt(Iasinh(e + fx) + a)) - IPolyLog(S(2), -Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(afS(2)sqrt(Iasinh(e + fx) + a)) + IPolyLog(S(2), Iexp(IPi/S(4) + e/S(2) + fx/S(2)))cosh(IPi/S(4) + e/S(2) + fx/S(2))/(afS(2)sqrt(Iasinh(e + fx) + a)) + S(1)/(af*S(2)sqrt(Iasinh(e + fx) + a)), expand=True, _diff=True, _numerical=True) ''' assert rubi_test(rubi_integrate(S(1)/(x(Iasinh(e + fx) + a)(S(3)/2)), x), x, Integrate(S(1)/(x(Iasinh(e + fx) + a)(S(3)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(xS(2)(Iasinh(e + fx) + a)(S(3)/2)), x), x, Integrate(S(1)/(xS(2)(Iasinh(e + fx) + a)(S(3)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(x(Iasinh(c + dx) + a)(S(5)/2)), x), x, Integrate(S(1)/(x(Iasinh(c + dx) + a)(S(5)/2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((Iasinh(e + fx) + a)*(S(1)/3)/x, x), x, Integrate((Iasinh(e + fx) + a)*(S(1)/3)/x, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*m(Iasinh(e + fx) + a)n, x), x, Integrate((c + dx)*m(Iasinh(e + fx) + a)n, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*m(Iasinh(e + fx) + a)S(3), x), x, -S(3)S(2)*(-m + S(-3))a*S(3)(-f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), -S(2)f(c + dx)/d)exp(-S(2)cf/d + S(2)e)/f + S(3)S(2)*(-m + S(-3))a*S(3)(f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), S(2)f(c + dx)/d)exp(S(2)cf/d - S(2)e)/f - S(3)(-m + S(-1))IaS(3)(-f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), -S(3)f(c + dx)/d)exp(-S(3)cf/d + S(3)e)/(S(8)f) - S(3)*(-m + S(-1))IaS(3)(f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), S(3)f(c + dx)/d)exp(S(3)cf/d - S(3)e)/(S(8)f) + S(15)Ia*S(3)(-f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), -f(c + dx)/d)exp(-cf/d + e)/(S(8)f) + S(15)IaS(3)(f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), f(c + dx)/d)exp(cf/d - e)/(S(8)f) + S(5)a*S(3)(c + dx)(m + S(1))/(S(2)d(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*m(Iasinh(e + fx) + a)S(2), x), x, -S(2)(-m + S(-3))a*S(2)(-f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), -S(2)f(c + dx)/d)exp(-S(2)cf/d + S(2)e)/f + S(2)(-m + S(-3))a*S(2)(f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), S(2)f(c + dx)/d)exp(S(2)cf/d - S(2)e)/f + Ia*S(2)(-f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), -f(c + dx)/d)exp(-cf/d + e)/f + IaS(2)(f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), f(c + dx)/d)exp(cf/d - e)/f + S(3)aS(2)(c + dx)(m + S(1))/(S(2)d(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*m(Iasinh(e + fx) + a), x), x, Ia(-f(c + dx)/d)(-m)(c + dx)mGamma(m + S(1), -f(c + dx)/d)exp(-cf/d + e)/(S(2)f) + Ia(f(c + dx)/d)(-m)(c + dx)mGamma(m + S(1), f(c + dx)/d)exp(cf/d - e)/(S(2)f) + a(c + dx)(m + S(1))/(d(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)m/(Iasinh(e + fx) + a), x), x, Integrate((c + dx)m/cosh(IPi/S(4) + e/S(2) + fx/S(2))S(2), x)/(S(2)a), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)m/(Iasinh(e + fx) + a)*S(2), x), x, Integrate((c + dx)*m/cosh(IPi/S(4) + e/S(2) + fx/S(2))S(4), x)/(S(4)a*S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))(c + dx)S(3), x), x, a(c + dx)S(4)/(S(4)d) - S(6)bd*S(3)sinh(e + fx)/fS(4) + S(6)bdS(2)(c + dx)cosh(e + fx)/fS(3) - S(3)bd(c + dx)S(2)sinh(e + fx)/fS(2) + b(c + dx)S(3)cosh(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))(c + dx)S(2), x), x, a(c + dx)S(3)/(S(3)d) + S(2)bd*S(2)cosh(e + fx)/fS(3) - S(2)bd(c + dx)sinh(e + fx)/fS(2) + b(c + dx)S(2)cosh(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))(c + dx), x), x, a(c + dx)S(2)/(S(2)d) - bdsinh(e + fx)/fS(2) + b(c + dx)cosh(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))/(c + dx), x), x, alog(c + dx)/d + bCoshIntegral(cf/d + fx)sinh(-cf/d + e)/d + bSinhIntegral(cf/d + fx)cosh(-cf/d + e)/d, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))/(c + dx)S(2), x), x, -a/(d(c + dx)) - bsinh(e + fx)/(d(c + dx)) + bfCoshIntegral(cf/d + fx)cosh(-cf/d + e)/dS(2) + bfSinhIntegral(cf/d + fx)sinh(-cf/d + e)/dS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))/(c + dx)*S(3), x), x, -a/(S(2)d(c + dx)*S(2)) - bsinh(e + fx)/(S(2)d(c + dx)*S(2)) - bfcosh(e + fx)/(S(2)dS(2)(c + dx)) + bf*S(2)CoshIntegral(cf/d + fx)sinh(-cf/d + e)/(S(2)dS(3)) + bf*S(2)SinhIntegral(cf/d + fx)cosh(-cf/d + e)/(S(2)dS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))S(2)(c + dx)S(3), x), x, aS(2)(c + dx)S(4)/(S(4)d) - S(12)abdS(3)sinh(e + fx)/fS(4) + S(12)abd*S(2)(c + dx)cosh(e + fx)/fS(3) - S(6)abd(c + dx)*S(2)sinh(e + fx)/fS(2) + S(2)ab(c + dx)S(3)cosh(e + fx)/f - S(3)b*S(2)cdS(2)x/(S(4)fS(2)) - S(3)b*S(2)d*S(3)x*S(2)/(S(8)f*S(2)) - S(3)b*S(2)d*S(3)sinh(e + fx)S(2)/(S(8)f*S(4)) + S(3)b*S(2)d*S(2)(c + dx)sinh(e + fx)cosh(e + fx)/(S(4)f*S(3)) - S(3)b*S(2)d(c + dx)*S(2)sinh(e + fx)S(2)/(S(4)fS(2)) + bS(2)(c + dx)*S(3)sinh(e + fx)cosh(e + fx)/(S(2)f) - b*S(2)(c + dx)S(4)/(S(8)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))*S(2)(c + dx)S(2), x), x, aS(2)(c + dx)S(3)/(S(3)d) + S(4)abdS(2)cosh(e + fx)/fS(3) - S(4)abd(c + dx)sinh(e + fx)/f*S(2) + S(2)ab(c + dx)S(2)cosh(e + fx)/f - bS(2)d*S(2)x/(S(4)fS(2)) + bS(2)d*S(2)sinh(e + fx)cosh(e + fx)/(S(4)fS(3)) - bS(2)d(c + dx)sinh(e + fx)S(2)/(S(2)fS(2)) + bS(2)(c + dx)*S(2)sinh(e + fx)cosh(e + fx)/(S(2)f) - b*S(2)(c + dx)S(3)/(S(6)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))*S(2)(c + dx), x), x, aS(2)(c + dx)S(2)/(S(2)d) - S(2)abdsinh(e + fx)/fS(2) + S(2)ab(c + dx)cosh(e + fx)/f - bS(2)cx/S(2) - bS(2)dxS(2)/S(4) - bS(2)dsinh(e + fx)*S(2)/(S(4)fS(2)) + bS(2)(c + dx)sinh(e + fx)cosh(e + fx)/(S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))S(2)/(c + dx), x), x, a*S(2)log(c + dx)/d + S(2)abCoshIntegral(cf/d + fx)sinh(-cf/d + e)/d + S(2)abSinhIntegral(cf/d + fx)cosh(-cf/d + e)/d + bS(2)CoshIntegral(S(2)cf/d + S(2)fx)cosh(-S(2)cf/d + S(2)e)/(S(2)d) + bS(2)SinhIntegral(S(2)cf/d + S(2)fx)sinh(-S(2)cf/d + S(2)e)/(S(2)d) - bS(2)log(c + dx)/(S(2)d), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))*S(2)/(c + dx)S(2), x), x, -aS(2)/(d(c + dx)) - S(2)absinh(e + fx)/(d(c + dx)) + S(2)abfCoshIntegral(cf/d + fx)cosh(-cf/d + e)/d*S(2) + S(2)abfSinhIntegral(cf/d + fx)sinh(-cf/d + e)/dS(2) - bS(2)sinh(e + fx)S(2)/(d(c + dx)) + bS(2)fCoshIntegral(S(2)cf/d + S(2)fx)sinh(-S(2)cf/d + S(2)e)/dS(2) + bS(2)fSinhIntegral(S(2)cf/d + S(2)fx)cosh(-S(2)cf/d + S(2)e)/dS(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))S(2)/(c + dx)S(3), x), x, -aS(2)/(S(2)d(c + dx)S(2)) - absinh(e + fx)/(d(c + dx)*S(2)) - abfcosh(e + fx)/(dS(2)(c + dx)) + abfS(2)CoshIntegral(cf/d + fx)sinh(-cf/d + e)/d*S(3) + abfS(2)SinhIntegral(cf/d + fx)cosh(-cf/d + e)/dS(3) - bS(2)sinh(e + fx)*S(2)/(S(2)d(c + dx)S(2)) - bS(2)fsinh(e + fx)cosh(e + fx)/(dS(2)(c + dx)) + bS(2)f*S(2)CoshIntegral(S(2)cf/d + S(2)fx)cosh(-S(2)cf/d + S(2)e)/dS(3) + bS(2)fS(2)SinhIntegral(S(2)cf/d + S(2)fx)sinh(-S(2)cf/d + S(2)e)/d*S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)/(a + bsinh(e + fx)), x), x, S(6)d*S(3)PolyLog(S(4), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(4)sqrt(aS(2) + bS(2))) - S(6)dS(3)PolyLog(S(4), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(4)sqrt(aS(2) + bS(2))) - S(6)dS(2)(c + dx)PolyLog(S(3), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(3)sqrt(aS(2) + bS(2))) + S(6)dS(2)(c + dx)PolyLog(S(3), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(3)sqrt(aS(2) + bS(2))) + S(3)d(c + dx)S(2)PolyLog(S(2), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(2)sqrt(aS(2) + bS(2))) - S(3)d(c + dx)S(2)PolyLog(S(2), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(2)sqrt(aS(2) + bS(2))) + (c + dx)S(3)log(bexp(e + fx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(fsqrt(aS(2) + bS(2))) - (c + dx)*S(3)log(bexp(e + fx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(fsqrt(aS(2) + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(2)/(a + bsinh(e + fx)), x), x, -S(2)d*S(2)PolyLog(S(3), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(3)sqrt(aS(2) + bS(2))) + S(2)dS(2)PolyLog(S(3), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(3)sqrt(aS(2) + bS(2))) + S(2)d(c + dx)PolyLog(S(2), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(2)sqrt(aS(2) + bS(2))) - S(2)d(c + dx)PolyLog(S(2), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(2)sqrt(aS(2) + bS(2))) + (c + dx)S(2)log(bexp(e + fx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(fsqrt(aS(2) + bS(2))) - (c + dx)*S(2)log(bexp(e + fx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(fsqrt(aS(2) + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)/(a + bsinh(e + fx)), x), x, dPolyLog(S(2), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(fS(2)sqrt(aS(2) + bS(2))) - dPolyLog(S(2), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(fS(2)sqrt(aS(2) + bS(2))) + (c + dx)log(bexp(e + fx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(fsqrt(aS(2) + bS(2))) - (c + dx)log(bexp(e + fx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(fsqrt(aS(2) + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bsinh(e + fx))(c + dx)), x), x, Integrate(S(1)/((a + bsinh(e + fx))(c + dx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bsinh(e + fx))(c + dx)*S(2)), x), x, Integrate(S(1)/((a + bsinh(e + fx))(c + dx)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*S(3)/(a + bsinh(e + fx))S(2), x), x, S(6)adS(3)PolyLog(S(4), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(4)(aS(2) + bS(2))*(S(3)/2)) - S(6)adS(3)PolyLog(S(4), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(4)(aS(2) + bS(2))*(S(3)/2)) - S(6)adS(2)(c + dx)PolyLog(S(3), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(3)(aS(2) + bS(2))*(S(3)/2)) + S(6)adS(2)(c + dx)PolyLog(S(3), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(3)(aS(2) + bS(2))*(S(3)/2)) + S(3)ad(c + dx)S(2)PolyLog(S(2), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(2)(aS(2) + bS(2))*(S(3)/2)) - S(3)ad(c + dx)S(2)PolyLog(S(2), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(2)(aS(2) + bS(2))*(S(3)/2)) + a(c + dx)S(3)log(bexp(e + fx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(f(aS(2) + bS(2))(S(3)/2)) - a(c + dx)S(3)log(bexp(e + fx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(f(aS(2) + bS(2))(S(3)/2)) - b(c + dx)S(3)cosh(e + fx)/(f(a + bsinh(e + fx))(aS(2) + bS(2))) - S(6)d*S(3)PolyLog(S(3), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(4)(aS(2) + bS(2))) - S(6)dS(3)PolyLog(S(3), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(4)(aS(2) + bS(2))) + S(6)dS(2)(c + dx)PolyLog(S(2), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(3)(aS(2) + bS(2))) + S(6)dS(2)(c + dx)PolyLog(S(2), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(3)(aS(2) + bS(2))) + S(3)d(c + dx)S(2)log(bexp(e + fx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(f*S(2)(aS(2) + bS(2))) + S(3)d(c + dx)S(2)log(bexp(e + fx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(f*S(2)(aS(2) + bS(2))) - (c + dx)S(3)/(f(aS(2) + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)S(2)/(a + bsinh(e + fx))S(2), x), x, -S(2)adS(2)PolyLog(S(3), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(3)(aS(2) + bS(2))*(S(3)/2)) + S(2)adS(2)PolyLog(S(3), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(3)(aS(2) + bS(2))*(S(3)/2)) + S(2)ad(c + dx)PolyLog(S(2), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(2)(aS(2) + bS(2))*(S(3)/2)) - S(2)ad(c + dx)PolyLog(S(2), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(2)(aS(2) + bS(2))*(S(3)/2)) + a(c + dx)S(2)log(bexp(e + fx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(f(aS(2) + bS(2))(S(3)/2)) - a(c + dx)S(2)log(bexp(e + fx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(f(aS(2) + bS(2))(S(3)/2)) - b(c + dx)S(2)cosh(e + fx)/(f(a + bsinh(e + fx))(aS(2) + bS(2))) + S(2)d*S(2)PolyLog(S(2), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(f*S(3)(aS(2) + bS(2))) + S(2)dS(2)PolyLog(S(2), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(f*S(3)(aS(2) + bS(2))) + S(2)d(c + dx)log(bexp(e + fx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(f*S(2)(aS(2) + bS(2))) + S(2)d(c + dx)log(bexp(e + fx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(f*S(2)(aS(2) + bS(2))) - (c + dx)S(2)/(f(aS(2) + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)/(a + bsinh(e + fx))S(2), x), x, adPolyLog(S(2), -bexp(e + fx)/(a - sqrt(aS(2) + bS(2))))/(fS(2)(aS(2) + bS(2))*(S(3)/2)) - adPolyLog(S(2), -bexp(e + fx)/(a + sqrt(aS(2) + bS(2))))/(fS(2)(aS(2) + bS(2))*(S(3)/2)) + a(c + dx)log(bexp(e + fx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(f(aS(2) + bS(2))(S(3)/2)) - a(c + dx)log(bexp(e + fx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(f(aS(2) + bS(2))(S(3)/2)) - b(c + dx)cosh(e + fx)/(f(a + bsinh(e + fx))(aS(2) + bS(2))) + dlog(a + bsinh(e + fx))/(f*S(2)(aS(2) + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bsinh(e + fx))*S(2)(c + dx)), x), x, Integrate(S(1)/((a + bsinh(e + fx))S(2)(c + dx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bsinh(e + fx))S(2)(c + dx)S(2)), x), x, Integrate(S(1)/((a + bsinh(e + fx))S(2)(c + dx)S(2)), x), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate((e + fx)*S(2)/(a + bsinh(c + dx))S(3), x), x, S(3)a*S(2)(e + fx)S(2)log(bexp(c + dx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(S(2)d(aS(2) + bS(2))*(S(5)/2)) - S(3)a*S(2)(e + fx)S(2)log(bexp(c + dx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(S(2)d(aS(2) + bS(2))*(S(5)/2)) + S(3)a*S(2)f(e + fx)PolyLog(S(2), -bexp(c + dx)/(a - sqrt(aS(2) + bS(2))))/(dS(2)(aS(2) + bS(2))*(S(5)/2)) - S(3)a*S(2)f(e + fx)PolyLog(S(2), -bexp(c + dx)/(a + sqrt(aS(2) + bS(2))))/(dS(2)(aS(2) + bS(2))*(S(5)/2)) - S(3)a*S(2)f*S(2)PolyLog(S(3), -bexp(c + dx)/(a - sqrt(aS(2) + bS(2))))/(d*S(3)(aS(2) + bS(2))*(S(5)/2)) + S(3)a*S(2)f*S(2)PolyLog(S(3), -bexp(c + dx)/(a + sqrt(aS(2) + bS(2))))/(d*S(3)(aS(2) + bS(2))*(S(5)/2)) - S(3)ab(e + fx)S(2)cosh(c + dx)/(S(2)d(a + bsinh(c + dx))(aS(2) + bS(2))*S(2)) - S(3)a(e + fx)*S(2)/(S(2)d(aS(2) + bS(2))S(2)) + S(3)af(e + fx)log(bexp(c + dx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(d*S(2)(aS(2) + bS(2))*S(2)) + S(3)af(e + fx)log(bexp(c + dx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(d*S(2)(aS(2) + bS(2))*S(2)) + S(3)afS(2)PolyLog(S(2), -bexp(c + dx)/(a - sqrt(aS(2) + bS(2))))/(d*S(3)(aS(2) + bS(2))*S(2)) + S(3)afS(2)PolyLog(S(2), -bexp(c + dx)/(a + sqrt(aS(2) + bS(2))))/(d*S(3)(aS(2) + bS(2))*S(2)) - b(e + fx)S(2)cosh(c + dx)/(S(2)d(a + bsinh(c + dx))S(2)(aS(2) + bS(2))) - (e + fx)S(2)log(bexp(c + dx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(S(2)d(aS(2) + bS(2))*(S(3)/2)) + (e + fx)*S(2)log(bexp(c + dx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(S(2)d(aS(2) + bS(2))*(S(3)/2)) - f(e + fx)PolyLog(S(2), -bexp(c + dx)/(a - sqrt(aS(2) + bS(2))))/(d*S(2)(aS(2) + bS(2))*(S(3)/2)) + f(e + fx)PolyLog(S(2), -bexp(c + dx)/(a + sqrt(aS(2) + bS(2))))/(d*S(2)(aS(2) + bS(2))*(S(3)/2)) - f(e + fx)/(dS(2)(a + bsinh(c + dx))(aS(2) + bS(2))) + fS(2)PolyLog(S(3), -bexp(c + dx)/(a - sqrt(aS(2) + bS(2))))/(d*S(3)(aS(2) + bS(2))(S(3)/2)) - fS(2)PolyLog(S(3), -bexp(c + dx)/(a + sqrt(aS(2) + bS(2))))/(dS(3)(aS(2) + bS(2))*(S(3)/2)) - S(2)f*S(2)atanh((-atanh(c/S(2) + dx/S(2)) + b)/sqrt(aS(2) + bS(2)))/(d*S(3)(aS(2) + bS(2))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((e + fx)/(a + bsinh(c + dx))*S(3), x), x, S(3)a*S(2)(e + fx)log(bexp(c + dx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(S(2)d(aS(2) + bS(2))*(S(5)/2)) - S(3)a*S(2)(e + fx)log(bexp(c + dx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(S(2)d(aS(2) + bS(2))*(S(5)/2)) + S(3)a*S(2)fPolyLog(S(2), -bexp(c + dx)/(a - sqrt(aS(2) + bS(2))))/(S(2)d*S(2)(aS(2) + bS(2))*(S(5)/2)) - S(3)a*S(2)fPolyLog(S(2), -bexp(c + dx)/(a + sqrt(aS(2) + bS(2))))/(S(2)d*S(2)(aS(2) + bS(2))*(S(5)/2)) - S(3)ab(e + fx)cosh(c + dx)/(S(2)d(a + bsinh(c + dx))(aS(2) + bS(2))*S(2)) + S(3)aflog(a + bsinh(c + dx))/(S(2)dS(2)(aS(2) + bS(2))*S(2)) - b(e + fx)cosh(c + dx)/(S(2)d(a + bsinh(c + dx))S(2)(aS(2) + bS(2))) - (e + fx)log(bexp(c + dx)/(a - sqrt(aS(2) + bS(2))) + S(1))/(S(2)d(aS(2) + bS(2))*(S(3)/2)) + (e + fx)log(bexp(c + dx)/(a + sqrt(aS(2) + bS(2))) + S(1))/(S(2)d(aS(2) + bS(2))(S(3)/2)) - fPolyLog(S(2), -bexp(c + dx)/(a - sqrt(aS(2) + bS(2))))/(S(2)dS(2)(aS(2) + bS(2))*(S(3)/2)) + fPolyLog(S(2), -bexp(c + dx)/(a + sqrt(aS(2) + bS(2))))/(S(2)dS(2)(aS(2) + bS(2))*(S(3)/2)) - f/(S(2)d*S(2)(a + bsinh(c + dx))(aS(2) + bS(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bsinh(c + dx))S(3)(e + fx)), x), x, Integrate(S(1)/((a + bsinh(c + dx))S(3)(e + fx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/((a + bsinh(c + dx))S(3)(e + fx)S(2)), x), x, Integrate(S(1)/((a + bsinh(c + dx))S(3)(e + fx)S(2)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))n(c + dx)m, x), x, Integrate((a + bsinh(e + fx))n(c + dx)m, x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))S(3)(c + dx)m, x), x, S(3)S(2)*(-m + S(-3))abS(2)(-f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), -S(2)f(c + dx)/d)exp(-S(2)cf/d + S(2)e)/f - S(3)S(2)*(-m + S(-3))abS(2)(f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), S(2)f(c + dx)/d)exp(S(2)cf/d - S(2)e)/f + S(3)(-m + S(-1))b*S(3)(-f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), -S(3)f(c + dx)/d)exp(-S(3)cf/d + S(3)e)/(S(8)f) + S(3)*(-m + S(-1))b*S(3)(f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), S(3)f(c + dx)/d)exp(S(3)cf/d - S(3)e)/(S(8)f) + a*S(3)(c + dx)(m + S(1))/(d(m + S(1))) + S(3)aS(2)b(-f(c + dx)/d)(-m)(c + dx)mGamma(m + S(1), -f(c + dx)/d)exp(-cf/d + e)/(S(2)f) + S(3)a*S(2)b(f(c + dx)/d)(-m)(c + dx)mGamma(m + S(1), f(c + dx)/d)exp(cf/d - e)/(S(2)f) - S(3)abS(2)(c + dx)(m + S(1))/(S(2)d(m + S(1))) - S(3)b*S(3)(-f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), -f(c + dx)/d)exp(-cf/d + e)/(S(8)f) - S(3)b*S(3)(f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), f(c + dx)/d)exp(cf/d - e)/(S(8)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))S(2)(c + dx)m, x), x, S(2)(-m + S(-3))b*S(2)(-f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), -S(2)f(c + dx)/d)exp(-S(2)cf/d + S(2)e)/f - S(2)(-m + S(-3))b*S(2)(f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), S(2)f(c + dx)/d)exp(S(2)cf/d - S(2)e)/f + aS(2)(c + dx)(m + S(1))/(d(m + S(1))) + ab(-f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), -f(c + dx)/d)exp(-cf/d + e)/f + ab(f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), f(c + dx)/d)exp(cf/d - e)/f - b*S(2)(c + dx)(m + S(1))/(S(2)d(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a + bsinh(e + fx))(c + dx)m, x), x, a(c + dx)(m + S(1))/(d(m + S(1))) + b(-f(c + dx)/d)(-m)(c + dx)mGamma(m + S(1), -f(c + dx)/d)exp(-cf/d + e)/(S(2)f) + b(f(c + dx)/d)*(-m)(c + dx)mGamma(m + S(1), f(c + dx)/d)exp(cf/d - e)/(S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*m/(a + bsinh(e + fx)), x), x, Integrate((c + dx)*m/(a + bsinh(e + fx)), x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((c + dx)*m/(a + bsinh(e + fx))S(2), x), x, Integrate((c + dx)*m/(a + bsinh(e + fx))*S(2), x), expand=True, _diff=True, _numerical=True)
f5e36bffc8d3297d1e6a5e64088a7d9638934124b59aa123585582eca4c78310	import sys from sympy.external import import_module matchpy = import_module("matchpy") if not matchpy: #bin/test will not execute any tests now disabled = True if sys.version_info[:2] < (3, 6): disabled = True from sympy.integrals.rubi.utility_function import ( sympy_op_factory, Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ, ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ, PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ, NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest, SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient, Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan, ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec, ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less, Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ, InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ, EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, ExpandLinearProduct, GCD, ContentFactor, NumericFactor, NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, ExpandExpression, Apart, SmartApart, MatchQ, PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, RationalFunctionFactors, NonrationalFunctionFactors, Reverse, RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, GeneralizedBinomialParts, GeneralizedTrinomialDegree, GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, MinimumMonomialExponent, MonomialExponent, LinearMatchQ, PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, AlgebraicFunctionFactors, NonalgebraicFunctionFactors, QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, CombineExponents, FactorInteger, FactorAbsurdNumber, SubstForInverseFunction, SubstForFractionalPower, SubstForFractionalPowerOfQuotientOfLinears, FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, NormalizeIntegrandAux, NormalizeIntegrandFactor, NormalizeIntegrandFactorBase, NormalizeTogether, NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand, SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, KnownTangentIntegrandQ, KnownCotangentIntegrandQ, KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, FunctionOfLog, PowerVariableExpn, PowerVariableDegree, PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, ConstantFactor, SameQ, ReplacePart, CommonFactors, MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, _SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify, _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor, PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral, CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial, Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ ) from sympy.core.add import Add from sympy.core.mod import Mod from sympy.core.mul import Mul from sympy.core.numbers import (Float, I, Integer) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.functions.elementary.complexes import Abs from sympy.functions.elementary.miscellaneous import sqrt from sympy.integrals.integrals import Integral from sympy.logic.boolalg import (And, Or) from sympy.simplify.simplify import simplify from sympy.integrals.rubi.symbol import WC from sympy.core.symbol import symbols, Symbol from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf, exp, log) from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch) from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec) from sympy.integrals.rubi.rubimain import rubi_integrate from sympy.core.numbers import pi as Pi a, b, c, d, e, f, m, n, x, u , k, p, r, s, t, i, j= symbols('a b c d e f m n x u k p r s t i j') A, B, C, D, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B C D a b c d e f g h y z m n p q u v w F', ) def test_1(): assert rubi_test(rubi_integrate(sin(a + bx), x), x, -cos(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2), x), x, x/S(2) - sin(a + bx)cos(a + bx)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(3), x), x, cos(a + bx)*S(3)/(S(3)b) - cos(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(4), x), x, S(3)x/S(8) - sin(a + bx)S(3)cos(a + bx)/(S(4)b) - S(3)sin(a + bx)cos(a + bx)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(5), x), x, -cos(a + bx)*S(5)/(S(5)b) + S(2)cos(a + bx)*S(3)/(S(3)b) - cos(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(6), x), x, S(5)x/S(16) - sin(a + bx)S(5)cos(a + bx)/(S(6)b) - S(5)sin(a + bx)*S(3)cos(a + bx)/(S(24)b) - S(5)sin(a + bx)cos(a + bx)/(S(16)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(7), x), x, cos(a + bx)*S(7)/(S(7)b) - S(3)cos(a + bx)*S(5)/(S(5)b) + cos(a + bx)S(3)/b - cos(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(8), x), x, S(35)x/S(128) - sin(a + bx)S(7)cos(a + bx)/(S(8)b) - S(7)sin(a + bx)*S(5)cos(a + bx)/(S(48)b) - S(35)sin(a + bx)*S(3)cos(a + bx)/(S(192)b) - S(35)sin(a + bx)cos(a + bx)/(S(128)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*(S(7)/2), x), x, S(10)EllipticF(-Pi/S(4) + a/S(2) + bx/S(2), S(2))/(S(21)b) - S(2)sin(a + bx)*(S(5)/2)cos(a + bx)/(S(7)b) - S(10)sqrt(sin(a + bx))cos(a + bx)/(S(21)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*(S(5)/2), x), x, S(6)EllipticE(-Pi/S(4) + a/S(2) + bx/S(2), S(2))/(S(5)b) - S(2)sin(a + bx)*(S(3)/2)cos(a + bx)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)(S(3)/2), x), x, S(2)EllipticF(-Pi/S(4) + a/S(2) + bx/S(2), S(2))/(S(3)b) - S(2)sqrt(sin(a + bx))cos(a + bx)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sin(a + bx)), x), x, S(2)EllipticE(-Pi/S(4) + a/S(2) + bx/S(2), S(2))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(sin(a + bx)), x), x, S(2)EllipticF(-Pi/S(4) + a/S(2) + bx/S(2), S(2))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*(S(-3)/2), x), x, -S(2)EllipticE(-Pi/S(4) + a/S(2) + bx/S(2), S(2))/b - S(2)cos(a + bx)/(bsqrt(sin(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*(S(-5)/2), x), x, S(2)EllipticF(-Pi/S(4) + a/S(2) + bx/S(2), S(2))/(S(3)b) - S(2)cos(a + bx)/(S(3)bsin(a + bx)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*(S(-7)/2), x), x, -S(6)EllipticE(-Pi/S(4) + a/S(2) + bx/S(2), S(2))/(S(5)b) - S(6)cos(a + bx)/(S(5)bsqrt(sin(a + bx))) - S(2)cos(a + bx)/(S(5)bsin(a + bx)*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(7)/2), x), x, S(10)c*S(4)EllipticF(-Pi/S(4) + a/S(2) + bx/S(2), S(2))sqrt(sin(a + bx))/(S(21)bsqrt(csin(a + bx))) - S(10)c*S(3)sqrt(csin(a + bx))cos(a + bx)/(S(21)b) - S(2)c(csin(a + bx))(S(5)/2)cos(a + bx)/(S(7)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*(S(5)/2), x), x, S(6)c*S(2)sqrt(csin(a + bx))EllipticE(-Pi/S(4) + a/S(2) + bx/S(2), S(2))/(S(5)bsqrt(sin(a + bx))) - S(2)c(csin(a + bx))(S(3)/2)cos(a + bx)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*(S(3)/2), x), x, S(2)c*S(2)EllipticF(-Pi/S(4) + a/S(2) + bx/S(2), S(2))sqrt(sin(a + bx))/(S(3)bsqrt(csin(a + bx))) - S(2)csqrt(csin(a + bx))cos(a + bx)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx)), x), x, S(2)sqrt(csin(a + bx))EllipticE(-Pi/S(4) + a/S(2) + bx/S(2), S(2))/(bsqrt(sin(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(csin(a + bx)), x), x, S(2)EllipticF(-Pi/S(4) + a/S(2) + bx/S(2), S(2))sqrt(sin(a + bx))/(bsqrt(csin(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*(S(-3)/2), x), x, -S(2)cos(a + bx)/(bcsqrt(csin(a + bx))) - S(2)sqrt(csin(a + bx))EllipticE(-Pi/S(4) + a/S(2) + bx/S(2), S(2))/(bcS(2)sqrt(sin(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(-5)/2), x), x, -S(2)cos(a + bx)/(S(3)bc(csin(a + bx))*(S(3)/2)) + S(2)EllipticF(-Pi/S(4) + a/S(2) + bx/S(2), S(2))sqrt(sin(a + bx))/(S(3)bcS(2)sqrt(csin(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*(S(-7)/2), x), x, -S(2)cos(a + bx)/(S(5)bc(csin(a + bx))*(S(5)/2)) - S(6)cos(a + bx)/(S(5)bcS(3)sqrt(csin(a + bx))) - S(6)sqrt(csin(a + bx))EllipticE(-Pi/S(4) + a/S(2) + bx/S(2), S(2))/(S(5)bcS(4)sqrt(sin(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(4)/3), x), x, S(3)(csin(a + bx))*(S(7)/3)Hypergeometric2F1(S(1)/2, S(7)/6, S(13)/6, sin(a + bx)S(2))cos(a + bx)/(S(7)bcsqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(2)/3), x), x, S(3)(csin(a + bx))*(S(5)/3)Hypergeometric2F1(S(1)/2, S(5)/6, S(11)/6, sin(a + bx)S(2))cos(a + bx)/(S(5)bcsqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(1)/3), x), x, -S(3)c*(S(1)/3)sqrt(S(1) - (csin(a + bx))(S(2)/3)/c(S(2)/3))sqrt(S(9)/2 - S(3)sqrt(S(3))I/S(2))sqrt((-sqrt(S(3)) + I)/(-sqrt(S(3)) + S(3)I) + S(2)(csin(a + bx))(S(2)/3)/(c(S(2)/3)(S(3) + sqrt(S(3))I)))sqrt((sqrt(S(3)) + I)/(sqrt(S(3)) + S(3)I) + S(2)(csin(a + bx))(S(2)/3)/(c(S(2)/3)(S(3) - sqrt(S(3))I)))EllipticE(asin(sqrt(S(2))sqrt(S(1) - (csin(a + bx))(S(2)/3)/c(S(2)/3))/sqrt(S(3) + sqrt(S(3))I)), (-sqrt(S(3)) + S(3)I)/(sqrt(S(3)) + S(3)I))sec(a + bx)/b + S(3)sqrt(S(2))c*(S(1)/3)(S(1) - sqrt(S(3))I)sqrt(S(1) - (csin(a + bx))(S(2)/3)/c(S(2)/3))sqrt(S(3) - sqrt(S(3))I)sqrt((-sqrt(S(3)) + I)/(-sqrt(S(3)) + S(3)I) + S(2)(csin(a + bx))(S(2)/3)/(c(S(2)/3)(S(3) + sqrt(S(3))I)))sqrt((sqrt(S(3)) + I)/(sqrt(S(3)) + S(3)I) + S(2)(csin(a + bx))(S(2)/3)/(c(S(2)/3)(S(3) - sqrt(S(3))I)))EllipticF(asin(sqrt(S(2))sqrt(S(1) - (csin(a + bx))(S(2)/3)/c(S(2)/3))/sqrt(S(3) - sqrt(S(3))I)), (sqrt(S(3)) + S(3)I)/(-sqrt(S(3)) + S(3)I))sec(a + bx)/(S(4)b), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((csin(a + bx))*(S(1)/3), x), x, S(3)(csin(a + bx))*(S(4)/3)Hypergeometric2F1(S(1)/2, S(2)/3, S(5)/3, sin(a + bx)S(2))cos(a + bx)/(S(4)bcsqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(-1)/3), x), x, -S(3)sqrt(S(2))sqrt(S(1) - (csin(a + bx))(S(2)/3)/c(S(2)/3))sqrt(S(3) - sqrt(S(3))I)sqrt((-sqrt(S(3)) + I)/(-sqrt(S(3)) + S(3)I) + S(2)(csin(a + bx))(S(2)/3)/(c(S(2)/3)(S(3) + sqrt(S(3))I)))sqrt((sqrt(S(3)) + I)/(sqrt(S(3)) + S(3)I) + S(2)(csin(a + bx))(S(2)/3)/(c(S(2)/3)(S(3) - sqrt(S(3))I)))EllipticF(asin(sqrt(S(2))sqrt(S(1) - (csin(a + bx))(S(2)/3)/c(S(2)/3))/sqrt(S(3) - sqrt(S(3))I)), (sqrt(S(3)) + S(3)I)/(-sqrt(S(3)) + S(3)I))sec(a + bx)/(S(2)bc*(S(1)/3)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((csin(a + bx))(S(-1)/3), x), x, S(3)(csin(a + bx))*(S(2)/3)Hypergeometric2F1(S(1)/3, S(1)/2, S(4)/3, sin(a + bx)S(2))cos(a + bx)/(S(2)bcsqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(-2)/3), x), x, S(3)(S(3)/4)(csin(a + bx))*(S(1)/3)sqrt(c*(S(4)/3)(S(1) + (csin(a + bx))(S(2)/3)/c(S(2)/3) + (csin(a + bx))(S(4)/3)/c(S(4)/3))/(c*(S(2)/3) - (csin(a + bx))(S(2)/3)(S(1) + sqrt(S(3))))*S(2))(c*(S(2)/3) - (csin(a + bx))(S(2)/3))EllipticF(acos((c*(S(2)/3) - (csin(a + bx))(S(2)/3)(-sqrt(S(3)) + S(1)))/(c*(S(2)/3) - (csin(a + bx))(S(2)/3)(S(1) + sqrt(S(3))))), sqrt(S(3))/S(4) + S(1)/2)sec(a + bx)/(S(2)bc*(S(5)/3)sqrt(-(csin(a + bx))*(S(2)/3)(c*(S(2)/3) - (csin(a + bx))(S(2)/3))/(c(S(2)/3) - (csin(a + bx))(S(2)/3)(S(1) + sqrt(S(3))))*S(2))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((csin(a + bx))(S(-2)/3), x), x, S(3)(csin(a + bx))*(S(1)/3)Hypergeometric2F1(S(1)/6, S(1)/2, S(7)/6, sin(a + bx)S(2))cos(a + bx)/(bcsqrt(cos(a + bx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(-4)/3), x), x, -S(3)Hypergeometric2F1(S(-1)/6, S(1)/2, S(5)/6, sin(a + bx)S(2))cos(a + bx)/(bc(csin(a + bx))(S(1)/3)sqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*n, x), x, Hypergeometric2F1(S(1)/2, n/S(2) + S(1)/2, n/S(2) + S(3)/2, sin(a + bx)*S(2))sin(a + bx)(n + S(1))cos(a + bx)/(b(n + S(1))sqrt(cos(a + bx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))n, x), x, (csin(a + bx))(n + S(1))Hypergeometric2F1(S(1)/2, n/S(2) + S(1)/2, n/S(2) + S(3)/2, sin(a + bx)S(2))cos(a + bx)/(bc(n + S(1))sqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(x)S(2))(S(5)/2), x), x, -S(8)aS(2)sqrt(asin(x)S(2))cot(x)/S(15) - S(4)a(asin(x)S(2))(S(3)/2)cot(x)/S(15) - (asin(x)S(2))(S(5)/2)cot(x)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(x)S(2))(S(3)/2), x), x, -S(2)asqrt(asin(x)*S(2))cot(x)/S(3) - (asin(x)S(2))(S(3)/2)cot(x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(asin(x)S(2)), x), x, -sqrt(asin(x)*S(2))cot(x), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(asin(x)S(2)), x), x, -sin(x)atanh(cos(x))/sqrt(asin(x)S(2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(x)S(2))(S(-3)/2), x), x, -sin(x)atanh(cos(x))/(S(2)asqrt(asin(x)*S(2))) - cot(x)/(S(2)asqrt(asin(x)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(x)S(2))(S(-5)/2), x), x, -cot(x)/(S(4)a(asin(x)S(2))(S(3)/2)) - S(3)sin(x)atanh(cos(x))/(S(8)a*S(2)sqrt(asin(x)S(2))) - S(3)cot(x)/(S(8)aS(2)sqrt(asin(x)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(x)S(3))(S(5)/2), x), x, -S(26)aS(2)sqrt(asin(x)S(3))EllipticF(Pi/S(4) - x/S(2), S(2))/(S(77)sin(x)(S(3)/2)) - S(2)a*S(2)sqrt(asin(x)S(3))sin(x)*S(5)cos(x)/S(15) - S(26)aS(2)sqrt(asin(x)S(3))sin(x)*S(3)cos(x)/S(165) - S(78)aS(2)sqrt(asin(x)S(3))sin(x)cos(x)/S(385) - S(26)a*S(2)sqrt(asin(x)S(3))cot(x)/S(77), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(x)S(3))(S(3)/2), x), x, -S(14)asqrt(asin(x)*S(3))EllipticE(Pi/S(4) - x/S(2), S(2))/(S(15)sin(x)(S(3)/2)) - S(2)asqrt(asin(x)*S(3))sin(x)*S(2)cos(x)/S(9) - S(14)asqrt(asin(x)S(3))cos(x)/S(45), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(asin(x)S(3)), x), x, -S(2)sqrt(asin(x)S(3))EllipticF(Pi/S(4) - x/S(2), S(2))/(S(3)sin(x)(S(3)/2)) - S(2)sqrt(asin(x)S(3))cot(x)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(asin(x)S(3)), x), x, S(2)EllipticE(Pi/S(4) - x/S(2), S(2))sin(x)(S(3)/2)/sqrt(asin(x)*S(3)) - S(2)sin(x)cos(x)/sqrt(asin(x)*S(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(x)S(3))(S(-3)/2), x), x, -S(10)EllipticF(Pi/S(4) - x/S(2), S(2))sin(x)*(S(3)/2)/(S(21)asqrt(asin(x)*S(3))) - S(10)cos(x)/(S(21)asqrt(asin(x)S(3))) - S(2)cot(x)csc(x)/(S(7)asqrt(asin(x)*S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(x)S(3))(S(-5)/2), x), x, S(154)EllipticE(Pi/S(4) - x/S(2), S(2))sin(x)*(S(3)/2)/(S(195)a*S(2)sqrt(asin(x)S(3))) - S(154)sin(x)cos(x)/(S(195)a*S(2)sqrt(asin(x)S(3))) - S(2)cot(x)csc(x)S(4)/(S(13)a*S(2)sqrt(asin(x)S(3))) - S(22)cot(x)csc(x)S(2)/(S(117)a*S(2)sqrt(asin(x)S(3))) - S(154)cot(x)/(S(585)aS(2)sqrt(asin(x)S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(x)S(4))(S(5)/2), x), x, S(63)aS(2)xsqrt(asin(x)*S(4))csc(x)S(2)/S(256) - aS(2)sqrt(asin(x)*S(4))sin(x)*S(7)cos(x)/S(10) - S(9)aS(2)sqrt(asin(x)S(4))sin(x)*S(5)cos(x)/S(80) - S(21)aS(2)sqrt(asin(x)S(4))sin(x)*S(3)cos(x)/S(160) - S(21)aS(2)sqrt(asin(x)S(4))sin(x)cos(x)/S(128) - S(63)a*S(2)sqrt(asin(x)S(4))cot(x)/S(256), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(x)S(4))(S(3)/2), x), x, S(5)axsqrt(asin(x)S(4))csc(x)*S(2)/S(16) - asqrt(asin(x)S(4))sin(x)*S(3)cos(x)/S(6) - S(5)asqrt(asin(x)S(4))sin(x)cos(x)/S(24) - S(5)asqrt(asin(x)*S(4))cot(x)/S(16), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(asin(x)S(4)), x), x, xsqrt(asin(x)S(4))csc(x)*S(2)/S(2) - sqrt(asin(x)*S(4))cot(x)/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(asin(x)S(4)), x), x, -sin(x)cos(x)/sqrt(asin(x)S(4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(x)S(4))(S(-3)/2), x), x, -sin(x)cos(x)/(asqrt(asin(x)S(4))) - cos(x)S(2)cot(x)*S(3)/(S(5)asqrt(asin(x)*S(4))) - S(2)cos(x)*S(2)cot(x)/(S(3)asqrt(asin(x)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(x)S(4))(S(-5)/2), x), x, -sin(x)cos(x)/(aS(2)sqrt(asin(x)S(4))) - cos(x)S(2)cot(x)*S(7)/(S(9)a*S(2)sqrt(asin(x)S(4))) - S(4)cos(x)*S(2)cot(x)*S(5)/(S(7)a*S(2)sqrt(asin(x)S(4))) - S(6)cos(x)*S(2)cot(x)*S(3)/(S(5)a*S(2)sqrt(asin(x)S(4))) - S(4)cos(x)*S(2)cot(x)/(S(3)aS(2)sqrt(asin(x)S(4))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(c + dx)p)n, x), x, (bsin(c + dx)p)nHypergeometric2F1(S(1)/2, np/S(2) + S(1)/2, np/S(2) + S(3)/2, sin(c + dx)S(2))sin(c + dx)cos(c + dx)/(d(np + S(1))sqrt(cos(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx)S(2))n, x), x, (csin(a + bx)S(2))nHypergeometric2F1(S(1)/2, n + S(1)/2, n + S(3)/2, sin(a + bx)S(2))sin(a + bx)cos(a + bx)/(b(S(2)n + S(1))sqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx)S(3))n, x), x, (csin(a + bx)S(3))nHypergeometric2F1(S(1)/2, S(3)n/S(2) + S(1)/2, S(3)n/S(2) + S(3)/2, sin(a + bx)S(2))sin(a + bx)cos(a + bx)/(b(S(3)n + S(1))sqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx)S(4))n, x), x, (csin(a + bx)S(4))nHypergeometric2F1(S(1)/2, S(2)n + S(1)/2, S(2)n + S(3)/2, sin(a + bx)S(2))sin(a + bx)cos(a + bx)/(b(S(4)n + S(1))sqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx)m)(S(5)/2), x), x, S(2)c*S(2)sqrt(csin(a + bx)*m)Hypergeometric2F1(S(1)/2, S(5)m/S(4) + S(1)/2, S(5)m/S(4) + S(3)/2, sin(a + bx)S(2))sin(a + bx)(S(2)m + S(1))cos(a + bx)/(b(S(5)m + S(2))sqrt(cos(a + bx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx)m)(S(3)/2), x), x, S(2)csqrt(csin(a + bx)m)Hypergeometric2F1(S(1)/2, S(3)m/S(4) + S(1)/2, S(3)m/S(4) + S(3)/2, sin(a + bx)S(2))sin(a + bx)(m + S(1))cos(a + bx)/(b(S(3)m + S(2))sqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx)m), x), x, S(2)sqrt(csin(a + bx)*m)Hypergeometric2F1(S(1)/2, m/S(4) + S(1)/2, m/S(4) + S(3)/2, sin(a + bx)S(2))sin(a + bx)cos(a + bx)/(b(m + S(2))sqrt(cos(a + bx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(csin(a + bx)m), x), x, S(2)Hypergeometric2F1(S(1)/2, -m/S(4) + S(1)/2, -m/S(4) + S(3)/2, sin(a + bx)S(2))sin(a + bx)cos(a + bx)/(bsqrt(csin(a + bx)*m)(-m + S(2))sqrt(cos(a + bx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx)m)(S(-3)/2), x), x, S(2)Hypergeometric2F1(S(1)/2, -S(3)m/S(4) + S(1)/2, -S(3)m/S(4) + S(3)/2, sin(a + bx)S(2))sin(a + bx)(-m + S(1))cos(a + bx)/(bcsqrt(csin(a + bx)m)(-S(3)m + S(2))sqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx)m)(S(-5)/2), x), x, S(2)Hypergeometric2F1(S(1)/2, -S(5)m/S(4) + S(1)/2, -S(5)m/S(4) + S(3)/2, sin(a + bx)S(2))sin(a + bx)(-S(2)m + S(1))cos(a + bx)/(bcS(2)sqrt(csin(a + bx)*m)(-S(5)m + S(2))sqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx)m)(S(1)/m), x), x, -(csin(a + bx)m)(S(1)/m)cot(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((a(bsin(c + dx))p)n, x), x, (a(bsin(c + dx))p)nHypergeometric2F1(S(1)/2, np/S(2) + S(1)/2, np/S(2) + S(3)/2, sin(c + dx)S(2))sin(c + dx)cos(c + dx)/(d(np + S(1))sqrt(cos(c + dx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(e + fx))m(bsin(e + fx))*n, x), x, (asin(e + fx))(m + S(1))(bsin(e + fx))*nHypergeometric2F1(S(1)/2, m/S(2) + n/S(2) + S(1)/2, m/S(2) + n/S(2) + S(3)/2, sin(e + fx)S(2))cos(e + fx)/(af(m + n + S(1))sqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)cos(a + bx)*S(3), x), x, -cos(a + bx)*S(4)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)cos(a + bx)S(2), x), x, -cos(a + bx)*S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)cos(a + bx), x), x, sin(a + bx)*S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)sec(a + bx), x), x, -log(cos(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)sec(a + bx)S(2), x), x, sec(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)sec(a + bx)S(3), x), x, sec(a + bx)*S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)sec(a + bx)S(4), x), x, sec(a + bx)*S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)cos(a + bx)S(7), x), x, -sin(a + bx)*S(9)/(S(9)b) + S(3)sin(a + bx)*S(7)/(S(7)b) - S(3)sin(a + bx)*S(5)/(S(5)b) + sin(a + bx)S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)cos(a + bx)S(5), x), x, sin(a + bx)*S(7)/(S(7)b) - S(2)sin(a + bx)*S(5)/(S(5)b) + sin(a + bx)S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)cos(a + bx)S(3), x), x, -sin(a + bx)*S(5)/(S(5)b) + sin(a + bx)S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)cos(a + bx), x), x, sin(a + bx)*S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)sec(a + bx)S(2), x), x, -x + tan(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)sec(a + bx)S(4), x), x, tan(a + bx)*S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)sec(a + bx)S(6), x), x, tan(a + bx)*S(5)/(S(5)b) + tan(a + bx)S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)sec(a + bx)S(8), x), x, tan(a + bx)*S(7)/(S(7)b) + S(2)tan(a + bx)*S(5)/(S(5)b) + tan(a + bx)S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)sec(a + bx)S(10), x), x, tan(a + bx)*S(9)/(S(9)b) + S(3)tan(a + bx)*S(7)/(S(7)b) + S(3)tan(a + bx)*S(5)/(S(5)b) + tan(a + bx)S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)cos(a + bx)S(6), x), x, S(5)x/S(128) - sin(a + bx)cos(a + bx)S(7)/(S(8)b) + sin(a + bx)cos(a + bx)S(5)/(S(48)b) + S(5)sin(a + bx)cos(a + bx)*S(3)/(S(192)b) + S(5)sin(a + bx)cos(a + bx)/(S(128)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(2)cos(a + bx)S(4), x), x, x/S(16) - sin(a + bx)cos(a + bx)*S(5)/(S(6)b) + sin(a + bx)cos(a + bx)S(3)/(S(24)b) + sin(a + bx)cos(a + bx)/(S(16)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)cos(a + bx)S(2), x), x, x/S(8) - sin(a + bx)cos(a + bx)*S(3)/(S(4)b) + sin(a + bx)cos(a + bx)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2), x), x, x/S(2) - sin(a + bx)cos(a + bx)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(2)sec(a + bx), x), x, -sin(a + bx)/b + atanh(sin(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(2)sec(a + bx)S(3), x), x, tan(a + bx)sec(a + bx)/(S(2)b) - atanh(sin(a + bx))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(2)sec(a + bx)S(5), x), x, tan(a + bx)sec(a + bx)*S(3)/(S(4)b) - tan(a + bx)sec(a + bx)/(S(8)b) - atanh(sin(a + bx))/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)sec(a + bx)S(7), x), x, tan(a + bx)sec(a + bx)*S(5)/(S(6)b) - tan(a + bx)sec(a + bx)S(3)/(S(24)b) - tan(a + bx)sec(a + bx)/(S(16)b) - atanh(sin(a + bx))/(S(16)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)cos(a + bx)S(5), x), x, cos(a + bx)*S(8)/(S(8)b) - cos(a + bx)S(6)/(S(6)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)cos(a + bx)S(4), x), x, cos(a + bx)*S(7)/(S(7)b) - cos(a + bx)S(5)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)cos(a + bx)S(3), x), x, -sin(a + bx)*S(6)/(S(6)b) + sin(a + bx)S(4)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)cos(a + bx)S(2), x), x, cos(a + bx)*S(5)/(S(5)b) - cos(a + bx)S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)cos(a + bx), x), x, sin(a + bx)*S(4)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)sec(a + bx), x), x, -log(cos(a + bx))/b + cos(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)sec(a + bx)S(2), x), x, cos(a + bx)/b + sec(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(3)sec(a + bx)S(3), x), x, log(cos(a + bx))/b + tan(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)sec(a + bx)S(4), x), x, sec(a + bx)*S(3)/(S(3)b) - sec(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(3)sec(a + bx)S(5), x), x, tan(a + bx)*S(4)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)sec(a + bx)S(6), x), x, sec(a + bx)*S(5)/(S(5)b) - sec(a + bx)S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)sec(a + bx)S(7), x), x, sec(a + bx)*S(6)/(S(6)b) - sec(a + bx)S(4)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)sec(a + bx)S(8), x), x, sec(a + bx)*S(7)/(S(7)b) - sec(a + bx)S(5)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)sec(a + bx)S(9), x), x, sec(a + bx)*S(8)/(S(8)b) - sec(a + bx)S(6)/(S(6)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)cos(a + bx)S(7), x), x, -sin(a + bx)*S(11)/(S(11)b) + sin(a + bx)S(9)/(S(3)b) - S(3)sin(a + bx)*S(7)/(S(7)b) + sin(a + bx)S(5)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)cos(a + bx)S(5), x), x, sin(a + bx)*S(9)/(S(9)b) - S(2)sin(a + bx)*S(7)/(S(7)b) + sin(a + bx)S(5)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)cos(a + bx)S(3), x), x, -sin(a + bx)*S(7)/(S(7)b) + sin(a + bx)S(5)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)cos(a + bx), x), x, sin(a + bx)*S(5)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)sec(a + bx)S(2), x), x, -S(3)x/S(2) - sin(a + bx)S(2)tan(a + bx)/(S(2)b) + S(3)tan(a + bx)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(4)sec(a + bx)S(4), x), x, x + tan(a + bx)*S(3)/(S(3)b) - tan(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(4)sec(a + bx)S(6), x), x, tan(a + bx)*S(5)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)sec(a + bx)S(8), x), x, tan(a + bx)*S(7)/(S(7)b) + tan(a + bx)S(5)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)sec(a + bx)S(10), x), x, tan(a + bx)*S(9)/(S(9)b) + S(2)tan(a + bx)*S(7)/(S(7)b) + tan(a + bx)S(5)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)cos(a + bx)S(6), x), x, S(3)x/S(256) - sin(a + bx)S(3)cos(a + bx)S(7)/(S(10)b) - S(3)sin(a + bx)cos(a + bx)*S(7)/(S(80)b) + sin(a + bx)cos(a + bx)S(5)/(S(160)b) + sin(a + bx)cos(a + bx)S(3)/(S(128)b) + S(3)sin(a + bx)cos(a + bx)/(S(256)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(4)cos(a + bx)S(4), x), x, S(3)x/S(128) - sin(a + bx)S(3)cos(a + bx)S(5)/(S(8)b) - sin(a + bx)cos(a + bx)S(5)/(S(16)b) + sin(a + bx)cos(a + bx)S(3)/(S(64)b) + S(3)sin(a + bx)cos(a + bx)/(S(128)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(4)cos(a + bx)S(2), x), x, x/S(16) - sin(a + bx)*S(3)cos(a + bx)S(3)/(S(6)b) - sin(a + bx)cos(a + bx)S(3)/(S(8)b) + sin(a + bx)cos(a + bx)/(S(16)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4), x), x, S(3)x/S(8) - sin(a + bx)S(3)cos(a + bx)/(S(4)b) - S(3)sin(a + bx)cos(a + bx)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(4)sec(a + bx), x), x, -sin(a + bx)*S(3)/(S(3)b) - sin(a + bx)/b + atanh(sin(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)sec(a + bx)S(3), x), x, sin(a + bx)tan(a + bx)*S(2)/(S(2)b) + S(3)sin(a + bx)/(S(2)b) - S(3)atanh(sin(a + bx))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)sec(a + bx)S(5), x), x, tan(a + bx)*S(3)sec(a + bx)/(S(4)b) - S(3)tan(a + bx)sec(a + bx)/(S(8)b) + S(3)atanh(sin(a + bx))/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)sec(a + bx)S(7), x), x, tan(a + bx)*S(3)sec(a + bx)S(3)/(S(6)b) - tan(a + bx)sec(a + bx)S(3)/(S(8)b) + tan(a + bx)sec(a + bx)/(S(16)b) + atanh(sin(a + bx))/(S(16)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)sec(a + bx)S(9), x), x, tan(a + bx)*S(3)sec(a + bx)S(5)/(S(8)b) - tan(a + bx)sec(a + bx)S(5)/(S(16)b) + tan(a + bx)sec(a + bx)S(3)/(S(64)b) + S(3)tan(a + bx)sec(a + bx)/(S(128)b) + S(3)atanh(sin(a + bx))/(S(128)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)cos(a + bx)S(7), x), x, -cos(a + bx)*S(12)/(S(12)b) + cos(a + bx)S(10)/(S(5)b) - cos(a + bx)S(8)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)cos(a + bx)S(6), x), x, -cos(a + bx)*S(11)/(S(11)b) + S(2)cos(a + bx)*S(9)/(S(9)b) - cos(a + bx)S(7)/(S(7)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)cos(a + bx)S(5), x), x, sin(a + bx)*S(10)/(S(10)b) - sin(a + bx)S(8)/(S(4)b) + sin(a + bx)S(6)/(S(6)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)cos(a + bx)S(4), x), x, -cos(a + bx)*S(9)/(S(9)b) + S(2)cos(a + bx)*S(7)/(S(7)b) - cos(a + bx)S(5)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)cos(a + bx)S(3), x), x, -sin(a + bx)*S(8)/(S(8)b) + sin(a + bx)S(6)/(S(6)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)cos(a + bx)S(2), x), x, -cos(a + bx)*S(7)/(S(7)b) + S(2)cos(a + bx)*S(5)/(S(5)b) - cos(a + bx)S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)cos(a + bx), x), x, sin(a + bx)*S(6)/(S(6)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)sec(a + bx), x), x, -log(cos(a + bx))/b - cos(a + bx)S(4)/(S(4)b) + cos(a + bx)S(2)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(5)sec(a + bx)S(2), x), x, -cos(a + bx)*S(3)/(S(3)b) + S(2)cos(a + bx)/b + sec(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(5)sec(a + bx)S(3), x), x, S(2)log(cos(a + bx))/b - cos(a + bx)*S(2)/(S(2)b) + sec(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)sec(a + bx)S(4), x), x, -cos(a + bx)/b + sec(a + bx)S(3)/(S(3)b) - S(2)sec(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)sec(a + bx)S(5), x), x, -log(cos(a + bx))/b + tan(a + bx)S(4)/(S(4)b) - tan(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)sec(a + bx)S(6), x), x, sec(a + bx)*S(5)/(S(5)b) - S(2)sec(a + bx)*S(3)/(S(3)b) + sec(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(5)sec(a + bx)S(7), x), x, tan(a + bx)*S(6)/(S(6)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)sec(a + bx)S(8), x), x, sec(a + bx)*S(7)/(S(7)b) - S(2)sec(a + bx)*S(5)/(S(5)b) + sec(a + bx)S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)sec(a + bx)S(9), x), x, tan(a + bx)*S(8)/(S(8)b) + tan(a + bx)S(6)/(S(6)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)sec(a + bx)S(10), x), x, sec(a + bx)*S(9)/(S(9)b) - S(2)sec(a + bx)*S(7)/(S(7)b) + sec(a + bx)S(5)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)sec(a + bx)S(11), x), x, sec(a + bx)*S(10)/(S(10)b) - sec(a + bx)S(8)/(S(4)b) + sec(a + bx)S(6)/(S(6)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)sec(a + bx)S(12), x), x, sec(a + bx)*S(11)/(S(11)b) - S(2)sec(a + bx)*S(9)/(S(9)b) + sec(a + bx)S(7)/(S(7)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)sec(a + bx)S(13), x), x, sec(a + bx)*S(12)/(S(12)b) - sec(a + bx)S(10)/(S(5)b) + sec(a + bx)S(8)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(6)sec(a + bx)S(3), x), x, sin(a + bx)*S(3)tan(a + bx)S(2)/(S(2)b) + S(5)sin(a + bx)*S(3)/(S(6)b) + S(5)sin(a + bx)/(S(2)b) - S(5)atanh(sin(a + bx))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(7)sec(a + bx)S(6), x), x, cos(a + bx)/b + sec(a + bx)S(5)/(S(5)b) - sec(a + bx)S(3)/b + S(3)sec(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(6)/sin(a + bx), x), x, cos(a + bx)S(5)/(S(5)b) + cos(a + bx)S(3)/(S(3)b) + cos(a + bx)/b - atanh(cos(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(5)/sin(a + bx), x), x, log(sin(a + bx))/b + sin(a + bx)*S(4)/(S(4)b) - sin(a + bx)S(2)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(4)/sin(a + bx), x), x, cos(a + bx)S(3)/(S(3)b) + cos(a + bx)/b - atanh(cos(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(3)/sin(a + bx), x), x, log(sin(a + bx))/b - sin(a + bx)*S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(2)/sin(a + bx), x), x, cos(a + bx)/b - atanh(cos(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)/sin(a + bx), x), x, log(sin(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)/sin(a + bx), x), x, log(tan(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(2)/sin(a + bx), x), x, -atanh(cos(a + bx))/b + sec(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(3)/sin(a + bx), x), x, log(tan(a + bx))/b + tan(a + bx)*S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(4)/sin(a + bx), x), x, -atanh(cos(a + bx))/b + sec(a + bx)*S(3)/(S(3)b) + sec(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(5)/sin(a + bx), x), x, log(tan(a + bx))/b + tan(a + bx)*S(4)/(S(4)b) + tan(a + bx)S(2)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(6)/sin(a + bx), x), x, -atanh(cos(a + bx))/b + sec(a + bx)*S(5)/(S(5)b) + sec(a + bx)S(3)/(S(3)b) + sec(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(7)/sin(a + bx), x), x, log(tan(a + bx))/b + tan(a + bx)*S(6)/(S(6)b) + S(3)tan(a + bx)*S(4)/(S(4)b) + S(3)tan(a + bx)*S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(7)/sin(a + bx)*S(2), x), x, -sin(a + bx)*S(5)/(S(5)b) + sin(a + bx)S(3)/b - S(3)sin(a + bx)/b - csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(6)/sin(a + bx)*S(2), x), x, -S(15)x/S(8) + cos(a + bx)S(4)cot(a + bx)/(S(4)b) + S(5)cos(a + bx)*S(2)cot(a + bx)/(S(8)b) - S(15)cot(a + bx)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(5)/sin(a + bx)*S(2), x), x, sin(a + bx)*S(3)/(S(3)b) - S(2)sin(a + bx)/b - csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(4)/sin(a + bx)*S(2), x), x, -S(3)x/S(2) + cos(a + bx)S(2)cot(a + bx)/(S(2)b) - S(3)cot(a + bx)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(3)/sin(a + bx)*S(2), x), x, -sin(a + bx)/b - csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(2)/sin(a + bx)*S(2), x), x, -x - cot(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)/sin(a + bx)*S(2), x), x, -csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)/sin(a + bx)*S(2), x), x, atanh(sin(a + bx))/b - csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(2)/sin(a + bx)*S(2), x), x, tan(a + bx)/b - cot(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(3)/sin(a + bx)*S(2), x), x, S(3)atanh(sin(a + bx))/(S(2)b) + csc(a + bx)sec(a + bx)S(2)/(S(2)b) - S(3)csc(a + bx)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(4)/sin(a + bx)*S(2), x), x, tan(a + bx)*S(3)/(S(3)b) + S(2)tan(a + bx)/b - cot(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(5)/sin(a + bx)*S(2), x), x, S(15)atanh(sin(a + bx))/(S(8)b) + csc(a + bx)sec(a + bx)S(4)/(S(4)b) + S(5)csc(a + bx)sec(a + bx)*S(2)/(S(8)b) - S(15)csc(a + bx)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(7)/sin(a + bx)*S(3), x), x, -S(3)log(sin(a + bx))/b - sin(a + bx)*S(4)/(S(4)b) + S(3)sin(a + bx)*S(2)/(S(2)b) - csc(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(6)/sin(a + bx)*S(3), x), x, -cos(a + bx)*S(3)cot(a + bx)S(2)/(S(2)b) - S(5)cos(a + bx)*S(3)/(S(6)b) - S(5)cos(a + bx)/(S(2)b) + S(5)atanh(cos(a + bx))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(5)/sin(a + bx)*S(3), x), x, -S(2)log(sin(a + bx))/b + sin(a + bx)*S(2)/(S(2)b) - csc(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(4)/sin(a + bx)*S(3), x), x, -cos(a + bx)cot(a + bx)*S(2)/(S(2)b) - S(3)cos(a + bx)/(S(2)b) + S(3)atanh(cos(a + bx))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(3)/sin(a + bx)*S(3), x), x, -log(sin(a + bx))/b - cot(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(2)/sin(a + bx)*S(3), x), x, -cot(a + bx)csc(a + bx)/(S(2)b) + atanh(cos(a + bx))/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)/sin(a + bx)S(3), x), x, -csc(a + bx)*S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)/sin(a + bx)*S(3), x), x, log(tan(a + bx))/b - cot(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(2)/sin(a + bx)*S(3), x), x, -S(3)atanh(cos(a + bx))/(S(2)b) - csc(a + bx)S(2)sec(a + bx)/(S(2)b) + S(3)sec(a + bx)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(3)/sin(a + bx)*S(3), x), x, S(2)log(tan(a + bx))/b + tan(a + bx)*S(2)/(S(2)b) - cot(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(4)/sin(a + bx)*S(3), x), x, -S(5)atanh(cos(a + bx))/(S(2)b) - csc(a + bx)S(2)sec(a + bx)S(3)/(S(2)b) + S(5)sec(a + bx)*S(3)/(S(6)b) + S(5)sec(a + bx)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(5)/sin(a + bx)*S(3), x), x, S(3)log(tan(a + bx))/b + tan(a + bx)*S(4)/(S(4)b) + S(3)tan(a + bx)*S(2)/(S(2)b) - cot(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(9)/sin(a + bx)*S(4), x), x, sin(a + bx)*S(5)/(S(5)b) - S(4)sin(a + bx)*S(3)/(S(3)b) + S(6)sin(a + bx)/b - csc(a + bx)S(3)/(S(3)b) + S(4)csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(8)/sin(a + bx)*S(4), x), x, S(35)x/S(8) + cos(a + bx)S(4)cot(a + bx)S(3)/(S(4)b) + S(7)cos(a + bx)*S(2)cot(a + bx)S(3)/(S(8)b) - S(35)cot(a + bx)*S(3)/(S(24)b) + S(35)cot(a + bx)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(7)/sin(a + bx)*S(4), x), x, -sin(a + bx)*S(3)/(S(3)b) + S(3)sin(a + bx)/b - csc(a + bx)S(3)/(S(3)b) + S(3)csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(6)/sin(a + bx)*S(4), x), x, S(5)x/S(2) + cos(a + bx)S(2)cot(a + bx)S(3)/(S(2)b) - S(5)cot(a + bx)*S(3)/(S(6)b) + S(5)cot(a + bx)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(5)/sin(a + bx)*S(4), x), x, sin(a + bx)/b - csc(a + bx)S(3)/(S(3)b) + S(2)csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(4)/sin(a + bx)*S(4), x), x, x - cot(a + bx)*S(3)/(S(3)b) + cot(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(3)/sin(a + bx)*S(4), x), x, -csc(a + bx)*S(3)/(S(3)b) + csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(2)/sin(a + bx)*S(4), x), x, -cot(a + bx)*S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)/sin(a + bx)*S(4), x), x, -csc(a + bx)*S(3)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)/sin(a + bx)*S(4), x), x, atanh(sin(a + bx))/b - csc(a + bx)S(3)/(S(3)b) - csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(2)/sin(a + bx)*S(4), x), x, tan(a + bx)/b - cot(a + bx)S(3)/(S(3)b) - S(2)cot(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(3)/sin(a + bx)*S(4), x), x, S(5)atanh(sin(a + bx))/(S(2)b) + csc(a + bx)S(3)sec(a + bx)S(2)/(S(2)b) - S(5)csc(a + bx)*S(3)/(S(6)b) - S(5)csc(a + bx)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(4)/sin(a + bx)*S(4), x), x, tan(a + bx)*S(3)/(S(3)b) + S(3)tan(a + bx)/b - cot(a + bx)S(3)/(S(3)b) - S(3)cot(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(5)/sin(a + bx)*S(4), x), x, S(35)atanh(sin(a + bx))/(S(8)b) + csc(a + bx)S(3)sec(a + bx)S(4)/(S(4)b) + S(7)csc(a + bx)*S(3)sec(a + bx)S(2)/(S(8)b) - S(35)csc(a + bx)*S(3)/(S(24)b) - S(35)csc(a + bx)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(9)/sin(a + bx)*S(5), x), x, S(6)log(sin(a + bx))/b + sin(a + bx)*S(4)/(S(4)b) - S(2)sin(a + bx)*S(2)/b - csc(a + bx)*S(4)/(S(4)b) + S(2)csc(a + bx)*S(2)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*S(8)/sin(a + bx)*S(5), x), x, -cos(a + bx)*S(3)cot(a + bx)S(4)/(S(4)b) + S(7)cos(a + bx)*S(3)cot(a + bx)S(2)/(S(8)b) + S(35)cos(a + bx)*S(3)/(S(24)b) + S(35)cos(a + bx)/(S(8)b) - S(35)atanh(cos(a + bx))/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(7)/sin(a + bx)*S(5), x), x, S(3)log(sin(a + bx))/b - sin(a + bx)*S(2)/(S(2)b) - csc(a + bx)S(4)/(S(4)b) + S(3)csc(a + bx)*S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(6)/sin(a + bx)*S(5), x), x, -cos(a + bx)cot(a + bx)*S(4)/(S(4)b) + S(5)cos(a + bx)cot(a + bx)*S(2)/(S(8)b) + S(15)cos(a + bx)/(S(8)b) - S(15)atanh(cos(a + bx))/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(5)/sin(a + bx)*S(5), x), x, log(sin(a + bx))/b - cot(a + bx)S(4)/(S(4)b) + cot(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(4)/sin(a + bx)*S(5), x), x, -cot(a + bx)*S(3)csc(a + bx)/(S(4)b) + S(3)cot(a + bx)csc(a + bx)/(S(8)b) - S(3)atanh(cos(a + bx))/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(3)/sin(a + bx)*S(5), x), x, -cot(a + bx)*S(4)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)S(2)/sin(a + bx)*S(5), x), x, -cot(a + bx)csc(a + bx)*S(3)/(S(4)b) + cot(a + bx)csc(a + bx)/(S(8)b) + atanh(cos(a + bx))/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)/sin(a + bx)*S(5), x), x, -csc(a + bx)*S(4)/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)/sin(a + bx)*S(5), x), x, log(tan(a + bx))/b - cot(a + bx)S(4)/(S(4)b) - cot(a + bx)S(2)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(2)/sin(a + bx)*S(5), x), x, -S(15)atanh(cos(a + bx))/(S(8)b) - csc(a + bx)S(4)sec(a + bx)/(S(4)b) - S(5)csc(a + bx)*S(2)sec(a + bx)/(S(8)b) + S(15)sec(a + bx)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(3)/sin(a + bx)*S(5), x), x, S(3)log(tan(a + bx))/b + tan(a + bx)*S(2)/(S(2)b) - cot(a + bx)S(4)/(S(4)b) - S(3)cot(a + bx)*S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)S(4)/sin(a + bx)*S(5), x), x, -S(35)atanh(cos(a + bx))/(S(8)b) - csc(a + bx)S(4)sec(a + bx)S(3)/(S(4)b) - S(7)csc(a + bx)*S(2)sec(a + bx)S(3)/(S(8)b) + S(35)sec(a + bx)*S(3)/(S(24)b) + S(35)sec(a + bx)/(S(8)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(a + bx)*S(5)/sin(a + bx)*S(5), x), x, S(6)log(tan(a + bx))/b + tan(a + bx)*S(4)/(S(4)b) + S(2)tan(a + bx)*S(2)/b - cot(a + bx)*S(4)/(S(4)b) - S(2)cot(a + bx)S(2)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(x)S(2)/sin(x)S(6), x), x, -cot(x)S(5)/S(5) - cot(x)S(3)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(x)S(3)/sin(x)S(7), x), x, -csc(x)S(6)/S(6) + csc(x)*S(4)/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(3)/2)sin(a + bx), x), x, -S(2)(dcos(a + bx))*(S(5)/2)/(S(5)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcos(a + bx))sin(a + bx), x), x, -S(2)(dcos(a + bx))*(S(3)/2)/(S(3)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)/sqrt(dcos(a + bx)), x), x, -S(2)sqrt(dcos(a + bx))/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)/(dcos(a + bx))(S(3)/2), x), x, S(2)/(bdsqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)/(dcos(a + bx))*(S(5)/2), x), x, S(2)/(S(3)bd(dcos(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)/(dcos(a + bx))*(S(7)/2), x), x, S(2)/(S(5)bd(dcos(a + bx))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)/(dcos(a + bx))*(S(9)/2), x), x, S(2)/(S(7)bd(dcos(a + bx))*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(9)/2)sin(a + bx)S(2), x), x, S(28)d*S(4)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(S(195)bsqrt(cos(a + bx))) + S(28)d*S(3)(dcos(a + bx))*(S(3)/2)sin(a + bx)/(S(585)b) + S(4)d(dcos(a + bx))*(S(7)/2)sin(a + bx)/(S(117)b) - S(2)(dcos(a + bx))(S(11)/2)sin(a + bx)/(S(13)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(7)/2)sin(a + bx)S(2), x), x, S(20)d*S(4)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(231)bsqrt(dcos(a + bx))) + S(20)d*S(3)sqrt(dcos(a + bx))sin(a + bx)/(S(231)b) + S(4)d(dcos(a + bx))(S(5)/2)sin(a + bx)/(S(77)b) - S(2)(dcos(a + bx))(S(9)/2)sin(a + bx)/(S(11)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(5)/2)sin(a + bx)S(2), x), x, S(4)d*S(2)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(S(15)bsqrt(cos(a + bx))) + S(4)d(dcos(a + bx))(S(3)/2)sin(a + bx)/(S(45)b) - S(2)(dcos(a + bx))(S(7)/2)sin(a + bx)/(S(9)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(3)/2)sin(a + bx)S(2), x), x, S(4)d*S(2)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(21)bsqrt(dcos(a + bx))) + S(4)dsqrt(dcos(a + bx))sin(a + bx)/(S(21)b) - S(2)(dcos(a + bx))(S(5)/2)sin(a + bx)/(S(7)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcos(a + bx))sin(a + bx)S(2), x), x, S(4)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(S(5)bsqrt(cos(a + bx))) - S(2)(dcos(a + bx))*(S(3)/2)sin(a + bx)/(S(5)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(2)/sqrt(dcos(a + bx)), x), x, S(4)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(3)bsqrt(dcos(a + bx))) - S(2)sqrt(dcos(a + bx))sin(a + bx)/(S(3)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)/(dcos(a + bx))(S(3)/2), x), x, S(2)sin(a + bx)/(bdsqrt(dcos(a + bx))) - S(4)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(bdS(2)sqrt(cos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(2)/(dcos(a + bx))(S(5)/2), x), x, S(2)sin(a + bx)/(S(3)bd(dcos(a + bx))*(S(3)/2)) - S(4)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(3)bdS(2)sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(2)/(dcos(a + bx))(S(7)/2), x), x, S(2)sin(a + bx)/(S(5)bd(dcos(a + bx))*(S(5)/2)) - S(4)sin(a + bx)/(S(5)bdS(3)sqrt(dcos(a + bx))) + S(4)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(S(5)bdS(4)sqrt(cos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(2)/(dcos(a + bx))(S(9)/2), x), x, S(2)sin(a + bx)/(S(7)bd(dcos(a + bx))*(S(7)/2)) - S(4)sin(a + bx)/(S(21)bdS(3)(dcos(a + bx))*(S(3)/2)) - S(4)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(21)bdS(4)sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcos(a + bx))sin(a + bx)*S(3), x), x, -S(2)(dcos(a + bx))*(S(3)/2)/(S(3)bd) + S(2)(dcos(a + bx))*(S(7)/2)/(S(7)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(3)/sqrt(dcos(a + bx)), x), x, -S(2)sqrt(dcos(a + bx))/(bd) + S(2)(dcos(a + bx))*(S(5)/2)/(S(5)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(3)/(dcos(a + bx))(S(3)/2), x), x, S(2)/(bdsqrt(dcos(a + bx))) + S(2)(dcos(a + bx))*(S(3)/2)/(S(3)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(3)/(dcos(a + bx))(S(5)/2), x), x, S(2)/(S(3)bd(dcos(a + bx))*(S(3)/2)) + S(2)sqrt(dcos(a + bx))/(bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(3)/(dcos(a + bx))(S(7)/2), x), x, S(2)/(S(5)bd(dcos(a + bx))*(S(5)/2)) - S(2)/(bd*S(3)sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(3)/(dcos(a + bx))(S(9)/2), x), x, S(2)/(S(7)bd(dcos(a + bx))*(S(7)/2)) - S(2)/(S(3)bdS(3)(dcos(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(3)/(dcos(a + bx))(S(11)/2), x), x, S(2)/(S(9)bd(dcos(a + bx))*(S(9)/2)) - S(2)/(S(5)bdS(3)(dcos(a + bx))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(9)/2)sin(a + bx)S(4), x), x, S(56)d*S(4)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(S(1105)bsqrt(cos(a + bx))) + S(56)d*S(3)(dcos(a + bx))*(S(3)/2)sin(a + bx)/(S(3315)b) + S(8)d(dcos(a + bx))*(S(7)/2)sin(a + bx)/(S(663)b) - S(2)(dcos(a + bx))(S(11)/2)sin(a + bx)S(3)/(S(17)bd) - S(12)(dcos(a + bx))*(S(11)/2)sin(a + bx)/(S(221)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(7)/2)sin(a + bx)S(4), x), x, S(8)d*S(4)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(231)bsqrt(dcos(a + bx))) + S(8)d*S(3)sqrt(dcos(a + bx))sin(a + bx)/(S(231)b) + S(8)d(dcos(a + bx))(S(5)/2)sin(a + bx)/(S(385)b) - S(2)(dcos(a + bx))(S(9)/2)sin(a + bx)S(3)/(S(15)bd) - S(4)(dcos(a + bx))*(S(9)/2)sin(a + bx)/(S(55)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(5)/2)sin(a + bx)S(4), x), x, S(8)d*S(2)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(S(65)bsqrt(cos(a + bx))) + S(8)d(dcos(a + bx))(S(3)/2)sin(a + bx)/(S(195)b) - S(2)(dcos(a + bx))(S(7)/2)sin(a + bx)S(3)/(S(13)bd) - S(4)(dcos(a + bx))*(S(7)/2)sin(a + bx)/(S(39)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(3)/2)sin(a + bx)S(4), x), x, S(8)d*S(2)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(77)bsqrt(dcos(a + bx))) + S(8)dsqrt(dcos(a + bx))sin(a + bx)/(S(77)b) - S(2)(dcos(a + bx))(S(5)/2)sin(a + bx)S(3)/(S(11)bd) - S(12)(dcos(a + bx))*(S(5)/2)sin(a + bx)/(S(77)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcos(a + bx))sin(a + bx)S(4), x), x, S(8)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(S(15)bsqrt(cos(a + bx))) - S(2)(dcos(a + bx))*(S(3)/2)sin(a + bx)S(3)/(S(9)bd) - S(4)(dcos(a + bx))*(S(3)/2)sin(a + bx)/(S(15)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(4)/sqrt(dcos(a + bx)), x), x, S(8)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(7)bsqrt(dcos(a + bx))) - S(2)sqrt(dcos(a + bx))sin(a + bx)*S(3)/(S(7)bd) - S(4)sqrt(dcos(a + bx))sin(a + bx)/(S(7)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(4)/(dcos(a + bx))(S(3)/2), x), x, S(2)sin(a + bx)S(3)/(bdsqrt(dcos(a + bx))) - S(24)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(S(5)bd*S(2)sqrt(cos(a + bx))) + S(12)(dcos(a + bx))*(S(3)/2)sin(a + bx)/(S(5)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(4)/(dcos(a + bx))(S(5)/2), x), x, S(2)sin(a + bx)S(3)/(S(3)bd(dcos(a + bx))*(S(3)/2)) - S(8)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(3)bdS(2)sqrt(dcos(a + bx))) + S(4)sqrt(dcos(a + bx))sin(a + bx)/(S(3)bdS(3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(4)/(dcos(a + bx))(S(7)/2), x), x, S(2)sin(a + bx)S(3)/(S(5)bd(dcos(a + bx))*(S(5)/2)) - S(12)sin(a + bx)/(S(5)bdS(3)sqrt(dcos(a + bx))) + S(24)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(S(5)bdS(4)sqrt(cos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*S(4)/(dcos(a + bx))(S(9)/2), x), x, S(2)sin(a + bx)S(3)/(S(7)bd(dcos(a + bx))*(S(7)/2)) - S(4)sin(a + bx)/(S(7)bdS(3)(dcos(a + bx))*(S(3)/2)) + S(8)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(7)bdS(4)sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)S(5)cos(a + bx)(S(3)/2), x), x, -S(2)cos(a + bx)(S(13)/2)/(S(13)b) + S(4)cos(a + bx)*(S(9)/2)/(S(9)b) - S(2)cos(a + bx)*(S(5)/2)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*(S(9)/2)csc(a + bx), x), x, d(S(9)/2)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/b - d*(S(9)/2)atanh(sqrt(dcos(a + bx))/sqrt(d))/b + S(2)dS(3)(dcos(a + bx))*(S(3)/2)/(S(3)b) + S(2)d(dcos(a + bx))*(S(7)/2)/(S(7)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*(S(7)/2)csc(a + bx), x), x, -d(S(7)/2)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/b - d*(S(7)/2)atanh(sqrt(dcos(a + bx))/sqrt(d))/b + S(2)dS(3)sqrt(dcos(a + bx))/b + S(2)d(dcos(a + bx))*(S(5)/2)/(S(5)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*(S(5)/2)csc(a + bx), x), x, d(S(5)/2)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/b - d*(S(5)/2)atanh(sqrt(dcos(a + bx))/sqrt(d))/b + S(2)d(dcos(a + bx))*(S(3)/2)/(S(3)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*(S(3)/2)csc(a + bx), x), x, -d(S(3)/2)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/b - d*(S(3)/2)atanh(sqrt(dcos(a + bx))/sqrt(d))/b + S(2)dsqrt(dcos(a + bx))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcos(a + bx))csc(a + bx), x), x, sqrt(d)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/b - sqrt(d)atanh(sqrt(dcos(a + bx))/sqrt(d))/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)/sqrt(dcos(a + bx)), x), x, -ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(bsqrt(d)) - atanh(sqrt(dcos(a + bx))/sqrt(d))/(bsqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)/(dcos(a + bx))*(S(3)/2), x), x, S(2)/(bdsqrt(dcos(a + bx))) + ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(bd*(S(3)/2)) - atanh(sqrt(dcos(a + bx))/sqrt(d))/(bd*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)/(dcos(a + bx))*(S(5)/2), x), x, S(2)/(S(3)bd(dcos(a + bx))*(S(3)/2)) - ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(bd*(S(5)/2)) - atanh(sqrt(dcos(a + bx))/sqrt(d))/(bd*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)/(dcos(a + bx))*(S(7)/2), x), x, S(2)/(S(5)bd(dcos(a + bx))*(S(5)/2)) + S(2)/(bd*S(3)sqrt(dcos(a + bx))) + ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(bd(S(7)/2)) - atanh(sqrt(dcos(a + bx))/sqrt(d))/(bd*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)/(dcos(a + bx))*(S(9)/2), x), x, S(2)/(S(7)bd(dcos(a + bx))*(S(7)/2)) + S(2)/(S(3)bdS(3)(dcos(a + bx))*(S(3)/2)) - ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(bd*(S(9)/2)) - atanh(sqrt(dcos(a + bx))/sqrt(d))/(bd*(S(9)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(11)/2)csc(a + bx)S(2), x), x, -S(15)d*S(6)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(7)bsqrt(dcos(a + bx))) - S(15)d*S(5)sqrt(dcos(a + bx))sin(a + bx)/(S(7)b) - S(9)d*S(3)(dcos(a + bx))*(S(5)/2)sin(a + bx)/(S(7)b) - d(dcos(a + bx))(S(9)/2)csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(9)/2)csc(a + bx)S(2), x), x, -S(21)d*S(4)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(S(5)bsqrt(cos(a + bx))) - S(7)d*S(3)(dcos(a + bx))*(S(3)/2)sin(a + bx)/(S(5)b) - d(dcos(a + bx))(S(7)/2)csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(7)/2)csc(a + bx)S(2), x), x, -S(5)d*S(4)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(3)bsqrt(dcos(a + bx))) - S(5)d*S(3)sqrt(dcos(a + bx))sin(a + bx)/(S(3)b) - d(dcos(a + bx))*(S(5)/2)csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(5)/2)csc(a + bx)S(2), x), x, -S(3)d*S(2)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(bsqrt(cos(a + bx))) - d(dcos(a + bx))(S(3)/2)csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(3)/2)csc(a + bx)S(2), x), x, -dS(2)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(bsqrt(dcos(a + bx))) - dsqrt(dcos(a + bx))csc(a + bx)/b, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcos(a + bx))csc(a + bx)S(2), x), x, -sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(bsqrt(cos(a + bx))) - (dcos(a + bx))(S(3)/2)csc(a + bx)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)S(2)/sqrt(dcos(a + bx)), x), x, EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(bsqrt(dcos(a + bx))) - sqrt(dcos(a + bx))csc(a + bx)/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)S(2)/(dcos(a + bx))(S(3)/2), x), x, S(3)sin(a + bx)/(bdsqrt(dcos(a + bx))) - csc(a + bx)/(bdsqrt(dcos(a + bx))) - S(3)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(bd*S(2)sqrt(cos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(2)/(dcos(a + bx))(S(5)/2), x), x, S(5)sin(a + bx)/(S(3)bd(dcos(a + bx))*(S(3)/2)) - csc(a + bx)/(bd(dcos(a + bx))*(S(3)/2)) + S(5)EllipticF(a/S(2) + bx/S(2), S(2))sqrt(cos(a + bx))/(S(3)bdS(2)sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)S(2)/(dcos(a + bx))(S(7)/2), x), x, S(7)sin(a + bx)/(S(5)bd(dcos(a + bx))*(S(5)/2)) - csc(a + bx)/(bd(dcos(a + bx))*(S(5)/2)) + S(21)sin(a + bx)/(S(5)bdS(3)sqrt(dcos(a + bx))) - S(21)sqrt(dcos(a + bx))EllipticE(a/S(2) + bx/S(2), S(2))/(S(5)bdS(4)sqrt(cos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(11)/2)csc(a + bx)S(3), x), x, S(9)d*(S(11)/2)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(S(4)b) + S(9)d*(S(11)/2)atanh(sqrt(dcos(a + bx))/sqrt(d))/(S(4)b) - S(9)d*S(5)sqrt(dcos(a + bx))/(S(2)b) - S(9)d*S(3)(dcos(a + bx))*(S(5)/2)/(S(10)b) - d(dcos(a + bx))(S(9)/2)csc(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*(S(9)/2)csc(a + bx)S(3), x), x, -S(7)d*(S(9)/2)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(S(4)b) + S(7)d*(S(9)/2)atanh(sqrt(dcos(a + bx))/sqrt(d))/(S(4)b) - S(7)d*S(3)(dcos(a + bx))*(S(3)/2)/(S(6)b) - d(dcos(a + bx))(S(7)/2)csc(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*(S(7)/2)csc(a + bx)S(3), x), x, S(5)d*(S(7)/2)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(S(4)b) + S(5)d*(S(7)/2)atanh(sqrt(dcos(a + bx))/sqrt(d))/(S(4)b) - S(5)d*S(3)sqrt(dcos(a + bx))/(S(2)b) - d(dcos(a + bx))*(S(5)/2)csc(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*(S(5)/2)csc(a + bx)S(3), x), x, -S(3)d*(S(5)/2)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(S(4)b) + S(3)d*(S(5)/2)atanh(sqrt(dcos(a + bx))/sqrt(d))/(S(4)b) - d(dcos(a + bx))*(S(3)/2)csc(a + bx)S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*(S(3)/2)csc(a + bx)S(3), x), x, d(S(3)/2)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(S(4)b) + d(S(3)/2)atanh(sqrt(dcos(a + bx))/sqrt(d))/(S(4)b) - dsqrt(dcos(a + bx))csc(a + bx)*S(2)/(S(2)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcos(a + bx))csc(a + bx)*S(3), x), x, sqrt(d)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(S(4)b) - sqrt(d)atanh(sqrt(dcos(a + bx))/sqrt(d))/(S(4)b) - (dcos(a + bx))(S(3)/2)csc(a + bx)S(2)/(S(2)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(3)/sqrt(dcos(a + bx)), x), x, -sqrt(dcos(a + bx))csc(a + bx)S(2)/(S(2)bd) - S(3)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(S(4)bsqrt(d)) - S(3)atanh(sqrt(dcos(a + bx))/sqrt(d))/(S(4)bsqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(3)/(dcos(a + bx))(S(3)/2), x), x, -csc(a + bx)*S(2)/(S(2)bdsqrt(dcos(a + bx))) + S(5)/(S(2)bdsqrt(dcos(a + bx))) + S(5)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(S(4)bd*(S(3)/2)) - S(5)atanh(sqrt(dcos(a + bx))/sqrt(d))/(S(4)bd*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(3)/(dcos(a + bx))(S(5)/2), x), x, -csc(a + bx)*S(2)/(S(2)bd(dcos(a + bx))*(S(3)/2)) + S(7)/(S(6)bd(dcos(a + bx))*(S(3)/2)) - S(7)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(S(4)bd*(S(5)/2)) - S(7)atanh(sqrt(dcos(a + bx))/sqrt(d))/(S(4)bd*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(a + bx)*S(3)/(dcos(a + bx))(S(7)/2), x), x, -csc(a + bx)*S(2)/(S(2)bd(dcos(a + bx))*(S(5)/2)) + S(9)/(S(10)bd(dcos(a + bx))*(S(5)/2)) + S(9)/(S(2)bdS(3)sqrt(dcos(a + bx))) + S(9)ArcTan(sqrt(dcos(a + bx))/sqrt(d))/(S(4)bd(S(7)/2)) - S(9)atanh(sqrt(dcos(a + bx))/sqrt(d))/(S(4)bd*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(1)/5)sin(a + bx), x), x, -S(5)(dcos(a + bx))*(S(6)/5)/(S(6)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sin(x))cos(x)*S(3), x), x, -S(2)sin(x)*(S(7)/2)/S(7) + S(2)sin(x)(S(3)/2)/S(3), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(x)(S(3)/2)cos(x)S(3), x), x, -S(2)sin(x)*(S(9)/2)/S(9) + S(2)sin(x)(S(5)/2)/S(5), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(x)(S(5)/2)cos(x)S(3), x), x, -S(2)sin(x)*(S(11)/2)/S(11) + S(2)sin(x)(S(7)/2)/S(7), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(x)S(3)/sqrt(sin(x)), x), x, -S(2)sin(x)(S(5)/2)/S(5) + S(2)sqrt(sin(x)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx))(dcos(a + bx))(S(9)/2), x), x, S(7)d*S(4)sqrt(csin(a + bx))sqrt(dcos(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))/(S(20)bsqrt(sin(S(2)a + S(2)bx))) + S(7)dS(3)(csin(a + bx))*(S(3)/2)(dcos(a + bx))*(S(3)/2)/(S(30)bc) + d(csin(a + bx))*(S(3)/2)(dcos(a + bx))*(S(7)/2)/(S(5)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx))(dcos(a + bx))(S(5)/2), x), x, dS(2)sqrt(csin(a + bx))sqrt(dcos(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))/(S(2)bsqrt(sin(S(2)a + S(2)bx))) + d(csin(a + bx))*(S(3)/2)(dcos(a + bx))*(S(3)/2)/(S(3)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx))sqrt(dcos(a + bx)), x), x, sqrt(csin(a + bx))sqrt(dcos(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))/(bsqrt(sin(S(2)a + S(2)bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx))/(dcos(a + bx))(S(3)/2), x), x, -S(2)sqrt(csin(a + bx))sqrt(dcos(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))/(bd*S(2)sqrt(sin(S(2)a + S(2)bx))) + S(2)(csin(a + bx))*(S(3)/2)/(bcdsqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx))/(dcos(a + bx))*(S(7)/2), x), x, -S(4)sqrt(csin(a + bx))sqrt(dcos(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))/(S(5)bdS(4)sqrt(sin(S(2)a + S(2)bx))) + S(2)(csin(a + bx))*(S(3)/2)/(S(5)bcd(dcos(a + bx))(S(5)/2)) + S(4)(csin(a + bx))*(S(3)/2)/(S(5)bcd*S(3)sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx))(dcos(a + bx))(S(3)/2), x), x, -sqrt(S(2))sqrt(c)d(S(3)/2)ArcTan(S(1) - sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/(sqrt(c)sqrt(dcos(a + bx))))/(S(8)b) + sqrt(S(2))sqrt(c)d*(S(3)/2)ArcTan(S(1) + sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/(sqrt(c)sqrt(dcos(a + bx))))/(S(8)b) + sqrt(S(2))sqrt(c)d*(S(3)/2)log(sqrt(c)tan(a + bx) + sqrt(c) - sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/sqrt(dcos(a + bx)))/(S(16)b) - sqrt(S(2))sqrt(c)d(S(3)/2)log(sqrt(c)tan(a + bx) + sqrt(c) + sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/sqrt(dcos(a + bx)))/(S(16)b) + d(csin(a + bx))*(S(3)/2)sqrt(dcos(a + bx))/(S(2)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx))/sqrt(dcos(a + bx)), x), x, -sqrt(S(2))sqrt(c)ArcTan(S(1) - sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/(sqrt(c)sqrt(dcos(a + bx))))/(S(2)bsqrt(d)) + sqrt(S(2))sqrt(c)ArcTan(S(1) + sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/(sqrt(c)sqrt(dcos(a + bx))))/(S(2)bsqrt(d)) + sqrt(S(2))sqrt(c)log(sqrt(c)tan(a + bx) + sqrt(c) - sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/sqrt(dcos(a + bx)))/(S(4)bsqrt(d)) - sqrt(S(2))sqrt(c)log(sqrt(c)tan(a + bx) + sqrt(c) + sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/sqrt(dcos(a + bx)))/(S(4)bsqrt(d)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx))/(dcos(a + bx))*(S(5)/2), x), x, S(2)(csin(a + bx))*(S(3)/2)/(S(3)bcd(dcos(a + bx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx))/(dcos(a + bx))(S(9)/2), x), x, S(2)(csin(a + bx))*(S(3)/2)/(S(7)bcd(dcos(a + bx))(S(7)/2)) + S(8)(csin(a + bx))*(S(3)/2)/(S(21)bcd*S(3)(dcos(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx))/(dcos(a + bx))(S(13)/2), x), x, S(2)(csin(a + bx))*(S(3)/2)/(S(11)bcd(dcos(a + bx))(S(11)/2)) + S(16)(csin(a + bx))*(S(3)/2)/(S(77)bcd*S(3)(dcos(a + bx))*(S(7)/2)) + S(64)(csin(a + bx))*(S(3)/2)/(S(231)bcd*S(5)(dcos(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(3)/2)(dcos(a + bx))(S(3)/2), x), x, cS(2)dS(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(12)bsqrt(csin(a + bx))sqrt(dcos(a + bx))) + cdsqrt(csin(a + bx))sqrt(dcos(a + bx))/(S(6)b) - csqrt(csin(a + bx))(dcos(a + bx))*(S(5)/2)/(S(3)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(3)/2)/sqrt(dcos(a + bx)), x), x, cS(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(2)bsqrt(csin(a + bx))sqrt(dcos(a + bx))) - csqrt(csin(a + bx))sqrt(dcos(a + bx))/(bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(3)/2)/(dcos(a + bx))(S(5)/2), x), x, -cS(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(3)bdS(2)sqrt(csin(a + bx))sqrt(dcos(a + bx))) + S(2)csqrt(csin(a + bx))/(S(3)bd(dcos(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(3)/2)/(dcos(a + bx))(S(9)/2), x), x, -S(2)c*S(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(21)bdS(4)sqrt(csin(a + bx))sqrt(dcos(a + bx))) + S(2)csqrt(csin(a + bx))/(S(7)bd(dcos(a + bx))*(S(7)/2)) - S(2)csqrt(csin(a + bx))/(S(21)bdS(3)(dcos(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(3)/2)sqrt(dcos(a + bx)), x), x, sqrt(S(2))c(S(3)/2)sqrt(d)ArcTan(-sqrt(S(2))sqrt(c)sqrt(dcos(a + bx))/(sqrt(d)sqrt(csin(a + bx))) + S(1))/(S(8)b) - sqrt(S(2))c*(S(3)/2)sqrt(d)ArcTan(sqrt(S(2))sqrt(c)sqrt(dcos(a + bx))/(sqrt(d)sqrt(csin(a + bx))) + S(1))/(S(8)b) - sqrt(S(2))c*(S(3)/2)sqrt(d)log(-sqrt(S(2))sqrt(c)sqrt(dcos(a + bx))/sqrt(csin(a + bx)) + sqrt(d)cot(a + bx) + sqrt(d))/(S(16)b) + sqrt(S(2))c(S(3)/2)sqrt(d)log(sqrt(S(2))sqrt(c)sqrt(dcos(a + bx))/sqrt(csin(a + bx)) + sqrt(d)cot(a + bx) + sqrt(d))/(S(16)b) - csqrt(csin(a + bx))(dcos(a + bx))*(S(3)/2)/(S(2)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(3)/2)/(dcos(a + bx))(S(3)/2), x), x, -sqrt(S(2))c*(S(3)/2)ArcTan(-sqrt(S(2))sqrt(c)sqrt(dcos(a + bx))/(sqrt(d)sqrt(csin(a + bx))) + S(1))/(S(2)bd(S(3)/2)) + sqrt(S(2))c*(S(3)/2)ArcTan(sqrt(S(2))sqrt(c)sqrt(dcos(a + bx))/(sqrt(d)sqrt(csin(a + bx))) + S(1))/(S(2)bd(S(3)/2)) + sqrt(S(2))c*(S(3)/2)log(-sqrt(S(2))sqrt(c)sqrt(dcos(a + bx))/sqrt(csin(a + bx)) + sqrt(d)cot(a + bx) + sqrt(d))/(S(4)bd*(S(3)/2)) - sqrt(S(2))c*(S(3)/2)log(sqrt(S(2))sqrt(c)sqrt(dcos(a + bx))/sqrt(csin(a + bx)) + sqrt(d)cot(a + bx) + sqrt(d))/(S(4)bd*(S(3)/2)) + S(2)csqrt(csin(a + bx))/(bdsqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(3)/2)/(dcos(a + bx))(S(7)/2), x), x, S(2)(csin(a + bx))*(S(5)/2)/(S(5)bcd(dcos(a + bx))(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(3)/2)/(dcos(a + bx))(S(11)/2), x), x, S(2)csqrt(csin(a + bx))/(S(9)bd(dcos(a + bx))*(S(9)/2)) - S(2)csqrt(csin(a + bx))/(S(45)bdS(3)(dcos(a + bx))*(S(5)/2)) - S(8)csqrt(csin(a + bx))/(S(45)bdS(5)sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*(S(3)/2)/(dcos(a + bx))(S(15)/2), x), x, S(2)csqrt(csin(a + bx))/(S(13)bd(dcos(a + bx))*(S(13)/2)) - S(2)csqrt(csin(a + bx))/(S(117)bdS(3)(dcos(a + bx))*(S(9)/2)) - S(16)csqrt(csin(a + bx))/(S(585)bdS(5)(dcos(a + bx))*(S(5)/2)) - S(64)csqrt(csin(a + bx))/(S(585)bdS(7)sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*(S(5)/2)(dcos(a + bx))*(S(9)/2), x), x, S(3)c*S(2)d*S(4)sqrt(csin(a + bx))sqrt(dcos(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))/(S(40)bsqrt(sin(S(2)a + S(2)bx))) + cdS(3)(csin(a + bx))*(S(3)/2)(dcos(a + bx))*(S(3)/2)/(S(20)b) + S(3)cd(csin(a + bx))(S(3)/2)(dcos(a + bx))*(S(7)/2)/(S(70)b) - c(csin(a + bx))(S(3)/2)(dcos(a + bx))*(S(11)/2)/(S(7)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(5)/2)(dcos(a + bx))*(S(5)/2), x), x, S(3)c*S(2)d*S(2)sqrt(csin(a + bx))sqrt(dcos(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))/(S(20)bsqrt(sin(S(2)a + S(2)bx))) + cd(csin(a + bx))*(S(3)/2)(dcos(a + bx))*(S(3)/2)/(S(10)b) - c(csin(a + bx))(S(3)/2)(dcos(a + bx))*(S(7)/2)/(S(5)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(5)/2)sqrt(dcos(a + bx)), x), x, c*S(2)sqrt(csin(a + bx))sqrt(dcos(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))/(S(2)bsqrt(sin(S(2)a + S(2)bx))) - c(csin(a + bx))(S(3)/2)(dcos(a + bx))*(S(3)/2)/(S(3)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(5)/2)/(dcos(a + bx))(S(3)/2), x), x, -S(3)c*S(2)sqrt(csin(a + bx))sqrt(dcos(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))/(bd*S(2)sqrt(sin(S(2)a + S(2)bx))) + S(2)c(csin(a + bx))(S(3)/2)/(bdsqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(5)/2)/(dcos(a + bx))(S(7)/2), x), x, S(6)c*S(2)sqrt(csin(a + bx))sqrt(dcos(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))/(S(5)bdS(4)sqrt(sin(S(2)a + S(2)bx))) + S(2)c(csin(a + bx))(S(3)/2)/(S(5)bd(dcos(a + bx))*(S(5)/2)) - S(6)c(csin(a + bx))(S(3)/2)/(S(5)bdS(3)sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*(S(5)/2)/(dcos(a + bx))(S(11)/2), x), x, S(4)c*S(2)sqrt(csin(a + bx))sqrt(dcos(a + bx))EllipticE(-Pi/S(4) + a + bx, S(2))/(S(15)bdS(6)sqrt(sin(S(2)a + S(2)bx))) + S(2)c(csin(a + bx))(S(3)/2)/(S(9)bd(dcos(a + bx))*(S(9)/2)) - S(2)c(csin(a + bx))(S(3)/2)/(S(15)bdS(3)(dcos(a + bx))*(S(5)/2)) - S(4)c(csin(a + bx))(S(3)/2)/(S(15)bdS(5)sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*(S(5)/2)/sqrt(dcos(a + bx)), x), x, -S(3)sqrt(S(2))c(S(5)/2)ArcTan(S(1) - sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/(sqrt(c)sqrt(dcos(a + bx))))/(S(8)bsqrt(d)) + S(3)sqrt(S(2))c(S(5)/2)ArcTan(S(1) + sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/(sqrt(c)sqrt(dcos(a + bx))))/(S(8)bsqrt(d)) + S(3)sqrt(S(2))c(S(5)/2)log(sqrt(c)tan(a + bx) + sqrt(c) - sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/sqrt(dcos(a + bx)))/(S(16)bsqrt(d)) - S(3)sqrt(S(2))c*(S(5)/2)log(sqrt(c)tan(a + bx) + sqrt(c) + sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/sqrt(dcos(a + bx)))/(S(16)bsqrt(d)) - c(csin(a + bx))(S(3)/2)sqrt(dcos(a + bx))/(S(2)bd), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*(S(5)/2)/(dcos(a + bx))(S(5)/2), x), x, sqrt(S(2))c*(S(5)/2)ArcTan(S(1) - sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/(sqrt(c)sqrt(dcos(a + bx))))/(S(2)bd(S(5)/2)) - sqrt(S(2))c*(S(5)/2)ArcTan(S(1) + sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/(sqrt(c)sqrt(dcos(a + bx))))/(S(2)bd(S(5)/2)) - sqrt(S(2))c*(S(5)/2)log(sqrt(c)tan(a + bx) + sqrt(c) - sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/sqrt(dcos(a + bx)))/(S(4)bd*(S(5)/2)) + sqrt(S(2))c*(S(5)/2)log(sqrt(c)tan(a + bx) + sqrt(c) + sqrt(S(2))sqrt(d)sqrt(csin(a + bx))/sqrt(dcos(a + bx)))/(S(4)bd*(S(5)/2)) + S(2)c(csin(a + bx))(S(3)/2)/(S(3)bd(dcos(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(5)/2)/(dcos(a + bx))(S(9)/2), x), x, S(2)(csin(a + bx))*(S(7)/2)/(S(7)bcd(dcos(a + bx))(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(5)/2)/(dcos(a + bx))(S(13)/2), x), x, S(2)c(csin(a + bx))(S(3)/2)/(S(11)bd(dcos(a + bx))*(S(11)/2)) - S(6)c(csin(a + bx))(S(3)/2)/(S(77)bdS(3)(dcos(a + bx))*(S(7)/2)) - S(8)c(csin(a + bx))(S(3)/2)/(S(77)bdS(5)(dcos(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(5)/2)/(dcos(a + bx))(S(17)/2), x), x, S(2)c(csin(a + bx))(S(3)/2)/(S(15)bd(dcos(a + bx))*(S(15)/2)) - S(2)c(csin(a + bx))(S(3)/2)/(S(55)bdS(3)(dcos(a + bx))*(S(11)/2)) - S(16)c(csin(a + bx))(S(3)/2)/(S(385)bdS(5)(dcos(a + bx))*(S(7)/2)) - S(64)c(csin(a + bx))(S(3)/2)/(S(1155)bdS(7)(dcos(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*(S(7)/2)/cos(a + bx)*(S(7)/2), x), x, sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(cos(a + bx))/sqrt(sin(a + bx)))/(S(2)b) - sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(cos(a + bx))/sqrt(sin(a + bx)))/(S(2)b) - sqrt(S(2))log(cot(a + bx) + S(1) - sqrt(S(2))sqrt(cos(a + bx))/sqrt(sin(a + bx)))/(S(4)b) + sqrt(S(2))log(cot(a + bx) + S(1) + sqrt(S(2))sqrt(cos(a + bx))/sqrt(sin(a + bx)))/(S(4)b) + S(2)sin(a + bx)(S(5)/2)/(S(5)bcos(a + bx)*(S(5)/2)) - S(2)sqrt(sin(a + bx))/(bsqrt(cos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(x)(S(3)/2)/cos(x)(S(7)/2), x), x, S(2)sin(x)*(S(5)/2)/(S(5)cos(x)*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sin(x))/sqrt(cos(x)), x), x, -sqrt(S(2))ArcTan(-sqrt(S(2))sqrt(sin(x))/sqrt(cos(x)) + S(1))/S(2) + sqrt(S(2))ArcTan(sqrt(S(2))sqrt(sin(x))/sqrt(cos(x)) + S(1))/S(2) + sqrt(S(2))log(-sqrt(S(2))sqrt(sin(x))/sqrt(cos(x)) + tan(x) + S(1))/S(4) - sqrt(S(2))log(sqrt(S(2))sqrt(sin(x))/sqrt(cos(x)) + tan(x) + S(1))/S(4), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(x)(S(5)/2)/sqrt(cos(x)), x), x, -S(3)sqrt(S(2))ArcTan(-sqrt(S(2))sqrt(sin(x))/sqrt(cos(x)) + S(1))/S(8) + S(3)sqrt(S(2))ArcTan(sqrt(S(2))sqrt(sin(x))/sqrt(cos(x)) + S(1))/S(8) + S(3)sqrt(S(2))log(-sqrt(S(2))sqrt(sin(x))/sqrt(cos(x)) + tan(x) + S(1))/S(16) - S(3)sqrt(S(2))log(sqrt(S(2))sqrt(sin(x))/sqrt(cos(x)) + tan(x) + S(1))/S(16) - sin(x)(S(3)/2)sqrt(cos(x))/S(2), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*(S(7)/2)/sqrt(csin(a + bx)), x), x, S(5)d*S(4)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(12)bsqrt(csin(a + bx))sqrt(dcos(a + bx))) + S(5)dS(3)sqrt(csin(a + bx))sqrt(dcos(a + bx))/(S(6)bc) + dsqrt(csin(a + bx))(dcos(a + bx))(S(5)/2)/(S(3)bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))(S(3)/2)/sqrt(csin(a + bx)), x), x, dS(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(2)bsqrt(csin(a + bx))sqrt(dcos(a + bx))) + dsqrt(csin(a + bx))sqrt(dcos(a + bx))/(bc), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(csin(a + bx))sqrt(dcos(a + bx))), x), x, EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(bsqrt(csin(a + bx))sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(csin(a + bx))(dcos(a + bx))*(S(5)/2)), x), x, S(2)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(3)bdS(2)sqrt(csin(a + bx))sqrt(dcos(a + bx))) + S(2)sqrt(csin(a + bx))/(S(3)bcd(dcos(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(csin(a + bx))(dcos(a + bx))*(S(9)/2)), x), x, S(4)EllipticF(-Pi/S(4) + a + bx, S(2))sqrt(sin(S(2)a + S(2)bx))/(S(7)bdS(4)sqrt(csin(a + bx))sqrt(dcos(a + bx))) + S(2)sqrt(csin(a + bx))/(S(7)bcd(dcos(a + bx))*(S(7)/2)) + S(4)sqrt(csin(a + bx))/(S(7)bcdS(3)(dcos(a + bx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcos(a + bx))/sqrt(csin(a + bx)), x), x, sqrt(S(2))sqrt(d)ArcTan(-sqrt(S(2))sqrt(c)sqrt(dcos(a + bx))/(sqrt(d)sqrt(csin(a + bx))) + S(1))/(S(2)bsqrt(c)) - sqrt(S(2))sqrt(d)ArcTan(sqrt(S(2))sqrt(c)sqrt(dcos(a + bx))/(sqrt(d)sqrt(csin(a + bx))) + S(1))/(S(2)bsqrt(c)) - sqrt(S(2))sqrt(d)log(-sqrt(S(2))sqrt(c)sqrt(dcos(a + bx))/sqrt(csin(a + bx)) + sqrt(d)cot(a + bx) + sqrt(d))/(S(4)bsqrt(c)) + sqrt(S(2))sqrt(d)log(sqrt(S(2))sqrt(c)sqrt(dcos(a + bx))/sqrt(csin(a + bx)) + sqrt(d)cot(a + bx) + sqrt(d))/(S(4)bsqrt(c)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(csin(a + bx))(dcos(a + bx))*(S(3)/2)), x), x, S(2)sqrt(csin(a + bx))/(bcdsqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(csin(a + bx))(dcos(a + bx))*(S(7)/2)), x), x, S(2)sqrt(csin(a + bx))/(S(5)bcd(dcos(a + bx))*(S(5)/2)) + S(8)sqrt(csin(a + bx))/(S(5)bcdS(3)sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(csin(a + bx))(dcos(a + bx))(S(11)/2)), x), x, S(2)sqrt(csin(a + bx))/(S(9)bcd(dcos(a + bx))*(S(9)/2)) + S(16)sqrt(csin(a + bx))/(S(45)bcdS(3)(dcos(a + bx))*(S(5)/2)) + S(64)sqrt(csin(a + bx))/(S(45)bcdS(5)sqrt(dcos(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(cos(a + bx))/sqrt(sin(a + bx)), x), x, sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(cos(a + bx))/sqrt(sin(a + bx)))/(S(2)b) - sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(cos(a + bx))/sqrt(sin(a + bx)))/(S(2)b) - sqrt(S(2))log(cot(a + bx) + S(1) - sqrt(S(2))sqrt(cos(a + bx))/sqrt(sin(a + bx)))/(S(4)b) + sqrt(S(2))log(cot(a + bx) + S(1) + sqrt(S(2))sqrt(cos(a + bx))/sqrt(sin(a + bx)))/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)(S(3)/2)/sin(a + bx)*(S(3)/2), x), x, sqrt(S(2))ArcTan(-sqrt(S(2))sqrt(sin(a + bx))/sqrt(cos(a + bx)) + S(1))/(S(2)b) - sqrt(S(2))ArcTan(sqrt(S(2))sqrt(sin(a + bx))/sqrt(cos(a + bx)) + S(1))/(S(2)b) - sqrt(S(2))log(-sqrt(S(2))sqrt(sin(a + bx))/sqrt(cos(a + bx)) + tan(a + bx) + S(1))/(S(4)b) + sqrt(S(2))log(sqrt(S(2))sqrt(sin(a + bx))/sqrt(cos(a + bx)) + tan(a + bx) + S(1))/(S(4)b) - S(2)sqrt(cos(a + bx))/(bsqrt(sin(a + bx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*(S(5)/2)/sin(a + bx)*(S(5)/2), x), x, -sqrt(S(2))ArcTan(S(1) - sqrt(S(2))sqrt(cos(a + bx))/sqrt(sin(a + bx)))/(S(2)b) + sqrt(S(2))ArcTan(S(1) + sqrt(S(2))sqrt(cos(a + bx))/sqrt(sin(a + bx)))/(S(2)b) + sqrt(S(2))log(cot(a + bx) + S(1) - sqrt(S(2))sqrt(cos(a + bx))/sqrt(sin(a + bx)))/(S(4)b) - sqrt(S(2))log(cot(a + bx) + S(1) + sqrt(S(2))sqrt(cos(a + bx))/sqrt(sin(a + bx)))/(S(4)b) - S(2)cos(a + bx)(S(3)/2)/(S(3)bsin(a + bx)*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*(S(7)/2)/sin(a + bx)*(S(7)/2), x), x, -sqrt(S(2))ArcTan(-sqrt(S(2))sqrt(sin(a + bx))/sqrt(cos(a + bx)) + S(1))/(S(2)b) + sqrt(S(2))ArcTan(sqrt(S(2))sqrt(sin(a + bx))/sqrt(cos(a + bx)) + S(1))/(S(2)b) + sqrt(S(2))log(-sqrt(S(2))sqrt(sin(a + bx))/sqrt(cos(a + bx)) + tan(a + bx) + S(1))/(S(4)b) - sqrt(S(2))log(sqrt(S(2))sqrt(sin(a + bx))/sqrt(cos(a + bx)) + tan(a + bx) + S(1))/(S(4)b) + S(2)sqrt(cos(a + bx))/(bsqrt(sin(a + bx))) - S(2)cos(a + bx)(S(5)/2)/(S(5)bsin(a + bx)*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(1)/3)cos(e + fx)S(4), x), x, S(3)(bsin(e + fx))*(S(4)/3)Hypergeometric2F1(S(-3)/2, S(2)/3, S(5)/3, sin(e + fx)S(2))cos(e + fx)/(S(4)bfsqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(1)/3)cos(e + fx)S(2), x), x, S(3)(bsin(e + fx))*(S(4)/3)Hypergeometric2F1(S(-1)/2, S(2)/3, S(5)/3, sin(e + fx)S(2))cos(e + fx)/(S(4)bfsqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(1)/3), x), x, S(3)(bsin(e + fx))*(S(4)/3)Hypergeometric2F1(S(1)/2, S(2)/3, S(5)/3, sin(e + fx)S(2))cos(e + fx)/(S(4)bfsqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(1)/3)sec(e + fx)S(2), x), x, S(3)(bsin(e + fx))*(S(4)/3)sqrt(cos(e + fx)S(2))Hypergeometric2F1(S(2)/3, S(3)/2, S(5)/3, sin(e + fx)S(2))sec(e + fx)/(S(4)bf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(1)/3)sec(e + fx)S(4), x), x, S(3)(bsin(e + fx))*(S(4)/3)sqrt(cos(e + fx)S(2))Hypergeometric2F1(S(2)/3, S(5)/2, S(5)/3, sin(e + fx)S(2))sec(e + fx)/(S(4)bf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(5)/3)cos(e + fx)S(4), x), x, S(3)(bsin(e + fx))*(S(8)/3)Hypergeometric2F1(S(-3)/2, S(4)/3, S(7)/3, sin(e + fx)S(2))cos(e + fx)/(S(8)bfsqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(5)/3)cos(e + fx)S(2), x), x, S(3)(bsin(e + fx))*(S(8)/3)Hypergeometric2F1(S(-1)/2, S(4)/3, S(7)/3, sin(e + fx)S(2))cos(e + fx)/(S(8)bfsqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(5)/3), x), x, S(3)(bsin(e + fx))*(S(8)/3)Hypergeometric2F1(S(1)/2, S(4)/3, S(7)/3, sin(e + fx)S(2))cos(e + fx)/(S(8)bfsqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(5)/3)sec(e + fx)S(2), x), x, S(3)(bsin(e + fx))*(S(8)/3)sqrt(cos(e + fx)S(2))Hypergeometric2F1(S(4)/3, S(3)/2, S(7)/3, sin(e + fx)S(2))sec(e + fx)/(S(8)bf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(5)/3)sec(e + fx)S(4), x), x, S(3)(bsin(e + fx))*(S(8)/3)sqrt(cos(e + fx)S(2))Hypergeometric2F1(S(4)/3, S(5)/2, S(7)/3, sin(e + fx)S(2))sec(e + fx)/(S(8)bf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(e + fx)*S(4)/(bsin(e + fx))(S(1)/3), x), x, S(3)(bsin(e + fx))*(S(2)/3)Hypergeometric2F1(S(-3)/2, S(1)/3, S(4)/3, sin(e + fx)S(2))cos(e + fx)/(S(2)bfsqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(e + fx)*S(2)/(bsin(e + fx))(S(1)/3), x), x, S(3)(bsin(e + fx))*(S(2)/3)Hypergeometric2F1(S(-1)/2, S(1)/3, S(4)/3, sin(e + fx)S(2))cos(e + fx)/(S(2)bfsqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(-1)/3), x), x, S(3)(bsin(e + fx))*(S(2)/3)Hypergeometric2F1(S(1)/3, S(1)/2, S(4)/3, sin(e + fx)S(2))cos(e + fx)/(S(2)bfsqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(e + fx)*S(2)/(bsin(e + fx))(S(1)/3), x), x, S(3)(bsin(e + fx))*(S(2)/3)sqrt(cos(e + fx)S(2))Hypergeometric2F1(S(1)/3, S(3)/2, S(4)/3, sin(e + fx)S(2))sec(e + fx)/(S(2)bf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(e + fx)*S(4)/(bsin(e + fx))(S(1)/3), x), x, S(3)(bsin(e + fx))*(S(2)/3)sqrt(cos(e + fx)S(2))Hypergeometric2F1(S(1)/3, S(5)/2, S(4)/3, sin(e + fx)S(2))sec(e + fx)/(S(2)bf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(e + fx)*S(4)/(bsin(e + fx))(S(5)/3), x), x, -S(3)Hypergeometric2F1(S(-3)/2, S(-1)/3, S(2)/3, sin(e + fx)S(2))cos(e + fx)/(S(2)bf(bsin(e + fx))*(S(2)/3)sqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(e + fx)*S(2)/(bsin(e + fx))(S(5)/3), x), x, -S(3)Hypergeometric2F1(S(-1)/2, S(-1)/3, S(2)/3, sin(e + fx)S(2))cos(e + fx)/(S(2)bf(bsin(e + fx))*(S(2)/3)sqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))(S(-5)/3), x), x, -S(3)Hypergeometric2F1(S(-1)/3, S(1)/2, S(2)/3, sin(e + fx)S(2))cos(e + fx)/(S(2)bf(bsin(e + fx))*(S(2)/3)sqrt(cos(e + fx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(e + fx)*S(2)/(bsin(e + fx))(S(5)/3), x), x, -S(3)sqrt(cos(e + fx)S(2))Hypergeometric2F1(S(-1)/3, S(3)/2, S(2)/3, sin(e + fx)S(2))sec(e + fx)/(S(2)bf(bsin(e + fx))*(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sec(e + fx)*S(4)/(bsin(e + fx))(S(5)/3), x), x, -S(3)sqrt(cos(e + fx)S(2))Hypergeometric2F1(S(-1)/3, S(5)/2, S(2)/3, sin(e + fx)S(2))sec(e + fx)/(S(2)bf(bsin(e + fx))*(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*(S(1)/3)/cos(a + bx)*(S(1)/3), x), x, -sqrt(S(3))ArcTan(sqrt(S(3))(-S(2)sin(a + bx)(S(2)/3)/cos(a + bx)*(S(2)/3) + S(1))/S(3))/(S(2)b) - log(sin(a + bx)(S(2)/3)/cos(a + bx)*(S(2)/3) + S(1))/(S(2)b) + log(sin(a + bx)(S(4)/3)/cos(a + bx)*(S(4)/3) - sin(a + bx)*(S(2)/3)/cos(a + bx)*(S(2)/3) + S(1))/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)(S(2)/3)/cos(a + bx)*(S(2)/3), x), x, ArcTan(sin(a + bx)*(S(1)/3)/cos(a + bx)*(S(1)/3))/b - ArcTan(-S(2)sin(a + bx)(S(1)/3)/cos(a + bx)*(S(1)/3) + sqrt(S(3)))/(S(2)b) + ArcTan(S(2)sin(a + bx)*(S(1)/3)/cos(a + bx)*(S(1)/3) + sqrt(S(3)))/(S(2)b) + sqrt(S(3))log(sin(a + bx)*(S(2)/3)/cos(a + bx)*(S(2)/3) - sqrt(S(3))sin(a + bx)(S(1)/3)/cos(a + bx)*(S(1)/3) + S(1))/(S(4)b) - sqrt(S(3))log(sin(a + bx)*(S(2)/3)/cos(a + bx)*(S(2)/3) + sqrt(S(3))sin(a + bx)(S(1)/3)/cos(a + bx)*(S(1)/3) + S(1))/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)(S(4)/3)/cos(a + bx)*(S(4)/3), x), x, ArcTan(cos(a + bx)*(S(1)/3)/sin(a + bx)*(S(1)/3))/b - ArcTan(sqrt(S(3)) - S(2)cos(a + bx)(S(1)/3)/sin(a + bx)*(S(1)/3))/(S(2)b) + ArcTan(sqrt(S(3)) + S(2)cos(a + bx)*(S(1)/3)/sin(a + bx)*(S(1)/3))/(S(2)b) + sqrt(S(3))log(S(1) - sqrt(S(3))cos(a + bx)(S(1)/3)/sin(a + bx)*(S(1)/3) + cos(a + bx)*(S(2)/3)/sin(a + bx)*(S(2)/3))/(S(4)b) - sqrt(S(3))log(S(1) + sqrt(S(3))cos(a + bx)(S(1)/3)/sin(a + bx)*(S(1)/3) + cos(a + bx)*(S(2)/3)/sin(a + bx)*(S(2)/3))/(S(4)b) + S(3)sin(a + bx)*(S(1)/3)/(bcos(a + bx)(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*(S(5)/3)/cos(a + bx)*(S(5)/3), x), x, -sqrt(S(3))ArcTan(sqrt(S(3))(S(1) - S(2)cos(a + bx)(S(2)/3)/sin(a + bx)*(S(2)/3))/S(3))/(S(2)b) - log(S(1) + cos(a + bx)(S(2)/3)/sin(a + bx)*(S(2)/3))/(S(2)b) + log(S(1) - cos(a + bx)(S(2)/3)/sin(a + bx)*(S(2)/3) + cos(a + bx)*(S(4)/3)/sin(a + bx)*(S(4)/3))/(S(4)b) + S(3)sin(a + bx)*(S(2)/3)/(S(2)bcos(a + bx)*(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(a + bx)*(S(7)/3)/cos(a + bx)*(S(7)/3), x), x, sqrt(S(3))ArcTan(sqrt(S(3))(-S(2)sin(a + bx)(S(2)/3)/cos(a + bx)*(S(2)/3) + S(1))/S(3))/(S(2)b) + log(sin(a + bx)(S(2)/3)/cos(a + bx)*(S(2)/3) + S(1))/(S(2)b) - log(sin(a + bx)(S(4)/3)/cos(a + bx)*(S(4)/3) - sin(a + bx)*(S(2)/3)/cos(a + bx)*(S(2)/3) + S(1))/(S(4)b) + S(3)sin(a + bx)*(S(4)/3)/(S(4)bcos(a + bx)*(S(4)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*(S(1)/3)/sin(a + bx)*(S(1)/3), x), x, sqrt(S(3))ArcTan(sqrt(S(3))(S(1) - S(2)cos(a + bx)(S(2)/3)/sin(a + bx)*(S(2)/3))/S(3))/(S(2)b) + log(S(1) + cos(a + bx)(S(2)/3)/sin(a + bx)*(S(2)/3))/(S(2)b) - log(S(1) - cos(a + bx)(S(2)/3)/sin(a + bx)*(S(2)/3) + cos(a + bx)*(S(4)/3)/sin(a + bx)*(S(4)/3))/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)(S(2)/3)/sin(a + bx)*(S(2)/3), x), x, -ArcTan(cos(a + bx)*(S(1)/3)/sin(a + bx)*(S(1)/3))/b + ArcTan(sqrt(S(3)) - S(2)cos(a + bx)(S(1)/3)/sin(a + bx)*(S(1)/3))/(S(2)b) - ArcTan(sqrt(S(3)) + S(2)cos(a + bx)*(S(1)/3)/sin(a + bx)*(S(1)/3))/(S(2)b) - sqrt(S(3))log(S(1) - sqrt(S(3))cos(a + bx)(S(1)/3)/sin(a + bx)*(S(1)/3) + cos(a + bx)*(S(2)/3)/sin(a + bx)*(S(2)/3))/(S(4)b) + sqrt(S(3))log(S(1) + sqrt(S(3))cos(a + bx)(S(1)/3)/sin(a + bx)*(S(1)/3) + cos(a + bx)*(S(2)/3)/sin(a + bx)*(S(2)/3))/(S(4)b), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)(S(4)/3)/sin(a + bx)*(S(4)/3), x), x, -ArcTan(sin(a + bx)*(S(1)/3)/cos(a + bx)*(S(1)/3))/b + ArcTan(-S(2)sin(a + bx)(S(1)/3)/cos(a + bx)*(S(1)/3) + sqrt(S(3)))/(S(2)b) - ArcTan(S(2)sin(a + bx)*(S(1)/3)/cos(a + bx)*(S(1)/3) + sqrt(S(3)))/(S(2)b) - sqrt(S(3))log(sin(a + bx)*(S(2)/3)/cos(a + bx)*(S(2)/3) - sqrt(S(3))sin(a + bx)(S(1)/3)/cos(a + bx)*(S(1)/3) + S(1))/(S(4)b) + sqrt(S(3))log(sin(a + bx)*(S(2)/3)/cos(a + bx)*(S(2)/3) + sqrt(S(3))sin(a + bx)(S(1)/3)/cos(a + bx)*(S(1)/3) + S(1))/(S(4)b) - S(3)cos(a + bx)*(S(1)/3)/(bsin(a + bx)(S(1)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*(S(5)/3)/sin(a + bx)*(S(5)/3), x), x, sqrt(S(3))ArcTan(sqrt(S(3))(-S(2)sin(a + bx)(S(2)/3)/cos(a + bx)*(S(2)/3) + S(1))/S(3))/(S(2)b) + log(sin(a + bx)(S(2)/3)/cos(a + bx)*(S(2)/3) + S(1))/(S(2)b) - log(sin(a + bx)(S(4)/3)/cos(a + bx)*(S(4)/3) - sin(a + bx)*(S(2)/3)/cos(a + bx)*(S(2)/3) + S(1))/(S(4)b) - S(3)cos(a + bx)*(S(2)/3)/(S(2)bsin(a + bx)*(S(2)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(a + bx)*(S(7)/3)/sin(a + bx)*(S(7)/3), x), x, -sqrt(S(3))ArcTan(sqrt(S(3))(S(1) - S(2)cos(a + bx)(S(2)/3)/sin(a + bx)*(S(2)/3))/S(3))/(S(2)b) - log(S(1) + cos(a + bx)(S(2)/3)/sin(a + bx)*(S(2)/3))/(S(2)b) + log(S(1) - cos(a + bx)(S(2)/3)/sin(a + bx)*(S(2)/3) + cos(a + bx)*(S(4)/3)/sin(a + bx)*(S(4)/3))/(S(4)b) - S(3)cos(a + bx)*(S(4)/3)/(S(4)bsin(a + bx)(S(4)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(cos(x)(S(2)/3)/sin(x)*(S(8)/3), x), x, -S(3)cos(x)*(S(5)/3)/(S(5)sin(x)(S(5)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(x)(S(2)/3)/cos(x)*(S(8)/3), x), x, S(3)sin(x)*(S(5)/3)/(S(5)cos(x)*(S(5)/3)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*mcos(e + fx)n, x), x, (cos(e + fx)S(2))(-n/S(2) + S(1)/2)Hypergeometric2F1(m/S(2) + S(1)/2, -n/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(e + fx)*S(2))sin(e + fx)(m + S(1))cos(e + fx)(n + S(-1))/(f(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(e + fx))*nsin(e + fx)m, x), x, -(dcos(e + fx))(n + S(1))(sin(e + fx)S(2))(-m/S(2) + S(1)/2)Hypergeometric2F1(-m/S(2) + S(1)/2, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(e + fx)S(2))sin(e + fx)(m + S(-1))/(df(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))mcos(e + fx)n, x), x, (bsin(e + fx))(m + S(1))(cos(e + fx)S(2))(-n/S(2) + S(1)/2)Hypergeometric2F1(m/S(2) + S(1)/2, -n/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(e + fx)S(2))cos(e + fx)(n + S(-1))/(bf(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsin(e + fx))m(dcos(e + fx))*n, x), x, d(bsin(e + fx))*(m + S(1))(dcos(e + fx))*(n + S(-1))(cos(e + fx)S(2))(-n/S(2) + S(1)/2)Hypergeometric2F1(m/S(2) + S(1)/2, -n/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(e + fx)S(2))/(bf(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))mcos(a + bx)S(5), x), x, (csin(a + bx))(m + S(1))/(bc(m + S(1))) - S(2)(csin(a + bx))*(m + S(3))/(bc*S(3)(m + S(3))) + (csin(a + bx))*(m + S(5))/(bc*S(5)(m + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*mcos(a + bx)S(3), x), x, (csin(a + bx))(m + S(1))/(bc(m + S(1))) - (csin(a + bx))(m + S(3))/(bc*S(3)(m + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*mcos(a + bx), x), x, (csin(a + bx))(m + S(1))/(bc(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))msec(a + bx), x), x, (csin(a + bx))(m + S(1))Hypergeometric2F1(S(1), m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)S(2))/(bc(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))msec(a + bx)S(3), x), x, (csin(a + bx))(m + S(1))Hypergeometric2F1(S(2), m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)S(2))/(bc(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))mcos(a + bx)S(4), x), x, (csin(a + bx))(m + S(1))Hypergeometric2F1(S(-3)/2, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)S(2))cos(a + bx)/(bc(m + S(1))sqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))mcos(a + bx)S(2), x), x, (csin(a + bx))(m + S(1))Hypergeometric2F1(S(-1)/2, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)S(2))cos(a + bx)/(bc(m + S(1))sqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))m, x), x, (csin(a + bx))(m + S(1))Hypergeometric2F1(S(1)/2, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)S(2))cos(a + bx)/(bc(m + S(1))sqrt(cos(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))msec(a + bx)S(2), x), x, (csin(a + bx))(m + S(1))sqrt(cos(a + bx)S(2))Hypergeometric2F1(S(3)/2, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)S(2))sec(a + bx)/(bc(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))msec(a + bx)S(4), x), x, (csin(a + bx))(m + S(1))sqrt(cos(a + bx)S(2))Hypergeometric2F1(S(5)/2, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)S(2))sec(a + bx)/(bc(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))m(dcos(a + bx))*(S(3)/2), x), x, d(csin(a + bx))*(m + S(1))sqrt(dcos(a + bx))Hypergeometric2F1(S(-1)/4, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)*S(2))/(bc(m + S(1))(cos(a + bx)S(2))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))msqrt(dcos(a + bx)), x), x, d(csin(a + bx))(m + S(1))(cos(a + bx)S(2))(S(1)/4)Hypergeometric2F1(S(1)/4, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)S(2))/(bcsqrt(dcos(a + bx))(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*m/sqrt(dcos(a + bx)), x), x, d(csin(a + bx))*(m + S(1))(cos(a + bx)S(2))(S(3)/4)Hypergeometric2F1(S(3)/4, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)S(2))/(bc(dcos(a + bx))(S(3)/2)(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*m/(dcos(a + bx))(S(3)/2), x), x, (csin(a + bx))(m + S(1))(cos(a + bx)S(2))(S(1)/4)Hypergeometric2F1(S(5)/4, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)S(2))/(bcdsqrt(dcos(a + bx))(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))m/(dcos(a + bx))(S(5)/2), x), x, (csin(a + bx))(m + S(1))(cos(a + bx)S(2))(S(3)/4)Hypergeometric2F1(S(7)/4, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)S(2))/(bcd(dcos(a + bx))*(S(3)/2)(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*nsin(a + bx)S(5), x), x, -(dcos(a + bx))(n + S(1))/(bd(n + S(1))) + S(2)(dcos(a + bx))*(n + S(3))/(bd*S(3)(n + S(3))) - (dcos(a + bx))*(n + S(5))/(bd*S(5)(n + S(5))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*nsin(a + bx)S(3), x), x, -(dcos(a + bx))(n + S(1))/(bd(n + S(1))) + (dcos(a + bx))(n + S(3))/(bd*S(3)(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*nsin(a + bx), x), x, -(dcos(a + bx))(n + S(1))/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))ncsc(a + bx), x), x, -(dcos(a + bx))(n + S(1))Hypergeometric2F1(S(1), n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))ncsc(a + bx)S(3), x), x, -(dcos(a + bx))(n + S(1))Hypergeometric2F1(S(2), n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))ncsc(a + bx)S(5), x), x, -(dcos(a + bx))(n + S(1))Hypergeometric2F1(S(3), n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))nsin(a + bx)S(4), x), x, -(dcos(a + bx))(n + S(1))Hypergeometric2F1(S(-3)/2, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))sin(a + bx)/(bd(n + S(1))sqrt(sin(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))nsin(a + bx)S(2), x), x, -(dcos(a + bx))(n + S(1))Hypergeometric2F1(S(-1)/2, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))sin(a + bx)/(bd(n + S(1))sqrt(sin(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))n, x), x, -(dcos(a + bx))(n + S(1))Hypergeometric2F1(S(1)/2, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))sin(a + bx)/(bd(n + S(1))sqrt(sin(a + bx)S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))ncsc(a + bx)S(2), x), x, -(dcos(a + bx))(n + S(1))sqrt(sin(a + bx)S(2))Hypergeometric2F1(S(3)/2, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))csc(a + bx)/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))ncsc(a + bx)S(4), x), x, -(dcos(a + bx))(n + S(1))sqrt(sin(a + bx)S(2))Hypergeometric2F1(S(5)/2, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))csc(a + bx)/(bd(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(5)/2)(dcos(a + bx))*n, x), x, -c(csin(a + bx))*(S(3)/2)(dcos(a + bx))*(n + S(1))Hypergeometric2F1(S(-3)/4, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))/(bd(n + S(1))(sin(a + bx)S(2))(S(3)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))(S(3)/2)(dcos(a + bx))*n, x), x, -csqrt(csin(a + bx))(dcos(a + bx))(n + S(1))Hypergeometric2F1(S(-1)/4, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))/(bd(n + S(1))(sin(a + bx)S(2))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(csin(a + bx))(dcos(a + bx))*n, x), x, -c(dcos(a + bx))*(n + S(1))(sin(a + bx)S(2))(S(1)/4)Hypergeometric2F1(S(1)/4, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))/(bdsqrt(csin(a + bx))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*n/sqrt(csin(a + bx)), x), x, -c(dcos(a + bx))*(n + S(1))(sin(a + bx)S(2))(S(3)/4)Hypergeometric2F1(S(3)/4, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))/(bd(csin(a + bx))(S(3)/2)(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcos(a + bx))*n/(csin(a + bx))(S(3)/2), x), x, -(dcos(a + bx))(n + S(1))(sin(a + bx)S(2))(S(1)/4)Hypergeometric2F1(S(5)/4, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(a + bx)S(2))/(bcdsqrt(csin(a + bx))(n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))sin(e + fx)S(7), x), x, S(2)b*S(7)/(S(13)f(bsec(e + fx))(S(13)/2)) - S(2)b*S(5)/(S(3)f(bsec(e + fx))(S(9)/2)) + S(6)b*S(3)/(S(5)f(bsec(e + fx))(S(5)/2)) - S(2)b/(fsqrt(bsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))sin(e + fx)S(5), x), x, -S(2)b*S(5)/(S(9)f(bsec(e + fx))(S(9)/2)) + S(4)b*S(3)/(S(5)f(bsec(e + fx))(S(5)/2)) - S(2)b/(fsqrt(bsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))sin(e + fx)S(3), x), x, S(2)b*S(3)/(S(5)f(bsec(e + fx))(S(5)/2)) - S(2)b/(fsqrt(bsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))sin(e + fx), x), x, -S(2)b/(fsqrt(bsec(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))csc(e + fx), x), x, sqrt(b)ArcTan(sqrt(bsec(e + fx))/sqrt(b))/f - sqrt(b)atanh(sqrt(bsec(e + fx))/sqrt(b))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))csc(e + fx)S(3), x), x, S(3)sqrt(b)ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(S(4)f) - S(3)sqrt(b)atanh(sqrt(bsec(e + fx))/sqrt(b))/(S(4)f) - (bsec(e + fx))(S(3)/2)cot(e + fx)S(2)/(S(2)bf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))csc(e + fx)S(5), x), x, S(21)sqrt(b)ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(S(32)f) - S(21)sqrt(b)atanh(sqrt(bsec(e + fx))/sqrt(b))/(S(32)f) - S(7)(bsec(e + fx))*(S(3)/2)cot(e + fx)S(2)/(S(16)bf) - (bsec(e + fx))(S(7)/2)cot(e + fx)S(4)/(S(4)b*S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))sin(e + fx)*S(6), x), x, -S(2)bsin(e + fx)*S(5)/(S(11)fsqrt(bsec(e + fx))) - S(20)bsin(e + fx)*S(3)/(S(77)fsqrt(bsec(e + fx))) - S(40)bsin(e + fx)/(S(77)fsqrt(bsec(e + fx))) + S(80)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(77)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))sin(e + fx)*S(4), x), x, -S(2)bsin(e + fx)*S(3)/(S(7)fsqrt(bsec(e + fx))) - S(4)bsin(e + fx)/(S(7)fsqrt(bsec(e + fx))) + S(8)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(7)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))sin(e + fx)*S(2), x), x, -S(2)bsin(e + fx)/(S(3)fsqrt(bsec(e + fx))) + S(4)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx)), x), x, S(2)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))csc(e + fx)S(2), x), x, -bcsc(e + fx)/(fsqrt(bsec(e + fx))) + sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))csc(e + fx)*S(4), x), x, -bcsc(e + fx)S(3)/(S(3)fsqrt(bsec(e + fx))) - S(5)bcsc(e + fx)/(S(6)fsqrt(bsec(e + fx))) + S(5)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(6)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(bsec(e + fx))csc(e + fx)*S(6), x), x, -bcsc(e + fx)S(5)/(S(5)fsqrt(bsec(e + fx))) - S(3)bcsc(e + fx)*S(3)/(S(10)fsqrt(bsec(e + fx))) - S(3)bcsc(e + fx)/(S(4)fsqrt(bsec(e + fx))) + S(3)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(4)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(3)/2)sin(e + fx)S(7), x), x, S(2)b*S(7)/(S(11)f(bsec(e + fx))(S(11)/2)) - S(6)b*S(5)/(S(7)f(bsec(e + fx))(S(7)/2)) + S(2)b*S(3)/(f(bsec(e + fx))*(S(3)/2)) + S(2)bsqrt(bsec(e + fx))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(3)/2)sin(e + fx)S(5), x), x, -S(2)b*S(5)/(S(7)f(bsec(e + fx))(S(7)/2)) + S(4)b*S(3)/(S(3)f(bsec(e + fx))(S(3)/2)) + S(2)bsqrt(bsec(e + fx))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(3)/2)sin(e + fx)S(3), x), x, S(2)b*S(3)/(S(3)f(bsec(e + fx))(S(3)/2)) + S(2)bsqrt(bsec(e + fx))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(3)/2)sin(e + fx), x), x, S(2)bsqrt(bsec(e + fx))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(3)/2)csc(e + fx), x), x, -b(S(3)/2)ArcTan(sqrt(bsec(e + fx))/sqrt(b))/f - b*(S(3)/2)atanh(sqrt(bsec(e + fx))/sqrt(b))/f + S(2)bsqrt(bsec(e + fx))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(3)/2)csc(e + fx)S(3), x), x, -S(5)b*(S(3)/2)ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(S(4)f) - S(5)b*(S(3)/2)atanh(sqrt(bsec(e + fx))/sqrt(b))/(S(4)f) + S(5)bsqrt(bsec(e + fx))/(S(2)f) - (bsec(e + fx))*(S(5)/2)cot(e + fx)S(2)/(S(2)bf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(3)/2)sin(e + fx)S(6), x), x, S(20)b*S(3)sin(e + fx)S(3)/(S(9)f(bsec(e + fx))(S(3)/2)) + S(8)b*S(3)sin(e + fx)/(S(3)f(bsec(e + fx))(S(3)/2)) - S(16)b*S(2)EllipticE(e/S(2) + fx/S(2), S(2))/(S(3)fsqrt(bsec(e + fx))sqrt(cos(e + fx))) + S(2)bsqrt(bsec(e + fx))sin(e + fx)S(5)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(3)/2)sin(e + fx)S(4), x), x, S(12)b*S(3)sin(e + fx)/(S(5)f(bsec(e + fx))(S(3)/2)) - S(24)b*S(2)EllipticE(e/S(2) + fx/S(2), S(2))/(S(5)fsqrt(bsec(e + fx))sqrt(cos(e + fx))) + S(2)bsqrt(bsec(e + fx))sin(e + fx)S(3)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(3)/2)sin(e + fx)S(2), x), x, -S(4)b*S(2)EllipticE(e/S(2) + fx/S(2), S(2))/(fsqrt(bsec(e + fx))sqrt(cos(e + fx))) + S(2)bsqrt(bsec(e + fx))sin(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(3)/2), x), x, -S(2)b*S(2)EllipticE(e/S(2) + fx/S(2), S(2))/(fsqrt(bsec(e + fx))sqrt(cos(e + fx))) + S(2)bsqrt(bsec(e + fx))sin(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(3)/2)csc(e + fx)S(2), x), x, -S(3)b*S(2)EllipticE(e/S(2) + fx/S(2), S(2))/(fsqrt(bsec(e + fx))sqrt(cos(e + fx))) + S(3)bsqrt(bsec(e + fx))sin(e + fx)/f - bsqrt(bsec(e + fx))csc(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(3)/2)csc(e + fx)S(4), x), x, -S(7)b*S(2)EllipticE(e/S(2) + fx/S(2), S(2))/(S(2)fsqrt(bsec(e + fx))sqrt(cos(e + fx))) + S(7)bsqrt(bsec(e + fx))sin(e + fx)/(S(2)f) - bsqrt(bsec(e + fx))csc(e + fx)S(3)/(S(3)f) - S(7)bsqrt(bsec(e + fx))csc(e + fx)/(S(6)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(5)/2)sin(e + fx)S(7), x), x, S(2)b*S(7)/(S(9)f(bsec(e + fx))(S(9)/2)) - S(6)b*S(5)/(S(5)f(bsec(e + fx))(S(5)/2)) + S(6)b*S(3)/(fsqrt(bsec(e + fx))) + S(2)b(bsec(e + fx))*(S(3)/2)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(5)/2)sin(e + fx)S(5), x), x, -S(2)b*S(5)/(S(5)f(bsec(e + fx))(S(5)/2)) + S(4)b*S(3)/(fsqrt(bsec(e + fx))) + S(2)b(bsec(e + fx))*(S(3)/2)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(5)/2)sin(e + fx)S(3), x), x, S(2)b*S(3)/(fsqrt(bsec(e + fx))) + S(2)b(bsec(e + fx))*(S(3)/2)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(5)/2)sin(e + fx), x), x, S(2)b(bsec(e + fx))(S(3)/2)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(5)/2)csc(e + fx), x), x, b(S(5)/2)ArcTan(sqrt(bsec(e + fx))/sqrt(b))/f - b*(S(5)/2)atanh(sqrt(bsec(e + fx))/sqrt(b))/f + S(2)b(bsec(e + fx))*(S(3)/2)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(5)/2)csc(e + fx)S(3), x), x, S(7)b*(S(5)/2)ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(S(4)f) - S(7)b*(S(5)/2)atanh(sqrt(bsec(e + fx))/sqrt(b))/(S(4)f) + S(7)b(bsec(e + fx))(S(3)/2)/(S(6)f) - (bsec(e + fx))*(S(7)/2)cot(e + fx)S(2)/(S(2)bf), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(5)/2)csc(e + fx)S(5), x), x, S(77)b*(S(5)/2)ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(S(32)f) - S(77)b*(S(5)/2)atanh(sqrt(bsec(e + fx))/sqrt(b))/(S(32)f) + S(77)b(bsec(e + fx))(S(3)/2)/(S(48)f) - S(11)(bsec(e + fx))(S(7)/2)cot(e + fx)S(2)/(S(16)bf) - (bsec(e + fx))(S(11)/2)cot(e + fx)S(4)/(S(4)b*S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(5)/2)sin(e + fx)S(6), x), x, S(20)b*S(3)sin(e + fx)S(3)/(S(21)fsqrt(bsec(e + fx))) + S(40)b*S(3)sin(e + fx)/(S(21)fsqrt(bsec(e + fx))) - S(80)b*S(2)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(21)f) + S(2)b(bsec(e + fx))(S(3)/2)sin(e + fx)S(5)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(5)/2)sin(e + fx)S(4), x), x, S(4)b*S(3)sin(e + fx)/(S(3)fsqrt(bsec(e + fx))) - S(8)b*S(2)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(3)f) + S(2)b(bsec(e + fx))(S(3)/2)sin(e + fx)S(3)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(5)/2)sin(e + fx)S(2), x), x, -S(4)b*S(2)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(3)f) + S(2)b(bsec(e + fx))(S(3)/2)sin(e + fx)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(5)/2), x), x, S(2)b*S(2)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(3)f) + S(2)b(bsec(e + fx))(S(3)/2)sin(e + fx)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(5)/2)csc(e + fx)S(2), x), x, S(5)b*S(2)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(3)f) + S(5)b(bsec(e + fx))(S(3)/2)sin(e + fx)/(S(3)f) - b(bsec(e + fx))(S(3)/2)csc(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(5)/2)csc(e + fx)S(4), x), x, S(5)b*S(2)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(2)f) + S(5)b(bsec(e + fx))(S(3)/2)sin(e + fx)/(S(2)f) - b(bsec(e + fx))(S(3)/2)csc(e + fx)S(3)/(S(3)f) - S(3)b(bsec(e + fx))*(S(3)/2)csc(e + fx)/(S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)S(7)/sqrt(bsec(e + fx)), x), x, S(2)b*S(7)/(S(15)f(bsec(e + fx))(S(15)/2)) - S(6)b*S(5)/(S(11)f(bsec(e + fx))(S(11)/2)) + S(6)b*S(3)/(S(7)f(bsec(e + fx))(S(7)/2)) - S(2)b/(S(3)f(bsec(e + fx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*S(5)/sqrt(bsec(e + fx)), x), x, -S(2)b*S(5)/(S(11)f(bsec(e + fx))(S(11)/2)) + S(4)b*S(3)/(S(7)f(bsec(e + fx))(S(7)/2)) - S(2)b/(S(3)f(bsec(e + fx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*S(3)/sqrt(bsec(e + fx)), x), x, S(2)b*S(3)/(S(7)f(bsec(e + fx))(S(7)/2)) - S(2)b/(S(3)f(bsec(e + fx))*(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)/sqrt(bsec(e + fx)), x), x, -S(2)b/(S(3)f(bsec(e + fx))(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)/sqrt(bsec(e + fx)), x), x, -ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(sqrt(b)f) - atanh(sqrt(bsec(e + fx))/sqrt(b))/(sqrt(b)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(3)/sqrt(bsec(e + fx)), x), x, -sqrt(bsec(e + fx))cot(e + fx)S(2)/(S(2)bf) - ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(S(4)sqrt(b)f) - atanh(sqrt(bsec(e + fx))/sqrt(b))/(S(4)sqrt(b)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)*S(5)/sqrt(bsec(e + fx)), x), x, -S(5)sqrt(bsec(e + fx))cot(e + fx)*S(2)/(S(16)bf) - (bsec(e + fx))(S(5)/2)cot(e + fx)S(4)/(S(4)b*S(3)f) - S(5)ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(S(32)sqrt(b)f) - S(5)atanh(sqrt(bsec(e + fx))/sqrt(b))/(S(32)sqrt(b)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)S(6)/sqrt(bsec(e + fx)), x), x, -S(2)bsin(e + fx)*S(5)/(S(13)f(bsec(e + fx))(S(3)/2)) - S(20)bsin(e + fx)*S(3)/(S(117)f(bsec(e + fx))(S(3)/2)) - S(8)bsin(e + fx)/(S(39)f(bsec(e + fx))*(S(3)/2)) + S(16)EllipticE(e/S(2) + fx/S(2), S(2))/(S(39)fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*S(4)/sqrt(bsec(e + fx)), x), x, -S(2)bsin(e + fx)*S(3)/(S(9)f(bsec(e + fx))(S(3)/2)) - S(4)bsin(e + fx)/(S(15)f(bsec(e + fx))*(S(3)/2)) + S(8)EllipticE(e/S(2) + fx/S(2), S(2))/(S(15)fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*S(2)/sqrt(bsec(e + fx)), x), x, -S(2)bsin(e + fx)/(S(5)f(bsec(e + fx))*(S(3)/2)) + S(4)EllipticE(e/S(2) + fx/S(2), S(2))/(S(5)fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(bsec(e + fx)), x), x, S(2)EllipticE(e/S(2) + fx/S(2), S(2))/(fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(2)/sqrt(bsec(e + fx)), x), x, -bcsc(e + fx)/(f(bsec(e + fx))*(S(3)/2)) - EllipticE(e/S(2) + fx/S(2), S(2))/(fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)*S(4)/sqrt(bsec(e + fx)), x), x, -bcsc(e + fx)S(3)/(S(3)f(bsec(e + fx))(S(3)/2)) - bcsc(e + fx)/(S(2)f(bsec(e + fx))(S(3)/2)) - EllipticE(e/S(2) + fx/S(2), S(2))/(S(2)fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(6)/sqrt(bsec(e + fx)), x), x, -bcsc(e + fx)S(5)/(S(5)f(bsec(e + fx))(S(3)/2)) - S(7)bcsc(e + fx)*S(3)/(S(30)f(bsec(e + fx))(S(3)/2)) - S(7)bcsc(e + fx)/(S(20)f(bsec(e + fx))*(S(3)/2)) - S(7)EllipticE(e/S(2) + fx/S(2), S(2))/(S(20)fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*S(7)/(bsec(e + fx))(S(3)/2), x), x, S(2)b*S(7)/(S(17)f(bsec(e + fx))(S(17)/2)) - S(6)b*S(5)/(S(13)f(bsec(e + fx))(S(13)/2)) + S(2)b*S(3)/(S(3)f(bsec(e + fx))(S(9)/2)) - S(2)b/(S(5)f(bsec(e + fx))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*S(5)/(bsec(e + fx))(S(3)/2), x), x, -S(2)b*S(5)/(S(13)f(bsec(e + fx))(S(13)/2)) + S(4)b*S(3)/(S(9)f(bsec(e + fx))(S(9)/2)) - S(2)b/(S(5)f(bsec(e + fx))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*S(3)/(bsec(e + fx))(S(3)/2), x), x, S(2)b*S(3)/(S(9)f(bsec(e + fx))(S(9)/2)) - S(2)b/(S(5)f(bsec(e + fx))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)/(bsec(e + fx))*(S(3)/2), x), x, -S(2)b/(S(5)f(bsec(e + fx))*(S(5)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)/(bsec(e + fx))*(S(3)/2), x), x, S(2)/(bfsqrt(bsec(e + fx))) + ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(b(S(3)/2)f) - atanh(sqrt(bsec(e + fx))/sqrt(b))/(b*(S(3)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(3)/(bsec(e + fx))(S(3)/2), x), x, -(bsec(e + fx))(S(3)/2)cot(e + fx)S(2)/(S(2)b*S(3)f) - ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(S(4)b(S(3)/2)f) + atanh(sqrt(bsec(e + fx))/sqrt(b))/(S(4)b(S(3)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(5)/(bsec(e + fx))(S(3)/2), x), x, -(bsec(e + fx))(S(3)/2)cot(e + fx)S(4)/(S(4)b*S(3)f) - S(3)(bsec(e + fx))(S(3)/2)cot(e + fx)S(2)/(S(16)b*S(3)f) - S(3)ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(S(32)b*(S(3)/2)f) + S(3)atanh(sqrt(bsec(e + fx))/sqrt(b))/(S(32)b*(S(3)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)S(4)/(bsec(e + fx))(S(3)/2), x), x, -S(2)bsin(e + fx)*S(3)/(S(11)f(bsec(e + fx))(S(5)/2)) - S(12)bsin(e + fx)/(S(77)f(bsec(e + fx))*(S(5)/2)) + S(8)sin(e + fx)/(S(77)bfsqrt(bsec(e + fx))) + S(8)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(77)b*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)S(2)/(bsec(e + fx))(S(3)/2), x), x, -S(2)bsin(e + fx)/(S(7)f(bsec(e + fx))*(S(5)/2)) + S(4)sin(e + fx)/(S(21)bfsqrt(bsec(e + fx))) + S(4)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(21)b*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*(S(-3)/2), x), x, S(2)sin(e + fx)/(S(3)bfsqrt(bsec(e + fx))) + S(2)sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(3)b*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(2)/(bsec(e + fx))(S(3)/2), x), x, -csc(e + fx)/(bfsqrt(bsec(e + fx))) - sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(b*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(4)/(bsec(e + fx))(S(3)/2), x), x, -csc(e + fx)*S(3)/(S(3)bfsqrt(bsec(e + fx))) + csc(e + fx)/(S(6)bfsqrt(bsec(e + fx))) - sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(6)bS(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(6)/(bsec(e + fx))(S(3)/2), x), x, -csc(e + fx)*S(5)/(S(5)bfsqrt(bsec(e + fx))) + csc(e + fx)S(3)/(S(30)bfsqrt(bsec(e + fx))) + csc(e + fx)/(S(12)bfsqrt(bsec(e + fx))) - sqrt(bsec(e + fx))EllipticF(e/S(2) + fx/S(2), S(2))sqrt(cos(e + fx))/(S(12)bS(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)S(7)/(bsec(e + fx))(S(5)/2), x), x, S(2)b*S(7)/(S(19)f(bsec(e + fx))(S(19)/2)) - S(2)b*S(5)/(S(5)f(bsec(e + fx))(S(15)/2)) + S(6)b*S(3)/(S(11)f(bsec(e + fx))(S(11)/2)) - S(2)b/(S(7)f(bsec(e + fx))*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*S(5)/(bsec(e + fx))(S(5)/2), x), x, -S(2)b*S(5)/(S(15)f(bsec(e + fx))(S(15)/2)) + S(4)b*S(3)/(S(11)f(bsec(e + fx))(S(11)/2)) - S(2)b/(S(7)f(bsec(e + fx))*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*S(3)/(bsec(e + fx))(S(5)/2), x), x, S(2)b*S(3)/(S(11)f(bsec(e + fx))(S(11)/2)) - S(2)b/(S(7)f(bsec(e + fx))*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)/(bsec(e + fx))*(S(5)/2), x), x, -S(2)b/(S(7)f(bsec(e + fx))*(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)/(bsec(e + fx))*(S(5)/2), x), x, S(2)/(S(3)bf(bsec(e + fx))*(S(3)/2)) - ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(b(S(5)/2)f) - atanh(sqrt(bsec(e + fx))/sqrt(b))/(b*(S(5)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(3)/(bsec(e + fx))(S(5)/2), x), x, -sqrt(bsec(e + fx))cot(e + fx)S(2)/(S(2)b*S(3)f) + S(3)ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(S(4)b*(S(5)/2)f) + S(3)atanh(sqrt(bsec(e + fx))/sqrt(b))/(S(4)b*(S(5)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(5)/(bsec(e + fx))(S(5)/2), x), x, -sqrt(bsec(e + fx))cot(e + fx)S(4)/(S(4)b*S(3)f) - sqrt(bsec(e + fx))cot(e + fx)*S(2)/(S(16)b*S(3)f) + S(3)ArcTan(sqrt(bsec(e + fx))/sqrt(b))/(S(32)b*(S(5)/2)f) + S(3)atanh(sqrt(bsec(e + fx))/sqrt(b))/(S(32)b*(S(5)/2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)S(4)/(bsec(e + fx))(S(5)/2), x), x, -S(2)bsin(e + fx)*S(3)/(S(13)f(bsec(e + fx))(S(7)/2)) - S(4)bsin(e + fx)/(S(39)f(bsec(e + fx))*(S(7)/2)) + S(8)sin(e + fx)/(S(195)bf(bsec(e + fx))*(S(3)/2)) + S(8)EllipticE(e/S(2) + fx/S(2), S(2))/(S(65)b*S(2)fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*S(2)/(bsec(e + fx))(S(5)/2), x), x, -S(2)bsin(e + fx)/(S(9)f(bsec(e + fx))*(S(7)/2)) + S(4)sin(e + fx)/(S(45)bf(bsec(e + fx))*(S(3)/2)) + S(4)EllipticE(e/S(2) + fx/S(2), S(2))/(S(15)b*S(2)fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))(S(-5)/2), x), x, S(2)sin(e + fx)/(S(5)bf(bsec(e + fx))*(S(3)/2)) + S(6)EllipticE(e/S(2) + fx/S(2), S(2))/(S(5)b*S(2)fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)*S(2)/(bsec(e + fx))(S(5)/2), x), x, -csc(e + fx)/(bf(bsec(e + fx))*(S(3)/2)) - S(3)EllipticE(e/S(2) + fx/S(2), S(2))/(bS(2)fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)*S(4)/(bsec(e + fx))(S(5)/2), x), x, -csc(e + fx)*S(3)/(S(3)bf(bsec(e + fx))*(S(3)/2)) + csc(e + fx)/(S(2)bf(bsec(e + fx))(S(3)/2)) + EllipticE(e/S(2) + fx/S(2), S(2))/(S(2)bS(2)fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)*S(6)/(bsec(e + fx))(S(5)/2), x), x, -csc(e + fx)*S(5)/(S(5)bf(bsec(e + fx))*(S(3)/2)) + csc(e + fx)*S(3)/(S(10)bf(bsec(e + fx))*(S(3)/2)) + S(3)csc(e + fx)/(S(20)bf(bsec(e + fx))*(S(3)/2)) + S(3)EllipticE(e/S(2) + fx/S(2), S(2))/(S(20)b*S(2)fsqrt(bsec(e + fx))sqrt(cos(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*(S(9)/2)/sqrt(bsec(e + fx)), x), x, -bsin(e + fx)(S(7)/2)/(S(5)f(bsec(e + fx))(S(3)/2)) - S(7)bsin(e + fx)*(S(3)/2)/(S(30)f(bsec(e + fx))(S(3)/2)) + S(7)sqrt(bsec(e + fx))EllipticE(-Pi/S(4) + e + fx, S(2))sqrt(sin(e + fx))cos(e + fx)/(S(20)bfsqrt(sin(S(2)e + S(2)fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)(S(5)/2)/sqrt(bsec(e + fx)), x), x, -bsin(e + fx)(S(3)/2)/(S(3)f(bsec(e + fx))(S(3)/2)) + sqrt(bsec(e + fx))EllipticE(-Pi/S(4) + e + fx, S(2))sqrt(sin(e + fx))cos(e + fx)/(S(2)bfsqrt(sin(S(2)e + S(2)fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(sin(e + fx))/sqrt(bsec(e + fx)), x), x, sqrt(bsec(e + fx))EllipticE(-Pi/S(4) + e + fx, S(2))sqrt(sin(e + fx))cos(e + fx)/(bfsqrt(sin(S(2)e + S(2)fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(bsec(e + fx))sin(e + fx)(S(3)/2)), x), x, -S(2)b/(f(bsec(e + fx))(S(3)/2)sqrt(sin(e + fx))) - S(2)sqrt(bsec(e + fx))EllipticE(-Pi/S(4) + e + fx, S(2))sqrt(sin(e + fx))cos(e + fx)/(bfsqrt(sin(S(2)e + S(2)fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(bsec(e + fx))sin(e + fx)(S(7)/2)), x), x, -S(4)b/(S(5)f(bsec(e + fx))*(S(3)/2)sqrt(sin(e + fx))) - S(2)b/(S(5)f(bsec(e + fx))*(S(3)/2)sin(e + fx)(S(5)/2)) - S(4)sqrt(bsec(e + fx))EllipticE(-Pi/S(4) + e + fx, S(2))sqrt(sin(e + fx))cos(e + fx)/(S(5)bfsqrt(sin(S(2)e + S(2)fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)(S(3)/2)/sqrt(bsec(e + fx)), x), x, -bsqrt(sin(e + fx))/(S(2)f(bsec(e + fx))(S(3)/2)) + sqrt(S(2))sqrt(cos(e + fx)/b)sqrt(bsec(e + fx))ArcTan(-sqrt(S(2))sqrt(b)sqrt(cos(e + fx)/b)/sqrt(sin(e + fx)) + S(1))/(S(8)sqrt(b)f) - sqrt(S(2))sqrt(cos(e + fx)/b)sqrt(bsec(e + fx))ArcTan(sqrt(S(2))sqrt(b)sqrt(cos(e + fx)/b)/sqrt(sin(e + fx)) + S(1))/(S(8)sqrt(b)f) - sqrt(S(2))sqrt(cos(e + fx)/b)sqrt(bsec(e + fx))log(-sqrt(S(2))sqrt(b)sqrt(cos(e + fx)/b)/sqrt(sin(e + fx)) + cot(e + fx) + S(1))/(S(16)sqrt(b)f) + sqrt(S(2))sqrt(cos(e + fx)/b)sqrt(bsec(e + fx))log(sqrt(S(2))sqrt(b)sqrt(cos(e + fx)/b)/sqrt(sin(e + fx)) + cot(e + fx) + S(1))/(S(16)sqrt(b)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(bsec(e + fx))sqrt(sin(e + fx))), x), x, sqrt(S(2))sqrt(cos(e + fx)/b)sqrt(bsec(e + fx))ArcTan(-sqrt(S(2))sqrt(b)sqrt(cos(e + fx)/b)/sqrt(sin(e + fx)) + S(1))/(S(2)sqrt(b)f) - sqrt(S(2))sqrt(cos(e + fx)/b)sqrt(bsec(e + fx))ArcTan(sqrt(S(2))sqrt(b)sqrt(cos(e + fx)/b)/sqrt(sin(e + fx)) + S(1))/(S(2)sqrt(b)f) - sqrt(S(2))sqrt(cos(e + fx)/b)sqrt(bsec(e + fx))log(-sqrt(S(2))sqrt(b)sqrt(cos(e + fx)/b)/sqrt(sin(e + fx)) + cot(e + fx) + S(1))/(S(4)sqrt(b)f) + sqrt(S(2))sqrt(cos(e + fx)/b)sqrt(bsec(e + fx))log(sqrt(S(2))sqrt(b)sqrt(cos(e + fx)/b)/sqrt(sin(e + fx)) + cot(e + fx) + S(1))/(S(4)sqrt(b)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(bsec(e + fx))sin(e + fx)(S(5)/2)), x), x, -S(2)b/(S(3)f(bsec(e + fx))*(S(3)/2)sin(e + fx)(S(3)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(bsec(e + fx))sin(e + fx)(S(9)/2)), x), x, -S(8)b/(S(21)f(bsec(e + fx))*(S(3)/2)sin(e + fx)(S(3)/2)) - S(2)b/(S(7)f(bsec(e + fx))*(S(3)/2)sin(e + fx)(S(7)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(bsec(e + fx))sin(e + fx)(S(13)/2)), x), x, -S(64)b/(S(231)f(bsec(e + fx))*(S(3)/2)sin(e + fx)(S(3)/2)) - S(16)b/(S(77)f(bsec(e + fx))*(S(3)/2)sin(e + fx)(S(7)/2)) - S(2)b/(S(11)f(bsec(e + fx))*(S(3)/2)sin(e + fx)(S(11)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/(sqrt(bsec(e + fx))sin(e + fx)(S(17)/2)), x), x, -S(256)b/(S(1155)f(bsec(e + fx))*(S(3)/2)sin(e + fx)(S(3)/2)) - S(64)b/(S(385)f(bsec(e + fx))*(S(3)/2)sin(e + fx)(S(7)/2)) - S(8)b/(S(55)f(bsec(e + fx))*(S(3)/2)sin(e + fx)(S(11)/2)) - S(2)b/(S(15)f(bsec(e + fx))*(S(3)/2)sin(e + fx)(S(15)/2)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))m(dsec(a + bx))(S(5)/2), x), x, dS(2)(csin(a + bx))(m + S(1))sqrt(dsec(a + bx))(cos(a + bx)S(2))(S(3)/4)Hypergeometric2F1(S(7)/4, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)*S(2))sec(a + bx)/(bc(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))m(dsec(a + bx))*(S(3)/2), x), x, d(csin(a + bx))*(m + S(1))sqrt(dsec(a + bx))(cos(a + bx)S(2))(S(1)/4)Hypergeometric2F1(S(5)/4, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)*S(2))/(bc(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))msqrt(dsec(a + bx)), x), x, (csin(a + bx))*(m + S(1))sqrt(dsec(a + bx))(cos(a + bx)S(2))(S(3)/4)Hypergeometric2F1(S(3)/4, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)*S(2))sec(a + bx)/(bc(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))m/sqrt(dsec(a + bx)), x), x, (csin(a + bx))(m + S(1))sqrt(dsec(a + bx))(cos(a + bx)S(2))(S(1)/4)Hypergeometric2F1(S(1)/4, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)*S(2))/(bcd(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((csin(a + bx))*m/(dsec(a + bx))(S(3)/2), x), x, (csin(a + bx))(m + S(1))sqrt(dsec(a + bx))Hypergeometric2F1(S(-1)/4, m/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(a + bx)*S(2))cos(a + bx)/(bcdS(2)(m + S(1))(cos(a + bx)S(2))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)msec(e + fx)n, x), x, (cos(e + fx)S(2))(n/S(2) + S(1)/2)Hypergeometric2F1(m/S(2) + S(1)/2, n/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(e + fx)*S(2))sin(e + fx)(m + S(1))sec(e + fx)(n + S(1))/(f(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(e + fx))*msec(e + fx)n, x), x, (asin(e + fx))(m + S(1))(cos(e + fx)S(2))(n/S(2) + S(1)/2)Hypergeometric2F1(m/S(2) + S(1)/2, n/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(e + fx)S(2))sec(e + fx)(n + S(1))/(af(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))nsin(e + fx)m, x), x, -b(bsec(e + fx))*(n + S(-1))(sin(e + fx)S(2))(-m/S(2) + S(1)/2)Hypergeometric2F1(-m/S(2) + S(1)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(e + fx)S(2))sin(e + fx)(m + S(-1))/(f(-n + S(1))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((bsec(e + fx))*nsin(e + fx)m, x), x, -(bsec(e + fx))n(sin(e + fx)S(2))(-m/S(2) + S(1)/2)Hypergeometric2F1(-m/S(2) + S(1)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(e + fx)S(2))sin(e + fx)(m + S(-1))cos(e + fx)/(f(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(e + fx))*m(bsec(e + fx))*n, x), x, (asin(e + fx))(m + S(1))(bsec(e + fx))*(n + S(1))(cos(e + fx)S(2))(n/S(2) + S(1)/2)Hypergeometric2F1(m/S(2) + S(1)/2, n/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(e + fx)S(2))/(abf(m + S(1))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((asin(e + fx))*m(bsec(e + fx))*n, x), x, (asin(e + fx))(m + S(1))(bsec(e + fx))*n(cos(e + fx)S(2))(n/S(2) + S(1)/2)Hypergeometric2F1(m/S(2) + S(1)/2, n/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(e + fx)S(2))sec(e + fx)/(af(m + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))nsin(e + fx)S(5), x), x, -bS(5)(bsec(e + fx))*(n + S(-5))/(f(-n + S(5))) + S(2)bS(3)(bsec(e + fx))*(n + S(-3))/(f(-n + S(3))) - b(bsec(e + fx))(n + S(-1))/(f(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*nsin(e + fx)S(3), x), x, bS(3)(bsec(e + fx))*(n + S(-3))/(f(-n + S(3))) - b(bsec(e + fx))(n + S(-1))/(f(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*nsin(e + fx), x), x, -b(bsec(e + fx))*(n + S(-1))/(f(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*ncsc(e + fx), x), x, -(bsec(e + fx))(n + S(1))Hypergeometric2F1(S(1), n/S(2) + S(1)/2, n/S(2) + S(3)/2, sec(e + fx)S(2))/(bf(n + S(1))), expand=True, _diff=True, _numerical=True) # long time assert rubi_test(rubi_integrate((bsec(e + fx))ncsc(e + fx)S(3), x), x, (bsec(e + fx))(n + S(3))Hypergeometric2F1(S(2), n/S(2) + S(3)/2, n/S(2) + S(5)/2, sec(e + fx)S(2))/(bS(3)f(n + S(3))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))nsin(e + fx)S(6), x), x, -(bsec(e + fx))nHypergeometric2F1(S(-5)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(e + fx)S(2))sin(e + fx)cos(e + fx)/(f(-n + S(1))sqrt(sin(e + fx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))nsin(e + fx)S(4), x), x, -(bsec(e + fx))nHypergeometric2F1(S(-3)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(e + fx)S(2))sin(e + fx)cos(e + fx)/(f(-n + S(1))sqrt(sin(e + fx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))nsin(e + fx)S(2), x), x, -(bsec(e + fx))nHypergeometric2F1(S(-1)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(e + fx)S(2))sin(e + fx)cos(e + fx)/(f(-n + S(1))sqrt(sin(e + fx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))n, x), x, -b(bsec(e + fx))*(n + S(-1))Hypergeometric2F1(S(1)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(e + fx)S(2))sin(e + fx)/(f(-n + S(1))sqrt(sin(e + fx)*S(2))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))ncsc(e + fx)S(2), x), x, -(bsec(e + fx))nsqrt(sin(e + fx)S(2))Hypergeometric2F1(S(3)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(e + fx)S(2))cot(e + fx)/(f(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(e + fx))*ncsc(e + fx)S(4), x), x, -(bsec(e + fx))nsqrt(sin(e + fx)S(2))Hypergeometric2F1(S(5)/2, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(e + fx)S(2))cot(e + fx)/(f(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(a + bx))*n(csin(a + bx))*(S(3)/2), x), x, -c(bsec(a + bx))*nsqrt(csin(a + bx))Hypergeometric2F1(S(-1)/4, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(a + bx)*S(2))cos(a + bx)/(b(-n + S(1))(sin(a + bx)S(2))(S(1)/4)), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(a + bx))*nsqrt(csin(a + bx)), x), x, -c(bsec(a + bx))n(sin(a + bx)S(2))(S(1)/4)Hypergeometric2F1(S(1)/4, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(a + bx)S(2))cos(a + bx)/(bsqrt(csin(a + bx))(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(a + bx))n/sqrt(csin(a + bx)), x), x, -c(bsec(a + bx))*n(sin(a + bx)S(2))(S(3)/4)Hypergeometric2F1(S(3)/4, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(a + bx)S(2))cos(a + bx)/(b(csin(a + bx))*(S(3)/2)(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((bsec(a + bx))*n/(csin(a + bx))(S(3)/2), x), x, -(bsec(a + bx))n(sin(a + bx)S(2))(S(1)/4)Hypergeometric2F1(S(5)/4, -n/S(2) + S(1)/2, -n/S(2) + S(3)/2, cos(a + bx)S(2))cos(a + bx)/(bcsqrt(csin(a + bx))(-n + S(1))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcsc(e + fx))sin(e + fx)*S(4), x), x, -S(2)d*S(3)cos(e + fx)/(S(7)f(dcsc(e + fx))(S(5)/2)) - S(10)dcos(e + fx)/(S(21)fsqrt(dcsc(e + fx))) + S(10)sqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(21)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcsc(e + fx))sin(e + fx)*S(3), x), x, -S(2)d*S(2)cos(e + fx)/(S(5)f(dcsc(e + fx))(S(3)/2)) + S(6)dEllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(5)fsqrt(dcsc(e + fx))sqrt(sin(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcsc(e + fx))sin(e + fx)*S(2), x), x, -S(2)dcos(e + fx)/(S(3)fsqrt(dcsc(e + fx))) + S(2)sqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcsc(e + fx))sin(e + fx), x), x, S(2)dEllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(fsqrt(dcsc(e + fx))sqrt(sin(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcsc(e + fx)), x), x, S(2)sqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcsc(e + fx))csc(e + fx), x), x, -S(2)dEllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(fsqrt(dcsc(e + fx))sqrt(sin(e + fx))) - S(2)sqrt(dcsc(e + fx))cos(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcsc(e + fx))csc(e + fx)*S(2), x), x, S(2)sqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)f) - S(2)(dcsc(e + fx))*(S(3)/2)cos(e + fx)/(S(3)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sqrt(dcsc(e + fx))csc(e + fx)S(3), x), x, -S(6)dEllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(5)fsqrt(dcsc(e + fx))sqrt(sin(e + fx))) - S(6)sqrt(dcsc(e + fx))cos(e + fx)/(S(5)f) - S(2)(dcsc(e + fx))(S(5)/2)cos(e + fx)/(S(5)d*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(e + fx))*(S(3)/2)sin(e + fx)S(5), x), x, -S(2)d*S(4)cos(e + fx)/(S(7)f(dcsc(e + fx))(S(5)/2)) - S(10)d*S(2)cos(e + fx)/(S(21)fsqrt(dcsc(e + fx))) + S(10)dsqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(21)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(e + fx))*(S(3)/2)sin(e + fx)S(4), x), x, -S(2)d*S(3)cos(e + fx)/(S(5)f(dcsc(e + fx))(S(3)/2)) + S(6)d*S(2)EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(5)fsqrt(dcsc(e + fx))sqrt(sin(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(e + fx))(S(3)/2)sin(e + fx)S(3), x), x, -S(2)d*S(2)cos(e + fx)/(S(3)fsqrt(dcsc(e + fx))) + S(2)dsqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(e + fx))*(S(3)/2)sin(e + fx)S(2), x), x, S(2)d*S(2)EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(fsqrt(dcsc(e + fx))sqrt(sin(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(e + fx))*(S(3)/2)sin(e + fx), x), x, S(2)dsqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(e + fx))(S(3)/2), x), x, -S(2)d*S(2)EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(fsqrt(dcsc(e + fx))sqrt(sin(e + fx))) - S(2)dsqrt(dcsc(e + fx))cos(e + fx)/f, expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(e + fx))*(S(3)/2)csc(e + fx), x), x, S(2)dsqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)f) - S(2)(dcsc(e + fx))(S(3)/2)cos(e + fx)/(S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(e + fx))*(S(3)/2)csc(e + fx)S(2), x), x, -S(6)d*S(2)EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(5)fsqrt(dcsc(e + fx))sqrt(sin(e + fx))) - S(6)dsqrt(dcsc(e + fx))cos(e + fx)/(S(5)f) - S(2)(dcsc(e + fx))(S(5)/2)cos(e + fx)/(S(5)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)*S(3)/sqrt(dcsc(e + fx)), x), x, -S(2)d*S(2)cos(e + fx)/(S(7)f(dcsc(e + fx))(S(5)/2)) - S(10)cos(e + fx)/(S(21)fsqrt(dcsc(e + fx))) + S(10)sqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(21)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)S(2)/sqrt(dcsc(e + fx)), x), x, -S(2)dcos(e + fx)/(S(5)f(dcsc(e + fx))*(S(3)/2)) + S(6)EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(5)fsqrt(dcsc(e + fx))sqrt(sin(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)/sqrt(dcsc(e + fx)), x), x, -S(2)cos(e + fx)/(S(3)fsqrt(dcsc(e + fx))) + S(2)sqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(S(1)/sqrt(dcsc(e + fx)), x), x, S(2)EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(fsqrt(dcsc(e + fx))sqrt(sin(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)/sqrt(dcsc(e + fx)), x), x, S(2)sqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)*S(2)/sqrt(dcsc(e + fx)), x), x, -S(2)EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(fsqrt(dcsc(e + fx))sqrt(sin(e + fx))) - S(2)sqrt(dcsc(e + fx))cos(e + fx)/(df), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(3)/sqrt(dcsc(e + fx)), x), x, S(2)sqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)df) - S(2)(dcsc(e + fx))(S(3)/2)cos(e + fx)/(S(3)d*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)S(2)/(dcsc(e + fx))(S(3)/2), x), x, -S(2)dcos(e + fx)/(S(7)f(dcsc(e + fx))*(S(5)/2)) - S(10)cos(e + fx)/(S(21)dfsqrt(dcsc(e + fx))) + S(10)sqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(21)d*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(sin(e + fx)/(dcsc(e + fx))(S(3)/2), x), x, -S(2)cos(e + fx)/(S(5)f(dcsc(e + fx))(S(3)/2)) + S(6)EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(5)dfsqrt(dcsc(e + fx))sqrt(sin(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((dcsc(e + fx))*(S(-3)/2), x), x, -S(2)cos(e + fx)/(S(3)dfsqrt(dcsc(e + fx))) + S(2)sqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)d*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)/(dcsc(e + fx))(S(3)/2), x), x, S(2)EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(dfsqrt(dcsc(e + fx))sqrt(sin(e + fx))), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)*S(2)/(dcsc(e + fx))(S(3)/2), x), x, S(2)sqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(d*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(3)/(dcsc(e + fx))(S(3)/2), x), x, -S(2)EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(dfsqrt(dcsc(e + fx))sqrt(sin(e + fx))) - S(2)sqrt(dcsc(e + fx))cos(e + fx)/(d*S(2)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(4)/(dcsc(e + fx))(S(3)/2), x), x, S(2)sqrt(dcsc(e + fx))EllipticF(-Pi/S(4) + e/S(2) + fx/S(2), S(2))sqrt(sin(e + fx))/(S(3)dS(2)f) - S(2)(dcsc(e + fx))(S(3)/2)cos(e + fx)/(S(3)d*S(3)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate(csc(e + fx)S(5)/(dcsc(e + fx))(S(3)/2), x), x, -S(6)EllipticE(-Pi/S(4) + e/S(2) + fx/S(2), S(2))/(S(5)dfsqrt(dcsc(e + fx))sqrt(sin(e + fx))) - S(6)sqrt(dcsc(e + fx))cos(e + fx)/(S(5)d*S(2)f) - S(2)(dcsc(e + fx))(S(5)/2)cos(e + fx)/(S(5)d*S(4)f), expand=True, _diff=True, _numerical=True) assert rubi_test(rubi_integrate((asin(e + fx))*m(bcsc(e + fx))*n, x), x, (asin(e + fx))(m + S(1))(bcsc(e + fx))*nHypergeometric2F1(S(1)/2, m/S(2) - n/S(2) + S(1)/2, m/S(2) - n/S(2) + S(3)/2, sin(e + fx)S(2))cos(e + fx)/(af(m - n + S(1))sqrt(cos(e + fx)*S(2))), expand=True, _diff=True, _numerical=True)
1cc4d79aec35a133215e5e984c82b726561579504263e16bde5ba6ac46034079	from sympy.assumptions.refine import refine from sympy.concrete.summations import Sum from sympy.core.add import Add from sympy.core.basic import Basic from sympy.core.containers import Tuple from sympy.core.expr import (ExprBuilder, unchanged, Expr, UnevaluatedExpr) from sympy.core.function import (Function, expand, WildFunction, AppliedUndef, Derivative, diff) from sympy.core.mul import Mul from sympy.core.numbers import (NumberSymbol, E, zoo, oo, Float, I, Rational, nan, Integer, Number, pi) from sympy.core.power import Pow from sympy.core.relational import Ge, Lt, Gt, Le from sympy.core.singleton import S from sympy.core.sorting import default_sort_key from sympy.core.symbol import Symbol, symbols, Dummy, Wild from sympy.core.sympify import sympify from sympy.functions.combinatorial.factorials import factorial from sympy.functions.elementary.exponential import exp_polar, exp, log from sympy.functions.elementary.miscellaneous import sqrt, Max from sympy.functions.elementary.piecewise import Piecewise from sympy.functions.elementary.trigonometric import tan, sin, cos from sympy.functions.special.delta_functions import (Heaviside, DiracDelta) from sympy.functions.special.error_functions import Si from sympy.functions.special.gamma_functions import gamma from sympy.integrals.integrals import integrate, Integral from sympy.physics.secondquant import FockState from sympy.polys.partfrac import apart from sympy.polys.polytools import factor, cancel, Poly from sympy.polys.rationaltools import together from sympy.series.order import O from sympy.simplify.combsimp import combsimp from sympy.simplify.gammasimp import gammasimp from sympy.simplify.powsimp import powsimp from sympy.simplify.radsimp import collect, radsimp from sympy.simplify.ratsimp import ratsimp from sympy.simplify.simplify import simplify, nsimplify from sympy.simplify.trigsimp import trigsimp from sympy.physics.units import meter from sympy.testing.pytest import raises, XFAIL from sympy.abc import a, b, c, n, t, u, x, y, z f, g, h = symbols('f,g,h', cls=Function) class DummyNumber: """ Minimal implementation of a number that works with SymPy. If one has a Number class (e.g. Sage Integer, or some other custom class) that one wants to work well with SymPy, one has to implement at least the methods of this class DummyNumber, resp. its subclasses I5 and F1_1. Basically, one just needs to implement either __int__() or __float__() and then one needs to make sure that the class works with Python integers and with itself. """ def __radd__(self, a): if isinstance(a, (int, float)): return a + self.number return NotImplemented def __add__(self, a): if isinstance(a, (int, float, DummyNumber)): return self.number + a return NotImplemented def __rsub__(self, a): if isinstance(a, (int, float)): return a - self.number return NotImplemented def __sub__(self, a): if isinstance(a, (int, float, DummyNumber)): return self.number - a return NotImplemented def __rmul__(self, a): if isinstance(a, (int, float)): return a * self.number return NotImplemented def __mul__(self, a): if isinstance(a, (int, float, DummyNumber)): return self.number * a return NotImplemented def __rtruediv__(self, a): if isinstance(a, (int, float)): return a / self.number return NotImplemented def __truediv__(self, a): if isinstance(a, (int, float, DummyNumber)): return self.number / a return NotImplemented def __rpow__(self, a): if isinstance(a, (int, float)): return a self.number return NotImplemented def __pow__(self, a): if isinstance(a, (int, float, DummyNumber)): return self.number a return NotImplemented def __pos__(self): return self.number def __neg__(self): return - self.number class I5(DummyNumber): number = 5 def __int__(self): return self.number class F1_1(DummyNumber): number = 1.1 def __float__(self): return self.number i5 = I5() f1_1 = F1_1() # basic SymPy objects basic_objs = [ Rational(2), Float("1.3"), x, y, pow(x, y)y, ] # all supported objects all_objs = basic_objs + [ 5, 5.5, i5, f1_1 ] def dotest(s): for xo in all_objs: for yo in all_objs: s(xo, yo) return True def test_basic(): def j(a, b): x = a x = +a x = -a x = a + b x = a - b x = ab x = a/b x = a*b del x assert dotest(j) def test_ibasic(): def s(a, b): x = a x += b x = a x -= b x = a x = b x = a x /= b assert dotest(s) class NonBasic: '''This class represents an object that knows how to implement binary operations like +, -, etc with Expr but is not a subclass of Basic itself. The NonExpr subclass below does subclass Basic but not Expr. For both NonBasic and NonExpr it should be possible for them to override Expr.__add__ etc because Expr.__add__ should be returning NotImplemented for non Expr classes. Otherwise Expr.__add__ would create meaningless objects like Add(Integer(1), FiniteSet(2)) and it wouldn't be possible for other classes to override these operations when interacting with Expr. ''' def __add__(self, other): return SpecialOp('+', self, other) def __radd__(self, other): return SpecialOp('+', other, self) def __sub__(self, other): return SpecialOp('-', self, other) def __rsub__(self, other): return SpecialOp('-', other, self) def __mul__(self, other): return SpecialOp('', self, other) def __rmul__(self, other): return SpecialOp('', other, self) def __truediv__(self, other): return SpecialOp('/', self, other) def __rtruediv__(self, other): return SpecialOp('/', other, self) def __floordiv__(self, other): return SpecialOp('//', self, other) def __rfloordiv__(self, other): return SpecialOp('//', other, self) def __mod__(self, other): return SpecialOp('%', self, other) def __rmod__(self, other): return SpecialOp('%', other, self) def __divmod__(self, other): return SpecialOp('divmod', self, other) def __rdivmod__(self, other): return SpecialOp('divmod', other, self) def __pow__(self, other): return SpecialOp('', self, other) def __rpow__(self, other): return SpecialOp('', other, self) def __lt__(self, other): return SpecialOp('<', self, other) def __gt__(self, other): return SpecialOp('>', self, other) def __le__(self, other): return SpecialOp('<=', self, other) def __ge__(self, other): return SpecialOp('>=', self, other) class NonExpr(Basic, NonBasic): '''Like NonBasic above except this is a subclass of Basic but not Expr''' pass class SpecialOp(): '''Represents the results of operations with NonBasic and NonExpr''' def __new__(cls, op, arg1, arg2): obj = object.__new__(cls) obj.args = (op, arg1, arg2) return obj class NonArithmetic(Basic): '''Represents a Basic subclass that does not support arithmetic operations''' pass def test_cooperative_operations(): '''Tests that Expr uses binary operations cooperatively. In particular it should be possible for non-Expr classes to override binary operators like +, - etc when used with Expr instances. This should work for non-Expr classes whether they are Basic subclasses or not. Also non-Expr classes that do not define binary operators with Expr should give TypeError. ''' # A bunch of instances of Expr subclasses exprs = [ Expr(), S.Zero, S.One, S.Infinity, S.NegativeInfinity, S.ComplexInfinity, S.Half, Float(0.5), Integer(2), Symbol('x'), Mul(2, Symbol('x')), Add(2, Symbol('x')), Pow(2, Symbol('x')), ] for e in exprs: # Test that these classes can override arithmetic operations in # combination with various Expr types. for ne in [NonBasic(), NonExpr()]: results = [ (ne + e, ('+', ne, e)), (e + ne, ('+', e, ne)), (ne - e, ('-', ne, e)), (e - ne, ('-', e, ne)), (ne * e, ('', ne, e)), (e ne, ('', e, ne)), (ne / e, ('/', ne, e)), (e / ne, ('/', e, ne)), (ne // e, ('//', ne, e)), (e // ne, ('//', e, ne)), (ne % e, ('%', ne, e)), (e % ne, ('%', e, ne)), (divmod(ne, e), ('divmod', ne, e)), (divmod(e, ne), ('divmod', e, ne)), (ne * e, ('', ne, e)), (e ne, ('*', e, ne)), (e < ne, ('>', ne, e)), (ne < e, ('<', ne, e)), (e > ne, ('<', ne, e)), (ne > e, ('>', ne, e)), (e <= ne, ('>=', ne, e)), (ne <= e, ('<=', ne, e)), (e >= ne, ('<=', ne, e)), (ne >= e, ('>=', ne, e)), ] for res, args in results: assert type(res) is SpecialOp and res.args == args # These classes do not support binary operators with Expr. Every # operation should raise in combination with any of the Expr types. for na in [NonArithmetic(), object()]: raises(TypeError, lambda : e + na) raises(TypeError, lambda : na + e) raises(TypeError, lambda : e - na) raises(TypeError, lambda : na - e) raises(TypeError, lambda : e na) raises(TypeError, lambda : na * e) raises(TypeError, lambda : e / na) raises(TypeError, lambda : na / e) raises(TypeError, lambda : e // na) raises(TypeError, lambda : na // e) raises(TypeError, lambda : e % na) raises(TypeError, lambda : na % e) raises(TypeError, lambda : divmod(e, na)) raises(TypeError, lambda : divmod(na, e)) raises(TypeError, lambda : e na) raises(TypeError, lambda : na e) raises(TypeError, lambda : e > na) raises(TypeError, lambda : na > e) raises(TypeError, lambda : e < na) raises(TypeError, lambda : na < e) raises(TypeError, lambda : e >= na) raises(TypeError, lambda : na >= e) raises(TypeError, lambda : e <= na) raises(TypeError, lambda : na <= e) def test_relational(): from sympy.core.relational import Lt assert (pi < 3) is S.false assert (pi <= 3) is S.false assert (pi > 3) is S.true assert (pi >= 3) is S.true assert (-pi < 3) is S.true assert (-pi <= 3) is S.true assert (-pi > 3) is S.false assert (-pi >= 3) is S.false r = Symbol('r', real=True) assert (r - 2 < r - 3) is S.false assert Lt(x + I, x + I + 2).func == Lt # issue 8288 def test_relational_assumptions(): m1 = Symbol("m1", nonnegative=False) m2 = Symbol("m2", positive=False) m3 = Symbol("m3", nonpositive=False) m4 = Symbol("m4", negative=False) assert (m1 < 0) == Lt(m1, 0) assert (m2 <= 0) == Le(m2, 0) assert (m3 > 0) == Gt(m3, 0) assert (m4 >= 0) == Ge(m4, 0) m1 = Symbol("m1", nonnegative=False, real=True) m2 = Symbol("m2", positive=False, real=True) m3 = Symbol("m3", nonpositive=False, real=True) m4 = Symbol("m4", negative=False, real=True) assert (m1 < 0) is S.true assert (m2 <= 0) is S.true assert (m3 > 0) is S.true assert (m4 >= 0) is S.true m1 = Symbol("m1", negative=True) m2 = Symbol("m2", nonpositive=True) m3 = Symbol("m3", positive=True) m4 = Symbol("m4", nonnegative=True) assert (m1 < 0) is S.true assert (m2 <= 0) is S.true assert (m3 > 0) is S.true assert (m4 >= 0) is S.true m1 = Symbol("m1", negative=False, real=True) m2 = Symbol("m2", nonpositive=False, real=True) m3 = Symbol("m3", positive=False, real=True) m4 = Symbol("m4", nonnegative=False, real=True) assert (m1 < 0) is S.false assert (m2 <= 0) is S.false assert (m3 > 0) is S.false assert (m4 >= 0) is S.false # See https://github.com/sympy/sympy/issues/17708 #def test_relational_noncommutative(): # from sympy import Lt, Gt, Le, Ge # A, B = symbols('A,B', commutative=False) # assert (A < B) == Lt(A, B) # assert (A <= B) == Le(A, B) # assert (A > B) == Gt(A, B) # assert (A >= B) == Ge(A, B) def test_basic_nostr(): for obj in basic_objs: raises(TypeError, lambda: obj + '1') raises(TypeError, lambda: obj - '1') if obj == 2: assert obj * '1' == '11' else: raises(TypeError, lambda: obj * '1') raises(TypeError, lambda: obj / '1') raises(TypeError, lambda: obj ** '1') def test_series_expansion_for_uniform_order(): assert (1/x + y + x).series(x, 0, 0) == 1/x + O(1, x) assert (1/x + y + x).series(x, 0, 1) == 1/x + y + O(x) assert (1/x + 1 + x).series(x, 0, 0) == 1/x + O(1, x) assert (1/x + 1 + x).series(x, 0, 1) == 1/x + 1 + O(x) assert (1/x + x).series(x, 0, 0) == 1/x + O(1, x) assert (1/x + y + yx + x).series(x, 0, 0) == 1/x + O(1, x) assert (1/x + y + yx + x).series(x, 0, 1) == 1/x + y + O(x) def test_leadterm(): assert (3 + 2x(log(3)/log(2) - 1)).leadterm(x) == (3, 0) assert (1/x2 + 1 + x + x2).leadterm(x)[1] == -2 assert (1/x + 1 + x + x2).leadterm(x)[1] == -1 assert (x2 + 1/x).leadterm(x)[1] == -1 assert (1 + x2).leadterm(x)[1] == 0 assert (x + 1).leadterm(x)[1] == 0 assert (x + x2).leadterm(x)[1] == 1 assert (x2).leadterm(x)[1] == 2 def test_as_leading_term(): assert (3 + 2x(log(3)/log(2) - 1)).as_leading_term(x) == 3 assert (1/x2 + 1 + x + x2).as_leading_term(x) == 1/x2 assert (1/x + 1 + x + x2).as_leading_term(x) == 1/x assert (x2 + 1/x).as_leading_term(x) == 1/x assert (1 + x2).as_leading_term(x) == 1 assert (x + 1).as_leading_term(x) == 1 assert (x + x2).as_leading_term(x) == x assert (x2).as_leading_term(x) == x2 assert (x + oo).as_leading_term(x) is oo raises(ValueError, lambda: (x + 1).as_leading_term(1)) # https://github.com/sympy/sympy/issues/21177 e = -3x + (x + Rational(3, 2) - sqrt(3)S.ImaginaryUnit/2)*2\ - Rational(3, 2) + 3sqrt(3)S.ImaginaryUnit/2 assert e.as_leading_term(x) == \ (12sqrt(3)x - 12S.ImaginaryUnitx)/(4sqrt(3) + 12S.ImaginaryUnit) # https://github.com/sympy/sympy/issues/21245 e = 1 - x - x2 d = (1 + sqrt(5))/2 assert e.subs(x, y + 1/d).as_leading_term(y) == \ (-576sqrt(5)y - 1280y)/(256sqrt(5) + 576) def test_leadterm2(): assert (xcos(1)cos(1 + sin(1)) + sin(1 + sin(1))).leadterm(x) == \ (sin(1 + sin(1)), 0) def test_leadterm3(): assert (y + z + x).leadterm(x) == (y + z, 0) def test_as_leading_term2(): assert (xcos(1)cos(1 + sin(1)) + sin(1 + sin(1))).as_leading_term(x) == \ sin(1 + sin(1)) def test_as_leading_term3(): assert (2 + pi + x).as_leading_term(x) == 2 + pi assert (2x + pix + x2).as_leading_term(x) == 2x + pix def test_as_leading_term4(): # see issue 6843 n = Symbol('n', integer=True, positive=True) r = -n3/(2n*2 + 4n + 2) - n2/(n2 + 2n + 1) + \ n2/(n + 1) - n/(2n*2 + 4n + 2) + n/(nx + x) + 2n/(n + 1) - \ 1 + 1/(nx + x) + 1/(n + 1) - 1/x assert r.as_leading_term(x).cancel() == n/2 def test_as_leading_term_stub(): class foo(Function): pass assert foo(1/x).as_leading_term(x) == foo(1/x) assert foo(1).as_leading_term(x) == foo(1) raises(NotImplementedError, lambda: foo(x).as_leading_term(x)) def test_as_leading_term_deriv_integral(): # related to issue 11313 assert Derivative(x * 3, x).as_leading_term(x) == 3x2 assert Derivative(x * 3, y).as_leading_term(x) == 0 assert Integral(x 3, x).as_leading_term(x) == x4/4 assert Integral(x ** 3, y).as_leading_term(x) == yx3 assert Derivative(exp(x), x).as_leading_term(x) == 1 assert Derivative(log(x), x).as_leading_term(x) == (1/x).as_leading_term(x) def test_atoms(): assert x.atoms() == {x} assert (1 + x).atoms() == {x, S.One} assert (1 + 2cos(x)).atoms(Symbol) == {x} assert (1 + 2cos(x)).atoms(Symbol, Number) == {S.One, S(2), x} assert (2(x(yx))).atoms() == {S(2), x, y} assert S.Half.atoms() == {S.Half} assert S.Half.atoms(Symbol) == set() assert sin(oo).atoms(oo) == set() assert Poly(0, x).atoms() == {S.Zero, x} assert Poly(1, x).atoms() == {S.One, x} assert Poly(x, x).atoms() == {x} assert Poly(x, x, y).atoms() == {x, y} assert Poly(x + y, x, y).atoms() == {x, y} assert Poly(x + y, x, y, z).atoms() == {x, y, z} assert Poly(x + yt, x, y, z).atoms() == {t, x, y, z} assert (Ipi).atoms(NumberSymbol) == {pi} assert (Ipi).atoms(NumberSymbol, I) == \ (Ipi).atoms(I, NumberSymbol) == {pi, I} assert exp(exp(x)).atoms(exp) == {exp(exp(x)), exp(x)} assert (1 + x(2 + y) + exp(3 + z)).atoms(Add) == \ {1 + x(2 + y) + exp(3 + z), 2 + y, 3 + z} # issue 6132 e = (f(x) + sin(x) + 2) assert e.atoms(AppliedUndef) == \ {f(x)} assert e.atoms(AppliedUndef, Function) == \ {f(x), sin(x)} assert e.atoms(Function) == \ {f(x), sin(x)} assert e.atoms(AppliedUndef, Number) == \ {f(x), S(2)} assert e.atoms(Function, Number) == \ {S(2), sin(x), f(x)} def test_is_polynomial(): k = Symbol('k', nonnegative=True, integer=True) assert Rational(2).is_polynomial(x, y, z) is True assert (S.Pi).is_polynomial(x, y, z) is True assert x.is_polynomial(x) is True assert x.is_polynomial(y) is True assert (x2).is_polynomial(x) is True assert (x2).is_polynomial(y) is True assert (x(-2)).is_polynomial(x) is False assert (x(-2)).is_polynomial(y) is True assert (2x).is_polynomial(x) is False assert (2x).is_polynomial(y) is True assert (xk).is_polynomial(x) is False assert (xk).is_polynomial(k) is False assert (xx).is_polynomial(x) is False assert (kk).is_polynomial(k) is False assert (kx).is_polynomial(k) is False assert (x(-k)).is_polynomial(x) is False assert ((2x)k).is_polynomial(x) is False assert (x2 + 3x - 8).is_polynomial(x) is True assert (x*2 + 3x - 8).is_polynomial(y) is True assert (x*2 + 3x - 8).is_polynomial() is True assert sqrt(x).is_polynomial(x) is False assert (sqrt(x)3).is_polynomial(x) is False assert (x2 + 3xsqrt(y) - 8).is_polynomial(x) is True assert (x*2 + 3xsqrt(y) - 8).is_polynomial(y) is False assert ((x2)(y*2) + x(y*2) + yx + exp(2)).is_polynomial() is True assert ((x*2)(y*2) + x(y*2) + yx + exp(x)).is_polynomial() is False assert ( (x*2)(y*2) + x(y*2) + yx + exp(2)).is_polynomial(x, y) is True assert ( (x*2)(y*2) + x(y*2) + yx + exp(x)).is_polynomial(x, y) is False assert (1/f(x) + 1).is_polynomial(f(x)) is False def test_is_rational_function(): assert Integer(1).is_rational_function() is True assert Integer(1).is_rational_function(x) is True assert Rational(17, 54).is_rational_function() is True assert Rational(17, 54).is_rational_function(x) is True assert (12/x).is_rational_function() is True assert (12/x).is_rational_function(x) is True assert (x/y).is_rational_function() is True assert (x/y).is_rational_function(x) is True assert (x/y).is_rational_function(x, y) is True assert (x2 + 1/x/y).is_rational_function() is True assert (x2 + 1/x/y).is_rational_function(x) is True assert (x2 + 1/x/y).is_rational_function(x, y) is True assert (sin(y)/x).is_rational_function() is False assert (sin(y)/x).is_rational_function(y) is False assert (sin(y)/x).is_rational_function(x) is True assert (sin(y)/x).is_rational_function(x, y) is False assert (S.NaN).is_rational_function() is False assert (S.Infinity).is_rational_function() is False assert (S.NegativeInfinity).is_rational_function() is False assert (S.ComplexInfinity).is_rational_function() is False def test_is_meromorphic(): f = a/x2 + b + x + cx2 assert f.is_meromorphic(x, 0) is True assert f.is_meromorphic(x, 1) is True assert f.is_meromorphic(x, zoo) is True g = 3 + 2x(log(3)/log(2) - 1) assert g.is_meromorphic(x, 0) is False assert g.is_meromorphic(x, 1) is True assert g.is_meromorphic(x, zoo) is False n = Symbol('n', integer=True) e = sin(1/x)nx assert e.is_meromorphic(x, 0) is False assert e.is_meromorphic(x, 1) is True assert e.is_meromorphic(x, zoo) is False e = log(x)pi assert e.is_meromorphic(x, 0) is False assert e.is_meromorphic(x, 1) is False assert e.is_meromorphic(x, 2) is True assert e.is_meromorphic(x, zoo) is False assert (log(x)a).is_meromorphic(x, 0) is False assert (log(x)a).is_meromorphic(x, 1) is False assert (alog(x)).is_meromorphic(x, 0) is None assert (3log(x)).is_meromorphic(x, 0) is False assert (3log(x)).is_meromorphic(x, 1) is True def test_is_algebraic_expr(): assert sqrt(3).is_algebraic_expr(x) is True assert sqrt(3).is_algebraic_expr() is True eq = ((1 + x2)/(1 - y2))(S.One/3) assert eq.is_algebraic_expr(x) is True assert eq.is_algebraic_expr(y) is True assert (sqrt(x) + y(S(2)/3)).is_algebraic_expr(x) is True assert (sqrt(x) + y(S(2)/3)).is_algebraic_expr(y) is True assert (sqrt(x) + y(S(2)/3)).is_algebraic_expr() is True assert (cos(y)/sqrt(x)).is_algebraic_expr() is False assert (cos(y)/sqrt(x)).is_algebraic_expr(x) is True assert (cos(y)/sqrt(x)).is_algebraic_expr(y) is False assert (cos(y)/sqrt(x)).is_algebraic_expr(x, y) is False def test_SAGE1(): #see https://github.com/sympy/sympy/issues/3346 class MyInt: def _sympy_(self): return Integer(5) m = MyInt() e = Rational(2)m assert e == 10 raises(TypeError, lambda: Rational(2)MyInt) def test_SAGE2(): class MyInt: def __int__(self): return 5 assert sympify(MyInt()) == 5 e = Rational(2)MyInt() assert e == 10 raises(TypeError, lambda: Rational(2)MyInt) def test_SAGE3(): class MySymbol: def __rmul__(self, other): return ('mys', other, self) o = MySymbol() e = xo assert e == ('mys', x, o) def test_len(): e = xy assert len(e.args) == 2 e = x + y + z assert len(e.args) == 3 def test_doit(): a = Integral(x2, x) assert isinstance(a.doit(), Integral) is False assert isinstance(a.doit(integrals=True), Integral) is False assert isinstance(a.doit(integrals=False), Integral) is True assert (2Integral(x, x)).doit() == x*2 def test_attribute_error(): raises(AttributeError, lambda: x.cos()) raises(AttributeError, lambda: x.sin()) raises(AttributeError, lambda: x.exp()) def test_args(): assert (xy).args in ((x, y), (y, x)) assert (x + y).args in ((x, y), (y, x)) assert (xy + 1).args in ((xy, 1), (1, xy)) assert sin(xy).args == (xy,) assert sin(xy).args[0] == xy assert (xy).args == (x, y) assert (xy).args[0] == x assert (xy).args[1] == y def test_noncommutative_expand_issue_3757(): A, B, C = symbols('A,B,C', commutative=False) assert AB - BA != 0 assert (A(A + B)B).expand() == A2B + AB2 assert (A(A + B + C)B).expand() == A2B + AB2 + ACB def test_as_numer_denom(): a, b, c = symbols('a, b, c') assert nan.as_numer_denom() == (nan, 1) assert oo.as_numer_denom() == (oo, 1) assert (-oo).as_numer_denom() == (-oo, 1) assert zoo.as_numer_denom() == (zoo, 1) assert (-zoo).as_numer_denom() == (zoo, 1) assert x.as_numer_denom() == (x, 1) assert (1/x).as_numer_denom() == (1, x) assert (x/y).as_numer_denom() == (x, y) assert (x/2).as_numer_denom() == (x, 2) assert (xy/z).as_numer_denom() == (xy, z) assert (x/(yz)).as_numer_denom() == (x, yz) assert S.Half.as_numer_denom() == (1, 2) assert (1/y2).as_numer_denom() == (1, y2) assert (x/y2).as_numer_denom() == (x, y2) assert ((x2 + 1)/y).as_numer_denom() == (x2 + 1, y) assert (x(y + 1)/y*7).as_numer_denom() == (x(y + 1), y7) assert (x-2).as_numer_denom() == (1, x*2) assert (a/x + b/2/x + c/3/x).as_numer_denom() == \ (6a + 3b + 2c, 6x) assert (a/x + b/2/x + c/3/y).as_numer_denom() == \ (2cx + y(6a + 3b), 6xy) assert (a/x + b/2/x + c/.5/x).as_numer_denom() == \ (2a + b + 4.0c, 2x) # this should take no more than a few seconds assert int(log(Add([Dummy()/i/x for i in range(1, 705)] ).as_numer_denom()[1]/x).n(4)) == 705 for i in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]: assert (i + x/3).as_numer_denom() == \ (x + i, 3) assert (S.Infinity + x/3 + y/4).as_numer_denom() == \ (4x + 3y + S.Infinity, 12) assert (oox + zooy).as_numer_denom() == \ (zooy + oox, 1) A, B, C = symbols('A,B,C', commutative=False) assert (ABC*-1).as_numer_denom() == (ABC-1, 1) assert (ABC-1/x).as_numer_denom() == (ABC-1, x) assert (C-1AB).as_numer_denom() == (C-1AB, 1) assert (C-1AB/x).as_numer_denom() == (C-1AB, x) assert ((ABC)-1).as_numer_denom() == ((ABC)-1, 1) assert ((ABC)-1/x).as_numer_denom() == ((ABC)-1, x) # the following morphs from Add to Mul during processing assert Add(0, (x + y)/z/-2, evaluate=False).as_numer_denom( ) == (-x - y, 2z) def test_trunc(): import math x, y = symbols('x y') assert math.trunc(2) == 2 assert math.trunc(4.57) == 4 assert math.trunc(-5.79) == -5 assert math.trunc(pi) == 3 assert math.trunc(log(7)) == 1 assert math.trunc(exp(5)) == 148 assert math.trunc(cos(pi)) == -1 assert math.trunc(sin(5)) == 0 raises(TypeError, lambda: math.trunc(x)) raises(TypeError, lambda: math.trunc(x + y*2)) raises(TypeError, lambda: math.trunc(oo)) def test_as_independent(): assert S.Zero.as_independent(x, as_Add=True) == (0, 0) assert S.Zero.as_independent(x, as_Add=False) == (0, 0) assert (2xsin(x) + y + x).as_independent(x) == (y, x + 2xsin(x)) assert (2xsin(x) + y + x).as_independent(y) == (x + 2xsin(x), y) assert (2xsin(x) + y + x).as_independent(x, y) == (0, y + x + 2xsin(x)) assert (xsin(x)cos(y)).as_independent(x) == (cos(y), xsin(x)) assert (xsin(x)cos(y)).as_independent(y) == (xsin(x), cos(y)) assert (xsin(x)cos(y)).as_independent(x, y) == (1, xsin(x)cos(y)) assert (sin(x)).as_independent(x) == (1, sin(x)) assert (sin(x)).as_independent(y) == (sin(x), 1) assert (2sin(x)).as_independent(x) == (2, sin(x)) assert (2sin(x)).as_independent(y) == (2sin(x), 1) # issue 4903 = 1766b n1, n2, n3 = symbols('n1 n2 n3', commutative=False) assert (n1 + n1n2).as_independent(n2) == (n1, n1n2) assert (n2n1 + n1n2).as_independent(n2) == (0, n1n2 + n2n1) assert (n1n2n1).as_independent(n2) == (n1, n2n1) assert (n1n2n1).as_independent(n1) == (1, n1n2n1) assert (3x).as_independent(x, as_Add=True) == (0, 3x) assert (3x).as_independent(x, as_Add=False) == (3, x) assert (3 + x).as_independent(x, as_Add=True) == (3, x) assert (3 + x).as_independent(x, as_Add=False) == (1, 3 + x) # issue 5479 assert (3x).as_independent(Symbol) == (3, x) # issue 5648 assert (n1xy).as_independent(x) == (n1y, x) assert ((x + n1)(x - y)).as_independent(x) == (1, (x + n1)(x - y)) assert ((x + n1)(x - y)).as_independent(y) == (x + n1, x - y) assert (DiracDelta(x - n1)DiracDelta(x - y)).as_independent(x) \ == (1, DiracDelta(x - n1)DiracDelta(x - y)) assert (xyn1n2n3).as_independent(n2) == (xyn1, n2n3) assert (xyn1n2n3).as_independent(n1) == (xy, n1n2n3) assert (xyn1n2n3).as_independent(n3) == (xyn1n2, n3) assert (DiracDelta(x - n1)DiracDelta(y - n1)DiracDelta(x - n2)).as_independent(y) == \ (DiracDelta(x - n1)DiracDelta(x - n2), DiracDelta(y - n1)) # issue 5784 assert (x + Integral(x, (x, 1, 2))).as_independent(x, strict=True) == \ (Integral(x, (x, 1, 2)), x) eq = Add(x, -x, 2, -3, evaluate=False) assert eq.as_independent(x) == (-1, Add(x, -x, evaluate=False)) eq = Mul(x, 1/x, 2, -3, evaluate=False) assert eq.as_independent(x) == (-6, Mul(x, 1/x, evaluate=False)) assert (xy).as_independent(z, as_Add=True) == (xy, 0) @XFAIL def test_call_2(): # TODO UndefinedFunction does not subclass Expr assert (2f)(x) == 2f(x) def test_replace(): e = log(sin(x)) + tan(sin(x2)) assert e.replace(sin, cos) == log(cos(x)) + tan(cos(x2)) assert e.replace( sin, lambda a: sin(2a)) == log(sin(2x)) + tan(sin(2x2)) a = Wild('a') b = Wild('b') assert e.replace(sin(a), cos(a)) == log(cos(x)) + tan(cos(x2)) assert e.replace( sin(a), lambda a: sin(2a)) == log(sin(2x)) + tan(sin(2x2)) # test exact assert (2x).replace(ax + b, b - a, exact=True) == 2x assert (2x).replace(ax + b, b - a) == 2x assert (2x).replace(ax + b, b - a, exact=False) == 2/x assert (2x).replace(ax + b, lambda a, b: b - a, exact=True) == 2x assert (2x).replace(ax + b, lambda a, b: b - a) == 2x assert (2x).replace(ax + b, lambda a, b: b - a, exact=False) == 2/x g = 2sin(x3) assert g.replace( lambda expr: expr.is_Number, lambda expr: expr2) == 4sin(x9) assert cos(x).replace(cos, sin, map=True) == (sin(x), {cos(x): sin(x)}) assert sin(x).replace(cos, sin) == sin(x) cond, func = lambda x: x.is_Mul, lambda x: 2x assert (xy).replace(cond, func, map=True) == (2xy, {xy: 2xy}) assert (x(1 + xy)).replace(cond, func, map=True) == \ (2x(2xy + 1), {x(2xy + 1): 2x(2xy + 1), xy: 2xy}) assert (ysin(x)).replace(sin, lambda expr: sin(expr)/y, map=True) == \ (sin(x), {sin(x): sin(x)/y}) # if not simultaneous then ysin(x) -> ysin(x)/y = sin(x) -> sin(x)/y assert (ysin(x)).replace(sin, lambda expr: sin(expr)/y, simultaneous=False) == sin(x)/y assert (x2 + O(x3)).replace(Pow, lambda b, e: be/e ) == x2/2 + O(x3) assert (x2 + O(x3)).replace(Pow, lambda b, e: be/e, simultaneous=False) == x2/2 + O(x3) assert (x(xy + 3)).replace(lambda x: x.is_Mul, lambda x: 2 + x) == \ x(xy + 5) + 2 e = (xy + 1)(2xy + 1) + 1 assert e.replace(cond, func, map=True) == ( 2((2xy + 1)(4xy + 1)) + 1, {2xy: 4xy, xy: 2xy, (2xy + 1)(4xy + 1): 2((2xy + 1)(4xy + 1))}) assert x.replace(x, y) == y assert (x + 1).replace(1, 2) == x + 2 # https://groups.google.com/forum/#!topic/sympy/8wCgeC95tz0 n1, n2, n3 = symbols('n1:4', commutative=False) assert (n1f(n2)).replace(f, lambda x: x) == n1n2 assert (n3f(n2)).replace(f, lambda x: x) == n3n2 # issue 16725 assert S.Zero.replace(Wild('x'), 1) == 1 # let the user override the default decision of False assert S.Zero.replace(Wild('x'), 1, exact=True) == 0 def test_find(): expr = (x + y + 2 + sin(3x)) assert expr.find(lambda u: u.is_Integer) == {S(2), S(3)} assert expr.find(lambda u: u.is_Symbol) == {x, y} assert expr.find(lambda u: u.is_Integer, group=True) == {S(2): 1, S(3): 1} assert expr.find(lambda u: u.is_Symbol, group=True) == {x: 2, y: 1} assert expr.find(Integer) == {S(2), S(3)} assert expr.find(Symbol) == {x, y} assert expr.find(Integer, group=True) == {S(2): 1, S(3): 1} assert expr.find(Symbol, group=True) == {x: 2, y: 1} a = Wild('a') expr = sin(sin(x)) + sin(x) + cos(x) + x assert expr.find(lambda u: type(u) is sin) == {sin(x), sin(sin(x))} assert expr.find( lambda u: type(u) is sin, group=True) == {sin(x): 2, sin(sin(x)): 1} assert expr.find(sin(a)) == {sin(x), sin(sin(x))} assert expr.find(sin(a), group=True) == {sin(x): 2, sin(sin(x)): 1} assert expr.find(sin) == {sin(x), sin(sin(x))} assert expr.find(sin, group=True) == {sin(x): 2, sin(sin(x)): 1} def test_count(): expr = (x + y + 2 + sin(3x)) assert expr.count(lambda u: u.is_Integer) == 2 assert expr.count(lambda u: u.is_Symbol) == 3 assert expr.count(Integer) == 2 assert expr.count(Symbol) == 3 assert expr.count(2) == 1 a = Wild('a') assert expr.count(sin) == 1 assert expr.count(sin(a)) == 1 assert expr.count(lambda u: type(u) is sin) == 1 assert f(x).count(f(x)) == 1 assert f(x).diff(x).count(f(x)) == 1 assert f(x).diff(x).count(x) == 2 def test_has_basics(): p = Wild('p') assert sin(x).has(x) assert sin(x).has(sin) assert not sin(x).has(y) assert not sin(x).has(cos) assert f(x).has(x) assert f(x).has(f) assert not f(x).has(y) assert not f(x).has(g) assert f(x).diff(x).has(x) assert f(x).diff(x).has(f) assert f(x).diff(x).has(Derivative) assert not f(x).diff(x).has(y) assert not f(x).diff(x).has(g) assert not f(x).diff(x).has(sin) assert (x2).has(Symbol) assert not (x2).has(Wild) assert (2p).has(Wild) assert not x.has() def test_has_multiple(): f = x2y + sin(2*t + log(z)) assert f.has(x) assert f.has(y) assert f.has(z) assert f.has(t) assert not f.has(u) assert f.has(x, y, z, t) assert f.has(x, y, z, t, u) i = Integer(4400) assert not i.has(x) assert (ix*i).has(x) assert not (iy*i).has(x) assert (iy*i).has(x, y) assert not (iy*i).has(x, z) def test_has_piecewise(): f = (xy + 3/y)*(3 + 2) p = Piecewise((g(x), x < -1), (1, x <= 1), (f, True)) assert p.has(x) assert p.has(y) assert not p.has(z) assert p.has(1) assert p.has(3) assert not p.has(4) assert p.has(f) assert p.has(g) assert not p.has(h) def test_has_iterative(): A, B, C = symbols('A,B,C', commutative=False) f = xgamma(x)sin(x)exp(xy)ABCcos(xAB) assert f.has(x) assert f.has(xy) assert f.has(xsin(x)) assert not f.has(xsin(y)) assert f.has(xA) assert f.has(xAB) assert not f.has(xAC) assert f.has(xABC) assert not f.has(xACB) assert f.has(xsin(x)ABC) assert not f.has(xsin(x)ACB) assert not f.has(xsin(y)ABC) assert f.has(xgamma(x)) assert not f.has(x + sin(x)) assert (x & y & z).has(x & z) def test_has_integrals(): f = Integral(x*2 + sin(xyz), (x, 0, x + y + z)) assert f.has(x + y) assert f.has(x + z) assert f.has(y + z) assert f.has(xy) assert f.has(xz) assert f.has(yz) assert not f.has(2x + y) assert not f.has(2xy) def test_has_tuple(): assert Tuple(x, y).has(x) assert not Tuple(x, y).has(z) assert Tuple(f(x), g(x)).has(x) assert not Tuple(f(x), g(x)).has(y) assert Tuple(f(x), g(x)).has(f) assert Tuple(f(x), g(x)).has(f(x)) # XXX to be deprecated #assert not Tuple(f, g).has(x) #assert Tuple(f, g).has(f) #assert not Tuple(f, g).has(h) assert Tuple(True).has(True) assert Tuple(True).has(S.true) assert not Tuple(True).has(1) def test_has_units(): from sympy.physics.units import m, s assert (xm/s).has(x) assert (xm/s).has(y, z) is False def test_has_polys(): poly = Poly(x2 + xysin(z), x, y, t) assert poly.has(x) assert poly.has(x, y, z) assert poly.has(x, y, z, t) def test_has_physics(): assert FockState((x, y)).has(x) def test_as_poly_as_expr(): f = x2 + 2xy assert f.as_poly().as_expr() == f assert f.as_poly(x, y).as_expr() == f assert (f + sin(x)).as_poly(x, y) is None p = Poly(f, x, y) assert p.as_poly() == p # https://github.com/sympy/sympy/issues/20610 assert S(2).as_poly() is None assert sqrt(2).as_poly(extension=True) is None raises(AttributeError, lambda: Tuple(x, x).as_poly(x)) raises(AttributeError, lambda: Tuple(x * 2, x, y).as_poly(x)) def test_nonzero(): assert bool(S.Zero) is False assert bool(S.One) is True assert bool(x) is True assert bool(x + y) is True assert bool(x - x) is False assert bool(xy) is True assert bool(x1) is True assert bool(x0) is False def test_is_number(): assert Float(3.14).is_number is True assert Integer(737).is_number is True assert Rational(3, 2).is_number is True assert Rational(8).is_number is True assert x.is_number is False assert (2x).is_number is False assert (x + y).is_number is False assert log(2).is_number is True assert log(x).is_number is False assert (2 + log(2)).is_number is True assert (8 + log(2)).is_number is True assert (2 + log(x)).is_number is False assert (8 + log(2) + x).is_number is False assert (1 + x*2/x - x).is_number is True assert Tuple(Integer(1)).is_number is False assert Add(2, x).is_number is False assert Mul(3, 4).is_number is True assert Pow(log(2), 2).is_number is True assert oo.is_number is True g = WildFunction('g') assert g.is_number is False assert (2g).is_number is False assert (x*2).subs(x, 3).is_number is True # test extensibility of .is_number # on subinstances of Basic class A(Basic): pass a = A() assert a.is_number is False def test_as_coeff_add(): assert S(2).as_coeff_add() == (2, ()) assert S(3.0).as_coeff_add() == (0, (S(3.0),)) assert S(-3.0).as_coeff_add() == (0, (S(-3.0),)) assert x.as_coeff_add() == (0, (x,)) assert (x - 1).as_coeff_add() == (-1, (x,)) assert (x + 1).as_coeff_add() == (1, (x,)) assert (x + 2).as_coeff_add() == (2, (x,)) assert (x + y).as_coeff_add(y) == (x, (y,)) assert (3x).as_coeff_add(y) == (3x, ()) # don't do expansion e = (x + y)2 assert e.as_coeff_add(y) == (0, (e,)) def test_as_coeff_mul(): assert S(2).as_coeff_mul() == (2, ()) assert S(3.0).as_coeff_mul() == (1, (S(3.0),)) assert S(-3.0).as_coeff_mul() == (-1, (S(3.0),)) assert S(-3.0).as_coeff_mul(rational=False) == (-S(3.0), ()) assert x.as_coeff_mul() == (1, (x,)) assert (-x).as_coeff_mul() == (-1, (x,)) assert (2x).as_coeff_mul() == (2, (x,)) assert (xy).as_coeff_mul(y) == (x, (y,)) assert (3 + x).as_coeff_mul() == (1, (3 + x,)) assert (3 + x).as_coeff_mul(y) == (3 + x, ()) # don't do expansion e = exp(x + y) assert e.as_coeff_mul(y) == (1, (e,)) e = 2(x + y) assert e.as_coeff_mul(y) == (1, (e,)) assert (1.1x).as_coeff_mul(rational=False) == (1.1, (x,)) assert (1.1x).as_coeff_mul() == (1, (1.1, x)) assert (-oox).as_coeff_mul(rational=True) == (-1, (oo, x)) def test_as_coeff_exponent(): assert (3x4).as_coeff_exponent(x) == (3, 4) assert (2x*3).as_coeff_exponent(x) == (2, 3) assert (4x*2).as_coeff_exponent(x) == (4, 2) assert (6x*1).as_coeff_exponent(x) == (6, 1) assert (3x*0).as_coeff_exponent(x) == (3, 0) assert (2x*0).as_coeff_exponent(x) == (2, 0) assert (1x*0).as_coeff_exponent(x) == (1, 0) assert (0x*0).as_coeff_exponent(x) == (0, 0) assert (-1x*0).as_coeff_exponent(x) == (-1, 0) assert (-2x*0).as_coeff_exponent(x) == (-2, 0) assert (2x*3 + pix*3).as_coeff_exponent(x) == (2 + pi, 3) assert (xlog(2)/(2x + pix)).as_coeff_exponent(x) == \ (log(2)/(2 + pi), 0) # issue 4784 D = Derivative fx = D(f(x), x) assert fx.as_coeff_exponent(f(x)) == (fx, 0) def test_extractions(): for base in (2, S.Exp1): assert Pow(basex, 3, evaluate=False ).extract_multiplicatively(basex) == base*(2x) assert (base*(5x)).extract_multiplicatively( base*(3x)) == base*(2x) assert ((xy)3).extract_multiplicatively(x2 y) == xy2 assert ((xy)3).extract_multiplicatively(x4 * y) is None assert (2x).extract_multiplicatively(2) == x assert (2x).extract_multiplicatively(3) is None assert (2x).extract_multiplicatively(-1) is None assert (S.Halfx).extract_multiplicatively(3) == x/6 assert (sqrt(x)).extract_multiplicatively(x) is None assert (sqrt(x)).extract_multiplicatively(1/x) is None assert x.extract_multiplicatively(-x) is None assert (-2 - 4I).extract_multiplicatively(-2) == 1 + 2I assert (-2 - 4I).extract_multiplicatively(3) is None assert (-2x - 4y - 8).extract_multiplicatively(-2) == x + 2y + 4 assert (-2xy - 4x2y).extract_multiplicatively(-2y) == 2x*2 + x assert (2xy + 4x*2y).extract_multiplicatively(2y) == 2x*2 + x assert (-4y*2x).extract_multiplicatively(-3y) is None assert (2x).extract_multiplicatively(1) == 2x assert (-oo).extract_multiplicatively(5) is -oo assert (oo).extract_multiplicatively(5) is oo assert ((xy)*3).extract_additively(1) is None assert (x + 1).extract_additively(x) == 1 assert (x + 1).extract_additively(2x) is None assert (x + 1).extract_additively(-x) is None assert (-x + 1).extract_additively(2x) is None assert (2x + 3).extract_additively(x) == x + 3 assert (2x + 3).extract_additively(2) == 2x + 1 assert (2x + 3).extract_additively(3) == 2x assert (2x + 3).extract_additively(-2) is None assert (2x + 3).extract_additively(3x) is None assert (2x + 3).extract_additively(2x) == 3 assert x.extract_additively(0) == x assert S(2).extract_additively(x) is None assert S(2.).extract_additively(2) is S.Zero assert S(2x + 3).extract_additively(x + 1) == x + 2 assert S(2x + 3).extract_additively(y + 1) is None assert S(2x - 3).extract_additively(x + 1) is None assert S(2x - 3).extract_additively(y + z) is None assert ((a + 1)x4 + y).extract_additively(x).expand() == \ 4ax + 3x + y assert ((a + 1)x4 + 3y).extract_additively(x + 2y).expand() == \ 4ax + 3x + y assert (y(x + 1)).extract_additively(x + 1) is None assert ((y + 1)(x + 1) + 3).extract_additively(x + 1) == \ y(x + 1) + 3 assert ((x + y)(x + 1) + x + y + 3).extract_additively(x + y) == \ x(x + y) + 3 assert (x + y + 2((x + y)(x + 1)) + 3).extract_additively((x + y)(x + 1)) == \ x + y + (x + 1)(x + y) + 3 assert ((y + 1)(x + 2y + 1) + 3).extract_additively(y + 1) == \ (x + 2y)(y + 1) + 3 assert (-x - xI).extract_additively(-x) == -Ix # extraction does not leave artificats, now assert (4x(y + 1) + y).extract_additively(x) == x(4y + 3) + y n = Symbol("n", integer=True) assert (Integer(-3)).could_extract_minus_sign() is True assert (-nx + x).could_extract_minus_sign() != \ (nx - x).could_extract_minus_sign() assert (x - y).could_extract_minus_sign() != \ (-x + y).could_extract_minus_sign() assert (1 - x - y).could_extract_minus_sign() is True assert (1 - x + y).could_extract_minus_sign() is False assert ((-x - xy)/y).could_extract_minus_sign() is False assert ((x + xy)/(-y)).could_extract_minus_sign() is True assert ((x + xy)/y).could_extract_minus_sign() is False assert ((-x - y)/(x + y)).could_extract_minus_sign() is False class sign_invariant(Function, Expr): nargs = 1 def __neg__(self): return self foo = sign_invariant(x) assert foo == -foo assert foo.could_extract_minus_sign() is False assert (x - y).could_extract_minus_sign() is False assert (-x + y).could_extract_minus_sign() is True assert (x - 1).could_extract_minus_sign() is False assert (1 - x).could_extract_minus_sign() is True assert (sqrt(2) - 1).could_extract_minus_sign() is True assert (1 - sqrt(2)).could_extract_minus_sign() is False # check that result is canonical eq = (3x + 15y).extract_multiplicatively(3) assert eq.args == eq.func(eq.args).args def test_nan_extractions(): for r in (1, 0, I, nan): assert nan.extract_additively(r) is None assert nan.extract_multiplicatively(r) is None def test_coeff(): assert (x + 1).coeff(x + 1) == 1 assert (3x).coeff(0) == 0 assert (z(1 + x)x2).coeff(1 + x) == zx*2 assert (1 + 2xx(1 + x)).coeff(xx*(1 + x)) == 2 assert (1 + 2x(y + z)).coeff(x(y + z)) == 2 assert (3 + 2x + 4x*2).coeff(1) == 0 assert (3 + 2x + 4x2).coeff(-1) == 0 assert (3 + 2x + 4x2).coeff(x) == 2 assert (3 + 2x + 4x2).coeff(x2) == 4 assert (3 + 2x + 4x2).coeff(x3) == 0 assert (-x/8 + xy).coeff(x) == Rational(-1, 8) + y assert (-x/8 + xy).coeff(-x) == S.One/8 assert (4x).coeff(2x) == 0 assert (2x).coeff(2x) == 1 assert (-oox).coeff(xoo) == -1 assert (10x).coeff(x, 0) == 0 assert (10x).coeff(10x, 0) == 0 n1, n2 = symbols('n1 n2', commutative=False) assert (n1n2).coeff(n1) == 1 assert (n1n2).coeff(n2) == n1 assert (n1n2 + xn1).coeff(n1) == 1 # 1n1(n2+x) assert (n2n1 + xn1).coeff(n1) == n2 + x assert (n2n1 + xn12).coeff(n1) == n2 assert (n1x).coeff(n1) == 0 assert (n1n2 + n2n1).coeff(n1) == 0 assert (2(n1 + n2)n2).coeff(n1 + n2, right=1) == n2 assert (2(n1 + n2)n2).coeff(n1 + n2, right=0) == 2 assert (2f(x) + 3f(x).diff(x)).coeff(f(x)) == 2 expr = z(x + y)2 expr2 = z(x + y)*2 + z(2x + 2y)2 assert expr.coeff(z) == (x + y)2 assert expr.coeff(x + y) == 0 assert expr2.coeff(z) == (x + y)*2 + (2x + 2y)2 assert (x + y + 3z).coeff(1) == x + y assert (-x + 2y).coeff(-1) == x assert (x - 2y).coeff(-1) == 2y assert (3 + 2x + 4x2).coeff(1) == 0 assert (-x - 2y).coeff(2) == -y assert (x + sqrt(2)x).coeff(sqrt(2)) == x assert (3 + 2x + 4x2).coeff(x) == 2 assert (3 + 2x + 4x2).coeff(x2) == 4 assert (3 + 2x + 4x2).coeff(x3) == 0 assert (z(x + y)2).coeff((x + y)2) == z assert (z(x + y)2).coeff(x + y) == 0 assert (2 + 2x + (x + 1)y).coeff(x + 1) == y assert (x + 2y + 3).coeff(1) == x assert (x + 2y + 3).coeff(x, 0) == 2y + 3 assert (x*2 + 2y + 3x).coeff(x2, 0) == 2y + 3x assert x.coeff(0, 0) == 0 assert x.coeff(x, 0) == 0 n, m, o, l = symbols('n m o l', commutative=False) assert n.coeff(n) == 1 assert y.coeff(n) == 0 assert (3n).coeff(n) == 3 assert (2 + n).coeff(xm) == 0 assert (2xnm).coeff(x) == 2nm assert (2 + n).coeff(xmn + y) == 0 assert (2xnm).coeff(3n) == 0 assert (nm + mnm).coeff(n) == 1 + m assert (nm + mnm).coeff(n, right=True) == m # = (1 + m)nm assert (nm + mn).coeff(n) == 0 assert (nm + omn).coeff(mn) == o assert (nm + omn).coeff(mn, right=True) == 1 assert (nm + nmn).coeff(nm, right=True) == 1 + n # = nm(n + 1) assert (xy).coeff(z, 0) == xy assert (xn + yn + zm).coeff(n) == x + y assert (nm + no + ol).coeff(n, right=True) == m + o assert (xnmn + ynmo + zl).coeff(m, right=True) == xn + yo assert (xnmn + xnmo + zl).coeff(m, right=True) == n + o assert (xnmn + xnmo + zl).coeff(m) == xn def test_coeff2(): r, kappa = symbols('r, kappa') psi = Function("psi") g = 1/r*2 (2rpsi(r).diff(r, 1) + r*2 psi(r).diff(r, 2)) g = g.expand() assert g.coeff(psi(r).diff(r)) == 2/r def test_coeff2_0(): r, kappa = symbols('r, kappa') psi = Function("psi") g = 1/r*2 (2rpsi(r).diff(r, 1) + r*2 psi(r).diff(r, 2)) g = g.expand() assert g.coeff(psi(r).diff(r, 2)) == 1 def test_coeff_expand(): expr = z(x + y)2 expr2 = z(x + y)*2 + z(2x + 2y)2 assert expr.coeff(z) == (x + y)2 assert expr2.coeff(z) == (x + y)*2 + (2x + 2y)2 def test_integrate(): assert x.integrate(x) == x2/2 assert x.integrate((x, 0, 1)) == S.Half def test_as_base_exp(): assert x.as_base_exp() == (x, S.One) assert (xyz).as_base_exp() == (xyz, S.One) assert (x + y + z).as_base_exp() == (x + y + z, S.One) assert ((x + y)z).as_base_exp() == (x + y, z) def test_issue_4963(): assert hasattr(Mul(x, y), "is_commutative") assert hasattr(Mul(x, y, evaluate=False), "is_commutative") assert hasattr(Pow(x, y), "is_commutative") assert hasattr(Pow(x, y, evaluate=False), "is_commutative") expr = Mul(Pow(2, 2, evaluate=False), 3, evaluate=False) + 1 assert hasattr(expr, "is_commutative") def test_action_verbs(): assert nsimplify(1/(exp(3pix/5) + 1)) == \ (1/(exp(3pix/5) + 1)).nsimplify() assert ratsimp(1/x + 1/y) == (1/x + 1/y).ratsimp() assert trigsimp(log(x), deep=True) == (log(x)).trigsimp(deep=True) assert radsimp(1/(2 + sqrt(2))) == (1/(2 + sqrt(2))).radsimp() assert radsimp(1/(a + bsqrt(c)), symbolic=False) == \ (1/(a + bsqrt(c))).radsimp(symbolic=False) assert powsimp(xyx*zyz, combine='all') == \ (xyxzyz).powsimp(combine='all') assert (xtyt).powsimp(force=True) == (xy)t assert simplify(xyxzyz) == (xyxzy*z).simplify() assert together(1/x + 1/y) == (1/x + 1/y).together() assert collect(ax*2 + bx*2 + ax - bx + c, x) == \ (ax*2 + bx*2 + ax - bx + c).collect(x) assert apart(y/(y + 2)/(y + 1), y) == (y/(y + 2)/(y + 1)).apart(y) assert combsimp(y/(x + 2)/(x + 1)) == (y/(x + 2)/(x + 1)).combsimp() assert gammasimp(gamma(x)/gamma(x-5)) == (gamma(x)/gamma(x-5)).gammasimp() assert factor(x2 + 5x + 6) == (x*2 + 5x + 6).factor() assert refine(sqrt(x2)) == sqrt(x2).refine() assert cancel((x*2 + 5x + 6)/(x + 2)) == ((x*2 + 5x + 6)/(x + 2)).cancel() def test_as_powers_dict(): assert x.as_powers_dict() == {x: 1} assert (x*yz).as_powers_dict() == {x: y, z: 1} assert Mul(2, 2, evaluate=False).as_powers_dict() == {S(2): S(2)} assert (xy).as_powers_dict()[z] == 0 assert (x + y).as_powers_dict()[z] == 0 def test_as_coefficients_dict(): check = [S.One, x, y, xy, 1] assert [Add(3x, 2x, y, 3).as_coefficients_dict()[i] for i in check] == \ [3, 5, 1, 0, 3] assert [Add(3x, 2x, y, 3, evaluate=False).as_coefficients_dict()[i] for i in check] == [3, 5, 1, 0, 3] assert [(3xy).as_coefficients_dict()[i] for i in check] == \ [0, 0, 0, 3, 0] assert [(3.0xy).as_coefficients_dict()[i] for i in check] == \ [0, 0, 0, 3.0, 0] assert (3.0xy).as_coefficients_dict()[3.0xy] == 0 def test_args_cnc(): A = symbols('A', commutative=False) assert (x + A).args_cnc() == \ [[], [x + A]] assert (x + a).args_cnc() == \ [[a + x], []] assert (xa).args_cnc() == \ [[a, x], []] assert (xyA(A + 1)).args_cnc(cset=True) == \ [{x, y}, [A, 1 + A]] assert Mul(x, x, evaluate=False).args_cnc(cset=True, warn=False) == \ [{x}, []] assert Mul(x, x2, evaluate=False).args_cnc(cset=True, warn=False) == \ [{x, x2}, []] raises(ValueError, lambda: Mul(x, x, evaluate=False).args_cnc(cset=True)) assert Mul(x, y, x, evaluate=False).args_cnc() == \ [[x, y, x], []] # always split -1 from leading number assert (-1.x).args_cnc() == [[-1, 1.0, x], []] def test_new_rawargs(): n = Symbol('n', commutative=False) a = x + n assert a.is_commutative is False assert a._new_rawargs(x).is_commutative assert a._new_rawargs(x, y).is_commutative assert a._new_rawargs(x, n).is_commutative is False assert a._new_rawargs(x, y, n).is_commutative is False m = xn assert m.is_commutative is False assert m._new_rawargs(x).is_commutative assert m._new_rawargs(n).is_commutative is False assert m._new_rawargs(x, y).is_commutative assert m._new_rawargs(x, n).is_commutative is False assert m._new_rawargs(x, y, n).is_commutative is False assert m._new_rawargs(x, n, reeval=False).is_commutative is False assert m._new_rawargs(S.One) is S.One def test_issue_5226(): assert Add(evaluate=False) == 0 assert Mul(evaluate=False) == 1 assert Mul(x + y, evaluate=False).is_Add def test_free_symbols(): # free_symbols should return the free symbols of an object assert S.One.free_symbols == set() assert x.free_symbols == {x} assert Integral(x, (x, 1, y)).free_symbols == {y} assert (-Integral(x, (x, 1, y))).free_symbols == {y} assert meter.free_symbols == set() assert (meter*x).free_symbols == {x} def test_has_free(): assert x.has_free(x) assert not x.has_free(y) assert (x + y).has_free(x) assert (x + y).has_free((x, z)) assert f(x).has_free(x) assert f(x).has_free(f(x)) assert Integral(f(x), (f(x), 1, y)).has_free(y) assert not Integral(f(x), (f(x), 1, y)).has_free(x) assert not Integral(f(x), (f(x), 1, y)).has_free(f(x)) def test_issue_5300(): x = Symbol('x', commutative=False) assert xsqrt(2)/sqrt(6) == xsqrt(3)/3 def test_floordiv(): from sympy.functions.elementary.integers import floor assert x // y == floor(x / y) def test_as_coeff_Mul(): assert Integer(3).as_coeff_Mul() == (Integer(3), Integer(1)) assert Rational(3, 4).as_coeff_Mul() == (Rational(3, 4), Integer(1)) assert Float(5.0).as_coeff_Mul() == (Float(5.0), Integer(1)) assert (Integer(3)x).as_coeff_Mul() == (Integer(3), x) assert (Rational(3, 4)x).as_coeff_Mul() == (Rational(3, 4), x) assert (Float(5.0)x).as_coeff_Mul() == (Float(5.0), x) assert (Integer(3)xy).as_coeff_Mul() == (Integer(3), xy) assert (Rational(3, 4)xy).as_coeff_Mul() == (Rational(3, 4), xy) assert (Float(5.0)xy).as_coeff_Mul() == (Float(5.0), xy) assert (x).as_coeff_Mul() == (S.One, x) assert (xy).as_coeff_Mul() == (S.One, xy) assert (-oox).as_coeff_Mul(rational=True) == (-1, oox) def test_as_coeff_Add(): assert Integer(3).as_coeff_Add() == (Integer(3), Integer(0)) assert Rational(3, 4).as_coeff_Add() == (Rational(3, 4), Integer(0)) assert Float(5.0).as_coeff_Add() == (Float(5.0), Integer(0)) assert (Integer(3) + x).as_coeff_Add() == (Integer(3), x) assert (Rational(3, 4) + x).as_coeff_Add() == (Rational(3, 4), x) assert (Float(5.0) + x).as_coeff_Add() == (Float(5.0), x) assert (Float(5.0) + x).as_coeff_Add(rational=True) == (0, Float(5.0) + x) assert (Integer(3) + x + y).as_coeff_Add() == (Integer(3), x + y) assert (Rational(3, 4) + x + y).as_coeff_Add() == (Rational(3, 4), x + y) assert (Float(5.0) + x + y).as_coeff_Add() == (Float(5.0), x + y) assert (x).as_coeff_Add() == (S.Zero, x) assert (xy).as_coeff_Add() == (S.Zero, xy) def test_expr_sorting(): exprs = [1/x2, 1/x, sqrt(sqrt(x)), sqrt(x), x, sqrt(x)3, x*2] assert sorted(exprs, key=default_sort_key) == exprs exprs = [x, 2x, 2x2, 2x3, xn, 2xn, sin(x), sin(x)n, sin(x2), cos(x), cos(x2), tan(x)] assert sorted(exprs, key=default_sort_key) == exprs exprs = [x + 1, x2 + x + 1, x3 + x2 + x + 1] assert sorted(exprs, key=default_sort_key) == exprs exprs = [S(4), x - 3I/2, x + 3I/2, x - 4I + 1, x + 4I + 1] assert sorted(exprs, key=default_sort_key) == exprs exprs = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)] assert sorted(exprs, key=default_sort_key) == exprs exprs = [f(x), g(x), exp(x), sin(x), cos(x), factorial(x)] assert sorted(exprs, key=default_sort_key) == exprs exprs = [Tuple(x, y), Tuple(x, z), Tuple(x, y, z)] assert sorted(exprs, key=default_sort_key) == exprs exprs = [[3], [1, 2]] assert sorted(exprs, key=default_sort_key) == exprs exprs = [[1, 2], [2, 3]] assert sorted(exprs, key=default_sort_key) == exprs exprs = [[1, 2], [1, 2, 3]] assert sorted(exprs, key=default_sort_key) == exprs exprs = [{x: -y}, {x: y}] assert sorted(exprs, key=default_sort_key) == exprs exprs = [{1}, {1, 2}] assert sorted(exprs, key=default_sort_key) == exprs a, b = exprs = [Dummy('x'), Dummy('x')] assert sorted([b, a], key=default_sort_key) == exprs def test_as_ordered_factors(): assert x.as_ordered_factors() == [x] assert (2xxnsin(x)cos(x)).as_ordered_factors() \ == [Integer(2), x, xn, sin(x), cos(x)] args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)] expr = Mul(args) assert expr.as_ordered_factors() == args A, B = symbols('A,B', commutative=False) assert (AB).as_ordered_factors() == [A, B] assert (BA).as_ordered_factors() == [B, A] def test_as_ordered_terms(): assert x.as_ordered_terms() == [x] assert (sin(x)*2cos(x) + sin(x)cos(x)2 + 1).as_ordered_terms() \ == [sin(x)2cos(x), sin(x)cos(x)2, 1] args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)] expr = Add(args) assert expr.as_ordered_terms() == args assert (1 + 4sqrt(3)pix).as_ordered_terms() == [4pixsqrt(3), 1] assert ( 2 + 3I).as_ordered_terms() == [2, 3I] assert (-2 + 3I).as_ordered_terms() == [-2, 3I] assert ( 2 - 3I).as_ordered_terms() == [2, -3I] assert (-2 - 3I).as_ordered_terms() == [-2, -3I] assert ( 4 + 3I).as_ordered_terms() == [4, 3I] assert (-4 + 3I).as_ordered_terms() == [-4, 3I] assert ( 4 - 3I).as_ordered_terms() == [4, -3I] assert (-4 - 3I).as_ordered_terms() == [-4, -3I] e = x*2y*2 + xy4 + y + 2 assert e.as_ordered_terms(order="lex") == [x2y2, xy*4, y, 2] assert e.as_ordered_terms(order="grlex") == [xy4, x2y2, y, 2] assert e.as_ordered_terms(order="rev-lex") == [2, y, xy4, x2y2] assert e.as_ordered_terms(order="rev-grlex") == [2, y, x2y*2, xy*4] k = symbols('k') assert k.as_ordered_terms(data=True) == ([(k, ((1.0, 0.0), (1,), ()))], [k]) def test_sort_key_atomic_expr(): from sympy.physics.units import m, s assert sorted([-m, s], key=lambda arg: arg.sort_key()) == [-m, s] def test_eval_interval(): assert exp(x)._eval_interval(Tuple(x, 0, 1)) == exp(1) - exp(0) # issue 4199 a = x/y raises(NotImplementedError, lambda: a._eval_interval(x, S.Zero, oo)._eval_interval(y, oo, S.Zero)) raises(NotImplementedError, lambda: a._eval_interval(x, S.Zero, oo)._eval_interval(y, S.Zero, oo)) a = x - y raises(NotImplementedError, lambda: a._eval_interval(x, S.One, oo)._eval_interval(y, oo, S.One)) raises(ValueError, lambda: x._eval_interval(x, None, None)) a = -yHeaviside(x - y) assert a._eval_interval(x, -oo, oo) == -y assert a._eval_interval(x, oo, -oo) == y def test_eval_interval_zoo(): # Test that limit is used when zoo is returned assert Si(1/x)._eval_interval(x, S.Zero, S.One) == -pi/2 + Si(1) def test_primitive(): assert (3(x + 1)2).primitive() == (3, (x + 1)2) assert (6x + 2).primitive() == (2, 3x + 1) assert (x/2 + 3).primitive() == (S.Half, x + 6) eq = (6x + 2)(x/2 + 3) assert eq.primitive()[0] == 1 eq = (2 + 2x)2 assert eq.primitive()[0] == 1 assert (4.0x).primitive() == (1, 4.0x) assert (4.0x + y/2).primitive() == (S.Half, 8.0x + y) assert (-2x).primitive() == (2, -x) assert Add(5z/7, 0.5x, 3y/2, evaluate=False).primitive() == \ (S.One/14, 7.0x + 21y + 10z) for i in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]: assert (i + x/3).primitive() == \ (S.One/3, i + x) assert (S.Infinity + 2x/3 + 4y/7).primitive() == \ (S.One/21, 14x + 12y + oo) assert S.Zero.primitive() == (S.One, S.Zero) def test_issue_5843(): a = 1 + x assert (2a).extract_multiplicatively(a) == 2 assert (4a).extract_multiplicatively(2a) == 2 assert ((3a)(2a)).extract_multiplicatively(a) == 6a def test_is_constant(): from sympy.solvers.solvers import checksol assert Sum(x, (x, 1, 10)).is_constant() is True assert Sum(x, (x, 1, n)).is_constant() is False assert Sum(x, (x, 1, n)).is_constant(y) is True assert Sum(x, (x, 1, n)).is_constant(n) is False assert Sum(x, (x, 1, n)).is_constant(x) is True eq = acos(x)*2 + asin(x)2 - a assert eq.is_constant() is True assert eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 assert x.is_constant() is False assert x.is_constant(y) is True assert log(x/y).is_constant() is False assert checksol(x, x, Sum(x, (x, 1, n))) is False assert checksol(x, x, Sum(x, (x, 1, n))) is False assert f(1).is_constant assert checksol(x, x, f(x)) is False assert Pow(x, S.Zero, evaluate=False).is_constant() is True # == 1 assert Pow(S.Zero, x, evaluate=False).is_constant() is False # == 0 or 1 assert (2x).is_constant() is False assert Pow(S(2), S(3), evaluate=False).is_constant() is True z1, z2 = symbols('z1 z2', zero=True) assert (z1 + 2z2).is_constant() is True assert meter.is_constant() is True assert (3meter).is_constant() is True assert (xmeter).is_constant() is False def test_equals(): assert (-3 - sqrt(5) + (-sqrt(10)/2 - sqrt(2)/2)2).equals(0) assert (x2 - 1).equals((x + 1)(x - 1)) assert (cos(x)2 + sin(x)2).equals(1) assert (acos(x)2 + asin(x)*2).equals(a) r = sqrt(2) assert (-1/(r + rx) + 1/r/(1 + x)).equals(0) assert factorial(x + 1).equals((x + 1)factorial(x)) assert sqrt(3).equals(2sqrt(3)) is False assert (sqrt(5)sqrt(3)).equals(sqrt(3)) is False assert (sqrt(5) + sqrt(3)).equals(0) is False assert (sqrt(5) + pi).equals(0) is False assert meter.equals(0) is False assert (3meter2).equals(0) is False eq = -(-1)(S(3)/4)6(S.One/4) + (-6)(S.One/4)I if eq != 0: # if canonicalization makes this zero, skip the test assert eq.equals(0) assert sqrt(x).equals(0) is False # from integrate(xsqrt(1 + 2x), x); # diff is zero only when assumptions allow i = 2sqrt(2)x*(S(5)/2)(1 + 1/(2x))(S(5)/2)/5 + \ 2sqrt(2)x(S(3)/2)(1 + 1/(2x))(S(5)/2)/(-6 - 3/x) ans = sqrt(2x + 1)(6x*2 + x - 1)/15 diff = i - ans assert diff.equals(0) is None # should be False, but previously this was False due to wrong intermediate result assert diff.subs(x, Rational(-1, 2)/2) == 7sqrt(2)/120 # there are regions for x for which the expression is True, for # example, when x < -1/2 or x > 0 the expression is zero p = Symbol('p', positive=True) assert diff.subs(x, p).equals(0) is True assert diff.subs(x, -1).equals(0) is True # prove via minimal_polynomial or self-consistency eq = sqrt(1 + sqrt(3)) + sqrt(3 + 3sqrt(3)) - sqrt(10 + 6sqrt(3)) assert eq.equals(0) q = 3Rational(1, 3) + 3 p = expand(q3)*Rational(1, 3) assert (p - q).equals(0) # issue 6829 # eq = qx + q/4 + x4 + x3 + 2x2 - S.One/3 # z = eq.subs(x, solve(eq, x)[0]) q = symbols('q') z = (q(-sqrt(-2(-(q - S(7)/8)S(2)/8 - S(2197)/13824)(S.One/3) - S(13)/12)/2 - sqrt((2q - S(7)/4)/sqrt(-2(-(q - S(7)/8)S(2)/8 - S(2197)/13824)(S.One/3) - S(13)/12) + 2(-(q - S(7)/8)S(2)/8 - S(2197)/13824)(S.One/3) - S(13)/6)/2 - S.One/4) + q/4 + (-sqrt(-2(-(q - S(7)/8)S(2)/8 - S(2197)/13824)(S.One/3) - S(13)/12)/2 - sqrt((2q - S(7)/4)/sqrt(-2(-(q - S(7)/8)S(2)/8 - S(2197)/13824)(S.One/3) - S(13)/12) + 2(-(q - S(7)/8)S(2)/8 - S(2197)/13824)(S.One/3) - S(13)/6)/2 - S.One/4)*4 + (-sqrt(-2(-(q - S(7)/8)S(2)/8 - S(2197)/13824)(S.One/3) - S(13)/12)/2 - sqrt((2q - S(7)/4)/sqrt(-2(-(q - S(7)/8)S(2)/8 - S(2197)/13824)(S.One/3) - S(13)/12) + 2(-(q - S(7)/8)S(2)/8 - S(2197)/13824)(S.One/3) - S(13)/6)/2 - S.One/4)3 + 2(-sqrt(-2(-(q - S(7)/8)S(2)/8 - S(2197)/13824)(S.One/3) - S(13)/12)/2 - sqrt((2q - S(7)/4)/sqrt(-2(-(q - S(7)/8)S(2)/8 - S(2197)/13824)(S.One/3) - S(13)/12) + 2(-(q - S(7)/8)S(2)/8 - S(2197)/13824)(S.One/3) - S(13)/6)/2 - S.One/4)*2 - Rational(1, 3)) assert z.equals(0) def test_random(): from sympy.functions.combinatorial.numbers import lucas from sympy.simplify.simplify import posify assert posify(x)[0]._random() is not None assert lucas(n)._random(2, -2, 0, -1, 1) is None # issue 8662 assert Piecewise((Max(x, y), z))._random() is None def test_round(): assert str(Float('0.1249999').round(2)) == '0.12' d20 = 12345678901234567890 ans = S(d20).round(2) assert ans.is_Integer and ans == d20 ans = S(d20).round(-2) assert ans.is_Integer and ans == 12345678901234567900 assert str(S('1/7').round(4)) == '0.1429' assert str(S('.[12345]').round(4)) == '0.1235' assert str(S('.1349').round(2)) == '0.13' n = S(12345) ans = n.round() assert ans.is_Integer assert ans == n ans = n.round(1) assert ans.is_Integer assert ans == n ans = n.round(4) assert ans.is_Integer assert ans == n assert n.round(-1) == 12340 r = Float(str(n)).round(-4) assert r == 10000 assert n.round(-5) == 0 assert str((pi + sqrt(2)).round(2)) == '4.56' assert (10(pi + sqrt(2))).round(-1) == 50 raises(TypeError, lambda: round(x + 2, 2)) assert str(S(2.3).round(1)) == '2.3' # rounding in SymPy (as in Decimal) should be # exact for the given precision; we check here # that when a 5 follows the last digit that # the rounded digit will be even. for i in range(-99, 100): # construct a decimal that ends in 5, e.g. 123 -> 0.1235 s = str(abs(i)) p = len(s) # we are going to round to the last digit of i n = '0.%s5' % s # put a 5 after i's digits j = p + 2 # 2 for '0.' if i < 0: # 1 for '-' j += 1 n = '-' + n v = str(Float(n).round(p))[:j] # pertinent digits if v.endswith('.'): continue # it ends with 0 which is even L = int(v[-1]) # last digit assert L % 2 == 0, (n, '->', v) assert (Float(.3, 3) + 2pi).round() == 7 assert (Float(.3, 3) + 2pi100).round() == 629 assert (pi + 2EI).round() == 3 + 5I # don't let request for extra precision give more than # what is known (in this case, only 3 digits) assert str((Float(.03, 3) + 2pi/100).round(5)) == '0.0928' assert str((Float(.03, 3) + 2pi/100).round(4)) == '0.0928' assert S.Zero.round() == 0 a = (Add(1, Float('1.' + '9'27, ''), evaluate=0)) assert a.round(10) == Float('3.0000000000', '') assert a.round(25) == Float('3.0000000000000000000000000', '') assert a.round(26) == Float('3.00000000000000000000000000', '') assert a.round(27) == Float('2.999999999999999999999999999', '') assert a.round(30) == Float('2.999999999999999999999999999', '') raises(TypeError, lambda: x.round()) raises(TypeError, lambda: f(1).round()) # exact magnitude of 10 assert str(S.One.round()) == '1' assert str(S(100).round()) == '100' # applied to real and imaginary portions assert (2pi + EI).round() == 6 + 3I assert (2pi + I/10).round() == 6 assert (pi/10 + 2I).round() == 2I # the lhs re and im parts are Float with dps of 2 # and those on the right have dps of 15 so they won't compare # equal unless we use string or compare components (which will # then coerce the floats to the same precision) or re-create # the floats assert str((pi/10 + EI).round(2)) == '0.31 + 2.72I' assert str((pi/10 + EI).round(2).as_real_imag()) == '(0.31, 2.72)' assert str((pi/10 + EI).round(2)) == '0.31 + 2.72I' # issue 6914 assert (I*(I + 3)).round(3) == Float('-0.208', '')I # issue 8720 assert S(-123.6).round() == -124 assert S(-1.5).round() == -2 assert S(-100.5).round() == -100 assert S(-1.5 - 10.5I).round() == -2 - 10I # issue 7961 assert str(S(0.006).round(2)) == '0.01' assert str(S(0.00106).round(4)) == '0.0011' # issue 8147 assert S.NaN.round() is S.NaN assert S.Infinity.round() is S.Infinity assert S.NegativeInfinity.round() is S.NegativeInfinity assert S.ComplexInfinity.round() is S.ComplexInfinity # check that types match for i in range(2): fi = float(i) # 2 args assert all(type(round(i, p)) is int for p in (-1, 0, 1)) assert all(S(i).round(p).is_Integer for p in (-1, 0, 1)) assert all(type(round(fi, p)) is float for p in (-1, 0, 1)) assert all(S(fi).round(p).is_Float for p in (-1, 0, 1)) # 1 arg (p is None) assert type(round(i)) is int assert S(i).round().is_Integer assert type(round(fi)) is int assert S(fi).round().is_Integer def test_held_expression_UnevaluatedExpr(): x = symbols("x") he = UnevaluatedExpr(1/x) e1 = xhe assert isinstance(e1, Mul) assert e1.args == (x, he) assert e1.doit() == 1 assert UnevaluatedExpr(Derivative(x, x)).doit(deep=False ) == Derivative(x, x) assert UnevaluatedExpr(Derivative(x, x)).doit() == 1 xx = Mul(x, x, evaluate=False) assert xx != x2 ue2 = UnevaluatedExpr(xx) assert isinstance(ue2, UnevaluatedExpr) assert ue2.args == (xx,) assert ue2.doit() == x2 assert ue2.doit(deep=False) == xx x2 = UnevaluatedExpr(2)2 assert type(x2) is Mul assert x2.args == (2, UnevaluatedExpr(2)) def test_round_exception_nostr(): # Don't use the string form of the expression in the round exception, as # it's too slow s = Symbol('bad') try: s.round() except TypeError as e: assert 'bad' not in str(e) else: # Did not raise raise AssertionError("Did not raise") def test_extract_branch_factor(): assert exp_polar(2.0Ipi).extract_branch_factor() == (1, 1) def test_identity_removal(): assert Add.make_args(x + 0) == (x,) assert Mul.make_args(x1) == (x,) def test_float_0(): assert Float(0.0) + 1 == Float(1.0) @XFAIL def test_float_0_fail(): assert Float(0.0)x == Float(0.0) assert (x + Float(0.0)).is_Add def test_issue_6325(): ans = (b2 + z2 - (b(a + bt) + z(c + tz))*2/( (a + bt)*2 + (c + tz)*2))/sqrt((a + bt)*2 + (c + tz)*2) e = sqrt((a + bt)*2 + (c + zt)*2) assert diff(e, t, 2) == ans assert e.diff(t, 2) == ans assert diff(e, t, 2, simplify=False) != ans def test_issue_7426(): f1 = a % c f2 = x % z assert f1.equals(f2) is None def test_issue_11122(): x = Symbol('x', extended_positive=False) assert unchanged(Gt, x, 0) # (x > 0) # (x > 0) should remain unevaluated after PR #16956 x = Symbol('x', positive=False, real=True) assert (x > 0) is S.false def test_issue_10651(): x = Symbol('x', real=True) e1 = (-1 + x)/(1 - x) e3 = (4x*2 - 4)/((1 - x)(1 + x)) e4 = 1/(cos(x)2) - (tan(x))2 x = Symbol('x', positive=True) e5 = (1 + x)/x assert e1.is_constant() is None assert e3.is_constant() is None assert e4.is_constant() is None assert e5.is_constant() is False def test_issue_10161(): x = symbols('x', real=True) assert xabs(x)abs(x) == x3 def test_issue_10755(): x = symbols('x') raises(TypeError, lambda: int(log(x))) raises(TypeError, lambda: log(x).round(2)) def test_issue_11877(): x = symbols('x') assert integrate(log(S.Half - x), (x, 0, S.Half)) == Rational(-1, 2) -log(2)/2 def test_normal(): x = symbols('x') e = Mul(S.Half, 1 + x, evaluate=False) assert e.normal() == e def test_expr(): x = symbols('x') raises(TypeError, lambda: tan(x).series(x, 2, oo, "+")) def test_ExprBuilder(): eb = ExprBuilder(Mul) eb.args.extend([x, x]) assert eb.build() == x2 def test_issue_22020(): from sympy.parsing.sympy_parser import parse_expr x = parse_expr("log((2V/3-V)/C)/-(R+r)C") y = parse_expr("log((2V/3-V)/C)/-(R+r)2") assert x.equals(y) is False def test_non_string_equality(): # Expressions should not compare equal to strings x = symbols('x') one = sympify(1) assert (x == 'x') is False assert (x != 'x') is True assert (one == '1') is False assert (one != '1') is True assert (x + 1 == 'x + 1') is False assert (x + 1 != 'x + 1') is True # Make sure == doesn't try to convert the resulting expression to a string # (e.g., by calling sympify() instead of _sympify()) class BadRepr: def __repr__(self): raise RuntimeError assert (x == BadRepr()) is False assert (x != BadRepr()) is True def test_21494(): from sympy.testing.pytest import warns_deprecated_sympy with warns_deprecated_sympy(): assert x.expr_free_symbols == {x} def test_Expr__eq__iterable_handling(): assert x != range(3)
e9b05abc384941d0a8861a9c6ce4dec9ddf40e1acd50edf260cf855de21b1c4f	"""Test whether all elements of cls.args are instances of Basic. """ # NOTE: keep tests sorted by (module, class name) key. If a class can't # be instantiated, add it here anyway with @SKIP("abstract class) (see # e.g. Function). import os import re from sympy.assumptions.ask import Q from sympy.core.basic import Basic from sympy.core.function import (Function, Lambda) from sympy.core.numbers import (Rational, oo, pi) from sympy.core.relational import Eq from sympy.core.singleton import S from sympy.core.symbol import symbols from sympy.functions.elementary.exponential import (exp, log) from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import sin from sympy.testing.pytest import XFAIL, SKIP a, b, c, x, y, z = symbols('a,b,c,x,y,z') whitelist = [ "sympy.assumptions.predicates", # tested by test_predicates() "sympy.assumptions.relation.equality", # tested by test_predicates() ] def test_all_classes_are_tested(): this = os.path.split(__file__)[0] path = os.path.join(this, os.pardir, os.pardir) sympy_path = os.path.abspath(path) prefix = os.path.split(sympy_path)[0] + os.sep re_cls = re.compile(r"^class ([A-Za-z][A-Za-z0-9_])\s\(", re.MULTILINE) modules = {} for root, dirs, files in os.walk(sympy_path): module = root.replace(prefix, "").replace(os.sep, ".") for file in files: if file.startswith(("_", "test_", "bench_")): continue if not file.endswith(".py"): continue with open(os.path.join(root, file), encoding='utf-8') as f: text = f.read() submodule = module + '.' + file[:-3] if any(submodule.startswith(wpath) for wpath in whitelist): continue names = re_cls.findall(text) if not names: continue try: mod = __import__(submodule, fromlist=names) except ImportError: continue def is_Basic(name): cls = getattr(mod, name) if hasattr(cls, '_sympy_deprecated_func'): cls = cls._sympy_deprecated_func if not isinstance(cls, type): # check instance of singleton class with same name cls = type(cls) return issubclass(cls, Basic) names = list(filter(is_Basic, names)) if names: modules[submodule] = names ns = globals() failed = [] for module, names in modules.items(): mod = module.replace('.', '__') for name in names: test = 'test_' + mod + '__' + name if test not in ns: failed.append(module + '.' + name) assert not failed, "Missing classes: %s. Please add tests for these to sympy/core/tests/test_args.py." % ", ".join(failed) def _test_args(obj): all_basic = all(isinstance(arg, Basic) for arg in obj.args) # Ideally obj.func(obj.args) would always recreate the object, but for # now, we only require it for objects with non-empty .args recreatable = not obj.args or obj.func(obj.args) == obj return all_basic and recreatable def test_sympy__algebras__quaternion__Quaternion(): from sympy.algebras.quaternion import Quaternion assert _test_args(Quaternion(x, 1, 2, 3)) def test_sympy__assumptions__assume__AppliedPredicate(): from sympy.assumptions.assume import AppliedPredicate, Predicate assert _test_args(AppliedPredicate(Predicate("test"), 2)) assert _test_args(Q.is_true(True)) @SKIP("abstract class") def test_sympy__assumptions__assume__Predicate(): pass def test_predicates(): predicates = [ getattr(Q, attr) for attr in Q.__class__.__dict__ if not attr.startswith('__')] for p in predicates: assert _test_args(p) def test_sympy__assumptions__assume__UndefinedPredicate(): from sympy.assumptions.assume import Predicate assert _test_args(Predicate("test")) @SKIP('abstract class') def test_sympy__assumptions__relation__binrel__BinaryRelation(): pass def test_sympy__assumptions__relation__binrel__AppliedBinaryRelation(): assert _test_args(Q.eq(1, 2)) def test_sympy__assumptions__wrapper__AssumptionsWrapper(): from sympy.assumptions.wrapper import AssumptionsWrapper assert _test_args(AssumptionsWrapper(x, Q.positive(x))) @SKIP("abstract Class") def test_sympy__codegen__ast__CodegenAST(): from sympy.codegen.ast import CodegenAST assert _test_args(CodegenAST()) @SKIP("abstract Class") def test_sympy__codegen__ast__AssignmentBase(): from sympy.codegen.ast import AssignmentBase assert _test_args(AssignmentBase(x, 1)) @SKIP("abstract Class") def test_sympy__codegen__ast__AugmentedAssignment(): from sympy.codegen.ast import AugmentedAssignment assert _test_args(AugmentedAssignment(x, 1)) def test_sympy__codegen__ast__AddAugmentedAssignment(): from sympy.codegen.ast import AddAugmentedAssignment assert _test_args(AddAugmentedAssignment(x, 1)) def test_sympy__codegen__ast__SubAugmentedAssignment(): from sympy.codegen.ast import SubAugmentedAssignment assert _test_args(SubAugmentedAssignment(x, 1)) def test_sympy__codegen__ast__MulAugmentedAssignment(): from sympy.codegen.ast import MulAugmentedAssignment assert _test_args(MulAugmentedAssignment(x, 1)) def test_sympy__codegen__ast__DivAugmentedAssignment(): from sympy.codegen.ast import DivAugmentedAssignment assert _test_args(DivAugmentedAssignment(x, 1)) def test_sympy__codegen__ast__ModAugmentedAssignment(): from sympy.codegen.ast import ModAugmentedAssignment assert _test_args(ModAugmentedAssignment(x, 1)) def test_sympy__codegen__ast__CodeBlock(): from sympy.codegen.ast import CodeBlock, Assignment assert _test_args(CodeBlock(Assignment(x, 1), Assignment(y, 2))) def test_sympy__codegen__ast__For(): from sympy.codegen.ast import For, CodeBlock, AddAugmentedAssignment from sympy.sets import Range assert _test_args(For(x, Range(10), CodeBlock(AddAugmentedAssignment(y, 1)))) def test_sympy__codegen__ast__Token(): from sympy.codegen.ast import Token assert _test_args(Token()) def test_sympy__codegen__ast__ContinueToken(): from sympy.codegen.ast import ContinueToken assert _test_args(ContinueToken()) def test_sympy__codegen__ast__BreakToken(): from sympy.codegen.ast import BreakToken assert _test_args(BreakToken()) def test_sympy__codegen__ast__NoneToken(): from sympy.codegen.ast import NoneToken assert _test_args(NoneToken()) def test_sympy__codegen__ast__String(): from sympy.codegen.ast import String assert _test_args(String('foobar')) def test_sympy__codegen__ast__QuotedString(): from sympy.codegen.ast import QuotedString assert _test_args(QuotedString('foobar')) def test_sympy__codegen__ast__Comment(): from sympy.codegen.ast import Comment assert _test_args(Comment('this is a comment')) def test_sympy__codegen__ast__Node(): from sympy.codegen.ast import Node assert _test_args(Node()) assert _test_args(Node(attrs={1, 2, 3})) def test_sympy__codegen__ast__Type(): from sympy.codegen.ast import Type assert _test_args(Type('float128')) def test_sympy__codegen__ast__IntBaseType(): from sympy.codegen.ast import IntBaseType assert _test_args(IntBaseType('bigint')) def test_sympy__codegen__ast___SizedIntType(): from sympy.codegen.ast import _SizedIntType assert _test_args(_SizedIntType('int128', 128)) def test_sympy__codegen__ast__SignedIntType(): from sympy.codegen.ast import SignedIntType assert _test_args(SignedIntType('int128_with_sign', 128)) def test_sympy__codegen__ast__UnsignedIntType(): from sympy.codegen.ast import UnsignedIntType assert _test_args(UnsignedIntType('unt128', 128)) def test_sympy__codegen__ast__FloatBaseType(): from sympy.codegen.ast import FloatBaseType assert _test_args(FloatBaseType('positive_real')) def test_sympy__codegen__ast__FloatType(): from sympy.codegen.ast import FloatType assert _test_args(FloatType('float242', 242, nmant=142, nexp=99)) def test_sympy__codegen__ast__ComplexBaseType(): from sympy.codegen.ast import ComplexBaseType assert _test_args(ComplexBaseType('positive_cmplx')) def test_sympy__codegen__ast__ComplexType(): from sympy.codegen.ast import ComplexType assert _test_args(ComplexType('complex42', 42, nmant=15, nexp=5)) def test_sympy__codegen__ast__Attribute(): from sympy.codegen.ast import Attribute assert _test_args(Attribute('noexcept')) def test_sympy__codegen__ast__Variable(): from sympy.codegen.ast import Variable, Type, value_const assert _test_args(Variable(x)) assert _test_args(Variable(y, Type('float32'), {value_const})) assert _test_args(Variable(z, type=Type('float64'))) def test_sympy__codegen__ast__Pointer(): from sympy.codegen.ast import Pointer, Type, pointer_const assert _test_args(Pointer(x)) assert _test_args(Pointer(y, type=Type('float32'))) assert _test_args(Pointer(z, Type('float64'), {pointer_const})) def test_sympy__codegen__ast__Declaration(): from sympy.codegen.ast import Declaration, Variable, Type vx = Variable(x, type=Type('float')) assert _test_args(Declaration(vx)) def test_sympy__codegen__ast__While(): from sympy.codegen.ast import While, AddAugmentedAssignment assert _test_args(While(abs(x) < 1, [AddAugmentedAssignment(x, -1)])) def test_sympy__codegen__ast__Scope(): from sympy.codegen.ast import Scope, AddAugmentedAssignment assert _test_args(Scope([AddAugmentedAssignment(x, -1)])) def test_sympy__codegen__ast__Stream(): from sympy.codegen.ast import Stream assert _test_args(Stream('stdin')) def test_sympy__codegen__ast__Print(): from sympy.codegen.ast import Print assert _test_args(Print([x, y])) assert _test_args(Print([x, y], "%d %d")) def test_sympy__codegen__ast__FunctionPrototype(): from sympy.codegen.ast import FunctionPrototype, real, Declaration, Variable inp_x = Declaration(Variable(x, type=real)) assert _test_args(FunctionPrototype(real, 'pwer', [inp_x])) def test_sympy__codegen__ast__FunctionDefinition(): from sympy.codegen.ast import FunctionDefinition, real, Declaration, Variable, Assignment inp_x = Declaration(Variable(x, type=real)) assert _test_args(FunctionDefinition(real, 'pwer', [inp_x], [Assignment(x, x*2)])) def test_sympy__codegen__ast__Return(): from sympy.codegen.ast import Return assert _test_args(Return(x)) def test_sympy__codegen__ast__FunctionCall(): from sympy.codegen.ast import FunctionCall assert _test_args(FunctionCall('pwer', [x])) def test_sympy__codegen__ast__Element(): from sympy.codegen.ast import Element assert _test_args(Element('x', range(3))) def test_sympy__codegen__cnodes__CommaOperator(): from sympy.codegen.cnodes import CommaOperator assert _test_args(CommaOperator(1, 2)) def test_sympy__codegen__cnodes__goto(): from sympy.codegen.cnodes import goto assert _test_args(goto('early_exit')) def test_sympy__codegen__cnodes__Label(): from sympy.codegen.cnodes import Label assert _test_args(Label('early_exit')) def test_sympy__codegen__cnodes__PreDecrement(): from sympy.codegen.cnodes import PreDecrement assert _test_args(PreDecrement(x)) def test_sympy__codegen__cnodes__PostDecrement(): from sympy.codegen.cnodes import PostDecrement assert _test_args(PostDecrement(x)) def test_sympy__codegen__cnodes__PreIncrement(): from sympy.codegen.cnodes import PreIncrement assert _test_args(PreIncrement(x)) def test_sympy__codegen__cnodes__PostIncrement(): from sympy.codegen.cnodes import PostIncrement assert _test_args(PostIncrement(x)) def test_sympy__codegen__cnodes__struct(): from sympy.codegen.ast import real, Variable from sympy.codegen.cnodes import struct assert _test_args(struct(declarations=[ Variable(x, type=real), Variable(y, type=real) ])) def test_sympy__codegen__cnodes__union(): from sympy.codegen.ast import float32, int32, Variable from sympy.codegen.cnodes import union assert _test_args(union(declarations=[ Variable(x, type=float32), Variable(y, type=int32) ])) def test_sympy__codegen__cxxnodes__using(): from sympy.codegen.cxxnodes import using assert _test_args(using('std::vector')) assert _test_args(using('std::vector', 'vec')) def test_sympy__codegen__fnodes__Program(): from sympy.codegen.fnodes import Program assert _test_args(Program('foobar', [])) def test_sympy__codegen__fnodes__Module(): from sympy.codegen.fnodes import Module assert _test_args(Module('foobar', [], [])) def test_sympy__codegen__fnodes__Subroutine(): from sympy.codegen.fnodes import Subroutine x = symbols('x', real=True) assert _test_args(Subroutine('foo', [x], [])) def test_sympy__codegen__fnodes__GoTo(): from sympy.codegen.fnodes import GoTo assert _test_args(GoTo([10])) assert _test_args(GoTo([10, 20], x > 1)) def test_sympy__codegen__fnodes__FortranReturn(): from sympy.codegen.fnodes import FortranReturn assert _test_args(FortranReturn(10)) def test_sympy__codegen__fnodes__Extent(): from sympy.codegen.fnodes import Extent assert _test_args(Extent()) assert _test_args(Extent(None)) assert _test_args(Extent(':')) assert _test_args(Extent(-3, 4)) assert _test_args(Extent(x, y)) def test_sympy__codegen__fnodes__use_rename(): from sympy.codegen.fnodes import use_rename assert _test_args(use_rename('loc', 'glob')) def test_sympy__codegen__fnodes__use(): from sympy.codegen.fnodes import use assert _test_args(use('modfoo', only='bar')) def test_sympy__codegen__fnodes__SubroutineCall(): from sympy.codegen.fnodes import SubroutineCall assert _test_args(SubroutineCall('foo', ['bar', 'baz'])) def test_sympy__codegen__fnodes__Do(): from sympy.codegen.fnodes import Do assert _test_args(Do([], 'i', 1, 42)) def test_sympy__codegen__fnodes__ImpliedDoLoop(): from sympy.codegen.fnodes import ImpliedDoLoop assert _test_args(ImpliedDoLoop('i', 'i', 1, 42)) def test_sympy__codegen__fnodes__ArrayConstructor(): from sympy.codegen.fnodes import ArrayConstructor assert _test_args(ArrayConstructor([1, 2, 3])) from sympy.codegen.fnodes import ImpliedDoLoop idl = ImpliedDoLoop('i', 'i', 1, 42) assert _test_args(ArrayConstructor([1, idl, 3])) def test_sympy__codegen__fnodes__sum_(): from sympy.codegen.fnodes import sum_ assert _test_args(sum_('arr')) def test_sympy__codegen__fnodes__product_(): from sympy.codegen.fnodes import product_ assert _test_args(product_('arr')) def test_sympy__codegen__numpy_nodes__logaddexp(): from sympy.codegen.numpy_nodes import logaddexp assert _test_args(logaddexp(x, y)) def test_sympy__codegen__numpy_nodes__logaddexp2(): from sympy.codegen.numpy_nodes import logaddexp2 assert _test_args(logaddexp2(x, y)) def test_sympy__codegen__pynodes__List(): from sympy.codegen.pynodes import List assert _test_args(List(1, 2, 3)) def test_sympy__codegen__scipy_nodes__cosm1(): from sympy.codegen.scipy_nodes import cosm1 assert _test_args(cosm1(x)) def test_sympy__codegen__abstract_nodes__List(): from sympy.codegen.abstract_nodes import List assert _test_args(List(1, 2, 3)) def test_sympy__combinatorics__graycode__GrayCode(): from sympy.combinatorics.graycode import GrayCode # an integer is given and returned from GrayCode as the arg assert _test_args(GrayCode(3, start='100')) assert _test_args(GrayCode(3, rank=1)) def test_sympy__combinatorics__permutations__Permutation(): from sympy.combinatorics.permutations import Permutation assert _test_args(Permutation([0, 1, 2, 3])) def test_sympy__combinatorics__permutations__AppliedPermutation(): from sympy.combinatorics.permutations import Permutation from sympy.combinatorics.permutations import AppliedPermutation p = Permutation([0, 1, 2, 3]) assert _test_args(AppliedPermutation(p, x)) def test_sympy__combinatorics__perm_groups__PermutationGroup(): from sympy.combinatorics.permutations import Permutation from sympy.combinatorics.perm_groups import PermutationGroup assert _test_args(PermutationGroup([Permutation([0, 1])])) def test_sympy__combinatorics__polyhedron__Polyhedron(): from sympy.combinatorics.permutations import Permutation from sympy.combinatorics.polyhedron import Polyhedron from sympy.abc import w, x, y, z pgroup = [Permutation([[0, 1, 2], [3]]), Permutation([[0, 1, 3], [2]]), Permutation([[0, 2, 3], [1]]), Permutation([[1, 2, 3], [0]]), Permutation([[0, 1], [2, 3]]), Permutation([[0, 2], [1, 3]]), Permutation([[0, 3], [1, 2]]), Permutation([[0, 1, 2, 3]])] corners = [w, x, y, z] faces = [(w, x, y), (w, y, z), (w, z, x), (x, y, z)] assert _test_args(Polyhedron(corners, faces, pgroup)) def test_sympy__combinatorics__prufer__Prufer(): from sympy.combinatorics.prufer import Prufer assert _test_args(Prufer([[0, 1], [0, 2], [0, 3]], 4)) def test_sympy__combinatorics__partitions__Partition(): from sympy.combinatorics.partitions import Partition assert _test_args(Partition([1])) def test_sympy__combinatorics__partitions__IntegerPartition(): from sympy.combinatorics.partitions import IntegerPartition assert _test_args(IntegerPartition([1])) def test_sympy__concrete__products__Product(): from sympy.concrete.products import Product assert _test_args(Product(x, (x, 0, 10))) assert _test_args(Product(x, (x, 0, y), (y, 0, 10))) @SKIP("abstract Class") def test_sympy__concrete__expr_with_limits__ExprWithLimits(): from sympy.concrete.expr_with_limits import ExprWithLimits assert _test_args(ExprWithLimits(x, (x, 0, 10))) assert _test_args(ExprWithLimits(xy, (x, 0, 10.),(y,1.,3))) @SKIP("abstract Class") def test_sympy__concrete__expr_with_limits__AddWithLimits(): from sympy.concrete.expr_with_limits import AddWithLimits assert _test_args(AddWithLimits(x, (x, 0, 10))) assert _test_args(AddWithLimits(xy, (x, 0, 10),(y,1,3))) @SKIP("abstract Class") def test_sympy__concrete__expr_with_intlimits__ExprWithIntLimits(): from sympy.concrete.expr_with_intlimits import ExprWithIntLimits assert _test_args(ExprWithIntLimits(x, (x, 0, 10))) assert _test_args(ExprWithIntLimits(xy, (x, 0, 10),(y,1,3))) def test_sympy__concrete__summations__Sum(): from sympy.concrete.summations import Sum assert _test_args(Sum(x, (x, 0, 10))) assert _test_args(Sum(x, (x, 0, y), (y, 0, 10))) def test_sympy__core__add__Add(): from sympy.core.add import Add assert _test_args(Add(x, y, z, 2)) def test_sympy__core__basic__Atom(): from sympy.core.basic import Atom assert _test_args(Atom()) def test_sympy__core__basic__Basic(): from sympy.core.basic import Basic assert _test_args(Basic()) def test_sympy__core__containers__Dict(): from sympy.core.containers import Dict assert _test_args(Dict({x: y, y: z})) def test_sympy__core__containers__Tuple(): from sympy.core.containers import Tuple assert _test_args(Tuple(x, y, z, 2)) def test_sympy__core__expr__AtomicExpr(): from sympy.core.expr import AtomicExpr assert _test_args(AtomicExpr()) def test_sympy__core__expr__Expr(): from sympy.core.expr import Expr assert _test_args(Expr()) def test_sympy__core__expr__UnevaluatedExpr(): from sympy.core.expr import UnevaluatedExpr from sympy.abc import x assert _test_args(UnevaluatedExpr(x)) def test_sympy__core__function__Application(): from sympy.core.function import Application assert _test_args(Application(1, 2, 3)) def test_sympy__core__function__AppliedUndef(): from sympy.core.function import AppliedUndef assert _test_args(AppliedUndef(1, 2, 3)) def test_sympy__core__function__Derivative(): from sympy.core.function import Derivative assert _test_args(Derivative(2, x, y, 3)) @SKIP("abstract class") def test_sympy__core__function__Function(): pass def test_sympy__core__function__Lambda(): assert _test_args(Lambda((x, y), x + y + z)) def test_sympy__core__function__Subs(): from sympy.core.function import Subs assert _test_args(Subs(x + y, x, 2)) def test_sympy__core__function__WildFunction(): from sympy.core.function import WildFunction assert _test_args(WildFunction('f')) def test_sympy__core__mod__Mod(): from sympy.core.mod import Mod assert _test_args(Mod(x, 2)) def test_sympy__core__mul__Mul(): from sympy.core.mul import Mul assert _test_args(Mul(2, x, y, z)) def test_sympy__core__numbers__Catalan(): from sympy.core.numbers import Catalan assert _test_args(Catalan()) def test_sympy__core__numbers__ComplexInfinity(): from sympy.core.numbers import ComplexInfinity assert _test_args(ComplexInfinity()) def test_sympy__core__numbers__EulerGamma(): from sympy.core.numbers import EulerGamma assert _test_args(EulerGamma()) def test_sympy__core__numbers__Exp1(): from sympy.core.numbers import Exp1 assert _test_args(Exp1()) def test_sympy__core__numbers__Float(): from sympy.core.numbers import Float assert _test_args(Float(1.23)) def test_sympy__core__numbers__GoldenRatio(): from sympy.core.numbers import GoldenRatio assert _test_args(GoldenRatio()) def test_sympy__core__numbers__TribonacciConstant(): from sympy.core.numbers import TribonacciConstant assert _test_args(TribonacciConstant()) def test_sympy__core__numbers__Half(): from sympy.core.numbers import Half assert _test_args(Half()) def test_sympy__core__numbers__ImaginaryUnit(): from sympy.core.numbers import ImaginaryUnit assert _test_args(ImaginaryUnit()) def test_sympy__core__numbers__Infinity(): from sympy.core.numbers import Infinity assert _test_args(Infinity()) def test_sympy__core__numbers__Integer(): from sympy.core.numbers import Integer assert _test_args(Integer(7)) @SKIP("abstract class") def test_sympy__core__numbers__IntegerConstant(): pass def test_sympy__core__numbers__NaN(): from sympy.core.numbers import NaN assert _test_args(NaN()) def test_sympy__core__numbers__NegativeInfinity(): from sympy.core.numbers import NegativeInfinity assert _test_args(NegativeInfinity()) def test_sympy__core__numbers__NegativeOne(): from sympy.core.numbers import NegativeOne assert _test_args(NegativeOne()) def test_sympy__core__numbers__Number(): from sympy.core.numbers import Number assert _test_args(Number(1, 7)) def test_sympy__core__numbers__NumberSymbol(): from sympy.core.numbers import NumberSymbol assert _test_args(NumberSymbol()) def test_sympy__core__numbers__One(): from sympy.core.numbers import One assert _test_args(One()) def test_sympy__core__numbers__Pi(): from sympy.core.numbers import Pi assert _test_args(Pi()) def test_sympy__core__numbers__Rational(): from sympy.core.numbers import Rational assert _test_args(Rational(1, 7)) @SKIP("abstract class") def test_sympy__core__numbers__RationalConstant(): pass def test_sympy__core__numbers__Zero(): from sympy.core.numbers import Zero assert _test_args(Zero()) @SKIP("abstract class") def test_sympy__core__operations__AssocOp(): pass @SKIP("abstract class") def test_sympy__core__operations__LatticeOp(): pass def test_sympy__core__power__Pow(): from sympy.core.power import Pow assert _test_args(Pow(x, 2)) def test_sympy__core__relational__Equality(): from sympy.core.relational import Equality assert _test_args(Equality(x, 2)) def test_sympy__core__relational__GreaterThan(): from sympy.core.relational import GreaterThan assert _test_args(GreaterThan(x, 2)) def test_sympy__core__relational__LessThan(): from sympy.core.relational import LessThan assert _test_args(LessThan(x, 2)) @SKIP("abstract class") def test_sympy__core__relational__Relational(): pass def test_sympy__core__relational__StrictGreaterThan(): from sympy.core.relational import StrictGreaterThan assert _test_args(StrictGreaterThan(x, 2)) def test_sympy__core__relational__StrictLessThan(): from sympy.core.relational import StrictLessThan assert _test_args(StrictLessThan(x, 2)) def test_sympy__core__relational__Unequality(): from sympy.core.relational import Unequality assert _test_args(Unequality(x, 2)) @SKIP("deprecated class") def test_sympy__core__trace__Tr(): pass def test_sympy__sandbox__indexed_integrals__IndexedIntegral(): from sympy.tensor import IndexedBase, Idx from sympy.sandbox.indexed_integrals import IndexedIntegral A = IndexedBase('A') i, j = symbols('i j', integer=True) a1, a2 = symbols('a1:3', cls=Idx) assert _test_args(IndexedIntegral(A[a1], A[a2])) assert _test_args(IndexedIntegral(A[i], A[j])) def test_sympy__calculus__accumulationbounds__AccumulationBounds(): from sympy.calculus.accumulationbounds import AccumulationBounds assert _test_args(AccumulationBounds(0, 1)) def test_sympy__sets__ordinals__OmegaPower(): from sympy.sets.ordinals import OmegaPower assert _test_args(OmegaPower(1, 1)) def test_sympy__sets__ordinals__Ordinal(): from sympy.sets.ordinals import Ordinal, OmegaPower assert _test_args(Ordinal(OmegaPower(2, 1))) def test_sympy__sets__ordinals__OrdinalOmega(): from sympy.sets.ordinals import OrdinalOmega assert _test_args(OrdinalOmega()) def test_sympy__sets__ordinals__OrdinalZero(): from sympy.sets.ordinals import OrdinalZero assert _test_args(OrdinalZero()) def test_sympy__sets__powerset__PowerSet(): from sympy.sets.powerset import PowerSet from sympy.core.singleton import S assert _test_args(PowerSet(S.EmptySet)) def test_sympy__sets__sets__EmptySet(): from sympy.sets.sets import EmptySet assert _test_args(EmptySet()) def test_sympy__sets__sets__UniversalSet(): from sympy.sets.sets import UniversalSet assert _test_args(UniversalSet()) def test_sympy__sets__sets__FiniteSet(): from sympy.sets.sets import FiniteSet assert _test_args(FiniteSet(x, y, z)) def test_sympy__sets__sets__Interval(): from sympy.sets.sets import Interval assert _test_args(Interval(0, 1)) def test_sympy__sets__sets__ProductSet(): from sympy.sets.sets import ProductSet, Interval assert _test_args(ProductSet(Interval(0, 1), Interval(0, 1))) @SKIP("does it make sense to test this?") def test_sympy__sets__sets__Set(): from sympy.sets.sets import Set assert _test_args(Set()) def test_sympy__sets__sets__Intersection(): from sympy.sets.sets import Intersection, Interval from sympy.core.symbol import Symbol x = Symbol('x') y = Symbol('y') S = Intersection(Interval(0, x), Interval(y, 1)) assert isinstance(S, Intersection) assert _test_args(S) def test_sympy__sets__sets__Union(): from sympy.sets.sets import Union, Interval assert _test_args(Union(Interval(0, 1), Interval(2, 3))) def test_sympy__sets__sets__Complement(): from sympy.sets.sets import Complement, Interval assert _test_args(Complement(Interval(0, 2), Interval(0, 1))) def test_sympy__sets__sets__SymmetricDifference(): from sympy.sets.sets import FiniteSet, SymmetricDifference assert _test_args(SymmetricDifference(FiniteSet(1, 2, 3), \ FiniteSet(2, 3, 4))) def test_sympy__sets__sets__DisjointUnion(): from sympy.sets.sets import FiniteSet, DisjointUnion assert _test_args(DisjointUnion(FiniteSet(1, 2, 3), \ FiniteSet(2, 3, 4))) def test_sympy__physics__quantum__trace__Tr(): from sympy.physics.quantum.trace import Tr a, b = symbols('a b', commutative=False) assert _test_args(Tr(a + b)) def test_sympy__sets__setexpr__SetExpr(): from sympy.sets.setexpr import SetExpr from sympy.sets.sets import Interval assert _test_args(SetExpr(Interval(0, 1))) def test_sympy__sets__fancysets__Rationals(): from sympy.sets.fancysets import Rationals assert _test_args(Rationals()) def test_sympy__sets__fancysets__Naturals(): from sympy.sets.fancysets import Naturals assert _test_args(Naturals()) def test_sympy__sets__fancysets__Naturals0(): from sympy.sets.fancysets import Naturals0 assert _test_args(Naturals0()) def test_sympy__sets__fancysets__Integers(): from sympy.sets.fancysets import Integers assert _test_args(Integers()) def test_sympy__sets__fancysets__Reals(): from sympy.sets.fancysets import Reals assert _test_args(Reals()) def test_sympy__sets__fancysets__Complexes(): from sympy.sets.fancysets import Complexes assert _test_args(Complexes()) def test_sympy__sets__fancysets__ComplexRegion(): from sympy.sets.fancysets import ComplexRegion from sympy.core.singleton import S from sympy.sets import Interval a = Interval(0, 1) b = Interval(2, 3) theta = Interval(0, 2S.Pi) assert _test_args(ComplexRegion(ab)) assert _test_args(ComplexRegion(atheta, polar=True)) def test_sympy__sets__fancysets__CartesianComplexRegion(): from sympy.sets.fancysets import CartesianComplexRegion from sympy.sets import Interval a = Interval(0, 1) b = Interval(2, 3) assert _test_args(CartesianComplexRegion(ab)) def test_sympy__sets__fancysets__PolarComplexRegion(): from sympy.sets.fancysets import PolarComplexRegion from sympy.core.singleton import S from sympy.sets import Interval a = Interval(0, 1) theta = Interval(0, 2S.Pi) assert _test_args(PolarComplexRegion(atheta)) def test_sympy__sets__fancysets__ImageSet(): from sympy.sets.fancysets import ImageSet from sympy.core.singleton import S from sympy.core.symbol import Symbol x = Symbol('x') assert _test_args(ImageSet(Lambda(x, x2), S.Naturals)) def test_sympy__sets__fancysets__Range(): from sympy.sets.fancysets import Range assert _test_args(Range(1, 5, 1)) def test_sympy__sets__conditionset__ConditionSet(): from sympy.sets.conditionset import ConditionSet from sympy.core.singleton import S from sympy.core.symbol import Symbol x = Symbol('x') assert _test_args(ConditionSet(x, Eq(x2, 1), S.Reals)) def test_sympy__sets__contains__Contains(): from sympy.sets.fancysets import Range from sympy.sets.contains import Contains assert _test_args(Contains(x, Range(0, 10, 2))) # STATS from sympy.stats.crv_types import NormalDistribution nd = NormalDistribution(0, 1) from sympy.stats.frv_types import DieDistribution die = DieDistribution(6) def test_sympy__stats__crv__ContinuousDomain(): from sympy.sets.sets import Interval from sympy.stats.crv import ContinuousDomain assert _test_args(ContinuousDomain({x}, Interval(-oo, oo))) def test_sympy__stats__crv__SingleContinuousDomain(): from sympy.sets.sets import Interval from sympy.stats.crv import SingleContinuousDomain assert _test_args(SingleContinuousDomain(x, Interval(-oo, oo))) def test_sympy__stats__crv__ProductContinuousDomain(): from sympy.sets.sets import Interval from sympy.stats.crv import SingleContinuousDomain, ProductContinuousDomain D = SingleContinuousDomain(x, Interval(-oo, oo)) E = SingleContinuousDomain(y, Interval(0, oo)) assert _test_args(ProductContinuousDomain(D, E)) def test_sympy__stats__crv__ConditionalContinuousDomain(): from sympy.sets.sets import Interval from sympy.stats.crv import (SingleContinuousDomain, ConditionalContinuousDomain) D = SingleContinuousDomain(x, Interval(-oo, oo)) assert _test_args(ConditionalContinuousDomain(D, x > 0)) def test_sympy__stats__crv__ContinuousPSpace(): from sympy.sets.sets import Interval from sympy.stats.crv import ContinuousPSpace, SingleContinuousDomain D = SingleContinuousDomain(x, Interval(-oo, oo)) assert _test_args(ContinuousPSpace(D, nd)) def test_sympy__stats__crv__SingleContinuousPSpace(): from sympy.stats.crv import SingleContinuousPSpace assert _test_args(SingleContinuousPSpace(x, nd)) @SKIP("abstract class") def test_sympy__stats__rv__Distribution(): pass @SKIP("abstract class") def test_sympy__stats__crv__SingleContinuousDistribution(): pass def test_sympy__stats__drv__SingleDiscreteDomain(): from sympy.stats.drv import SingleDiscreteDomain assert _test_args(SingleDiscreteDomain(x, S.Naturals)) def test_sympy__stats__drv__ProductDiscreteDomain(): from sympy.stats.drv import SingleDiscreteDomain, ProductDiscreteDomain X = SingleDiscreteDomain(x, S.Naturals) Y = SingleDiscreteDomain(y, S.Integers) assert _test_args(ProductDiscreteDomain(X, Y)) def test_sympy__stats__drv__SingleDiscretePSpace(): from sympy.stats.drv import SingleDiscretePSpace from sympy.stats.drv_types import PoissonDistribution assert _test_args(SingleDiscretePSpace(x, PoissonDistribution(1))) def test_sympy__stats__drv__DiscretePSpace(): from sympy.stats.drv import DiscretePSpace, SingleDiscreteDomain density = Lambda(x, 2(-x)) domain = SingleDiscreteDomain(x, S.Naturals) assert _test_args(DiscretePSpace(domain, density)) def test_sympy__stats__drv__ConditionalDiscreteDomain(): from sympy.stats.drv import ConditionalDiscreteDomain, SingleDiscreteDomain X = SingleDiscreteDomain(x, S.Naturals0) assert _test_args(ConditionalDiscreteDomain(X, x > 2)) def test_sympy__stats__joint_rv__JointPSpace(): from sympy.stats.joint_rv import JointPSpace, JointDistribution assert _test_args(JointPSpace('X', JointDistribution(1))) def test_sympy__stats__joint_rv__JointRandomSymbol(): from sympy.stats.joint_rv import JointRandomSymbol assert _test_args(JointRandomSymbol(x)) def test_sympy__stats__joint_rv_types__JointDistributionHandmade(): from sympy.tensor.indexed import Indexed from sympy.stats.joint_rv_types import JointDistributionHandmade x1, x2 = (Indexed('x', i) for i in (1, 2)) assert _test_args(JointDistributionHandmade(x1 + x2, S.Reals2)) def test_sympy__stats__joint_rv__MarginalDistribution(): from sympy.stats.rv import RandomSymbol from sympy.stats.joint_rv import MarginalDistribution r = RandomSymbol(S('r')) assert _test_args(MarginalDistribution(r, (r,))) def test_sympy__stats__compound_rv__CompoundDistribution(): from sympy.stats.compound_rv import CompoundDistribution from sympy.stats.drv_types import PoissonDistribution, Poisson r = Poisson('r', 10) assert _test_args(CompoundDistribution(PoissonDistribution(r))) def test_sympy__stats__compound_rv__CompoundPSpace(): from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution from sympy.stats.drv_types import PoissonDistribution, Poisson r = Poisson('r', 5) C = CompoundDistribution(PoissonDistribution(r)) assert _test_args(CompoundPSpace('C', C)) @SKIP("abstract class") def test_sympy__stats__drv__SingleDiscreteDistribution(): pass @SKIP("abstract class") def test_sympy__stats__drv__DiscreteDistribution(): pass @SKIP("abstract class") def test_sympy__stats__drv__DiscreteDomain(): pass def test_sympy__stats__rv__RandomDomain(): from sympy.stats.rv import RandomDomain from sympy.sets.sets import FiniteSet assert _test_args(RandomDomain(FiniteSet(x), FiniteSet(1, 2, 3))) def test_sympy__stats__rv__SingleDomain(): from sympy.stats.rv import SingleDomain from sympy.sets.sets import FiniteSet assert _test_args(SingleDomain(x, FiniteSet(1, 2, 3))) def test_sympy__stats__rv__ConditionalDomain(): from sympy.stats.rv import ConditionalDomain, RandomDomain from sympy.sets.sets import FiniteSet D = RandomDomain(FiniteSet(x), FiniteSet(1, 2)) assert _test_args(ConditionalDomain(D, x > 1)) def test_sympy__stats__rv__MatrixDomain(): from sympy.stats.rv import MatrixDomain from sympy.matrices import MatrixSet from sympy.core.singleton import S assert _test_args(MatrixDomain(x, MatrixSet(2, 2, S.Reals))) def test_sympy__stats__rv__PSpace(): from sympy.stats.rv import PSpace, RandomDomain from sympy.sets.sets import FiniteSet D = RandomDomain(FiniteSet(x), FiniteSet(1, 2, 3, 4, 5, 6)) assert _test_args(PSpace(D, die)) @SKIP("abstract Class") def test_sympy__stats__rv__SinglePSpace(): pass def test_sympy__stats__rv__RandomSymbol(): from sympy.stats.rv import RandomSymbol from sympy.stats.crv import SingleContinuousPSpace A = SingleContinuousPSpace(x, nd) assert _test_args(RandomSymbol(x, A)) @SKIP("abstract Class") def test_sympy__stats__rv__ProductPSpace(): pass def test_sympy__stats__rv__IndependentProductPSpace(): from sympy.stats.rv import IndependentProductPSpace from sympy.stats.crv import SingleContinuousPSpace A = SingleContinuousPSpace(x, nd) B = SingleContinuousPSpace(y, nd) assert _test_args(IndependentProductPSpace(A, B)) def test_sympy__stats__rv__ProductDomain(): from sympy.sets.sets import Interval from sympy.stats.rv import ProductDomain, SingleDomain D = SingleDomain(x, Interval(-oo, oo)) E = SingleDomain(y, Interval(0, oo)) assert _test_args(ProductDomain(D, E)) def test_sympy__stats__symbolic_probability__Probability(): from sympy.stats.symbolic_probability import Probability from sympy.stats import Normal X = Normal('X', 0, 1) assert _test_args(Probability(X > 0)) def test_sympy__stats__symbolic_probability__Expectation(): from sympy.stats.symbolic_probability import Expectation from sympy.stats import Normal X = Normal('X', 0, 1) assert _test_args(Expectation(X > 0)) def test_sympy__stats__symbolic_probability__Covariance(): from sympy.stats.symbolic_probability import Covariance from sympy.stats import Normal X = Normal('X', 0, 1) Y = Normal('Y', 0, 3) assert _test_args(Covariance(X, Y)) def test_sympy__stats__symbolic_probability__Variance(): from sympy.stats.symbolic_probability import Variance from sympy.stats import Normal X = Normal('X', 0, 1) assert _test_args(Variance(X)) def test_sympy__stats__symbolic_probability__Moment(): from sympy.stats.symbolic_probability import Moment from sympy.stats import Normal X = Normal('X', 0, 1) assert _test_args(Moment(X, 3, 2, X > 3)) def test_sympy__stats__symbolic_probability__CentralMoment(): from sympy.stats.symbolic_probability import CentralMoment from sympy.stats import Normal X = Normal('X', 0, 1) assert _test_args(CentralMoment(X, 2, X > 1)) def test_sympy__stats__frv_types__DiscreteUniformDistribution(): from sympy.stats.frv_types import DiscreteUniformDistribution from sympy.core.containers import Tuple assert _test_args(DiscreteUniformDistribution(Tuple(list(range(6))))) def test_sympy__stats__frv_types__DieDistribution(): assert _test_args(die) def test_sympy__stats__frv_types__BernoulliDistribution(): from sympy.stats.frv_types import BernoulliDistribution assert _test_args(BernoulliDistribution(S.Half, 0, 1)) def test_sympy__stats__frv_types__BinomialDistribution(): from sympy.stats.frv_types import BinomialDistribution assert _test_args(BinomialDistribution(5, S.Half, 1, 0)) def test_sympy__stats__frv_types__BetaBinomialDistribution(): from sympy.stats.frv_types import BetaBinomialDistribution assert _test_args(BetaBinomialDistribution(5, 1, 1)) def test_sympy__stats__frv_types__HypergeometricDistribution(): from sympy.stats.frv_types import HypergeometricDistribution assert _test_args(HypergeometricDistribution(10, 5, 3)) def test_sympy__stats__frv_types__RademacherDistribution(): from sympy.stats.frv_types import RademacherDistribution assert _test_args(RademacherDistribution()) def test_sympy__stats__frv_types__IdealSolitonDistribution(): from sympy.stats.frv_types import IdealSolitonDistribution assert _test_args(IdealSolitonDistribution(10)) def test_sympy__stats__frv_types__RobustSolitonDistribution(): from sympy.stats.frv_types import RobustSolitonDistribution assert _test_args(RobustSolitonDistribution(1000, 0.5, 0.1)) def test_sympy__stats__frv__FiniteDomain(): from sympy.stats.frv import FiniteDomain assert _test_args(FiniteDomain({(x, 1), (x, 2)})) # x can be 1 or 2 def test_sympy__stats__frv__SingleFiniteDomain(): from sympy.stats.frv import SingleFiniteDomain assert _test_args(SingleFiniteDomain(x, {1, 2})) # x can be 1 or 2 def test_sympy__stats__frv__ProductFiniteDomain(): from sympy.stats.frv import SingleFiniteDomain, ProductFiniteDomain xd = SingleFiniteDomain(x, {1, 2}) yd = SingleFiniteDomain(y, {1, 2}) assert _test_args(ProductFiniteDomain(xd, yd)) def test_sympy__stats__frv__ConditionalFiniteDomain(): from sympy.stats.frv import SingleFiniteDomain, ConditionalFiniteDomain xd = SingleFiniteDomain(x, {1, 2}) assert _test_args(ConditionalFiniteDomain(xd, x > 1)) def test_sympy__stats__frv__FinitePSpace(): from sympy.stats.frv import FinitePSpace, SingleFiniteDomain xd = SingleFiniteDomain(x, {1, 2, 3, 4, 5, 6}) assert _test_args(FinitePSpace(xd, {(x, 1): S.Half, (x, 2): S.Half})) xd = SingleFiniteDomain(x, {1, 2}) assert _test_args(FinitePSpace(xd, {(x, 1): S.Half, (x, 2): S.Half})) def test_sympy__stats__frv__SingleFinitePSpace(): from sympy.stats.frv import SingleFinitePSpace from sympy.core.symbol import Symbol assert _test_args(SingleFinitePSpace(Symbol('x'), die)) def test_sympy__stats__frv__ProductFinitePSpace(): from sympy.stats.frv import SingleFinitePSpace, ProductFinitePSpace from sympy.core.symbol import Symbol xp = SingleFinitePSpace(Symbol('x'), die) yp = SingleFinitePSpace(Symbol('y'), die) assert _test_args(ProductFinitePSpace(xp, yp)) @SKIP("abstract class") def test_sympy__stats__frv__SingleFiniteDistribution(): pass @SKIP("abstract class") def test_sympy__stats__crv__ContinuousDistribution(): pass def test_sympy__stats__frv_types__FiniteDistributionHandmade(): from sympy.stats.frv_types import FiniteDistributionHandmade from sympy.core.containers import Dict assert _test_args(FiniteDistributionHandmade(Dict({1: 1}))) def test_sympy__stats__crv_types__ContinuousDistributionHandmade(): from sympy.stats.crv_types import ContinuousDistributionHandmade from sympy.core.function import Lambda from sympy.sets.sets import Interval from sympy.abc import x assert _test_args(ContinuousDistributionHandmade(Lambda(x, 2x), Interval(0, 1))) def test_sympy__stats__drv_types__DiscreteDistributionHandmade(): from sympy.stats.drv_types import DiscreteDistributionHandmade from sympy.core.function import Lambda from sympy.sets.sets import FiniteSet from sympy.abc import x assert _test_args(DiscreteDistributionHandmade(Lambda(x, Rational(1, 10)), FiniteSet(range(10)))) def test_sympy__stats__rv__Density(): from sympy.stats.rv import Density from sympy.stats.crv_types import Normal assert _test_args(Density(Normal('x', 0, 1))) def test_sympy__stats__crv_types__ArcsinDistribution(): from sympy.stats.crv_types import ArcsinDistribution assert _test_args(ArcsinDistribution(0, 1)) def test_sympy__stats__crv_types__BeniniDistribution(): from sympy.stats.crv_types import BeniniDistribution assert _test_args(BeniniDistribution(1, 1, 1)) def test_sympy__stats__crv_types__BetaDistribution(): from sympy.stats.crv_types import BetaDistribution assert _test_args(BetaDistribution(1, 1)) def test_sympy__stats__crv_types__BetaNoncentralDistribution(): from sympy.stats.crv_types import BetaNoncentralDistribution assert _test_args(BetaNoncentralDistribution(1, 1, 1)) def test_sympy__stats__crv_types__BetaPrimeDistribution(): from sympy.stats.crv_types import BetaPrimeDistribution assert _test_args(BetaPrimeDistribution(1, 1)) def test_sympy__stats__crv_types__BoundedParetoDistribution(): from sympy.stats.crv_types import BoundedParetoDistribution assert _test_args(BoundedParetoDistribution(1, 1, 2)) def test_sympy__stats__crv_types__CauchyDistribution(): from sympy.stats.crv_types import CauchyDistribution assert _test_args(CauchyDistribution(0, 1)) def test_sympy__stats__crv_types__ChiDistribution(): from sympy.stats.crv_types import ChiDistribution assert _test_args(ChiDistribution(1)) def test_sympy__stats__crv_types__ChiNoncentralDistribution(): from sympy.stats.crv_types import ChiNoncentralDistribution assert _test_args(ChiNoncentralDistribution(1,1)) def test_sympy__stats__crv_types__ChiSquaredDistribution(): from sympy.stats.crv_types import ChiSquaredDistribution assert _test_args(ChiSquaredDistribution(1)) def test_sympy__stats__crv_types__DagumDistribution(): from sympy.stats.crv_types import DagumDistribution assert _test_args(DagumDistribution(1, 1, 1)) def test_sympy__stats__crv_types__ExGaussianDistribution(): from sympy.stats.crv_types import ExGaussianDistribution assert _test_args(ExGaussianDistribution(1, 1, 1)) def test_sympy__stats__crv_types__ExponentialDistribution(): from sympy.stats.crv_types import ExponentialDistribution assert _test_args(ExponentialDistribution(1)) def test_sympy__stats__crv_types__ExponentialPowerDistribution(): from sympy.stats.crv_types import ExponentialPowerDistribution assert _test_args(ExponentialPowerDistribution(0, 1, 1)) def test_sympy__stats__crv_types__FDistributionDistribution(): from sympy.stats.crv_types import FDistributionDistribution assert _test_args(FDistributionDistribution(1, 1)) def test_sympy__stats__crv_types__FisherZDistribution(): from sympy.stats.crv_types import FisherZDistribution assert _test_args(FisherZDistribution(1, 1)) def test_sympy__stats__crv_types__FrechetDistribution(): from sympy.stats.crv_types import FrechetDistribution assert _test_args(FrechetDistribution(1, 1, 1)) def test_sympy__stats__crv_types__GammaInverseDistribution(): from sympy.stats.crv_types import GammaInverseDistribution assert _test_args(GammaInverseDistribution(1, 1)) def test_sympy__stats__crv_types__GammaDistribution(): from sympy.stats.crv_types import GammaDistribution assert _test_args(GammaDistribution(1, 1)) def test_sympy__stats__crv_types__GumbelDistribution(): from sympy.stats.crv_types import GumbelDistribution assert _test_args(GumbelDistribution(1, 1, False)) def test_sympy__stats__crv_types__GompertzDistribution(): from sympy.stats.crv_types import GompertzDistribution assert _test_args(GompertzDistribution(1, 1)) def test_sympy__stats__crv_types__KumaraswamyDistribution(): from sympy.stats.crv_types import KumaraswamyDistribution assert _test_args(KumaraswamyDistribution(1, 1)) def test_sympy__stats__crv_types__LaplaceDistribution(): from sympy.stats.crv_types import LaplaceDistribution assert _test_args(LaplaceDistribution(0, 1)) def test_sympy__stats__crv_types__LevyDistribution(): from sympy.stats.crv_types import LevyDistribution assert _test_args(LevyDistribution(0, 1)) def test_sympy__stats__crv_types__LogCauchyDistribution(): from sympy.stats.crv_types import LogCauchyDistribution assert _test_args(LogCauchyDistribution(0, 1)) def test_sympy__stats__crv_types__LogisticDistribution(): from sympy.stats.crv_types import LogisticDistribution assert _test_args(LogisticDistribution(0, 1)) def test_sympy__stats__crv_types__LogLogisticDistribution(): from sympy.stats.crv_types import LogLogisticDistribution assert _test_args(LogLogisticDistribution(1, 1)) def test_sympy__stats__crv_types__LogitNormalDistribution(): from sympy.stats.crv_types import LogitNormalDistribution assert _test_args(LogitNormalDistribution(0, 1)) def test_sympy__stats__crv_types__LogNormalDistribution(): from sympy.stats.crv_types import LogNormalDistribution assert _test_args(LogNormalDistribution(0, 1)) def test_sympy__stats__crv_types__LomaxDistribution(): from sympy.stats.crv_types import LomaxDistribution assert _test_args(LomaxDistribution(1, 2)) def test_sympy__stats__crv_types__MaxwellDistribution(): from sympy.stats.crv_types import MaxwellDistribution assert _test_args(MaxwellDistribution(1)) def test_sympy__stats__crv_types__MoyalDistribution(): from sympy.stats.crv_types import MoyalDistribution assert _test_args(MoyalDistribution(1,2)) def test_sympy__stats__crv_types__NakagamiDistribution(): from sympy.stats.crv_types import NakagamiDistribution assert _test_args(NakagamiDistribution(1, 1)) def test_sympy__stats__crv_types__NormalDistribution(): from sympy.stats.crv_types import NormalDistribution assert _test_args(NormalDistribution(0, 1)) def test_sympy__stats__crv_types__GaussianInverseDistribution(): from sympy.stats.crv_types import GaussianInverseDistribution assert _test_args(GaussianInverseDistribution(1, 1)) def test_sympy__stats__crv_types__ParetoDistribution(): from sympy.stats.crv_types import ParetoDistribution assert _test_args(ParetoDistribution(1, 1)) def test_sympy__stats__crv_types__PowerFunctionDistribution(): from sympy.stats.crv_types import PowerFunctionDistribution assert _test_args(PowerFunctionDistribution(2,0,1)) def test_sympy__stats__crv_types__QuadraticUDistribution(): from sympy.stats.crv_types import QuadraticUDistribution assert _test_args(QuadraticUDistribution(1, 2)) def test_sympy__stats__crv_types__RaisedCosineDistribution(): from sympy.stats.crv_types import RaisedCosineDistribution assert _test_args(RaisedCosineDistribution(1, 1)) def test_sympy__stats__crv_types__RayleighDistribution(): from sympy.stats.crv_types import RayleighDistribution assert _test_args(RayleighDistribution(1)) def test_sympy__stats__crv_types__ReciprocalDistribution(): from sympy.stats.crv_types import ReciprocalDistribution assert _test_args(ReciprocalDistribution(5, 30)) def test_sympy__stats__crv_types__ShiftedGompertzDistribution(): from sympy.stats.crv_types import ShiftedGompertzDistribution assert _test_args(ShiftedGompertzDistribution(1, 1)) def test_sympy__stats__crv_types__StudentTDistribution(): from sympy.stats.crv_types import StudentTDistribution assert _test_args(StudentTDistribution(1)) def test_sympy__stats__crv_types__TrapezoidalDistribution(): from sympy.stats.crv_types import TrapezoidalDistribution assert _test_args(TrapezoidalDistribution(1, 2, 3, 4)) def test_sympy__stats__crv_types__TriangularDistribution(): from sympy.stats.crv_types import TriangularDistribution assert _test_args(TriangularDistribution(-1, 0, 1)) def test_sympy__stats__crv_types__UniformDistribution(): from sympy.stats.crv_types import UniformDistribution assert _test_args(UniformDistribution(0, 1)) def test_sympy__stats__crv_types__UniformSumDistribution(): from sympy.stats.crv_types import UniformSumDistribution assert _test_args(UniformSumDistribution(1)) def test_sympy__stats__crv_types__VonMisesDistribution(): from sympy.stats.crv_types import VonMisesDistribution assert _test_args(VonMisesDistribution(1, 1)) def test_sympy__stats__crv_types__WeibullDistribution(): from sympy.stats.crv_types import WeibullDistribution assert _test_args(WeibullDistribution(1, 1)) def test_sympy__stats__crv_types__WignerSemicircleDistribution(): from sympy.stats.crv_types import WignerSemicircleDistribution assert _test_args(WignerSemicircleDistribution(1)) def test_sympy__stats__drv_types__GeometricDistribution(): from sympy.stats.drv_types import GeometricDistribution assert _test_args(GeometricDistribution(.5)) def test_sympy__stats__drv_types__HermiteDistribution(): from sympy.stats.drv_types import HermiteDistribution assert _test_args(HermiteDistribution(1, 2)) def test_sympy__stats__drv_types__LogarithmicDistribution(): from sympy.stats.drv_types import LogarithmicDistribution assert _test_args(LogarithmicDistribution(.5)) def test_sympy__stats__drv_types__NegativeBinomialDistribution(): from sympy.stats.drv_types import NegativeBinomialDistribution assert _test_args(NegativeBinomialDistribution(.5, .5)) def test_sympy__stats__drv_types__FlorySchulzDistribution(): from sympy.stats.drv_types import FlorySchulzDistribution assert _test_args(FlorySchulzDistribution(.5)) def test_sympy__stats__drv_types__PoissonDistribution(): from sympy.stats.drv_types import PoissonDistribution assert _test_args(PoissonDistribution(1)) def test_sympy__stats__drv_types__SkellamDistribution(): from sympy.stats.drv_types import SkellamDistribution assert _test_args(SkellamDistribution(1, 1)) def test_sympy__stats__drv_types__YuleSimonDistribution(): from sympy.stats.drv_types import YuleSimonDistribution assert _test_args(YuleSimonDistribution(.5)) def test_sympy__stats__drv_types__ZetaDistribution(): from sympy.stats.drv_types import ZetaDistribution assert _test_args(ZetaDistribution(1.5)) def test_sympy__stats__joint_rv__JointDistribution(): from sympy.stats.joint_rv import JointDistribution assert _test_args(JointDistribution(1, 2, 3, 4)) def test_sympy__stats__joint_rv_types__MultivariateNormalDistribution(): from sympy.stats.joint_rv_types import MultivariateNormalDistribution assert _test_args( MultivariateNormalDistribution([0, 1], [[1, 0],[0, 1]])) def test_sympy__stats__joint_rv_types__MultivariateLaplaceDistribution(): from sympy.stats.joint_rv_types import MultivariateLaplaceDistribution assert _test_args(MultivariateLaplaceDistribution([0, 1], [[1, 0],[0, 1]])) def test_sympy__stats__joint_rv_types__MultivariateTDistribution(): from sympy.stats.joint_rv_types import MultivariateTDistribution assert _test_args(MultivariateTDistribution([0, 1], [[1, 0],[0, 1]], 1)) def test_sympy__stats__joint_rv_types__NormalGammaDistribution(): from sympy.stats.joint_rv_types import NormalGammaDistribution assert _test_args(NormalGammaDistribution(1, 2, 3, 4)) def test_sympy__stats__joint_rv_types__GeneralizedMultivariateLogGammaDistribution(): from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGammaDistribution v, l, mu = (4, [1, 2, 3, 4], [1, 2, 3, 4]) assert _test_args(GeneralizedMultivariateLogGammaDistribution(S.Half, v, l, mu)) def test_sympy__stats__joint_rv_types__MultivariateBetaDistribution(): from sympy.stats.joint_rv_types import MultivariateBetaDistribution assert _test_args(MultivariateBetaDistribution([1, 2, 3])) def test_sympy__stats__joint_rv_types__MultivariateEwensDistribution(): from sympy.stats.joint_rv_types import MultivariateEwensDistribution assert _test_args(MultivariateEwensDistribution(5, 1)) def test_sympy__stats__joint_rv_types__MultinomialDistribution(): from sympy.stats.joint_rv_types import MultinomialDistribution assert _test_args(MultinomialDistribution(5, [0.5, 0.1, 0.3])) def test_sympy__stats__joint_rv_types__NegativeMultinomialDistribution(): from sympy.stats.joint_rv_types import NegativeMultinomialDistribution assert _test_args(NegativeMultinomialDistribution(5, [0.5, 0.1, 0.3])) def test_sympy__stats__rv__RandomIndexedSymbol(): from sympy.stats.rv import RandomIndexedSymbol, pspace from sympy.stats.stochastic_process_types import DiscreteMarkovChain X = DiscreteMarkovChain("X") assert _test_args(RandomIndexedSymbol(X[0].symbol, pspace(X[0]))) def test_sympy__stats__rv__RandomMatrixSymbol(): from sympy.stats.rv import RandomMatrixSymbol from sympy.stats.random_matrix import RandomMatrixPSpace pspace = RandomMatrixPSpace('P') assert _test_args(RandomMatrixSymbol('M', 3, 3, pspace)) def test_sympy__stats__stochastic_process__StochasticPSpace(): from sympy.stats.stochastic_process import StochasticPSpace from sympy.stats.stochastic_process_types import StochasticProcess from sympy.stats.frv_types import BernoulliDistribution assert _test_args(StochasticPSpace("Y", StochasticProcess("Y", [1, 2, 3]), BernoulliDistribution(S.Half, 1, 0))) def test_sympy__stats__stochastic_process_types__StochasticProcess(): from sympy.stats.stochastic_process_types import StochasticProcess assert _test_args(StochasticProcess("Y", [1, 2, 3])) def test_sympy__stats__stochastic_process_types__MarkovProcess(): from sympy.stats.stochastic_process_types import MarkovProcess assert _test_args(MarkovProcess("Y", [1, 2, 3])) def test_sympy__stats__stochastic_process_types__DiscreteTimeStochasticProcess(): from sympy.stats.stochastic_process_types import DiscreteTimeStochasticProcess assert _test_args(DiscreteTimeStochasticProcess("Y", [1, 2, 3])) def test_sympy__stats__stochastic_process_types__ContinuousTimeStochasticProcess(): from sympy.stats.stochastic_process_types import ContinuousTimeStochasticProcess assert _test_args(ContinuousTimeStochasticProcess("Y", [1, 2, 3])) def test_sympy__stats__stochastic_process_types__TransitionMatrixOf(): from sympy.stats.stochastic_process_types import TransitionMatrixOf, DiscreteMarkovChain from sympy.matrices.expressions.matexpr import MatrixSymbol DMC = DiscreteMarkovChain("Y") assert _test_args(TransitionMatrixOf(DMC, MatrixSymbol('T', 3, 3))) def test_sympy__stats__stochastic_process_types__GeneratorMatrixOf(): from sympy.stats.stochastic_process_types import GeneratorMatrixOf, ContinuousMarkovChain from sympy.matrices.expressions.matexpr import MatrixSymbol DMC = ContinuousMarkovChain("Y") assert _test_args(GeneratorMatrixOf(DMC, MatrixSymbol('T', 3, 3))) def test_sympy__stats__stochastic_process_types__StochasticStateSpaceOf(): from sympy.stats.stochastic_process_types import StochasticStateSpaceOf, DiscreteMarkovChain DMC = DiscreteMarkovChain("Y") assert _test_args(StochasticStateSpaceOf(DMC, [0, 1, 2])) def test_sympy__stats__stochastic_process_types__DiscreteMarkovChain(): from sympy.stats.stochastic_process_types import DiscreteMarkovChain from sympy.matrices.expressions.matexpr import MatrixSymbol assert _test_args(DiscreteMarkovChain("Y", [0, 1, 2], MatrixSymbol('T', 3, 3))) def test_sympy__stats__stochastic_process_types__ContinuousMarkovChain(): from sympy.stats.stochastic_process_types import ContinuousMarkovChain from sympy.matrices.expressions.matexpr import MatrixSymbol assert _test_args(ContinuousMarkovChain("Y", [0, 1, 2], MatrixSymbol('T', 3, 3))) def test_sympy__stats__stochastic_process_types__BernoulliProcess(): from sympy.stats.stochastic_process_types import BernoulliProcess assert _test_args(BernoulliProcess("B", 0.5, 1, 0)) def test_sympy__stats__stochastic_process_types__CountingProcess(): from sympy.stats.stochastic_process_types import CountingProcess assert _test_args(CountingProcess("C")) def test_sympy__stats__stochastic_process_types__PoissonProcess(): from sympy.stats.stochastic_process_types import PoissonProcess assert _test_args(PoissonProcess("X", 2)) def test_sympy__stats__stochastic_process_types__WienerProcess(): from sympy.stats.stochastic_process_types import WienerProcess assert _test_args(WienerProcess("X")) def test_sympy__stats__stochastic_process_types__GammaProcess(): from sympy.stats.stochastic_process_types import GammaProcess assert _test_args(GammaProcess("X", 1, 2)) def test_sympy__stats__random_matrix__RandomMatrixPSpace(): from sympy.stats.random_matrix import RandomMatrixPSpace from sympy.stats.random_matrix_models import RandomMatrixEnsembleModel model = RandomMatrixEnsembleModel('R', 3) assert _test_args(RandomMatrixPSpace('P', model=model)) def test_sympy__stats__random_matrix_models__RandomMatrixEnsembleModel(): from sympy.stats.random_matrix_models import RandomMatrixEnsembleModel assert _test_args(RandomMatrixEnsembleModel('R', 3)) def test_sympy__stats__random_matrix_models__GaussianEnsembleModel(): from sympy.stats.random_matrix_models import GaussianEnsembleModel assert _test_args(GaussianEnsembleModel('G', 3)) def test_sympy__stats__random_matrix_models__GaussianUnitaryEnsembleModel(): from sympy.stats.random_matrix_models import GaussianUnitaryEnsembleModel assert _test_args(GaussianUnitaryEnsembleModel('U', 3)) def test_sympy__stats__random_matrix_models__GaussianOrthogonalEnsembleModel(): from sympy.stats.random_matrix_models import GaussianOrthogonalEnsembleModel assert _test_args(GaussianOrthogonalEnsembleModel('U', 3)) def test_sympy__stats__random_matrix_models__GaussianSymplecticEnsembleModel(): from sympy.stats.random_matrix_models import GaussianSymplecticEnsembleModel assert _test_args(GaussianSymplecticEnsembleModel('U', 3)) def test_sympy__stats__random_matrix_models__CircularEnsembleModel(): from sympy.stats.random_matrix_models import CircularEnsembleModel assert _test_args(CircularEnsembleModel('C', 3)) def test_sympy__stats__random_matrix_models__CircularUnitaryEnsembleModel(): from sympy.stats.random_matrix_models import CircularUnitaryEnsembleModel assert _test_args(CircularUnitaryEnsembleModel('U', 3)) def test_sympy__stats__random_matrix_models__CircularOrthogonalEnsembleModel(): from sympy.stats.random_matrix_models import CircularOrthogonalEnsembleModel assert _test_args(CircularOrthogonalEnsembleModel('O', 3)) def test_sympy__stats__random_matrix_models__CircularSymplecticEnsembleModel(): from sympy.stats.random_matrix_models import CircularSymplecticEnsembleModel assert _test_args(CircularSymplecticEnsembleModel('S', 3)) def test_sympy__stats__symbolic_multivariate_probability__ExpectationMatrix(): from sympy.stats import ExpectationMatrix from sympy.stats.rv import RandomMatrixSymbol assert _test_args(ExpectationMatrix(RandomMatrixSymbol('R', 2, 1))) def test_sympy__stats__symbolic_multivariate_probability__VarianceMatrix(): from sympy.stats import VarianceMatrix from sympy.stats.rv import RandomMatrixSymbol assert _test_args(VarianceMatrix(RandomMatrixSymbol('R', 3, 1))) def test_sympy__stats__symbolic_multivariate_probability__CrossCovarianceMatrix(): from sympy.stats import CrossCovarianceMatrix from sympy.stats.rv import RandomMatrixSymbol assert _test_args(CrossCovarianceMatrix(RandomMatrixSymbol('R', 3, 1), RandomMatrixSymbol('X', 3, 1))) def test_sympy__stats__matrix_distributions__MatrixPSpace(): from sympy.stats.matrix_distributions import MatrixDistribution, MatrixPSpace from sympy.matrices.dense import Matrix M = MatrixDistribution(1, Matrix([[1, 0], [0, 1]])) assert _test_args(MatrixPSpace('M', M, 2, 2)) def test_sympy__stats__matrix_distributions__MatrixDistribution(): from sympy.stats.matrix_distributions import MatrixDistribution from sympy.matrices.dense import Matrix assert _test_args(MatrixDistribution(1, Matrix([[1, 0], [0, 1]]))) def test_sympy__stats__matrix_distributions__MatrixGammaDistribution(): from sympy.stats.matrix_distributions import MatrixGammaDistribution from sympy.matrices.dense import Matrix assert _test_args(MatrixGammaDistribution(3, 4, Matrix([[1, 0], [0, 1]]))) def test_sympy__stats__matrix_distributions__WishartDistribution(): from sympy.stats.matrix_distributions import WishartDistribution from sympy.matrices.dense import Matrix assert _test_args(WishartDistribution(3, Matrix([[1, 0], [0, 1]]))) def test_sympy__stats__matrix_distributions__MatrixNormalDistribution(): from sympy.stats.matrix_distributions import MatrixNormalDistribution from sympy.matrices.expressions.matexpr import MatrixSymbol L = MatrixSymbol('L', 1, 2) S1 = MatrixSymbol('S1', 1, 1) S2 = MatrixSymbol('S2', 2, 2) assert _test_args(MatrixNormalDistribution(L, S1, S2)) def test_sympy__stats__matrix_distributions__MatrixStudentTDistribution(): from sympy.stats.matrix_distributions import MatrixStudentTDistribution from sympy.matrices.expressions.matexpr import MatrixSymbol v = symbols('v', positive=True) Omega = MatrixSymbol('Omega', 3, 3) Sigma = MatrixSymbol('Sigma', 1, 1) Location = MatrixSymbol('Location', 1, 3) assert _test_args(MatrixStudentTDistribution(v, Location, Omega, Sigma)) def test_sympy__utilities__matchpy_connector__WildDot(): from sympy.utilities.matchpy_connector import WildDot assert _test_args(WildDot("w_")) def test_sympy__utilities__matchpy_connector__WildPlus(): from sympy.utilities.matchpy_connector import WildPlus assert _test_args(WildPlus("w__")) def test_sympy__utilities__matchpy_connector__WildStar(): from sympy.utilities.matchpy_connector import WildStar assert _test_args(WildStar("w___")) def test_sympy__core__symbol__Str(): from sympy.core.symbol import Str assert _test_args(Str('t')) def test_sympy__core__symbol__Dummy(): from sympy.core.symbol import Dummy assert _test_args(Dummy('t')) def test_sympy__core__symbol__Symbol(): from sympy.core.symbol import Symbol assert _test_args(Symbol('t')) def test_sympy__core__symbol__Wild(): from sympy.core.symbol import Wild assert _test_args(Wild('x', exclude=[x])) @SKIP("abstract class") def test_sympy__functions__combinatorial__factorials__CombinatorialFunction(): pass def test_sympy__functions__combinatorial__factorials__FallingFactorial(): from sympy.functions.combinatorial.factorials import FallingFactorial assert _test_args(FallingFactorial(2, x)) def test_sympy__functions__combinatorial__factorials__MultiFactorial(): from sympy.functions.combinatorial.factorials import MultiFactorial assert _test_args(MultiFactorial(x)) def test_sympy__functions__combinatorial__factorials__RisingFactorial(): from sympy.functions.combinatorial.factorials import RisingFactorial assert _test_args(RisingFactorial(2, x)) def test_sympy__functions__combinatorial__factorials__binomial(): from sympy.functions.combinatorial.factorials import binomial assert _test_args(binomial(2, x)) def test_sympy__functions__combinatorial__factorials__subfactorial(): from sympy.functions.combinatorial.factorials import subfactorial assert _test_args(subfactorial(x)) def test_sympy__functions__combinatorial__factorials__factorial(): from sympy.functions.combinatorial.factorials import factorial assert _test_args(factorial(x)) def test_sympy__functions__combinatorial__factorials__factorial2(): from sympy.functions.combinatorial.factorials import factorial2 assert _test_args(factorial2(x)) def test_sympy__functions__combinatorial__numbers__bell(): from sympy.functions.combinatorial.numbers import bell assert _test_args(bell(x, y)) def test_sympy__functions__combinatorial__numbers__bernoulli(): from sympy.functions.combinatorial.numbers import bernoulli assert _test_args(bernoulli(x)) def test_sympy__functions__combinatorial__numbers__catalan(): from sympy.functions.combinatorial.numbers import catalan assert _test_args(catalan(x)) def test_sympy__functions__combinatorial__numbers__genocchi(): from sympy.functions.combinatorial.numbers import genocchi assert _test_args(genocchi(x)) def test_sympy__functions__combinatorial__numbers__euler(): from sympy.functions.combinatorial.numbers import euler assert _test_args(euler(x)) def test_sympy__functions__combinatorial__numbers__carmichael(): from sympy.functions.combinatorial.numbers import carmichael assert _test_args(carmichael(x)) def test_sympy__functions__combinatorial__numbers__motzkin(): from sympy.functions.combinatorial.numbers import motzkin assert _test_args(motzkin(5)) def test_sympy__functions__combinatorial__numbers__fibonacci(): from sympy.functions.combinatorial.numbers import fibonacci assert _test_args(fibonacci(x)) def test_sympy__functions__combinatorial__numbers__tribonacci(): from sympy.functions.combinatorial.numbers import tribonacci assert _test_args(tribonacci(x)) def test_sympy__functions__combinatorial__numbers__harmonic(): from sympy.functions.combinatorial.numbers import harmonic assert _test_args(harmonic(x, 2)) def test_sympy__functions__combinatorial__numbers__lucas(): from sympy.functions.combinatorial.numbers import lucas assert _test_args(lucas(x)) def test_sympy__functions__combinatorial__numbers__partition(): from sympy.core.symbol import Symbol from sympy.functions.combinatorial.numbers import partition assert _test_args(partition(Symbol('a', integer=True))) def test_sympy__functions__elementary__complexes__Abs(): from sympy.functions.elementary.complexes import Abs assert _test_args(Abs(x)) def test_sympy__functions__elementary__complexes__adjoint(): from sympy.functions.elementary.complexes import adjoint assert _test_args(adjoint(x)) def test_sympy__functions__elementary__complexes__arg(): from sympy.functions.elementary.complexes import arg assert _test_args(arg(x)) def test_sympy__functions__elementary__complexes__conjugate(): from sympy.functions.elementary.complexes import conjugate assert _test_args(conjugate(x)) def test_sympy__functions__elementary__complexes__im(): from sympy.functions.elementary.complexes import im assert _test_args(im(x)) def test_sympy__functions__elementary__complexes__re(): from sympy.functions.elementary.complexes import re assert _test_args(re(x)) def test_sympy__functions__elementary__complexes__sign(): from sympy.functions.elementary.complexes import sign assert _test_args(sign(x)) def test_sympy__functions__elementary__complexes__polar_lift(): from sympy.functions.elementary.complexes import polar_lift assert _test_args(polar_lift(x)) def test_sympy__functions__elementary__complexes__periodic_argument(): from sympy.functions.elementary.complexes import periodic_argument assert _test_args(periodic_argument(x, y)) def test_sympy__functions__elementary__complexes__principal_branch(): from sympy.functions.elementary.complexes import principal_branch assert _test_args(principal_branch(x, y)) def test_sympy__functions__elementary__complexes__transpose(): from sympy.functions.elementary.complexes import transpose assert _test_args(transpose(x)) def test_sympy__functions__elementary__exponential__LambertW(): from sympy.functions.elementary.exponential import LambertW assert _test_args(LambertW(2)) @SKIP("abstract class") def test_sympy__functions__elementary__exponential__ExpBase(): pass def test_sympy__functions__elementary__exponential__exp(): from sympy.functions.elementary.exponential import exp assert _test_args(exp(2)) def test_sympy__functions__elementary__exponential__exp_polar(): from sympy.functions.elementary.exponential import exp_polar assert _test_args(exp_polar(2)) def test_sympy__functions__elementary__exponential__log(): from sympy.functions.elementary.exponential import log assert _test_args(log(2)) @SKIP("abstract class") def test_sympy__functions__elementary__hyperbolic__HyperbolicFunction(): pass @SKIP("abstract class") def test_sympy__functions__elementary__hyperbolic__ReciprocalHyperbolicFunction(): pass @SKIP("abstract class") def test_sympy__functions__elementary__hyperbolic__InverseHyperbolicFunction(): pass def test_sympy__functions__elementary__hyperbolic__acosh(): from sympy.functions.elementary.hyperbolic import acosh assert _test_args(acosh(2)) def test_sympy__functions__elementary__hyperbolic__acoth(): from sympy.functions.elementary.hyperbolic import acoth assert _test_args(acoth(2)) def test_sympy__functions__elementary__hyperbolic__asinh(): from sympy.functions.elementary.hyperbolic import asinh assert _test_args(asinh(2)) def test_sympy__functions__elementary__hyperbolic__atanh(): from sympy.functions.elementary.hyperbolic import atanh assert _test_args(atanh(2)) def test_sympy__functions__elementary__hyperbolic__asech(): from sympy.functions.elementary.hyperbolic import asech assert _test_args(asech(x)) def test_sympy__functions__elementary__hyperbolic__acsch(): from sympy.functions.elementary.hyperbolic import acsch assert _test_args(acsch(x)) def test_sympy__functions__elementary__hyperbolic__cosh(): from sympy.functions.elementary.hyperbolic import cosh assert _test_args(cosh(2)) def test_sympy__functions__elementary__hyperbolic__coth(): from sympy.functions.elementary.hyperbolic import coth assert _test_args(coth(2)) def test_sympy__functions__elementary__hyperbolic__csch(): from sympy.functions.elementary.hyperbolic import csch assert _test_args(csch(2)) def test_sympy__functions__elementary__hyperbolic__sech(): from sympy.functions.elementary.hyperbolic import sech assert _test_args(sech(2)) def test_sympy__functions__elementary__hyperbolic__sinh(): from sympy.functions.elementary.hyperbolic import sinh assert _test_args(sinh(2)) def test_sympy__functions__elementary__hyperbolic__tanh(): from sympy.functions.elementary.hyperbolic import tanh assert _test_args(tanh(2)) @SKIP("abstract class") def test_sympy__functions__elementary__integers__RoundFunction(): pass def test_sympy__functions__elementary__integers__ceiling(): from sympy.functions.elementary.integers import ceiling assert _test_args(ceiling(x)) def test_sympy__functions__elementary__integers__floor(): from sympy.functions.elementary.integers import floor assert _test_args(floor(x)) def test_sympy__functions__elementary__integers__frac(): from sympy.functions.elementary.integers import frac assert _test_args(frac(x)) def test_sympy__functions__elementary__miscellaneous__IdentityFunction(): from sympy.functions.elementary.miscellaneous import IdentityFunction assert _test_args(IdentityFunction()) def test_sympy__functions__elementary__miscellaneous__Max(): from sympy.functions.elementary.miscellaneous import Max assert _test_args(Max(x, 2)) def test_sympy__functions__elementary__miscellaneous__Min(): from sympy.functions.elementary.miscellaneous import Min assert _test_args(Min(x, 2)) @SKIP("abstract class") def test_sympy__functions__elementary__miscellaneous__MinMaxBase(): pass def test_sympy__functions__elementary__miscellaneous__Rem(): from sympy.functions.elementary.miscellaneous import Rem assert _test_args(Rem(x, 2)) def test_sympy__functions__elementary__piecewise__ExprCondPair(): from sympy.functions.elementary.piecewise import ExprCondPair assert _test_args(ExprCondPair(1, True)) def test_sympy__functions__elementary__piecewise__Piecewise(): from sympy.functions.elementary.piecewise import Piecewise assert _test_args(Piecewise((1, x >= 0), (0, True))) @SKIP("abstract class") def test_sympy__functions__elementary__trigonometric__TrigonometricFunction(): pass @SKIP("abstract class") def test_sympy__functions__elementary__trigonometric__ReciprocalTrigonometricFunction(): pass @SKIP("abstract class") def test_sympy__functions__elementary__trigonometric__InverseTrigonometricFunction(): pass def test_sympy__functions__elementary__trigonometric__acos(): from sympy.functions.elementary.trigonometric import acos assert _test_args(acos(2)) def test_sympy__functions__elementary__trigonometric__acot(): from sympy.functions.elementary.trigonometric import acot assert _test_args(acot(2)) def test_sympy__functions__elementary__trigonometric__asin(): from sympy.functions.elementary.trigonometric import asin assert _test_args(asin(2)) def test_sympy__functions__elementary__trigonometric__asec(): from sympy.functions.elementary.trigonometric import asec assert _test_args(asec(x)) def test_sympy__functions__elementary__trigonometric__acsc(): from sympy.functions.elementary.trigonometric import acsc assert _test_args(acsc(x)) def test_sympy__functions__elementary__trigonometric__atan(): from sympy.functions.elementary.trigonometric import atan assert _test_args(atan(2)) def test_sympy__functions__elementary__trigonometric__atan2(): from sympy.functions.elementary.trigonometric import atan2 assert _test_args(atan2(2, 3)) def test_sympy__functions__elementary__trigonometric__cos(): from sympy.functions.elementary.trigonometric import cos assert _test_args(cos(2)) def test_sympy__functions__elementary__trigonometric__csc(): from sympy.functions.elementary.trigonometric import csc assert _test_args(csc(2)) def test_sympy__functions__elementary__trigonometric__cot(): from sympy.functions.elementary.trigonometric import cot assert _test_args(cot(2)) def test_sympy__functions__elementary__trigonometric__sin(): assert _test_args(sin(2)) def test_sympy__functions__elementary__trigonometric__sinc(): from sympy.functions.elementary.trigonometric import sinc assert _test_args(sinc(2)) def test_sympy__functions__elementary__trigonometric__sec(): from sympy.functions.elementary.trigonometric import sec assert _test_args(sec(2)) def test_sympy__functions__elementary__trigonometric__tan(): from sympy.functions.elementary.trigonometric import tan assert _test_args(tan(2)) @SKIP("abstract class") def test_sympy__functions__special__bessel__BesselBase(): pass @SKIP("abstract class") def test_sympy__functions__special__bessel__SphericalBesselBase(): pass @SKIP("abstract class") def test_sympy__functions__special__bessel__SphericalHankelBase(): pass def test_sympy__functions__special__bessel__besseli(): from sympy.functions.special.bessel import besseli assert _test_args(besseli(x, 1)) def test_sympy__functions__special__bessel__besselj(): from sympy.functions.special.bessel import besselj assert _test_args(besselj(x, 1)) def test_sympy__functions__special__bessel__besselk(): from sympy.functions.special.bessel import besselk assert _test_args(besselk(x, 1)) def test_sympy__functions__special__bessel__bessely(): from sympy.functions.special.bessel import bessely assert _test_args(bessely(x, 1)) def test_sympy__functions__special__bessel__hankel1(): from sympy.functions.special.bessel import hankel1 assert _test_args(hankel1(x, 1)) def test_sympy__functions__special__bessel__hankel2(): from sympy.functions.special.bessel import hankel2 assert _test_args(hankel2(x, 1)) def test_sympy__functions__special__bessel__jn(): from sympy.functions.special.bessel import jn assert _test_args(jn(0, x)) def test_sympy__functions__special__bessel__yn(): from sympy.functions.special.bessel import yn assert _test_args(yn(0, x)) def test_sympy__functions__special__bessel__hn1(): from sympy.functions.special.bessel import hn1 assert _test_args(hn1(0, x)) def test_sympy__functions__special__bessel__hn2(): from sympy.functions.special.bessel import hn2 assert _test_args(hn2(0, x)) def test_sympy__functions__special__bessel__AiryBase(): pass def test_sympy__functions__special__bessel__airyai(): from sympy.functions.special.bessel import airyai assert _test_args(airyai(2)) def test_sympy__functions__special__bessel__airybi(): from sympy.functions.special.bessel import airybi assert _test_args(airybi(2)) def test_sympy__functions__special__bessel__airyaiprime(): from sympy.functions.special.bessel import airyaiprime assert _test_args(airyaiprime(2)) def test_sympy__functions__special__bessel__airybiprime(): from sympy.functions.special.bessel import airybiprime assert _test_args(airybiprime(2)) def test_sympy__functions__special__bessel__marcumq(): from sympy.functions.special.bessel import marcumq assert _test_args(marcumq(x, y, z)) def test_sympy__functions__special__elliptic_integrals__elliptic_k(): from sympy.functions.special.elliptic_integrals import elliptic_k as K assert _test_args(K(x)) def test_sympy__functions__special__elliptic_integrals__elliptic_f(): from sympy.functions.special.elliptic_integrals import elliptic_f as F assert _test_args(F(x, y)) def test_sympy__functions__special__elliptic_integrals__elliptic_e(): from sympy.functions.special.elliptic_integrals import elliptic_e as E assert _test_args(E(x)) assert _test_args(E(x, y)) def test_sympy__functions__special__elliptic_integrals__elliptic_pi(): from sympy.functions.special.elliptic_integrals import elliptic_pi as P assert _test_args(P(x, y)) assert _test_args(P(x, y, z)) def test_sympy__functions__special__delta_functions__DiracDelta(): from sympy.functions.special.delta_functions import DiracDelta assert _test_args(DiracDelta(x, 1)) def test_sympy__functions__special__singularity_functions__SingularityFunction(): from sympy.functions.special.singularity_functions import SingularityFunction assert _test_args(SingularityFunction(x, y, z)) def test_sympy__functions__special__delta_functions__Heaviside(): from sympy.functions.special.delta_functions import Heaviside assert _test_args(Heaviside(x)) def test_sympy__functions__special__error_functions__erf(): from sympy.functions.special.error_functions import erf assert _test_args(erf(2)) def test_sympy__functions__special__error_functions__erfc(): from sympy.functions.special.error_functions import erfc assert _test_args(erfc(2)) def test_sympy__functions__special__error_functions__erfi(): from sympy.functions.special.error_functions import erfi assert _test_args(erfi(2)) def test_sympy__functions__special__error_functions__erf2(): from sympy.functions.special.error_functions import erf2 assert _test_args(erf2(2, 3)) def test_sympy__functions__special__error_functions__erfinv(): from sympy.functions.special.error_functions import erfinv assert _test_args(erfinv(2)) def test_sympy__functions__special__error_functions__erfcinv(): from sympy.functions.special.error_functions import erfcinv assert _test_args(erfcinv(2)) def test_sympy__functions__special__error_functions__erf2inv(): from sympy.functions.special.error_functions import erf2inv assert _test_args(erf2inv(2, 3)) @SKIP("abstract class") def test_sympy__functions__special__error_functions__FresnelIntegral(): pass def test_sympy__functions__special__error_functions__fresnels(): from sympy.functions.special.error_functions import fresnels assert _test_args(fresnels(2)) def test_sympy__functions__special__error_functions__fresnelc(): from sympy.functions.special.error_functions import fresnelc assert _test_args(fresnelc(2)) def test_sympy__functions__special__error_functions__erfs(): from sympy.functions.special.error_functions import _erfs assert _test_args(_erfs(2)) def test_sympy__functions__special__error_functions__Ei(): from sympy.functions.special.error_functions import Ei assert _test_args(Ei(2)) def test_sympy__functions__special__error_functions__li(): from sympy.functions.special.error_functions import li assert _test_args(li(2)) def test_sympy__functions__special__error_functions__Li(): from sympy.functions.special.error_functions import Li assert _test_args(Li(5)) @SKIP("abstract class") def test_sympy__functions__special__error_functions__TrigonometricIntegral(): pass def test_sympy__functions__special__error_functions__Si(): from sympy.functions.special.error_functions import Si assert _test_args(Si(2)) def test_sympy__functions__special__error_functions__Ci(): from sympy.functions.special.error_functions import Ci assert _test_args(Ci(2)) def test_sympy__functions__special__error_functions__Shi(): from sympy.functions.special.error_functions import Shi assert _test_args(Shi(2)) def test_sympy__functions__special__error_functions__Chi(): from sympy.functions.special.error_functions import Chi assert _test_args(Chi(2)) def test_sympy__functions__special__error_functions__expint(): from sympy.functions.special.error_functions import expint assert _test_args(expint(y, x)) def test_sympy__functions__special__gamma_functions__gamma(): from sympy.functions.special.gamma_functions import gamma assert _test_args(gamma(x)) def test_sympy__functions__special__gamma_functions__loggamma(): from sympy.functions.special.gamma_functions import loggamma assert _test_args(loggamma(x)) def test_sympy__functions__special__gamma_functions__lowergamma(): from sympy.functions.special.gamma_functions import lowergamma assert _test_args(lowergamma(x, 2)) def test_sympy__functions__special__gamma_functions__polygamma(): from sympy.functions.special.gamma_functions import polygamma assert _test_args(polygamma(x, 2)) def test_sympy__functions__special__gamma_functions__digamma(): from sympy.functions.special.gamma_functions import digamma assert _test_args(digamma(x)) def test_sympy__functions__special__gamma_functions__trigamma(): from sympy.functions.special.gamma_functions import trigamma assert _test_args(trigamma(x)) def test_sympy__functions__special__gamma_functions__uppergamma(): from sympy.functions.special.gamma_functions import uppergamma assert _test_args(uppergamma(x, 2)) def test_sympy__functions__special__gamma_functions__multigamma(): from sympy.functions.special.gamma_functions import multigamma assert _test_args(multigamma(x, 1)) def test_sympy__functions__special__beta_functions__beta(): from sympy.functions.special.beta_functions import beta assert _test_args(beta(x)) assert _test_args(beta(x, x)) def test_sympy__functions__special__beta_functions__betainc(): from sympy.functions.special.beta_functions import betainc assert _test_args(betainc(a, b, x, y)) def test_sympy__functions__special__beta_functions__betainc_regularized(): from sympy.functions.special.beta_functions import betainc_regularized assert _test_args(betainc_regularized(a, b, x, y)) def test_sympy__functions__special__mathieu_functions__MathieuBase(): pass def test_sympy__functions__special__mathieu_functions__mathieus(): from sympy.functions.special.mathieu_functions import mathieus assert _test_args(mathieus(1, 1, 1)) def test_sympy__functions__special__mathieu_functions__mathieuc(): from sympy.functions.special.mathieu_functions import mathieuc assert _test_args(mathieuc(1, 1, 1)) def test_sympy__functions__special__mathieu_functions__mathieusprime(): from sympy.functions.special.mathieu_functions import mathieusprime assert _test_args(mathieusprime(1, 1, 1)) def test_sympy__functions__special__mathieu_functions__mathieucprime(): from sympy.functions.special.mathieu_functions import mathieucprime assert _test_args(mathieucprime(1, 1, 1)) @SKIP("abstract class") def test_sympy__functions__special__hyper__TupleParametersBase(): pass @SKIP("abstract class") def test_sympy__functions__special__hyper__TupleArg(): pass def test_sympy__functions__special__hyper__hyper(): from sympy.functions.special.hyper import hyper assert _test_args(hyper([1, 2, 3], [4, 5], x)) def test_sympy__functions__special__hyper__meijerg(): from sympy.functions.special.hyper import meijerg assert _test_args(meijerg([1, 2, 3], [4, 5], [6], [], x)) @SKIP("abstract class") def test_sympy__functions__special__hyper__HyperRep(): pass def test_sympy__functions__special__hyper__HyperRep_power1(): from sympy.functions.special.hyper import HyperRep_power1 assert _test_args(HyperRep_power1(x, y)) def test_sympy__functions__special__hyper__HyperRep_power2(): from sympy.functions.special.hyper import HyperRep_power2 assert _test_args(HyperRep_power2(x, y)) def test_sympy__functions__special__hyper__HyperRep_log1(): from sympy.functions.special.hyper import HyperRep_log1 assert _test_args(HyperRep_log1(x)) def test_sympy__functions__special__hyper__HyperRep_atanh(): from sympy.functions.special.hyper import HyperRep_atanh assert _test_args(HyperRep_atanh(x)) def test_sympy__functions__special__hyper__HyperRep_asin1(): from sympy.functions.special.hyper import HyperRep_asin1 assert _test_args(HyperRep_asin1(x)) def test_sympy__functions__special__hyper__HyperRep_asin2(): from sympy.functions.special.hyper import HyperRep_asin2 assert _test_args(HyperRep_asin2(x)) def test_sympy__functions__special__hyper__HyperRep_sqrts1(): from sympy.functions.special.hyper import HyperRep_sqrts1 assert _test_args(HyperRep_sqrts1(x, y)) def test_sympy__functions__special__hyper__HyperRep_sqrts2(): from sympy.functions.special.hyper import HyperRep_sqrts2 assert _test_args(HyperRep_sqrts2(x, y)) def test_sympy__functions__special__hyper__HyperRep_log2(): from sympy.functions.special.hyper import HyperRep_log2 assert _test_args(HyperRep_log2(x)) def test_sympy__functions__special__hyper__HyperRep_cosasin(): from sympy.functions.special.hyper import HyperRep_cosasin assert _test_args(HyperRep_cosasin(x, y)) def test_sympy__functions__special__hyper__HyperRep_sinasin(): from sympy.functions.special.hyper import HyperRep_sinasin assert _test_args(HyperRep_sinasin(x, y)) def test_sympy__functions__special__hyper__appellf1(): from sympy.functions.special.hyper import appellf1 a, b1, b2, c, x, y = symbols('a b1 b2 c x y') assert _test_args(appellf1(a, b1, b2, c, x, y)) @SKIP("abstract class") def test_sympy__functions__special__polynomials__OrthogonalPolynomial(): pass def test_sympy__functions__special__polynomials__jacobi(): from sympy.functions.special.polynomials import jacobi assert _test_args(jacobi(x, y, 2, 2)) def test_sympy__functions__special__polynomials__gegenbauer(): from sympy.functions.special.polynomials import gegenbauer assert _test_args(gegenbauer(x, 2, 2)) def test_sympy__functions__special__polynomials__chebyshevt(): from sympy.functions.special.polynomials import chebyshevt assert _test_args(chebyshevt(x, 2)) def test_sympy__functions__special__polynomials__chebyshevt_root(): from sympy.functions.special.polynomials import chebyshevt_root assert _test_args(chebyshevt_root(3, 2)) def test_sympy__functions__special__polynomials__chebyshevu(): from sympy.functions.special.polynomials import chebyshevu assert _test_args(chebyshevu(x, 2)) def test_sympy__functions__special__polynomials__chebyshevu_root(): from sympy.functions.special.polynomials import chebyshevu_root assert _test_args(chebyshevu_root(3, 2)) def test_sympy__functions__special__polynomials__hermite(): from sympy.functions.special.polynomials import hermite assert _test_args(hermite(x, 2)) def test_sympy__functions__special__polynomials__legendre(): from sympy.functions.special.polynomials import legendre assert _test_args(legendre(x, 2)) def test_sympy__functions__special__polynomials__assoc_legendre(): from sympy.functions.special.polynomials import assoc_legendre assert _test_args(assoc_legendre(x, 0, y)) def test_sympy__functions__special__polynomials__laguerre(): from sympy.functions.special.polynomials import laguerre assert _test_args(laguerre(x, 2)) def test_sympy__functions__special__polynomials__assoc_laguerre(): from sympy.functions.special.polynomials import assoc_laguerre assert _test_args(assoc_laguerre(x, 0, y)) def test_sympy__functions__special__spherical_harmonics__Ynm(): from sympy.functions.special.spherical_harmonics import Ynm assert _test_args(Ynm(1, 1, x, y)) def test_sympy__functions__special__spherical_harmonics__Znm(): from sympy.functions.special.spherical_harmonics import Znm assert _test_args(Znm(x, y, 1, 1)) def test_sympy__functions__special__tensor_functions__LeviCivita(): from sympy.functions.special.tensor_functions import LeviCivita assert _test_args(LeviCivita(x, y, 2)) def test_sympy__functions__special__tensor_functions__KroneckerDelta(): from sympy.functions.special.tensor_functions import KroneckerDelta assert _test_args(KroneckerDelta(x, y)) def test_sympy__functions__special__zeta_functions__dirichlet_eta(): from sympy.functions.special.zeta_functions import dirichlet_eta assert _test_args(dirichlet_eta(x)) def test_sympy__functions__special__zeta_functions__riemann_xi(): from sympy.functions.special.zeta_functions import riemann_xi assert _test_args(riemann_xi(x)) def test_sympy__functions__special__zeta_functions__zeta(): from sympy.functions.special.zeta_functions import zeta assert _test_args(zeta(101)) def test_sympy__functions__special__zeta_functions__lerchphi(): from sympy.functions.special.zeta_functions import lerchphi assert _test_args(lerchphi(x, y, z)) def test_sympy__functions__special__zeta_functions__polylog(): from sympy.functions.special.zeta_functions import polylog assert _test_args(polylog(x, y)) def test_sympy__functions__special__zeta_functions__stieltjes(): from sympy.functions.special.zeta_functions import stieltjes assert _test_args(stieltjes(x, y)) def test_sympy__integrals__integrals__Integral(): from sympy.integrals.integrals import Integral assert _test_args(Integral(2, (x, 0, 1))) def test_sympy__integrals__risch__NonElementaryIntegral(): from sympy.integrals.risch import NonElementaryIntegral assert _test_args(NonElementaryIntegral(exp(-x2), x)) @SKIP("abstract class") def test_sympy__integrals__transforms__IntegralTransform(): pass def test_sympy__integrals__transforms__MellinTransform(): from sympy.integrals.transforms import MellinTransform assert _test_args(MellinTransform(2, x, y)) def test_sympy__integrals__transforms__InverseMellinTransform(): from sympy.integrals.transforms import InverseMellinTransform assert _test_args(InverseMellinTransform(2, x, y, 0, 1)) def test_sympy__integrals__transforms__LaplaceTransform(): from sympy.integrals.transforms import LaplaceTransform assert _test_args(LaplaceTransform(2, x, y)) def test_sympy__integrals__transforms__InverseLaplaceTransform(): from sympy.integrals.transforms import InverseLaplaceTransform assert _test_args(InverseLaplaceTransform(2, x, y, 0)) @SKIP("abstract class") def test_sympy__integrals__transforms__FourierTypeTransform(): pass def test_sympy__integrals__transforms__InverseFourierTransform(): from sympy.integrals.transforms import InverseFourierTransform assert _test_args(InverseFourierTransform(2, x, y)) def test_sympy__integrals__transforms__FourierTransform(): from sympy.integrals.transforms import FourierTransform assert _test_args(FourierTransform(2, x, y)) @SKIP("abstract class") def test_sympy__integrals__transforms__SineCosineTypeTransform(): pass def test_sympy__integrals__transforms__InverseSineTransform(): from sympy.integrals.transforms import InverseSineTransform assert _test_args(InverseSineTransform(2, x, y)) def test_sympy__integrals__transforms__SineTransform(): from sympy.integrals.transforms import SineTransform assert _test_args(SineTransform(2, x, y)) def test_sympy__integrals__transforms__InverseCosineTransform(): from sympy.integrals.transforms import InverseCosineTransform assert _test_args(InverseCosineTransform(2, x, y)) def test_sympy__integrals__transforms__CosineTransform(): from sympy.integrals.transforms import CosineTransform assert _test_args(CosineTransform(2, x, y)) @SKIP("abstract class") def test_sympy__integrals__transforms__HankelTypeTransform(): pass def test_sympy__integrals__transforms__InverseHankelTransform(): from sympy.integrals.transforms import InverseHankelTransform assert _test_args(InverseHankelTransform(2, x, y, 0)) def test_sympy__integrals__transforms__HankelTransform(): from sympy.integrals.transforms import HankelTransform assert _test_args(HankelTransform(2, x, y, 0)) @XFAIL def test_sympy__liealgebras__cartan_type__CartanType_generator(): from sympy.liealgebras.cartan_type import CartanType_generator assert _test_args(CartanType_generator("A2")) def test_sympy__liealgebras__cartan_type__Standard_Cartan(): from sympy.liealgebras.cartan_type import Standard_Cartan assert _test_args(Standard_Cartan("A", 2)) def test_sympy__liealgebras__weyl_group__WeylGroup(): from sympy.liealgebras.weyl_group import WeylGroup assert _test_args(WeylGroup("B4")) def test_sympy__liealgebras__root_system__RootSystem(): from sympy.liealgebras.root_system import RootSystem assert _test_args(RootSystem("A2")) def test_sympy__liealgebras__type_a__TypeA(): from sympy.liealgebras.type_a import TypeA assert _test_args(TypeA(2)) def test_sympy__liealgebras__type_b__TypeB(): from sympy.liealgebras.type_b import TypeB assert _test_args(TypeB(4)) def test_sympy__liealgebras__type_c__TypeC(): from sympy.liealgebras.type_c import TypeC assert _test_args(TypeC(4)) def test_sympy__liealgebras__type_d__TypeD(): from sympy.liealgebras.type_d import TypeD assert _test_args(TypeD(4)) def test_sympy__liealgebras__type_e__TypeE(): from sympy.liealgebras.type_e import TypeE assert _test_args(TypeE(6)) def test_sympy__liealgebras__type_f__TypeF(): from sympy.liealgebras.type_f import TypeF assert _test_args(TypeF(4)) def test_sympy__liealgebras__type_g__TypeG(): from sympy.liealgebras.type_g import TypeG assert _test_args(TypeG(2)) def test_sympy__logic__boolalg__And(): from sympy.logic.boolalg import And assert _test_args(And(x, y, 1)) @SKIP("abstract class") def test_sympy__logic__boolalg__Boolean(): pass def test_sympy__logic__boolalg__BooleanFunction(): from sympy.logic.boolalg import BooleanFunction assert _test_args(BooleanFunction(1, 2, 3)) @SKIP("abstract class") def test_sympy__logic__boolalg__BooleanAtom(): pass def test_sympy__logic__boolalg__BooleanTrue(): from sympy.logic.boolalg import true assert _test_args(true) def test_sympy__logic__boolalg__BooleanFalse(): from sympy.logic.boolalg import false assert _test_args(false) def test_sympy__logic__boolalg__Equivalent(): from sympy.logic.boolalg import Equivalent assert _test_args(Equivalent(x, 2)) def test_sympy__logic__boolalg__ITE(): from sympy.logic.boolalg import ITE assert _test_args(ITE(x, y, 1)) def test_sympy__logic__boolalg__Implies(): from sympy.logic.boolalg import Implies assert _test_args(Implies(x, y)) def test_sympy__logic__boolalg__Nand(): from sympy.logic.boolalg import Nand assert _test_args(Nand(x, y, 1)) def test_sympy__logic__boolalg__Nor(): from sympy.logic.boolalg import Nor assert _test_args(Nor(x, y)) def test_sympy__logic__boolalg__Not(): from sympy.logic.boolalg import Not assert _test_args(Not(x)) def test_sympy__logic__boolalg__Or(): from sympy.logic.boolalg import Or assert _test_args(Or(x, y)) def test_sympy__logic__boolalg__Xor(): from sympy.logic.boolalg import Xor assert _test_args(Xor(x, y, 2)) def test_sympy__logic__boolalg__Xnor(): from sympy.logic.boolalg import Xnor assert _test_args(Xnor(x, y, 2)) def test_sympy__logic__boolalg__Exclusive(): from sympy.logic.boolalg import Exclusive assert _test_args(Exclusive(x, y, z)) def test_sympy__matrices__matrices__DeferredVector(): from sympy.matrices.matrices import DeferredVector assert _test_args(DeferredVector("X")) @SKIP("abstract class") def test_sympy__matrices__expressions__matexpr__MatrixBase(): pass @SKIP("abstract class") def test_sympy__matrices__immutable__ImmutableRepMatrix(): pass def test_sympy__matrices__immutable__ImmutableDenseMatrix(): from sympy.matrices.immutable import ImmutableDenseMatrix m = ImmutableDenseMatrix([[1, 2], [3, 4]]) assert _test_args(m) assert _test_args(Basic(list(m))) m = ImmutableDenseMatrix(1, 1, [1]) assert _test_args(m) assert _test_args(Basic(list(m))) m = ImmutableDenseMatrix(2, 2, lambda i, j: 1) assert m[0, 0] is S.One m = ImmutableDenseMatrix(2, 2, lambda i, j: 1/(1 + i) + 1/(1 + j)) assert m[1, 1] is S.One # true div. will give 1.0 if i,j not sympified assert _test_args(m) assert _test_args(Basic(list(m))) def test_sympy__matrices__immutable__ImmutableSparseMatrix(): from sympy.matrices.immutable import ImmutableSparseMatrix m = ImmutableSparseMatrix([[1, 2], [3, 4]]) assert _test_args(m) assert _test_args(Basic(list(m))) m = ImmutableSparseMatrix(1, 1, {(0, 0): 1}) assert _test_args(m) assert _test_args(Basic(list(m))) m = ImmutableSparseMatrix(1, 1, [1]) assert _test_args(m) assert _test_args(Basic(list(m))) m = ImmutableSparseMatrix(2, 2, lambda i, j: 1) assert m[0, 0] is S.One m = ImmutableSparseMatrix(2, 2, lambda i, j: 1/(1 + i) + 1/(1 + j)) assert m[1, 1] is S.One # true div. will give 1.0 if i,j not sympified assert _test_args(m) assert _test_args(Basic(list(m))) def test_sympy__matrices__expressions__slice__MatrixSlice(): from sympy.matrices.expressions.slice import MatrixSlice from sympy.matrices.expressions import MatrixSymbol X = MatrixSymbol('X', 4, 4) assert _test_args(MatrixSlice(X, (0, 2), (0, 2))) def test_sympy__matrices__expressions__applyfunc__ElementwiseApplyFunction(): from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction from sympy.matrices.expressions import MatrixSymbol X = MatrixSymbol("X", x, x) func = Lambda(x, x*2) assert _test_args(ElementwiseApplyFunction(func, X)) def test_sympy__matrices__expressions__blockmatrix__BlockDiagMatrix(): from sympy.matrices.expressions.blockmatrix import BlockDiagMatrix from sympy.matrices.expressions import MatrixSymbol X = MatrixSymbol('X', x, x) Y = MatrixSymbol('Y', y, y) assert _test_args(BlockDiagMatrix(X, Y)) def test_sympy__matrices__expressions__blockmatrix__BlockMatrix(): from sympy.matrices.expressions.blockmatrix import BlockMatrix from sympy.matrices.expressions import MatrixSymbol, ZeroMatrix X = MatrixSymbol('X', x, x) Y = MatrixSymbol('Y', y, y) Z = MatrixSymbol('Z', x, y) O = ZeroMatrix(y, x) assert _test_args(BlockMatrix([[X, Z], [O, Y]])) def test_sympy__matrices__expressions__inverse__Inverse(): from sympy.matrices.expressions.inverse import Inverse from sympy.matrices.expressions import MatrixSymbol assert _test_args(Inverse(MatrixSymbol('A', 3, 3))) def test_sympy__matrices__expressions__matadd__MatAdd(): from sympy.matrices.expressions.matadd import MatAdd from sympy.matrices.expressions import MatrixSymbol X = MatrixSymbol('X', x, y) Y = MatrixSymbol('Y', x, y) assert _test_args(MatAdd(X, Y)) @SKIP("abstract class") def test_sympy__matrices__expressions__matexpr__MatrixExpr(): pass def test_sympy__matrices__expressions__matexpr__MatrixElement(): from sympy.matrices.expressions.matexpr import MatrixSymbol, MatrixElement from sympy.core.singleton import S assert _test_args(MatrixElement(MatrixSymbol('A', 3, 5), S(2), S(3))) def test_sympy__matrices__expressions__matexpr__MatrixSymbol(): from sympy.matrices.expressions.matexpr import MatrixSymbol assert _test_args(MatrixSymbol('A', 3, 5)) def test_sympy__matrices__expressions__special__OneMatrix(): from sympy.matrices.expressions.special import OneMatrix assert _test_args(OneMatrix(3, 5)) def test_sympy__matrices__expressions__special__ZeroMatrix(): from sympy.matrices.expressions.special import ZeroMatrix assert _test_args(ZeroMatrix(3, 5)) def test_sympy__matrices__expressions__special__GenericZeroMatrix(): from sympy.matrices.expressions.special import GenericZeroMatrix assert _test_args(GenericZeroMatrix()) def test_sympy__matrices__expressions__special__Identity(): from sympy.matrices.expressions.special import Identity assert _test_args(Identity(3)) def test_sympy__matrices__expressions__special__GenericIdentity(): from sympy.matrices.expressions.special import GenericIdentity assert _test_args(GenericIdentity()) def test_sympy__matrices__expressions__sets__MatrixSet(): from sympy.matrices.expressions.sets import MatrixSet from sympy.core.singleton import S assert _test_args(MatrixSet(2, 2, S.Reals)) def test_sympy__matrices__expressions__matmul__MatMul(): from sympy.matrices.expressions.matmul import MatMul from sympy.matrices.expressions import MatrixSymbol X = MatrixSymbol('X', x, y) Y = MatrixSymbol('Y', y, x) assert _test_args(MatMul(X, Y)) def test_sympy__matrices__expressions__dotproduct__DotProduct(): from sympy.matrices.expressions.dotproduct import DotProduct from sympy.matrices.expressions import MatrixSymbol X = MatrixSymbol('X', x, 1) Y = MatrixSymbol('Y', x, 1) assert _test_args(DotProduct(X, Y)) def test_sympy__matrices__expressions__diagonal__DiagonalMatrix(): from sympy.matrices.expressions.diagonal import DiagonalMatrix from sympy.matrices.expressions import MatrixSymbol x = MatrixSymbol('x', 10, 1) assert _test_args(DiagonalMatrix(x)) def test_sympy__matrices__expressions__diagonal__DiagonalOf(): from sympy.matrices.expressions.diagonal import DiagonalOf from sympy.matrices.expressions import MatrixSymbol X = MatrixSymbol('x', 10, 10) assert _test_args(DiagonalOf(X)) def test_sympy__matrices__expressions__diagonal__DiagMatrix(): from sympy.matrices.expressions.diagonal import DiagMatrix from sympy.matrices.expressions import MatrixSymbol x = MatrixSymbol('x', 10, 1) assert _test_args(DiagMatrix(x)) def test_sympy__matrices__expressions__hadamard__HadamardProduct(): from sympy.matrices.expressions.hadamard import HadamardProduct from sympy.matrices.expressions import MatrixSymbol X = MatrixSymbol('X', x, y) Y = MatrixSymbol('Y', x, y) assert _test_args(HadamardProduct(X, Y)) def test_sympy__matrices__expressions__hadamard__HadamardPower(): from sympy.matrices.expressions.hadamard import HadamardPower from sympy.matrices.expressions import MatrixSymbol from sympy.core.symbol import Symbol X = MatrixSymbol('X', x, y) n = Symbol("n") assert _test_args(HadamardPower(X, n)) def test_sympy__matrices__expressions__kronecker__KroneckerProduct(): from sympy.matrices.expressions.kronecker import KroneckerProduct from sympy.matrices.expressions import MatrixSymbol X = MatrixSymbol('X', x, y) Y = MatrixSymbol('Y', x, y) assert _test_args(KroneckerProduct(X, Y)) def test_sympy__matrices__expressions__matpow__MatPow(): from sympy.matrices.expressions.matpow import MatPow from sympy.matrices.expressions import MatrixSymbol X = MatrixSymbol('X', x, x) assert _test_args(MatPow(X, 2)) def test_sympy__matrices__expressions__transpose__Transpose(): from sympy.matrices.expressions.transpose import Transpose from sympy.matrices.expressions import MatrixSymbol assert _test_args(Transpose(MatrixSymbol('A', 3, 5))) def test_sympy__matrices__expressions__adjoint__Adjoint(): from sympy.matrices.expressions.adjoint import Adjoint from sympy.matrices.expressions import MatrixSymbol assert _test_args(Adjoint(MatrixSymbol('A', 3, 5))) def test_sympy__matrices__expressions__trace__Trace(): from sympy.matrices.expressions.trace import Trace from sympy.matrices.expressions import MatrixSymbol assert _test_args(Trace(MatrixSymbol('A', 3, 3))) def test_sympy__matrices__expressions__determinant__Determinant(): from sympy.matrices.expressions.determinant import Determinant from sympy.matrices.expressions import MatrixSymbol assert _test_args(Determinant(MatrixSymbol('A', 3, 3))) def test_sympy__matrices__expressions__determinant__Permanent(): from sympy.matrices.expressions.determinant import Permanent from sympy.matrices.expressions import MatrixSymbol assert _test_args(Permanent(MatrixSymbol('A', 3, 4))) def test_sympy__matrices__expressions__funcmatrix__FunctionMatrix(): from sympy.matrices.expressions.funcmatrix import FunctionMatrix from sympy.core.symbol import symbols i, j = symbols('i,j') assert _test_args(FunctionMatrix(3, 3, Lambda((i, j), i - j) )) def test_sympy__matrices__expressions__fourier__DFT(): from sympy.matrices.expressions.fourier import DFT from sympy.core.singleton import S assert _test_args(DFT(S(2))) def test_sympy__matrices__expressions__fourier__IDFT(): from sympy.matrices.expressions.fourier import IDFT from sympy.core.singleton import S assert _test_args(IDFT(S(2))) from sympy.matrices.expressions import MatrixSymbol X = MatrixSymbol('X', 10, 10) def test_sympy__matrices__expressions__factorizations__LofLU(): from sympy.matrices.expressions.factorizations import LofLU assert _test_args(LofLU(X)) def test_sympy__matrices__expressions__factorizations__UofLU(): from sympy.matrices.expressions.factorizations import UofLU assert _test_args(UofLU(X)) def test_sympy__matrices__expressions__factorizations__QofQR(): from sympy.matrices.expressions.factorizations import QofQR assert _test_args(QofQR(X)) def test_sympy__matrices__expressions__factorizations__RofQR(): from sympy.matrices.expressions.factorizations import RofQR assert _test_args(RofQR(X)) def test_sympy__matrices__expressions__factorizations__LofCholesky(): from sympy.matrices.expressions.factorizations import LofCholesky assert _test_args(LofCholesky(X)) def test_sympy__matrices__expressions__factorizations__UofCholesky(): from sympy.matrices.expressions.factorizations import UofCholesky assert _test_args(UofCholesky(X)) def test_sympy__matrices__expressions__factorizations__EigenVectors(): from sympy.matrices.expressions.factorizations import EigenVectors assert _test_args(EigenVectors(X)) def test_sympy__matrices__expressions__factorizations__EigenValues(): from sympy.matrices.expressions.factorizations import EigenValues assert _test_args(EigenValues(X)) def test_sympy__matrices__expressions__factorizations__UofSVD(): from sympy.matrices.expressions.factorizations import UofSVD assert _test_args(UofSVD(X)) def test_sympy__matrices__expressions__factorizations__VofSVD(): from sympy.matrices.expressions.factorizations import VofSVD assert _test_args(VofSVD(X)) def test_sympy__matrices__expressions__factorizations__SofSVD(): from sympy.matrices.expressions.factorizations import SofSVD assert _test_args(SofSVD(X)) @SKIP("abstract class") def test_sympy__matrices__expressions__factorizations__Factorization(): pass def test_sympy__matrices__expressions__permutation__PermutationMatrix(): from sympy.combinatorics import Permutation from sympy.matrices.expressions.permutation import PermutationMatrix assert _test_args(PermutationMatrix(Permutation([2, 0, 1]))) def test_sympy__matrices__expressions__permutation__MatrixPermute(): from sympy.combinatorics import Permutation from sympy.matrices.expressions.matexpr import MatrixSymbol from sympy.matrices.expressions.permutation import MatrixPermute A = MatrixSymbol('A', 3, 3) assert _test_args(MatrixPermute(A, Permutation([2, 0, 1]))) def test_sympy__matrices__expressions__companion__CompanionMatrix(): from sympy.core.symbol import Symbol from sympy.matrices.expressions.companion import CompanionMatrix from sympy.polys.polytools import Poly x = Symbol('x') p = Poly([1, 2, 3], x) assert _test_args(CompanionMatrix(p)) def test_sympy__physics__vector__frame__CoordinateSym(): from sympy.physics.vector import CoordinateSym from sympy.physics.vector import ReferenceFrame assert _test_args(CoordinateSym('R_x', ReferenceFrame('R'), 0)) def test_sympy__physics__paulialgebra__Pauli(): from sympy.physics.paulialgebra import Pauli assert _test_args(Pauli(1)) def test_sympy__physics__quantum__anticommutator__AntiCommutator(): from sympy.physics.quantum.anticommutator import AntiCommutator assert _test_args(AntiCommutator(x, y)) def test_sympy__physics__quantum__cartesian__PositionBra3D(): from sympy.physics.quantum.cartesian import PositionBra3D assert _test_args(PositionBra3D(x, y, z)) def test_sympy__physics__quantum__cartesian__PositionKet3D(): from sympy.physics.quantum.cartesian import PositionKet3D assert _test_args(PositionKet3D(x, y, z)) def test_sympy__physics__quantum__cartesian__PositionState3D(): from sympy.physics.quantum.cartesian import PositionState3D assert _test_args(PositionState3D(x, y, z)) def test_sympy__physics__quantum__cartesian__PxBra(): from sympy.physics.quantum.cartesian import PxBra assert _test_args(PxBra(x, y, z)) def test_sympy__physics__quantum__cartesian__PxKet(): from sympy.physics.quantum.cartesian import PxKet assert _test_args(PxKet(x, y, z)) def test_sympy__physics__quantum__cartesian__PxOp(): from sympy.physics.quantum.cartesian import PxOp assert _test_args(PxOp(x, y, z)) def test_sympy__physics__quantum__cartesian__XBra(): from sympy.physics.quantum.cartesian import XBra assert _test_args(XBra(x)) def test_sympy__physics__quantum__cartesian__XKet(): from sympy.physics.quantum.cartesian import XKet assert _test_args(XKet(x)) def test_sympy__physics__quantum__cartesian__XOp(): from sympy.physics.quantum.cartesian import XOp assert _test_args(XOp(x)) def test_sympy__physics__quantum__cartesian__YOp(): from sympy.physics.quantum.cartesian import YOp assert _test_args(YOp(x)) def test_sympy__physics__quantum__cartesian__ZOp(): from sympy.physics.quantum.cartesian import ZOp assert _test_args(ZOp(x)) def test_sympy__physics__quantum__cg__CG(): from sympy.physics.quantum.cg import CG from sympy.core.singleton import S assert _test_args(CG(Rational(3, 2), Rational(3, 2), S.Half, Rational(-1, 2), 1, 1)) def test_sympy__physics__quantum__cg__Wigner3j(): from sympy.physics.quantum.cg import Wigner3j assert _test_args(Wigner3j(6, 0, 4, 0, 2, 0)) def test_sympy__physics__quantum__cg__Wigner6j(): from sympy.physics.quantum.cg import Wigner6j assert _test_args(Wigner6j(1, 2, 3, 2, 1, 2)) def test_sympy__physics__quantum__cg__Wigner9j(): from sympy.physics.quantum.cg import Wigner9j assert _test_args(Wigner9j(2, 1, 1, Rational(3, 2), S.Half, 1, S.Half, S.Half, 0)) def test_sympy__physics__quantum__circuitplot__Mz(): from sympy.physics.quantum.circuitplot import Mz assert _test_args(Mz(0)) def test_sympy__physics__quantum__circuitplot__Mx(): from sympy.physics.quantum.circuitplot import Mx assert _test_args(Mx(0)) def test_sympy__physics__quantum__commutator__Commutator(): from sympy.physics.quantum.commutator import Commutator A, B = symbols('A,B', commutative=False) assert _test_args(Commutator(A, B)) def test_sympy__physics__quantum__constants__HBar(): from sympy.physics.quantum.constants import HBar assert _test_args(HBar()) def test_sympy__physics__quantum__dagger__Dagger(): from sympy.physics.quantum.dagger import Dagger from sympy.physics.quantum.state import Ket assert _test_args(Dagger(Dagger(Ket('psi')))) def test_sympy__physics__quantum__gate__CGate(): from sympy.physics.quantum.gate import CGate, Gate assert _test_args(CGate((0, 1), Gate(2))) def test_sympy__physics__quantum__gate__CGateS(): from sympy.physics.quantum.gate import CGateS, Gate assert _test_args(CGateS((0, 1), Gate(2))) def test_sympy__physics__quantum__gate__CNotGate(): from sympy.physics.quantum.gate import CNotGate assert _test_args(CNotGate(0, 1)) def test_sympy__physics__quantum__gate__Gate(): from sympy.physics.quantum.gate import Gate assert _test_args(Gate(0)) def test_sympy__physics__quantum__gate__HadamardGate(): from sympy.physics.quantum.gate import HadamardGate assert _test_args(HadamardGate(0)) def test_sympy__physics__quantum__gate__IdentityGate(): from sympy.physics.quantum.gate import IdentityGate assert _test_args(IdentityGate(0)) def test_sympy__physics__quantum__gate__OneQubitGate(): from sympy.physics.quantum.gate import OneQubitGate assert _test_args(OneQubitGate(0)) def test_sympy__physics__quantum__gate__PhaseGate(): from sympy.physics.quantum.gate import PhaseGate assert _test_args(PhaseGate(0)) def test_sympy__physics__quantum__gate__SwapGate(): from sympy.physics.quantum.gate import SwapGate assert _test_args(SwapGate(0, 1)) def test_sympy__physics__quantum__gate__TGate(): from sympy.physics.quantum.gate import TGate assert _test_args(TGate(0)) def test_sympy__physics__quantum__gate__TwoQubitGate(): from sympy.physics.quantum.gate import TwoQubitGate assert _test_args(TwoQubitGate(0)) def test_sympy__physics__quantum__gate__UGate(): from sympy.physics.quantum.gate import UGate from sympy.matrices.immutable import ImmutableDenseMatrix from sympy.core.containers import Tuple from sympy.core.numbers import Integer assert _test_args( UGate(Tuple(Integer(1)), ImmutableDenseMatrix([[1, 0], [0, 2]]))) def test_sympy__physics__quantum__gate__XGate(): from sympy.physics.quantum.gate import XGate assert _test_args(XGate(0)) def test_sympy__physics__quantum__gate__YGate(): from sympy.physics.quantum.gate import YGate assert _test_args(YGate(0)) def test_sympy__physics__quantum__gate__ZGate(): from sympy.physics.quantum.gate import ZGate assert _test_args(ZGate(0)) @SKIP("TODO: sympy.physics") def test_sympy__physics__quantum__grover__OracleGate(): from sympy.physics.quantum.grover import OracleGate assert _test_args(OracleGate()) def test_sympy__physics__quantum__grover__WGate(): from sympy.physics.quantum.grover import WGate assert _test_args(WGate(1)) def test_sympy__physics__quantum__hilbert__ComplexSpace(): from sympy.physics.quantum.hilbert import ComplexSpace assert _test_args(ComplexSpace(x)) def test_sympy__physics__quantum__hilbert__DirectSumHilbertSpace(): from sympy.physics.quantum.hilbert import DirectSumHilbertSpace, ComplexSpace, FockSpace c = ComplexSpace(2) f = FockSpace() assert _test_args(DirectSumHilbertSpace(c, f)) def test_sympy__physics__quantum__hilbert__FockSpace(): from sympy.physics.quantum.hilbert import FockSpace assert _test_args(FockSpace()) def test_sympy__physics__quantum__hilbert__HilbertSpace(): from sympy.physics.quantum.hilbert import HilbertSpace assert _test_args(HilbertSpace()) def test_sympy__physics__quantum__hilbert__L2(): from sympy.physics.quantum.hilbert import L2 from sympy.core.numbers import oo from sympy.sets.sets import Interval assert _test_args(L2(Interval(0, oo))) def test_sympy__physics__quantum__hilbert__TensorPowerHilbertSpace(): from sympy.physics.quantum.hilbert import TensorPowerHilbertSpace, FockSpace f = FockSpace() assert _test_args(TensorPowerHilbertSpace(f, 2)) def test_sympy__physics__quantum__hilbert__TensorProductHilbertSpace(): from sympy.physics.quantum.hilbert import TensorProductHilbertSpace, FockSpace, ComplexSpace c = ComplexSpace(2) f = FockSpace() assert _test_args(TensorProductHilbertSpace(f, c)) def test_sympy__physics__quantum__innerproduct__InnerProduct(): from sympy.physics.quantum import Bra, Ket, InnerProduct b = Bra('b') k = Ket('k') assert _test_args(InnerProduct(b, k)) def test_sympy__physics__quantum__operator__DifferentialOperator(): from sympy.physics.quantum.operator import DifferentialOperator from sympy.core.function import (Derivative, Function) f = Function('f') assert _test_args(DifferentialOperator(1/xDerivative(f(x), x), f(x))) def test_sympy__physics__quantum__operator__HermitianOperator(): from sympy.physics.quantum.operator import HermitianOperator assert _test_args(HermitianOperator('H')) def test_sympy__physics__quantum__operator__IdentityOperator(): from sympy.physics.quantum.operator import IdentityOperator assert _test_args(IdentityOperator(5)) def test_sympy__physics__quantum__operator__Operator(): from sympy.physics.quantum.operator import Operator assert _test_args(Operator('A')) def test_sympy__physics__quantum__operator__OuterProduct(): from sympy.physics.quantum.operator import OuterProduct from sympy.physics.quantum import Ket, Bra b = Bra('b') k = Ket('k') assert _test_args(OuterProduct(k, b)) def test_sympy__physics__quantum__operator__UnitaryOperator(): from sympy.physics.quantum.operator import UnitaryOperator assert _test_args(UnitaryOperator('U')) def test_sympy__physics__quantum__piab__PIABBra(): from sympy.physics.quantum.piab import PIABBra assert _test_args(PIABBra('B')) def test_sympy__physics__quantum__boson__BosonOp(): from sympy.physics.quantum.boson import BosonOp assert _test_args(BosonOp('a')) assert _test_args(BosonOp('a', False)) def test_sympy__physics__quantum__boson__BosonFockKet(): from sympy.physics.quantum.boson import BosonFockKet assert _test_args(BosonFockKet(1)) def test_sympy__physics__quantum__boson__BosonFockBra(): from sympy.physics.quantum.boson import BosonFockBra assert _test_args(BosonFockBra(1)) def test_sympy__physics__quantum__boson__BosonCoherentKet(): from sympy.physics.quantum.boson import BosonCoherentKet assert _test_args(BosonCoherentKet(1)) def test_sympy__physics__quantum__boson__BosonCoherentBra(): from sympy.physics.quantum.boson import BosonCoherentBra assert _test_args(BosonCoherentBra(1)) def test_sympy__physics__quantum__fermion__FermionOp(): from sympy.physics.quantum.fermion import FermionOp assert _test_args(FermionOp('c')) assert _test_args(FermionOp('c', False)) def test_sympy__physics__quantum__fermion__FermionFockKet(): from sympy.physics.quantum.fermion import FermionFockKet assert _test_args(FermionFockKet(1)) def test_sympy__physics__quantum__fermion__FermionFockBra(): from sympy.physics.quantum.fermion import FermionFockBra assert _test_args(FermionFockBra(1)) def test_sympy__physics__quantum__pauli__SigmaOpBase(): from sympy.physics.quantum.pauli import SigmaOpBase assert _test_args(SigmaOpBase()) def test_sympy__physics__quantum__pauli__SigmaX(): from sympy.physics.quantum.pauli import SigmaX assert _test_args(SigmaX()) def test_sympy__physics__quantum__pauli__SigmaY(): from sympy.physics.quantum.pauli import SigmaY assert _test_args(SigmaY()) def test_sympy__physics__quantum__pauli__SigmaZ(): from sympy.physics.quantum.pauli import SigmaZ assert _test_args(SigmaZ()) def test_sympy__physics__quantum__pauli__SigmaMinus(): from sympy.physics.quantum.pauli import SigmaMinus assert _test_args(SigmaMinus()) def test_sympy__physics__quantum__pauli__SigmaPlus(): from sympy.physics.quantum.pauli import SigmaPlus assert _test_args(SigmaPlus()) def test_sympy__physics__quantum__pauli__SigmaZKet(): from sympy.physics.quantum.pauli import SigmaZKet assert _test_args(SigmaZKet(0)) def test_sympy__physics__quantum__pauli__SigmaZBra(): from sympy.physics.quantum.pauli import SigmaZBra assert _test_args(SigmaZBra(0)) def test_sympy__physics__quantum__piab__PIABHamiltonian(): from sympy.physics.quantum.piab import PIABHamiltonian assert _test_args(PIABHamiltonian('P')) def test_sympy__physics__quantum__piab__PIABKet(): from sympy.physics.quantum.piab import PIABKet assert _test_args(PIABKet('K')) def test_sympy__physics__quantum__qexpr__QExpr(): from sympy.physics.quantum.qexpr import QExpr assert _test_args(QExpr(0)) def test_sympy__physics__quantum__qft__Fourier(): from sympy.physics.quantum.qft import Fourier assert _test_args(Fourier(0, 1)) def test_sympy__physics__quantum__qft__IQFT(): from sympy.physics.quantum.qft import IQFT assert _test_args(IQFT(0, 1)) def test_sympy__physics__quantum__qft__QFT(): from sympy.physics.quantum.qft import QFT assert _test_args(QFT(0, 1)) def test_sympy__physics__quantum__qft__RkGate(): from sympy.physics.quantum.qft import RkGate assert _test_args(RkGate(0, 1)) def test_sympy__physics__quantum__qubit__IntQubit(): from sympy.physics.quantum.qubit import IntQubit assert _test_args(IntQubit(0)) def test_sympy__physics__quantum__qubit__IntQubitBra(): from sympy.physics.quantum.qubit import IntQubitBra assert _test_args(IntQubitBra(0)) def test_sympy__physics__quantum__qubit__IntQubitState(): from sympy.physics.quantum.qubit import IntQubitState, QubitState assert _test_args(IntQubitState(QubitState(0, 1))) def test_sympy__physics__quantum__qubit__Qubit(): from sympy.physics.quantum.qubit import Qubit assert _test_args(Qubit(0, 0, 0)) def test_sympy__physics__quantum__qubit__QubitBra(): from sympy.physics.quantum.qubit import QubitBra assert _test_args(QubitBra('1', 0)) def test_sympy__physics__quantum__qubit__QubitState(): from sympy.physics.quantum.qubit import QubitState assert _test_args(QubitState(0, 1)) def test_sympy__physics__quantum__density__Density(): from sympy.physics.quantum.density import Density from sympy.physics.quantum.state import Ket assert _test_args(Density([Ket(0), 0.5], [Ket(1), 0.5])) @SKIP("TODO: sympy.physics.quantum.shor: Cmod Not Implemented") def test_sympy__physics__quantum__shor__CMod(): from sympy.physics.quantum.shor import CMod assert _test_args(CMod()) def test_sympy__physics__quantum__spin__CoupledSpinState(): from sympy.physics.quantum.spin import CoupledSpinState assert _test_args(CoupledSpinState(1, 0, (1, 1))) assert _test_args(CoupledSpinState(1, 0, (1, S.Half, S.Half))) assert _test_args(CoupledSpinState( 1, 0, (1, S.Half, S.Half), ((2, 3, S.Half), (1, 2, 1)) )) j, m, j1, j2, j3, j12, x = symbols('j m j1:4 j12 x') assert CoupledSpinState( j, m, (j1, j2, j3)).subs(j2, x) == CoupledSpinState(j, m, (j1, x, j3)) assert CoupledSpinState(j, m, (j1, j2, j3), ((1, 3, j12), (1, 2, j)) ).subs(j12, x) == \ CoupledSpinState(j, m, (j1, j2, j3), ((1, 3, x), (1, 2, j)) ) def test_sympy__physics__quantum__spin__J2Op(): from sympy.physics.quantum.spin import J2Op assert _test_args(J2Op('J')) def test_sympy__physics__quantum__spin__JminusOp(): from sympy.physics.quantum.spin import JminusOp assert _test_args(JminusOp('J')) def test_sympy__physics__quantum__spin__JplusOp(): from sympy.physics.quantum.spin import JplusOp assert _test_args(JplusOp('J')) def test_sympy__physics__quantum__spin__JxBra(): from sympy.physics.quantum.spin import JxBra assert _test_args(JxBra(1, 0)) def test_sympy__physics__quantum__spin__JxBraCoupled(): from sympy.physics.quantum.spin import JxBraCoupled assert _test_args(JxBraCoupled(1, 0, (1, 1))) def test_sympy__physics__quantum__spin__JxKet(): from sympy.physics.quantum.spin import JxKet assert _test_args(JxKet(1, 0)) def test_sympy__physics__quantum__spin__JxKetCoupled(): from sympy.physics.quantum.spin import JxKetCoupled assert _test_args(JxKetCoupled(1, 0, (1, 1))) def test_sympy__physics__quantum__spin__JxOp(): from sympy.physics.quantum.spin import JxOp assert _test_args(JxOp('J')) def test_sympy__physics__quantum__spin__JyBra(): from sympy.physics.quantum.spin import JyBra assert _test_args(JyBra(1, 0)) def test_sympy__physics__quantum__spin__JyBraCoupled(): from sympy.physics.quantum.spin import JyBraCoupled assert _test_args(JyBraCoupled(1, 0, (1, 1))) def test_sympy__physics__quantum__spin__JyKet(): from sympy.physics.quantum.spin import JyKet assert _test_args(JyKet(1, 0)) def test_sympy__physics__quantum__spin__JyKetCoupled(): from sympy.physics.quantum.spin import JyKetCoupled assert _test_args(JyKetCoupled(1, 0, (1, 1))) def test_sympy__physics__quantum__spin__JyOp(): from sympy.physics.quantum.spin import JyOp assert _test_args(JyOp('J')) def test_sympy__physics__quantum__spin__JzBra(): from sympy.physics.quantum.spin import JzBra assert _test_args(JzBra(1, 0)) def test_sympy__physics__quantum__spin__JzBraCoupled(): from sympy.physics.quantum.spin import JzBraCoupled assert _test_args(JzBraCoupled(1, 0, (1, 1))) def test_sympy__physics__quantum__spin__JzKet(): from sympy.physics.quantum.spin import JzKet assert _test_args(JzKet(1, 0)) def test_sympy__physics__quantum__spin__JzKetCoupled(): from sympy.physics.quantum.spin import JzKetCoupled assert _test_args(JzKetCoupled(1, 0, (1, 1))) def test_sympy__physics__quantum__spin__JzOp(): from sympy.physics.quantum.spin import JzOp assert _test_args(JzOp('J')) def test_sympy__physics__quantum__spin__Rotation(): from sympy.physics.quantum.spin import Rotation assert _test_args(Rotation(pi, 0, pi/2)) def test_sympy__physics__quantum__spin__SpinState(): from sympy.physics.quantum.spin import SpinState assert _test_args(SpinState(1, 0)) def test_sympy__physics__quantum__spin__WignerD(): from sympy.physics.quantum.spin import WignerD assert _test_args(WignerD(0, 1, 2, 3, 4, 5)) def test_sympy__physics__quantum__state__Bra(): from sympy.physics.quantum.state import Bra assert _test_args(Bra(0)) def test_sympy__physics__quantum__state__BraBase(): from sympy.physics.quantum.state import BraBase assert _test_args(BraBase(0)) def test_sympy__physics__quantum__state__Ket(): from sympy.physics.quantum.state import Ket assert _test_args(Ket(0)) def test_sympy__physics__quantum__state__KetBase(): from sympy.physics.quantum.state import KetBase assert _test_args(KetBase(0)) def test_sympy__physics__quantum__state__State(): from sympy.physics.quantum.state import State assert _test_args(State(0)) def test_sympy__physics__quantum__state__StateBase(): from sympy.physics.quantum.state import StateBase assert _test_args(StateBase(0)) def test_sympy__physics__quantum__state__OrthogonalBra(): from sympy.physics.quantum.state import OrthogonalBra assert _test_args(OrthogonalBra(0)) def test_sympy__physics__quantum__state__OrthogonalKet(): from sympy.physics.quantum.state import OrthogonalKet assert _test_args(OrthogonalKet(0)) def test_sympy__physics__quantum__state__OrthogonalState(): from sympy.physics.quantum.state import OrthogonalState assert _test_args(OrthogonalState(0)) def test_sympy__physics__quantum__state__TimeDepBra(): from sympy.physics.quantum.state import TimeDepBra assert _test_args(TimeDepBra('psi', 't')) def test_sympy__physics__quantum__state__TimeDepKet(): from sympy.physics.quantum.state import TimeDepKet assert _test_args(TimeDepKet('psi', 't')) def test_sympy__physics__quantum__state__TimeDepState(): from sympy.physics.quantum.state import TimeDepState assert _test_args(TimeDepState('psi', 't')) def test_sympy__physics__quantum__state__Wavefunction(): from sympy.physics.quantum.state import Wavefunction from sympy.functions import sin from sympy.functions.elementary.piecewise import Piecewise n = 1 L = 1 g = Piecewise((0, x < 0), (0, x > L), (sqrt(2//L)sin(npix/L), True)) assert _test_args(Wavefunction(g, x)) def test_sympy__physics__quantum__tensorproduct__TensorProduct(): from sympy.physics.quantum.tensorproduct import TensorProduct x, y = symbols("x y", commutative=False) assert _test_args(TensorProduct(x, y)) def test_sympy__physics__quantum__identitysearch__GateIdentity(): from sympy.physics.quantum.gate import X from sympy.physics.quantum.identitysearch import GateIdentity assert _test_args(GateIdentity(X(0), X(0))) def test_sympy__physics__quantum__sho1d__SHOOp(): from sympy.physics.quantum.sho1d import SHOOp assert _test_args(SHOOp('a')) def test_sympy__physics__quantum__sho1d__RaisingOp(): from sympy.physics.quantum.sho1d import RaisingOp assert _test_args(RaisingOp('a')) def test_sympy__physics__quantum__sho1d__LoweringOp(): from sympy.physics.quantum.sho1d import LoweringOp assert _test_args(LoweringOp('a')) def test_sympy__physics__quantum__sho1d__NumberOp(): from sympy.physics.quantum.sho1d import NumberOp assert _test_args(NumberOp('N')) def test_sympy__physics__quantum__sho1d__Hamiltonian(): from sympy.physics.quantum.sho1d import Hamiltonian assert _test_args(Hamiltonian('H')) def test_sympy__physics__quantum__sho1d__SHOState(): from sympy.physics.quantum.sho1d import SHOState assert _test_args(SHOState(0)) def test_sympy__physics__quantum__sho1d__SHOKet(): from sympy.physics.quantum.sho1d import SHOKet assert _test_args(SHOKet(0)) def test_sympy__physics__quantum__sho1d__SHOBra(): from sympy.physics.quantum.sho1d import SHOBra assert _test_args(SHOBra(0)) def test_sympy__physics__secondquant__AnnihilateBoson(): from sympy.physics.secondquant import AnnihilateBoson assert _test_args(AnnihilateBoson(0)) def test_sympy__physics__secondquant__AnnihilateFermion(): from sympy.physics.secondquant import AnnihilateFermion assert _test_args(AnnihilateFermion(0)) @SKIP("abstract class") def test_sympy__physics__secondquant__Annihilator(): pass def test_sympy__physics__secondquant__AntiSymmetricTensor(): from sympy.physics.secondquant import AntiSymmetricTensor i, j = symbols('i j', below_fermi=True) a, b = symbols('a b', above_fermi=True) assert _test_args(AntiSymmetricTensor('v', (a, i), (b, j))) def test_sympy__physics__secondquant__BosonState(): from sympy.physics.secondquant import BosonState assert _test_args(BosonState((0, 1))) @SKIP("abstract class") def test_sympy__physics__secondquant__BosonicOperator(): pass def test_sympy__physics__secondquant__Commutator(): from sympy.physics.secondquant import Commutator x, y = symbols('x y', commutative=False) assert _test_args(Commutator(x, y)) def test_sympy__physics__secondquant__CreateBoson(): from sympy.physics.secondquant import CreateBoson assert _test_args(CreateBoson(0)) def test_sympy__physics__secondquant__CreateFermion(): from sympy.physics.secondquant import CreateFermion assert _test_args(CreateFermion(0)) @SKIP("abstract class") def test_sympy__physics__secondquant__Creator(): pass def test_sympy__physics__secondquant__Dagger(): from sympy.physics.secondquant import Dagger assert _test_args(Dagger(x)) def test_sympy__physics__secondquant__FermionState(): from sympy.physics.secondquant import FermionState assert _test_args(FermionState((0, 1))) def test_sympy__physics__secondquant__FermionicOperator(): from sympy.physics.secondquant import FermionicOperator assert _test_args(FermionicOperator(0)) def test_sympy__physics__secondquant__FockState(): from sympy.physics.secondquant import FockState assert _test_args(FockState((0, 1))) def test_sympy__physics__secondquant__FockStateBosonBra(): from sympy.physics.secondquant import FockStateBosonBra assert _test_args(FockStateBosonBra((0, 1))) def test_sympy__physics__secondquant__FockStateBosonKet(): from sympy.physics.secondquant import FockStateBosonKet assert _test_args(FockStateBosonKet((0, 1))) def test_sympy__physics__secondquant__FockStateBra(): from sympy.physics.secondquant import FockStateBra assert _test_args(FockStateBra((0, 1))) def test_sympy__physics__secondquant__FockStateFermionBra(): from sympy.physics.secondquant import FockStateFermionBra assert _test_args(FockStateFermionBra((0, 1))) def test_sympy__physics__secondquant__FockStateFermionKet(): from sympy.physics.secondquant import FockStateFermionKet assert _test_args(FockStateFermionKet((0, 1))) def test_sympy__physics__secondquant__FockStateKet(): from sympy.physics.secondquant import FockStateKet assert _test_args(FockStateKet((0, 1))) def test_sympy__physics__secondquant__InnerProduct(): from sympy.physics.secondquant import InnerProduct from sympy.physics.secondquant import FockStateKet, FockStateBra assert _test_args(InnerProduct(FockStateBra((0, 1)), FockStateKet((0, 1)))) def test_sympy__physics__secondquant__NO(): from sympy.physics.secondquant import NO, F, Fd assert _test_args(NO(Fd(x)F(y))) def test_sympy__physics__secondquant__PermutationOperator(): from sympy.physics.secondquant import PermutationOperator assert _test_args(PermutationOperator(0, 1)) def test_sympy__physics__secondquant__SqOperator(): from sympy.physics.secondquant import SqOperator assert _test_args(SqOperator(0)) def test_sympy__physics__secondquant__TensorSymbol(): from sympy.physics.secondquant import TensorSymbol assert _test_args(TensorSymbol(x)) def test_sympy__physics__control__lti__LinearTimeInvariant(): # Direct instances of LinearTimeInvariant class are not allowed. # func(args) tests for its derived classes (TransferFunction, # Series, Parallel and TransferFunctionMatrix) should pass. pass def test_sympy__physics__control__lti__SISOLinearTimeInvariant(): # Direct instances of SISOLinearTimeInvariant class are not allowed. pass def test_sympy__physics__control__lti__MIMOLinearTimeInvariant(): # Direct instances of MIMOLinearTimeInvariant class are not allowed. pass def test_sympy__physics__control__lti__TransferFunction(): from sympy.physics.control.lti import TransferFunction assert _test_args(TransferFunction(2, 3, x)) def test_sympy__physics__control__lti__Series(): from sympy.physics.control import Series, TransferFunction tf1 = TransferFunction(x2 - y3, y - z, x) tf2 = TransferFunction(y - x, z + y, x) assert _test_args(Series(tf1, tf2)) def test_sympy__physics__control__lti__MIMOSeries(): from sympy.physics.control import MIMOSeries, TransferFunction, TransferFunctionMatrix tf1 = TransferFunction(x2 - y3, y - z, x) tf2 = TransferFunction(y - x, z + y, x) tfm_1 = TransferFunctionMatrix([[tf2, tf1]]) tfm_2 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) tfm_3 = TransferFunctionMatrix([[tf1], [tf2]]) assert _test_args(MIMOSeries(tfm_3, tfm_2, tfm_1)) def test_sympy__physics__control__lti__Parallel(): from sympy.physics.control import Parallel, TransferFunction tf1 = TransferFunction(x2 - y3, y - z, x) tf2 = TransferFunction(y - x, z + y, x) assert _test_args(Parallel(tf1, tf2)) def test_sympy__physics__control__lti__MIMOParallel(): from sympy.physics.control import MIMOParallel, TransferFunction, TransferFunctionMatrix tf1 = TransferFunction(x2 - y3, y - z, x) tf2 = TransferFunction(y - x, z + y, x) tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) tfm_2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]]) assert _test_args(MIMOParallel(tfm_1, tfm_2)) def test_sympy__physics__control__lti__Feedback(): from sympy.physics.control import TransferFunction, Feedback tf1 = TransferFunction(x2 - y3, y - z, x) tf2 = TransferFunction(y - x, z + y, x) assert _test_args(Feedback(tf1, tf2)) assert _test_args(Feedback(tf1, tf2, 1)) def test_sympy__physics__control__lti__MIMOFeedback(): from sympy.physics.control import TransferFunction, MIMOFeedback, TransferFunctionMatrix tf1 = TransferFunction(x2 - y3, y - z, x) tf2 = TransferFunction(y - x, z + y, x) tfm_1 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]]) tfm_2 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) assert _test_args(MIMOFeedback(tfm_1, tfm_2)) assert _test_args(MIMOFeedback(tfm_1, tfm_2, 1)) def test_sympy__physics__control__lti__TransferFunctionMatrix(): from sympy.physics.control import TransferFunction, TransferFunctionMatrix tf1 = TransferFunction(x2 - y3, y - z, x) tf2 = TransferFunction(y - x, z + y, x) assert _test_args(TransferFunctionMatrix([[tf1, tf2]])) def test_sympy__physics__units__dimensions__Dimension(): from sympy.physics.units.dimensions import Dimension assert _test_args(Dimension("length", "L")) def test_sympy__physics__units__dimensions__DimensionSystem(): from sympy.physics.units.dimensions import DimensionSystem from sympy.physics.units.definitions.dimension_definitions import length, time, velocity assert _test_args(DimensionSystem((length, time), (velocity,))) def test_sympy__physics__units__quantities__Quantity(): from sympy.physics.units.quantities import Quantity assert _test_args(Quantity("dam")) def test_sympy__physics__units__prefixes__Prefix(): from sympy.physics.units.prefixes import Prefix assert _test_args(Prefix('kilo', 'k', 3)) def test_sympy__core__numbers__AlgebraicNumber(): from sympy.core.numbers import AlgebraicNumber assert _test_args(AlgebraicNumber(sqrt(2), [1, 2, 3])) def test_sympy__polys__polytools__GroebnerBasis(): from sympy.polys.polytools import GroebnerBasis assert _test_args(GroebnerBasis([x, y, z], x, y, z)) def test_sympy__polys__polytools__Poly(): from sympy.polys.polytools import Poly assert _test_args(Poly(2, x, y)) def test_sympy__polys__polytools__PurePoly(): from sympy.polys.polytools import PurePoly assert _test_args(PurePoly(2, x, y)) @SKIP('abstract class') def test_sympy__polys__rootoftools__RootOf(): pass def test_sympy__polys__rootoftools__ComplexRootOf(): from sympy.polys.rootoftools import ComplexRootOf assert _test_args(ComplexRootOf(x3 + x + 1, 0)) def test_sympy__polys__rootoftools__RootSum(): from sympy.polys.rootoftools import RootSum assert _test_args(RootSum(x3 + x + 1, sin)) def test_sympy__series__limits__Limit(): from sympy.series.limits import Limit assert _test_args(Limit(x, x, 0, dir='-')) def test_sympy__series__order__Order(): from sympy.series.order import Order assert _test_args(Order(1, x, y)) @SKIP('Abstract Class') def test_sympy__series__sequences__SeqBase(): pass def test_sympy__series__sequences__EmptySequence(): # Need to imort the instance from series not the class from # series.sequence from sympy.series import EmptySequence assert _test_args(EmptySequence) @SKIP('Abstract Class') def test_sympy__series__sequences__SeqExpr(): pass def test_sympy__series__sequences__SeqPer(): from sympy.series.sequences import SeqPer assert _test_args(SeqPer((1, 2, 3), (0, 10))) def test_sympy__series__sequences__SeqFormula(): from sympy.series.sequences import SeqFormula assert _test_args(SeqFormula(x2, (0, 10))) def test_sympy__series__sequences__RecursiveSeq(): from sympy.series.sequences import RecursiveSeq y = Function("y") n = symbols("n") assert _test_args(RecursiveSeq(y(n - 1) + y(n - 2), y(n), n, (0, 1))) assert _test_args(RecursiveSeq(y(n - 1) + y(n - 2), y(n), n)) def test_sympy__series__sequences__SeqExprOp(): from sympy.series.sequences import SeqExprOp, sequence s1 = sequence((1, 2, 3)) s2 = sequence(x2) assert _test_args(SeqExprOp(s1, s2)) def test_sympy__series__sequences__SeqAdd(): from sympy.series.sequences import SeqAdd, sequence s1 = sequence((1, 2, 3)) s2 = sequence(x2) assert _test_args(SeqAdd(s1, s2)) def test_sympy__series__sequences__SeqMul(): from sympy.series.sequences import SeqMul, sequence s1 = sequence((1, 2, 3)) s2 = sequence(x2) assert _test_args(SeqMul(s1, s2)) @SKIP('Abstract Class') def test_sympy__series__series_class__SeriesBase(): pass def test_sympy__series__fourier__FourierSeries(): from sympy.series.fourier import fourier_series assert _test_args(fourier_series(x, (x, -pi, pi))) def test_sympy__series__fourier__FiniteFourierSeries(): from sympy.series.fourier import fourier_series assert _test_args(fourier_series(sin(pix), (x, -1, 1))) def test_sympy__series__formal__FormalPowerSeries(): from sympy.series.formal import fps assert _test_args(fps(log(1 + x), x)) def test_sympy__series__formal__Coeff(): from sympy.series.formal import fps assert _test_args(fps(x*2 + x + 1, x)) @SKIP('Abstract Class') def test_sympy__series__formal__FiniteFormalPowerSeries(): pass def test_sympy__series__formal__FormalPowerSeriesProduct(): from sympy.series.formal import fps f1, f2 = fps(sin(x)), fps(exp(x)) assert _test_args(f1.product(f2, x)) def test_sympy__series__formal__FormalPowerSeriesCompose(): from sympy.series.formal import fps f1, f2 = fps(exp(x)), fps(sin(x)) assert _test_args(f1.compose(f2, x)) def test_sympy__series__formal__FormalPowerSeriesInverse(): from sympy.series.formal import fps f1 = fps(exp(x)) assert _test_args(f1.inverse(x)) def test_sympy__simplify__hyperexpand__Hyper_Function(): from sympy.simplify.hyperexpand import Hyper_Function assert _test_args(Hyper_Function([2], [1])) def test_sympy__simplify__hyperexpand__G_Function(): from sympy.simplify.hyperexpand import G_Function assert _test_args(G_Function([2], [1], [], [])) @SKIP("abstract class") def test_sympy__tensor__array__ndim_array__ImmutableNDimArray(): pass def test_sympy__tensor__array__dense_ndim_array__ImmutableDenseNDimArray(): from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray densarr = ImmutableDenseNDimArray(range(10, 34), (2, 3, 4)) assert _test_args(densarr) def test_sympy__tensor__array__sparse_ndim_array__ImmutableSparseNDimArray(): from sympy.tensor.array.sparse_ndim_array import ImmutableSparseNDimArray sparr = ImmutableSparseNDimArray(range(10, 34), (2, 3, 4)) assert _test_args(sparr) def test_sympy__tensor__array__array_comprehension__ArrayComprehension(): from sympy.tensor.array.array_comprehension import ArrayComprehension arrcom = ArrayComprehension(x, (x, 1, 5)) assert _test_args(arrcom) def test_sympy__tensor__array__array_comprehension__ArrayComprehensionMap(): from sympy.tensor.array.array_comprehension import ArrayComprehensionMap arrcomma = ArrayComprehensionMap(lambda: 0, (x, 1, 5)) assert _test_args(arrcomma) def test_sympy__tensor__array__array_derivatives__ArrayDerivative(): from sympy.tensor.array.array_derivatives import ArrayDerivative A = MatrixSymbol("A", 2, 2) arrder = ArrayDerivative(A, A, evaluate=False) assert _test_args(arrder) def test_sympy__tensor__array__expressions__array_expressions__ArraySymbol(): from sympy.tensor.array.expressions.array_expressions import ArraySymbol m, n, k = symbols("m n k") array = ArraySymbol("A", (m, n, k, 2)) assert _test_args(array) def test_sympy__tensor__array__expressions__array_expressions__ArrayElement(): from sympy.tensor.array.expressions.array_expressions import ArrayElement m, n, k = symbols("m n k") ae = ArrayElement("A", (m, n, k, 2)) assert _test_args(ae) def test_sympy__tensor__array__expressions__array_expressions__ZeroArray(): from sympy.tensor.array.expressions.array_expressions import ZeroArray m, n, k = symbols("m n k") za = ZeroArray(m, n, k, 2) assert _test_args(za) def test_sympy__tensor__array__expressions__array_expressions__OneArray(): from sympy.tensor.array.expressions.array_expressions import OneArray m, n, k = symbols("m n k") za = OneArray(m, n, k, 2) assert _test_args(za) def test_sympy__tensor__functions__TensorProduct(): from sympy.tensor.functions import TensorProduct A = MatrixSymbol('A', 3, 3) B = MatrixSymbol('B', 3, 3) tp = TensorProduct(A, B) assert _test_args(tp) def test_sympy__tensor__indexed__Idx(): from sympy.tensor.indexed import Idx assert _test_args(Idx('test')) assert _test_args(Idx('test', (0, 10))) assert _test_args(Idx('test', 2)) assert _test_args(Idx('test', x)) def test_sympy__tensor__indexed__Indexed(): from sympy.tensor.indexed import Indexed, Idx assert _test_args(Indexed('A', Idx('i'), Idx('j'))) def test_sympy__tensor__indexed__IndexedBase(): from sympy.tensor.indexed import IndexedBase assert _test_args(IndexedBase('A', shape=(x, y))) assert _test_args(IndexedBase('A', 1)) assert _test_args(IndexedBase('A')[0, 1]) def test_sympy__tensor__tensor__TensorIndexType(): from sympy.tensor.tensor import TensorIndexType assert _test_args(TensorIndexType('Lorentz')) @SKIP("deprecated class") def test_sympy__tensor__tensor__TensorType(): pass def test_sympy__tensor__tensor__TensorSymmetry(): from sympy.tensor.tensor import TensorSymmetry, get_symmetric_group_sgs assert _test_args(TensorSymmetry(get_symmetric_group_sgs(2))) def test_sympy__tensor__tensor__TensorHead(): from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, TensorHead Lorentz = TensorIndexType('Lorentz', dummy_name='L') sym = TensorSymmetry(get_symmetric_group_sgs(1)) assert _test_args(TensorHead('p', [Lorentz], sym, 0)) def test_sympy__tensor__tensor__TensorIndex(): from sympy.tensor.tensor import TensorIndexType, TensorIndex Lorentz = TensorIndexType('Lorentz', dummy_name='L') assert _test_args(TensorIndex('i', Lorentz)) @SKIP("abstract class") def test_sympy__tensor__tensor__TensExpr(): pass def test_sympy__tensor__tensor__TensAdd(): from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, tensor_indices, TensAdd, tensor_heads Lorentz = TensorIndexType('Lorentz', dummy_name='L') a, b = tensor_indices('a,b', Lorentz) sym = TensorSymmetry(get_symmetric_group_sgs(1)) p, q = tensor_heads('p,q', [Lorentz], sym) t1 = p(a) t2 = q(a) assert _test_args(TensAdd(t1, t2)) def test_sympy__tensor__tensor__Tensor(): from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, tensor_indices, TensorHead Lorentz = TensorIndexType('Lorentz', dummy_name='L') a, b = tensor_indices('a,b', Lorentz) sym = TensorSymmetry(get_symmetric_group_sgs(1)) p = TensorHead('p', [Lorentz], sym) assert _test_args(p(a)) def test_sympy__tensor__tensor__TensMul(): from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, tensor_indices, tensor_heads Lorentz = TensorIndexType('Lorentz', dummy_name='L') a, b = tensor_indices('a,b', Lorentz) sym = TensorSymmetry(get_symmetric_group_sgs(1)) p, q = tensor_heads('p, q', [Lorentz], sym) assert _test_args(3p(a)q(b)) def test_sympy__tensor__tensor__TensorElement(): from sympy.tensor.tensor import TensorIndexType, TensorHead, TensorElement L = TensorIndexType("L") A = TensorHead("A", [L, L]) telem = TensorElement(A(x, y), {x: 1}) assert _test_args(telem) def test_sympy__tensor__toperators__PartialDerivative(): from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead from sympy.tensor.toperators import PartialDerivative Lorentz = TensorIndexType('Lorentz', dummy_name='L') a, b = tensor_indices('a,b', Lorentz) A = TensorHead("A", [Lorentz]) assert _test_args(PartialDerivative(A(a), A(b))) def test_as_coeff_add(): assert (7, (3x, 4x2)) == (7 + 3x + 4x2).as_coeff_add() def test_sympy__geometry__curve__Curve(): from sympy.geometry.curve import Curve assert _test_args(Curve((x, 1), (x, 0, 1))) def test_sympy__geometry__point__Point(): from sympy.geometry.point import Point assert _test_args(Point(0, 1)) def test_sympy__geometry__point__Point2D(): from sympy.geometry.point import Point2D assert _test_args(Point2D(0, 1)) def test_sympy__geometry__point__Point3D(): from sympy.geometry.point import Point3D assert _test_args(Point3D(0, 1, 2)) def test_sympy__geometry__ellipse__Ellipse(): from sympy.geometry.ellipse import Ellipse assert _test_args(Ellipse((0, 1), 2, 3)) def test_sympy__geometry__ellipse__Circle(): from sympy.geometry.ellipse import Circle assert _test_args(Circle((0, 1), 2)) def test_sympy__geometry__parabola__Parabola(): from sympy.geometry.parabola import Parabola from sympy.geometry.line import Line assert _test_args(Parabola((0, 0), Line((2, 3), (4, 3)))) @SKIP("abstract class") def test_sympy__geometry__line__LinearEntity(): pass def test_sympy__geometry__line__Line(): from sympy.geometry.line import Line assert _test_args(Line((0, 1), (2, 3))) def test_sympy__geometry__line__Ray(): from sympy.geometry.line import Ray assert _test_args(Ray((0, 1), (2, 3))) def test_sympy__geometry__line__Segment(): from sympy.geometry.line import Segment assert _test_args(Segment((0, 1), (2, 3))) @SKIP("abstract class") def test_sympy__geometry__line__LinearEntity2D(): pass def test_sympy__geometry__line__Line2D(): from sympy.geometry.line import Line2D assert _test_args(Line2D((0, 1), (2, 3))) def test_sympy__geometry__line__Ray2D(): from sympy.geometry.line import Ray2D assert _test_args(Ray2D((0, 1), (2, 3))) def test_sympy__geometry__line__Segment2D(): from sympy.geometry.line import Segment2D assert _test_args(Segment2D((0, 1), (2, 3))) @SKIP("abstract class") def test_sympy__geometry__line__LinearEntity3D(): pass def test_sympy__geometry__line__Line3D(): from sympy.geometry.line import Line3D assert _test_args(Line3D((0, 1, 1), (2, 3, 4))) def test_sympy__geometry__line__Segment3D(): from sympy.geometry.line import Segment3D assert _test_args(Segment3D((0, 1, 1), (2, 3, 4))) def test_sympy__geometry__line__Ray3D(): from sympy.geometry.line import Ray3D assert _test_args(Ray3D((0, 1, 1), (2, 3, 4))) def test_sympy__geometry__plane__Plane(): from sympy.geometry.plane import Plane assert _test_args(Plane((1, 1, 1), (-3, 4, -2), (1, 2, 3))) def test_sympy__geometry__polygon__Polygon(): from sympy.geometry.polygon import Polygon assert _test_args(Polygon((0, 1), (2, 3), (4, 5), (6, 7))) def test_sympy__geometry__polygon__RegularPolygon(): from sympy.geometry.polygon import RegularPolygon assert _test_args(RegularPolygon((0, 1), 2, 3, 4)) def test_sympy__geometry__polygon__Triangle(): from sympy.geometry.polygon import Triangle assert _test_args(Triangle((0, 1), (2, 3), (4, 5))) def test_sympy__geometry__entity__GeometryEntity(): from sympy.geometry.entity import GeometryEntity from sympy.geometry.point import Point assert _test_args(GeometryEntity(Point(1, 0), 1, [1, 2])) @SKIP("abstract class") def test_sympy__geometry__entity__GeometrySet(): pass def test_sympy__diffgeom__diffgeom__Manifold(): from sympy.diffgeom import Manifold assert _test_args(Manifold('name', 3)) def test_sympy__diffgeom__diffgeom__Patch(): from sympy.diffgeom import Manifold, Patch assert _test_args(Patch('name', Manifold('name', 3))) def test_sympy__diffgeom__diffgeom__CoordSystem(): from sympy.diffgeom import Manifold, Patch, CoordSystem assert _test_args(CoordSystem('name', Patch('name', Manifold('name', 3)))) assert _test_args(CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c])) def test_sympy__diffgeom__diffgeom__CoordinateSymbol(): from sympy.diffgeom import Manifold, Patch, CoordSystem, CoordinateSymbol assert _test_args(CoordinateSymbol(CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]), 0)) def test_sympy__diffgeom__diffgeom__Point(): from sympy.diffgeom import Manifold, Patch, CoordSystem, Point assert _test_args(Point( CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]), [x, y])) def test_sympy__diffgeom__diffgeom__BaseScalarField(): from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]) assert _test_args(BaseScalarField(cs, 0)) def test_sympy__diffgeom__diffgeom__BaseVectorField(): from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]) assert _test_args(BaseVectorField(cs, 0)) def test_sympy__diffgeom__diffgeom__Differential(): from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]) assert _test_args(Differential(BaseScalarField(cs, 0))) def test_sympy__diffgeom__diffgeom__Commutator(): from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField, Commutator cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]) cs1 = CoordSystem('name1', Patch('name', Manifold('name', 3)), [a, b, c]) v = BaseVectorField(cs, 0) v1 = BaseVectorField(cs1, 0) assert _test_args(Commutator(v, v1)) def test_sympy__diffgeom__diffgeom__TensorProduct(): from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, TensorProduct cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]) d = Differential(BaseScalarField(cs, 0)) assert _test_args(TensorProduct(d, d)) def test_sympy__diffgeom__diffgeom__WedgeProduct(): from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, WedgeProduct cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]) d = Differential(BaseScalarField(cs, 0)) d1 = Differential(BaseScalarField(cs, 1)) assert _test_args(WedgeProduct(d, d1)) def test_sympy__diffgeom__diffgeom__LieDerivative(): from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, BaseVectorField, LieDerivative cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]) d = Differential(BaseScalarField(cs, 0)) v = BaseVectorField(cs, 0) assert _test_args(LieDerivative(v, d)) def test_sympy__diffgeom__diffgeom__BaseCovarDerivativeOp(): from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseCovarDerivativeOp cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]) assert _test_args(BaseCovarDerivativeOp(cs, 0, [[[0, ]3, ]3, ]3)) def test_sympy__diffgeom__diffgeom__CovarDerivativeOp(): from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField, CovarDerivativeOp cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]) v = BaseVectorField(cs, 0) _test_args(CovarDerivativeOp(v, [[[0, ]3, ]3, ]3)) def test_sympy__categories__baseclasses__Class(): from sympy.categories.baseclasses import Class assert _test_args(Class()) def test_sympy__categories__baseclasses__Object(): from sympy.categories import Object assert _test_args(Object("A")) @SKIP("abstract class") def test_sympy__categories__baseclasses__Morphism(): pass def test_sympy__categories__baseclasses__IdentityMorphism(): from sympy.categories import Object, IdentityMorphism assert _test_args(IdentityMorphism(Object("A"))) def test_sympy__categories__baseclasses__NamedMorphism(): from sympy.categories import Object, NamedMorphism assert _test_args(NamedMorphism(Object("A"), Object("B"), "f")) def test_sympy__categories__baseclasses__CompositeMorphism(): from sympy.categories import Object, NamedMorphism, CompositeMorphism A = Object("A") B = Object("B") C = Object("C") f = NamedMorphism(A, B, "f") g = NamedMorphism(B, C, "g") assert _test_args(CompositeMorphism(f, g)) def test_sympy__categories__baseclasses__Diagram(): from sympy.categories import Object, NamedMorphism, Diagram A = Object("A") B = Object("B") f = NamedMorphism(A, B, "f") d = Diagram([f]) assert _test_args(d) def test_sympy__categories__baseclasses__Category(): from sympy.categories import Object, NamedMorphism, Diagram, Category A = Object("A") B = Object("B") C = Object("C") f = NamedMorphism(A, B, "f") g = NamedMorphism(B, C, "g") d1 = Diagram([f, g]) d2 = Diagram([f]) K = Category("K", commutative_diagrams=[d1, d2]) assert _test_args(K) def test_sympy__ntheory__factor___totient(): from sympy.ntheory.factor_ import totient k = symbols('k', integer=True) t = totient(k) assert _test_args(t) def test_sympy__ntheory__factor___reduced_totient(): from sympy.ntheory.factor_ import reduced_totient k = symbols('k', integer=True) t = reduced_totient(k) assert _test_args(t) def test_sympy__ntheory__factor___divisor_sigma(): from sympy.ntheory.factor_ import divisor_sigma k = symbols('k', integer=True) n = symbols('n', integer=True) t = divisor_sigma(n, k) assert _test_args(t) def test_sympy__ntheory__factor___udivisor_sigma(): from sympy.ntheory.factor_ import udivisor_sigma k = symbols('k', integer=True) n = symbols('n', integer=True) t = udivisor_sigma(n, k) assert _test_args(t) def test_sympy__ntheory__factor___primenu(): from sympy.ntheory.factor_ import primenu n = symbols('n', integer=True) t = primenu(n) assert _test_args(t) def test_sympy__ntheory__factor___primeomega(): from sympy.ntheory.factor_ import primeomega n = symbols('n', integer=True) t = primeomega(n) assert _test_args(t) def test_sympy__ntheory__residue_ntheory__mobius(): from sympy.ntheory import mobius assert _test_args(mobius(2)) def test_sympy__ntheory__generate__primepi(): from sympy.ntheory import primepi n = symbols('n') t = primepi(n) assert _test_args(t) def test_sympy__physics__optics__waves__TWave(): from sympy.physics.optics import TWave A, f, phi = symbols('A, f, phi') assert _test_args(TWave(A, f, phi)) def test_sympy__physics__optics__gaussopt__BeamParameter(): from sympy.physics.optics import BeamParameter assert _test_args(BeamParameter(530e-9, 1, w=1e-3)) def test_sympy__physics__optics__medium__Medium(): from sympy.physics.optics import Medium assert _test_args(Medium('m')) def test_sympy__physics__optics__medium__MediumN(): from sympy.physics.optics.medium import Medium assert _test_args(Medium('m', n=2)) def test_sympy__physics__optics__medium__MediumPP(): from sympy.physics.optics.medium import Medium assert _test_args(Medium('m', permittivity=2, permeability=2)) def test_sympy__tensor__array__expressions__array_expressions__ArrayContraction(): from sympy.tensor.array.expressions.array_expressions import ArrayContraction from sympy.tensor.indexed import IndexedBase A = symbols("A", cls=IndexedBase) assert _test_args(ArrayContraction(A, (0, 1))) def test_sympy__tensor__array__expressions__array_expressions__ArrayDiagonal(): from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal from sympy.tensor.indexed import IndexedBase A = symbols("A", cls=IndexedBase) assert _test_args(ArrayDiagonal(A, (0, 1))) def test_sympy__tensor__array__expressions__array_expressions__ArrayTensorProduct(): from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct from sympy.tensor.indexed import IndexedBase A, B = symbols("A B", cls=IndexedBase) assert _test_args(ArrayTensorProduct(A, B)) def test_sympy__tensor__array__expressions__array_expressions__ArrayAdd(): from sympy.tensor.array.expressions.array_expressions import ArrayAdd from sympy.tensor.indexed import IndexedBase A, B = symbols("A B", cls=IndexedBase) assert _test_args(ArrayAdd(A, B)) def test_sympy__tensor__array__expressions__array_expressions__PermuteDims(): from sympy.tensor.array.expressions.array_expressions import PermuteDims A = MatrixSymbol("A", 4, 4) assert _test_args(PermuteDims(A, (1, 0))) def test_sympy__tensor__array__expressions__array_expressions__ArrayElementwiseApplyFunc(): from sympy.tensor.array.expressions.array_expressions import ArraySymbol, ArrayElementwiseApplyFunc A = ArraySymbol("A", (4,)) assert _test_args(ArrayElementwiseApplyFunc(exp, A)) def test_sympy__tensor__array__expressions__array_expressions__Reshape(): from sympy.tensor.array.expressions.array_expressions import ArraySymbol, Reshape A = ArraySymbol("A", (4,)) assert _test_args(Reshape(A, (2, 2))) def test_sympy__codegen__ast__Assignment(): from sympy.codegen.ast import Assignment assert _test_args(Assignment(x, y)) def test_sympy__codegen__cfunctions__expm1(): from sympy.codegen.cfunctions import expm1 assert _test_args(expm1(x)) def test_sympy__codegen__cfunctions__log1p(): from sympy.codegen.cfunctions import log1p assert _test_args(log1p(x)) def test_sympy__codegen__cfunctions__exp2(): from sympy.codegen.cfunctions import exp2 assert _test_args(exp2(x)) def test_sympy__codegen__cfunctions__log2(): from sympy.codegen.cfunctions import log2 assert _test_args(log2(x)) def test_sympy__codegen__cfunctions__fma(): from sympy.codegen.cfunctions import fma assert _test_args(fma(x, y, z)) def test_sympy__codegen__cfunctions__log10(): from sympy.codegen.cfunctions import log10 assert _test_args(log10(x)) def test_sympy__codegen__cfunctions__Sqrt(): from sympy.codegen.cfunctions import Sqrt assert _test_args(Sqrt(x)) def test_sympy__codegen__cfunctions__Cbrt(): from sympy.codegen.cfunctions import Cbrt assert _test_args(Cbrt(x)) def test_sympy__codegen__cfunctions__hypot(): from sympy.codegen.cfunctions import hypot assert _test_args(hypot(x, y)) def test_sympy__codegen__fnodes__FFunction(): from sympy.codegen.fnodes import FFunction assert _test_args(FFunction('f')) def test_sympy__codegen__fnodes__F95Function(): from sympy.codegen.fnodes import F95Function assert _test_args(F95Function('f')) def test_sympy__codegen__fnodes__isign(): from sympy.codegen.fnodes import isign assert _test_args(isign(1, x)) def test_sympy__codegen__fnodes__dsign(): from sympy.codegen.fnodes import dsign assert _test_args(dsign(1, x)) def test_sympy__codegen__fnodes__cmplx(): from sympy.codegen.fnodes import cmplx assert _test_args(cmplx(x, y)) def test_sympy__codegen__fnodes__kind(): from sympy.codegen.fnodes import kind assert _test_args(kind(x)) def test_sympy__codegen__fnodes__merge(): from sympy.codegen.fnodes import merge assert _test_args(merge(1, 2, Eq(x, 0))) def test_sympy__codegen__fnodes___literal(): from sympy.codegen.fnodes import _literal assert _test_args(_literal(1)) def test_sympy__codegen__fnodes__literal_sp(): from sympy.codegen.fnodes import literal_sp assert _test_args(literal_sp(1)) def test_sympy__codegen__fnodes__literal_dp(): from sympy.codegen.fnodes import literal_dp assert _test_args(literal_dp(1)) def test_sympy__codegen__matrix_nodes__MatrixSolve(): from sympy.matrices import MatrixSymbol from sympy.codegen.matrix_nodes import MatrixSolve A = MatrixSymbol('A', 3, 3) v = MatrixSymbol('x', 3, 1) assert _test_args(MatrixSolve(A, v)) def test_sympy__vector__coordsysrect__CoordSys3D(): from sympy.vector.coordsysrect import CoordSys3D assert _test_args(CoordSys3D('C')) def test_sympy__vector__point__Point(): from sympy.vector.point import Point assert _test_args(Point('P')) def test_sympy__vector__basisdependent__BasisDependent(): #from sympy.vector.basisdependent import BasisDependent #These classes have been created to maintain an OOP hierarchy #for Vectors and Dyadics. Are NOT meant to be initialized pass def test_sympy__vector__basisdependent__BasisDependentMul(): #from sympy.vector.basisdependent import BasisDependentMul #These classes have been created to maintain an OOP hierarchy #for Vectors and Dyadics. Are NOT meant to be initialized pass def test_sympy__vector__basisdependent__BasisDependentAdd(): #from sympy.vector.basisdependent import BasisDependentAdd #These classes have been created to maintain an OOP hierarchy #for Vectors and Dyadics. Are NOT meant to be initialized pass def test_sympy__vector__basisdependent__BasisDependentZero(): #from sympy.vector.basisdependent import BasisDependentZero #These classes have been created to maintain an OOP hierarchy #for Vectors and Dyadics. Are NOT meant to be initialized pass def test_sympy__vector__vector__BaseVector(): from sympy.vector.vector import BaseVector from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') assert _test_args(BaseVector(0, C, ' ', ' ')) def test_sympy__vector__vector__VectorAdd(): from sympy.vector.vector import VectorAdd, VectorMul from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') from sympy.abc import a, b, c, x, y, z v1 = aC.i + bC.j + cC.k v2 = xC.i + yC.j + zC.k assert _test_args(VectorAdd(v1, v2)) assert _test_args(VectorMul(x, v1)) def test_sympy__vector__vector__VectorMul(): from sympy.vector.vector import VectorMul from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') from sympy.abc import a assert _test_args(VectorMul(a, C.i)) def test_sympy__vector__vector__VectorZero(): from sympy.vector.vector import VectorZero assert _test_args(VectorZero()) def test_sympy__vector__vector__Vector(): #from sympy.vector.vector import Vector #Vector is never to be initialized using args pass def test_sympy__vector__vector__Cross(): from sympy.vector.vector import Cross from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') _test_args(Cross(C.i, C.j)) def test_sympy__vector__vector__Dot(): from sympy.vector.vector import Dot from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') _test_args(Dot(C.i, C.j)) def test_sympy__vector__dyadic__Dyadic(): #from sympy.vector.dyadic import Dyadic #Dyadic is never to be initialized using args pass def test_sympy__vector__dyadic__BaseDyadic(): from sympy.vector.dyadic import BaseDyadic from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') assert _test_args(BaseDyadic(C.i, C.j)) def test_sympy__vector__dyadic__DyadicMul(): from sympy.vector.dyadic import BaseDyadic, DyadicMul from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') assert _test_args(DyadicMul(3, BaseDyadic(C.i, C.j))) def test_sympy__vector__dyadic__DyadicAdd(): from sympy.vector.dyadic import BaseDyadic, DyadicAdd from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') assert _test_args(2 DyadicAdd(BaseDyadic(C.i, C.i), BaseDyadic(C.i, C.j))) def test_sympy__vector__dyadic__DyadicZero(): from sympy.vector.dyadic import DyadicZero assert _test_args(DyadicZero()) def test_sympy__vector__deloperator__Del(): from sympy.vector.deloperator import Del assert _test_args(Del()) def test_sympy__vector__implicitregion__ImplicitRegion(): from sympy.vector.implicitregion import ImplicitRegion from sympy.abc import x, y assert _test_args(ImplicitRegion((x, y), y*3 - 4x)) def test_sympy__vector__integrals__ParametricIntegral(): from sympy.vector.integrals import ParametricIntegral from sympy.vector.parametricregion import ParametricRegion from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') assert _test_args(ParametricIntegral(C.yC.i - 10C.j,\ ParametricRegion((x, y), (x, 1, 3), (y, -2, 2)))) def test_sympy__vector__operators__Curl(): from sympy.vector.operators import Curl from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') assert _test_args(Curl(C.i)) def test_sympy__vector__operators__Laplacian(): from sympy.vector.operators import Laplacian from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') assert _test_args(Laplacian(C.i)) def test_sympy__vector__operators__Divergence(): from sympy.vector.operators import Divergence from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') assert _test_args(Divergence(C.i)) def test_sympy__vector__operators__Gradient(): from sympy.vector.operators import Gradient from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') assert _test_args(Gradient(C.x)) def test_sympy__vector__orienters__Orienter(): #from sympy.vector.orienters import Orienter #Not to be initialized pass def test_sympy__vector__orienters__ThreeAngleOrienter(): #from sympy.vector.orienters import ThreeAngleOrienter #Not to be initialized pass def test_sympy__vector__orienters__AxisOrienter(): from sympy.vector.orienters import AxisOrienter from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') assert _test_args(AxisOrienter(x, C.i)) def test_sympy__vector__orienters__BodyOrienter(): from sympy.vector.orienters import BodyOrienter assert _test_args(BodyOrienter(x, y, z, '123')) def test_sympy__vector__orienters__SpaceOrienter(): from sympy.vector.orienters import SpaceOrienter assert _test_args(SpaceOrienter(x, y, z, '123')) def test_sympy__vector__orienters__QuaternionOrienter(): from sympy.vector.orienters import QuaternionOrienter a, b, c, d = symbols('a b c d') assert _test_args(QuaternionOrienter(a, b, c, d)) def test_sympy__vector__parametricregion__ParametricRegion(): from sympy.abc import t from sympy.vector.parametricregion import ParametricRegion assert _test_args(ParametricRegion((t, t**3), (t, 0, 2))) def test_sympy__vector__scalar__BaseScalar(): from sympy.vector.scalar import BaseScalar from sympy.vector.coordsysrect import CoordSys3D C = CoordSys3D('C') assert _test_args(BaseScalar(0, C, ' ', ' ')) def test_sympy__physics__wigner__Wigner3j(): from sympy.physics.wigner import Wigner3j assert _test_args(Wigner3j(0, 0, 0, 0, 0, 0)) def test_sympy__integrals__rubi__symbol__matchpyWC(): from sympy.integrals.rubi.symbol import matchpyWC assert _test_args(matchpyWC(1, True, 'a')) def test_sympy__integrals__rubi__utility_function__rubi_unevaluated_expr(): from sympy.integrals.rubi.utility_function import rubi_unevaluated_expr a = symbols('a') assert _test_args(rubi_unevaluated_expr(a)) def test_sympy__integrals__rubi__utility_function__rubi_exp(): from sympy.integrals.rubi.utility_function import rubi_exp assert _test_args(rubi_exp(5)) def test_sympy__integrals__rubi__utility_function__rubi_log(): from sympy.integrals.rubi.utility_function import rubi_log assert _test_args(rubi_log(5)) def test_sympy__integrals__rubi__utility_function__Int(): from sympy.integrals.rubi.utility_function import Int assert _test_args(Int(5, x)) def test_sympy__integrals__rubi__utility_function__Util_Coefficient(): from sympy.integrals.rubi.utility_function import Util_Coefficient a, x = symbols('a x') assert _test_args(Util_Coefficient(a, x)) def test_sympy__integrals__rubi__utility_function__Gamma(): from sympy.integrals.rubi.utility_function import Gamma assert _test_args(Gamma(x)) def test_sympy__integrals__rubi__utility_function__Util_Part(): from sympy.integrals.rubi.utility_function import Util_Part a, b = symbols('a b') assert _test_args(Util_Part(a + b, 0)) def test_sympy__integrals__rubi__utility_function__PolyGamma(): from sympy.integrals.rubi.utility_function import PolyGamma assert _test_args(PolyGamma(1, x)) def test_sympy__integrals__rubi__utility_function__ProductLog(): from sympy.integrals.rubi.utility_function import ProductLog assert _test_args(ProductLog(1)) def test_sympy__combinatorics__schur_number__SchurNumber(): from sympy.combinatorics.schur_number import SchurNumber assert _test_args(SchurNumber(x)) def test_sympy__combinatorics__perm_groups__SymmetricPermutationGroup(): from sympy.combinatorics.perm_groups import SymmetricPermutationGroup assert _test_args(SymmetricPermutationGroup(5)) def test_sympy__combinatorics__perm_groups__Coset(): from sympy.combinatorics.permutations import Permutation from sympy.combinatorics.perm_groups import PermutationGroup, Coset a = Permutation(1, 2) b = Permutation(0, 1) G = PermutationGroup([a, b]) assert _test_args(Coset(a, G))
704fac0f23a172288e4854066850cfb8fb83219b9e3c14683fed746b8ac42005	from sympy.core.logic import fuzzy_and from sympy.core.sympify import _sympify from sympy.multipledispatch import dispatch from sympy.testing.pytest import XFAIL, raises, warns_deprecated_sympy from sympy.assumptions.ask import Q from sympy.core.add import Add from sympy.core.basic import Basic from sympy.core.expr import Expr from sympy.core.function import Function from sympy.core.mul import Mul from sympy.core.numbers import (Float, I, Rational, nan, oo, pi, zoo) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.core.symbol import (Symbol, symbols) from sympy.functions.elementary.exponential import (exp, exp_polar, log) from sympy.functions.elementary.integers import (ceiling, floor) from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (cos, sin) from sympy.logic.boolalg import (And, Implies, Not, Or, Xor) from sympy.sets import Reals from sympy.simplify.simplify import simplify from sympy.simplify.trigsimp import trigsimp from sympy.core.relational import (Relational, Equality, Unequality, GreaterThan, LessThan, StrictGreaterThan, StrictLessThan, Rel, Eq, Lt, Le, Gt, Ge, Ne, is_le, is_gt, is_ge, is_lt, is_eq, is_neq) from sympy.sets.sets import Interval, FiniteSet from itertools import combinations x, y, z, t = symbols('x,y,z,t') def rel_check(a, b): from sympy.testing.pytest import raises assert a.is_number and b.is_number for do in range(len({type(a), type(b)})): if S.NaN in (a, b): v = [(a == b), (a != b)] assert len(set(v)) == 1 and v[0] == False assert not (a != b) and not (a == b) assert raises(TypeError, lambda: a < b) assert raises(TypeError, lambda: a <= b) assert raises(TypeError, lambda: a > b) assert raises(TypeError, lambda: a >= b) else: E = [(a == b), (a != b)] assert len(set(E)) == 2 v = [ (a < b), (a <= b), (a > b), (a >= b)] i = [ [True, True, False, False], [False, True, False, True], # <-- i == 1 [False, False, True, True]].index(v) if i == 1: assert E[0] or (a.is_Float != b.is_Float) # ugh else: assert E[1] a, b = b, a return True def test_rel_ne(): assert Relational(x, y, '!=') == Ne(x, y) # issue 6116 p = Symbol('p', positive=True) assert Ne(p, 0) is S.true def test_rel_subs(): e = Relational(x, y, '==') e = e.subs(x, z) assert isinstance(e, Equality) assert e.lhs == z assert e.rhs == y e = Relational(x, y, '>=') e = e.subs(x, z) assert isinstance(e, GreaterThan) assert e.lhs == z assert e.rhs == y e = Relational(x, y, '<=') e = e.subs(x, z) assert isinstance(e, LessThan) assert e.lhs == z assert e.rhs == y e = Relational(x, y, '>') e = e.subs(x, z) assert isinstance(e, StrictGreaterThan) assert e.lhs == z assert e.rhs == y e = Relational(x, y, '<') e = e.subs(x, z) assert isinstance(e, StrictLessThan) assert e.lhs == z assert e.rhs == y e = Eq(x, 0) assert e.subs(x, 0) is S.true assert e.subs(x, 1) is S.false def test_wrappers(): e = x + x2 res = Relational(y, e, '==') assert Rel(y, x + x2, '==') == res assert Eq(y, x + x2) == res res = Relational(y, e, '<') assert Lt(y, x + x2) == res res = Relational(y, e, '<=') assert Le(y, x + x2) == res res = Relational(y, e, '>') assert Gt(y, x + x2) == res res = Relational(y, e, '>=') assert Ge(y, x + x2) == res res = Relational(y, e, '!=') assert Ne(y, x + x2) == res def test_Eq_Ne(): assert Eq(x, x) # issue 5719 with warns_deprecated_sympy(): assert Eq(x) == Eq(x, 0) # issue 6116 p = Symbol('p', positive=True) assert Eq(p, 0) is S.false # issue 13348; 19048 # SymPy is strict about 0 and 1 not being # interpreted as Booleans assert Eq(True, 1) is S.false assert Eq(False, 0) is S.false assert Eq(~x, 0) is S.false assert Eq(~x, 1) is S.false assert Ne(True, 1) is S.true assert Ne(False, 0) is S.true assert Ne(~x, 0) is S.true assert Ne(~x, 1) is S.true assert Eq((), 1) is S.false assert Ne((), 1) is S.true def test_as_poly(): from sympy.polys.polytools import Poly # Only Eq should have an as_poly method: assert Eq(x, 1).as_poly() == Poly(x - 1, x, domain='ZZ') raises(AttributeError, lambda: Ne(x, 1).as_poly()) raises(AttributeError, lambda: Ge(x, 1).as_poly()) raises(AttributeError, lambda: Gt(x, 1).as_poly()) raises(AttributeError, lambda: Le(x, 1).as_poly()) raises(AttributeError, lambda: Lt(x, 1).as_poly()) def test_rel_Infinity(): # NOTE: All of these are actually handled by sympy.core.Number, and do # not create Relational objects. assert (oo > oo) is S.false assert (oo > -oo) is S.true assert (oo > 1) is S.true assert (oo < oo) is S.false assert (oo < -oo) is S.false assert (oo < 1) is S.false assert (oo >= oo) is S.true assert (oo >= -oo) is S.true assert (oo >= 1) is S.true assert (oo <= oo) is S.true assert (oo <= -oo) is S.false assert (oo <= 1) is S.false assert (-oo > oo) is S.false assert (-oo > -oo) is S.false assert (-oo > 1) is S.false assert (-oo < oo) is S.true assert (-oo < -oo) is S.false assert (-oo < 1) is S.true assert (-oo >= oo) is S.false assert (-oo >= -oo) is S.true assert (-oo >= 1) is S.false assert (-oo <= oo) is S.true assert (-oo <= -oo) is S.true assert (-oo <= 1) is S.true def test_infinite_symbol_inequalities(): x = Symbol('x', extended_positive=True, infinite=True) y = Symbol('y', extended_positive=True, infinite=True) z = Symbol('z', extended_negative=True, infinite=True) w = Symbol('w', extended_negative=True, infinite=True) inf_set = (x, y, oo) ninf_set = (z, w, -oo) for inf1 in inf_set: assert (inf1 < 1) is S.false assert (inf1 > 1) is S.true assert (inf1 <= 1) is S.false assert (inf1 >= 1) is S.true for inf2 in inf_set: assert (inf1 < inf2) is S.false assert (inf1 > inf2) is S.false assert (inf1 <= inf2) is S.true assert (inf1 >= inf2) is S.true for ninf1 in ninf_set: assert (inf1 < ninf1) is S.false assert (inf1 > ninf1) is S.true assert (inf1 <= ninf1) is S.false assert (inf1 >= ninf1) is S.true assert (ninf1 < inf1) is S.true assert (ninf1 > inf1) is S.false assert (ninf1 <= inf1) is S.true assert (ninf1 >= inf1) is S.false for ninf1 in ninf_set: assert (ninf1 < 1) is S.true assert (ninf1 > 1) is S.false assert (ninf1 <= 1) is S.true assert (ninf1 >= 1) is S.false for ninf2 in ninf_set: assert (ninf1 < ninf2) is S.false assert (ninf1 > ninf2) is S.false assert (ninf1 <= ninf2) is S.true assert (ninf1 >= ninf2) is S.true def test_bool(): assert Eq(0, 0) is S.true assert Eq(1, 0) is S.false assert Ne(0, 0) is S.false assert Ne(1, 0) is S.true assert Lt(0, 1) is S.true assert Lt(1, 0) is S.false assert Le(0, 1) is S.true assert Le(1, 0) is S.false assert Le(0, 0) is S.true assert Gt(1, 0) is S.true assert Gt(0, 1) is S.false assert Ge(1, 0) is S.true assert Ge(0, 1) is S.false assert Ge(1, 1) is S.true assert Eq(I, 2) is S.false assert Ne(I, 2) is S.true raises(TypeError, lambda: Gt(I, 2)) raises(TypeError, lambda: Ge(I, 2)) raises(TypeError, lambda: Lt(I, 2)) raises(TypeError, lambda: Le(I, 2)) a = Float('.000000000000000000001', '') b = Float('.0000000000000000000001', '') assert Eq(pi + a, pi + b) is S.false def test_rich_cmp(): assert (x < y) == Lt(x, y) assert (x <= y) == Le(x, y) assert (x > y) == Gt(x, y) assert (x >= y) == Ge(x, y) def test_doit(): from sympy.core.symbol import Symbol p = Symbol('p', positive=True) n = Symbol('n', negative=True) np = Symbol('np', nonpositive=True) nn = Symbol('nn', nonnegative=True) assert Gt(p, 0).doit() is S.true assert Gt(p, 1).doit() == Gt(p, 1) assert Ge(p, 0).doit() is S.true assert Le(p, 0).doit() is S.false assert Lt(n, 0).doit() is S.true assert Le(np, 0).doit() is S.true assert Gt(nn, 0).doit() == Gt(nn, 0) assert Lt(nn, 0).doit() is S.false assert Eq(x, 0).doit() == Eq(x, 0) def test_new_relational(): x = Symbol('x') assert Eq(x, 0) == Relational(x, 0) # None ==> Equality assert Eq(x, 0) == Relational(x, 0, '==') assert Eq(x, 0) == Relational(x, 0, 'eq') assert Eq(x, 0) == Equality(x, 0) assert Eq(x, 0) != Relational(x, 1) # None ==> Equality assert Eq(x, 0) != Relational(x, 1, '==') assert Eq(x, 0) != Relational(x, 1, 'eq') assert Eq(x, 0) != Equality(x, 1) assert Eq(x, -1) == Relational(x, -1) # None ==> Equality assert Eq(x, -1) == Relational(x, -1, '==') assert Eq(x, -1) == Relational(x, -1, 'eq') assert Eq(x, -1) == Equality(x, -1) assert Eq(x, -1) != Relational(x, 1) # None ==> Equality assert Eq(x, -1) != Relational(x, 1, '==') assert Eq(x, -1) != Relational(x, 1, 'eq') assert Eq(x, -1) != Equality(x, 1) assert Ne(x, 0) == Relational(x, 0, '!=') assert Ne(x, 0) == Relational(x, 0, '<>') assert Ne(x, 0) == Relational(x, 0, 'ne') assert Ne(x, 0) == Unequality(x, 0) assert Ne(x, 0) != Relational(x, 1, '!=') assert Ne(x, 0) != Relational(x, 1, '<>') assert Ne(x, 0) != Relational(x, 1, 'ne') assert Ne(x, 0) != Unequality(x, 1) assert Ge(x, 0) == Relational(x, 0, '>=') assert Ge(x, 0) == Relational(x, 0, 'ge') assert Ge(x, 0) == GreaterThan(x, 0) assert Ge(x, 1) != Relational(x, 0, '>=') assert Ge(x, 1) != Relational(x, 0, 'ge') assert Ge(x, 1) != GreaterThan(x, 0) assert (x >= 1) == Relational(x, 1, '>=') assert (x >= 1) == Relational(x, 1, 'ge') assert (x >= 1) == GreaterThan(x, 1) assert (x >= 0) != Relational(x, 1, '>=') assert (x >= 0) != Relational(x, 1, 'ge') assert (x >= 0) != GreaterThan(x, 1) assert Le(x, 0) == Relational(x, 0, '<=') assert Le(x, 0) == Relational(x, 0, 'le') assert Le(x, 0) == LessThan(x, 0) assert Le(x, 1) != Relational(x, 0, '<=') assert Le(x, 1) != Relational(x, 0, 'le') assert Le(x, 1) != LessThan(x, 0) assert (x <= 1) == Relational(x, 1, '<=') assert (x <= 1) == Relational(x, 1, 'le') assert (x <= 1) == LessThan(x, 1) assert (x <= 0) != Relational(x, 1, '<=') assert (x <= 0) != Relational(x, 1, 'le') assert (x <= 0) != LessThan(x, 1) assert Gt(x, 0) == Relational(x, 0, '>') assert Gt(x, 0) == Relational(x, 0, 'gt') assert Gt(x, 0) == StrictGreaterThan(x, 0) assert Gt(x, 1) != Relational(x, 0, '>') assert Gt(x, 1) != Relational(x, 0, 'gt') assert Gt(x, 1) != StrictGreaterThan(x, 0) assert (x > 1) == Relational(x, 1, '>') assert (x > 1) == Relational(x, 1, 'gt') assert (x > 1) == StrictGreaterThan(x, 1) assert (x > 0) != Relational(x, 1, '>') assert (x > 0) != Relational(x, 1, 'gt') assert (x > 0) != StrictGreaterThan(x, 1) assert Lt(x, 0) == Relational(x, 0, '<') assert Lt(x, 0) == Relational(x, 0, 'lt') assert Lt(x, 0) == StrictLessThan(x, 0) assert Lt(x, 1) != Relational(x, 0, '<') assert Lt(x, 1) != Relational(x, 0, 'lt') assert Lt(x, 1) != StrictLessThan(x, 0) assert (x < 1) == Relational(x, 1, '<') assert (x < 1) == Relational(x, 1, 'lt') assert (x < 1) == StrictLessThan(x, 1) assert (x < 0) != Relational(x, 1, '<') assert (x < 0) != Relational(x, 1, 'lt') assert (x < 0) != StrictLessThan(x, 1) # finally, some fuzz testing from sympy.core.random import randint for i in range(100): while 1: strtype, length = (chr, 65535) if randint(0, 1) else (chr, 255) relation_type = strtype(randint(0, length)) if randint(0, 1): relation_type += strtype(randint(0, length)) if relation_type not in ('==', 'eq', '!=', '<>', 'ne', '>=', 'ge', '<=', 'le', '>', 'gt', '<', 'lt', ':=', '+=', '-=', '=', '/=', '%='): break raises(ValueError, lambda: Relational(x, 1, relation_type)) assert all(Relational(x, 0, op).rel_op == '==' for op in ('eq', '==')) assert all(Relational(x, 0, op).rel_op == '!=' for op in ('ne', '<>', '!=')) assert all(Relational(x, 0, op).rel_op == '>' for op in ('gt', '>')) assert all(Relational(x, 0, op).rel_op == '<' for op in ('lt', '<')) assert all(Relational(x, 0, op).rel_op == '>=' for op in ('ge', '>=')) assert all(Relational(x, 0, op).rel_op == '<=' for op in ('le', '<=')) def test_relational_arithmetic(): for cls in [Eq, Ne, Le, Lt, Ge, Gt]: rel = cls(x, y) raises(TypeError, lambda: 0+rel) raises(TypeError, lambda: 1rel) raises(TypeError, lambda: 1rel) raises(TypeError, lambda: rel1) raises(TypeError, lambda: Add(0, rel)) raises(TypeError, lambda: Mul(1, rel)) raises(TypeError, lambda: Pow(1, rel)) raises(TypeError, lambda: Pow(rel, 1)) def test_relational_bool_output(): # https://github.com/sympy/sympy/issues/5931 raises(TypeError, lambda: bool(x > 3)) raises(TypeError, lambda: bool(x >= 3)) raises(TypeError, lambda: bool(x < 3)) raises(TypeError, lambda: bool(x <= 3)) raises(TypeError, lambda: bool(Eq(x, 3))) raises(TypeError, lambda: bool(Ne(x, 3))) def test_relational_logic_symbols(): # See issue 6204 assert (x < y) & (z < t) == And(x < y, z < t) assert (x < y) \| (z < t) == Or(x < y, z < t) assert ~(x < y) == Not(x < y) assert (x < y) >> (z < t) == Implies(x < y, z < t) assert (x < y) << (z < t) == Implies(z < t, x < y) assert (x < y) ^ (z < t) == Xor(x < y, z < t) assert isinstance((x < y) & (z < t), And) assert isinstance((x < y) \| (z < t), Or) assert isinstance(~(x < y), GreaterThan) assert isinstance((x < y) >> (z < t), Implies) assert isinstance((x < y) << (z < t), Implies) assert isinstance((x < y) ^ (z < t), (Or, Xor)) def test_univariate_relational_as_set(): assert (x > 0).as_set() == Interval(0, oo, True, True) assert (x >= 0).as_set() == Interval(0, oo) assert (x < 0).as_set() == Interval(-oo, 0, True, True) assert (x <= 0).as_set() == Interval(-oo, 0) assert Eq(x, 0).as_set() == FiniteSet(0) assert Ne(x, 0).as_set() == Interval(-oo, 0, True, True) + \ Interval(0, oo, True, True) assert (x*2 >= 4).as_set() == Interval(-oo, -2) + Interval(2, oo) @XFAIL def test_multivariate_relational_as_set(): assert (xy >= 0).as_set() == Interval(0, oo)Interval(0, oo) + \ Interval(-oo, 0)Interval(-oo, 0) def test_Not(): assert Not(Equality(x, y)) == Unequality(x, y) assert Not(Unequality(x, y)) == Equality(x, y) assert Not(StrictGreaterThan(x, y)) == LessThan(x, y) assert Not(StrictLessThan(x, y)) == GreaterThan(x, y) assert Not(GreaterThan(x, y)) == StrictLessThan(x, y) assert Not(LessThan(x, y)) == StrictGreaterThan(x, y) def test_evaluate(): assert str(Eq(x, x, evaluate=False)) == 'Eq(x, x)' assert Eq(x, x, evaluate=False).doit() == S.true assert str(Ne(x, x, evaluate=False)) == 'Ne(x, x)' assert Ne(x, x, evaluate=False).doit() == S.false assert str(Ge(x, x, evaluate=False)) == 'x >= x' assert str(Le(x, x, evaluate=False)) == 'x <= x' assert str(Gt(x, x, evaluate=False)) == 'x > x' assert str(Lt(x, x, evaluate=False)) == 'x < x' def assert_all_ineq_raise_TypeError(a, b): raises(TypeError, lambda: a > b) raises(TypeError, lambda: a >= b) raises(TypeError, lambda: a < b) raises(TypeError, lambda: a <= b) raises(TypeError, lambda: b > a) raises(TypeError, lambda: b >= a) raises(TypeError, lambda: b < a) raises(TypeError, lambda: b <= a) def assert_all_ineq_give_class_Inequality(a, b): """All inequality operations on `a` and `b` result in class Inequality.""" from sympy.core.relational import _Inequality as Inequality assert isinstance(a > b, Inequality) assert isinstance(a >= b, Inequality) assert isinstance(a < b, Inequality) assert isinstance(a <= b, Inequality) assert isinstance(b > a, Inequality) assert isinstance(b >= a, Inequality) assert isinstance(b < a, Inequality) assert isinstance(b <= a, Inequality) def test_imaginary_compare_raises_TypeError(): # See issue #5724 assert_all_ineq_raise_TypeError(I, x) def test_complex_compare_not_real(): # two cases which are not real y = Symbol('y', imaginary=True) z = Symbol('z', complex=True, extended_real=False) for w in (y, z): assert_all_ineq_raise_TypeError(2, w) # some cases which should remain un-evaluated t = Symbol('t') x = Symbol('x', real=True) z = Symbol('z', complex=True) for w in (x, z, t): assert_all_ineq_give_class_Inequality(2, w) def test_imaginary_and_inf_compare_raises_TypeError(): # See pull request #7835 y = Symbol('y', imaginary=True) assert_all_ineq_raise_TypeError(oo, y) assert_all_ineq_raise_TypeError(-oo, y) def test_complex_pure_imag_not_ordered(): raises(TypeError, lambda: 2I < 3I) # more generally x = Symbol('x', real=True, nonzero=True) y = Symbol('y', imaginary=True) z = Symbol('z', complex=True) assert_all_ineq_raise_TypeError(I, y) t = Ix # an imaginary number, should raise errors assert_all_ineq_raise_TypeError(2, t) t = -Iy # a real number, so no errors assert_all_ineq_give_class_Inequality(2, t) t = Iz # unknown, should be unevaluated assert_all_ineq_give_class_Inequality(2, t) def test_x_minus_y_not_same_as_x_lt_y(): """ A consequence of pull request #7792 is that `x - y < 0` and `x < y` are not synonymous. """ x = I + 2 y = I + 3 raises(TypeError, lambda: x < y) assert x - y < 0 ineq = Lt(x, y, evaluate=False) raises(TypeError, lambda: ineq.doit()) assert ineq.lhs - ineq.rhs < 0 t = Symbol('t', imaginary=True) x = 2 + t y = 3 + t ineq = Lt(x, y, evaluate=False) raises(TypeError, lambda: ineq.doit()) assert ineq.lhs - ineq.rhs < 0 # this one should give error either way x = I + 2 y = 2I + 3 raises(TypeError, lambda: x < y) raises(TypeError, lambda: x - y < 0) def test_nan_equality_exceptions(): # See issue #7774 import random assert Equality(nan, nan) is S.false assert Unequality(nan, nan) is S.true # See issue #7773 A = (x, S.Zero, S.One/3, pi, oo, -oo) assert Equality(nan, random.choice(A)) is S.false assert Equality(random.choice(A), nan) is S.false assert Unequality(nan, random.choice(A)) is S.true assert Unequality(random.choice(A), nan) is S.true def test_nan_inequality_raise_errors(): # See discussion in pull request #7776. We test inequalities with # a set including examples of various classes. for q in (x, S.Zero, S(10), S.One/3, pi, S(1.3), oo, -oo, nan): assert_all_ineq_raise_TypeError(q, nan) def test_nan_complex_inequalities(): # Comparisons of NaN with non-real raise errors, we're not too # fussy whether its the NaN error or complex error. for r in (I, zoo, Symbol('z', imaginary=True)): assert_all_ineq_raise_TypeError(r, nan) def test_complex_infinity_inequalities(): raises(TypeError, lambda: zoo > 0) raises(TypeError, lambda: zoo >= 0) raises(TypeError, lambda: zoo < 0) raises(TypeError, lambda: zoo <= 0) def test_inequalities_symbol_name_same(): """Using the operator and functional forms should give same results.""" # We test all combinations from a set # FIXME: could replace with random selection after test passes A = (x, y, S.Zero, S.One/3, pi, oo, -oo) for a in A: for b in A: assert Gt(a, b) == (a > b) assert Lt(a, b) == (a < b) assert Ge(a, b) == (a >= b) assert Le(a, b) == (a <= b) for b in (y, S.Zero, S.One/3, pi, oo, -oo): assert Gt(x, b, evaluate=False) == (x > b) assert Lt(x, b, evaluate=False) == (x < b) assert Ge(x, b, evaluate=False) == (x >= b) assert Le(x, b, evaluate=False) == (x <= b) for b in (y, S.Zero, S.One/3, pi, oo, -oo): assert Gt(b, x, evaluate=False) == (b > x) assert Lt(b, x, evaluate=False) == (b < x) assert Ge(b, x, evaluate=False) == (b >= x) assert Le(b, x, evaluate=False) == (b <= x) def test_inequalities_symbol_name_same_complex(): """Using the operator and functional forms should give same results. With complex non-real numbers, both should raise errors. """ # FIXME: could replace with random selection after test passes for a in (x, S.Zero, S.One/3, pi, oo, Rational(1, 3)): raises(TypeError, lambda: Gt(a, I)) raises(TypeError, lambda: a > I) raises(TypeError, lambda: Lt(a, I)) raises(TypeError, lambda: a < I) raises(TypeError, lambda: Ge(a, I)) raises(TypeError, lambda: a >= I) raises(TypeError, lambda: Le(a, I)) raises(TypeError, lambda: a <= I) def test_inequalities_cant_sympify_other(): # see issue 7833 from operator import gt, lt, ge, le bar = "foo" for a in (x, S.Zero, S.One/3, pi, I, zoo, oo, -oo, nan, Rational(1, 3)): for op in (lt, gt, le, ge): raises(TypeError, lambda: op(a, bar)) def test_ineq_avoid_wild_symbol_flip(): # see issue #7951, we try to avoid this internally, e.g., by using # __lt__ instead of "<". from sympy.core.symbol import Wild p = symbols('p', cls=Wild) # x > p might flip, but Gt should not: assert Gt(x, p) == Gt(x, p, evaluate=False) # Previously failed as 'p > x': e = Lt(x, y).subs({y: p}) assert e == Lt(x, p, evaluate=False) # Previously failed as 'p <= x': e = Ge(x, p).doit() assert e == Ge(x, p, evaluate=False) def test_issue_8245(): a = S("6506833320952669167898688709329/5070602400912917605986812821504") assert rel_check(a, a.n(10)) assert rel_check(a, a.n(20)) assert rel_check(a, a.n()) # prec of 30 is enough to fully capture a as mpf assert Float(a, 30) == Float(str(a.p), '')/Float(str(a.q), '') for i in range(31): r = Rational(Float(a, i)) f = Float(r) assert (f < a) == (Rational(f) < a) # test sign handling assert (-f < -a) == (Rational(-f) < -a) # test equivalence handling isa = Float(a.p,'')/Float(a.q,'') assert isa <= a assert not isa < a assert isa >= a assert not isa > a assert isa > 0 a = sqrt(2) r = Rational(str(a.n(30))) assert rel_check(a, r) a = sqrt(2) r = Rational(str(a.n(29))) assert rel_check(a, r) assert Eq(log(cos(2)2 + sin(2)2), 0) is S.true def test_issue_8449(): p = Symbol('p', nonnegative=True) assert Lt(-oo, p) assert Ge(-oo, p) is S.false assert Gt(oo, -p) assert Le(oo, -p) is S.false def test_simplify_relational(): assert simplify(x(y + 1) - xy - x + 1 < x) == (x > 1) assert simplify(x(y + 1) - xy - x - 1 < x) == (x > -1) assert simplify(x < x(y + 1) - xy - x + 1) == (x < 1) q, r = symbols("q r") assert (((-q + r) - (q - r)) <= 0).simplify() == (q >= r) root2 = sqrt(2) equation = ((root2 * (-q + r) - root2 * (q - r)) <= 0).simplify() assert equation == (q >= r) r = S.One < x # canonical operations are not the same as simplification, # so if there is no simplification, canonicalization will # be done unless the measure forbids it assert simplify(r) == r.canonical assert simplify(r, ratio=0) != r.canonical # this is not a random test; in _eval_simplify # this will simplify to S.false and that is the # reason for the 'if r.is_Relational' in Relational's # _eval_simplify routine assert simplify(-(2*(piRational(3, 2)) + 6pi)(1/pi) + 2(2(pi/2) + 3pi)(1/pi) < 0) is S.false # canonical at least assert Eq(y, x).simplify() == Eq(x, y) assert Eq(x - 1, 0).simplify() == Eq(x, 1) assert Eq(x - 1, x).simplify() == S.false assert Eq(2x - 1, x).simplify() == Eq(x, 1) assert Eq(2x, 4).simplify() == Eq(x, 2) z = cos(1)2 + sin(1)2 - 1 # z.is_zero is None assert Eq(zx, 0).simplify() == S.true assert Ne(y, x).simplify() == Ne(x, y) assert Ne(x - 1, 0).simplify() == Ne(x, 1) assert Ne(x - 1, x).simplify() == S.true assert Ne(2x - 1, x).simplify() == Ne(x, 1) assert Ne(2x, 4).simplify() == Ne(x, 2) assert Ne(zx, 0).simplify() == S.false # No real-valued assumptions assert Ge(y, x).simplify() == Le(x, y) assert Ge(x - 1, 0).simplify() == Ge(x, 1) assert Ge(x - 1, x).simplify() == S.false assert Ge(2x - 1, x).simplify() == Ge(x, 1) assert Ge(2x, 4).simplify() == Ge(x, 2) assert Ge(zx, 0).simplify() == S.true assert Ge(x, -2).simplify() == Ge(x, -2) assert Ge(-x, -2).simplify() == Le(x, 2) assert Ge(x, 2).simplify() == Ge(x, 2) assert Ge(-x, 2).simplify() == Le(x, -2) assert Le(y, x).simplify() == Ge(x, y) assert Le(x - 1, 0).simplify() == Le(x, 1) assert Le(x - 1, x).simplify() == S.true assert Le(2x - 1, x).simplify() == Le(x, 1) assert Le(2x, 4).simplify() == Le(x, 2) assert Le(zx, 0).simplify() == S.true assert Le(x, -2).simplify() == Le(x, -2) assert Le(-x, -2).simplify() == Ge(x, 2) assert Le(x, 2).simplify() == Le(x, 2) assert Le(-x, 2).simplify() == Ge(x, -2) assert Gt(y, x).simplify() == Lt(x, y) assert Gt(x - 1, 0).simplify() == Gt(x, 1) assert Gt(x - 1, x).simplify() == S.false assert Gt(2x - 1, x).simplify() == Gt(x, 1) assert Gt(2x, 4).simplify() == Gt(x, 2) assert Gt(zx, 0).simplify() == S.false assert Gt(x, -2).simplify() == Gt(x, -2) assert Gt(-x, -2).simplify() == Lt(x, 2) assert Gt(x, 2).simplify() == Gt(x, 2) assert Gt(-x, 2).simplify() == Lt(x, -2) assert Lt(y, x).simplify() == Gt(x, y) assert Lt(x - 1, 0).simplify() == Lt(x, 1) assert Lt(x - 1, x).simplify() == S.true assert Lt(2x - 1, x).simplify() == Lt(x, 1) assert Lt(2x, 4).simplify() == Lt(x, 2) assert Lt(zx, 0).simplify() == S.false assert Lt(x, -2).simplify() == Lt(x, -2) assert Lt(-x, -2).simplify() == Gt(x, 2) assert Lt(x, 2).simplify() == Lt(x, 2) assert Lt(-x, 2).simplify() == Gt(x, -2) # Test particulat branches of _eval_simplify m = exp(1) - exp_polar(1) assert simplify(mx > 1) is S.false # These two tests the same branch assert simplify(mx + 2my > 1) is S.false assert simplify(mx + y > 1 + y) is S.false def test_equals(): w, x, y, z = symbols('w:z') f = Function('f') assert Eq(x, 1).equals(Eq(x(y + 1) - xy - x + 1, x)) assert Eq(x, y).equals(x < y, True) == False assert Eq(x, f(1)).equals(Eq(x, f(2)), True) == f(1) - f(2) assert Eq(f(1), y).equals(Eq(f(2), y), True) == f(1) - f(2) assert Eq(x, f(1)).equals(Eq(f(2), x), True) == f(1) - f(2) assert Eq(f(1), x).equals(Eq(x, f(2)), True) == f(1) - f(2) assert Eq(w, x).equals(Eq(y, z), True) == False assert Eq(f(1), f(2)).equals(Eq(f(3), f(4)), True) == f(1) - f(3) assert (x < y).equals(y > x, True) == True assert (x < y).equals(y >= x, True) == False assert (x < y).equals(z < y, True) == False assert (x < y).equals(x < z, True) == False assert (x < f(1)).equals(x < f(2), True) == f(1) - f(2) assert (f(1) < x).equals(f(2) < x, True) == f(1) - f(2) def test_reversed(): assert (x < y).reversed == (y > x) assert (x <= y).reversed == (y >= x) assert Eq(x, y, evaluate=False).reversed == Eq(y, x, evaluate=False) assert Ne(x, y, evaluate=False).reversed == Ne(y, x, evaluate=False) assert (x >= y).reversed == (y <= x) assert (x > y).reversed == (y < x) def test_canonical(): c = [i.canonical for i in ( x + y < z, x + 2 > 3, x < 2, S(2) > x, x2 > -x/y, Gt(3, 2, evaluate=False) )] assert [i.canonical for i in c] == c assert [i.reversed.canonical for i in c] == c assert not any(i.lhs.is_Number and not i.rhs.is_Number for i in c) c = [i.reversed.func(i.rhs, i.lhs, evaluate=False).canonical for i in c] assert [i.canonical for i in c] == c assert [i.reversed.canonical for i in c] == c assert not any(i.lhs.is_Number and not i.rhs.is_Number for i in c) assert Eq(y < x, x > y).canonical is S.true @XFAIL def test_issue_8444_nonworkingtests(): x = symbols('x', real=True) assert (x <= oo) == (x >= -oo) == True x = symbols('x') assert x >= floor(x) assert (x < floor(x)) == False assert x <= ceiling(x) assert (x > ceiling(x)) == False def test_issue_8444_workingtests(): x = symbols('x') assert Gt(x, floor(x)) == Gt(x, floor(x), evaluate=False) assert Ge(x, floor(x)) == Ge(x, floor(x), evaluate=False) assert Lt(x, ceiling(x)) == Lt(x, ceiling(x), evaluate=False) assert Le(x, ceiling(x)) == Le(x, ceiling(x), evaluate=False) i = symbols('i', integer=True) assert (i > floor(i)) == False assert (i < ceiling(i)) == False def test_issue_10304(): d = cos(1)2 + sin(1)*2 - 1 assert d.is_comparable is False # if this fails, find a new d e = 1 + dI assert simplify(Eq(e, 0)) is S.false def test_issue_18412(): d = (Rational(1, 6) + z / 4 / y) assert Eq(x, pi * y*3 d).replace(y*3, z) == Eq(x, pi z * d) def test_issue_10401(): x = symbols('x') fin = symbols('inf', finite=True) inf = symbols('inf', infinite=True) inf2 = symbols('inf2', infinite=True) infx = symbols('infx', infinite=True, extended_real=True) # Used in the commented tests below: #infx2 = symbols('infx2', infinite=True, extended_real=True) infnx = symbols('inf~x', infinite=True, extended_real=False) infnx2 = symbols('inf~x2', infinite=True, extended_real=False) infp = symbols('infp', infinite=True, extended_positive=True) infp1 = symbols('infp1', infinite=True, extended_positive=True) infn = symbols('infn', infinite=True, extended_negative=True) zero = symbols('z', zero=True) nonzero = symbols('nz', zero=False, finite=True) assert Eq(1/(1/x + 1), 1).func is Eq assert Eq(1/(1/x + 1), 1).subs(x, S.ComplexInfinity) is S.true assert Eq(1/(1/fin + 1), 1) is S.false T, F = S.true, S.false assert Eq(fin, inf) is F assert Eq(inf, inf2) not in (T, F) and inf != inf2 assert Eq(1 + inf, 2 + inf2) not in (T, F) and inf != inf2 assert Eq(infp, infp1) is T assert Eq(infp, infn) is F assert Eq(1 + Ioo, Ioo) is F assert Eq(Ioo, 1 + Ioo) is F assert Eq(1 + Ioo, 2 + Ioo) is F assert Eq(1 + Ioo, 2 + Iinfx) is F assert Eq(1 + Ioo, 2 + infx) is F # FIXME: The test below fails because (-infx).is_extended_positive is True # (should be None) #assert Eq(1 + Iinfx, 1 + Iinfx2) not in (T, F) and infx != infx2 # assert Eq(zoo, sqrt(2) + Ioo) is F assert Eq(zoo, oo) is F r = Symbol('r', real=True) i = Symbol('i', imaginary=True) assert Eq(iI, r) not in (T, F) assert Eq(infx, infnx) is F assert Eq(infnx, infnx2) not in (T, F) and infnx != infnx2 assert Eq(zoo, oo) is F assert Eq(inf/inf2, 0) is F assert Eq(inf/fin, 0) is F assert Eq(fin/inf, 0) is T assert Eq(zero/nonzero, 0) is T and ((zero/nonzero) != 0) # The commented out test below is incorrect because: assert zoo == -zoo assert Eq(zoo, -zoo) is T assert Eq(oo, -oo) is F assert Eq(inf, -inf) not in (T, F) assert Eq(fin/(fin + 1), 1) is S.false o = symbols('o', odd=True) assert Eq(o, 2o) is S.false p = symbols('p', positive=True) assert Eq(p/(p - 1), 1) is F def test_issue_10633(): assert Eq(True, False) == False assert Eq(False, True) == False assert Eq(True, True) == True assert Eq(False, False) == True def test_issue_10927(): x = symbols('x') assert str(Eq(x, oo)) == 'Eq(x, oo)' assert str(Eq(x, -oo)) == 'Eq(x, -oo)' def test_issues_13081_12583_12534(): # 13081 r = Rational('905502432259640373/288230376151711744') assert (r < pi) is S.false assert (r > pi) is S.true # 12583 v = sqrt(2) u = sqrt(v) + 2/sqrt(10 - 8/sqrt(2 - v) + 4v(1/sqrt(2 - v) - 1)) assert (u >= 0) is S.true # 12534; Rational vs NumberSymbol # here are some precisions for which Rational forms # at a lower and higher precision bracket the value of pi # e.g. for p = 20: # Rational(pi.n(p + 1)).n(25) = 3.14159265358979323846 2834 # pi.n(25) = 3.14159265358979323846 2643 # Rational(pi.n(p )).n(25) = 3.14159265358979323846 1987 assert [p for p in range(20, 50) if (Rational(pi.n(p)) < pi) and (pi < Rational(pi.n(p + 1)))] == [20, 24, 27, 33, 37, 43, 48] # pick one such precision and affirm that the reversed operation # gives the opposite result, i.e. if x < y is true then x > y # must be false for i in (20, 21): v = pi.n(i) assert rel_check(Rational(v), pi) assert rel_check(v, pi) assert rel_check(pi.n(20), pi.n(21)) # Float vs Rational # the rational form is less than the floating representation # at the same precision assert [i for i in range(15, 50) if Rational(pi.n(i)) > pi.n(i)] == [] # this should be the same if we reverse the relational assert [i for i in range(15, 50) if pi.n(i) < Rational(pi.n(i))] == [] def test_issue_18188(): from sympy.sets.conditionset import ConditionSet result1 = Eq(xcos(x) - 3sin(x), 0) assert result1.as_set() == ConditionSet(x, Eq(xcos(x) - 3sin(x), 0), Reals) result2 = Eq(x*2 + sqrt(x2) + sin(x), 0) assert result2.as_set() == ConditionSet(x, Eq(sqrt(2)sqrt(x) + x2 + sin(x), 0), Reals) def test_binary_symbols(): ans = {x} for f in Eq, Ne: for t in S.true, S.false: eq = f(x, S.true) assert eq.binary_symbols == ans assert eq.reversed.binary_symbols == ans assert f(x, 1).binary_symbols == set() def test_rel_args(): # can't have Boolean args; this is automatic for True/False # with Python 3 and we confirm that SymPy does the same # for true/false for op in ['<', '<=', '>', '>=']: for b in (S.true, x < 1, And(x, y)): for v in (0.1, 1, 232, t, S.One): raises(TypeError, lambda: Relational(b, v, op)) def test_Equality_rewrite_as_Add(): eq = Eq(x + y, y - x) assert eq.rewrite(Add) == 2x assert eq.rewrite(Add, evaluate=None).args == (x, x, y, -y) assert eq.rewrite(Add, evaluate=False).args == (x, y, x, -y) for e in (True, False, None): assert Eq(x, 0, evaluate=e).rewrite(Add) == x assert Eq(0, x, evaluate=e).rewrite(Add) == x def test_issue_15847(): a = Ne(x(x+y), x2 + xy) assert simplify(a) == False def test_negated_property(): eq = Eq(x, y) assert eq.negated == Ne(x, y) eq = Ne(x, y) assert eq.negated == Eq(x, y) eq = Ge(x + y, y - x) assert eq.negated == Lt(x + y, y - x) for f in (Eq, Ne, Ge, Gt, Le, Lt): assert f(x, y).negated.negated == f(x, y) def test_reversedsign_property(): eq = Eq(x, y) assert eq.reversedsign == Eq(-x, -y) eq = Ne(x, y) assert eq.reversedsign == Ne(-x, -y) eq = Ge(x + y, y - x) assert eq.reversedsign == Le(-x - y, x - y) for f in (Eq, Ne, Ge, Gt, Le, Lt): assert f(x, y).reversedsign.reversedsign == f(x, y) for f in (Eq, Ne, Ge, Gt, Le, Lt): assert f(-x, y).reversedsign.reversedsign == f(-x, y) for f in (Eq, Ne, Ge, Gt, Le, Lt): assert f(x, -y).reversedsign.reversedsign == f(x, -y) for f in (Eq, Ne, Ge, Gt, Le, Lt): assert f(-x, -y).reversedsign.reversedsign == f(-x, -y) def test_reversed_reversedsign_property(): for f in (Eq, Ne, Ge, Gt, Le, Lt): assert f(x, y).reversed.reversedsign == f(x, y).reversedsign.reversed for f in (Eq, Ne, Ge, Gt, Le, Lt): assert f(-x, y).reversed.reversedsign == f(-x, y).reversedsign.reversed for f in (Eq, Ne, Ge, Gt, Le, Lt): assert f(x, -y).reversed.reversedsign == f(x, -y).reversedsign.reversed for f in (Eq, Ne, Ge, Gt, Le, Lt): assert f(-x, -y).reversed.reversedsign == \ f(-x, -y).reversedsign.reversed def test_improved_canonical(): def test_different_forms(listofforms): for form1, form2 in combinations(listofforms, 2): assert form1.canonical == form2.canonical def generate_forms(expr): return [expr, expr.reversed, expr.reversedsign, expr.reversed.reversedsign] test_different_forms(generate_forms(x > -y)) test_different_forms(generate_forms(x >= -y)) test_different_forms(generate_forms(Eq(x, -y))) test_different_forms(generate_forms(Ne(x, -y))) test_different_forms(generate_forms(pi < x)) test_different_forms(generate_forms(pi - 5y < -x + 2y*2 - 7)) assert (pi >= x).canonical == (x <= pi) def test_set_equality_canonical(): a, b, c = symbols('a b c') A = Eq(FiniteSet(a, b, c), FiniteSet(1, 2, 3)) B = Ne(FiniteSet(a, b, c), FiniteSet(4, 5, 6)) assert A.canonical == A.reversed assert B.canonical == B.reversed def test_trigsimp(): # issue 16736 s, c = sin(2x), cos(2x) eq = Eq(s, c) assert trigsimp(eq) == eq # no rearrangement of sides # simplification of sides might result in # an unevaluated Eq changed = trigsimp(Eq(s + c, sqrt(2))) assert isinstance(changed, Eq) assert changed.subs(x, pi/8) is S.true # or an evaluated one assert trigsimp(Eq(cos(x)2 + sin(x)2, 1)) is S.true def test_polynomial_relation_simplification(): assert Ge(3x(x + 1) + 4, 3x).simplify() in [Ge(x2, -Rational(4,3)), Le(-x2, Rational(4, 3))] assert Le(-(3x(x + 1) + 4), -3x).simplify() in [Ge(x2, -Rational(4,3)), Le(-x2, Rational(4, 3))] assert ((x2+3)(x*2-1)+3x >= 2x2).simplify() in [(x4 + 3x >= 3), (-x*4 - 3x <= -3)] def test_multivariate_linear_function_simplification(): assert Ge(x + y, x - y).simplify() == Ge(y, 0) assert Le(-x + y, -x - y).simplify() == Le(y, 0) assert Eq(2x + y, 2x + y - 3).simplify() == False assert (2x + y > 2x + y - 3).simplify() == True assert (2x + y < 2x + y - 3).simplify() == False assert (2x + y < 2x + y + 3).simplify() == True a, b, c, d, e, f, g = symbols('a b c d e f g') assert Lt(a + b + c + 2d, 3d - f + g). simplify() == Lt(a, -b - c + d - f + g) def test_nonpolymonial_relations(): assert Eq(cos(x), 0).simplify() == Eq(cos(x), 0) def test_18778(): raises(TypeError, lambda: is_le(Basic(), Basic())) raises(TypeError, lambda: is_gt(Basic(), Basic())) raises(TypeError, lambda: is_ge(Basic(), Basic())) raises(TypeError, lambda: is_lt(Basic(), Basic())) def test_EvalEq(): """ This test exists to ensure backwards compatibility. The method to use is _eval_is_eq """ from sympy.core.expr import Expr class PowTest(Expr): def __new__(cls, base, exp): return Basic.__new__(PowTest, _sympify(base), _sympify(exp)) def _eval_Eq(lhs, rhs): if type(lhs) == PowTest and type(rhs) == PowTest: return lhs.args[0] == rhs.args[0] and lhs.args[1] == rhs.args[1] assert is_eq(PowTest(3, 4), PowTest(3,4)) assert is_eq(PowTest(3, 4), _sympify(4)) is None assert is_neq(PowTest(3, 4), PowTest(3,7)) def test_is_eq(): # test assumptions assert is_eq(x, y, Q.infinite(x) & Q.finite(y)) is False assert is_eq(x, y, Q.infinite(x) & Q.infinite(y) & Q.extended_real(x) & ~Q.extended_real(y)) is False assert is_eq(x, y, Q.infinite(x) & Q.infinite(y) & Q.extended_positive(x) & Q.extended_negative(y)) is False assert is_eq(x+I, y+I, Q.infinite(x) & Q.finite(y)) is False assert is_eq(1+xI, 1+yI, Q.infinite(x) & Q.finite(y)) is False assert is_eq(x, S(0), assumptions=Q.zero(x)) assert is_eq(x, S(0), assumptions=~Q.zero(x)) is False assert is_eq(x, S(0), assumptions=Q.nonzero(x)) is False assert is_neq(x, S(0), assumptions=Q.zero(x)) is False assert is_neq(x, S(0), assumptions=~Q.zero(x)) assert is_neq(x, S(0), assumptions=Q.nonzero(x)) # test registration class PowTest(Expr): def __new__(cls, base, exp): return Basic.__new__(cls, _sympify(base), _sympify(exp)) @dispatch(PowTest, PowTest) def _eval_is_eq(lhs, rhs): if type(lhs) == PowTest and type(rhs) == PowTest: return fuzzy_and([is_eq(lhs.args[0], rhs.args[0]), is_eq(lhs.args[1], rhs.args[1])]) assert is_eq(PowTest(3, 4), PowTest(3,4)) assert is_eq(PowTest(3, 4), _sympify(4)) is None assert is_neq(PowTest(3, 4), PowTest(3,7)) def test_is_ge_le(): # test assumptions assert is_ge(x, S(0), Q.nonnegative(x)) is True assert is_ge(x, S(0), Q.negative(x)) is False # test registration class PowTest(Expr): def __new__(cls, base, exp): return Basic.__new__(cls, _sympify(base), _sympify(exp)) @dispatch(PowTest, PowTest) def _eval_is_ge(lhs, rhs): if type(lhs) == PowTest and type(rhs) == PowTest: return fuzzy_and([is_ge(lhs.args[0], rhs.args[0]), is_ge(lhs.args[1], rhs.args[1])]) assert is_ge(PowTest(3, 9), PowTest(3,2)) assert is_gt(PowTest(3, 9), PowTest(3,2)) assert is_le(PowTest(3, 2), PowTest(3,9)) assert is_lt(PowTest(3, 2), PowTest(3,9)) def test_weak_strict(): for func in (Eq, Ne): eq = func(x, 1) assert eq.strict == eq.weak == eq eq = Gt(x, 1) assert eq.weak == Ge(x, 1) assert eq.strict == eq eq = Lt(x, 1) assert eq.weak == Le(x, 1) assert eq.strict == eq eq = Ge(x, 1) assert eq.strict == Gt(x, 1) assert eq.weak == eq eq = Le(x, 1) assert eq.strict == Lt(x, 1) assert eq.weak == eq
a5cff47dccac05a8fdddce622d6f7fa4864ba7a02d79576ea82ae1462bb2052c	from sympy.core.add import Add from sympy.core.containers import Tuple from sympy.core.function import (Function, Lambda) from sympy.core.mul import Mul from sympy.core.numbers import (Float, I, Integer, Rational, pi) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.core.symbol import Symbol from sympy.functions.elementary.complexes import Abs from sympy.functions.elementary.exponential import exp from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (cos, sin) from sympy.logic.boolalg import (false, Or, true, Xor) from sympy.matrices.dense import Matrix from sympy.parsing.sympy_parser import null from sympy.polys.polytools import Poly from sympy.printing.repr import srepr from sympy.sets.fancysets import Range from sympy.sets.sets import Interval from sympy.abc import x, y from sympy.core.sympify import (sympify, _sympify, SympifyError, kernS, CantSympify, converter) from sympy.core.decorators import _sympifyit from sympy.external import import_module from sympy.testing.pytest import raises, XFAIL, skip, warns_deprecated_sympy from sympy.utilities.decorator import conserve_mpmath_dps from sympy.geometry import Point, Line from sympy.functions.combinatorial.factorials import factorial, factorial2 from sympy.abc import _clash, _clash1, _clash2 from sympy.external.gmpy import HAS_GMPY from sympy.sets import FiniteSet, EmptySet from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray import mpmath from collections import defaultdict, OrderedDict from mpmath.rational import mpq numpy = import_module('numpy') def test_issue_3538(): v = sympify("exp(x)") assert v == exp(x) assert type(v) == type(exp(x)) assert str(type(v)) == str(type(exp(x))) def test_sympify1(): assert sympify("x") == Symbol("x") assert sympify(" x") == Symbol("x") assert sympify(" x ") == Symbol("x") # issue 4877 n1 = S.Half assert sympify('--.5') == n1 assert sympify('-1/2') == -n1 assert sympify('-+--.5') == -n1 assert sympify('-.[3]') == Rational(-1, 3) assert sympify('.[3]') == Rational(1, 3) assert sympify('+.[3]') == Rational(1, 3) assert sympify('+0.[3]10-2') == Rational(1, 300) assert sympify('.[052631578947368421]') == Rational(1, 19) assert sympify('.0[526315789473684210]') == Rational(1, 19) assert sympify('.034[56]') == Rational(1711, 49500) # options to make reals into rationals assert sympify('1.22[345]', rational=True) == \ 1 + Rational(22, 100) + Rational(345, 99900) assert sympify('2/2.6', rational=True) == Rational(10, 13) assert sympify('2.6/2', rational=True) == Rational(13, 10) assert sympify('2.6e2/17', rational=True) == Rational(260, 17) assert sympify('2.6e+2/17', rational=True) == Rational(260, 17) assert sympify('2.6e-2/17', rational=True) == Rational(26, 17000) assert sympify('2.1+3/4', rational=True) == \ Rational(21, 10) + Rational(3, 4) assert sympify('2.234456', rational=True) == Rational(279307, 125000) assert sympify('2.234456e23', rational=True) == 223445600000000000000000 assert sympify('2.234456e-23', rational=True) == \ Rational(279307, 12500000000000000000000000000) assert sympify('-2.234456e-23', rational=True) == \ Rational(-279307, 12500000000000000000000000000) assert sympify('12345678901/17', rational=True) == \ Rational(12345678901, 17) assert sympify('1/.3 + x', rational=True) == Rational(10, 3) + x # make sure longs in fractions work assert sympify('222222222222/11111111111') == \ Rational(222222222222, 11111111111) # ... even if they come from repetend notation assert sympify('1/.2[123456789012]') == Rational(333333333333, 70781892967) # ... or from high precision reals assert sympify('.1234567890123456', rational=True) == \ Rational(19290123283179, 156250000000000) def test_sympify_Fraction(): try: import fractions except ImportError: pass else: value = sympify(fractions.Fraction(101, 127)) assert value == Rational(101, 127) and type(value) is Rational def test_sympify_gmpy(): if HAS_GMPY: if HAS_GMPY == 2: import gmpy2 as gmpy elif HAS_GMPY == 1: import gmpy value = sympify(gmpy.mpz(1000001)) assert value == Integer(1000001) and type(value) is Integer value = sympify(gmpy.mpq(101, 127)) assert value == Rational(101, 127) and type(value) is Rational @conserve_mpmath_dps def test_sympify_mpmath(): value = sympify(mpmath.mpf(1.0)) assert value == Float(1.0) and type(value) is Float mpmath.mp.dps = 12 assert sympify( mpmath.pi).epsilon_eq(Float("3.14159265359"), Float("1e-12")) == True assert sympify( mpmath.pi).epsilon_eq(Float("3.14159265359"), Float("1e-13")) == False mpmath.mp.dps = 6 assert sympify( mpmath.pi).epsilon_eq(Float("3.14159"), Float("1e-5")) == True assert sympify( mpmath.pi).epsilon_eq(Float("3.14159"), Float("1e-6")) == False assert sympify(mpmath.mpc(1.0 + 2.0j)) == Float(1.0) + Float(2.0)I assert sympify(mpq(1, 2)) == S.Half def test_sympify2(): class A: def _sympy_(self): return Symbol("x")3 a = A() assert _sympify(a) == x3 assert sympify(a) == x3 assert a == x3 def test_sympify3(): assert sympify("x3") == x3 assert sympify("x^3") == x3 assert sympify("1/2") == Integer(1)/2 raises(SympifyError, lambda: _sympify('x3')) raises(SympifyError, lambda: _sympify('1/2')) def test_sympify_keywords(): raises(SympifyError, lambda: sympify('if')) raises(SympifyError, lambda: sympify('for')) raises(SympifyError, lambda: sympify('while')) raises(SympifyError, lambda: sympify('lambda')) def test_sympify_float(): assert sympify("1e-64") != 0 assert sympify("1e-20000") != 0 def test_sympify_bool(): assert sympify(True) is true assert sympify(False) is false def test_sympyify_iterables(): ans = [Rational(3, 10), Rational(1, 5)] assert sympify(['.3', '.2'], rational=True) == ans assert sympify(dict(x=0, y=1)) == {x: 0, y: 1} assert sympify(['1', '2', ['3', '4']]) == [S(1), S(2), [S(3), S(4)]] @XFAIL def test_issue_16772(): # because there is a converter for tuple, the # args are only sympified without the flags being passed # along; list, on the other hand, is not converted # with a converter so its args are traversed later ans = [Rational(3, 10), Rational(1, 5)] assert sympify(tuple(['.3', '.2']), rational=True) == Tuple(ans) def test_issue_16859(): class no(float, CantSympify): pass raises(SympifyError, lambda: sympify(no(1.2))) def test_sympify4(): class A: def _sympy_(self): return Symbol("x") a = A() assert _sympify(a)3 == x3 assert sympify(a)3 == x3 assert a == x def test_sympify_text(): assert sympify('some') == Symbol('some') assert sympify('core') == Symbol('core') assert sympify('True') is True assert sympify('False') is False assert sympify('Poly') == Poly assert sympify('sin') == sin def test_sympify_function(): assert sympify('factor(x2-1, x)') == -(1 - x)(x + 1) assert sympify('sin(pi/2)cos(pi)') == -Integer(1) def test_sympify_poly(): p = Poly(x2 + x + 1, x) assert _sympify(p) is p assert sympify(p) is p def test_sympify_factorial(): assert sympify('x!') == factorial(x) assert sympify('(x+1)!') == factorial(x + 1) assert sympify('(1 + y(x + 1))!') == factorial(1 + y(x + 1)) assert sympify('(1 + y(x + 1)!)^2') == (1 + yfactorial(x + 1))2 assert sympify('yx!') == yfactorial(x) assert sympify('x!!') == factorial2(x) assert sympify('(x+1)!!') == factorial2(x + 1) assert sympify('(1 + y(x + 1))!!') == factorial2(1 + y(x + 1)) assert sympify('(1 + y(x + 1)!!)^2') == (1 + yfactorial2(x + 1))2 assert sympify('yx!!') == yfactorial2(x) assert sympify('factorial2(x)!') == factorial(factorial2(x)) raises(SympifyError, lambda: sympify("+!!")) raises(SympifyError, lambda: sympify(")!!")) raises(SympifyError, lambda: sympify("!")) raises(SympifyError, lambda: sympify("(!)")) raises(SympifyError, lambda: sympify("x!!!")) def test_issue_3595(): assert sympify("a_") == Symbol("a_") assert sympify("_a") == Symbol("_a") def test_lambda(): x = Symbol('x') assert sympify('lambda: 1') == Lambda((), 1) assert sympify('lambda x: x') == Lambda(x, x) assert sympify('lambda x: 2x') == Lambda(x, 2x) assert sympify('lambda x, y: 2x+y') == Lambda((x, y), 2x + y) def test_lambda_raises(): raises(SympifyError, lambda: sympify("lambda args: args")) # args argument error raises(SympifyError, lambda: sympify("lambda *kwargs: kwargs[0]")) # kwargs argument error raises(SympifyError, lambda: sympify("lambda x = 1: x")) # Keyword argument error with raises(SympifyError): _sympify('lambda: 1') def test_sympify_raises(): raises(SympifyError, lambda: sympify("fx)")) class A: def __str__(self): return 'x' with warns_deprecated_sympy(): assert sympify(A()) == Symbol('x') def test__sympify(): x = Symbol('x') f = Function('f') # positive _sympify assert _sympify(x) is x assert _sympify(1) == Integer(1) assert _sympify(0.5) == Float("0.5") assert _sympify(1 + 1j) == 1.0 + I1.0 # Function f is not Basic and can't sympify to Basic. We allow it to pass # with sympify but not with _sympify. # https://github.com/sympy/sympy/issues/20124 assert sympify(f) is f raises(SympifyError, lambda: _sympify(f)) class A: def _sympy_(self): return Integer(5) a = A() assert _sympify(a) == Integer(5) # negative _sympify raises(SympifyError, lambda: _sympify('1')) raises(SympifyError, lambda: _sympify([1, 2, 3])) def test_sympifyit(): x = Symbol('x') y = Symbol('y') @_sympifyit('b', NotImplemented) def add(a, b): return a + b assert add(x, 1) == x + 1 assert add(x, 0.5) == x + Float('0.5') assert add(x, y) == x + y assert add(x, '1') == NotImplemented @_sympifyit('b') def add_raises(a, b): return a + b assert add_raises(x, 1) == x + 1 assert add_raises(x, 0.5) == x + Float('0.5') assert add_raises(x, y) == x + y raises(SympifyError, lambda: add_raises(x, '1')) def test_int_float(): class F1_1: def __float__(self): return 1.1 class F1_1b: """ This class is still a float, even though it also implements __int__(). """ def __float__(self): return 1.1 def __int__(self): return 1 class F1_1c: """ This class is still a float, because it implements _sympy_() """ def __float__(self): return 1.1 def __int__(self): return 1 def _sympy_(self): return Float(1.1) class I5: def __int__(self): return 5 class I5b: """ This class implements both __int__() and __float__(), so it will be treated as Float in SymPy. One could change this behavior, by using float(a) == int(a), but deciding that integer-valued floats represent exact numbers is arbitrary and often not correct, so we do not do it. If, in the future, we decide to do it anyway, the tests for I5b need to be changed. """ def __float__(self): return 5.0 def __int__(self): return 5 class I5c: """ This class implements both __int__() and __float__(), but also a _sympy_() method, so it will be Integer. """ def __float__(self): return 5.0 def __int__(self): return 5 def _sympy_(self): return Integer(5) i5 = I5() i5b = I5b() i5c = I5c() f1_1 = F1_1() f1_1b = F1_1b() f1_1c = F1_1c() assert sympify(i5) == 5 assert isinstance(sympify(i5), Integer) assert sympify(i5b) == 5 assert isinstance(sympify(i5b), Float) assert sympify(i5c) == 5 assert isinstance(sympify(i5c), Integer) assert abs(sympify(f1_1) - 1.1) < 1e-5 assert abs(sympify(f1_1b) - 1.1) < 1e-5 assert abs(sympify(f1_1c) - 1.1) < 1e-5 assert _sympify(i5) == 5 assert isinstance(_sympify(i5), Integer) assert _sympify(i5b) == 5 assert isinstance(_sympify(i5b), Float) assert _sympify(i5c) == 5 assert isinstance(_sympify(i5c), Integer) assert abs(_sympify(f1_1) - 1.1) < 1e-5 assert abs(_sympify(f1_1b) - 1.1) < 1e-5 assert abs(_sympify(f1_1c) - 1.1) < 1e-5 def test_evaluate_false(): cases = { '2 + 3': Add(2, 3, evaluate=False), '2*2 / 3': Mul(Pow(2, 2, evaluate=False), Pow(3, -1, evaluate=False), evaluate=False), '2 + 3 5': Add(2, Mul(3, 5, evaluate=False), evaluate=False), '2 - 3 * 5': Add(2, Mul(-1, Mul(3, 5,evaluate=False), evaluate=False), evaluate=False), '1 / 3': Mul(1, Pow(3, -1, evaluate=False), evaluate=False), 'True \| False': Or(True, False, evaluate=False), '1 + 2 + 3 + 53 + integrate(x)': Add(1, 2, 3, Mul(5, 3, evaluate=False), x2/2, evaluate=False), '2 4 * 6 + 8': Add(Mul(2, 4, 6, evaluate=False), 8, evaluate=False), '2 - 8 / 4': Add(2, Mul(-1, Mul(8, Pow(4, -1, evaluate=False), evaluate=False), evaluate=False), evaluate=False), '2 - 22': Add(2, Mul(-1, Pow(2, 2, evaluate=False), evaluate=False), evaluate=False), } for case, result in cases.items(): assert sympify(case, evaluate=False) == result def test_issue_4133(): a = sympify('Integer(4)') assert a == Integer(4) assert a.is_Integer def test_issue_3982(): a = [3, 2.0] assert sympify(a) == [Integer(3), Float(2.0)] assert sympify(tuple(a)) == Tuple(Integer(3), Float(2.0)) assert sympify(set(a)) == FiniteSet(Integer(3), Float(2.0)) def test_S_sympify(): assert S(1)/2 == sympify(1)/2 == S.Half assert (-2)(S(1)/2) == sqrt(2)I def test_issue_4788(): assert srepr(S(1.0 + 0J)) == srepr(S(1.0)) == srepr(Float(1.0)) def test_issue_4798_None(): assert S(None) is None def test_issue_3218(): assert sympify("x+\ny") == x + y def test_issue_4988_builtins(): C = Symbol('C') vars = {'C': C} exp1 = sympify('C') assert exp1 == C # Make sure it did not get mixed up with sympy.C exp2 = sympify('C', vars) assert exp2 == C # Make sure it did not get mixed up with sympy.C def test_geometry(): p = sympify(Point(0, 1)) assert p == Point(0, 1) and isinstance(p, Point) L = sympify(Line(p, (1, 0))) assert L == Line((0, 1), (1, 0)) and isinstance(L, Line) def test_kernS(): s = '-1 - 2(-(-x + 1/x)/(x(x - 1/x)2) - 1/(x(x - 1/x)))' # when 1497 is fixed, this no longer should pass: the expression # should be unchanged assert -1 - 2(-(-x + 1/x)/(x(x - 1/x)*2) - 1/(x(x - 1/x))) == -1 # sympification should not allow the constant to enter a Mul # or else the structure can change dramatically ss = kernS(s) assert ss != -1 and ss.simplify() == -1 s = '-1 - 2(-(-x + 1/x)/(x(x - 1/x)*2) - 1/(x(x - 1/x)))'.replace( 'x', '_kern') ss = kernS(s) assert ss != -1 and ss.simplify() == -1 # issue 6687 assert (kernS('Interval(-1,-2 - 4(-3))') == Interval(-1, Add(-2, Mul(12, 1, evaluate=False), evaluate=False))) assert kernS('_kern') == Symbol('_kern') assert kernS('E-(x)') == exp(-x) e = 2(x + y)y assert kernS(['2(x + y)y', ('2(x + y)y',)]) == [e, (e,)] assert kernS('-(2sin(x)*2 + 2sin(x)cos(x))y/2') == \ -y(2sin(x)*2 + 2sin(x)cos(x))/2 # issue 15132 assert kernS('(1 - x)/(1 - x(1-y))') == kernS('(1-x)/(1-(1-y)x)') assert kernS('(1-2-(4+1)(1-y)x)') == (1 - x(1 - y)/32) assert kernS('(1-2*(4+1)(1-y)x)') == (1 - 32x(1 - y)) assert kernS('(1-2.(1-y)x)') == 1 - 2.x(1 - y) one = kernS('x - (x - 1)') assert one != 1 and one.expand() == 1 assert kernS("(2x)/(x-1)") == 2x/(x-1) def test_issue_6540_6552(): assert S('[[1/3,2], (2/5,)]') == [[Rational(1, 3), 2], (Rational(2, 5),)] assert S('[[2/6,2], (2/4,)]') == [[Rational(1, 3), 2], (S.Half,)] assert S('[[[2(1)]]]') == [[[2]]] assert S('Matrix([2(1)])') == Matrix([2]) def test_issue_6046(): assert str(S("Q & C", locals=_clash1)) == 'C & Q' assert str(S('pi(x)', locals=_clash2)) == 'pi(x)' locals = {} exec("from sympy.abc import Q, C", locals) assert str(S('C&Q', locals)) == 'C & Q' # clash can act as Symbol or Function assert str(S('pi(C, Q)', locals=_clash)) == 'pi(C, Q)' assert len(S('pi + x', locals=_clash2).free_symbols) == 2 # but not both raises(TypeError, lambda: S('pi + pi(x)', locals=_clash2)) assert all(set(i.values()) == {null} for i in ( _clash, _clash1, _clash2)) def test_issue_8821_highprec_from_str(): s = str(pi.evalf(128)) p = sympify(s) assert Abs(sin(p)) < 1e-127 def test_issue_10295(): if not numpy: skip("numpy not installed.") A = numpy.array([[1, 3, -1], [0, 1, 7]]) sA = S(A) assert sA.shape == (2, 3) for (ri, ci), val in numpy.ndenumerate(A): assert sA[ri, ci] == val B = numpy.array([-7, x, 3y*2]) sB = S(B) assert sB.shape == (3,) assert B[0] == sB[0] == -7 assert B[1] == sB[1] == x assert B[2] == sB[2] == 3y*2 C = numpy.arange(0, 24) C.resize(2,3,4) sC = S(C) assert sC[0, 0, 0].is_integer assert sC[0, 0, 0] == 0 a1 = numpy.array([1, 2, 3]) a2 = numpy.array([i for i in range(24)]) a2.resize(2, 4, 3) assert sympify(a1) == ImmutableDenseNDimArray([1, 2, 3]) assert sympify(a2) == ImmutableDenseNDimArray([i for i in range(24)], (2, 4, 3)) def test_Range(): # Only works in Python 3 where range returns a range type assert sympify(range(10)) == Range(10) assert _sympify(range(10)) == Range(10) def test_sympify_set(): n = Symbol('n') assert sympify({n}) == FiniteSet(n) assert sympify(set()) == EmptySet def test_sympify_numpy(): if not numpy: skip('numpy not installed. Abort numpy tests.') np = numpy def equal(x, y): return x == y and type(x) == type(y) assert sympify(np.bool_(1)) is S(True) try: assert equal( sympify(np.int_(1234567891234567891)), S(1234567891234567891)) assert equal( sympify(np.intp(1234567891234567891)), S(1234567891234567891)) except OverflowError: # May fail on 32-bit systems: Python int too large to convert to C long pass assert equal(sympify(np.intc(1234567891)), S(1234567891)) assert equal(sympify(np.int8(-123)), S(-123)) assert equal(sympify(np.int16(-12345)), S(-12345)) assert equal(sympify(np.int32(-1234567891)), S(-1234567891)) assert equal( sympify(np.int64(-1234567891234567891)), S(-1234567891234567891)) assert equal(sympify(np.uint8(123)), S(123)) assert equal(sympify(np.uint16(12345)), S(12345)) assert equal(sympify(np.uint32(1234567891)), S(1234567891)) assert equal( sympify(np.uint64(1234567891234567891)), S(1234567891234567891)) assert equal(sympify(np.float32(1.123456)), Float(1.123456, precision=24)) assert equal(sympify(np.float64(1.1234567891234)), Float(1.1234567891234, precision=53)) assert equal(sympify(np.longdouble(1.123456789)), Float(1.123456789, precision=80)) assert equal(sympify(np.complex64(1 + 2j)), S(1.0 + 2.0I)) assert equal(sympify(np.complex128(1 + 2j)), S(1.0 + 2.0I)) assert equal(sympify(np.longcomplex(1 + 2j)), S(1.0 + 2.0I)) #float96 does not exist on all platforms if hasattr(np, 'float96'): assert equal(sympify(np.float96(1.123456789)), Float(1.123456789, precision=80)) #float128 does not exist on all platforms if hasattr(np, 'float128'): assert equal(sympify(np.float128(1.123456789123)), Float(1.123456789123, precision=80)) @XFAIL def test_sympify_rational_numbers_set(): ans = [Rational(3, 10), Rational(1, 5)] assert sympify({'.3', '.2'}, rational=True) == FiniteSet(ans) def test_sympify_mro(): """Tests the resolution order for classes that implement _sympy_""" class a: def _sympy_(self): return Integer(1) class b(a): def _sympy_(self): return Integer(2) class c(a): pass assert sympify(a()) == Integer(1) assert sympify(b()) == Integer(2) assert sympify(c()) == Integer(1) def test_sympify_converter(): """Tests the resolution order for classes in converter""" class a: pass class b(a): pass class c(a): pass converter[a] = lambda x: Integer(1) converter[b] = lambda x: Integer(2) assert sympify(a()) == Integer(1) assert sympify(b()) == Integer(2) assert sympify(c()) == Integer(1) class MyInteger(Integer): pass if int in converter: int_converter = converter[int] else: int_converter = None try: converter[int] = MyInteger assert sympify(1) == MyInteger(1) finally: if int_converter is None: del converter[int] else: converter[int] = int_converter def test_issue_13924(): if not numpy: skip("numpy not installed.") a = sympify(numpy.array([1])) assert isinstance(a, ImmutableDenseNDimArray) assert a[0] == 1 def test_numpy_sympify_args(): # Issue 15098. Make sure sympify args work with numpy types (like numpy.str_) if not numpy: skip("numpy not installed.") a = sympify(numpy.str_('a')) assert type(a) is Symbol assert a == Symbol('a') class CustomSymbol(Symbol): pass a = sympify(numpy.str_('a'), {"Symbol": CustomSymbol}) assert isinstance(a, CustomSymbol) a = sympify(numpy.str_('x^y')) assert a == xy a = sympify(numpy.str_('x^y'), convert_xor=False) assert a == Xor(x, y) raises(SympifyError, lambda: sympify(numpy.str_('x'), strict=True)) a = sympify(numpy.str_('1.1')) assert isinstance(a, Float) assert a == 1.1 a = sympify(numpy.str_('1.1'), rational=True) assert isinstance(a, Rational) assert a == Rational(11, 10) a = sympify(numpy.str_('x + x')) assert isinstance(a, Mul) assert a == 2x a = sympify(numpy.str_('x + x'), evaluate=False) assert isinstance(a, Add) assert a == Add(x, x, evaluate=False) def test_issue_5939(): a = Symbol('a') b = Symbol('b') assert sympify('''a+\nb''') == a + b def test_issue_16759(): d = sympify({.5: 1}) assert S.Half not in d assert Float(.5) in d assert d[.5] is S.One d = sympify(OrderedDict({.5: 1})) assert S.Half not in d assert Float(.5) in d assert d[.5] is S.One d = sympify(defaultdict(int, {.5: 1})) assert S.Half not in d assert Float(.5) in d assert d[.5] is S.One def test_issue_17811(): a = Function('a') assert sympify('a(x)5', evaluate=False) == Mul(a(x), 5, evaluate=False) def test_issue_14706(): if not numpy: skip("numpy not installed.") z1 = numpy.zeros((1, 1), dtype=numpy.float64) z2 = numpy.zeros((2, 2), dtype=numpy.float64) z3 = numpy.zeros((), dtype=numpy.float64) y1 = numpy.ones((1, 1), dtype=numpy.float64) y2 = numpy.ones((2, 2), dtype=numpy.float64) y3 = numpy.ones((), dtype=numpy.float64) assert numpy.all(x + z1 == numpy.full((1, 1), x)) assert numpy.all(x + z2 == numpy.full((2, 2), x)) assert numpy.all(z1 + x == numpy.full((1, 1), x)) assert numpy.all(z2 + x == numpy.full((2, 2), x)) for z in [z3, numpy.int64(0), numpy.float64(0), numpy.complex64(0)]: assert x + z == x assert z + x == x assert isinstance(x + z, Symbol) assert isinstance(z + x, Symbol) # If these tests fail, then it means that numpy has finally # fixed the issue of scalar conversion for rank>0 arrays # which is mentioned in numpy/numpy#10404. In that case, # some changes have to be made in sympify.py. # Note: For future reference, for anyone who takes up this # issue when numpy has finally fixed their side of the problem, # the changes for this temporary fix were introduced in PR 18651 assert numpy.all(x + y1 == numpy.full((1, 1), x + 1.0)) assert numpy.all(x + y2 == numpy.full((2, 2), x + 1.0)) assert numpy.all(y1 + x == numpy.full((1, 1), x + 1.0)) assert numpy.all(y2 + x == numpy.full((2, 2), x + 1.0)) for y_ in [y3, numpy.int64(1), numpy.float64(1), numpy.complex64(1)]: assert x + y_ == y_ + x assert isinstance(x + y_, Add) assert isinstance(y_ + x, Add) assert x + numpy.array(x) == 2 x assert x + numpy.array([x]) == numpy.array([2x], dtype=object) assert sympify(numpy.array([1])) == ImmutableDenseNDimArray([1], 1) assert sympify(numpy.array([[[1]]])) == ImmutableDenseNDimArray([1], (1, 1, 1)) assert sympify(z1) == ImmutableDenseNDimArray([0], (1, 1)) assert sympify(z2) == ImmutableDenseNDimArray([0, 0, 0, 0], (2, 2)) assert sympify(z3) == ImmutableDenseNDimArray([0], ()) assert sympify(z3, strict=True) == 0.0 raises(SympifyError, lambda: sympify(numpy.array([1]), strict=True)) raises(SympifyError, lambda: sympify(z1, strict=True)) raises(SympifyError, lambda: sympify(z2, strict=True)) def test_issue_21536(): #test to check evaluate=False in case of iterable input u = sympify("x+3x+2", evaluate=False) v = sympify("2x+4x+2+4", evaluate=False) assert u.is_Add and set(u.args) == {x, 3x, 2} assert v.is_Add and set(v.args) == {2x, 4x, 2, 4} assert sympify(["x+3x+2", "2x+4x+2+4"], evaluate=False) == [u, v] #test to check evaluate=True in case of iterable input u = sympify("x+3x+2", evaluate=True) v = sympify("2x+4x+2+4", evaluate=True) assert u.is_Add and set(u.args) == {4x, 2} assert v.is_Add and set(v.args) == {6x, 6} assert sympify(["x+3x+2", "2x+4x+2+4"], evaluate=True) == [u, v] #test to check evaluate with no input in case of iterable input u = sympify("x+3x+2") v = sympify("2x+4x+2+4") assert u.is_Add and set(u.args) == {4x, 2} assert v.is_Add and set(v.args) == {6x, 6} assert sympify(["x+3x+2", "2x+4x+2+4"]) == [u, v]
d9ec03e00a3d71836f6edd9300877ca19094d0608d5ac483c78a8ee7efd78707	from sympy import abc from sympy.concrete.summations import Sum from sympy.core.add import Add from sympy.core.function import (Derivative, Function, diff) from sympy.core.mul import Mul from sympy.core.numbers import (Float, I, Integer, Rational, oo, pi) from sympy.core.singleton import S from sympy.core.symbol import (Symbol, Wild, symbols) from sympy.functions.elementary.exponential import (exp, log) from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (cos, sin) from sympy.functions.special.hyper import meijerg from sympy.polys.polytools import Poly from sympy.simplify.radsimp import collect from sympy.simplify.simplify import signsimp from sympy.testing.pytest import XFAIL def test_symbol(): x = Symbol('x') a, b, c, p, q = map(Wild, 'abcpq') e = x assert e.match(x) == {} assert e.matches(x) == {} assert e.match(a) == {a: x} e = Rational(5) assert e.match(c) == {c: 5} assert e.match(e) == {} assert e.match(e + 1) is None def test_add(): x, y, a, b, c = map(Symbol, 'xyabc') p, q, r = map(Wild, 'pqr') e = a + b assert e.match(p + b) == {p: a} assert e.match(p + a) == {p: b} e = 1 + b assert e.match(p + b) == {p: 1} e = a + b + c assert e.match(a + p + c) == {p: b} assert e.match(b + p + c) == {p: a} e = a + b + c + x assert e.match(a + p + x + c) == {p: b} assert e.match(b + p + c + x) == {p: a} assert e.match(b) is None assert e.match(b + p) == {p: a + c + x} assert e.match(a + p + c) == {p: b + x} assert e.match(b + p + c) == {p: a + x} e = 4x + 5 assert e.match(4x + p) == {p: 5} assert e.match(3x + p) == {p: x + 5} assert e.match(px + 5) == {p: 4} def test_power(): x, y, a, b, c = map(Symbol, 'xyabc') p, q, r = map(Wild, 'pqr') e = (x + y)a assert e.match(pq) == {p: x + y, q: a} assert e.match(pp) is None e = (x + y)(x + y) assert e.match(pp) == {p: x + y} assert e.match(pq) == {p: x + y, q: x + y} e = (2x)2 assert e.match(pqr) == {p: 4, q: x, r: 2} e = Integer(1) assert e.match(xp) == {p: 0} def test_match_exclude(): x = Symbol('x') y = Symbol('y') p = Wild("p") q = Wild("q") r = Wild("r") e = Rational(6) assert e.match(2p) == {p: 3} e = 3/(4x + 5) assert e.match(3/(px + q)) == {p: 4, q: 5} e = 3/(4x + 5) assert e.match(p/(qx + r)) == {p: 3, q: 4, r: 5} e = 2/(x + 1) assert e.match(p/(qx + r)) == {p: 2, q: 1, r: 1} e = 1/(x + 1) assert e.match(p/(qx + r)) == {p: 1, q: 1, r: 1} e = 4x + 5 assert e.match(px + q) == {p: 4, q: 5} e = 4x + 5y + 6 assert e.match(px + qy + r) == {p: 4, q: 5, r: 6} a = Wild('a', exclude=[x]) e = 3x assert e.match(px) == {p: 3} assert e.match(ax) == {a: 3} e = 3x2 assert e.match(px) == {p: 3x} assert e.match(ax) is None e = 3x + 3 + 6/x assert e.match(px*2 + px + 2p) == {p: 3/x} assert e.match(ax*2 + ax + 2a) is None def test_mul(): x, y, a, b, c = map(Symbol, 'xyabc') p, q = map(Wild, 'pq') e = 4x assert e.match(px) == {p: 4} assert e.match(py) is None assert e.match(e + py) == {p: 0} e = axbc assert e.match(px) == {p: abc} assert e.match(cpx) == {p: ab} e = (a + b)(a + c) assert e.match((p + b)(p + c)) == {p: a} e = x assert e.match(px) == {p: 1} e = exp(x) assert e.match(xpexp(xq)) == {p: 0, q: 1} e = IPoly(x, x) assert e.match(Ip) == {p: x} def test_mul_noncommutative(): x, y = symbols('x y') A, B, C = symbols('A B C', commutative=False) u, v = symbols('u v', cls=Wild) w, z = symbols('w z', cls=Wild, commutative=False) assert (uv).matches(x) in ({v: x, u: 1}, {u: x, v: 1}) assert (uv).matches(xy) in ({v: y, u: x}, {u: y, v: x}) assert (uv).matches(A) is None assert (uv).matches(AB) is None assert (uv).matches(xA) is None assert (uv).matches(xyA) is None assert (uv).matches(xAB) is None assert (uv).matches(xyAB) is None assert (vw).matches(x) is None assert (vw).matches(xy) is None assert (vw).matches(A) == {w: A, v: 1} assert (vw).matches(AB) == {w: AB, v: 1} assert (vw).matches(xA) == {w: A, v: x} assert (vw).matches(xyA) == {w: A, v: xy} assert (vw).matches(xAB) == {w: AB, v: x} assert (vw).matches(xyAB) == {w: AB, v: xy} assert (vw).matches(-x) is None assert (vw).matches(-xy) is None assert (vw).matches(-A) == {w: A, v: -1} assert (vw).matches(-AB) == {w: AB, v: -1} assert (vw).matches(-xA) == {w: A, v: -x} assert (vw).matches(-xyA) == {w: A, v: -xy} assert (vw).matches(-xAB) == {w: AB, v: -x} assert (vw).matches(-xyAB) == {w: AB, v: -xy} assert (wz).matches(x) is None assert (wz).matches(xy) is None assert (wz).matches(A) is None assert (wz).matches(AB) == {w: A, z: B} assert (wz).matches(BA) == {w: B, z: A} assert (wz).matches(ABC) in [{w: A, z: BC}, {w: AB, z: C}] assert (wz).matches(xA) is None assert (wz).matches(xyA) is None assert (wz).matches(xAB) is None assert (wz).matches(xyAB) is None assert (wA).matches(A) is None assert (AwB).matches(AB) is None assert (uwz).matches(x) is None assert (uwz).matches(xy) is None assert (uwz).matches(A) is None assert (uwz).matches(AB) == {u: 1, w: A, z: B} assert (uwz).matches(BA) == {u: 1, w: B, z: A} assert (uwz).matches(xA) is None assert (uwz).matches(xyA) is None assert (uwz).matches(xAB) == {u: x, w: A, z: B} assert (uwz).matches(xBA) == {u: x, w: B, z: A} assert (uwz).matches(xyAB) == {u: xy, w: A, z: B} assert (uwz).matches(xyBA) == {u: xy, w: B, z: A} assert (uA).matches(xA) == {u: x} assert (uA).matches(xAB) is None assert (uB).matches(xA) is None assert (uAB).matches(xAB) == {u: x} assert (uAB).matches(xBA) is None assert (uAB).matches(xA) is None assert (uwA).matches(xAB) is None assert (uwB).matches(xAB) == {u: x, w: A} assert (uvAB).matches(xAB) in [{u: x, v: 1}, {v: x, u: 1}] assert (uvAB).matches(xBA) is None assert (uvAB).matches(uvAC) is None def test_mul_noncommutative_mismatch(): A, B, C = symbols('A B C', commutative=False) w = symbols('w', cls=Wild, commutative=False) assert (wBw).matches(ABA) == {w: A} assert (wBw).matches(ACBAC) == {w: AC} assert (wBw).matches(ACBAB) is None assert (wBw).matches(ABC) is None assert (wwC).matches(ABC) is None def test_mul_noncommutative_pow(): A, B, C = symbols('A B C', commutative=False) w = symbols('w', cls=Wild, commutative=False) assert (ABw).matches(AB*2) == {w: B} assert (A(B*2)w(B3)).matches(AB8) == {w: B3} assert (ABwC).matches(A(B*4)C) == {w: B*3} assert (AB(w(-1))).matches(AB(C(-1))) == {w: C} assert (A(Bw)(-1)C).matches(A(BC)*(-1)C) == {w: C} assert ((w*2)BC).matches((A2)BC) == {w: A} assert ((w2)B(w3)).matches((A2)B(A3)) == {w: A} assert ((w2)B(w4)).matches((A2)B(A2)) is None def test_complex(): a, b, c = map(Symbol, 'abc') x, y = map(Wild, 'xy') assert (1 + I).match(x + I) == {x: 1} assert (a + I).match(x + I) == {x: a} assert (2I).match(xI) == {x: 2} assert (aI).match(xI) == {x: a} assert (aI).match(xy) == {x: I, y: a} assert (2I).match(xy) == {x: 2, y: I} assert (a + bI).match(x + yI) == {x: a, y: b} def test_functions(): from sympy.core.function import WildFunction x = Symbol('x') g = WildFunction('g') p = Wild('p') q = Wild('q') f = cos(5x) notf = x assert f.match(pcos(qx)) == {p: 1, q: 5} assert f.match(pg) == {p: 1, g: cos(5x)} assert notf.match(g) is None @XFAIL def test_functions_X1(): from sympy.core.function import WildFunction x = Symbol('x') g = WildFunction('g') p = Wild('p') q = Wild('q') f = cos(5x) assert f.match(pg(qx)) == {p: 1, g: cos, q: 5} def test_interface(): x, y = map(Symbol, 'xy') p, q = map(Wild, 'pq') assert (x + 1).match(p + 1) == {p: x} assert (x3).match(p3) == {p: x} assert (x3).match(p3) == {p: x} assert (xcos(y)).match(pcos(q)) == {p: x, q: y} assert (xy).match(pq) in [{p:x, q:y}, {p:y, q:x}] assert (x + y).match(p + q) in [{p:x, q:y}, {p:y, q:x}] assert (xy + 1).match(pq) in [{p:1, q:1 + xy}, {p:1 + xy, q:1}] def test_derivative1(): x, y = map(Symbol, 'xy') p, q = map(Wild, 'pq') f = Function('f', nargs=1) fd = Derivative(f(x), x) assert fd.match(p) == {p: fd} assert (fd + 1).match(p + 1) == {p: fd} assert (fd).match(fd) == {} assert (3fd).match(pfd) is not None assert (3fd - 1).match(pfd + q) == {p: 3, q: -1} def test_derivative_bug1(): f = Function("f") x = Symbol("x") a = Wild("a", exclude=[f, x]) b = Wild("b", exclude=[f]) pattern = a Derivative(f(x), x, x) + b expr = Derivative(f(x), x) + x2 d1 = {b: x2} d2 = pattern.xreplace(d1).matches(expr, d1) assert d2 is None def test_derivative2(): f = Function("f") x = Symbol("x") a = Wild("a", exclude=[f, x]) b = Wild("b", exclude=[f]) e = Derivative(f(x), x) assert e.match(Derivative(f(x), x)) == {} assert e.match(Derivative(f(x), x, x)) is None e = Derivative(f(x), x, x) assert e.match(Derivative(f(x), x)) is None assert e.match(Derivative(f(x), x, x)) == {} e = Derivative(f(x), x) + x*2 assert e.match(aDerivative(f(x), x) + b) == {a: 1, b: x*2} assert e.match(aDerivative(f(x), x, x) + b) is None e = Derivative(f(x), x, x) + x*2 assert e.match(aDerivative(f(x), x) + b) is None assert e.match(aDerivative(f(x), x, x) + b) == {a: 1, b: x2} def test_match_deriv_bug1(): n = Function('n') l = Function('l') x = Symbol('x') p = Wild('p') e = diff(l(x), x)/x - diff(diff(n(x), x), x)/2 - \ diff(n(x), x)2/4 + diff(n(x), x)diff(l(x), x)/4 e = e.subs(n(x), -l(x)).doit() t = xexp(-l(x)) t2 = t.diff(x, x)/t assert e.match( (pt2).expand() ) == {p: Rational(-1, 2)} def test_match_bug2(): x, y = map(Symbol, 'xy') p, q, r = map(Wild, 'pqr') res = (x + y).match(p + q + r) assert (p + q + r).subs(res) == x + y def test_match_bug3(): x, a, b = map(Symbol, 'xab') p = Wild('p') assert (bxexp(ax)).match(xexp(px)) is None def test_match_bug4(): x = Symbol('x') p = Wild('p') e = x assert e.match(-px) == {p: -1} def test_match_bug5(): x = Symbol('x') p = Wild('p') e = -x assert e.match(-px) == {p: 1} def test_match_bug6(): x = Symbol('x') p = Wild('p') e = x assert e.match(3px) == {p: Rational(1)/3} def test_match_polynomial(): x = Symbol('x') a = Wild('a', exclude=[x]) b = Wild('b', exclude=[x]) c = Wild('c', exclude=[x]) d = Wild('d', exclude=[x]) eq = 4x*3 + 3x*2 + 2x + 1 pattern = ax3 + bx*2 + cx + d assert eq.match(pattern) == {a: 4, b: 3, c: 2, d: 1} assert (eq - 3x2).match(pattern) == {a: 4, b: 0, c: 2, d: 1} assert (x + sqrt(2) + 3).match(a + bx + cx2) == \ {b: 1, a: sqrt(2) + 3, c: 0} def test_exclude(): x, y, a = map(Symbol, 'xya') p = Wild('p', exclude=[1, x]) q = Wild('q') r = Wild('r', exclude=[sin, y]) assert sin(x).match(r) is None assert cos(y).match(r) is None e = 3x*2 + yx + a assert e.match(px2 + qx + r) == {p: 3, q: y, r: a} e = x + 1 assert e.match(x + p) is None assert e.match(p + 1) is None assert e.match(x + 1 + p) == {p: 0} e = cos(x) + 5sin(y) assert e.match(r) is None assert e.match(cos(y) + r) is None assert e.match(r + psin(q)) == {r: cos(x), p: 5, q: y} def test_floats(): a, b = map(Wild, 'ab') e = cos(0.12345, evaluate=False)*2 r = e.match(acos(b)*2) assert r == {a: 1, b: Float(0.12345)} def test_Derivative_bug1(): f = Function("f") x = abc.x a = Wild("a", exclude=[f(x)]) b = Wild("b", exclude=[f(x)]) eq = f(x).diff(x) assert eq.match(aDerivative(f(x), x) + b) == {a: 1, b: 0} def test_match_wild_wild(): p = Wild('p') q = Wild('q') r = Wild('r') assert p.match(q + r) in [ {q: p, r: 0}, {q: 0, r: p} ] assert p.match(qr) in [ {q: p, r: 1}, {q: 1, r: p} ] p = Wild('p') q = Wild('q', exclude=[p]) r = Wild('r') assert p.match(q + r) == {q: 0, r: p} assert p.match(qr) == {q: 1, r: p} p = Wild('p') q = Wild('q', exclude=[p]) r = Wild('r', exclude=[p]) assert p.match(q + r) is None assert p.match(qr) is None def test__combine_inverse(): x, y = symbols("x y") assert Mul._combine_inverse(xIy, xI) == y assert Mul._combine_inverse(xx(1 + y), x(1 + y)) == x assert Mul._combine_inverse(xIy, yI) == x assert Mul._combine_inverse(ooIy, yI) is oo assert Mul._combine_inverse(ooIy, ooI) == y assert Mul._combine_inverse(ooIy, ooI) == y assert Mul._combine_inverse(ooy, -oo) == -y assert Mul._combine_inverse(-ooy, oo) == -y assert Mul._combine_inverse((1-exp(x/y)),(exp(x/y)-1)) == -1 assert Add._combine_inverse(oo, oo) is S.Zero assert Add._combine_inverse(ooI, ooI) is S.Zero assert Add._combine_inverse(xoo, xoo) is S.Zero assert Add._combine_inverse(-xoo, -xoo) is S.Zero assert Add._combine_inverse((x - oo)(x + oo), -oo) def test_issue_3773(): x = symbols('x') z, phi, r = symbols('z phi r') c, A, B, N = symbols('c A B N', cls=Wild) l = Wild('l', exclude=(0,)) eq = z * sin(2phi) r*7 matcher = c sin(phiN)l r*A log(r)*B assert eq.match(matcher) == {c: z, l: 1, N: 2, A: 7, B: 0} assert (-eq).match(matcher) == {c: -z, l: 1, N: 2, A: 7, B: 0} assert (xeq).match(matcher) == {c: xz, l: 1, N: 2, A: 7, B: 0} assert (-7xeq).match(matcher) == {c: -7xz, l: 1, N: 2, A: 7, B: 0} matcher = csin(phiN)l r*A assert eq.match(matcher) == {c: z, l: 1, N: 2, A: 7} assert (-eq).match(matcher) == {c: -z, l: 1, N: 2, A: 7} assert (xeq).match(matcher) == {c: xz, l: 1, N: 2, A: 7} assert (-7xeq).match(matcher) == {c: -7xz, l: 1, N: 2, A: 7} def test_issue_3883(): from sympy.abc import gamma, mu, x f = (-gamma (x - mu)*2 - log(gamma) + log(2pi))/2 a, b, c = symbols('a b c', cls=Wild, exclude=(gamma,)) assert f.match(a * log(gamma) + b * gamma + c) == \ {a: Rational(-1, 2), b: -(-mu + x)*2/2, c: log(2pi)/2} assert f.expand().collect(gamma).match(a * log(gamma) + b * gamma + c) == \ {a: Rational(-1, 2), b: (-(x - mu)*2/2).expand(), c: (log(2pi)/2).expand()} g1 = Wild('g1', exclude=[gamma]) g2 = Wild('g2', exclude=[gamma]) g3 = Wild('g3', exclude=[gamma]) assert f.expand().match(g1 * log(gamma) + g2 * gamma + g3) == \ {g3: log(2)/2 + log(pi)/2, g1: Rational(-1, 2), g2: -mu*2/2 + mux - x*2/2} def test_issue_4418(): x = Symbol('x') a, b, c = symbols('a b c', cls=Wild, exclude=(x,)) f, g = symbols('f g', cls=Function) eq = diff(g(x)f(x).diff(x), x) assert eq.match( g(x).diff(x)f(x).diff(x) + g(x)f(x).diff(x, x) + c) == {c: 0} assert eq.match(ag(x).diff( x)f(x).diff(x) + bg(x)f(x).diff(x, x) + c) == {a: 1, b: 1, c: 0} def test_issue_4700(): f = Function('f') x = Symbol('x') a, b = symbols('a b', cls=Wild, exclude=(f(x),)) p = af(x) + b eq1 = sin(x) eq2 = f(x) + sin(x) eq3 = f(x) + x + sin(x) eq4 = x + sin(x) assert eq1.match(p) == {a: 0, b: sin(x)} assert eq2.match(p) == {a: 1, b: sin(x)} assert eq3.match(p) == {a: 1, b: x + sin(x)} assert eq4.match(p) == {a: 0, b: x + sin(x)} def test_issue_5168(): a, b, c = symbols('a b c', cls=Wild) x = Symbol('x') f = Function('f') assert x.match(a) == {a: x} assert x.match(af(x)*c) == {a: x, c: 0} assert x.match(ab) == {a: 1, b: x} assert x.match(abf(x)*c) == {a: 1, b: x, c: 0} assert (-x).match(a) == {a: -x} assert (-x).match(af(x)*c) == {a: -x, c: 0} assert (-x).match(ab) == {a: -1, b: x} assert (-x).match(abf(x)*c) == {a: -1, b: x, c: 0} assert (2x).match(a) == {a: 2x} assert (2x).match(af(x)c) == {a: 2x, c: 0} assert (2x).match(ab) == {a: 2, b: x} assert (2x).match(abf(x)c) == {a: 2, b: x, c: 0} assert (-2x).match(a) == {a: -2x} assert (-2x).match(af(x)c) == {a: -2x, c: 0} assert (-2x).match(ab) == {a: -2, b: x} assert (-2x).match(abf(x)c) == {a: -2, b: x, c: 0} def test_issue_4559(): x = Symbol('x') e = Symbol('e') w = Wild('w', exclude=[x]) y = Wild('y') # this is as it should be assert (3/x).match(w/y) == {w: 3, y: x} assert (3x).match(wy) == {w: 3, y: x} assert (x/3).match(y/w) == {w: 3, y: x} assert (3x).match(y/w) == {w: S.One/3, y: x} assert (3x).match(y/w) == {w: Rational(1, 3), y: x} # these could be allowed to fail assert (x/3).match(w/y) == {w: S.One/3, y: 1/x} assert (3x).match(w/y) == {w: 3, y: 1/x} assert (3/x).match(wy) == {w: 3, y: 1/x} # Note that solve will give # multiple roots but match only gives one: # # >>> solve(xr-y2,y) # [-x(r/2), x(r/2)] r = Symbol('r', rational=True) assert (xr).match(y2) == {y: x(r/2)} assert (xe).match(y2) == {y: sqrt(xe)} # since (xi = y) -> x = y(1/i) where i is an integer # the following should also be valid as long as y is not # zero when i is negative. a = Wild('a') e = S.Zero assert e.match(a) == {a: e} assert e.match(1/a) is None assert e.match(a.3) is None e = S(3) assert e.match(1/a) == {a: 1/e} assert e.match(1/a2) == {a: 1/sqrt(e)} e = pi assert e.match(1/a) == {a: 1/e} assert e.match(1/a2) == {a: 1/sqrt(e)} assert (-e).match(sqrt(a)) is None assert (-e).match(a2) == {a: Isqrt(pi)} # The pattern matcher doesn't know how to handle (x - a)2 == (a - x)2. To # avoid ambiguity in actual applications, don't put a coefficient (including a # minus sign) in front of a wild. @XFAIL def test_issue_4883(): a = Wild('a') x = Symbol('x') e = [i2 for i in (x - 2, 2 - x)] p = [i2 for i in (x - a, a- x)] for eq in e: for pat in p: assert eq.match(pat) == {a: 2} def test_issue_4319(): x, y = symbols('x y') p = -x(S.One/8 - y) ans = {S.Zero, y - S.One/8} def ok(pat): assert set(p.match(pat).values()) == ans ok(Wild("coeff", exclude=[x])x + Wild("rest")) ok(Wild("w", exclude=[x])x + Wild("rest")) ok(Wild("coeff", exclude=[x])x + Wild("rest")) ok(Wild("w", exclude=[x])x + Wild("rest")) ok(Wild("e", exclude=[x])x + Wild("rest")) ok(Wild("ress", exclude=[x])x + Wild("rest")) ok(Wild("resu", exclude=[x])x + Wild("rest")) def test_issue_3778(): p, c, q = symbols('p c q', cls=Wild) x = Symbol('x') assert (sin(x)*2).match(sin(p)sin(q)c) == {q: x, c: 1, p: x} assert (2sin(x)).match(sin(p) + sin(q) + c) == {q: x, c: 0, p: x} def test_issue_6103(): x = Symbol('x') a = Wild('a') assert (-Ixoo).match(Iaoo) == {a: -x} def test_issue_3539(): a = Wild('a') x = Symbol('x') assert (x - 2).match(a - x) is None assert (6/x).match(ax) is None assert (6/x2).match(a/x) == {a: 6/x} def test_gh_issue_2711(): x = Symbol('x') f = meijerg(((), ()), ((0,), ()), x) a = Wild('a') b = Wild('b') assert f.find(a) == {(S.Zero,), ((), ()), ((S.Zero,), ()), x, S.Zero, (), meijerg(((), ()), ((S.Zero,), ()), x)} assert f.find(a + b) == \ {meijerg(((), ()), ((S.Zero,), ()), x), x, S.Zero} assert f.find(a2) == {meijerg(((), ()), ((S.Zero,), ()), x), x} def test_issue_17354(): from sympy.core.symbol import (Wild, symbols) x, y = symbols("x y", real=True) a, b = symbols("a b", cls=Wild) assert ((0 <= x).reversed \| (y <= x)).match((1/a <= b) \| (a <= b)) is None def test_match_issue_17397(): f = Function("f") x = Symbol("x") a3 = Wild('a3', exclude=[f(x), f(x).diff(x), f(x).diff(x, 2)]) b3 = Wild('b3', exclude=[f(x), f(x).diff(x), f(x).diff(x, 2)]) c3 = Wild('c3', exclude=[f(x), f(x).diff(x), f(x).diff(x, 2)]) deq = a3(f(x).diff(x, 2)) + b3f(x).diff(x) + c3f(x) eq = (x-2)*2(f(x).diff(x, 2)) + (x-2)(f(x).diff(x)) + ((x-2)2 - 4)f(x) r = collect(eq, [f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq) assert r == {a3: (x - 2)2, c3: (x - 2)2 - 4, b3: x - 2} eq =xf(x) + xDerivative(f(x), (x, 2)) - 4f(x) + Derivative(f(x), x) \ - 4Derivative(f(x), (x, 2)) - 2Derivative(f(x), x)/x + 4Derivative(f(x), (x, 2))/x r = collect(eq, [f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq) assert r == {a3: x - 4 + 4/x, b3: 1 - 2/x, c3: x - 4} def test_match_issue_21942(): a, r, w = symbols('a, r, w', nonnegative=True) p = symbols('p', positive=True) g_ = Wild('g') pattern = g_ ** (1 / (1 - p)) eq = (a * r (1 - p) + w (1 - p) * (1 - a)) ** (1 / (1 - p)) m = {g_: a * r (1 - p) + w (1 - p) * (1 - a)} assert pattern.matches(eq) == m assert (-pattern).matches(-eq) == m assert pattern.matches(signsimp(eq)) is None def test_match_terms(): X, Y = map(Wild, "XY") x, y, z = symbols('x y z') assert (5y - x).match(5X - Y) == {X: y, Y: x} # 15907 assert (x + (y - 1)z).match(x + Xz) == {X: y - 1} # 20747 assert (x - log(x/y)(1-exp(x/y))).match(x - log(X/y)(1-exp(x/y))) == {X: x} def test_match_bound(): V, W = map(Wild, "VW") x, y = symbols('x y') assert Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, W))) == {W: 2} assert Sum(x, (x, 1, 2)).match(Sum(V, (V, 1, W))) == {W: 2, V:x} assert Sum(x, (x, 1, 2)).match(Sum(V, (V, 1, 2))) == {V:x} def test_issue_22462(): x, f = symbols('x'), Function('f') n, Q = symbols('n Q', cls=Wild) pattern = -Qf(x)n eq = 5f(x)**2 assert pattern.matches(eq) == {n: 2, Q: -5}
898366b844e3bde1c15693b2c32bff8d2b505a41707f51ae1018e8654561c4ae	"""Tests of tools for setting up interactive IPython sessions. """ from sympy.interactive.session import (init_ipython_session, enable_automatic_symbols, enable_automatic_int_sympification) from sympy.core import Symbol, Rational, Integer from sympy.external import import_module from sympy.testing.pytest import raises # TODO: The code below could be made more granular with something like: # # @requires('IPython', version=">=0.11") # def test_automatic_symbols(ipython): ipython = import_module("IPython", min_module_version="0.11") if not ipython: #bin/test will not execute any tests now disabled = True # WARNING: These tests will modify the existing IPython environment. IPython # uses a single instance for its interpreter, so there is no way to isolate # the test from another IPython session. It also means that if this test is # run twice in the same Python session it will fail. This isn't usually a # problem because the test suite is run in a subprocess by default, but if the # tests are run with subprocess=False it can pollute the current IPython # session. See the discussion in issue #15149. def test_automatic_symbols(): # NOTE: Because of the way the hook works, you have to use run_cell(code, # True). This means that the code must have no Out, or it will be printed # during the tests. app = init_ipython_session() app.run_cell("from sympy import ") enable_automatic_symbols(app) symbol = "verylongsymbolname" assert symbol not in app.user_ns app.run_cell("a = %s" % symbol, True) assert symbol not in app.user_ns app.run_cell("a = type(%s)" % symbol, True) assert app.user_ns['a'] == Symbol app.run_cell("%s = Symbol('%s')" % (symbol, symbol), True) assert symbol in app.user_ns # Check that built-in names aren't overridden app.run_cell("a = all == __builtin__.all", True) assert "all" not in app.user_ns assert app.user_ns['a'] is True # Check that SymPy names aren't overridden app.run_cell("import sympy") app.run_cell("a = factorial == sympy.factorial", True) assert app.user_ns['a'] is True def test_int_to_Integer(): # XXX: Warning, don't test with == here. 0.5 == Rational(1, 2) is True! app = init_ipython_session() app.run_cell("from sympy import Integer") app.run_cell("a = 1") assert isinstance(app.user_ns['a'], int) enable_automatic_int_sympification(app) app.run_cell("a = 1/2") assert isinstance(app.user_ns['a'], Rational) app.run_cell("a = 1") assert isinstance(app.user_ns['a'], Integer) app.run_cell("a = int(1)") assert isinstance(app.user_ns['a'], int) app.run_cell("a = (1/\n2)") assert app.user_ns['a'] == Rational(1, 2) # TODO: How can we test that the output of a SyntaxError is the original # input, not the transformed input? def test_ipythonprinting(): # Initialize and setup IPython session app = init_ipython_session() app.run_cell("ip = get_ipython()") app.run_cell("inst = ip.instance()") app.run_cell("format = inst.display_formatter.format") app.run_cell("from sympy import Symbol") # Printing without printing extension app.run_cell("a = format(Symbol('pi'))") app.run_cell("a2 = format(Symbol('pi')2)") # Deal with API change starting at IPython 1.0 if int(ipython.__version__.split(".")[0]) < 1: assert app.user_ns['a']['text/plain'] == "pi" assert app.user_ns['a2']['text/plain'] == "pi2" else: assert app.user_ns['a'][0]['text/plain'] == "pi" assert app.user_ns['a2'][0]['text/plain'] == "pi2" # Load printing extension app.run_cell("from sympy import init_printing") app.run_cell("init_printing()") # Printing with printing extension app.run_cell("a = format(Symbol('pi'))") app.run_cell("a2 = format(Symbol('pi')*2)") # Deal with API change starting at IPython 1.0 if int(ipython.__version__.split(".")[0]) < 1: assert app.user_ns['a']['text/plain'] in ('\N{GREEK SMALL LETTER PI}', 'pi') assert app.user_ns['a2']['text/plain'] in (' 2\n\N{GREEK SMALL LETTER PI} ', ' 2\npi ') else: assert app.user_ns['a'][0]['text/plain'] in ('\N{GREEK SMALL LETTER PI}', 'pi') assert app.user_ns['a2'][0]['text/plain'] in (' 2\n\N{GREEK SMALL LETTER PI} ', ' 2\npi ') def test_print_builtin_option(): # Initialize and setup IPython session app = init_ipython_session() app.run_cell("ip = get_ipython()") app.run_cell("inst = ip.instance()") app.run_cell("format = inst.display_formatter.format") app.run_cell("from sympy import Symbol") app.run_cell("from sympy import init_printing") app.run_cell("a = format({Symbol('pi'): 3.14, Symbol('n_i'): 3})") # Deal with API change starting at IPython 1.0 if int(ipython.__version__.split(".")[0]) < 1: text = app.user_ns['a']['text/plain'] raises(KeyError, lambda: app.user_ns['a']['text/latex']) else: text = app.user_ns['a'][0]['text/plain'] raises(KeyError, lambda: app.user_ns['a'][0]['text/latex']) # XXX: How can we make this ignore the terminal width? This test fails if # the terminal is too narrow. assert text in ("{pi: 3.14, n_i: 3}", '{n\N{LATIN SUBSCRIPT SMALL LETTER I}: 3, \N{GREEK SMALL LETTER PI}: 3.14}', "{n_i: 3, pi: 3.14}", '{\N{GREEK SMALL LETTER PI}: 3.14, n\N{LATIN SUBSCRIPT SMALL LETTER I}: 3}') # If we enable the default printing, then the dictionary's should render # as a LaTeX version of the whole dict: ${\pi: 3.14, n_i: 3}$ app.run_cell("inst.display_formatter.formatters['text/latex'].enabled = True") app.run_cell("init_printing(use_latex=True)") app.run_cell("a = format({Symbol('pi'): 3.14, Symbol('n_i'): 3})") # Deal with API change starting at IPython 1.0 if int(ipython.__version__.split(".")[0]) < 1: text = app.user_ns['a']['text/plain'] latex = app.user_ns['a']['text/latex'] else: text = app.user_ns['a'][0]['text/plain'] latex = app.user_ns['a'][0]['text/latex'] assert text in ("{pi: 3.14, n_i: 3}", '{n\N{LATIN SUBSCRIPT SMALL LETTER I}: 3, \N{GREEK SMALL LETTER PI}: 3.14}', "{n_i: 3, pi: 3.14}", '{\N{GREEK SMALL LETTER PI}: 3.14, n\N{LATIN SUBSCRIPT SMALL LETTER I}: 3}') assert latex == r'$\displaystyle \left\{ n_{i} : 3, \ \pi : 3.14\right\}$' # Objects with an _latex overload should also be handled by our tuple # printer. app.run_cell("""\ class WithOverload: def _latex(self, printer): return r"\\LaTeX" """) app.run_cell("a = format((WithOverload(),))") # Deal with API change starting at IPython 1.0 if int(ipython.__version__.split(".")[0]) < 1: latex = app.user_ns['a']['text/latex'] else: latex = app.user_ns['a'][0]['text/latex'] assert latex == r'$\displaystyle \left( \LaTeX,\right)$' app.run_cell("inst.display_formatter.formatters['text/latex'].enabled = True") app.run_cell("init_printing(use_latex=True, print_builtin=False)") app.run_cell("a = format({Symbol('pi'): 3.14, Symbol('n_i'): 3})") # Deal with API change starting at IPython 1.0 if int(ipython.__version__.split(".")[0]) < 1: text = app.user_ns['a']['text/plain'] raises(KeyError, lambda: app.user_ns['a']['text/latex']) else: text = app.user_ns['a'][0]['text/plain'] raises(KeyError, lambda: app.user_ns['a'][0]['text/latex']) # Note : In Python 3 we have one text type: str which holds Unicode data # and two byte types bytes and bytearray. # Python 3.3.3 + IPython 0.13.2 gives: '{n_i: 3, pi: 3.14}' # Python 3.3.3 + IPython 1.1.0 gives: '{n_i: 3, pi: 3.14}' assert text in ("{pi: 3.14, n_i: 3}", "{n_i: 3, pi: 3.14}") def test_builtin_containers(): # Initialize and setup IPython session app = init_ipython_session() app.run_cell("ip = get_ipython()") app.run_cell("inst = ip.instance()") app.run_cell("format = inst.display_formatter.format") app.run_cell("inst.display_formatter.formatters['text/latex'].enabled = True") app.run_cell("from sympy import init_printing, Matrix") app.run_cell('init_printing(use_latex=True, use_unicode=False)') # Make sure containers that shouldn't pretty print don't. app.run_cell('a = format((True, False))') app.run_cell('import sys') app.run_cell('b = format(sys.flags)') app.run_cell('c = format((Matrix([1, 2]),))') # Deal with API change starting at IPython 1.0 if int(ipython.__version__.split(".")[0]) < 1: assert app.user_ns['a']['text/plain'] == '(True, False)' assert 'text/latex' not in app.user_ns['a'] assert app.user_ns['b']['text/plain'][:10] == 'sys.flags(' assert 'text/latex' not in app.user_ns['b'] assert app.user_ns['c']['text/plain'] == \ """\ [1] \n\ ([ ],) [2] \ """ assert app.user_ns['c']['text/latex'] == '$\\displaystyle \\left( \\left[\\begin{matrix}1\\\\2\\end{matrix}\\right],\\right)$' else: assert app.user_ns['a'][0]['text/plain'] == '(True, False)' assert 'text/latex' not in app.user_ns['a'][0] assert app.user_ns['b'][0]['text/plain'][:10] == 'sys.flags(' assert 'text/latex' not in app.user_ns['b'][0] assert app.user_ns['c'][0]['text/plain'] == \ """\ [1] \n\ ([ ],) [2] \ """ assert app.user_ns['c'][0]['text/latex'] == '$\\displaystyle \\left( \\left[\\begin{matrix}1\\\\2\\end{matrix}\\right],\\right)$' def test_matplotlib_bad_latex(): # Initialize and setup IPython session app = init_ipython_session() app.run_cell("import IPython") app.run_cell("ip = get_ipython()") app.run_cell("inst = ip.instance()") app.run_cell("format = inst.display_formatter.format") app.run_cell("from sympy import init_printing, Matrix") app.run_cell("init_printing(use_latex='matplotlib')") # The png formatter is not enabled by default in this context app.run_cell("inst.display_formatter.formatters['image/png'].enabled = True") # Make sure no warnings are raised by IPython app.run_cell("import warnings") # IPython.core.formatters.FormatterWarning was introduced in IPython 2.0 if int(ipython.__version__.split(".")[0]) < 2: app.run_cell("warnings.simplefilter('error')") else: app.run_cell("warnings.simplefilter('error', IPython.core.formatters.FormatterWarning)") # This should not raise an exception app.run_cell("a = format(Matrix([1, 2, 3]))") # issue 9799 app.run_cell("from sympy import Piecewise, Symbol, Eq") app.run_cell("x = Symbol('x'); pw = format(Piecewise((1, Eq(x, 0)), (0, True)))") def test_override_repr_latex(): # Initialize and setup IPython session app = init_ipython_session() app.run_cell("import IPython") app.run_cell("ip = get_ipython()") app.run_cell("inst = ip.instance()") app.run_cell("format = inst.display_formatter.format") app.run_cell("inst.display_formatter.formatters['text/latex'].enabled = True") app.run_cell("from sympy import init_printing") app.run_cell("from sympy import Symbol") app.run_cell("init_printing(use_latex=True)") app.run_cell("""\ class SymbolWithOverload(Symbol): def _repr_latex_(self): return r"Hello " + super()._repr_latex_() + " world" """) app.run_cell("a = format(SymbolWithOverload('s'))") if int(ipython.__version__.split(".")[0]) < 1: latex = app.user_ns['a']['text/latex'] else: latex = app.user_ns['a'][0]['text/latex'] assert latex == r'Hello $\displaystyle s$ world'
33755418e0e1dcc2c495e3e362d8edf5bd3645e313870f975e9b904e4a2180db	""" Rational number type based on Python integers. The PythonRational class from here has been moved to sympy.external.pythonmpq This module is just left here for backwards compatibility. """ from sympy.core.numbers import Rational from sympy.core.sympify import _sympy_converter from sympy.utilities import public from sympy.external.pythonmpq import PythonMPQ PythonRational = public(PythonMPQ) def sympify_pythonrational(arg): return Rational(arg.numerator, arg.denominator) _sympy_converter[PythonRational] = sympify_pythonrational
7cc294dd6c1df977e7d1f82b8dd3232fa06be835624a30d5a89f06f008b5aca9	# coding=utf-8 from os import walk, sep, pardir from os.path import split, join, abspath, exists, isfile from glob import glob import re import random import ast from sympy.testing.pytest import raises from sympy.testing.quality_unicode import _test_this_file_encoding # System path separator (usually slash or backslash) to be # used with excluded files, e.g. # exclude = set([ # "%(sep)smpmath%(sep)s" % sepd, # ]) sepd = {"sep": sep} # path and sympy_path SYMPY_PATH = abspath(join(split(__file__)[0], pardir, pardir)) # go to sympy/ assert exists(SYMPY_PATH) TOP_PATH = abspath(join(SYMPY_PATH, pardir)) BIN_PATH = join(TOP_PATH, "bin") EXAMPLES_PATH = join(TOP_PATH, "examples") # Error messages message_space = "File contains trailing whitespace: %s, line %s." message_implicit = "File contains an implicit import: %s, line %s." message_tabs = "File contains tabs instead of spaces: %s, line %s." message_carriage = "File contains carriage returns at end of line: %s, line %s" message_str_raise = "File contains string exception: %s, line %s" message_gen_raise = "File contains generic exception: %s, line %s" message_old_raise = "File contains old-style raise statement: %s, line %s, \"%s\"" message_eof = "File does not end with a newline: %s, line %s" message_multi_eof = "File ends with more than 1 newline: %s, line %s" message_test_suite_def = "Function should start with 'test_' or '_': %s, line %s" message_duplicate_test = "This is a duplicate test function: %s, line %s" message_self_assignments = "File contains assignments to self/cls: %s, line %s." message_func_is = "File contains '.func is': %s, line %s." message_bare_expr = "File contains bare expression: %s, line %s." implicit_test_re = re.compile(r'^\s(>>> )?(\.\.\. )?from . import .\') str_raise_re = re.compile( r'^\s(>>> )?(\.\.\. )?raise(\s+(\'\|\")\|\s($\s)+(\'\|\"))') gen_raise_re = re.compile( r'^\s(>>> )?(\.\.\. )?raise(\s+Exception\|\s(\(\s)+Exception)') old_raise_re = re.compile(r'^\s(>>> )?(\.\.\. )?raise((\s\(\s)\|\s+)\w+\s,') test_suite_def_re = re.compile(r'^def\s+(?!(_\|test))[^(]\(\s$\s:$') test_ok_def_re = re.compile(r'^def\s+test_.:$') test_file_re = re.compile(r'.[/\\]test_.\.py$') func_is_re = re.compile(r'\.\sfunc\s+is') def tab_in_leading(s): """Returns True if there are tabs in the leading whitespace of a line, including the whitespace of docstring code samples.""" n = len(s) - len(s.lstrip()) if not s[n:n + 3] in ['...', '>>>']: check = s[:n] else: smore = s[n + 3:] check = s[:n] + smore[:len(smore) - len(smore.lstrip())] return not (check.expandtabs() == check) def find_self_assignments(s): """Returns a list of "bad" assignments: if there are instances of assigning to the first argument of the class method (except for staticmethod's). """ t = [n for n in ast.parse(s).body if isinstance(n, ast.ClassDef)] bad = [] for c in t: for n in c.body: if not isinstance(n, ast.FunctionDef): continue if any(d.id == 'staticmethod' for d in n.decorator_list if isinstance(d, ast.Name)): continue if n.name == '__new__': continue if not n.args.args: continue first_arg = n.args.args[0].arg for m in ast.walk(n): if isinstance(m, ast.Assign): for a in m.targets: if isinstance(a, ast.Name) and a.id == first_arg: bad.append(m) elif (isinstance(a, ast.Tuple) and any(q.id == first_arg for q in a.elts if isinstance(q, ast.Name))): bad.append(m) return bad def check_directory_tree(base_path, file_check, exclusions=set(), pattern=".py"): """ Checks all files in the directory tree (with base_path as starting point) with the file_check function provided, skipping files that contain any of the strings in the set provided by exclusions. """ if not base_path: return for root, dirs, files in walk(base_path): check_files(glob(join(root, pattern)), file_check, exclusions) def check_files(files, file_check, exclusions=set(), pattern=None): """ Checks all files with the file_check function provided, skipping files that contain any of the strings in the set provided by exclusions. """ if not files: return for fname in files: if not exists(fname) or not isfile(fname): continue if any(ex in fname for ex in exclusions): continue if pattern is None or re.match(pattern, fname): file_check(fname) class _Visit(ast.NodeVisitor): """return the line number corresponding to the line on which a bare expression appears if it is a binary op or a comparison that is not in a with block. EXAMPLES ======== >>> import ast >>> class _Visit(ast.NodeVisitor): ... def visit_Expr(self, node): ... if isinstance(node.value, (ast.BinOp, ast.Compare)): ... print(node.lineno) ... def visit_With(self, node): ... pass # no checking there ... >>> code='''x = 1 # line 1 ... for i in range(3): ... x == 2 # <-- 3 ... if x == 2: ... x == 3 # <-- 5 ... x + 1 # <-- 6 ... x = 1 ... if x == 1: ... print(1) ... while x != 1: ... x == 1 # <-- 11 ... with raises(TypeError): ... c == 1 ... raise TypeError ... assert x == 1 ... ''' >>> _Visit().visit(ast.parse(code)) 3 5 6 11 """ def visit_Expr(self, node): if isinstance(node.value, (ast.BinOp, ast.Compare)): assert None, message_bare_expr % ('', node.lineno) def visit_With(self, node): pass BareExpr = _Visit() def line_with_bare_expr(code): """return None or else 0-based line number of code on which a bare expression appeared. """ tree = ast.parse(code) try: BareExpr.visit(tree) except AssertionError as msg: assert msg.args msg = msg.args[0] assert msg.startswith(message_bare_expr.split(':', 1)[0]) return int(msg.rsplit(' ', 1)[1].rstrip('.')) # the line number def test_files(): """ This test tests all files in SymPy and checks that: o no lines contains a trailing whitespace o no lines end with \r\n o no line uses tabs instead of spaces o that the file ends with a single newline o there are no general or string exceptions o there are no old style raise statements o name of arg-less test suite functions start with _ or test_ o no duplicate function names that start with test_ o no assignments to self variable in class methods o no lines contain ".func is" except in the test suite o there is no do-nothing expression like `a == b` or `x + 1` """ def test(fname): with open(fname, encoding="utf8") as test_file: test_this_file(fname, test_file) with open(fname, encoding='utf8') as test_file: _test_this_file_encoding(fname, test_file) def test_this_file(fname, test_file): idx = None code = test_file.read() test_file.seek(0) # restore reader to head py = fname if sep not in fname else fname.rsplit(sep, 1)[-1] if py.startswith('test_'): idx = line_with_bare_expr(code) if idx is not None: assert False, message_bare_expr % (fname, idx + 1) line = None # to flag the case where there were no lines in file tests = 0 test_set = set() for idx, line in enumerate(test_file): if test_file_re.match(fname): if test_suite_def_re.match(line): assert False, message_test_suite_def % (fname, idx + 1) if test_ok_def_re.match(line): tests += 1 test_set.add(line[3:].split('(')[0].strip()) if len(test_set) != tests: assert False, message_duplicate_test % (fname, idx + 1) if line.endswith(" \n") or line.endswith("\t\n"): assert False, message_space % (fname, idx + 1) if line.endswith("\r\n"): assert False, message_carriage % (fname, idx + 1) if tab_in_leading(line): assert False, message_tabs % (fname, idx + 1) if str_raise_re.search(line): assert False, message_str_raise % (fname, idx + 1) if gen_raise_re.search(line): assert False, message_gen_raise % (fname, idx + 1) if (implicit_test_re.search(line) and not list(filter(lambda ex: ex in fname, import_exclude))): assert False, message_implicit % (fname, idx + 1) if func_is_re.search(line) and not test_file_re.search(fname): assert False, message_func_is % (fname, idx + 1) result = old_raise_re.search(line) if result is not None: assert False, message_old_raise % ( fname, idx + 1, result.group(2)) if line is not None: if line == '\n' and idx > 0: assert False, message_multi_eof % (fname, idx + 1) elif not line.endswith('\n'): # eof newline check assert False, message_eof % (fname, idx + 1) # Files to test at top level top_level_files = [join(TOP_PATH, file) for file in [ "isympy.py", "build.py", "setup.py", "setupegg.py", ]] # Files to exclude from all tests exclude = { "%(sep)ssympy%(sep)sparsing%(sep)sautolev%(sep)s_antlr%(sep)sautolevparser.py" % sepd, "%(sep)ssympy%(sep)sparsing%(sep)sautolev%(sep)s_antlr%(sep)sautolevlexer.py" % sepd, "%(sep)ssympy%(sep)sparsing%(sep)sautolev%(sep)s_antlr%(sep)sautolevlistener.py" % sepd, "%(sep)ssympy%(sep)sparsing%(sep)slatex%(sep)s_antlr%(sep)slatexparser.py" % sepd, "%(sep)ssympy%(sep)sparsing%(sep)slatex%(sep)s_antlr%(sep)slatexlexer.py" % sepd, } # Files to exclude from the implicit import test import_exclude = { # glob imports are allowed in top-level __init__.py: "%(sep)ssympy%(sep)s__init__.py" % sepd, # these __init__.py should be fixed: # XXX: not really, they use useful import pattern (DRY) "%(sep)svector%(sep)s__init__.py" % sepd, "%(sep)smechanics%(sep)s__init__.py" % sepd, "%(sep)squantum%(sep)s__init__.py" % sepd, "%(sep)spolys%(sep)s__init__.py" % sepd, "%(sep)spolys%(sep)sdomains%(sep)s__init__.py" % sepd, # interactive SymPy executes ``from sympy import ``: "%(sep)sinteractive%(sep)ssession.py" % sepd, # isympy.py executes ``from sympy import ``: "%(sep)sisympy.py" % sepd, # these two are import timing tests: "%(sep)sbin%(sep)ssympy_time.py" % sepd, "%(sep)sbin%(sep)ssympy_time_cache.py" % sepd, # Taken from Python stdlib: "%(sep)sparsing%(sep)ssympy_tokenize.py" % sepd, # this one should be fixed: "%(sep)splotting%(sep)spygletplot%(sep)s" % sepd, # False positive in the docstring "%(sep)sbin%(sep)stest_external_imports.py" % sepd, "%(sep)sbin%(sep)stest_submodule_imports.py" % sepd, # These are deprecated stubs that can be removed at some point: "%(sep)sutilities%(sep)sruntests.py" % sepd, "%(sep)sutilities%(sep)spytest.py" % sepd, "%(sep)sutilities%(sep)srandtest.py" % sepd, "%(sep)sutilities%(sep)stmpfiles.py" % sepd, "%(sep)sutilities%(sep)squality_unicode.py" % sepd, "%(sep)sutilities%(sep)sbenchmarking.py" % sepd, } check_files(top_level_files, test) check_directory_tree(BIN_PATH, test, {"~", ".pyc", ".sh"}, "") check_directory_tree(SYMPY_PATH, test, exclude) check_directory_tree(EXAMPLES_PATH, test, exclude) def _with_space(c): # return c with a random amount of leading space return random.randint(0, 10)' ' + c def test_raise_statement_regular_expression(): candidates_ok = [ "some text # raise Exception, 'text'", "raise ValueError('text') # raise Exception, 'text'", "raise ValueError('text')", "raise ValueError", "raise ValueError('text')", "raise ValueError('text') #,", # Talking about an exception in a docstring ''''"""This function will raise ValueError, except when it doesn't"""''', "raise (ValueError('text')", ] str_candidates_fail = [ "raise 'exception'", "raise 'Exception'", 'raise "exception"', 'raise "Exception"', "raise 'ValueError'", ] gen_candidates_fail = [ "raise Exception('text') # raise Exception, 'text'", "raise Exception('text')", "raise Exception", "raise Exception('text')", "raise Exception('text') #,", "raise Exception, 'text'", "raise Exception, 'text' # raise Exception('text')", "raise Exception, 'text' # raise Exception, 'text'", ">>> raise Exception, 'text'", ">>> raise Exception, 'text' # raise Exception('text')", ">>> raise Exception, 'text' # raise Exception, 'text'", ] old_candidates_fail = [ "raise Exception, 'text'", "raise Exception, 'text' # raise Exception('text')", "raise Exception, 'text' # raise Exception, 'text'", ">>> raise Exception, 'text'", ">>> raise Exception, 'text' # raise Exception('text')", ">>> raise Exception, 'text' # raise Exception, 'text'", "raise ValueError, 'text'", "raise ValueError, 'text' # raise Exception('text')", "raise ValueError, 'text' # raise Exception, 'text'", ">>> raise ValueError, 'text'", ">>> raise ValueError, 'text' # raise Exception('text')", ">>> raise ValueError, 'text' # raise Exception, 'text'", "raise(ValueError,", "raise (ValueError,", "raise( ValueError,", "raise ( ValueError,", "raise(ValueError ,", "raise (ValueError ,", "raise( ValueError ,", "raise ( ValueError ,", ] for c in candidates_ok: assert str_raise_re.search(_with_space(c)) is None, c assert gen_raise_re.search(_with_space(c)) is None, c assert old_raise_re.search(_with_space(c)) is None, c for c in str_candidates_fail: assert str_raise_re.search(_with_space(c)) is not None, c for c in gen_candidates_fail: assert gen_raise_re.search(_with_space(c)) is not None, c for c in old_candidates_fail: assert old_raise_re.search(_with_space(c)) is not None, c def test_implicit_imports_regular_expression(): candidates_ok = [ "from sympy import something", ">>> from sympy import something", "from sympy.somewhere import something", ">>> from sympy.somewhere import something", "import sympy", ">>> import sympy", "import sympy.something.something", "... import sympy", "... import sympy.something.something", "... from sympy import something", "... from sympy.somewhere import something", ">> from sympy import ", # To allow 'fake' docstrings "# from sympy import ", "some text # from sympy import ", ] candidates_fail = [ "from sympy import ", ">>> from sympy import ", "from sympy.somewhere import ", ">>> from sympy.somewhere import ", "... from sympy import ", "... from sympy.somewhere import *", ] for c in candidates_ok: assert implicit_test_re.search(_with_space(c)) is None, c for c in candidates_fail: assert implicit_test_re.search(_with_space(c)) is not None, c def test_test_suite_defs(): candidates_ok = [ " def foo():\n", "def foo(arg):\n", "def _foo():\n", "def test_foo():\n", ] candidates_fail = [ "def foo():\n", "def foo() :\n", "def foo( ):\n", "def foo():\n", ] for c in candidates_ok: assert test_suite_def_re.search(c) is None, c for c in candidates_fail: assert test_suite_def_re.search(c) is not None, c def test_test_duplicate_defs(): candidates_ok = [ "def foo():\ndef foo():\n", "def test():\ndef test_():\n", "def test_():\ndef test__():\n", ] candidates_fail = [ "def test_():\ndef test_ ():\n", "def test_1():\ndef test_1():\n", ] ok = (None, 'check') def check(file): tests = 0 test_set = set() for idx, line in enumerate(file.splitlines()): if test_ok_def_re.match(line): tests += 1 test_set.add(line[3:].split('(')[0].strip()) if len(test_set) != tests: return False, message_duplicate_test % ('check', idx + 1) return None, 'check' for c in candidates_ok: assert check(c) == ok for c in candidates_fail: assert check(c) != ok def test_find_self_assignments(): candidates_ok = [ "class A(object):\n def foo(self, arg): arg = self\n", "class A(object):\n def foo(self, arg): self.prop = arg\n", "class A(object):\n def foo(self, arg): obj, obj2 = arg, self\n", "class A(object):\n @classmethod\n def bar(cls, arg): arg = cls\n", "class A(object):\n def foo(var, arg): arg = var\n", ] candidates_fail = [ "class A(object):\n def foo(self, arg): self = arg\n", "class A(object):\n def foo(self, arg): obj, self = arg, arg\n", "class A(object):\n def foo(self, arg):\n if arg: self = arg", "class A(object):\n @classmethod\n def foo(cls, arg): cls = arg\n", "class A(object):\n def foo(var, arg): var = arg\n", ] for c in candidates_ok: assert find_self_assignments(c) == [] for c in candidates_fail: assert find_self_assignments(c) != [] def test_test_unicode_encoding(): unicode_whitelist = ['foo'] unicode_strict_whitelist = ['bar'] fname = 'abc' test_file = ['α'] raises(AssertionError, lambda: _test_this_file_encoding( fname, test_file, unicode_whitelist, unicode_strict_whitelist)) fname = 'abc' test_file = ['abc'] _test_this_file_encoding( fname, test_file, unicode_whitelist, unicode_strict_whitelist) fname = 'foo' test_file = ['abc'] raises(AssertionError, lambda: _test_this_file_encoding( fname, test_file, unicode_whitelist, unicode_strict_whitelist)) fname = 'bar' test_file = ['abc'] _test_this_file_encoding( fname, test_file, unicode_whitelist, unicode_strict_whitelist)
bcf647015a83075fabe368581f73fcc6bc89eab8c520213bade1fe5ae2ee659c	from sympy.core.numbers import (Float, Rational, oo, pi) from sympy.core.relational import Eq from sympy.core.singleton import S from sympy.core.symbol import (Symbol, symbols) from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (acos, cos, sin) from sympy.sets import EmptySet from sympy.simplify.simplify import simplify from sympy.functions.elementary.trigonometric import tan from sympy.geometry import (Circle, GeometryError, Line, Point, Ray, Segment, Triangle, intersection, Point3D, Line3D, Ray3D, Segment3D, Point2D, Line2D) from sympy.geometry.line import Undecidable from sympy.geometry.polygon import _asa as asa from sympy.utilities.iterables import cartes from sympy.testing.pytest import raises, warns x = Symbol('x', real=True) y = Symbol('y', real=True) z = Symbol('z', real=True) k = Symbol('k', real=True) x1 = Symbol('x1', real=True) y1 = Symbol('y1', real=True) t = Symbol('t', real=True) a, b = symbols('a,b', real=True) m = symbols('m', real=True) def test_object_from_equation(): from sympy.abc import x, y, a, b assert Line(3x + y + 18) == Line2D(Point2D(0, -18), Point2D(1, -21)) assert Line(3x + 5 * y + 1) == Line2D( Point2D(0, Rational(-1, 5)), Point2D(1, Rational(-4, 5))) assert Line(3a + b + 18, x="a", y="b") == Line2D( Point2D(0, -18), Point2D(1, -21)) assert Line(3x + y) == Line2D(Point2D(0, 0), Point2D(1, -3)) assert Line(x + y) == Line2D(Point2D(0, 0), Point2D(1, -1)) assert Line(Eq(3a + b, -18), x="a", y=b) == Line2D( Point2D(0, -18), Point2D(1, -21)) # issue 22361 assert Line(x - 1) == Line2D(Point2D(1, 0), Point2D(1, 1)) assert Line(2x - 2, y=x) == Line2D(Point2D(0, 1), Point2D(1, 1)) assert Line(y) == Line2D(Point2D(0, 0), Point2D(1, 0)) assert Line(2y, x=y) == Line2D(Point2D(0, 0), Point2D(0, 1)) assert Line(y, x=y) == Line2D(Point2D(0, 0), Point2D(0, 1)) raises(ValueError, lambda: Line(x / y)) raises(ValueError, lambda: Line(a / b, x='a', y='b')) raises(ValueError, lambda: Line(y / x)) raises(ValueError, lambda: Line(b / a, x='a', y='b')) raises(ValueError, lambda: Line((x + 1)2 + y)) def feq(a, b): """Test if two floating point values are 'equal'.""" t_float = Float("1.0E-10") return -t_float < a - b < t_float def test_angle_between(): a = Point(1, 2, 3, 4) b = a.orthogonal_direction o = a.origin assert feq(Line.angle_between(Line(Point(0, 0), Point(1, 1)), Line(Point(0, 0), Point(5, 0))).evalf(), pi.evalf() / 4) assert Line(a, o).angle_between(Line(b, o)) == pi / 2 assert Line3D.angle_between(Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1)), Line3D(Point3D(0, 0, 0), Point3D(5, 0, 0))) == acos(sqrt(3) / 3) def test_closing_angle(): a = Ray((0, 0), angle=0) b = Ray((1, 2), angle=pi/2) assert a.closing_angle(b) == -pi/2 assert b.closing_angle(a) == pi/2 assert a.closing_angle(a) == 0 def test_smallest_angle(): a = Line(Point(1, 1), Point(1, 2)) b = Line(Point(1, 1),Point(2, 3)) assert a.smallest_angle_between(b) == acos(2sqrt(5)/5) def test_svg(): a = Line(Point(1, 1),Point(1, 2)) assert a._svg() == '<path fill-rule="evenodd" fill="#66cc99" stroke="#555555" stroke-width="2.0" opacity="0.6" d="M 1.00000000000000,1.00000000000000 L 1.00000000000000,2.00000000000000" marker-start="url(#markerReverseArrow)" marker-end="url(#markerArrow)"/>' a = Segment(Point(1, 0),Point(1, 1)) assert a._svg() == '<path fill-rule="evenodd" fill="#66cc99" stroke="#555555" stroke-width="2.0" opacity="0.6" d="M 1.00000000000000,0 L 1.00000000000000,1.00000000000000" />' a = Ray(Point(2, 3), Point(3, 5)) assert a._svg() == '<path fill-rule="evenodd" fill="#66cc99" stroke="#555555" stroke-width="2.0" opacity="0.6" d="M 2.00000000000000,3.00000000000000 L 3.00000000000000,5.00000000000000" marker-start="url(#markerCircle)" marker-end="url(#markerArrow)"/>' def test_arbitrary_point(): l1 = Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1)) l2 = Line(Point(x1, x1), Point(y1, y1)) assert l2.arbitrary_point() in l2 assert Ray((1, 1), angle=pi / 4).arbitrary_point() == \ Point(t + 1, t + 1) assert Segment((1, 1), (2, 3)).arbitrary_point() == Point(1 + t, 1 + 2 * t) assert l1.perpendicular_segment(l1.arbitrary_point()) == l1.arbitrary_point() assert Ray3D((1, 1, 1), direction_ratio=[1, 2, 3]).arbitrary_point() == \ Point3D(t + 1, 2 * t + 1, 3 * t + 1) assert Segment3D(Point3D(0, 0, 0), Point3D(1, 1, 1)).midpoint == \ Point3D(S.Half, S.Half, S.Half) assert Segment3D(Point3D(x1, x1, x1), Point3D(y1, y1, y1)).length == sqrt(3) * sqrt((x1 - y1) ** 2) assert Segment3D((1, 1, 1), (2, 3, 4)).arbitrary_point() == \ Point3D(t + 1, 2 * t + 1, 3 * t + 1) raises(ValueError, (lambda: Line((x, 1), (2, 3)).arbitrary_point(x))) def test_are_concurrent_2d(): l1 = Line(Point(0, 0), Point(1, 1)) l2 = Line(Point(x1, x1), Point(x1, 1 + x1)) assert Line.are_concurrent(l1) is False assert Line.are_concurrent(l1, l2) assert Line.are_concurrent(l1, l1, l1, l2) assert Line.are_concurrent(l1, l2, Line(Point(5, x1), Point(Rational(-3, 5), x1))) assert Line.are_concurrent(l1, Line(Point(0, 0), Point(-x1, x1)), l2) is False def test_are_concurrent_3d(): p1 = Point3D(0, 0, 0) l1 = Line(p1, Point3D(1, 1, 1)) parallel_1 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)) parallel_2 = Line3D(Point3D(0, 1, 0), Point3D(1, 1, 0)) assert Line3D.are_concurrent(l1) is False assert Line3D.are_concurrent(l1, Line(Point3D(x1, x1, x1), Point3D(y1, y1, y1))) is False assert Line3D.are_concurrent(l1, Line3D(p1, Point3D(x1, x1, x1)), Line(Point3D(x1, x1, x1), Point3D(x1, 1 + x1, 1))) is True assert Line3D.are_concurrent(parallel_1, parallel_2) is False def test_arguments(): """Functions accepting `Point` objects in `geometry` should also accept tuples, lists, and generators and automatically convert them to points.""" from sympy.utilities.iterables import subsets singles2d = ((1, 2), [1, 3], Point(1, 5)) doubles2d = subsets(singles2d, 2) l2d = Line(Point2D(1, 2), Point2D(2, 3)) singles3d = ((1, 2, 3), [1, 2, 4], Point(1, 2, 6)) doubles3d = subsets(singles3d, 2) l3d = Line(Point3D(1, 2, 3), Point3D(1, 1, 2)) singles4d = ((1, 2, 3, 4), [1, 2, 3, 5], Point(1, 2, 3, 7)) doubles4d = subsets(singles4d, 2) l4d = Line(Point(1, 2, 3, 4), Point(2, 2, 2, 2)) # test 2D test_single = ['contains', 'distance', 'equals', 'parallel_line', 'perpendicular_line', 'perpendicular_segment', 'projection', 'intersection'] for p in doubles2d: Line2D(p) for func in test_single: for p in singles2d: getattr(l2d, func)(p) # test 3D for p in doubles3d: Line3D(p) for func in test_single: for p in singles3d: getattr(l3d, func)(p) # test 4D for p in doubles4d: Line(p) for func in test_single: for p in singles4d: getattr(l4d, func)(p) def test_basic_properties_2d(): p1 = Point(0, 0) p2 = Point(1, 1) p10 = Point(2000, 2000) p_r3 = Ray(p1, p2).random_point() p_r4 = Ray(p2, p1).random_point() l1 = Line(p1, p2) l3 = Line(Point(x1, x1), Point(x1, 1 + x1)) l4 = Line(p1, Point(1, 0)) r1 = Ray(p1, Point(0, 1)) r2 = Ray(Point(0, 1), p1) s1 = Segment(p1, p10) p_s1 = s1.random_point() assert Line((1, 1), slope=1) == Line((1, 1), (2, 2)) assert Line((1, 1), slope=oo) == Line((1, 1), (1, 2)) assert Line((1, 1), slope=oo).bounds == (1, 1, 1, 2) assert Line((1, 1), slope=-oo) == Line((1, 1), (1, 2)) assert Line(p1, p2).scale(2, 1) == Line(p1, Point(2, 1)) assert Line(p1, p2) == Line(p1, p2) assert Line(p1, p2) != Line(p2, p1) assert l1 != Line(Point(x1, x1), Point(y1, y1)) assert l1 != l3 assert Line(p1, p10) != Line(p10, p1) assert Line(p1, p10) != p1 assert p1 in l1 # is p1 on the line l1? assert p1 not in l3 assert s1 in Line(p1, p10) assert Ray(Point(0, 0), Point(0, 1)) in Ray(Point(0, 0), Point(0, 2)) assert Ray(Point(0, 0), Point(0, 2)) in Ray(Point(0, 0), Point(0, 1)) assert Ray(Point(0, 0), Point(0, 2)).xdirection == S.Zero assert Ray(Point(0, 0), Point(1, 2)).xdirection == S.Infinity assert Ray(Point(0, 0), Point(-1, 2)).xdirection == S.NegativeInfinity assert Ray(Point(0, 0), Point(2, 0)).ydirection == S.Zero assert Ray(Point(0, 0), Point(2, 2)).ydirection == S.Infinity assert Ray(Point(0, 0), Point(2, -2)).ydirection == S.NegativeInfinity assert (r1 in s1) is False assert Segment(p1, p2) in s1 assert Ray(Point(x1, x1), Point(x1, 1 + x1)) != Ray(p1, Point(-1, 5)) assert Segment(p1, p2).midpoint == Point(S.Half, S.Half) assert Segment(p1, Point(-x1, x1)).length == sqrt(2 (x1 ** 2)) assert l1.slope == 1 assert l3.slope is oo assert l4.slope == 0 assert Line(p1, Point(0, 1)).slope is oo assert Line(r1.source, r1.random_point()).slope == r1.slope assert Line(r2.source, r2.random_point()).slope == r2.slope assert Segment(Point(0, -1), Segment(p1, Point(0, 1)).random_point()).slope == Segment(p1, Point(0, 1)).slope assert l4.coefficients == (0, 1, 0) assert Line((-x, x), (-x + 1, x - 1)).coefficients == (1, 1, 0) assert Line(p1, Point(0, 1)).coefficients == (1, 0, 0) # issue 7963 r = Ray((0, 0), angle=x) assert r.subs(x, 3 * pi / 4) == Ray((0, 0), (-1, 1)) assert r.subs(x, 5 * pi / 4) == Ray((0, 0), (-1, -1)) assert r.subs(x, -pi / 4) == Ray((0, 0), (1, -1)) assert r.subs(x, pi / 2) == Ray((0, 0), (0, 1)) assert r.subs(x, -pi / 2) == Ray((0, 0), (0, -1)) for ind in range(0, 5): assert l3.random_point() in l3 assert p_r3.x >= p1.x and p_r3.y >= p1.y assert p_r4.x <= p2.x and p_r4.y <= p2.y assert p1.x <= p_s1.x <= p10.x and p1.y <= p_s1.y <= p10.y assert hash(s1) != hash(Segment(p10, p1)) assert s1.plot_interval() == [t, 0, 1] assert Line(p1, p10).plot_interval() == [t, -5, 5] assert Ray((0, 0), angle=pi / 4).plot_interval() == [t, 0, 10] def test_basic_properties_3d(): p1 = Point3D(0, 0, 0) p2 = Point3D(1, 1, 1) p3 = Point3D(x1, x1, x1) p5 = Point3D(x1, 1 + x1, 1) l1 = Line3D(p1, p2) l3 = Line3D(p3, p5) r1 = Ray3D(p1, Point3D(-1, 5, 0)) r3 = Ray3D(p1, p2) s1 = Segment3D(p1, p2) assert Line3D((1, 1, 1), direction_ratio=[2, 3, 4]) == Line3D(Point3D(1, 1, 1), Point3D(3, 4, 5)) assert Line3D((1, 1, 1), direction_ratio=[1, 5, 7]) == Line3D(Point3D(1, 1, 1), Point3D(2, 6, 8)) assert Line3D((1, 1, 1), direction_ratio=[1, 2, 3]) == Line3D(Point3D(1, 1, 1), Point3D(2, 3, 4)) assert Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).direction_cosine == [1, 0, 0] assert Line3D(Line3D(p1, Point3D(0, 1, 0))) == Line3D(p1, Point3D(0, 1, 0)) assert Ray3D(Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))) == Ray3D(p1, Point3D(1, 0, 0)) assert Line3D(p1, p2) != Line3D(p2, p1) assert l1 != l3 assert l1 != Line3D(p3, Point3D(y1, y1, y1)) assert r3 != r1 assert Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 1)) in Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)) assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)) in Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 1)) assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)).xdirection == S.Infinity assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)).ydirection == S.Infinity assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)).zdirection == S.Infinity assert Ray3D(Point3D(0, 0, 0), Point3D(-2, 2, 2)).xdirection == S.NegativeInfinity assert Ray3D(Point3D(0, 0, 0), Point3D(2, -2, 2)).ydirection == S.NegativeInfinity assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, -2)).zdirection == S.NegativeInfinity assert Ray3D(Point3D(0, 0, 0), Point3D(0, 2, 2)).xdirection == S.Zero assert Ray3D(Point3D(0, 0, 0), Point3D(2, 0, 2)).ydirection == S.Zero assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 0)).zdirection == S.Zero assert p1 in l1 assert p1 not in l3 assert l1.direction_ratio == [1, 1, 1] assert s1.midpoint == Point3D(S.Half, S.Half, S.Half) # Test zdirection assert Ray3D(p1, Point3D(0, 0, -1)).zdirection is S.NegativeInfinity def test_contains(): p1 = Point(0, 0) r = Ray(p1, Point(4, 4)) r1 = Ray3D(p1, Point3D(0, 0, -1)) r2 = Ray3D(p1, Point3D(0, 1, 0)) r3 = Ray3D(p1, Point3D(0, 0, 1)) l = Line(Point(0, 1), Point(3, 4)) # Segment contains assert Point(0, (a + b) / 2) in Segment((0, a), (0, b)) assert Point((a + b) / 2, 0) in Segment((a, 0), (b, 0)) assert Point3D(0, 1, 0) in Segment3D((0, 1, 0), (0, 1, 0)) assert Point3D(1, 0, 0) in Segment3D((1, 0, 0), (1, 0, 0)) assert Segment3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).contains([]) is True assert Segment3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).contains( Segment3D(Point3D(2, 2, 2), Point3D(3, 2, 2))) is False # Line contains assert l.contains(Point(0, 1)) is True assert l.contains((0, 1)) is True assert l.contains((0, 0)) is False # Ray contains assert r.contains(p1) is True assert r.contains((1, 1)) is True assert r.contains((1, 3)) is False assert r.contains(Segment((1, 1), (2, 2))) is True assert r.contains(Segment((1, 2), (2, 5))) is False assert r.contains(Ray((2, 2), (3, 3))) is True assert r.contains(Ray((2, 2), (3, 5))) is False assert r1.contains(Segment3D(p1, Point3D(0, 0, -10))) is True assert r1.contains(Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))) is False assert r2.contains(Point3D(0, 0, 0)) is True assert r3.contains(Point3D(0, 0, 0)) is True assert Ray3D(Point3D(1, 1, 1), Point3D(1, 0, 0)).contains([]) is False assert Line3D((0, 0, 0), (x, y, z)).contains((2 * x, 2 * y, 2 * z)) with warns(UserWarning): assert Line3D(p1, Point3D(0, 1, 0)).contains(Point(1.0, 1.0)) is False with warns(UserWarning): assert r3.contains(Point(1.0, 1.0)) is False def test_contains_nonreal_symbols(): u, v, w, z = symbols('u, v, w, z') l = Segment(Point(u, w), Point(v, z)) p = Point(uRational(2, 3) + v/3, wRational(2, 3) + z/3) assert l.contains(p) def test_distance_2d(): p1 = Point(0, 0) p2 = Point(1, 1) half = S.Half s1 = Segment(Point(0, 0), Point(1, 1)) s2 = Segment(Point(half, half), Point(1, 0)) r = Ray(p1, p2) assert s1.distance(Point(0, 0)) == 0 assert s1.distance((0, 0)) == 0 assert s2.distance(Point(0, 0)) == 2 half / 2 assert s2.distance(Point(Rational(3) / 2, Rational(3) / 2)) == 2 half assert Line(p1, p2).distance(Point(-1, 1)) == sqrt(2) assert Line(p1, p2).distance(Point(1, -1)) == sqrt(2) assert Line(p1, p2).distance(Point(2, 2)) == 0 assert Line(p1, p2).distance((-1, 1)) == sqrt(2) assert Line((0, 0), (0, 1)).distance(p1) == 0 assert Line((0, 0), (0, 1)).distance(p2) == 1 assert Line((0, 0), (1, 0)).distance(p1) == 0 assert Line((0, 0), (1, 0)).distance(p2) == 1 assert r.distance(Point(-1, -1)) == sqrt(2) assert r.distance(Point(1, 1)) == 0 assert r.distance(Point(-1, 1)) == sqrt(2) assert Ray((1, 1), (2, 2)).distance(Point(1.5, 3)) == 3 * sqrt(2) / 4 assert r.distance((1, 1)) == 0 def test_dimension_normalization(): with warns(UserWarning): assert Ray((1, 1), (2, 1, 2)) == Ray((1, 1, 0), (2, 1, 2)) def test_distance_3d(): p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1) p3 = Point3D(Rational(3) / 2, Rational(3) / 2, Rational(3) / 2) s1 = Segment3D(Point3D(0, 0, 0), Point3D(1, 1, 1)) s2 = Segment3D(Point3D(S.Half, S.Half, S.Half), Point3D(1, 0, 1)) r = Ray3D(p1, p2) assert s1.distance(p1) == 0 assert s2.distance(p1) == sqrt(3) / 2 assert s2.distance(p3) == 2 * sqrt(6) / 3 assert s1.distance((0, 0, 0)) == 0 assert s2.distance((0, 0, 0)) == sqrt(3) / 2 assert s1.distance(p1) == 0 assert s2.distance(p1) == sqrt(3) / 2 assert s2.distance(p3) == 2 * sqrt(6) / 3 assert s1.distance((0, 0, 0)) == 0 assert s2.distance((0, 0, 0)) == sqrt(3) / 2 # Line to point assert Line3D(p1, p2).distance(Point3D(-1, 1, 1)) == 2 * sqrt(6) / 3 assert Line3D(p1, p2).distance(Point3D(1, -1, 1)) == 2 * sqrt(6) / 3 assert Line3D(p1, p2).distance(Point3D(2, 2, 2)) == 0 assert Line3D(p1, p2).distance((2, 2, 2)) == 0 assert Line3D(p1, p2).distance((1, -1, 1)) == 2 * sqrt(6) / 3 assert Line3D((0, 0, 0), (0, 1, 0)).distance(p1) == 0 assert Line3D((0, 0, 0), (0, 1, 0)).distance(p2) == sqrt(2) assert Line3D((0, 0, 0), (1, 0, 0)).distance(p1) == 0 assert Line3D((0, 0, 0), (1, 0, 0)).distance(p2) == sqrt(2) # Ray to point assert r.distance(Point3D(-1, -1, -1)) == sqrt(3) assert r.distance(Point3D(1, 1, 1)) == 0 assert r.distance((-1, -1, -1)) == sqrt(3) assert r.distance((1, 1, 1)) == 0 assert Ray3D((0, 0, 0), (1, 1, 2)).distance((-1, -1, 2)) == 4 * sqrt(3) / 3 assert Ray3D((1, 1, 1), (2, 2, 2)).distance(Point3D(1.5, -3, -1)) == Rational(9) / 2 assert Ray3D((1, 1, 1), (2, 2, 2)).distance(Point3D(1.5, 3, 1)) == sqrt(78) / 6 def test_equals(): p1 = Point(0, 0) p2 = Point(1, 1) l1 = Line(p1, p2) l2 = Line((0, 5), slope=m) l3 = Line(Point(x1, x1), Point(x1, 1 + x1)) assert l1.perpendicular_line(p1.args).equals(Line(Point(0, 0), Point(1, -1))) assert l1.perpendicular_line(p1).equals(Line(Point(0, 0), Point(1, -1))) assert Line(Point(x1, x1), Point(y1, y1)).parallel_line(Point(-x1, x1)). \ equals(Line(Point(-x1, x1), Point(-y1, 2 * x1 - y1))) assert l3.parallel_line(p1.args).equals(Line(Point(0, 0), Point(0, -1))) assert l3.parallel_line(p1).equals(Line(Point(0, 0), Point(0, -1))) assert (l2.distance(Point(2, 3)) - 2 * abs(m + 1) / sqrt(m ** 2 + 1)).equals(0) assert Line3D(p1, Point3D(0, 1, 0)).equals(Point(1.0, 1.0)) is False assert Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).equals(Line3D(Point3D(-5, 0, 0), Point3D(-1, 0, 0))) is True assert Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).equals(Line3D(p1, Point3D(0, 1, 0))) is False assert Ray3D(p1, Point3D(0, 0, -1)).equals(Point(1.0, 1.0)) is False assert Ray3D(p1, Point3D(0, 0, -1)).equals(Ray3D(p1, Point3D(0, 0, -1))) is True assert Line3D((0, 0), (t, t)).perpendicular_line(Point(0, 1, 0)).equals( Line3D(Point3D(0, 1, 0), Point3D(S.Half, S.Half, 0))) assert Line3D((0, 0), (t, t)).perpendicular_segment(Point(0, 1, 0)).equals(Segment3D((0, 1), (S.Half, S.Half))) assert Line3D(p1, Point3D(0, 1, 0)).equals(Point(1.0, 1.0)) is False def test_equation(): p1 = Point(0, 0) p2 = Point(1, 1) l1 = Line(p1, p2) l3 = Line(Point(x1, x1), Point(x1, 1 + x1)) assert simplify(l1.equation()) in (x - y, y - x) assert simplify(l3.equation()) in (x - x1, x1 - x) assert simplify(l1.equation()) in (x - y, y - x) assert simplify(l3.equation()) in (x - x1, x1 - x) assert Line(p1, Point(1, 0)).equation(x=x, y=y) == y assert Line(p1, Point(0, 1)).equation() == x assert Line(Point(2, 0), Point(2, 1)).equation() == x - 2 assert Line(p2, Point(2, 1)).equation() == y - 1 assert Line3D(Point(x1, x1, x1), Point(y1, y1, y1) ).equation() == (-x + y, -x + z) assert Line3D(Point(1, 2, 3), Point(2, 3, 4) ).equation() == (-x + y - 1, -x + z - 2) assert Line3D(Point(1, 2, 3), Point(1, 3, 4) ).equation() == (x - 1, -y + z - 1) assert Line3D(Point(1, 2, 3), Point(2, 2, 4) ).equation() == (y - 2, -x + z - 2) assert Line3D(Point(1, 2, 3), Point(2, 3, 3) ).equation() == (-x + y - 1, z - 3) assert Line3D(Point(1, 2, 3), Point(1, 2, 4) ).equation() == (x - 1, y - 2) assert Line3D(Point(1, 2, 3), Point(1, 3, 3) ).equation() == (x - 1, z - 3) assert Line3D(Point(1, 2, 3), Point(2, 2, 3) ).equation() == (y - 2, z - 3) def test_intersection_2d(): p1 = Point(0, 0) p2 = Point(1, 1) p3 = Point(x1, x1) p4 = Point(y1, y1) l1 = Line(p1, p2) l3 = Line(Point(0, 0), Point(3, 4)) r1 = Ray(Point(1, 1), Point(2, 2)) r2 = Ray(Point(0, 0), Point(3, 4)) r4 = Ray(p1, p2) r6 = Ray(Point(0, 1), Point(1, 2)) r7 = Ray(Point(0.5, 0.5), Point(1, 1)) s1 = Segment(p1, p2) s2 = Segment(Point(0.25, 0.25), Point(0.5, 0.5)) s3 = Segment(Point(0, 0), Point(3, 4)) assert intersection(l1, p1) == [p1] assert intersection(l1, Point(x1, 1 + x1)) == [] assert intersection(l1, Line(p3, p4)) in [[l1], [Line(p3, p4)]] assert intersection(l1, l1.parallel_line(Point(x1, 1 + x1))) == [] assert intersection(l3, l3) == [l3] assert intersection(l3, r2) == [r2] assert intersection(l3, s3) == [s3] assert intersection(s3, l3) == [s3] assert intersection(Segment(Point(-10, 10), Point(10, 10)), Segment(Point(-5, -5), Point(-5, 5))) == [] assert intersection(r2, l3) == [r2] assert intersection(r1, Ray(Point(2, 2), Point(0, 0))) == [Segment(Point(1, 1), Point(2, 2))] assert intersection(r1, Ray(Point(1, 1), Point(-1, -1))) == [Point(1, 1)] assert intersection(r1, Segment(Point(0, 0), Point(2, 2))) == [Segment(Point(1, 1), Point(2, 2))] assert r4.intersection(s2) == [s2] assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == [] assert r4.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))] assert r4.intersection(Ray(p2, p1)) == [s1] assert Ray(p2, p1).intersection(r6) == [] assert r4.intersection(r7) == r7.intersection(r4) == [r7] assert Ray3D((0, 0), (3, 0)).intersection(Ray3D((1, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))] assert Ray3D((1, 0), (3, 0)).intersection(Ray3D((0, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))] assert Ray(Point(0, 0), Point(0, 4)).intersection(Ray(Point(0, 1), Point(0, -1))) == \ [Segment(Point(0, 0), Point(0, 1))] assert Segment3D((0, 0), (3, 0)).intersection( Segment3D((1, 0), (2, 0))) == [Segment3D((1, 0), (2, 0))] assert Segment3D((1, 0), (2, 0)).intersection( Segment3D((0, 0), (3, 0))) == [Segment3D((1, 0), (2, 0))] assert Segment3D((0, 0), (3, 0)).intersection( Segment3D((3, 0), (4, 0))) == [Point3D((3, 0))] assert Segment3D((0, 0), (3, 0)).intersection( Segment3D((2, 0), (5, 0))) == [Segment3D((2, 0), (3, 0))] assert Segment3D((0, 0), (3, 0)).intersection( Segment3D((-2, 0), (1, 0))) == [Segment3D((0, 0), (1, 0))] assert Segment3D((0, 0), (3, 0)).intersection( Segment3D((-2, 0), (0, 0))) == [Point3D(0, 0)] assert s1.intersection(Segment(Point(1, 1), Point(2, 2))) == [Point(1, 1)] assert s1.intersection(Segment(Point(0.5, 0.5), Point(1.5, 1.5))) == [Segment(Point(0.5, 0.5), p2)] assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == [] assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1] assert s1.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))] assert s1.intersection(Line(Point(1, 0), Point(2, 1))) == [] assert s1.intersection(s2) == [s2] assert s2.intersection(s1) == [s2] assert asa(120, 8, 52) == \ Triangle( Point(0, 0), Point(8, 0), Point(-4 * cos(19 * pi / 90) / sin(2 * pi / 45), 4 * sqrt(3) * cos(19 * pi / 90) / sin(2 * pi / 45))) assert Line((0, 0), (1, 1)).intersection(Ray((1, 0), (1, 2))) == [Point(1, 1)] assert Line((0, 0), (1, 1)).intersection(Segment((1, 0), (1, 2))) == [Point(1, 1)] assert Ray((0, 0), (1, 1)).intersection(Ray((1, 0), (1, 2))) == [Point(1, 1)] assert Ray((0, 0), (1, 1)).intersection(Segment((1, 0), (1, 2))) == [Point(1, 1)] assert Ray((0, 0), (10, 10)).contains(Segment((1, 1), (2, 2))) is True assert Segment((1, 1), (2, 2)) in Line((0, 0), (10, 10)) assert s1.intersection(Ray((1, 1), (4, 4))) == [Point(1, 1)] # This test is disabled because it hangs after rref changes which simplify # intermediate results and return a different representation from when the # test was written. # # 16628 - this should be fast # p0 = Point2D(Rational(249, 5), Rational(497999, 10000)) # p1 = Point2D((-58977084786sqrt(405639795226) + 2030690077184193 + # 20112207807sqrt(630547164901) + 99600sqrt(255775022850776494562626)) # /(2000sqrt(255775022850776494562626) + 1991998000sqrt(405639795226) # + 1991998000sqrt(630547164901) + 1622561172902000), # (-498000sqrt(255775022850776494562626) - 995999sqrt(630547164901) + # 90004251917891999 + # 496005510002sqrt(405639795226))/(10000sqrt(255775022850776494562626) # + 9959990000sqrt(405639795226) + 9959990000sqrt(630547164901) + # 8112805864510000)) # p2 = Point2D(Rational(497, 10), Rational(-497, 10)) # p3 = Point2D(Rational(-497, 10), Rational(-497, 10)) # l = Line(p0, p1) # s = Segment(p2, p3) # n = (-52673223862sqrt(405639795226) - 15764156209307469 - # 9803028531sqrt(630547164901) + # 33200sqrt(255775022850776494562626)) # d = sqrt(405639795226) + 315274080450 + 498000sqrt( # 630547164901) + sqrt(255775022850776494562626) # assert intersection(l, s) == [ # Point2D(n/dRational(3, 2000), Rational(-497, 10))] def test_line_intersection(): # see also test_issue_11238 in test_matrices.py x0 = tan(piRational(13, 45)) x1 = sqrt(3) x2 = x0*2 x, y = [8x0/(x0 + x1), (24x0 - 8x1x2)/(x2 - 3)] assert Line(Point(0, 0), Point(1, -sqrt(3))).contains(Point(x, y)) is True def test_intersection_3d(): p1 = Point3D(0, 0, 0) p2 = Point3D(1, 1, 1) l1 = Line3D(p1, p2) l2 = Line3D(Point3D(0, 0, 0), Point3D(3, 4, 0)) r1 = Ray3D(Point3D(1, 1, 1), Point3D(2, 2, 2)) r2 = Ray3D(Point3D(0, 0, 0), Point3D(3, 4, 0)) s1 = Segment3D(Point3D(0, 0, 0), Point3D(3, 4, 0)) assert intersection(l1, p1) == [p1] assert intersection(l1, Point3D(x1, 1 + x1, 1)) == [] assert intersection(l1, l1.parallel_line(p1)) == [Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1))] assert intersection(l2, r2) == [r2] assert intersection(l2, s1) == [s1] assert intersection(r2, l2) == [r2] assert intersection(r1, Ray3D(Point3D(1, 1, 1), Point3D(-1, -1, -1))) == [Point3D(1, 1, 1)] assert intersection(r1, Segment3D(Point3D(0, 0, 0), Point3D(2, 2, 2))) == [ Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))] assert intersection(Ray3D(Point3D(1, 0, 0), Point3D(-1, 0, 0)), Ray3D(Point3D(0, 1, 0), Point3D(0, -1, 0))) \ == [Point3D(0, 0, 0)] assert intersection(r1, Ray3D(Point3D(2, 2, 2), Point3D(0, 0, 0))) == \ [Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))] assert intersection(s1, r2) == [s1] assert Line3D(Point3D(4, 0, 1), Point3D(0, 4, 1)).intersection(Line3D(Point3D(0, 0, 1), Point3D(4, 4, 1))) == \ [Point3D(2, 2, 1)] assert Line3D((0, 1, 2), (0, 2, 3)).intersection(Line3D((0, 1, 2), (0, 1, 1))) == [Point3D(0, 1, 2)] assert Line3D((0, 0), (t, t)).intersection(Line3D((0, 1), (t, t))) == \ [Point3D(t, t)] assert Ray3D(Point3D(0, 0, 0), Point3D(0, 4, 0)).intersection(Ray3D(Point3D(0, 1, 1), Point3D(0, -1, 1))) == [] def test_is_parallel(): p1 = Point3D(0, 0, 0) p2 = Point3D(1, 1, 1) p3 = Point3D(x1, x1, x1) l2 = Line(Point(x1, x1), Point(y1, y1)) l2_1 = Line(Point(x1, x1), Point(x1, 1 + x1)) assert Line.is_parallel(Line(Point(0, 0), Point(1, 1)), l2) assert Line.is_parallel(l2, Line(Point(x1, x1), Point(x1, 1 + x1))) is False assert Line.is_parallel(l2, l2.parallel_line(Point(-x1, x1))) assert Line.is_parallel(l2_1, l2_1.parallel_line(Point(0, 0))) assert Line3D(p1, p2).is_parallel(Line3D(p1, p2)) # same as in 2D assert Line3D(Point3D(4, 0, 1), Point3D(0, 4, 1)).is_parallel(Line3D(Point3D(0, 0, 1), Point3D(4, 4, 1))) is False assert Line3D(p1, p2).parallel_line(p3) == Line3D(Point3D(x1, x1, x1), Point3D(x1 + 1, x1 + 1, x1 + 1)) assert Line3D(p1, p2).parallel_line(p3.args) == \ Line3D(Point3D(x1, x1, x1), Point3D(x1 + 1, x1 + 1, x1 + 1)) assert Line3D(Point3D(4, 0, 1), Point3D(0, 4, 1)).is_parallel(Line3D(Point3D(0, 0, 1), Point3D(4, 4, 1))) is False def test_is_perpendicular(): p1 = Point(0, 0) p2 = Point(1, 1) l1 = Line(p1, p2) l2 = Line(Point(x1, x1), Point(y1, y1)) l1_1 = Line(p1, Point(-x1, x1)) # 2D assert Line.is_perpendicular(l1, l1_1) assert Line.is_perpendicular(l1, l2) is False p = l1.random_point() assert l1.perpendicular_segment(p) == p # 3D assert Line3D.is_perpendicular(Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)), Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0))) is True assert Line3D.is_perpendicular(Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)), Line3D(Point3D(0, 1, 0), Point3D(1, 1, 0))) is False assert Line3D.is_perpendicular(Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1)), Line3D(Point3D(x1, x1, x1), Point3D(y1, y1, y1))) is False def test_is_similar(): p1 = Point(2000, 2000) p2 = p1.scale(2, 2) r1 = Ray3D(Point3D(1, 1, 1), Point3D(1, 0, 0)) r2 = Ray(Point(0, 0), Point(0, 1)) s1 = Segment(Point(0, 0), p1) assert s1.is_similar(Segment(p1, p2)) assert s1.is_similar(r2) is False assert r1.is_similar(Line3D(Point3D(1, 1, 1), Point3D(1, 0, 0))) is True assert r1.is_similar(Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0))) is False def test_length(): s2 = Segment3D(Point3D(x1, x1, x1), Point3D(y1, y1, y1)) assert Line(Point(0, 0), Point(1, 1)).length is oo assert s2.length == sqrt(3) sqrt((x1 - y1) ** 2) assert Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1)).length is oo def test_projection(): p1 = Point(0, 0) p2 = Point3D(0, 0, 0) p3 = Point(-x1, x1) l1 = Line(p1, Point(1, 1)) l2 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)) l3 = Line3D(p2, Point3D(1, 1, 1)) r1 = Ray(Point(1, 1), Point(2, 2)) s1 = Segment(Point2D(0, 0), Point2D(0, 1)) s2 = Segment(Point2D(1, 0), Point2D(2, 1/2)) assert Line(Point(x1, x1), Point(y1, y1)).projection(Point(y1, y1)) == Point(y1, y1) assert Line(Point(x1, x1), Point(x1, 1 + x1)).projection(Point(1, 1)) == Point(x1, 1) assert Segment(Point(-2, 2), Point(0, 4)).projection(r1) == Segment(Point(-1, 3), Point(0, 4)) assert Segment(Point(0, 4), Point(-2, 2)).projection(r1) == Segment(Point(0, 4), Point(-1, 3)) assert s2.projection(s1) == EmptySet assert l1.projection(p3) == p1 assert l1.projection(Ray(p1, Point(-1, 5))) == Ray(Point(0, 0), Point(2, 2)) assert l1.projection(Ray(p1, Point(-1, 1))) == p1 assert r1.projection(Ray(Point(1, 1), Point(-1, -1))) == Point(1, 1) assert r1.projection(Ray(Point(0, 4), Point(-1, -5))) == Segment(Point(1, 1), Point(2, 2)) assert r1.projection(Segment(Point(-1, 5), Point(-5, -10))) == Segment(Point(1, 1), Point(2, 2)) assert r1.projection(Ray(Point(1, 1), Point(-1, -1))) == Point(1, 1) assert r1.projection(Ray(Point(0, 4), Point(-1, -5))) == Segment(Point(1, 1), Point(2, 2)) assert r1.projection(Segment(Point(-1, 5), Point(-5, -10))) == Segment(Point(1, 1), Point(2, 2)) assert l3.projection(Ray3D(p2, Point3D(-1, 5, 0))) == Ray3D(Point3D(0, 0, 0), Point3D(Rational(4, 3), Rational(4, 3), Rational(4, 3))) assert l3.projection(Ray3D(p2, Point3D(-1, 1, 1))) == Ray3D(Point3D(0, 0, 0), Point3D(Rational(1, 3), Rational(1, 3), Rational(1, 3))) assert l2.projection(Point3D(5, 5, 0)) == Point3D(5, 0) assert l2.projection(Line3D(Point3D(0, 1, 0), Point3D(1, 1, 0))).equals(l2) def test_perpendicular_bisector(): s1 = Segment(Point(0, 0), Point(1, 1)) aline = Line(Point(S.Half, S.Half), Point(Rational(3, 2), Rational(-1, 2))) on_line = Segment(Point(S.Half, S.Half), Point(Rational(3, 2), Rational(-1, 2))).midpoint assert s1.perpendicular_bisector().equals(aline) assert s1.perpendicular_bisector(on_line).equals(Segment(s1.midpoint, on_line)) assert s1.perpendicular_bisector(on_line + (1, 0)).equals(aline) def test_raises(): d, e = symbols('a,b', real=True) s = Segment((d, 0), (e, 0)) raises(TypeError, lambda: Line((1, 1), 1)) raises(ValueError, lambda: Line(Point(0, 0), Point(0, 0))) raises(Undecidable, lambda: Point(2 * d, 0) in s) raises(ValueError, lambda: Ray3D(Point(1.0, 1.0))) raises(ValueError, lambda: Line3D(Point3D(0, 0, 0), Point3D(0, 0, 0))) raises(TypeError, lambda: Line3D((1, 1), 1)) raises(ValueError, lambda: Line3D(Point3D(0, 0, 0))) raises(TypeError, lambda: Ray((1, 1), 1)) raises(GeometryError, lambda: Line(Point(0, 0), Point(1, 0)) .projection(Circle(Point(0, 0), 1))) def test_ray_generation(): assert Ray((1, 1), angle=pi / 4) == Ray((1, 1), (2, 2)) assert Ray((1, 1), angle=pi / 2) == Ray((1, 1), (1, 2)) assert Ray((1, 1), angle=-pi / 2) == Ray((1, 1), (1, 0)) assert Ray((1, 1), angle=-3 * pi / 2) == Ray((1, 1), (1, 2)) assert Ray((1, 1), angle=5 * pi / 2) == Ray((1, 1), (1, 2)) assert Ray((1, 1), angle=5.0 * pi / 2) == Ray((1, 1), (1, 2)) assert Ray((1, 1), angle=pi) == Ray((1, 1), (0, 1)) assert Ray((1, 1), angle=3.0 * pi) == Ray((1, 1), (0, 1)) assert Ray((1, 1), angle=4.0 * pi) == Ray((1, 1), (2, 1)) assert Ray((1, 1), angle=0) == Ray((1, 1), (2, 1)) assert Ray((1, 1), angle=4.05 * pi) == Ray(Point(1, 1), Point(2, -sqrt(5) * sqrt(2 * sqrt(5) + 10) / 4 - sqrt( 2 * sqrt(5) + 10) / 4 + 2 + sqrt(5))) assert Ray((1, 1), angle=4.02 * pi) == Ray(Point(1, 1), Point(2, 1 + tan(4.02 * pi))) assert Ray((1, 1), angle=5) == Ray((1, 1), (2, 1 + tan(5))) assert Ray3D((1, 1, 1), direction_ratio=[4, 4, 4]) == Ray3D(Point3D(1, 1, 1), Point3D(5, 5, 5)) assert Ray3D((1, 1, 1), direction_ratio=[1, 2, 3]) == Ray3D(Point3D(1, 1, 1), Point3D(2, 3, 4)) assert Ray3D((1, 1, 1), direction_ratio=[1, 1, 1]) == Ray3D(Point3D(1, 1, 1), Point3D(2, 2, 2)) def test_symbolic_intersect(): # Issue 7814. circle = Circle(Point(x, 0), y) line = Line(Point(k, z), slope=0) assert line.intersection(circle) == [Point(x + sqrt((y - z) * (y + z)), z), Point(x - sqrt((y - z) * (y + z)), z)] def test_issue_2941(): def _check(): for f, g in cartes([(Line, Ray, Segment)] 2): l1 = f(a, b) l2 = g(c, d) assert l1.intersection(l2) == l2.intersection(l1) # intersect at end point c, d = (-2, -2), (-2, 0) a, b = (0, 0), (1, 1) _check() # midline intersection c, d = (-2, -3), (-2, 0) _check() def test_parameter_value(): t = Symbol('t') p1, p2 = Point(0, 1), Point(5, 6) l = Line(p1, p2) assert l.parameter_value((5, 6), t) == {t: 1} raises(ValueError, lambda: l.parameter_value((0, 0), t)) def test_bisectors(): r1 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)) r2 = Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0)) bisections = r1.bisectors(r2) assert bisections == [Line3D(Point3D(0, 0, 0), Point3D(1, 1, 0)), Line3D(Point3D(0, 0, 0), Point3D(1, -1, 0))] ans = [Line3D(Point3D(0, 0, 0), Point3D(1, 0, 1)), Line3D(Point3D(0, 0, 0), Point3D(-1, 0, 1))] l1 = (0, 0, 0), (0, 0, 1) l2 = (0, 0), (1, 0) for a, b in cartes((Line, Segment, Ray), repeat=2): assert a(l1).bisectors(b(l2)) == ans def test_issue_8615(): a = Line3D(Point3D(6, 5, 0), Point3D(6, -6, 0)) b = Line3D(Point3D(6, -1, 19/10), Point3D(6, -1, 0)) assert a.intersection(b) == [Point3D(6, -1, 0)]
659bf642cbe92a7c983b1d33e5fb8c3917e0f8b0e93c1eb69e21edd9aa7d6db5	from sympy.core.basic import Basic from sympy.core.numbers import (I, Rational, pi) from sympy.core.parameters import evaluate from sympy.core.singleton import S from sympy.core.symbol import Symbol from sympy.core.sympify import sympify from sympy.functions.elementary.miscellaneous import sqrt from sympy.geometry import Line, Point, Point2D, Point3D, Line3D, Plane from sympy.geometry.entity import rotate, scale, translate, GeometryEntity from sympy.matrices import Matrix from sympy.utilities.iterables import subsets, permutations, cartes from sympy.utilities.misc import Undecidable from sympy.testing.pytest import raises, warns def test_point(): x = Symbol('x', real=True) y = Symbol('y', real=True) x1 = Symbol('x1', real=True) x2 = Symbol('x2', real=True) y1 = Symbol('y1', real=True) y2 = Symbol('y2', real=True) half = S.Half p1 = Point(x1, x2) p2 = Point(y1, y2) p3 = Point(0, 0) p4 = Point(1, 1) p5 = Point(0, 1) line = Line(Point(1, 0), slope=1) assert p1 in p1 assert p1 not in p2 assert p2.y == y2 assert (p3 + p4) == p4 assert (p2 - p1) == Point(y1 - x1, y2 - x2) assert -p2 == Point(-y1, -y2) raises(TypeError, lambda: Point(1)) raises(ValueError, lambda: Point([1])) raises(ValueError, lambda: Point(3, I)) raises(ValueError, lambda: Point(2I, I)) raises(ValueError, lambda: Point(3 + I, I)) assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3)) assert Point.midpoint(p3, p4) == Point(half, half) assert Point.midpoint(p1, p4) == Point(half + halfx1, half + halfx2) assert Point.midpoint(p2, p2) == p2 assert p2.midpoint(p2) == p2 assert p1.origin == Point(0, 0) assert Point.distance(p3, p4) == sqrt(2) assert Point.distance(p1, p1) == 0 assert Point.distance(p3, p2) == sqrt(p2.x2 + p2.y2) raises(TypeError, lambda: Point.distance(p1, 0)) raises(TypeError, lambda: Point.distance(p1, GeometryEntity())) # distance should be symmetric assert p1.distance(line) == line.distance(p1) assert p4.distance(line) == line.distance(p4) assert Point.taxicab_distance(p4, p3) == 2 assert Point.canberra_distance(p4, p5) == 1 raises(ValueError, lambda: Point.canberra_distance(p3, p3)) p1_1 = Point(x1, x1) p1_2 = Point(y2, y2) p1_3 = Point(x1 + 1, x1) assert Point.is_collinear(p3) with warns(UserWarning): assert Point.is_collinear(p3, Point(p3, dim=4)) assert p3.is_collinear() assert Point.is_collinear(p3, p4) assert Point.is_collinear(p3, p4, p1_1, p1_2) assert Point.is_collinear(p3, p4, p1_1, p1_3) is False assert Point.is_collinear(p3, p3, p4, p5) is False raises(TypeError, lambda: Point.is_collinear(line)) raises(TypeError, lambda: p1_1.is_collinear(line)) assert p3.intersection(Point(0, 0)) == [p3] assert p3.intersection(p4) == [] assert p3.intersection(line) == [] assert Point.intersection(Point(0, 0, 0), Point(0, 0)) == [Point(0, 0, 0)] x_pos = Symbol('x', positive=True) p2_1 = Point(x_pos, 0) p2_2 = Point(0, x_pos) p2_3 = Point(-x_pos, 0) p2_4 = Point(0, -x_pos) p2_5 = Point(x_pos, 5) assert Point.is_concyclic(p2_1) assert Point.is_concyclic(p2_1, p2_2) assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_4) for pts in permutations((p2_1, p2_2, p2_3, p2_5)): assert Point.is_concyclic(pts) is False assert Point.is_concyclic(p4, p4 * 2, p4 * 3) is False assert Point(0, 0).is_concyclic((1, 1), (2, 2), (2, 1)) is False assert Point.is_concyclic(Point(0, 0, 0, 0), Point(1, 0, 0, 0), Point(1, 1, 0, 0), Point(1, 1, 1, 0)) is False assert p1.is_scalar_multiple(p1) assert p1.is_scalar_multiple(2p1) assert not p1.is_scalar_multiple(p2) assert Point.is_scalar_multiple(Point(1, 1), (-1, -1)) assert Point.is_scalar_multiple(Point(0, 0), (0, -1)) # test when is_scalar_multiple can't be determined raises(Undecidable, lambda: Point.is_scalar_multiple(Point(sympify("x1%y1"), sympify("x2%y2")), Point(0, 1))) assert Point(0, 1).orthogonal_direction == Point(1, 0) assert Point(1, 0).orthogonal_direction == Point(0, 1) assert p1.is_zero is None assert p3.is_zero assert p4.is_zero is False assert p1.is_nonzero is None assert p3.is_nonzero is False assert p4.is_nonzero assert p4.scale(2, 3) == Point(2, 3) assert p3.scale(2, 3) == p3 assert p4.rotate(pi, Point(0.5, 0.5)) == p3 assert p1.__radd__(p2) == p1.midpoint(p2).scale(2, 2) assert (-p3).__rsub__(p4) == p3.midpoint(p4).scale(2, 2) assert p4 5 == Point(5, 5) assert p4 / 5 == Point(0.2, 0.2) assert 5 * p4 == Point(5, 5) raises(ValueError, lambda: Point(0, 0) + 10) # Point differences should be simplified assert Point(x(x - 1), y) - Point(x2 - x, y + 1) == Point(0, -1) a, b = S.Half, Rational(1, 3) assert Point(a, b).evalf(2) == \ Point(a.n(2), b.n(2), evaluate=False) raises(ValueError, lambda: Point(1, 2) + 1) # test project assert Point.project((0, 1), (1, 0)) == Point(0, 0) assert Point.project((1, 1), (1, 0)) == Point(1, 0) raises(ValueError, lambda: Point.project(p1, Point(0, 0))) # test transformations p = Point(1, 0) assert p.rotate(pi/2) == Point(0, 1) assert p.rotate(pi/2, p) == p p = Point(1, 1) assert p.scale(2, 3) == Point(2, 3) assert p.translate(1, 2) == Point(2, 3) assert p.translate(1) == Point(2, 1) assert p.translate(y=1) == Point(1, 2) assert p.translate(p.args) == Point(2, 2) # Check invalid input for transform raises(ValueError, lambda: p3.transform(p3)) raises(ValueError, lambda: p.transform(Matrix([[1, 0], [0, 1]]))) # test __contains__ assert 0 in Point(0, 0, 0, 0) assert 1 not in Point(0, 0, 0, 0) # test affine_rank assert Point.affine_rank() == -1 def test_point3D(): x = Symbol('x', real=True) y = Symbol('y', real=True) x1 = Symbol('x1', real=True) x2 = Symbol('x2', real=True) x3 = Symbol('x3', real=True) y1 = Symbol('y1', real=True) y2 = Symbol('y2', real=True) y3 = Symbol('y3', real=True) half = S.Half p1 = Point3D(x1, x2, x3) p2 = Point3D(y1, y2, y3) p3 = Point3D(0, 0, 0) p4 = Point3D(1, 1, 1) p5 = Point3D(0, 1, 2) assert p1 in p1 assert p1 not in p2 assert p2.y == y2 assert (p3 + p4) == p4 assert (p2 - p1) == Point3D(y1 - x1, y2 - x2, y3 - x3) assert -p2 == Point3D(-y1, -y2, -y3) assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3)) assert Point3D.midpoint(p3, p4) == Point3D(half, half, half) assert Point3D.midpoint(p1, p4) == Point3D(half + halfx1, half + halfx2, half + halfx3) assert Point3D.midpoint(p2, p2) == p2 assert p2.midpoint(p2) == p2 assert Point3D.distance(p3, p4) == sqrt(3) assert Point3D.distance(p1, p1) == 0 assert Point3D.distance(p3, p2) == sqrt(p2.x2 + p2.y2 + p2.z2) p1_1 = Point3D(x1, x1, x1) p1_2 = Point3D(y2, y2, y2) p1_3 = Point3D(x1 + 1, x1, x1) Point3D.are_collinear(p3) assert Point3D.are_collinear(p3, p4) assert Point3D.are_collinear(p3, p4, p1_1, p1_2) assert Point3D.are_collinear(p3, p4, p1_1, p1_3) is False assert Point3D.are_collinear(p3, p3, p4, p5) is False assert p3.intersection(Point3D(0, 0, 0)) == [p3] assert p3.intersection(p4) == [] assert p4 5 == Point3D(5, 5, 5) assert p4 / 5 == Point3D(0.2, 0.2, 0.2) assert 5 * p4 == Point3D(5, 5, 5) raises(ValueError, lambda: Point3D(0, 0, 0) + 10) # Test coordinate properties assert p1.coordinates == (x1, x2, x3) assert p2.coordinates == (y1, y2, y3) assert p3.coordinates == (0, 0, 0) assert p4.coordinates == (1, 1, 1) assert p5.coordinates == (0, 1, 2) assert p5.x == 0 assert p5.y == 1 assert p5.z == 2 # Point differences should be simplified assert Point3D(x(x - 1), y, 2) - Point3D(x2 - x, y + 1, 1) == \ Point3D(0, -1, 1) a, b, c = S.Half, Rational(1, 3), Rational(1, 4) assert Point3D(a, b, c).evalf(2) == \ Point(a.n(2), b.n(2), c.n(2), evaluate=False) raises(ValueError, lambda: Point3D(1, 2, 3) + 1) # test transformations p = Point3D(1, 1, 1) assert p.scale(2, 3) == Point3D(2, 3, 1) assert p.translate(1, 2) == Point3D(2, 3, 1) assert p.translate(1) == Point3D(2, 1, 1) assert p.translate(z=1) == Point3D(1, 1, 2) assert p.translate(p.args) == Point3D(2, 2, 2) # Test __new__ assert Point3D(0.1, 0.2, evaluate=False, on_morph='ignore').args[0].is_Float # Test length property returns correctly assert p.length == 0 assert p1_1.length == 0 assert p1_2.length == 0 # Test are_colinear type error raises(TypeError, lambda: Point3D.are_collinear(p, x)) # Test are_coplanar assert Point.are_coplanar() assert Point.are_coplanar((1, 2, 0), (1, 2, 0), (1, 3, 0)) assert Point.are_coplanar((1, 2, 0), (1, 2, 3)) with warns(UserWarning): raises(ValueError, lambda: Point2D.are_coplanar((1, 2), (1, 2, 3))) assert Point3D.are_coplanar((1, 2, 0), (1, 2, 3)) assert Point.are_coplanar((0, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 1)) is False planar2 = Point3D(1, -1, 1) planar3 = Point3D(-1, 1, 1) assert Point3D.are_coplanar(p, planar2, planar3) == True assert Point3D.are_coplanar(p, planar2, planar3, p3) == False assert Point.are_coplanar(p, planar2) planar2 = Point3D(1, 1, 2) planar3 = Point3D(1, 1, 3) assert Point3D.are_coplanar(p, planar2, planar3) # line, not plane plane = Plane((1, 2, 1), (2, 1, 0), (3, 1, 2)) assert Point.are_coplanar([plane.projection(((-1)i, i)) for i in range(4)]) # all 2D points are coplanar assert Point.are_coplanar(Point(x, y), Point(x, x + y), Point(y, x + 2)) is True # Test Intersection assert planar2.intersection(Line3D(p, planar3)) == [Point3D(1, 1, 2)] # Test Scale assert planar2.scale(1, 1, 1) == planar2 assert planar2.scale(2, 2, 2, planar3) == Point3D(1, 1, 1) assert planar2.scale(1, 1, 1, p3) == planar2 # Test Transform identity = Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]) assert p.transform(identity) == p trans = Matrix([[1, 0, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0, 0, 0, 1]]) assert p.transform(trans) == Point3D(2, 2, 2) raises(ValueError, lambda: p.transform(p)) raises(ValueError, lambda: p.transform(Matrix([[1, 0], [0, 1]]))) # Test Equals assert p.equals(x1) == False # Test __sub__ p_4d = Point(0, 0, 0, 1) with warns(UserWarning): assert p - p_4d == Point(1, 1, 1, -1) p_4d3d = Point(0, 0, 1, 0) with warns(UserWarning): assert p - p_4d3d == Point(1, 1, 0, 0) def test_Point2D(): # Test Distance p1 = Point2D(1, 5) p2 = Point2D(4, 2.5) p3 = (6, 3) assert p1.distance(p2) == sqrt(61)/2 assert p2.distance(p3) == sqrt(17)/2 # Test coordinates assert p1.x == 1 assert p1.y == 5 assert p2.x == 4 assert p2.y == 2.5 assert p1.coordinates == (1, 5) assert p2.coordinates == (4, 2.5) # test bounds assert p1.bounds == (1, 5, 1, 5) def test_issue_9214(): p1 = Point3D(4, -2, 6) p2 = Point3D(1, 2, 3) p3 = Point3D(7, 2, 3) assert Point3D.are_collinear(p1, p2, p3) is False def test_issue_11617(): p1 = Point3D(1,0,2) p2 = Point2D(2,0) with warns(UserWarning): assert p1.distance(p2) == sqrt(5) def test_transform(): p = Point(1, 1) assert p.transform(rotate(pi/2)) == Point(-1, 1) assert p.transform(scale(3, 2)) == Point(3, 2) assert p.transform(translate(1, 2)) == Point(2, 3) assert Point(1, 1).scale(2, 3, (4, 5)) == \ Point(-2, -7) assert Point(1, 1).translate(4, 5) == \ Point(5, 6) def test_concyclic_doctest_bug(): p1, p2 = Point(-1, 0), Point(1, 0) p3, p4 = Point(0, 1), Point(-1, 2) assert Point.is_concyclic(p1, p2, p3) assert not Point.is_concyclic(p1, p2, p3, p4) def test_arguments(): """Functions accepting `Point` objects in `geometry` should also accept tuples and lists and automatically convert them to points.""" singles2d = ((1,2), [1,2], Point(1,2)) singles2d2 = ((1,3), [1,3], Point(1,3)) doubles2d = cartes(singles2d, singles2d2) p2d = Point2D(1,2) singles3d = ((1,2,3), [1,2,3], Point(1,2,3)) doubles3d = subsets(singles3d, 2) p3d = Point3D(1,2,3) singles4d = ((1,2,3,4), [1,2,3,4], Point(1,2,3,4)) doubles4d = subsets(singles4d, 2) p4d = Point(1,2,3,4) # test 2D test_single = ['distance', 'is_scalar_multiple', 'taxicab_distance', 'midpoint', 'intersection', 'dot', 'equals', '__add__', '__sub__'] test_double = ['is_concyclic', 'is_collinear'] for p in singles2d: Point2D(p) for func in test_single: for p in singles2d: getattr(p2d, func)(p) for func in test_double: for p in doubles2d: getattr(p2d, func)(p) # test 3D test_double = ['is_collinear'] for p in singles3d: Point3D(p) for func in test_single: for p in singles3d: getattr(p3d, func)(p) for func in test_double: for p in doubles3d: getattr(p3d, func)(p) # test 4D test_double = ['is_collinear'] for p in singles4d: Point(p) for func in test_single: for p in singles4d: getattr(p4d, func)(p) for func in test_double: for p in doubles4d: getattr(p4d, func)(p) # test evaluate=False for ops x = Symbol('x') a = Point(0, 1) assert a + (0.1, x) == Point(0.1, 1 + x, evaluate=False) a = Point(0, 1) assert a/10.0 == Point(0, 0.1, evaluate=False) a = Point(0, 1) assert a10.0 == Point(0.0, 10.0, evaluate=False) # test evaluate=False when changing dimensions u = Point(.1, .2, evaluate=False) u4 = Point(u, dim=4, on_morph='ignore') assert u4.args == (.1, .2, 0, 0) assert all(i.is_Float for i in u4.args[:2]) # and even when not* changing dimensions assert all(i.is_Float for i in Point(u).args) # never raise error if creating an origin assert Point(dim=3, on_morph='error') # raise error with unmatched dimension raises(ValueError, lambda: Point(1, 1, dim=3, on_morph='error')) # test unknown on_morph raises(ValueError, lambda: Point(1, 1, dim=3, on_morph='unknown')) # test invalid expressions raises(TypeError, lambda: Point(Basic(), Basic())) def test_unit(): assert Point(1, 1).unit == Point(sqrt(2)/2, sqrt(2)/2) def test_dot(): raises(TypeError, lambda: Point(1, 2).dot(Line((0, 0), (1, 1)))) def test__normalize_dimension(): assert Point._normalize_dimension(Point(1, 2), Point(3, 4)) == [ Point(1, 2), Point(3, 4)] assert Point._normalize_dimension( Point(1, 2), Point(3, 4, 0), on_morph='ignore') == [ Point(1, 2, 0), Point(3, 4, 0)] def test_issue_22684(): # Used to give an error with evaluate(False): Point(1, 2) def test_direction_cosine(): p1 = Point3D(0, 0, 0) p2 = Point3D(1, 1, 1) assert p1.direction_cosine(Point3D(1, 0, 0)) == [1, 0, 0] assert p1.direction_cosine(Point3D(0, 1, 0)) == [0, 1, 0] assert p1.direction_cosine(Point3D(0, 0, pi)) == [0, 0, 1] assert p1.direction_cosine(Point3D(5, 0, 0)) == [1, 0, 0] assert p1.direction_cosine(Point3D(0, sqrt(3), 0)) == [0, 1, 0] assert p1.direction_cosine(Point3D(0, 0, 5)) == [0, 0, 1] assert p1.direction_cosine(Point3D(2.4, 2.4, 0)) == [sqrt(2)/2, sqrt(2)/2, 0] assert p1.direction_cosine(Point3D(1, 1, 1)) == [sqrt(3) / 3, sqrt(3) / 3, sqrt(3) / 3] assert p1.direction_cosine(Point3D(-12, 0 -15)) == [-4sqrt(41)/41, -5sqrt(41)/41, 0] assert p2.direction_cosine(Point3D(0, 0, 0)) == [-sqrt(3) / 3, -sqrt(3) / 3, -sqrt(3) / 3] assert p2.direction_cosine(Point3D(1, 1, 12)) == [0, 0, 1] assert p2.direction_cosine(Point3D(12, 1, 12)) == [sqrt(2) / 2, 0, sqrt(2) / 2]
9cc84fe4d8d05737c2f35116eb13cd13bf84b8be04f8231955bfe5b1bf6b7406	# -- coding: utf-8 -- import sys from sympy.assumptions import Q from sympy.core import Symbol, Function, Float, Rational, Integer, I, Mul, Pow, Eq from sympy.functions import exp, factorial, factorial2, sin from sympy.logic import And from sympy.series import Limit from sympy.testing.pytest import raises, skip from sympy.parsing.sympy_parser import ( parse_expr, standard_transformations, rationalize, TokenError, split_symbols, implicit_multiplication, convert_equals_signs, convert_xor, function_exponentiation, lambda_notation, auto_symbol, repeated_decimals, implicit_multiplication_application, auto_number, factorial_notation, implicit_application, _transformation, T ) def test_sympy_parser(): x = Symbol('x') inputs = { '2x': 2 x, '3.00': Float(3), '22/7': Rational(22, 7), '2+3j': 2 + 3I, 'exp(x)': exp(x), 'x!': factorial(x), 'x!!': factorial2(x), '(x + 1)! - 1': factorial(x + 1) - 1, '3.[3]': Rational(10, 3), '.0[3]': Rational(1, 30), '3.2[3]': Rational(97, 30), '1.3[12]': Rational(433, 330), '1 + 3.[3]': Rational(13, 3), '1 + .0[3]': Rational(31, 30), '1 + 3.2[3]': Rational(127, 30), '.[0011]': Rational(1, 909), '0.1[00102] + 1': Rational(366697, 333330), '1.[0191]': Rational(10190, 9999), '10!': 3628800, '-(2)': -Integer(2), '[-1, -2, 3]': [Integer(-1), Integer(-2), Integer(3)], 'Symbol("x").free_symbols': x.free_symbols, "S('S(3).n(n=3)')": 3.00, 'factorint(12, visual=True)': Mul( Pow(2, 2, evaluate=False), Pow(3, 1, evaluate=False), evaluate=False), 'Limit(sin(x), x, 0, dir="-")': Limit(sin(x), x, 0, dir='-'), 'Q.even(x)': Q.even(x), } for text, result in inputs.items(): assert parse_expr(text) == result raises(TypeError, lambda: parse_expr('x', standard_transformations)) raises(TypeError, lambda: parse_expr('x', transformations=lambda x,y: 1)) raises(TypeError, lambda: parse_expr('x', transformations=(lambda x,y: 1,))) raises(TypeError, lambda: parse_expr('x', transformations=((),))) raises(TypeError, lambda: parse_expr('x', {}, [], [])) raises(TypeError, lambda: parse_expr('x', [], [], {})) raises(TypeError, lambda: parse_expr('x', [], [], {})) def test_rationalize(): inputs = { '0.123': Rational(123, 1000) } transformations = standard_transformations + (rationalize,) for text, result in inputs.items(): assert parse_expr(text, transformations=transformations) == result def test_factorial_fail(): inputs = ['x!!!', 'x!!!!', '(!)'] for text in inputs: try: parse_expr(text) assert False except TokenError: assert True def test_repeated_fail(): inputs = ['1[1]', '.1e1[1]', '0x1[1]', '1.1j[1]', '1.1[1 + 1]', '0.1[[1]]', '0x1.1[1]'] # All are valid Python, so only raise TypeError for invalid indexing for text in inputs: raises(TypeError, lambda: parse_expr(text)) inputs = ['0.1[', '0.1[1', '0.1[]'] for text in inputs: raises((TokenError, SyntaxError), lambda: parse_expr(text)) def test_repeated_dot_only(): assert parse_expr('.[1]') == Rational(1, 9) assert parse_expr('1 + .[1]') == Rational(10, 9) def test_local_dict(): local_dict = { 'my_function': lambda x: x + 2 } inputs = { 'my_function(2)': Integer(4) } for text, result in inputs.items(): assert parse_expr(text, local_dict=local_dict) == result def test_local_dict_split_implmult(): t = standard_transformations + (split_symbols, implicit_multiplication,) w = Symbol('w', real=True) y = Symbol('y') assert parse_expr('yx', local_dict={'x':w}, transformations=t) == yw def test_local_dict_symbol_to_fcn(): x = Symbol('x') d = {'foo': Function('bar')} assert parse_expr('foo(x)', local_dict=d) == d['foo'](x) d = {'foo': Symbol('baz')} raises(TypeError, lambda: parse_expr('foo(x)', local_dict=d)) def test_global_dict(): global_dict = { 'Symbol': Symbol } inputs = { 'Q & S': And(Symbol('Q'), Symbol('S')) } for text, result in inputs.items(): assert parse_expr(text, global_dict=global_dict) == result def test_issue_2515(): raises(TokenError, lambda: parse_expr('(()')) raises(TokenError, lambda: parse_expr('"""')) def test_issue_7663(): x = Symbol('x') e = '2(x+1)' assert parse_expr(e, evaluate=0) == parse_expr(e, evaluate=False) assert parse_expr(e, evaluate=0).equals(2(x+1)) def test_recursive_evaluate_false_10560(): inputs = { '4-3' : '4-3', '-43' : '(-4)3', "-2xy": '(-2)xy', "x-4x": "x(-4)x" } for text, result in inputs.items(): assert parse_expr(text, evaluate=False) == parse_expr(result, evaluate=False) def test_function_evaluate_false(): inputs = [ 'Abs(0)', 'im(0)', 're(0)', 'sign(0)', 'arg(0)', 'conjugate(0)', 'acos(0)', 'acot(0)', 'acsc(0)', 'asec(0)', 'asin(0)', 'atan(0)', 'acosh(0)', 'acoth(0)', 'acsch(0)', 'asech(0)', 'asinh(0)', 'atanh(0)', 'cos(0)', 'cot(0)', 'csc(0)', 'sec(0)', 'sin(0)', 'tan(0)', 'cosh(0)', 'coth(0)', 'csch(0)', 'sech(0)', 'sinh(0)', 'tanh(0)', 'exp(0)', 'log(0)', 'sqrt(0)', ] for case in inputs: expr = parse_expr(case, evaluate=False) assert case == str(expr) != str(expr.doit()) assert str(parse_expr('ln(0)', evaluate=False)) == 'log(0)' assert str(parse_expr('cbrt(0)', evaluate=False)) == '0*(1/3)' def test_issue_10773(): inputs = { '-10/5': '(-10)/5', '-10/-5' : '(-10)/(-5)', } for text, result in inputs.items(): assert parse_expr(text, evaluate=False) == parse_expr(result, evaluate=False) def test_split_symbols(): transformations = standard_transformations + \ (split_symbols, implicit_multiplication,) x = Symbol('x') y = Symbol('y') xy = Symbol('xy') assert parse_expr("xy") == xy assert parse_expr("xy", transformations=transformations) == xy def test_split_symbols_function(): transformations = standard_transformations + \ (split_symbols, implicit_multiplication,) x = Symbol('x') y = Symbol('y') a = Symbol('a') f = Function('f') assert parse_expr("ay(x+1)", transformations=transformations) == ay(x+1) assert parse_expr("af(x+1)", transformations=transformations, local_dict={'f':f}) == af(x+1) def test_functional_exponent(): t = standard_transformations + (convert_xor, function_exponentiation) x = Symbol('x') y = Symbol('y') a = Symbol('a') yfcn = Function('y') assert parse_expr("sin^2(x)", transformations=t) == (sin(x))2 assert parse_expr("sin^y(x)", transformations=t) == (sin(x))y assert parse_expr("exp^y(x)", transformations=t) == (exp(x))y assert parse_expr("E^y(x)", transformations=t) == exp(yfcn(x)) assert parse_expr("a^y(x)", transformations=t) == a(yfcn(x)) def test_match_parentheses_implicit_multiplication(): transformations = standard_transformations + \ (implicit_multiplication,) raises(TokenError, lambda: parse_expr('(1,2),(3,4]',transformations=transformations)) def test_convert_equals_signs(): transformations = standard_transformations + \ (convert_equals_signs, ) x = Symbol('x') y = Symbol('y') assert parse_expr("12=x", transformations=transformations) == Eq(2, x) assert parse_expr("y = x", transformations=transformations) == Eq(y, x) assert parse_expr("(2y = x) = False", transformations=transformations) == Eq(Eq(2y, x), False) def test_parse_function_issue_3539(): x = Symbol('x') f = Function('f') assert parse_expr('f(x)') == f(x) def test_split_symbols_numeric(): transformations = ( standard_transformations + (implicit_multiplication_application,)) n = Symbol('n') expr1 = parse_expr('2*n 3n') expr2 = parse_expr('2n3n', transformations=transformations) assert expr1 == expr2 == 2n3n expr1 = parse_expr('n12n34', transformations=transformations) assert expr1 == n12n34 def test_unicode_names(): assert parse_expr('α') == Symbol('α') def test_python3_features(): # Make sure the tokenizer can handle Python 3-only features if sys.version_info < (3, 7): skip("test_python3_features requires Python 3.7 or newer") assert parse_expr("123_456") == 123456 assert parse_expr("1.2[3_4]") == parse_expr("1.2[34]") == Rational(611, 495) assert parse_expr("1.2[012_012]") == parse_expr("1.2[012012]") == Rational(400, 333) assert parse_expr('.[3_4]') == parse_expr('.[34]') == Rational(34, 99) assert parse_expr('.1[3_4]') == parse_expr('.1[34]') == Rational(133, 990) assert parse_expr('123_123.123_123[3_4]') == parse_expr('123123.123123[34]') == Rational(12189189189211, 99000000) def test_issue_19501(): x = Symbol('x') eq = parse_expr('E*x(1+x)', local_dict={'x': x}, transformations=( standard_transformations + (implicit_multiplication_application,))) assert eq.free_symbols == {x} def test_parsing_definitions(): from sympy.abc import x assert len(_transformation) == 12 # if this changes, extend below assert _transformation[0] == lambda_notation assert _transformation[1] == auto_symbol assert _transformation[2] == repeated_decimals assert _transformation[3] == auto_number assert _transformation[4] == factorial_notation assert _transformation[5] == implicit_multiplication_application assert _transformation[6] == convert_xor assert _transformation[7] == implicit_application assert _transformation[8] == implicit_multiplication assert _transformation[9] == convert_equals_signs assert _transformation[10] == function_exponentiation assert _transformation[11] == rationalize assert T[:5] == T[0,1,2,3,4] == standard_transformations t = _transformation assert T[-1, 0] == (t[len(t) - 1], t[0]) assert T[:5, 8] == standard_transformations + (t[8],) assert parse_expr('0.3x^2', transformations='all') == 3x*2/10 assert parse_expr('sin 3x', transformations='implicit') == sin(3x) def test_builtins(): cases = [ ('abs(x)', 'Abs(x)'), ('max(x, y)', 'Max(x, y)'), ('min(x, y)', 'Min(x, y)'), ('pow(x, y)', 'Pow(x, y)'), ] for built_in_func_call, sympy_func_call in cases: assert parse_expr(built_in_func_call) == parse_expr(sympy_func_call) assert str(parse_expr('pow(38, -1, 97)')) == '23' def test_issue_22822(): raises(ValueError, lambda: parse_expr('x', {'': 1})) data = {'some_parameter': None} assert parse_expr('some_parameter is None', data) is True
33adf4a113c5798c9493f6dbf4c0bf65aecae5215a1dffb3de77ba3569d38ee2	# Ported from latex2sympy by @augustt198 # https://github.com/augustt198/latex2sympy # See license in LICENSE.txt import sympy from sympy.external import import_module from sympy.printing.str import StrPrinter from sympy.physics.quantum.state import Bra, Ket from .errors import LaTeXParsingError LaTeXParser = LaTeXLexer = MathErrorListener = None try: LaTeXParser = import_module('sympy.parsing.latex._antlr.latexparser', import_kwargs={'fromlist': ['LaTeXParser']}).LaTeXParser LaTeXLexer = import_module('sympy.parsing.latex._antlr.latexlexer', import_kwargs={'fromlist': ['LaTeXLexer']}).LaTeXLexer except Exception: pass ErrorListener = import_module('antlr4.error.ErrorListener', warn_not_installed=True, import_kwargs={'fromlist': ['ErrorListener']} ) if ErrorListener: class MathErrorListener(ErrorListener.ErrorListener): # type: ignore def __init__(self, src): super(ErrorListener.ErrorListener, self).__init__() self.src = src def syntaxError(self, recog, symbol, line, col, msg, e): fmt = "%s\n%s\n%s" marker = "~" * col + "^" if msg.startswith("missing"): err = fmt % (msg, self.src, marker) elif msg.startswith("no viable"): err = fmt % ("I expected something else here", self.src, marker) elif msg.startswith("mismatched"): names = LaTeXParser.literalNames expected = [ names[i] for i in e.getExpectedTokens() if i < len(names) ] if len(expected) < 10: expected = " ".join(expected) err = (fmt % ("I expected one of these: " + expected, self.src, marker)) else: err = (fmt % ("I expected something else here", self.src, marker)) else: err = fmt % ("I don't understand this", self.src, marker) raise LaTeXParsingError(err) def parse_latex(sympy): antlr4 = import_module('antlr4', warn_not_installed=True) if None in [antlr4, MathErrorListener]: raise ImportError("LaTeX parsing requires the antlr4 Python package," " provided by pip (antlr4-python2-runtime or" " antlr4-python3-runtime) or" " conda (antlr-python-runtime)") matherror = MathErrorListener(sympy) stream = antlr4.InputStream(sympy) lex = LaTeXLexer(stream) lex.removeErrorListeners() lex.addErrorListener(matherror) tokens = antlr4.CommonTokenStream(lex) parser = LaTeXParser(tokens) # remove default console error listener parser.removeErrorListeners() parser.addErrorListener(matherror) relation = parser.math().relation() expr = convert_relation(relation) return expr def convert_relation(rel): if rel.expr(): return convert_expr(rel.expr()) lh = convert_relation(rel.relation(0)) rh = convert_relation(rel.relation(1)) if rel.LT(): return sympy.StrictLessThan(lh, rh) elif rel.LTE(): return sympy.LessThan(lh, rh) elif rel.GT(): return sympy.StrictGreaterThan(lh, rh) elif rel.GTE(): return sympy.GreaterThan(lh, rh) elif rel.EQUAL(): return sympy.Eq(lh, rh) elif rel.NEQ(): return sympy.Ne(lh, rh) def convert_expr(expr): return convert_add(expr.additive()) def convert_add(add): if add.ADD(): lh = convert_add(add.additive(0)) rh = convert_add(add.additive(1)) return sympy.Add(lh, rh, evaluate=False) elif add.SUB(): lh = convert_add(add.additive(0)) rh = convert_add(add.additive(1)) return sympy.Add(lh, sympy.Mul(-1, rh, evaluate=False), evaluate=False) else: return convert_mp(add.mp()) def convert_mp(mp): if hasattr(mp, 'mp'): mp_left = mp.mp(0) mp_right = mp.mp(1) else: mp_left = mp.mp_nofunc(0) mp_right = mp.mp_nofunc(1) if mp.MUL() or mp.CMD_TIMES() or mp.CMD_CDOT(): lh = convert_mp(mp_left) rh = convert_mp(mp_right) return sympy.Mul(lh, rh, evaluate=False) elif mp.DIV() or mp.CMD_DIV() or mp.COLON(): lh = convert_mp(mp_left) rh = convert_mp(mp_right) return sympy.Mul(lh, sympy.Pow(rh, -1, evaluate=False), evaluate=False) else: if hasattr(mp, 'unary'): return convert_unary(mp.unary()) else: return convert_unary(mp.unary_nofunc()) def convert_unary(unary): if hasattr(unary, 'unary'): nested_unary = unary.unary() else: nested_unary = unary.unary_nofunc() if hasattr(unary, 'postfix_nofunc'): first = unary.postfix() tail = unary.postfix_nofunc() postfix = [first] + tail else: postfix = unary.postfix() if unary.ADD(): return convert_unary(nested_unary) elif unary.SUB(): numabs = convert_unary(nested_unary) # Use Integer(-n) instead of Mul(-1, n) return -numabs elif postfix: return convert_postfix_list(postfix) def convert_postfix_list(arr, i=0): if i >= len(arr): raise LaTeXParsingError("Index out of bounds") res = convert_postfix(arr[i]) if isinstance(res, sympy.Expr): if i == len(arr) - 1: return res # nothing to multiply by else: if i > 0: left = convert_postfix(arr[i - 1]) right = convert_postfix(arr[i + 1]) if isinstance(left, sympy.Expr) and isinstance( right, sympy.Expr): left_syms = convert_postfix(arr[i - 1]).atoms(sympy.Symbol) right_syms = convert_postfix(arr[i + 1]).atoms( sympy.Symbol) # if the left and right sides contain no variables and the # symbol in between is 'x', treat as multiplication. if not (left_syms or right_syms) and str(res) == 'x': return convert_postfix_list(arr, i + 1) # multiply by next return sympy.Mul( res, convert_postfix_list(arr, i + 1), evaluate=False) else: # must be derivative wrt = res[0] if i == len(arr) - 1: raise LaTeXParsingError("Expected expression for derivative") else: expr = convert_postfix_list(arr, i + 1) return sympy.Derivative(expr, wrt) def do_subs(expr, at): if at.expr(): at_expr = convert_expr(at.expr()) syms = at_expr.atoms(sympy.Symbol) if len(syms) == 0: return expr elif len(syms) > 0: sym = next(iter(syms)) return expr.subs(sym, at_expr) elif at.equality(): lh = convert_expr(at.equality().expr(0)) rh = convert_expr(at.equality().expr(1)) return expr.subs(lh, rh) def convert_postfix(postfix): if hasattr(postfix, 'exp'): exp_nested = postfix.exp() else: exp_nested = postfix.exp_nofunc() exp = convert_exp(exp_nested) for op in postfix.postfix_op(): if op.BANG(): if isinstance(exp, list): raise LaTeXParsingError("Cannot apply postfix to derivative") exp = sympy.factorial(exp, evaluate=False) elif op.eval_at(): ev = op.eval_at() at_b = None at_a = None if ev.eval_at_sup(): at_b = do_subs(exp, ev.eval_at_sup()) if ev.eval_at_sub(): at_a = do_subs(exp, ev.eval_at_sub()) if at_b is not None and at_a is not None: exp = sympy.Add(at_b, -1 * at_a, evaluate=False) elif at_b is not None: exp = at_b elif at_a is not None: exp = at_a return exp def convert_exp(exp): if hasattr(exp, 'exp'): exp_nested = exp.exp() else: exp_nested = exp.exp_nofunc() if exp_nested: base = convert_exp(exp_nested) if isinstance(base, list): raise LaTeXParsingError("Cannot raise derivative to power") if exp.atom(): exponent = convert_atom(exp.atom()) elif exp.expr(): exponent = convert_expr(exp.expr()) return sympy.Pow(base, exponent, evaluate=False) else: if hasattr(exp, 'comp'): return convert_comp(exp.comp()) else: return convert_comp(exp.comp_nofunc()) def convert_comp(comp): if comp.group(): return convert_expr(comp.group().expr()) elif comp.abs_group(): return sympy.Abs(convert_expr(comp.abs_group().expr()), evaluate=False) elif comp.atom(): return convert_atom(comp.atom()) elif comp.frac(): return convert_frac(comp.frac()) elif comp.binom(): return convert_binom(comp.binom()) elif comp.floor(): return convert_floor(comp.floor()) elif comp.ceil(): return convert_ceil(comp.ceil()) elif comp.func(): return convert_func(comp.func()) def convert_atom(atom): if atom.LETTER(): subscriptName = '' if atom.subexpr(): subscript = None if atom.subexpr().expr(): # subscript is expr subscript = convert_expr(atom.subexpr().expr()) else: # subscript is atom subscript = convert_atom(atom.subexpr().atom()) subscriptName = '_{' + StrPrinter().doprint(subscript) + '}' return sympy.Symbol(atom.LETTER().getText() + subscriptName) elif atom.SYMBOL(): s = atom.SYMBOL().getText()[1:] if s == "infty": return sympy.oo else: if atom.subexpr(): subscript = None if atom.subexpr().expr(): # subscript is expr subscript = convert_expr(atom.subexpr().expr()) else: # subscript is atom subscript = convert_atom(atom.subexpr().atom()) subscriptName = StrPrinter().doprint(subscript) s += '_{' + subscriptName + '}' return sympy.Symbol(s) elif atom.NUMBER(): s = atom.NUMBER().getText().replace(",", "") return sympy.Number(s) elif atom.DIFFERENTIAL(): var = get_differential_var(atom.DIFFERENTIAL()) return sympy.Symbol('d' + var.name) elif atom.mathit(): text = rule2text(atom.mathit().mathit_text()) return sympy.Symbol(text) elif atom.bra(): val = convert_expr(atom.bra().expr()) return Bra(val) elif atom.ket(): val = convert_expr(atom.ket().expr()) return Ket(val) def rule2text(ctx): stream = ctx.start.getInputStream() # starting index of starting token startIdx = ctx.start.start # stopping index of stopping token stopIdx = ctx.stop.stop return stream.getText(startIdx, stopIdx) def convert_frac(frac): diff_op = False partial_op = False lower_itv = frac.lower.getSourceInterval() lower_itv_len = lower_itv[1] - lower_itv[0] + 1 if (frac.lower.start == frac.lower.stop and frac.lower.start.type == LaTeXLexer.DIFFERENTIAL): wrt = get_differential_var_str(frac.lower.start.text) diff_op = True elif (lower_itv_len == 2 and frac.lower.start.type == LaTeXLexer.SYMBOL and frac.lower.start.text == '\\partial' and (frac.lower.stop.type == LaTeXLexer.LETTER or frac.lower.stop.type == LaTeXLexer.SYMBOL)): partial_op = True wrt = frac.lower.stop.text if frac.lower.stop.type == LaTeXLexer.SYMBOL: wrt = wrt[1:] if diff_op or partial_op: wrt = sympy.Symbol(wrt) if (diff_op and frac.upper.start == frac.upper.stop and frac.upper.start.type == LaTeXLexer.LETTER and frac.upper.start.text == 'd'): return [wrt] elif (partial_op and frac.upper.start == frac.upper.stop and frac.upper.start.type == LaTeXLexer.SYMBOL and frac.upper.start.text == '\\partial'): return [wrt] upper_text = rule2text(frac.upper) expr_top = None if diff_op and upper_text.startswith('d'): expr_top = parse_latex(upper_text[1:]) elif partial_op and frac.upper.start.text == '\\partial': expr_top = parse_latex(upper_text[len('\\partial'):]) if expr_top: return sympy.Derivative(expr_top, wrt) expr_top = convert_expr(frac.upper) expr_bot = convert_expr(frac.lower) inverse_denom = sympy.Pow(expr_bot, -1, evaluate=False) if expr_top == 1: return inverse_denom else: return sympy.Mul(expr_top, inverse_denom, evaluate=False) def convert_binom(binom): expr_n = convert_expr(binom.n) expr_k = convert_expr(binom.k) return sympy.binomial(expr_n, expr_k, evaluate=False) def convert_floor(floor): val = convert_expr(floor.val) return sympy.floor(val, evaluate=False) def convert_ceil(ceil): val = convert_expr(ceil.val) return sympy.ceiling(val, evaluate=False) def convert_func(func): if func.func_normal(): if func.L_PAREN(): # function called with parenthesis arg = convert_func_arg(func.func_arg()) else: arg = convert_func_arg(func.func_arg_noparens()) name = func.func_normal().start.text[1:] # change arc<trig> -> a<trig> if name in [ "arcsin", "arccos", "arctan", "arccsc", "arcsec", "arccot" ]: name = "a" + name[3:] expr = getattr(sympy.functions, name)(arg, evaluate=False) if name in ["arsinh", "arcosh", "artanh"]: name = "a" + name[2:] expr = getattr(sympy.functions, name)(arg, evaluate=False) if name == "exp": expr = sympy.exp(arg, evaluate=False) if (name == "log" or name == "ln"): if func.subexpr(): if func.subexpr().expr(): base = convert_expr(func.subexpr().expr()) else: base = convert_atom(func.subexpr().atom()) elif name == "log": base = 10 elif name == "ln": base = sympy.E expr = sympy.log(arg, base, evaluate=False) func_pow = None should_pow = True if func.supexpr(): if func.supexpr().expr(): func_pow = convert_expr(func.supexpr().expr()) else: func_pow = convert_atom(func.supexpr().atom()) if name in [ "sin", "cos", "tan", "csc", "sec", "cot", "sinh", "cosh", "tanh" ]: if func_pow == -1: name = "a" + name should_pow = False expr = getattr(sympy.functions, name)(arg, evaluate=False) if func_pow and should_pow: expr = sympy.Pow(expr, func_pow, evaluate=False) return expr elif func.LETTER() or func.SYMBOL(): if func.LETTER(): fname = func.LETTER().getText() elif func.SYMBOL(): fname = func.SYMBOL().getText()[1:] fname = str(fname) # can't be unicode if func.subexpr(): subscript = None if func.subexpr().expr(): # subscript is expr subscript = convert_expr(func.subexpr().expr()) else: # subscript is atom subscript = convert_atom(func.subexpr().atom()) subscriptName = StrPrinter().doprint(subscript) fname += '_{' + subscriptName + '}' input_args = func.args() output_args = [] while input_args.args(): # handle multiple arguments to function output_args.append(convert_expr(input_args.expr())) input_args = input_args.args() output_args.append(convert_expr(input_args.expr())) return sympy.Function(fname)(*output_args) elif func.FUNC_INT(): return handle_integral(func) elif func.FUNC_SQRT(): expr = convert_expr(func.base) if func.root: r = convert_expr(func.root) return sympy.root(expr, r, evaluate=False) else: return sympy.sqrt(expr, evaluate=False) elif func.FUNC_OVERLINE(): expr = convert_expr(func.base) return sympy.conjugate(expr, evaluate=False) elif func.FUNC_SUM(): return handle_sum_or_prod(func, "summation") elif func.FUNC_PROD(): return handle_sum_or_prod(func, "product") elif func.FUNC_LIM(): return handle_limit(func) def convert_func_arg(arg): if hasattr(arg, 'expr'): return convert_expr(arg.expr()) else: return convert_mp(arg.mp_nofunc()) def handle_integral(func): if func.additive(): integrand = convert_add(func.additive()) elif func.frac(): integrand = convert_frac(func.frac()) else: integrand = 1 int_var = None if func.DIFFERENTIAL(): int_var = get_differential_var(func.DIFFERENTIAL()) else: for sym in integrand.atoms(sympy.Symbol): s = str(sym) if len(s) > 1 and s[0] == 'd': if s[1] == '\\': int_var = sympy.Symbol(s[2:]) else: int_var = sympy.Symbol(s[1:]) int_sym = sym if int_var: integrand = integrand.subs(int_sym, 1) else: # Assume dx by default int_var = sympy.Symbol('x') if func.subexpr(): if func.subexpr().atom(): lower = convert_atom(func.subexpr().atom()) else: lower = convert_expr(func.subexpr().expr()) if func.supexpr().atom(): upper = convert_atom(func.supexpr().atom()) else: upper = convert_expr(func.supexpr().expr()) return sympy.Integral(integrand, (int_var, lower, upper)) else: return sympy.Integral(integrand, int_var) def handle_sum_or_prod(func, name): val = convert_mp(func.mp()) iter_var = convert_expr(func.subeq().equality().expr(0)) start = convert_expr(func.subeq().equality().expr(1)) if func.supexpr().expr(): # ^{expr} end = convert_expr(func.supexpr().expr()) else: # ^atom end = convert_atom(func.supexpr().atom()) if name == "summation": return sympy.Sum(val, (iter_var, start, end)) elif name == "product": return sympy.Product(val, (iter_var, start, end)) def handle_limit(func): sub = func.limit_sub() if sub.LETTER(): var = sympy.Symbol(sub.LETTER().getText()) elif sub.SYMBOL(): var = sympy.Symbol(sub.SYMBOL().getText()[1:]) else: var = sympy.Symbol('x') if sub.SUB(): direction = "-" else: direction = "+" approaching = convert_expr(sub.expr()) content = convert_mp(func.mp()) return sympy.Limit(content, var, approaching, direction) def get_differential_var(d): text = get_differential_var_str(d.getText()) return sympy.Symbol(text) def get_differential_var_str(text): for i in range(1, len(text)): c = text[i] if not (c == " " or c == "\r" or c == "\n" or c == "\t"): idx = i break text = text[idx:] if text[0] == "\\": text = text[1:] return text
3cb033171fefd08b5946f90b0695977dc5464001a5b11fcea747094b7ff51e25	from sympy.core.numbers import I from sympy.functions.elementary.exponential import (exp, log) from sympy.polys.partfrac import apart from sympy.core.symbol import Dummy from sympy.external import import_module from sympy.functions import arg, Abs from sympy.integrals.transforms import _fast_inverse_laplace from sympy.physics.control.lti import SISOLinearTimeInvariant from sympy.plotting.plot import LineOver1DRangeSeries from sympy.polys.polytools import Poly from sympy.printing.latex import latex __all__ = ['pole_zero_numerical_data', 'pole_zero_plot', 'step_response_numerical_data', 'step_response_plot', 'impulse_response_numerical_data', 'impulse_response_plot', 'ramp_response_numerical_data', 'ramp_response_plot', 'bode_magnitude_numerical_data', 'bode_phase_numerical_data', 'bode_magnitude_plot', 'bode_phase_plot', 'bode_plot'] matplotlib = import_module( 'matplotlib', import_kwargs={'fromlist': ['pyplot']}, catch=(RuntimeError,)) numpy = import_module('numpy') if matplotlib: plt = matplotlib.pyplot if numpy: np = numpy # Matplotlib already has numpy as a compulsory dependency. No need to install it separately. def _check_system(system): """Function to check whether the dynamical system passed for plots is compatible or not.""" if not isinstance(system, SISOLinearTimeInvariant): raise NotImplementedError("Only SISO LTI systems are currently supported.") sys = system.to_expr() len_free_symbols = len(sys.free_symbols) if len_free_symbols > 1: raise ValueError("Extra degree of freedom found. Make sure" " that there are no free symbols in the dynamical system other" " than the variable of Laplace transform.") if sys.has(exp): # Should test that exp is not part of a constant, in which case # no exception is required, compare exp(s) with sexp(1) raise NotImplementedError("Time delay terms are not supported.") def pole_zero_numerical_data(system): """ Returns the numerical data of poles and zeros of the system. It is internally used by ``pole_zero_plot`` to get the data for plotting poles and zeros. Users can use this data to further analyse the dynamics of the system or plot using a different backend/plotting-module. Parameters ========== system : SISOLinearTimeInvariant The system for which the pole-zero data is to be computed. Returns ======= tuple : (zeros, poles) zeros = Zeros of the system. NumPy array of complex numbers. poles = Poles of the system. NumPy array of complex numbers. Raises ====== NotImplementedError When a SISO LTI system is not passed. When time delay terms are present in the system. ValueError When more than one free symbol is present in the system. The only variable in the transfer function should be the variable of the Laplace transform. Examples ======== >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction >>> from sympy.physics.control.control_plots import pole_zero_numerical_data >>> tf1 = TransferFunction(s2 + 1, s4 + 4s*3 + 6s*2 + 5s + 2, s) >>> pole_zero_numerical_data(tf1) # doctest: +SKIP ([-0.+1.j 0.-1.j], [-2. +0.j -0.5+0.8660254j -0.5-0.8660254j -1. +0.j ]) See Also ======== pole_zero_plot """ _check_system(system) system = system.doit() # Get the equivalent TransferFunction object. num_poly = Poly(system.num, system.var).all_coeffs() den_poly = Poly(system.den, system.var).all_coeffs() num_poly = np.array(num_poly, dtype=np.complex128) den_poly = np.array(den_poly, dtype=np.complex128) zeros = np.roots(num_poly) poles = np.roots(den_poly) return zeros, poles def pole_zero_plot(system, pole_color='blue', pole_markersize=10, zero_color='orange', zero_markersize=7, grid=True, show_axes=True, show=True, kwargs): r""" Returns the Pole-Zero plot (also known as PZ Plot or PZ Map) of a system. A Pole-Zero plot is a graphical representation of a system's poles and zeros. It is plotted on a complex plane, with circular markers representing the system's zeros and 'x' shaped markers representing the system's poles. Parameters ========== system : SISOLinearTimeInvariant type systems The system for which the pole-zero plot is to be computed. pole_color : str, tuple, optional The color of the pole points on the plot. Default color is blue. The color can be provided as a matplotlib color string, or a 3-tuple of floats each in the 0-1 range. pole_markersize : Number, optional The size of the markers used to mark the poles in the plot. Default pole markersize is 10. zero_color : str, tuple, optional The color of the zero points on the plot. Default color is orange. The color can be provided as a matplotlib color string, or a 3-tuple of floats each in the 0-1 range. zero_markersize : Number, optional The size of the markers used to mark the zeros in the plot. Default zero markersize is 7. grid : boolean, optional If ``True``, the plot will have a grid. Defaults to True. show_axes : boolean, optional If ``True``, the coordinate axes will be shown. Defaults to False. show : boolean, optional If ``True``, the plot will be displayed otherwise the equivalent matplotlib ``plot`` object will be returned. Defaults to True. Examples ======== .. plot:: :context: close-figs :format: doctest :include-source: True >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction >>> from sympy.physics.control.control_plots import pole_zero_plot >>> tf1 = TransferFunction(s2 + 1, s*4 + 4s*3 + 6s*2 + 5s + 2, s) >>> pole_zero_plot(tf1) # doctest: +SKIP See Also ======== pole_zero_numerical_data References ========== .. [1] https://en.wikipedia.org/wiki/Pole%E2%80%93zero_plot """ zeros, poles = pole_zero_numerical_data(system) zero_real = np.real(zeros) zero_imag = np.imag(zeros) pole_real = np.real(poles) pole_imag = np.imag(poles) plt.plot(pole_real, pole_imag, 'x', mfc='none', markersize=pole_markersize, color=pole_color) plt.plot(zero_real, zero_imag, 'o', markersize=zero_markersize, color=zero_color) plt.xlabel('Real Axis') plt.ylabel('Imaginary Axis') plt.title(f'Poles and Zeros of ${latex(system)}$', pad=20) if grid: plt.grid() if show_axes: plt.axhline(0, color='black') plt.axvline(0, color='black') if show: plt.show() return return plt def step_response_numerical_data(system, prec=8, lower_limit=0, upper_limit=10, kwargs): """ Returns the numerical values of the points in the step response plot of a SISO continuous-time system. By default, adaptive sampling is used. If the user wants to instead get an uniformly sampled response, then ``adaptive`` kwarg should be passed ``False`` and ``nb_of_points`` must be passed as additional kwargs. Refer to the parameters of class :class:`sympy.plotting.plot.LineOver1DRangeSeries` for more details. Parameters ========== system : SISOLinearTimeInvariant The system for which the unit step response data is to be computed. prec : int, optional The decimal point precision for the point coordinate values. Defaults to 8. lower_limit : Number, optional The lower limit of the plot range. Defaults to 0. upper_limit : Number, optional The upper limit of the plot range. Defaults to 10. kwargs : Additional keyword arguments are passed to the underlying :class:`sympy.plotting.plot.LineOver1DRangeSeries` class. Returns ======= tuple : (x, y) x = Time-axis values of the points in the step response. NumPy array. y = Amplitude-axis values of the points in the step response. NumPy array. Raises ====== NotImplementedError When a SISO LTI system is not passed. When time delay terms are present in the system. ValueError When more than one free symbol is present in the system. The only variable in the transfer function should be the variable of the Laplace transform. When ``lower_limit`` parameter is less than 0. Examples ======== >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction >>> from sympy.physics.control.control_plots import step_response_numerical_data >>> tf1 = TransferFunction(s, s2 + 5s + 8, s) >>> step_response_numerical_data(tf1) # doctest: +SKIP ([0.0, 0.025413462339411542, 0.0484508722725343, ... , 9.670250533855183, 9.844291913708725, 10.0], [0.0, 0.023844582399907256, 0.042894276802320226, ..., 6.828770759094287e-12, 6.456457160755703e-12]) See Also ======== step_response_plot """ if lower_limit < 0: raise ValueError("Lower limit of time must be greater " "than or equal to zero.") _check_system(system) _x = Dummy("x") expr = system.to_expr()/(system.var) expr = apart(expr, system.var, full=True) _y = _fast_inverse_laplace(expr, system.var, _x).evalf(prec) return LineOver1DRangeSeries(_y, (_x, lower_limit, upper_limit), kwargs).get_points() def step_response_plot(system, color='b', prec=8, lower_limit=0, upper_limit=10, show_axes=False, grid=True, show=True, kwargs): r""" Returns the unit step response of a continuous-time system. It is the response of the system when the input signal is a step function. Parameters ========== system : SISOLinearTimeInvariant type The LTI SISO system for which the Step Response is to be computed. color : str, tuple, optional The color of the line. Default is Blue. show : boolean, optional If ``True``, the plot will be displayed otherwise the equivalent matplotlib ``plot`` object will be returned. Defaults to True. lower_limit : Number, optional The lower limit of the plot range. Defaults to 0. upper_limit : Number, optional The upper limit of the plot range. Defaults to 10. prec : int, optional The decimal point precision for the point coordinate values. Defaults to 8. show_axes : boolean, optional If ``True``, the coordinate axes will be shown. Defaults to False. grid : boolean, optional If ``True``, the plot will have a grid. Defaults to True. Examples ======== .. plot:: :context: close-figs :format: doctest :include-source: True >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction >>> from sympy.physics.control.control_plots import step_response_plot >>> tf1 = TransferFunction(8s*2 + 18s + 32, s*3 + 6s*2 + 14s + 24, s) >>> step_response_plot(tf1) # doctest: +SKIP See Also ======== impulse_response_plot, ramp_response_plot References ========== .. [1] https://www.mathworks.com/help/control/ref/lti.step.html """ x, y = step_response_numerical_data(system, prec=prec, lower_limit=lower_limit, upper_limit=upper_limit, kwargs) plt.plot(x, y, color=color) plt.xlabel('Time (s)') plt.ylabel('Amplitude') plt.title(f'Unit Step Response of ${latex(system)}$', pad=20) if grid: plt.grid() if show_axes: plt.axhline(0, color='black') plt.axvline(0, color='black') if show: plt.show() return return plt def impulse_response_numerical_data(system, prec=8, lower_limit=0, upper_limit=10, kwargs): """ Returns the numerical values of the points in the impulse response plot of a SISO continuous-time system. By default, adaptive sampling is used. If the user wants to instead get an uniformly sampled response, then ``adaptive`` kwarg should be passed ``False`` and ``nb_of_points`` must be passed as additional kwargs. Refer to the parameters of class :class:`sympy.plotting.plot.LineOver1DRangeSeries` for more details. Parameters ========== system : SISOLinearTimeInvariant The system for which the impulse response data is to be computed. prec : int, optional The decimal point precision for the point coordinate values. Defaults to 8. lower_limit : Number, optional The lower limit of the plot range. Defaults to 0. upper_limit : Number, optional The upper limit of the plot range. Defaults to 10. kwargs : Additional keyword arguments are passed to the underlying :class:`sympy.plotting.plot.LineOver1DRangeSeries` class. Returns ======= tuple : (x, y) x = Time-axis values of the points in the impulse response. NumPy array. y = Amplitude-axis values of the points in the impulse response. NumPy array. Raises ====== NotImplementedError When a SISO LTI system is not passed. When time delay terms are present in the system. ValueError When more than one free symbol is present in the system. The only variable in the transfer function should be the variable of the Laplace transform. When ``lower_limit`` parameter is less than 0. Examples ======== >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction >>> from sympy.physics.control.control_plots import impulse_response_numerical_data >>> tf1 = TransferFunction(s, s*2 + 5s + 8, s) >>> impulse_response_numerical_data(tf1) # doctest: +SKIP ([0.0, 0.06616480200395854,... , 9.854500743565858, 10.0], [0.9999999799999999, 0.7042848373025861,...,7.170748906965121e-13, -5.1901263495547205e-12]) See Also ======== impulse_response_plot """ if lower_limit < 0: raise ValueError("Lower limit of time must be greater " "than or equal to zero.") _check_system(system) _x = Dummy("x") expr = system.to_expr() expr = apart(expr, system.var, full=True) _y = _fast_inverse_laplace(expr, system.var, _x).evalf(prec) return LineOver1DRangeSeries(_y, (_x, lower_limit, upper_limit), kwargs).get_points() def impulse_response_plot(system, color='b', prec=8, lower_limit=0, upper_limit=10, show_axes=False, grid=True, show=True, kwargs): r""" Returns the unit impulse response (Input is the Dirac-Delta Function) of a continuous-time system. Parameters ========== system : SISOLinearTimeInvariant type The LTI SISO system for which the Impulse Response is to be computed. color : str, tuple, optional The color of the line. Default is Blue. show : boolean, optional If ``True``, the plot will be displayed otherwise the equivalent matplotlib ``plot`` object will be returned. Defaults to True. lower_limit : Number, optional The lower limit of the plot range. Defaults to 0. upper_limit : Number, optional The upper limit of the plot range. Defaults to 10. prec : int, optional The decimal point precision for the point coordinate values. Defaults to 8. show_axes : boolean, optional If ``True``, the coordinate axes will be shown. Defaults to False. grid : boolean, optional If ``True``, the plot will have a grid. Defaults to True. Examples ======== .. plot:: :context: close-figs :format: doctest :include-source: True >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction >>> from sympy.physics.control.control_plots import impulse_response_plot >>> tf1 = TransferFunction(8s2 + 18s + 32, s*3 + 6s*2 + 14s + 24, s) >>> impulse_response_plot(tf1) # doctest: +SKIP See Also ======== step_response_plot, ramp_response_plot References ========== .. [1] https://www.mathworks.com/help/control/ref/lti.impulse.html """ x, y = impulse_response_numerical_data(system, prec=prec, lower_limit=lower_limit, upper_limit=upper_limit, kwargs) plt.plot(x, y, color=color) plt.xlabel('Time (s)') plt.ylabel('Amplitude') plt.title(f'Impulse Response of ${latex(system)}$', pad=20) if grid: plt.grid() if show_axes: plt.axhline(0, color='black') plt.axvline(0, color='black') if show: plt.show() return return plt def ramp_response_numerical_data(system, slope=1, prec=8, lower_limit=0, upper_limit=10, kwargs): """ Returns the numerical values of the points in the ramp response plot of a SISO continuous-time system. By default, adaptive sampling is used. If the user wants to instead get an uniformly sampled response, then ``adaptive`` kwarg should be passed ``False`` and ``nb_of_points`` must be passed as additional kwargs. Refer to the parameters of class :class:`sympy.plotting.plot.LineOver1DRangeSeries` for more details. Parameters ========== system : SISOLinearTimeInvariant The system for which the ramp response data is to be computed. slope : Number, optional The slope of the input ramp function. Defaults to 1. prec : int, optional The decimal point precision for the point coordinate values. Defaults to 8. lower_limit : Number, optional The lower limit of the plot range. Defaults to 0. upper_limit : Number, optional The upper limit of the plot range. Defaults to 10. kwargs : Additional keyword arguments are passed to the underlying :class:`sympy.plotting.plot.LineOver1DRangeSeries` class. Returns ======= tuple : (x, y) x = Time-axis values of the points in the ramp response plot. NumPy array. y = Amplitude-axis values of the points in the ramp response plot. NumPy array. Raises ====== NotImplementedError When a SISO LTI system is not passed. When time delay terms are present in the system. ValueError When more than one free symbol is present in the system. The only variable in the transfer function should be the variable of the Laplace transform. When ``lower_limit`` parameter is less than 0. When ``slope`` is negative. Examples ======== >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction >>> from sympy.physics.control.control_plots import ramp_response_numerical_data >>> tf1 = TransferFunction(s, s*2 + 5s + 8, s) >>> ramp_response_numerical_data(tf1) # doctest: +SKIP (([0.0, 0.12166980856813935,..., 9.861246379582118, 10.0], [1.4504508011325967e-09, 0.006046440489058766,..., 0.12499999999568202, 0.12499999999661349])) See Also ======== ramp_response_plot """ if slope < 0: raise ValueError("Slope must be greater than or equal" " to zero.") if lower_limit < 0: raise ValueError("Lower limit of time must be greater " "than or equal to zero.") _check_system(system) _x = Dummy("x") expr = (slopesystem.to_expr())/((system.var)2) expr = apart(expr, system.var, full=True) _y = _fast_inverse_laplace(expr, system.var, _x).evalf(prec) return LineOver1DRangeSeries(_y, (_x, lower_limit, upper_limit), kwargs).get_points() def ramp_response_plot(system, slope=1, color='b', prec=8, lower_limit=0, upper_limit=10, show_axes=False, grid=True, show=True, kwargs): r""" Returns the ramp response of a continuous-time system. Ramp function is defined as the straight line passing through origin ($f(x) = mx$). The slope of the ramp function can be varied by the user and the default value is 1. Parameters ========== system : SISOLinearTimeInvariant type The LTI SISO system for which the Ramp Response is to be computed. slope : Number, optional The slope of the input ramp function. Defaults to 1. color : str, tuple, optional The color of the line. Default is Blue. show : boolean, optional If ``True``, the plot will be displayed otherwise the equivalent matplotlib ``plot`` object will be returned. Defaults to True. lower_limit : Number, optional The lower limit of the plot range. Defaults to 0. upper_limit : Number, optional The upper limit of the plot range. Defaults to 10. prec : int, optional The decimal point precision for the point coordinate values. Defaults to 8. show_axes : boolean, optional If ``True``, the coordinate axes will be shown. Defaults to False. grid : boolean, optional If ``True``, the plot will have a grid. Defaults to True. Examples ======== .. plot:: :context: close-figs :format: doctest :include-source: True >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction >>> from sympy.physics.control.control_plots import ramp_response_plot >>> tf1 = TransferFunction(s, (s+4)(s+8), s) >>> ramp_response_plot(tf1, upper_limit=2) # doctest: +SKIP See Also ======== step_response_plot, ramp_response_plot References ========== .. [1] https://en.wikipedia.org/wiki/Ramp_function """ x, y = ramp_response_numerical_data(system, slope=slope, prec=prec, lower_limit=lower_limit, upper_limit=upper_limit, kwargs) plt.plot(x, y, color=color) plt.xlabel('Time (s)') plt.ylabel('Amplitude') plt.title(f'Ramp Response of ${latex(system)}$ [Slope = {slope}]', pad=20) if grid: plt.grid() if show_axes: plt.axhline(0, color='black') plt.axvline(0, color='black') if show: plt.show() return return plt def bode_magnitude_numerical_data(system, initial_exp=-5, final_exp=5, kwargs): """ Returns the numerical data of the Bode magnitude plot of the system. It is internally used by ``bode_magnitude_plot`` to get the data for plotting Bode magnitude plot. Users can use this data to further analyse the dynamics of the system or plot using a different backend/plotting-module. Parameters ========== system : SISOLinearTimeInvariant The system for which the data is to be computed. initial_exp : Number, optional The initial exponent of 10 of the semilog plot. Defaults to -5. final_exp : Number, optional The final exponent of 10 of the semilog plot. Defaults to 5. Returns ======= tuple : (x, y) x = x-axis values of the Bode magnitude plot. y = y-axis values of the Bode magnitude plot. Raises ====== NotImplementedError When a SISO LTI system is not passed. When time delay terms are present in the system. ValueError When more than one free symbol is present in the system. The only variable in the transfer function should be the variable of the Laplace transform. Examples ======== >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction >>> from sympy.physics.control.control_plots import bode_magnitude_numerical_data >>> tf1 = TransferFunction(s2 + 1, s4 + 4s3 + 6s*2 + 5s + 2, s) >>> bode_magnitude_numerical_data(tf1) # doctest: +SKIP ([1e-05, 1.5148378120533502e-05,..., 68437.36188804005, 100000.0], [-6.020599914256786, -6.0205999155219505,..., -193.4117304087953, -200.00000000260573]) See Also ======== bode_magnitude_plot, bode_phase_numerical_data """ _check_system(system) expr = system.to_expr() _w = Dummy("w", real=True) w_expr = expr.subs({system.var: I_w}) mag = 20log(Abs(w_expr), 10) return LineOver1DRangeSeries(mag, (_w, 10initial_exp, 10final_exp), xscale='log', kwargs).get_points() def bode_magnitude_plot(system, initial_exp=-5, final_exp=5, color='b', show_axes=False, grid=True, show=True, kwargs): r""" Returns the Bode magnitude plot of a continuous-time system. See ``bode_plot`` for all the parameters. """ x, y = bode_magnitude_numerical_data(system, initial_exp=initial_exp, final_exp=final_exp) plt.plot(x, y, color=color, kwargs) plt.xscale('log') plt.xlabel('Frequency (Hz) [Log Scale]') plt.ylabel('Magnitude (dB)') plt.title(f'Bode Plot (Magnitude) of ${latex(system)}$', pad=20) if grid: plt.grid(True) if show_axes: plt.axhline(0, color='black') plt.axvline(0, color='black') if show: plt.show() return return plt def bode_phase_numerical_data(system, initial_exp=-5, final_exp=5, kwargs): """ Returns the numerical data of the Bode phase plot of the system. It is internally used by ``bode_phase_plot`` to get the data for plotting Bode phase plot. Users can use this data to further analyse the dynamics of the system or plot using a different backend/plotting-module. Parameters ========== system : SISOLinearTimeInvariant The system for which the Bode phase plot data is to be computed. initial_exp : Number, optional The initial exponent of 10 of the semilog plot. Defaults to -5. final_exp : Number, optional The final exponent of 10 of the semilog plot. Defaults to 5. Returns ======= tuple : (x, y) x = x-axis values of the Bode phase plot. y = y-axis values of the Bode phase plot. Raises ====== NotImplementedError When a SISO LTI system is not passed. When time delay terms are present in the system. ValueError When more than one free symbol is present in the system. The only variable in the transfer function should be the variable of the Laplace transform. Examples ======== >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction >>> from sympy.physics.control.control_plots import bode_phase_numerical_data >>> tf1 = TransferFunction(s2 + 1, s4 + 4s3 + 6s*2 + 5s + 2, s) >>> bode_phase_numerical_data(tf1) # doctest: +SKIP ([1e-05, 1.4472354033813751e-05, 2.035581932165858e-05,..., 47577.3248186011, 67884.09326036123, 100000.0], [-2.5000000000291665e-05, -3.6180885085e-05, -5.08895483066e-05,...,-3.1415085799262523, -3.14155265358979]) See Also ======== bode_magnitude_plot, bode_phase_numerical_data """ _check_system(system) expr = system.to_expr() _w = Dummy("w", real=True) w_expr = expr.subs({system.var: I_w}) phase = arg(w_expr) return LineOver1DRangeSeries(phase, (_w, 10initial_exp, 10final_exp), xscale='log', kwargs).get_points() def bode_phase_plot(system, initial_exp=-5, final_exp=5, color='b', show_axes=False, grid=True, show=True, kwargs): r""" Returns the Bode phase plot of a continuous-time system. See ``bode_plot`` for all the parameters. """ x, y = bode_phase_numerical_data(system, initial_exp=initial_exp, final_exp=final_exp) plt.plot(x, y, color=color, kwargs) plt.xscale('log') plt.xlabel('Frequency (Hz) [Log Scale]') plt.ylabel('Phase (rad)') plt.title(f'Bode Plot (Phase) of ${latex(system)}$', pad=20) if grid: plt.grid(True) if show_axes: plt.axhline(0, color='black') plt.axvline(0, color='black') if show: plt.show() return return plt def bode_plot(system, initial_exp=-5, final_exp=5, grid=True, show_axes=False, show=True, kwargs): r""" Returns the Bode phase and magnitude plots of a continuous-time system. Parameters ========== system : SISOLinearTimeInvariant type The LTI SISO system for which the Bode Plot is to be computed. initial_exp : Number, optional The initial exponent of 10 of the semilog plot. Defaults to -5. final_exp : Number, optional The final exponent of 10 of the semilog plot. Defaults to 5. show : boolean, optional If ``True``, the plot will be displayed otherwise the equivalent matplotlib ``plot`` object will be returned. Defaults to True. prec : int, optional The decimal point precision for the point coordinate values. Defaults to 8. grid : boolean, optional If ``True``, the plot will have a grid. Defaults to True. show_axes : boolean, optional If ``True``, the coordinate axes will be shown. Defaults to False. Examples ======== .. plot:: :context: close-figs :format: doctest :include-source: True >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction >>> from sympy.physics.control.control_plots import bode_plot >>> tf1 = TransferFunction(1s*2 + 0.1s + 7.5, 1s4 + 0.12s*3 + 9s2, s) >>> bode_plot(tf1, initial_exp=0.2, final_exp=0.7) # doctest: +SKIP See Also ======== bode_magnitude_plot, bode_phase_plot """ plt.subplot(211) bode_magnitude_plot(system, initial_exp=initial_exp, final_exp=final_exp, show=False, grid=grid, show_axes=show_axes, kwargs).title(f'Bode Plot of ${latex(system)}$', pad=20) plt.subplot(212) bode_phase_plot(system, initial_exp=initial_exp, final_exp=final_exp, show=False, grid=grid, show_axes=show_axes, **kwargs).title(None) if show: plt.show() return return plt
d85fd7497cca2ed0d09bd2e468635fd950063bd0ef05af456d998006fdc3d701	from sympy.core.backend import zeros, Matrix, diff, eye from sympy.core.sorting import default_sort_key from sympy.physics.vector import (ReferenceFrame, dynamicsymbols, partial_velocity) from sympy.physics.mechanics.method import _Methods from sympy.physics.mechanics.particle import Particle from sympy.physics.mechanics.rigidbody import RigidBody from sympy.physics.mechanics.functions import (msubs, find_dynamicsymbols, _f_list_parser) from sympy.physics.mechanics.linearize import Linearizer from sympy.utilities.iterables import iterable __all__ = ['KanesMethod'] class KanesMethod(_Methods): """Kane's method object. Explanation =========== This object is used to do the "book-keeping" as you go through and form equations of motion in the way Kane presents in: Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill The attributes are for equations in the form [M] udot = forcing. Attributes ========== q, u : Matrix Matrices of the generalized coordinates and speeds bodies : iterable Iterable of Point and RigidBody objects in the system. loads : iterable Iterable of (Point, vector) or (ReferenceFrame, vector) tuples describing the forces on the system. auxiliary : Matrix If applicable, the set of auxiliary Kane's equations used to solve for non-contributing forces. mass_matrix : Matrix The system's mass matrix forcing : Matrix The system's forcing vector mass_matrix_full : Matrix The "mass matrix" for the u's and q's forcing_full : Matrix The "forcing vector" for the u's and q's Examples ======== This is a simple example for a one degree of freedom translational spring-mass-damper. In this example, we first need to do the kinematics. This involves creating generalized speeds and coordinates and their derivatives. Then we create a point and set its velocity in a frame. >>> from sympy import symbols >>> from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame >>> from sympy.physics.mechanics import Point, Particle, KanesMethod >>> q, u = dynamicsymbols('q u') >>> qd, ud = dynamicsymbols('q u', 1) >>> m, c, k = symbols('m c k') >>> N = ReferenceFrame('N') >>> P = Point('P') >>> P.set_vel(N, u * N.x) Next we need to arrange/store information in the way that KanesMethod requires. The kinematic differential equations need to be stored in a dict. A list of forces/torques must be constructed, where each entry in the list is a (Point, Vector) or (ReferenceFrame, Vector) tuple, where the Vectors represent the Force or Torque. Next a particle needs to be created, and it needs to have a point and mass assigned to it. Finally, a list of all bodies and particles needs to be created. >>> kd = [qd - u] >>> FL = [(P, (-k * q - c * u) * N.x)] >>> pa = Particle('pa', P, m) >>> BL = [pa] Finally we can generate the equations of motion. First we create the KanesMethod object and supply an inertial frame, coordinates, generalized speeds, and the kinematic differential equations. Additional quantities such as configuration and motion constraints, dependent coordinates and speeds, and auxiliary speeds are also supplied here (see the online documentation). Next we form FR* and FR to complete: Fr + Fr* = 0. We have the equations of motion at this point. It makes sense to rearrange them though, so we calculate the mass matrix and the forcing terms, for E.o.M. in the form: [MM] udot = forcing, where MM is the mass matrix, udot is a vector of the time derivatives of the generalized speeds, and forcing is a vector representing "forcing" terms. >>> KM = KanesMethod(N, q_ind=[q], u_ind=[u], kd_eqs=kd) >>> (fr, frstar) = KM.kanes_equations(BL, FL) >>> MM = KM.mass_matrix >>> forcing = KM.forcing >>> rhs = MM.inv() * forcing >>> rhs Matrix([[(-cu(t) - kq(t))/m]]) >>> KM.linearize(A_and_B=True)[0] Matrix([ [ 0, 1], [-k/m, -c/m]]) Please look at the documentation pages for more information on how to perform linearization and how to deal with dependent coordinates & speeds, and how do deal with bringing non-contributing forces into evidence. """ def __init__(self, frame, q_ind, u_ind, kd_eqs=None, q_dependent=None, configuration_constraints=None, u_dependent=None, velocity_constraints=None, acceleration_constraints=None, u_auxiliary=None, bodies=None, forcelist=None): """Please read the online documentation. """ if not q_ind: q_ind = [dynamicsymbols('dummy_q')] kd_eqs = [dynamicsymbols('dummy_kd')] if not isinstance(frame, ReferenceFrame): raise TypeError('An inertial ReferenceFrame must be supplied') self._inertial = frame self._fr = None self._frstar = None self._forcelist = forcelist self._bodylist = bodies self._initialize_vectors(q_ind, q_dependent, u_ind, u_dependent, u_auxiliary) self._initialize_kindiffeq_matrices(kd_eqs) self._initialize_constraint_matrices(configuration_constraints, velocity_constraints, acceleration_constraints) def _initialize_vectors(self, q_ind, q_dep, u_ind, u_dep, u_aux): """Initialize the coordinate and speed vectors.""" none_handler = lambda x: Matrix(x) if x else Matrix() # Initialize generalized coordinates q_dep = none_handler(q_dep) if not iterable(q_ind): raise TypeError('Generalized coordinates must be an iterable.') if not iterable(q_dep): raise TypeError('Dependent coordinates must be an iterable.') q_ind = Matrix(q_ind) self._qdep = q_dep self._q = Matrix([q_ind, q_dep]) self._qdot = self.q.diff(dynamicsymbols._t) # Initialize generalized speeds u_dep = none_handler(u_dep) if not iterable(u_ind): raise TypeError('Generalized speeds must be an iterable.') if not iterable(u_dep): raise TypeError('Dependent speeds must be an iterable.') u_ind = Matrix(u_ind) self._udep = u_dep self._u = Matrix([u_ind, u_dep]) self._udot = self.u.diff(dynamicsymbols._t) self._uaux = none_handler(u_aux) def _initialize_constraint_matrices(self, config, vel, acc): """Initializes constraint matrices.""" # Define vector dimensions o = len(self.u) m = len(self._udep) p = o - m none_handler = lambda x: Matrix(x) if x else Matrix() # Initialize configuration constraints config = none_handler(config) if len(self._qdep) != len(config): raise ValueError('There must be an equal number of dependent ' 'coordinates and configuration constraints.') self._f_h = none_handler(config) # Initialize velocity and acceleration constraints vel = none_handler(vel) acc = none_handler(acc) if len(vel) != m: raise ValueError('There must be an equal number of dependent ' 'speeds and velocity constraints.') if acc and (len(acc) != m): raise ValueError('There must be an equal number of dependent ' 'speeds and acceleration constraints.') if vel: u_zero = {i: 0 for i in self.u} udot_zero = {i: 0 for i in self._udot} # When calling kanes_equations, another class instance will be # created if auxiliary u's are present. In this case, the # computation of kinetic differential equation matrices will be # skipped as this was computed during the original KanesMethod # object, and the qd_u_map will not be available. if self._qdot_u_map is not None: vel = msubs(vel, self._qdot_u_map) self._f_nh = msubs(vel, u_zero) self._k_nh = (vel - self._f_nh).jacobian(self.u) # If no acceleration constraints given, calculate them. if not acc: _f_dnh = (self._k_nh.diff(dynamicsymbols._t) * self.u + self._f_nh.diff(dynamicsymbols._t)) if self._qdot_u_map is not None: _f_dnh = msubs(_f_dnh, self._qdot_u_map) self._f_dnh = _f_dnh self._k_dnh = self._k_nh else: if self._qdot_u_map is not None: acc = msubs(acc, self._qdot_u_map) self._f_dnh = msubs(acc, udot_zero) self._k_dnh = (acc - self._f_dnh).jacobian(self._udot) # Form of non-holonomic constraints is Bu + C = 0. # We partition B into independent and dependent columns: # Ars is then -B_dep.inv() B_ind, and it relates dependent speeds # to independent speeds as: udep = Arsuind, neglecting the C term. B_ind = self._k_nh[:, :p] B_dep = self._k_nh[:, p:o] self._Ars = -B_dep.LUsolve(B_ind) else: self._f_nh = Matrix() self._k_nh = Matrix() self._f_dnh = Matrix() self._k_dnh = Matrix() self._Ars = Matrix() def _initialize_kindiffeq_matrices(self, kdeqs): """Initialize the kinematic differential equation matrices. Parameters ========== kdeqs : sequence of sympy expressions Kinematic differential equations in the form of f(u,q',q,t) where f() = 0. The equations have to be linear in the generalized coordinates and generalized speeds. """ if kdeqs: if len(self.q) != len(kdeqs): raise ValueError('There must be an equal number of kinematic ' 'differential equations and coordinates.') u = self.u qdot = self._qdot kdeqs = Matrix(kdeqs) u_zero = {ui: 0 for ui in u} uaux_zero = {uai: 0 for uai in self._uaux} qdot_zero = {qdi: 0 for qdi in qdot} # Extract the linear coefficient matrices as per the following # equation: # # k_ku(q,t)u(t) + k_kqdot(q,t)q'(t) + f_k(q,t) = 0 # k_ku = kdeqs.jacobian(u) k_kqdot = kdeqs.jacobian(qdot) f_k = kdeqs.xreplace(u_zero).xreplace(qdot_zero) # The kinematic differential equations should be linear in both q' # and u, so check for u and q' in the components. dy_syms = find_dynamicsymbols(k_ku.row_join(k_kqdot).row_join(f_k)) nonlin_vars = [vari for vari in u[:] + qdot[:] if vari in dy_syms] if nonlin_vars: msg = ('The provided kinematic differential equations are ' 'nonlinear in {}. They must be linear in the ' 'generalized speeds and derivatives of the generalized ' 'coordinates.') raise ValueError(msg.format(nonlin_vars)) # Solve for q'(t) such that the coefficient matrices are now in # this form: # # k_kqdot^-1k_kuu(t) + Iq'(t) + k_kqdot^-1f_k = 0 # # NOTE : Solving the kinematic differential equations here is not # necessary and prevents the equations from being provided in fully # implicit form. f_k = k_kqdot.LUsolve(f_k) k_ku = k_kqdot.LUsolve(k_ku) k_kqdot = eye(len(qdot)) self._qdot_u_map = dict(zip(qdot, -(k_kuu + f_k))) self._f_k = f_k.xreplace(uaux_zero) self._k_ku = k_ku.xreplace(uaux_zero) self._k_kqdot = k_kqdot else: self._qdot_u_map = None self._f_k = Matrix() self._k_ku = Matrix() self._k_kqdot = Matrix() def _form_fr(self, fl): """Form the generalized active force.""" if fl is not None and (len(fl) == 0 or not iterable(fl)): raise ValueError('Force pairs must be supplied in an ' 'non-empty iterable or None.') N = self._inertial # pull out relevant velocities for constructing partial velocities vel_list, f_list = _f_list_parser(fl, N) vel_list = [msubs(i, self._qdot_u_map) for i in vel_list] f_list = [msubs(i, self._qdot_u_map) for i in f_list] # Fill Fr with dot product of partial velocities and forces o = len(self.u) b = len(f_list) FR = zeros(o, 1) partials = partial_velocity(vel_list, self.u, N) for i in range(o): FR[i] = sum(partials[j][i] & f_list[j] for j in range(b)) # In case there are dependent speeds if self._udep: p = o - len(self._udep) FRtilde = FR[:p, 0] FRold = FR[p:o, 0] FRtilde += self._Ars.T * FRold FR = FRtilde self._forcelist = fl self._fr = FR return FR def _form_frstar(self, bl): """Form the generalized inertia force.""" if not iterable(bl): raise TypeError('Bodies must be supplied in an iterable.') t = dynamicsymbols._t N = self._inertial # Dicts setting things to zero udot_zero = {i: 0 for i in self._udot} uaux_zero = {i: 0 for i in self._uaux} uauxdot = [diff(i, t) for i in self._uaux] uauxdot_zero = {i: 0 for i in uauxdot} # Dictionary of q' and q'' to u and u' q_ddot_u_map = {k.diff(t): v.diff(t) for (k, v) in self._qdot_u_map.items()} q_ddot_u_map.update(self._qdot_u_map) # Fill up the list of partials: format is a list with num elements # equal to number of entries in body list. Each of these elements is a # list - either of length 1 for the translational components of # particles or of length 2 for the translational and rotational # components of rigid bodies. The inner most list is the list of # partial velocities. def get_partial_velocity(body): if isinstance(body, RigidBody): vlist = [body.masscenter.vel(N), body.frame.ang_vel_in(N)] elif isinstance(body, Particle): vlist = [body.point.vel(N),] else: raise TypeError('The body list may only contain either ' 'RigidBody or Particle as list elements.') v = [msubs(vel, self._qdot_u_map) for vel in vlist] return partial_velocity(v, self.u, N) partials = [get_partial_velocity(body) for body in bl] # Compute fr_star in two components: # fr_star = -(MMu' + nonMM) o = len(self.u) MM = zeros(o, o) nonMM = zeros(o, 1) zero_uaux = lambda expr: msubs(expr, uaux_zero) zero_udot_uaux = lambda expr: msubs(msubs(expr, udot_zero), uaux_zero) for i, body in enumerate(bl): if isinstance(body, RigidBody): M = zero_uaux(body.mass) I = zero_uaux(body.central_inertia) vel = zero_uaux(body.masscenter.vel(N)) omega = zero_uaux(body.frame.ang_vel_in(N)) acc = zero_udot_uaux(body.masscenter.acc(N)) inertial_force = (M.diff(t) vel + M * acc) inertial_torque = zero_uaux((I.dt(body.frame) & omega) + msubs(I & body.frame.ang_acc_in(N), udot_zero) + (omega ^ (I & omega))) for j in range(o): tmp_vel = zero_uaux(partials[i][0][j]) tmp_ang = zero_uaux(I & partials[i][1][j]) for k in range(o): # translational MM[j, k] += M * (tmp_vel & partials[i][0][k]) # rotational MM[j, k] += (tmp_ang & partials[i][1][k]) nonMM[j] += inertial_force & partials[i][0][j] nonMM[j] += inertial_torque & partials[i][1][j] else: M = zero_uaux(body.mass) vel = zero_uaux(body.point.vel(N)) acc = zero_udot_uaux(body.point.acc(N)) inertial_force = (M.diff(t) * vel + M * acc) for j in range(o): temp = zero_uaux(partials[i][0][j]) for k in range(o): MM[j, k] += M * (temp & partials[i][0][k]) nonMM[j] += inertial_force & partials[i][0][j] # Compose fr_star out of MM and nonMM MM = zero_uaux(msubs(MM, q_ddot_u_map)) nonMM = msubs(msubs(nonMM, q_ddot_u_map), udot_zero, uauxdot_zero, uaux_zero) fr_star = -(MM * msubs(Matrix(self._udot), uauxdot_zero) + nonMM) # If there are dependent speeds, we need to find fr_star_tilde if self._udep: p = o - len(self._udep) fr_star_ind = fr_star[:p, 0] fr_star_dep = fr_star[p:o, 0] fr_star = fr_star_ind + (self._Ars.T * fr_star_dep) # Apply the same to MM MMi = MM[:p, :] MMd = MM[p:o, :] MM = MMi + (self._Ars.T * MMd) self._bodylist = bl self._frstar = fr_star self._k_d = MM self._f_d = -msubs(self._fr + self._frstar, udot_zero) return fr_star def to_linearizer(self): """Returns an instance of the Linearizer class, initiated from the data in the KanesMethod class. This may be more desirable than using the linearize class method, as the Linearizer object will allow more efficient recalculation (i.e. about varying operating points).""" if (self._fr is None) or (self._frstar is None): raise ValueError('Need to compute Fr, Fr* first.') # Get required equation components. The Kane's method class breaks # these into pieces. Need to reassemble f_c = self._f_h if self._f_nh and self._k_nh: f_v = self._f_nh + self._k_nhMatrix(self.u) else: f_v = Matrix() if self._f_dnh and self._k_dnh: f_a = self._f_dnh + self._k_dnhMatrix(self._udot) else: f_a = Matrix() # Dicts to sub to zero, for splitting up expressions u_zero = {i: 0 for i in self.u} ud_zero = {i: 0 for i in self._udot} qd_zero = {i: 0 for i in self._qdot} qd_u_zero = {i: 0 for i in Matrix([self._qdot, self.u])} # Break the kinematic differential eqs apart into f_0 and f_1 f_0 = msubs(self._f_k, u_zero) + self._k_kqdotMatrix(self._qdot) f_1 = msubs(self._f_k, qd_zero) + self._k_kuMatrix(self.u) # Break the dynamic differential eqs into f_2 and f_3 f_2 = msubs(self._frstar, qd_u_zero) f_3 = msubs(self._frstar, ud_zero) + self._fr f_4 = zeros(len(f_2), 1) # Get the required vector components q = self.q u = self.u if self._qdep: q_i = q[:-len(self._qdep)] else: q_i = q q_d = self._qdep if self._udep: u_i = u[:-len(self._udep)] else: u_i = u u_d = self._udep # Form dictionary to set auxiliary speeds & their derivatives to 0. uaux = self._uaux uauxdot = uaux.diff(dynamicsymbols._t) uaux_zero = {i: 0 for i in Matrix([uaux, uauxdot])} # Checking for dynamic symbols outside the dynamic differential # equations; throws error if there is. sym_list = set(Matrix([q, self._qdot, u, self._udot, uaux, uauxdot])) if any(find_dynamicsymbols(i, sym_list) for i in [self._k_kqdot, self._k_ku, self._f_k, self._k_dnh, self._f_dnh, self._k_d]): raise ValueError('Cannot have dynamicsymbols outside dynamic \ forcing vector.') # Find all other dynamic symbols, forming the forcing vector r. # Sort r to make it canonical. r = list(find_dynamicsymbols(msubs(self._f_d, uaux_zero), sym_list)) r.sort(key=default_sort_key) # Check for any derivatives of variables in r that are also found in r. for i in r: if diff(i, dynamicsymbols._t) in r: raise ValueError('Cannot have derivatives of specified \ quantities when linearizing forcing terms.') return Linearizer(f_0, f_1, f_2, f_3, f_4, f_c, f_v, f_a, q, u, q_i, q_d, u_i, u_d, r) # TODO : Remove `new_method` after 1.1 has been released. def linearize(self, , new_method=None, kwargs): """ Linearize the equations of motion about a symbolic operating point. Explanation =========== If kwarg A_and_B is False (default), returns M, A, B, r for the linearized form, M[q', u']^T = A[q_ind, u_ind]^T + Br. If kwarg A_and_B is True, returns A, B, r for the linearized form dx = Ax + Br, where x = [q_ind, u_ind]^T. Note that this is computationally intensive if there are many symbolic parameters. For this reason, it may be more desirable to use the default A_and_B=False, returning M, A, and B. Values may then be substituted in to these matrices, and the state space form found as A = P.TM.inv()A, B = P.TM.inv()B, where P = Linearizer.perm_mat. In both cases, r is found as all dynamicsymbols in the equations of motion that are not part of q, u, q', or u'. They are sorted in canonical form. The operating points may be also entered using the ``op_point`` kwarg. This takes a dictionary of {symbol: value}, or a an iterable of such dictionaries. The values may be numeric or symbolic. The more values you can specify beforehand, the faster this computation will run. For more documentation, please see the ``Linearizer`` class.""" linearizer = self.to_linearizer() result = linearizer.linearize(*kwargs) return result + (linearizer.r,) def kanes_equations(self, bodies=None, loads=None): """ Method to form Kane's equations, Fr + Fr = 0. Explanation =========== Returns (Fr, Fr). In the case where auxiliary generalized speeds are present (say, s auxiliary speeds, o generalized speeds, and m motion constraints) the length of the returned vectors will be o - m + s in length. The first o - m equations will be the constrained Kane's equations, then the s auxiliary Kane's equations. These auxiliary equations can be accessed with the auxiliary_eqs(). Parameters ========== bodies : iterable An iterable of all RigidBody's and Particle's in the system. A system must have at least one body. loads : iterable Takes in an iterable of (Particle, Vector) or (ReferenceFrame, Vector) tuples which represent the force at a point or torque on a frame. Must be either a non-empty iterable of tuples or None which corresponds to a system with no constraints. """ if bodies is None: bodies = self.bodies if loads is None and self._forcelist is not None: loads = self._forcelist if loads == []: loads = None if not self._k_kqdot: raise AttributeError('Create an instance of KanesMethod with ' 'kinematic differential equations to use this method.') fr = self._form_fr(loads) frstar = self._form_frstar(bodies) if self._uaux: if not self._udep: km = KanesMethod(self._inertial, self.q, self._uaux, u_auxiliary=self._uaux) else: km = KanesMethod(self._inertial, self.q, self._uaux, u_auxiliary=self._uaux, u_dependent=self._udep, velocity_constraints=(self._k_nh self.u + self._f_nh)) km._qdot_u_map = self._qdot_u_map self._km = km fraux = km._form_fr(loads) frstaraux = km._form_frstar(bodies) self._aux_eq = fraux + frstaraux self._fr = fr.col_join(fraux) self._frstar = frstar.col_join(frstaraux) return (self._fr, self._frstar) def _form_eoms(self): fr, frstar = self.kanes_equations(self.bodylist, self.forcelist) return fr + frstar def rhs(self, inv_method=None): """Returns the system's equations of motion in first order form. The output is the right hand side of:: x' = \|q'\| =: f(q, u, r, p, t) \|u'\| The right hand side is what is needed by most numerical ODE integrators. Parameters ========== inv_method : str The specific sympy inverse matrix calculation method to use. For a list of valid methods, see :meth:`~sympy.matrices.matrices.MatrixBase.inv` """ rhs = zeros(len(self.q) + len(self.u), 1) kdes = self.kindiffdict() for i, q_i in enumerate(self.q): rhs[i] = kdes[q_i.diff()] if inv_method is None: rhs[len(self.q):, 0] = self.mass_matrix.LUsolve(self.forcing) else: rhs[len(self.q):, 0] = (self.mass_matrix.inv(inv_method, try_block_diag=True) * self.forcing) return rhs def kindiffdict(self): """Returns a dictionary mapping q' to u.""" if not self._qdot_u_map: raise AttributeError('Create an instance of KanesMethod with ' 'kinematic differential equations to use this method.') return self._qdot_u_map @property def auxiliary_eqs(self): """A matrix containing the auxiliary equations.""" if not self._fr or not self._frstar: raise ValueError('Need to compute Fr, Fr* first.') if not self._uaux: raise ValueError('No auxiliary speeds have been declared.') return self._aux_eq @property def mass_matrix(self): """The mass matrix of the system.""" if not self._fr or not self._frstar: raise ValueError('Need to compute Fr, Fr* first.') return Matrix([self._k_d, self._k_dnh]) @property def mass_matrix_full(self): """The mass matrix of the system, augmented by the kinematic differential equations.""" if not self._fr or not self._frstar: raise ValueError('Need to compute Fr, Fr* first.') o = len(self.u) n = len(self.q) return ((self._k_kqdot).row_join(zeros(n, o))).col_join((zeros(o, n)).row_join(self.mass_matrix)) @property def forcing(self): """The forcing vector of the system.""" if not self._fr or not self._frstar: raise ValueError('Need to compute Fr, Fr* first.') return -Matrix([self._f_d, self._f_dnh]) @property def forcing_full(self): """The forcing vector of the system, augmented by the kinematic differential equations.""" if not self._fr or not self._frstar: raise ValueError('Need to compute Fr, Fr* first.') f1 = self._k_ku * Matrix(self.u) + self._f_k return -Matrix([f1, self._f_d, self._f_dnh]) @property def q(self): return self._q @property def u(self): return self._u @property def bodylist(self): return self._bodylist @property def forcelist(self): return self._forcelist @property def bodies(self): return self._bodylist @property def loads(self): return self._forcelist
66a63f9719d9cbf63057c19900db84a14da8c065a0b8ab9318aff20a8b498e33	# isort:skip_file """ Dimensional analysis and unit systems. This module defines dimension/unit systems and physical quantities. It is based on a group-theoretical construction where dimensions are represented as vectors (coefficients being the exponents), and units are defined as a dimension to which we added a scale. Quantities are built from a factor and a unit, and are the basic objects that one will use when doing computations. All objects except systems and prefixes can be used in SymPy expressions. Note that as part of a CAS, various objects do not combine automatically under operations. Details about the implementation can be found in the documentation, and we will not repeat all the explanations we gave there concerning our approach. Ideas about future developments can be found on the `Github wiki <https://github.com/sympy/sympy/wiki/Unit-systems>`_, and you should consult this page if you are willing to help. Useful functions: - ``find_unit``: easily lookup pre-defined units. - ``convert_to(expr, newunit)``: converts an expression into the same expression expressed in another unit. """ from .dimensions import Dimension, DimensionSystem from .unitsystem import UnitSystem from .util import convert_to from .quantities import Quantity from .definitions.dimension_definitions import ( amount_of_substance, acceleration, action, capacitance, charge, conductance, current, energy, force, frequency, impedance, inductance, length, luminous_intensity, magnetic_density, magnetic_flux, mass, momentum, power, pressure, temperature, time, velocity, voltage, volume ) Unit = Quantity speed = velocity luminosity = luminous_intensity magnetic_flux_density = magnetic_density amount = amount_of_substance from .prefixes import ( # 10-power based: yotta, zetta, exa, peta, tera, giga, mega, kilo, hecto, deca, deci, centi, milli, micro, nano, pico, femto, atto, zepto, yocto, # 2-power based: kibi, mebi, gibi, tebi, pebi, exbi, ) from .definitions import ( percent, percents, permille, rad, radian, radians, deg, degree, degrees, sr, steradian, steradians, mil, angular_mil, angular_mils, m, meter, meters, kg, kilogram, kilograms, s, second, seconds, A, ampere, amperes, K, kelvin, kelvins, mol, mole, moles, cd, candela, candelas, g, gram, grams, mg, milligram, milligrams, ug, microgram, micrograms, newton, newtons, N, joule, joules, J, watt, watts, W, pascal, pascals, Pa, pa, hertz, hz, Hz, coulomb, coulombs, C, volt, volts, v, V, ohm, ohms, siemens, S, mho, mhos, farad, farads, F, henry, henrys, H, tesla, teslas, T, weber, webers, Wb, wb, optical_power, dioptre, D, lux, lx, katal, kat, gray, Gy, becquerel, Bq, km, kilometer, kilometers, dm, decimeter, decimeters, cm, centimeter, centimeters, mm, millimeter, millimeters, um, micrometer, micrometers, micron, microns, nm, nanometer, nanometers, pm, picometer, picometers, ft, foot, feet, inch, inches, yd, yard, yards, mi, mile, miles, nmi, nautical_mile, nautical_miles, l, L, liter, liters, dl, dL, deciliter, deciliters, cl, cL, centiliter, centiliters, ml, mL, milliliter, milliliters, ms, millisecond, milliseconds, us, microsecond, microseconds, ns, nanosecond, nanoseconds, ps, picosecond, picoseconds, minute, minutes, h, hour, hours, day, days, anomalistic_year, anomalistic_years, sidereal_year, sidereal_years, tropical_year, tropical_years, common_year, common_years, julian_year, julian_years, draconic_year, draconic_years, gaussian_year, gaussian_years, full_moon_cycle, full_moon_cycles, year, years, G, gravitational_constant, c, speed_of_light, elementary_charge, hbar, planck, eV, electronvolt, electronvolts, avogadro_number, avogadro, avogadro_constant, boltzmann, boltzmann_constant, stefan, stefan_boltzmann_constant, R, molar_gas_constant, faraday_constant, josephson_constant, von_klitzing_constant, amu, amus, atomic_mass_unit, atomic_mass_constant, gee, gees, acceleration_due_to_gravity, u0, magnetic_constant, vacuum_permeability, e0, electric_constant, vacuum_permittivity, Z0, vacuum_impedance, coulomb_constant, electric_force_constant, atmosphere, atmospheres, atm, kPa, bar, bars, pound, pounds, psi, dHg0, mmHg, torr, mmu, mmus, milli_mass_unit, quart, quarts, ly, lightyear, lightyears, au, astronomical_unit, astronomical_units, planck_mass, planck_time, planck_temperature, planck_length, planck_charge, planck_area, planck_volume, planck_momentum, planck_energy, planck_force, planck_power, planck_density, planck_energy_density, planck_intensity, planck_angular_frequency, planck_pressure, planck_current, planck_voltage, planck_impedance, planck_acceleration, bit, bits, byte, kibibyte, kibibytes, mebibyte, mebibytes, gibibyte, gibibytes, tebibyte, tebibytes, pebibyte, pebibytes, exbibyte, exbibytes, ) from .systems import ( mks, mksa, si ) def find_unit(quantity, unit_system="SI"): """ Return a list of matching units or dimension names. - If ``quantity`` is a string -- units/dimensions containing the string `quantity`. - If ``quantity`` is a unit or dimension -- units having matching base units or dimensions. Examples ======== >>> from sympy.physics import units as u >>> u.find_unit('charge') ['C', 'coulomb', 'coulombs', 'planck_charge', 'elementary_charge'] >>> u.find_unit(u.charge) ['C', 'coulomb', 'coulombs', 'planck_charge', 'elementary_charge'] >>> u.find_unit("ampere") ['ampere', 'amperes'] >>> u.find_unit('volt') ['volt', 'volts', 'electronvolt', 'electronvolts', 'planck_voltage'] >>> u.find_unit(u.inch**3)[:9] ['L', 'l', 'cL', 'cl', 'dL', 'dl', 'mL', 'ml', 'liter'] """ unit_system = UnitSystem.get_unit_system(unit_system) import sympy.physics.units as u rv = [] if isinstance(quantity, str): rv = [i for i in dir(u) if quantity in i and isinstance(getattr(u, i), Quantity)] dim = getattr(u, quantity) if isinstance(dim, Dimension): rv.extend(find_unit(dim)) else: for i in sorted(dir(u)): other = getattr(u, i) if not isinstance(other, Quantity): continue if isinstance(quantity, Quantity): if quantity.dimension == other.dimension: rv.append(str(i)) elif isinstance(quantity, Dimension): if other.dimension == quantity: rv.append(str(i)) elif other.dimension == Dimension(unit_system.get_dimensional_expr(quantity)): rv.append(str(i)) return sorted(set(rv), key=lambda x: (len(x), x)) # NOTE: the old units module had additional variables: # 'density', 'illuminance', 'resistance'. # They were not dimensions, but units (old Unit class). __all__ = [ 'Dimension', 'DimensionSystem', 'UnitSystem', 'convert_to', 'Quantity', 'amount_of_substance', 'acceleration', 'action', 'capacitance', 'charge', 'conductance', 'current', 'energy', 'force', 'frequency', 'impedance', 'inductance', 'length', 'luminous_intensity', 'magnetic_density', 'magnetic_flux', 'mass', 'momentum', 'power', 'pressure', 'temperature', 'time', 'velocity', 'voltage', 'volume', 'Unit', 'speed', 'luminosity', 'magnetic_flux_density', 'amount', 'yotta', 'zetta', 'exa', 'peta', 'tera', 'giga', 'mega', 'kilo', 'hecto', 'deca', 'deci', 'centi', 'milli', 'micro', 'nano', 'pico', 'femto', 'atto', 'zepto', 'yocto', 'kibi', 'mebi', 'gibi', 'tebi', 'pebi', 'exbi', 'percent', 'percents', 'permille', 'rad', 'radian', 'radians', 'deg', 'degree', 'degrees', 'sr', 'steradian', 'steradians', 'mil', 'angular_mil', 'angular_mils', 'm', 'meter', 'meters', 'kg', 'kilogram', 'kilograms', 's', 'second', 'seconds', 'A', 'ampere', 'amperes', 'K', 'kelvin', 'kelvins', 'mol', 'mole', 'moles', 'cd', 'candela', 'candelas', 'g', 'gram', 'grams', 'mg', 'milligram', 'milligrams', 'ug', 'microgram', 'micrograms', 'newton', 'newtons', 'N', 'joule', 'joules', 'J', 'watt', 'watts', 'W', 'pascal', 'pascals', 'Pa', 'pa', 'hertz', 'hz', 'Hz', 'coulomb', 'coulombs', 'C', 'volt', 'volts', 'v', 'V', 'ohm', 'ohms', 'siemens', 'S', 'mho', 'mhos', 'farad', 'farads', 'F', 'henry', 'henrys', 'H', 'tesla', 'teslas', 'T', 'weber', 'webers', 'Wb', 'wb', 'optical_power', 'dioptre', 'D', 'lux', 'lx', 'katal', 'kat', 'gray', 'Gy', 'becquerel', 'Bq', 'km', 'kilometer', 'kilometers', 'dm', 'decimeter', 'decimeters', 'cm', 'centimeter', 'centimeters', 'mm', 'millimeter', 'millimeters', 'um', 'micrometer', 'micrometers', 'micron', 'microns', 'nm', 'nanometer', 'nanometers', 'pm', 'picometer', 'picometers', 'ft', 'foot', 'feet', 'inch', 'inches', 'yd', 'yard', 'yards', 'mi', 'mile', 'miles', 'nmi', 'nautical_mile', 'nautical_miles', 'l', 'L', 'liter', 'liters', 'dl', 'dL', 'deciliter', 'deciliters', 'cl', 'cL', 'centiliter', 'centiliters', 'ml', 'mL', 'milliliter', 'milliliters', 'ms', 'millisecond', 'milliseconds', 'us', 'microsecond', 'microseconds', 'ns', 'nanosecond', 'nanoseconds', 'ps', 'picosecond', 'picoseconds', 'minute', 'minutes', 'h', 'hour', 'hours', 'day', 'days', 'anomalistic_year', 'anomalistic_years', 'sidereal_year', 'sidereal_years', 'tropical_year', 'tropical_years', 'common_year', 'common_years', 'julian_year', 'julian_years', 'draconic_year', 'draconic_years', 'gaussian_year', 'gaussian_years', 'full_moon_cycle', 'full_moon_cycles', 'year', 'years', 'G', 'gravitational_constant', 'c', 'speed_of_light', 'elementary_charge', 'hbar', 'planck', 'eV', 'electronvolt', 'electronvolts', 'avogadro_number', 'avogadro', 'avogadro_constant', 'boltzmann', 'boltzmann_constant', 'stefan', 'stefan_boltzmann_constant', 'R', 'molar_gas_constant', 'faraday_constant', 'josephson_constant', 'von_klitzing_constant', 'amu', 'amus', 'atomic_mass_unit', 'atomic_mass_constant', 'gee', 'gees', 'acceleration_due_to_gravity', 'u0', 'magnetic_constant', 'vacuum_permeability', 'e0', 'electric_constant', 'vacuum_permittivity', 'Z0', 'vacuum_impedance', 'coulomb_constant', 'electric_force_constant', 'atmosphere', 'atmospheres', 'atm', 'kPa', 'bar', 'bars', 'pound', 'pounds', 'psi', 'dHg0', 'mmHg', 'torr', 'mmu', 'mmus', 'milli_mass_unit', 'quart', 'quarts', 'ly', 'lightyear', 'lightyears', 'au', 'astronomical_unit', 'astronomical_units', 'planck_mass', 'planck_time', 'planck_temperature', 'planck_length', 'planck_charge', 'planck_area', 'planck_volume', 'planck_momentum', 'planck_energy', 'planck_force', 'planck_power', 'planck_density', 'planck_energy_density', 'planck_intensity', 'planck_angular_frequency', 'planck_pressure', 'planck_current', 'planck_voltage', 'planck_impedance', 'planck_acceleration', 'bit', 'bits', 'byte', 'kibibyte', 'kibibytes', 'mebibyte', 'mebibytes', 'gibibyte', 'gibibytes', 'tebibyte', 'tebibytes', 'pebibyte', 'pebibytes', 'exbibyte', 'exbibytes', 'mks', 'mksa', 'si', ]
732b34b561a2d5508f58c4d9b8a69d70dfdb4e8cff6ea2ac0c7484573b95dc21	""" Contains * Medium """ from sympy.physics.units import second, meter, kilogram, ampere __all__ = ['Medium'] from sympy.core.basic import Basic from sympy.core.symbol import Str from sympy.core.sympify import _sympify from sympy.functions.elementary.miscellaneous import sqrt from sympy.physics.units import speed_of_light, u0, e0 c = speed_of_light.convert_to(meter/second) _e0mksa = e0.convert_to(ampere*2second*4/(kilogrammeter*3)) _u0mksa = u0.convert_to(meterkilogram/(ampere*2second*2)) class Medium(Basic): """ This class represents an optical medium. The prime reason to implement this is to facilitate refraction, Fermat's principle, etc. Explanation =========== An optical medium is a material through which electromagnetic waves propagate. The permittivity and permeability of the medium define how electromagnetic waves propagate in it. Parameters ========== name: string The display name of the Medium. permittivity: Sympifyable Electric permittivity of the space. permeability: Sympifyable Magnetic permeability of the space. n: Sympifyable Index of refraction of the medium. Examples ======== >>> from sympy.abc import epsilon, mu >>> from sympy.physics.optics import Medium >>> m1 = Medium('m1') >>> m2 = Medium('m2', epsilon, mu) >>> m1.intrinsic_impedance 149896229pikilogrammeter*2/(1250000ampere*2second*3) >>> m2.refractive_index 299792458metersqrt(epsilonmu)/second References ========== .. [1] https://en.wikipedia.org/wiki/Optical_medium """ def __new__(cls, name, permittivity=None, permeability=None, n=None): if not isinstance(name, Str): name = Str(name) permittivity = _sympify(permittivity) if permittivity is not None else permittivity permeability = _sympify(permeability) if permeability is not None else permeability n = _sympify(n) if n is not None else n if n is not None: if permittivity is not None and permeability is None: permeability = n2/(c2permittivity) return MediumPP(name, permittivity, permeability) elif permeability is not None and permittivity is None: permittivity = n2/(c2permeability) return MediumPP(name, permittivity, permeability) elif permittivity is not None and permittivity is not None: raise ValueError("Specifying all of permittivity, permeability, and n is not allowed") else: return MediumN(name, n) elif permittivity is not None and permeability is not None: return MediumPP(name, permittivity, permeability) elif permittivity is None and permeability is None: return MediumPP(name, _e0mksa, _u0mksa) else: raise ValueError("Arguments are underspecified. Either specify n or any two of permittivity, " "permeability, and n") @property def name(self): return self.args[0] @property def speed(self): """ Returns speed of the electromagnetic wave travelling in the medium. Examples ======== >>> from sympy.physics.optics import Medium >>> m = Medium('m') >>> m.speed 299792458meter/second >>> m2 = Medium('m2', n=1) >>> m.speed == m2.speed True """ return c / self.n @property def refractive_index(self): """ Returns refractive index of the medium. Examples ======== >>> from sympy.physics.optics import Medium >>> m = Medium('m') >>> m.refractive_index 1 """ return (c/self.speed) class MediumN(Medium): """ Represents an optical medium for which only the refractive index is known. Useful for simple ray optics. This class should never be instantiated directly. Instead it should be instantiated indirectly by instantiating Medium with only n specified. Examples ======== >>> from sympy.physics.optics import Medium >>> m = Medium('m', n=2) >>> m MediumN(Str('m'), 2) """ def __new__(cls, name, n): obj = super(Medium, cls).__new__(cls, name, n) return obj @property def n(self): return self.args[1] class MediumPP(Medium): """ Represents an optical medium for which the permittivity and permeability are known. This class should never be instantiated directly. Instead it should be instantiated indirectly by instantiating Medium with any two of permittivity, permeability, and n specified, or by not specifying any of permittivity, permeability, or n, in which case default values for permittivity and permeability will be used. Examples ======== >>> from sympy.physics.optics import Medium >>> from sympy.abc import epsilon, mu >>> m1 = Medium('m1', permittivity=epsilon, permeability=mu) >>> m1 MediumPP(Str('m1'), epsilon, mu) >>> m2 = Medium('m2') >>> m2 MediumPP(Str('m2'), 625000ampere*2second*4/(22468879468420441pikilogrammeter*3), pikilogrammeter/(2500000ampere*2second*2)) """ def __new__(cls, name, permittivity, permeability): obj = super(Medium, cls).__new__(cls, name, permittivity, permeability) return obj @property def intrinsic_impedance(self): """ Returns intrinsic impedance of the medium. Explanation =========== The intrinsic impedance of a medium is the ratio of the transverse components of the electric and magnetic fields of the electromagnetic wave travelling in the medium. In a region with no electrical conductivity it simplifies to the square root of ratio of magnetic permeability to electric permittivity. Examples ======== >>> from sympy.physics.optics import Medium >>> m = Medium('m') >>> m.intrinsic_impedance 149896229pikilogrammeter*2/(1250000ampere*2second*3) """ return sqrt(self.permeability / self.permittivity) @property def permittivity(self): """ Returns electric permittivity of the medium. Examples ======== >>> from sympy.physics.optics import Medium >>> m = Medium('m') >>> m.permittivity 625000ampere*2second*4/(22468879468420441pikilogrammeter*3) """ return self.args[1] @property def permeability(self): """ Returns magnetic permeability of the medium. Examples ======== >>> from sympy.physics.optics import Medium >>> m = Medium('m') >>> m.permeability pikilogrammeter/(2500000ampere*2second*2) """ return self.args[2] @property def n(self): return csqrt(self.permittivity*self.permeability)
acdb2fa9b1fd19623704c54d204537d5fdcbcb00c72a4530d6afc4e3bd37f3a8	from sympy.core.numbers import I from sympy.core.symbol import Dummy from sympy.functions.elementary.complexes import (Abs, arg) from sympy.functions.elementary.exponential import log from sympy.abc import s, p, a from sympy.external import import_module from sympy.physics.control.control_plots import \ (pole_zero_numerical_data, pole_zero_plot, step_response_numerical_data, step_response_plot, impulse_response_numerical_data, impulse_response_plot, ramp_response_numerical_data, ramp_response_plot, bode_magnitude_numerical_data, bode_phase_numerical_data, bode_plot) from sympy.physics.control.lti import (TransferFunction, Series, Parallel, TransferFunctionMatrix) from sympy.testing.pytest import raises, skip matplotlib = import_module( 'matplotlib', import_kwargs={'fromlist': ['pyplot']}, catch=(RuntimeError,)) numpy = import_module('numpy') tf1 = TransferFunction(1, p*2 + 0.5p + 2, p) tf2 = TransferFunction(p, 6p2 + 3p + 1, p) tf3 = TransferFunction(p, p3 - 1, p) tf4 = TransferFunction(10, p3, p) tf5 = TransferFunction(5, s*2 + 2s + 10, s) tf6 = TransferFunction(1, 1, s) tf7 = TransferFunction(4s3 + 9s2 + 0.1s + 11, 8s6 + 9s4 + 11, s) tf8 = TransferFunction(5, s2 + (2+I)s + 10, s) ser1 = Series(tf4, TransferFunction(1, p - 5, p)) ser2 = Series(tf3, TransferFunction(p, p + 2, p)) par1 = Parallel(tf1, tf2) par2 = Parallel(tf1, tf2, tf3) def _to_tuple(a, b): return tuple(a), tuple(b) def _trim_tuple(a, b): a, b = _to_tuple(a, b) return tuple(a[0: 2] + a[len(a)//2 : len(a)//2 + 1] + a[-2:]), \ tuple(b[0: 2] + b[len(b)//2 : len(b)//2 + 1] + b[-2:]) def y_coordinate_equality(plot_data_func, evalf_func, system): """Checks whether the y-coordinate value of the plotted data point is equal to the value of the function at a particular x.""" x, y = plot_data_func(system) x, y = _trim_tuple(x, y) y_exp = tuple(evalf_func(system, x_i) for x_i in x) return all(Abs(y_exp_i - y_i) < 1e-8 for y_exp_i, y_i in zip(y_exp, y)) def test_errors(): if not matplotlib: skip("Matplotlib not the default backend") # Invalid `system` check tfm = TransferFunctionMatrix([[tf6, tf5], [tf5, tf6]]) expr = 1/(s2 - 1) raises(NotImplementedError, lambda: pole_zero_plot(tfm)) raises(NotImplementedError, lambda: pole_zero_numerical_data(expr)) raises(NotImplementedError, lambda: impulse_response_plot(expr)) raises(NotImplementedError, lambda: impulse_response_numerical_data(tfm)) raises(NotImplementedError, lambda: step_response_plot(tfm)) raises(NotImplementedError, lambda: step_response_numerical_data(expr)) raises(NotImplementedError, lambda: ramp_response_plot(expr)) raises(NotImplementedError, lambda: ramp_response_numerical_data(tfm)) raises(NotImplementedError, lambda: bode_plot(tfm)) # More than 1 variables tf_a = TransferFunction(a, s + 1, s) raises(ValueError, lambda: pole_zero_plot(tf_a)) raises(ValueError, lambda: pole_zero_numerical_data(tf_a)) raises(ValueError, lambda: impulse_response_plot(tf_a)) raises(ValueError, lambda: impulse_response_numerical_data(tf_a)) raises(ValueError, lambda: step_response_plot(tf_a)) raises(ValueError, lambda: step_response_numerical_data(tf_a)) raises(ValueError, lambda: ramp_response_plot(tf_a)) raises(ValueError, lambda: ramp_response_numerical_data(tf_a)) raises(ValueError, lambda: bode_plot(tf_a)) # lower_limit > 0 for response plots raises(ValueError, lambda: impulse_response_plot(tf1, lower_limit=-1)) raises(ValueError, lambda: step_response_plot(tf1, lower_limit=-0.1)) raises(ValueError, lambda: ramp_response_plot(tf1, lower_limit=-4/3)) # slope in ramp_response_plot() is negative raises(ValueError, lambda: ramp_response_plot(tf1, slope=-0.1)) def test_pole_zero(): if not matplotlib: skip("Matplotlib not the default backend") assert _to_tuple(pole_zero_numerical_data(tf1)) == \ ((), ((-0.24999999999999994+1.3919410907075054j), (-0.24999999999999994-1.3919410907075054j))) assert _to_tuple(pole_zero_numerical_data(tf2)) == \ ((0.0,), ((-0.25+0.3227486121839514j), (-0.25-0.3227486121839514j))) assert _to_tuple(pole_zero_numerical_data(tf3)) == \ ((0.0,), ((-0.5000000000000004+0.8660254037844395j), (-0.5000000000000004-0.8660254037844395j), (0.9999999999999998+0j))) assert _to_tuple(pole_zero_numerical_data(tf7)) == \ (((-0.6722222222222222+0.8776898690157247j), (-0.6722222222222222-0.8776898690157247j)), ((2.220446049250313e-16+1.2797182176061541j), (2.220446049250313e-16-1.2797182176061541j), (-0.7657146670186428+0.5744385024099056j), (-0.7657146670186428-0.5744385024099056j), (0.7657146670186427+0.5744385024099052j), (0.7657146670186427-0.5744385024099052j))) assert _to_tuple(pole_zero_numerical_data(ser1)) == \ ((), (5.0, 0.0, 0.0, 0.0)) assert _to_tuple(pole_zero_numerical_data(par1)) == \ ((-5.645751311064592, -0.5000000000000008, -0.3542486889354093), ((-0.24999999999999986+1.3919410907075052j), (-0.24999999999999986-1.3919410907075052j), (-0.2499999999999998+0.32274861218395134j), (-0.2499999999999998-0.32274861218395134j))) assert _to_tuple(pole_zero_numerical_data(tf8)) == \ ((), ((-1.1641600331447917-3.545808351896439j), (-0.8358399668552097+2.5458083518964383j))) def test_bode(): if not matplotlib: skip("Matplotlib not the default backend") def bode_phase_evalf(system, point): expr = system.to_expr() _w = Dummy("w", real=True) w_expr = expr.subs({system.var: I_w}) return arg(w_expr).subs({_w: point}).evalf() def bode_mag_evalf(system, point): expr = system.to_expr() _w = Dummy("w", real=True) w_expr = expr.subs({system.var: I_w}) return 20log(Abs(w_expr), 10).subs({_w: point}).evalf() def test_bode_data(sys): return y_coordinate_equality(bode_magnitude_numerical_data, bode_mag_evalf, sys) \ and y_coordinate_equality(bode_phase_numerical_data, bode_phase_evalf, sys) assert test_bode_data(tf1) assert test_bode_data(tf2) assert test_bode_data(tf3) assert test_bode_data(tf4) assert test_bode_data(tf5) def check_point_accuracy(a, b): return all(Abs(a_i - b_i) < 1e-12 for \ a_i, b_i in zip(a, b)) def test_impulse_response(): if not matplotlib: skip("Matplotlib not the default backend") def impulse_res_tester(sys, expected_value): x, y = _to_tuple(impulse_response_numerical_data(sys, adaptive=False, nb_of_points=10)) x_check = check_point_accuracy(x, expected_value[0]) y_check = check_point_accuracy(y, expected_value[1]) return x_check and y_check exp1 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 0.544019738507865, 0.01993849743234938, -0.31140243360893216, -0.022852779906491996, 0.1778306498155759, 0.01962941084328499, -0.1013115194573652, -0.014975541213105696, 0.0575789724730714)) exp2 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.1666666675, 0.08389223412935855, 0.02338051973475047, -0.014966807776379383, -0.034645954223054234, -0.040560075735512804, -0.037658628907103885, -0.030149507719590022, -0.021162090730736834, -0.012721292737437523)) exp3 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (4.369893391586999e-09, 1.1750333000630964, 3.2922404058312473, 9.432290008148343, 28.37098083007151, 86.18577464367974, 261.90356653762115, 795.6538758627842, 2416.9920942096983, 7342.159505206647)) exp4 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 6.17283950617284, 24.69135802469136, 55.555555555555564, 98.76543209876544, 154.320987654321, 222.22222222222226, 302.46913580246917, 395.0617283950618, 500.0)) exp5 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, -0.10455606138085417, 0.06757671513476461, -0.03234567568833768, 0.013582514927757873, -0.005273419510705473, 0.0019364083003354075, -0.000680070134067832, 0.00022969845960406913, -7.476094359583917e-05)) exp6 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (-6.016699583000218e-09, 0.35039802056107394, 3.3728423827689884, 12.119846079276684, 25.86101014293389, 29.352480635282088, -30.49475907497664, -273.8717189554019, -863.2381702029659, -1747.0262164682233)) exp7 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 18.934638095560974, 5346.93244680907, 1384609.8718249386, 358161126.65801865, 92645770015.70108, 23964739753087.42, 6198974342083139.0, 1.603492601616059e+18, 4.147764422869658e+20)) assert impulse_res_tester(tf1, exp1) assert impulse_res_tester(tf2, exp2) assert impulse_res_tester(tf3, exp3) assert impulse_res_tester(tf4, exp4) assert impulse_res_tester(tf5, exp5) assert impulse_res_tester(tf7, exp6) assert impulse_res_tester(ser1, exp7) def test_step_response(): if not matplotlib: skip("Matplotlib not the default backend") def step_res_tester(sys, expected_value): x, y = _to_tuple(step_response_numerical_data(sys, adaptive=False, nb_of_points=10)) x_check = check_point_accuracy(x, expected_value[0]) y_check = check_point_accuracy(y, expected_value[1]) return x_check and y_check exp1 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (-1.9193285738516863e-08, 0.42283495488246126, 0.7840485977945262, 0.5546841805655717, 0.33903033806932087, 0.4627251747410237, 0.5909907598988051, 0.5247213989553071, 0.4486997874319281, 0.4839358435839171)) exp2 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 0.13728409095645816, 0.19474559355325086, 0.1974909129243011, 0.16841657696573073, 0.12559777736159378, 0.08153828016664713, 0.04360471317348958, 0.015072994568868221, -0.003636420058445484)) exp3 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 0.6314542141914303, 2.9356520038101035, 9.37731009663807, 28.452300356688376, 86.25721933273988, 261.9236645044672, 795.6435410577224, 2416.9786984578764, 7342.154119725917)) exp4 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 2.286236899862826, 18.28989519890261, 61.72839629629631, 146.31916159122088, 285.7796124828532, 493.8271703703705, 784.1792566529494, 1170.553292729767, 1666.6667)) exp5 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (-3.999999997894577e-09, 0.6720357068882895, 0.4429938256137113, 0.5182010838004518, 0.4944139147159695, 0.5016379853883338, 0.4995466896527733, 0.5001154784851325, 0.49997448824584123, 0.5000039745919259)) exp6 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (-1.5433688493882158e-09, 0.3428705539937336, 1.1253619102202777, 3.1849962651016517, 9.47532757182671, 28.727231099148135, 87.29426924860557, 265.2138681048606, 805.6636260007757, 2447.387582370878)) assert step_res_tester(tf1, exp1) assert step_res_tester(tf2, exp2) assert step_res_tester(tf3, exp3) assert step_res_tester(tf4, exp4) assert step_res_tester(tf5, exp5) assert step_res_tester(ser2, exp6) def test_ramp_response(): if not matplotlib: skip("Matplotlib not the default backend") def ramp_res_tester(sys, num_points, expected_value, slope=1): x, y = _to_tuple(ramp_response_numerical_data(sys, slope=slope, adaptive=False, nb_of_points=num_points)) x_check = check_point_accuracy(x, expected_value[0]) y_check = check_point_accuracy(y, expected_value[1]) return x_check and y_check exp1 = ((0.0, 2.0, 4.0, 6.0, 8.0, 10.0), (0.0, 0.7324667795033895, 1.9909720978650398, 2.7956587704217783, 3.9224897567931514, 4.85022655284895)) exp2 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (2.4360213402019326e-08, 0.10175320182493253, 0.33057612497658406, 0.5967937263298935, 0.8431511866718248, 1.0398805391471613, 1.1776043125035738, 1.2600994825747305, 1.2981042689274653, 1.304684417610106)) exp3 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (-3.9329040468771836e-08, 0.34686634635794555, 2.9998828170537903, 12.33303690737476, 40.993913948137795, 127.84145222317912, 391.41713691996, 1192.0006858708389, 3623.9808672503405, 11011.728034546572)) exp4 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 1.9051973784484078, 30.483158055174524, 154.32098765432104, 487.7305288827924, 1190.7483615302544, 2469.1358024691367, 4574.3789056546275, 7803.688462124678, 12500.0)) exp5 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 3.8844361856975635, 9.141792069209865, 14.096349157657231, 19.09783068994694, 24.10179770390321, 29.09907319114121, 34.10040420185154, 39.09983919254265, 44.10006013058409)) exp6 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0)) assert ramp_res_tester(tf1, 6, exp1) assert ramp_res_tester(tf2, 10, exp2, 1.2) assert ramp_res_tester(tf3, 10, exp3, 1.5) assert ramp_res_tester(tf4, 10, exp4, 3) assert ramp_res_tester(tf5, 10, exp5, 9) assert ramp_res_tester(tf6, 10, exp6)
359967e816f06ad1d20b6d3c480e735e1e720a3b7e100d4de4f83a72ff389ab6	from sympy.core.backend import (cos, expand, Matrix, sin, symbols, tan, sqrt, S, zeros) from sympy.simplify.simplify import simplify from sympy.physics.mechanics import (dynamicsymbols, ReferenceFrame, Point, RigidBody, KanesMethod, inertia, Particle, dot) from sympy.testing.pytest import raises def test_one_dof(): # This is for a 1 dof spring-mass-damper case. # It is described in more detail in the KanesMethod docstring. q, u = dynamicsymbols('q u') qd, ud = dynamicsymbols('q u', 1) m, c, k = symbols('m c k') N = ReferenceFrame('N') P = Point('P') P.set_vel(N, u * N.x) kd = [qd - u] FL = [(P, (-k * q - c * u) * N.x)] pa = Particle('pa', P, m) BL = [pa] KM = KanesMethod(N, [q], [u], kd) KM.kanes_equations(BL, FL) assert KM.bodies == BL assert KM.loads == FL MM = KM.mass_matrix forcing = KM.forcing rhs = MM.inv() * forcing assert expand(rhs[0]) == expand(-(q * k + u * c) / m) assert simplify(KM.rhs() - KM.mass_matrix_full.LUsolve(KM.forcing_full)) == zeros(2, 1) assert (KM.linearize(A_and_B=True, )[0] == Matrix([[0, 1], [-k/m, -c/m]])) def test_two_dof(): # This is for a 2 d.o.f., 2 particle spring-mass-damper. # The first coordinate is the displacement of the first particle, and the # second is the relative displacement between the first and second # particles. Speeds are defined as the time derivatives of the particles. q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2') q1d, q2d, u1d, u2d = dynamicsymbols('q1 q2 u1 u2', 1) m, c1, c2, k1, k2 = symbols('m c1 c2 k1 k2') N = ReferenceFrame('N') P1 = Point('P1') P2 = Point('P2') P1.set_vel(N, u1 * N.x) P2.set_vel(N, (u1 + u2) * N.x) kd = [q1d - u1, q2d - u2] # Now we create the list of forces, then assign properties to each # particle, then create a list of all particles. FL = [(P1, (-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2) * N.x), (P2, (-k2 * q2 - c2 * u2) * N.x)] pa1 = Particle('pa1', P1, m) pa2 = Particle('pa2', P2, m) BL = [pa1, pa2] # Finally we create the KanesMethod object, specify the inertial frame, # pass relevant information, and form Fr & Fr. Then we calculate the mass # matrix and forcing terms, and finally solve for the udots. KM = KanesMethod(N, q_ind=[q1, q2], u_ind=[u1, u2], kd_eqs=kd) KM.kanes_equations(BL, FL) MM = KM.mass_matrix forcing = KM.forcing rhs = MM.inv() forcing assert expand(rhs[0]) == expand((-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2)/m) assert expand(rhs[1]) == expand((k1 * q1 + c1 * u1 - 2 * k2 * q2 - 2 * c2 * u2) / m) assert simplify(KM.rhs() - KM.mass_matrix_full.LUsolve(KM.forcing_full)) == zeros(4, 1) # Make sure an error is raised if nonlinear kinematic differential # equations are supplied. kd = [q1d - u1*2, sin(q2d) - cos(u2)] raises(ValueError, lambda: KanesMethod(N, q_ind=[q1, q2], u_ind=[u1, u2], kd_eqs=kd)) def test_pend(): q, u = dynamicsymbols('q u') qd, ud = dynamicsymbols('q u', 1) m, l, g = symbols('m l g') N = ReferenceFrame('N') P = Point('P') P.set_vel(N, -l u * sin(q) * N.x + l * u * cos(q) * N.y) kd = [qd - u] FL = [(P, m * g * N.x)] pa = Particle('pa', P, m) BL = [pa] KM = KanesMethod(N, [q], [u], kd) KM.kanes_equations(BL, FL) MM = KM.mass_matrix forcing = KM.forcing rhs = MM.inv() * forcing rhs.simplify() assert expand(rhs[0]) == expand(-g / l * sin(q)) assert simplify(KM.rhs() - KM.mass_matrix_full.LUsolve(KM.forcing_full)) == zeros(2, 1) def test_rolling_disc(): # Rolling Disc Example # Here the rolling disc is formed from the contact point up, removing the # need to introduce generalized speeds. Only 3 configuration and three # speed variables are need to describe this system, along with the disc's # mass and radius, and the local gravity (note that mass will drop out). q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1 q2 q3 u1 u2 u3') q1d, q2d, q3d, u1d, u2d, u3d = dynamicsymbols('q1 q2 q3 u1 u2 u3', 1) r, m, g = symbols('r m g') # The kinematics are formed by a series of simple rotations. Each simple # rotation creates a new frame, and the next rotation is defined by the new # frame's basis vectors. This example uses a 3-1-2 series of rotations, or # Z, X, Y series of rotations. Angular velocity for this is defined using # the second frame's basis (the lean frame). N = ReferenceFrame('N') Y = N.orientnew('Y', 'Axis', [q1, N.z]) L = Y.orientnew('L', 'Axis', [q2, Y.x]) R = L.orientnew('R', 'Axis', [q3, L.y]) w_R_N_qd = R.ang_vel_in(N) R.set_ang_vel(N, u1 * L.x + u2 * L.y + u3 * L.z) # This is the translational kinematics. We create a point with no velocity # in N; this is the contact point between the disc and ground. Next we form # the position vector from the contact point to the disc's center of mass. # Finally we form the velocity and acceleration of the disc. C = Point('C') C.set_vel(N, 0) Dmc = C.locatenew('Dmc', r * L.z) Dmc.v2pt_theory(C, N, R) # This is a simple way to form the inertia dyadic. I = inertia(L, m / 4 * r*2, m / 2 r*2, m / 4 r*2) # Kinematic differential equations; how the generalized coordinate time # derivatives relate to generalized speeds. kd = [dot(R.ang_vel_in(N) - w_R_N_qd, uv) for uv in L] # Creation of the force list; it is the gravitational force at the mass # center of the disc. Then we create the disc by assigning a Point to the # center of mass attribute, a ReferenceFrame to the frame attribute, and mass # and inertia. Then we form the body list. ForceList = [(Dmc, - m g * Y.z)] BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc)) BodyList = [BodyD] # Finally we form the equations of motion, using the same steps we did # before. Specify inertial frame, supply generalized speeds, supply # kinematic differential equation dictionary, compute Fr from the force # list and Fr* from the body list, compute the mass matrix and forcing # terms, then solve for the u dots (time derivatives of the generalized # speeds). KM = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3], kd_eqs=kd) KM.kanes_equations(BodyList, ForceList) MM = KM.mass_matrix forcing = KM.forcing rhs = MM.inv() * forcing kdd = KM.kindiffdict() rhs = rhs.subs(kdd) rhs.simplify() assert rhs.expand() == Matrix([(6u2u3r - u32rtan(q2) + 4gsin(q2))/(5r), -2u1u3/3, u1(-2u2 + u3tan(q2))]).expand() assert simplify(KM.rhs() - KM.mass_matrix_full.LUsolve(KM.forcing_full)) == zeros(6, 1) # This code tests our output vs. benchmark values. When r=g=m=1, the # critical speed (where all eigenvalues of the linearized equations are 0) # is 1 / sqrt(3) for the upright case. A = KM.linearize(A_and_B=True)[0] A_upright = A.subs({r: 1, g: 1, m: 1}).subs({q1: 0, q2: 0, q3: 0, u1: 0, u3: 0}) import sympy assert sympy.sympify(A_upright.subs({u2: 1 / sqrt(3)})).eigenvals() == {S.Zero: 6} def test_aux(): # Same as above, except we have 2 auxiliary speeds for the ground contact # point, which is known to be zero. In one case, we go through then # substitute the aux. speeds in at the end (they are zero, as well as their # derivative), in the other case, we use the built-in auxiliary speed part # of KanesMethod. The equations from each should be the same. q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1 q2 q3 u1 u2 u3') q1d, q2d, q3d, u1d, u2d, u3d = dynamicsymbols('q1 q2 q3 u1 u2 u3', 1) u4, u5, f1, f2 = dynamicsymbols('u4, u5, f1, f2') u4d, u5d = dynamicsymbols('u4, u5', 1) r, m, g = symbols('r m g') N = ReferenceFrame('N') Y = N.orientnew('Y', 'Axis', [q1, N.z]) L = Y.orientnew('L', 'Axis', [q2, Y.x]) R = L.orientnew('R', 'Axis', [q3, L.y]) w_R_N_qd = R.ang_vel_in(N) R.set_ang_vel(N, u1 L.x + u2 * L.y + u3 * L.z) C = Point('C') C.set_vel(N, u4 * L.x + u5 * (Y.z ^ L.x)) Dmc = C.locatenew('Dmc', r * L.z) Dmc.v2pt_theory(C, N, R) Dmc.a2pt_theory(C, N, R) I = inertia(L, m / 4 * r*2, m / 2 r*2, m / 4 r*2) kd = [dot(R.ang_vel_in(N) - w_R_N_qd, uv) for uv in L] ForceList = [(Dmc, - m g * Y.z), (C, f1 * L.x + f2 * (Y.z ^ L.x))] BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc)) BodyList = [BodyD] KM = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3, u4, u5], kd_eqs=kd) (fr, frstar) = KM.kanes_equations(BodyList, ForceList) fr = fr.subs({u4d: 0, u5d: 0}).subs({u4: 0, u5: 0}) frstar = frstar.subs({u4d: 0, u5d: 0}).subs({u4: 0, u5: 0}) KM2 = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3], kd_eqs=kd, u_auxiliary=[u4, u5]) (fr2, frstar2) = KM2.kanes_equations(BodyList, ForceList) fr2 = fr2.subs({u4d: 0, u5d: 0}).subs({u4: 0, u5: 0}) frstar2 = frstar2.subs({u4d: 0, u5d: 0}).subs({u4: 0, u5: 0}) frstar.simplify() frstar2.simplify() assert (fr - fr2).expand() == Matrix([0, 0, 0, 0, 0]) assert (frstar - frstar2).expand() == Matrix([0, 0, 0, 0, 0]) def test_parallel_axis(): # This is for a 2 dof inverted pendulum on a cart. # This tests the parallel axis code in KanesMethod. The inertia of the # pendulum is defined about the hinge, not about the center of mass. # Defining the constants and knowns of the system gravity = symbols('g') k, ls = symbols('k ls') a, mA, mC = symbols('a mA mC') F = dynamicsymbols('F') Ix, Iy, Iz = symbols('Ix Iy Iz') # Declaring the Generalized coordinates and speeds q1, q2 = dynamicsymbols('q1 q2') q1d, q2d = dynamicsymbols('q1 q2', 1) u1, u2 = dynamicsymbols('u1 u2') u1d, u2d = dynamicsymbols('u1 u2', 1) # Creating reference frames N = ReferenceFrame('N') A = ReferenceFrame('A') A.orient(N, 'Axis', [-q2, N.z]) A.set_ang_vel(N, -u2 * N.z) # Origin of Newtonian reference frame O = Point('O') # Creating and Locating the positions of the cart, C, and the # center of mass of the pendulum, A C = O.locatenew('C', q1 * N.x) Ao = C.locatenew('Ao', a * A.y) # Defining velocities of the points O.set_vel(N, 0) C.set_vel(N, u1 * N.x) Ao.v2pt_theory(C, N, A) Cart = Particle('Cart', C, mC) Pendulum = RigidBody('Pendulum', Ao, A, mA, (inertia(A, Ix, Iy, Iz), C)) # kinematical differential equations kindiffs = [q1d - u1, q2d - u2] bodyList = [Cart, Pendulum] forceList = [(Ao, -N.y * gravity * mA), (C, -N.y * gravity * mC), (C, -N.x * k * (q1 - ls)), (C, N.x * F)] km = KanesMethod(N, [q1, q2], [u1, u2], kindiffs) (fr, frstar) = km.kanes_equations(bodyList, forceList) mm = km.mass_matrix_full assert mm[3, 3] == Iz def test_input_format(): # 1 dof problem from test_one_dof q, u = dynamicsymbols('q u') qd, ud = dynamicsymbols('q u', 1) m, c, k = symbols('m c k') N = ReferenceFrame('N') P = Point('P') P.set_vel(N, u * N.x) kd = [qd - u] FL = [(P, (-k * q - c * u) * N.x)] pa = Particle('pa', P, m) BL = [pa] KM = KanesMethod(N, [q], [u], kd) # test for input format kane.kanes_equations((body1, body2, particle1)) assert KM.kanes_equations(BL)[0] == Matrix([0]) # test for input format kane.kanes_equations(bodies=(body1, body 2), loads=(load1,load2)) assert KM.kanes_equations(bodies=BL, loads=None)[0] == Matrix([0]) # test for input format kane.kanes_equations(bodies=(body1, body 2), loads=None) assert KM.kanes_equations(BL, loads=None)[0] == Matrix([0]) # test for input format kane.kanes_equations(bodies=(body1, body 2)) assert KM.kanes_equations(BL)[0] == Matrix([0]) # test for input format kane.kanes_equations(bodies=(body1, body2), loads=[]) assert KM.kanes_equations(BL, [])[0] == Matrix([0]) # test for error raised when a wrong force list (in this case a string) is provided raises(ValueError, lambda: KM._form_fr('bad input')) # 1 dof problem from test_one_dof with FL & BL in instance KM = KanesMethod(N, [q], [u], kd, bodies=BL, forcelist=FL) assert KM.kanes_equations()[0] == Matrix([-cu - kq]) # 2 dof problem from test_two_dof q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2') q1d, q2d, u1d, u2d = dynamicsymbols('q1 q2 u1 u2', 1) m, c1, c2, k1, k2 = symbols('m c1 c2 k1 k2') N = ReferenceFrame('N') P1 = Point('P1') P2 = Point('P2') P1.set_vel(N, u1 * N.x) P2.set_vel(N, (u1 + u2) * N.x) kd = [q1d - u1, q2d - u2] FL = ((P1, (-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2) * N.x), (P2, (-k2 * q2 - c2 * u2) * N.x)) pa1 = Particle('pa1', P1, m) pa2 = Particle('pa2', P2, m) BL = (pa1, pa2) KM = KanesMethod(N, q_ind=[q1, q2], u_ind=[u1, u2], kd_eqs=kd) # test for input format # kane.kanes_equations((body1, body2), (load1, load2)) KM.kanes_equations(BL, FL) MM = KM.mass_matrix forcing = KM.forcing rhs = MM.inv() * forcing assert expand(rhs[0]) == expand((-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2)/m) assert expand(rhs[1]) == expand((k1 * q1 + c1 * u1 - 2 * k2 * q2 - 2 * c2 * u2) / m)
cb5a5a7e59006feef6d9c3ed67b3e4bf605b1c522d9b6885435e0eaa70fd4d2b	from .unit_definitions import ( percent, percents, permille, rad, radian, radians, deg, degree, degrees, sr, steradian, steradians, mil, angular_mil, angular_mils, m, meter, meters, kg, kilogram, kilograms, s, second, seconds, A, ampere, amperes, K, kelvin, kelvins, mol, mole, moles, cd, candela, candelas, g, gram, grams, mg, milligram, milligrams, ug, microgram, micrograms, newton, newtons, N, joule, joules, J, watt, watts, W, pascal, pascals, Pa, pa, hertz, hz, Hz, coulomb, coulombs, C, volt, volts, v, V, ohm, ohms, siemens, S, mho, mhos, farad, farads, F, henry, henrys, H, tesla, teslas, T, weber, webers, Wb, wb, optical_power, dioptre, D, lux, lx, katal, kat, gray, Gy, becquerel, Bq, km, kilometer, kilometers, dm, decimeter, decimeters, cm, centimeter, centimeters, mm, millimeter, millimeters, um, micrometer, micrometers, micron, microns, nm, nanometer, nanometers, pm, picometer, picometers, ft, foot, feet, inch, inches, yd, yard, yards, mi, mile, miles, nmi, nautical_mile, nautical_miles, l, L, liter, liters, dl, dL, deciliter, deciliters, cl, cL, centiliter, centiliters, ml, mL, milliliter, milliliters, ms, millisecond, milliseconds, us, microsecond, microseconds, ns, nanosecond, nanoseconds, ps, picosecond, picoseconds, minute, minutes, h, hour, hours, day, days, anomalistic_year, anomalistic_years, sidereal_year, sidereal_years, tropical_year, tropical_years, common_year, common_years, julian_year, julian_years, draconic_year, draconic_years, gaussian_year, gaussian_years, full_moon_cycle, full_moon_cycles, year, years, G, gravitational_constant, c, speed_of_light, elementary_charge, hbar, planck, eV, electronvolt, electronvolts, avogadro_number, avogadro, avogadro_constant, boltzmann, boltzmann_constant, stefan, stefan_boltzmann_constant, R, molar_gas_constant, faraday_constant, josephson_constant, von_klitzing_constant, amu, amus, atomic_mass_unit, atomic_mass_constant, gee, gees, acceleration_due_to_gravity, u0, magnetic_constant, vacuum_permeability, e0, electric_constant, vacuum_permittivity, Z0, vacuum_impedance, coulomb_constant, coulombs_constant, electric_force_constant, atmosphere, atmospheres, atm, kPa, kilopascal, bar, bars, pound, pounds, psi, dHg0, mmHg, torr, mmu, mmus, milli_mass_unit, quart, quarts, ly, lightyear, lightyears, au, astronomical_unit, astronomical_units, planck_mass, planck_time, planck_temperature, planck_length, planck_charge, planck_area, planck_volume, planck_momentum, planck_energy, planck_force, planck_power, planck_density, planck_energy_density, planck_intensity, planck_angular_frequency, planck_pressure, planck_current, planck_voltage, planck_impedance, planck_acceleration, bit, bits, byte, kibibyte, kibibytes, mebibyte, mebibytes, gibibyte, gibibytes, tebibyte, tebibytes, pebibyte, pebibytes, exbibyte, exbibytes, curie, rutherford ) __all__ = [ 'percent', 'percents', 'permille', 'rad', 'radian', 'radians', 'deg', 'degree', 'degrees', 'sr', 'steradian', 'steradians', 'mil', 'angular_mil', 'angular_mils', 'm', 'meter', 'meters', 'kg', 'kilogram', 'kilograms', 's', 'second', 'seconds', 'A', 'ampere', 'amperes', 'K', 'kelvin', 'kelvins', 'mol', 'mole', 'moles', 'cd', 'candela', 'candelas', 'g', 'gram', 'grams', 'mg', 'milligram', 'milligrams', 'ug', 'microgram', 'micrograms', 'newton', 'newtons', 'N', 'joule', 'joules', 'J', 'watt', 'watts', 'W', 'pascal', 'pascals', 'Pa', 'pa', 'hertz', 'hz', 'Hz', 'coulomb', 'coulombs', 'C', 'volt', 'volts', 'v', 'V', 'ohm', 'ohms', 'siemens', 'S', 'mho', 'mhos', 'farad', 'farads', 'F', 'henry', 'henrys', 'H', 'tesla', 'teslas', 'T', 'weber', 'webers', 'Wb', 'wb', 'optical_power', 'dioptre', 'D', 'lux', 'lx', 'katal', 'kat', 'gray', 'Gy', 'becquerel', 'Bq', 'km', 'kilometer', 'kilometers', 'dm', 'decimeter', 'decimeters', 'cm', 'centimeter', 'centimeters', 'mm', 'millimeter', 'millimeters', 'um', 'micrometer', 'micrometers', 'micron', 'microns', 'nm', 'nanometer', 'nanometers', 'pm', 'picometer', 'picometers', 'ft', 'foot', 'feet', 'inch', 'inches', 'yd', 'yard', 'yards', 'mi', 'mile', 'miles', 'nmi', 'nautical_mile', 'nautical_miles', 'l', 'L', 'liter', 'liters', 'dl', 'dL', 'deciliter', 'deciliters', 'cl', 'cL', 'centiliter', 'centiliters', 'ml', 'mL', 'milliliter', 'milliliters', 'ms', 'millisecond', 'milliseconds', 'us', 'microsecond', 'microseconds', 'ns', 'nanosecond', 'nanoseconds', 'ps', 'picosecond', 'picoseconds', 'minute', 'minutes', 'h', 'hour', 'hours', 'day', 'days', 'anomalistic_year', 'anomalistic_years', 'sidereal_year', 'sidereal_years', 'tropical_year', 'tropical_years', 'common_year', 'common_years', 'julian_year', 'julian_years', 'draconic_year', 'draconic_years', 'gaussian_year', 'gaussian_years', 'full_moon_cycle', 'full_moon_cycles', 'year', 'years', 'G', 'gravitational_constant', 'c', 'speed_of_light', 'elementary_charge', 'hbar', 'planck', 'eV', 'electronvolt', 'electronvolts', 'avogadro_number', 'avogadro', 'avogadro_constant', 'boltzmann', 'boltzmann_constant', 'stefan', 'stefan_boltzmann_constant', 'R', 'molar_gas_constant', 'faraday_constant', 'josephson_constant', 'von_klitzing_constant', 'amu', 'amus', 'atomic_mass_unit', 'atomic_mass_constant', 'gee', 'gees', 'acceleration_due_to_gravity', 'u0', 'magnetic_constant', 'vacuum_permeability', 'e0', 'electric_constant', 'vacuum_permittivity', 'Z0', 'vacuum_impedance', 'coulomb_constant', 'coulombs_constant', 'electric_force_constant', 'atmosphere', 'atmospheres', 'atm', 'kPa', 'kilopascal', 'bar', 'bars', 'pound', 'pounds', 'psi', 'dHg0', 'mmHg', 'torr', 'mmu', 'mmus', 'milli_mass_unit', 'quart', 'quarts', 'ly', 'lightyear', 'lightyears', 'au', 'astronomical_unit', 'astronomical_units', 'planck_mass', 'planck_time', 'planck_temperature', 'planck_length', 'planck_charge', 'planck_area', 'planck_volume', 'planck_momentum', 'planck_energy', 'planck_force', 'planck_power', 'planck_density', 'planck_energy_density', 'planck_intensity', 'planck_angular_frequency', 'planck_pressure', 'planck_current', 'planck_voltage', 'planck_impedance', 'planck_acceleration', 'bit', 'bits', 'byte', 'kibibyte', 'kibibytes', 'mebibyte', 'mebibytes', 'gibibyte', 'gibibytes', 'tebibyte', 'tebibytes', 'pebibyte', 'pebibytes', 'exbibyte', 'exbibytes', 'curie', 'rutherford', ]
ffda17253c1649801fe8e5a37f6e6782ec2f01fa42018be96c723861aee4d120	from sympy.physics.units.definitions.dimension_definitions import current, temperature, amount_of_substance, \ luminous_intensity, angle, charge, voltage, impedance, conductance, capacitance, inductance, magnetic_density, \ magnetic_flux, information from sympy.core.numbers import (Rational, pi) from sympy.core.singleton import S as S_singleton from sympy.physics.units.prefixes import kilo, milli, micro, deci, centi, nano, pico, kibi, mebi, gibi, tebi, pebi, exbi from sympy.physics.units.quantities import Quantity One = S_singleton.One #### UNITS #### # Dimensionless: percent = percents = Quantity("percent", latex_repr=r"\%") percent.set_global_relative_scale_factor(Rational(1, 100), One) permille = Quantity("permille") permille.set_global_relative_scale_factor(Rational(1, 1000), One) # Angular units (dimensionless) rad = radian = radians = Quantity("radian", abbrev="rad") radian.set_global_dimension(angle) deg = degree = degrees = Quantity("degree", abbrev="deg", latex_repr=r"^\circ") degree.set_global_relative_scale_factor(pi/180, radian) sr = steradian = steradians = Quantity("steradian", abbrev="sr") mil = angular_mil = angular_mils = Quantity("angular_mil", abbrev="mil") # Base units: m = meter = meters = Quantity("meter", abbrev="m") # gram; used to define its prefixed units g = gram = grams = Quantity("gram", abbrev="g") # NOTE: the `kilogram` has scale factor 1000. In SI, kg is a base unit, but # nonetheless we are trying to be compatible with the `kilo` prefix. In a # similar manner, people using CGS or gaussian units could argue that the # `centimeter` rather than `meter` is the fundamental unit for length, but the # scale factor of `centimeter` will be kept as 1/100 to be compatible with the # `centi` prefix. The current state of the code assumes SI unit dimensions, in # the future this module will be modified in order to be unit system-neutral # (that is, support all kinds of unit systems). kg = kilogram = kilograms = Quantity("kilogram", abbrev="kg") kg.set_global_relative_scale_factor(kilo, gram) s = second = seconds = Quantity("second", abbrev="s") A = ampere = amperes = Quantity("ampere", abbrev='A') ampere.set_global_dimension(current) K = kelvin = kelvins = Quantity("kelvin", abbrev='K') kelvin.set_global_dimension(temperature) mol = mole = moles = Quantity("mole", abbrev="mol") mole.set_global_dimension(amount_of_substance) cd = candela = candelas = Quantity("candela", abbrev="cd") candela.set_global_dimension(luminous_intensity) mg = milligram = milligrams = Quantity("milligram", abbrev="mg") mg.set_global_relative_scale_factor(milli, gram) ug = microgram = micrograms = Quantity("microgram", abbrev="ug", latex_repr=r"\mu\text{g}") ug.set_global_relative_scale_factor(micro, gram) # derived units newton = newtons = N = Quantity("newton", abbrev="N") joule = joules = J = Quantity("joule", abbrev="J") watt = watts = W = Quantity("watt", abbrev="W") pascal = pascals = Pa = pa = Quantity("pascal", abbrev="Pa") hertz = hz = Hz = Quantity("hertz", abbrev="Hz") # CGS derived units: dyne = Quantity("dyne") dyne.set_global_relative_scale_factor(One/105, newton) erg = Quantity("erg") erg.set_global_relative_scale_factor(One/107, joule) # MKSA extension to MKS: derived units coulomb = coulombs = C = Quantity("coulomb", abbrev='C') coulomb.set_global_dimension(charge) volt = volts = v = V = Quantity("volt", abbrev='V') volt.set_global_dimension(voltage) ohm = ohms = Quantity("ohm", abbrev='ohm', latex_repr=r"\Omega") ohm.set_global_dimension(impedance) siemens = S = mho = mhos = Quantity("siemens", abbrev='S') siemens.set_global_dimension(conductance) farad = farads = F = Quantity("farad", abbrev='F') farad.set_global_dimension(capacitance) henry = henrys = H = Quantity("henry", abbrev='H') henry.set_global_dimension(inductance) tesla = teslas = T = Quantity("tesla", abbrev='T') tesla.set_global_dimension(magnetic_density) weber = webers = Wb = wb = Quantity("weber", abbrev='Wb') weber.set_global_dimension(magnetic_flux) # CGS units for electromagnetic quantities: statampere = Quantity("statampere") statcoulomb = statC = franklin = Quantity("statcoulomb", abbrev="statC") statvolt = Quantity("statvolt") gauss = Quantity("gauss") maxwell = Quantity("maxwell") debye = Quantity("debye") oersted = Quantity("oersted") # Other derived units: optical_power = dioptre = diopter = D = Quantity("dioptre") lux = lx = Quantity("lux", abbrev="lx") # katal is the SI unit of catalytic activity katal = kat = Quantity("katal", abbrev="kat") # gray is the SI unit of absorbed dose gray = Gy = Quantity("gray") # becquerel is the SI unit of radioactivity becquerel = Bq = Quantity("becquerel", abbrev="Bq") # Common length units km = kilometer = kilometers = Quantity("kilometer", abbrev="km") km.set_global_relative_scale_factor(kilo, meter) dm = decimeter = decimeters = Quantity("decimeter", abbrev="dm") dm.set_global_relative_scale_factor(deci, meter) cm = centimeter = centimeters = Quantity("centimeter", abbrev="cm") cm.set_global_relative_scale_factor(centi, meter) mm = millimeter = millimeters = Quantity("millimeter", abbrev="mm") mm.set_global_relative_scale_factor(milli, meter) um = micrometer = micrometers = micron = microns = \ Quantity("micrometer", abbrev="um", latex_repr=r'\mu\text{m}') um.set_global_relative_scale_factor(micro, meter) nm = nanometer = nanometers = Quantity("nanometer", abbrev="nm") nm.set_global_relative_scale_factor(nano, meter) pm = picometer = picometers = Quantity("picometer", abbrev="pm") pm.set_global_relative_scale_factor(pico, meter) ft = foot = feet = Quantity("foot", abbrev="ft") ft.set_global_relative_scale_factor(Rational(3048, 10000), meter) inch = inches = Quantity("inch") inch.set_global_relative_scale_factor(Rational(1, 12), foot) yd = yard = yards = Quantity("yard", abbrev="yd") yd.set_global_relative_scale_factor(3, feet) mi = mile = miles = Quantity("mile") mi.set_global_relative_scale_factor(5280, feet) nmi = nautical_mile = nautical_miles = Quantity("nautical_mile") nmi.set_global_relative_scale_factor(6076, feet) # Common volume and area units l = L = liter = liters = Quantity("liter") dl = dL = deciliter = deciliters = Quantity("deciliter") dl.set_global_relative_scale_factor(Rational(1, 10), liter) cl = cL = centiliter = centiliters = Quantity("centiliter") cl.set_global_relative_scale_factor(Rational(1, 100), liter) ml = mL = milliliter = milliliters = Quantity("milliliter") ml.set_global_relative_scale_factor(Rational(1, 1000), liter) # Common time units ms = millisecond = milliseconds = Quantity("millisecond", abbrev="ms") millisecond.set_global_relative_scale_factor(milli, second) us = microsecond = microseconds = Quantity("microsecond", abbrev="us", latex_repr=r'\mu\text{s}') microsecond.set_global_relative_scale_factor(micro, second) ns = nanosecond = nanoseconds = Quantity("nanosecond", abbrev="ns") nanosecond.set_global_relative_scale_factor(nano, second) ps = picosecond = picoseconds = Quantity("picosecond", abbrev="ps") picosecond.set_global_relative_scale_factor(pico, second) minute = minutes = Quantity("minute") minute.set_global_relative_scale_factor(60, second) h = hour = hours = Quantity("hour") hour.set_global_relative_scale_factor(60, minute) day = days = Quantity("day") day.set_global_relative_scale_factor(24, hour) anomalistic_year = anomalistic_years = Quantity("anomalistic_year") anomalistic_year.set_global_relative_scale_factor(365.259636, day) sidereal_year = sidereal_years = Quantity("sidereal_year") sidereal_year.set_global_relative_scale_factor(31558149.540, seconds) tropical_year = tropical_years = Quantity("tropical_year") tropical_year.set_global_relative_scale_factor(365.24219, day) common_year = common_years = Quantity("common_year") common_year.set_global_relative_scale_factor(365, day) julian_year = julian_years = Quantity("julian_year") julian_year.set_global_relative_scale_factor((365 + One/4), day) draconic_year = draconic_years = Quantity("draconic_year") draconic_year.set_global_relative_scale_factor(346.62, day) gaussian_year = gaussian_years = Quantity("gaussian_year") gaussian_year.set_global_relative_scale_factor(365.2568983, day) full_moon_cycle = full_moon_cycles = Quantity("full_moon_cycle") full_moon_cycle.set_global_relative_scale_factor(411.78443029, day) year = years = tropical_year #### CONSTANTS #### # Newton constant G = gravitational_constant = Quantity("gravitational_constant", abbrev="G") # speed of light c = speed_of_light = Quantity("speed_of_light", abbrev="c") # elementary charge elementary_charge = Quantity("elementary_charge", abbrev="e") # Planck constant planck = Quantity("planck", abbrev="h") # Reduced Planck constant hbar = Quantity("hbar", abbrev="hbar") # Electronvolt eV = electronvolt = electronvolts = Quantity("electronvolt", abbrev="eV") # Avogadro number avogadro_number = Quantity("avogadro_number") # Avogadro constant avogadro = avogadro_constant = Quantity("avogadro_constant") # Boltzmann constant boltzmann = boltzmann_constant = Quantity("boltzmann_constant") # Stefan-Boltzmann constant stefan = stefan_boltzmann_constant = Quantity("stefan_boltzmann_constant") # Atomic mass amu = amus = atomic_mass_unit = atomic_mass_constant = Quantity("atomic_mass_constant") # Molar gas constant R = molar_gas_constant = Quantity("molar_gas_constant", abbrev="R") # Faraday constant faraday_constant = Quantity("faraday_constant") # Josephson constant josephson_constant = Quantity("josephson_constant", abbrev="K_j") # Von Klitzing constant von_klitzing_constant = Quantity("von_klitzing_constant", abbrev="R_k") # Acceleration due to gravity (on the Earth surface) gee = gees = acceleration_due_to_gravity = Quantity("acceleration_due_to_gravity", abbrev="g") # magnetic constant: u0 = magnetic_constant = vacuum_permeability = Quantity("magnetic_constant") # electric constat: e0 = electric_constant = vacuum_permittivity = Quantity("vacuum_permittivity") # vacuum impedance: Z0 = vacuum_impedance = Quantity("vacuum_impedance", abbrev='Z_0', latex_repr=r'Z_{0}') # Coulomb's constant: coulomb_constant = coulombs_constant = electric_force_constant = \ Quantity("coulomb_constant", abbrev="k_e") atmosphere = atmospheres = atm = Quantity("atmosphere", abbrev="atm") kPa = kilopascal = Quantity("kilopascal", abbrev="kPa") kilopascal.set_global_relative_scale_factor(kilo, Pa) bar = bars = Quantity("bar", abbrev="bar") pound = pounds = Quantity("pound") # exact psi = Quantity("psi") dHg0 = 13.5951 # approx value at 0 C mmHg = torr = Quantity("mmHg") atmosphere.set_global_relative_scale_factor(101325, pascal) bar.set_global_relative_scale_factor(100, kPa) pound.set_global_relative_scale_factor(Rational(45359237, 100000000), kg) mmu = mmus = milli_mass_unit = Quantity("milli_mass_unit") quart = quarts = Quantity("quart") # Other convenient units and magnitudes ly = lightyear = lightyears = Quantity("lightyear", abbrev="ly") au = astronomical_unit = astronomical_units = Quantity("astronomical_unit", abbrev="AU") # Fundamental Planck units: planck_mass = Quantity("planck_mass", abbrev="m_P", latex_repr=r'm_\text{P}') planck_time = Quantity("planck_time", abbrev="t_P", latex_repr=r't_\text{P}') planck_temperature = Quantity("planck_temperature", abbrev="T_P", latex_repr=r'T_\text{P}') planck_length = Quantity("planck_length", abbrev="l_P", latex_repr=r'l_\text{P}') planck_charge = Quantity("planck_charge", abbrev="q_P", latex_repr=r'q_\text{P}') # Derived Planck units: planck_area = Quantity("planck_area") planck_volume = Quantity("planck_volume") planck_momentum = Quantity("planck_momentum") planck_energy = Quantity("planck_energy", abbrev="E_P", latex_repr=r'E_\text{P}') planck_force = Quantity("planck_force", abbrev="F_P", latex_repr=r'F_\text{P}') planck_power = Quantity("planck_power", abbrev="P_P", latex_repr=r'P_\text{P}') planck_density = Quantity("planck_density", abbrev="rho_P", latex_repr=r'\rho_\text{P}') planck_energy_density = Quantity("planck_energy_density", abbrev="rho^E_P") planck_intensity = Quantity("planck_intensity", abbrev="I_P", latex_repr=r'I_\text{P}') planck_angular_frequency = Quantity("planck_angular_frequency", abbrev="omega_P", latex_repr=r'\omega_\text{P}') planck_pressure = Quantity("planck_pressure", abbrev="p_P", latex_repr=r'p_\text{P}') planck_current = Quantity("planck_current", abbrev="I_P", latex_repr=r'I_\text{P}') planck_voltage = Quantity("planck_voltage", abbrev="V_P", latex_repr=r'V_\text{P}') planck_impedance = Quantity("planck_impedance", abbrev="Z_P", latex_repr=r'Z_\text{P}') planck_acceleration = Quantity("planck_acceleration", abbrev="a_P", latex_repr=r'a_\text{P}') # Information theory units: bit = bits = Quantity("bit") bit.set_global_dimension(information) byte = bytes = Quantity("byte") kibibyte = kibibytes = Quantity("kibibyte") mebibyte = mebibytes = Quantity("mebibyte") gibibyte = gibibytes = Quantity("gibibyte") tebibyte = tebibytes = Quantity("tebibyte") pebibyte = pebibytes = Quantity("pebibyte") exbibyte = exbibytes = Quantity("exbibyte") byte.set_global_relative_scale_factor(8, bit) kibibyte.set_global_relative_scale_factor(kibi, byte) mebibyte.set_global_relative_scale_factor(mebi, byte) gibibyte.set_global_relative_scale_factor(gibi, byte) tebibyte.set_global_relative_scale_factor(tebi, byte) pebibyte.set_global_relative_scale_factor(pebi, byte) exbibyte.set_global_relative_scale_factor(exbi, byte) # Older units for radioactivity curie = Ci = Quantity("curie", abbrev="Ci") rutherford = Rd = Quantity("rutherford", abbrev="Rd")
fafbe108abc844d14f296ff3754f6211d2f5aa1a89c1fda78262368e7648d85a	import warnings from sympy.core.add import Add from sympy.core.function import (Function, diff) from sympy.core.numbers import (Number, Rational) from sympy.core.singleton import S from sympy.core.symbol import (Symbol, symbols) from sympy.functions.elementary.complexes import Abs from sympy.functions.elementary.exponential import (exp, log) from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import sin from sympy.integrals.integrals import integrate from sympy.physics.units import (amount_of_substance, convert_to, find_unit, volume, kilometer, joule) from sympy.physics.units.definitions import (amu, au, centimeter, coulomb, day, foot, grams, hour, inch, kg, km, m, meter, millimeter, minute, quart, s, second, speed_of_light, bit, byte, kibibyte, mebibyte, gibibyte, tebibyte, pebibyte, exbibyte, kilogram, gravitational_constant) from sympy.physics.units.definitions.dimension_definitions import ( Dimension, charge, length, time, temperature, pressure, energy ) from sympy.physics.units.prefixes import PREFIXES, kilo from sympy.physics.units.quantities import Quantity from sympy.physics.units.systems import SI from sympy.testing.pytest import XFAIL, raises, warns_deprecated_sympy k = PREFIXES["k"] def test_str_repr(): assert str(kg) == "kilogram" def test_eq(): # simple test assert 10m == 10m assert 10m != 10s def test_convert_to(): q = Quantity("q1") q.set_global_relative_scale_factor(S(5000), meter) assert q.convert_to(m) == 5000m assert speed_of_light.convert_to(m / s) == 299792458 m / s # TODO: eventually support this kind of conversion: # assert (2speed_of_light).convert_to(m / s) == 2 299792458 * m / s assert day.convert_to(s) == 86400s # Wrong dimension to convert: assert q.convert_to(s) == q assert speed_of_light.convert_to(m) == speed_of_light expr = joulesecond conv = convert_to(expr, joule) assert conv == joulesecond def test_Quantity_definition(): q = Quantity("s10", abbrev="sabbr") q.set_global_relative_scale_factor(10, second) u = Quantity("u", abbrev="dam") u.set_global_relative_scale_factor(10, meter) km = Quantity("km") km.set_global_relative_scale_factor(kilo, meter) v = Quantity("u") v.set_global_relative_scale_factor(5kilo, meter) assert q.scale_factor == 10 assert q.dimension == time assert q.abbrev == Symbol("sabbr") assert u.dimension == length assert u.scale_factor == 10 assert u.abbrev == Symbol("dam") assert km.scale_factor == 1000 assert km.func(km.args) == km assert km.func(km.args).args == km.args assert v.dimension == length assert v.scale_factor == 5000 with warns_deprecated_sympy(): Quantity('invalid', 'dimension', 1) with warns_deprecated_sympy(): Quantity('mismatch', dimension=length, scale_factor=kg) def test_abbrev(): u = Quantity("u") u.set_global_relative_scale_factor(S.One, meter) assert u.name == Symbol("u") assert u.abbrev == Symbol("u") u = Quantity("u", abbrev="om") u.set_global_relative_scale_factor(S(2), meter) assert u.name == Symbol("u") assert u.abbrev == Symbol("om") assert u.scale_factor == 2 assert isinstance(u.scale_factor, Number) u = Quantity("u", abbrev="ikm") u.set_global_relative_scale_factor(3kilo, meter) assert u.abbrev == Symbol("ikm") assert u.scale_factor == 3000 def test_print(): u = Quantity("unitname", abbrev="dam") assert repr(u) == "unitname" assert str(u) == "unitname" def test_Quantity_eq(): u = Quantity("u", abbrev="dam") v = Quantity("v1") assert u != v v = Quantity("v2", abbrev="ds") assert u != v v = Quantity("v3", abbrev="dm") assert u != v def test_add_sub(): u = Quantity("u") v = Quantity("v") w = Quantity("w") u.set_global_relative_scale_factor(S(10), meter) v.set_global_relative_scale_factor(S(5), meter) w.set_global_relative_scale_factor(S(2), second) assert isinstance(u + v, Add) assert (u + v.convert_to(u)) == (1 + S.Half)u # TODO: eventually add this: # assert (u + v).convert_to(u) == (1 + S.Half)u assert isinstance(u - v, Add) assert (u - v.convert_to(u)) == S.Halfu # TODO: eventually add this: # assert (u - v).convert_to(u) == S.Halfu def test_quantity_abs(): v_w1 = Quantity('v_w1') v_w2 = Quantity('v_w2') v_w3 = Quantity('v_w3') v_w1.set_global_relative_scale_factor(1, meter/second) v_w2.set_global_relative_scale_factor(1, meter/second) v_w3.set_global_relative_scale_factor(1, meter/second) expr = v_w3 - Abs(v_w1 - v_w2) assert SI.get_dimensional_expr(v_w1) == (length/time).name Dq = Dimension(SI.get_dimensional_expr(expr)) with warns_deprecated_sympy(): Dq1 = Dimension(Quantity.get_dimensional_expr(expr)) assert Dq == Dq1 assert SI.get_dimension_system().get_dimensional_dependencies(Dq) == { 'length': 1, 'time': -1, } assert meter == sqrt(meter2) def test_check_unit_consistency(): u = Quantity("u") v = Quantity("v") w = Quantity("w") u.set_global_relative_scale_factor(S(10), meter) v.set_global_relative_scale_factor(S(5), meter) w.set_global_relative_scale_factor(S(2), second) def check_unit_consistency(expr): SI._collect_factor_and_dimension(expr) raises(ValueError, lambda: check_unit_consistency(u + w)) raises(ValueError, lambda: check_unit_consistency(u - w)) raises(ValueError, lambda: check_unit_consistency(u + 1)) raises(ValueError, lambda: check_unit_consistency(u - 1)) raises(ValueError, lambda: check_unit_consistency(1 - exp(u / w))) def test_mul_div(): u = Quantity("u") v = Quantity("v") t = Quantity("t") ut = Quantity("ut") v2 = Quantity("v") u.set_global_relative_scale_factor(S(10), meter) v.set_global_relative_scale_factor(S(5), meter) t.set_global_relative_scale_factor(S(2), second) ut.set_global_relative_scale_factor(S(20), metersecond) v2.set_global_relative_scale_factor(S(5), meter/second) assert 1 / u == u*(-1) assert u / 1 == u v1 = u / t v2 = v # Pow only supports structural equality: assert v1 != v2 assert v1 == v2.convert_to(v1) # TODO: decide whether to allow such expression in the future # (requires somehow manipulating the core). # assert u / Quantity('l2', dimension=length, scale_factor=2) == 5 assert u 1 == u ut1 = u * t ut2 = ut # Mul only supports structural equality: assert ut1 != ut2 assert ut1 == ut2.convert_to(ut1) # Mul only supports structural equality: lp1 = Quantity("lp1") lp1.set_global_relative_scale_factor(S(2), 1/meter) assert u * lp1 != 20 assert u0 == 1 assert u1 == u # TODO: Pow only support structural equality: u2 = Quantity("u2") u3 = Quantity("u3") u2.set_global_relative_scale_factor(S(100), meter2) u3.set_global_relative_scale_factor(Rational(1, 10), 1/meter) assert u 2 != u2 assert u -1 != u3 assert u 2 == u2.convert_to(u) assert u ** -1 == u3.convert_to(u) def test_units(): assert convert_to((5m/s day) / km, 1) == 432 assert convert_to(foot / meter, meter) == Rational(3048, 10000) # amu is a pure mass so mass/mass gives a number, not an amount (mol) # TODO: need better simplification routine: assert str(convert_to(grams/amu, grams).n(2)) == '6.0e+23' # Light from the sun needs about 8.3 minutes to reach earth t = (1au / speed_of_light) / minute # TODO: need a better way to simplify expressions containing units: t = convert_to(convert_to(t, meter / minute), meter) assert t.simplify() == Rational(49865956897, 5995849160) # TODO: fix this, it should give `m` without `Abs` assert sqrt(m2) == m assert (sqrt(m))2 == m t = Symbol('t') assert integrate(tm/s, (t, 1s, 5s)) == 12ms assert (t * m/s).integrate((t, 1s, 5s)) == 12ms def test_issue_quart(): assert convert_to(4 * quart / inch ** 3, meter) == 231 assert convert_to(4 * quart / inch 3, millimeter) == 231 def test_issue_5565(): assert (m < s).is_Relational def test_find_unit(): assert find_unit('coulomb') == ['coulomb', 'coulombs', 'coulomb_constant'] assert find_unit(coulomb) == ['C', 'coulomb', 'coulombs', 'planck_charge', 'elementary_charge'] assert find_unit(charge) == ['C', 'coulomb', 'coulombs', 'planck_charge', 'elementary_charge'] assert find_unit(inch) == [ 'm', 'au', 'cm', 'dm', 'ft', 'km', 'ly', 'mi', 'mm', 'nm', 'pm', 'um', 'yd', 'nmi', 'feet', 'foot', 'inch', 'mile', 'yard', 'meter', 'miles', 'yards', 'inches', 'meters', 'micron', 'microns', 'decimeter', 'kilometer', 'lightyear', 'nanometer', 'picometer', 'centimeter', 'decimeters', 'kilometers', 'lightyears', 'micrometer', 'millimeter', 'nanometers', 'picometers', 'centimeters', 'micrometers', 'millimeters', 'nautical_mile', 'planck_length', 'nautical_miles', 'astronomical_unit', 'astronomical_units'] assert find_unit(inch-1) == ['D', 'dioptre', 'optical_power'] assert find_unit(length-1) == ['D', 'dioptre', 'optical_power'] assert find_unit(inch 3) == [ 'L', 'l', 'cL', 'cl', 'dL', 'dl', 'mL', 'ml', 'liter', 'quart', 'liters', 'quarts', 'deciliter', 'centiliter', 'deciliters', 'milliliter', 'centiliters', 'milliliters', 'planck_volume'] assert find_unit('voltage') == ['V', 'v', 'volt', 'volts', 'planck_voltage'] def test_Quantity_derivative(): x = symbols("x") assert diff(xmeter, x) == meter assert diff(x3meter*2, x) == 3x*2meter2 assert diff(meter, meter) == 1 assert diff(meter2, meter) == 2meter def test_quantity_postprocessing(): q1 = Quantity('q1') q2 = Quantity('q2') SI.set_quantity_dimension(q1, lengthpressure*2temperature/time) SI.set_quantity_dimension(q2, energypressuretemperature/(length*2time)) assert q1 + q2 q = q1 + q2 Dq = Dimension(SI.get_dimensional_expr(q)) assert SI.get_dimension_system().get_dimensional_dependencies(Dq) == { 'length': -1, 'mass': 2, 'temperature': 1, 'time': -5, } def test_factor_and_dimension(): assert (3000, Dimension(1)) == SI._collect_factor_and_dimension(3000) assert (1001, length) == SI._collect_factor_and_dimension(meter + km) assert (2, length/time) == SI._collect_factor_and_dimension( meter/second + 36km/(10hour)) x, y = symbols('x y') assert (x + y/100, length) == SI._collect_factor_and_dimension( xm + ycentimeter) cH = Quantity('cH') SI.set_quantity_dimension(cH, amount_of_substance/volume) pH = -log(cH) assert (1, volume/amount_of_substance) == SI._collect_factor_and_dimension( exp(pH)) v_w1 = Quantity('v_w1') v_w2 = Quantity('v_w2') v_w1.set_global_relative_scale_factor(Rational(3, 2), meter/second) v_w2.set_global_relative_scale_factor(2, meter/second) expr = Abs(v_w1/2 - v_w2) assert (Rational(5, 4), length/time) == \ SI._collect_factor_and_dimension(expr) expr = Rational(5, 2)second/meterv_w1 - 3000 assert (-(2996 + Rational(1, 4)), Dimension(1)) == \ SI._collect_factor_and_dimension(expr) expr = v_w1(v_w2/v_w1) assert ((Rational(3, 2))Rational(4, 3), (length/time)*Rational(4, 3)) == \ SI._collect_factor_and_dimension(expr) with warns_deprecated_sympy(): assert (3000, Dimension(1)) == Quantity._collect_factor_and_dimension(3000) @XFAIL def test_factor_and_dimension_with_Abs(): with warns_deprecated_sympy(): v_w1 = Quantity('v_w1', length/time, Rational(3, 2)meter/second) v_w1.set_global_relative_scale_factor(Rational(3, 2), meter/second) expr = v_w1 - Abs(v_w1) assert (0, length/time) == Quantity._collect_factor_and_dimension(expr) def test_dimensional_expr_of_derivative(): l = Quantity('l') t = Quantity('t') t1 = Quantity('t1') l.set_global_relative_scale_factor(36, km) t.set_global_relative_scale_factor(1, hour) t1.set_global_relative_scale_factor(1, second) x = Symbol('x') y = Symbol('y') f = Function('f') dfdx = f(x, y).diff(x, y) dl_dt = dfdx.subs({f(x, y): l, x: t, y: t1}) assert SI.get_dimensional_expr(dl_dt) ==\ SI.get_dimensional_expr(l / t / t1) ==\ Symbol("length")/Symbol("time")2 assert SI._collect_factor_and_dimension(dl_dt) ==\ SI._collect_factor_and_dimension(l / t / t1) ==\ (10, length/time2) def test_get_dimensional_expr_with_function(): v_w1 = Quantity('v_w1') v_w2 = Quantity('v_w2') v_w1.set_global_relative_scale_factor(1, meter/second) v_w2.set_global_relative_scale_factor(1, meter/second) assert SI.get_dimensional_expr(sin(v_w1)) == \ sin(SI.get_dimensional_expr(v_w1)) assert SI.get_dimensional_expr(sin(v_w1/v_w2)) == 1 def test_binary_information(): assert convert_to(kibibyte, byte) == 1024byte assert convert_to(mebibyte, byte) == 10242byte assert convert_to(gibibyte, byte) == 1024*3byte assert convert_to(tebibyte, byte) == 1024*4byte assert convert_to(pebibyte, byte) == 1024*5byte assert convert_to(exbibyte, byte) == 1024*6byte assert kibibyte.convert_to(bit) == 81024bit assert byte.convert_to(bit) == 8bit a = 10kibibytehour assert convert_to(a, byte) == 10240bytehour assert convert_to(a, minute) == 600kibibyteminute assert convert_to(a, [byte, minute]) == 614400byteminute def test_conversion_with_2_nonstandard_dimensions(): good_grade = Quantity("good_grade") kilo_good_grade = Quantity("kilo_good_grade") centi_good_grade = Quantity("centi_good_grade") kilo_good_grade.set_global_relative_scale_factor(1000, good_grade) centi_good_grade.set_global_relative_scale_factor(S.One/105, kilo_good_grade) charity_points = Quantity("charity_points") milli_charity_points = Quantity("milli_charity_points") missions = Quantity("missions") milli_charity_points.set_global_relative_scale_factor(S.One/1000, charity_points) missions.set_global_relative_scale_factor(251, charity_points) assert convert_to( kilo_good_grademilli_charity_pointsmillimeter, [centi_good_grade, missions, centimeter] ) == S.One 10*5 / (2511000) / 10 * centi_good_grademissionscentimeter def test_eval_subs(): energy, mass, force = symbols('energy mass force') expr1 = energy/mass units = {energy: kilogrammeter2/second2, mass: kilogram} assert expr1.subs(units) == meter2/second2 expr2 = force/mass units = {force:gravitational_constantkilogram2/meter2, mass:kilogram} assert expr2.subs(units) == gravitational_constantkilogram/meter2 def test_issue_14932(): assert (log(inch) - log(2)).simplify() == log(inch/2) assert (log(inch) - log(foot)).simplify() == -log(12) p = symbols('p', positive=True) assert (log(inch) - log(p)).simplify() == log(inch/p) def test_issue_14547(): # the root issue is that an argument with dimensions should # not raise an error when the `arg - 1` calculation is # performed in the assumptions system from sympy.physics.units import foot, inch from sympy.core.relational import Eq assert log(foot).is_zero is None assert log(foot).is_positive is None assert log(foot).is_nonnegative is None assert log(foot).is_negative is None assert log(foot).is_algebraic is None assert log(foot).is_rational is None # doesn't raise error assert Eq(log(foot), log(inch)) is not None # might be False or unevaluated x = Symbol('x') e = foot + x assert e.is_Add and set(e.args) == {foot, x} e = foot + 1 assert e.is_Add and set(e.args) == {foot, 1} def test_deprecated_quantity_methods(): step = Quantity("step") with warns_deprecated_sympy(): step.set_dimension(length) step.set_scale_factor(2meter) assert convert_to(step, centimeter) == 200centimeter assert convert_to(1000step/second, kilometer/second) == 2kilometer/second def test_issue_22164(): warnings.simplefilter("error") dm = Quantity("dm") SI.set_quantity_dimension(dm, length) SI.set_quantity_scale_factor(dm, 1) bad_exp = Quantity("bad_exp") SI.set_quantity_dimension(bad_exp, length) SI.set_quantity_scale_factor(bad_exp, 1) expr = dm * bad_exp # deprecation warning is not expected here SI._collect_factor_and_dimension(expr)
4711bb3779739823bff212282b23e70ca7df9a4f1017303dc9a9aed7f9abfdd1	from sympy.functions.elementary.miscellaneous import sqrt from sympy.physics.optics import Medium from sympy.abc import epsilon, mu, n from sympy.physics.units import speed_of_light, u0, e0, m, kg, s, A from sympy.testing.pytest import raises c = speed_of_light.convert_to(m/s) e0 = e0.convert_to(A*2s*4/(kgm*3)) u0 = u0.convert_to(mkg/(A*2s*2)) def test_medium(): m1 = Medium('m1') assert m1.intrinsic_impedance == sqrt(u0/e0) assert m1.speed == 1/sqrt(e0u0) assert m1.refractive_index == csqrt(e0u0) assert m1.permittivity == e0 assert m1.permeability == u0 m2 = Medium('m2', epsilon, mu) assert m2.intrinsic_impedance == sqrt(mu/epsilon) assert m2.speed == 1/sqrt(epsilonmu) assert m2.refractive_index == csqrt(epsilonmu) assert m2.permittivity == epsilon assert m2.permeability == mu # Increasing electric permittivity and magnetic permeability # by small amount from its value in vacuum. m3 = Medium('m3', 9.010*(-12)s*4A2/(m3kg), 1.4510*(-6)kgm/(A2s*2)) assert m3.refractive_index > m1.refractive_index assert m3 != m1 # Decreasing electric permittivity and magnetic permeability # by small amount from its value in vacuum. m4 = Medium('m4', 7.010*(-12)s*4A2/(m3kg), 1.1510*(-6)kgm/(A2s*2)) assert m4.refractive_index < m1.refractive_index m5 = Medium('m5', permittivity=71010*(-12)s*4A2/(m3kg), n=1.33) assert abs(m5.intrinsic_impedance - 6.24845417765552kgm2/(A2s*3)) \ < 1e-12kgm2/(A2s*3) assert abs(m5.speed - 225407863.157895m/s) < 1e-6m/s assert abs(m5.refractive_index - 1.33000000000000) < 1e-12 assert abs(m5.permittivity - 7.1e-10A*2s*4/(kgm*3)) \ < 1e-20A*2s*4/(kgm*3) assert abs(m5.permeability - 2.77206575232851e-8kgm/(A2s*2)) \ < 1e-20kgm/(A2s2) m6 = Medium('m6', None, mu, n) assert m6.permittivity == n2/(c*2mu) # test for equality of refractive indices assert Medium('m7').refractive_index == Medium('m8', e0, u0).refractive_index raises(ValueError, lambda:Medium('m9', e0, u0, 2))
ac10478c9214036a11f37520109fc620048359f4d95c3cce96599c3bdd176035	from sympy.tensor.functions import TensorProduct from sympy.matrices.dense import Matrix from sympy.matrices.expressions.matexpr import MatrixSymbol from sympy.tensor.array import Array from sympy.abc import x, y, z from sympy.abc import i, j, k, l A = MatrixSymbol("A", 3, 3) B = MatrixSymbol("B", 3, 3) C = MatrixSymbol("C", 3, 3) def test_TensorProduct_construction(): assert TensorProduct(3, 4) == 12 assert isinstance(TensorProduct(A, A), TensorProduct) expr = TensorProduct(TensorProduct(x, y), z) assert expr == xyz expr = TensorProduct(TensorProduct(A, B), C) assert expr == TensorProduct(A, B, C) expr = TensorProduct(Matrix.eye(2), Array([[0, -1], [1, 0]])) assert expr == Array([ [ [[0, -1], [1, 0]], [[0, 0], [0, 0]] ], [ [[0, 0], [0, 0]], [[0, -1], [1, 0]] ] ]) def test_TensorProduct_shape(): expr = TensorProduct(3, 4, evaluate=False) assert expr.shape == () assert expr.rank() == 0 expr = TensorProduct(Array([1, 2]), Array([x, y]), evaluate=False) assert expr.shape == (2, 2) assert expr.rank() == 2 expr = TensorProduct(expr, expr, evaluate=False) assert expr.shape == (2, 2, 2, 2) assert expr.rank() == 4 expr = TensorProduct(Matrix.eye(2), Array([[0, -1], [1, 0]]), evaluate=False) assert expr.shape == (2, 2, 2, 2) assert expr.rank() == 4 def test_TensorProduct_getitem(): expr = TensorProduct(A, B) assert expr[i, j, k, l] == A[i, j]*B[k, l]
7ee75b5c1f2c6389f5e29994dd7cee8b5443aa1a633573f6cd3bc6a0696239bf	import itertools from collections import defaultdict from typing import Tuple as tTuple, Union as tUnion, FrozenSet, Dict as tDict, List, Optional from functools import singledispatch from itertools import accumulate from sympy import MatMul, Basic, Wild, KroneckerProduct from sympy.assumptions.ask import (Q, ask) from sympy.core.mul import Mul from sympy.core.singleton import S from sympy.matrices.expressions.diagonal import DiagMatrix from sympy.matrices.expressions.hadamard import hadamard_product, HadamardPower from sympy.matrices.expressions.matexpr import MatrixExpr from sympy.matrices.expressions.special import (Identity, ZeroMatrix, OneMatrix) from sympy.matrices.expressions.trace import Trace from sympy.matrices.expressions.transpose import Transpose from sympy.combinatorics.permutations import _af_invert, Permutation from sympy.matrices.common import MatrixCommon from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction from sympy.matrices.expressions.matexpr import MatrixElement from sympy.tensor.array.expressions.array_expressions import PermuteDims, ArrayDiagonal, \ ArrayTensorProduct, OneArray, get_rank, _get_subrank, ZeroArray, ArrayContraction, \ ArrayAdd, _CodegenArrayAbstract, get_shape, ArrayElementwiseApplyFunc, _ArrayExpr, _EditArrayContraction, _ArgE, \ ArrayElement, _array_tensor_product, _array_contraction, _array_diagonal, _array_add, _permute_dims from sympy.tensor.array.expressions.utils import _get_mapping_from_subranks def _get_candidate_for_matmul_from_contraction(scan_indices: List[Optional[int]], remaining_args: List[_ArgE]) -> tTuple[Optional[_ArgE], bool, int]: scan_indices_int: List[int] = [i for i in scan_indices if i is not None] if len(scan_indices_int) == 0: return None, False, -1 transpose: bool = False candidate: Optional[_ArgE] = None candidate_index: int = -1 for arg_with_ind2 in remaining_args: if not isinstance(arg_with_ind2.element, MatrixExpr): continue for index in scan_indices_int: if candidate_index != -1 and candidate_index != index: # A candidate index has already been selected, check # repetitions only for that index: continue if index in arg_with_ind2.indices: if set(arg_with_ind2.indices) == {index}: # Index repeated twice in arg_with_ind2 candidate = None break if candidate is None: candidate = arg_with_ind2 candidate_index = index transpose = (index == arg_with_ind2.indices[1]) else: # Index repeated more than twice, break candidate = None break return candidate, transpose, candidate_index def _insert_candidate_into_editor(editor: _EditArrayContraction, arg_with_ind: _ArgE, candidate: _ArgE, transpose1: bool, transpose2: bool): other = candidate.element other_index: Optional[int] if transpose2: other = Transpose(other) other_index = candidate.indices[0] else: other_index = candidate.indices[1] new_element = (Transpose(arg_with_ind.element) if transpose1 else arg_with_ind.element) * other editor.args_with_ind.remove(candidate) new_arge = _ArgE(new_element) return new_arge, other_index def _support_function_tp1_recognize(contraction_indices, args): if len(contraction_indices) == 0: return _a2m_tensor_product(args) ac = _array_contraction(_array_tensor_product(args), contraction_indices) editor = _EditArrayContraction(ac) editor.track_permutation_start() while True: flag_stop: bool = True for i, arg_with_ind in enumerate(editor.args_with_ind): if not isinstance(arg_with_ind.element, MatrixExpr): continue first_index = arg_with_ind.indices[0] second_index = arg_with_ind.indices[1] first_frequency = editor.count_args_with_index(first_index) second_frequency = editor.count_args_with_index(second_index) if first_index is not None and first_frequency == 1 and first_index == second_index: flag_stop = False arg_with_ind.element = Trace(arg_with_ind.element)._normalize() arg_with_ind.indices = [] break scan_indices = [] if first_frequency == 2: scan_indices.append(first_index) if second_frequency == 2: scan_indices.append(second_index) candidate, transpose, found_index = _get_candidate_for_matmul_from_contraction(scan_indices, editor.args_with_ind[i+1:]) if candidate is not None: flag_stop = False editor.track_permutation_merge(arg_with_ind, candidate) transpose1 = found_index == first_index new_arge, other_index = _insert_candidate_into_editor(editor, arg_with_ind, candidate, transpose1, transpose) if found_index == first_index: new_arge.indices = [second_index, other_index] else: new_arge.indices = [first_index, other_index] set_indices = set(new_arge.indices) if len(set_indices) == 1 and set_indices != {None}: # This is a trace: new_arge.element = Trace(new_arge.element)._normalize() new_arge.indices = [] editor.args_with_ind[i] = new_arge # TODO: is this break necessary? break if flag_stop: break editor.refresh_indices() return editor.to_array_contraction() def _find_trivial_matrices_rewrite(expr: ArrayTensorProduct): # If there are matrices of trivial shape in the tensor product (i.e. shape # (1, 1)), try to check if there is a suitable non-trivial MatMul where the # expression can be inserted. # For example, if "a" has shape (1, 1) and "b" has shape (k, 1), the # expressions "_array_tensor_product(a, bb.T)" can be rewritten as # "bab.T" trivial_matrices = [] pos: Optional[int] = None first: Optional[MatrixExpr] = None second: Optional[MatrixExpr] = None removed: List[int] = [] counter: int = 0 args: List[Optional[Basic]] = [i for i in expr.args] for i, arg in enumerate(expr.args): if isinstance(arg, MatrixExpr): if arg.shape == (1, 1): trivial_matrices.append(arg) args[i] = None removed.extend([counter, counter+1]) elif pos is None and isinstance(arg, MatMul): margs = arg.args for j, e in enumerate(margs): if isinstance(e, MatrixExpr) and e.shape[1] == 1: pos = i first = MatMul.fromiter(margs[:j+1]) second = MatMul.fromiter(margs[j+1:]) break counter += get_rank(arg) if pos is None: return expr, [] args[pos] = (firstMatMul.fromiter(i for i in trivial_matrices)second).doit() return _array_tensor_product([i for i in args if i is not None]), removed def _find_trivial_kronecker_products_broadcast(expr: ArrayTensorProduct): newargs: List[Basic] = [] removed = [] count_dims = 0 for i, arg in enumerate(expr.args): count_dims += get_rank(arg) shape = get_shape(arg) current_range = [count_dims-i for i in range(len(shape), 0, -1)] if (shape == (1, 1) and len(newargs) > 0 and 1 not in get_shape(newargs[-1]) and isinstance(newargs[-1], MatrixExpr) and isinstance(arg, MatrixExpr)): # KroneckerProduct object allows the trick of broadcasting: newargs[-1] = KroneckerProduct(newargs[-1], arg) removed.extend(current_range) elif 1 not in shape and len(newargs) > 0 and get_shape(newargs[-1]) == (1, 1): # Broadcast: newargs[-1] = KroneckerProduct(newargs[-1], arg) prev_range = [i for i in range(min(current_range)) if i not in removed] removed.extend(prev_range[-2:]) else: newargs.append(arg) return _array_tensor_product(newargs), removed @singledispatch def _array2matrix(expr): return expr @_array2matrix.register(ZeroArray) def _(expr: ZeroArray): if get_rank(expr) == 2: return ZeroMatrix(expr.shape) else: return expr @_array2matrix.register(ArrayTensorProduct) def _(expr: ArrayTensorProduct): return _a2m_tensor_product([_array2matrix(arg) for arg in expr.args]) @_array2matrix.register(ArrayContraction) def _(expr: ArrayContraction): expr = expr.flatten_contraction_of_diagonal() expr = identify_removable_identity_matrices(expr) expr = expr.split_multiple_contractions() expr = identify_hadamard_products(expr) if not isinstance(expr, ArrayContraction): return _array2matrix(expr) subexpr = expr.expr contraction_indices: tTuple[tTuple[int]] = expr.contraction_indices if contraction_indices == ((0,), (1,)) or ( contraction_indices == ((0,),) and subexpr.shape[1] == 1 ) or ( contraction_indices == ((1,),) and subexpr.shape[0] == 1 ): shape = subexpr.shape subexpr = _array2matrix(subexpr) if isinstance(subexpr, MatrixExpr): return OneMatrix(1, shape[0])subexprOneMatrix(shape[1], 1) if isinstance(subexpr, ArrayTensorProduct): newexpr = _array_contraction(_array2matrix(subexpr), contraction_indices) contraction_indices = newexpr.contraction_indices if any(i > 2 for i in newexpr.subranks): addends = _array_add([_a2m_tensor_product(j) for j in itertools.product([i.args if isinstance(i, ArrayAdd) else [i] for i in expr.expr.args])]) newexpr = _array_contraction(addends, contraction_indices) if isinstance(newexpr, ArrayAdd): ret = _array2matrix(newexpr) return ret assert isinstance(newexpr, ArrayContraction) ret = _support_function_tp1_recognize(contraction_indices, list(newexpr.expr.args)) return ret elif not isinstance(subexpr, _CodegenArrayAbstract): ret = _array2matrix(subexpr) if isinstance(ret, MatrixExpr): assert expr.contraction_indices == ((0, 1),) return _a2m_trace(ret) else: return _array_contraction(ret, expr.contraction_indices) @_array2matrix.register(ArrayDiagonal) def _(expr: ArrayDiagonal): pexpr = _array_diagonal(_array2matrix(expr.expr), expr.diagonal_indices) pexpr = identify_hadamard_products(pexpr) if isinstance(pexpr, ArrayDiagonal): pexpr = _array_diag2contr_diagmatrix(pexpr) if expr == pexpr: return expr return _array2matrix(pexpr) @_array2matrix.register(PermuteDims) def _(expr: PermuteDims): if expr.permutation.array_form == [1, 0]: return _a2m_transpose(_array2matrix(expr.expr)) elif isinstance(expr.expr, ArrayTensorProduct): ranks = expr.expr.subranks inv_permutation = expr.permutation(-1) newrange = [inv_permutation(i) for i in range(sum(ranks))] newpos = [] counter = 0 for rank in ranks: newpos.append(newrange[counter:counter+rank]) counter += rank newargs = [] newperm = [] scalars = [] for pos, arg in zip(newpos, expr.expr.args): if len(pos) == 0: scalars.append(_array2matrix(arg)) elif pos == sorted(pos): newargs.append((_array2matrix(arg), pos[0])) newperm.extend(pos) elif len(pos) == 2: newargs.append((_a2m_transpose(_array2matrix(arg)), pos[0])) newperm.extend(reversed(pos)) else: raise NotImplementedError() newargs = [i[0] for i in newargs] return _permute_dims(_a2m_tensor_product(scalars, newargs), _af_invert(newperm)) elif isinstance(expr.expr, ArrayContraction): mat_mul_lines = _array2matrix(expr.expr) if not isinstance(mat_mul_lines, ArrayTensorProduct): return _permute_dims(mat_mul_lines, expr.permutation) # TODO: this assumes that all arguments are matrices, it may not be the case: permutation = Permutation(2len(mat_mul_lines.args)-1)expr.permutation permuted = [permutation(i) for i in range(2len(mat_mul_lines.args))] args_array = [None for i in mat_mul_lines.args] for i in range(len(mat_mul_lines.args)): p1 = permuted[2i] p2 = permuted[2i+1] if p1 // 2 != p2 // 2: return _permute_dims(mat_mul_lines, permutation) if p1 > p2: args_array[i] = _a2m_transpose(mat_mul_lines.args[p1 // 2]) else: args_array[i] = mat_mul_lines.args[p1 // 2] return _a2m_tensor_product(args_array) else: return expr @_array2matrix.register(ArrayAdd) def _(expr: ArrayAdd): addends = [_array2matrix(arg) for arg in expr.args] return _a2m_add(addends) @_array2matrix.register(ArrayElementwiseApplyFunc) def _(expr: ArrayElementwiseApplyFunc): subexpr = _array2matrix(expr.expr) if isinstance(subexpr, MatrixExpr): if subexpr.shape != (1, 1): d = expr.function.bound_symbols[0] w = Wild("w", exclude=[d]) p = Wild("p", exclude=[d]) m = expr.function.expr.match(wdp) if m is not None: return m[w]HadamardPower(subexpr, m[p]) return ElementwiseApplyFunction(expr.function, subexpr) else: return ArrayElementwiseApplyFunc(expr.function, subexpr) @_array2matrix.register(ArrayElement) def _(expr: ArrayElement): ret = _array2matrix(expr.name) if isinstance(ret, MatrixExpr): return MatrixElement(ret, expr.indices) return ArrayElement(ret, expr.indices) @singledispatch def _remove_trivial_dims(expr): return expr, [] @_remove_trivial_dims.register(ArrayTensorProduct) def _(expr: ArrayTensorProduct): # Recognize expressions like [x, y] with shape (k, 1, k, 1) as `xy.T`. # The matrix expression has to be equivalent to the tensor product of the # matrices, with trivial dimensions (i.e. dim=1) dropped. # That is, add contractions over trivial dimensions: removed = [] newargs = [] cumul = list(accumulate([0] + [get_rank(arg) for arg in expr.args])) pending = None prev_i = None for i, arg in enumerate(expr.args): current_range = list(range(cumul[i], cumul[i+1])) if isinstance(arg, OneArray): removed.extend(current_range) continue if not isinstance(arg, (MatrixExpr, MatrixCommon)): rarg, rem = _remove_trivial_dims(arg) removed.extend(rem) newargs.append(rarg) continue elif getattr(arg, "is_Identity", False) and arg.shape == (1, 1): if arg.shape == (1, 1): # Ignore identity matrices of shape (1, 1) - they are equivalent to scalar 1. removed.extend(current_range) continue elif arg.shape == (1, 1): arg, _ = _remove_trivial_dims(arg) # Matrix is equivalent to scalar: if len(newargs) == 0: newargs.append(arg) elif 1 in get_shape(newargs[-1]): if newargs[-1].shape[1] == 1: newargs[-1] = newargs[-1]arg else: newargs[-1] = argnewargs[-1] removed.extend(current_range) else: newargs.append(arg) elif 1 in arg.shape: k = [i for i in arg.shape if i != 1][0] if pending is None: pending = k prev_i = i newargs.append(arg) elif pending == k: prev = newargs[-1] if prev.shape[0] == 1: d1 = cumul[prev_i] prev = _a2m_transpose(prev) else: d1 = cumul[prev_i] + 1 if arg.shape[1] == 1: d2 = cumul[i] + 1 arg = _a2m_transpose(arg) else: d2 = cumul[i] newargs[-1] = prevarg pending = None removed.extend([d1, d2]) else: newargs.append(arg) pending = k prev_i = i else: newargs.append(arg) pending = None newexpr, newremoved = _a2m_tensor_product(newargs), sorted(removed) if isinstance(newexpr, ArrayTensorProduct): newexpr, newremoved2 = _find_trivial_matrices_rewrite(newexpr) newremoved = _combine_removed(-1, newremoved, newremoved2) if isinstance(newexpr, ArrayTensorProduct): newexpr, newremoved2 = _find_trivial_kronecker_products_broadcast(newexpr) newremoved = _combine_removed(-1, newremoved, newremoved2) return newexpr, newremoved @_remove_trivial_dims.register(ArrayAdd) def _(expr: ArrayAdd): rec = [_remove_trivial_dims(arg) for arg in expr.args] newargs, removed = zip(rec) if len(set([get_shape(i) for i in newargs])) > 1: return expr, [] if len(removed) == 0: return expr, removed removed1 = removed[0] return _a2m_add(newargs), removed1 @_remove_trivial_dims.register(PermuteDims) def _(expr: PermuteDims): subexpr, subremoved = _remove_trivial_dims(expr.expr) p = expr.permutation.array_form pinv = _af_invert(expr.permutation.array_form) shift = list(accumulate([1 if i in subremoved else 0 for i in range(len(p))])) premoved = [pinv[i] for i in subremoved] p2 = [e - shift[e] for i, e in enumerate(p) if e not in subremoved] # TODO: check if subremoved should be permuted as well... newexpr = _permute_dims(subexpr, p2) premoved = sorted(premoved) if newexpr != expr: newexpr, removed2 = _remove_trivial_dims(_array2matrix(newexpr)) premoved = _combine_removed(-1, premoved, removed2) return newexpr, premoved @_remove_trivial_dims.register(ArrayContraction) def _(expr: ArrayContraction): new_expr, removed0 = _array_contraction_to_diagonal_multiple_identity(expr) if new_expr != expr: new_expr2, removed1 = _remove_trivial_dims(_array2matrix(new_expr)) removed = _combine_removed(-1, removed0, removed1) return new_expr2, removed rank1 = get_rank(expr) expr, removed1 = remove_identity_matrices(expr) if not isinstance(expr, ArrayContraction): expr2, removed2 = _remove_trivial_dims(expr) return expr2, _combine_removed(rank1, removed1, removed2) newexpr, removed2 = _remove_trivial_dims(expr.expr) shifts = list(accumulate([1 if i in removed2 else 0 for i in range(get_rank(expr.expr))])) new_contraction_indices = [tuple(j for j in i if j not in removed2) for i in expr.contraction_indices] # Remove possible empty tuples "()": new_contraction_indices = [i for i in new_contraction_indices if len(i) > 0] contraction_indices_flat = [j for i in expr.contraction_indices for j in i] removed2 = [i for i in removed2 if i not in contraction_indices_flat] new_contraction_indices = [tuple(j - shifts[j] for j in i) for i in new_contraction_indices] # Shift removed2: removed2 = ArrayContraction._push_indices_up(expr.contraction_indices, removed2) removed = _combine_removed(rank1, removed1, removed2) return _array_contraction(newexpr, new_contraction_indices), list(removed) def _remove_diagonalized_identity_matrices(expr: ArrayDiagonal): assert isinstance(expr, ArrayDiagonal) editor = _EditArrayContraction(expr) mapping = {i: {j for j in editor.args_with_ind if i in j.indices} for i in range(-1, -1-editor.number_of_diagonal_indices, -1)} removed = [] counter: int = 0 for i, arg_with_ind in enumerate(editor.args_with_ind): counter += len(arg_with_ind.indices) if isinstance(arg_with_ind.element, Identity): if None in arg_with_ind.indices and any(i is not None and (i < 0) == True for i in arg_with_ind.indices): diag_ind = [j for j in arg_with_ind.indices if j is not None][0] other = [j for j in mapping[diag_ind] if j != arg_with_ind][0] if not isinstance(other.element, MatrixExpr): continue if 1 not in other.element.shape: continue if None not in other.indices: continue editor.args_with_ind[i].element = None none_index = other.indices.index(None) other.element = DiagMatrix(other.element) other_range = editor.get_absolute_range(other) removed.extend([other_range[0] + none_index]) editor.args_with_ind = [i for i in editor.args_with_ind if i.element is not None] removed = ArrayDiagonal._push_indices_up(expr.diagonal_indices, removed, get_rank(expr.expr)) return editor.to_array_contraction(), removed @_remove_trivial_dims.register(ArrayDiagonal) def _(expr: ArrayDiagonal): newexpr, removed = _remove_trivial_dims(expr.expr) shifts = list(accumulate([0] + [1 if i in removed else 0 for i in range(get_rank(expr.expr))])) new_diag_indices_map = {i: tuple(j for j in i if j not in removed) for i in expr.diagonal_indices} for old_diag_tuple, new_diag_tuple in new_diag_indices_map.items(): if len(new_diag_tuple) == 1: removed = [i for i in removed if i not in old_diag_tuple] new_diag_indices = [tuple(j - shifts[j] for j in i) for i in new_diag_indices_map.values()] rank = get_rank(expr.expr) removed = ArrayDiagonal._push_indices_up(expr.diagonal_indices, removed, rank) removed = sorted({i for i in removed}) # If there are single axes to diagonalize remaining, it means that their # corresponding dimension has been removed, they no longer need diagonalization: new_diag_indices = [i for i in new_diag_indices if len(i) > 0] if len(new_diag_indices) > 0: newexpr2 = _array_diagonal(newexpr, new_diag_indices, allow_trivial_diags=True) else: newexpr2 = newexpr if isinstance(newexpr2, ArrayDiagonal): newexpr3, removed2 = _remove_diagonalized_identity_matrices(newexpr2) removed = _combine_removed(-1, removed, removed2) return newexpr3, removed else: return newexpr2, removed @_remove_trivial_dims.register(ElementwiseApplyFunction) def _(expr: ElementwiseApplyFunction): subexpr, removed = _remove_trivial_dims(expr.expr) if subexpr.shape == (1, 1): # TODO: move this to ElementwiseApplyFunction return expr.function(subexpr), removed + [0, 1] return ElementwiseApplyFunction(expr.function, subexpr), [] @_remove_trivial_dims.register(ArrayElementwiseApplyFunc) def _(expr: ArrayElementwiseApplyFunc): subexpr, removed = _remove_trivial_dims(expr.expr) return ArrayElementwiseApplyFunc(expr.function, subexpr), removed def convert_array_to_matrix(expr): r""" Recognize matrix expressions in codegen objects. If more than one matrix multiplication line have been detected, return a list with the matrix expressions. Examples ======== >>> from sympy.tensor.array.expressions.conv_indexed_to_array import convert_indexed_to_array >>> from sympy.tensor.array import tensorcontraction, tensorproduct >>> from sympy import MatrixSymbol, Sum >>> from sympy.abc import i, j, k, l, N >>> from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array >>> from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix >>> A = MatrixSymbol("A", N, N) >>> B = MatrixSymbol("B", N, N) >>> C = MatrixSymbol("C", N, N) >>> D = MatrixSymbol("D", N, N) >>> expr = Sum(A[i, j]B[j, k], (j, 0, N-1)) >>> cg = convert_indexed_to_array(expr) >>> convert_array_to_matrix(cg) AB >>> cg = convert_indexed_to_array(expr, first_indices=[k]) >>> convert_array_to_matrix(cg) B.TA.T Transposition is detected: >>> expr = Sum(A[j, i]B[j, k], (j, 0, N-1)) >>> cg = convert_indexed_to_array(expr) >>> convert_array_to_matrix(cg) A.TB >>> cg = convert_indexed_to_array(expr, first_indices=[k]) >>> convert_array_to_matrix(cg) B.TA Detect the trace: >>> expr = Sum(A[i, i], (i, 0, N-1)) >>> cg = convert_indexed_to_array(expr) >>> convert_array_to_matrix(cg) Trace(A) Recognize some more complex traces: >>> expr = Sum(A[i, j]B[j, i], (i, 0, N-1), (j, 0, N-1)) >>> cg = convert_indexed_to_array(expr) >>> convert_array_to_matrix(cg) Trace(AB) More complicated expressions: >>> expr = Sum(A[i, j]B[k, j]A[l, k], (j, 0, N-1), (k, 0, N-1)) >>> cg = convert_indexed_to_array(expr) >>> convert_array_to_matrix(cg) AB.TA.T Expressions constructed from matrix expressions do not contain literal indices, the positions of free indices are returned instead: >>> expr = AB >>> cg = convert_matrix_to_array(expr) >>> convert_array_to_matrix(cg) AB If more than one line of matrix multiplications is detected, return separate matrix multiplication factors embedded in a tensor product object: >>> cg = tensorcontraction(tensorproduct(A, B, C, D), (1, 2), (5, 6)) >>> convert_array_to_matrix(cg) ArrayTensorProduct(AB, CD) The two lines have free indices at axes 0, 3 and 4, 7, respectively. """ rec = _array2matrix(expr) rec, removed = _remove_trivial_dims(rec) return rec def _array_diag2contr_diagmatrix(expr: ArrayDiagonal): if isinstance(expr.expr, ArrayTensorProduct): args = list(expr.expr.args) diag_indices = list(expr.diagonal_indices) mapping = _get_mapping_from_subranks([_get_subrank(arg) for arg in args]) tuple_links = [[mapping[j] for j in i] for i in diag_indices] contr_indices = [] total_rank = get_rank(expr) replaced = [False for arg in args] for i, (abs_pos, rel_pos) in enumerate(zip(diag_indices, tuple_links)): if len(abs_pos) != 2: continue (pos1_outer, pos1_inner), (pos2_outer, pos2_inner) = rel_pos arg1 = args[pos1_outer] arg2 = args[pos2_outer] if get_rank(arg1) != 2 or get_rank(arg2) != 2: if replaced[pos1_outer]: diag_indices[i] = None if replaced[pos2_outer]: diag_indices[i] = None continue pos1_in2 = 1 - pos1_inner pos2_in2 = 1 - pos2_inner if arg1.shape[pos1_in2] == 1: if arg1.shape[pos1_inner] != 1: darg1 = DiagMatrix(arg1) else: darg1 = arg1 args.append(darg1) contr_indices.append(((pos2_outer, pos2_inner), (len(args)-1, pos1_inner))) total_rank += 1 diag_indices[i] = None args[pos1_outer] = OneArray(arg1.shape[pos1_in2]) replaced[pos1_outer] = True elif arg2.shape[pos2_in2] == 1: if arg2.shape[pos2_inner] != 1: darg2 = DiagMatrix(arg2) else: darg2 = arg2 args.append(darg2) contr_indices.append(((pos1_outer, pos1_inner), (len(args)-1, pos2_inner))) total_rank += 1 diag_indices[i] = None args[pos2_outer] = OneArray(arg2.shape[pos2_in2]) replaced[pos2_outer] = True diag_indices_new = [i for i in diag_indices if i is not None] cumul = list(accumulate([0] + [get_rank(arg) for arg in args])) contr_indices2 = [tuple(cumul[a] + b for a, b in i) for i in contr_indices] tc = _array_contraction( _array_tensor_product(args), contr_indices2 ) td = _array_diagonal(tc, diag_indices_new) return td return expr def _a2m_mul(args): if not any(isinstance(i, _CodegenArrayAbstract) for i in args): from sympy.matrices.expressions.matmul import MatMul return MatMul(args).doit() else: return _array_contraction( _array_tensor_product(args), [(2i-1, 2i) for i in range(1, len(args))] ) def _a2m_tensor_product(args): scalars = [] arrays = [] for arg in args: if isinstance(arg, (MatrixExpr, _ArrayExpr, _CodegenArrayAbstract)): arrays.append(arg) else: scalars.append(arg) scalar = Mul.fromiter(scalars) if len(arrays) == 0: return scalar if scalar != 1: if isinstance(arrays[0], _CodegenArrayAbstract): arrays = [scalar] + arrays else: arrays[0] = scalar return _array_tensor_product(arrays) def _a2m_add(args): if not any(isinstance(i, _CodegenArrayAbstract) for i in args): from sympy.matrices.expressions.matadd import MatAdd return MatAdd(args).doit() else: return _array_add(args) def _a2m_trace(arg): if isinstance(arg, _CodegenArrayAbstract): return _array_contraction(arg, (0, 1)) else: from sympy.matrices.expressions.trace import Trace return Trace(arg) def _a2m_transpose(arg): if isinstance(arg, _CodegenArrayAbstract): return _permute_dims(arg, [1, 0]) else: from sympy.matrices.expressions.transpose import Transpose return Transpose(arg).doit() def identify_hadamard_products(expr: tUnion[ArrayContraction, ArrayDiagonal]): editor: _EditArrayContraction = _EditArrayContraction(expr) map_contr_to_args: tDict[FrozenSet, List[_ArgE]] = defaultdict(list) map_ind_to_inds: tDict[Optional[int], int] = defaultdict(int) for arg_with_ind in editor.args_with_ind: for ind in arg_with_ind.indices: map_ind_to_inds[ind] += 1 if None in arg_with_ind.indices: continue map_contr_to_args[frozenset(arg_with_ind.indices)].append(arg_with_ind) k: FrozenSet[int] v: List[_ArgE] for k, v in map_contr_to_args.items(): make_trace: bool = False if len(k) == 1 and next(iter(k)) >= 0 and sum([next(iter(k)) in i for i in map_contr_to_args]) == 1: # This is a trace: the arguments are fully contracted with only one # index, and the index isn't used anywhere else: make_trace = True first_element = S.One elif len(k) != 2: # Hadamard product only defined for matrices: continue if len(v) == 1: # Hadamard product with a single argument makes no sense: continue for ind in k: if map_ind_to_inds[ind] <= 2: # There is no other contraction, skip: continue def check_transpose(x): x = [i if i >= 0 else -1-i for i in x] return x == sorted(x) # Check if expression is a trace: if all([map_ind_to_inds[j] == len(v) and j >= 0 for j in k]) and all([j >= 0 for j in k]): # This is a trace make_trace = True first_element = v[0].element if not check_transpose(v[0].indices): first_element = first_element.T hadamard_factors = v[1:] else: hadamard_factors = v # This is a Hadamard product: hp = hadamard_product([i.element if check_transpose(i.indices) else Transpose(i.element) for i in hadamard_factors]) hp_indices = v[0].indices if not check_transpose(hadamard_factors[0].indices): hp_indices = list(reversed(hp_indices)) if make_trace: hp = Trace(first_elementhp.T)._normalize() hp_indices = [] editor.insert_after(v[0], _ArgE(hp, hp_indices)) for i in v: editor.args_with_ind.remove(i) return editor.to_array_contraction() def identify_removable_identity_matrices(expr): editor = _EditArrayContraction(expr) flag: bool = True while flag: flag = False for arg_with_ind in editor.args_with_ind: if isinstance(arg_with_ind.element, Identity): k = arg_with_ind.element.shape[0] # Candidate for removal: if arg_with_ind.indices == [None, None]: # Free identity matrix, will be cleared by _remove_trivial_dims: continue elif None in arg_with_ind.indices: ind = [j for j in arg_with_ind.indices if j is not None][0] counted = editor.count_args_with_index(ind) if counted == 1: # Identity matrix contracted only on one index with itself, # transform to a OneArray(k) element: editor.insert_after(arg_with_ind, OneArray(k)) editor.args_with_ind.remove(arg_with_ind) flag = True break elif counted > 2: # Case counted = 2 is a matrix multiplication by identity matrix, skip it. # Case counted > 2 is a multiple contraction, # this is a case where the contraction becomes a diagonalization if the # identity matrix is dropped. continue elif arg_with_ind.indices[0] == arg_with_ind.indices[1]: ind = arg_with_ind.indices[0] counted = editor.count_args_with_index(ind) if counted > 1: editor.args_with_ind.remove(arg_with_ind) flag = True break else: # This is a trace, skip it as it will be recognized somewhere else: pass elif ask(Q.diagonal(arg_with_ind.element)): if arg_with_ind.indices == [None, None]: continue elif None in arg_with_ind.indices: pass elif arg_with_ind.indices[0] == arg_with_ind.indices[1]: ind = arg_with_ind.indices[0] counted = editor.count_args_with_index(ind) if counted == 3: # A_ai B_bi D_ii ==> A_ai D_ij B_bj ind_new = editor.get_new_contraction_index() other_args = [j for j in editor.args_with_ind if j != arg_with_ind] other_args[1].indices = [ind_new if j == ind else j for j in other_args[1].indices] arg_with_ind.indices = [ind, ind_new] flag = True break return editor.to_array_contraction() def remove_identity_matrices(expr: ArrayContraction): editor = _EditArrayContraction(expr) removed: List[int] = [] permutation_map = {} free_indices = list(accumulate([0] + [sum([i is None for i in arg.indices]) for arg in editor.args_with_ind])) free_map = {k: v for k, v in zip(editor.args_with_ind, free_indices[:-1])} update_pairs = {} for ind in range(editor.number_of_contraction_indices): args = editor.get_args_with_index(ind) identity_matrices = [i for i in args if isinstance(i.element, Identity)] number_identity_matrices = len(identity_matrices) # If the contraction involves a non-identity matrix and multiple identity matrices: if number_identity_matrices != len(args) - 1 or number_identity_matrices == 0: continue # Get the non-identity element: non_identity = [i for i in args if not isinstance(i.element, Identity)][0] # Check that all identity matrices have at least one free index # (otherwise they would be contractions to some other elements) if any([None not in i.indices for i in identity_matrices]): continue # Mark the identity matrices for removal: for i in identity_matrices: i.element = None removed.extend(range(free_map[i], free_map[i] + len([j for j in i.indices if j is None]))) last_removed = removed.pop(-1) update_pairs[last_removed, ind] = non_identity.indices[:] # Remove the indices from the non-identity matrix, as the contraction # no longer exists: non_identity.indices = [None if i == ind else i for i in non_identity.indices] removed.sort() shifts = list(accumulate([1 if i in removed else 0 for i in range(get_rank(expr))])) for (last_removed, ind), non_identity_indices in update_pairs.items(): pos = [free_map[non_identity] + i for i, e in enumerate(non_identity_indices) if e == ind] assert len(pos) == 1 for j in pos: permutation_map[j] = last_removed editor.args_with_ind = [i for i in editor.args_with_ind if i.element is not None] ret_expr = editor.to_array_contraction() permutation = [] counter = 0 counter2 = 0 for j in range(get_rank(expr)): if j in removed: continue if counter2 in permutation_map: target = permutation_map[counter2] permutation.append(target - shifts[target]) counter2 += 1 else: while counter in permutation_map.values(): counter += 1 permutation.append(counter) counter += 1 counter2 += 1 ret_expr2 = _permute_dims(ret_expr, _af_invert(permutation)) return ret_expr2, removed def _combine_removed(dim: int, removed1: List[int], removed2: List[int]) -> List[int]: # Concatenate two axis removal operations as performed by # _remove_trivial_dims, removed1 = sorted(removed1) removed2 = sorted(removed2) i = 0 j = 0 removed = [] while True: if j >= len(removed2): while i < len(removed1): removed.append(removed1[i]) i += 1 break elif i < len(removed1) and removed1[i] <= i + removed2[j]: removed.append(removed1[i]) i += 1 else: removed.append(i + removed2[j]) j += 1 return removed def _array_contraction_to_diagonal_multiple_identity(expr: ArrayContraction): editor = _EditArrayContraction(expr) editor.track_permutation_start() removed: List[int] = [] diag_index_counter: int = 0 for i in range(editor.number_of_contraction_indices): identities = [] args = [] for j, arg in enumerate(editor.args_with_ind): if i not in arg.indices: continue if isinstance(arg.element, Identity): identities.append(arg) else: args.append(arg) if len(identities) == 0: continue if len(args) + len(identities) < 3: continue new_diag_ind = -1 - diag_index_counter diag_index_counter += 1 # Variable "flag" to control whether to skip this contraction set: flag: bool = True for i1, id1 in enumerate(identities): if None not in id1.indices: flag = True break free_pos = list(range(editor.get_absolute_free_range(id1)))[0] editor._track_permutation[-1].append(free_pos) # type: ignore id1.element = None flag = False break if flag: continue for arg in identities[:i1] + identities[i1+1:]: arg.element = None removed.extend(range(*editor.get_absolute_free_range(arg))) for arg in args: arg.indices = [new_diag_ind if j == i else j for j in arg.indices] for j, e in enumerate(editor.args_with_ind): if e.element is None: editor._track_permutation[j] = None # type: ignore editor._track_permutation = [i for i in editor._track_permutation if i is not None] # type: ignore # Renumber permutation array form in order to deal with deleted positions: remap = {e: i for i, e in enumerate(sorted({k for j in editor._track_permutation for k in j}))} editor._track_permutation = [[remap[j] for j in i] for i in editor._track_permutation] editor.args_with_ind = [i for i in editor.args_with_ind if i.element is not None] new_expr = editor.to_array_contraction() return new_expr, removed
c8863546ea45f7e1555778d92eca96c59d5c928135f87910c8cf1c4089b8fc6b	from sympy import KroneckerProduct from sympy.core.basic import Basic from sympy.core.function import Lambda from sympy.core.mul import Mul from sympy.core.numbers import Integer from sympy.core.power import Pow from sympy.core.singleton import S from sympy.core.symbol import (Dummy, symbols) from sympy.matrices.expressions.hadamard import (HadamardPower, HadamardProduct) from sympy.matrices.expressions.matadd import MatAdd from sympy.matrices.expressions.matmul import MatMul from sympy.matrices.expressions.matpow import MatPow from sympy.matrices.expressions.trace import Trace from sympy.matrices.expressions.transpose import Transpose from sympy.matrices.expressions.matexpr import MatrixExpr from sympy.tensor.array.expressions.array_expressions import \ ArrayElementwiseApplyFunc, _array_tensor_product, _array_contraction, \ _array_diagonal, _array_add, _permute_dims, Reshape def convert_matrix_to_array(expr: Basic) -> Basic: if isinstance(expr, MatMul): args_nonmat = [] args = [] for arg in expr.args: if isinstance(arg, MatrixExpr): args.append(arg) else: args_nonmat.append(convert_matrix_to_array(arg)) contractions = [(2i+1, 2i+2) for i in range(len(args)-1)] scalar = _array_tensor_product(args_nonmat) if args_nonmat else S.One if scalar == 1: tprod = _array_tensor_product( [convert_matrix_to_array(arg) for arg in args]) else: tprod = _array_tensor_product( scalar, [convert_matrix_to_array(arg) for arg in args]) return _array_contraction( tprod, contractions ) elif isinstance(expr, MatAdd): return _array_add( [convert_matrix_to_array(arg) for arg in expr.args] ) elif isinstance(expr, Transpose): return _permute_dims( convert_matrix_to_array(expr.args[0]), [1, 0] ) elif isinstance(expr, Trace): inner_expr: MatrixExpr = convert_matrix_to_array(expr.arg) # type: ignore return _array_contraction(inner_expr, (0, len(inner_expr.shape) - 1)) elif isinstance(expr, Mul): return _array_tensor_product([convert_matrix_to_array(i) for i in expr.args]) elif isinstance(expr, Pow): base = convert_matrix_to_array(expr.base) if (expr.exp > 0) == True: return _array_tensor_product([base for i in range(expr.exp)]) else: return expr elif isinstance(expr, MatPow): base = convert_matrix_to_array(expr.base) if expr.exp.is_Integer != True: b = symbols("b", cls=Dummy) return ArrayElementwiseApplyFunc(Lambda(b, bexpr.exp), convert_matrix_to_array(base)) elif (expr.exp > 0) == True: return convert_matrix_to_array(MatMul.fromiter(base for i in range(expr.exp))) else: return expr elif isinstance(expr, HadamardProduct): tp = _array_tensor_product([convert_matrix_to_array(arg) for arg in expr.args]) diag = [[2i for i in range(len(expr.args))], [2i+1 for i in range(len(expr.args))]] return _array_diagonal(tp, diag) elif isinstance(expr, HadamardPower): base, exp = expr.args if isinstance(exp, Integer) and exp > 0: return convert_matrix_to_array(HadamardProduct.fromiter(base for i in range(exp))) else: d = Dummy("d") return ArrayElementwiseApplyFunc(Lambda(d, dexp), base) elif isinstance(expr, KroneckerProduct): kp_args = [convert_matrix_to_array(arg) for arg in expr.args] permutation = [2i for i in range(len(kp_args))] + [2i + 1 for i in range(len(kp_args))] return Reshape(_permute_dims(_array_tensor_product(kp_args), permutation), expr.shape) else: return expr
358e653b7dc650bd722c318007cfdce5dd6fa4b8b2f6b9015fd34401cfc7327b	import operator from collections import defaultdict, Counter from functools import reduce import itertools from itertools import accumulate from typing import Optional, List, Dict as tDict, Tuple as tTuple import typing from sympy import Integer, KroneckerDelta, Equality from sympy.core.basic import Basic from sympy.core.containers import Tuple from sympy.core.expr import Expr from sympy.core.function import (Function, Lambda) from sympy.core.mul import Mul from sympy.core.singleton import S from sympy.core.sorting import default_sort_key from sympy.core.symbol import (Dummy, Symbol) from sympy.matrices.common import MatrixCommon from sympy.matrices.expressions.diagonal import diagonalize_vector from sympy.matrices.expressions.matexpr import MatrixExpr from sympy.matrices.expressions.special import ZeroMatrix from sympy.tensor.array.arrayop import (permutedims, tensorcontraction, tensordiagonal, tensorproduct) from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray from sympy.tensor.array.ndim_array import NDimArray from sympy.tensor.indexed import (Indexed, IndexedBase) from sympy.matrices.expressions.matexpr import MatrixElement from sympy.tensor.array.expressions.utils import _apply_recursively_over_nested_lists, _sort_contraction_indices, \ _get_mapping_from_subranks, _build_push_indices_up_func_transformation, _get_contraction_links, \ _build_push_indices_down_func_transformation from sympy.combinatorics import Permutation from sympy.combinatorics.permutations import _af_invert from sympy.core.sympify import _sympify class _ArrayExpr(Expr): shape : tTuple[Expr, ...] class ArraySymbol(_ArrayExpr): """ Symbol representing an array expression """ def __new__(cls, symbol, shape: typing.Iterable) -> "ArraySymbol": if isinstance(symbol, str): symbol = Symbol(symbol) # symbol = _sympify(symbol) shape = Tuple(map(_sympify, shape)) obj = Expr.__new__(cls, symbol, shape) return obj @property def name(self): return self._args[0] @property def shape(self): return self._args[1] def __getitem__(self, item): return ArrayElement(self, item) def as_explicit(self): if not all(i.is_Integer for i in self.shape): raise ValueError("cannot express explicit array with symbolic shape") data = [self[i] for i in itertools.product([range(j) for j in self.shape])] return ImmutableDenseNDimArray(data).reshape(self.shape) class ArrayElement(_ArrayExpr): """ An element of an array. """ _diff_wrt = True is_symbol = True is_commutative = True def __new__(cls, name, indices): if isinstance(name, str): name = Symbol(name) name = _sympify(name) indices = _sympify(tuple(indices)) if hasattr(name, "shape"): if any((i >= s) == True for i, s in zip(indices, name.shape)): raise ValueError("shape is out of bounds") if any((i < 0) == True for i in indices): raise ValueError("shape contains negative values") obj = Expr.__new__(cls, name, indices) return obj @property def name(self): return self._args[0] @property def indices(self): return self._args[1] def _eval_derivative(self, s): if not isinstance(s, ArrayElement): return S.Zero if s == self: return S.One if s.name != self.name: return S.Zero return Mul.fromiter(KroneckerDelta(i, j) for i, j in zip(self.indices, s.indices)) class ZeroArray(_ArrayExpr): """ Symbolic array of zeros. Equivalent to ``ZeroMatrix`` for matrices. """ def __new__(cls, shape): if len(shape) == 0: return S.Zero shape = map(_sympify, shape) obj = Expr.__new__(cls, shape) return obj @property def shape(self): return self._args def as_explicit(self): if not all(i.is_Integer for i in self.shape): raise ValueError("Cannot return explicit form for symbolic shape.") return ImmutableDenseNDimArray.zeros(self.shape) class OneArray(_ArrayExpr): """ Symbolic array of ones. """ def __new__(cls, shape): if len(shape) == 0: return S.One shape = map(_sympify, shape) obj = Expr.__new__(cls, shape) return obj @property def shape(self): return self._args def as_explicit(self): if not all(i.is_Integer for i in self.shape): raise ValueError("Cannot return explicit form for symbolic shape.") return ImmutableDenseNDimArray([S.One for i in range(reduce(operator.mul, self.shape))]).reshape(self.shape) class _CodegenArrayAbstract(Basic): @property def subranks(self): """ Returns the ranks of the objects in the uppermost tensor product inside the current object. In case no tensor products are contained, return the atomic ranks. Examples ======== >>> from sympy.tensor.array import tensorproduct, tensorcontraction >>> from sympy import MatrixSymbol >>> M = MatrixSymbol("M", 3, 3) >>> N = MatrixSymbol("N", 3, 3) >>> P = MatrixSymbol("P", 3, 3) Important: do not confuse the rank of the matrix with the rank of an array. >>> tp = tensorproduct(M, N, P) >>> tp.subranks [2, 2, 2] >>> co = tensorcontraction(tp, (1, 2), (3, 4)) >>> co.subranks [2, 2, 2] """ return self._subranks[:] def subrank(self): """ The sum of ``subranks``. """ return sum(self.subranks) @property def shape(self): return self._shape class ArrayTensorProduct(_CodegenArrayAbstract): r""" Class to represent the tensor product of array-like objects. """ def __new__(cls, args, *kwargs): args = [_sympify(arg) for arg in args] canonicalize = kwargs.pop("canonicalize", False) ranks = [get_rank(arg) for arg in args] obj = Basic.__new__(cls, args) obj._subranks = ranks shapes = [get_shape(i) for i in args] if any(i is None for i in shapes): obj._shape = None else: obj._shape = tuple(j for i in shapes for j in i) if canonicalize: return obj._canonicalize() return obj def _canonicalize(self): args = self.args args = self._flatten(args) ranks = [get_rank(arg) for arg in args] # Check if there are nested permutation and lift them up: permutation_cycles = [] for i, arg in enumerate(args): if not isinstance(arg, PermuteDims): continue permutation_cycles.extend([[k + sum(ranks[:i]) for k in j] for j in arg.permutation.cyclic_form]) args[i] = arg.expr if permutation_cycles: return _permute_dims(_array_tensor_product(args), Permutation(sum(ranks)-1)Permutation(permutation_cycles)) if len(args) == 1: return args[0] # If any object is a ZeroArray, return a ZeroArray: if any(isinstance(arg, (ZeroArray, ZeroMatrix)) for arg in args): shapes = reduce(operator.add, [get_shape(i) for i in args], ()) return ZeroArray(shapes) # If there are contraction objects inside, transform the whole # expression into `ArrayContraction`: contractions = {i: arg for i, arg in enumerate(args) if isinstance(arg, ArrayContraction)} if contractions: ranks = [_get_subrank(arg) if isinstance(arg, ArrayContraction) else get_rank(arg) for arg in args] cumulative_ranks = list(accumulate([0] + ranks))[:-1] tp = _array_tensor_product([arg.expr if isinstance(arg, ArrayContraction) else arg for arg in args]) contraction_indices = [tuple(cumulative_ranks[i] + k for k in j) for i, arg in contractions.items() for j in arg.contraction_indices] return _array_contraction(tp, contraction_indices) diagonals = {i: arg for i, arg in enumerate(args) if isinstance(arg, ArrayDiagonal)} if diagonals: inverse_permutation = [] last_perm = [] ranks = [get_rank(arg) for arg in args] cumulative_ranks = list(accumulate([0] + ranks))[:-1] for i, arg in enumerate(args): if isinstance(arg, ArrayDiagonal): i1 = get_rank(arg) - len(arg.diagonal_indices) i2 = len(arg.diagonal_indices) inverse_permutation.extend([cumulative_ranks[i] + j for j in range(i1)]) last_perm.extend([cumulative_ranks[i] + j for j in range(i1, i1 + i2)]) else: inverse_permutation.extend([cumulative_ranks[i] + j for j in range(get_rank(arg))]) inverse_permutation.extend(last_perm) tp = _array_tensor_product([arg.expr if isinstance(arg, ArrayDiagonal) else arg for arg in args]) ranks2 = [_get_subrank(arg) if isinstance(arg, ArrayDiagonal) else get_rank(arg) for arg in args] cumulative_ranks2 = list(accumulate([0] + ranks2))[:-1] diagonal_indices = [tuple(cumulative_ranks2[i] + k for k in j) for i, arg in diagonals.items() for j in arg.diagonal_indices] return _permute_dims(_array_diagonal(tp, diagonal_indices), _af_invert(inverse_permutation)) return self.func(args, canonicalize=False) def doit(self, *kwargs): deep = kwargs.get("deep", True) if deep: return self.func([arg.doit(*kwargs) for arg in self.args])._canonicalize() else: return self._canonicalize() @classmethod def _flatten(cls, args): args = [i for arg in args for i in (arg.args if isinstance(arg, cls) else [arg])] return args def as_explicit(self): return tensorproduct([arg.as_explicit() if hasattr(arg, "as_explicit") else arg for arg in self.args]) class ArrayAdd(_CodegenArrayAbstract): r""" Class for elementwise array additions. """ def __new__(cls, args, kwargs): args = [_sympify(arg) for arg in args] ranks = [get_rank(arg) for arg in args] ranks = list(set(ranks)) if len(ranks) != 1: raise ValueError("summing arrays of different ranks") shapes = [arg.shape for arg in args] if len({i for i in shapes if i is not None}) > 1: raise ValueError("mismatching shapes in addition") canonicalize = kwargs.pop("canonicalize", False) obj = Basic.__new__(cls, args) obj._subranks = ranks if any(i is None for i in shapes): obj._shape = None else: obj._shape = shapes[0] if canonicalize: return obj._canonicalize() return obj def _canonicalize(self): args = self.args # Flatten: args = self._flatten_args(args) shapes = [get_shape(arg) for arg in args] args = [arg for arg in args if not isinstance(arg, (ZeroArray, ZeroMatrix))] if len(args) == 0: if any(i for i in shapes if i is None): raise NotImplementedError("cannot handle addition of ZeroMatrix/ZeroArray and undefined shape object") return ZeroArray(shapes[0]) elif len(args) == 1: return args[0] return self.func(args, canonicalize=False) def doit(self, *kwargs): deep = kwargs.get("deep", True) if deep: return self.func([arg.doit(kwargs) for arg in self.args])._canonicalize() else: return self._canonicalize() @classmethod def _flatten_args(cls, args): new_args = [] for arg in args: if isinstance(arg, ArrayAdd): new_args.extend(arg.args) else: new_args.append(arg) return new_args def as_explicit(self): return reduce(operator.add, [arg.as_explicit() for arg in self.args]) class PermuteDims(_CodegenArrayAbstract): r""" Class to represent permutation of axes of arrays. Examples ======== >>> from sympy.tensor.array import permutedims >>> from sympy import MatrixSymbol >>> M = MatrixSymbol("M", 3, 3) >>> cg = permutedims(M, [1, 0]) The object ``cg`` represents the transposition of ``M``, as the permutation ``[1, 0]`` will act on its indices by switching them: `M_{ij} \Rightarrow M_{ji}` This is evident when transforming back to matrix form: >>> from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix >>> convert_array_to_matrix(cg) M.T >>> N = MatrixSymbol("N", 3, 2) >>> cg = permutedims(N, [1, 0]) >>> cg.shape (2, 3) Permutations of tensor products are simplified in order to achieve a standard form: >>> from sympy.tensor.array import tensorproduct >>> M = MatrixSymbol("M", 4, 5) >>> tp = tensorproduct(M, N) >>> tp.shape (4, 5, 3, 2) >>> perm1 = permutedims(tp, [2, 3, 1, 0]) The args ``(M, N)`` have been sorted and the permutation has been simplified, the expression is equivalent: >>> perm1.expr.args (N, M) >>> perm1.shape (3, 2, 5, 4) >>> perm1.permutation (2 3) The permutation in its array form has been simplified from ``[2, 3, 1, 0]`` to ``[0, 1, 3, 2]``, as the arguments of the tensor product `M` and `N` have been switched: >>> perm1.permutation.array_form [0, 1, 3, 2] We can nest a second permutation: >>> perm2 = permutedims(perm1, [1, 0, 2, 3]) >>> perm2.shape (2, 3, 5, 4) >>> perm2.permutation.array_form [1, 0, 3, 2] """ def __new__(cls, expr, permutation, kwargs): from sympy.combinatorics import Permutation expr = _sympify(expr) permutation = Permutation(permutation) permutation_size = permutation.size expr_rank = get_rank(expr) if permutation_size != expr_rank: raise ValueError("Permutation size must be the length of the shape of expr") canonicalize = kwargs.pop("canonicalize", False) obj = Basic.__new__(cls, expr, permutation) obj._subranks = [get_rank(expr)] shape = get_shape(expr) if shape is None: obj._shape = None else: obj._shape = tuple(shape[permutation(i)] for i in range(len(shape))) if canonicalize: return obj._canonicalize() return obj def _canonicalize(self): expr = self.expr permutation = self.permutation if isinstance(expr, PermuteDims): subexpr = expr.expr subperm = expr.permutation permutation = permutation * subperm expr = subexpr if isinstance(expr, ArrayContraction): expr, permutation = self._PermuteDims_denestarg_ArrayContraction(expr, permutation) if isinstance(expr, ArrayTensorProduct): expr, permutation = self._PermuteDims_denestarg_ArrayTensorProduct(expr, permutation) if isinstance(expr, (ZeroArray, ZeroMatrix)): return ZeroArray([expr.shape[i] for i in permutation.array_form]) plist = permutation.array_form if plist == sorted(plist): return expr return self.func(expr, permutation, canonicalize=False) def doit(self, kwargs): deep = kwargs.get("deep", True) if deep: return self.func([arg.doit(*kwargs) for arg in self.args])._canonicalize() else: return self._canonicalize() @property def expr(self): return self.args[0] @property def permutation(self): return self.args[1] @classmethod def _PermuteDims_denestarg_ArrayTensorProduct(cls, expr, permutation): # Get the permutation in its image-form: perm_image_form = _af_invert(permutation.array_form) args = list(expr.args) # Starting index global position for every arg: cumul = list(accumulate([0] + expr.subranks)) # Split `perm_image_form` into a list of list corresponding to the indices # of every argument: perm_image_form_in_components = [perm_image_form[cumul[i]:cumul[i+1]] for i in range(len(args))] # Create an index, target-position-key array: ps = [(i, sorted(comp)) for i, comp in enumerate(perm_image_form_in_components)] # Sort the array according to the target-position-key: # In this way, we define a canonical way to sort the arguments according # to the permutation. ps.sort(key=lambda x: x[1]) # Read the inverse-permutation (i.e. image-form) of the args: perm_args_image_form = [i[0] for i in ps] # Apply the args-permutation to the `args`: args_sorted = [args[i] for i in perm_args_image_form] # Apply the args-permutation to the array-form of the permutation of the axes (of `expr`): perm_image_form_sorted_args = [perm_image_form_in_components[i] for i in perm_args_image_form] new_permutation = Permutation(_af_invert([j for i in perm_image_form_sorted_args for j in i])) return _array_tensor_product(args_sorted), new_permutation @classmethod def _PermuteDims_denestarg_ArrayContraction(cls, expr, permutation): if not isinstance(expr, ArrayContraction): return expr, permutation if not isinstance(expr.expr, ArrayTensorProduct): return expr, permutation args = expr.expr.args subranks = [get_rank(arg) for arg in expr.expr.args] contraction_indices = expr.contraction_indices contraction_indices_flat = [j for i in contraction_indices for j in i] cumul = list(accumulate([0] + subranks)) # Spread the permutation in its array form across the args in the corresponding # tensor-product arguments with free indices: permutation_array_blocks_up = [] image_form = _af_invert(permutation.array_form) counter = 0 for i, e in enumerate(subranks): current = [] for j in range(cumul[i], cumul[i+1]): if j in contraction_indices_flat: continue current.append(image_form[counter]) counter += 1 permutation_array_blocks_up.append(current) # Get the map of axis repositioning for every argument of tensor-product: index_blocks = [[j for j in range(cumul[i], cumul[i+1])] for i, e in enumerate(expr.subranks)] index_blocks_up = expr._push_indices_up(expr.contraction_indices, index_blocks) inverse_permutation = permutation*(-1) index_blocks_up_permuted = [[inverse_permutation(j) for j in i if j is not None] for i in index_blocks_up] # Sorting key is a list of tuple, first element is the index of `args`, second element of # the tuple is the sorting key to sort `args` of the tensor product: sorting_keys = list(enumerate(index_blocks_up_permuted)) sorting_keys.sort(key=lambda x: x[1]) # Now we can get the permutation acting on the args in its image-form: new_perm_image_form = [i[0] for i in sorting_keys] # Apply the args-level permutation to various elements: new_index_blocks = [index_blocks[i] for i in new_perm_image_form] new_index_perm_array_form = _af_invert([j for i in new_index_blocks for j in i]) new_args = [args[i] for i in new_perm_image_form] new_contraction_indices = [tuple(new_index_perm_array_form[j] for j in i) for i in contraction_indices] new_expr = _array_contraction(_array_tensor_product(new_args), new_contraction_indices) new_permutation = Permutation(_af_invert([j for i in [permutation_array_blocks_up[k] for k in new_perm_image_form] for j in i])) return new_expr, new_permutation @classmethod def _check_permutation_mapping(cls, expr, permutation): subranks = expr.subranks index2arg = [i for i, arg in enumerate(expr.args) for j in range(expr.subranks[i])] permuted_indices = [permutation(i) for i in range(expr.subrank())] new_args = list(expr.args) arg_candidate_index = index2arg[permuted_indices[0]] current_indices = [] new_permutation = [] inserted_arg_cand_indices = set([]) for i, idx in enumerate(permuted_indices): if index2arg[idx] != arg_candidate_index: new_permutation.extend(current_indices) current_indices = [] arg_candidate_index = index2arg[idx] current_indices.append(idx) arg_candidate_rank = subranks[arg_candidate_index] if len(current_indices) == arg_candidate_rank: new_permutation.extend(sorted(current_indices)) local_current_indices = [j - min(current_indices) for j in current_indices] i1 = index2arg[i] new_args[i1] = _permute_dims(new_args[i1], Permutation(local_current_indices)) inserted_arg_cand_indices.add(arg_candidate_index) current_indices = [] new_permutation.extend(current_indices) # TODO: swap args positions in order to simplify the expression: # TODO: this should be in a function args_positions = list(range(len(new_args))) # Get possible shifts: maps = {} cumulative_subranks = [0] + list(accumulate(subranks)) for i in range(0, len(subranks)): s = set([index2arg[new_permutation[j]] for j in range(cumulative_subranks[i], cumulative_subranks[i+1])]) if len(s) != 1: continue elem = next(iter(s)) if i != elem: maps[i] = elem # Find cycles in the map: lines = [] current_line = [] while maps: if len(current_line) == 0: k, v = maps.popitem() current_line.append(k) else: k = current_line[-1] if k not in maps: current_line = [] continue v = maps.pop(k) if v in current_line: lines.append(current_line) current_line = [] continue current_line.append(v) for line in lines: for i, e in enumerate(line): args_positions[line[(i + 1) % len(line)]] = e # TODO: function in order to permute the args: permutation_blocks = [[new_permutation[cumulative_subranks[i] + j] for j in range(e)] for i, e in enumerate(subranks)] new_args = [new_args[i] for i in args_positions] new_permutation_blocks = [permutation_blocks[i] for i in args_positions] new_permutation2 = [j for i in new_permutation_blocks for j in i] return _array_tensor_product(new_args), Permutation(new_permutation2) # *(-1) @classmethod def _check_if_there_are_closed_cycles(cls, expr, permutation): args = list(expr.args) subranks = expr.subranks cyclic_form = permutation.cyclic_form cumulative_subranks = [0] + list(accumulate(subranks)) cyclic_min = [min(i) for i in cyclic_form] cyclic_max = [max(i) for i in cyclic_form] cyclic_keep = [] for i, cycle in enumerate(cyclic_form): flag = True for j in range(0, len(cumulative_subranks) - 1): if cyclic_min[i] >= cumulative_subranks[j] and cyclic_max[i] < cumulative_subranks[j+1]: # Found a sinkable cycle. args[j] = _permute_dims(args[j], Permutation([[k - cumulative_subranks[j] for k in cyclic_form[i]]])) flag = False break if flag: cyclic_keep.append(cyclic_form[i]) return _array_tensor_product(args), Permutation(cyclic_keep, size=permutation.size) def nest_permutation(self): r""" DEPRECATED. """ ret = self._nest_permutation(self.expr, self.permutation) if ret is None: return self return ret @classmethod def _nest_permutation(cls, expr, permutation): if isinstance(expr, ArrayTensorProduct): return _permute_dims(cls._check_if_there_are_closed_cycles(expr, permutation)) elif isinstance(expr, ArrayContraction): # Invert tree hierarchy: put the contraction above. cycles = permutation.cyclic_form newcycles = ArrayContraction._convert_outer_indices_to_inner_indices(expr, cycles) newpermutation = Permutation(newcycles) new_contr_indices = [tuple(newpermutation(j) for j in i) for i in expr.contraction_indices] return _array_contraction(PermuteDims(expr.expr, newpermutation), new_contr_indices) elif isinstance(expr, ArrayAdd): return _array_add([PermuteDims(arg, permutation) for arg in expr.args]) return None def as_explicit(self): return permutedims(self.expr.as_explicit(), self.permutation) class ArrayDiagonal(_CodegenArrayAbstract): r""" Class to represent the diagonal operator. Explanation =========== In a 2-dimensional array it returns the diagonal, this looks like the operation: `A_{ij} \rightarrow A_{ii}` The diagonal over axes 1 and 2 (the second and third) of the tensor product of two 2-dimensional arrays `A \otimes B` is `\Big[ A_{ab} B_{cd} \Big]_{abcd} \rightarrow \Big[ A_{ai} B_{id} \Big]_{adi}` In this last example the array expression has been reduced from 4-dimensional to 3-dimensional. Notice that no contraction has occurred, rather there is a new index `i` for the diagonal, contraction would have reduced the array to 2 dimensions. Notice that the diagonalized out dimensions are added as new dimensions at the end of the indices. """ def __new__(cls, expr, diagonal_indices, kwargs): expr = _sympify(expr) diagonal_indices = [Tuple(sorted(i)) for i in diagonal_indices] canonicalize = kwargs.get("canonicalize", False) shape = get_shape(expr) if shape is not None: cls._validate(expr, diagonal_indices, kwargs) # Get new shape: positions, shape = cls._get_positions_shape(shape, diagonal_indices) else: positions = None if len(diagonal_indices) == 0: return expr obj = Basic.__new__(cls, expr, diagonal_indices) obj._positions = positions obj._subranks = _get_subranks(expr) obj._shape = shape if canonicalize: return obj._canonicalize() return obj def _canonicalize(self): expr = self.expr diagonal_indices = self.diagonal_indices trivial_diags = [i for i in diagonal_indices if len(i) == 1] if len(trivial_diags) > 0: trivial_pos = {e[0]: i for i, e in enumerate(diagonal_indices) if len(e) == 1} diag_pos = {e: i for i, e in enumerate(diagonal_indices) if len(e) > 1} diagonal_indices_short = [i for i in diagonal_indices if len(i) > 1] rank1 = get_rank(self) rank2 = len(diagonal_indices) rank3 = rank1 - rank2 inv_permutation = [] counter1: int = 0 indices_down = ArrayDiagonal._push_indices_down(diagonal_indices_short, list(range(rank1)), get_rank(expr)) for i in indices_down: if i in trivial_pos: inv_permutation.append(rank3 + trivial_pos[i]) elif isinstance(i, (Integer, int)): inv_permutation.append(counter1) counter1 += 1 else: inv_permutation.append(rank3 + diag_pos[i]) permutation = _af_invert(inv_permutation) if len(diagonal_indices_short) > 0: return _permute_dims(_array_diagonal(expr, diagonal_indices_short), permutation) else: return _permute_dims(expr, permutation) if isinstance(expr, ArrayAdd): return self._ArrayDiagonal_denest_ArrayAdd(expr, diagonal_indices) if isinstance(expr, ArrayDiagonal): return self._ArrayDiagonal_denest_ArrayDiagonal(expr, diagonal_indices) if isinstance(expr, PermuteDims): return self._ArrayDiagonal_denest_PermuteDims(expr, diagonal_indices) if isinstance(expr, (ZeroArray, ZeroMatrix)): positions, shape = self._get_positions_shape(expr.shape, diagonal_indices) return ZeroArray(shape) return self.func(expr, diagonal_indices, canonicalize=False) def doit(self, *kwargs): deep = kwargs.get("deep", True) if deep: return self.func([arg.doit(*kwargs) for arg in self.args])._canonicalize() else: return self._canonicalize() @staticmethod def _validate(expr, diagonal_indices, *kwargs): # Check that no diagonalization happens on indices with mismatched # dimensions: shape = get_shape(expr) for i in diagonal_indices: if any(j >= len(shape) for j in i): raise ValueError("index is larger than expression shape") if len({shape[j] for j in i}) != 1: raise ValueError("diagonalizing indices of different dimensions") if not kwargs.get("allow_trivial_diags", False) and len(i) <= 1: raise ValueError("need at least two axes to diagonalize") if len(set(i)) != len(i): raise ValueError("axis index cannot be repeated") @staticmethod def _remove_trivial_dimensions(shape, diagonal_indices): return [tuple(j for j in i) for i in diagonal_indices if shape[i[0]] != 1] @property def expr(self): return self.args[0] @property def diagonal_indices(self): return self.args[1:] @staticmethod def _flatten(expr, outer_diagonal_indices): inner_diagonal_indices = expr.diagonal_indices all_inner = [j for i in inner_diagonal_indices for j in i] all_inner.sort() # TODO: add API for total rank and cumulative rank: total_rank = _get_subrank(expr) inner_rank = len(all_inner) outer_rank = total_rank - inner_rank shifts = [0 for i in range(outer_rank)] counter = 0 pointer = 0 for i in range(outer_rank): while pointer < inner_rank and counter >= all_inner[pointer]: counter += 1 pointer += 1 shifts[i] += pointer counter += 1 outer_diagonal_indices = tuple(tuple(shifts[j] + j for j in i) for i in outer_diagonal_indices) diagonal_indices = inner_diagonal_indices + outer_diagonal_indices return _array_diagonal(expr.expr, diagonal_indices) @classmethod def _ArrayDiagonal_denest_ArrayAdd(cls, expr, diagonal_indices): return _array_add([_array_diagonal(arg, diagonal_indices) for arg in expr.args]) @classmethod def _ArrayDiagonal_denest_ArrayDiagonal(cls, expr, diagonal_indices): return cls._flatten(expr, diagonal_indices) @classmethod def _ArrayDiagonal_denest_PermuteDims(cls, expr: PermuteDims, diagonal_indices): back_diagonal_indices = [[expr.permutation(j) for j in i] for i in diagonal_indices] nondiag = [i for i in range(get_rank(expr)) if not any(i in j for j in diagonal_indices)] back_nondiag = [expr.permutation(i) for i in nondiag] remap = {e: i for i, e in enumerate(sorted(back_nondiag))} new_permutation1 = [remap[i] for i in back_nondiag] shift = len(new_permutation1) diag_block_perm = [i + shift for i in range(len(back_diagonal_indices))] new_permutation = new_permutation1 + diag_block_perm return _permute_dims( _array_diagonal( expr.expr, back_diagonal_indices ), new_permutation ) def _push_indices_down_nonstatic(self, indices): transform = lambda x: self._positions[x] if x < len(self._positions) else None return _apply_recursively_over_nested_lists(transform, indices) def _push_indices_up_nonstatic(self, indices): def transform(x): for i, e in enumerate(self._positions): if (isinstance(e, int) and x == e) or (isinstance(e, tuple) and x in e): return i return _apply_recursively_over_nested_lists(transform, indices) @classmethod def _push_indices_down(cls, diagonal_indices, indices, rank): positions, shape = cls._get_positions_shape(range(rank), diagonal_indices) transform = lambda x: positions[x] if x < len(positions) else None return _apply_recursively_over_nested_lists(transform, indices) @classmethod def _push_indices_up(cls, diagonal_indices, indices, rank): positions, shape = cls._get_positions_shape(range(rank), diagonal_indices) def transform(x): for i, e in enumerate(positions): if (isinstance(e, int) and x == e) or (isinstance(e, (tuple, Tuple)) and (x in e)): return i return _apply_recursively_over_nested_lists(transform, indices) @classmethod def _get_positions_shape(cls, shape, diagonal_indices): data1 = tuple((i, shp) for i, shp in enumerate(shape) if not any(i in j for j in diagonal_indices)) pos1, shp1 = zip(data1) if data1 else ((), ()) data2 = tuple((i, shape[i[0]]) for i in diagonal_indices) pos2, shp2 = zip(data2) if data2 else ((), ()) positions = pos1 + pos2 shape = shp1 + shp2 return positions, shape def as_explicit(self): return tensordiagonal(self.expr.as_explicit(), self.diagonal_indices) class ArrayElementwiseApplyFunc(_CodegenArrayAbstract): def __new__(cls, function, element): if not isinstance(function, Lambda): d = Dummy('d') function = Lambda(d, function(d)) obj = _CodegenArrayAbstract.__new__(cls, function, element) obj._subranks = _get_subranks(element) return obj @property def function(self): return self.args[0] @property def expr(self): return self.args[1] @property def shape(self): return self.expr.shape def _get_function_fdiff(self): d = Dummy("d") function = self.function(d) fdiff = function.diff(d) if isinstance(fdiff, Function): fdiff = type(fdiff) else: fdiff = Lambda(d, fdiff) return fdiff class ArrayContraction(_CodegenArrayAbstract): r""" This class is meant to represent contractions of arrays in a form easily processable by the code printers. """ def __new__(cls, expr, contraction_indices, kwargs): contraction_indices = _sort_contraction_indices(contraction_indices) expr = _sympify(expr) canonicalize = kwargs.get("canonicalize", False) obj = Basic.__new__(cls, expr, contraction_indices) obj._subranks = _get_subranks(expr) obj._mapping = _get_mapping_from_subranks(obj._subranks) free_indices_to_position = {i: i for i in range(sum(obj._subranks)) if all(i not in cind for cind in contraction_indices)} obj._free_indices_to_position = free_indices_to_position shape = get_shape(expr) cls._validate(expr, contraction_indices) if shape: shape = tuple(shp for i, shp in enumerate(shape) if not any(i in j for j in contraction_indices)) obj._shape = shape if canonicalize: return obj._canonicalize() return obj def _canonicalize(self): expr = self.expr contraction_indices = self.contraction_indices if len(contraction_indices) == 0: return expr if isinstance(expr, ArrayContraction): return self._ArrayContraction_denest_ArrayContraction(expr, contraction_indices) if isinstance(expr, (ZeroArray, ZeroMatrix)): return self._ArrayContraction_denest_ZeroArray(expr, contraction_indices) if isinstance(expr, PermuteDims): return self._ArrayContraction_denest_PermuteDims(expr, contraction_indices) if isinstance(expr, ArrayTensorProduct): expr, contraction_indices = self._sort_fully_contracted_args(expr, contraction_indices) expr, contraction_indices = self._lower_contraction_to_addends(expr, contraction_indices) if len(contraction_indices) == 0: return expr if isinstance(expr, ArrayDiagonal): return self._ArrayContraction_denest_ArrayDiagonal(expr, contraction_indices) if isinstance(expr, ArrayAdd): return self._ArrayContraction_denest_ArrayAdd(expr, contraction_indices) # Check single index contractions on 1-dimensional axes: contraction_indices = [i for i in contraction_indices if len(i) > 1 or get_shape(expr)[i[0]] != 1] if len(contraction_indices) == 0: return expr return self.func(expr, contraction_indices, canonicalize=False) def doit(self, kwargs): deep = kwargs.get("deep", True) if deep: return self.func([arg.doit(*kwargs) for arg in self.args])._canonicalize() else: return self._canonicalize() def __mul__(self, other): if other == 1: return self else: raise NotImplementedError("Product of N-dim arrays is not uniquely defined. Use another method.") def __rmul__(self, other): if other == 1: return self else: raise NotImplementedError("Product of N-dim arrays is not uniquely defined. Use another method.") @staticmethod def _validate(expr, contraction_indices): shape = get_shape(expr) if shape is None: return # Check that no contraction happens when the shape is mismatched: for i in contraction_indices: if len({shape[j] for j in i if shape[j] != -1}) != 1: raise ValueError("contracting indices of different dimensions") @classmethod def _push_indices_down(cls, contraction_indices, indices): flattened_contraction_indices = [j for i in contraction_indices for j in i] flattened_contraction_indices.sort() transform = _build_push_indices_down_func_transformation(flattened_contraction_indices) return _apply_recursively_over_nested_lists(transform, indices) @classmethod def _push_indices_up(cls, contraction_indices, indices): flattened_contraction_indices = [j for i in contraction_indices for j in i] flattened_contraction_indices.sort() transform = _build_push_indices_up_func_transformation(flattened_contraction_indices) return _apply_recursively_over_nested_lists(transform, indices) @classmethod def _lower_contraction_to_addends(cls, expr, contraction_indices): if isinstance(expr, ArrayAdd): raise NotImplementedError() if not isinstance(expr, ArrayTensorProduct): return expr, contraction_indices subranks = expr.subranks cumranks = list(accumulate([0] + subranks)) contraction_indices_remaining = [] contraction_indices_args = [[] for i in expr.args] backshift = set([]) for i, contraction_group in enumerate(contraction_indices): for j in range(len(expr.args)): if not isinstance(expr.args[j], ArrayAdd): continue if all(cumranks[j] <= k < cumranks[j+1] for k in contraction_group): contraction_indices_args[j].append([k - cumranks[j] for k in contraction_group]) backshift.update(contraction_group) break else: contraction_indices_remaining.append(contraction_group) if len(contraction_indices_remaining) == len(contraction_indices): return expr, contraction_indices total_rank = get_rank(expr) shifts = list(accumulate([1 if i in backshift else 0 for i in range(total_rank)])) contraction_indices_remaining = [Tuple.fromiter(j - shifts[j] for j in i) for i in contraction_indices_remaining] ret = _array_tensor_product([ _array_contraction(arg, contr) for arg, contr in zip(expr.args, contraction_indices_args) ]) return ret, contraction_indices_remaining def split_multiple_contractions(self): """ Recognize multiple contractions and attempt at rewriting them as paired-contractions. This allows some contractions involving more than two indices to be rewritten as multiple contractions involving two indices, thus allowing the expression to be rewritten as a matrix multiplication line. Examples: * `A_ij b_j0 C_jk` ===> `ADiagMatrix(b)C` Care for: - matrix being diagonalized (i.e. `A_ii`) - vectors being diagonalized (i.e. `a_i0`) Multiple contractions can be split into matrix multiplications if not more than two arguments are non-diagonals or non-vectors. Vectors get diagonalized while diagonal matrices remain diagonal. The non-diagonal matrices can be at the beginning or at the end of the final matrix multiplication line. """ editor = _EditArrayContraction(self) contraction_indices = self.contraction_indices onearray_insert = [] for indl, links in enumerate(contraction_indices): if len(links) <= 2: continue # Check multiple contractions: # # Examples: # # * `A_ij b_j0 C_jk` ===> `ADiagMatrix(b)C \otimes OneArray(1)` with permutation (1 2) # # Care for: # - matrix being diagonalized (i.e. `A_ii`) # - vectors being diagonalized (i.e. `a_i0`) # Multiple contractions can be split into matrix multiplications if # not more than three arguments are non-diagonals or non-vectors. # # Vectors get diagonalized while diagonal matrices remain diagonal. # The non-diagonal matrices can be at the beginning or at the end # of the final matrix multiplication line. positions = editor.get_mapping_for_index(indl) # Also consider the case of diagonal matrices being contracted: current_dimension = self.expr.shape[links[0]] not_vectors: tTuple[_ArgE, int] = [] vectors: tTuple[_ArgE, int] = [] for arg_ind, rel_ind in positions: arg = editor.args_with_ind[arg_ind] mat = arg.element abs_arg_start, abs_arg_end = editor.get_absolute_range(arg) other_arg_pos = 1-rel_ind other_arg_abs = abs_arg_start + other_arg_pos if ((1 not in mat.shape) or ((current_dimension == 1) is True and mat.shape != (1, 1)) or any(other_arg_abs in l for li, l in enumerate(contraction_indices) if li != indl) ): not_vectors.append((arg, rel_ind)) else: vectors.append((arg, rel_ind)) if len(not_vectors) > 2: # If more than two arguments in the multiple contraction are # non-vectors and non-diagonal matrices, we cannot find a way # to split this contraction into a matrix multiplication line: continue # Three cases to handle: # - zero non-vectors # - one non-vector # - two non-vectors for v, rel_ind in vectors: v.element = diagonalize_vector(v.element) vectors_to_loop = not_vectors[:1] + vectors + not_vectors[1:] first_not_vector, rel_ind = vectors_to_loop[0] new_index = first_not_vector.indices[rel_ind] for v, rel_ind in vectors_to_loop[1:-1]: v.indices[rel_ind] = new_index new_index = editor.get_new_contraction_index() assert v.indices.index(None) == 1 - rel_ind v.indices[v.indices.index(None)] = new_index onearray_insert.append(v) last_vec, rel_ind = vectors_to_loop[-1] last_vec.indices[rel_ind] = new_index for v in onearray_insert: editor.insert_after(v, _ArgE(OneArray(1), [None])) return editor.to_array_contraction() def flatten_contraction_of_diagonal(self): if not isinstance(self.expr, ArrayDiagonal): return self contraction_down = self.expr._push_indices_down(self.expr.diagonal_indices, self.contraction_indices) new_contraction_indices = [] diagonal_indices = self.expr.diagonal_indices[:] for i in contraction_down: contraction_group = list(i) for j in i: diagonal_with = [k for k in diagonal_indices if j in k] contraction_group.extend([l for k in diagonal_with for l in k]) diagonal_indices = [k for k in diagonal_indices if k not in diagonal_with] new_contraction_indices.append(sorted(set(contraction_group))) new_contraction_indices = ArrayDiagonal._push_indices_up(diagonal_indices, new_contraction_indices) return _array_contraction( _array_diagonal( self.expr.expr, diagonal_indices ), new_contraction_indices ) @staticmethod def _get_free_indices_to_position_map(free_indices, contraction_indices): free_indices_to_position = {} flattened_contraction_indices = [j for i in contraction_indices for j in i] counter = 0 for ind in free_indices: while counter in flattened_contraction_indices: counter += 1 free_indices_to_position[ind] = counter counter += 1 return free_indices_to_position @staticmethod def _get_index_shifts(expr): """ Get the mapping of indices at the positions before the contraction occurs. Examples ======== >>> from sympy.tensor.array import tensorproduct, tensorcontraction >>> from sympy import MatrixSymbol >>> M = MatrixSymbol("M", 3, 3) >>> N = MatrixSymbol("N", 3, 3) >>> cg = tensorcontraction(tensorproduct(M, N), [1, 2]) >>> cg._get_index_shifts(cg) [0, 2] Indeed, ``cg`` after the contraction has two dimensions, 0 and 1. They need to be shifted by 0 and 2 to get the corresponding positions before the contraction (that is, 0 and 3). """ inner_contraction_indices = expr.contraction_indices all_inner = [j for i in inner_contraction_indices for j in i] all_inner.sort() # TODO: add API for total rank and cumulative rank: total_rank = _get_subrank(expr) inner_rank = len(all_inner) outer_rank = total_rank - inner_rank shifts = [0 for i in range(outer_rank)] counter = 0 pointer = 0 for i in range(outer_rank): while pointer < inner_rank and counter >= all_inner[pointer]: counter += 1 pointer += 1 shifts[i] += pointer counter += 1 return shifts @staticmethod def _convert_outer_indices_to_inner_indices(expr, outer_contraction_indices): shifts = ArrayContraction._get_index_shifts(expr) outer_contraction_indices = tuple(tuple(shifts[j] + j for j in i) for i in outer_contraction_indices) return outer_contraction_indices @staticmethod def _flatten(expr, outer_contraction_indices): inner_contraction_indices = expr.contraction_indices outer_contraction_indices = ArrayContraction._convert_outer_indices_to_inner_indices(expr, outer_contraction_indices) contraction_indices = inner_contraction_indices + outer_contraction_indices return _array_contraction(expr.expr, contraction_indices) @classmethod def _ArrayContraction_denest_ArrayContraction(cls, expr, contraction_indices): return cls._flatten(expr, contraction_indices) @classmethod def _ArrayContraction_denest_ZeroArray(cls, expr, contraction_indices): contraction_indices_flat = [j for i in contraction_indices for j in i] shape = [e for i, e in enumerate(expr.shape) if i not in contraction_indices_flat] return ZeroArray(shape) @classmethod def _ArrayContraction_denest_ArrayAdd(cls, expr, contraction_indices): return _array_add([_array_contraction(i, contraction_indices) for i in expr.args]) @classmethod def _ArrayContraction_denest_PermuteDims(cls, expr, contraction_indices): permutation = expr.permutation plist = permutation.array_form new_contraction_indices = [tuple(permutation(j) for j in i) for i in contraction_indices] new_plist = [i for i in plist if not any(i in j for j in new_contraction_indices)] new_plist = cls._push_indices_up(new_contraction_indices, new_plist) return _permute_dims( _array_contraction(expr.expr, new_contraction_indices), Permutation(new_plist) ) @classmethod def _ArrayContraction_denest_ArrayDiagonal(cls, expr: 'ArrayDiagonal', contraction_indices): diagonal_indices = list(expr.diagonal_indices) down_contraction_indices = expr._push_indices_down(expr.diagonal_indices, contraction_indices, get_rank(expr.expr)) # Flatten diagonally contracted indices: down_contraction_indices = [[k for j in i for k in (j if isinstance(j, (tuple, Tuple)) else [j])] for i in down_contraction_indices] new_contraction_indices = [] for contr_indgrp in down_contraction_indices: ind = contr_indgrp[:] for j, diag_indgrp in enumerate(diagonal_indices): if diag_indgrp is None: continue if any(i in diag_indgrp for i in contr_indgrp): ind.extend(diag_indgrp) diagonal_indices[j] = None new_contraction_indices.append(sorted(set(ind))) new_diagonal_indices_down = [i for i in diagonal_indices if i is not None] new_diagonal_indices = ArrayContraction._push_indices_up(new_contraction_indices, new_diagonal_indices_down) return _array_diagonal( _array_contraction(expr.expr, new_contraction_indices), new_diagonal_indices ) @classmethod def _sort_fully_contracted_args(cls, expr, contraction_indices): if expr.shape is None: return expr, contraction_indices cumul = list(accumulate([0] + expr.subranks)) index_blocks = [list(range(cumul[i], cumul[i+1])) for i in range(len(expr.args))] contraction_indices_flat = {j for i in contraction_indices for j in i} fully_contracted = [all(j in contraction_indices_flat for j in range(cumul[i], cumul[i+1])) for i, arg in enumerate(expr.args)] new_pos = sorted(range(len(expr.args)), key=lambda x: (0, default_sort_key(expr.args[x])) if fully_contracted[x] else (1,)) new_args = [expr.args[i] for i in new_pos] new_index_blocks_flat = [j for i in new_pos for j in index_blocks[i]] index_permutation_array_form = _af_invert(new_index_blocks_flat) new_contraction_indices = [tuple(index_permutation_array_form[j] for j in i) for i in contraction_indices] new_contraction_indices = _sort_contraction_indices(new_contraction_indices) return _array_tensor_product(new_args), new_contraction_indices def _get_contraction_tuples(self): r""" Return tuples containing the argument index and position within the argument of the index position. Examples ======== >>> from sympy import MatrixSymbol >>> from sympy.abc import N >>> from sympy.tensor.array import tensorproduct, tensorcontraction >>> A = MatrixSymbol("A", N, N) >>> B = MatrixSymbol("B", N, N) >>> cg = tensorcontraction(tensorproduct(A, B), (1, 2)) >>> cg._get_contraction_tuples() [[(0, 1), (1, 0)]] Notes ===== Here the contraction pair `(1, 2)` meaning that the 2nd and 3rd indices of the tensor product `A\otimes B` are contracted, has been transformed into `(0, 1)` and `(1, 0)`, identifying the same indices in a different notation. `(0, 1)` is the second index (1) of the first argument (i.e. 0 or `A`). `(1, 0)` is the first index (i.e. 0) of the second argument (i.e. 1 or `B`). """ mapping = self._mapping return [[mapping[j] for j in i] for i in self.contraction_indices] @staticmethod def _contraction_tuples_to_contraction_indices(expr, contraction_tuples): # TODO: check that `expr` has `.subranks`: ranks = expr.subranks cumulative_ranks = [0] + list(accumulate(ranks)) return [tuple(cumulative_ranks[j]+k for j, k in i) for i in contraction_tuples] @property def free_indices(self): return self._free_indices[:] @property def free_indices_to_position(self): return dict(self._free_indices_to_position) @property def expr(self): return self.args[0] @property def contraction_indices(self): return self.args[1:] def _contraction_indices_to_components(self): expr = self.expr if not isinstance(expr, ArrayTensorProduct): raise NotImplementedError("only for contractions of tensor products") ranks = expr.subranks mapping = {} counter = 0 for i, rank in enumerate(ranks): for j in range(rank): mapping[counter] = (i, j) counter += 1 return mapping def sort_args_by_name(self): """ Sort arguments in the tensor product so that their order is lexicographical. Examples ======== >>> from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array >>> from sympy import MatrixSymbol >>> from sympy.abc import N >>> A = MatrixSymbol("A", N, N) >>> B = MatrixSymbol("B", N, N) >>> C = MatrixSymbol("C", N, N) >>> D = MatrixSymbol("D", N, N) >>> cg = convert_matrix_to_array(CDAB) >>> cg ArrayContraction(ArrayTensorProduct(A, D, C, B), (0, 3), (1, 6), (2, 5)) >>> cg.sort_args_by_name() ArrayContraction(ArrayTensorProduct(A, D, B, C), (0, 3), (1, 4), (2, 7)) """ expr = self.expr if not isinstance(expr, ArrayTensorProduct): return self args = expr.args sorted_data = sorted(enumerate(args), key=lambda x: default_sort_key(x[1])) pos_sorted, args_sorted = zip(sorted_data) reordering_map = {i: pos_sorted.index(i) for i, arg in enumerate(args)} contraction_tuples = self._get_contraction_tuples() contraction_tuples = [[(reordering_map[j], k) for j, k in i] for i in contraction_tuples] c_tp = _array_tensor_product(args_sorted) new_contr_indices = self._contraction_tuples_to_contraction_indices( c_tp, contraction_tuples ) return _array_contraction(c_tp, new_contr_indices) def _get_contraction_links(self): r""" Returns a dictionary of links between arguments in the tensor product being contracted. See the example for an explanation of the values. Examples ======== >>> from sympy import MatrixSymbol >>> from sympy.abc import N >>> from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array >>> A = MatrixSymbol("A", N, N) >>> B = MatrixSymbol("B", N, N) >>> C = MatrixSymbol("C", N, N) >>> D = MatrixSymbol("D", N, N) Matrix multiplications are pairwise contractions between neighboring matrices: `A_{ij} B_{jk} C_{kl} D_{lm}` >>> cg = convert_matrix_to_array(ABCD) >>> cg ArrayContraction(ArrayTensorProduct(B, C, A, D), (0, 5), (1, 2), (3, 6)) >>> cg._get_contraction_links() {0: {0: (2, 1), 1: (1, 0)}, 1: {0: (0, 1), 1: (3, 0)}, 2: {1: (0, 0)}, 3: {0: (1, 1)}} This dictionary is interpreted as follows: argument in position 0 (i.e. matrix `A`) has its second index (i.e. 1) contracted to `(1, 0)`, that is argument in position 1 (matrix `B`) on the first index slot of `B`, this is the contraction provided by the index `j` from `A`. The argument in position 1 (that is, matrix `B`) has two contractions, the ones provided by the indices `j` and `k`, respectively the first and second indices (0 and 1 in the sub-dict). The link `(0, 1)` and `(2, 0)` respectively. `(0, 1)` is the index slot 1 (the 2nd) of argument in position 0 (that is, `A_{\ldot j}`), and so on. """ args, dlinks = _get_contraction_links([self], self.subranks, self.contraction_indices) return dlinks def as_explicit(self): return tensorcontraction(self.expr.as_explicit(), self.contraction_indices) class Reshape(_CodegenArrayAbstract): def __new__(cls, expr, shape): expr = _sympify(expr) if not isinstance(shape, Tuple): shape = Tuple(shape) if Equality(Mul.fromiter(expr.shape), Mul.fromiter(shape)) == False: raise ValueError("shape mismatch") obj = Expr.__new__(cls, expr, shape) obj._shape = tuple(shape) obj._expr = expr return obj @property def shape(self): return self._shape @property def expr(self): return self._expr def doit(self, args, *kwargs): if kwargs.get("deep", True): expr = self.expr.doit(args, *kwargs) else: expr = self.expr if isinstance(expr, (MatrixCommon, NDimArray)): return expr.reshape(self.shape) return Reshape(expr, self.shape) def as_explicit(self): ee = self.expr.as_explicit() if isinstance(ee, MatrixCommon): from sympy import Array ee = Array(ee) elif isinstance(ee, MatrixExpr): return self return ee.reshape(self.shape) class _ArgE: """ The ``_ArgE`` object contains references to the array expression (``.element``) and a list containing the information about index contractions (``.indices``). Index contractions are numbered and contracted indices show the number of the contraction. Uncontracted indices have ``None`` value. For example: ``_ArgE(M, [None, 3])`` This object means that expression ``M`` is part of an array contraction and has two indices, the first is not contracted (value ``None``), the second index is contracted to the 4th (i.e. number ``3``) group of the array contraction object. """ indices: List[Optional[int]] def __init__(self, element, indices: Optional[List[Optional[int]]] = None): self.element = element if indices is None: self.indices = [None for i in range(get_rank(element))] else: self.indices = indices def __str__(self): return "_ArgE(%s, %s)" % (self.element, self.indices) __repr__ = __str__ class _IndPos: """ Index position, requiring two integers in the constructor: - arg: the position of the argument in the tensor product, - rel: the relative position of the index inside the argument. """ def __init__(self, arg: int, rel: int): self.arg = arg self.rel = rel def __str__(self): return "_IndPos(%i, %i)" % (self.arg, self.rel) __repr__ = __str__ def __iter__(self): yield from [self.arg, self.rel] class _EditArrayContraction: """ Utility class to help manipulate array contraction objects. This class takes as input an ``ArrayContraction`` object and turns it into an editable object. The field ``args_with_ind`` of this class is a list of ``_ArgE`` objects which can be used to easily edit the contraction structure of the expression. Once editing is finished, the ``ArrayContraction`` object may be recreated by calling the ``.to_array_contraction()`` method. """ def __init__(self, base_array: typing.Union[ArrayContraction, ArrayDiagonal, ArrayTensorProduct]): expr: Basic diagonalized: tTuple[tTuple[int, ...], ...] contraction_indices: List[tTuple[int]] if isinstance(base_array, ArrayContraction): mapping = _get_mapping_from_subranks(base_array.subranks) expr = base_array.expr contraction_indices = base_array.contraction_indices diagonalized = () elif isinstance(base_array, ArrayDiagonal): if isinstance(base_array.expr, ArrayContraction): mapping = _get_mapping_from_subranks(base_array.expr.subranks) expr = base_array.expr.expr diagonalized = ArrayContraction._push_indices_down(base_array.expr.contraction_indices, base_array.diagonal_indices) contraction_indices = base_array.expr.contraction_indices elif isinstance(base_array.expr, ArrayTensorProduct): mapping = {} expr = base_array.expr diagonalized = base_array.diagonal_indices contraction_indices = [] else: mapping = {} expr = base_array.expr diagonalized = base_array.diagonal_indices contraction_indices = [] elif isinstance(base_array, ArrayTensorProduct): expr = base_array contraction_indices = [] diagonalized = () else: raise NotImplementedError() if isinstance(expr, ArrayTensorProduct): args = list(expr.args) else: args = [expr] args_with_ind: List[_ArgE] = [_ArgE(arg) for arg in args] for i, contraction_tuple in enumerate(contraction_indices): for j in contraction_tuple: arg_pos, rel_pos = mapping[j] args_with_ind[arg_pos].indices[rel_pos] = i self.args_with_ind: List[_ArgE] = args_with_ind self.number_of_contraction_indices: int = len(contraction_indices) self._track_permutation: Optional[List[List[int]]] = None mapping = _get_mapping_from_subranks(base_array.subranks) # Trick: add diagonalized indices as negative indices into the editor object: for i, e in enumerate(diagonalized): for j in e: arg_pos, rel_pos = mapping[j] self.args_with_ind[arg_pos].indices[rel_pos] = -1 - i def insert_after(self, arg: _ArgE, new_arg: _ArgE): pos = self.args_with_ind.index(arg) self.args_with_ind.insert(pos + 1, new_arg) def get_new_contraction_index(self): self.number_of_contraction_indices += 1 return self.number_of_contraction_indices - 1 def refresh_indices(self): updates: tDict[int, int] = {} for arg_with_ind in self.args_with_ind: updates.update({i: -1 for i in arg_with_ind.indices if i is not None}) for i, e in enumerate(sorted(updates)): updates[e] = i self.number_of_contraction_indices: int = len(updates) for arg_with_ind in self.args_with_ind: arg_with_ind.indices = [updates.get(i, None) for i in arg_with_ind.indices] def merge_scalars(self): scalars = [] for arg_with_ind in self.args_with_ind: if len(arg_with_ind.indices) == 0: scalars.append(arg_with_ind) for i in scalars: self.args_with_ind.remove(i) scalar = Mul.fromiter([i.element for i in scalars]) if len(self.args_with_ind) == 0: self.args_with_ind.append(_ArgE(scalar)) else: from sympy.tensor.array.expressions.conv_array_to_matrix import _a2m_tensor_product self.args_with_ind[0].element = _a2m_tensor_product(scalar, self.args_with_ind[0].element) def to_array_contraction(self): # Count the ranks of the arguments: counter = 0 # Create a collector for the new diagonal indices: diag_indices = defaultdict(list) count_index_freq = Counter() for arg_with_ind in self.args_with_ind: count_index_freq.update(Counter(arg_with_ind.indices)) free_index_count = count_index_freq[None] # Construct the inverse permutation: inv_perm1 = [] inv_perm2 = [] # Keep track of which diagonal indices have already been processed: done = set([]) # Counter for the diagonal indices: counter4 = 0 for arg_with_ind in self.args_with_ind: # If some diagonalization axes have been removed, they should be # permuted in order to keep the permutation. # Add permutation here counter2 = 0 # counter for the indices for i in arg_with_ind.indices: if i is None: inv_perm1.append(counter4) counter2 += 1 counter4 += 1 continue if i >= 0: continue # Reconstruct the diagonal indices: diag_indices[-1 - i].append(counter + counter2) if count_index_freq[i] == 1 and i not in done: inv_perm1.append(free_index_count - 1 - i) done.add(i) elif i not in done: inv_perm2.append(free_index_count - 1 - i) done.add(i) counter2 += 1 # Remove negative indices to restore a proper editor object: arg_with_ind.indices = [i if i is not None and i >= 0 else None for i in arg_with_ind.indices] counter += len([i for i in arg_with_ind.indices if i is None or i < 0]) inverse_permutation = inv_perm1 + inv_perm2 permutation = _af_invert(inverse_permutation) # Get the diagonal indices after the detection of HadamardProduct in the expression: diag_indices_filtered = [tuple(v) for v in diag_indices.values() if len(v) > 1] self.merge_scalars() self.refresh_indices() args = [arg.element for arg in self.args_with_ind] contraction_indices = self.get_contraction_indices() expr = _array_contraction(_array_tensor_product(args), contraction_indices) expr2 = _array_diagonal(expr, diag_indices_filtered) if self._track_permutation is not None: permutation2 = _af_invert([j for i in self._track_permutation for j in i]) expr2 = _permute_dims(expr2, permutation2) expr3 = _permute_dims(expr2, permutation) return expr3 def get_contraction_indices(self) -> List[List[int]]: contraction_indices: List[List[int]] = [[] for i in range(self.number_of_contraction_indices)] current_position: int = 0 for i, arg_with_ind in enumerate(self.args_with_ind): for j in arg_with_ind.indices: if j is not None: contraction_indices[j].append(current_position) current_position += 1 return contraction_indices def get_mapping_for_index(self, ind) -> List[_IndPos]: if ind >= self.number_of_contraction_indices: raise ValueError("index value exceeding the index range") positions: List[_IndPos] = [] for i, arg_with_ind in enumerate(self.args_with_ind): for j, arg_ind in enumerate(arg_with_ind.indices): if ind == arg_ind: positions.append(_IndPos(i, j)) return positions def get_contraction_indices_to_ind_rel_pos(self) -> List[List[_IndPos]]: contraction_indices: List[List[_IndPos]] = [[] for i in range(self.number_of_contraction_indices)] for i, arg_with_ind in enumerate(self.args_with_ind): for j, ind in enumerate(arg_with_ind.indices): if ind is not None: contraction_indices[ind].append(_IndPos(i, j)) return contraction_indices def count_args_with_index(self, index: int) -> int: """ Count the number of arguments that have the given index. """ counter: int = 0 for arg_with_ind in self.args_with_ind: if index in arg_with_ind.indices: counter += 1 return counter def get_args_with_index(self, index: int) -> List[_ArgE]: """ Get a list of arguments having the given index. """ ret: List[_ArgE] = [i for i in self.args_with_ind if index in i.indices] return ret @property def number_of_diagonal_indices(self): data = set([]) for arg in self.args_with_ind: data.update({i for i in arg.indices if i is not None and i < 0}) return len(data) def track_permutation_start(self): permutation = [] perm_diag = [] counter: int = 0 counter2: int = -1 for arg_with_ind in self.args_with_ind: perm = [] for i in arg_with_ind.indices: if i is not None: if i < 0: perm_diag.append(counter2) counter2 -= 1 continue perm.append(counter) counter += 1 permutation.append(perm) max_ind = max([max(i) if i else -1 for i in permutation]) if permutation else -1 perm_diag = [max_ind - i for i in perm_diag] self._track_permutation = permutation + [perm_diag] def track_permutation_merge(self, destination: _ArgE, from_element: _ArgE): index_destination = self.args_with_ind.index(destination) index_element = self.args_with_ind.index(from_element) self._track_permutation[index_destination].extend(self._track_permutation[index_element]) # type: ignore self._track_permutation.pop(index_element) # type: ignore def get_absolute_free_range(self, arg: _ArgE) -> typing.Tuple[int, int]: """ Return the range of the free indices of the arg as absolute positions among all free indices. """ counter = 0 for arg_with_ind in self.args_with_ind: number_free_indices = len([i for i in arg_with_ind.indices if i is None]) if arg_with_ind == arg: return counter, counter + number_free_indices counter += number_free_indices raise IndexError("argument not found") def get_absolute_range(self, arg: _ArgE) -> typing.Tuple[int, int]: """ Return the absolute range of indices for arg, disregarding dummy indices. """ counter = 0 for arg_with_ind in self.args_with_ind: number_indices = len(arg_with_ind.indices) if arg_with_ind == arg: return counter, counter + number_indices counter += number_indices raise IndexError("argument not found") def get_rank(expr): if isinstance(expr, (MatrixExpr, MatrixElement)): return 2 if isinstance(expr, _CodegenArrayAbstract): return len(expr.shape) if isinstance(expr, NDimArray): return expr.rank() if isinstance(expr, Indexed): return expr.rank if isinstance(expr, IndexedBase): shape = expr.shape if shape is None: return -1 else: return len(shape) if hasattr(expr, "shape"): return len(expr.shape) return 0 def _get_subrank(expr): if isinstance(expr, _CodegenArrayAbstract): return expr.subrank() return get_rank(expr) def _get_subranks(expr): if isinstance(expr, _CodegenArrayAbstract): return expr.subranks else: return [get_rank(expr)] def get_shape(expr): if hasattr(expr, "shape"): return expr.shape return () def nest_permutation(expr): if isinstance(expr, PermuteDims): return expr.nest_permutation() else: return expr def _array_tensor_product(args, kwargs): return ArrayTensorProduct(args, canonicalize=True, *kwargs) def _array_contraction(expr, contraction_indices, *kwargs): return ArrayContraction(expr, contraction_indices, canonicalize=True, *kwargs) def _array_diagonal(expr, diagonal_indices, *kwargs): return ArrayDiagonal(expr, diagonal_indices, canonicalize=True, kwargs) def _permute_dims(expr, permutation, kwargs): return PermuteDims(expr, permutation, canonicalize=True, *kwargs) def _array_add(args, *kwargs): return ArrayAdd(args, canonicalize=True, **kwargs)
2def2bffee978b9b0ff9d0a49c234da2a44d35eb4fef554b383fe78167bfbdb3	import operator from functools import reduce, singledispatch from sympy.core.expr import Expr from sympy.core.singleton import S from sympy.matrices.expressions.hadamard import HadamardProduct from sympy.matrices.expressions.inverse import Inverse from sympy.matrices.expressions.matexpr import (MatrixExpr, MatrixSymbol) from sympy.matrices.expressions.special import Identity from sympy.matrices.expressions.transpose import Transpose from sympy.combinatorics.permutations import _af_invert from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction from sympy.tensor.array.expressions.array_expressions import ( _ArrayExpr, ZeroArray, ArraySymbol, ArrayTensorProduct, ArrayAdd, PermuteDims, ArrayDiagonal, ArrayElementwiseApplyFunc, get_rank, get_shape, ArrayContraction, _array_tensor_product, _array_contraction, _array_diagonal, _array_add, _permute_dims, Reshape) from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array @singledispatch def array_derive(expr, x): raise NotImplementedError(f"not implemented for type {type(expr)}") @array_derive.register(Expr) def _(expr: Expr, x: _ArrayExpr): return ZeroArray(x.shape) @array_derive.register(ArrayTensorProduct) def _(expr: ArrayTensorProduct, x: Expr): args = expr.args addend_list = [] for i, arg in enumerate(expr.args): darg = array_derive(arg, x) if darg == 0: continue args_prev = args[:i] args_succ = args[i+1:] shape_prev = reduce(operator.add, map(get_shape, args_prev), ()) shape_succ = reduce(operator.add, map(get_shape, args_succ), ()) addend = _array_tensor_product(args_prev, darg, args_succ) tot1 = len(get_shape(x)) tot2 = tot1 + len(shape_prev) tot3 = tot2 + len(get_shape(arg)) tot4 = tot3 + len(shape_succ) perm = [i for i in range(tot1, tot2)] + \ [i for i in range(tot1)] + [i for i in range(tot2, tot3)] + \ [i for i in range(tot3, tot4)] addend = _permute_dims(addend, _af_invert(perm)) addend_list.append(addend) if len(addend_list) == 1: return addend_list[0] elif len(addend_list) == 0: return S.Zero else: return _array_add(addend_list) @array_derive.register(ArraySymbol) def _(expr: ArraySymbol, x: _ArrayExpr): if expr == x: return _permute_dims( ArrayTensorProduct.fromiter(Identity(i) for i in expr.shape), [2i for i in range(len(expr.shape))] + [2i+1 for i in range(len(expr.shape))] ) return ZeroArray((x.shape + expr.shape)) @array_derive.register(MatrixSymbol) def _(expr: MatrixSymbol, x: _ArrayExpr): m, n = expr.shape if expr == x: return _permute_dims( _array_tensor_product(Identity(m), Identity(n)), [0, 2, 1, 3] ) return ZeroArray((x.shape + expr.shape)) @array_derive.register(Identity) def _(expr: Identity, x: _ArrayExpr): return ZeroArray((x.shape + expr.shape)) @array_derive.register(Transpose) def _(expr: Transpose, x: Expr): # D(A.T, A) ==> (m,n,i,j) ==> D(A_ji, A_mn) = d_mj d_ni # D(B.T, A) ==> (m,n,i,j) ==> D(B_ji, A_mn) fd = array_derive(expr.arg, x) return _permute_dims(fd, [0, 1, 3, 2]) @array_derive.register(Inverse) def _(expr: Inverse, x: Expr): mat = expr.I dexpr = array_derive(mat, x) tp = _array_tensor_product(-expr, dexpr, expr) mp = _array_contraction(tp, (1, 4), (5, 6)) pp = _permute_dims(mp, [1, 2, 0, 3]) return pp @array_derive.register(ElementwiseApplyFunction) def _(expr: ElementwiseApplyFunction, x: Expr): assert get_rank(expr) == 2 assert get_rank(x) == 2 fdiff = expr._get_function_fdiff() dexpr = array_derive(expr.expr, x) tp = _array_tensor_product( ElementwiseApplyFunction(fdiff, expr.expr), dexpr ) td = _array_diagonal( tp, (0, 4), (1, 5) ) return td @array_derive.register(ArrayElementwiseApplyFunc) def _(expr: ArrayElementwiseApplyFunc, x: Expr): fdiff = expr._get_function_fdiff() subexpr = expr.expr dsubexpr = array_derive(subexpr, x) tp = _array_tensor_product( dsubexpr, ArrayElementwiseApplyFunc(fdiff, subexpr) ) b = get_rank(x) c = get_rank(expr) diag_indices = [(b + i, b + c + i) for i in range(c)] return _array_diagonal(tp, diag_indices) @array_derive.register(MatrixExpr) def _(expr: MatrixExpr, x: Expr): cg = convert_matrix_to_array(expr) return array_derive(cg, x) @array_derive.register(HadamardProduct) def _(expr: HadamardProduct, x: Expr): raise NotImplementedError() @array_derive.register(ArrayContraction) def _(expr: ArrayContraction, x: Expr): fd = array_derive(expr.expr, x) rank_x = len(get_shape(x)) contraction_indices = expr.contraction_indices new_contraction_indices = [tuple(j + rank_x for j in i) for i in contraction_indices] return _array_contraction(fd, new_contraction_indices) @array_derive.register(ArrayDiagonal) def _(expr: ArrayDiagonal, x: Expr): dsubexpr = array_derive(expr.expr, x) rank_x = len(get_shape(x)) diag_indices = [[j + rank_x for j in i] for i in expr.diagonal_indices] return _array_diagonal(dsubexpr, diag_indices) @array_derive.register(ArrayAdd) def _(expr: ArrayAdd, x: Expr): return _array_add(*[array_derive(arg, x) for arg in expr.args]) @array_derive.register(PermuteDims) def _(expr: PermuteDims, x: Expr): de = array_derive(expr.expr, x) perm = [0, 1] + [i + 2 for i in expr.permutation.array_form] return _permute_dims(de, perm) @array_derive.register(Reshape) def _(expr: Reshape, x: Expr): de = array_derive(expr.expr, x) return Reshape(de, get_shape(x) + expr.shape) def matrix_derive(expr, x): from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix ce = convert_matrix_to_array(expr) dce = array_derive(ce, x) return convert_array_to_matrix(dce).doit()
447911cd60c00f42fdd7c46d165d95a02bd31a20ff82808c55bc8da0ce5670b9	from sympy import Lambda, S, Dummy, KroneckerProduct from sympy.core.symbol import symbols from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import cos, sin from sympy.matrices.expressions.hadamard import HadamardProduct, HadamardPower from sympy.matrices.expressions.special import (Identity, OneMatrix, ZeroMatrix) from sympy.matrices.expressions.matexpr import MatrixElement from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array from sympy.tensor.array.expressions.conv_array_to_matrix import _support_function_tp1_recognize, \ _array_diag2contr_diagmatrix, convert_array_to_matrix, _remove_trivial_dims, _array2matrix, \ _combine_removed, identify_removable_identity_matrices, _array_contraction_to_diagonal_multiple_identity from sympy.matrices.expressions.matexpr import MatrixSymbol from sympy.combinatorics import Permutation from sympy.matrices.expressions.diagonal import DiagMatrix, DiagonalMatrix from sympy.matrices import Trace, MatMul, Transpose from sympy.tensor.array.expressions.array_expressions import ZeroArray, OneArray, \ ArrayElement, ArraySymbol, ArrayElementwiseApplyFunc, _array_tensor_product, _array_contraction, \ _array_diagonal, _permute_dims, PermuteDims, ArrayAdd, ArrayDiagonal, ArrayContraction, ArrayTensorProduct from sympy.testing.pytest import raises i, j, k, l, m, n = symbols("i j k l m n") I = Identity(k) I1 = Identity(1) M = MatrixSymbol("M", k, k) N = MatrixSymbol("N", k, k) P = MatrixSymbol("P", k, k) Q = MatrixSymbol("Q", k, k) A = MatrixSymbol("A", k, k) B = MatrixSymbol("B", k, k) C = MatrixSymbol("C", k, k) D = MatrixSymbol("D", k, k) X = MatrixSymbol("X", k, k) Y = MatrixSymbol("Y", k, k) a = MatrixSymbol("a", k, 1) b = MatrixSymbol("b", k, 1) c = MatrixSymbol("c", k, 1) d = MatrixSymbol("d", k, 1) x = MatrixSymbol("x", k, 1) y = MatrixSymbol("y", k, 1) def test_arrayexpr_convert_array_to_matrix(): cg = _array_contraction(_array_tensor_product(M), (0, 1)) assert convert_array_to_matrix(cg) == Trace(M) cg = _array_contraction(_array_tensor_product(M, N), (0, 1), (2, 3)) assert convert_array_to_matrix(cg) == Trace(M) * Trace(N) cg = _array_contraction(_array_tensor_product(M, N), (0, 3), (1, 2)) assert convert_array_to_matrix(cg) == Trace(M * N) cg = _array_contraction(_array_tensor_product(M, N), (0, 2), (1, 3)) assert convert_array_to_matrix(cg) == Trace(M * N.T) cg = convert_matrix_to_array(M * N * P) assert convert_array_to_matrix(cg) == M * N * P cg = convert_matrix_to_array(M * N.T * P) assert convert_array_to_matrix(cg) == M * N.T * P cg = _array_contraction(_array_tensor_product(M,N,P,Q), (1, 2), (5, 6)) assert convert_array_to_matrix(cg) == _array_tensor_product(M * N, P * Q) cg = _array_contraction(_array_tensor_product(-2, M, N), (1, 2)) assert convert_array_to_matrix(cg) == -2 * M * N a = MatrixSymbol("a", k, 1) b = MatrixSymbol("b", k, 1) c = MatrixSymbol("c", k, 1) cg = PermuteDims( _array_contraction( _array_tensor_product( a, ArrayAdd( _array_tensor_product(b, c), _array_tensor_product(c, b), ) ), (2, 4)), [0, 1, 3, 2]) assert convert_array_to_matrix(cg) == a * (b.T * c + c.T * b) za = ZeroArray(m, n) assert convert_array_to_matrix(za) == ZeroMatrix(m, n) cg = _array_tensor_product(3, M) assert convert_array_to_matrix(cg) == 3 * M # Partial conversion to matrix multiplication: expr = _array_contraction(_array_tensor_product(M, N, P, Q), (0, 2), (1, 4, 6)) assert convert_array_to_matrix(expr) == _array_contraction(_array_tensor_product(M.TN, P, Q), (0, 2, 4)) x = MatrixSymbol("x", k, 1) cg = PermuteDims( _array_contraction(_array_tensor_product(OneArray(1), x, OneArray(1), DiagMatrix(Identity(1))), (0, 5)), Permutation(1, 2, 3)) assert convert_array_to_matrix(cg) == x expr = ArrayAdd(M, PermuteDims(M, [1, 0])) assert convert_array_to_matrix(expr) == M + Transpose(M) def test_arrayexpr_convert_array_to_matrix2(): cg = _array_contraction(_array_tensor_product(M, N), (1, 3)) assert convert_array_to_matrix(cg) == M N.T cg = PermuteDims(_array_tensor_product(M, N), Permutation([0, 1, 3, 2])) assert convert_array_to_matrix(cg) == _array_tensor_product(M, N.T) cg = _array_tensor_product(M, PermuteDims(N, Permutation([1, 0]))) assert convert_array_to_matrix(cg) == _array_tensor_product(M, N.T) cg = _array_contraction( PermuteDims( _array_tensor_product(M, N, P, Q), Permutation([0, 2, 3, 1, 4, 5, 7, 6])), (1, 2), (3, 5) ) assert convert_array_to_matrix(cg) == _array_tensor_product(M * P.T * Trace(N), Q.T) cg = _array_contraction( _array_tensor_product(M, N, P, PermuteDims(Q, Permutation([1, 0]))), (1, 5), (2, 3) ) assert convert_array_to_matrix(cg) == _array_tensor_product(M * P.T * Trace(N), Q.T) cg = _array_tensor_product(M, PermuteDims(N, [1, 0])) assert convert_array_to_matrix(cg) == _array_tensor_product(M, N.T) cg = _array_tensor_product(PermuteDims(M, [1, 0]), PermuteDims(N, [1, 0])) assert convert_array_to_matrix(cg) == _array_tensor_product(M.T, N.T) cg = _array_tensor_product(PermuteDims(N, [1, 0]), PermuteDims(M, [1, 0])) assert convert_array_to_matrix(cg) == _array_tensor_product(N.T, M.T) cg = _array_contraction(M, (0,), (1,)) assert convert_array_to_matrix(cg) == OneMatrix(1, k)MOneMatrix(k, 1) cg = _array_contraction(x, (0,), (1,)) assert convert_array_to_matrix(cg) == OneMatrix(1, k)x Xm = MatrixSymbol("Xm", m, n) cg = _array_contraction(Xm, (0,), (1,)) assert convert_array_to_matrix(cg) == OneMatrix(1, m)XmOneMatrix(n, 1) def test_arrayexpr_convert_array_to_diagonalized_vector(): # Check matrix recognition over trivial dimensions: cg = _array_tensor_product(a, b) assert convert_array_to_matrix(cg) == a b.T cg = _array_tensor_product(I1, a, b) assert convert_array_to_matrix(cg) == a * b.T # Recognize trace inside a tensor product: cg = _array_contraction(_array_tensor_product(A, B, C), (0, 3), (1, 2)) assert convert_array_to_matrix(cg) == Trace(A * B) * C # Transform diagonal operator to contraction: cg = _array_diagonal(_array_tensor_product(A, a), (1, 2)) assert _array_diag2contr_diagmatrix(cg) == _array_contraction(_array_tensor_product(A, OneArray(1), DiagMatrix(a)), (1, 3)) assert convert_array_to_matrix(cg) == A * DiagMatrix(a) cg = _array_diagonal(_array_tensor_product(a, b), (0, 2)) assert _array_diag2contr_diagmatrix(cg) == _permute_dims( _array_contraction(_array_tensor_product(DiagMatrix(a), OneArray(1), b), (0, 3)), [1, 2, 0] ) assert convert_array_to_matrix(cg) == b.T * DiagMatrix(a) cg = _array_diagonal(_array_tensor_product(A, a), (0, 2)) assert _array_diag2contr_diagmatrix(cg) == _array_contraction(_array_tensor_product(A, OneArray(1), DiagMatrix(a)), (0, 3)) assert convert_array_to_matrix(cg) == A.T * DiagMatrix(a) cg = _array_diagonal(_array_tensor_product(I, x, I1), (0, 2), (3, 5)) assert _array_diag2contr_diagmatrix(cg) == _array_contraction(_array_tensor_product(I, OneArray(1), I1, DiagMatrix(x)), (0, 5)) assert convert_array_to_matrix(cg) == DiagMatrix(x) cg = _array_diagonal(_array_tensor_product(I, x, A, B), (1, 2), (5, 6)) assert _array_diag2contr_diagmatrix(cg) == _array_diagonal(_array_contraction(_array_tensor_product(I, OneArray(1), A, B, DiagMatrix(x)), (1, 7)), (5, 6)) # TODO: this is returning a wrong result: # convert_array_to_matrix(cg) cg = _array_diagonal(_array_tensor_product(I1, a, b), (1, 3, 5)) assert convert_array_to_matrix(cg) == ab.T cg = _array_diagonal(_array_tensor_product(I1, a, b), (1, 3)) assert _array_diag2contr_diagmatrix(cg) == _array_contraction(_array_tensor_product(OneArray(1), a, b, I1), (2, 6)) assert convert_array_to_matrix(cg) == ab.T cg = _array_diagonal(_array_tensor_product(x, I1), (1, 2)) assert isinstance(cg, ArrayDiagonal) assert cg.diagonal_indices == ((1, 2),) assert convert_array_to_matrix(cg) == x cg = _array_diagonal(_array_tensor_product(x, I), (0, 2)) assert _array_diag2contr_diagmatrix(cg) == _array_contraction(_array_tensor_product(OneArray(1), I, DiagMatrix(x)), (1, 3)) assert convert_array_to_matrix(cg).doit() == DiagMatrix(x) raises(ValueError, lambda: _array_diagonal(x, (1,))) # Ignore identity matrices with contractions: cg = _array_contraction(_array_tensor_product(I, A, I, I), (0, 2), (1, 3), (5, 7)) assert cg.split_multiple_contractions() == cg assert convert_array_to_matrix(cg) == Trace(A) * I cg = _array_contraction(_array_tensor_product(Trace(A) * I, I, I), (1, 5), (3, 4)) assert cg.split_multiple_contractions() == cg assert convert_array_to_matrix(cg).doit() == Trace(A) * I # Add DiagMatrix when required: cg = _array_contraction(_array_tensor_product(A, a), (1, 2)) assert cg.split_multiple_contractions() == cg assert convert_array_to_matrix(cg) == A * a cg = _array_contraction(_array_tensor_product(A, a, B), (1, 2, 4)) assert cg.split_multiple_contractions() == _array_contraction(_array_tensor_product(A, DiagMatrix(a), OneArray(1), B), (1, 2), (3, 5)) assert convert_array_to_matrix(cg) == A * DiagMatrix(a) * B cg = _array_contraction(_array_tensor_product(A, a, B), (0, 2, 4)) assert cg.split_multiple_contractions() == _array_contraction(_array_tensor_product(A, DiagMatrix(a), OneArray(1), B), (0, 2), (3, 5)) assert convert_array_to_matrix(cg) == A.T * DiagMatrix(a) * B cg = _array_contraction(_array_tensor_product(A, a, b, a.T, B), (0, 2, 4, 7, 9)) assert cg.split_multiple_contractions() == _array_contraction(_array_tensor_product(A, DiagMatrix(a), OneArray(1), DiagMatrix(b), OneArray(1), DiagMatrix(a), OneArray(1), B), (0, 2), (3, 5), (6, 9), (8, 12)) assert convert_array_to_matrix(cg) == A.T * DiagMatrix(a) * DiagMatrix(b) * DiagMatrix(a) * B.T cg = _array_contraction(_array_tensor_product(I1, I1, I1), (1, 2, 4)) assert cg.split_multiple_contractions() == _array_contraction(_array_tensor_product(I1, I1, OneArray(1), I1), (1, 2), (3, 5)) assert convert_array_to_matrix(cg) == 1 cg = _array_contraction(_array_tensor_product(I, I, I, I, A), (1, 2, 8), (5, 6, 9)) assert convert_array_to_matrix(cg.split_multiple_contractions()).doit() == A cg = _array_contraction(_array_tensor_product(A, a, C, a, B), (1, 2, 4), (5, 6, 8)) expected = _array_contraction(_array_tensor_product(A, DiagMatrix(a), OneArray(1), C, DiagMatrix(a), OneArray(1), B), (1, 3), (2, 5), (6, 7), (8, 10)) assert cg.split_multiple_contractions() == expected assert convert_array_to_matrix(cg) == A * DiagMatrix(a) * C * DiagMatrix(a) * B cg = _array_contraction(_array_tensor_product(a, I1, b, I1, (a.Tb).applyfunc(cos)), (1, 2, 8), (5, 6, 9)) expected = _array_contraction(_array_tensor_product(a, I1, OneArray(1), b, I1, OneArray(1), (a.Tb).applyfunc(cos)), (1, 3), (2, 10), (6, 8), (7, 11)) assert cg.split_multiple_contractions().dummy_eq(expected) assert convert_array_to_matrix(cg).doit().dummy_eq(MatMul(a, (a.T * b).applyfunc(cos), b.T)) def test_arrayexpr_convert_array_contraction_tp_additions(): a = ArrayAdd( _array_tensor_product(M, N), _array_tensor_product(N, M) ) tp = _array_tensor_product(P, a, Q) expr = _array_contraction(tp, (3, 4)) expected = _array_tensor_product( P, ArrayAdd( _array_contraction(_array_tensor_product(M, N), (1, 2)), _array_contraction(_array_tensor_product(N, M), (1, 2)), ), Q ) assert expr == expected assert convert_array_to_matrix(expr) == _array_tensor_product(P, M * N + N * M, Q) expr = _array_contraction(tp, (1, 2), (3, 4), (5, 6)) result = _array_contraction( _array_tensor_product( P, ArrayAdd( _array_contraction(_array_tensor_product(M, N), (1, 2)), _array_contraction(_array_tensor_product(N, M), (1, 2)), ), Q ), (1, 2), (3, 4)) assert expr == result assert convert_array_to_matrix(expr) == P * (M * N + N * M) * Q def test_arrayexpr_convert_array_to_implicit_matmul(): # Trivial dimensions are suppressed, so the result can be expressed in matrix form: cg = _array_tensor_product(a, b) assert convert_array_to_matrix(cg) == a * b.T cg = _array_tensor_product(a, b, I) assert convert_array_to_matrix(cg) == _array_tensor_product(ab.T, I) cg = _array_tensor_product(I, a, b) assert convert_array_to_matrix(cg) == _array_tensor_product(I, ab.T) cg = _array_tensor_product(a, I, b) assert convert_array_to_matrix(cg) == _array_tensor_product(a, I, b) cg = _array_contraction(_array_tensor_product(I, I), (1, 2)) assert convert_array_to_matrix(cg) == I cg = PermuteDims(_array_tensor_product(I, Identity(1)), [0, 2, 1, 3]) assert convert_array_to_matrix(cg) == I def test_arrayexpr_convert_array_to_matrix_remove_trivial_dims(): # Tensor Product: assert _remove_trivial_dims(_array_tensor_product(a, b)) == (a * b.T, [1, 3]) assert _remove_trivial_dims(_array_tensor_product(a.T, b)) == (a * b.T, [0, 3]) assert _remove_trivial_dims(_array_tensor_product(a, b.T)) == (a * b.T, [1, 2]) assert _remove_trivial_dims(_array_tensor_product(a.T, b.T)) == (a * b.T, [0, 2]) assert _remove_trivial_dims(_array_tensor_product(I, a.T, b.T)) == (_array_tensor_product(I, a * b.T), [2, 4]) assert _remove_trivial_dims(_array_tensor_product(a.T, I, b.T)) == (_array_tensor_product(a.T, I, b.T), []) assert _remove_trivial_dims(_array_tensor_product(a, I)) == (_array_tensor_product(a, I), []) assert _remove_trivial_dims(_array_tensor_product(I, a)) == (_array_tensor_product(I, a), []) assert _remove_trivial_dims(_array_tensor_product(a.T, b.T, c, d)) == ( _array_tensor_product(a * b.T, c * d.T), [0, 2, 5, 7]) assert _remove_trivial_dims(_array_tensor_product(a.T, I, b.T, c, d, I)) == ( _array_tensor_product(a.T, I, bc.T, d, I), [4, 7]) # Addition: cg = ArrayAdd(_array_tensor_product(a, b), _array_tensor_product(c, d)) assert _remove_trivial_dims(cg) == (a b.T + c * d.T, [1, 3]) # Permute Dims: cg = PermuteDims(_array_tensor_product(a, b), Permutation(3)(1, 2)) assert _remove_trivial_dims(cg) == (a * b.T, [2, 3]) cg = PermuteDims(_array_tensor_product(a, I, b), Permutation(5)(1, 2, 3, 4)) assert _remove_trivial_dims(cg) == (cg, []) cg = PermuteDims(_array_tensor_product(I, b, a), Permutation(5)(1, 2, 4, 5, 3)) assert _remove_trivial_dims(cg) == (PermuteDims(_array_tensor_product(I, b * a.T), [0, 2, 3, 1]), [4, 5]) # Diagonal: cg = _array_diagonal(_array_tensor_product(M, a), (1, 2)) assert _remove_trivial_dims(cg) == (cg, []) # Contraction: cg = _array_contraction(_array_tensor_product(M, a), (1, 2)) assert _remove_trivial_dims(cg) == (cg, []) # A few more cases to test the removal and shift of nested removed axes # with array contractions and array diagonals: tp = _array_tensor_product( OneMatrix(1, 1), M, x, OneMatrix(1, 1), Identity(1), ) expr = _array_contraction(tp, (1, 8)) rexpr, removed = _remove_trivial_dims(expr) assert removed == [0, 5, 6, 7] expr = _array_contraction(tp, (1, 8), (3, 4)) rexpr, removed = _remove_trivial_dims(expr) assert removed == [0, 3, 4, 5] expr = _array_diagonal(tp, (1, 8)) rexpr, removed = _remove_trivial_dims(expr) assert removed == [0, 5, 6, 7, 8] expr = _array_diagonal(tp, (1, 8), (3, 4)) rexpr, removed = _remove_trivial_dims(expr) assert removed == [0, 3, 4, 5, 6] expr = _array_diagonal(_array_contraction(_array_tensor_product(A, x, I, I1), (1, 2, 5)), (1, 4)) rexpr, removed = _remove_trivial_dims(expr) assert removed == [2, 3] cg = _array_diagonal(_array_tensor_product(PermuteDims(_array_tensor_product(x, I1), Permutation(1, 2, 3)), (x.Tx).applyfunc(sqrt)), (2, 4), (3, 5)) rexpr, removed = _remove_trivial_dims(cg) assert removed == [1, 2] # Contractions with identity matrices need to be followed by a permutation # in order cg = _array_contraction(_array_tensor_product(A, B, C, M, I), (1, 8)) ret, removed = _remove_trivial_dims(cg) assert ret == PermuteDims(_array_tensor_product(A, B, C, M), [0, 2, 3, 4, 5, 6, 7, 1]) assert removed == [] cg = _array_contraction(_array_tensor_product(A, B, C, M, I), (1, 8), (3, 4)) ret, removed = _remove_trivial_dims(cg) assert ret == PermuteDims(_array_contraction(_array_tensor_product(A, B, C, M), (3, 4)), [0, 2, 3, 4, 5, 1]) assert removed == [] # Trivial matrices are sometimes inserted into MatMul expressions: cg = _array_tensor_product(bb.T, a.Ta) ret, removed = _remove_trivial_dims(cg) assert ret == ba.Tab.T assert removed == [2, 3] Xs = ArraySymbol("X", (3, 2, k)) cg = _array_tensor_product(M, Xs, b.Tc, aa.T, bb.T, c.Td) ret, removed = _remove_trivial_dims(cg) assert ret == _array_tensor_product(M, Xs, ab.Tcc.Tda.T, bb.T) assert removed == [5, 6, 11, 12] cg = _array_diagonal(_array_tensor_product(I, I1, x), (1, 4), (3, 5)) assert _remove_trivial_dims(cg) == (PermuteDims(_array_diagonal(_array_tensor_product(I, x), (1, 2)), Permutation(1, 2)), [1]) expr = _array_diagonal(_array_tensor_product(x, I, y), (0, 2)) assert _remove_trivial_dims(expr) == (PermuteDims(_array_tensor_product(DiagMatrix(x), y), [1, 2, 3, 0]), [0]) expr = _array_diagonal(_array_tensor_product(x, I, y), (0, 2), (3, 4)) assert _remove_trivial_dims(expr) == (expr, []) def test_arrayexpr_convert_array_to_matrix_diag2contraction_diagmatrix(): cg = _array_diagonal(_array_tensor_product(M, a), (1, 2)) res = _array_diag2contr_diagmatrix(cg) assert res.shape == cg.shape assert res == _array_contraction(_array_tensor_product(M, OneArray(1), DiagMatrix(a)), (1, 3)) raises(ValueError, lambda: _array_diagonal(_array_tensor_product(a, M), (1, 2))) cg = _array_diagonal(_array_tensor_product(a.T, M), (1, 2)) res = _array_diag2contr_diagmatrix(cg) assert res.shape == cg.shape assert res == _array_contraction(_array_tensor_product(OneArray(1), M, DiagMatrix(a.T)), (1, 4)) cg = _array_diagonal(_array_tensor_product(a.T, M, N, b.T), (1, 2), (4, 7)) res = _array_diag2contr_diagmatrix(cg) assert res.shape == cg.shape assert res == _array_contraction( _array_tensor_product(OneArray(1), M, N, OneArray(1), DiagMatrix(a.T), DiagMatrix(b.T)), (1, 7), (3, 9)) cg = _array_diagonal(_array_tensor_product(a, M, N, b.T), (0, 2), (4, 7)) res = _array_diag2contr_diagmatrix(cg) assert res.shape == cg.shape assert res == _array_contraction( _array_tensor_product(OneArray(1), M, N, OneArray(1), DiagMatrix(a), DiagMatrix(b.T)), (1, 6), (3, 9)) cg = _array_diagonal(_array_tensor_product(a, M, N, b.T), (0, 4), (3, 7)) res = _array_diag2contr_diagmatrix(cg) assert res.shape == cg.shape assert res == _array_contraction( _array_tensor_product(OneArray(1), M, N, OneArray(1), DiagMatrix(a), DiagMatrix(b.T)), (3, 6), (2, 9)) I1 = Identity(1) x = MatrixSymbol("x", k, 1) A = MatrixSymbol("A", k, k) cg = _array_diagonal(_array_tensor_product(x, A.T, I1), (0, 2)) assert _array_diag2contr_diagmatrix(cg).shape == cg.shape assert _array2matrix(cg).shape == cg.shape def test_arrayexpr_convert_array_to_matrix_support_function(): assert _support_function_tp1_recognize([], [2 * k]) == 2 * k assert _support_function_tp1_recognize([(1, 2)], [A, 2 * k, B, 3]) == 6 * k * A * B assert _support_function_tp1_recognize([(0, 3), (1, 2)], [A, B]) == Trace(A * B) assert _support_function_tp1_recognize([(1, 2)], [A, B]) == A * B assert _support_function_tp1_recognize([(0, 2)], [A, B]) == A.T * B assert _support_function_tp1_recognize([(1, 3)], [A, B]) == A * B.T assert _support_function_tp1_recognize([(0, 3)], [A, B]) == A.T * B.T assert _support_function_tp1_recognize([(1, 2), (5, 6)], [A, B, C, D]) == _array_tensor_product(A * B, C * D) assert _support_function_tp1_recognize([(1, 4), (3, 6)], [A, B, C, D]) == PermuteDims( _array_tensor_product(A * C, B * D), [0, 2, 1, 3]) assert _support_function_tp1_recognize([(0, 3), (1, 4)], [A, B, C]) == B * A * C assert _support_function_tp1_recognize([(9, 10), (1, 2), (5, 6), (3, 4), (7, 8)], [X, Y, A, B, C, D]) == X * Y * A * B * C * D assert _support_function_tp1_recognize([(9, 10), (1, 2), (5, 6), (3, 4)], [X, Y, A, B, C, D]) == _array_tensor_product(X * Y * A * B, C * D) assert _support_function_tp1_recognize([(1, 7), (3, 8), (4, 11)], [X, Y, A, B, C, D]) == PermuteDims( _array_tensor_product(X * B.T, Y * C, A.T * D.T), [0, 2, 4, 1, 3, 5] ) assert _support_function_tp1_recognize([(0, 1), (3, 6), (5, 8)], [X, A, B, C, D]) == PermuteDims( _array_tensor_product(Trace(X) * A * C, B * D), [0, 2, 1, 3]) assert _support_function_tp1_recognize([(1, 2), (3, 4), (5, 6), (7, 8)], [A, A, B, C, D]) == A ** 2 * B * C * D assert _support_function_tp1_recognize([(1, 2), (3, 4), (5, 6), (7, 8)], [X, A, B, C, D]) == X * A * B * C * D assert _support_function_tp1_recognize([(1, 6), (3, 8), (5, 10)], [X, Y, A, B, C, D]) == PermuteDims( _array_tensor_product(X * B, Y * C, A * D), [0, 2, 4, 1, 3, 5] ) assert _support_function_tp1_recognize([(1, 4), (3, 6)], [A, B, C, D]) == PermuteDims( _array_tensor_product(A * C, B * D), [0, 2, 1, 3]) assert _support_function_tp1_recognize([(0, 4), (1, 7), (2, 5), (3, 8)], [X, A, B, C, D]) == CX.TBAD assert _support_function_tp1_recognize([(0, 4), (1, 7), (2, 5), (3, 8)], [X, A, B, C, D]) == CX.TBAD def test_convert_array_to_hadamard_products(): expr = HadamardProduct(M, N) cg = convert_matrix_to_array(expr) ret = convert_array_to_matrix(cg) assert ret == expr expr = HadamardProduct(M, N)P cg = convert_matrix_to_array(expr) ret = convert_array_to_matrix(cg) assert ret == expr expr = QHadamardProduct(M, N)P cg = convert_matrix_to_array(expr) ret = convert_array_to_matrix(cg) assert ret == expr expr = QHadamardProduct(M, N.T)P cg = convert_matrix_to_array(expr) ret = convert_array_to_matrix(cg) assert ret == expr expr = HadamardProduct(M, N)HadamardProduct(Q, P) cg = convert_matrix_to_array(expr) ret = convert_array_to_matrix(cg) assert expr == ret expr = P.THadamardProduct(M, N)HadamardProduct(Q, P) cg = convert_matrix_to_array(expr) ret = convert_array_to_matrix(cg) assert expr == ret # ArrayDiagonal should be converted cg = _array_diagonal(_array_tensor_product(M, N, Q), (1, 3), (0, 2, 4)) ret = convert_array_to_matrix(cg) expected = PermuteDims(_array_diagonal(_array_tensor_product(HadamardProduct(M.T, N.T), Q), (1, 2)), [1, 0, 2]) assert expected == ret # Special case that should return the same expression: cg = _array_diagonal(_array_tensor_product(HadamardProduct(M, N), Q), (0, 2)) ret = convert_array_to_matrix(cg) assert ret == cg # Hadamard products with traces: expr = Trace(HadamardProduct(M, N)) cg = convert_matrix_to_array(expr) ret = convert_array_to_matrix(cg) assert ret == Trace(HadamardProduct(M.T, N.T)) expr = Trace(AHadamardProduct(M, N)) cg = convert_matrix_to_array(expr) ret = convert_array_to_matrix(cg) assert ret == Trace(HadamardProduct(M, N)A) expr = Trace(HadamardProduct(A, M)N) cg = convert_matrix_to_array(expr) ret = convert_array_to_matrix(cg) assert ret == Trace(HadamardProduct(M.T, N)A) # These should not be converted into Hadamard products: cg = _array_diagonal(_array_tensor_product(M, N), (0, 1, 2, 3)) ret = convert_array_to_matrix(cg) assert ret == cg cg = _array_diagonal(_array_tensor_product(A), (0, 1)) ret = convert_array_to_matrix(cg) assert ret == cg cg = _array_diagonal(_array_tensor_product(M, N, P), (0, 2, 4), (1, 3, 5)) assert convert_array_to_matrix(cg) == HadamardProduct(M, N, P) cg = _array_diagonal(_array_tensor_product(M, N, P), (0, 3, 4), (1, 2, 5)) assert convert_array_to_matrix(cg) == HadamardProduct(M, P, N.T) cg = _array_diagonal(_array_tensor_product(I, I1, x), (1, 4), (3, 5)) assert convert_array_to_matrix(cg) == DiagMatrix(x) def test_identify_removable_identity_matrices(): D = DiagonalMatrix(MatrixSymbol("D", k, k)) cg = _array_contraction(_array_tensor_product(A, B, I), (1, 2, 4, 5)) expected = _array_contraction(_array_tensor_product(A, B), (1, 2)) assert identify_removable_identity_matrices(cg) == expected cg = _array_contraction(_array_tensor_product(A, B, C, I), (1, 3, 5, 6, 7)) expected = _array_contraction(_array_tensor_product(A, B, C), (1, 3, 5)) assert identify_removable_identity_matrices(cg) == expected # Tests with diagonal matrices: cg = _array_contraction(_array_tensor_product(A, B, D), (1, 2, 4, 5)) ret = identify_removable_identity_matrices(cg) expected = _array_contraction(_array_tensor_product(A, B, D), (1, 4), (2, 5)) assert ret == expected cg = _array_contraction(_array_tensor_product(A, B, D, M, N), (1, 2, 4, 5, 6, 8)) ret = identify_removable_identity_matrices(cg) assert ret == cg def test_combine_removed(): assert _combine_removed(6, [0, 1, 2], [0, 1, 2]) == [0, 1, 2, 3, 4, 5] assert _combine_removed(8, [2, 5], [1, 3, 4]) == [1, 2, 4, 5, 6] assert _combine_removed(8, [7], []) == [7] def test_array_contraction_to_diagonal_multiple_identities(): expr = _array_contraction(_array_tensor_product(A, B, I, C), (1, 2, 4), (5, 6)) assert _array_contraction_to_diagonal_multiple_identity(expr) == (expr, []) assert convert_array_to_matrix(expr) == _array_contraction(_array_tensor_product(A, B, C), (1, 2, 4)) expr = _array_contraction(_array_tensor_product(A, I, I), (1, 2, 4)) assert _array_contraction_to_diagonal_multiple_identity(expr) == (A, [2]) assert convert_array_to_matrix(expr) == A expr = _array_contraction(_array_tensor_product(A, I, I, B), (1, 2, 4), (3, 6)) assert _array_contraction_to_diagonal_multiple_identity(expr) == (expr, []) expr = _array_contraction(_array_tensor_product(A, I, I, B), (1, 2, 3, 4, 6)) assert _array_contraction_to_diagonal_multiple_identity(expr) == (expr, []) def test_convert_array_element_to_matrix(): expr = ArrayElement(M, (i, j)) assert convert_array_to_matrix(expr) == MatrixElement(M, i, j) expr = ArrayElement(_array_contraction(_array_tensor_product(M, N), (1, 3)), (i, j)) assert convert_array_to_matrix(expr) == MatrixElement(MN.T, i, j) expr = ArrayElement(_array_tensor_product(M, N), (i, j, m, n)) assert convert_array_to_matrix(expr) == expr def test_convert_array_elementwise_function_to_matrix(): d = Dummy("d") expr = ArrayElementwiseApplyFunc(Lambda(d, sin(d)), x.Ty) assert convert_array_to_matrix(expr) == sin(x.Ty) expr = ArrayElementwiseApplyFunc(Lambda(d, d2), x.Ty) assert convert_array_to_matrix(expr) == (x.Ty)2 expr = ArrayElementwiseApplyFunc(Lambda(d, sin(d)), x) assert convert_array_to_matrix(expr).dummy_eq(x.applyfunc(sin)) expr = ArrayElementwiseApplyFunc(Lambda(d, 1 / (2 sqrt(d))), x) assert convert_array_to_matrix(expr) == S.Half * HadamardPower(x, -S.Half) def test_array2matrix(): # See issue https://github.com/sympy/sympy/pull/22877 expr = PermuteDims(ArrayContraction(ArrayTensorProduct(x, I, I1, x), (0, 3), (1, 7)), Permutation(2, 3)) expected = PermuteDims(ArrayTensorProduct(xx.T, I1), Permutation(3)(1, 2)) assert _array2matrix(expr) == expected def test_recognize_broadcasting(): expr = ArrayTensorProduct(x.Tx, A) assert _remove_trivial_dims(expr) == (KroneckerProduct(x.Tx, A), [0, 1]) expr = ArrayTensorProduct(A, x.Tx) assert _remove_trivial_dims(expr) == (KroneckerProduct(A, x.Tx), [2, 3]) expr = ArrayTensorProduct(A, B, x.Tx, C) assert _remove_trivial_dims(expr) == (ArrayTensorProduct(A, KroneckerProduct(B, x.Tx), C), [4, 5]) # Always prefer matrix multiplication to Kronecker product, if possible: expr = ArrayTensorProduct(a, b, x.Tx) assert _remove_trivial_dims(expr) == (ax.Tx*b.T, [1, 3, 4, 5])
f3f7b1b12349be4ccfbe210638dc1ee7b1c89ade4e4a9e6ed0df33344647a063	from sympy import Lambda, KroneckerProduct from sympy.core.symbol import symbols, Dummy from sympy.matrices.expressions.hadamard import (HadamardPower, HadamardProduct) from sympy.matrices.expressions.inverse import Inverse from sympy.matrices.expressions.matexpr import MatrixSymbol from sympy.matrices.expressions.matpow import MatPow from sympy.matrices.expressions.special import Identity from sympy.matrices.expressions.trace import Trace from sympy.matrices.expressions.transpose import Transpose from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayContraction, \ PermuteDims, ArrayDiagonal, ArrayElementwiseApplyFunc, _array_contraction, _array_tensor_product, Reshape from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array i, j, k, l, m, n = symbols("i j k l m n") I = Identity(k) M = MatrixSymbol("M", k, k) N = MatrixSymbol("N", k, k) P = MatrixSymbol("P", k, k) Q = MatrixSymbol("Q", k, k) A = MatrixSymbol("A", k, k) B = MatrixSymbol("B", k, k) C = MatrixSymbol("C", k, k) D = MatrixSymbol("D", k, k) X = MatrixSymbol("X", k, k) Y = MatrixSymbol("Y", k, k) a = MatrixSymbol("a", k, 1) b = MatrixSymbol("b", k, 1) c = MatrixSymbol("c", k, 1) d = MatrixSymbol("d", k, 1) def test_arrayexpr_convert_matrix_to_array(): expr = MN result = ArrayContraction(ArrayTensorProduct(M, N), (1, 2)) assert convert_matrix_to_array(expr) == result expr = MNM result = _array_contraction(ArrayTensorProduct(M, N, M), (1, 2), (3, 4)) assert convert_matrix_to_array(expr) == result expr = Transpose(M) assert convert_matrix_to_array(expr) == PermuteDims(M, [1, 0]) expr = MTranspose(N) assert convert_matrix_to_array(expr) == _array_contraction(_array_tensor_product(M, PermuteDims(N, [1, 0])), (1, 2)) expr = 3MN res = convert_matrix_to_array(expr) rexpr = convert_array_to_matrix(res) assert expr == rexpr expr = 3M + NM.TM + 4kN res = convert_matrix_to_array(expr) rexpr = convert_array_to_matrix(res) assert expr == rexpr expr = Inverse(M)N rexpr = convert_array_to_matrix(convert_matrix_to_array(expr)) assert expr == rexpr expr = M*2 rexpr = convert_array_to_matrix(convert_matrix_to_array(expr)) assert expr == rexpr expr = M(2N + 3M) res = convert_matrix_to_array(expr) rexpr = convert_array_to_matrix(res) assert expr == rexpr expr = Trace(M) result = ArrayContraction(M, (0, 1)) assert convert_matrix_to_array(expr) == result expr = 3Trace(M) result = ArrayContraction(ArrayTensorProduct(3, M), (0, 1)) assert convert_matrix_to_array(expr) == result expr = 3Trace(Trace(M) * M) result = ArrayContraction(ArrayTensorProduct(3, M, M), (0, 1), (2, 3)) assert convert_matrix_to_array(expr) == result expr = 3Trace(M)2 result = ArrayContraction(ArrayTensorProduct(3, M, M), (0, 1), (2, 3)) assert convert_matrix_to_array(expr) == result expr = HadamardProduct(M, N) result = ArrayDiagonal(ArrayTensorProduct(M, N), (0, 2), (1, 3)) assert convert_matrix_to_array(expr) == result expr = HadamardProduct(MN, NM) result = ArrayDiagonal(ArrayContraction(ArrayTensorProduct(M, N, N, M), (1, 2), (5, 6)), (0, 2), (1, 3)) assert convert_matrix_to_array(expr) == result expr = HadamardPower(M, 2) result = ArrayDiagonal(ArrayTensorProduct(M, M), (0, 2), (1, 3)) assert convert_matrix_to_array(expr) == result expr = HadamardPower(MN, 2) result = ArrayDiagonal(ArrayContraction(ArrayTensorProduct(M, N, M, N), (1, 2), (5, 6)), (0, 2), (1, 3)) assert convert_matrix_to_array(expr) == result expr = HadamardPower(M, n) d0 = Dummy("d0") result = ArrayElementwiseApplyFunc(Lambda(d0, d0n), M) assert convert_matrix_to_array(expr).dummy_eq(result) expr = M2 assert isinstance(expr, MatPow) assert convert_matrix_to_array(expr) == ArrayContraction(ArrayTensorProduct(M, M), (1, 2)) expr = a.Tb cg = convert_matrix_to_array(expr) assert cg == ArrayContraction(ArrayTensorProduct(a, b), (0, 2)) expr = KroneckerProduct(A, B) cg = convert_matrix_to_array(expr) assert cg == Reshape(PermuteDims(ArrayTensorProduct(A, B), [0, 2, 1, 3]), (k2, k2)) expr = KroneckerProduct(A, B, C, D) cg = convert_matrix_to_array(expr) assert cg == Reshape(PermuteDims(ArrayTensorProduct(A, B, C, D), [0, 2, 4, 6, 1, 3, 5, 7]), (k4, k*4))
9171c6cf1d51e079aea1b6bc2ee373fba347718456c87ebe47a572bb1b589281	import random from sympy import tensordiagonal, eye, KroneckerDelta, Array from sympy.core.symbol import symbols from sympy.functions.elementary.trigonometric import (cos, sin) from sympy.matrices.expressions.diagonal import DiagMatrix from sympy.matrices.expressions.matexpr import MatrixSymbol from sympy.matrices.expressions.special import ZeroMatrix from sympy.tensor.array.arrayop import (permutedims, tensorcontraction, tensorproduct) from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray from sympy.combinatorics import Permutation from sympy.tensor.array.expressions.array_expressions import ZeroArray, OneArray, ArraySymbol, ArrayElement, \ PermuteDims, ArrayContraction, ArrayTensorProduct, ArrayDiagonal, \ ArrayAdd, nest_permutation, ArrayElementwiseApplyFunc, _EditArrayContraction, _ArgE, _array_tensor_product, \ _array_contraction, _array_diagonal, _array_add, _permute_dims, Reshape from sympy.testing.pytest import raises i, j, k, l, m, n = symbols("i j k l m n") M = ArraySymbol("M", (k, k)) N = ArraySymbol("N", (k, k)) P = ArraySymbol("P", (k, k)) Q = ArraySymbol("Q", (k, k)) A = ArraySymbol("A", (k, k)) B = ArraySymbol("B", (k, k)) C = ArraySymbol("C", (k, k)) D = ArraySymbol("D", (k, k)) X = ArraySymbol("X", (k, k)) Y = ArraySymbol("Y", (k, k)) a = ArraySymbol("a", (k, 1)) b = ArraySymbol("b", (k, 1)) c = ArraySymbol("c", (k, 1)) d = ArraySymbol("d", (k, 1)) def test_array_symbol_and_element(): A = ArraySymbol("A", (2,)) A0 = ArrayElement(A, (0,)) A1 = ArrayElement(A, (1,)) assert A.as_explicit() == ImmutableDenseNDimArray([A0, A1]) A2 = tensorproduct(A, A) assert A2.shape == (2, 2) # TODO: not yet supported: # assert A2.as_explicit() == Array([[A[0]A[0], A[1]A[0]], [A[0]A[1], A[1]A[1]]]) A3 = tensorcontraction(A2, (0, 1)) assert A3.shape == () # TODO: not yet supported: # assert A3.as_explicit() == Array([]) A = ArraySymbol("A", (2, 3, 4)) Ae = A.as_explicit() assert Ae == ImmutableDenseNDimArray( [[[ArrayElement(A, (i, j, k)) for k in range(4)] for j in range(3)] for i in range(2)]) p = _permute_dims(A, Permutation(0, 2, 1)) assert isinstance(p, PermuteDims) def test_zero_array(): assert ZeroArray() == 0 assert ZeroArray().is_Integer za = ZeroArray(3, 2, 4) assert za.shape == (3, 2, 4) za_e = za.as_explicit() assert za_e.shape == (3, 2, 4) m, n, k = symbols("m n k") za = ZeroArray(m, n, k, 2) assert za.shape == (m, n, k, 2) raises(ValueError, lambda: za.as_explicit()) def test_one_array(): assert OneArray() == 1 assert OneArray().is_Integer oa = OneArray(3, 2, 4) assert oa.shape == (3, 2, 4) oa_e = oa.as_explicit() assert oa_e.shape == (3, 2, 4) m, n, k = symbols("m n k") oa = OneArray(m, n, k, 2) assert oa.shape == (m, n, k, 2) raises(ValueError, lambda: oa.as_explicit()) def test_arrayexpr_contraction_construction(): cg = _array_contraction(A) assert cg == A cg = _array_contraction(_array_tensor_product(A, B), (1, 0)) assert cg == _array_contraction(_array_tensor_product(A, B), (0, 1)) cg = _array_contraction(_array_tensor_product(M, N), (0, 1)) indtup = cg._get_contraction_tuples() assert indtup == [[(0, 0), (0, 1)]] assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(0, 1)] cg = _array_contraction(_array_tensor_product(M, N), (1, 2)) indtup = cg._get_contraction_tuples() assert indtup == [[(0, 1), (1, 0)]] assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(1, 2)] cg = _array_contraction(_array_tensor_product(M, M, N), (1, 4), (2, 5)) indtup = cg._get_contraction_tuples() assert indtup == [[(0, 0), (1, 1)], [(0, 1), (2, 0)]] assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(0, 3), (1, 4)] # Test removal of trivial contraction: assert _array_contraction(a, (1,)) == a assert _array_contraction( _array_tensor_product(a, b), (0, 2), (1,), (3,)) == _array_contraction( _array_tensor_product(a, b), (0, 2)) def test_arrayexpr_array_flatten(): # Flatten nested ArrayTensorProduct objects: expr1 = _array_tensor_product(M, N) expr2 = _array_tensor_product(P, Q) expr = _array_tensor_product(expr1, expr2) assert expr == _array_tensor_product(M, N, P, Q) assert expr.args == (M, N, P, Q) # Flatten mixed ArrayTensorProduct and ArrayContraction objects: cg1 = _array_contraction(expr1, (1, 2)) cg2 = _array_contraction(expr2, (0, 3)) expr = _array_tensor_product(cg1, cg2) assert expr == _array_contraction(_array_tensor_product(M, N, P, Q), (1, 2), (4, 7)) expr = _array_tensor_product(M, cg1) assert expr == _array_contraction(_array_tensor_product(M, M, N), (3, 4)) # Flatten nested ArrayContraction objects: cgnested = _array_contraction(cg1, (0, 1)) assert cgnested == _array_contraction(_array_tensor_product(M, N), (0, 3), (1, 2)) cgnested = _array_contraction(_array_tensor_product(cg1, cg2), (0, 3)) assert cgnested == _array_contraction(_array_tensor_product(M, N, P, Q), (0, 6), (1, 2), (4, 7)) cg3 = _array_contraction(_array_tensor_product(M, N, P, Q), (1, 3), (2, 4)) cgnested = _array_contraction(cg3, (0, 1)) assert cgnested == _array_contraction(_array_tensor_product(M, N, P, Q), (0, 5), (1, 3), (2, 4)) cgnested = _array_contraction(cg3, (0, 3), (1, 2)) assert cgnested == _array_contraction(_array_tensor_product(M, N, P, Q), (0, 7), (1, 3), (2, 4), (5, 6)) cg4 = _array_contraction(_array_tensor_product(M, N, P, Q), (1, 5), (3, 7)) cgnested = _array_contraction(cg4, (0, 1)) assert cgnested == _array_contraction(_array_tensor_product(M, N, P, Q), (0, 2), (1, 5), (3, 7)) cgnested = _array_contraction(cg4, (0, 1), (2, 3)) assert cgnested == _array_contraction(_array_tensor_product(M, N, P, Q), (0, 2), (1, 5), (3, 7), (4, 6)) cg = _array_diagonal(cg4) assert cg == cg4 assert isinstance(cg, type(cg4)) # Flatten nested ArrayDiagonal objects: cg1 = _array_diagonal(expr1, (1, 2)) cg2 = _array_diagonal(expr2, (0, 3)) cg3 = _array_diagonal(_array_tensor_product(M, N, P, Q), (1, 3), (2, 4)) cg4 = _array_diagonal(_array_tensor_product(M, N, P, Q), (1, 5), (3, 7)) cgnested = _array_diagonal(cg1, (0, 1)) assert cgnested == _array_diagonal(_array_tensor_product(M, N), (1, 2), (0, 3)) cgnested = _array_diagonal(cg3, (1, 2)) assert cgnested == _array_diagonal(_array_tensor_product(M, N, P, Q), (1, 3), (2, 4), (5, 6)) cgnested = _array_diagonal(cg4, (1, 2)) assert cgnested == _array_diagonal(_array_tensor_product(M, N, P, Q), (1, 5), (3, 7), (2, 4)) cg = _array_add(M, N) cg2 = _array_add(cg, P) assert isinstance(cg2, ArrayAdd) assert cg2.args == (M, N, P) assert cg2.shape == (k, k) expr = _array_tensor_product(_array_diagonal(X, (0, 1)), _array_diagonal(A, (0, 1))) assert expr == _array_diagonal(_array_tensor_product(X, A), (0, 1), (2, 3)) expr1 = _array_diagonal(_array_tensor_product(X, A), (1, 2)) expr2 = _array_tensor_product(expr1, a) assert expr2 == _permute_dims(_array_diagonal(_array_tensor_product(X, A, a), (1, 2)), [0, 1, 4, 2, 3]) expr1 = _array_contraction(_array_tensor_product(X, A), (1, 2)) expr2 = _array_tensor_product(expr1, a) assert isinstance(expr2, ArrayContraction) assert isinstance(expr2.expr, ArrayTensorProduct) cg = _array_tensor_product(_array_diagonal(_array_tensor_product(A, X, Y), (0, 3), (1, 5)), a, b) assert cg == _permute_dims(_array_diagonal(_array_tensor_product(A, X, Y, a, b), (0, 3), (1, 5)), [0, 1, 6, 7, 2, 3, 4, 5]) def test_arrayexpr_array_diagonal(): cg = _array_diagonal(M, (1, 0)) assert cg == _array_diagonal(M, (0, 1)) cg = _array_diagonal(_array_tensor_product(M, N, P), (4, 1), (2, 0)) assert cg == _array_diagonal(_array_tensor_product(M, N, P), (1, 4), (0, 2)) cg = _array_diagonal(_array_tensor_product(M, N), (1, 2), (3,), allow_trivial_diags=True) assert cg == _permute_dims(_array_diagonal(_array_tensor_product(M, N), (1, 2)), [0, 2, 1]) Ax = ArraySymbol("Ax", shape=(1, 2, 3, 4, 3, 5, 6, 2, 7)) cg = _array_diagonal(Ax, (1, 7), (3,), (2, 4), (6,), allow_trivial_diags=True) assert cg == _permute_dims(_array_diagonal(Ax, (1, 7), (2, 4)), [0, 2, 4, 5, 1, 6, 3]) cg = _array_diagonal(M, (0,), allow_trivial_diags=True) assert cg == _permute_dims(M, [1, 0]) raises(ValueError, lambda: _array_diagonal(M, (0, 0))) def test_arrayexpr_array_shape(): expr = _array_tensor_product(M, N, P, Q) assert expr.shape == (k, k, k, k, k, k, k, k) Z = MatrixSymbol("Z", m, n) expr = _array_tensor_product(M, Z) assert expr.shape == (k, k, m, n) expr2 = _array_contraction(expr, (0, 1)) assert expr2.shape == (m, n) expr2 = _array_diagonal(expr, (0, 1)) assert expr2.shape == (m, n, k) exprp = _permute_dims(expr, [2, 1, 3, 0]) assert exprp.shape == (m, k, n, k) expr3 = _array_tensor_product(N, Z) expr2 = _array_add(expr, expr3) assert expr2.shape == (k, k, m, n) # Contraction along axes with discordant dimensions: raises(ValueError, lambda: _array_contraction(expr, (1, 2))) # Also diagonal needs the same dimensions: raises(ValueError, lambda: _array_diagonal(expr, (1, 2))) # Diagonal requires at least to axes to compute the diagonal: raises(ValueError, lambda: _array_diagonal(expr, (1,))) def test_arrayexpr_permutedims_sink(): cg = _permute_dims(_array_tensor_product(M, N), [0, 1, 3, 2], nest_permutation=False) sunk = nest_permutation(cg) assert sunk == _array_tensor_product(M, _permute_dims(N, [1, 0])) cg = _permute_dims(_array_tensor_product(M, N), [1, 0, 3, 2], nest_permutation=False) sunk = nest_permutation(cg) assert sunk == _array_tensor_product(_permute_dims(M, [1, 0]), _permute_dims(N, [1, 0])) cg = _permute_dims(_array_tensor_product(M, N), [3, 2, 1, 0], nest_permutation=False) sunk = nest_permutation(cg) assert sunk == _array_tensor_product(_permute_dims(N, [1, 0]), _permute_dims(M, [1, 0])) cg = _permute_dims(_array_contraction(_array_tensor_product(M, N), (1, 2)), [1, 0], nest_permutation=False) sunk = nest_permutation(cg) assert sunk == _array_contraction(_permute_dims(_array_tensor_product(M, N), [[0, 3]]), (1, 2)) cg = _permute_dims(_array_tensor_product(M, N), [1, 0, 3, 2], nest_permutation=False) sunk = nest_permutation(cg) assert sunk == _array_tensor_product(_permute_dims(M, [1, 0]), _permute_dims(N, [1, 0])) cg = _permute_dims(_array_contraction(_array_tensor_product(M, N, P), (1, 2), (3, 4)), [1, 0], nest_permutation=False) sunk = nest_permutation(cg) assert sunk == _array_contraction(_permute_dims(_array_tensor_product(M, N, P), [[0, 5]]), (1, 2), (3, 4)) def test_arrayexpr_push_indices_up_and_down(): indices = list(range(12)) contr_diag_indices = [(0, 6), (2, 8)] assert ArrayContraction._push_indices_down(contr_diag_indices, indices) == (1, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15) assert ArrayContraction._push_indices_up(contr_diag_indices, indices) == (None, 0, None, 1, 2, 3, None, 4, None, 5, 6, 7) assert ArrayDiagonal._push_indices_down(contr_diag_indices, indices, 10) == (1, 3, 4, 5, 7, 9, (0, 6), (2, 8), None, None, None, None) assert ArrayDiagonal._push_indices_up(contr_diag_indices, indices, 10) == (6, 0, 7, 1, 2, 3, 6, 4, 7, 5, None, None) contr_diag_indices = [(1, 2), (7, 8)] assert ArrayContraction._push_indices_down(contr_diag_indices, indices) == (0, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15) assert ArrayContraction._push_indices_up(contr_diag_indices, indices) == (0, None, None, 1, 2, 3, 4, None, None, 5, 6, 7) assert ArrayDiagonal._push_indices_down(contr_diag_indices, indices, 10) == (0, 3, 4, 5, 6, 9, (1, 2), (7, 8), None, None, None, None) assert ArrayDiagonal._push_indices_up(contr_diag_indices, indices, 10) == (0, 6, 6, 1, 2, 3, 4, 7, 7, 5, None, None) def test_arrayexpr_split_multiple_contractions(): a = MatrixSymbol("a", k, 1) b = MatrixSymbol("b", k, 1) A = MatrixSymbol("A", k, k) B = MatrixSymbol("B", k, k) C = MatrixSymbol("C", k, k) X = MatrixSymbol("X", k, k) cg = _array_contraction(_array_tensor_product(A.T, a, b, b.T, (AXb).applyfunc(cos)), (1, 2, 8), (5, 6, 9)) expected = _array_contraction(_array_tensor_product(A.T, DiagMatrix(a), OneArray(1), b, b.T, (AXb).applyfunc(cos)), (1, 3), (2, 9), (6, 7, 10)) assert cg.split_multiple_contractions().dummy_eq(expected) # Check no overlap of lines: cg = _array_contraction(_array_tensor_product(A, a, C, a, B), (1, 2, 4), (5, 6, 8), (3, 7)) assert cg.split_multiple_contractions() == cg cg = _array_contraction(_array_tensor_product(a, b, A), (0, 2, 4), (1, 3)) assert cg.split_multiple_contractions() == cg def test_arrayexpr_nested_permutations(): cg = _permute_dims(_permute_dims(M, (1, 0)), (1, 0)) assert cg == M times = 3 plist1 = [list(range(6)) for i in range(times)] plist2 = [list(range(6)) for i in range(times)] for i in range(times): random.shuffle(plist1[i]) random.shuffle(plist2[i]) plist1.append([2, 5, 4, 1, 0, 3]) plist2.append([3, 5, 0, 4, 1, 2]) plist1.append([2, 5, 4, 0, 3, 1]) plist2.append([3, 0, 5, 1, 2, 4]) plist1.append([5, 4, 2, 0, 3, 1]) plist2.append([4, 5, 0, 2, 3, 1]) Me = M.subs(k, 3).as_explicit() Ne = N.subs(k, 3).as_explicit() Pe = P.subs(k, 3).as_explicit() cge = tensorproduct(Me, Ne, Pe) for permutation_array1, permutation_array2 in zip(plist1, plist2): p1 = Permutation(permutation_array1) p2 = Permutation(permutation_array2) cg = _permute_dims( _permute_dims( _array_tensor_product(M, N, P), p1), p2 ) result = _permute_dims( _array_tensor_product(M, N, P), p2p1 ) assert cg == result # Check that `permutedims` behaves the same way with explicit-component arrays: result1 = _permute_dims(_permute_dims(cge, p1), p2) result2 = _permute_dims(cge, p2p1) assert result1 == result2 def test_arrayexpr_contraction_permutation_mix(): Me = M.subs(k, 3).as_explicit() Ne = N.subs(k, 3).as_explicit() cg1 = _array_contraction(PermuteDims(_array_tensor_product(M, N), Permutation([0, 2, 1, 3])), (2, 3)) cg2 = _array_contraction(_array_tensor_product(M, N), (1, 3)) assert cg1 == cg2 cge1 = tensorcontraction(permutedims(tensorproduct(Me, Ne), Permutation([0, 2, 1, 3])), (2, 3)) cge2 = tensorcontraction(tensorproduct(Me, Ne), (1, 3)) assert cge1 == cge2 cg1 = _permute_dims(_array_tensor_product(M, N), Permutation([0, 1, 3, 2])) cg2 = _array_tensor_product(M, _permute_dims(N, Permutation([1, 0]))) assert cg1 == cg2 cg1 = _array_contraction( _permute_dims( _array_tensor_product(M, N, P, Q), Permutation([0, 2, 3, 1, 4, 5, 7, 6])), (1, 2), (3, 5) ) cg2 = _array_contraction( _array_tensor_product(M, N, P, _permute_dims(Q, Permutation([1, 0]))), (1, 5), (2, 3) ) assert cg1 == cg2 cg1 = _array_contraction( _permute_dims( _array_tensor_product(M, N, P, Q), Permutation([1, 0, 4, 6, 2, 7, 5, 3])), (0, 1), (2, 6), (3, 7) ) cg2 = _permute_dims( _array_contraction( _array_tensor_product(M, P, Q, N), (0, 1), (2, 3), (4, 7)), [1, 0] ) assert cg1 == cg2 cg1 = _array_contraction( _permute_dims( _array_tensor_product(M, N, P, Q), Permutation([1, 0, 4, 6, 7, 2, 5, 3])), (0, 1), (2, 6), (3, 7) ) cg2 = _permute_dims( _array_contraction( _array_tensor_product(_permute_dims(M, [1, 0]), N, P, Q), (0, 1), (3, 6), (4, 5) ), Permutation([1, 0]) ) assert cg1 == cg2 def test_arrayexpr_permute_tensor_product(): cg1 = _permute_dims(_array_tensor_product(M, N, P, Q), Permutation([2, 3, 1, 0, 5, 4, 6, 7])) cg2 = _array_tensor_product(N, _permute_dims(M, [1, 0]), _permute_dims(P, [1, 0]), Q) assert cg1 == cg2 # TODO: reverse operation starting with `PermuteDims` and getting down to `bb`... cg1 = _permute_dims(_array_tensor_product(M, N, P, Q), Permutation([2, 3, 4, 5, 0, 1, 6, 7])) cg2 = _array_tensor_product(N, P, M, Q) assert cg1 == cg2 cg1 = _permute_dims(_array_tensor_product(M, N, P, Q), Permutation([2, 3, 4, 6, 5, 7, 0, 1])) assert cg1.expr == _array_tensor_product(N, P, Q, M) assert cg1.permutation == Permutation([0, 1, 2, 4, 3, 5, 6, 7]) cg1 = _array_contraction( _permute_dims( _array_tensor_product(N, Q, Q, M), [2, 1, 5, 4, 0, 3, 6, 7]), [1, 2, 6]) cg2 = _permute_dims(_array_contraction(_array_tensor_product(Q, Q, N, M), (3, 5, 6)), [0, 2, 3, 1, 4]) assert cg1 == cg2 cg1 = _array_contraction( _array_contraction( _array_contraction( _array_contraction( _permute_dims( _array_tensor_product(N, Q, Q, M), [2, 1, 5, 4, 0, 3, 6, 7]), [1, 2, 6]), [1, 3, 4]), [1]), [0]) cg2 = _array_contraction(_array_tensor_product(M, N, Q, Q), (0, 3, 5), (1, 4, 7), (2,), (6,)) assert cg1 == cg2 def test_arrayexpr_canonicalize_diagonal__permute_dims(): tp = _array_tensor_product(M, Q, N, P) expr = _array_diagonal( _permute_dims(tp, [0, 1, 2, 4, 7, 6, 3, 5]), (2, 4, 5), (6, 7), (0, 3)) result = _array_diagonal(tp, (2, 6, 7), (3, 5), (0, 4)) assert expr == result tp = _array_tensor_product(M, N, P, Q) expr = _array_diagonal(_permute_dims(tp, [0, 5, 2, 4, 1, 6, 3, 7]), (1, 2, 6), (3, 4)) result = _array_diagonal(_array_tensor_product(M, P, N, Q), (3, 4, 5), (1, 2)) assert expr == result def test_arrayexpr_canonicalize_diagonal_contraction(): tp = _array_tensor_product(M, N, P, Q) expr = _array_contraction(_array_diagonal(tp, (1, 3, 4)), (0, 3)) result = _array_diagonal(_array_contraction(_array_tensor_product(M, N, P, Q), (0, 6)), (0, 2, 3)) assert expr == result expr = _array_contraction(_array_diagonal(tp, (0, 1, 2, 3, 7)), (1, 2, 3)) result = _array_contraction(_array_tensor_product(M, N, P, Q), (0, 1, 2, 3, 5, 6, 7)) assert expr == result expr = _array_contraction(_array_diagonal(tp, (0, 2, 6, 7)), (1, 2, 3)) result = _array_diagonal(_array_contraction(tp, (3, 4, 5)), (0, 2, 3, 4)) assert expr == result td = _array_diagonal(_array_tensor_product(M, N, P, Q), (0, 3)) expr = _array_contraction(td, (2, 1), (0, 4, 6, 5, 3)) result = _array_contraction(_array_tensor_product(M, N, P, Q), (0, 1, 3, 5, 6, 7), (2, 4)) assert expr == result def test_arrayexpr_array_wrong_permutation_size(): cg = _array_tensor_product(M, N) raises(ValueError, lambda: _permute_dims(cg, [1, 0])) raises(ValueError, lambda: _permute_dims(cg, [1, 0, 2, 3, 5, 4])) def test_arrayexpr_nested_array_elementwise_add(): cg = _array_contraction(_array_add( _array_tensor_product(M, N), _array_tensor_product(N, M) ), (1, 2)) result = _array_add( _array_contraction(_array_tensor_product(M, N), (1, 2)), _array_contraction(_array_tensor_product(N, M), (1, 2)) ) assert cg == result cg = _array_diagonal(_array_add( _array_tensor_product(M, N), _array_tensor_product(N, M) ), (1, 2)) result = _array_add( _array_diagonal(_array_tensor_product(M, N), (1, 2)), _array_diagonal(_array_tensor_product(N, M), (1, 2)) ) assert cg == result def test_arrayexpr_array_expr_zero_array(): za1 = ZeroArray(k, l, m, n) zm1 = ZeroMatrix(m, n) za2 = ZeroArray(k, m, m, n) zm2 = ZeroMatrix(m, m) zm3 = ZeroMatrix(k, k) assert _array_tensor_product(M, N, za1) == ZeroArray(k, k, k, k, k, l, m, n) assert _array_tensor_product(M, N, zm1) == ZeroArray(k, k, k, k, m, n) assert _array_contraction(za1, (3,)) == ZeroArray(k, l, m) assert _array_contraction(zm1, (1,)) == ZeroArray(m) assert _array_contraction(za2, (1, 2)) == ZeroArray(k, n) assert _array_contraction(zm2, (0, 1)) == 0 assert _array_diagonal(za2, (1, 2)) == ZeroArray(k, n, m) assert _array_diagonal(zm2, (0, 1)) == ZeroArray(m) assert _permute_dims(za1, [2, 1, 3, 0]) == ZeroArray(m, l, n, k) assert _permute_dims(zm1, [1, 0]) == ZeroArray(n, m) assert _array_add(za1) == za1 assert _array_add(zm1) == ZeroArray(m, n) tp1 = _array_tensor_product(MatrixSymbol("A", k, l), MatrixSymbol("B", m, n)) assert _array_add(tp1, za1) == tp1 tp2 = _array_tensor_product(MatrixSymbol("C", k, l), MatrixSymbol("D", m, n)) assert _array_add(tp1, za1, tp2) == _array_add(tp1, tp2) assert _array_add(M, zm3) == M assert _array_add(M, N, zm3) == _array_add(M, N) def test_arrayexpr_array_expr_applyfunc(): A = ArraySymbol("A", (3, k, 2)) aaf = ArrayElementwiseApplyFunc(sin, A) assert aaf.shape == (3, k, 2) def test_edit_array_contraction(): cg = _array_contraction(_array_tensor_product(A, B, C, D), (1, 2, 5)) ecg = _EditArrayContraction(cg) assert ecg.to_array_contraction() == cg ecg.args_with_ind[1], ecg.args_with_ind[2] = ecg.args_with_ind[2], ecg.args_with_ind[1] assert ecg.to_array_contraction() == _array_contraction(_array_tensor_product(A, C, B, D), (1, 3, 4)) ci = ecg.get_new_contraction_index() new_arg = _ArgE(X) new_arg.indices = [ci, ci] ecg.args_with_ind.insert(2, new_arg) assert ecg.to_array_contraction() == _array_contraction(_array_tensor_product(A, C, X, B, D), (1, 3, 6), (4, 5)) assert ecg.get_contraction_indices() == [[1, 3, 6], [4, 5]] assert [[tuple(j) for j in i] for i in ecg.get_contraction_indices_to_ind_rel_pos()] == [[(0, 1), (1, 1), (3, 0)], [(2, 0), (2, 1)]] assert [list(i) for i in ecg.get_mapping_for_index(0)] == [[0, 1], [1, 1], [3, 0]] assert [list(i) for i in ecg.get_mapping_for_index(1)] == [[2, 0], [2, 1]] raises(ValueError, lambda: ecg.get_mapping_for_index(2)) ecg.args_with_ind.pop(1) assert ecg.to_array_contraction() == _array_contraction(_array_tensor_product(A, X, B, D), (1, 4), (2, 3)) ecg.args_with_ind[0].indices[1] = ecg.args_with_ind[1].indices[0] ecg.args_with_ind[1].indices[1] = ecg.args_with_ind[2].indices[0] assert ecg.to_array_contraction() == _array_contraction(_array_tensor_product(A, X, B, D), (1, 2), (3, 4)) ecg.insert_after(ecg.args_with_ind[1], _ArgE(C)) assert ecg.to_array_contraction() == _array_contraction(_array_tensor_product(A, X, C, B, D), (1, 2), (3, 6)) def test_array_expressions_no_canonicalization(): tp = _array_tensor_product(M, N, P) # ArrayTensorProduct: expr = ArrayTensorProduct(tp, N) assert str(expr) == "ArrayTensorProduct(ArrayTensorProduct(M, N, P), N)" assert expr.doit() == ArrayTensorProduct(M, N, P, N) expr = ArrayTensorProduct(ArrayContraction(M, (0, 1)), N) assert str(expr) == "ArrayTensorProduct(ArrayContraction(M, (0, 1)), N)" assert expr.doit() == ArrayContraction(ArrayTensorProduct(M, N), (0, 1)) expr = ArrayTensorProduct(ArrayDiagonal(M, (0, 1)), N) assert str(expr) == "ArrayTensorProduct(ArrayDiagonal(M, (0, 1)), N)" assert expr.doit() == PermuteDims(ArrayDiagonal(ArrayTensorProduct(M, N), (0, 1)), [2, 0, 1]) expr = ArrayTensorProduct(PermuteDims(M, [1, 0]), N) assert str(expr) == "ArrayTensorProduct(PermuteDims(M, (0 1)), N)" assert expr.doit() == PermuteDims(ArrayTensorProduct(M, N), [1, 0, 2, 3]) # ArrayContraction: expr = ArrayContraction(_array_contraction(tp, (0, 2)), (0, 1)) assert isinstance(expr, ArrayContraction) assert isinstance(expr.expr, ArrayContraction) assert str(expr) == "ArrayContraction(ArrayContraction(ArrayTensorProduct(M, N, P), (0, 2)), (0, 1))" assert expr.doit() == ArrayContraction(tp, (0, 2), (1, 3)) expr = ArrayContraction(ArrayContraction(ArrayContraction(tp, (0, 1)), (0, 1)), (0, 1)) assert expr.doit() == ArrayContraction(tp, (0, 1), (2, 3), (4, 5)) # assert expr._canonicalize() == ArrayContraction(ArrayContraction(tp, (0, 1)), (0, 1), (2, 3)) expr = ArrayContraction(ArrayDiagonal(tp, (0, 1)), (0, 1)) assert str(expr) == "ArrayContraction(ArrayDiagonal(ArrayTensorProduct(M, N, P), (0, 1)), (0, 1))" assert expr.doit() == ArrayDiagonal(ArrayContraction(ArrayTensorProduct(N, M, P), (0, 1)), (0, 1)) expr = ArrayContraction(PermuteDims(M, [1, 0]), (0, 1)) assert str(expr) == "ArrayContraction(PermuteDims(M, (0 1)), (0, 1))" assert expr.doit() == ArrayContraction(M, (0, 1)) # ArrayDiagonal: expr = ArrayDiagonal(ArrayDiagonal(tp, (0, 2)), (0, 1)) assert str(expr) == "ArrayDiagonal(ArrayDiagonal(ArrayTensorProduct(M, N, P), (0, 2)), (0, 1))" assert expr.doit() == ArrayDiagonal(tp, (0, 2), (1, 3)) expr = ArrayDiagonal(ArrayDiagonal(ArrayDiagonal(tp, (0, 1)), (0, 1)), (0, 1)) assert expr.doit() == ArrayDiagonal(tp, (0, 1), (2, 3), (4, 5)) assert expr._canonicalize() == expr.doit() expr = ArrayDiagonal(ArrayContraction(tp, (0, 1)), (0, 1)) assert str(expr) == "ArrayDiagonal(ArrayContraction(ArrayTensorProduct(M, N, P), (0, 1)), (0, 1))" assert expr.doit() == expr expr = ArrayDiagonal(PermuteDims(M, [1, 0]), (0, 1)) assert str(expr) == "ArrayDiagonal(PermuteDims(M, (0 1)), (0, 1))" assert expr.doit() == ArrayDiagonal(M, (0, 1)) # ArrayAdd: expr = ArrayAdd(M) assert isinstance(expr, ArrayAdd) assert expr.doit() == M expr = ArrayAdd(ArrayAdd(M, N), P) assert str(expr) == "ArrayAdd(ArrayAdd(M, N), P)" assert expr.doit() == ArrayAdd(M, N, P) expr = ArrayAdd(M, ArrayAdd(N, ArrayAdd(P, M))) assert expr.doit() == ArrayAdd(M, N, P, M) assert expr._canonicalize() == ArrayAdd(M, N, ArrayAdd(P, M)) expr = ArrayAdd(M, ZeroArray(k, k), N) assert str(expr) == "ArrayAdd(M, ZeroArray(k, k), N)" assert expr.doit() == ArrayAdd(M, N) # PermuteDims: expr = PermuteDims(PermuteDims(M, [1, 0]), [1, 0]) assert str(expr) == "PermuteDims(PermuteDims(M, (0 1)), (0 1))" assert expr.doit() == M expr = PermuteDims(PermuteDims(PermuteDims(M, [1, 0]), [1, 0]), [1, 0]) assert expr.doit() == PermuteDims(M, [1, 0]) assert expr._canonicalize() == expr.doit() # Reshape expr = Reshape(A, (k2,)) assert expr.shape == (k2,) assert isinstance(expr, Reshape) def test_array_expr_construction_with_functions(): tp = tensorproduct(M, N) assert tp == ArrayTensorProduct(M, N) expr = tensorproduct(A, eye(2)) assert expr == ArrayTensorProduct(A, eye(2)) # Contraction: expr = tensorcontraction(M, (0, 1)) assert expr == ArrayContraction(M, (0, 1)) expr = tensorcontraction(tp, (1, 2)) assert expr == ArrayContraction(tp, (1, 2)) expr = tensorcontraction(tensorcontraction(tp, (1, 2)), (0, 1)) assert expr == ArrayContraction(tp, (0, 3), (1, 2)) # Diagonalization: expr = tensordiagonal(M, (0, 1)) assert expr == ArrayDiagonal(M, (0, 1)) expr = tensordiagonal(tensordiagonal(tp, (0, 1)), (0, 1)) assert expr == ArrayDiagonal(tp, (0, 1), (2, 3)) # Permutation of dimensions: expr = permutedims(M, [1, 0]) assert expr == PermuteDims(M, [1, 0]) expr = permutedims(PermuteDims(tp, [1, 0, 2, 3]), [0, 1, 3, 2]) assert expr == PermuteDims(tp, [1, 0, 3, 2]) def test_array_element_expressions(): # Check commutative property: assert M[0, 0]N[0, 0] == N[0, 0]M[0, 0] # Check derivatives: assert M[0, 0].diff(M[0, 0]) == 1 assert M[0, 0].diff(M[1, 0]) == 0 assert M[0, 0].diff(N[0, 0]) == 0 assert M[0, 1].diff(M[i, j]) == KroneckerDelta(i, 0)KroneckerDelta(j, 1) assert M[0, 1].diff(N[i, j]) == 0 K4 = ArraySymbol("K4", shape=(k, k, k, k)) assert K4[i, j, k, l].diff(K4[1, 2, 3, 4]) == ( KroneckerDelta(i, 1)KroneckerDelta(j, 2)KroneckerDelta(k, 3)KroneckerDelta(l, 4) ) def test_array_expr_reshape(): A = MatrixSymbol("A", 2, 2) B = ArraySymbol("B", (2, 2, 2)) C = Array([1, 2, 3, 4]) expr = Reshape(A, (4,)) assert expr.expr == A assert expr.shape == (4,) assert expr.as_explicit() == Array([A[0, 0], A[0, 1], A[1, 0], A[1, 1]]) expr = Reshape(B, (2, 4)) assert expr.expr == B assert expr.shape == (2, 4) ee = expr.as_explicit() assert isinstance(ee, ImmutableDenseNDimArray) assert ee.shape == (2, 4) assert ee == Array([[B[0, 0, 0], B[0, 0, 1], B[0, 1, 0], B[0, 1, 1]], [B[1, 0, 0], B[1, 0, 1], B[1, 1, 0], B[1, 1, 1]]]) expr = Reshape(A, (k, 2)) assert expr.shape == (k, 2) raises(ValueError, lambda: Reshape(A, (2, 3))) raises(ValueError, lambda: Reshape(A, (3,))) expr = Reshape(C, (2, 2)) assert expr.expr == C assert expr.shape == (2, 2) assert expr.doit() == Array([[1, 2], [3, 4]])
9690b8972e8c24b0d105c56ddc670fdc7ea47bd6c8fa8dd550bc34c7217941b6	from sympy.core.symbol import symbols from sympy.functions.elementary.trigonometric import (cos, sin) from sympy.matrices.expressions.matexpr import MatrixSymbol from sympy.matrices.expressions.special import Identity from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction from sympy.tensor.array.expressions.array_expressions import ArraySymbol, ArrayTensorProduct, \ PermuteDims, ArrayDiagonal, ArrayElementwiseApplyFunc, ArrayContraction, _permute_dims, Reshape from sympy.tensor.array.expressions.arrayexpr_derivatives import array_derive k = symbols("k") I = Identity(k) X = MatrixSymbol("X", k, k) x = MatrixSymbol("x", k, 1) A = MatrixSymbol("A", k, k) B = MatrixSymbol("B", k, k) C = MatrixSymbol("C", k, k) D = MatrixSymbol("D", k, k) A1 = ArraySymbol("A", (3, 2, k)) def test_arrayexpr_derivatives1(): res = array_derive(X, X) assert res == PermuteDims(ArrayTensorProduct(I, I), [0, 2, 1, 3]) cg = ArrayTensorProduct(A, X, B) res = array_derive(cg, X) assert res == _permute_dims( ArrayTensorProduct(I, A, I, B), [0, 4, 2, 3, 1, 5, 6, 7]) cg = ArrayContraction(X, (0, 1)) res = array_derive(cg, X) assert res == ArrayContraction(ArrayTensorProduct(I, I), (1, 3)) cg = ArrayDiagonal(X, (0, 1)) res = array_derive(cg, X) assert res == ArrayDiagonal(ArrayTensorProduct(I, I), (1, 3)) cg = ElementwiseApplyFunction(sin, X) res = array_derive(cg, X) assert res.dummy_eq(ArrayDiagonal( ArrayTensorProduct( ElementwiseApplyFunction(cos, X), I, I ), (0, 3), (1, 5))) cg = ArrayElementwiseApplyFunc(sin, X) res = array_derive(cg, X) assert res.dummy_eq(ArrayDiagonal( ArrayTensorProduct( I, I, ArrayElementwiseApplyFunc(cos, X) ), (1, 4), (3, 5))) res = array_derive(A1, A1) assert res == PermuteDims( ArrayTensorProduct(Identity(3), Identity(2), Identity(k)), [0, 2, 4, 1, 3, 5] ) cg = ArrayElementwiseApplyFunc(sin, A1) res = array_derive(cg, A1) assert res.dummy_eq(ArrayDiagonal( ArrayTensorProduct( Identity(3), Identity(2), Identity(k), ArrayElementwiseApplyFunc(cos, A1) ), (1, 6), (3, 7), (5, 8) )) cg = Reshape(A, (k2,)) res = array_derive(cg, A) assert res == Reshape(PermuteDims(ArrayTensorProduct(I, I), [0, 2, 1, 3]), (k, k, k2))
0b7d05621e96d2045da9cc57670c7ad71aade8e1f86bebc600625dd367975e04	from sympy.assumptions.ask import Q from sympy.assumptions.refine import refine from sympy.core.numbers import oo from sympy.core.relational import Equality, Eq, Ne from sympy.core.singleton import S from sympy.core.symbol import (Dummy, symbols) from sympy.functions import Piecewise from sympy.functions.elementary.trigonometric import cos, sin from sympy.sets.sets import (Interval, Union) from sympy.simplify.simplify import simplify from sympy.logic.boolalg import ( And, Boolean, Equivalent, ITE, Implies, Nand, Nor, Not, Or, POSform, SOPform, Xor, Xnor, conjuncts, disjuncts, distribute_or_over_and, distribute_and_over_or, eliminate_implications, is_nnf, is_cnf, is_dnf, simplify_logic, to_nnf, to_cnf, to_dnf, to_int_repr, bool_map, true, false, BooleanAtom, is_literal, term_to_integer, truth_table, as_Boolean, to_anf, is_anf, distribute_xor_over_and, anf_coeffs, ANFform, bool_minterm, bool_maxterm, bool_monomial, _check_pair, _convert_to_varsSOP, _convert_to_varsPOS, Exclusive, gateinputcount) from sympy.assumptions.cnf import CNF from sympy.testing.pytest import raises, XFAIL, slow from itertools import combinations, permutations, product A, B, C, D = symbols('A:D') a, b, c, d, e, w, x, y, z = symbols('a:e w:z') def test_overloading(): """Test that \|, & are overloaded as expected""" assert A & B == And(A, B) assert A \| B == Or(A, B) assert (A & B) \| C == Or(And(A, B), C) assert A >> B == Implies(A, B) assert A << B == Implies(B, A) assert ~A == Not(A) assert A ^ B == Xor(A, B) def test_And(): assert And() is true assert And(A) == A assert And(True) is true assert And(False) is false assert And(True, True) is true assert And(True, False) is false assert And(False, False) is false assert And(True, A) == A assert And(False, A) is false assert And(True, True, True) is true assert And(True, True, A) == A assert And(True, False, A) is false assert And(1, A) == A raises(TypeError, lambda: And(2, A)) raises(TypeError, lambda: And(A < 2, A)) assert And(A < 1, A >= 1) is false e = A > 1 assert And(e, e.canonical) == e.canonical g, l, ge, le = A > B, B < A, A >= B, B <= A assert And(g, l, ge, le) == And(ge, g) assert {And(i) for i in permutations((l,g,le,ge))} == {And(ge, g)} assert And(And(Eq(a, 0), Eq(b, 0)), And(Ne(a, 0), Eq(c, 0))) is false def test_Or(): assert Or() is false assert Or(A) == A assert Or(True) is true assert Or(False) is false assert Or(True, True) is true assert Or(True, False) is true assert Or(False, False) is false assert Or(True, A) is true assert Or(False, A) == A assert Or(True, False, False) is true assert Or(True, False, A) is true assert Or(False, False, A) == A assert Or(1, A) is true raises(TypeError, lambda: Or(2, A)) raises(TypeError, lambda: Or(A < 2, A)) assert Or(A < 1, A >= 1) is true e = A > 1 assert Or(e, e.canonical) == e g, l, ge, le = A > B, B < A, A >= B, B <= A assert Or(g, l, ge, le) == Or(g, ge) def test_Xor(): assert Xor() is false assert Xor(A) == A assert Xor(A, A) is false assert Xor(True, A, A) is true assert Xor(A, A, A, A, A) == A assert Xor(True, False, False, A, B) == ~Xor(A, B) assert Xor(True) is true assert Xor(False) is false assert Xor(True, True) is false assert Xor(True, False) is true assert Xor(False, False) is false assert Xor(True, A) == ~A assert Xor(False, A) == A assert Xor(True, False, False) is true assert Xor(True, False, A) == ~A assert Xor(False, False, A) == A assert isinstance(Xor(A, B), Xor) assert Xor(A, B, Xor(C, D)) == Xor(A, B, C, D) assert Xor(A, B, Xor(B, C)) == Xor(A, C) assert Xor(A < 1, A >= 1, B) == Xor(0, 1, B) == Xor(1, 0, B) e = A > 1 assert Xor(e, e.canonical) == Xor(0, 0) == Xor(1, 1) def test_rewrite_as_And(): expr = x ^ y assert expr.rewrite(And) == (x \| y) & (~x \| ~y) def test_rewrite_as_Or(): expr = x ^ y assert expr.rewrite(Or) == (x & ~y) \| (y & ~x) def test_rewrite_as_Nand(): expr = (y & z) \| (z & ~w) assert expr.rewrite(Nand) == ~(~(y & z) & ~(z & ~w)) def test_rewrite_as_Nor(): expr = z & (y \| ~w) assert expr.rewrite(Nor) == ~(~z \| ~(y \| ~w)) def test_Not(): raises(TypeError, lambda: Not(True, False)) assert Not(True) is false assert Not(False) is true assert Not(0) is true assert Not(1) is false assert Not(2) is false def test_Nand(): assert Nand() is false assert Nand(A) == ~A assert Nand(True) is false assert Nand(False) is true assert Nand(True, True) is false assert Nand(True, False) is true assert Nand(False, False) is true assert Nand(True, A) == ~A assert Nand(False, A) is true assert Nand(True, True, True) is false assert Nand(True, True, A) == ~A assert Nand(True, False, A) is true def test_Nor(): assert Nor() is true assert Nor(A) == ~A assert Nor(True) is false assert Nor(False) is true assert Nor(True, True) is false assert Nor(True, False) is false assert Nor(False, False) is true assert Nor(True, A) is false assert Nor(False, A) == ~A assert Nor(True, True, True) is false assert Nor(True, True, A) is false assert Nor(True, False, A) is false def test_Xnor(): assert Xnor() is true assert Xnor(A) == ~A assert Xnor(A, A) is true assert Xnor(True, A, A) is false assert Xnor(A, A, A, A, A) == ~A assert Xnor(True) is false assert Xnor(False) is true assert Xnor(True, True) is true assert Xnor(True, False) is false assert Xnor(False, False) is true assert Xnor(True, A) == A assert Xnor(False, A) == ~A assert Xnor(True, False, False) is false assert Xnor(True, False, A) == A assert Xnor(False, False, A) == ~A def test_Implies(): raises(ValueError, lambda: Implies(A, B, C)) assert Implies(True, True) is true assert Implies(True, False) is false assert Implies(False, True) is true assert Implies(False, False) is true assert Implies(0, A) is true assert Implies(1, 1) is true assert Implies(1, 0) is false assert A >> B == B << A assert (A < 1) >> (A >= 1) == (A >= 1) assert (A < 1) >> (S.One > A) is true assert A >> A is true def test_Equivalent(): assert Equivalent(A, B) == Equivalent(B, A) == Equivalent(A, B, A) assert Equivalent() is true assert Equivalent(A, A) == Equivalent(A) is true assert Equivalent(True, True) == Equivalent(False, False) is true assert Equivalent(True, False) == Equivalent(False, True) is false assert Equivalent(A, True) == A assert Equivalent(A, False) == Not(A) assert Equivalent(A, B, True) == A & B assert Equivalent(A, B, False) == ~A & ~B assert Equivalent(1, A) == A assert Equivalent(0, A) == Not(A) assert Equivalent(A, Equivalent(B, C)) != Equivalent(Equivalent(A, B), C) assert Equivalent(A < 1, A >= 1) is false assert Equivalent(A < 1, A >= 1, 0) is false assert Equivalent(A < 1, A >= 1, 1) is false assert Equivalent(A < 1, S.One > A) == Equivalent(1, 1) == Equivalent(0, 0) assert Equivalent(Equality(A, B), Equality(B, A)) is true def test_Exclusive(): assert Exclusive(False, False, False) is true assert Exclusive(True, False, False) is true assert Exclusive(True, True, False) is false assert Exclusive(True, True, True) is false def test_equals(): assert Not(Or(A, B)).equals(And(Not(A), Not(B))) is True assert Equivalent(A, B).equals((A >> B) & (B >> A)) is True assert ((A \| ~B) & (~A \| B)).equals((~A & ~B) \| (A & B)) is True assert (A >> B).equals(~A >> ~B) is False assert (A >> (B >> A)).equals(A >> (C >> A)) is False raises(NotImplementedError, lambda: (A & B).equals(A > B)) def test_simplification_boolalg(): """ Test working of simplification methods. """ set1 = [[0, 0, 1], [0, 1, 1], [1, 0, 0], [1, 1, 0]] set2 = [[0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1]] assert SOPform([x, y, z], set1) == Or(And(Not(x), z), And(Not(z), x)) assert Not(SOPform([x, y, z], set2)) == \ Not(Or(And(Not(x), Not(z)), And(x, z))) assert POSform([x, y, z], set1 + set2) is true assert SOPform([x, y, z], set1 + set2) is true assert SOPform([Dummy(), Dummy(), Dummy()], set1 + set2) is true minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1], [1, 1, 1, 1]] dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]] assert ( SOPform([w, x, y, z], minterms, dontcares) == Or(And(y, z), And(Not(w), Not(x)))) assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z) minterms = [1, 3, 7, 11, 15] dontcares = [0, 2, 5] assert ( SOPform([w, x, y, z], minterms, dontcares) == Or(And(y, z), And(Not(w), Not(x)))) assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z) minterms = [1, [0, 0, 1, 1], 7, [1, 0, 1, 1], [1, 1, 1, 1]] dontcares = [0, [0, 0, 1, 0], 5] assert ( SOPform([w, x, y, z], minterms, dontcares) == Or(And(y, z), And(Not(w), Not(x)))) assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z) minterms = [1, {y: 1, z: 1}] dontcares = [0, [0, 0, 1, 0], 5] assert ( SOPform([w, x, y, z], minterms, dontcares) == Or(And(y, z), And(Not(w), Not(x)))) assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z) minterms = [{y: 1, z: 1}, 1] dontcares = [[0, 0, 0, 0]] minterms = [[0, 0, 0]] raises(ValueError, lambda: SOPform([w, x, y, z], minterms)) raises(ValueError, lambda: POSform([w, x, y, z], minterms)) raises(TypeError, lambda: POSform([w, x, y, z], ["abcdefg"])) # test simplification ans = And(A, Or(B, C)) assert simplify_logic(A & (B \| C)) == ans assert simplify_logic((A & B) \| (A & C)) == ans assert simplify_logic(Implies(A, B)) == Or(Not(A), B) assert simplify_logic(Equivalent(A, B)) == \ Or(And(A, B), And(Not(A), Not(B))) assert simplify_logic(And(Equality(A, 2), C)) == And(Equality(A, 2), C) assert simplify_logic(And(Equality(A, 2), A)) is S.false assert simplify_logic(And(Equality(A, 2), A)) == And(Equality(A, 2), A) assert simplify_logic(And(Equality(A, B), C)) == And(Equality(A, B), C) assert simplify_logic(Or(And(Equality(A, 3), B), And(Equality(A, 3), C))) \ == And(Equality(A, 3), Or(B, C)) b = (~x & ~y & ~z) \| (~x & ~y & z) e = And(A, b) assert simplify_logic(e) == A & ~x & ~y raises(ValueError, lambda: simplify_logic(A & (B \| C), form='blabla')) assert simplify(Or(x <= y, And(x < y, z))) == (x <= y) assert simplify(Or(x <= y, And(y > x, z))) == (x <= y) assert simplify(Or(x >= y, And(y < x, z))) == (x >= y) # Check that expressions with nine variables or more are not simplified # (without the force-flag) a, b, c, d, e, f, g, h, j = symbols('a b c d e f g h j') expr = a & b & c & d & e & f & g & h & j \| \ a & b & c & d & e & f & g & h & ~j # This expression can be simplified to get rid of the j variables assert simplify_logic(expr) == expr # check input ans = SOPform([x, y], [[1, 0]]) assert SOPform([x, y], [[1, 0]]) == ans assert POSform([x, y], [[1, 0]]) == ans raises(ValueError, lambda: SOPform([x], [[1]], [[1]])) assert SOPform([x], [[1]], [[0]]) is true assert SOPform([x], [[0]], [[1]]) is true assert SOPform([x], [], []) is false raises(ValueError, lambda: POSform([x], [[1]], [[1]])) assert POSform([x], [[1]], [[0]]) is true assert POSform([x], [[0]], [[1]]) is true assert POSform([x], [], []) is false # check working of simplify assert simplify((A & B) \| (A & C)) == And(A, Or(B, C)) assert simplify(And(x, Not(x))) == False assert simplify(Or(x, Not(x))) == True assert simplify(And(Eq(x, 0), Eq(x, y))) == And(Eq(x, 0), Eq(y, 0)) assert And(Eq(x - 1, 0), Eq(x, y)).simplify() == And(Eq(x, 1), Eq(y, 1)) assert And(Ne(x - 1, 0), Ne(x, y)).simplify() == And(Ne(x, 1), Ne(x, y)) assert And(Eq(x - 1, 0), Ne(x, y)).simplify() == And(Eq(x, 1), Ne(y, 1)) assert And(Eq(x - 1, 0), Eq(x, z + y), Eq(y + x, 0)).simplify( ) == And(Eq(x, 1), Eq(y, -1), Eq(z, 2)) assert And(Eq(x - 1, 0), Eq(x + 2, 3)).simplify() == Eq(x, 1) assert And(Ne(x - 1, 0), Ne(x + 2, 3)).simplify() == Ne(x, 1) assert And(Eq(x - 1, 0), Eq(x + 2, 2)).simplify() == False assert And(Ne(x - 1, 0), Ne(x + 2, 2)).simplify( ) == And(Ne(x, 1), Ne(x, 0)) def test_bool_map(): """ Test working of bool_map function. """ minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1], [1, 1, 1, 1]] assert bool_map(Not(Not(a)), a) == (a, {a: a}) assert bool_map(SOPform([w, x, y, z], minterms), POSform([w, x, y, z], minterms)) == \ (And(Or(Not(w), y), Or(Not(x), y), z), {x: x, w: w, z: z, y: y}) assert bool_map(SOPform([x, z, y], [[1, 0, 1]]), SOPform([a, b, c], [[1, 0, 1]])) != False function1 = SOPform([x, z, y], [[1, 0, 1], [0, 0, 1]]) function2 = SOPform([a, b, c], [[1, 0, 1], [1, 0, 0]]) assert bool_map(function1, function2) == \ (function1, {y: a, z: b}) assert bool_map(Xor(x, y), ~Xor(x, y)) == False assert bool_map(And(x, y), Or(x, y)) is None assert bool_map(And(x, y), And(x, y, z)) is None # issue 16179 assert bool_map(Xor(x, y, z), ~Xor(x, y, z)) == False assert bool_map(Xor(a, x, y, z), ~Xor(a, x, y, z)) == False def test_bool_symbol(): """Test that mixing symbols with boolean values works as expected""" assert And(A, True) == A assert And(A, True, True) == A assert And(A, False) is false assert And(A, True, False) is false assert Or(A, True) is true assert Or(A, False) == A def test_is_boolean(): assert isinstance(True, Boolean) is False assert isinstance(true, Boolean) is True assert 1 == True assert 1 != true assert (1 == true) is False assert 0 == False assert 0 != false assert (0 == false) is False assert true.is_Boolean is True assert (A & B).is_Boolean assert (A \| B).is_Boolean assert (~A).is_Boolean assert (A ^ B).is_Boolean assert A.is_Boolean != isinstance(A, Boolean) assert isinstance(A, Boolean) def test_subs(): assert (A & B).subs(A, True) == B assert (A & B).subs(A, False) is false assert (A & B).subs(B, True) == A assert (A & B).subs(B, False) is false assert (A & B).subs({A: True, B: True}) is true assert (A \| B).subs(A, True) is true assert (A \| B).subs(A, False) == B assert (A \| B).subs(B, True) is true assert (A \| B).subs(B, False) == A assert (A \| B).subs({A: True, B: True}) is true """ we test for axioms of boolean algebra see https://en.wikipedia.org/wiki/Boolean_algebra_(structure) """ def test_commutative(): """Test for commutativity of And and Or""" A, B = map(Boolean, symbols('A,B')) assert A & B == B & A assert A \| B == B \| A def test_and_associativity(): """Test for associativity of And""" assert (A & B) & C == A & (B & C) def test_or_assicativity(): assert ((A \| B) \| C) == (A \| (B \| C)) def test_double_negation(): a = Boolean() assert ~(~a) == a # test methods def test_eliminate_implications(): assert eliminate_implications(Implies(A, B, evaluate=False)) == (~A) \| B assert eliminate_implications( A >> (C >> Not(B))) == Or(Or(Not(B), Not(C)), Not(A)) assert eliminate_implications(Equivalent(A, B, C, D)) == \ (~A \| B) & (~B \| C) & (~C \| D) & (~D \| A) def test_conjuncts(): assert conjuncts(A & B & C) == {A, B, C} assert conjuncts((A \| B) & C) == {A \| B, C} assert conjuncts(A) == {A} assert conjuncts(True) == {True} assert conjuncts(False) == {False} def test_disjuncts(): assert disjuncts(A \| B \| C) == {A, B, C} assert disjuncts((A \| B) & C) == {(A \| B) & C} assert disjuncts(A) == {A} assert disjuncts(True) == {True} assert disjuncts(False) == {False} def test_distribute(): assert distribute_and_over_or(Or(And(A, B), C)) == And(Or(A, C), Or(B, C)) assert distribute_or_over_and(And(A, Or(B, C))) == Or(And(A, B), And(A, C)) assert distribute_xor_over_and(And(A, Xor(B, C))) == Xor(And(A, B), And(A, C)) def test_to_anf(): x, y, z = symbols('x,y,z') assert to_anf(And(x, y)) == And(x, y) assert to_anf(Or(x, y)) == Xor(x, y, And(x, y)) assert to_anf(Or(Implies(x, y), And(x, y), y)) == \ Xor(x, True, x & y, remove_true=False) assert to_anf(Or(Nand(x, y), Nor(x, y), Xnor(x, y), Implies(x, y))) == True assert to_anf(Or(x, Not(y), Nor(x,z), And(x, y), Nand(y, z))) == \ Xor(True, And(y, z), And(x, y, z), remove_true=False) assert to_anf(Xor(x, y)) == Xor(x, y) assert to_anf(Not(x)) == Xor(x, True, remove_true=False) assert to_anf(Nand(x, y)) == Xor(True, And(x, y), remove_true=False) assert to_anf(Nor(x, y)) == Xor(x, y, True, And(x, y), remove_true=False) assert to_anf(Implies(x, y)) == Xor(x, True, And(x, y), remove_true=False) assert to_anf(Equivalent(x, y)) == Xor(x, y, True, remove_true=False) assert to_anf(Nand(x \| y, x >> y), deep=False) == \ Xor(True, And(Or(x, y), Implies(x, y)), remove_true=False) assert to_anf(Nor(x ^ y, x & y), deep=False) == \ Xor(True, Or(Xor(x, y), And(x, y)), remove_true=False) def test_to_nnf(): assert to_nnf(true) is true assert to_nnf(false) is false assert to_nnf(A) == A assert to_nnf(A \| ~A \| B) is true assert to_nnf(A & ~A & B) is false assert to_nnf(A >> B) == ~A \| B assert to_nnf(Equivalent(A, B, C)) == (~A \| B) & (~B \| C) & (~C \| A) assert to_nnf(A ^ B ^ C) == \ (A \| B \| C) & (~A \| ~B \| C) & (A \| ~B \| ~C) & (~A \| B \| ~C) assert to_nnf(ITE(A, B, C)) == (~A \| B) & (A \| C) assert to_nnf(Not(A \| B \| C)) == ~A & ~B & ~C assert to_nnf(Not(A & B & C)) == ~A \| ~B \| ~C assert to_nnf(Not(A >> B)) == A & ~B assert to_nnf(Not(Equivalent(A, B, C))) == And(Or(A, B, C), Or(~A, ~B, ~C)) assert to_nnf(Not(A ^ B ^ C)) == \ (~A \| B \| C) & (A \| ~B \| C) & (A \| B \| ~C) & (~A \| ~B \| ~C) assert to_nnf(Not(ITE(A, B, C))) == (~A \| ~B) & (A \| ~C) assert to_nnf((A >> B) ^ (B >> A)) == (A & ~B) \| (~A & B) assert to_nnf((A >> B) ^ (B >> A), False) == \ (~A \| ~B \| A \| B) & ((A & ~B) \| (~A & B)) assert ITE(A, 1, 0).to_nnf() == A assert ITE(A, 0, 1).to_nnf() == ~A # although ITE can hold non-Boolean, it will complain if # an attempt is made to convert the ITE to Boolean nnf raises(TypeError, lambda: ITE(A < 1, [1], B).to_nnf()) def test_to_cnf(): assert to_cnf(~(B \| C)) == And(Not(B), Not(C)) assert to_cnf((A & B) \| C) == And(Or(A, C), Or(B, C)) assert to_cnf(A >> B) == (~A) \| B assert to_cnf(A >> (B & C)) == (~A \| B) & (~A \| C) assert to_cnf(A & (B \| C) \| ~A & (B \| C), True) == B \| C assert to_cnf(A & B) == And(A, B) assert to_cnf(Equivalent(A, B)) == And(Or(A, Not(B)), Or(B, Not(A))) assert to_cnf(Equivalent(A, B & C)) == \ (~A \| B) & (~A \| C) & (~B \| ~C \| A) assert to_cnf(Equivalent(A, B \| C), True) == \ And(Or(Not(B), A), Or(Not(C), A), Or(B, C, Not(A))) assert to_cnf(A + 1) == A + 1 def test_issue_18904(): x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15 = symbols('x1:16') eq = (( x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 ) \| ( x1 & x2 & x3 & x4 & x5 & x6 & x7 & x10 & x9 ) \| ( x1 & x11 & x3 & x12 & x5 & x13 & x14 & x15 & x9 )) assert is_cnf(to_cnf(eq)) raises(ValueError, lambda: to_cnf(eq, simplify=True)) for f, t in zip((And, Or), (to_cnf, to_dnf)): eq = f(x1, x2, x3, x4, x5, x6, x7, x8, x9) raises(ValueError, lambda: to_cnf(eq, simplify=True)) assert t(eq, simplify=True, force=True) == eq def test_issue_9949(): assert is_cnf(to_cnf((b > -5) \| (a > 2) & (a < 4))) def test_to_CNF(): assert CNF.CNF_to_cnf(CNF.to_CNF(~(B \| C))) == to_cnf(~(B \| C)) assert CNF.CNF_to_cnf(CNF.to_CNF((A & B) \| C)) == to_cnf((A & B) \| C) assert CNF.CNF_to_cnf(CNF.to_CNF(A >> B)) == to_cnf(A >> B) assert CNF.CNF_to_cnf(CNF.to_CNF(A >> (B & C))) == to_cnf(A >> (B & C)) assert CNF.CNF_to_cnf(CNF.to_CNF(A & (B \| C) \| ~A & (B \| C))) == to_cnf(A & (B \| C) \| ~A & (B \| C)) assert CNF.CNF_to_cnf(CNF.to_CNF(A & B)) == to_cnf(A & B) def test_to_dnf(): assert to_dnf(~(B \| C)) == And(Not(B), Not(C)) assert to_dnf(A & (B \| C)) == Or(And(A, B), And(A, C)) assert to_dnf(A >> B) == (~A) \| B assert to_dnf(A >> (B & C)) == (~A) \| (B & C) assert to_dnf(A \| B) == A \| B assert to_dnf(Equivalent(A, B), True) == \ Or(And(A, B), And(Not(A), Not(B))) assert to_dnf(Equivalent(A, B & C), True) == \ Or(And(A, B, C), And(Not(A), Not(B)), And(Not(A), Not(C))) assert to_dnf(A + 1) == A + 1 def test_to_int_repr(): x, y, z = map(Boolean, symbols('x,y,z')) def sorted_recursive(arg): try: return sorted(sorted_recursive(x) for x in arg) except TypeError: # arg is not a sequence return arg assert sorted_recursive(to_int_repr([x \| y, z \| x], [x, y, z])) == \ sorted_recursive([[1, 2], [1, 3]]) assert sorted_recursive(to_int_repr([x \| y, z \| ~x], [x, y, z])) == \ sorted_recursive([[1, 2], [3, -1]]) def test_is_anf(): x, y = symbols('x,y') assert is_anf(true) is True assert is_anf(false) is True assert is_anf(x) is True assert is_anf(And(x, y)) is True assert is_anf(Xor(x, y, And(x, y))) is True assert is_anf(Xor(x, y, Or(x, y))) is False assert is_anf(Xor(Not(x), y)) is False def test_is_nnf(): assert is_nnf(true) is True assert is_nnf(A) is True assert is_nnf(~A) is True assert is_nnf(A & B) is True assert is_nnf((A & B) \| (~A & A) \| (~B & B) \| (~A & ~B), False) is True assert is_nnf((A \| B) & (~A \| ~B)) is True assert is_nnf(Not(Or(A, B))) is False assert is_nnf(A ^ B) is False assert is_nnf((A & B) \| (~A & A) \| (~B & B) \| (~A & ~B), True) is False def test_is_cnf(): assert is_cnf(x) is True assert is_cnf(x \| y \| z) is True assert is_cnf(x & y & z) is True assert is_cnf((x \| y) & z) is True assert is_cnf((x & y) \| z) is False assert is_cnf(~(x & y) \| z) is False def test_is_dnf(): assert is_dnf(x) is True assert is_dnf(x \| y \| z) is True assert is_dnf(x & y & z) is True assert is_dnf((x & y) \| z) is True assert is_dnf((x \| y) & z) is False assert is_dnf(~(x \| y) & z) is False def test_ITE(): A, B, C = symbols('A:C') assert ITE(True, False, True) is false assert ITE(True, True, False) is true assert ITE(False, True, False) is false assert ITE(False, False, True) is true assert isinstance(ITE(A, B, C), ITE) A = True assert ITE(A, B, C) == B A = False assert ITE(A, B, C) == C B = True assert ITE(And(A, B), B, C) == C assert ITE(Or(A, False), And(B, True), False) is false assert ITE(x, A, B) == Not(x) assert ITE(x, B, A) == x assert ITE(1, x, y) == x assert ITE(0, x, y) == y raises(TypeError, lambda: ITE(2, x, y)) raises(TypeError, lambda: ITE(1, [], y)) raises(TypeError, lambda: ITE(1, (), y)) raises(TypeError, lambda: ITE(1, y, [])) assert ITE(1, 1, 1) is S.true assert isinstance(ITE(1, 1, 1, evaluate=False), ITE) raises(TypeError, lambda: ITE(x > 1, y, x)) assert ITE(Eq(x, True), y, x) == ITE(x, y, x) assert ITE(Eq(x, False), y, x) == ITE(~x, y, x) assert ITE(Ne(x, True), y, x) == ITE(~x, y, x) assert ITE(Ne(x, False), y, x) == ITE(x, y, x) assert ITE(Eq(S. true, x), y, x) == ITE(x, y, x) assert ITE(Eq(S.false, x), y, x) == ITE(~x, y, x) assert ITE(Ne(S.true, x), y, x) == ITE(~x, y, x) assert ITE(Ne(S.false, x), y, x) == ITE(x, y, x) # 0 and 1 in the context are not treated as True/False # so the equality must always be False since dissimilar # objects cannot be equal assert ITE(Eq(x, 0), y, x) == x assert ITE(Eq(x, 1), y, x) == x assert ITE(Ne(x, 0), y, x) == y assert ITE(Ne(x, 1), y, x) == y assert ITE(Eq(x, 0), y, z).subs(x, 0) == y assert ITE(Eq(x, 0), y, z).subs(x, 1) == z raises(ValueError, lambda: ITE(x > 1, y, x, z)) def test_is_literal(): assert is_literal(True) is True assert is_literal(False) is True assert is_literal(A) is True assert is_literal(~A) is True assert is_literal(Or(A, B)) is False assert is_literal(Q.zero(A)) is True assert is_literal(Not(Q.zero(A))) is True assert is_literal(Or(A, B)) is False assert is_literal(And(Q.zero(A), Q.zero(B))) is False assert is_literal(x < 3) assert not is_literal(x + y < 3) def test_operators(): # Mostly test __and__, __rand__, and so on assert True & A == A & True == A assert False & A == A & False == False assert A & B == And(A, B) assert True \| A == A \| True == True assert False \| A == A \| False == A assert A \| B == Or(A, B) assert ~A == Not(A) assert True >> A == A << True == A assert False >> A == A << False == True assert A >> True == True << A == True assert A >> False == False << A == ~A assert A >> B == B << A == Implies(A, B) assert True ^ A == A ^ True == ~A assert False ^ A == A ^ False == A assert A ^ B == Xor(A, B) def test_true_false(): assert true is S.true assert false is S.false assert true is not True assert false is not False assert true assert not false assert true == True assert false == False assert not (true == False) assert not (false == True) assert not (true == false) assert hash(true) == hash(True) assert hash(false) == hash(False) assert len({true, True}) == len({false, False}) == 1 assert isinstance(true, BooleanAtom) assert isinstance(false, BooleanAtom) # We don't want to subclass from bool, because bool subclasses from # int. But operators like &, \|, ^, <<, >>, and ~ act differently on 0 and # 1 then we want them to on true and false. See the docstrings of the # various And, Or, etc. functions for examples. assert not isinstance(true, bool) assert not isinstance(false, bool) # Note: using 'is' comparison is important here. We want these to return # true and false, not True and False assert Not(true) is false assert Not(True) is false assert Not(false) is true assert Not(False) is true assert ~true is false assert ~false is true for T, F in product((True, true), (False, false)): assert And(T, F) is false assert And(F, T) is false assert And(F, F) is false assert And(T, T) is true assert And(T, x) == x assert And(F, x) is false if not (T is True and F is False): assert T & F is false assert F & T is false if F is not False: assert F & F is false if T is not True: assert T & T is true assert Or(T, F) is true assert Or(F, T) is true assert Or(F, F) is false assert Or(T, T) is true assert Or(T, x) is true assert Or(F, x) == x if not (T is True and F is False): assert T \| F is true assert F \| T is true if F is not False: assert F \| F is false if T is not True: assert T \| T is true assert Xor(T, F) is true assert Xor(F, T) is true assert Xor(F, F) is false assert Xor(T, T) is false assert Xor(T, x) == ~x assert Xor(F, x) == x if not (T is True and F is False): assert T ^ F is true assert F ^ T is true if F is not False: assert F ^ F is false if T is not True: assert T ^ T is false assert Nand(T, F) is true assert Nand(F, T) is true assert Nand(F, F) is true assert Nand(T, T) is false assert Nand(T, x) == ~x assert Nand(F, x) is true assert Nor(T, F) is false assert Nor(F, T) is false assert Nor(F, F) is true assert Nor(T, T) is false assert Nor(T, x) is false assert Nor(F, x) == ~x assert Implies(T, F) is false assert Implies(F, T) is true assert Implies(F, F) is true assert Implies(T, T) is true assert Implies(T, x) == x assert Implies(F, x) is true assert Implies(x, T) is true assert Implies(x, F) == ~x if not (T is True and F is False): assert T >> F is false assert F << T is false assert F >> T is true assert T << F is true if F is not False: assert F >> F is true assert F << F is true if T is not True: assert T >> T is true assert T << T is true assert Equivalent(T, F) is false assert Equivalent(F, T) is false assert Equivalent(F, F) is true assert Equivalent(T, T) is true assert Equivalent(T, x) == x assert Equivalent(F, x) == ~x assert Equivalent(x, T) == x assert Equivalent(x, F) == ~x assert ITE(T, T, T) is true assert ITE(T, T, F) is true assert ITE(T, F, T) is false assert ITE(T, F, F) is false assert ITE(F, T, T) is true assert ITE(F, T, F) is false assert ITE(F, F, T) is true assert ITE(F, F, F) is false assert all(i.simplify(1, 2) is i for i in (S.true, S.false)) def test_bool_as_set(): assert ITE(y <= 0, False, y >= 1).as_set() == Interval(1, oo) assert And(x <= 2, x >= -2).as_set() == Interval(-2, 2) assert Or(x >= 2, x <= -2).as_set() == Interval(-oo, -2) + Interval(2, oo) assert Not(x > 2).as_set() == Interval(-oo, 2) # issue 10240 assert Not(And(x > 2, x < 3)).as_set() == \ Union(Interval(-oo, 2), Interval(3, oo)) assert true.as_set() == S.UniversalSet assert false.as_set() is S.EmptySet assert x.as_set() == S.UniversalSet assert And(Or(x < 1, x > 3), x < 2).as_set() == Interval.open(-oo, 1) assert And(x < 1, sin(x) < 3).as_set() == (x < 1).as_set() raises(NotImplementedError, lambda: (sin(x) < 1).as_set()) # watch for object morph in as_set assert Eq(-1, cos(2x)*2/sin(2x)*2).as_set() is S.EmptySet @XFAIL def test_multivariate_bool_as_set(): x, y = symbols('x,y') assert And(x >= 0, y >= 0).as_set() == Interval(0, oo)Interval(0, oo) assert Or(x >= 0, y >= 0).as_set() == S.RealsS.Reals - \ Interval(-oo, 0, True, True)Interval(-oo, 0, True, True) def test_all_or_nothing(): x = symbols('x', extended_real=True) args = x >= -oo, x <= oo v = And(args) if v.func is And: assert len(v.args) == len(args) - args.count(S.true) else: assert v == True v = Or(args) if v.func is Or: assert len(v.args) == 2 else: assert v == True def test_canonical_atoms(): assert true.canonical == true assert false.canonical == false def test_negated_atoms(): assert true.negated == false assert false.negated == true def test_issue_8777(): assert And(x > 2, x < oo).as_set() == Interval(2, oo, left_open=True) assert And(x >= 1, x < oo).as_set() == Interval(1, oo) assert (x < oo).as_set() == Interval(-oo, oo) assert (x > -oo).as_set() == Interval(-oo, oo) def test_issue_8975(): assert Or(And(-oo < x, x <= -2), And(2 <= x, x < oo)).as_set() == \ Interval(-oo, -2) + Interval(2, oo) def test_term_to_integer(): assert term_to_integer([1, 0, 1, 0, 0, 1, 0]) == 82 assert term_to_integer('0010101000111001') == 10809 def test_issue_21971(): a, b, c, d = symbols('a b c d') f = a & b & c \| a & c assert f.subs(a & c, d) == b & d \| d assert f.subs(a & b & c, d) == a & c \| d f = (a \| b \| c) & (a \| c) assert f.subs(a \| c, d) == (b \| d) & d assert f.subs(a \| b \| c, d) == (a \| c) & d f = (a ^ b ^ c) & (a ^ c) assert f.subs(a ^ c, d) == (b ^ d) & d assert f.subs(a ^ b ^ c, d) == (a ^ c) & d def test_truth_table(): assert list(truth_table(And(x, y), [x, y], input=False)) == \ [False, False, False, True] assert list(truth_table(x \| y, [x, y], input=False)) == \ [False, True, True, True] assert list(truth_table(x >> y, [x, y], input=False)) == \ [True, True, False, True] assert list(truth_table(And(x, y), [x, y])) == \ [([0, 0], False), ([0, 1], False), ([1, 0], False), ([1, 1], True)] def test_issue_8571(): for t in (S.true, S.false): raises(TypeError, lambda: +t) raises(TypeError, lambda: -t) raises(TypeError, lambda: abs(t)) # use int(bool(t)) to get 0 or 1 raises(TypeError, lambda: int(t)) for o in [S.Zero, S.One, x]: for _ in range(2): raises(TypeError, lambda: o + t) raises(TypeError, lambda: o - t) raises(TypeError, lambda: o % t) raises(TypeError, lambda: ot) raises(TypeError, lambda: o/t) raises(TypeError, lambda: ot) o, t = t, o # do again in reversed order def test_expand_relational(): n = symbols('n', negative=True) p, q = symbols('p q', positive=True) r = ((n + q(-n/q + 1))/(q(-n/q + 1)) < 0) assert r is not S.false assert r.expand() is S.false assert (q > 0).expand() is S.true def test_issue_12717(): assert S.true.is_Atom == True assert S.false.is_Atom == True def test_as_Boolean(): nz = symbols('nz', nonzero=True) assert all(as_Boolean(i) is S.true for i in (True, S.true, 1, nz)) z = symbols('z', zero=True) assert all(as_Boolean(i) is S.false for i in (False, S.false, 0, z)) assert all(as_Boolean(i) == i for i in (x, x < 0)) for i in (2, S(2), x + 1, []): raises(TypeError, lambda: as_Boolean(i)) def test_binary_symbols(): assert ITE(x < 1, y, z).binary_symbols == {y, z} for f in (Eq, Ne): assert f(x, 1).binary_symbols == set() assert f(x, True).binary_symbols == {x} assert f(x, False).binary_symbols == {x} assert S.true.binary_symbols == set() assert S.false.binary_symbols == set() assert x.binary_symbols == {x} assert And(x, Eq(y, False), Eq(z, 1)).binary_symbols == {x, y} assert Q.prime(x).binary_symbols == set() assert Q.lt(x, 1).binary_symbols == set() assert Q.is_true(x).binary_symbols == {x} assert Q.eq(x, True).binary_symbols == {x} assert Q.prime(x).binary_symbols == set() def test_BooleanFunction_diff(): assert And(x, y).diff(x) == Piecewise((0, Eq(y, False)), (1, True)) def test_issue_14700(): A, B, C, D, E, F, G, H = symbols('A B C D E F G H') q = ((B & D & H & ~F) \| (B & H & ~C & ~D) \| (B & H & ~C & ~F) \| (B & H & ~D & ~G) \| (B & H & ~F & ~G) \| (C & G & ~B & ~D) \| (C & G & ~D & ~H) \| (C & G & ~F & ~H) \| (D & F & H & ~B) \| (D & F & ~G & ~H) \| (B & D & F & ~C & ~H) \| (D & E & F & ~B & ~C) \| (D & F & ~A & ~B & ~C) \| (D & F & ~A & ~C & ~H) \| (A & B & D & F & ~E & ~H)) soldnf = ((B & D & H & ~F) \| (D & F & H & ~B) \| (B & H & ~C & ~D) \| (B & H & ~D & ~G) \| (C & G & ~B & ~D) \| (C & G & ~D & ~H) \| (C & G & ~F & ~H) \| (D & F & ~G & ~H) \| (D & E & F & ~C & ~H) \| (D & F & ~A & ~C & ~H) \| (A & B & D & F & ~E & ~H)) solcnf = ((B \| C \| D) & (B \| D \| G) & (C \| D \| H) & (C \| F \| H) & (D \| G \| H) & (F \| G \| H) & (B \| F \| ~D \| ~H) & (~B \| ~D \| ~F \| ~H) & (D \| ~B \| ~C \| ~G \| ~H) & (A \| H \| ~C \| ~D \| ~F \| ~G) & (H \| ~C \| ~D \| ~E \| ~F \| ~G) & (B \| E \| H \| ~A \| ~D \| ~F \| ~G)) assert simplify_logic(q, "dnf") == soldnf assert simplify_logic(q, "cnf") == solcnf minterms = [[0, 1, 0, 0], [0, 1, 0, 1], [0, 1, 1, 0], [0, 1, 1, 1], [0, 0, 1, 1], [1, 0, 1, 1]] dontcares = [[1, 0, 0, 0], [1, 0, 0, 1], [1, 1, 0, 0], [1, 1, 0, 1]] assert SOPform([w, x, y, z], minterms) == (x & ~w) \| (y & z & ~x) # Should not be more complicated with don't cares assert SOPform([w, x, y, z], minterms, dontcares) == \ (x & ~w) \| (y & z & ~x) def test_relational_simplification(): w, x, y, z = symbols('w x y z', real=True) d, e = symbols('d e', real=False) # Test all combinations or sign and order assert Or(x >= y, x < y).simplify() == S.true assert Or(x >= y, y > x).simplify() == S.true assert Or(x >= y, -x > -y).simplify() == S.true assert Or(x >= y, -y < -x).simplify() == S.true assert Or(-x <= -y, x < y).simplify() == S.true assert Or(-x <= -y, -x > -y).simplify() == S.true assert Or(-x <= -y, y > x).simplify() == S.true assert Or(-x <= -y, -y < -x).simplify() == S.true assert Or(y <= x, x < y).simplify() == S.true assert Or(y <= x, y > x).simplify() == S.true assert Or(y <= x, -x > -y).simplify() == S.true assert Or(y <= x, -y < -x).simplify() == S.true assert Or(-y >= -x, x < y).simplify() == S.true assert Or(-y >= -x, y > x).simplify() == S.true assert Or(-y >= -x, -x > -y).simplify() == S.true assert Or(-y >= -x, -y < -x).simplify() == S.true assert Or(x < y, x >= y).simplify() == S.true assert Or(y > x, x >= y).simplify() == S.true assert Or(-x > -y, x >= y).simplify() == S.true assert Or(-y < -x, x >= y).simplify() == S.true assert Or(x < y, -x <= -y).simplify() == S.true assert Or(-x > -y, -x <= -y).simplify() == S.true assert Or(y > x, -x <= -y).simplify() == S.true assert Or(-y < -x, -x <= -y).simplify() == S.true assert Or(x < y, y <= x).simplify() == S.true assert Or(y > x, y <= x).simplify() == S.true assert Or(-x > -y, y <= x).simplify() == S.true assert Or(-y < -x, y <= x).simplify() == S.true assert Or(x < y, -y >= -x).simplify() == S.true assert Or(y > x, -y >= -x).simplify() == S.true assert Or(-x > -y, -y >= -x).simplify() == S.true assert Or(-y < -x, -y >= -x).simplify() == S.true # Some other tests assert Or(x >= y, w < z, x <= y).simplify() == S.true assert And(x >= y, x < y).simplify() == S.false assert Or(x >= y, Eq(y, x)).simplify() == (x >= y) assert And(x >= y, Eq(y, x)).simplify() == Eq(x, y) assert And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \ (Eq(x, y) & (x >= 1) & (y >= 5) & (y > z)) assert Or(Eq(x, y), x >= y, w < y, z < y).simplify() == \ (x >= y) \| (y > z) \| (w < y) assert And(Eq(x, y), x >= y, w < y, y >= z, z < y).simplify() == \ Eq(x, y) & (y > z) & (w < y) # assert And(Eq(x, y), x >= y, w < y, y >= z, z < y).simplify(relational_minmax=True) == \ # And(Eq(x, y), y > Max(w, z)) # assert Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify(relational_minmax=True) == \ # (Eq(x, y) \| (x >= 1) \| (y > Min(2, z))) assert And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \ (Eq(x, y) & (x >= 1) & (y >= 5) & (y > z)) assert (Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)).simplify() == \ (Eq(x, y) & Eq(d, e) & (d >= e)) assert And(Eq(x, y), Eq(x, -y)).simplify() == And(Eq(x, 0), Eq(y, 0)) assert Xor(x >= y, x <= y).simplify() == Ne(x, y) assert And(x > 1, x < -1, Eq(x, y)).simplify() == S.false # From #16690 assert And(x >= y, Eq(y, 0)).simplify() == And(x >= 0, Eq(y, 0)) def test_issue_8373(): x = symbols('x', real=True) assert Or(x < 1, x > -1).simplify() == S.true assert Or(x < 1, x >= 1).simplify() == S.true assert And(x < 1, x >= 1).simplify() == S.false assert Or(x <= 1, x >= 1).simplify() == S.true def test_issue_7950(): x = symbols('x', real=True) assert And(Eq(x, 1), Eq(x, 2)).simplify() == S.false @slow def test_relational_simplification_numerically(): def test_simplification_numerically_function(original, simplified): symb = original.free_symbols n = len(symb) valuelist = list(set(list(combinations(list(range(-(n-1), n))n, n)))) for values in valuelist: sublist = dict(zip(symb, values)) originalvalue = original.subs(sublist) simplifiedvalue = simplified.subs(sublist) assert originalvalue == simplifiedvalue, "Original: {}\nand"\ " simplified: {}\ndo not evaluate to the same value for {}"\ "".format(original, simplified, sublist) w, x, y, z = symbols('w x y z', real=True) d, e = symbols('d e', real=False) expressions = (And(Eq(x, y), x >= y, w < y, y >= z, z < y), And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y), Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y), And(x >= y, Eq(y, x)), Or(And(Eq(x, y), x >= y, w < y, Or(y >= z, z < y)), And(Eq(x, y), x >= 1, 2 < y, y >= -1, z < y)), (Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)), ) for expression in expressions: test_simplification_numerically_function(expression, expression.simplify()) def test_relational_simplification_patterns_numerically(): from sympy.core import Wild from sympy.logic.boolalg import _simplify_patterns_and, \ _simplify_patterns_or, _simplify_patterns_xor a = Wild('a') b = Wild('b') c = Wild('c') symb = [a, b, c] patternlists = [[And, _simplify_patterns_and()], [Or, _simplify_patterns_or()], [Xor, _simplify_patterns_xor()]] valuelist = list(set(list(combinations(list(range(-2, 3))3, 3)))) # Skip combinations of +/-2 and 0, except for all 0 valuelist = [v for v in valuelist if any([w % 2 for w in v]) or not any(v)] for func, patternlist in patternlists: for pattern in patternlist: original = func(pattern[0].args) simplified = pattern[1] for values in valuelist: sublist = dict(zip(symb, values)) originalvalue = original.xreplace(sublist) simplifiedvalue = simplified.xreplace(sublist) assert originalvalue == simplifiedvalue, "Original: {}\nand"\ " simplified: {}\ndo not evaluate to the same value for"\ "{}".format(pattern[0], simplified, sublist) def test_issue_16803(): n = symbols('n') # No simplification done, but should not raise an exception assert ((n > 3) \| (n < 0) \| ((n > 0) & (n < 3))).simplify() == \ (n > 3) \| (n < 0) \| ((n > 0) & (n < 3)) def test_issue_17530(): r = {x: oo, y: oo} assert Or(x + y > 0, x - y < 0).subs(r) assert not And(x + y < 0, x - y < 0).subs(r) raises(TypeError, lambda: Or(x + y < 0, x - y < 0).subs(r)) raises(TypeError, lambda: And(x + y > 0, x - y < 0).subs(r)) raises(TypeError, lambda: And(x + y > 0, x - y < 0).subs(r)) def test_anf_coeffs(): assert anf_coeffs([1, 0]) == [1, 1] assert anf_coeffs([0, 0, 0, 1]) == [0, 0, 0, 1] assert anf_coeffs([0, 1, 1, 1]) == [0, 1, 1, 1] assert anf_coeffs([1, 1, 1, 0]) == [1, 0, 0, 1] assert anf_coeffs([1, 0, 0, 0]) == [1, 1, 1, 1] assert anf_coeffs([1, 0, 0, 1]) == [1, 1, 1, 0] assert anf_coeffs([1, 1, 0, 1]) == [1, 0, 1, 1] def test_ANFform(): x, y = symbols('x,y') assert ANFform([x], [1, 1]) == True assert ANFform([x], [0, 0]) == False assert ANFform([x], [1, 0]) == Xor(x, True, remove_true=False) assert ANFform([x, y], [1, 1, 1, 0]) == \ Xor(True, And(x, y), remove_true=False) def test_bool_minterm(): x, y = symbols('x,y') assert bool_minterm(3, [x, y]) == And(x, y) assert bool_minterm([1, 0], [x, y]) == And(Not(y), x) def test_bool_maxterm(): x, y = symbols('x,y') assert bool_maxterm(2, [x, y]) == Or(Not(x), y) assert bool_maxterm([0, 1], [x, y]) == Or(Not(y), x) def test_bool_monomial(): x, y = symbols('x,y') assert bool_monomial(1, [x, y]) == y assert bool_monomial([1, 1], [x, y]) == And(x, y) def test_check_pair(): assert _check_pair([0, 1, 0], [0, 1, 1]) == 2 assert _check_pair([0, 1, 0], [1, 1, 1]) == -1 def test_issue_19114(): expr = (B & C) \| (A & ~C) \| (~A & ~B) # Expression is minimal, but there are multiple minimal forms possible res1 = (A & B) \| (C & ~A) \| (~B & ~C) result = to_dnf(expr, simplify=True) assert result in (expr, res1) def test_issue_20870(): result = SOPform([a, b, c, d], [1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15]) expected = ((d & ~b) \| (a & b & c) \| (a & ~c & ~d) \| (b & ~a & ~c) \| (c & ~a & ~d)) assert result == expected def test_convert_to_varsSOP(): assert _convert_to_varsSOP([0, 1, 0], [x, y, z]) == And(Not(x), y, Not(z)) assert _convert_to_varsSOP([3, 1, 0], [x, y, z]) == And(y, Not(z)) def test_convert_to_varsPOS(): assert _convert_to_varsPOS([0, 1, 0], [x, y, z]) == Or(x, Not(y), z) assert _convert_to_varsPOS([3, 1, 0], [x, y, z]) == Or(Not(y), z) def test_gateinputcount(): a, b, c, d, e = symbols('a:e') assert gateinputcount(And(a, b)) == 2 assert gateinputcount(a \| b & c & d ^ (e \| a)) == 9 assert gateinputcount(And(a, True)) == 0 raises(TypeError, lambda: gateinputcount(ab)) def test_refine(): # relational assert not refine(x < 0, ~(x < 0)) assert refine(x < 0, (x < 0)) assert refine(x < 0, (0 > x)) is S.true assert refine(x < 0, (y < 0)) == (x < 0) assert not refine(x <= 0, ~(x <= 0)) assert refine(x <= 0, (x <= 0)) assert refine(x <= 0, (0 >= x)) is S.true assert refine(x <= 0, (y <= 0)) == (x <= 0) assert not refine(x > 0, ~(x > 0)) assert refine(x > 0, (x > 0)) assert refine(x > 0, (0 < x)) is S.true assert refine(x > 0, (y > 0)) == (x > 0) assert not refine(x >= 0, ~(x >= 0)) assert refine(x >= 0, (x >= 0)) assert refine(x >= 0, (0 <= x)) is S.true assert refine(x >= 0, (y >= 0)) == (x >= 0) assert not refine(Eq(x, 0), ~(Eq(x, 0))) assert refine(Eq(x, 0), (Eq(x, 0))) assert refine(Eq(x, 0), (Eq(0, x))) is S.true assert refine(Eq(x, 0), (Eq(y, 0))) == Eq(x, 0) assert not refine(Ne(x, 0), ~(Ne(x, 0))) assert refine(Ne(x, 0), (Ne(0, x))) is S.true assert refine(Ne(x, 0), (Ne(x, 0))) assert refine(Ne(x, 0), (Ne(y, 0))) == (Ne(x, 0)) # boolean functions assert refine(And(x > 0, y > 0), (x > 0)) == (y > 0) assert refine(And(x > 0, y > 0), (x > 0) & (y > 0)) is S.true # predicates assert refine(Q.positive(x), Q.positive(x)) is S.true assert refine(Q.positive(x), Q.negative(x)) is S.false assert refine(Q.positive(x), Q.real(x)) == Q.positive(x) def test_relational_threeterm_simplification_patterns_numerically(): from sympy.core import Wild from sympy.logic.boolalg import _simplify_patterns_and3 a = Wild('a') b = Wild('b') c = Wild('c') symb = [a, b, c] patternlists = [[And, _simplify_patterns_and3()]] valuelist = list(set(list(combinations(list(range(-2, 3))3, 3)))) # Skip combinations of +/-2 and 0, except for all 0 valuelist = [v for v in valuelist if any([w % 2 for w in v]) or not any(v)] for func, patternlist in patternlists: for pattern in patternlist: original = func(*pattern[0].args) simplified = pattern[1] for values in valuelist: sublist = dict(zip(symb, values)) originalvalue = original.xreplace(sublist) simplifiedvalue = simplified.xreplace(sublist) assert originalvalue == simplifiedvalue, "Original: {}\nand"\ " simplified: {}\ndo not evaluate to the same value for"\ "{}".format(pattern[0], simplified, sublist)
7b9532a94724f51e12956cd5d3034965853bbcc1c5b8cf53df5f2cde997058fe	""" Some examples have been taken from: http://www.math.uwaterloo.ca/~hwolkowi//matrixcookbook.pdf """ from sympy import KroneckerProduct from sympy.combinatorics import Permutation from sympy.concrete.summations import Sum from sympy.core.numbers import Rational from sympy.core.singleton import S from sympy.core.symbol import symbols from sympy.functions.elementary.exponential import (exp, log) from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (cos, sin, tan) from sympy.functions.special.tensor_functions import KroneckerDelta from sympy.matrices.expressions.determinant import Determinant from sympy.matrices.expressions.diagonal import DiagMatrix from sympy.matrices.expressions.hadamard import (HadamardPower, HadamardProduct, hadamard_product) from sympy.matrices.expressions.inverse import Inverse from sympy.matrices.expressions.matexpr import MatrixSymbol from sympy.matrices.expressions.special import OneMatrix from sympy.matrices.expressions.trace import Trace from sympy.matrices.expressions.matadd import MatAdd from sympy.matrices.expressions.matmul import MatMul from sympy.matrices.expressions.special import (Identity, ZeroMatrix) from sympy.tensor.array.array_derivatives import ArrayDerivative from sympy.matrices.expressions import hadamard_power from sympy.tensor.array.expressions.array_expressions import ArrayAdd, ArrayTensorProduct, PermuteDims k = symbols("k") i, j = symbols("i j") m, n = symbols("m n") X = MatrixSymbol("X", k, k) x = MatrixSymbol("x", k, 1) y = MatrixSymbol("y", k, 1) A = MatrixSymbol("A", k, k) B = MatrixSymbol("B", k, k) C = MatrixSymbol("C", k, k) D = MatrixSymbol("D", k, k) a = MatrixSymbol("a", k, 1) b = MatrixSymbol("b", k, 1) c = MatrixSymbol("c", k, 1) d = MatrixSymbol("d", k, 1) KDelta = lambda i, j: KroneckerDelta(i, j, (0, k-1)) def _check_derivative_with_explicit_matrix(expr, x, diffexpr, dim=2): # TODO: this is commented because it slows down the tests. return expr = expr.xreplace({k: dim}) x = x.xreplace({k: dim}) diffexpr = diffexpr.xreplace({k: dim}) expr = expr.as_explicit() x = x.as_explicit() diffexpr = diffexpr.as_explicit() assert expr.diff(x).reshape(diffexpr.shape).tomatrix() == diffexpr def test_matrix_derivative_by_scalar(): assert A.diff(i) == ZeroMatrix(k, k) assert (A(X + B)c).diff(i) == ZeroMatrix(k, 1) assert x.diff(i) == ZeroMatrix(k, 1) assert (x.Ty).diff(i) == ZeroMatrix(1, 1) assert (xx.T).diff(i) == ZeroMatrix(k, k) assert (x + y).diff(i) == ZeroMatrix(k, 1) assert hadamard_power(x, 2).diff(i) == ZeroMatrix(k, 1) assert hadamard_power(x, i).diff(i).dummy_eq( HadamardProduct(x.applyfunc(log), HadamardPower(x, i))) assert hadamard_product(x, y).diff(i) == ZeroMatrix(k, 1) assert hadamard_product(iOneMatrix(k, 1), x, y).diff(i) == hadamard_product(x, y) assert (ix).diff(i) == x assert (sin(i)ABx).diff(i) == cos(i)ABx assert x.applyfunc(sin).diff(i) == ZeroMatrix(k, 1) assert Trace(i2X).diff(i) == 2iTrace(X) mu = symbols("mu") expr = (2mux) assert expr.diff(x) == 2muIdentity(k) def test_matrix_derivative_non_matrix_result(): # This is a 4-dimensional array: I = Identity(k) AdA = PermuteDims(ArrayTensorProduct(I, I), Permutation(3)(1, 2)) assert A.diff(A) == AdA assert A.T.diff(A) == PermuteDims(ArrayTensorProduct(I, I), Permutation(3)(1, 2, 3)) assert (2A).diff(A) == PermuteDims(ArrayTensorProduct(2I, I), Permutation(3)(1, 2)) assert MatAdd(A, A).diff(A) == ArrayAdd(AdA, AdA) assert (A + B).diff(A) == AdA def test_matrix_derivative_trivial_cases(): # Cookbook example 33: # TODO: find a way to represent a four-dimensional zero-array: assert X.diff(A) == ArrayDerivative(X, A) def test_matrix_derivative_with_inverse(): # Cookbook example 61: expr = a.TInverse(X)b assert expr.diff(X) == -Inverse(X).Tab.TInverse(X).T # Cookbook example 62: expr = Determinant(Inverse(X)) # Not implemented yet: # assert expr.diff(X) == -Determinant(X.inv())(X.inv()).T # Cookbook example 63: expr = Trace(AInverse(X)B) assert expr.diff(X) == -(X*(-1)BAX(-1)).T # Cookbook example 64: expr = Trace(Inverse(X + A)) assert expr.diff(X) == -(Inverse(X + A)).T2 def test_matrix_derivative_vectors_and_scalars(): assert x.diff(x) == Identity(k) assert x[i, 0].diff(x[m, 0]).doit() == KDelta(m, i) assert x.T.diff(x) == Identity(k) # Cookbook example 69: expr = x.Ta assert expr.diff(x) == a assert expr[0, 0].diff(x[m, 0]).doit() == a[m, 0] expr = a.Tx assert expr.diff(x) == a # Cookbook example 70: expr = a.TXb assert expr.diff(X) == ab.T # Cookbook example 71: expr = a.TX.Tb assert expr.diff(X) == ba.T # Cookbook example 72: expr = a.TXa assert expr.diff(X) == aa.T expr = a.TX.Ta assert expr.diff(X) == aa.T # Cookbook example 77: expr = b.TX.TXc assert expr.diff(X) == Xbc.T + Xcb.T # Cookbook example 78: expr = (Bx + b).TC(Dx + d) assert expr.diff(x) == B.TC(Dx + d) + D.TC.T(Bx + b) # Cookbook example 81: expr = x.TBx assert expr.diff(x) == Bx + B.Tx # Cookbook example 82: expr = b.TX.TDXc assert expr.diff(X) == D.TXbc.T + DXcb.T # Cookbook example 83: expr = (Xb + c).TD(Xb + c) assert expr.diff(X) == D(Xb + c)b.T + D.T(Xb + c)b.T assert str(expr[0, 0].diff(X[m, n]).doit()) == \ 'b[n, 0]Sum((c[_i_1, 0] + Sum(X[_i_1, _i_3]b[_i_3, 0], (_i_3, 0, k - 1)))D[_i_1, m], (_i_1, 0, k - 1)) + Sum((c[_i_2, 0] + Sum(X[_i_2, _i_4]b[_i_4, 0], (_i_4, 0, k - 1)))D[m, _i_2]b[n, 0], (_i_2, 0, k - 1))' # See https://github.com/sympy/sympy/issues/16504#issuecomment-1018339957 expr = xx.Tx I = Identity(k) assert expr.diff(x) == KroneckerProduct(I, x.Tx) + 2xx.T def test_matrix_derivatives_of_traces(): expr = Trace(A)A I = Identity(k) assert expr.diff(A) == ArrayAdd(ArrayTensorProduct(I, A), PermuteDims(ArrayTensorProduct(Trace(A)I, I), Permutation(3)(1, 2))) assert expr[i, j].diff(A[m, n]).doit() == ( KDelta(i, m)KDelta(j, n)Trace(A) + KDelta(m, n)A[i, j] ) ## First order: # Cookbook example 99: expr = Trace(X) assert expr.diff(X) == Identity(k) assert expr.rewrite(Sum).diff(X[m, n]).doit() == KDelta(m, n) # Cookbook example 100: expr = Trace(XA) assert expr.diff(X) == A.T assert expr.rewrite(Sum).diff(X[m, n]).doit() == A[n, m] # Cookbook example 101: expr = Trace(AXB) assert expr.diff(X) == A.TB.T assert expr.rewrite(Sum).diff(X[m, n]).doit().dummy_eq((A.TB.T)[m, n]) # Cookbook example 102: expr = Trace(AX.TB) assert expr.diff(X) == BA # Cookbook example 103: expr = Trace(X.TA) assert expr.diff(X) == A # Cookbook example 104: expr = Trace(AX.T) assert expr.diff(X) == A # Cookbook example 105: # TODO: TensorProduct is not supported #expr = Trace(TensorProduct(A, X)) #assert expr.diff(X) == Trace(A)Identity(k) ## Second order: # Cookbook example 106: expr = Trace(X*2) assert expr.diff(X) == 2X.T # Cookbook example 107: expr = Trace(X*2B) assert expr.diff(X) == (XB + BX).T expr = Trace(MatMul(X, X, B)) assert expr.diff(X) == (XB + BX).T # Cookbook example 108: expr = Trace(X.TBX) assert expr.diff(X) == BX + B.TX # Cookbook example 109: expr = Trace(BXX.T) assert expr.diff(X) == BX + B.TX # Cookbook example 110: expr = Trace(XX.TB) assert expr.diff(X) == BX + B.TX # Cookbook example 111: expr = Trace(XBX.T) assert expr.diff(X) == XB.T + XB # Cookbook example 112: expr = Trace(BX.TX) assert expr.diff(X) == XB.T + XB # Cookbook example 113: expr = Trace(X.TXB) assert expr.diff(X) == XB.T + XB # Cookbook example 114: expr = Trace(AXBX) assert expr.diff(X) == A.TX.TB.T + B.TX.TA.T # Cookbook example 115: expr = Trace(X.TX) assert expr.diff(X) == 2X expr = Trace(XX.T) assert expr.diff(X) == 2X # Cookbook example 116: expr = Trace(B.TX.TCXB) assert expr.diff(X) == C.TXBB.T + CXBB.T # Cookbook example 117: expr = Trace(X.TBXC) assert expr.diff(X) == BXC + B.TXC.T # Cookbook example 118: expr = Trace(AXBX.TC) assert expr.diff(X) == A.TC.TXB.T + CAXB # Cookbook example 119: expr = Trace((AXB + C)(AXB + C).T) assert expr.diff(X) == 2A.T(AXB + C)B.T # Cookbook example 120: # TODO: no support for TensorProduct. # expr = Trace(TensorProduct(X, X)) # expr = Trace(X)Trace(X) # expr.diff(X) == 2Trace(X)Identity(k) # Higher Order # Cookbook example 121: expr = Trace(Xk) #assert expr.diff(X) == k(X*(k-1)).T # Cookbook example 122: expr = Trace(AX*k) #assert expr.diff(X) == # Needs indices # Cookbook example 123: expr = Trace(B.TX.TCXX.TCXB) assert expr.diff(X) == CXX.TCXBB.T + C.TXBB.TX.TC.TX + CXBB.TX.TCX + C.TXX.TC.TXBB.T # Other # Cookbook example 124: expr = Trace(AX(-1)B) assert expr.diff(X) == -Inverse(X).TA.TB.TInverse(X).T # Cookbook example 125: expr = Trace(Inverse(X.TCX)A) # Warning: result in the cookbook is equivalent if B and C are symmetric: assert expr.diff(X) == - X.inv().TA.TX.inv()C.inv().TX.inv().T - X.inv().TAX.inv()C.inv()X.inv().T # Cookbook example 126: expr = Trace((X.TCX).inv()(X.TBX)) assert expr.diff(X) == -2CX(X.TCX).inv()X.TBX(X.TCX).inv() + 2BX(X.TCX).inv() # Cookbook example 127: expr = Trace((A + X.TCX).inv()(X.TBX)) # Warning: result in the cookbook is equivalent if B and C are symmetric: assert expr.diff(X) == BXInverse(A + X.TCX) - CXInverse(A + X.TCX)X.TBXInverse(A + X.TCX) - C.TXInverse(A.T + (CX).TX)X.TB.TXInverse(A.T + (CX).TX) + B.TXInverse(A.T + (CX).TX) def test_derivatives_of_complicated_matrix_expr(): expr = a.T(AX(X.TB + XA) + B.TX.T(ab.T(XDX.T + X(X.TB + AX)DB - X.TC.TA)B + B(XD.T + BAXA.T - 3XD))B + 42XBX.TA.T(X + X.T))b result = (B(BAXA.T - 3XD + XD.T) + ab.T(X(AX + X.TB)DB + XDX.T - X.TC.TA)B)Bba.TB.T + B*2ba.TB.TX.Tab.TXD + 42AXB.TX.Tab.T + BDB3ba.TB.TX.Tab.TX + Bba.TAX + ab.T(42X + 42X.T)AXB.T + ba.TXBab.TB.T2XD.T + ba.TXBab.TB.T3D.T(B.TX + X.TA.T) + 42ba.TXBX.TA.T + A.T(42X + 42X.T)ba.TXB + A.TB.T2XBab.TB.TA + A.Tab.T(A.TX.T + B.TX) + A.TX.Tba.TXBab.TB.T*3D.T + B.TXBab.TB.TD - 3B.TXBab.TB.TD.T - C.TAB2ba.TB.TX.Tab.T + X.TA.Tab.TA.T assert expr.diff(X) == result def test_mixed_deriv_mixed_expressions(): expr = 3Trace(A) assert expr.diff(A) == 3Identity(k) expr = k deriv = expr.diff(A) assert isinstance(deriv, ZeroMatrix) assert deriv == ZeroMatrix(k, k) expr = Trace(A)2 assert expr.diff(A) == (2Trace(A))Identity(k) expr = Trace(A)A I = Identity(k) assert expr.diff(A) == ArrayAdd(ArrayTensorProduct(I, A), PermuteDims(ArrayTensorProduct(Trace(A)I, I), Permutation(3)(1, 2))) expr = Trace(Trace(A)A) assert expr.diff(A) == (2Trace(A))Identity(k) expr = Trace(Trace(Trace(A)A)A) assert expr.diff(A) == (3Trace(A)2)Identity(k) def test_derivatives_matrix_norms(): expr = x.Ty assert expr.diff(x) == y assert expr[0, 0].diff(x[m, 0]).doit() == y[m, 0] expr = (x.Ty)*S.Half assert expr.diff(x) == y/(2sqrt(x.Ty)) expr = (x.Tx)*S.Half assert expr.diff(x) == x(x.Tx)Rational(-1, 2) expr = (c.Tax.Tb)*S.Half assert expr.diff(x) == ba.Tc/sqrt(c.Tax.Tb)/2 expr = (c.Tax.Tb)Rational(1, 3) assert expr.diff(x) == ba.Tc(c.Tax.Tb)Rational(-2, 3)/3 expr = (a.TXb)S.Half assert expr.diff(X) == a/(2sqrt(a.TXb))b.T expr = d.Tx(a.TXb)S.Halfy.Tc assert expr.diff(X) == a/(2sqrt(a.TXb))x.Tdy.Tcb.T def test_derivatives_elementwise_applyfunc(): expr = x.applyfunc(tan) assert expr.diff(x).dummy_eq( DiagMatrix(x.applyfunc(lambda x: tan(x)2 + 1))) assert expr[i, 0].diff(x[m, 0]).doit() == (tan(x[i, 0])2 + 1)KDelta(i, m) _check_derivative_with_explicit_matrix(expr, x, expr.diff(x)) expr = (i*2x).applyfunc(sin) assert expr.diff(i).dummy_eq( HadamardProduct((2i)x, (i*2x).applyfunc(cos))) assert expr[i, 0].diff(i).doit() == 2ix[i, 0]cos(i2x[i, 0]) _check_derivative_with_explicit_matrix(expr, i, expr.diff(i)) expr = (log(i)AB).applyfunc(sin) assert expr.diff(i).dummy_eq( HadamardProduct(AB/i, (log(i)AB).applyfunc(cos))) _check_derivative_with_explicit_matrix(expr, i, expr.diff(i)) expr = Ax.applyfunc(exp) # TODO: restore this result (currently returning the transpose): # assert expr.diff(x).dummy_eq(DiagMatrix(x.applyfunc(exp))A.T) _check_derivative_with_explicit_matrix(expr, x, expr.diff(x)) expr = x.TAx + ky.applyfunc(sin).Tx assert expr.diff(x).dummy_eq(A.Tx + Ax + ky.applyfunc(sin)) _check_derivative_with_explicit_matrix(expr, x, expr.diff(x)) expr = x.applyfunc(sin).Ty # TODO: restore (currently returning the traspose): # assert expr.diff(x).dummy_eq(DiagMatrix(x.applyfunc(cos))y) _check_derivative_with_explicit_matrix(expr, x, expr.diff(x)) expr = (a.T * X * b).applyfunc(sin) assert expr.diff(X).dummy_eq(a(a.TXb).applyfunc(cos)b.T) _check_derivative_with_explicit_matrix(expr, X, expr.diff(X)) expr = a.T * X.applyfunc(sin) * b assert expr.diff(X).dummy_eq( DiagMatrix(a)X.applyfunc(cos)DiagMatrix(b)) _check_derivative_with_explicit_matrix(expr, X, expr.diff(X)) expr = a.T * (AXB).applyfunc(sin) * b assert expr.diff(X).dummy_eq( A.TDiagMatrix(a)(AXB).applyfunc(cos)DiagMatrix(b)B.T) _check_derivative_with_explicit_matrix(expr, X, expr.diff(X)) expr = a.T * (AXb).applyfunc(sin) * b.T # TODO: not implemented #assert expr.diff(X) == ... #_check_derivative_with_explicit_matrix(expr, X, expr.diff(X)) expr = a.TAX.applyfunc(sin)Bb assert expr.diff(X).dummy_eq( HadamardProduct(A.T * a * b.T * B.T, X.applyfunc(cos))) expr = a.T * (AX.applyfunc(sin)B).applyfunc(log) * b # TODO: wrong # assert expr.diff(X) == A.TDiagMatrix(a)(AX.applyfunc(sin)B).applyfunc(Lambda(k, 1/k))DiagMatrix(b)B.T expr = a.T * (X.applyfunc(sin)).applyfunc(log) * b # TODO: wrong # assert expr.diff(X) == DiagMatrix(a)X.applyfunc(sin).applyfunc(Lambda(k, 1/k))DiagMatrix(b) def test_derivatives_of_hadamard_expressions(): # Hadamard Product expr = hadamard_product(a, x, b) assert expr.diff(x) == DiagMatrix(hadamard_product(b, a)) expr = a.Thadamard_product(A, X, B)b assert expr.diff(X) == HadamardProduct(ab.T, A, B) # Hadamard Power expr = hadamard_power(x, 2) assert expr.diff(x).doit() == 2DiagMatrix(x) expr = hadamard_power(x.T, 2) assert expr.diff(x).doit() == 2DiagMatrix(x) expr = hadamard_power(x, S.Half) assert expr.diff(x) == S.HalfDiagMatrix(hadamard_power(x, Rational(-1, 2))) expr = hadamard_power(a.TXb, 2) assert expr.diff(X) == 2aa.TXbb.T expr = hadamard_power(a.TXb, S.Half) assert expr.diff(X) == a/(2sqrt(a.TXb))*b.T
f38377748baceebd3d5d5ecf17f462129151c35cb9b4e183df858acd233d17ee	from sympy.core import Basic, Expr from sympy.core.function import Lambda from sympy.core.numbers import oo, Infinity, NegativeInfinity, Zero, Integer from sympy.core.singleton import S from sympy.core.symbol import symbols from sympy.functions.elementary.miscellaneous import (Max, Min) from sympy.sets.fancysets import ImageSet from sympy.sets.setexpr import set_div from sympy.sets.sets import Set, Interval, FiniteSet, Union from sympy.multipledispatch import Dispatcher _x, _y = symbols("x y") _set_pow = Dispatcher('_set_pow') @_set_pow.register(Basic, Basic) def _(x, y): return None @_set_pow.register(Set, Set) def _(x, y): return ImageSet(Lambda((_x, _y), (_x _y)), x, y) @_set_pow.register(Expr, Expr) def _(x, y): return xy @_set_pow.register(Interval, Zero) def _(x, z): return FiniteSet(S.One) @_set_pow.register(Interval, Integer) def _(x, exponent): """ Powers in interval arithmetic https://en.wikipedia.org/wiki/Interval_arithmetic """ s1 = x.startexponent s2 = x.endexponent if ((s2 > s1) if exponent > 0 else (x.end > -x.start)) == True: left_open = x.left_open right_open = x.right_open # TODO: handle unevaluated condition. sleft = s2 else: # TODO: `s2 > s1` could be unevaluated. left_open = x.right_open right_open = x.left_open sleft = s1 if x.start.is_positive: return Interval( Min(s1, s2), Max(s1, s2), left_open, right_open) elif x.end.is_negative: return Interval( Min(s1, s2), Max(s1, s2), left_open, right_open) # Case where x.start < 0 and x.end > 0: if exponent.is_odd: if exponent.is_negative: if x.start.is_zero: return Interval(s2, oo, x.right_open) if x.end.is_zero: return Interval(-oo, s1, True, x.left_open) return Union(Interval(-oo, s1, True, x.left_open), Interval(s2, oo, x.right_open)) else: return Interval(s1, s2, x.left_open, x.right_open) elif exponent.is_even: if exponent.is_negative: if x.start.is_zero: return Interval(s2, oo, x.right_open) if x.end.is_zero: return Interval(s1, oo, x.left_open) return Interval(0, oo) else: return Interval(S.Zero, sleft, S.Zero not in x, left_open) @_set_pow.register(Interval, Infinity) def _(b, e): # TODO: add logic for open intervals? if b.start.is_nonnegative: if b.end < 1: return FiniteSet(S.Zero) if b.start > 1: return FiniteSet(S.Infinity) return Interval(0, oo) elif b.end.is_negative: if b.start > -1: return FiniteSet(S.Zero) if b.end < -1: return FiniteSet(-oo, oo) return Interval(-oo, oo) else: if b.start > -1: if b.end < 1: return FiniteSet(S.Zero) return Interval(0, oo) return Interval(-oo, oo) @_set_pow.register(Interval, NegativeInfinity) def _(b, e): return _set_pow(set_div(S.One, b), oo)
412b2e432df13e54b58f3135a087e41f5e5e3866976b66f02376232a72d57f3b	from sympy.core.function import Lambda, expand_complex from sympy.core.mul import Mul from sympy.core.numbers import ilcm from sympy.core.relational import Eq from sympy.core.singleton import S from sympy.core.symbol import (Dummy, symbols) from sympy.core.sorting import ordered from sympy.sets.fancysets import ComplexRegion from sympy.sets.sets import (FiniteSet, Intersection, Interval, Set, Union) from sympy.multipledispatch import Dispatcher from sympy.sets.conditionset import ConditionSet from sympy.sets.fancysets import (Integers, Naturals, Reals, Range, ImageSet, Rationals) from sympy.sets.sets import EmptySet, UniversalSet, imageset, ProductSet from sympy.simplify.radsimp import numer intersection_sets = Dispatcher('intersection_sets') @intersection_sets.register(ConditionSet, ConditionSet) def _(a, b): return None @intersection_sets.register(ConditionSet, Set) def _(a, b): return ConditionSet(a.sym, a.condition, Intersection(a.base_set, b)) @intersection_sets.register(Naturals, Integers) def _(a, b): return a @intersection_sets.register(Naturals, Naturals) def _(a, b): return a if a is S.Naturals else b @intersection_sets.register(Interval, Naturals) def _(a, b): return intersection_sets(b, a) @intersection_sets.register(ComplexRegion, Set) def _(self, other): if other.is_ComplexRegion: # self in rectangular form if (not self.polar) and (not other.polar): return ComplexRegion(Intersection(self.sets, other.sets)) # self in polar form elif self.polar and other.polar: r1, theta1 = self.a_interval, self.b_interval r2, theta2 = other.a_interval, other.b_interval new_r_interval = Intersection(r1, r2) new_theta_interval = Intersection(theta1, theta2) # 0 and 2Pi means the same if ((2S.Pi in theta1 and S.Zero in theta2) or (2S.Pi in theta2 and S.Zero in theta1)): new_theta_interval = Union(new_theta_interval, FiniteSet(0)) return ComplexRegion(new_r_intervalnew_theta_interval, polar=True) if other.is_subset(S.Reals): new_interval = [] x = symbols("x", cls=Dummy, real=True) # self in rectangular form if not self.polar: for element in self.psets: if S.Zero in element.args[1]: new_interval.append(element.args[0]) new_interval = Union(new_interval) return Intersection(new_interval, other) # self in polar form elif self.polar: for element in self.psets: if S.Zero in element.args[1]: new_interval.append(element.args[0]) if S.Pi in element.args[1]: new_interval.append(ImageSet(Lambda(x, -x), element.args[0])) if S.Zero in element.args[0]: new_interval.append(FiniteSet(0)) new_interval = Union(new_interval) return Intersection(new_interval, other) @intersection_sets.register(Integers, Reals) def _(a, b): return a @intersection_sets.register(Range, Interval) def _(a, b): # Check that there are no symbolic arguments if not all(i.is_number for i in a.args + b.args[:2]): return # In case of null Range, return an EmptySet. if a.size == 0: return S.EmptySet from sympy.functions.elementary.integers import floor, ceiling # trim down to self's size, and represent # as a Range with step 1. start = ceiling(max(b.inf, a.inf)) if start not in b: start += 1 end = floor(min(b.sup, a.sup)) if end not in b: end -= 1 return intersection_sets(a, Range(start, end + 1)) @intersection_sets.register(Range, Naturals) def _(a, b): return intersection_sets(a, Interval(b.inf, S.Infinity)) @intersection_sets.register(Range, Range) def _(a, b): # Check that there are no symbolic range arguments if not all(all(v.is_number for v in r.args) for r in [a, b]): return None # non-overlap quick exits if not b: return S.EmptySet if not a: return S.EmptySet if b.sup < a.inf: return S.EmptySet if b.inf > a.sup: return S.EmptySet # work with finite end at the start r1 = a if r1.start.is_infinite: r1 = r1.reversed r2 = b if r2.start.is_infinite: r2 = r2.reversed # If both ends are infinite then it means that one Range is just the set # of all integers (the step must be 1). if r1.start.is_infinite: return b if r2.start.is_infinite: return a from sympy.solvers.diophantine.diophantine import diop_linear from sympy.functions.elementary.complexes import sign # this equation represents the values of the Range; # it's a linear equation eq = lambda r, i: r.start + ir.step # we want to know when the two equations might # have integer solutions so we use the diophantine # solver va, vb = diop_linear(eq(r1, Dummy('a')) - eq(r2, Dummy('b'))) # check for no solution no_solution = va is None and vb is None if no_solution: return S.EmptySet # there is a solution # ------------------- # find the coincident point, c a0 = va.as_coeff_Add()[0] c = eq(r1, a0) # find the first point, if possible, in each range # since c may not be that point def _first_finite_point(r1, c): if c == r1.start: return c # st is the signed step we need to take to # get from c to r1.start st = sign(r1.start - c)step # use Range to calculate the first point: # we want to get as close as possible to # r1.start; the Range will not be null since # it will at least contain c s1 = Range(c, r1.start + st, st)[-1] if s1 == r1.start: pass else: # if we didn't hit r1.start then, if the # sign of st didn't match the sign of r1.step # we are off by one and s1 is not in r1 if sign(r1.step) != sign(st): s1 -= st if s1 not in r1: return return s1 # calculate the step size of the new Range step = abs(ilcm(r1.step, r2.step)) s1 = _first_finite_point(r1, c) if s1 is None: return S.EmptySet s2 = _first_finite_point(r2, c) if s2 is None: return S.EmptySet # replace the corresponding start or stop in # the original Ranges with these points; the # result must have at least one point since # we know that s1 and s2 are in the Ranges def _updated_range(r, first): st = sign(r.step)step if r.start.is_finite: rv = Range(first, r.stop, st) else: rv = Range(r.start, first + st, st) return rv r1 = _updated_range(a, s1) r2 = _updated_range(b, s2) # work with them both in the increasing direction if sign(r1.step) < 0: r1 = r1.reversed if sign(r2.step) < 0: r2 = r2.reversed # return clipped Range with positive step; it # can't be empty at this point start = max(r1.start, r2.start) stop = min(r1.stop, r2.stop) return Range(start, stop, step) @intersection_sets.register(Range, Integers) def _(a, b): return a @intersection_sets.register(ImageSet, Set) def _(self, other): from sympy.solvers.diophantine import diophantine # Only handle the straight-forward univariate case if (len(self.lamda.variables) > 1 or self.lamda.signature != self.lamda.variables): return None base_set = self.base_sets[0] # Intersection between ImageSets with Integers as base set # For {f(n) : n in Integers} & {g(m) : m in Integers} we solve the # diophantine equations f(n)=g(m). # If the solutions for n are {h(t) : t in Integers} then we return # {f(h(t)) : t in integers}. # If the solutions for n are {n_1, n_2, ..., n_k} then we return # {f(n_i) : 1 <= i <= k}. if base_set is S.Integers: gm = None if isinstance(other, ImageSet) and other.base_sets == (S.Integers,): gm = other.lamda.expr var = other.lamda.variables[0] # Symbol of second ImageSet lambda must be distinct from first m = Dummy('m') gm = gm.subs(var, m) elif other is S.Integers: m = gm = Dummy('m') if gm is not None: fn = self.lamda.expr n = self.lamda.variables[0] try: solns = list(diophantine(fn - gm, syms=(n, m), permute=True)) except (TypeError, NotImplementedError): # TypeError if equation not polynomial with rational coeff. # NotImplementedError if correct format but no solver. return # 3 cases are possible for solns: # - empty set, # - one or more parametric (infinite) solutions, # - a finite number of (non-parametric) solution couples. # Among those, there is one type of solution set that is # not helpful here: multiple parametric solutions. if len(solns) == 0: return S.EmptySet elif any(s.free_symbols for tupl in solns for s in tupl): if len(solns) == 1: soln, solm = solns[0] (t,) = soln.free_symbols expr = fn.subs(n, soln.subs(t, n)).expand() return imageset(Lambda(n, expr), S.Integers) else: return else: return FiniteSet((fn.subs(n, s[0]) for s in solns)) if other == S.Reals: from sympy.solvers.solvers import denoms, solve_linear def _solution_union(exprs, sym): # return a union of linear solutions to i in expr; # if i cannot be solved, use a ConditionSet for solution sols = [] for i in exprs: x, xis = solve_linear(i, 0, [sym]) if x == sym: sols.append(FiniteSet(xis)) else: sols.append(ConditionSet(sym, Eq(i, 0))) return Union(sols) f = self.lamda.expr n = self.lamda.variables[0] n_ = Dummy(n.name, real=True) f_ = f.subs(n, n_) re, im = f_.as_real_imag() im = expand_complex(im) re = re.subs(n_, n) im = im.subs(n_, n) ifree = im.free_symbols lam = Lambda(n, re) if im.is_zero: # allow re-evaluation # of self in this case to make # the result canonical pass elif im.is_zero is False: return S.EmptySet elif ifree != {n}: return None else: # univarite imaginary part in same variable; # use numer instead of as_numer_denom to keep # this as fast as possible while still handling # simple cases base_set &= _solution_union( Mul.make_args(numer(im)), n) # exclude values that make denominators 0 base_set -= _solution_union(denoms(f), n) return imageset(lam, base_set) elif isinstance(other, Interval): from sympy.solvers.solveset import (invert_real, invert_complex, solveset) f = self.lamda.expr n = self.lamda.variables[0] new_inf, new_sup = None, None new_lopen, new_ropen = other.left_open, other.right_open if f.is_real: inverter = invert_real else: inverter = invert_complex g1, h1 = inverter(f, other.inf, n) g2, h2 = inverter(f, other.sup, n) if all(isinstance(i, FiniteSet) for i in (h1, h2)): if g1 == n: if len(h1) == 1: new_inf = h1.args[0] if g2 == n: if len(h2) == 1: new_sup = h2.args[0] # TODO: Design a technique to handle multiple-inverse # functions # Any of the new boundary values cannot be determined if any(i is None for i in (new_sup, new_inf)): return range_set = S.EmptySet if all(i.is_real for i in (new_sup, new_inf)): # this assumes continuity of underlying function # however fixes the case when it is decreasing if new_inf > new_sup: new_inf, new_sup = new_sup, new_inf new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen) range_set = base_set.intersect(new_interval) else: if other.is_subset(S.Reals): solutions = solveset(f, n, S.Reals) if not isinstance(range_set, (ImageSet, ConditionSet)): range_set = solutions.intersect(other) else: return if range_set is S.EmptySet: return S.EmptySet elif isinstance(range_set, Range) and range_set.size is not S.Infinity: range_set = FiniteSet(list(range_set)) if range_set is not None: return imageset(Lambda(n, f), range_set) return else: return @intersection_sets.register(ProductSet, ProductSet) def _(a, b): if len(b.args) != len(a.args): return S.EmptySet return ProductSet((i.intersect(j) for i, j in zip(a.sets, b.sets))) @intersection_sets.register(Interval, Interval) def _(a, b): # handle (-oo, oo) infty = S.NegativeInfinity, S.Infinity if a == Interval(infty): l, r = a.left, a.right if l.is_real or l in infty or r.is_real or r in infty: return b # We can't intersect [0,3] with [x,6] -- we don't know if x>0 or x<0 if not a._is_comparable(b): return None empty = False if a.start <= b.end and b.start <= a.end: # Get topology right. if a.start < b.start: start = b.start left_open = b.left_open elif a.start > b.start: start = a.start left_open = a.left_open else: #this is to ensure that if Eq(a.start,b.start) but #type(a.start) != type(b.start) the order of a and b #does not matter for the result start = list(ordered([a,b]))[0].start left_open = a.left_open or b.left_open if a.end < b.end: end = a.end right_open = a.right_open elif a.end > b.end: end = b.end right_open = b.right_open else: end = list(ordered([a,b]))[0].end right_open = a.right_open or b.right_open if end - start == 0 and (left_open or right_open): empty = True else: empty = True if empty: return S.EmptySet return Interval(start, end, left_open, right_open) @intersection_sets.register(EmptySet, Set) def _(a, b): return S.EmptySet @intersection_sets.register(UniversalSet, Set) def _(a, b): return b @intersection_sets.register(FiniteSet, FiniteSet) def _(a, b): return FiniteSet((a._elements & b._elements)) @intersection_sets.register(FiniteSet, Set) def _(a, b): try: return FiniteSet([el for el in a if el in b]) except TypeError: return None # could not evaluate `el in b` due to symbolic ranges. @intersection_sets.register(Set, Set) def _(a, b): return None @intersection_sets.register(Integers, Rationals) def _(a, b): return a @intersection_sets.register(Naturals, Rationals) def _(a, b): return a @intersection_sets.register(Rationals, Reals) def _(a, b): return a def _intlike_interval(a, b): try: from sympy.functions.elementary.integers import floor, ceiling if b._inf is S.NegativeInfinity and b._sup is S.Infinity: return a s = Range(max(a.inf, ceiling(b.left)), floor(b.right) + 1) return intersection_sets(s, b) # take out endpoints if open interval except ValueError: return None @intersection_sets.register(Integers, Interval) def _(a, b): return _intlike_interval(a, b) @intersection_sets.register(Naturals, Interval) def _(a, b): return _intlike_interval(a, b)
fee705bf1b83bfdc2c98ac469020d4111b5893723fb50c63663637664ae85acd	from sympy.core.numbers import oo, Infinity, NegativeInfinity from sympy.core.singleton import S from sympy.core import Basic, Expr from sympy.multipledispatch import Dispatcher from sympy.sets import Interval, FiniteSet # XXX: The functions in this module are clearly not tested and are broken in a # number of ways. _set_add = Dispatcher('_set_add') _set_sub = Dispatcher('_set_sub') @_set_add.register(Basic, Basic) def _(x, y): return None @_set_add.register(Expr, Expr) def _(x, y): return x+y @_set_add.register(Interval, Interval) def _(x, y): """ Additions in interval arithmetic https://en.wikipedia.org/wiki/Interval_arithmetic """ return Interval(x.start + y.start, x.end + y.end, x.left_open or y.left_open, x.right_open or y.right_open) @_set_add.register(Interval, Infinity) def _(x, y): if x.start is S.NegativeInfinity: return Interval(-oo, oo) return FiniteSet({S.Infinity}) @_set_add.register(Interval, NegativeInfinity) def _(x, y): if x.end is S.Infinity: return Interval(-oo, oo) return FiniteSet({S.NegativeInfinity}) @_set_sub.register(Basic, Basic) def _(x, y): return None @_set_sub.register(Expr, Expr) def _(x, y): return x-y @_set_sub.register(Interval, Interval) def _(x, y): """ Subtractions in interval arithmetic https://en.wikipedia.org/wiki/Interval_arithmetic """ return Interval(x.start - y.end, x.end - y.start, x.left_open or y.right_open, x.right_open or y.left_open) @_set_sub.register(Interval, Infinity) def _(x, y): if x.start is S.NegativeInfinity: return Interval(-oo, oo) return FiniteSet(-oo) @_set_sub.register(Interval, NegativeInfinity) def _(x, y): if x.start is S.NegativeInfinity: return Interval(-oo, oo) return FiniteSet(-oo)
15dba35b5e1cd399bd889377fa8d8643fec6da20c4f627c5614c4ae142d8cbe5	from sympy.core.singleton import S from sympy.core.sympify import sympify from sympy.sets.sets import (EmptySet, FiniteSet, Intersection, Interval, ProductSet, Set, Union, UniversalSet) from sympy.sets.fancysets import (ComplexRegion, Naturals, Naturals0, Integers, Rationals, Reals) from sympy.multipledispatch import Dispatcher union_sets = Dispatcher('union_sets') @union_sets.register(Naturals0, Naturals) def _(a, b): return a @union_sets.register(Rationals, Naturals) def _(a, b): return a @union_sets.register(Rationals, Naturals0) def _(a, b): return a @union_sets.register(Reals, Naturals) def _(a, b): return a @union_sets.register(Reals, Naturals0) def _(a, b): return a @union_sets.register(Reals, Rationals) def _(a, b): return a @union_sets.register(Integers, Set) def _(a, b): intersect = Intersection(a, b) if intersect == a: return b elif intersect == b: return a @union_sets.register(ComplexRegion, Set) def _(a, b): if b.is_subset(S.Reals): # treat a subset of reals as a complex region b = ComplexRegion.from_real(b) if b.is_ComplexRegion: # a in rectangular form if (not a.polar) and (not b.polar): return ComplexRegion(Union(a.sets, b.sets)) # a in polar form elif a.polar and b.polar: return ComplexRegion(Union(a.sets, b.sets), polar=True) return None @union_sets.register(EmptySet, Set) def _(a, b): return b @union_sets.register(UniversalSet, Set) def _(a, b): return a @union_sets.register(ProductSet, ProductSet) def _(a, b): if b.is_subset(a): return a if len(b.sets) != len(a.sets): return None if len(a.sets) == 2: a1, a2 = a.sets b1, b2 = b.sets if a1 == b1: return a1 * Union(a2, b2) if a2 == b2: return Union(a1, b1) * a2 return None @union_sets.register(ProductSet, Set) def _(a, b): if b.is_subset(a): return a return None @union_sets.register(Interval, Interval) def _(a, b): if a._is_comparable(b): from sympy.functions.elementary.miscellaneous import Min, Max # Non-overlapping intervals end = Min(a.end, b.end) start = Max(a.start, b.start) if (end < start or (end == start and (end not in a and end not in b))): return None else: start = Min(a.start, b.start) end = Max(a.end, b.end) left_open = ((a.start != start or a.left_open) and (b.start != start or b.left_open)) right_open = ((a.end != end or a.right_open) and (b.end != end or b.right_open)) return Interval(start, end, left_open, right_open) @union_sets.register(Interval, UniversalSet) def _(a, b): return S.UniversalSet @union_sets.register(Interval, Set) def _(a, b): # If I have open end points and these endpoints are contained in b # But only in case, when endpoints are finite. Because # interval does not contain oo or -oo. open_left_in_b_and_finite = (a.left_open and sympify(b.contains(a.start)) is S.true and a.start.is_finite) open_right_in_b_and_finite = (a.right_open and sympify(b.contains(a.end)) is S.true and a.end.is_finite) if open_left_in_b_and_finite or open_right_in_b_and_finite: # Fill in my end points and return open_left = a.left_open and a.start not in b open_right = a.right_open and a.end not in b new_a = Interval(a.start, a.end, open_left, open_right) return {new_a, b} return None @union_sets.register(FiniteSet, FiniteSet) def _(a, b): return FiniteSet((a._elements \| b._elements)) @union_sets.register(FiniteSet, Set) def _(a, b): # If `b` set contains one of my elements, remove it from `a` if any(b.contains(x) == True for x in a): return { FiniteSet([x for x in a if b.contains(x) != True]), b} return None @union_sets.register(Set, Set) def _(a, b): return None
8d8485a85347ea65db2853e0bc8008198f0185b2cfa35740bdc360e0cc6609a2	from sympy.core.singleton import S from sympy.sets.sets import Set from sympy.calculus.singularities import singularities from sympy.core import Expr, Add from sympy.core.function import Lambda, FunctionClass, diff, expand_mul from sympy.core.numbers import Float, oo from sympy.core.symbol import Dummy, symbols, Wild from sympy.functions.elementary.exponential import exp, log from sympy.functions.elementary.miscellaneous import Min, Max from sympy.logic.boolalg import true from sympy.multipledispatch import Dispatcher from sympy.sets import (imageset, Interval, FiniteSet, Union, ImageSet, Intersection, Range, Complement) from sympy.sets.sets import EmptySet, is_function_invertible_in_set from sympy.sets.fancysets import Integers, Naturals, Reals from sympy.functions.elementary.exponential import match_real_imag _x, _y = symbols("x y") FunctionUnion = (FunctionClass, Lambda) _set_function = Dispatcher('_set_function') @_set_function.register(FunctionClass, Set) def _(f, x): return None @_set_function.register(FunctionUnion, FiniteSet) def _(f, x): return FiniteSet(map(f, x)) @_set_function.register(Lambda, Interval) def _(f, x): from sympy.solvers.solveset import solveset from sympy.series import limit # TODO: handle functions with infinitely many solutions (eg, sin, tan) # TODO: handle multivariate functions expr = f.expr if len(expr.free_symbols) > 1 or len(f.variables) != 1: return var = f.variables[0] if not var.is_real: if expr.subs(var, Dummy(real=True)).is_real is False: return if expr.is_Piecewise: result = S.EmptySet domain_set = x for (p_expr, p_cond) in expr.args: if p_cond is true: intrvl = domain_set else: intrvl = p_cond.as_set() intrvl = Intersection(domain_set, intrvl) if p_expr.is_Number: image = FiniteSet(p_expr) else: image = imageset(Lambda(var, p_expr), intrvl) result = Union(result, image) # remove the part which has been `imaged` domain_set = Complement(domain_set, intrvl) if domain_set is S.EmptySet: break return result if not x.start.is_comparable or not x.end.is_comparable: return try: from sympy.polys.polyutils import _nsort sing = list(singularities(expr, var, x)) if len(sing) > 1: sing = _nsort(sing) except NotImplementedError: return if x.left_open: _start = limit(expr, var, x.start, dir="+") elif x.start not in sing: _start = f(x.start) if x.right_open: _end = limit(expr, var, x.end, dir="-") elif x.end not in sing: _end = f(x.end) if len(sing) == 0: soln_expr = solveset(diff(expr, var), var) if not (isinstance(soln_expr, FiniteSet) or soln_expr is S.EmptySet): return solns = list(soln_expr) extr = [_start, _end] + [f(i) for i in solns if i.is_real and i in x] start, end = Min(extr), Max(extr) left_open, right_open = False, False if _start <= _end: # the minimum or maximum value can occur simultaneously # on both the edge of the interval and in some interior # point if start == _start and start not in solns: left_open = x.left_open if end == _end and end not in solns: right_open = x.right_open else: if start == _end and start not in solns: left_open = x.right_open if end == _start and end not in solns: right_open = x.left_open return Interval(start, end, left_open, right_open) else: return imageset(f, Interval(x.start, sing[0], x.left_open, True)) + \ Union([imageset(f, Interval(sing[i], sing[i + 1], True, True)) for i in range(0, len(sing) - 1)]) + \ imageset(f, Interval(sing[-1], x.end, True, x.right_open)) @_set_function.register(FunctionClass, Interval) def _(f, x): if f == exp: return Interval(exp(x.start), exp(x.end), x.left_open, x.right_open) elif f == log: return Interval(log(x.start), log(x.end), x.left_open, x.right_open) return ImageSet(Lambda(_x, f(_x)), x) @_set_function.register(FunctionUnion, Union) def _(f, x): return Union((imageset(f, arg) for arg in x.args)) @_set_function.register(FunctionUnion, Intersection) def _(f, x): # If the function is invertible, intersect the maps of the sets. if is_function_invertible_in_set(f, x): return Intersection((imageset(f, arg) for arg in x.args)) else: return ImageSet(Lambda(_x, f(_x)), x) @_set_function.register(FunctionUnion, EmptySet) def _(f, x): return x @_set_function.register(FunctionUnion, Set) def _(f, x): return ImageSet(Lambda(_x, f(_x)), x) @_set_function.register(FunctionUnion, Range) def _(f, self): if not self: return S.EmptySet if not isinstance(f.expr, Expr): return if self.size == 1: return FiniteSet(f(self[0])) if f is S.IdentityFunction: return self x = f.variables[0] expr = f.expr # handle f that is linear in f's variable if x not in expr.free_symbols or x in expr.diff(x).free_symbols: return if self.start.is_finite: F = f(self.stepx + self.start) # for i in range(len(self)) else: F = f(-self.stepx + self[-1]) F = expand_mul(F) if F != expr: return imageset(x, F, Range(self.size)) @_set_function.register(FunctionUnion, Integers) def _(f, self): expr = f.expr if not isinstance(expr, Expr): return n = f.variables[0] if expr == abs(n): return S.Naturals0 # f(x) + c and f(-x) + c cover the same integers # so choose the form that has the fewest negatives c = f(0) fx = f(n) - c f_x = f(-n) - c neg_count = lambda e: sum(_.could_extract_minus_sign() for _ in Add.make_args(e)) if neg_count(f_x) < neg_count(fx): expr = f_x + c a = Wild('a', exclude=[n]) b = Wild('b', exclude=[n]) match = expr.match(an + b) if match and match[a] and ( not match[a].atoms(Float) and not match[b].atoms(Float)): # canonical shift a, b = match[a], match[b] if a in [1, -1]: # drop integer addends in b nonint = [] for bi in Add.make_args(b): if not bi.is_integer: nonint.append(bi) b = Add(nonint) if b.is_number and a.is_real: # avoid Mod for complex numbers, #11391 br, bi = match_real_imag(b) if br and br.is_comparable and a.is_comparable: br %= a b = br + S.ImaginaryUnitbi elif b.is_number and a.is_imaginary: br, bi = match_real_imag(b) ai = a/S.ImaginaryUnit if bi and bi.is_comparable and ai.is_comparable: bi %= ai b = br + S.ImaginaryUnitbi expr = an + b if expr != f.expr: return ImageSet(Lambda(n, expr), S.Integers) @_set_function.register(FunctionUnion, Naturals) def _(f, self): expr = f.expr if not isinstance(expr, Expr): return x = f.variables[0] if not expr.free_symbols - {x}: if expr == abs(x): if self is S.Naturals: return self return S.Naturals0 step = expr.coeff(x) c = expr.subs(x, 0) if c.is_Integer and step.is_Integer and expr == stepx + c: if self is S.Naturals: c += step if step > 0: if step == 1: if c == 0: return S.Naturals0 elif c == 1: return S.Naturals return Range(c, oo, step) return Range(c, -oo, step) @_set_function.register(FunctionUnion, Reals) def _(f, self): expr = f.expr if not isinstance(expr, Expr): return return _set_function(f, Interval(-oo, oo))
6b391aff978e694ae8debdefef63e6113d2863cf9a6a8264cd510aee4baa5c20	from sympy.core.singleton import S from sympy.core.symbol import Symbol from sympy.core.logic import fuzzy_and, fuzzy_bool, fuzzy_not, fuzzy_or from sympy.core.relational import Eq from sympy.sets.sets import FiniteSet, Interval, Set, Union, ProductSet from sympy.sets.fancysets import Complexes, Reals, Range, Rationals from sympy.multipledispatch import Dispatcher _inf_sets = [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals, S.Complexes] is_subset_sets = Dispatcher('is_subset_sets') @is_subset_sets.register(Set, Set) def _(a, b): return None @is_subset_sets.register(Interval, Interval) def _(a, b): # This is correct but can be made more comprehensive... if fuzzy_bool(a.start < b.start): return False if fuzzy_bool(a.end > b.end): return False if (b.left_open and not a.left_open and fuzzy_bool(Eq(a.start, b.start))): return False if (b.right_open and not a.right_open and fuzzy_bool(Eq(a.end, b.end))): return False @is_subset_sets.register(Interval, FiniteSet) def _(a_interval, b_fs): # An Interval can only be a subset of a finite set if it is finite # which can only happen if it has zero measure. if fuzzy_not(a_interval.measure.is_zero): return False @is_subset_sets.register(Interval, Union) def _(a_interval, b_u): if all(isinstance(s, (Interval, FiniteSet)) for s in b_u.args): intervals = [s for s in b_u.args if isinstance(s, Interval)] if all(fuzzy_bool(a_interval.start < s.start) for s in intervals): return False if all(fuzzy_bool(a_interval.end > s.end) for s in intervals): return False if a_interval.measure.is_nonzero: no_overlap = lambda s1, s2: fuzzy_or([ fuzzy_bool(s1.end <= s2.start), fuzzy_bool(s1.start >= s2.end), ]) if all(no_overlap(s, a_interval) for s in intervals): return False @is_subset_sets.register(Range, Range) def _(a, b): if a.step == b.step == 1: return fuzzy_and([fuzzy_bool(a.start >= b.start), fuzzy_bool(a.stop <= b.stop)]) @is_subset_sets.register(Range, Interval) def _(a_range, b_interval): if a_range.step.is_positive: if b_interval.left_open and a_range.inf.is_finite: cond_left = a_range.inf > b_interval.left else: cond_left = a_range.inf >= b_interval.left if b_interval.right_open and a_range.sup.is_finite: cond_right = a_range.sup < b_interval.right else: cond_right = a_range.sup <= b_interval.right return fuzzy_and([cond_left, cond_right]) @is_subset_sets.register(Range, FiniteSet) def _(a_range, b_finiteset): try: a_size = a_range.size except ValueError: # symbolic Range of unknown size return None if a_size > len(b_finiteset): return False elif any(arg.has(Symbol) for arg in a_range.args): return fuzzy_and(b_finiteset.contains(x) for x in a_range) else: # Checking A \ B == EmptySet is more efficient than repeated naive # membership checks on an arbitrary FiniteSet. a_set = set(a_range) b_remaining = len(b_finiteset) # Symbolic expressions and numbers of unknown type (integer or not) are # all counted as "candidates", i.e. potentially matching some a in # a_range. cnt_candidate = 0 for b in b_finiteset: if b.is_Integer: a_set.discard(b) elif fuzzy_not(b.is_integer): pass else: cnt_candidate += 1 b_remaining -= 1 if len(a_set) > b_remaining + cnt_candidate: return False if len(a_set) == 0: return True return None @is_subset_sets.register(Interval, Range) def _(a_interval, b_range): if a_interval.measure.is_extended_nonzero: return False @is_subset_sets.register(Interval, Rationals) def _(a_interval, b_rationals): if a_interval.measure.is_extended_nonzero: return False @is_subset_sets.register(Range, Complexes) def _(a, b): return True @is_subset_sets.register(Complexes, Interval) def _(a, b): return False @is_subset_sets.register(Complexes, Range) def _(a, b): return False @is_subset_sets.register(Complexes, Rationals) def _(a, b): return False @is_subset_sets.register(Rationals, Reals) def _(a, b): return True @is_subset_sets.register(Rationals, Range) def _(a, b): return False @is_subset_sets.register(ProductSet, FiniteSet) def _(a_ps, b_fs): return fuzzy_and(b_fs.contains(x) for x in a_ps)
5c56e43b0e0f0d0ba668e1149477902ef8e12289b47f78a789262ffca6aefb1c	from sympy.core import Basic, Expr from sympy.core.numbers import oo from sympy.core.symbol import symbols from sympy.multipledispatch import Dispatcher from sympy.sets.setexpr import set_mul from sympy.sets.sets import Interval, Set _x, _y = symbols("x y") _set_mul = Dispatcher('_set_mul') _set_div = Dispatcher('_set_div') @_set_mul.register(Basic, Basic) def _(x, y): return None @_set_mul.register(Set, Set) def _(x, y): return None @_set_mul.register(Expr, Expr) def _(x, y): return xy @_set_mul.register(Interval, Interval) def _(x, y): """ Multiplications in interval arithmetic https://en.wikipedia.org/wiki/Interval_arithmetic """ # TODO: some intervals containing 0 and oo will fail as 0oo returns nan. comvals = ( (x.start * y.start, bool(x.left_open or y.left_open)), (x.start * y.end, bool(x.left_open or y.right_open)), (x.end * y.start, bool(x.right_open or y.left_open)), (x.end * y.end, bool(x.right_open or y.right_open)), ) # TODO: handle symbolic intervals minval, minopen = min(comvals) maxval, maxopen = max(comvals) return Interval( minval, maxval, minopen, maxopen ) @_set_div.register(Basic, Basic) def _(x, y): return None @_set_div.register(Expr, Expr) def _(x, y): return x/y @_set_div.register(Set, Set) def _(x, y): return None @_set_div.register(Interval, Interval) def _(x, y): """ Divisions in interval arithmetic https://en.wikipedia.org/wiki/Interval_arithmetic """ if (y.start*y.end).is_negative: return Interval(-oo, oo) if y.start == 0: s2 = oo else: s2 = 1/y.start if y.end == 0: s1 = -oo else: s1 = 1/y.end return set_mul(x, Interval(s1, s2, y.right_open, y.left_open))
18d3246df33abce22aaeb6a1273f302e5ed013dc7e8ce594694b18ee79113de5	from sympy.concrete.summations import Sum from sympy.core.add import Add from sympy.core.function import Lambda from sympy.core.numbers import (Float, I, Rational, nan, oo, pi, zoo) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.core.symbol import (Symbol, symbols) from sympy.core.sympify import sympify from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt) from sympy.functions.elementary.piecewise import Piecewise from sympy.functions.elementary.trigonometric import (cos, sin) from sympy.logic.boolalg import (false, true) from sympy.matrices.dense import Matrix from sympy.polys.rootoftools import rootof from sympy.sets.contains import Contains from sympy.sets.fancysets import (ImageSet, Range) from sympy.sets.sets import (Complement, DisjointUnion, FiniteSet, Intersection, Interval, ProductSet, Set, SymmetricDifference, Union, imageset) from mpmath import mpi from sympy.core.expr import unchanged from sympy.core.relational import Eq, Ne, Le, Lt, LessThan from sympy.logic import And, Or, Xor from sympy.testing.pytest import raises, XFAIL, warns_deprecated_sympy from sympy.abc import x, y, z, m, n EmptySet = S.EmptySet def test_imageset(): ints = S.Integers assert imageset(x, x - 1, S.Naturals) is S.Naturals0 assert imageset(x, x + 1, S.Naturals0) is S.Naturals assert imageset(x, abs(x), S.Naturals0) is S.Naturals0 assert imageset(x, abs(x), S.Naturals) is S.Naturals assert imageset(x, abs(x), S.Integers) is S.Naturals0 # issue 16878a r = symbols('r', real=True) assert imageset(x, (x, x), S.Reals)._contains((1, r)) == None assert imageset(x, (x, x), S.Reals)._contains((1, 2)) == False assert (r, r) in imageset(x, (x, x), S.Reals) assert 1 + I in imageset(x, x + I, S.Reals) assert {1} not in imageset(x, (x,), S.Reals) assert (1, 1) not in imageset(x, (x,), S.Reals) raises(TypeError, lambda: imageset(x, ints)) raises(ValueError, lambda: imageset(x, y, z, ints)) raises(ValueError, lambda: imageset(Lambda(x, cos(x)), y)) assert (1, 2) in imageset(Lambda((x, y), (x, y)), ints, ints) raises(ValueError, lambda: imageset(Lambda(x, x), ints, ints)) assert imageset(cos, ints) == ImageSet(Lambda(x, cos(x)), ints) def f(x): return cos(x) assert imageset(f, ints) == imageset(x, cos(x), ints) f = lambda x: cos(x) assert imageset(f, ints) == ImageSet(Lambda(x, cos(x)), ints) assert imageset(x, 1, ints) == FiniteSet(1) assert imageset(x, y, ints) == {y} assert imageset((x, y), (1, z), ints, S.Reals) == {(1, z)} clash = Symbol('x', integer=true) assert (str(imageset(lambda x: x + clash, Interval(-2, 1)).lamda.expr) in ('x0 + x', 'x + x0')) x1, x2 = symbols("x1, x2") assert imageset(lambda x, y: Add(x, y), Interval(1, 2), Interval(2, 3)).dummy_eq( ImageSet(Lambda((x1, x2), x1 + x2), Interval(1, 2), Interval(2, 3))) def test_is_empty(): for s in [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals, S.UniversalSet]: assert s.is_empty is False assert S.EmptySet.is_empty is True def test_is_finiteset(): for s in [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals, S.UniversalSet]: assert s.is_finite_set is False assert S.EmptySet.is_finite_set is True assert FiniteSet(1, 2).is_finite_set is True assert Interval(1, 2).is_finite_set is False assert Interval(x, y).is_finite_set is None assert ProductSet(FiniteSet(1), FiniteSet(2)).is_finite_set is True assert ProductSet(FiniteSet(1), Interval(1, 2)).is_finite_set is False assert ProductSet(FiniteSet(1), Interval(x, y)).is_finite_set is None assert Union(Interval(0, 1), Interval(2, 3)).is_finite_set is False assert Union(FiniteSet(1), Interval(2, 3)).is_finite_set is False assert Union(FiniteSet(1), FiniteSet(2)).is_finite_set is True assert Union(FiniteSet(1), Interval(x, y)).is_finite_set is None assert Intersection(Interval(x, y), FiniteSet(1)).is_finite_set is True assert Intersection(Interval(x, y), Interval(1, 2)).is_finite_set is None assert Intersection(FiniteSet(x), FiniteSet(y)).is_finite_set is True assert Complement(FiniteSet(1), Interval(x, y)).is_finite_set is True assert Complement(Interval(x, y), FiniteSet(1)).is_finite_set is None assert Complement(Interval(1, 2), FiniteSet(x)).is_finite_set is False assert DisjointUnion(Interval(-5, 3), FiniteSet(x, y)).is_finite_set is False assert DisjointUnion(S.EmptySet, FiniteSet(x, y), S.EmptySet).is_finite_set is True def test_deprecated_is_EmptySet(): with warns_deprecated_sympy(): S.EmptySet.is_EmptySet def test_interval_arguments(): assert Interval(0, oo) == Interval(0, oo, False, True) assert Interval(0, oo).right_open is true assert Interval(-oo, 0) == Interval(-oo, 0, True, False) assert Interval(-oo, 0).left_open is true assert Interval(oo, -oo) == S.EmptySet assert Interval(oo, oo) == S.EmptySet assert Interval(-oo, -oo) == S.EmptySet assert Interval(oo, x) == S.EmptySet assert Interval(oo, oo) == S.EmptySet assert Interval(x, -oo) == S.EmptySet assert Interval(x, x) == {x} assert isinstance(Interval(1, 1), FiniteSet) e = Sum(x, (x, 1, 3)) assert isinstance(Interval(e, e), FiniteSet) assert Interval(1, 0) == S.EmptySet assert Interval(1, 1).measure == 0 assert Interval(1, 1, False, True) == S.EmptySet assert Interval(1, 1, True, False) == S.EmptySet assert Interval(1, 1, True, True) == S.EmptySet assert isinstance(Interval(0, Symbol('a')), Interval) assert Interval(Symbol('a', positive=True), 0) == S.EmptySet raises(ValueError, lambda: Interval(0, S.ImaginaryUnit)) raises(ValueError, lambda: Interval(0, Symbol('z', extended_real=False))) raises(ValueError, lambda: Interval(x, x + S.ImaginaryUnit)) raises(NotImplementedError, lambda: Interval(0, 1, And(x, y))) raises(NotImplementedError, lambda: Interval(0, 1, False, And(x, y))) raises(NotImplementedError, lambda: Interval(0, 1, z, And(x, y))) def test_interval_symbolic_end_points(): a = Symbol('a', real=True) assert Union(Interval(0, a), Interval(0, 3)).sup == Max(a, 3) assert Union(Interval(a, 0), Interval(-3, 0)).inf == Min(-3, a) assert Interval(0, a).contains(1) == LessThan(1, a) def test_interval_is_empty(): x, y = symbols('x, y') r = Symbol('r', real=True) p = Symbol('p', positive=True) n = Symbol('n', negative=True) nn = Symbol('nn', nonnegative=True) assert Interval(1, 2).is_empty == False assert Interval(3, 3).is_empty == False # FiniteSet assert Interval(r, r).is_empty == False # FiniteSet assert Interval(r, r + nn).is_empty == False assert Interval(x, x).is_empty == False assert Interval(1, oo).is_empty == False assert Interval(-oo, oo).is_empty == False assert Interval(-oo, 1).is_empty == False assert Interval(x, y).is_empty == None assert Interval(r, oo).is_empty == False # real implies finite assert Interval(n, 0).is_empty == False assert Interval(n, 0, left_open=True).is_empty == False assert Interval(p, 0).is_empty == True # EmptySet assert Interval(nn, 0).is_empty == None assert Interval(n, p).is_empty == False assert Interval(0, p, left_open=True).is_empty == False assert Interval(0, p, right_open=True).is_empty == False assert Interval(0, nn, left_open=True).is_empty == None assert Interval(0, nn, right_open=True).is_empty == None def test_union(): assert Union(Interval(1, 2), Interval(2, 3)) == Interval(1, 3) assert Union(Interval(1, 2), Interval(2, 3, True)) == Interval(1, 3) assert Union(Interval(1, 3), Interval(2, 4)) == Interval(1, 4) assert Union(Interval(1, 2), Interval(1, 3)) == Interval(1, 3) assert Union(Interval(1, 3), Interval(1, 2)) == Interval(1, 3) assert Union(Interval(1, 3, False, True), Interval(1, 2)) == \ Interval(1, 3, False, True) assert Union(Interval(1, 3), Interval(1, 2, False, True)) == Interval(1, 3) assert Union(Interval(1, 2, True), Interval(1, 3)) == Interval(1, 3) assert Union(Interval(1, 2, True), Interval(1, 3, True)) == \ Interval(1, 3, True) assert Union(Interval(1, 2, True), Interval(1, 3, True, True)) == \ Interval(1, 3, True, True) assert Union(Interval(1, 2, True, True), Interval(1, 3, True)) == \ Interval(1, 3, True) assert Union(Interval(1, 3), Interval(2, 3)) == Interval(1, 3) assert Union(Interval(1, 3, False, True), Interval(2, 3)) == \ Interval(1, 3) assert Union(Interval(1, 2, False, True), Interval(2, 3, True)) != \ Interval(1, 3) assert Union(Interval(1, 2), S.EmptySet) == Interval(1, 2) assert Union(S.EmptySet) == S.EmptySet assert Union(Interval(0, 1), [FiniteSet(1.0/n) for n in range(1, 10)]) == \ Interval(0, 1) # issue #18241: x = Symbol('x') assert Union(Interval(0, 1), FiniteSet(1, x)) == Union( Interval(0, 1), FiniteSet(x)) assert unchanged(Union, Interval(0, 1), FiniteSet(2, x)) assert Interval(1, 2).union(Interval(2, 3)) == \ Interval(1, 2) + Interval(2, 3) assert Interval(1, 2).union(Interval(2, 3)) == Interval(1, 3) assert Union(Set()) == Set() assert FiniteSet(1) + FiniteSet(2) + FiniteSet(3) == FiniteSet(1, 2, 3) assert FiniteSet('ham') + FiniteSet('eggs') == FiniteSet('ham', 'eggs') assert FiniteSet(1, 2, 3) + S.EmptySet == FiniteSet(1, 2, 3) assert FiniteSet(1, 2, 3) & FiniteSet(2, 3, 4) == FiniteSet(2, 3) assert FiniteSet(1, 2, 3) \| FiniteSet(2, 3, 4) == FiniteSet(1, 2, 3, 4) assert FiniteSet(1, 2, 3) & S.EmptySet == S.EmptySet assert FiniteSet(1, 2, 3) \| S.EmptySet == FiniteSet(1, 2, 3) x = Symbol("x") y = Symbol("y") z = Symbol("z") assert S.EmptySet \| FiniteSet(x, FiniteSet(y, z)) == \ FiniteSet(x, FiniteSet(y, z)) # Test that Intervals and FiniteSets play nicely assert Interval(1, 3) + FiniteSet(2) == Interval(1, 3) assert Interval(1, 3, True, True) + FiniteSet(3) == \ Interval(1, 3, True, False) X = Interval(1, 3) + FiniteSet(5) Y = Interval(1, 2) + FiniteSet(3) XandY = X.intersect(Y) assert 2 in X and 3 in X and 3 in XandY assert XandY.is_subset(X) and XandY.is_subset(Y) raises(TypeError, lambda: Union(1, 2, 3)) assert X.is_iterable is False # issue 7843 assert Union(S.EmptySet, FiniteSet(-sqrt(-I), sqrt(-I))) == \ FiniteSet(-sqrt(-I), sqrt(-I)) assert Union(S.Reals, S.Integers) == S.Reals def test_union_iter(): # Use Range because it is ordered u = Union(Range(3), Range(5), Range(4), evaluate=False) # Round robin assert list(u) == [0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4] def test_union_is_empty(): assert (Interval(x, y) + FiniteSet(1)).is_empty == False assert (Interval(x, y) + Interval(-x, y)).is_empty == None def test_difference(): assert Interval(1, 3) - Interval(1, 2) == Interval(2, 3, True) assert Interval(1, 3) - Interval(2, 3) == Interval(1, 2, False, True) assert Interval(1, 3, True) - Interval(2, 3) == Interval(1, 2, True, True) assert Interval(1, 3, True) - Interval(2, 3, True) == \ Interval(1, 2, True, False) assert Interval(0, 2) - FiniteSet(1) == \ Union(Interval(0, 1, False, True), Interval(1, 2, True, False)) # issue #18119 assert S.Reals - FiniteSet(I) == S.Reals assert S.Reals - FiniteSet(-I, I) == S.Reals assert Interval(0, 10) - FiniteSet(-I, I) == Interval(0, 10) assert Interval(0, 10) - FiniteSet(1, I) == Union( Interval.Ropen(0, 1), Interval.Lopen(1, 10)) assert S.Reals - FiniteSet(1, 2 + I, x, y2) == Complement( Union(Interval.open(-oo, 1), Interval.open(1, oo)), FiniteSet(x, y2), evaluate=False) assert FiniteSet(1, 2, 3) - FiniteSet(2) == FiniteSet(1, 3) assert FiniteSet('ham', 'eggs') - FiniteSet('eggs') == FiniteSet('ham') assert FiniteSet(1, 2, 3, 4) - Interval(2, 10, True, False) == \ FiniteSet(1, 2) assert FiniteSet(1, 2, 3, 4) - S.EmptySet == FiniteSet(1, 2, 3, 4) assert Union(Interval(0, 2), FiniteSet(2, 3, 4)) - Interval(1, 3) == \ Union(Interval(0, 1, False, True), FiniteSet(4)) assert -1 in S.Reals - S.Naturals def test_Complement(): A = FiniteSet(1, 3, 4) B = FiniteSet(3, 4) C = Interval(1, 3) D = Interval(1, 2) assert Complement(A, B, evaluate=False).is_iterable is True assert Complement(A, C, evaluate=False).is_iterable is True assert Complement(C, D, evaluate=False).is_iterable is None assert FiniteSet(Complement(A, B, evaluate=False)) == FiniteSet(1) assert FiniteSet(Complement(A, C, evaluate=False)) == FiniteSet(4) raises(TypeError, lambda: FiniteSet(Complement(C, A, evaluate=False))) assert Complement(Interval(1, 3), Interval(1, 2)) == Interval(2, 3, True) assert Complement(FiniteSet(1, 3, 4), FiniteSet(3, 4)) == FiniteSet(1) assert Complement(Union(Interval(0, 2), FiniteSet(2, 3, 4)), Interval(1, 3)) == \ Union(Interval(0, 1, False, True), FiniteSet(4)) assert 3 not in Complement(Interval(0, 5), Interval(1, 4), evaluate=False) assert -1 in Complement(S.Reals, S.Naturals, evaluate=False) assert 1 not in Complement(S.Reals, S.Naturals, evaluate=False) assert Complement(S.Integers, S.UniversalSet) == EmptySet assert S.UniversalSet.complement(S.Integers) == EmptySet assert (0 not in S.Reals.intersect(S.Integers - FiniteSet(0))) assert S.EmptySet - S.Integers == S.EmptySet assert (S.Integers - FiniteSet(0)) - FiniteSet(1) == S.Integers - FiniteSet(0, 1) assert S.Reals - Union(S.Naturals, FiniteSet(pi)) == \ Intersection(S.Reals - S.Naturals, S.Reals - FiniteSet(pi)) # issue 12712 assert Complement(FiniteSet(x, y, 2), Interval(-10, 10)) == \ Complement(FiniteSet(x, y), Interval(-10, 10)) A = FiniteSet(symbols('a:c')) B = FiniteSet(symbols('d:f')) assert unchanged(Complement, ProductSet(A, A), B) A2 = ProductSet(A, A) B3 = ProductSet(B, B, B) assert A2 - B3 == A2 assert B3 - A2 == B3 def test_set_operations_nonsets(): '''Tests that e.g. FiniteSet(1) * 2 raises TypeError''' ops = [ lambda a, b: a + b, lambda a, b: a - b, lambda a, b: a * b, lambda a, b: a / b, lambda a, b: a // b, lambda a, b: a \| b, lambda a, b: a & b, lambda a, b: a ^ b, # FiniteSet(1) 2 gives a ProductSet #lambda a, b: a b, ] Sx = FiniteSet(x) Sy = FiniteSet(y) sets = [ {1}, FiniteSet(1), Interval(1, 2), Union(Sx, Interval(1, 2)), Intersection(Sx, Sy), Complement(Sx, Sy), ProductSet(Sx, Sy), S.EmptySet, ] nums = [0, 1, 2, S(0), S(1), S(2)] for si in sets: for ni in nums: for op in ops: raises(TypeError, lambda : op(si, ni)) raises(TypeError, lambda : op(ni, si)) raises(TypeError, lambda: si object()) raises(TypeError, lambda: si {1}) def test_complement(): assert Complement({1, 2}, {1}) == {2} assert Interval(0, 1).complement(S.Reals) == \ Union(Interval(-oo, 0, True, True), Interval(1, oo, True, True)) assert Interval(0, 1, True, False).complement(S.Reals) == \ Union(Interval(-oo, 0, True, False), Interval(1, oo, True, True)) assert Interval(0, 1, False, True).complement(S.Reals) == \ Union(Interval(-oo, 0, True, True), Interval(1, oo, False, True)) assert Interval(0, 1, True, True).complement(S.Reals) == \ Union(Interval(-oo, 0, True, False), Interval(1, oo, False, True)) assert S.UniversalSet.complement(S.EmptySet) == S.EmptySet assert S.UniversalSet.complement(S.Reals) == S.EmptySet assert S.UniversalSet.complement(S.UniversalSet) == S.EmptySet assert S.EmptySet.complement(S.Reals) == S.Reals assert Union(Interval(0, 1), Interval(2, 3)).complement(S.Reals) == \ Union(Interval(-oo, 0, True, True), Interval(1, 2, True, True), Interval(3, oo, True, True)) assert FiniteSet(0).complement(S.Reals) == \ Union(Interval(-oo, 0, True, True), Interval(0, oo, True, True)) assert (FiniteSet(5) + Interval(S.NegativeInfinity, 0)).complement(S.Reals) == \ Interval(0, 5, True, True) + Interval(5, S.Infinity, True, True) assert FiniteSet(1, 2, 3).complement(S.Reals) == \ Interval(S.NegativeInfinity, 1, True, True) + \ Interval(1, 2, True, True) + Interval(2, 3, True, True) +\ Interval(3, S.Infinity, True, True) assert FiniteSet(x).complement(S.Reals) == Complement(S.Reals, FiniteSet(x)) assert FiniteSet(0, x).complement(S.Reals) == Complement(Interval(-oo, 0, True, True) + Interval(0, oo, True, True) , FiniteSet(x), evaluate=False) square = Interval(0, 1) * Interval(0, 1) notsquare = square.complement(S.RealsS.Reals) assert all(pt in square for pt in [(0, 0), (.5, .5), (1, 0), (1, 1)]) assert not any( pt in notsquare for pt in [(0, 0), (.5, .5), (1, 0), (1, 1)]) assert not any(pt in square for pt in [(-1, 0), (1.5, .5), (10, 10)]) assert all(pt in notsquare for pt in [(-1, 0), (1.5, .5), (10, 10)]) def test_intersect1(): assert all(S.Integers.intersection(i) is i for i in (S.Naturals, S.Naturals0)) assert all(i.intersection(S.Integers) is i for i in (S.Naturals, S.Naturals0)) s = S.Naturals0 assert S.Naturals.intersection(s) is S.Naturals assert s.intersection(S.Naturals) is S.Naturals x = Symbol('x') assert Interval(0, 2).intersect(Interval(1, 2)) == Interval(1, 2) assert Interval(0, 2).intersect(Interval(1, 2, True)) == \ Interval(1, 2, True) assert Interval(0, 2, True).intersect(Interval(1, 2)) == \ Interval(1, 2, False, False) assert Interval(0, 2, True, True).intersect(Interval(1, 2)) == \ Interval(1, 2, False, True) assert Interval(0, 2).intersect(Union(Interval(0, 1), Interval(2, 3))) == \ Union(Interval(0, 1), Interval(2, 2)) assert FiniteSet(1, 2).intersect(FiniteSet(1, 2, 3)) == FiniteSet(1, 2) assert FiniteSet(1, 2, x).intersect(FiniteSet(x)) == FiniteSet(x) assert FiniteSet('ham', 'eggs').intersect(FiniteSet('ham')) == \ FiniteSet('ham') assert FiniteSet(1, 2, 3, 4, 5).intersect(S.EmptySet) == S.EmptySet assert Interval(0, 5).intersect(FiniteSet(1, 3)) == FiniteSet(1, 3) assert Interval(0, 1, True, True).intersect(FiniteSet(1)) == S.EmptySet assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(1, 2)) == \ Union(Interval(1, 1), Interval(2, 2)) assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(0, 2)) == \ Union(Interval(0, 1), Interval(2, 2)) assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(1, 2, True, True)) == \ S.EmptySet assert Union(Interval(0, 1), Interval(2, 3)).intersect(S.EmptySet) == \ S.EmptySet assert Union(Interval(0, 5), FiniteSet('ham')).intersect(FiniteSet(2, 3, 4, 5, 6)) == \ Intersection(FiniteSet(2, 3, 4, 5, 6), Union(FiniteSet('ham'), Interval(0, 5))) assert Intersection(FiniteSet(1, 2, 3), Interval(2, x), Interval(3, y)) == \ Intersection(FiniteSet(3), Interval(2, x), Interval(3, y), evaluate=False) assert Intersection(FiniteSet(1, 2), Interval(0, 3), Interval(x, y)) == \ Intersection({1, 2}, Interval(x, y), evaluate=False) assert Intersection(FiniteSet(1, 2, 4), Interval(0, 3), Interval(x, y)) == \ Intersection({1, 2}, Interval(x, y), evaluate=False) # XXX: Is the real=True necessary here? # https://github.com/sympy/sympy/issues/17532 m, n = symbols('m, n', real=True) assert Intersection(FiniteSet(m), FiniteSet(m, n), Interval(m, m+1)) == \ FiniteSet(m) # issue 8217 assert Intersection(FiniteSet(x), FiniteSet(y)) == \ Intersection(FiniteSet(x), FiniteSet(y), evaluate=False) assert FiniteSet(x).intersect(S.Reals) == \ Intersection(S.Reals, FiniteSet(x), evaluate=False) # tests for the intersection alias assert Interval(0, 5).intersection(FiniteSet(1, 3)) == FiniteSet(1, 3) assert Interval(0, 1, True, True).intersection(FiniteSet(1)) == S.EmptySet assert Union(Interval(0, 1), Interval(2, 3)).intersection(Interval(1, 2)) == \ Union(Interval(1, 1), Interval(2, 2)) def test_intersection(): # iterable i = Intersection(FiniteSet(1, 2, 3), Interval(2, 5), evaluate=False) assert i.is_iterable assert set(i) == {S(2), S(3)} # challenging intervals x = Symbol('x', real=True) i = Intersection(Interval(0, 3), Interval(x, 6)) assert (5 in i) is False raises(TypeError, lambda: 2 in i) # Singleton special cases assert Intersection(Interval(0, 1), S.EmptySet) == S.EmptySet assert Intersection(Interval(-oo, oo), Interval(-oo, x)) == Interval(-oo, x) # Products line = Interval(0, 5) i = Intersection(line2, line3, evaluate=False) assert (2, 2) not in i assert (2, 2, 2) not in i raises(TypeError, lambda: list(i)) a = Intersection(Intersection(S.Integers, S.Naturals, evaluate=False), S.Reals, evaluate=False) assert a._argset == frozenset([Intersection(S.Naturals, S.Integers, evaluate=False), S.Reals]) assert Intersection(S.Complexes, FiniteSet(S.ComplexInfinity)) == S.EmptySet # issue 12178 assert Intersection() == S.UniversalSet # issue 16987 assert Intersection({1}, {1}, {x}) == Intersection({1}, {x}) def test_issue_9623(): n = Symbol('n') a = S.Reals b = Interval(0, oo) c = FiniteSet(n) assert Intersection(a, b, c) == Intersection(b, c) assert Intersection(Interval(1, 2), Interval(3, 4), FiniteSet(n)) == EmptySet def test_is_disjoint(): assert Interval(0, 2).is_disjoint(Interval(1, 2)) == False assert Interval(0, 2).is_disjoint(Interval(3, 4)) == True def test_ProductSet__len__(): A = FiniteSet(1, 2) B = FiniteSet(1, 2, 3) assert ProductSet(A).__len__() == 2 assert ProductSet(A).__len__() is not S(2) assert ProductSet(A, B).__len__() == 6 assert ProductSet(A, B).__len__() is not S(6) def test_ProductSet(): # ProductSet is always a set of Tuples assert ProductSet(S.Reals) == S.Reals * 1 assert ProductSet(S.Reals, S.Reals) == S.Reals 2 assert ProductSet(S.Reals, S.Reals, S.Reals) == S.Reals 3 assert ProductSet(S.Reals) != S.Reals assert ProductSet(S.Reals, S.Reals) == S.Reals * S.Reals assert ProductSet(S.Reals, S.Reals, S.Reals) != S.Reals * S.Reals * S.Reals assert ProductSet(S.Reals, S.Reals, S.Reals) == (S.Reals * S.Reals * S.Reals).flatten() assert 1 not in ProductSet(S.Reals) assert (1,) in ProductSet(S.Reals) assert 1 not in ProductSet(S.Reals, S.Reals) assert (1, 2) in ProductSet(S.Reals, S.Reals) assert (1, I) not in ProductSet(S.Reals, S.Reals) assert (1, 2, 3) in ProductSet(S.Reals, S.Reals, S.Reals) assert (1, 2, 3) in S.Reals ** 3 assert (1, 2, 3) not in S.Reals * S.Reals * S.Reals assert ((1, 2), 3) in S.Reals * S.Reals * S.Reals assert (1, (2, 3)) not in S.Reals * S.Reals * S.Reals assert (1, (2, 3)) in S.Reals * (S.Reals * S.Reals) assert ProductSet() == FiniteSet(()) assert ProductSet(S.Reals, S.EmptySet) == S.EmptySet # See GH-17458 for ni in range(5): Rn = ProductSet((S.Reals,) ni) assert (1,) * ni in Rn assert 1 not in Rn assert (S.Reals * S.Reals) * S.Reals != S.Reals * (S.Reals * S.Reals) S1 = S.Reals S2 = S.Integers x1 = pi x2 = 3 assert x1 in S1 assert x2 in S2 assert (x1, x2) in S1 * S2 S3 = S1 * S2 x3 = (x1, x2) assert x3 in S3 assert (x3, x3) in S3 * S3 assert x3 + x3 not in S3 * S3 raises(ValueError, lambda: S.Reals*-1) with warns_deprecated_sympy(): ProductSet(FiniteSet(s) for s in range(2)) raises(TypeError, lambda: ProductSet(None)) S1 = FiniteSet(1, 2) S2 = FiniteSet(3, 4) S3 = ProductSet(S1, S2) assert (S3.as_relational(x, y) == And(S1.as_relational(x), S2.as_relational(y)) == And(Or(Eq(x, 1), Eq(x, 2)), Or(Eq(y, 3), Eq(y, 4)))) raises(ValueError, lambda: S3.as_relational(x)) raises(ValueError, lambda: S3.as_relational(x, 1)) raises(ValueError, lambda: ProductSet(Interval(0, 1)).as_relational(x, y)) Z2 = ProductSet(S.Integers, S.Integers) assert Z2.contains((1, 2)) is S.true assert Z2.contains((1,)) is S.false assert Z2.contains(x) == Contains(x, Z2, evaluate=False) assert Z2.contains(x).subs(x, 1) is S.false assert Z2.contains((x, 1)).subs(x, 2) is S.true assert Z2.contains((x, y)) == Contains((x, y), Z2, evaluate=False) assert unchanged(Contains, (x, y), Z2) assert Contains((1, 2), Z2) is S.true def test_ProductSet_of_single_arg_is_not_arg(): assert unchanged(ProductSet, Interval(0, 1)) assert unchanged(ProductSet, ProductSet(Interval(0, 1))) def test_ProductSet_is_empty(): assert ProductSet(S.Integers, S.Reals).is_empty == False assert ProductSet(Interval(x, 1), S.Reals).is_empty == None def test_interval_subs(): a = Symbol('a', real=True) assert Interval(0, a).subs(a, 2) == Interval(0, 2) assert Interval(a, 0).subs(a, 2) == S.EmptySet def test_interval_to_mpi(): assert Interval(0, 1).to_mpi() == mpi(0, 1) assert Interval(0, 1, True, False).to_mpi() == mpi(0, 1) assert type(Interval(0, 1).to_mpi()) == type(mpi(0, 1)) def test_set_evalf(): assert Interval(S(11)/64, S.Half).evalf() == Interval( Float('0.171875'), Float('0.5')) assert Interval(x, S.Half, right_open=True).evalf() == Interval( x, Float('0.5'), right_open=True) assert Interval(-oo, S.Half).evalf() == Interval(-oo, Float('0.5')) assert FiniteSet(2, x).evalf() == FiniteSet(Float('2.0'), x) def test_measure(): a = Symbol('a', real=True) assert Interval(1, 3).measure == 2 assert Interval(0, a).measure == a assert Interval(1, a).measure == a - 1 assert Union(Interval(1, 2), Interval(3, 4)).measure == 2 assert Union(Interval(1, 2), Interval(3, 4), FiniteSet(5, 6, 7)).measure \ == 2 assert FiniteSet(1, 2, oo, a, -oo, -5).measure == 0 assert S.EmptySet.measure == 0 square = Interval(0, 10) Interval(0, 10) offsetsquare = Interval(5, 15) * Interval(5, 15) band = Interval(-oo, oo) * Interval(2, 4) assert square.measure == offsetsquare.measure == 100 assert (square + offsetsquare).measure == 175 # there is some overlap assert (square - offsetsquare).measure == 75 assert (square * FiniteSet(1, 2, 3)).measure == 0 assert (square.intersect(band)).measure == 20 assert (square + band).measure is oo assert (band * FiniteSet(1, 2, 3)).measure is nan def test_is_subset(): assert Interval(0, 1).is_subset(Interval(0, 2)) is True assert Interval(0, 3).is_subset(Interval(0, 2)) is False assert Interval(0, 1).is_subset(FiniteSet(0, 1)) is False assert FiniteSet(1, 2).is_subset(FiniteSet(1, 2, 3, 4)) assert FiniteSet(4, 5).is_subset(FiniteSet(1, 2, 3, 4)) is False assert FiniteSet(1).is_subset(Interval(0, 2)) assert FiniteSet(1, 2).is_subset(Interval(0, 2, True, True)) is False assert (Interval(1, 2) + FiniteSet(3)).is_subset( Interval(0, 2, False, True) + FiniteSet(2, 3)) assert Interval(3, 4).is_subset(Union(Interval(0, 1), Interval(2, 5))) is True assert Interval(3, 6).is_subset(Union(Interval(0, 1), Interval(2, 5))) is False assert FiniteSet(1, 2, 3, 4).is_subset(Interval(0, 5)) is True assert S.EmptySet.is_subset(FiniteSet(1, 2, 3)) is True assert Interval(0, 1).is_subset(S.EmptySet) is False assert S.EmptySet.is_subset(S.EmptySet) is True raises(ValueError, lambda: S.EmptySet.is_subset(1)) # tests for the issubset alias assert FiniteSet(1, 2, 3, 4).issubset(Interval(0, 5)) is True assert S.EmptySet.issubset(FiniteSet(1, 2, 3)) is True assert S.Naturals.is_subset(S.Integers) assert S.Naturals0.is_subset(S.Integers) assert FiniteSet(x).is_subset(FiniteSet(y)) is None assert FiniteSet(x).is_subset(FiniteSet(y).subs(y, x)) is True assert FiniteSet(x).is_subset(FiniteSet(y).subs(y, x+1)) is False assert Interval(0, 1).is_subset(Interval(0, 1, left_open=True)) is False assert Interval(-2, 3).is_subset(Union(Interval(-oo, -2), Interval(3, oo))) is False n = Symbol('n', integer=True) assert Range(-3, 4, 1).is_subset(FiniteSet(-10, 10)) is False assert Range(S(10)*100).is_subset(FiniteSet(0, 1, 2)) is False assert Range(6, 0, -2).is_subset(FiniteSet(2, 4, 6)) is True assert Range(1, oo).is_subset(FiniteSet(1, 2)) is False assert Range(-oo, 1).is_subset(FiniteSet(1)) is False assert Range(3).is_subset(FiniteSet(0, 1, n)) is None assert Range(n, n + 2).is_subset(FiniteSet(n, n + 1)) is True assert Range(5).is_subset(Interval(0, 4, right_open=True)) is False #issue 19513 assert imageset(Lambda(n, 1/n), S.Integers).is_subset(S.Reals) is None def test_is_proper_subset(): assert Interval(0, 1).is_proper_subset(Interval(0, 2)) is True assert Interval(0, 3).is_proper_subset(Interval(0, 2)) is False assert S.EmptySet.is_proper_subset(FiniteSet(1, 2, 3)) is True raises(ValueError, lambda: Interval(0, 1).is_proper_subset(0)) def test_is_superset(): assert Interval(0, 1).is_superset(Interval(0, 2)) == False assert Interval(0, 3).is_superset(Interval(0, 2)) assert FiniteSet(1, 2).is_superset(FiniteSet(1, 2, 3, 4)) == False assert FiniteSet(4, 5).is_superset(FiniteSet(1, 2, 3, 4)) == False assert FiniteSet(1).is_superset(Interval(0, 2)) == False assert FiniteSet(1, 2).is_superset(Interval(0, 2, True, True)) == False assert (Interval(1, 2) + FiniteSet(3)).is_superset( Interval(0, 2, False, True) + FiniteSet(2, 3)) == False assert Interval(3, 4).is_superset(Union(Interval(0, 1), Interval(2, 5))) == False assert FiniteSet(1, 2, 3, 4).is_superset(Interval(0, 5)) == False assert S.EmptySet.is_superset(FiniteSet(1, 2, 3)) == False assert Interval(0, 1).is_superset(S.EmptySet) == True assert S.EmptySet.is_superset(S.EmptySet) == True raises(ValueError, lambda: S.EmptySet.is_superset(1)) # tests for the issuperset alias assert Interval(0, 1).issuperset(S.EmptySet) == True assert S.EmptySet.issuperset(S.EmptySet) == True def test_is_proper_superset(): assert Interval(0, 1).is_proper_superset(Interval(0, 2)) is False assert Interval(0, 3).is_proper_superset(Interval(0, 2)) is True assert FiniteSet(1, 2, 3).is_proper_superset(S.EmptySet) is True raises(ValueError, lambda: Interval(0, 1).is_proper_superset(0)) def test_contains(): assert Interval(0, 2).contains(1) is S.true assert Interval(0, 2).contains(3) is S.false assert Interval(0, 2, True, False).contains(0) is S.false assert Interval(0, 2, True, False).contains(2) is S.true assert Interval(0, 2, False, True).contains(0) is S.true assert Interval(0, 2, False, True).contains(2) is S.false assert Interval(0, 2, True, True).contains(0) is S.false assert Interval(0, 2, True, True).contains(2) is S.false assert (Interval(0, 2) in Interval(0, 2)) is False assert FiniteSet(1, 2, 3).contains(2) is S.true assert FiniteSet(1, 2, Symbol('x')).contains(Symbol('x')) is S.true assert FiniteSet(y)._contains(x) is None raises(TypeError, lambda: x in FiniteSet(y)) assert FiniteSet({x, y})._contains({x}) is None assert FiniteSet({x, y}).subs(y, x)._contains({x}) is True assert FiniteSet({x, y}).subs(y, x+1)._contains({x}) is False # issue 8197 from sympy.abc import a, b assert isinstance(FiniteSet(b).contains(-a), Contains) assert isinstance(FiniteSet(b).contains(a), Contains) assert isinstance(FiniteSet(a).contains(1), Contains) raises(TypeError, lambda: 1 in FiniteSet(a)) # issue 8209 rad1 = Pow(Pow(2, Rational(1, 3)) - 1, Rational(1, 3)) rad2 = Pow(Rational(1, 9), Rational(1, 3)) - Pow(Rational(2, 9), Rational(1, 3)) + Pow(Rational(4, 9), Rational(1, 3)) s1 = FiniteSet(rad1) s2 = FiniteSet(rad2) assert s1 - s2 == S.EmptySet items = [1, 2, S.Infinity, S('ham'), -1.1] fset = FiniteSet(items) assert all(item in fset for item in items) assert all(fset.contains(item) is S.true for item in items) assert Union(Interval(0, 1), Interval(2, 5)).contains(3) is S.true assert Union(Interval(0, 1), Interval(2, 5)).contains(6) is S.false assert Union(Interval(0, 1), FiniteSet(2, 5)).contains(3) is S.false assert S.EmptySet.contains(1) is S.false assert FiniteSet(rootof(x3 + x - 1, 0)).contains(S.Infinity) is S.false assert rootof(x5 + x3 + 1, 0) in S.Reals assert not rootof(x5 + x3 + 1, 1) in S.Reals # non-bool results assert Union(Interval(1, 2), Interval(3, 4)).contains(x) == \ Or(And(S.One <= x, x <= 2), And(S(3) <= x, x <= 4)) assert Intersection(Interval(1, x), Interval(2, 3)).contains(y) == \ And(y <= 3, y <= x, S.One <= y, S(2) <= y) assert (S.Complexes).contains(S.ComplexInfinity) == S.false def test_interval_symbolic(): x = Symbol('x') e = Interval(0, 1) assert e.contains(x) == And(S.Zero <= x, x <= 1) raises(TypeError, lambda: x in e) e = Interval(0, 1, True, True) assert e.contains(x) == And(S.Zero < x, x < 1) c = Symbol('c', real=False) assert Interval(x, x + 1).contains(c) == False e = Symbol('e', extended_real=True) assert Interval(-oo, oo).contains(e) == And( S.NegativeInfinity < e, e < S.Infinity) def test_union_contains(): x = Symbol('x') i1 = Interval(0, 1) i2 = Interval(2, 3) i3 = Union(i1, i2) assert i3.as_relational(x) == Or(And(S.Zero <= x, x <= 1), And(S(2) <= x, x <= 3)) raises(TypeError, lambda: x in i3) e = i3.contains(x) assert e == i3.as_relational(x) assert e.subs(x, -0.5) is false assert e.subs(x, 0.5) is true assert e.subs(x, 1.5) is false assert e.subs(x, 2.5) is true assert e.subs(x, 3.5) is false U = Interval(0, 2, True, True) + Interval(10, oo) + FiniteSet(-1, 2, 5, 6) assert all(el not in U for el in [0, 4, -oo]) assert all(el in U for el in [2, 5, 10]) def test_is_number(): assert Interval(0, 1).is_number is False assert Set().is_number is False def test_Interval_is_left_unbounded(): assert Interval(3, 4).is_left_unbounded is False assert Interval(-oo, 3).is_left_unbounded is True assert Interval(Float("-inf"), 3).is_left_unbounded is True def test_Interval_is_right_unbounded(): assert Interval(3, 4).is_right_unbounded is False assert Interval(3, oo).is_right_unbounded is True assert Interval(3, Float("+inf")).is_right_unbounded is True def test_Interval_as_relational(): x = Symbol('x') assert Interval(-1, 2, False, False).as_relational(x) == \ And(Le(-1, x), Le(x, 2)) assert Interval(-1, 2, True, False).as_relational(x) == \ And(Lt(-1, x), Le(x, 2)) assert Interval(-1, 2, False, True).as_relational(x) == \ And(Le(-1, x), Lt(x, 2)) assert Interval(-1, 2, True, True).as_relational(x) == \ And(Lt(-1, x), Lt(x, 2)) assert Interval(-oo, 2, right_open=False).as_relational(x) == And(Lt(-oo, x), Le(x, 2)) assert Interval(-oo, 2, right_open=True).as_relational(x) == And(Lt(-oo, x), Lt(x, 2)) assert Interval(-2, oo, left_open=False).as_relational(x) == And(Le(-2, x), Lt(x, oo)) assert Interval(-2, oo, left_open=True).as_relational(x) == And(Lt(-2, x), Lt(x, oo)) assert Interval(-oo, oo).as_relational(x) == And(Lt(-oo, x), Lt(x, oo)) x = Symbol('x', real=True) y = Symbol('y', real=True) assert Interval(x, y).as_relational(x) == (x <= y) assert Interval(y, x).as_relational(x) == (y <= x) def test_Finite_as_relational(): x = Symbol('x') y = Symbol('y') assert FiniteSet(1, 2).as_relational(x) == Or(Eq(x, 1), Eq(x, 2)) assert FiniteSet(y, -5).as_relational(x) == Or(Eq(x, y), Eq(x, -5)) def test_Union_as_relational(): x = Symbol('x') assert (Interval(0, 1) + FiniteSet(2)).as_relational(x) == \ Or(And(Le(0, x), Le(x, 1)), Eq(x, 2)) assert (Interval(0, 1, True, True) + FiniteSet(1)).as_relational(x) == \ And(Lt(0, x), Le(x, 1)) assert Or(x < 0, x > 0).as_set().as_relational(x) == \ And((x > -oo), (x < oo), Ne(x, 0)) assert (Interval.Ropen(1, 3) + Interval.Lopen(3, 5) ).as_relational(x) == And((x > 1), (x < 5), Ne(x, 3)) def test_Intersection_as_relational(): x = Symbol('x') assert (Intersection(Interval(0, 1), FiniteSet(2), evaluate=False).as_relational(x) == And(And(Le(0, x), Le(x, 1)), Eq(x, 2))) def test_Complement_as_relational(): x = Symbol('x') expr = Complement(Interval(0, 1), FiniteSet(2), evaluate=False) assert expr.as_relational(x) == \ And(Le(0, x), Le(x, 1), Ne(x, 2)) @XFAIL def test_Complement_as_relational_fail(): x = Symbol('x') expr = Complement(Interval(0, 1), FiniteSet(2), evaluate=False) # XXX This example fails because 0 <= x changes to x >= 0 # during the evaluation. assert expr.as_relational(x) == \ (0 <= x) & (x <= 1) & Ne(x, 2) def test_SymmetricDifference_as_relational(): x = Symbol('x') expr = SymmetricDifference(Interval(0, 1), FiniteSet(2), evaluate=False) assert expr.as_relational(x) == Xor(Eq(x, 2), Le(0, x) & Le(x, 1)) def test_EmptySet(): assert S.EmptySet.as_relational(Symbol('x')) is S.false assert S.EmptySet.intersect(S.UniversalSet) == S.EmptySet assert S.EmptySet.boundary == S.EmptySet def test_finite_basic(): x = Symbol('x') A = FiniteSet(1, 2, 3) B = FiniteSet(3, 4, 5) AorB = Union(A, B) AandB = A.intersect(B) assert A.is_subset(AorB) and B.is_subset(AorB) assert AandB.is_subset(A) assert AandB == FiniteSet(3) assert A.inf == 1 and A.sup == 3 assert AorB.inf == 1 and AorB.sup == 5 assert FiniteSet(x, 1, 5).sup == Max(x, 5) assert FiniteSet(x, 1, 5).inf == Min(x, 1) # issue 7335 assert FiniteSet(S.EmptySet) != S.EmptySet assert FiniteSet(FiniteSet(1, 2, 3)) != FiniteSet(1, 2, 3) assert FiniteSet((1, 2, 3)) != FiniteSet(1, 2, 3) # Ensure a variety of types can exist in a FiniteSet assert FiniteSet((1, 2), A, -5, x, 'eggs', x2) assert (A > B) is False assert (A >= B) is False assert (A < B) is False assert (A <= B) is False assert AorB > A and AorB > B assert AorB >= A and AorB >= B assert A >= A and A <= A assert A >= AandB and B >= AandB assert A > AandB and B > AandB def test_product_basic(): H, T = 'H', 'T' unit_line = Interval(0, 1) d6 = FiniteSet(1, 2, 3, 4, 5, 6) d4 = FiniteSet(1, 2, 3, 4) coin = FiniteSet(H, T) square = unit_line * unit_line assert (0, 0) in square assert 0 not in square assert (H, T) in coin ** 2 assert (.5, .5, .5) in (square * unit_line).flatten() assert ((.5, .5), .5) in square * unit_line assert (H, 3, 3) in (coin * d6 * d6).flatten() assert ((H, 3), 3) in coin * d6 * d6 HH, TT = sympify(H), sympify(T) assert set(coin*2) == {(HH, HH), (HH, TT), (TT, HH), (TT, TT)} assert (d4d4).is_subset(d6d6) assert square.complement(Interval(-oo, oo)Interval(-oo, oo)) == Union( (Interval(-oo, 0, True, True) + Interval(1, oo, True, True))Interval(-oo, oo), Interval(-oo, oo)(Interval(-oo, 0, True, True) + Interval(1, oo, True, True))) assert (Interval(-5, 5)3).is_subset(Interval(-10, 10)3) assert not (Interval(-10, 10)3).is_subset(Interval(-5, 5)3) assert not (Interval(-5, 5)2).is_subset(Interval(-10, 10)3) assert (Interval(.2, .5)FiniteSet(.5)).is_subset(square) # segment in square assert len(coincoincoin) == 8 assert len(S.EmptySetS.EmptySet) == 0 assert len(S.EmptySetcoin) == 0 raises(TypeError, lambda: len(coinInterval(0, 2))) def test_real(): x = Symbol('x', real=True) I = Interval(0, 5) J = Interval(10, 20) A = FiniteSet(1, 2, 30, x, S.Pi) B = FiniteSet(-4, 0) C = FiniteSet(100) D = FiniteSet('Ham', 'Eggs') assert all(s.is_subset(S.Reals) for s in [I, J, A, B, C]) assert not D.is_subset(S.Reals) assert all((a + b).is_subset(S.Reals) for a in [I, J, A, B, C] for b in [I, J, A, B, C]) assert not any((a + D).is_subset(S.Reals) for a in [I, J, A, B, C, D]) assert not (I + A + D).is_subset(S.Reals) def test_supinf(): x = Symbol('x', real=True) y = Symbol('y', real=True) assert (Interval(0, 1) + FiniteSet(2)).sup == 2 assert (Interval(0, 1) + FiniteSet(2)).inf == 0 assert (Interval(0, 1) + FiniteSet(x)).sup == Max(1, x) assert (Interval(0, 1) + FiniteSet(x)).inf == Min(0, x) assert FiniteSet(5, 1, x).sup == Max(5, x) assert FiniteSet(5, 1, x).inf == Min(1, x) assert FiniteSet(5, 1, x, y).sup == Max(5, x, y) assert FiniteSet(5, 1, x, y).inf == Min(1, x, y) assert FiniteSet(5, 1, x, y, S.Infinity, S.NegativeInfinity).sup == \ S.Infinity assert FiniteSet(5, 1, x, y, S.Infinity, S.NegativeInfinity).inf == \ S.NegativeInfinity assert FiniteSet('Ham', 'Eggs').sup == Max('Ham', 'Eggs') def test_universalset(): U = S.UniversalSet x = Symbol('x') assert U.as_relational(x) is S.true assert U.union(Interval(2, 4)) == U assert U.intersect(Interval(2, 4)) == Interval(2, 4) assert U.measure is S.Infinity assert U.boundary == S.EmptySet assert U.contains(0) is S.true def test_Union_of_ProductSets_shares(): line = Interval(0, 2) points = FiniteSet(0, 1, 2) assert Union(line * line, line * points) == line * line def test_Interval_free_symbols(): # issue 6211 assert Interval(0, 1).free_symbols == set() x = Symbol('x', real=True) assert Interval(0, x).free_symbols == {x} def test_image_interval(): x = Symbol('x', real=True) a = Symbol('a', real=True) assert imageset(x, 2x, Interval(-2, 1)) == Interval(-4, 2) assert imageset(x, 2x, Interval(-2, 1, True, False)) == \ Interval(-4, 2, True, False) assert imageset(x, x2, Interval(-2, 1, True, False)) == \ Interval(0, 4, False, True) assert imageset(x, x2, Interval(-2, 1)) == Interval(0, 4) assert imageset(x, x2, Interval(-2, 1, True, False)) == \ Interval(0, 4, False, True) assert imageset(x, x2, Interval(-2, 1, True, True)) == \ Interval(0, 4, False, True) assert imageset(x, (x - 2)*2, Interval(1, 3)) == Interval(0, 1) assert imageset(x, 3x*4 - 26x*3 + 78x*2 - 90x, Interval(0, 4)) == \ Interval(-35, 0) # Multiple Maxima assert imageset(x, x + 1/x, Interval(-oo, oo)) == Interval(-oo, -2) \ + Interval(2, oo) # Single Infinite discontinuity assert imageset(x, 1/x + 1/(x-1)*2, Interval(0, 2, True, False)) == \ Interval(Rational(3, 2), oo, False) # Multiple Infinite discontinuities # Test for Python lambda assert imageset(lambda x: 2x, Interval(-2, 1)) == Interval(-4, 2) assert imageset(Lambda(x, ax), Interval(0, 1)) == \ ImageSet(Lambda(x, ax), Interval(0, 1)) assert imageset(Lambda(x, sin(cos(x))), Interval(0, 1)) == \ ImageSet(Lambda(x, sin(cos(x))), Interval(0, 1)) def test_image_piecewise(): f = Piecewise((x, x <= -1), (1/x2, x <= 5), (x3, True)) f1 = Piecewise((0, x <= 1), (1, x <= 2), (2, True)) assert imageset(x, f, Interval(-5, 5)) == Union(Interval(-5, -1), Interval(Rational(1, 25), oo)) assert imageset(x, f1, Interval(1, 2)) == FiniteSet(0, 1) @XFAIL # See: https://github.com/sympy/sympy/pull/2723#discussion_r8659826 def test_image_Intersection(): x = Symbol('x', real=True) y = Symbol('y', real=True) assert imageset(x, x2, Interval(-2, 0).intersect(Interval(x, y))) == \ Interval(0, 4).intersect(Interval(Min(x2, y2), Max(x2, y*2))) def test_image_FiniteSet(): x = Symbol('x', real=True) assert imageset(x, 2x, FiniteSet(1, 2, 3)) == FiniteSet(2, 4, 6) def test_image_Union(): x = Symbol('x', real=True) assert imageset(x, x*2, Interval(-2, 0) + FiniteSet(1, 2, 3)) == \ (Interval(0, 4) + FiniteSet(9)) def test_image_EmptySet(): x = Symbol('x', real=True) assert imageset(x, 2x, S.EmptySet) == S.EmptySet def test_issue_5724_7680(): assert I not in S.Reals # issue 7680 assert Interval(-oo, oo).contains(I) is S.false def test_boundary(): assert FiniteSet(1).boundary == FiniteSet(1) assert all(Interval(0, 1, left_open, right_open).boundary == FiniteSet(0, 1) for left_open in (true, false) for right_open in (true, false)) def test_boundary_Union(): assert (Interval(0, 1) + Interval(2, 3)).boundary == FiniteSet(0, 1, 2, 3) assert ((Interval(0, 1, False, True) + Interval(1, 2, True, False)).boundary == FiniteSet(0, 1, 2)) assert (Interval(0, 1) + FiniteSet(2)).boundary == FiniteSet(0, 1, 2) assert Union(Interval(0, 10), Interval(5, 15), evaluate=False).boundary \ == FiniteSet(0, 15) assert Union(Interval(0, 10), Interval(0, 1), evaluate=False).boundary \ == FiniteSet(0, 10) assert Union(Interval(0, 10, True, True), Interval(10, 15, True, True), evaluate=False).boundary \ == FiniteSet(0, 10, 15) @XFAIL def test_union_boundary_of_joining_sets(): """ Testing the boundary of unions is a hard problem """ assert Union(Interval(0, 10), Interval(10, 15), evaluate=False).boundary \ == FiniteSet(0, 15) def test_boundary_ProductSet(): open_square = Interval(0, 1, True, True) ** 2 assert open_square.boundary == (FiniteSet(0, 1) * Interval(0, 1) + Interval(0, 1) * FiniteSet(0, 1)) second_square = Interval(1, 2, True, True) * Interval(0, 1, True, True) assert (open_square + second_square).boundary == ( FiniteSet(0, 1) * Interval(0, 1) + FiniteSet(1, 2) * Interval(0, 1) + Interval(0, 1) * FiniteSet(0, 1) + Interval(1, 2) * FiniteSet(0, 1)) def test_boundary_ProductSet_line(): line_in_r2 = Interval(0, 1) * FiniteSet(0) assert line_in_r2.boundary == line_in_r2 def test_is_open(): assert Interval(0, 1, False, False).is_open is False assert Interval(0, 1, True, False).is_open is False assert Interval(0, 1, True, True).is_open is True assert FiniteSet(1, 2, 3).is_open is False def test_is_closed(): assert Interval(0, 1, False, False).is_closed is True assert Interval(0, 1, True, False).is_closed is False assert FiniteSet(1, 2, 3).is_closed is True def test_closure(): assert Interval(0, 1, False, True).closure == Interval(0, 1, False, False) def test_interior(): assert Interval(0, 1, False, True).interior == Interval(0, 1, True, True) def test_issue_7841(): raises(TypeError, lambda: x in S.Reals) def test_Eq(): assert Eq(Interval(0, 1), Interval(0, 1)) assert Eq(Interval(0, 1), Interval(0, 2)) == False s1 = FiniteSet(0, 1) s2 = FiniteSet(1, 2) assert Eq(s1, s1) assert Eq(s1, s2) == False assert Eq(s1s2, s1s2) assert Eq(s1s2, s2s1) == False assert unchanged(Eq, FiniteSet({x, y}), FiniteSet({x})) assert Eq(FiniteSet({x, y}).subs(y, x), FiniteSet({x})) is S.true assert Eq(FiniteSet({x, y}), FiniteSet({x})).subs(y, x) is S.true assert Eq(FiniteSet({x, y}).subs(y, x+1), FiniteSet({x})) is S.false assert Eq(FiniteSet({x, y}), FiniteSet({x})).subs(y, x+1) is S.false assert Eq(ProductSet({1}, {2}), Interval(1, 2)) is S.false assert Eq(ProductSet({1}), ProductSet({1}, {2})) is S.false assert Eq(FiniteSet(()), FiniteSet(1)) is S.false assert Eq(ProductSet(), FiniteSet(1)) is S.false i1 = Interval(0, 1) i2 = Interval(x, y) assert unchanged(Eq, ProductSet(i1, i1), ProductSet(i2, i2)) def test_SymmetricDifference(): A = FiniteSet(0, 1, 2, 3, 4, 5) B = FiniteSet(2, 4, 6, 8, 10) C = Interval(8, 10) assert SymmetricDifference(A, B, evaluate=False).is_iterable is True assert SymmetricDifference(A, C, evaluate=False).is_iterable is None assert FiniteSet(SymmetricDifference(A, B, evaluate=False)) == \ FiniteSet(0, 1, 3, 5, 6, 8, 10) raises(TypeError, lambda: FiniteSet(SymmetricDifference(A, C, evaluate=False))) assert SymmetricDifference(FiniteSet(0, 1, 2, 3, 4, 5), \ FiniteSet(2, 4, 6, 8, 10)) == FiniteSet(0, 1, 3, 5, 6, 8, 10) assert SymmetricDifference(FiniteSet(2, 3, 4), FiniteSet(2, 3, 4 ,5)) \ == FiniteSet(5) assert FiniteSet(1, 2, 3, 4, 5) ^ FiniteSet(1, 2, 5, 6) == \ FiniteSet(3, 4, 6) assert Set(S(1), S(2), S(3)) ^ Set(S(2), S(3), S(4)) == Union(Set(S(1), S(2), S(3)) - Set(S(2), S(3), S(4)), \ Set(S(2), S(3), S(4)) - Set(S(1), S(2), S(3))) assert Interval(0, 4) ^ Interval(2, 5) == Union(Interval(0, 4) - \ Interval(2, 5), Interval(2, 5) - Interval(0, 4)) def test_issue_9536(): from sympy.functions.elementary.exponential import log a = Symbol('a', real=True) assert FiniteSet(log(a)).intersect(S.Reals) == Intersection(S.Reals, FiniteSet(log(a))) def test_issue_9637(): n = Symbol('n') a = FiniteSet(n) b = FiniteSet(2, n) assert Complement(S.Reals, a) == Complement(S.Reals, a, evaluate=False) assert Complement(Interval(1, 3), a) == Complement(Interval(1, 3), a, evaluate=False) assert Complement(Interval(1, 3), b) == \ Complement(Union(Interval(1, 2, False, True), Interval(2, 3, True, False)), a) assert Complement(a, S.Reals) == Complement(a, S.Reals, evaluate=False) assert Complement(a, Interval(1, 3)) == Complement(a, Interval(1, 3), evaluate=False) def test_issue_9808(): # See https://github.com/sympy/sympy/issues/16342 assert Complement(FiniteSet(y), FiniteSet(1)) == Complement(FiniteSet(y), FiniteSet(1), evaluate=False) assert Complement(FiniteSet(1, 2, x), FiniteSet(x, y, 2, 3)) == \ Complement(FiniteSet(1), FiniteSet(y), evaluate=False) def test_issue_9956(): assert Union(Interval(-oo, oo), FiniteSet(1)) == Interval(-oo, oo) assert Interval(-oo, oo).contains(1) is S.true def test_issue_Symbol_inter(): i = Interval(0, oo) r = S.Reals mat = Matrix([0, 0, 0]) assert Intersection(r, i, FiniteSet(m), FiniteSet(m, n)) == \ Intersection(i, FiniteSet(m)) assert Intersection(FiniteSet(1, m, n), FiniteSet(m, n, 2), i) == \ Intersection(i, FiniteSet(m, n)) assert Intersection(FiniteSet(m, n, x), FiniteSet(m, z), r) == \ Intersection(Intersection({m, z}, {m, n, x}), r) assert Intersection(FiniteSet(m, n, 3), FiniteSet(m, n, x), r) == \ Intersection(FiniteSet(3, m, n), FiniteSet(m, n, x), r, evaluate=False) assert Intersection(FiniteSet(m, n, 3), FiniteSet(m, n, 2, 3), r) == \ Intersection(FiniteSet(3, m, n), r) assert Intersection(r, FiniteSet(mat, 2, n), FiniteSet(0, mat, n)) == \ Intersection(r, FiniteSet(n)) assert Intersection(FiniteSet(sin(x), cos(x)), FiniteSet(sin(x), cos(x), 1), r) == \ Intersection(r, FiniteSet(sin(x), cos(x))) assert Intersection(FiniteSet(x2, 1, sin(x)), FiniteSet(x2, 2, sin(x)), r) == \ Intersection(r, FiniteSet(x2, sin(x))) def test_issue_11827(): assert S.Naturals04 def test_issue_10113(): f = x2/(x2 - 4) assert imageset(x, f, S.Reals) == Union(Interval(-oo, 0), Interval(1, oo, True, True)) assert imageset(x, f, Interval(-2, 2)) == Interval(-oo, 0) assert imageset(x, f, Interval(-2, 3)) == Union(Interval(-oo, 0), Interval(Rational(9, 5), oo)) def test_issue_10248(): raises( TypeError, lambda: list(Intersection(S.Reals, FiniteSet(x))) ) A = Symbol('A', real=True) assert list(Intersection(S.Reals, FiniteSet(A))) == [A] def test_issue_9447(): a = Interval(0, 1) + Interval(2, 3) assert Complement(S.UniversalSet, a) == Complement( S.UniversalSet, Union(Interval(0, 1), Interval(2, 3)), evaluate=False) assert Complement(S.Naturals, a) == Complement( S.Naturals, Union(Interval(0, 1), Interval(2, 3)), evaluate=False) def test_issue_10337(): assert (FiniteSet(2) == 3) is False assert (FiniteSet(2) != 3) is True raises(TypeError, lambda: FiniteSet(2) < 3) raises(TypeError, lambda: FiniteSet(2) <= 3) raises(TypeError, lambda: FiniteSet(2) > 3) raises(TypeError, lambda: FiniteSet(2) >= 3) def test_issue_10326(): bad = [ EmptySet, FiniteSet(1), Interval(1, 2), S.ComplexInfinity, S.ImaginaryUnit, S.Infinity, S.NaN, S.NegativeInfinity, ] interval = Interval(0, 5) for i in bad: assert i not in interval x = Symbol('x', real=True) nr = Symbol('nr', extended_real=False) assert x + 1 in Interval(x, x + 4) assert nr not in Interval(x, x + 4) assert Interval(1, 2) in FiniteSet(Interval(0, 5), Interval(1, 2)) assert Interval(-oo, oo).contains(oo) is S.false assert Interval(-oo, oo).contains(-oo) is S.false def test_issue_2799(): U = S.UniversalSet a = Symbol('a', real=True) inf_interval = Interval(a, oo) R = S.Reals assert U + inf_interval == inf_interval + U assert U + R == R + U assert R + inf_interval == inf_interval + R def test_issue_9706(): assert Interval(-oo, 0).closure == Interval(-oo, 0, True, False) assert Interval(0, oo).closure == Interval(0, oo, False, True) assert Interval(-oo, oo).closure == Interval(-oo, oo) def test_issue_8257(): reals_plus_infinity = Union(Interval(-oo, oo), FiniteSet(oo)) reals_plus_negativeinfinity = Union(Interval(-oo, oo), FiniteSet(-oo)) assert Interval(-oo, oo) + FiniteSet(oo) == reals_plus_infinity assert FiniteSet(oo) + Interval(-oo, oo) == reals_plus_infinity assert Interval(-oo, oo) + FiniteSet(-oo) == reals_plus_negativeinfinity assert FiniteSet(-oo) + Interval(-oo, oo) == reals_plus_negativeinfinity def test_issue_10931(): assert S.Integers - S.Integers == EmptySet assert S.Integers - S.Reals == EmptySet def test_issue_11174(): soln = Intersection(Interval(-oo, oo), FiniteSet(-x), evaluate=False) assert Intersection(FiniteSet(-x), S.Reals) == soln soln = Intersection(S.Reals, FiniteSet(x), evaluate=False) assert Intersection(FiniteSet(x), S.Reals) == soln def test_issue_18505(): assert ImageSet(Lambda(n, sqrt(pin/2 - 1 + pi/2)), S.Integers).contains(0) == \ Contains(0, ImageSet(Lambda(n, sqrt(pin/2 - 1 + pi/2)), S.Integers)) def test_finite_set_intersection(): # The following should not produce recursion errors # Note: some of these are not completely correct. See # https://github.com/sympy/sympy/issues/16342. assert Intersection(FiniteSet(-oo, x), FiniteSet(x)) == FiniteSet(x) assert Intersection._handle_finite_sets([FiniteSet(-oo, x), FiniteSet(0, x)]) == FiniteSet(x) assert Intersection._handle_finite_sets([FiniteSet(-oo, x), FiniteSet(x)]) == FiniteSet(x) assert Intersection._handle_finite_sets([FiniteSet(2, 3, x, y), FiniteSet(1, 2, x)]) == \ Intersection._handle_finite_sets([FiniteSet(1, 2, x), FiniteSet(2, 3, x, y)]) == \ Intersection(FiniteSet(1, 2, x), FiniteSet(2, 3, x, y)) == \ Intersection(FiniteSet(1, 2, x), FiniteSet(2, x, y)) assert FiniteSet(1+x-y) & FiniteSet(1) == \ FiniteSet(1) & FiniteSet(1+x-y) == \ Intersection(FiniteSet(1+x-y), FiniteSet(1), evaluate=False) assert FiniteSet(1) & FiniteSet(x) == FiniteSet(x) & FiniteSet(1) == \ Intersection(FiniteSet(1), FiniteSet(x), evaluate=False) assert FiniteSet({x}) & FiniteSet({x, y}) == \ Intersection(FiniteSet({x}), FiniteSet({x, y}), evaluate=False) def test_union_intersection_constructor(): # The actual exception does not matter here, so long as these fail sets = [FiniteSet(1), FiniteSet(2)] raises(Exception, lambda: Union(sets)) raises(Exception, lambda: Intersection(sets)) raises(Exception, lambda: Union(tuple(sets))) raises(Exception, lambda: Intersection(tuple(sets))) raises(Exception, lambda: Union(i for i in sets)) raises(Exception, lambda: Intersection(i for i in sets)) # Python sets are treated the same as FiniteSet # The union of a single set (of sets) is the set (of sets) itself assert Union(set(sets)) == FiniteSet(sets) assert Intersection(set(sets)) == FiniteSet(sets) assert Union({1}, {2}) == FiniteSet(1, 2) assert Intersection({1, 2}, {2, 3}) == FiniteSet(2) def test_Union_contains(): assert zoo not in Union( Interval.open(-oo, 0), Interval.open(0, oo)) @XFAIL def test_issue_16878b(): # in intersection_sets for (ImageSet, Set) there is no code # that handles the base_set of S.Reals like there is # for Integers assert imageset(x, (x, x), S.Reals).is_subset(S.Reals*2) is True def test_DisjointUnion(): assert DisjointUnion(FiniteSet(1, 2, 3), FiniteSet(1, 2, 3), FiniteSet(1, 2, 3)).rewrite(Union) == (FiniteSet(1, 2, 3) FiniteSet(0, 1, 2)) assert DisjointUnion(Interval(1, 3), Interval(2, 4)).rewrite(Union) == Union(Interval(1, 3) * FiniteSet(0), Interval(2, 4) * FiniteSet(1)) assert DisjointUnion(Interval(0, 5), Interval(0, 5)).rewrite(Union) == Union(Interval(0, 5) * FiniteSet(0), Interval(0, 5) * FiniteSet(1)) assert DisjointUnion(Interval(-1, 2), S.EmptySet, S.EmptySet).rewrite(Union) == Interval(-1, 2) * FiniteSet(0) assert DisjointUnion(Interval(-1, 2)).rewrite(Union) == Interval(-1, 2) * FiniteSet(0) assert DisjointUnion(S.EmptySet, Interval(-1, 2), S.EmptySet).rewrite(Union) == Interval(-1, 2) * FiniteSet(1) assert DisjointUnion(Interval(-oo, oo)).rewrite(Union) == Interval(-oo, oo) * FiniteSet(0) assert DisjointUnion(S.EmptySet).rewrite(Union) == S.EmptySet assert DisjointUnion().rewrite(Union) == S.EmptySet raises(TypeError, lambda: DisjointUnion(Symbol('n'))) x = Symbol("x") y = Symbol("y") z = Symbol("z") assert DisjointUnion(FiniteSet(x), FiniteSet(y, z)).rewrite(Union) == (FiniteSet(x) * FiniteSet(0)) + (FiniteSet(y, z) * FiniteSet(1)) def test_DisjointUnion_is_empty(): assert DisjointUnion(S.EmptySet).is_empty is True assert DisjointUnion(S.EmptySet, S.EmptySet).is_empty is True assert DisjointUnion(S.EmptySet, FiniteSet(1, 2, 3)).is_empty is False def test_DisjointUnion_is_iterable(): assert DisjointUnion(S.Integers, S.Naturals, S.Rationals).is_iterable is True assert DisjointUnion(S.EmptySet, S.Reals).is_iterable is False assert DisjointUnion(FiniteSet(1, 2, 3), S.EmptySet, FiniteSet(x, y)).is_iterable is True assert DisjointUnion(S.EmptySet, S.EmptySet).is_iterable is False def test_DisjointUnion_contains(): assert (0, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) assert (0, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) assert (0, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) assert (1, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) assert (1, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) assert (1, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) assert (2, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) assert (2, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) assert (2, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) assert (0, 1, 2) not in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) assert (0, 0.5) not in DisjointUnion(FiniteSet(0.5)) assert (0, 5) not in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) assert (x, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y)) assert (y, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y)) assert (z, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y)) assert (y, 2) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y)) assert (0.5, 0) in DisjointUnion(Interval(0, 1), Interval(0, 2)) assert (0.5, 1) in DisjointUnion(Interval(0, 1), Interval(0, 2)) assert (1.5, 0) not in DisjointUnion(Interval(0, 1), Interval(0, 2)) assert (1.5, 1) in DisjointUnion(Interval(0, 1), Interval(0, 2)) def test_DisjointUnion_iter(): D = DisjointUnion(FiniteSet(3, 5, 7, 9), FiniteSet(x, y, z)) it = iter(D) L1 = [(x, 1), (y, 1), (z, 1)] L2 = [(3, 0), (5, 0), (7, 0), (9, 0)] nxt = next(it) assert nxt in L2 L2.remove(nxt) nxt = next(it) assert nxt in L1 L1.remove(nxt) nxt = next(it) assert nxt in L2 L2.remove(nxt) nxt = next(it) assert nxt in L1 L1.remove(nxt) nxt = next(it) assert nxt in L2 L2.remove(nxt) nxt = next(it) assert nxt in L1 L1.remove(nxt) nxt = next(it) assert nxt in L2 L2.remove(nxt) raises(StopIteration, lambda: next(it)) raises(ValueError, lambda: iter(DisjointUnion(Interval(0, 1), S.EmptySet))) def test_DisjointUnion_len(): assert len(DisjointUnion(FiniteSet(3, 5, 7, 9), FiniteSet(x, y, z))) == 7 assert len(DisjointUnion(S.EmptySet, S.EmptySet, FiniteSet(x, y, z), S.EmptySet)) == 3 raises(ValueError, lambda: len(DisjointUnion(Interval(0, 1), S.EmptySet))) def test_issue_20089(): B = FiniteSet(FiniteSet(1, 2), FiniteSet(1)) assert 1 not in B assert 1.0 not in B assert not Eq(1, FiniteSet(1, 2)) assert FiniteSet(1) in B A = FiniteSet(1, 2) assert A in B assert B.issubset(B) assert not A.issubset(B) assert 1 in A C = FiniteSet(FiniteSet(1, 2), FiniteSet(1), 1, 2) assert A.issubset(C) assert B.issubset(C) def test_issue_19378(): a = FiniteSet(1, 2) b = ProductSet(a, a) c = FiniteSet((1, 1), (1, 2), (2, 1), (2, 2)) assert b.is_subset(c) is True d = FiniteSet(1) assert b.is_subset(d) is False assert Eq(c, b).simplify() is S.true assert Eq(a, c).simplify() is S.false assert Eq({1}, {x}).simplify() == Eq({1}, {x}) def test_intersection_symbolic(): n = Symbol('n') # These should not throw an error assert isinstance(Intersection(Range(n), Range(100)), Intersection) assert isinstance(Intersection(Range(n), Interval(1, 100)), Intersection) assert isinstance(Intersection(Range(100), Interval(1, n)), Intersection) @XFAIL def test_intersection_symbolic_failing(): n = Symbol('n', integer=True, positive=True) assert Intersection(Range(10, n), Range(4, 500, 5)) == Intersection( Range(14, n), Range(14, 500, 5)) assert Intersection(Interval(10, n), Range(4, 500, 5)) == Intersection( Interval(14, n), Range(14, 500, 5)) def test_issue_20379(): #https://github.com/sympy/sympy/issues/20379 x = pi - 3.14159265358979 assert FiniteSet(x).evalf(2) == FiniteSet(Float('3.23108914886517e-15', 2)) def test_finiteset_simplify(): S = FiniteSet(1, cos(1)2 + sin(1)2) assert S.simplify() == {1}
e5ba02d85926938ebe8161aa5b375390f99330d1c73a90340a707dabde3a1f07	from time import perf_counter import pyglet.gl as pgl from sympy.plotting.pygletplot.managed_window import ManagedWindow from sympy.plotting.pygletplot.plot_camera import PlotCamera from sympy.plotting.pygletplot.plot_controller import PlotController class PlotWindow(ManagedWindow): def __init__(self, plot, antialiasing=True, ortho=False, invert_mouse_zoom=False, linewidth=1.5, caption="SymPy Plot", kwargs): """ Named Arguments =============== antialiasing = True True OR False ortho = False True OR False invert_mouse_zoom = False True OR False """ self.plot = plot self.camera = None self._calculating = False self.antialiasing = antialiasing self.ortho = ortho self.invert_mouse_zoom = invert_mouse_zoom self.linewidth = linewidth self.title = caption self.last_caption_update = 0 self.caption_update_interval = 0.2 self.drawing_first_object = True super().__init__(kwargs) def setup(self): self.camera = PlotCamera(self, ortho=self.ortho) self.controller = PlotController(self, invert_mouse_zoom=self.invert_mouse_zoom) self.push_handlers(self.controller) pgl.glClearColor(1.0, 1.0, 1.0, 0.0) pgl.glClearDepth(1.0) pgl.glDepthFunc(pgl.GL_LESS) pgl.glEnable(pgl.GL_DEPTH_TEST) pgl.glEnable(pgl.GL_LINE_SMOOTH) pgl.glShadeModel(pgl.GL_SMOOTH) pgl.glLineWidth(self.linewidth) pgl.glEnable(pgl.GL_BLEND) pgl.glBlendFunc(pgl.GL_SRC_ALPHA, pgl.GL_ONE_MINUS_SRC_ALPHA) if self.antialiasing: pgl.glHint(pgl.GL_LINE_SMOOTH_HINT, pgl.GL_NICEST) pgl.glHint(pgl.GL_POLYGON_SMOOTH_HINT, pgl.GL_NICEST) self.camera.setup_projection() def on_resize(self, w, h): super().on_resize(w, h) if self.camera is not None: self.camera.setup_projection() def update(self, dt): self.controller.update(dt) def draw(self): self.plot._render_lock.acquire() self.camera.apply_transformation() calc_verts_pos, calc_verts_len = 0, 0 calc_cverts_pos, calc_cverts_len = 0, 0 should_update_caption = (perf_counter() - self.last_caption_update > self.caption_update_interval) if len(self.plot._functions.values()) == 0: self.drawing_first_object = True iterfunctions = iter(self.plot._functions.values()) for r in iterfunctions: if self.drawing_first_object: self.camera.set_rot_preset(r.default_rot_preset) self.drawing_first_object = False pgl.glPushMatrix() r._draw() pgl.glPopMatrix() # might as well do this while we are # iterating and have the lock rather # than locking and iterating twice # per frame: if should_update_caption: try: if r.calculating_verts: calc_verts_pos += r.calculating_verts_pos calc_verts_len += r.calculating_verts_len if r.calculating_cverts: calc_cverts_pos += r.calculating_cverts_pos calc_cverts_len += r.calculating_cverts_len except ValueError: pass for r in self.plot._pobjects: pgl.glPushMatrix() r._draw() pgl.glPopMatrix() if should_update_caption: self.update_caption(calc_verts_pos, calc_verts_len, calc_cverts_pos, calc_cverts_len) self.last_caption_update = perf_counter() if self.plot._screenshot: self.plot._screenshot._execute_saving() self.plot._render_lock.release() def update_caption(self, calc_verts_pos, calc_verts_len, calc_cverts_pos, calc_cverts_len): caption = self.title if calc_verts_len or calc_cverts_len: caption += " (calculating" if calc_verts_len > 0: p = (calc_verts_pos / calc_verts_len) * 100 caption += " vertices %i%%" % (p) if calc_cverts_len > 0: p = (calc_cverts_pos / calc_cverts_len) * 100 caption += " colors %i%%" % (p) caption += ")" if self.caption != caption: self.set_caption(caption)
f15a2df14b1373a3176b125f88540b852ea0b91b459a988d04158dacbb62a58a	#!/usr/bin/env python """Distutils based setup script for SymPy. This uses Distutils (https://python.org/sigs/distutils-sig/) the standard python mechanism for installing packages. Optionally, you can use Setuptools (https://setuptools.readthedocs.io/en/latest/) to automatically handle dependencies. For the easiest installation just type the command (you'll probably need root privileges for that): python setup.py install This will install the library in the default location. For instructions on how to customize the install procedure read the output of: python setup.py --help install In addition, there are some other commands: python setup.py clean -> will clean all trash (.pyc and stuff) python setup.py test -> will run the complete test suite python setup.py bench -> will run the complete benchmark suite python setup.py audit -> will run pyflakes checker on source code To get a full list of available commands, read the output of: python setup.py --help-commands Or, if all else fails, feel free to write to the sympy list at [email protected] and ask for help. """ import sys import os import shutil import glob import subprocess from distutils.command.sdist import sdist min_mpmath_version = '0.19' # This directory dir_setup = os.path.dirname(os.path.realpath(__file__)) extra_kwargs = {} try: from setuptools import setup, Command extra_kwargs['zip_safe'] = False extra_kwargs['entry_points'] = { 'console_scripts': [ 'isympy = isympy:main', ] } except ImportError: from distutils.core import setup, Command extra_kwargs['scripts'] = ['bin/isympy'] # handle mpmath deps in the hard way: from sympy.external.importtools import version_tuple try: import mpmath if version_tuple(mpmath.__version__) < version_tuple(min_mpmath_version): raise ImportError except ImportError: print("Please install the mpmath package with a version >= %s" % min_mpmath_version) sys.exit(-1) if sys.version_info < (3, 7): print("SymPy requires Python 3.7 or newer. Python %d.%d detected" % sys.version_info[:2]) sys.exit(-1) # Check that this list is uptodate against the result of the command: # python bin/generate_module_list.py modules = [ 'sympy.algebras', 'sympy.assumptions', 'sympy.assumptions.handlers', 'sympy.assumptions.predicates', 'sympy.assumptions.relation', 'sympy.benchmarks', 'sympy.calculus', 'sympy.categories', 'sympy.codegen', 'sympy.combinatorics', 'sympy.concrete', 'sympy.core', 'sympy.core.benchmarks', 'sympy.crypto', 'sympy.diffgeom', 'sympy.discrete', 'sympy.external', 'sympy.functions', 'sympy.functions.combinatorial', 'sympy.functions.elementary', 'sympy.functions.elementary.benchmarks', 'sympy.functions.special', 'sympy.functions.special.benchmarks', 'sympy.geometry', 'sympy.holonomic', 'sympy.integrals', 'sympy.integrals.benchmarks', 'sympy.integrals.rubi', 'sympy.integrals.rubi.parsetools', 'sympy.integrals.rubi.rubi_tests', 'sympy.integrals.rubi.rules', 'sympy.interactive', 'sympy.liealgebras', 'sympy.logic', 'sympy.logic.algorithms', 'sympy.logic.utilities', 'sympy.matrices', 'sympy.matrices.benchmarks', 'sympy.matrices.expressions', 'sympy.multipledispatch', 'sympy.ntheory', 'sympy.parsing', 'sympy.parsing.autolev', 'sympy.parsing.autolev._antlr', 'sympy.parsing.c', 'sympy.parsing.fortran', 'sympy.parsing.latex', 'sympy.parsing.latex._antlr', 'sympy.physics', 'sympy.physics.continuum_mechanics', 'sympy.physics.control', 'sympy.physics.hep', 'sympy.physics.mechanics', 'sympy.physics.optics', 'sympy.physics.quantum', 'sympy.physics.units', 'sympy.physics.units.definitions', 'sympy.physics.units.systems', 'sympy.physics.vector', 'sympy.plotting', 'sympy.plotting.intervalmath', 'sympy.plotting.pygletplot', 'sympy.polys', 'sympy.polys.agca', 'sympy.polys.benchmarks', 'sympy.polys.domains', 'sympy.polys.matrices', 'sympy.polys.numberfields', 'sympy.printing', 'sympy.printing.pretty', 'sympy.sandbox', 'sympy.series', 'sympy.series.benchmarks', 'sympy.sets', 'sympy.sets.handlers', 'sympy.simplify', 'sympy.solvers', 'sympy.solvers.benchmarks', 'sympy.solvers.diophantine', 'sympy.solvers.ode', 'sympy.stats', 'sympy.stats.sampling', 'sympy.strategies', 'sympy.strategies.branch', 'sympy.tensor', 'sympy.tensor.array', 'sympy.tensor.array.expressions', 'sympy.testing', 'sympy.unify', 'sympy.utilities', 'sympy.utilities._compilation', 'sympy.utilities.mathml', 'sympy.vector', ] class audit(Command): """Audits SymPy's source code for following issues: - Names which are used but not defined or used before they are defined. - Names which are redefined without having been used. """ description = "Audit SymPy source with PyFlakes" user_options = [] def initialize_options(self): self.all = None def finalize_options(self): pass def run(self): try: import pyflakes.scripts.pyflakes as flakes except ImportError: print("In order to run the audit, you need to have PyFlakes installed.") sys.exit(-1) dirs = (os.path.join(d) for d in (m.split('.') for m in modules)) warns = 0 for dir in dirs: for filename in os.listdir(dir): if filename.endswith('.py') and filename != '__init__.py': warns += flakes.checkPath(os.path.join(dir, filename)) if warns > 0: print("Audit finished with total %d warnings" % warns) class clean(Command): """Cleans .pyc and debian trashs, so you should get the same copy as is in the VCS. """ description = "remove build files" user_options = [("all", "a", "the same")] def initialize_options(self): self.all = None def finalize_options(self): pass def run(self): curr_dir = os.getcwd() for root, dirs, files in os.walk(dir_setup): for file in files: if file.endswith('.pyc') and os.path.isfile: os.remove(os.path.join(root, file)) os.chdir(dir_setup) names = ["python-build-stamp-2.4", "MANIFEST", "build", "dist", "doc/_build", "sample.tex"] for f in names: if os.path.isfile(f): os.remove(f) elif os.path.isdir(f): shutil.rmtree(f) for name in glob.glob(os.path.join(dir_setup, "doc", "src", "modules", "physics", "vector", ".pdf")): if os.path.isfile(name): os.remove(name) os.chdir(curr_dir) class test_sympy(Command): """Runs all tests under the sympy/ folder """ description = "run all tests and doctests; also see bin/test and bin/doctest" user_options = [] # distutils complains if this is not here. def __init__(self, args): self.args = args[0] # so we can pass it to other classes Command.__init__(self, args) def initialize_options(self): # distutils wants this pass def finalize_options(self): # this too pass def run(self): from sympy.testing import runtests runtests.run_all_tests() class run_benchmarks(Command): """Runs all SymPy benchmarks""" description = "run all benchmarks" user_options = [] # distutils complains if this is not here. def __init__(self, args): self.args = args[0] # so we can pass it to other classes Command.__init__(self, args) def initialize_options(self): # distutils wants this pass def finalize_options(self): # this too pass # we use py.test like architecture: # # o collector -- collects benchmarks # o runner -- executes benchmarks # o presenter -- displays benchmarks results # # this is done in sympy.utilities.benchmarking on top of py.test def run(self): from sympy.utilities import benchmarking benchmarking.main(['sympy']) class antlr(Command): """Generate code with antlr4""" description = "generate parser code from antlr grammars" user_options = [] # distutils complains if this is not here. def __init__(self, args): self.args = args[0] # so we can pass it to other classes Command.__init__(self, args) def initialize_options(self): # distutils wants this pass def finalize_options(self): # this too pass def run(self): from sympy.parsing.latex._build_latex_antlr import build_parser if not build_parser(): sys.exit(-1) class sdist_sympy(sdist): def run(self): # Fetch git commit hash and write down to commit_hash.txt before # shipped in tarball. commit_hash = None commit_hash_filepath = 'doc/commit_hash.txt' try: commit_hash = \ subprocess.check_output(['git', 'rev-parse', 'HEAD']) commit_hash = commit_hash.decode('ascii') commit_hash = commit_hash.rstrip() print('Commit hash found : {}.'.format(commit_hash)) print('Writing it to {}.'.format(commit_hash_filepath)) except: pass if commit_hash: with open(commit_hash_filepath, 'w') as f: f.write(commit_hash) super(sdist_sympy, self).run() try: os.remove(commit_hash_filepath) print( 'Successfully removed temporary file {}.' .format(commit_hash_filepath)) except OSError as e: print("Error deleting %s - %s." % (e.filename, e.strerror)) # Check that this list is uptodate against the result of the command: # python bin/generate_test_list.py tests = [ 'sympy.algebras.tests', 'sympy.assumptions.tests', 'sympy.calculus.tests', 'sympy.categories.tests', 'sympy.codegen.tests', 'sympy.combinatorics.tests', 'sympy.concrete.tests', 'sympy.core.tests', 'sympy.crypto.tests', 'sympy.diffgeom.tests', 'sympy.discrete.tests', 'sympy.external.tests', 'sympy.functions.combinatorial.tests', 'sympy.functions.elementary.tests', 'sympy.functions.special.tests', 'sympy.geometry.tests', 'sympy.holonomic.tests', 'sympy.integrals.rubi.parsetools.tests', 'sympy.integrals.rubi.rubi_tests.tests', 'sympy.integrals.rubi.tests', 'sympy.integrals.tests', 'sympy.interactive.tests', 'sympy.liealgebras.tests', 'sympy.logic.tests', 'sympy.matrices.expressions.tests', 'sympy.matrices.tests', 'sympy.multipledispatch.tests', 'sympy.ntheory.tests', 'sympy.parsing.tests', 'sympy.physics.continuum_mechanics.tests', 'sympy.physics.control.tests', 'sympy.physics.hep.tests', 'sympy.physics.mechanics.tests', 'sympy.physics.optics.tests', 'sympy.physics.quantum.tests', 'sympy.physics.tests', 'sympy.physics.units.tests', 'sympy.physics.vector.tests', 'sympy.plotting.intervalmath.tests', 'sympy.plotting.pygletplot.tests', 'sympy.plotting.tests', 'sympy.polys.agca.tests', 'sympy.polys.domains.tests', 'sympy.polys.matrices.tests', 'sympy.polys.numberfields.tests', 'sympy.polys.tests', 'sympy.printing.pretty.tests', 'sympy.printing.tests', 'sympy.sandbox.tests', 'sympy.series.tests', 'sympy.sets.tests', 'sympy.simplify.tests', 'sympy.solvers.diophantine.tests', 'sympy.solvers.ode.tests', 'sympy.solvers.tests', 'sympy.stats.sampling.tests', 'sympy.stats.tests', 'sympy.strategies.branch.tests', 'sympy.strategies.tests', 'sympy.tensor.array.expressions.tests', 'sympy.tensor.array.tests', 'sympy.tensor.tests', 'sympy.testing.tests', 'sympy.unify.tests', 'sympy.utilities._compilation.tests', 'sympy.utilities.tests', 'sympy.vector.tests', ] with open(os.path.join(dir_setup, 'sympy', 'release.py')) as f: # Defines __version__ exec(f.read()) if __name__ == '__main__': setup(name='sympy', version=__version__, description='Computer algebra system (CAS) in Python', author='SymPy development team', author_email='[email protected]', license='BSD', keywords="Math CAS", url='https://sympy.org', project_urls={ 'Source': 'https://github.com/sympy/sympy', }, py_modules=['isympy'], packages=['sympy'] + modules + tests, ext_modules=[], package_data={ 'sympy.utilities.mathml': ['data/.xsl'], 'sympy.logic.benchmarks': ['input/.cnf'], 'sympy.parsing.autolev': [ '.g4', 'test-examples/.al', 'test-examples/.py', 'test-examples/pydy-example-repo/.al', 'test-examples/pydy-example-repo/.py', 'test-examples/README.txt', ], 'sympy.parsing.latex': ['.txt', '.g4'], 'sympy.integrals.rubi.parsetools': ['header.py.txt'], 'sympy.plotting.tests': ['test_region_.png'], 'sympy': ['py.typed'] }, data_files=[('share/man/man1', ['doc/man/isympy.1'])], cmdclass={'test': test_sympy, 'bench': run_benchmarks, 'clean': clean, 'audit': audit, 'antlr': antlr, 'sdist': sdist_sympy, }, python_requires='>=3.7', classifiers=[ 'License :: OSI Approved :: BSD License', 'Operating System :: OS Independent', 'Programming Language :: Python', 'Topic :: Scientific/Engineering', 'Topic :: Scientific/Engineering :: Mathematics', 'Topic :: Scientific/Engineering :: Physics', 'Programming Language :: Python :: 3', 'Programming Language :: Python :: 3.7', 'Programming Language :: Python :: 3.8', 'Programming Language :: Python :: 3.9', 'Programming Language :: Python :: 3.10', 'Programming Language :: Python :: 3 :: Only', 'Programming Language :: Python :: Implementation :: CPython', 'Programming Language :: Python :: Implementation :: PyPy', ], install_requires=[ 'mpmath>=%s' % min_mpmath_version, ], **extra_kwargs )
914b7fa711027a28e15bcd6ca8a37dc0060ee8cd1e12b658ffe2810c7d7e275c	#!/usr/bin/env python # -- coding: utf-8 -- """ A tool to generate AUTHORS. We started tracking authors before moving to git, so we have to do some manual rearrangement of the git history authors in order to get the order in AUTHORS. bin/mailmap_check.py should be run before committing the results. See here for instructions on using this script: https://github.com/sympy/sympy/wiki/Development-workflow#update-mailmap """ from __future__ import unicode_literals from __future__ import print_function import sys import os from pathlib import Path from subprocess import run, PIPE from collections import OrderedDict, defaultdict from argparse import ArgumentParser if sys.version_info < (3, 7): sys.exit("This script requires Python 3.7 or newer") def sympy_dir(): return Path(__file__).resolve().parent.parent # put sympy on the path sys.path.insert(0, str(sympy_dir())) import sympy from sympy.utilities.misc import filldedent from sympy.external.importtools import version_tuple def main(args): parser = ArgumentParser(description='Update the .mailmap and/or AUTHORS files') parser.add_argument('--update-authors', action='store_true', help=filldedent(""" Also update the AUTHORS file. Note that it should only necessary for the release manager to do this as part of the release process for SymPy.""")) args = parser.parse_args(args) if not check_git_version(): return 1 # find who git knows ahout try: git_people = get_authors_from_git() except AssertionError as msg: print(red(msg)) return 1 lines_mailmap = read_lines(mailmap_path()) def key(line): # return lower case first address on line or # raise an error if not an entry if '#' in line: line = line.split('#')[0] L, R = line.count("<"), line.count(">") assert L == R and L in (1, 2) return line.split(">", 1)[0].split("<")[1].lower() who = OrderedDict() for i, line in enumerate(lines_mailmap): try: who.setdefault(key(line), []).append(line) except AssertionError: who[i] = [line] problems = False missing = False ambiguous = False dups = defaultdict(list) for person in git_people: email = key(person) dups[email].append(person) if email not in who: print(red("This author is not included in the .mailmap file:")) print(person) missing = True elif not any(p.startswith(person) for p in who[email]): print(red("Ambiguous names in .mailmap")) print(red("This email address appears for multiple entries:")) print('Person:', person) print('Mailmap entries:') for line in who[email]: print(line) ambiguous = True if missing: print(red(filldedent(""" The .mailmap file needs to be updated because there are commits with unrecognised author/email metadata. """))) problems = True if ambiguous: print(red(filldedent(""" Lines should be added to .mailmap to indicate the correct name and email aliases for all commits. """))) problems = True for email, commitauthors in dups.items(): if len(commitauthors) > 2: print(red(filldedent(""" The following commits are recorded with different metadata but the same/ambiguous email address. The .mailmap file will need to be updated."""))) for author in commitauthors: print(author) problems = True lines_mailmap_sorted = sort_lines_mailmap(lines_mailmap) write_lines(mailmap_path(), lines_mailmap_sorted) if lines_mailmap_sorted != lines_mailmap: problems = True print(red("The mailmap file was reordered")) # Check if changes to AUTHORS file are also needed lines_authors = make_authors_file_lines(git_people) old_lines_authors = read_lines(authors_path()) for person in old_lines_authors[8:]: if person not in git_people: print(red("This author is in the AUTHORS file but not .mailmap:")) print(person) problems = True if problems: print(red(filldedent(""" For instructions on updating the .mailmap file see: https://github.com/sympy/sympy/wiki/Development-workflow#add-your-name-and-email-address-to-the-mailmap-file""", break_on_hyphens=False, break_long_words=False))) else: print(green("No changes needed in .mailmap")) # Actually update the AUTHORS file (if --update-authors was passed) authors_changed = update_authors_file(lines_authors, old_lines_authors, args.update_authors) return int(problems) + int(authors_changed) def update_authors_file(lines, old_lines, update_yesno): if old_lines == lines: print(green('No changes needed in AUTHORS.')) return 0 # Actually write changes to the file? if update_yesno: write_lines(authors_path(), lines) print(red("Changes were made in the authors file")) # check for new additions new_authors = [] for i in sorted(set(lines) - set(old_lines)): try: author_name(i) new_authors.append(i) except AssertionError: continue if new_authors: if update_yesno: print(yellow("The following authors were added to AUTHORS.")) else: print(green(filldedent(""" The following authors will be added to the AUTHORS file at the time of the next SymPy release."""))) print() for i in sorted(new_authors, key=lambda x: x.lower()): print('\t%s' % i) if new_authors and update_yesno: return 1 else: return 0 def check_git_version(): # check git version minimal = '1.8.4.2' git_ver = run(['git', '--version'], stdout=PIPE, encoding='utf-8').stdout[12:] if version_tuple(git_ver) < version_tuple(minimal): print(yellow("Please use a git version >= %s" % minimal)) return False else: return True def authors_path(): return sympy_dir() / 'AUTHORS' def mailmap_path(): return sympy_dir() / '.mailmap' def red(text): return "\033[31m%s\033[0m" % text def yellow(text): return "\033[33m%s\033[0m" % text def green(text): return "\033[32m%s\033[0m" % text def author_name(line): assert line.count("<") == line.count(">") == 1 assert line.endswith(">") return line.split("<", 1)[0].strip() def get_authors_from_git(): git_command = ["git", "log", "--topo-order", "--reverse", "--format=%aN <%aE>"] git_people = run(git_command, stdout=PIPE, encoding='utf-8').stdout.strip().split("\n") # remove duplicates, keeping the original order git_people = list(OrderedDict.fromkeys(git_people)) # Do the few changes necessary in order to reproduce AUTHORS: def move(l, i1, i2, who): x = l.pop(i1) # this will fail if the .mailmap is not right assert who == author_name(x), \ '%s was not found at line %i' % (who, i1) l.insert(i2, x) move(git_people, 2, 0, 'Ondřej Čertík') move(git_people, 42, 1, 'Fabian Pedregosa') move(git_people, 22, 2, 'Jurjen N.E. Bos') git_people.insert(4, "Marc-Etienne M.Leveille <[email protected]>") move(git_people, 10, 5, 'Brian Jorgensen') git_people.insert(11, "Ulrich Hecht <[email protected]>") # this will fail if the .mailmap is not right assert 'Kirill Smelkov' == author_name(git_people.pop(12) ), 'Kirill Smelkov was not found at line 12' move(git_people, 12, 32, 'Sebastian Krämer') move(git_people, 227, 35, 'Case Van Horsen') git_people.insert(43, "Dan <[email protected]>") move(git_people, 57, 59, 'Aaron Meurer') move(git_people, 58, 57, 'Andrew Docherty') move(git_people, 67, 66, 'Chris Smith') move(git_people, 79, 76, 'Kevin Goodsell') git_people.insert(84, "Chu-Ching Huang <[email protected]>") move(git_people, 93, 92, 'James Pearson') # this will fail if the .mailmap is not right assert 'Sergey B Kirpichev' == author_name(git_people.pop(226) ), 'Sergey B Kirpichev was not found at line 226.' index = git_people.index( "azure-pipelines[bot] " + "<azure-pipelines[bot]@users.noreply.github.com>") git_people.pop(index) index = git_people.index( "whitesource-bolt-for-github[bot] " + "<whitesource-bolt-for-github[bot]@users.noreply.github.com>") git_people.pop(index) return git_people def make_authors_file_lines(git_people): # define new lines for the file header = filldedent(""" All people who contributed to SymPy by sending at least a patch or more (in the order of the date of their first contribution), except those who explicitly didn't want to be mentioned. People with a next to their names are not found in the metadata of the git history. This file is generated automatically by running `./bin/authors_update.py`. """).lstrip() header_extra = f"There are a total of {len(git_people)} authors.""" lines = header.splitlines() lines.append('') lines.append(header_extra) lines.append('') lines.extend(git_people) return lines def sort_lines_mailmap(lines): for n, line in enumerate(lines): if not line.startswith('#'): header_end = n break header = lines[:header_end] mailmap_lines = lines[header_end:] return header + sorted(mailmap_lines) def read_lines(path): with open(path, 'r', encoding='utf-8') as fin: return [line.strip() for line in fin.readlines()] def write_lines(path, lines): with open(path, 'w', encoding='utf-8') as fout: fout.write('\n'.join(lines)) fout.write('\n') if __name__ == "__main__": import sys sys.exit(main(*sys.argv[1:]))
3e01690453fd756778f006d02b418066e98b5702a05a3804639896c041ccc7fa	#!/usr/bin/env python import os import json from subprocess import check_output from collections import OrderedDict, defaultdict from collections.abc import Mapping import glob from contextlib import contextmanager import requests from requests_oauthlib import OAuth2 def main(version, push=None): """ WARNING: If push is given as --push then this will push the release to github. """ push = push == '--push' _GitHub_release(version, push) def error(msg): raise ValueError(msg) def blue(text): return "\033[34m%s\033[0m" % text def red(text): return "\033[31m%s\033[0m" % text def green(text): return "\033[32m%s\033[0m" % text def _GitHub_release(version, push, username=None, user='sympy', token=None, token_file_path="~/.sympy/release-token", repo='sympy', draft=False): """ Upload the release files to GitHub. The tag must be pushed up first. You can test on another repo by changing user and repo. """ if not requests: error("requests and requests-oauthlib must be installed to upload to GitHub") release_text = GitHub_release_text(version) short_version = get_sympy_short_version(version) tag = 'sympy-' + version prerelease = short_version != version urls = URLs(user=user, repo=repo) if not username: username = input("GitHub username: ") token = load_token_file(token_file_path) if not token: username, password, token = GitHub_authenticate(urls, username, token) # If the tag in question is not pushed up yet, then GitHub will just # create it off of master automatically, which is not what we want. We # could make it create it off the release branch, but even then, we would # not be sure that the correct commit is tagged. So we require that the # tag exist first. if not check_tag_exists(version): sys.exit(red("The tag for this version has not been pushed yet. Cannot upload the release.")) # See https://developer.github.com/v3/repos/releases/#create-a-release # First, create the release post = {} post['tag_name'] = tag post['name'] = "SymPy " + version post['body'] = release_text post['draft'] = draft post['prerelease'] = prerelease print("Creating release for tag", tag, end=' ') if push: result = query_GitHub(urls.releases_url, username, password=None, token=token, data=json.dumps(post)).json() release_id = result['id'] else: print(green("Not pushing!")) print(green("Done")) # Then, upload all the files to it. for key in descriptions: tarball = get_tarball_name(key, version) params = {} params['name'] = tarball if tarball.endswith('gz'): headers = {'Content-Type':'application/gzip'} elif tarball.endswith('pdf'): headers = {'Content-Type':'application/pdf'} elif tarball.endswith('zip'): headers = {'Content-Type':'application/zip'} else: headers = {'Content-Type':'application/octet-stream'} print("Uploading", tarball, end=' ') sys.stdout.flush() with open(os.path.join('release/release-' + version, tarball), 'rb') as f: if push: result = query_GitHub(urls.release_uploads_url % release_id, username, password=None, token=token, data=f, params=params, headers=headers).json() else: print(green("Not uploading!")) print(green("Done")) # TODO: download the files and check that they have the right sha256 sum def GitHub_release_text(version): """ Generate text to put in the GitHub release Markdown box """ shortversion = get_sympy_short_version(version) htmltable = table(version) out = """\ See https://github.com/sympy/sympy/wiki/release-notes-for-{shortversion} for the release notes. {htmltable} Note: Do not download the Source code (zip) or the Source code (tar.gz) files below. """ out = out.format(shortversion=shortversion, htmltable=htmltable) print(blue("Here are the release notes to copy into the GitHub release " "Markdown form:")) print() print(out) return out def get_sympy_short_version(version): """ Get the short version of SymPy being released, not including any rc tags (like 0.7.3) """ parts = version.split('.') # Remove rc tags e.g. 1.10rc1 -> [1, 10] lastpart = '' for dig in parts[-1]: if dig.isdigit(): lastpart += dig else: break parts[-1] = lastpart return '.'.join(parts) class URLs(object): """ This class contains URLs and templates which used in requests to GitHub API """ def __init__(self, user="sympy", repo="sympy", api_url="https://api.github.com", authorize_url="https://api.github.com/authorizations", uploads_url='https://uploads.github.com', main_url='https://github.com'): """Generates all URLs and templates""" self.user = user self.repo = repo self.api_url = api_url self.authorize_url = authorize_url self.uploads_url = uploads_url self.main_url = main_url self.pull_list_url = api_url + "/repos" + "/" + user + "/" + repo + "/pulls" self.issue_list_url = api_url + "/repos/" + user + "/" + repo + "/issues" self.releases_url = api_url + "/repos/" + user + "/" + repo + "/releases" self.single_issue_template = self.issue_list_url + "/%d" self.single_pull_template = self.pull_list_url + "/%d" self.user_info_template = api_url + "/users/%s" self.user_repos_template = api_url + "/users/%s/repos" self.issue_comment_template = (api_url + "/repos" + "/" + user + "/" + repo + "/issues/%d" + "/comments") self.release_uploads_url = (uploads_url + "/repos/" + user + "/" + repo + "/releases/%d" + "/assets") self.release_download_url = (main_url + "/" + user + "/" + repo + "/releases/download/%s/%s") def load_token_file(path="~/.sympy/release-token"): print("> Using token file %s" % path) path = os.path.expanduser(path) path = os.path.abspath(path) if os.path.isfile(path): try: with open(path) as f: token = f.readline() except IOError: print("> Unable to read token file") return else: print("> Token file does not exist") return return token.strip() def GitHub_authenticate(urls, username, token=None): _login_message = """\ Enter your GitHub username & password or press ^C to quit. The password will be kept as a Python variable as long as this script is running and https to authenticate with GitHub, otherwise not saved anywhere else:\ """ if username: print("> Authenticating as %s" % username) else: print(_login_message) username = input("Username: ") authenticated = False if token: print("> Authenticating using token") try: GitHub_check_authentication(urls, username, None, token) except AuthenticationFailed: print("> Authentication failed") else: print("> OK") password = None authenticated = True while not authenticated: password = getpass("Password: ") try: print("> Checking username and password ...") GitHub_check_authentication(urls, username, password, None) except AuthenticationFailed: print("> Authentication failed") else: print("> OK.") authenticated = True if password: generate = input("> Generate API token? [Y/n] ") if generate.lower() in ["y", "ye", "yes", ""]: name = input("> Name of token on GitHub? [SymPy Release] ") if name == "": name = "SymPy Release" token = generate_token(urls, username, password, name=name) print("Your token is", token) print("Use this token from now on as GitHub_release:token=" + token + ",username=" + username) print(red("DO NOT share this token with anyone")) save = input("Do you want to save this token to a file [yes]? ") if save.lower().strip() in ['y', 'yes', 'ye', '']: save_token_file(token) return username, password, token def run(cmdline, cwd=None): """ Run command in subprocess and get lines of output """ return check_output(cmdline, encoding='utf-8', cwd=cwd).splitlines() def check_tag_exists(version): """ Check if the tag for this release has been uploaded yet. """ tag = 'sympy-' + version all_tag_lines = run('git', 'ls-remote', '--tags', 'origin') return any(tag in tag_line for tag_line in all_tag_lines) def generate_token(urls, username, password, OTP=None, name="SymPy Release"): enc_data = json.dumps( { "scopes": ["public_repo"], "note": name } ) url = urls.authorize_url rep = query_GitHub(url, username=username, password=password, data=enc_data).json() return rep["token"] def GitHub_check_authentication(urls, username, password, token): """ Checks that username & password is valid. """ query_GitHub(urls.api_url, username, password, token) class AuthenticationFailed(Exception): pass def query_GitHub(url, username=None, password=None, token=None, data=None, OTP=None, headers=None, params=None, files=None): """ Query GitHub API. In case of a multipage result, DOES NOT query the next page. """ headers = headers or {} if OTP: headers['X-GitHub-OTP'] = OTP if token: auth = OAuth2(client_id=username, token=dict(access_token=token, token_type='bearer')) else: auth = HTTPBasicAuth(username, password) if data: r = requests.post(url, auth=auth, data=data, headers=headers, params=params, files=files) else: r = requests.get(url, auth=auth, headers=headers, params=params, stream=True) if r.status_code == 401: two_factor = r.headers.get('X-GitHub-OTP') if two_factor: print("A two-factor authentication code is required:", two_factor.split(';')[1].strip()) OTP = input("Authentication code: ") return query_GitHub(url, username=username, password=password, token=token, data=data, OTP=OTP) raise AuthenticationFailed("invalid username or password") r.raise_for_status() return r def save_token_file(token): token_file = input("> Enter token file location [~/.sympy/release-token] ") token_file = token_file or "~/.sympy/release-token" token_file_expand = os.path.expanduser(token_file) token_file_expand = os.path.abspath(token_file_expand) token_folder, _ = os.path.split(token_file_expand) try: if not os.path.isdir(token_folder): os.mkdir(token_folder, 0o700) with open(token_file_expand, 'w') as f: f.write(token + '\n') os.chmod(token_file_expand, stat.S_IREAD \| stat.S_IWRITE) except OSError as e: print("> Unable to create folder for token file: ", e) return except IOError as e: print("> Unable to save token file: ", e) return return token_file def table(version): """ Make an html table of the downloads. This is for pasting into the GitHub releases page. See GitHub_release(). """ tarball_formatter_dict = dict(_tarball_format(version)) shortversion = get_sympy_short_version(version) tarball_formatter_dict['version'] = shortversion sha256s = [i.split('\t') for i in _sha256(version, print_=False, local=True).split('\n')] sha256s_dict = {name: sha256 for sha256, name in sha256s} sizes = [i.split('\t') for i in _size(version, print_=False).split('\n')] sizes_dict = {name: size for size, name in sizes} table = [] # https://docs.python.org/2/library/contextlib.html#contextlib.contextmanager. Not # recommended as a real way to generate html, but it works better than # anything else I've tried. @contextmanager def tag(name): table.append("<%s>" % name) yield table.append("</%s>" % name) @contextmanager def a_href(link): table.append("<a href=\"%s\">" % link) yield table.append("</a>") with tag('table'): with tag('tr'): for headname in ["Filename", "Description", "size", "sha256"]: with tag("th"): table.append(headname) for key in descriptions: name = get_tarball_name(key, version) with tag('tr'): with tag('td'): with a_href('https://github.com/sympy/sympy/releases/download/sympy-%s/%s' % (version, name)): with tag('b'): table.append(name) with tag('td'): table.append(descriptions[key].format(tarball_formatter_dict)) with tag('td'): table.append(sizes_dict[name]) with tag('td'): table.append(sha256s_dict[name]) out = ' '.join(table) return out descriptions = OrderedDict([ ('source', "The SymPy source installer.",), ('wheel', "A wheel of the package.",), ('html', '''Html documentation. This is the same as the <a href="https://docs.sympy.org/latest/index.html">online documentation</a>.''',), ('pdf', '''Pdf version of the <a href="https://docs.sympy.org/latest/index.html"> html documentation</a>.''',), ]) def _size(version, print_=True): """ Print the sizes of the release files. Run locally. """ out = run((['du', '-h'] + release_files(version))) out = [i.split() for i in out] out = '\n'.join(["%s\t%s" % (i, os.path.split(j)[1]) for i, j in out]) if print_: print(out) return out def _sha256(version, print_=True, local=False): if local: out = run((['shasum', '-a', '256'] + release_files(version))) else: raise ValueError('Should not get here...') # out = run((['shasum', '-a', '256', '/root/release/'])) # Remove the release/ part for printing. Useful for copy-pasting into the # release notes. out = [i.split() for i in out] out = '\n'.join(["%s\t%s" % (i, os.path.split(j)[1]) for i, j in out]) if print_: print(out) return out def get_tarball_name(file, version): """ Get the name of a tarball file should be one of source-orig: The original name of the source tarball source-orig-notar: The name of the untarred directory source: The source tarball (after renaming) wheel: The wheel html: The name of the html zip html-nozip: The name of the html, without ".zip" pdf-orig: The original name of the pdf file pdf: The name of the pdf file (after renaming) """ doctypename = defaultdict(str, {'html': 'zip', 'pdf': 'pdf'}) if file in {'source-orig', 'source'}: name = 'sympy-{version}.tar.gz' elif file == 'source-orig-notar': name = "sympy-{version}" elif file in {'html', 'pdf', 'html-nozip'}: name = "sympy-docs-{type}-{version}" if file == 'html-nozip': # zip files keep the name of the original zipped directory. See # https://github.com/sympy/sympy/issues/7087. file = 'html' else: name += ".{extension}" elif file == 'pdf-orig': name = "sympy-{version}.pdf" elif file == 'wheel': name = 'sympy-{version}-py3-none-any.whl' else: raise ValueError(file + " is not a recognized argument") ret = name.format(version=version, type=file, extension=doctypename[file]) return ret def release_files(version): """ Returns the list of local release files """ paths = glob.glob('release/release-' + version + '/') if not paths: raise ValueError("No release files found") return paths tarball_name_types = { 'source-orig', 'source-orig-notar', 'source', 'wheel', 'html', 'html-nozip', 'pdf-orig', 'pdf', } # Have to make this lazy so that version can be defined. class _tarball_format(Mapping): def __init__(self, version): self.version = version def __getitem__(self, name): return get_tarball_name(name, self.version) def __iter__(self): return iter(tarball_name_types) def __len__(self): return len(tarball_name_types) if __name__ == "__main__": import sys main(*sys.argv[1:])
abc0d5072086d044d0a72e07da10ac71f49bc233a7b9e497fb0cd627a4ecdd44	#!/usr/bin/env python3 import os from pathlib import Path from subprocess import check_output import unicodedata def main(version, prevversion, outdir): """ Print authors text to put at the bottom of the release notes """ outdir = Path(outdir) authors, authorcount, newauthorcount = get_authors(version, prevversion) authors_text = f"""## Authors The following people contributed at least one patch to this release (names are given in alphabetical order by last name). A total of {authorcount} people contributed to this release. People with a * by their names contributed a patch for the first time for this release; {newauthorcount} people contributed for the first time for this release. Thanks to everyone who contributed to this release! """ authors_lines = [] for name in authors: authors_lines.append("- " + name) authors_text += '\n'.join(authors_lines) # Output to file and to screen with open(outdir / 'authors.txt', 'w') as authorsfile: authorsfile.write(authors_text) print() print(blue("Here are the authors to put at the bottom of the release notes.")) print() print(authors_text) def blue(text): return "\033[34m%s\033[0m" % text def get_authors(version, prevversion): """ Get the list of authors since the previous release Returns the list in alphabetical order by last name. Authors who contributed for the first time for this release will have a star appended to the end of their names. Note: it's a good idea to use ./bin/mailmap_update.py (from the base sympy directory) to make AUTHORS and .mailmap up-to-date first before using this. fab vagrant release does this automatically. """ def lastnamekey(name): """ Sort key to sort by last name Note, we decided to sort based on the last name, because that way is fair. We used to sort by commit count or line number count, but that bumps up people who made lots of maintenance changes like updating mpmath or moving some files around. """ # Note, this will do the wrong thing for people who have multi-word # last names, but there are also people with middle initials. I don't # know of a perfect way to handle everyone. Feel free to fix up the # list by hand. text = name.strip().split()[-1].lower() # Convert things like Čertík to Certik return unicodedata.normalize('NFKD', text).encode('ascii', 'ignore') # The get_previous_version function can be flakey so we require the # previous version to be provided explicitly by the caller. # #old_release_tag = get_previous_version_tag(version) old_release_tag = 'sympy-' + prevversion out = check_output(['git', '--no-pager', 'log', old_release_tag + '..', '--format=%aN']) releaseauthors = set(out.decode('utf-8').strip().split('\n')) out = check_output(['git', '--no-pager', 'log', old_release_tag, '--format=%aN']) priorauthors = set(out.decode('utf-8').strip().split('\n')) releaseauthors = {name.strip() for name in releaseauthors if name.strip()} priorauthors = {name.strip() for name in priorauthors if name.strip()} newauthors = releaseauthors - priorauthors starred_newauthors = {name + "" for name in newauthors} authors = releaseauthors - newauthors \| starred_newauthors return (sorted(authors, key=lastnamekey), len(releaseauthors), len(newauthors)) def get_previous_version_tag(version): """ Get the version of the previous release """ # We try, probably too hard, to portably get the number of the previous # release of SymPy. Our strategy is to look at the git tags. The # following assumptions are made about the git tags: # - The only tags are for releases # - The tags are given the consistent naming: # sympy-major.minor.micro[.rcnumber] # (e.g., sympy-0.7.2 or sympy-0.7.2.rc1) # In particular, it goes back in the tag history and finds the most recent # tag that doesn't contain the current short version number as a substring. shortversion = get_sympy_short_version(version) curcommit = "HEAD" while True: cmdline = f'git describe --abbrev=0 --tags {curcommit}' print(cmdline) curtag = check_output(cmdline.split()).decode('utf-8').strip() if shortversion in curtag: # If the tagged commit is a merge commit, we cannot be sure # that it will go back in the right direction. This almost # never happens, so just error cmdline = f'git rev-list --parents -n 1 {curtag}' print(cmdline) parents = check_output(cmdline.split()).decode('utf-8').strip().split() # rev-list prints the current commit and then all its parents # If the tagged commit is* a merge commit, just comment this # out, and manually make sure `get_previous_version_tag` is correct # assert len(parents) == 2, curtag curcommit = curtag + "^" # The parent of the tagged commit else: print(blue("Using {tag} as the tag for the previous " "release.".format(tag=curtag))) return curtag sys.exit(red("Could not find the tag for the previous release.")) def get_sympy_short_version(version): """ Get the short version of SymPy being released, not including any rc tags (like 0.7.3) """ parts = version.split('.') # Remove rc tags # Handle both 1.0.rc1 and 1.1rc1 if not parts[-1].isdigit(): if parts[-1][0].isdigit(): parts[-1] = parts[-1][0] else: parts.pop(-1) return '.'.join(parts) if __name__ == "__main__": import sys sys.exit(main(*sys.argv[1:]))
66953c3ad451e0fecb0d516f6a5fb525e1cf5b6f9131d83dcdc6bf2377e0cda1	#!/usr/bin/env python3 from os.path import join, basename, normpath from subprocess import check_call def main(version, prevversion, outdir): check_version(version, outdir) run_stage(['bin/mailmap_check.py', '--update-authors']) run_stage(['mkdir', '-p', outdir]) build_release_files('bdist_wheel', 'sympy-%s-py3-none-any.whl', outdir, version) build_release_files('sdist', 'sympy-%s.tar.gz', outdir, version) run_stage(['release/compare_tar_against_git.py', join(outdir, 'sympy-%s.tar.gz' % (version,)), '.']) run_stage(['release/test_install.py', version, outdir]) run_stage(['release/build_docs.py', version, outdir]) run_stage(['release/sha256.py', version, outdir]) run_stage(['release/authors.py', version, prevversion, outdir]) def green(text): return "\033[32m%s\033[0m" % text def red(text): return "\033[31m%s\033[0m" % text def print_header(color, msgs): newlines = '\n' vline = '-' 80 print(color(newlines + vline)) for msg in msgs: print(color(msg)) print(color(vline + newlines)) def run_stage(cmd): cmdline = ' $ %s' % (' '.join(cmd),) print_header(green, 'running:', cmdline) try: check_call(cmd) except Exception as e: print_header(red, 'failed:', cmdline) raise e from None else: print_header(green, 'completed:', cmdline) def build_release_files(cmd, fname, outdir, version): fname = fname % (version,) run_stage(['python', 'setup.py', '-q', cmd]) src = join('dist', fname) dst = join(outdir, fname) run_stage(['mv', src, dst]) def check_version(version, outdir): from sympy.release import __version__ as checked_out_version if version != checked_out_version: msg = "version %s does not match checkout %s" raise AssertionError(msg % (version, checked_out_version)) if basename(normpath(outdir)) != 'release-%s' % (version,): msg = "version %s does not match output directory %s" raise AssertionError(msg % (version, outdir)) if __name__ == "__main__": import sys main(*sys.argv[1:])
7b1c682ce2143ec1e4b8f7967d264864ccf8e3db929b2a8315777766f1179d75	#!/usr/bin/env python3 from subprocess import check_output import sys import os.path def main(tarname, gitroot): """Run this as ./compare_tar_against_git.py TARFILE GITROOT Args ==== TARFILE: Path to the built sdist (sympy-xx.tar.gz) GITROOT: Path ro root of git (dir containing .git) """ compare_tar_against_git(tarname, gitroot) ## TARBALL WHITELISTS # If a file does not end up in the tarball that should, add it to setup.py if # it is Python, or MANIFEST.in if it is not. (There is a command at the top # of setup.py to gather all the things that should be there). # TODO: Also check that this whitelist isn't growing out of date from files # removed from git. # Files that are in git that should not be in the tarball git_whitelist = { # Git specific dotfiles '.gitattributes', '.gitignore', '.mailmap', # Travis and CI '.travis.yml', '.github/workflows/runtests.yml', '.github/workflows/ci-sage.yml', '.github/workflows/comment-on-pr.yml', '.github/workflows/release.yml', '.ci/durations.json', '.ci/generate_durations_log.sh', '.ci/parse_durations_log.py', '.ci/blacklisted.json', '.ci/README.rst', '.github/FUNDING.yml', '.editorconfig', '.coveragerc', 'CODEOWNERS', 'asv.conf.actions.json', 'asv.conf.travis.json', 'coveragerc_travis', 'codecov.yml', 'pytest.ini', 'MANIFEST.in', 'banner.svg', # Code of conduct 'CODE_OF_CONDUCT.md', # Pull request template 'PULL_REQUEST_TEMPLATE.md', # Contributing guide 'CONTRIBUTING.md', # Nothing from bin/ should be shipped unless we intend to install it. Most # of this stuff is for development anyway. To run the tests from the # tarball, use setup.py test, or import sympy and run sympy.test() or # sympy.doctest(). 'bin/adapt_paths.py', 'bin/ask_update.py', 'bin/authors_update.py', 'bin/build_doc.sh', 'bin/coverage_doctest.py', 'bin/coverage_report.py', 'bin/deploy_doc.sh', 'bin/diagnose_imports', 'bin/doctest', 'bin/generate_module_list.py', 'bin/generate_test_list.py', 'bin/get_sympy.py', 'bin/mailmap_update.py', 'bin/py.bench', 'bin/strip_whitespace', 'bin/sympy_time.py', 'bin/sympy_time_cache.py', 'bin/test', 'bin/test_external_imports.py', 'bin/test_executable.py', 'bin/test_import', 'bin/test_import.py', 'bin/test_isolated', 'bin/test_py2_import.py', 'bin/test_setup.py', 'bin/test_submodule_imports.py', 'bin/test_travis.sh', 'bin/test_optional_dependencies.py', 'bin/test_sphinx.sh', 'bin/mailmap_check.py', 'bin/test_symengine.py', 'bin/test_tensorflow.py', # The notebooks are not ready for shipping yet. They need to be cleaned # up, and preferably doctested. See also # https://github.com/sympy/sympy/issues/6039. 'examples/advanced/identitysearch_example.ipynb', 'examples/beginner/plot_advanced.ipynb', 'examples/beginner/plot_colors.ipynb', 'examples/beginner/plot_discont.ipynb', 'examples/beginner/plot_gallery.ipynb', 'examples/beginner/plot_intro.ipynb', 'examples/intermediate/limit_examples_advanced.ipynb', 'examples/intermediate/schwarzschild.ipynb', 'examples/notebooks/density.ipynb', 'examples/notebooks/fidelity.ipynb', 'examples/notebooks/fresnel_integrals.ipynb', 'examples/notebooks/qubits.ipynb', 'examples/notebooks/sho1d_example.ipynb', 'examples/notebooks/spin.ipynb', 'examples/notebooks/trace.ipynb', 'examples/notebooks/Bezout_Dixon_resultant.ipynb', 'examples/notebooks/IntegrationOverPolytopes.ipynb', 'examples/notebooks/Macaulay_resultant.ipynb', 'examples/notebooks/Sylvester_resultant.ipynb', 'examples/notebooks/README.txt', # This stuff :) 'release/.gitignore', 'release/README.md', 'release/Vagrantfile', 'release/fabfile.py', 'release/Dockerfile', 'release/Dockerfile-base', 'release/release.sh', 'release/rever.xsh', 'release/pull_and_run_rever.sh', 'release/compare_tar_against_git.py', 'release/update_docs.py', 'release/aptinstall.sh', 'release/build_docs.py', 'release/github_release.py', 'release/helpers.py', 'release/releasecheck.py', 'release/requirements.txt', 'release/update_requirements.sh', 'release/test_install.py', 'release/sha256.py', 'release/authors.py', 'release/ci_release_script.sh', # This is just a distribute version of setup.py. Used mainly for setup.py # develop, which we don't care about in the release tarball 'setupegg.py', # pytest stuff 'conftest.py', # Encrypted deploy key for deploying dev docs to GitHub 'github_deploy_key.enc', } # Files that should be in the tarball should not be in git tarball_whitelist = { # Generated by setup.py. Contains metadata for PyPI. "PKG-INFO", # Generated by setuptools. More metadata. 'setup.cfg', 'sympy.egg-info/PKG-INFO', 'sympy.egg-info/SOURCES.txt', 'sympy.egg-info/dependency_links.txt', 'sympy.egg-info/requires.txt', 'sympy.egg-info/top_level.txt', 'sympy.egg-info/not-zip-safe', 'sympy.egg-info/entry_points.txt', # Not sure where this is generated from... 'doc/commit_hash.txt', } def blue(text): return "\033[34m%s\033[0m" % text def red(text): return "\033[31m%s\033[0m" % text def run(cmdline, cwd=None): """ Run command in subprocess and get lines of output """ return check_output(cmdline, encoding='utf-8', cwd=cwd).splitlines() def full_path_split(path): """ Function to do a full split on a path. """ # Based on https://stackoverflow.com/a/13505966/161801 rest, tail = os.path.split(path) if not rest or rest == os.path.sep: return (tail,) return full_path_split(rest) + (tail,) def compare_tar_against_git(tarname, gitroot): """ Compare the contents of the tarball against git ls-files See the bottom of the file for the whitelists. """ git_lsfiles = set(i.strip() for i in run('git', 'ls-files', cwd=gitroot)) tar_output_orig = set(run('tar', 'tf', tarname)) tar_output = set() for file in tar_output_orig: # The tar files are like sympy-0.7.3/sympy/__init__.py, and the git # files are like sympy/__init__.py. split_path = full_path_split(file) if split_path[-1]: # Exclude directories, as git ls-files does not include them tar_output.add(os.path.join(split_path[1:])) # print tar_output # print git_lsfiles fail = False print() print(blue("Files in the tarball from git that should not be there:")) print() for line in sorted(tar_output.intersection(git_whitelist)): fail = True print(line) print() print(blue("Files in git but not in the tarball:")) print() for line in sorted(git_lsfiles - tar_output - git_whitelist): fail = True print(line) print() print(blue("Files in the tarball but not in git:")) print() for line in sorted(tar_output - git_lsfiles - tarball_whitelist): fail = True print(line) print() if fail: sys.exit(red("Non-whitelisted files found or not found in the tarball")) if __name__ == "__main__": main(*sys.argv[1:])
2a339ea1920916014ad39fb938207a2643d7735b65c1ff66f9206a4dc9b9add6	""" SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python. It depends on mpmath, and other external libraries may be optionally for things like plotting support. See the webpage for more information and documentation: https://sympy.org """ import sys if sys.version_info < (3, 7): raise ImportError("Python version 3.7 or above is required for SymPy.") del sys try: import mpmath except ImportError: raise ImportError("SymPy now depends on mpmath as an external library. " "See https://docs.sympy.org/latest/install.html#mpmath for more information.") del mpmath from sympy.release import __version__ if 'dev' in __version__: def enable_warnings(): import warnings warnings.filterwarnings('default', '.', DeprecationWarning, module='sympy.') del warnings enable_warnings() del enable_warnings def __sympy_debug(): # helper function so we don't import os globally import os debug_str = os.getenv('SYMPY_DEBUG', 'False') if debug_str in ('True', 'False'): return eval(debug_str) else: raise RuntimeError("unrecognized value for SYMPY_DEBUG: %s" % debug_str) SYMPY_DEBUG = __sympy_debug() # type: bool from .core import (sympify, SympifyError, cacheit, Basic, Atom, preorder_traversal, S, Expr, AtomicExpr, UnevaluatedExpr, Symbol, Wild, Dummy, symbols, var, Number, Float, Rational, Integer, NumberSymbol, RealNumber, igcd, ilcm, seterr, E, I, nan, oo, pi, zoo, AlgebraicNumber, comp, mod_inverse, Pow, integer_nthroot, integer_log, Mul, prod, Add, Mod, Rel, Eq, Ne, Lt, Le, Gt, Ge, Equality, GreaterThan, LessThan, Unequality, StrictGreaterThan, StrictLessThan, vectorize, Lambda, WildFunction, Derivative, diff, FunctionClass, Function, Subs, expand, PoleError, count_ops, expand_mul, expand_log, expand_func, expand_trig, expand_complex, expand_multinomial, nfloat, expand_power_base, expand_power_exp, arity, PrecisionExhausted, N, evalf, Tuple, Dict, gcd_terms, factor_terms, factor_nc, evaluate, Catalan, EulerGamma, GoldenRatio, TribonacciConstant, bottom_up, use, postorder_traversal, default_sort_key, ordered) from .logic import (to_cnf, to_dnf, to_nnf, And, Or, Not, Xor, Nand, Nor, Implies, Equivalent, ITE, POSform, SOPform, simplify_logic, bool_map, true, false, satisfiable) from .assumptions import (AppliedPredicate, Predicate, AssumptionsContext, assuming, Q, ask, register_handler, remove_handler, refine) from .polys import (Poly, PurePoly, poly_from_expr, parallel_poly_from_expr, degree, total_degree, degree_list, LC, LM, LT, pdiv, prem, pquo, pexquo, div, rem, quo, exquo, half_gcdex, gcdex, invert, subresultants, resultant, discriminant, cofactors, gcd_list, gcd, lcm_list, lcm, terms_gcd, trunc, monic, content, primitive, compose, decompose, sturm, gff_list, gff, sqf_norm, sqf_part, sqf_list, sqf, factor_list, factor, intervals, refine_root, count_roots, real_roots, nroots, ground_roots, nth_power_roots_poly, cancel, reduced, groebner, is_zero_dimensional, GroebnerBasis, poly, symmetrize, horner, interpolate, rational_interpolate, viete, together, BasePolynomialError, ExactQuotientFailed, PolynomialDivisionFailed, OperationNotSupported, HeuristicGCDFailed, HomomorphismFailed, IsomorphismFailed, ExtraneousFactors, EvaluationFailed, RefinementFailed, CoercionFailed, NotInvertible, NotReversible, NotAlgebraic, DomainError, PolynomialError, UnificationFailed, GeneratorsError, GeneratorsNeeded, ComputationFailed, UnivariatePolynomialError, MultivariatePolynomialError, PolificationFailed, OptionError, FlagError, minpoly, minimal_polynomial, primitive_element, field_isomorphism, to_number_field, isolate, round_two, prime_decomp, prime_valuation, itermonomials, Monomial, lex, grlex, grevlex, ilex, igrlex, igrevlex, CRootOf, rootof, RootOf, ComplexRootOf, RootSum, roots, Domain, FiniteField, IntegerRing, RationalField, RealField, ComplexField, PythonFiniteField, GMPYFiniteField, PythonIntegerRing, GMPYIntegerRing, PythonRational, GMPYRationalField, AlgebraicField, PolynomialRing, FractionField, ExpressionDomain, FF_python, FF_gmpy, ZZ_python, ZZ_gmpy, QQ_python, QQ_gmpy, GF, FF, ZZ, QQ, ZZ_I, QQ_I, RR, CC, EX, EXRAW, construct_domain, swinnerton_dyer_poly, cyclotomic_poly, symmetric_poly, random_poly, interpolating_poly, jacobi_poly, chebyshevt_poly, chebyshevu_poly, hermite_poly, legendre_poly, laguerre_poly, apart, apart_list, assemble_partfrac_list, Options, ring, xring, vring, sring, field, xfield, vfield, sfield) from .series import (Order, O, limit, Limit, gruntz, series, approximants, residue, EmptySequence, SeqPer, SeqFormula, sequence, SeqAdd, SeqMul, fourier_series, fps, difference_delta, limit_seq) from .functions import (factorial, factorial2, rf, ff, binomial, RisingFactorial, FallingFactorial, subfactorial, carmichael, fibonacci, lucas, motzkin, tribonacci, harmonic, bernoulli, bell, euler, catalan, genocchi, partition, sqrt, root, Min, Max, Id, real_root, Rem, cbrt, re, im, sign, Abs, conjugate, arg, polar_lift, periodic_argument, unbranched_argument, principal_branch, transpose, adjoint, polarify, unpolarify, sin, cos, tan, sec, csc, cot, sinc, asin, acos, atan, asec, acsc, acot, atan2, exp_polar, exp, ln, log, LambertW, sinh, cosh, tanh, coth, sech, csch, asinh, acosh, atanh, acoth, asech, acsch, floor, ceiling, frac, Piecewise, piecewise_fold, erf, erfc, erfi, erf2, erfinv, erfcinv, erf2inv, Ei, expint, E1, li, Li, Si, Ci, Shi, Chi, fresnels, fresnelc, gamma, lowergamma, uppergamma, polygamma, loggamma, digamma, trigamma, multigamma, dirichlet_eta, zeta, lerchphi, polylog, stieltjes, Eijk, LeviCivita, KroneckerDelta, SingularityFunction, DiracDelta, Heaviside, bspline_basis, bspline_basis_set, interpolating_spline, besselj, bessely, besseli, besselk, hankel1, hankel2, jn, yn, jn_zeros, hn1, hn2, airyai, airybi, airyaiprime, airybiprime, marcumq, hyper, meijerg, appellf1, legendre, assoc_legendre, hermite, chebyshevt, chebyshevu, chebyshevu_root, chebyshevt_root, laguerre, assoc_laguerre, gegenbauer, jacobi, jacobi_normalized, Ynm, Ynm_c, Znm, elliptic_k, elliptic_f, elliptic_e, elliptic_pi, beta, mathieus, mathieuc, mathieusprime, mathieucprime, riemann_xi, betainc, betainc_regularized) from .ntheory import (nextprime, prevprime, prime, primepi, primerange, randprime, Sieve, sieve, primorial, cycle_length, composite, compositepi, isprime, divisors, proper_divisors, factorint, multiplicity, perfect_power, pollard_pm1, pollard_rho, primefactors, totient, trailing, divisor_count, proper_divisor_count, divisor_sigma, factorrat, reduced_totient, primenu, primeomega, mersenne_prime_exponent, is_perfect, is_mersenne_prime, is_abundant, is_deficient, is_amicable, abundance, npartitions, is_primitive_root, is_quad_residue, legendre_symbol, jacobi_symbol, n_order, sqrt_mod, quadratic_residues, primitive_root, nthroot_mod, is_nthpow_residue, sqrt_mod_iter, mobius, discrete_log, quadratic_congruence, binomial_coefficients, binomial_coefficients_list, multinomial_coefficients, continued_fraction_periodic, continued_fraction_iterator, continued_fraction_reduce, continued_fraction_convergents, continued_fraction, egyptian_fraction) from .concrete import product, Product, summation, Sum from .discrete import (fft, ifft, ntt, intt, fwht, ifwht, mobius_transform, inverse_mobius_transform, convolution, covering_product, intersecting_product) from .simplify import (simplify, hypersimp, hypersimilar, logcombine, separatevars, posify, besselsimp, kroneckersimp, signsimp, nsimplify, FU, fu, sqrtdenest, cse, epath, EPath, hyperexpand, collect, rcollect, radsimp, collect_const, fraction, numer, denom, trigsimp, exptrigsimp, powsimp, powdenest, combsimp, gammasimp, ratsimp, ratsimpmodprime) from .sets import (Set, Interval, Union, EmptySet, FiniteSet, ProductSet, Intersection, DisjointUnion, imageset, Complement, SymmetricDifference, ImageSet, Range, ComplexRegion, Complexes, Reals, Contains, ConditionSet, Ordinal, OmegaPower, ord0, PowerSet, Naturals, Naturals0, UniversalSet, Integers, Rationals) from .solvers import (solve, solve_linear_system, solve_linear_system_LU, solve_undetermined_coeffs, nsolve, solve_linear, checksol, det_quick, inv_quick, check_assumptions, failing_assumptions, diophantine, rsolve, rsolve_poly, rsolve_ratio, rsolve_hyper, checkodesol, classify_ode, dsolve, homogeneous_order, solve_poly_system, solve_triangulated, pde_separate, pde_separate_add, pde_separate_mul, pdsolve, classify_pde, checkpdesol, ode_order, reduce_inequalities, reduce_abs_inequality, reduce_abs_inequalities, solve_poly_inequality, solve_rational_inequalities, solve_univariate_inequality, decompogen, solveset, linsolve, linear_eq_to_matrix, nonlinsolve, substitution) from .matrices import (ShapeError, NonSquareMatrixError, GramSchmidt, casoratian, diag, eye, hessian, jordan_cell, list2numpy, matrix2numpy, matrix_multiply_elementwise, ones, randMatrix, rot_axis1, rot_axis2, rot_axis3, symarray, wronskian, zeros, MutableDenseMatrix, DeferredVector, MatrixBase, Matrix, MutableMatrix, MutableSparseMatrix, banded, ImmutableDenseMatrix, ImmutableSparseMatrix, ImmutableMatrix, SparseMatrix, MatrixSlice, BlockDiagMatrix, BlockMatrix, FunctionMatrix, Identity, Inverse, MatAdd, MatMul, MatPow, MatrixExpr, MatrixSymbol, Trace, Transpose, ZeroMatrix, OneMatrix, blockcut, block_collapse, matrix_symbols, Adjoint, hadamard_product, HadamardProduct, HadamardPower, Determinant, det, diagonalize_vector, DiagMatrix, DiagonalMatrix, DiagonalOf, trace, DotProduct, kronecker_product, KroneckerProduct, PermutationMatrix, MatrixPermute, Permanent, per) from .geometry import (Point, Point2D, Point3D, Line, Ray, Segment, Line2D, Segment2D, Ray2D, Line3D, Segment3D, Ray3D, Plane, Ellipse, Circle, Polygon, RegularPolygon, Triangle, rad, deg, are_similar, centroid, convex_hull, idiff, intersection, closest_points, farthest_points, GeometryError, Curve, Parabola) from .utilities import (flatten, group, take, subsets, variations, numbered_symbols, cartes, capture, dict_merge, prefixes, postfixes, sift, topological_sort, unflatten, has_dups, has_variety, reshape, rotations, filldedent, lambdify, source, threaded, xthreaded, public, memoize_property, timed) from .integrals import (integrate, Integral, line_integrate, mellin_transform, inverse_mellin_transform, MellinTransform, InverseMellinTransform, laplace_transform, inverse_laplace_transform, LaplaceTransform, InverseLaplaceTransform, fourier_transform, inverse_fourier_transform, FourierTransform, InverseFourierTransform, sine_transform, inverse_sine_transform, SineTransform, InverseSineTransform, cosine_transform, inverse_cosine_transform, CosineTransform, InverseCosineTransform, hankel_transform, inverse_hankel_transform, HankelTransform, InverseHankelTransform, singularityintegrate) from .tensor import (IndexedBase, Idx, Indexed, get_contraction_structure, get_indices, shape, MutableDenseNDimArray, ImmutableDenseNDimArray, MutableSparseNDimArray, ImmutableSparseNDimArray, NDimArray, tensorproduct, tensorcontraction, tensordiagonal, derive_by_array, permutedims, Array, DenseNDimArray, SparseNDimArray) from .parsing import parse_expr from .calculus import (euler_equations, singularities, is_increasing, is_strictly_increasing, is_decreasing, is_strictly_decreasing, is_monotonic, finite_diff_weights, apply_finite_diff, differentiate_finite, periodicity, not_empty_in, AccumBounds, is_convex, stationary_points, minimum, maximum) from .algebras import Quaternion from .printing import (pager_print, pretty, pretty_print, pprint, pprint_use_unicode, pprint_try_use_unicode, latex, print_latex, multiline_latex, mathml, print_mathml, python, print_python, pycode, ccode, print_ccode, glsl_code, print_glsl, cxxcode, fcode, print_fcode, rcode, print_rcode, jscode, print_jscode, julia_code, mathematica_code, octave_code, rust_code, print_gtk, preview, srepr, print_tree, StrPrinter, sstr, sstrrepr, TableForm, dotprint, maple_code, print_maple_code) from .testing import test, doctest # This module causes conflicts with other modules: # from .stats import * # Adds about .04-.05 seconds of import time # from combinatorics import * # This module is slow to import: #from physics import units from .plotting import plot, textplot, plot_backends, plot_implicit, plot_parametric from .interactive import init_session, init_printing, interactive_traversal evalf._create_evalf_table() __all__ = [ # sympy.core 'sympify', 'SympifyError', 'cacheit', 'Basic', 'Atom', 'preorder_traversal', 'S', 'Expr', 'AtomicExpr', 'UnevaluatedExpr', 'Symbol', 'Wild', 'Dummy', 'symbols', 'var', 'Number', 'Float', 'Rational', 'Integer', 'NumberSymbol', 'RealNumber', 'igcd', 'ilcm', 'seterr', 'E', 'I', 'nan', 'oo', 'pi', 'zoo', 'AlgebraicNumber', 'comp', 'mod_inverse', 'Pow', 'integer_nthroot', 'integer_log', 'Mul', 'prod', 'Add', 'Mod', 'Rel', 'Eq', 'Ne', 'Lt', 'Le', 'Gt', 'Ge', 'Equality', 'GreaterThan', 'LessThan', 'Unequality', 'StrictGreaterThan', 'StrictLessThan', 'vectorize', 'Lambda', 'WildFunction', 'Derivative', 'diff', 'FunctionClass', 'Function', 'Subs', 'expand', 'PoleError', 'count_ops', 'expand_mul', 'expand_log', 'expand_func', 'expand_trig', 'expand_complex', 'expand_multinomial', 'nfloat', 'expand_power_base', 'expand_power_exp', 'arity', 'PrecisionExhausted', 'N', 'evalf', 'Tuple', 'Dict', 'gcd_terms', 'factor_terms', 'factor_nc', 'evaluate', 'Catalan', 'EulerGamma', 'GoldenRatio', 'TribonacciConstant', 'bottom_up', 'use', 'postorder_traversal', 'default_sort_key', 'ordered', # sympy.logic 'to_cnf', 'to_dnf', 'to_nnf', 'And', 'Or', 'Not', 'Xor', 'Nand', 'Nor', 'Implies', 'Equivalent', 'ITE', 'POSform', 'SOPform', 'simplify_logic', 'bool_map', 'true', 'false', 'satisfiable', # sympy.assumptions 'AppliedPredicate', 'Predicate', 'AssumptionsContext', 'assuming', 'Q', 'ask', 'register_handler', 'remove_handler', 'refine', # sympy.polys 'Poly', 'PurePoly', 'poly_from_expr', 'parallel_poly_from_expr', 'degree', 'total_degree', 'degree_list', 'LC', 'LM', 'LT', 'pdiv', 'prem', 'pquo', 'pexquo', 'div', 'rem', 'quo', 'exquo', 'half_gcdex', 'gcdex', 'invert', 'subresultants', 'resultant', 'discriminant', 'cofactors', 'gcd_list', 'gcd', 'lcm_list', 'lcm', 'terms_gcd', 'trunc', 'monic', 'content', 'primitive', 'compose', 'decompose', 'sturm', 'gff_list', 'gff', 'sqf_norm', 'sqf_part', 'sqf_list', 'sqf', 'factor_list', 'factor', 'intervals', 'refine_root', 'count_roots', 'real_roots', 'nroots', 'ground_roots', 'nth_power_roots_poly', 'cancel', 'reduced', 'groebner', 'is_zero_dimensional', 'GroebnerBasis', 'poly', 'symmetrize', 'horner', 'interpolate', 'rational_interpolate', 'viete', 'together', 'BasePolynomialError', 'ExactQuotientFailed', 'PolynomialDivisionFailed', 'OperationNotSupported', 'HeuristicGCDFailed', 'HomomorphismFailed', 'IsomorphismFailed', 'ExtraneousFactors', 'EvaluationFailed', 'RefinementFailed', 'CoercionFailed', 'NotInvertible', 'NotReversible', 'NotAlgebraic', 'DomainError', 'PolynomialError', 'UnificationFailed', 'GeneratorsError', 'GeneratorsNeeded', 'ComputationFailed', 'UnivariatePolynomialError', 'MultivariatePolynomialError', 'PolificationFailed', 'OptionError', 'FlagError', 'minpoly', 'minimal_polynomial', 'primitive_element', 'field_isomorphism', 'to_number_field', 'isolate', 'round_two', 'prime_decomp', 'prime_valuation', 'itermonomials', 'Monomial', 'lex', 'grlex', 'grevlex', 'ilex', 'igrlex', 'igrevlex', 'CRootOf', 'rootof', 'RootOf', 'ComplexRootOf', 'RootSum', 'roots', 'Domain', 'FiniteField', 'IntegerRing', 'RationalField', 'RealField', 'ComplexField', 'PythonFiniteField', 'GMPYFiniteField', 'PythonIntegerRing', 'GMPYIntegerRing', 'PythonRational', 'GMPYRationalField', 'AlgebraicField', 'PolynomialRing', 'FractionField', 'ExpressionDomain', 'FF_python', 'FF_gmpy', 'ZZ_python', 'ZZ_gmpy', 'QQ_python', 'QQ_gmpy', 'GF', 'FF', 'ZZ', 'QQ', 'ZZ_I', 'QQ_I', 'RR', 'CC', 'EX', 'EXRAW', 'construct_domain', 'swinnerton_dyer_poly', 'cyclotomic_poly', 'symmetric_poly', 'random_poly', 'interpolating_poly', 'jacobi_poly', 'chebyshevt_poly', 'chebyshevu_poly', 'hermite_poly', 'legendre_poly', 'laguerre_poly', 'apart', 'apart_list', 'assemble_partfrac_list', 'Options', 'ring', 'xring', 'vring', 'sring', 'field', 'xfield', 'vfield', 'sfield', # sympy.series 'Order', 'O', 'limit', 'Limit', 'gruntz', 'series', 'approximants', 'residue', 'EmptySequence', 'SeqPer', 'SeqFormula', 'sequence', 'SeqAdd', 'SeqMul', 'fourier_series', 'fps', 'difference_delta', 'limit_seq', # sympy.functions 'factorial', 'factorial2', 'rf', 'ff', 'binomial', 'RisingFactorial', 'FallingFactorial', 'subfactorial', 'carmichael', 'fibonacci', 'lucas', 'motzkin', 'tribonacci', 'harmonic', 'bernoulli', 'bell', 'euler', 'catalan', 'genocchi', 'partition', 'sqrt', 'root', 'Min', 'Max', 'Id', 'real_root', 'Rem', 'cbrt', 're', 'im', 'sign', 'Abs', 'conjugate', 'arg', 'polar_lift', 'periodic_argument', 'unbranched_argument', 'principal_branch', 'transpose', 'adjoint', 'polarify', 'unpolarify', 'sin', 'cos', 'tan', 'sec', 'csc', 'cot', 'sinc', 'asin', 'acos', 'atan', 'asec', 'acsc', 'acot', 'atan2', 'exp_polar', 'exp', 'ln', 'log', 'LambertW', 'sinh', 'cosh', 'tanh', 'coth', 'sech', 'csch', 'asinh', 'acosh', 'atanh', 'acoth', 'asech', 'acsch', 'floor', 'ceiling', 'frac', 'Piecewise', 'piecewise_fold', 'erf', 'erfc', 'erfi', 'erf2', 'erfinv', 'erfcinv', 'erf2inv', 'Ei', 'expint', 'E1', 'li', 'Li', 'Si', 'Ci', 'Shi', 'Chi', 'fresnels', 'fresnelc', 'gamma', 'lowergamma', 'uppergamma', 'polygamma', 'loggamma', 'digamma', 'trigamma', 'multigamma', 'dirichlet_eta', 'zeta', 'lerchphi', 'polylog', 'stieltjes', 'Eijk', 'LeviCivita', 'KroneckerDelta', 'SingularityFunction', 'DiracDelta', 'Heaviside', 'bspline_basis', 'bspline_basis_set', 'interpolating_spline', 'besselj', 'bessely', 'besseli', 'besselk', 'hankel1', 'hankel2', 'jn', 'yn', 'jn_zeros', 'hn1', 'hn2', 'airyai', 'airybi', 'airyaiprime', 'airybiprime', 'marcumq', 'hyper', 'meijerg', 'appellf1', 'legendre', 'assoc_legendre', 'hermite', 'chebyshevt', 'chebyshevu', 'chebyshevu_root', 'chebyshevt_root', 'laguerre', 'assoc_laguerre', 'gegenbauer', 'jacobi', 'jacobi_normalized', 'Ynm', 'Ynm_c', 'Znm', 'elliptic_k', 'elliptic_f', 'elliptic_e', 'elliptic_pi', 'beta', 'mathieus', 'mathieuc', 'mathieusprime', 'mathieucprime', 'riemann_xi','betainc', 'betainc_regularized', # sympy.ntheory 'nextprime', 'prevprime', 'prime', 'primepi', 'primerange', 'randprime', 'Sieve', 'sieve', 'primorial', 'cycle_length', 'composite', 'compositepi', 'isprime', 'divisors', 'proper_divisors', 'factorint', 'multiplicity', 'perfect_power', 'pollard_pm1', 'pollard_rho', 'primefactors', 'totient', 'trailing', 'divisor_count', 'proper_divisor_count', 'divisor_sigma', 'factorrat', 'reduced_totient', 'primenu', 'primeomega', 'mersenne_prime_exponent', 'is_perfect', 'is_mersenne_prime', 'is_abundant', 'is_deficient', 'is_amicable', 'abundance', 'npartitions', 'is_primitive_root', 'is_quad_residue', 'legendre_symbol', 'jacobi_symbol', 'n_order', 'sqrt_mod', 'quadratic_residues', 'primitive_root', 'nthroot_mod', 'is_nthpow_residue', 'sqrt_mod_iter', 'mobius', 'discrete_log', 'quadratic_congruence', 'binomial_coefficients', 'binomial_coefficients_list', 'multinomial_coefficients', 'continued_fraction_periodic', 'continued_fraction_iterator', 'continued_fraction_reduce', 'continued_fraction_convergents', 'continued_fraction', 'egyptian_fraction', # sympy.concrete 'product', 'Product', 'summation', 'Sum', # sympy.discrete 'fft', 'ifft', 'ntt', 'intt', 'fwht', 'ifwht', 'mobius_transform', 'inverse_mobius_transform', 'convolution', 'covering_product', 'intersecting_product', # sympy.simplify 'simplify', 'hypersimp', 'hypersimilar', 'logcombine', 'separatevars', 'posify', 'besselsimp', 'kroneckersimp', 'signsimp', 'nsimplify', 'FU', 'fu', 'sqrtdenest', 'cse', 'epath', 'EPath', 'hyperexpand', 'collect', 'rcollect', 'radsimp', 'collect_const', 'fraction', 'numer', 'denom', 'trigsimp', 'exptrigsimp', 'powsimp', 'powdenest', 'combsimp', 'gammasimp', 'ratsimp', 'ratsimpmodprime', # sympy.sets 'Set', 'Interval', 'Union', 'EmptySet', 'FiniteSet', 'ProductSet', 'Intersection', 'imageset', 'DisjointUnion', 'Complement', 'SymmetricDifference', 'ImageSet', 'Range', 'ComplexRegion', 'Reals', 'Contains', 'ConditionSet', 'Ordinal', 'OmegaPower', 'ord0', 'PowerSet', 'Naturals', 'Naturals0', 'UniversalSet', 'Integers', 'Rationals', 'Complexes', # sympy.solvers 'solve', 'solve_linear_system', 'solve_linear_system_LU', 'solve_undetermined_coeffs', 'nsolve', 'solve_linear', 'checksol', 'det_quick', 'inv_quick', 'check_assumptions', 'failing_assumptions', 'diophantine', 'rsolve', 'rsolve_poly', 'rsolve_ratio', 'rsolve_hyper', 'checkodesol', 'classify_ode', 'dsolve', 'homogeneous_order', 'solve_poly_system', 'solve_triangulated', 'pde_separate', 'pde_separate_add', 'pde_separate_mul', 'pdsolve', 'classify_pde', 'checkpdesol', 'ode_order', 'reduce_inequalities', 'reduce_abs_inequality', 'reduce_abs_inequalities', 'solve_poly_inequality', 'solve_rational_inequalities', 'solve_univariate_inequality', 'decompogen', 'solveset', 'linsolve', 'linear_eq_to_matrix', 'nonlinsolve', 'substitution', # sympy.matrices 'ShapeError', 'NonSquareMatrixError', 'GramSchmidt', 'casoratian', 'diag', 'eye', 'hessian', 'jordan_cell', 'list2numpy', 'matrix2numpy', 'matrix_multiply_elementwise', 'ones', 'randMatrix', 'rot_axis1', 'rot_axis2', 'rot_axis3', 'symarray', 'wronskian', 'zeros', 'MutableDenseMatrix', 'DeferredVector', 'MatrixBase', 'Matrix', 'MutableMatrix', 'MutableSparseMatrix', 'banded', 'ImmutableDenseMatrix', 'ImmutableSparseMatrix', 'ImmutableMatrix', 'SparseMatrix', 'MatrixSlice', 'BlockDiagMatrix', 'BlockMatrix', 'FunctionMatrix', 'Identity', 'Inverse', 'MatAdd', 'MatMul', 'MatPow', 'MatrixExpr', 'MatrixSymbol', 'Trace', 'Transpose', 'ZeroMatrix', 'OneMatrix', 'blockcut', 'block_collapse', 'matrix_symbols', 'Adjoint', 'hadamard_product', 'HadamardProduct', 'HadamardPower', 'Determinant', 'det', 'diagonalize_vector', 'DiagMatrix', 'DiagonalMatrix', 'DiagonalOf', 'trace', 'DotProduct', 'kronecker_product', 'KroneckerProduct', 'PermutationMatrix', 'MatrixPermute', 'Permanent', 'per', # sympy.geometry 'Point', 'Point2D', 'Point3D', 'Line', 'Ray', 'Segment', 'Line2D', 'Segment2D', 'Ray2D', 'Line3D', 'Segment3D', 'Ray3D', 'Plane', 'Ellipse', 'Circle', 'Polygon', 'RegularPolygon', 'Triangle', 'rad', 'deg', 'are_similar', 'centroid', 'convex_hull', 'idiff', 'intersection', 'closest_points', 'farthest_points', 'GeometryError', 'Curve', 'Parabola', # sympy.utilities 'flatten', 'group', 'take', 'subsets', 'variations', 'numbered_symbols', 'cartes', 'capture', 'dict_merge', 'prefixes', 'postfixes', 'sift', 'topological_sort', 'unflatten', 'has_dups', 'has_variety', 'reshape', 'rotations', 'filldedent', 'lambdify', 'source', 'threaded', 'xthreaded', 'public', 'memoize_property', 'timed', # sympy.integrals 'integrate', 'Integral', 'line_integrate', 'mellin_transform', 'inverse_mellin_transform', 'MellinTransform', 'InverseMellinTransform', 'laplace_transform', 'inverse_laplace_transform', 'LaplaceTransform', 'InverseLaplaceTransform', 'fourier_transform', 'inverse_fourier_transform', 'FourierTransform', 'InverseFourierTransform', 'sine_transform', 'inverse_sine_transform', 'SineTransform', 'InverseSineTransform', 'cosine_transform', 'inverse_cosine_transform', 'CosineTransform', 'InverseCosineTransform', 'hankel_transform', 'inverse_hankel_transform', 'HankelTransform', 'InverseHankelTransform', 'singularityintegrate', # sympy.tensor 'IndexedBase', 'Idx', 'Indexed', 'get_contraction_structure', 'get_indices', 'shape', 'MutableDenseNDimArray', 'ImmutableDenseNDimArray', 'MutableSparseNDimArray', 'ImmutableSparseNDimArray', 'NDimArray', 'tensorproduct', 'tensorcontraction', 'tensordiagonal', 'derive_by_array', 'permutedims', 'Array', 'DenseNDimArray', 'SparseNDimArray', # sympy.parsing 'parse_expr', # sympy.calculus 'euler_equations', 'singularities', 'is_increasing', 'is_strictly_increasing', 'is_decreasing', 'is_strictly_decreasing', 'is_monotonic', 'finite_diff_weights', 'apply_finite_diff', 'differentiate_finite', 'periodicity', 'not_empty_in', 'AccumBounds', 'is_convex', 'stationary_points', 'minimum', 'maximum', # sympy.algebras 'Quaternion', # sympy.printing 'pager_print', 'pretty', 'pretty_print', 'pprint', 'pprint_use_unicode', 'pprint_try_use_unicode', 'latex', 'print_latex', 'multiline_latex', 'mathml', 'print_mathml', 'python', 'print_python', 'pycode', 'ccode', 'print_ccode', 'glsl_code', 'print_glsl', 'cxxcode', 'fcode', 'print_fcode', 'rcode', 'print_rcode', 'jscode', 'print_jscode', 'julia_code', 'mathematica_code', 'octave_code', 'rust_code', 'print_gtk', 'preview', 'srepr', 'print_tree', 'StrPrinter', 'sstr', 'sstrrepr', 'TableForm', 'dotprint', 'maple_code', 'print_maple_code', # sympy.plotting 'plot', 'textplot', 'plot_backends', 'plot_implicit', 'plot_parametric', # sympy.interactive 'init_session', 'init_printing', 'interactive_traversal', # sympy.testing 'test', 'doctest', ] #===========================================================================# # # # XXX: The names below were importable before SymPy 1.6 using # # # # from sympy import * # # # # This happened implicitly because there was no __all__ defined in this # # __init__.py file. Not every package is imported. The list matches what # # would have been imported before. It is possible that these packages will # # not be imported by a star-import from sympy in future. # # # #===========================================================================# __all__.extend(( 'algebras', 'assumptions', 'calculus', 'concrete', 'discrete', 'external', 'functions', 'geometry', 'interactive', 'multipledispatch', 'ntheory', 'parsing', 'plotting', 'polys', 'printing', 'release', 'strategies', 'tensor', 'utilities', ))
0016f6df12d1c936c34d5a4c17022bd8ef29f87b03f4d665c43d6cb526fc7e59	__version__ = "1.11.dev"
85f8d08246eb3dbe61b0bf909acee460c2f89308b358b268140cee612eb95f55	#!/usr/bin/env python """Matplotlib 2D plotting example Demonstrates plotting with matplotlib. """ import sys from sample import sample from sympy import sqrt, Symbol from sympy.utilities.iterables import is_sequence from sympy.external import import_module def mplot2d(f, var, , show=True): """ Plot a 2d function using matplotlib/Tk. """ import warnings warnings.filterwarnings("ignore", r"Could not match \S") p = import_module('pylab') if not p: sys.exit("Matplotlib is required to use mplot2d.") if not is_sequence(f): f = [f, ] for f_i in f: x, y = sample(f_i, var) p.plot(x, y) p.draw() if show: p.show() def main(): x = Symbol('x') # mplot2d(log(x), (x, 0, 2, 100)) # mplot2d([sin(x), -sin(x)], (x, float(-2pi), float(2*pi), 50)) mplot2d([sqrt(x), -sqrt(x), sqrt(-x), -sqrt(-x)], (x, -40.0, 40.0, 80)) if __name__ == "__main__": main()
caa6fcbd1132bc892780705a3402d82193d829de0e6bcaba1c2256949f13f404	# # SymPy documentation build configuration file, created by # sphinx-quickstart.py on Sat Mar 22 19:34:32 2008. # # This file is execfile()d with the current directory set to its containing dir. # # The contents of this file are pickled, so don't put values in the namespace # that aren't pickleable (module imports are okay, they're removed automatically). # # All configuration values have a default value; values that are commented out # serve to show the default value. import sys import inspect import os import subprocess from datetime import datetime # Make sure we import sympy from git sys.path.insert(0, os.path.abspath('../..')) import sympy # If your extensions are in another directory, add it here. sys.path = ['ext'] + sys.path # General configuration # --------------------- # Add any Sphinx extension module names here, as strings. They can be extensions # coming with Sphinx (named 'sphinx.addons.*') or your custom ones. extensions = ['sphinx.ext.autodoc', 'sphinx.ext.linkcode', 'sphinx_math_dollar', 'sphinx.ext.mathjax', 'numpydoc', 'sympylive', 'sphinx_reredirects', 'sphinx.ext.graphviz', 'matplotlib.sphinxext.plot_directive', 'myst_parser', 'sphinx.ext.intersphinx' ] redirects = { "install.rst": "guides/getting_started/install.html", "documentation-style-guide.rst": "guides/contributing/documentation-style-guide.html", "gotchas.rst": "explanation/gotchas.html", "special_topics/classification.rst": "explanation/classification.html", "special_topics/finite_diff_derivatives.rst": "explanation/finite_diff_derivatives.html", "special_topics/intro.rst": "explanation/index.html", "special_topics/index.rst": "explanation/index.html", "modules/index.rst": "reference/public/index.html", "modules/physics/index.rst": "reference/physics/index.html", } # Use this to use pngmath instead #extensions = ['sphinx.ext.autodoc', 'sphinx.ext.viewcode', 'sphinx.ext.pngmath', ] # Enable warnings for all bad cross references. These are turned into errors # with the -W flag in the Makefile. nitpicky = True nitpick_ignore = [ ('py:class', 'sympy.logic.boolalg.Boolean') ] # To stop docstrings inheritance. autodoc_inherit_docstrings = False # See https://www.sympy.org/sphinx-math-dollar/ mathjax3_config = { "tex": { "inlineMath": [['\$', '\$']], "displayMath": [["\\[", "\\]"]], } } # Myst configuration (for .md files). See # https://myst-parser.readthedocs.io/en/latest/syntax/optional.html myst_enable_extensions = ["dollarmath", "linkify"] myst_heading_anchors = 2 # myst_update_mathjax = False # Add any paths that contain templates here, relative to this directory. templates_path = ['_templates'] # The suffix of source filenames. source_suffix = '.rst' # The master toctree document. master_doc = 'index' suppress_warnings = ['ref.citation', 'ref.footnote'] # General substitutions. project = 'SymPy' copyright = '{} SymPy Development Team'.format(datetime.utcnow().year) # The default replacements for \|version\| and \|release\|, also used in various # other places throughout the built documents. # # The short X.Y version. version = sympy.__version__ # The full version, including alpha/beta/rc tags. release = version # There are two options for replacing \|today\|: either, you set today to some # non-false value, then it is used: #today = '' # Else, today_fmt is used as the format for a strftime call. today_fmt = '%B %d, %Y' # List of documents that shouldn't be included in the build. #unused_docs = [] # If true, '()' will be appended to :func: etc. cross-reference text. #add_function_parentheses = True # If true, the current module name will be prepended to all description # unit titles (such as .. function::). #add_module_names = True # If true, sectionauthor and moduleauthor directives will be shown in the # output. They are ignored by default. #show_authors = False # The name of the Pygments (syntax highlighting) style to use. pygments_style = 'sphinx' # Don't show the source code hyperlinks when using matplotlib plot directive. plot_html_show_source_link = False # Options for HTML output # ----------------------- # The style sheet to use for HTML and HTML Help pages. A file of that name # must exist either in Sphinx' static/ path, or in one of the custom paths # given in html_static_path. html_style = 'default.css' # Add any paths that contain custom static files (such as style sheets) here, # relative to this directory. They are copied after the builtin static files, # so a file named "default.css" will overwrite the builtin "default.css". html_static_path = ['_static'] # If not '', a 'Last updated on:' timestamp is inserted at every page bottom, # using the given strftime format. html_last_updated_fmt = '%b %d, %Y' # was classic html_theme = "classic" html_logo = '_static/sympylogo.png' html_favicon = '../_build/logo/sympy-notailtext-favicon.ico' # See http://www.sphinx-doc.org/en/master/theming.html#builtin-themes # If true, SmartyPants will be used to convert quotes and dashes to # typographically correct entities. #html_use_smartypants = True # Content template for the index page. #html_index = '' # Custom sidebar templates, maps document names to template names. #html_sidebars = {} # Additional templates that should be rendered to pages, maps page names to # template names. #html_additional_pages = {} # If false, no module index is generated. #html_use_modindex = True html_domain_indices = ['py-modindex'] # If true, the reST sources are included in the HTML build as _sources/<name>. #html_copy_source = True # Output file base name for HTML help builder. htmlhelp_basename = 'SymPydoc' language = 'en' # Options for LaTeX output # ------------------------ # The paper size ('letter' or 'a4'). #latex_paper_size = 'letter' # The font size ('10pt', '11pt' or '12pt'). #latex_font_size = '10pt' # Grouping the document tree into LaTeX files. List of tuples # (source start file, target name, title, author, document class [howto/manual], toctree_only). # toctree_only is set to True so that the start file document itself is not included in the # output, only the documents referenced by it via TOC trees. The extra stuff in the master # document is intended to show up in the HTML, but doesn't really belong in the LaTeX output. latex_documents = [('index', 'sympy-%s.tex' % release, 'SymPy Documentation', 'SymPy Development Team', 'manual', True)] # Additional stuff for the LaTeX preamble. # Tweaked to work with XeTeX. latex_elements = { 'babel': '', 'fontenc': r''' % Define version of \LaTeX that is usable in math mode \let\OldLaTeX\LaTeX \renewcommand{\LaTeX}{\text{\OldLaTeX}} \usepackage{bm} \usepackage{amssymb} \usepackage{fontspec} \usepackage[english]{babel} \defaultfontfeatures{Mapping=tex-text} \setmainfont{DejaVu Serif} \setsansfont{DejaVu Sans} \setmonofont{DejaVu Sans Mono} ''', 'fontpkg': '', 'inputenc': '', 'utf8extra': '', 'preamble': r''' ''' } # SymPy logo on title page html_logo = '_static/sympylogo.png' latex_logo = '_static/sympylogo_big.png' # Documents to append as an appendix to all manuals. #latex_appendices = [] # Show page numbers next to internal references latex_show_pagerefs = True # We use False otherwise the module index gets generated twice. latex_use_modindex = False default_role = 'math' pngmath_divpng_args = ['-gamma 1.5', '-D 110'] # Note, this is ignored by the mathjax extension # Any \newcommand should be defined in the file pngmath_latex_preamble = '\\usepackage{amsmath}\n' \ '\\usepackage{bm}\n' \ '\\usepackage{amsfonts}\n' \ '\\usepackage{amssymb}\n' \ '\\setlength{\\parindent}{0pt}\n' texinfo_documents = [ (master_doc, 'sympy', 'SymPy Documentation', 'SymPy Development Team', 'SymPy', 'Computer algebra system (CAS) in Python', 'Programming', 1), ] # Use svg for graphviz graphviz_output_format = 'svg' # Enable links to other packages intersphinx_mapping = { 'matplotlib': ('https://matplotlib.org/stable/', None) } # Requried for linkcode extension. # Get commit hash from the external file. commit_hash_filepath = '../commit_hash.txt' commit_hash = None if os.path.isfile(commit_hash_filepath): with open(commit_hash_filepath) as f: commit_hash = f.readline() # Get commit hash from the external file. if not commit_hash: try: commit_hash = subprocess.check_output(['git', 'rev-parse', 'HEAD']) commit_hash = commit_hash.decode('ascii') commit_hash = commit_hash.rstrip() except: import warnings warnings.warn( "Failed to get the git commit hash as the command " \ "'git rev-parse HEAD' is not working. The commit hash will be " \ "assumed as the SymPy master, but the lines may be misleading " \ "or nonexistent as it is not the correct branch the doc is " \ "built with. Check your installation of 'git' if you want to " \ "resolve this warning.") commit_hash = 'master' fork = 'sympy' blobpath = \ "https://github.com/{}/sympy/blob/{}/sympy/".format(fork, commit_hash) def linkcode_resolve(domain, info): """Determine the URL corresponding to Python object.""" if domain != 'py': return modname = info['module'] fullname = info['fullname'] submod = sys.modules.get(modname) if submod is None: return obj = submod for part in fullname.split('.'): try: obj = getattr(obj, part) except Exception: return # strip decorators, which would resolve to the source of the decorator # possibly an upstream bug in getsourcefile, bpo-1764286 try: unwrap = inspect.unwrap except AttributeError: pass else: obj = unwrap(obj) try: fn = inspect.getsourcefile(obj) except Exception: fn = None if not fn: return try: source, lineno = inspect.getsourcelines(obj) except Exception: lineno = None if lineno: linespec = "#L%d-L%d" % (lineno, lineno + len(source) - 1) else: linespec = "" fn = os.path.relpath(fn, start=os.path.dirname(sympy.__file__)) return blobpath + fn + linespec
7dd094998411a88c6d803ffb63bee4121017e06f520bb2340b80c4117d2cfa65	""" Continuous Random Variables - Prebuilt variables Contains ======== Arcsin Benini Beta BetaNoncentral BetaPrime BoundedPareto Cauchy Chi ChiNoncentral ChiSquared Dagum Erlang ExGaussian Exponential ExponentialPower FDistribution FisherZ Frechet Gamma GammaInverse Gumbel Gompertz Kumaraswamy Laplace Levy LogCauchy Logistic LogLogistic LogitNormal LogNormal Lomax Maxwell Moyal Nakagami Normal Pareto PowerFunction QuadraticU RaisedCosine Rayleigh Reciprocal ShiftedGompertz StudentT Trapezoidal Triangular Uniform UniformSum VonMises Wald Weibull WignerSemicircle """ from sympy.functions.elementary.exponential import exp from sympy.functions.elementary.trigonometric import (atan, cos, sin, tan) from sympy.functions.special.bessel import (besseli, besselj, besselk) from sympy.functions.special.beta_functions import beta as beta_fn from sympy.concrete.summations import Sum from sympy.core.basic import Basic from sympy.core.function import Lambda from sympy.core.numbers import (I, Rational, pi) from sympy.core.relational import (Eq, Ne) from sympy.core.singleton import S from sympy.core.symbol import Dummy from sympy.core.sympify import sympify from sympy.functions.combinatorial.factorials import (binomial, factorial) from sympy.functions.elementary.complexes import (Abs, sign) from sympy.functions.elementary.exponential import log from sympy.functions.elementary.hyperbolic import sinh from sympy.functions.elementary.integers import floor from sympy.functions.elementary.miscellaneous import sqrt, Max, Min from sympy.functions.elementary.piecewise import Piecewise from sympy.functions.elementary.trigonometric import asin from sympy.functions.special.error_functions import (erf, erfc, erfi, erfinv, expint) from sympy.functions.special.gamma_functions import (gamma, lowergamma, uppergamma) from sympy.functions.special.hyper import hyper from sympy.integrals.integrals import integrate from sympy.logic.boolalg import And from sympy.sets.sets import Interval from sympy.matrices import MatrixBase from sympy.stats.crv import SingleContinuousPSpace, SingleContinuousDistribution from sympy.stats.rv import _value_check, is_random oo = S.Infinity __all__ = ['ContinuousRV', 'Arcsin', 'Benini', 'Beta', 'BetaNoncentral', 'BetaPrime', 'BoundedPareto', 'Cauchy', 'Chi', 'ChiNoncentral', 'ChiSquared', 'Dagum', 'Erlang', 'ExGaussian', 'Exponential', 'ExponentialPower', 'FDistribution', 'FisherZ', 'Frechet', 'Gamma', 'GammaInverse', 'Gompertz', 'Gumbel', 'Kumaraswamy', 'Laplace', 'Levy', 'LogCauchy', 'Logistic', 'LogLogistic', 'LogitNormal', 'LogNormal', 'Lomax', 'Maxwell', 'Moyal', 'Nakagami', 'Normal', 'GaussianInverse', 'Pareto', 'PowerFunction', 'QuadraticU', 'RaisedCosine', 'Rayleigh', 'Reciprocal', 'StudentT', 'ShiftedGompertz', 'Trapezoidal', 'Triangular', 'Uniform', 'UniformSum', 'VonMises', 'Wald', 'Weibull', 'WignerSemicircle', ] @is_random.register(MatrixBase) def _(x): return any(is_random(i) for i in x) def rv(symbol, cls, args, *kwargs): args = list(map(sympify, args)) dist = cls(args) if kwargs.pop('check', True): dist.check(args) pspace = SingleContinuousPSpace(symbol, dist) if any(is_random(arg) for arg in args): from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution pspace = CompoundPSpace(symbol, CompoundDistribution(dist)) return pspace.value class ContinuousDistributionHandmade(SingleContinuousDistribution): _argnames = ('pdf',) def __new__(cls, pdf, set=Interval(-oo, oo)): return Basic.__new__(cls, pdf, set) @property def set(self): return self.args[1] @staticmethod def check(pdf, set): x = Dummy('x') val = integrate(pdf(x), (x, set)) _value_check(Eq(val, 1) != S.false, "The pdf on the given set is incorrect.") def ContinuousRV(symbol, density, set=Interval(-oo, oo), kwargs): """ Create a Continuous Random Variable given the following: Parameters ========== symbol : Symbol Represents name of the random variable. density : Expression containing symbol Represents probability density function. set : set/Interval Represents the region where the pdf is valid, by default is real line. check : bool If True, it will check whether the given density integrates to 1 over the given set. If False, it will not perform this check. Default is False. Returns ======= RandomSymbol Many common continuous random variable types are already implemented. This function should be necessary only very rarely. Examples ======== >>> from sympy import Symbol, sqrt, exp, pi >>> from sympy.stats import ContinuousRV, P, E >>> x = Symbol("x") >>> pdf = sqrt(2)exp(-x*2/2)/(2sqrt(pi)) # Normal distribution >>> X = ContinuousRV(x, pdf) >>> E(X) 0 >>> P(X>0) 1/2 """ pdf = Piecewise((density, set.as_relational(symbol)), (0, True)) pdf = Lambda(symbol, pdf) # have a default of False while `rv` should have a default of True kwargs['check'] = kwargs.pop('check', False) return rv(symbol.name, ContinuousDistributionHandmade, (pdf, set), *kwargs) ######################################## # Continuous Probability Distributions # ######################################## #------------------------------------------------------------------------------- # Arcsin distribution ---------------------------------------------------------- class ArcsinDistribution(SingleContinuousDistribution): _argnames = ('a', 'b') @property def set(self): return Interval(self.a, self.b) def pdf(self, x): a, b = self.a, self.b return 1/(pisqrt((x - a)(b - x))) def _cdf(self, x): a, b = self.a, self.b return Piecewise( (S.Zero, x < a), (2asin(sqrt((x - a)/(b - a)))/pi, x <= b), (S.One, True)) def Arcsin(name, a=0, b=1): r""" Create a Continuous Random Variable with an arcsin distribution. The density of the arcsin distribution is given by .. math:: f(x) := \frac{1}{\pi\sqrt{(x-a)(b-x)}} with :math:`x \in (a,b)`. It must hold that :math:`-\infty < a < b < \infty`. Parameters ========== a : Real number, the left interval boundary b : Real number, the right interval boundary Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Arcsin, density, cdf >>> from sympy import Symbol >>> a = Symbol("a", real=True) >>> b = Symbol("b", real=True) >>> z = Symbol("z") >>> X = Arcsin("x", a, b) >>> density(X)(z) 1/(pisqrt((-a + z)(b - z))) >>> cdf(X)(z) Piecewise((0, a > z), (2asin(sqrt((-a + z)/(-a + b)))/pi, b >= z), (1, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Arcsine_distribution """ return rv(name, ArcsinDistribution, (a, b)) #------------------------------------------------------------------------------- # Benini distribution ---------------------------------------------------------- class BeniniDistribution(SingleContinuousDistribution): _argnames = ('alpha', 'beta', 'sigma') @staticmethod def check(alpha, beta, sigma): _value_check(alpha > 0, "Shape parameter Alpha must be positive.") _value_check(beta > 0, "Shape parameter Beta must be positive.") _value_check(sigma > 0, "Scale parameter Sigma must be positive.") @property def set(self): return Interval(self.sigma, oo) def pdf(self, x): alpha, beta, sigma = self.alpha, self.beta, self.sigma return (exp(-alphalog(x/sigma) - betalog(x/sigma)2) (alpha/x + 2betalog(x/sigma)/x)) def _moment_generating_function(self, t): raise NotImplementedError('The moment generating function of the ' 'Benini distribution does not exist.') def Benini(name, alpha, beta, sigma): r""" Create a Continuous Random Variable with a Benini distribution. The density of the Benini distribution is given by .. math:: f(x) := e^{-\alpha\log{\frac{x}{\sigma}} -\beta\log^2\left[{\frac{x}{\sigma}}\right]} \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right) This is a heavy-tailed distribution and is also known as the log-Rayleigh distribution. Parameters ========== alpha : Real number, `\alpha > 0`, a shape beta : Real number, `\beta > 0`, a shape sigma : Real number, `\sigma > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Benini, density, cdf >>> from sympy import Symbol, pprint >>> alpha = Symbol("alpha", positive=True) >>> beta = Symbol("beta", positive=True) >>> sigma = Symbol("sigma", positive=True) >>> z = Symbol("z") >>> X = Benini("x", alpha, beta, sigma) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) / / z \\ / z \ 2/ z \ \| 2betalog\|-----\|\| - alphalog\|-----\| - betalog \|-----\| \|alpha \sigma/\| \sigma/ \sigma/ \|----- + -----------------\|e \ z z / >>> cdf(X)(z) Piecewise((1 - exp(-alphalog(z/sigma) - betalog(z/sigma)2), sigma <= z), (0, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Benini_distribution .. [2] http://reference.wolfram.com/legacy/v8/ref/BeniniDistribution.html """ return rv(name, BeniniDistribution, (alpha, beta, sigma)) #------------------------------------------------------------------------------- # Beta distribution ------------------------------------------------------------ class BetaDistribution(SingleContinuousDistribution): _argnames = ('alpha', 'beta') set = Interval(0, 1) @staticmethod def check(alpha, beta): _value_check(alpha > 0, "Shape parameter Alpha must be positive.") _value_check(beta > 0, "Shape parameter Beta must be positive.") def pdf(self, x): alpha, beta = self.alpha, self.beta return x(alpha - 1) (1 - x)*(beta - 1) / beta_fn(alpha, beta) def _characteristic_function(self, t): return hyper((self.alpha,), (self.alpha + self.beta,), It) def _moment_generating_function(self, t): return hyper((self.alpha,), (self.alpha + self.beta,), t) def Beta(name, alpha, beta): r""" Create a Continuous Random Variable with a Beta distribution. The density of the Beta distribution is given by .. math:: f(x) := \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)} with :math:`x \in [0,1]`. Parameters ========== alpha : Real number, `\alpha > 0`, a shape beta : Real number, `\beta > 0`, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Beta, density, E, variance >>> from sympy import Symbol, simplify, pprint, factor >>> alpha = Symbol("alpha", positive=True) >>> beta = Symbol("beta", positive=True) >>> z = Symbol("z") >>> X = Beta("x", alpha, beta) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) alpha - 1 beta - 1 z (1 - z) -------------------------- B(alpha, beta) >>> simplify(E(X)) alpha/(alpha + beta) >>> factor(simplify(variance(X))) alphabeta/((alpha + beta)*2(alpha + beta + 1)) References ========== .. [1] https://en.wikipedia.org/wiki/Beta_distribution .. [2] http://mathworld.wolfram.com/BetaDistribution.html """ return rv(name, BetaDistribution, (alpha, beta)) #------------------------------------------------------------------------------- # Noncentral Beta distribution ------------------------------------------------------------ class BetaNoncentralDistribution(SingleContinuousDistribution): _argnames = ('alpha', 'beta', 'lamda') set = Interval(0, 1) @staticmethod def check(alpha, beta, lamda): _value_check(alpha > 0, "Shape parameter Alpha must be positive.") _value_check(beta > 0, "Shape parameter Beta must be positive.") _value_check(lamda >= 0, "Noncentrality parameter Lambda must be positive") def pdf(self, x): alpha, beta, lamda = self.alpha, self.beta, self.lamda k = Dummy("k") return Sum(exp(-lamda / 2) * (lamda / 2)*k x*(alpha + k - 1) ( 1 - x)*(beta - 1) / (factorial(k) beta_fn(alpha + k, beta)), (k, 0, oo)) def BetaNoncentral(name, alpha, beta, lamda): r""" Create a Continuous Random Variable with a Type I Noncentral Beta distribution. The density of the Noncentral Beta distribution is given by .. math:: f(x) := \sum_{k=0}^\infty e^{-\lambda/2}\frac{(\lambda/2)^k}{k!} \frac{x^{\alpha+k-1}(1-x)^{\beta-1}}{\mathrm{B}(\alpha+k,\beta)} with :math:`x \in [0,1]`. Parameters ========== alpha : Real number, `\alpha > 0`, a shape beta : Real number, `\beta > 0`, a shape lamda : Real number, `\lambda \geq 0`, noncentrality parameter Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import BetaNoncentral, density, cdf >>> from sympy import Symbol, pprint >>> alpha = Symbol("alpha", positive=True) >>> beta = Symbol("beta", positive=True) >>> lamda = Symbol("lamda", nonnegative=True) >>> z = Symbol("z") >>> X = BetaNoncentral("x", alpha, beta, lamda) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) oo _____ \ ` \ -lamda \ k ------- \ k + alpha - 1 /lamda\ beta - 1 2 ) z \|-----\| (1 - z) e / \ 2 / / ------------------------------------------------ / B(k + alpha, beta)k! /____, k = 0 Compute cdf with specific 'x', 'alpha', 'beta' and 'lamda' values as follows: >>> cdf(BetaNoncentral("x", 1, 1, 1), evaluate=False)(2).doit() 2exp(1/2) The argument evaluate=False prevents an attempt at evaluation of the sum for general x, before the argument 2 is passed. References ========== .. [1] https://en.wikipedia.org/wiki/Noncentral_beta_distribution .. [2] https://reference.wolfram.com/language/ref/NoncentralBetaDistribution.html """ return rv(name, BetaNoncentralDistribution, (alpha, beta, lamda)) #------------------------------------------------------------------------------- # Beta prime distribution ------------------------------------------------------ class BetaPrimeDistribution(SingleContinuousDistribution): _argnames = ('alpha', 'beta') @staticmethod def check(alpha, beta): _value_check(alpha > 0, "Shape parameter Alpha must be positive.") _value_check(beta > 0, "Shape parameter Beta must be positive.") set = Interval(0, oo) def pdf(self, x): alpha, beta = self.alpha, self.beta return x(alpha - 1)(1 + x)*(-alpha - beta)/beta_fn(alpha, beta) def BetaPrime(name, alpha, beta): r""" Create a continuous random variable with a Beta prime distribution. The density of the Beta prime distribution is given by .. math:: f(x) := \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)} with :math:`x > 0`. Parameters ========== alpha : Real number, `\alpha > 0`, a shape beta : Real number, `\beta > 0`, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import BetaPrime, density >>> from sympy import Symbol, pprint >>> alpha = Symbol("alpha", positive=True) >>> beta = Symbol("beta", positive=True) >>> z = Symbol("z") >>> X = BetaPrime("x", alpha, beta) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) alpha - 1 -alpha - beta z (z + 1) ------------------------------- B(alpha, beta) References ========== .. [1] https://en.wikipedia.org/wiki/Beta_prime_distribution .. [2] http://mathworld.wolfram.com/BetaPrimeDistribution.html """ return rv(name, BetaPrimeDistribution, (alpha, beta)) #------------------------------------------------------------------------------- # Bounded Pareto Distribution -------------------------------------------------- class BoundedParetoDistribution(SingleContinuousDistribution): _argnames = ('alpha', 'left', 'right') @property def set(self): return Interval(self.left, self.right) @staticmethod def check(alpha, left, right): _value_check (alpha.is_positive, "Shape must be positive.") _value_check (left.is_positive, "Left value should be positive.") _value_check (right > left, "Right should be greater than left.") def pdf(self, x): alpha, left, right = self.alpha, self.left, self.right num = alpha * (left*alpha) x(- alpha -1) den = 1 - (left/right)alpha return num/den def BoundedPareto(name, alpha, left, right): r""" Create a continuous random variable with a Bounded Pareto distribution. The density of the Bounded Pareto distribution is given by .. math:: f(x) := \frac{\alpha L^{\alpha}x^{-\alpha-1}}{1-(\frac{L}{H})^{\alpha}} Parameters ========== alpha : Real Number, `\alpha > 0` Shape parameter left : Real Number, `left > 0` Location parameter right : Real Number, `right > left` Location parameter Examples ======== >>> from sympy.stats import BoundedPareto, density, cdf, E >>> from sympy import symbols >>> L, H = symbols('L, H', positive=True) >>> X = BoundedPareto('X', 2, L, H) >>> x = symbols('x') >>> density(X)(x) 2L2/(x3(1 - L2/H2)) >>> cdf(X)(x) Piecewise((-H*2L2/(x2(H2 - L2)) + H2/(H2 - L2), L <= x), (0, True)) >>> E(X).simplify() 2HL/(H + L) Returns ======= RandomSymbol References ========== .. [1] https://en.wikipedia.org/wiki/Pareto_distribution#Bounded_Pareto_distribution """ return rv (name, BoundedParetoDistribution, (alpha, left, right)) # ------------------------------------------------------------------------------ # Cauchy distribution ---------------------------------------------------------- class CauchyDistribution(SingleContinuousDistribution): _argnames = ('x0', 'gamma') @staticmethod def check(x0, gamma): _value_check(gamma > 0, "Scale parameter Gamma must be positive.") _value_check(x0.is_real, "Location parameter must be real.") def pdf(self, x): return 1/(piself.gamma(1 + ((x - self.x0)/self.gamma)2)) def _cdf(self, x): x0, gamma = self.x0, self.gamma return (1/pi)atan((x - x0)/gamma) + S.Half def _characteristic_function(self, t): return exp(self.x0 * I * t - self.gamma * Abs(t)) def _moment_generating_function(self, t): raise NotImplementedError("The moment generating function for the " "Cauchy distribution does not exist.") def _quantile(self, p): return self.x0 + self.gammatan(pi(p - S.Half)) def Cauchy(name, x0, gamma): r""" Create a continuous random variable with a Cauchy distribution. The density of the Cauchy distribution is given by .. math:: f(x) := \frac{1}{\pi \gamma [1 + {(\frac{x-x_0}{\gamma})}^2]} Parameters ========== x0 : Real number, the location gamma : Real number, `\gamma > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Cauchy, density >>> from sympy import Symbol >>> x0 = Symbol("x0") >>> gamma = Symbol("gamma", positive=True) >>> z = Symbol("z") >>> X = Cauchy("x", x0, gamma) >>> density(X)(z) 1/(pigamma(1 + (-x0 + z)2/gamma2)) References ========== .. [1] https://en.wikipedia.org/wiki/Cauchy_distribution .. [2] http://mathworld.wolfram.com/CauchyDistribution.html """ return rv(name, CauchyDistribution, (x0, gamma)) #------------------------------------------------------------------------------- # Chi distribution ------------------------------------------------------------- class ChiDistribution(SingleContinuousDistribution): _argnames = ('k',) @staticmethod def check(k): _value_check(k > 0, "Number of degrees of freedom (k) must be positive.") _value_check(k.is_integer, "Number of degrees of freedom (k) must be an integer.") set = Interval(0, oo) def pdf(self, x): return 2*(1 - self.k/2)x*(self.k - 1)exp(-x2/2)/gamma(self.k/2) def _characteristic_function(self, t): k = self.k part_1 = hyper((k/2,), (S.Half,), -t2/2) part_2 = Itsqrt(2)gamma((k+1)/2)/gamma(k/2) part_3 = hyper(((k+1)/2,), (Rational(3, 2),), -t2/2) return part_1 + part_2part_3 def _moment_generating_function(self, t): k = self.k part_1 = hyper((k / 2,), (S.Half,), t ** 2 / 2) part_2 = t * sqrt(2) * gamma((k + 1) / 2) / gamma(k / 2) part_3 = hyper(((k + 1) / 2,), (S(3) / 2,), t ** 2 / 2) return part_1 + part_2 * part_3 def Chi(name, k): r""" Create a continuous random variable with a Chi distribution. The density of the Chi distribution is given by .. math:: f(x) := \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)} with :math:`x \geq 0`. Parameters ========== k : Positive integer, The number of degrees of freedom Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Chi, density, E >>> from sympy import Symbol, simplify >>> k = Symbol("k", integer=True) >>> z = Symbol("z") >>> X = Chi("x", k) >>> density(X)(z) 2*(1 - k/2)z*(k - 1)exp(-z*2/2)/gamma(k/2) >>> simplify(E(X)) sqrt(2)gamma(k/2 + 1/2)/gamma(k/2) References ========== .. [1] https://en.wikipedia.org/wiki/Chi_distribution .. [2] http://mathworld.wolfram.com/ChiDistribution.html """ return rv(name, ChiDistribution, (k,)) #------------------------------------------------------------------------------- # Non-central Chi distribution ------------------------------------------------- class ChiNoncentralDistribution(SingleContinuousDistribution): _argnames = ('k', 'l') @staticmethod def check(k, l): _value_check(k > 0, "Number of degrees of freedom (k) must be positive.") _value_check(k.is_integer, "Number of degrees of freedom (k) must be an integer.") _value_check(l > 0, "Shift parameter Lambda must be positive.") set = Interval(0, oo) def pdf(self, x): k, l = self.k, self.l return exp(-(x2+l2)/2)xkl / (lx)(k/2) besseli(k/2-1, lx) def ChiNoncentral(name, k, l): r""" Create a continuous random variable with a non-central Chi distribution. Explanation =========== The density of the non-central Chi distribution is given by .. math:: f(x) := \frac{e^{-(x^2+\lambda^2)/2} x^k\lambda} {(\lambda x)^{k/2}} I_{k/2-1}(\lambda x) with `x \geq 0`. Here, `I_\nu (x)` is the :ref:`modified Bessel function of the first kind <besseli>`. Parameters ========== k : A positive Integer, $k > 0$ The number of degrees of freedom. lambda : Real number, `\lambda > 0` Shift parameter. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import ChiNoncentral, density >>> from sympy import Symbol >>> k = Symbol("k", integer=True) >>> l = Symbol("l") >>> z = Symbol("z") >>> X = ChiNoncentral("x", k, l) >>> density(X)(z) lz*kexp(-l2/2 - z2/2)besseli(k/2 - 1, lz)/(lz)(k/2) References ========== .. [1] https://en.wikipedia.org/wiki/Noncentral_chi_distribution """ return rv(name, ChiNoncentralDistribution, (k, l)) #------------------------------------------------------------------------------- # Chi squared distribution ----------------------------------------------------- class ChiSquaredDistribution(SingleContinuousDistribution): _argnames = ('k',) @staticmethod def check(k): _value_check(k > 0, "Number of degrees of freedom (k) must be positive.") _value_check(k.is_integer, "Number of degrees of freedom (k) must be an integer.") set = Interval(0, oo) def pdf(self, x): k = self.k return 1/(2(k/2)gamma(k/2))x(k/2 - 1)exp(-x/2) def _cdf(self, x): k = self.k return Piecewise( (S.One/gamma(k/2)lowergamma(k/2, x/2), x >= 0), (0, True) ) def _characteristic_function(self, t): return (1 - 2It)(-self.k/2) def _moment_generating_function(self, t): return (1 - 2t)(-self.k/2) def ChiSquared(name, k): r""" Create a continuous random variable with a Chi-squared distribution. Explanation =========== The density of the Chi-squared distribution is given by .. math:: f(x) := \frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)} x^{\frac{k}{2}-1} e^{-\frac{x}{2}} with :math:`x \geq 0`. Parameters ========== k : Positive integer The number of degrees of freedom. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import ChiSquared, density, E, variance, moment >>> from sympy import Symbol >>> k = Symbol("k", integer=True, positive=True) >>> z = Symbol("z") >>> X = ChiSquared("x", k) >>> density(X)(z) z(k/2 - 1)exp(-z/2)/(2(k/2)gamma(k/2)) >>> E(X) k >>> variance(X) 2k >>> moment(X, 3) k3 + 6k*2 + 8k References ========== .. [1] https://en.wikipedia.org/wiki/Chi_squared_distribution .. [2] http://mathworld.wolfram.com/Chi-SquaredDistribution.html """ return rv(name, ChiSquaredDistribution, (k, )) #------------------------------------------------------------------------------- # Dagum distribution ----------------------------------------------------------- class DagumDistribution(SingleContinuousDistribution): _argnames = ('p', 'a', 'b') set = Interval(0, oo) @staticmethod def check(p, a, b): _value_check(p > 0, "Shape parameter p must be positive.") _value_check(a > 0, "Shape parameter a must be positive.") _value_check(b > 0, "Scale parameter b must be positive.") def pdf(self, x): p, a, b = self.p, self.a, self.b return ap/x((x/b)*(ap)/(((x/b)a + 1)(p + 1))) def _cdf(self, x): p, a, b = self.p, self.a, self.b return Piecewise(((S.One + (S(x)/b)-a)-p, x>=0), (S.Zero, True)) def Dagum(name, p, a, b): r""" Create a continuous random variable with a Dagum distribution. Explanation =========== The density of the Dagum distribution is given by .. math:: f(x) := \frac{a p}{x} \left( \frac{\left(\tfrac{x}{b}\right)^{a p}} {\left(\left(\tfrac{x}{b}\right)^a + 1 \right)^{p+1}} \right) with :math:`x > 0`. Parameters ========== p : Real number `p > 0`, a shape. a : Real number `a > 0`, a shape. b : Real number `b > 0`, a scale. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Dagum, density, cdf >>> from sympy import Symbol >>> p = Symbol("p", positive=True) >>> a = Symbol("a", positive=True) >>> b = Symbol("b", positive=True) >>> z = Symbol("z") >>> X = Dagum("x", p, a, b) >>> density(X)(z) ap(z/b)*(ap)((z/b)a + 1)(-p - 1)/z >>> cdf(X)(z) Piecewise(((1 + (z/b)(-a))(-p), z >= 0), (0, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Dagum_distribution """ return rv(name, DagumDistribution, (p, a, b)) #------------------------------------------------------------------------------- # Erlang distribution ---------------------------------------------------------- def Erlang(name, k, l): r""" Create a continuous random variable with an Erlang distribution. Explanation =========== The density of the Erlang distribution is given by .. math:: f(x) := \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!} with :math:`x \in [0,\infty]`. Parameters ========== k : Positive integer l : Real number, `\lambda > 0`, the rate Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Erlang, density, cdf, E, variance >>> from sympy import Symbol, simplify, pprint >>> k = Symbol("k", integer=True, positive=True) >>> l = Symbol("l", positive=True) >>> z = Symbol("z") >>> X = Erlang("x", k, l) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) k k - 1 -lz l z e --------------- Gamma(k) >>> C = cdf(X)(z) >>> pprint(C, use_unicode=False) /lowergamma(k, lz) \|------------------ for z > 0 < Gamma(k) \| \ 0 otherwise >>> E(X) k/l >>> simplify(variance(X)) k/l2 References ========== .. [1] https://en.wikipedia.org/wiki/Erlang_distribution .. [2] http://mathworld.wolfram.com/ErlangDistribution.html """ return rv(name, GammaDistribution, (k, S.One/l)) # ------------------------------------------------------------------------------- # ExGaussian distribution ----------------------------------------------------- class ExGaussianDistribution(SingleContinuousDistribution): _argnames = ('mean', 'std', 'rate') set = Interval(-oo, oo) @staticmethod def check(mean, std, rate): _value_check( std > 0, "Standard deviation of ExGaussian must be positive.") _value_check(rate > 0, "Rate of ExGaussian must be positive.") def pdf(self, x): mean, std, rate = self.mean, self.std, self.rate term1 = rate/2 term2 = exp(rate (2 * mean + rate * std*2 - 2x)/2) term3 = erfc((mean + ratestd2 - x)/(sqrt(2)std)) return term1term2term3 def _cdf(self, x): from sympy.stats import cdf mean, std, rate = self.mean, self.std, self.rate u = rate(x - mean) v = ratestd GaussianCDF1 = cdf(Normal('x', 0, v))(u) GaussianCDF2 = cdf(Normal('x', v2, v))(u) return GaussianCDF1 - exp(-u + (v2/2) + log(GaussianCDF2)) def _characteristic_function(self, t): mean, std, rate = self.mean, self.std, self.rate term1 = (1 - It/rate)(-1) term2 = exp(Imeant - std2t*2/2) return term1 term2 def _moment_generating_function(self, t): mean, std, rate = self.mean, self.std, self.rate term1 = (1 - t/rate)*(-1) term2 = exp(meant + std*2t*2/2) return term1term2 def ExGaussian(name, mean, std, rate): r""" Create a continuous random variable with an Exponentially modified Gaussian (EMG) distribution. Explanation =========== The density of the exponentially modified Gaussian distribution is given by .. math:: f(x) := \frac{\lambda}{2}e^{\frac{\lambda}{2}(2\mu+\lambda\sigma^2-2x)} \text{erfc}(\frac{\mu + \lambda\sigma^2 - x}{\sqrt{2}\sigma}) with $x > 0$. Note that the expected value is `1/\lambda`. Parameters ========== name : A string giving a name for this distribution mean : A Real number, the mean of Gaussian component std : A positive Real number, :math: `\sigma^2 > 0` the variance of Gaussian component rate : A positive Real number, :math: `\lambda > 0` the rate of Exponential component Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import ExGaussian, density, cdf, E >>> from sympy.stats import variance, skewness >>> from sympy import Symbol, pprint, simplify >>> mean = Symbol("mu") >>> std = Symbol("sigma", positive=True) >>> rate = Symbol("lamda", positive=True) >>> z = Symbol("z") >>> X = ExGaussian("x", mean, std, rate) >>> pprint(density(X)(z), use_unicode=False) / 2 \ lamda\lamdasigma + 2mu - 2z/ --------------------------------- / ___ / 2 \\ 2 \|\/ 2 \lamdasigma + mu - z/\| lamdae erfc\|-----------------------------\| \ 2sigma / ---------------------------------------------------------------------------- 2 >>> cdf(X)(z) -(erf(sqrt(2)(-lamda*2sigma*2 + lamda(-mu + z))/(2lamdasigma))/2 + 1/2)exp(lamda2sigma*2/2 - lamda(-mu + z)) + erf(sqrt(2)(-mu + z)/(2sigma))/2 + 1/2 >>> E(X) (lamdamu + 1)/lamda >>> simplify(variance(X)) sigma2 + lamda(-2) >>> simplify(skewness(X)) 2/(lamda2sigma2 + 1)(3/2) References ========== .. [1] https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution """ return rv(name, ExGaussianDistribution, (mean, std, rate)) #------------------------------------------------------------------------------- # Exponential distribution ----------------------------------------------------- class ExponentialDistribution(SingleContinuousDistribution): _argnames = ('rate',) set = Interval(0, oo) @staticmethod def check(rate): _value_check(rate > 0, "Rate must be positive.") def pdf(self, x): return self.rate * exp(-self.ratex) def _cdf(self, x): return Piecewise( (S.One - exp(-self.ratex), x >= 0), (0, True), ) def _characteristic_function(self, t): rate = self.rate return rate / (rate - It) def _moment_generating_function(self, t): rate = self.rate return rate / (rate - t) def _quantile(self, p): return -log(1-p)/self.rate def Exponential(name, rate): r""" Create a continuous random variable with an Exponential distribution. Explanation =========== The density of the exponential distribution is given by .. math:: f(x) := \lambda \exp(-\lambda x) with $x > 0$. Note that the expected value is `1/\lambda`. Parameters ========== rate : A positive Real number, `\lambda > 0`, the rate (or inverse scale/inverse mean) Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Exponential, density, cdf, E >>> from sympy.stats import variance, std, skewness, quantile >>> from sympy import Symbol >>> l = Symbol("lambda", positive=True) >>> z = Symbol("z") >>> p = Symbol("p") >>> X = Exponential("x", l) >>> density(X)(z) lambdaexp(-lambdaz) >>> cdf(X)(z) Piecewise((1 - exp(-lambdaz), z >= 0), (0, True)) >>> quantile(X)(p) -log(1 - p)/lambda >>> E(X) 1/lambda >>> variance(X) lambda*(-2) >>> skewness(X) 2 >>> X = Exponential('x', 10) >>> density(X)(z) 10exp(-10z) >>> E(X) 1/10 >>> std(X) 1/10 References ========== .. [1] https://en.wikipedia.org/wiki/Exponential_distribution .. [2] http://mathworld.wolfram.com/ExponentialDistribution.html """ return rv(name, ExponentialDistribution, (rate, )) # ------------------------------------------------------------------------------- # Exponential Power distribution ----------------------------------------------------- class ExponentialPowerDistribution(SingleContinuousDistribution): _argnames = ('mu', 'alpha', 'beta') set = Interval(-oo, oo) @staticmethod def check(mu, alpha, beta): _value_check(alpha > 0, "Scale parameter alpha must be positive.") _value_check(beta > 0, "Shape parameter beta must be positive.") def pdf(self, x): mu, alpha, beta = self.mu, self.alpha, self.beta num = betaexp(-(Abs(x - mu)/alpha)*beta) den = 2alphagamma(1/beta) return num/den def _cdf(self, x): mu, alpha, beta = self.mu, self.alpha, self.beta num = lowergamma(1/beta, (Abs(x - mu) / alpha)beta) den = 2gamma(1/beta) return sign(x - mu)num/den + S.Half def ExponentialPower(name, mu, alpha, beta): r""" Create a Continuous Random Variable with Exponential Power distribution. This distribution is known also as Generalized Normal distribution version 1. Explanation =========== The density of the Exponential Power distribution is given by .. math:: f(x) := \frac{\beta}{2\alpha\Gamma(\frac{1}{\beta})} e^{{-(\frac{\|x - \mu\|}{\alpha})^{\beta}}} with :math:`x \in [ - \infty, \infty ]`. Parameters ========== mu : Real number A location. alpha : Real number,`\alpha > 0` A scale. beta : Real number, `\beta > 0` A shape. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import ExponentialPower, density, cdf >>> from sympy import Symbol, pprint >>> z = Symbol("z") >>> mu = Symbol("mu") >>> alpha = Symbol("alpha", positive=True) >>> beta = Symbol("beta", positive=True) >>> X = ExponentialPower("x", mu, alpha, beta) >>> pprint(density(X)(z), use_unicode=False) beta /\|mu - z\|\ -\|--------\| \ alpha / betae --------------------- / 1 \ 2alphaGamma\|----\| \beta/ >>> cdf(X)(z) 1/2 + lowergamma(1/beta, (Abs(mu - z)/alpha)*beta)sign(-mu + z)/(2gamma(1/beta)) References ========== .. [1] https://reference.wolfram.com/language/ref/ExponentialPowerDistribution.html .. [2] https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1 """ return rv(name, ExponentialPowerDistribution, (mu, alpha, beta)) #------------------------------------------------------------------------------- # F distribution --------------------------------------------------------------- class FDistributionDistribution(SingleContinuousDistribution): _argnames = ('d1', 'd2') set = Interval(0, oo) @staticmethod def check(d1, d2): _value_check((d1 > 0, d1.is_integer), "Degrees of freedom d1 must be positive integer.") _value_check((d2 > 0, d2.is_integer), "Degrees of freedom d2 must be positive integer.") def pdf(self, x): d1, d2 = self.d1, self.d2 return (sqrt((d1x)*d1d2*d2 / (d1x+d2)*(d1+d2)) / (x beta_fn(d1/2, d2/2))) def _moment_generating_function(self, t): raise NotImplementedError('The moment generating function for the ' 'F-distribution does not exist.') def FDistribution(name, d1, d2): r""" Create a continuous random variable with a F distribution. Explanation =========== The density of the F distribution is given by .. math:: f(x) := \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}} {(d_1 x + d_2)^{d_1 + d_2}}}} {x \mathrm{B} \left(\frac{d_1}{2}, \frac{d_2}{2}\right)} with :math:`x > 0`. Parameters ========== d1 : `d_1 > 0`, where `d_1` is the degrees of freedom (`n_1 - 1`) d2 : `d_2 > 0`, where `d_2` is the degrees of freedom (`n_2 - 1`) Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import FDistribution, density >>> from sympy import Symbol, pprint >>> d1 = Symbol("d1", positive=True) >>> d2 = Symbol("d2", positive=True) >>> z = Symbol("z") >>> X = FDistribution("x", d1, d2) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) d2 -- ______________________________ 2 / d1 -d1 - d2 d2 \/ (d1z) (d1z + d2) -------------------------------------- /d1 d2\ zB\|--, --\| \2 2 / References ========== .. [1] https://en.wikipedia.org/wiki/F-distribution .. [2] http://mathworld.wolfram.com/F-Distribution.html """ return rv(name, FDistributionDistribution, (d1, d2)) #------------------------------------------------------------------------------- # Fisher Z distribution -------------------------------------------------------- class FisherZDistribution(SingleContinuousDistribution): _argnames = ('d1', 'd2') set = Interval(-oo, oo) @staticmethod def check(d1, d2): _value_check(d1 > 0, "Degree of freedom d1 must be positive.") _value_check(d2 > 0, "Degree of freedom d2 must be positive.") def pdf(self, x): d1, d2 = self.d1, self.d2 return (2d1*(d1/2)d2*(d2/2) / beta_fn(d1/2, d2/2) exp(d1x) / (d1exp(2x)+d2)((d1+d2)/2)) def FisherZ(name, d1, d2): r""" Create a Continuous Random Variable with an Fisher's Z distribution. Explanation =========== The density of the Fisher's Z distribution is given by .. math:: f(x) := \frac{2d_1^{d_1/2} d_2^{d_2/2}} {\mathrm{B}(d_1/2, d_2/2)} \frac{e^{d_1z}}{\left(d_1e^{2z}+d_2\right)^{\left(d_1+d_2\right)/2}} .. TODO - What is the difference between these degrees of freedom? Parameters ========== d1 : `d_1 > 0` Degree of freedom. d2 : `d_2 > 0` Degree of freedom. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import FisherZ, density >>> from sympy import Symbol, pprint >>> d1 = Symbol("d1", positive=True) >>> d2 = Symbol("d2", positive=True) >>> z = Symbol("z") >>> X = FisherZ("x", d1, d2) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) d1 d2 d1 d2 - -- - -- -- -- 2 2 2 2 / 2z \ d1z 2d1 d2 \d1e + d2/ e ----------------------------------------- /d1 d2\ B\|--, --\| \2 2 / References ========== .. [1] https://en.wikipedia.org/wiki/Fisher%27s_z-distribution .. [2] http://mathworld.wolfram.com/Fishersz-Distribution.html """ return rv(name, FisherZDistribution, (d1, d2)) #------------------------------------------------------------------------------- # Frechet distribution --------------------------------------------------------- class FrechetDistribution(SingleContinuousDistribution): _argnames = ('a', 's', 'm') set = Interval(0, oo) @staticmethod def check(a, s, m): _value_check(a > 0, "Shape parameter alpha must be positive.") _value_check(s > 0, "Scale parameter s must be positive.") def __new__(cls, a, s=1, m=0): a, s, m = list(map(sympify, (a, s, m))) return Basic.__new__(cls, a, s, m) def pdf(self, x): a, s, m = self.a, self.s, self.m return a/s * ((x-m)/s)*(-1-a) exp(-((x-m)/s)(-a)) def _cdf(self, x): a, s, m = self.a, self.s, self.m return Piecewise((exp(-((x-m)/s)(-a)), x >= m), (S.Zero, True)) def Frechet(name, a, s=1, m=0): r""" Create a continuous random variable with a Frechet distribution. Explanation =========== The density of the Frechet distribution is given by .. math:: f(x) := \frac{\alpha}{s} \left(\frac{x-m}{s}\right)^{-1-\alpha} e^{-(\frac{x-m}{s})^{-\alpha}} with :math:`x \geq m`. Parameters ========== a : Real number, :math:`a \in \left(0, \infty\right)` the shape s : Real number, :math:`s \in \left(0, \infty\right)` the scale m : Real number, :math:`m \in \left(-\infty, \infty\right)` the minimum Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Frechet, density, cdf >>> from sympy import Symbol >>> a = Symbol("a", positive=True) >>> s = Symbol("s", positive=True) >>> m = Symbol("m", real=True) >>> z = Symbol("z") >>> X = Frechet("x", a, s, m) >>> density(X)(z) a((-m + z)/s)(-a - 1)exp(-1/((-m + z)/s)a)/s >>> cdf(X)(z) Piecewise((exp(-1/((-m + z)/s)a), m <= z), (0, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution """ return rv(name, FrechetDistribution, (a, s, m)) #------------------------------------------------------------------------------- # Gamma distribution ----------------------------------------------------------- class GammaDistribution(SingleContinuousDistribution): _argnames = ('k', 'theta') set = Interval(0, oo) @staticmethod def check(k, theta): _value_check(k > 0, "k must be positive") _value_check(theta > 0, "Theta must be positive") def pdf(self, x): k, theta = self.k, self.theta return x*(k - 1) exp(-x/theta) / (gamma(k)thetak) def _cdf(self, x): k, theta = self.k, self.theta return Piecewise( (lowergamma(k, S(x)/theta)/gamma(k), x > 0), (S.Zero, True)) def _characteristic_function(self, t): return (1 - self.thetaIt)(-self.k) def _moment_generating_function(self, t): return (1- self.thetat)*(-self.k) def Gamma(name, k, theta): r""" Create a continuous random variable with a Gamma distribution. Explanation =========== The density of the Gamma distribution is given by .. math:: f(x) := \frac{1}{\Gamma(k) \theta^k} x^{k - 1} e^{-\frac{x}{\theta}} with :math:`x \in [0,1]`. Parameters ========== k : Real number, `k > 0`, a shape theta : Real number, `\theta > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Gamma, density, cdf, E, variance >>> from sympy import Symbol, pprint, simplify >>> k = Symbol("k", positive=True) >>> theta = Symbol("theta", positive=True) >>> z = Symbol("z") >>> X = Gamma("x", k, theta) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) -z ----- -k k - 1 theta theta z e --------------------- Gamma(k) >>> C = cdf(X, meijerg=True)(z) >>> pprint(C, use_unicode=False) / / z \ \|klowergamma\|k, -----\| \| \ theta/ <---------------------- for z >= 0 \| Gamma(k + 1) \| \ 0 otherwise >>> E(X) ktheta >>> V = simplify(variance(X)) >>> pprint(V, use_unicode=False) 2 ktheta References ========== .. [1] https://en.wikipedia.org/wiki/Gamma_distribution .. [2] http://mathworld.wolfram.com/GammaDistribution.html """ return rv(name, GammaDistribution, (k, theta)) #------------------------------------------------------------------------------- # Inverse Gamma distribution --------------------------------------------------- class GammaInverseDistribution(SingleContinuousDistribution): _argnames = ('a', 'b') set = Interval(0, oo) @staticmethod def check(a, b): _value_check(a > 0, "alpha must be positive") _value_check(b > 0, "beta must be positive") def pdf(self, x): a, b = self.a, self.b return b*a/gamma(a) x*(-a-1) exp(-b/x) def _cdf(self, x): a, b = self.a, self.b return Piecewise((uppergamma(a,b/x)/gamma(a), x > 0), (S.Zero, True)) def _characteristic_function(self, t): a, b = self.a, self.b return 2 * (-Ibt)*(a/2) besselk(a, sqrt(-4Ibt)) / gamma(a) def _moment_generating_function(self, t): raise NotImplementedError('The moment generating function for the ' 'gamma inverse distribution does not exist.') def GammaInverse(name, a, b): r""" Create a continuous random variable with an inverse Gamma distribution. Explanation =========== The density of the inverse Gamma distribution is given by .. math:: f(x) := \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp\left(\frac{-\beta}{x}\right) with :math:`x > 0`. Parameters ========== a : Real number, `a > 0`, a shape b : Real number, `b > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import GammaInverse, density, cdf >>> from sympy import Symbol, pprint >>> a = Symbol("a", positive=True) >>> b = Symbol("b", positive=True) >>> z = Symbol("z") >>> X = GammaInverse("x", a, b) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) -b --- a -a - 1 z b z e --------------- Gamma(a) >>> cdf(X)(z) Piecewise((uppergamma(a, b/z)/gamma(a), z > 0), (0, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Inverse-gamma_distribution """ return rv(name, GammaInverseDistribution, (a, b)) #------------------------------------------------------------------------------- # Gumbel distribution (Maximum and Minimum) -------------------------------------------------------- class GumbelDistribution(SingleContinuousDistribution): _argnames = ('beta', 'mu', 'minimum') set = Interval(-oo, oo) @staticmethod def check(beta, mu, minimum): _value_check(beta > 0, "Scale parameter beta must be positive.") def pdf(self, x): beta, mu = self.beta, self.mu z = (x - mu)/beta f_max = (1/beta)exp(-z - exp(-z)) f_min = (1/beta)exp(z - exp(z)) return Piecewise((f_min, self.minimum), (f_max, not self.minimum)) def _cdf(self, x): beta, mu = self.beta, self.mu z = (x - mu)/beta F_max = exp(-exp(-z)) F_min = 1 - exp(-exp(z)) return Piecewise((F_min, self.minimum), (F_max, not self.minimum)) def _characteristic_function(self, t): cf_max = gamma(1 - Iself.betat) exp(Iself.mut) cf_min = gamma(1 + Iself.betat) * exp(Iself.mut) return Piecewise((cf_min, self.minimum), (cf_max, not self.minimum)) def _moment_generating_function(self, t): mgf_max = gamma(1 - self.betat) exp(self.mut) mgf_min = gamma(1 + self.betat) * exp(self.mut) return Piecewise((mgf_min, self.minimum), (mgf_max, not self.minimum)) def Gumbel(name, beta, mu, minimum=False): r""" Create a Continuous Random Variable with Gumbel distribution. Explanation =========== The density of the Gumbel distribution is given by For Maximum .. math:: f(x) := \dfrac{1}{\beta} \exp \left( -\dfrac{x-\mu}{\beta} - \exp \left( -\dfrac{x - \mu}{\beta} \right) \right) with :math:`x \in [ - \infty, \infty ]`. For Minimum .. math:: f(x) := \frac{e^{- e^{\frac{- \mu + x}{\beta}} + \frac{- \mu + x}{\beta}}}{\beta} with :math:`x \in [ - \infty, \infty ]`. Parameters ========== mu : Real number, `\mu`, a location beta : Real number, `\beta > 0`, a scale minimum : Boolean, by default ``False``, set to ``True`` for enabling minimum distribution Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Gumbel, density, cdf >>> from sympy import Symbol >>> x = Symbol("x") >>> mu = Symbol("mu") >>> beta = Symbol("beta", positive=True) >>> X = Gumbel("x", beta, mu) >>> density(X)(x) exp(-exp(-(-mu + x)/beta) - (-mu + x)/beta)/beta >>> cdf(X)(x) exp(-exp(-(-mu + x)/beta)) References ========== .. [1] http://mathworld.wolfram.com/GumbelDistribution.html .. [2] https://en.wikipedia.org/wiki/Gumbel_distribution .. [3] http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_max.html .. [4] http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_min.html """ return rv(name, GumbelDistribution, (beta, mu, minimum)) #------------------------------------------------------------------------------- # Gompertz distribution -------------------------------------------------------- class GompertzDistribution(SingleContinuousDistribution): _argnames = ('b', 'eta') set = Interval(0, oo) @staticmethod def check(b, eta): _value_check(b > 0, "b must be positive") _value_check(eta > 0, "eta must be positive") def pdf(self, x): eta, b = self.eta, self.b return betaexp(bx)exp(eta)exp(-etaexp(bx)) def _cdf(self, x): eta, b = self.eta, self.b return 1 - exp(eta)exp(-etaexp(bx)) def _moment_generating_function(self, t): eta, b = self.eta, self.b return eta exp(eta) * expint(t/b, eta) def Gompertz(name, b, eta): r""" Create a Continuous Random Variable with Gompertz distribution. Explanation =========== The density of the Gompertz distribution is given by .. math:: f(x) := b \eta e^{b x} e^{\eta} \exp \left(-\eta e^{bx} \right) with :math:`x \in [0, \infty)`. Parameters ========== b : Real number, `b > 0`, a scale eta : Real number, `\eta > 0`, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Gompertz, density >>> from sympy import Symbol >>> b = Symbol("b", positive=True) >>> eta = Symbol("eta", positive=True) >>> z = Symbol("z") >>> X = Gompertz("x", b, eta) >>> density(X)(z) betaexp(eta)exp(bz)exp(-etaexp(bz)) References ========== .. [1] https://en.wikipedia.org/wiki/Gompertz_distribution """ return rv(name, GompertzDistribution, (b, eta)) #------------------------------------------------------------------------------- # Kumaraswamy distribution ----------------------------------------------------- class KumaraswamyDistribution(SingleContinuousDistribution): _argnames = ('a', 'b') set = Interval(0, oo) @staticmethod def check(a, b): _value_check(a > 0, "a must be positive") _value_check(b > 0, "b must be positive") def pdf(self, x): a, b = self.a, self.b return a b * x*(a-1) (1-xa)(b-1) def _cdf(self, x): a, b = self.a, self.b return Piecewise( (S.Zero, x < S.Zero), (1 - (1 - xa)b, x <= S.One), (S.One, True)) def Kumaraswamy(name, a, b): r""" Create a Continuous Random Variable with a Kumaraswamy distribution. Explanation =========== The density of the Kumaraswamy distribution is given by .. math:: f(x) := a b x^{a-1} (1-x^a)^{b-1} with :math:`x \in [0,1]`. Parameters ========== a : Real number, `a > 0`, a shape b : Real number, `b > 0`, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Kumaraswamy, density, cdf >>> from sympy import Symbol, pprint >>> a = Symbol("a", positive=True) >>> b = Symbol("b", positive=True) >>> z = Symbol("z") >>> X = Kumaraswamy("x", a, b) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) b - 1 a - 1 / a\ abz \1 - z / >>> cdf(X)(z) Piecewise((0, z < 0), (1 - (1 - za)b, z <= 1), (1, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Kumaraswamy_distribution """ return rv(name, KumaraswamyDistribution, (a, b)) #------------------------------------------------------------------------------- # Laplace distribution --------------------------------------------------------- class LaplaceDistribution(SingleContinuousDistribution): _argnames = ('mu', 'b') set = Interval(-oo, oo) @staticmethod def check(mu, b): _value_check(b > 0, "Scale parameter b must be positive.") _value_check(mu.is_real, "Location parameter mu should be real") def pdf(self, x): mu, b = self.mu, self.b return 1/(2b)exp(-Abs(x - mu)/b) def _cdf(self, x): mu, b = self.mu, self.b return Piecewise( (S.Halfexp((x - mu)/b), x < mu), (S.One - S.Halfexp(-(x - mu)/b), x >= mu) ) def _characteristic_function(self, t): return exp(self.muIt) / (1 + self.b2t*2) def _moment_generating_function(self, t): return exp(self.mut) / (1 - self.b*2t*2) def Laplace(name, mu, b): r""" Create a continuous random variable with a Laplace distribution. Explanation =========== The density of the Laplace distribution is given by .. math:: f(x) := \frac{1}{2 b} \exp \left(-\frac{\|x-\mu\|}b \right) Parameters ========== mu : Real number or a list/matrix, the location (mean) or the location vector b : Real number or a positive definite matrix, representing a scale or the covariance matrix. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Laplace, density, cdf >>> from sympy import Symbol, pprint >>> mu = Symbol("mu") >>> b = Symbol("b", positive=True) >>> z = Symbol("z") >>> X = Laplace("x", mu, b) >>> density(X)(z) exp(-Abs(mu - z)/b)/(2b) >>> cdf(X)(z) Piecewise((exp((-mu + z)/b)/2, mu > z), (1 - exp((mu - z)/b)/2, True)) >>> L = Laplace('L', [1, 2], [[1, 0], [0, 1]]) >>> pprint(density(L)(1, 2), use_unicode=False) 5 / ____\ e besselk\0, \/ 35 / --------------------- pi References ========== .. [1] https://en.wikipedia.org/wiki/Laplace_distribution .. [2] http://mathworld.wolfram.com/LaplaceDistribution.html """ if isinstance(mu, (list, MatrixBase)) and\ isinstance(b, (list, MatrixBase)): from sympy.stats.joint_rv_types import MultivariateLaplace return MultivariateLaplace(name, mu, b) return rv(name, LaplaceDistribution, (mu, b)) #------------------------------------------------------------------------------- # Levy distribution --------------------------------------------------------- class LevyDistribution(SingleContinuousDistribution): _argnames = ('mu', 'c') @property def set(self): return Interval(self.mu, oo) @staticmethod def check(mu, c): _value_check(c > 0, "c (scale parameter) must be positive") _value_check(mu.is_real, "mu (location paramater) must be real") def pdf(self, x): mu, c = self.mu, self.c return sqrt(c/(2pi))exp(-c/(2(x - mu)))/((x - mu)*(S.One + S.Half)) def _cdf(self, x): mu, c = self.mu, self.c return erfc(sqrt(c/(2(x - mu)))) def _characteristic_function(self, t): mu, c = self.mu, self.c return exp(I * mu * t - sqrt(-2 * I * c * t)) def _moment_generating_function(self, t): raise NotImplementedError('The moment generating function of Levy distribution does not exist.') def Levy(name, mu, c): r""" Create a continuous random variable with a Levy distribution. The density of the Levy distribution is given by .. math:: f(x) := \sqrt(\frac{c}{2 \pi}) \frac{\exp -\frac{c}{2 (x - \mu)}}{(x - \mu)^{3/2}} Parameters ========== mu : Real number The location parameter. c : Real number, `c > 0` A scale parameter. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Levy, density, cdf >>> from sympy import Symbol >>> mu = Symbol("mu", real=True) >>> c = Symbol("c", positive=True) >>> z = Symbol("z") >>> X = Levy("x", mu, c) >>> density(X)(z) sqrt(2)sqrt(c)exp(-c/(-2mu + 2z))/(2sqrt(pi)(-mu + z)*(3/2)) >>> cdf(X)(z) erfc(sqrt(c)sqrt(1/(-2mu + 2z))) References ========== .. [1] https://en.wikipedia.org/wiki/L%C3%A9vy_distribution .. [2] http://mathworld.wolfram.com/LevyDistribution.html """ return rv(name, LevyDistribution, (mu, c)) #------------------------------------------------------------------------------- # Log-Cauchy distribution -------------------------------------------------------- class LogCauchyDistribution(SingleContinuousDistribution): _argnames = ('mu', 'sigma') set = Interval.open(0, oo) @staticmethod def check(mu, sigma): _value_check((sigma > 0) != False, "Scale parameter Gamma must be positive.") _value_check(mu.is_real != False, "Location parameter must be real.") def pdf(self, x): mu, sigma = self.mu, self.sigma return 1/(xpi)(sigma/((log(x) - mu)2 + sigma2)) def _cdf(self, x): mu, sigma = self.mu, self.sigma return (1/pi)atan((log(x) - mu)/sigma) + S.Half def _characteristic_function(self, t): raise NotImplementedError("The characteristic function for the " "Log-Cauchy distribution does not exist.") def _moment_generating_function(self, t): raise NotImplementedError("The moment generating function for the " "Log-Cauchy distribution does not exist.") def LogCauchy(name, mu, sigma): r""" Create a continuous random variable with a Log-Cauchy distribution. The density of the Log-Cauchy distribution is given by .. math:: f(x) := \frac{1}{\pi x} \frac{\sigma}{(log(x)-\mu^2) + \sigma^2} Parameters ========== mu : Real number, the location sigma : Real number, `\sigma > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import LogCauchy, density, cdf >>> from sympy import Symbol, S >>> mu = 2 >>> sigma = S.One / 5 >>> z = Symbol("z") >>> X = LogCauchy("x", mu, sigma) >>> density(X)(z) 1/(5piz((log(z) - 2)*2 + 1/25)) >>> cdf(X)(z) atan(5log(z) - 10)/pi + 1/2 References ========== .. [1] https://en.wikipedia.org/wiki/Log-Cauchy_distribution """ return rv(name, LogCauchyDistribution, (mu, sigma)) #------------------------------------------------------------------------------- # Logistic distribution -------------------------------------------------------- class LogisticDistribution(SingleContinuousDistribution): _argnames = ('mu', 's') set = Interval(-oo, oo) @staticmethod def check(mu, s): _value_check(s > 0, "Scale parameter s must be positive.") def pdf(self, x): mu, s = self.mu, self.s return exp(-(x - mu)/s)/(s(1 + exp(-(x - mu)/s))2) def _cdf(self, x): mu, s = self.mu, self.s return S.One/(1 + exp(-(x - mu)/s)) def _characteristic_function(self, t): return Piecewise((exp(Itself.mu) piself.st / sinh(piself.st), Ne(t, 0)), (S.One, True)) def _moment_generating_function(self, t): return exp(self.mut) beta_fn(1 - self.st, 1 + self.st) def _quantile(self, p): return self.mu - self.slog(-S.One + S.One/p) def Logistic(name, mu, s): r""" Create a continuous random variable with a logistic distribution. Explanation =========== The density of the logistic distribution is given by .. math:: f(x) := \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2} Parameters ========== mu : Real number, the location (mean) s : Real number, `s > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Logistic, density, cdf >>> from sympy import Symbol >>> mu = Symbol("mu", real=True) >>> s = Symbol("s", positive=True) >>> z = Symbol("z") >>> X = Logistic("x", mu, s) >>> density(X)(z) exp((mu - z)/s)/(s(exp((mu - z)/s) + 1)*2) >>> cdf(X)(z) 1/(exp((mu - z)/s) + 1) References ========== .. [1] https://en.wikipedia.org/wiki/Logistic_distribution .. [2] http://mathworld.wolfram.com/LogisticDistribution.html """ return rv(name, LogisticDistribution, (mu, s)) #------------------------------------------------------------------------------- # Log-logistic distribution -------------------------------------------------------- class LogLogisticDistribution(SingleContinuousDistribution): _argnames = ('alpha', 'beta') set = Interval(0, oo) @staticmethod def check(alpha, beta): _value_check(alpha > 0, "Scale parameter Alpha must be positive.") _value_check(beta > 0, "Shape parameter Beta must be positive.") def pdf(self, x): a, b = self.alpha, self.beta return ((b/a)(x/a)(b - 1))/(1 + (x/a)b)2 def _cdf(self, x): a, b = self.alpha, self.beta return 1/(1 + (x/a)(-b)) def _quantile(self, p): a, b = self.alpha, self.beta return a((p/(1 - p))(1/b)) def expectation(self, expr, var, kwargs): a, b = self.args return Piecewise((S.NaN, b <= 1), (pia/(bsin(pi/b)), True)) def LogLogistic(name, alpha, beta): r""" Create a continuous random variable with a log-logistic distribution. The distribution is unimodal when ``beta > 1``. Explanation =========== The density of the log-logistic distribution is given by .. math:: f(x) := \frac{(\frac{\beta}{\alpha})(\frac{x}{\alpha})^{\beta - 1}} {(1 + (\frac{x}{\alpha})^{\beta})^2} Parameters ========== alpha : Real number, `\alpha > 0`, scale parameter and median of distribution beta : Real number, `\beta > 0`, a shape parameter Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import LogLogistic, density, cdf, quantile >>> from sympy import Symbol, pprint >>> alpha = Symbol("alpha", positive=True) >>> beta = Symbol("beta", positive=True) >>> p = Symbol("p") >>> z = Symbol("z", positive=True) >>> X = LogLogistic("x", alpha, beta) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) beta - 1 / z \ beta\|-----\| \alpha/ ------------------------ 2 / beta \ \|/ z \ \| alpha\|\|-----\| + 1\| \\alpha/ / >>> cdf(X)(z) 1/(1 + (z/alpha)(-beta)) >>> quantile(X)(p) alpha(p/(1 - p))(1/beta) References ========== .. [1] https://en.wikipedia.org/wiki/Log-logistic_distribution """ return rv(name, LogLogisticDistribution, (alpha, beta)) #------------------------------------------------------------------------------- #Logit-Normal distribution------------------------------------------------------ class LogitNormalDistribution(SingleContinuousDistribution): _argnames = ('mu', 's') set = Interval.open(0, 1) @staticmethod def check(mu, s): _value_check((s 2).is_real is not False and s 2 > 0, "Squared scale parameter s must be positive.") _value_check(mu.is_real is not False, "Location parameter must be real") def _logit(self, x): return log(x / (1 - x)) def pdf(self, x): mu, s = self.mu, self.s return exp(-(self._logit(x) - mu)2/(2s2))(S.One/sqrt(2pi(s*2)))(1/(x(1 - x))) def _cdf(self, x): mu, s = self.mu, self.s return (S.One/2)(1 + erf((self._logit(x) - mu)/(sqrt(2s2)))) def LogitNormal(name, mu, s): r""" Create a continuous random variable with a Logit-Normal distribution. The density of the logistic distribution is given by .. math:: f(x) := \frac{1}{s \sqrt{2 \pi}} \frac{1}{x(1 - x)} e^{- \frac{(logit(x) - \mu)^2}{s^2}} where logit(x) = \log(\frac{x}{1 - x}) Parameters ========== mu : Real number, the location (mean) s : Real number, `s > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import LogitNormal, density, cdf >>> from sympy import Symbol,pprint >>> mu = Symbol("mu", real=True) >>> s = Symbol("s", positive=True) >>> z = Symbol("z") >>> X = LogitNormal("x",mu,s) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) 2 / / z \\ -\|-mu + log\|-----\|\| \ \1 - z// --------------------- 2 ___ 2s \/ 2 e ---------------------------- ____ 2\/ pi sz(1 - z) >>> density(X)(z) sqrt(2)exp(-(-mu + log(z/(1 - z)))*2/(2s*2))/(2sqrt(pi)sz(1 - z)) >>> cdf(X)(z) erf(sqrt(2)(-mu + log(z/(1 - z)))/(2s))/2 + 1/2 References ========== .. [1] https://en.wikipedia.org/wiki/Logit-normal_distribution """ return rv(name, LogitNormalDistribution, (mu, s)) #------------------------------------------------------------------------------- # Log Normal distribution ------------------------------------------------------ class LogNormalDistribution(SingleContinuousDistribution): _argnames = ('mean', 'std') set = Interval(0, oo) @staticmethod def check(mean, std): _value_check(std > 0, "Parameter std must be positive.") def pdf(self, x): mean, std = self.mean, self.std return exp(-(log(x) - mean)2 / (2std*2)) / (xsqrt(2pi)std) def _cdf(self, x): mean, std = self.mean, self.std return Piecewise( (S.Half + S.Halferf((log(x) - mean)/sqrt(2)/std), x > 0), (S.Zero, True) ) def _moment_generating_function(self, t): raise NotImplementedError('Moment generating function of the log-normal distribution is not defined.') def LogNormal(name, mean, std): r""" Create a continuous random variable with a log-normal distribution. Explanation =========== The density of the log-normal distribution is given by .. math:: f(x) := \frac{1}{x\sqrt{2\pi\sigma^2}} e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}} with :math:`x \geq 0`. Parameters ========== mu : Real number The log-scale. sigma : Real number A shape. ($\sigma^2 > 0$) Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import LogNormal, density >>> from sympy import Symbol, pprint >>> mu = Symbol("mu", real=True) >>> sigma = Symbol("sigma", positive=True) >>> z = Symbol("z") >>> X = LogNormal("x", mu, sigma) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) 2 -(-mu + log(z)) ----------------- 2 ___ 2sigma \/ 2 e ------------------------ ____ 2\/ pi sigmaz >>> X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1 >>> density(X)(z) sqrt(2)exp(-log(z)2/2)/(2sqrt(pi)z) References ========== .. [1] https://en.wikipedia.org/wiki/Lognormal .. [2] http://mathworld.wolfram.com/LogNormalDistribution.html """ return rv(name, LogNormalDistribution, (mean, std)) #------------------------------------------------------------------------------- # Lomax Distribution ----------------------------------------------------------- class LomaxDistribution(SingleContinuousDistribution): _argnames = ('alpha', 'lamda',) set = Interval(0, oo) @staticmethod def check(alpha, lamda): _value_check(alpha.is_real, "Shape parameter should be real.") _value_check(lamda.is_real, "Scale parameter should be real.") _value_check(alpha.is_positive, "Shape parameter should be positive.") _value_check(lamda.is_positive, "Scale parameter should be positive.") def pdf(self, x): lamba, alpha = self.lamda, self.alpha return (alpha/lamba) (S.One + x/lamba)*(-alpha-1) def Lomax(name, alpha, lamda): r""" Create a continuous random variable with a Lomax distribution. Explanation =========== The density of the Lomax distribution is given by .. math:: f(x) := \frac{\alpha}{\lambda}\left[1+\frac{x}{\lambda}\right]^{-(\alpha+1)} Parameters ========== alpha : Real Number, `\alpha > 0` Shape parameter lamda : Real Number, `\lambda > 0` Scale parameter Examples ======== >>> from sympy.stats import Lomax, density, cdf, E >>> from sympy import symbols >>> a, l = symbols('a, l', positive=True) >>> X = Lomax('X', a, l) >>> x = symbols('x') >>> density(X)(x) a(1 + x/l)(-a - 1)/l >>> cdf(X)(x) Piecewise((1 - 1/(1 + x/l)a, x >= 0), (0, True)) >>> a = 2 >>> X = Lomax('X', a, l) >>> E(X) l Returns ======= RandomSymbol References ========== .. [1] https://en.wikipedia.org/wiki/Lomax_distribution """ return rv(name, LomaxDistribution, (alpha, lamda)) #------------------------------------------------------------------------------- # Maxwell distribution --------------------------------------------------------- class MaxwellDistribution(SingleContinuousDistribution): _argnames = ('a',) set = Interval(0, oo) @staticmethod def check(a): _value_check(a > 0, "Parameter a must be positive.") def pdf(self, x): a = self.a return sqrt(2/pi)x2exp(-x*2/(2a2))/a3 def _cdf(self, x): a = self.a return erf(sqrt(2)x/(2a)) - sqrt(2)xexp(-x*2/(2a*2))/(sqrt(pi)a) def Maxwell(name, a): r""" Create a continuous random variable with a Maxwell distribution. Explanation =========== The density of the Maxwell distribution is given by .. math:: f(x) := \sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/(2a^2)}}{a^3} with :math:`x \geq 0`. .. TODO - what does the parameter mean? Parameters ========== a : Real number, `a > 0` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Maxwell, density, E, variance >>> from sympy import Symbol, simplify >>> a = Symbol("a", positive=True) >>> z = Symbol("z") >>> X = Maxwell("x", a) >>> density(X)(z) sqrt(2)z2exp(-z*2/(2a*2))/(sqrt(pi)a*3) >>> E(X) 2sqrt(2)a/sqrt(pi) >>> simplify(variance(X)) a2(-8 + 3pi)/pi References ========== .. [1] https://en.wikipedia.org/wiki/Maxwell_distribution .. [2] http://mathworld.wolfram.com/MaxwellDistribution.html """ return rv(name, MaxwellDistribution, (a, )) #------------------------------------------------------------------------------- # Moyal Distribution ----------------------------------------------------------- class MoyalDistribution(SingleContinuousDistribution): _argnames = ('mu', 'sigma') @staticmethod def check(mu, sigma): _value_check(mu.is_real, "Location parameter must be real.") _value_check(sigma.is_real and sigma > 0, "Scale parameter must be real\ and positive.") def pdf(self, x): mu, sigma = self.mu, self.sigma num = exp(-(exp(-(x - mu)/sigma) + (x - mu)/(sigma))/2) den = (sqrt(2pi) * sigma) return num/den def _characteristic_function(self, t): mu, sigma = self.mu, self.sigma term1 = exp(Itmu) term2 = (2*(-Isigmat) gamma(Rational(1, 2) - Itsigma)) return (term1 * term2)/sqrt(pi) def _moment_generating_function(self, t): mu, sigma = self.mu, self.sigma term1 = exp(tmu) term2 = (2(-1sigmat) gamma(Rational(1, 2) - tsigma)) return (term1 term2)/sqrt(pi) def Moyal(name, mu, sigma): r""" Create a continuous random variable with a Moyal distribution. Explanation =========== The density of the Moyal distribution is given by .. math:: f(x) := \frac{\exp-\frac{1}{2}\exp-\frac{x-\mu}{\sigma}-\frac{x-\mu}{2\sigma}}{\sqrt{2\pi}\sigma} with :math:`x \in \mathbb{R}`. Parameters ========== mu : Real number Location parameter sigma : Real positive number Scale parameter Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Moyal, density, cdf >>> from sympy import Symbol, simplify >>> mu = Symbol("mu", real=True) >>> sigma = Symbol("sigma", positive=True, real=True) >>> z = Symbol("z") >>> X = Moyal("x", mu, sigma) >>> density(X)(z) sqrt(2)exp(-exp((mu - z)/sigma)/2 - (-mu + z)/(2sigma))/(2sqrt(pi)sigma) >>> simplify(cdf(X)(z)) 1 - erf(sqrt(2)exp((mu - z)/(2sigma))/2) References ========== .. [1] https://reference.wolfram.com/language/ref/MoyalDistribution.html .. [2] http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf """ return rv(name, MoyalDistribution, (mu, sigma)) #------------------------------------------------------------------------------- # Nakagami distribution -------------------------------------------------------- class NakagamiDistribution(SingleContinuousDistribution): _argnames = ('mu', 'omega') set = Interval(0, oo) @staticmethod def check(mu, omega): _value_check(mu >= S.Half, "Shape parameter mu must be greater than equal to 1/2.") _value_check(omega > 0, "Spread parameter omega must be positive.") def pdf(self, x): mu, omega = self.mu, self.omega return 2mumu/(gamma(mu)omega*mu)x*(2mu - 1)exp(-mu/omegax*2) def _cdf(self, x): mu, omega = self.mu, self.omega return Piecewise( (lowergamma(mu, (mu/omega)x*2)/gamma(mu), x > 0), (S.Zero, True)) def Nakagami(name, mu, omega): r""" Create a continuous random variable with a Nakagami distribution. Explanation =========== The density of the Nakagami distribution is given by .. math:: f(x) := \frac{2\mu^\mu}{\Gamma(\mu)\omega^\mu} x^{2\mu-1} \exp\left(-\frac{\mu}{\omega}x^2 \right) with :math:`x > 0`. Parameters ========== mu : Real number, `\mu \geq \frac{1}{2}`, a shape omega : Real number, `\omega > 0`, the spread Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Nakagami, density, E, variance, cdf >>> from sympy import Symbol, simplify, pprint >>> mu = Symbol("mu", positive=True) >>> omega = Symbol("omega", positive=True) >>> z = Symbol("z") >>> X = Nakagami("x", mu, omega) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) 2 -muz ------- mu -mu 2mu - 1 omega 2mu omega z e ---------------------------------- Gamma(mu) >>> simplify(E(X)) sqrt(mu)sqrt(omega)gamma(mu + 1/2)/gamma(mu + 1) >>> V = simplify(variance(X)) >>> pprint(V, use_unicode=False) 2 omegaGamma (mu + 1/2) omega - ----------------------- Gamma(mu)Gamma(mu + 1) >>> cdf(X)(z) Piecewise((lowergamma(mu, muz2/omega)/gamma(mu), z > 0), (0, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Nakagami_distribution """ return rv(name, NakagamiDistribution, (mu, omega)) #------------------------------------------------------------------------------- # Normal distribution ---------------------------------------------------------- class NormalDistribution(SingleContinuousDistribution): _argnames = ('mean', 'std') @staticmethod def check(mean, std): _value_check(std > 0, "Standard deviation must be positive") def pdf(self, x): return exp(-(x - self.mean)2 / (2self.std2)) / (sqrt(2pi)self.std) def _cdf(self, x): mean, std = self.mean, self.std return erf(sqrt(2)(-mean + x)/(2std))/2 + S.Half def _characteristic_function(self, t): mean, std = self.mean, self.std return exp(Imeant - std2t*2/2) def _moment_generating_function(self, t): mean, std = self.mean, self.std return exp(meant + std*2t*2/2) def _quantile(self, p): mean, std = self.mean, self.std return mean + stdsqrt(2)erfinv(2p - 1) def Normal(name, mean, std): r""" Create a continuous random variable with a Normal distribution. Explanation =========== The density of the Normal distribution is given by .. math:: f(x) := \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} } Parameters ========== mu : Real number or a list representing the mean or the mean vector sigma : Real number or a positive definite square matrix, :math:`\sigma^2 > 0`, the variance Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Normal, density, E, std, cdf, skewness, quantile, marginal_distribution >>> from sympy import Symbol, simplify, pprint >>> mu = Symbol("mu") >>> sigma = Symbol("sigma", positive=True) >>> z = Symbol("z") >>> y = Symbol("y") >>> p = Symbol("p") >>> X = Normal("x", mu, sigma) >>> density(X)(z) sqrt(2)exp(-(-mu + z)2/(2sigma*2))/(2sqrt(pi)sigma) >>> C = simplify(cdf(X))(z) # it needs a little more help... >>> pprint(C, use_unicode=False) / ___ \ \|\/ 2 (-mu + z)\| erf\|---------------\| \ 2sigma / 1 -------------------- + - 2 2 >>> quantile(X)(p) mu + sqrt(2)sigmaerfinv(2p - 1) >>> simplify(skewness(X)) 0 >>> X = Normal("x", 0, 1) # Mean 0, standard deviation 1 >>> density(X)(z) sqrt(2)exp(-z2/2)/(2sqrt(pi)) >>> E(2X + 1) 1 >>> simplify(std(2X + 1)) 2 >>> m = Normal('X', [1, 2], [[2, 1], [1, 2]]) >>> pprint(density(m)(y, z), use_unicode=False) 2 2 y yz z - -- + --- - -- + z - 1 ___ 3 3 3 \/ 3 e ------------------------------ 6pi >>> marginal_distribution(m, m[0])(1) 1/(2sqrt(pi)) References ========== .. [1] https://en.wikipedia.org/wiki/Normal_distribution .. [2] http://mathworld.wolfram.com/NormalDistributionFunction.html """ if isinstance(mean, list) or getattr(mean, 'is_Matrix', False) and\ isinstance(std, list) or getattr(std, 'is_Matrix', False): from sympy.stats.joint_rv_types import MultivariateNormal return MultivariateNormal(name, mean, std) return rv(name, NormalDistribution, (mean, std)) #------------------------------------------------------------------------------- # Inverse Gaussian distribution ---------------------------------------------------------- class GaussianInverseDistribution(SingleContinuousDistribution): _argnames = ('mean', 'shape') @property def set(self): return Interval(0, oo) @staticmethod def check(mean, shape): _value_check(shape > 0, "Shape parameter must be positive") _value_check(mean > 0, "Mean must be positive") def pdf(self, x): mu, s = self.mean, self.shape return exp(-s(x - mu)2 / (2xmu2)) sqrt(s/(2pix*3)) def _cdf(self, x): from sympy.stats import cdf mu, s = self.mean, self.shape stdNormalcdf = cdf(Normal('x', 0, 1)) first_term = stdNormalcdf(sqrt(s/x) ((x/mu) - S.One)) second_term = exp(2s/mu) stdNormalcdf(-sqrt(s/x)(x/mu + S.One)) return first_term + second_term def _characteristic_function(self, t): mu, s = self.mean, self.shape return exp((s/mu)(1 - sqrt(1 - (2mu2It)/s))) def _moment_generating_function(self, t): mu, s = self.mean, self.shape return exp((s/mu)(1 - sqrt(1 - (2mu2t)/s))) def GaussianInverse(name, mean, shape): r""" Create a continuous random variable with an Inverse Gaussian distribution. Inverse Gaussian distribution is also known as Wald distribution. Explanation =========== The density of the Inverse Gaussian distribution is given by .. math:: f(x) := \sqrt{\frac{\lambda}{2\pi x^3}} e^{-\frac{\lambda(x-\mu)^2}{2x\mu^2}} Parameters ========== mu : Positive number representing the mean. lambda : Positive number representing the shape parameter. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import GaussianInverse, density, E, std, skewness >>> from sympy import Symbol, pprint >>> mu = Symbol("mu", positive=True) >>> lamda = Symbol("lambda", positive=True) >>> z = Symbol("z", positive=True) >>> X = GaussianInverse("x", mu, lamda) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) 2 -lambda(-mu + z) ------------------- 2 ___ ________ 2mu z \/ 2 \/ lambda e ------------------------------------- ____ 3/2 2\/ pi z >>> E(X) mu >>> std(X).expand() mu(3/2)/sqrt(lambda) >>> skewness(X).expand() 3sqrt(mu)/sqrt(lambda) References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution .. [2] http://mathworld.wolfram.com/InverseGaussianDistribution.html """ return rv(name, GaussianInverseDistribution, (mean, shape)) Wald = GaussianInverse #------------------------------------------------------------------------------- # Pareto distribution ---------------------------------------------------------- class ParetoDistribution(SingleContinuousDistribution): _argnames = ('xm', 'alpha') @property def set(self): return Interval(self.xm, oo) @staticmethod def check(xm, alpha): _value_check(xm > 0, "Xm must be positive") _value_check(alpha > 0, "Alpha must be positive") def pdf(self, x): xm, alpha = self.xm, self.alpha return alpha * xmalpha / x(alpha + 1) def _cdf(self, x): xm, alpha = self.xm, self.alpha return Piecewise( (S.One - xmalpha/xalpha, x>=xm), (0, True), ) def _moment_generating_function(self, t): xm, alpha = self.xm, self.alpha return alpha * (-xmt)alpha uppergamma(-alpha, -xmt) def _characteristic_function(self, t): xm, alpha = self.xm, self.alpha return alpha (-I * xm * t) ** alpha * uppergamma(-alpha, -I * xm * t) def Pareto(name, xm, alpha): r""" Create a continuous random variable with the Pareto distribution. Explanation =========== The density of the Pareto distribution is given by .. math:: f(x) := \frac{\alpha\,x_m^\alpha}{x^{\alpha+1}} with :math:`x \in [x_m,\infty]`. Parameters ========== xm : Real number, `x_m > 0`, a scale alpha : Real number, `\alpha > 0`, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Pareto, density >>> from sympy import Symbol >>> xm = Symbol("xm", positive=True) >>> beta = Symbol("beta", positive=True) >>> z = Symbol("z") >>> X = Pareto("x", xm, beta) >>> density(X)(z) betaxmbetaz*(-beta - 1) References ========== .. [1] https://en.wikipedia.org/wiki/Pareto_distribution .. [2] http://mathworld.wolfram.com/ParetoDistribution.html """ return rv(name, ParetoDistribution, (xm, alpha)) #------------------------------------------------------------------------------- # PowerFunction distribution --------------------------------------------------- class PowerFunctionDistribution(SingleContinuousDistribution): _argnames=('alpha','a','b') @property def set(self): return Interval(self.a, self.b) @staticmethod def check(alpha, a, b): _value_check(a.is_real, "Continuous Boundary parameter should be real.") _value_check(b.is_real, "Continuous Boundary parameter should be real.") _value_check(a < b, " 'a' the left Boundary must be smaller than 'b' the right Boundary." ) _value_check(alpha.is_positive, "Continuous Shape parameter should be positive.") def pdf(self, x): alpha, a, b = self.alpha, self.a, self.b num = alpha(x - a)(alpha - 1) den = (b - a)alpha return num/den def PowerFunction(name, alpha, a, b): r""" Creates a continuous random variable with a Power Function Distribution. Explanation =========== The density of PowerFunction distribution is given by .. math:: f(x) := \frac{{\alpha}(x - a)^{\alpha - 1}}{(b - a)^{\alpha}} with :math:`x \in [a,b]`. Parameters ========== alpha : Positive number, `0 < \alpha`, the shape paramater a : Real number, :math:`-\infty < a`, the left boundary b : Real number, :math:`a < b < \infty`, the right boundary Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import PowerFunction, density, cdf, E, variance >>> from sympy import Symbol >>> alpha = Symbol("alpha", positive=True) >>> a = Symbol("a", real=True) >>> b = Symbol("b", real=True) >>> z = Symbol("z") >>> X = PowerFunction("X", 2, a, b) >>> density(X)(z) (-2a + 2z)/(-a + b)2 >>> cdf(X)(z) Piecewise((a2/(a*2 - 2ab + b2) - 2az/(a2 - 2ab + b2) + z2/(a2 - 2ab + b2), a <= z), (0, True)) >>> alpha = 2 >>> a = 0 >>> b = 1 >>> Y = PowerFunction("Y", alpha, a, b) >>> E(Y) 2/3 >>> variance(Y) 1/18 References ========== .. [1] http://www.mathwave.com/help/easyfit/html/analyses/distributions/power_func.html """ return rv(name, PowerFunctionDistribution, (alpha, a, b)) #------------------------------------------------------------------------------- # QuadraticU distribution ------------------------------------------------------ class QuadraticUDistribution(SingleContinuousDistribution): _argnames = ('a', 'b') @property def set(self): return Interval(self.a, self.b) @staticmethod def check(a, b): _value_check(b > a, "Parameter b must be in range (%s, oo)."%(a)) def pdf(self, x): a, b = self.a, self.b alpha = 12 / (b-a)3 beta = (a+b) / 2 return Piecewise( (alpha (x-beta)*2, And(a<=x, x<=b)), (S.Zero, True)) def _moment_generating_function(self, t): a, b = self.a, self.b return -3 (exp(at) (4 + (a*2 + 2a(-2 + b) + b2) t) \ - exp(bt) (4 + (-4b + (a + b)2) t)) / ((a-b)*3 t*2) def _characteristic_function(self, t): a, b = self.a, self.b return -3I(exp(Iatexp(Ibt)) * (4I - (-4b + (a+b)*2)t)) \ / ((a-b)*3 t*2) def QuadraticU(name, a, b): r""" Create a Continuous Random Variable with a U-quadratic distribution. Explanation =========== The density of the U-quadratic distribution is given by .. math:: f(x) := \alpha (x-\beta)^2 with :math:`x \in [a,b]`. Parameters ========== a : Real number b : Real number, :math:`a < b` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import QuadraticU, density >>> from sympy import Symbol, pprint >>> a = Symbol("a", real=True) >>> b = Symbol("b", real=True) >>> z = Symbol("z") >>> X = QuadraticU("x", a, b) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) / 2 \| / a b \ \|12\|- - - - + z\| \| \ 2 2 / <----------------- for And(b >= z, a <= z) \| 3 \| (-a + b) \| \ 0 otherwise References ========== .. [1] https://en.wikipedia.org/wiki/U-quadratic_distribution """ return rv(name, QuadraticUDistribution, (a, b)) #------------------------------------------------------------------------------- # RaisedCosine distribution ---------------------------------------------------- class RaisedCosineDistribution(SingleContinuousDistribution): _argnames = ('mu', 's') @property def set(self): return Interval(self.mu - self.s, self.mu + self.s) @staticmethod def check(mu, s): _value_check(s > 0, "s must be positive") def pdf(self, x): mu, s = self.mu, self.s return Piecewise( ((1+cos(pi(x-mu)/s)) / (2s), And(mu-s<=x, x<=mu+s)), (S.Zero, True)) def _characteristic_function(self, t): mu, s = self.mu, self.s return Piecewise((exp(-Ipimu/s)/2, Eq(t, -pi/s)), (exp(Ipimu/s)/2, Eq(t, pi/s)), (pi*2sin(st)exp(Imut) / (st(pi2 - s2t2)), True)) def _moment_generating_function(self, t): mu, s = self.mu, self.s return pi2 sinh(st) exp(mut) / (st(pi2 + s2t*2)) def RaisedCosine(name, mu, s): r""" Create a Continuous Random Variable with a raised cosine distribution. Explanation =========== The density of the raised cosine distribution is given by .. math:: f(x) := \frac{1}{2s}\left(1+\cos\left(\frac{x-\mu}{s}\pi\right)\right) with :math:`x \in [\mu-s,\mu+s]`. Parameters ========== mu : Real number s : Real number, `s > 0` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import RaisedCosine, density >>> from sympy import Symbol, pprint >>> mu = Symbol("mu", real=True) >>> s = Symbol("s", positive=True) >>> z = Symbol("z") >>> X = RaisedCosine("x", mu, s) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) / /pi(-mu + z)\ \|cos\|------------\| + 1 \| \ s / <--------------------- for And(z >= mu - s, z <= mu + s) \| 2s \| \ 0 otherwise References ========== .. [1] https://en.wikipedia.org/wiki/Raised_cosine_distribution """ return rv(name, RaisedCosineDistribution, (mu, s)) #------------------------------------------------------------------------------- # Rayleigh distribution -------------------------------------------------------- class RayleighDistribution(SingleContinuousDistribution): _argnames = ('sigma',) set = Interval(0, oo) @staticmethod def check(sigma): _value_check(sigma > 0, "Scale parameter sigma must be positive.") def pdf(self, x): sigma = self.sigma return x/sigma2exp(-x*2/(2sigma2)) def _cdf(self, x): sigma = self.sigma return 1 - exp(-(x2/(2sigma2))) def _characteristic_function(self, t): sigma = self.sigma return 1 - sigmatexp(-sigma2t*2/2) sqrt(pi/2) * (erfi(sigmat/sqrt(2)) - I) def _moment_generating_function(self, t): sigma = self.sigma return 1 + sigmatexp(sigma2t*2/2) sqrt(pi/2) * (erf(sigmat/sqrt(2)) + 1) def Rayleigh(name, sigma): r""" Create a continuous random variable with a Rayleigh distribution. Explanation =========== The density of the Rayleigh distribution is given by .. math :: f(x) := \frac{x}{\sigma^2} e^{-x^2/2\sigma^2} with :math:`x > 0`. Parameters ========== sigma : Real number, `\sigma > 0` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Rayleigh, density, E, variance >>> from sympy import Symbol >>> sigma = Symbol("sigma", positive=True) >>> z = Symbol("z") >>> X = Rayleigh("x", sigma) >>> density(X)(z) zexp(-z*2/(2sigma2))/sigma2 >>> E(X) sqrt(2)sqrt(pi)sigma/2 >>> variance(X) -pisigma2/2 + 2sigma*2 References ========== .. [1] https://en.wikipedia.org/wiki/Rayleigh_distribution .. [2] http://mathworld.wolfram.com/RayleighDistribution.html """ return rv(name, RayleighDistribution, (sigma, )) #------------------------------------------------------------------------------- # Reciprocal distribution -------------------------------------------------------- class ReciprocalDistribution(SingleContinuousDistribution): _argnames = ('a', 'b') @property def set(self): return Interval(self.a, self.b) @staticmethod def check(a, b): _value_check(a > 0, "Parameter > 0. a = %s"%a) _value_check((a < b), "Parameter b must be in range (%s, +oo]. b = %s"%(a, b)) def pdf(self, x): a, b = self.a, self.b return 1/(x(log(b) - log(a))) def Reciprocal(name, a, b): r"""Creates a continuous random variable with a reciprocal distribution. Parameters ========== a : Real number, :math:`0 < a` b : Real number, :math:`a < b` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Reciprocal, density, cdf >>> from sympy import symbols >>> a, b, x = symbols('a, b, x', positive=True) >>> R = Reciprocal('R', a, b) >>> density(R)(x) 1/(x(-log(a) + log(b))) >>> cdf(R)(x) Piecewise((log(a)/(log(a) - log(b)) - log(x)/(log(a) - log(b)), a <= x), (0, True)) Reference ========= .. [1] https://en.wikipedia.org/wiki/Reciprocal_distribution """ return rv(name, ReciprocalDistribution, (a, b)) #------------------------------------------------------------------------------- # Shifted Gompertz distribution ------------------------------------------------ class ShiftedGompertzDistribution(SingleContinuousDistribution): _argnames = ('b', 'eta') set = Interval(0, oo) @staticmethod def check(b, eta): _value_check(b > 0, "b must be positive") _value_check(eta > 0, "eta must be positive") def pdf(self, x): b, eta = self.b, self.eta return bexp(-bx)exp(-etaexp(-bx))(1+eta(1-exp(-bx))) def ShiftedGompertz(name, b, eta): r""" Create a continuous random variable with a Shifted Gompertz distribution. Explanation =========== The density of the Shifted Gompertz distribution is given by .. math:: f(x) := b e^{-b x} e^{-\eta \exp(-b x)} \left[1 + \eta(1 - e^(-bx)) \right] with :math:`x \in [0, \infty)`. Parameters ========== b : Real number, `b > 0`, a scale eta : Real number, `\eta > 0`, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import ShiftedGompertz, density >>> from sympy import Symbol >>> b = Symbol("b", positive=True) >>> eta = Symbol("eta", positive=True) >>> x = Symbol("x") >>> X = ShiftedGompertz("x", b, eta) >>> density(X)(x) b(eta(1 - exp(-bx)) + 1)exp(-bx)exp(-etaexp(-bx)) References ========== .. [1] https://en.wikipedia.org/wiki/Shifted_Gompertz_distribution """ return rv(name, ShiftedGompertzDistribution, (b, eta)) #------------------------------------------------------------------------------- # StudentT distribution -------------------------------------------------------- class StudentTDistribution(SingleContinuousDistribution): _argnames = ('nu',) set = Interval(-oo, oo) @staticmethod def check(nu): _value_check(nu > 0, "Degrees of freedom nu must be positive.") def pdf(self, x): nu = self.nu return 1/(sqrt(nu)beta_fn(S.Half, nu/2))(1 + x2/nu)(-(nu + 1)/2) def _cdf(self, x): nu = self.nu return S.Half + xgamma((nu+1)/2)hyper((S.Half, (nu+1)/2), (Rational(3, 2),), -x2/nu)/(sqrt(pinu)gamma(nu/2)) def _moment_generating_function(self, t): raise NotImplementedError('The moment generating function for the Student-T distribution is undefined.') def StudentT(name, nu): r""" Create a continuous random variable with a student's t distribution. Explanation =========== The density of the student's t distribution is given by .. math:: f(x) := \frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}} Parameters ========== nu : Real number, `\nu > 0`, the degrees of freedom Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import StudentT, density, cdf >>> from sympy import Symbol, pprint >>> nu = Symbol("nu", positive=True) >>> z = Symbol("z") >>> X = StudentT("x", nu) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) nu 1 - -- - - 2 2 / 2\ \| z \| \|1 + --\| \ nu/ ----------------- ____ / nu\ \/ nu B\|1/2, --\| \ 2 / >>> cdf(X)(z) 1/2 + zgamma(nu/2 + 1/2)hyper((1/2, nu/2 + 1/2), (3/2,), -z*2/nu)/(sqrt(pi)sqrt(nu)gamma(nu/2)) References ========== .. [1] https://en.wikipedia.org/wiki/Student_t-distribution .. [2] http://mathworld.wolfram.com/Studentst-Distribution.html """ return rv(name, StudentTDistribution, (nu, )) #------------------------------------------------------------------------------- # Trapezoidal distribution ------------------------------------------------------ class TrapezoidalDistribution(SingleContinuousDistribution): _argnames = ('a', 'b', 'c', 'd') @property def set(self): return Interval(self.a, self.d) @staticmethod def check(a, b, c, d): _value_check(a < d, "Lower bound parameter a < %s. a = %s"%(d, a)) _value_check((a <= b, b < c), "Level start parameter b must be in range [%s, %s). b = %s"%(a, c, b)) _value_check((b < c, c <= d), "Level end parameter c must be in range (%s, %s]. c = %s"%(b, d, c)) _value_check(d >= c, "Upper bound parameter d > %s. d = %s"%(c, d)) def pdf(self, x): a, b, c, d = self.a, self.b, self.c, self.d return Piecewise( (2(x-a) / ((b-a)(d+c-a-b)), And(a <= x, x < b)), (2 / (d+c-a-b), And(b <= x, x < c)), (2(d-x) / ((d-c)(d+c-a-b)), And(c <= x, x <= d)), (S.Zero, True)) def Trapezoidal(name, a, b, c, d): r""" Create a continuous random variable with a trapezoidal distribution. Explanation =========== The density of the trapezoidal distribution is given by .. math:: f(x) := \begin{cases} 0 & \mathrm{for\ } x < a, \\ \frac{2(x-a)}{(b-a)(d+c-a-b)} & \mathrm{for\ } a \le x < b, \\ \frac{2}{d+c-a-b} & \mathrm{for\ } b \le x < c, \\ \frac{2(d-x)}{(d-c)(d+c-a-b)} & \mathrm{for\ } c \le x < d, \\ 0 & \mathrm{for\ } d < x. \end{cases} Parameters ========== a : Real number, :math:`a < d` b : Real number, :math:`a \le b < c` c : Real number, :math:`b < c \le d` d : Real number Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Trapezoidal, density >>> from sympy import Symbol, pprint >>> a = Symbol("a") >>> b = Symbol("b") >>> c = Symbol("c") >>> d = Symbol("d") >>> z = Symbol("z") >>> X = Trapezoidal("x", a,b,c,d) >>> pprint(density(X)(z), use_unicode=False) / -2a + 2z \|------------------------- for And(a <= z, b > z) \|(-a + b)(-a - b + c + d) \| \| 2 \| -------------- for And(b <= z, c > z) < -a - b + c + d \| \| 2d - 2z \|------------------------- for And(d >= z, c <= z) \|(-c + d)(-a - b + c + d) \| \ 0 otherwise References ========== .. [1] https://en.wikipedia.org/wiki/Trapezoidal_distribution """ return rv(name, TrapezoidalDistribution, (a, b, c, d)) #------------------------------------------------------------------------------- # Triangular distribution ------------------------------------------------------ class TriangularDistribution(SingleContinuousDistribution): _argnames = ('a', 'b', 'c') @property def set(self): return Interval(self.a, self.b) @staticmethod def check(a, b, c): _value_check(b > a, "Parameter b > %s. b = %s"%(a, b)) _value_check((a <= c, c <= b), "Parameter c must be in range [%s, %s]. c = %s"%(a, b, c)) def pdf(self, x): a, b, c = self.a, self.b, self.c return Piecewise( (2(x - a)/((b - a)(c - a)), And(a <= x, x < c)), (2/(b - a), Eq(x, c)), (2(b - x)/((b - a)(b - c)), And(c < x, x <= b)), (S.Zero, True)) def _characteristic_function(self, t): a, b, c = self.a, self.b, self.c return -2 ((b-c) * exp(Iat) - (b-a) * exp(Ict) + (c-a) * exp(Ibt)) / ((b-a)(c-a)(b-c)t2) def _moment_generating_function(self, t): a, b, c = self.a, self.b, self.c return 2 ((b - c) * exp(a * t) - (b - a) * exp(c * t) + (c - a) * exp(b * t)) / ( (b - a) * (c - a) * (b - c) * t ** 2) def Triangular(name, a, b, c): r""" Create a continuous random variable with a triangular distribution. Explanation =========== The density of the triangular distribution is given by .. math:: f(x) := \begin{cases} 0 & \mathrm{for\ } x < a, \\ \frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x < c, \\ \frac{2}{b-a} & \mathrm{for\ } x = c, \\ \frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\ 0 & \mathrm{for\ } b < x. \end{cases} Parameters ========== a : Real number, :math:`a \in \left(-\infty, \infty\right)` b : Real number, :math:`a < b` c : Real number, :math:`a \leq c \leq b` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Triangular, density >>> from sympy import Symbol, pprint >>> a = Symbol("a") >>> b = Symbol("b") >>> c = Symbol("c") >>> z = Symbol("z") >>> X = Triangular("x", a,b,c) >>> pprint(density(X)(z), use_unicode=False) / -2a + 2z \|----------------- for And(a <= z, c > z) \|(-a + b)(-a + c) \| \| 2 \| ------ for c = z < -a + b \| \| 2b - 2z \|---------------- for And(b >= z, c < z) \|(-a + b)(b - c) \| \ 0 otherwise References ========== .. [1] https://en.wikipedia.org/wiki/Triangular_distribution .. [2] http://mathworld.wolfram.com/TriangularDistribution.html """ return rv(name, TriangularDistribution, (a, b, c)) #------------------------------------------------------------------------------- # Uniform distribution --------------------------------------------------------- class UniformDistribution(SingleContinuousDistribution): _argnames = ('left', 'right') @property def set(self): return Interval(self.left, self.right) @staticmethod def check(left, right): _value_check(left < right, "Lower limit should be less than Upper limit.") def pdf(self, x): left, right = self.left, self.right return Piecewise( (S.One/(right - left), And(left <= x, x <= right)), (S.Zero, True) ) def _cdf(self, x): left, right = self.left, self.right return Piecewise( (S.Zero, x < left), ((x - left)/(right - left), x <= right), (S.One, True) ) def _characteristic_function(self, t): left, right = self.left, self.right return Piecewise(((exp(Itright) - exp(Itleft)) / (It(right - left)), Ne(t, 0)), (S.One, True)) def _moment_generating_function(self, t): left, right = self.left, self.right return Piecewise(((exp(tright) - exp(tleft)) / (t * (right - left)), Ne(t, 0)), (S.One, True)) def expectation(self, expr, var, kwargs): kwargs['evaluate'] = True result = SingleContinuousDistribution.expectation(self, expr, var, kwargs) result = result.subs({Max(self.left, self.right): self.right, Min(self.left, self.right): self.left}) return result def Uniform(name, left, right): r""" Create a continuous random variable with a uniform distribution. Explanation =========== The density of the uniform distribution is given by .. math:: f(x) := \begin{cases} \frac{1}{b - a} & \text{for } x \in [a,b] \\ 0 & \text{otherwise} \end{cases} with :math:`x \in [a,b]`. Parameters ========== a : Real number, :math:`-\infty < a`, the left boundary b : Real number, :math:`a < b < \infty`, the right boundary Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Uniform, density, cdf, E, variance >>> from sympy import Symbol, simplify >>> a = Symbol("a", negative=True) >>> b = Symbol("b", positive=True) >>> z = Symbol("z") >>> X = Uniform("x", a, b) >>> density(X)(z) Piecewise((1/(-a + b), (b >= z) & (a <= z)), (0, True)) >>> cdf(X)(z) Piecewise((0, a > z), ((-a + z)/(-a + b), b >= z), (1, True)) >>> E(X) a/2 + b/2 >>> simplify(variance(X)) a*2/12 - ab/6 + b*2/12 References ========== .. [1] https://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 .. [2] http://mathworld.wolfram.com/UniformDistribution.html """ return rv(name, UniformDistribution, (left, right)) #------------------------------------------------------------------------------- # UniformSum distribution ------------------------------------------------------ class UniformSumDistribution(SingleContinuousDistribution): _argnames = ('n',) @property def set(self): return Interval(0, self.n) @staticmethod def check(n): _value_check((n > 0, n.is_integer), "Parameter n must be positive integer.") def pdf(self, x): n = self.n k = Dummy("k") return 1/factorial( n - 1)Sum((-1)*kbinomial(n, k)(x - k)(n - 1), (k, 0, floor(x))) def _cdf(self, x): n = self.n k = Dummy("k") return Piecewise((S.Zero, x < 0), (1/factorial(n)Sum((-1)*kbinomial(n, k)(x - k)(n), (k, 0, floor(x))), x <= n), (S.One, True)) def _characteristic_function(self, t): return ((exp(It) - 1) / (It))self.n def _moment_generating_function(self, t): return ((exp(t) - 1) / t)self.n def UniformSum(name, n): r""" Create a continuous random variable with an Irwin-Hall distribution. Explanation =========== The probability distribution function depends on a single parameter $n$ which is an integer. The density of the Irwin-Hall distribution is given by .. math :: f(x) := \frac{1}{(n-1)!}\sum_{k=0}^{\left\lfloor x\right\rfloor}(-1)^k \binom{n}{k}(x-k)^{n-1} Parameters ========== n : A positive integer, `n > 0` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import UniformSum, density, cdf >>> from sympy import Symbol, pprint >>> n = Symbol("n", integer=True) >>> z = Symbol("z") >>> X = UniformSum("x", n) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) floor(z) ___ \ ` \ k n - 1 /n\ ) (-1) (-k + z) \| \| / \k/ /__, k = 0 -------------------------------- (n - 1)! >>> cdf(X)(z) Piecewise((0, z < 0), (Sum((-1)_k(-_k + z)*nbinomial(n, _k), (_k, 0, floor(z)))/factorial(n), n >= z), (1, True)) Compute cdf with specific 'x' and 'n' values as follows : >>> cdf(UniformSum("x", 5), evaluate=False)(2).doit() 9/40 The argument evaluate=False prevents an attempt at evaluation of the sum for general n, before the argument 2 is passed. References ========== .. [1] https://en.wikipedia.org/wiki/Uniform_sum_distribution .. [2] http://mathworld.wolfram.com/UniformSumDistribution.html """ return rv(name, UniformSumDistribution, (n, )) #------------------------------------------------------------------------------- # VonMises distribution -------------------------------------------------------- class VonMisesDistribution(SingleContinuousDistribution): _argnames = ('mu', 'k') set = Interval(0, 2pi) @staticmethod def check(mu, k): _value_check(k > 0, "k must be positive") def pdf(self, x): mu, k = self.mu, self.k return exp(kcos(x-mu)) / (2pibesseli(0, k)) def VonMises(name, mu, k): r""" Create a Continuous Random Variable with a von Mises distribution. Explanation =========== The density of the von Mises distribution is given by .. math:: f(x) := \frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)} with :math:`x \in [0,2\pi]`. Parameters ========== mu : Real number Measure of location. k : Real number Measure of concentration. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import VonMises, density >>> from sympy import Symbol, pprint >>> mu = Symbol("mu") >>> k = Symbol("k", positive=True) >>> z = Symbol("z") >>> X = VonMises("x", mu, k) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) kcos(mu - z) e ------------------ 2pibesseli(0, k) References ========== .. [1] https://en.wikipedia.org/wiki/Von_Mises_distribution .. [2] http://mathworld.wolfram.com/vonMisesDistribution.html """ return rv(name, VonMisesDistribution, (mu, k)) #------------------------------------------------------------------------------- # Weibull distribution --------------------------------------------------------- class WeibullDistribution(SingleContinuousDistribution): _argnames = ('alpha', 'beta') set = Interval(0, oo) @staticmethod def check(alpha, beta): _value_check(alpha > 0, "Alpha must be positive") _value_check(beta > 0, "Beta must be positive") def pdf(self, x): alpha, beta = self.alpha, self.beta return beta (x/alpha)*(beta - 1) exp(-(x/alpha)*beta) / alpha def Weibull(name, alpha, beta): r""" Create a continuous random variable with a Weibull distribution. Explanation =========== The density of the Weibull distribution is given by .. math:: f(x) := \begin{cases} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^{k}} & x\geq0\\ 0 & x<0 \end{cases} Parameters ========== lambda : Real number, $\lambda > 0$, a scale k : Real number, $k > 0$, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Weibull, density, E, variance >>> from sympy import Symbol, simplify >>> l = Symbol("lambda", positive=True) >>> k = Symbol("k", positive=True) >>> z = Symbol("z") >>> X = Weibull("x", l, k) >>> density(X)(z) k(z/lambda)*(k - 1)exp(-(z/lambda)*k)/lambda >>> simplify(E(X)) lambdagamma(1 + 1/k) >>> simplify(variance(X)) lambda*2(-gamma(1 + 1/k)*2 + gamma(1 + 2/k)) References ========== .. [1] https://en.wikipedia.org/wiki/Weibull_distribution .. [2] http://mathworld.wolfram.com/WeibullDistribution.html """ return rv(name, WeibullDistribution, (alpha, beta)) #------------------------------------------------------------------------------- # Wigner semicircle distribution ----------------------------------------------- class WignerSemicircleDistribution(SingleContinuousDistribution): _argnames = ('R',) @property def set(self): return Interval(-self.R, self.R) @staticmethod def check(R): _value_check(R > 0, "Radius R must be positive.") def pdf(self, x): R = self.R return 2/(piR*2)sqrt(R2 - x2) def _characteristic_function(self, t): return Piecewise((2 * besselj(1, self.Rt) / (self.Rt), Ne(t, 0)), (S.One, True)) def _moment_generating_function(self, t): return Piecewise((2 * besseli(1, self.Rt) / (self.Rt), Ne(t, 0)), (S.One, True)) def WignerSemicircle(name, R): r""" Create a continuous random variable with a Wigner semicircle distribution. Explanation =========== The density of the Wigner semicircle distribution is given by .. math:: f(x) := \frac2{\pi R^2}\,\sqrt{R^2-x^2} with :math:`x \in [-R,R]`. Parameters ========== R : Real number, `R > 0`, the radius Returns ======= A RandomSymbol. Examples ======== >>> from sympy.stats import WignerSemicircle, density, E >>> from sympy import Symbol >>> R = Symbol("R", positive=True) >>> z = Symbol("z") >>> X = WignerSemicircle("x", R) >>> density(X)(z) 2sqrt(R2 - z2)/(piR**2) >>> E(X) 0 References ========== .. [1] https://en.wikipedia.org/wiki/Wigner_semicircle_distribution .. [2] http://mathworld.wolfram.com/WignersSemicircleLaw.html """ return rv(name, WignerSemicircleDistribution, (R,))
22591fdf0e907ae3663a9dea126f66abaf8a8dc4bb2e1237a4d8ce390ec7fa5c	""" Finite Discrete Random Variables - Prebuilt variable types Contains ======== FiniteRV DiscreteUniform Die Bernoulli Coin Binomial BetaBinomial Hypergeometric Rademacher IdealSoliton RobustSoliton """ from sympy.core.cache import cacheit from sympy.core.function import Lambda from sympy.core.numbers import (Integer, Rational) from sympy.core.relational import (Eq, Ge, Gt, Le, Lt) from sympy.core.singleton import S from sympy.core.symbol import (Dummy, Symbol) from sympy.core.sympify import sympify from sympy.functions.combinatorial.factorials import binomial from sympy.functions.elementary.exponential import log from sympy.functions.elementary.piecewise import Piecewise from sympy.logic.boolalg import Or from sympy.sets.contains import Contains from sympy.sets.fancysets import Range from sympy.sets.sets import (Intersection, Interval) from sympy.functions.special.beta_functions import beta as beta_fn from sympy.stats.frv import (SingleFiniteDistribution, SingleFinitePSpace) from sympy.stats.rv import _value_check, Density, is_random from sympy.utilities.iterables import multiset from sympy.utilities.misc import filldedent __all__ = ['FiniteRV', 'DiscreteUniform', 'Die', 'Bernoulli', 'Coin', 'Binomial', 'BetaBinomial', 'Hypergeometric', 'Rademacher', 'IdealSoliton', 'RobustSoliton', ] def rv(name, cls, args, kwargs): args = list(map(sympify, args)) dist = cls(args) if kwargs.pop('check', True): dist.check(args) pspace = SingleFinitePSpace(name, dist) if any(is_random(arg) for arg in args): from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution pspace = CompoundPSpace(name, CompoundDistribution(dist)) return pspace.value class FiniteDistributionHandmade(SingleFiniteDistribution): @property def dict(self): return self.args[0] def pmf(self, x): x = Symbol('x') return Lambda(x, Piecewise(( [(v, Eq(k, x)) for k, v in self.dict.items()] + [(S.Zero, True)]))) @property def set(self): return set(self.dict.keys()) @staticmethod def check(density): for p in density.values(): _value_check((p >= 0, p <= 1), "Probability at a point must be between 0 and 1.") val = sum(density.values()) _value_check(Eq(val, 1) != S.false, "Total Probability must be 1.") def FiniteRV(name, density, kwargs): r""" Create a Finite Random Variable given a dict representing the density. Parameters ========== name : Symbol Represents name of the random variable. density : dict Dictionary containing the pdf of finite distribution check : bool If True, it will check whether the given density integrates to 1 over the given set. If False, it will not perform this check. Default is False. Examples ======== >>> from sympy.stats import FiniteRV, P, E >>> density = {0: .1, 1: .2, 2: .3, 3: .4} >>> X = FiniteRV('X', density) >>> E(X) 2.00000000000000 >>> P(X >= 2) 0.700000000000000 Returns ======= RandomSymbol """ # have a default of False while `rv` should have a default of True kwargs['check'] = kwargs.pop('check', False) return rv(name, FiniteDistributionHandmade, density, kwargs) class DiscreteUniformDistribution(SingleFiniteDistribution): @staticmethod def check(args): # not using _value_check since there is a # suggestion for the user if len(set(args)) != len(args): weights = multiset(args) n = Integer(len(args)) for k in weights: weights[k] /= n raise ValueError(filldedent(""" Repeated args detected but set expected. For a distribution having different weights for each item use the following:""") + ( '\nS("FiniteRV(%s, %s)")' % ("'X'", weights))) @property def p(self): return Rational(1, len(self.args)) @property # type: ignore @cacheit def dict(self): return {k: self.p for k in self.set} @property def set(self): return set(self.args) def pmf(self, x): if x in self.args: return self.p else: return S.Zero def DiscreteUniform(name, items): r""" Create a Finite Random Variable representing a uniform distribution over the input set. Parameters ========== items : list/tuple Items over which Uniform distribution is to be made Examples ======== >>> from sympy.stats import DiscreteUniform, density >>> from sympy import symbols >>> X = DiscreteUniform('X', symbols('a b c')) # equally likely over a, b, c >>> density(X).dict {a: 1/3, b: 1/3, c: 1/3} >>> Y = DiscreteUniform('Y', list(range(5))) # distribution over a range >>> density(Y).dict {0: 1/5, 1: 1/5, 2: 1/5, 3: 1/5, 4: 1/5} Returns ======= RandomSymbol References ========== .. [1] https://en.wikipedia.org/wiki/Discrete_uniform_distribution .. [2] http://mathworld.wolfram.com/DiscreteUniformDistribution.html """ return rv(name, DiscreteUniformDistribution, items) class DieDistribution(SingleFiniteDistribution): _argnames = ('sides',) @staticmethod def check(sides): _value_check((sides.is_positive, sides.is_integer), "number of sides must be a positive integer.") @property def is_symbolic(self): return not self.sides.is_number @property def high(self): return self.sides @property def low(self): return S.One @property def set(self): if self.is_symbolic: return Intersection(S.Naturals0, Interval(0, self.sides)) return set(map(Integer, list(range(1, self.sides + 1)))) def pmf(self, x): x = sympify(x) if not (x.is_number or x.is_Symbol or is_random(x)): raise ValueError("'x' expected as an argument of type 'number', 'Symbol', or " "'RandomSymbol' not %s" % (type(x))) cond = Ge(x, 1) & Le(x, self.sides) & Contains(x, S.Integers) return Piecewise((S.One/self.sides, cond), (S.Zero, True)) def Die(name, sides=6): r""" Create a Finite Random Variable representing a fair die. Parameters ========== sides : Integer Represents the number of sides of the Die, by default is 6 Examples ======== >>> from sympy.stats import Die, density >>> from sympy import Symbol >>> D6 = Die('D6', 6) # Six sided Die >>> density(D6).dict {1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6} >>> D4 = Die('D4', 4) # Four sided Die >>> density(D4).dict {1: 1/4, 2: 1/4, 3: 1/4, 4: 1/4} >>> n = Symbol('n', positive=True, integer=True) >>> Dn = Die('Dn', n) # n sided Die >>> density(Dn).dict Density(DieDistribution(n)) >>> density(Dn).dict.subs(n, 4).doit() {1: 1/4, 2: 1/4, 3: 1/4, 4: 1/4} Returns ======= RandomSymbol """ return rv(name, DieDistribution, sides) class BernoulliDistribution(SingleFiniteDistribution): _argnames = ('p', 'succ', 'fail') @staticmethod def check(p, succ, fail): _value_check((p >= 0, p <= 1), "p should be in range [0, 1].") @property def set(self): return {self.succ, self.fail} def pmf(self, x): if isinstance(self.succ, Symbol) and isinstance(self.fail, Symbol): return Piecewise((self.p, x == self.succ), (1 - self.p, x == self.fail), (S.Zero, True)) return Piecewise((self.p, Eq(x, self.succ)), (1 - self.p, Eq(x, self.fail)), (S.Zero, True)) def Bernoulli(name, p, succ=1, fail=0): r""" Create a Finite Random Variable representing a Bernoulli process. Parameters ========== p : Rational number between 0 and 1 Represents probability of success succ : Integer/symbol/string Represents event of success fail : Integer/symbol/string Represents event of failure Examples ======== >>> from sympy.stats import Bernoulli, density >>> from sympy import S >>> X = Bernoulli('X', S(3)/4) # 1-0 Bernoulli variable, probability = 3/4 >>> density(X).dict {0: 1/4, 1: 3/4} >>> X = Bernoulli('X', S.Half, 'Heads', 'Tails') # A fair coin toss >>> density(X).dict {Heads: 1/2, Tails: 1/2} Returns ======= RandomSymbol References ========== .. [1] https://en.wikipedia.org/wiki/Bernoulli_distribution .. [2] http://mathworld.wolfram.com/BernoulliDistribution.html """ return rv(name, BernoulliDistribution, p, succ, fail) def Coin(name, p=S.Half): r""" Create a Finite Random Variable representing a Coin toss. Parameters ========== p : Rational Numeber between 0 and 1 Represents probability of getting "Heads", by default is Half Examples ======== >>> from sympy.stats import Coin, density >>> from sympy import Rational >>> C = Coin('C') # A fair coin toss >>> density(C).dict {H: 1/2, T: 1/2} >>> C2 = Coin('C2', Rational(3, 5)) # An unfair coin >>> density(C2).dict {H: 3/5, T: 2/5} Returns ======= RandomSymbol See Also ======== sympy.stats.Binomial References ========== .. [1] https://en.wikipedia.org/wiki/Coin_flipping """ return rv(name, BernoulliDistribution, p, 'H', 'T') class BinomialDistribution(SingleFiniteDistribution): _argnames = ('n', 'p', 'succ', 'fail') @staticmethod def check(n, p, succ, fail): _value_check((n.is_integer, n.is_nonnegative), "'n' must be nonnegative integer.") _value_check((p <= 1, p >= 0), "p should be in range [0, 1].") @property def high(self): return self.n @property def low(self): return S.Zero @property def is_symbolic(self): return not self.n.is_number @property def set(self): if self.is_symbolic: return Intersection(S.Naturals0, Interval(0, self.n)) return set(self.dict.keys()) def pmf(self, x): n, p = self.n, self.p x = sympify(x) if not (x.is_number or x.is_Symbol or is_random(x)): raise ValueError("'x' expected as an argument of type 'number', 'Symbol', or " "'RandomSymbol' not %s" % (type(x))) cond = Ge(x, 0) & Le(x, n) & Contains(x, S.Integers) return Piecewise((binomial(n, x) * p*x (1 - p)*(n - x), cond), (S.Zero, True)) @property # type: ignore @cacheit def dict(self): if self.is_symbolic: return Density(self) return {kself.succ + (self.n-k)self.fail: self.pmf(k) for k in range(0, self.n + 1)} def Binomial(name, n, p, succ=1, fail=0): r""" Create a Finite Random Variable representing a binomial distribution. Parameters ========== n : Positive Integer Represents number of trials p : Rational Number between 0 and 1 Represents probability of success succ : Integer/symbol/string Represents event of success, by default is 1 fail : Integer/symbol/string Represents event of failure, by default is 0 Examples ======== >>> from sympy.stats import Binomial, density >>> from sympy import S, Symbol >>> X = Binomial('X', 4, S.Half) # Four "coin flips" >>> density(X).dict {0: 1/16, 1: 1/4, 2: 3/8, 3: 1/4, 4: 1/16} >>> n = Symbol('n', positive=True, integer=True) >>> p = Symbol('p', positive=True) >>> X = Binomial('X', n, S.Half) # n "coin flips" >>> density(X).dict Density(BinomialDistribution(n, 1/2, 1, 0)) >>> density(X).dict.subs(n, 4).doit() {0: 1/16, 1: 1/4, 2: 3/8, 3: 1/4, 4: 1/16} Returns ======= RandomSymbol References ========== .. [1] https://en.wikipedia.org/wiki/Binomial_distribution .. [2] http://mathworld.wolfram.com/BinomialDistribution.html """ return rv(name, BinomialDistribution, n, p, succ, fail) #------------------------------------------------------------------------------- # Beta-binomial distribution ---------------------------------------------------------- class BetaBinomialDistribution(SingleFiniteDistribution): _argnames = ('n', 'alpha', 'beta') @staticmethod def check(n, alpha, beta): _value_check((n.is_integer, n.is_nonnegative), "'n' must be nonnegative integer. n = %s." % str(n)) _value_check((alpha > 0), "'alpha' must be: alpha > 0 . alpha = %s" % str(alpha)) _value_check((beta > 0), "'beta' must be: beta > 0 . beta = %s" % str(beta)) @property def high(self): return self.n @property def low(self): return S.Zero @property def is_symbolic(self): return not self.n.is_number @property def set(self): if self.is_symbolic: return Intersection(S.Naturals0, Interval(0, self.n)) return set(map(Integer, list(range(0, self.n + 1)))) def pmf(self, k): n, a, b = self.n, self.alpha, self.beta return binomial(n, k) beta_fn(k + a, n - k + b) / beta_fn(a, b) def BetaBinomial(name, n, alpha, beta): r""" Create a Finite Random Variable representing a Beta-binomial distribution. Parameters ========== n : Positive Integer Represents number of trials alpha : Real positive number beta : Real positive number Examples ======== >>> from sympy.stats import BetaBinomial, density >>> X = BetaBinomial('X', 2, 1, 1) >>> density(X).dict {0: 1/3, 1: 2beta(2, 2), 2: 1/3} Returns ======= RandomSymbol References ========== .. [1] https://en.wikipedia.org/wiki/Beta-binomial_distribution .. [2] http://mathworld.wolfram.com/BetaBinomialDistribution.html """ return rv(name, BetaBinomialDistribution, n, alpha, beta) class HypergeometricDistribution(SingleFiniteDistribution): _argnames = ('N', 'm', 'n') @staticmethod def check(n, N, m): _value_check((N.is_integer, N.is_nonnegative), "'N' must be nonnegative integer. N = %s." % str(n)) _value_check((n.is_integer, n.is_nonnegative), "'n' must be nonnegative integer. n = %s." % str(n)) _value_check((m.is_integer, m.is_nonnegative), "'m' must be nonnegative integer. m = %s." % str(n)) @property def is_symbolic(self): return not all(x.is_number for x in (self.N, self.m, self.n)) @property def high(self): return Piecewise((self.n, Lt(self.n, self.m) != False), (self.m, True)) @property def low(self): return Piecewise((0, Gt(0, self.n + self.m - self.N) != False), (self.n + self.m - self.N, True)) @property def set(self): N, m, n = self.N, self.m, self.n if self.is_symbolic: return Intersection(S.Naturals0, Interval(self.low, self.high)) return {i for i in range(max(0, n + m - N), min(n, m) + 1)} def pmf(self, k): N, m, n = self.N, self.m, self.n return S(binomial(m, k) binomial(N - m, n - k))/binomial(N, n) def Hypergeometric(name, N, m, n): r""" Create a Finite Random Variable representing a hypergeometric distribution. Parameters ========== N : Positive Integer Represents finite population of size N. m : Positive Integer Represents number of trials with required feature. n : Positive Integer Represents numbers of draws. Examples ======== >>> from sympy.stats import Hypergeometric, density >>> X = Hypergeometric('X', 10, 5, 3) # 10 marbles, 5 white (success), 3 draws >>> density(X).dict {0: 1/12, 1: 5/12, 2: 5/12, 3: 1/12} Returns ======= RandomSymbol References ========== .. [1] https://en.wikipedia.org/wiki/Hypergeometric_distribution .. [2] http://mathworld.wolfram.com/HypergeometricDistribution.html """ return rv(name, HypergeometricDistribution, N, m, n) class RademacherDistribution(SingleFiniteDistribution): @property def set(self): return {-1, 1} @property def pmf(self): k = Dummy('k') return Lambda(k, Piecewise((S.Half, Or(Eq(k, -1), Eq(k, 1))), (S.Zero, True))) def Rademacher(name): r""" Create a Finite Random Variable representing a Rademacher distribution. Examples ======== >>> from sympy.stats import Rademacher, density >>> X = Rademacher('X') >>> density(X).dict {-1: 1/2, 1: 1/2} Returns ======= RandomSymbol See Also ======== sympy.stats.Bernoulli References ========== .. [1] https://en.wikipedia.org/wiki/Rademacher_distribution """ return rv(name, RademacherDistribution) class IdealSolitonDistribution(SingleFiniteDistribution): _argnames = ('k',) @staticmethod def check(k): _value_check(k.is_integer and k.is_positive, "'k' must be a positive integer.") @property def low(self): return S.One @property def high(self): return self.k @property def set(self): return set(list(Range(1, self.k+1))) @property # type: ignore @cacheit def dict(self): if self.k.is_Symbol: return Density(self) d = {1: Rational(1, self.k)} d.update(dict((i, Rational(1, i(i - 1))) for i in range(2, self.k + 1))) return d def pmf(self, x): x = sympify(x) if not (x.is_number or x.is_Symbol or is_random(x)): raise ValueError("'x' expected as an argument of type 'number', 'Symbol', or " "'RandomSymbol' not %s" % (type(x))) cond1 = Eq(x, 1) & x.is_integer cond2 = Ge(x, 1) & Le(x, self.k) & x.is_integer return Piecewise((1/self.k, cond1), (1/(x(x - 1)), cond2), (S.Zero, True)) def IdealSoliton(name, k): r""" Create a Finite Random Variable of Ideal Soliton Distribution Parameters ========== k : Positive Integer Represents the number of input symbols in an LT (Luby Transform) code. Examples ======== >>> from sympy.stats import IdealSoliton, density, P, E >>> sol = IdealSoliton('sol', 5) >>> density(sol).dict {1: 1/5, 2: 1/2, 3: 1/6, 4: 1/12, 5: 1/20} >>> density(sol).set {1, 2, 3, 4, 5} >>> from sympy import Symbol >>> k = Symbol('k', positive=True, integer=True) >>> sol = IdealSoliton('sol', k) >>> density(sol).dict Density(IdealSolitonDistribution(k)) >>> density(sol).dict.subs(k, 10).doit() {1: 1/10, 2: 1/2, 3: 1/6, 4: 1/12, 5: 1/20, 6: 1/30, 7: 1/42, 8: 1/56, 9: 1/72, 10: 1/90} >>> E(sol.subs(k, 10)) 7381/2520 >>> P(sol.subs(k, 4) > 2) 1/4 Returns ======= RandomSymbol References ========== .. [1] https://en.wikipedia.org/wiki/Soliton_distribution#Ideal_distribution .. [2] http://pages.cs.wisc.edu/~suman/courses/740/papers/luby02lt.pdf """ return rv(name, IdealSolitonDistribution, k) class RobustSolitonDistribution(SingleFiniteDistribution): _argnames= ('k', 'delta', 'c') @staticmethod def check(k, delta, c): _value_check(k.is_integer and k.is_positive, "'k' must be a positive integer") _value_check(Gt(delta, 0) and Le(delta, 1), "'delta' must be a real number in the interval (0,1)") _value_check(c.is_positive, "'c' must be a positive real number.") @property def R(self): return self.c * log(self.k/self.delta) * self.k*0.5 @property def Z(self): z = 0 for i in Range(1, round(self.k/self.R)): z += (1/i) z += log(self.R/self.delta) return 1 + z self.R/self.k @property def low(self): return S.One @property def high(self): return self.k @property def set(self): return set(list(Range(1, self.k+1))) @property def is_symbolic(self): return not (self.k.is_number and self.c.is_number and self.delta.is_number) def pmf(self, x): x = sympify(x) if not (x.is_number or x.is_Symbol or is_random(x)): raise ValueError("'x' expected as an argument of type 'number', 'Symbol', or " "'RandomSymbol' not %s" % (type(x))) cond1 = Eq(x, 1) & x.is_integer cond2 = Ge(x, 1) & Le(x, self.k) & x.is_integer rho = Piecewise((Rational(1, self.k), cond1), (Rational(1, x(x-1)), cond2), (S.Zero, True)) cond1 = Ge(x, 1) & Le(x, round(self.k/self.R)-1) cond2 = Eq(x, round(self.k/self.R)) tau = Piecewise((self.R/(self.k x), cond1), (self.R * log(self.R/self.delta)/self.k, cond2), (S.Zero, True)) return (rho + tau)/self.Z def RobustSoliton(name, k, delta, c): r''' Create a Finite Random Variable of Robust Soliton Distribution Parameters ========== k : Positive Integer Represents the number of input symbols in an LT (Luby Transform) code. delta : Positive Rational Number Represents the failure probability. Must be in the interval (0,1). c : Positive Rational Number Constant of proportionality. Values close to 1 are recommended Examples ======== >>> from sympy.stats import RobustSoliton, density, P, E >>> robSol = RobustSoliton('robSol', 5, 0.5, 0.01) >>> density(robSol).dict {1: 0.204253668152708, 2: 0.490631107897393, 3: 0.165210624506162, 4: 0.0834387731899302, 5: 0.0505633404760675} >>> density(robSol).set {1, 2, 3, 4, 5} >>> from sympy import Symbol >>> k = Symbol('k', positive=True, integer=True) >>> c = Symbol('c', positive=True) >>> robSol = RobustSoliton('robSol', k, 0.5, c) >>> density(robSol).dict Density(RobustSolitonDistribution(k, 0.5, c)) >>> density(robSol).dict.subs(k, 10).subs(c, 0.03).doit() {1: 0.116641095387194, 2: 0.467045731687165, 3: 0.159984123349381, 4: 0.0821431680681869, 5: 0.0505765646770100, 6: 0.0345781523420719, 7: 0.0253132820710503, 8: 0.0194459129233227, 9: 0.0154831166726115, 10: 0.0126733075238887} >>> E(robSol.subs(k, 10).subs(c, 0.05)) 2.91358846104106 >>> P(robSol.subs(k, 4).subs(c, 0.1) > 2) 0.243650614389834 Returns ======= RandomSymbol References ========== .. [1] https://en.wikipedia.org/wiki/Soliton_distribution#Robust_distribution .. [2] http://www.inference.org.uk/mackay/itprnn/ps/588.596.pdf .. [3] http://pages.cs.wisc.edu/~suman/courses/740/papers/luby02lt.pdf ''' return rv(name, RobustSolitonDistribution, k, delta, c)
81f0e752e002fc217809034841f0664d7757b0498ed9d75eda9f5d95fea12ad1	import random import itertools from typing import (Sequence as tSequence, Union as tUnion, List as tList, Tuple as tTuple, Set as tSet) from sympy.concrete.summations import Sum from sympy.core.add import Add from sympy.core.basic import Basic from sympy.core.cache import cacheit from sympy.core.containers import Tuple from sympy.core.expr import Expr from sympy.core.function import (Function, Lambda) from sympy.core.mul import Mul from sympy.core.numbers import (Integer, Rational, igcd, oo, pi) from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne) from sympy.core.singleton import S from sympy.core.symbol import (Dummy, Symbol) from sympy.functions.combinatorial.factorials import factorial from sympy.functions.elementary.exponential import exp from sympy.functions.elementary.integers import ceiling from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.piecewise import Piecewise from sympy.functions.special.gamma_functions import gamma from sympy.logic.boolalg import (And, Not, Or) from sympy.matrices.common import NonSquareMatrixError from sympy.matrices.dense import (Matrix, eye, ones, zeros) from sympy.matrices.expressions.blockmatrix import BlockMatrix from sympy.matrices.expressions.matexpr import MatrixSymbol from sympy.matrices.expressions.special import Identity from sympy.matrices.immutable import ImmutableMatrix from sympy.sets.conditionset import ConditionSet from sympy.sets.contains import Contains from sympy.sets.fancysets import Range from sympy.sets.sets import (FiniteSet, Intersection, Interval, Set, Union) from sympy.solvers.solveset import linsolve from sympy.tensor.indexed import (Indexed, IndexedBase) from sympy.core.relational import Relational from sympy.logic.boolalg import Boolean from sympy.utilities.exceptions import sympy_deprecation_warning from sympy.utilities.iterables import strongly_connected_components from sympy.stats.joint_rv import JointDistribution from sympy.stats.joint_rv_types import JointDistributionHandmade from sympy.stats.rv import (RandomIndexedSymbol, random_symbols, RandomSymbol, _symbol_converter, _value_check, pspace, given, dependent, is_random, sample_iter, Distribution, Density) from sympy.stats.stochastic_process import StochasticPSpace from sympy.stats.symbolic_probability import Probability, Expectation from sympy.stats.frv_types import Bernoulli, BernoulliDistribution, FiniteRV from sympy.stats.drv_types import Poisson, PoissonDistribution from sympy.stats.crv_types import Normal, NormalDistribution, Gamma, GammaDistribution from sympy.core.sympify import _sympify, sympify EmptySet = S.EmptySet __all__ = [ 'StochasticProcess', 'DiscreteTimeStochasticProcess', 'DiscreteMarkovChain', 'TransitionMatrixOf', 'StochasticStateSpaceOf', 'GeneratorMatrixOf', 'ContinuousMarkovChain', 'BernoulliProcess', 'PoissonProcess', 'WienerProcess', 'GammaProcess' ] @is_random.register(Indexed) def _(x): return is_random(x.base) @is_random.register(RandomIndexedSymbol) # type: ignore def _(x): return True def _set_converter(itr): """ Helper function for converting list/tuple/set to Set. If parameter is not an instance of list/tuple/set then no operation is performed. Returns ======= Set The argument converted to Set. Raises ====== TypeError If the argument is not an instance of list/tuple/set. """ if isinstance(itr, (list, tuple, set)): itr = FiniteSet(itr) if not isinstance(itr, Set): raise TypeError("%s is not an instance of list/tuple/set."%(itr)) return itr def _state_converter(itr: tSequence) -> tUnion[Tuple, Range]: """ Helper function for converting list/tuple/set/Range/Tuple/FiniteSet to tuple/Range. """ itr_ret: tUnion[Tuple, Range] if isinstance(itr, (Tuple, set, FiniteSet)): itr_ret = Tuple((sympify(i) if isinstance(i, str) else i for i in itr)) elif isinstance(itr, (list, tuple)): # check if states are unique if len(set(itr)) != len(itr): raise ValueError('The state space must have unique elements.') itr_ret = Tuple((sympify(i) if isinstance(i, str) else i for i in itr)) elif isinstance(itr, Range): # the only ordered set in SymPy I know of # try to convert to tuple try: itr_ret = Tuple((sympify(i) if isinstance(i, str) else i for i in itr)) except (TypeError, ValueError): itr_ret = itr else: raise TypeError("%s is not an instance of list/tuple/set/Range/Tuple/FiniteSet." % (itr)) return itr_ret def _sym_sympify(arg): """ Converts an arbitrary expression to a type that can be used inside SymPy. As generally strings are unwise to use in the expressions, it returns the Symbol of argument if the string type argument is passed. Parameters ========= arg: The parameter to be converted to be used in SymPy. Returns ======= The converted parameter. """ if isinstance(arg, str): return Symbol(arg) else: return _sympify(arg) def _matrix_checks(matrix): if not isinstance(matrix, (Matrix, MatrixSymbol, ImmutableMatrix)): raise TypeError("Transition probabilities either should " "be a Matrix or a MatrixSymbol.") if matrix.shape[0] != matrix.shape[1]: raise NonSquareMatrixError("%s is not a square matrix"%(matrix)) if isinstance(matrix, Matrix): matrix = ImmutableMatrix(matrix.tolist()) return matrix class StochasticProcess(Basic): """ Base class for all the stochastic processes whether discrete or continuous. Parameters ========== sym: Symbol or str state_space: Set The state space of the stochastic process, by default S.Reals. For discrete sets it is zero indexed. See Also ======== DiscreteTimeStochasticProcess """ index_set = S.Reals def __new__(cls, sym, state_space=S.Reals, *kwargs): sym = _symbol_converter(sym) state_space = _set_converter(state_space) return Basic.__new__(cls, sym, state_space) @property def symbol(self): return self.args[0] @property def state_space(self) -> tUnion[FiniteSet, Range]: if not isinstance(self.args[1], (FiniteSet, Range)): assert isinstance(self.args[1], Tuple) return FiniteSet(self.args[1]) return self.args[1] def _deprecation_warn_distribution(self): sympy_deprecation_warning( """ Calling the distribution method with a RandomIndexedSymbol argument, like X.distribution(X(t)) is deprecated. Instead, call distribution() with the given timestamp, like X.distribution(t) """, deprecated_since_version="1.7.1", active_deprecations_target="deprecated-distribution-randomindexedsymbol", stacklevel=4, ) def distribution(self, key=None): if key is None: self._deprecation_warn_distribution() return Distribution() def density(self, x): return Density() def __call__(self, time): """ Overridden in ContinuousTimeStochasticProcess. """ raise NotImplementedError("Use [] for indexing discrete time stochastic process.") def __getitem__(self, time): """ Overridden in DiscreteTimeStochasticProcess. """ raise NotImplementedError("Use () for indexing continuous time stochastic process.") def probability(self, condition): raise NotImplementedError() def joint_distribution(self, args): """ Computes the joint distribution of the random indexed variables. Parameters ========== args: iterable The finite list of random indexed variables/the key of a stochastic process whose joint distribution has to be computed. Returns ======= JointDistribution The joint distribution of the list of random indexed variables. An unevaluated object is returned if it is not possible to compute the joint distribution. Raises ====== ValueError: When the arguments passed are not of type RandomIndexSymbol or Number. """ args = list(args) for i, arg in enumerate(args): if S(arg).is_Number: if self.index_set.is_subset(S.Integers): args[i] = self.__getitem__(arg) else: args[i] = self.__call__(arg) elif not isinstance(arg, RandomIndexedSymbol): raise ValueError("Expected a RandomIndexedSymbol or " "key not %s"%(type(arg))) if args[0].pspace.distribution == Distribution(): return JointDistribution(args) density = Lambda(tuple(args), expr=Mul.fromiter(arg.pspace.process.density(arg) for arg in args)) return JointDistributionHandmade(density) def expectation(self, condition, given_condition): raise NotImplementedError("Abstract method for expectation queries.") def sample(self): raise NotImplementedError("Abstract method for sampling queries.") class DiscreteTimeStochasticProcess(StochasticProcess): """ Base class for all discrete stochastic processes. """ def __getitem__(self, time): """ For indexing discrete time stochastic processes. Returns ======= RandomIndexedSymbol """ time = sympify(time) if not time.is_symbol and time not in self.index_set: raise IndexError("%s is not in the index set of %s"%(time, self.symbol)) idx_obj = Indexed(self.symbol, time) pspace_obj = StochasticPSpace(self.symbol, self, self.distribution(time)) return RandomIndexedSymbol(idx_obj, pspace_obj) class ContinuousTimeStochasticProcess(StochasticProcess): """ Base class for all continuous time stochastic process. """ def __call__(self, time): """ For indexing continuous time stochastic processes. Returns ======= RandomIndexedSymbol """ time = sympify(time) if not time.is_symbol and time not in self.index_set: raise IndexError("%s is not in the index set of %s"%(time, self.symbol)) func_obj = Function(self.symbol)(time) pspace_obj = StochasticPSpace(self.symbol, self, self.distribution(time)) return RandomIndexedSymbol(func_obj, pspace_obj) class TransitionMatrixOf(Boolean): """ Assumes that the matrix is the transition matrix of the process. """ def __new__(cls, process, matrix): if not isinstance(process, DiscreteMarkovChain): raise ValueError("Currently only DiscreteMarkovChain " "support TransitionMatrixOf.") matrix = _matrix_checks(matrix) return Basic.__new__(cls, process, matrix) process = property(lambda self: self.args[0]) matrix = property(lambda self: self.args[1]) class GeneratorMatrixOf(TransitionMatrixOf): """ Assumes that the matrix is the generator matrix of the process. """ def __new__(cls, process, matrix): if not isinstance(process, ContinuousMarkovChain): raise ValueError("Currently only ContinuousMarkovChain " "support GeneratorMatrixOf.") matrix = _matrix_checks(matrix) return Basic.__new__(cls, process, matrix) class StochasticStateSpaceOf(Boolean): def __new__(cls, process, state_space): if not isinstance(process, (DiscreteMarkovChain, ContinuousMarkovChain)): raise ValueError("Currently only DiscreteMarkovChain and ContinuousMarkovChain " "support StochasticStateSpaceOf.") state_space = _state_converter(state_space) if isinstance(state_space, Range): ss_size = ceiling((state_space.stop - state_space.start) / state_space.step) else: ss_size = len(state_space) state_index = Range(ss_size) return Basic.__new__(cls, process, state_index) process = property(lambda self: self.args[0]) state_index = property(lambda self: self.args[1]) class MarkovProcess(StochasticProcess): """ Contains methods that handle queries common to Markov processes. """ @property def number_of_states(self) -> tUnion[Integer, Symbol]: """ The number of states in the Markov Chain. """ return _sympify(self.args[2].shape[0]) # type: ignore @property def _state_index(self): """ Returns state index as Range. """ return self.args[1] @classmethod def _sanity_checks(cls, state_space, trans_probs): # Try to never have None as state_space or trans_probs. # This helps a lot if we get it done at the start. if (state_space is None) and (trans_probs is None): _n = Dummy('n', integer=True, nonnegative=True) state_space = _state_converter(Range(_n)) trans_probs = _matrix_checks(MatrixSymbol('_T', _n, _n)) elif state_space is None: trans_probs = _matrix_checks(trans_probs) state_space = _state_converter(Range(trans_probs.shape[0])) elif trans_probs is None: state_space = _state_converter(state_space) if isinstance(state_space, Range): _n = ceiling((state_space.stop - state_space.start) / state_space.step) else: _n = len(state_space) trans_probs = MatrixSymbol('_T', _n, _n) else: state_space = _state_converter(state_space) trans_probs = _matrix_checks(trans_probs) # Range object doesn't want to give a symbolic size # so we do it ourselves. if isinstance(state_space, Range): ss_size = ceiling((state_space.stop - state_space.start) / state_space.step) else: ss_size = len(state_space) if ss_size != trans_probs.shape[0]: raise ValueError('The size of the state space and the number of ' 'rows of the transition matrix must be the same.') return state_space, trans_probs def _extract_information(self, given_condition): """ Helper function to extract information, like, transition matrix/generator matrix, state space, etc. """ if isinstance(self, DiscreteMarkovChain): trans_probs = self.transition_probabilities state_index = self._state_index elif isinstance(self, ContinuousMarkovChain): trans_probs = self.generator_matrix state_index = self._state_index if isinstance(given_condition, And): gcs = given_condition.args given_condition = S.true for gc in gcs: if isinstance(gc, TransitionMatrixOf): trans_probs = gc.matrix if isinstance(gc, StochasticStateSpaceOf): state_index = gc.state_index if isinstance(gc, Relational): given_condition = given_condition & gc if isinstance(given_condition, TransitionMatrixOf): trans_probs = given_condition.matrix given_condition = S.true if isinstance(given_condition, StochasticStateSpaceOf): state_index = given_condition.state_index given_condition = S.true return trans_probs, state_index, given_condition def _check_trans_probs(self, trans_probs, row_sum=1): """ Helper function for checking the validity of transition probabilities. """ if not isinstance(trans_probs, MatrixSymbol): rows = trans_probs.tolist() for row in rows: if (sum(row) - row_sum) != 0: raise ValueError("Values in a row must sum to %s. " "If you are using Float or floats then please use Rational."%(row_sum)) def _work_out_state_index(self, state_index, given_condition, trans_probs): """ Helper function to extract state space if there is a random symbol in the given condition. """ # if given condition is None, then there is no need to work out # state_space from random variables if given_condition != None: rand_var = list(given_condition.atoms(RandomSymbol) - given_condition.atoms(RandomIndexedSymbol)) if len(rand_var) == 1: state_index = rand_var[0].pspace.set # `not None` is `True`. So the old test fails for symbolic sizes. # Need to build the statement differently. sym_cond = not self.number_of_states.is_Integer cond1 = not sym_cond and len(state_index) != trans_probs.shape[0] if cond1: raise ValueError("state space is not compatible with the transition probabilities.") if not isinstance(trans_probs.shape[0], Symbol): state_index = FiniteSet([i for i in range(trans_probs.shape[0])]) return state_index @cacheit def _preprocess(self, given_condition, evaluate): """ Helper function for pre-processing the information. """ is_insufficient = False if not evaluate: # avoid pre-processing if the result is not to be evaluated return (True, None, None, None) # extracting transition matrix and state space trans_probs, state_index, given_condition = self._extract_information(given_condition) # given_condition does not have sufficient information # for computations if trans_probs is None or \ given_condition is None: is_insufficient = True else: # checking transition probabilities if isinstance(self, DiscreteMarkovChain): self._check_trans_probs(trans_probs, row_sum=1) elif isinstance(self, ContinuousMarkovChain): self._check_trans_probs(trans_probs, row_sum=0) # working out state space state_index = self._work_out_state_index(state_index, given_condition, trans_probs) return is_insufficient, trans_probs, state_index, given_condition def replace_with_index(self, condition): if isinstance(condition, Relational): lhs, rhs = condition.lhs, condition.rhs if not isinstance(lhs, RandomIndexedSymbol): lhs, rhs = rhs, lhs condition = type(condition)(self.index_of.get(lhs, lhs), self.index_of.get(rhs, rhs)) return condition def probability(self, condition, given_condition=None, evaluate=True, kwargs): """ Handles probability queries for Markov process. Parameters ========== condition: Relational given_condition: Relational/And Returns ======= Probability If the information is not sufficient. Expr In all other cases. Note ==== Any information passed at the time of query overrides any information passed at the time of object creation like transition probabilities, state space. Pass the transition matrix using TransitionMatrixOf, generator matrix using GeneratorMatrixOf and state space using StochasticStateSpaceOf in given_condition using & or And. """ check, mat, state_index, new_given_condition = \ self._preprocess(given_condition, evaluate) rv = list(condition.atoms(RandomIndexedSymbol)) symbolic = False for sym in rv: if sym.key.is_symbol: symbolic = True break if check: return Probability(condition, new_given_condition) if isinstance(self, ContinuousMarkovChain): trans_probs = self.transition_probabilities(mat) elif isinstance(self, DiscreteMarkovChain): trans_probs = mat condition = self.replace_with_index(condition) given_condition = self.replace_with_index(given_condition) new_given_condition = self.replace_with_index(new_given_condition) if isinstance(condition, Relational): if isinstance(new_given_condition, And): gcs = new_given_condition.args else: gcs = (new_given_condition, ) min_key_rv = list(new_given_condition.atoms(RandomIndexedSymbol)) if len(min_key_rv): min_key_rv = min_key_rv[0] for r in rv: if min_key_rv.key.is_symbol or r.key.is_symbol: continue if min_key_rv.key > r.key: return Probability(condition) else: min_key_rv = None return Probability(condition) if symbolic: return self._symbolic_probability(condition, new_given_condition, rv, min_key_rv) if len(rv) > 1: rv[0] = condition.lhs rv[1] = condition.rhs if rv[0].key < rv[1].key: rv[0], rv[1] = rv[1], rv[0] if isinstance(condition, Gt): condition = Lt(condition.lhs, condition.rhs) elif isinstance(condition, Lt): condition = Gt(condition.lhs, condition.rhs) elif isinstance(condition, Ge): condition = Le(condition.lhs, condition.rhs) elif isinstance(condition, Le): condition = Ge(condition.lhs, condition.rhs) s = Rational(0, 1) n = len(self.state_space) if isinstance(condition, (Eq, Ne)): for i in range(0, n): s += self.probability(Eq(rv[0], i), Eq(rv[1], i)) self.probability(Eq(rv[1], i), new_given_condition) return s if isinstance(condition, Eq) else 1 - s else: upper = 0 greater = False if isinstance(condition, (Ge, Lt)): upper = 1 if isinstance(condition, (Ge, Gt)): greater = True for i in range(0, n): if i <= n//2: for j in range(0, i + upper): s += self.probability(Eq(rv[0], i), Eq(rv[1], j)) * self.probability(Eq(rv[1], j), new_given_condition) else: s += self.probability(Eq(rv[0], i), new_given_condition) for j in range(i + upper, n): s -= self.probability(Eq(rv[0], i), Eq(rv[1], j)) * self.probability(Eq(rv[1], j), new_given_condition) return s if greater else 1 - s rv = rv[0] states = condition.as_set() prob, gstate = dict(), None for gc in gcs: if gc.has(min_key_rv): if gc.has(Probability): p, gp = (gc.rhs, gc.lhs) if isinstance(gc.lhs, Probability) \ else (gc.lhs, gc.rhs) gr = gp.args[0] gset = Intersection(gr.as_set(), state_index) gstate = list(gset)[0] prob[gset] = p else: _, gstate = (gc.lhs.key, gc.rhs) if isinstance(gc.lhs, RandomIndexedSymbol) \ else (gc.rhs.key, gc.lhs) if not all(k in self.index_set for k in (rv.key, min_key_rv.key)): raise IndexError("The timestamps of the process are not in it's index set.") states = Intersection(states, state_index) if not isinstance(self.number_of_states, Symbol) else states for state in Union(states, FiniteSet(gstate)): if not state.is_Integer or Ge(state, mat.shape[0]) is True: raise IndexError("No information is available for (%s, %s) in " "transition probabilities of shape, (%s, %s). " "State space is zero indexed." %(gstate, state, mat.shape[0], mat.shape[1])) if prob: gstates = Union(prob.keys()) if len(gstates) == 1: gstate = list(gstates)[0] gprob = list(prob.values())[0] prob[gstates] = gprob elif len(gstates) == len(state_index) - 1: gstate = list(state_index - gstates)[0] gprob = S.One - sum(prob.values()) prob[state_index - gstates] = gprob else: raise ValueError("Conflicting information.") else: gprob = S.One if min_key_rv == rv: return sum([prob[FiniteSet(state)] for state in states]) if isinstance(self, ContinuousMarkovChain): return gprob sum([trans_probs(rv.key - min_key_rv.key).__getitem__((gstate, state)) for state in states]) if isinstance(self, DiscreteMarkovChain): return gprob * sum([(trans_probs(rv.key - min_key_rv.key)).__getitem__((gstate, state)) for state in states]) if isinstance(condition, Not): expr = condition.args[0] return S.One - self.probability(expr, given_condition, evaluate, kwargs) if isinstance(condition, And): compute_later, state2cond, conds = [], dict(), condition.args for expr in conds: if isinstance(expr, Relational): ris = list(expr.atoms(RandomIndexedSymbol))[0] if state2cond.get(ris, None) is None: state2cond[ris] = S.true state2cond[ris] &= expr else: compute_later.append(expr) ris = [] for ri in state2cond: ris.append(ri) cset = Intersection(state2cond[ri].as_set(), state_index) if len(cset) == 0: return S.Zero state2cond[ri] = cset.as_relational(ri) sorted_ris = sorted(ris, key=lambda ri: ri.key) prod = self.probability(state2cond[sorted_ris[0]], given_condition, evaluate, *kwargs) for i in range(1, len(sorted_ris)): ri, prev_ri = sorted_ris[i], sorted_ris[i-1] if not isinstance(state2cond[ri], Eq): raise ValueError("The process is in multiple states at %s, unable to determine the probability."%(ri)) mat_of = TransitionMatrixOf(self, mat) if isinstance(self, DiscreteMarkovChain) else GeneratorMatrixOf(self, mat) prod = self.probability(state2cond[ri], state2cond[prev_ri] & mat_of & StochasticStateSpaceOf(self, state_index), evaluate, *kwargs) for expr in compute_later: prod = self.probability(expr, given_condition, evaluate, kwargs) return prod if isinstance(condition, Or): return sum([self.probability(expr, given_condition, evaluate, kwargs) for expr in condition.args]) raise NotImplementedError("Mechanism for handling (%s, %s) queries hasn't been " "implemented yet."%(condition, given_condition)) def _symbolic_probability(self, condition, new_given_condition, rv, min_key_rv): #Function to calculate probability for queries with symbols if isinstance(condition, Relational): curr_state = new_given_condition.rhs if isinstance(new_given_condition.lhs, RandomIndexedSymbol) \ else new_given_condition.lhs next_state = condition.rhs if isinstance(condition.lhs, RandomIndexedSymbol) \ else condition.lhs if isinstance(condition, (Eq, Ne)): if isinstance(self, DiscreteMarkovChain): P = self.transition_probabilities*(rv[0].key - min_key_rv.key) else: P = exp(self.generator_matrix(rv[0].key - min_key_rv.key)) prob = P[curr_state, next_state] if isinstance(condition, Eq) else 1 - P[curr_state, next_state] return Piecewise((prob, rv[0].key > min_key_rv.key), (Probability(condition), True)) else: upper = 1 greater = False if isinstance(condition, (Ge, Lt)): upper = 0 if isinstance(condition, (Ge, Gt)): greater = True k = Dummy('k') condition = Eq(condition.lhs, k) if isinstance(condition.lhs, RandomIndexedSymbol)\ else Eq(condition.rhs, k) total = Sum(self.probability(condition, new_given_condition), (k, next_state + upper, self.state_space._sup)) return Piecewise((total, rv[0].key > min_key_rv.key), (Probability(condition), True)) if greater\ else Piecewise((1 - total, rv[0].key > min_key_rv.key), (Probability(condition), True)) else: return Probability(condition, new_given_condition) def expectation(self, expr, condition=None, evaluate=True, *kwargs): """ Handles expectation queries for markov process. Parameters ========== expr: RandomIndexedSymbol, Relational, Logic Condition for which expectation has to be computed. Must contain a RandomIndexedSymbol of the process. condition: Relational, Logic The given conditions under which computations should be done. Returns ======= Expectation Unevaluated object if computations cannot be done due to insufficient information. Expr In all other cases when the computations are successful. Note ==== Any information passed at the time of query overrides any information passed at the time of object creation like transition probabilities, state space. Pass the transition matrix using TransitionMatrixOf, generator matrix using GeneratorMatrixOf and state space using StochasticStateSpaceOf in given_condition using & or And. """ check, mat, state_index, condition = \ self._preprocess(condition, evaluate) if check: return Expectation(expr, condition) rvs = random_symbols(expr) if isinstance(expr, Expr) and isinstance(condition, Eq) \ and len(rvs) == 1: # handle queries similar to E(f(X[i]), Eq(X[i-m], <some-state>)) condition=self.replace_with_index(condition) state_index=self.replace_with_index(state_index) rv = list(rvs)[0] lhsg, rhsg = condition.lhs, condition.rhs if not isinstance(lhsg, RandomIndexedSymbol): lhsg, rhsg = (rhsg, lhsg) if rhsg not in state_index: raise ValueError("%s state is not in the state space."%(rhsg)) if rv.key < lhsg.key: raise ValueError("Incorrect given condition is given, expectation " "time %s < time %s"%(rv.key, rv.key)) mat_of = TransitionMatrixOf(self, mat) if isinstance(self, DiscreteMarkovChain) else GeneratorMatrixOf(self, mat) cond = condition & mat_of & \ StochasticStateSpaceOf(self, state_index) func = lambda s: self.probability(Eq(rv, s), cond) expr.subs(rv, self._state_index[s]) return sum([func(s) for s in state_index]) raise NotImplementedError("Mechanism for handling (%s, %s) queries hasn't been " "implemented yet."%(expr, condition)) class DiscreteMarkovChain(DiscreteTimeStochasticProcess, MarkovProcess): """ Represents a finite discrete time-homogeneous Markov chain. This type of Markov Chain can be uniquely characterised by its (ordered) state space and its one-step transition probability matrix. Parameters ========== sym: The name given to the Markov Chain state_space: Optional, by default, Range(n) trans_probs: Optional, by default, MatrixSymbol('_T', n, n) Examples ======== >>> from sympy.stats import DiscreteMarkovChain, TransitionMatrixOf, P, E >>> from sympy import Matrix, MatrixSymbol, Eq, symbols >>> T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]]) >>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T) >>> YS = DiscreteMarkovChain("Y") >>> Y.state_space {0, 1, 2} >>> Y.transition_probabilities Matrix([ [0.5, 0.2, 0.3], [0.2, 0.5, 0.3], [0.2, 0.3, 0.5]]) >>> TS = MatrixSymbol('T', 3, 3) >>> P(Eq(YS[3], 2), Eq(YS[1], 1) & TransitionMatrixOf(YS, TS)) T[0, 2]T[1, 0] + T[1, 1]T[1, 2] + T[1, 2]T[2, 2] >>> P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2) 0.36 Probabilities will be calculated based on indexes rather than state names. For example, with the Sunny-Cloudy-Rainy model with string state names: >>> from sympy.core.symbol import Str >>> Y = DiscreteMarkovChain("Y", [Str('Sunny'), Str('Cloudy'), Str('Rainy')], T) >>> P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2) 0.36 This gives the same answer as the ``[0, 1, 2]`` state space. Currently, there is no support for state names within probability and expectation statements. Here is a work-around using ``Str``: >>> P(Eq(Str('Rainy'), Y[3]), Eq(Y[1], Str('Cloudy'))).round(2) 0.36 Symbol state names can also be used: >>> sunny, cloudy, rainy = symbols('Sunny, Cloudy, Rainy') >>> Y = DiscreteMarkovChain("Y", [sunny, cloudy, rainy], T) >>> P(Eq(Y[3], rainy), Eq(Y[1], cloudy)).round(2) 0.36 Expectations will be calculated as follows: >>> E(Y[3], Eq(Y[1], cloudy)) 0.38Cloudy + 0.36Rainy + 0.26Sunny Probability of expressions with multiple RandomIndexedSymbols can also be calculated provided there is only 1 RandomIndexedSymbol in the given condition. It is always better to use Rational instead of floating point numbers for the probabilities in the transition matrix to avoid errors. >>> from sympy import Gt, Le, Rational >>> T = Matrix([[Rational(5, 10), Rational(3, 10), Rational(2, 10)], [Rational(2, 10), Rational(7, 10), Rational(1, 10)], [Rational(3, 10), Rational(3, 10), Rational(4, 10)]]) >>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T) >>> P(Eq(Y[3], Y[1]), Eq(Y[0], 0)).round(3) 0.409 >>> P(Gt(Y[3], Y[1]), Eq(Y[0], 0)).round(2) 0.36 >>> P(Le(Y[15], Y[10]), Eq(Y[8], 2)).round(7) 0.6963328 Symbolic probability queries are also supported >>> a, b, c, d = symbols('a b c d') >>> T = Matrix([[Rational(1, 10), Rational(4, 10), Rational(5, 10)], [Rational(3, 10), Rational(4, 10), Rational(3, 10)], [Rational(7, 10), Rational(2, 10), Rational(1, 10)]]) >>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T) >>> query = P(Eq(Y[a], b), Eq(Y[c], d)) >>> query.subs({a:10, b:2, c:5, d:1}).round(4) 0.3096 >>> P(Eq(Y[10], 2), Eq(Y[5], 1)).evalf().round(4) 0.3096 >>> query_gt = P(Gt(Y[a], b), Eq(Y[c], d)) >>> query_gt.subs({a:21, b:0, c:5, d:0}).evalf().round(5) 0.64705 >>> P(Gt(Y[21], 0), Eq(Y[5], 0)).round(5) 0.64705 There is limited support for arbitrarily sized states: >>> n = symbols('n', nonnegative=True, integer=True) >>> T = MatrixSymbol('T', n, n) >>> Y = DiscreteMarkovChain("Y", trans_probs=T) >>> Y.state_space Range(0, n, 1) >>> query = P(Eq(Y[a], b), Eq(Y[c], d)) >>> query.subs({a:10, b:2, c:5, d:1}) (T5)[1, 2] References ========== .. [1] https://en.wikipedia.org/wiki/Markov_chain#Discrete-time_Markov_chain .. [2] https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf """ index_set = S.Naturals0 def __new__(cls, sym, state_space=None, trans_probs=None): sym = _symbol_converter(sym) state_space, trans_probs = MarkovProcess._sanity_checks(state_space, trans_probs) obj = Basic.__new__(cls, sym, state_space, trans_probs) # type: ignore indices = dict() if isinstance(obj.number_of_states, Integer): for index, state in enumerate(obj._state_index): indices[state] = index obj.index_of = indices return obj @property def transition_probabilities(self): """ Transition probabilities of discrete Markov chain, either an instance of Matrix or MatrixSymbol. """ return self.args[2] def communication_classes(self) -> tList[tTuple[tList[Basic], Boolean, Integer]]: """ Returns the list of communication classes that partition the states of the markov chain. A communication class is defined to be a set of states such that every state in that set is reachable from every other state in that set. Due to its properties this forms a class in the mathematical sense. Communication classes are also known as recurrence classes. Returns ======= classes The ``classes`` are a list of tuples. Each tuple represents a single communication class with its properties. The first element in the tuple is the list of states in the class, the second element is whether the class is recurrent and the third element is the period of the communication class. Examples ======== >>> from sympy.stats import DiscreteMarkovChain >>> from sympy import Matrix >>> T = Matrix([[0, 1, 0], ... [1, 0, 0], ... [1, 0, 0]]) >>> X = DiscreteMarkovChain('X', [1, 2, 3], T) >>> classes = X.communication_classes() >>> for states, is_recurrent, period in classes: ... states, is_recurrent, period ([1, 2], True, 2) ([3], False, 1) From this we can see that states ``1`` and ``2`` communicate, are recurrent and have a period of 2. We can also see state ``3`` is transient with a period of 1. Notes ===== The algorithm used is of order ``O(n2)`` where ``n`` is the number of states in the markov chain. It uses Tarjan's algorithm to find the classes themselves and then it uses a breadth-first search algorithm to find each class's periodicity. Most of the algorithm's components approach ``O(n)`` as the matrix becomes more and more sparse. References ========== .. [1] http://www.columbia.edu/~ww2040/4701Sum07/4701-06-Notes-MCII.pdf .. [2] http://cecas.clemson.edu/~shierd/Shier/markov.pdf .. [3] https://ujcontent.uj.ac.za/vital/access/services/Download/uj:7506/CONTENT1 .. [4] https://www.mathworks.com/help/econ/dtmc.classify.html """ n = self.number_of_states T = self.transition_probabilities if isinstance(T, MatrixSymbol): raise NotImplementedError("Cannot perform the operation with a symbolic matrix.") # begin Tarjan's algorithm V = Range(n) # don't use state names. Rather use state # indexes since we use them for matrix # indexing here and later onward E = [(i, j) for i in V for j in V if T[i, j] != 0] classes = strongly_connected_components((V, E)) # end Tarjan's algorithm recurrence = [] periods = [] for class_ in classes: # begin recurrent check (similar to self._check_trans_probs()) submatrix = T[class_, class_] # get the submatrix with those states is_recurrent = S.true rows = submatrix.tolist() for row in rows: if (sum(row) - 1) != 0: is_recurrent = S.false break recurrence.append(is_recurrent) # end recurrent check # begin breadth-first search non_tree_edge_values: tSet[int] = set() visited = {class_[0]} newly_visited = {class_[0]} level = {class_[0]: 0} current_level = 0 done = False # imitate a do-while loop while not done: # runs at most len(class_) times done = len(visited) == len(class_) current_level += 1 # this loop and the while loop above run a combined len(class_) number of times. # so this triple nested loop runs through each of the n states once. for i in newly_visited: # the loop below runs len(class_) number of times # complexity is around about O(n * avg(len(class_))) newly_visited = {j for j in class_ if T[i, j] != 0} new_tree_edges = newly_visited.difference(visited) for j in new_tree_edges: level[j] = current_level new_non_tree_edges = newly_visited.intersection(visited) new_non_tree_edge_values = {level[i]-level[j]+1 for j in new_non_tree_edges} non_tree_edge_values = non_tree_edge_values.union(new_non_tree_edge_values) visited = visited.union(new_tree_edges) # igcd needs at least 2 arguments positive_ntev = {val_e for val_e in non_tree_edge_values if val_e > 0} if len(positive_ntev) == 0: periods.append(len(class_)) elif len(positive_ntev) == 1: periods.append(positive_ntev.pop()) else: periods.append(igcd(positive_ntev)) # end breadth-first search # convert back to the user's state names classes = [[_sympify(self._state_index[i]) for i in class_] for class_ in classes] return list(zip(classes, recurrence, map(Integer,periods))) def fundamental_matrix(self): """ Each entry fundamental matrix can be interpreted as the expected number of times the chains is in state j if it started in state i. References ========== .. [1] https://lips.cs.princeton.edu/the-fundamental-matrix-of-a-finite-markov-chain/ """ _, _, _, Q = self.decompose() if Q.shape[0] > 0: # if non-ergodic I = eye(Q.shape[0]) if (I - Q).det() == 0: raise ValueError("The fundamental matrix doesn't exist.") return (I - Q).inv().as_immutable() else: # if ergodic P = self.transition_probabilities I = eye(P.shape[0]) w = self.fixed_row_vector() W = Matrix([list(w) for i in range(0, P.shape[0])]) if (I - P + W).det() == 0: raise ValueError("The fundamental matrix doesn't exist.") return (I - P + W).inv().as_immutable() def absorbing_probabilities(self): """ Computes the absorbing probabilities, i.e. the ij-th entry of the matrix denotes the probability of Markov chain being absorbed in state j starting from state i. """ _, _, R, _ = self.decompose() N = self.fundamental_matrix() if R is None or N is None: return None return NR def absorbing_probabilites(self): sympy_deprecation_warning( """ DiscreteMarkovChain.absorbing_probabilites() is deprecated. Use absorbing_probabilities() instead (note the spelling difference). """, deprecated_since_version="1.7", active_deprecations_target="deprecated-absorbing_probabilites", ) return self.absorbing_probabilities() def is_regular(self): tuples = self.communication_classes() if len(tuples) == 0: return S.false # not defined for a 0x0 matrix classes, _, periods = list(zip(tuples)) return And(len(classes) == 1, periods[0] == 1) def is_ergodic(self): tuples = self.communication_classes() if len(tuples) == 0: return S.false # not defined for a 0x0 matrix classes, _, _ = list(zip(tuples)) return S(len(classes) == 1) def is_absorbing_state(self, state): trans_probs = self.transition_probabilities if isinstance(trans_probs, ImmutableMatrix) and \ state < trans_probs.shape[0]: return S(trans_probs[state, state]) is S.One def is_absorbing_chain(self): states, A, B, C = self.decompose() r = A.shape[0] return And(r > 0, A == Identity(r).as_explicit()) def stationary_distribution(self, condition_set=False) -> tUnion[ImmutableMatrix, ConditionSet, Lambda]: r""" The stationary distribution is any row vector, p, that solves p = pP, is row stochastic and each element in p must be nonnegative. That means in matrix form: :math:`(P-I)^T p^T = 0` and :math:`(1, \dots, 1) p = 1` where ``P`` is the one-step transition matrix. All time-homogeneous Markov Chains with a finite state space have at least one stationary distribution. In addition, if a finite time-homogeneous Markov Chain is irreducible, the stationary distribution is unique. Parameters ========== condition_set : bool If the chain has a symbolic size or transition matrix, it will return a ``Lambda`` if ``False`` and return a ``ConditionSet`` if ``True``. Examples ======== >>> from sympy.stats import DiscreteMarkovChain >>> from sympy import Matrix, S An irreducible Markov Chain >>> T = Matrix([[S(1)/2, S(1)/2, 0], ... [S(4)/5, S(1)/5, 0], ... [1, 0, 0]]) >>> X = DiscreteMarkovChain('X', trans_probs=T) >>> X.stationary_distribution() Matrix([[8/13, 5/13, 0]]) A reducible Markov Chain >>> T = Matrix([[S(1)/2, S(1)/2, 0], ... [S(4)/5, S(1)/5, 0], ... [0, 0, 1]]) >>> X = DiscreteMarkovChain('X', trans_probs=T) >>> X.stationary_distribution() Matrix([[8/13 - 8tau0/13, 5/13 - 5tau0/13, tau0]]) >>> Y = DiscreteMarkovChain('Y') >>> Y.stationary_distribution() Lambda((wm, _T), Eq(wm_T, wm)) >>> Y.stationary_distribution(condition_set=True) ConditionSet(wm, Eq(wm_T, wm)) References ========== .. [1] https://www.probabilitycourse.com/chapter11/11_2_6_stationary_and_limiting_distributions.php .. [2] https://galton.uchicago.edu/~yibi/teaching/stat317/2014/Lectures/Lecture4_6up.pdf See Also ======== sympy.stats.DiscreteMarkovChain.limiting_distribution """ trans_probs = self.transition_probabilities n = self.number_of_states if n == 0: return ImmutableMatrix(Matrix([[]])) # symbolic matrix version if isinstance(trans_probs, MatrixSymbol): wm = MatrixSymbol('wm', 1, n) if condition_set: return ConditionSet(wm, Eq(wm * trans_probs, wm)) else: return Lambda((wm, trans_probs), Eq(wm * trans_probs, wm)) # numeric matrix version a = Matrix(trans_probs - Identity(n)).T a[0, 0:n] = ones(1, n) # type: ignore b = zeros(n, 1) b[0, 0] = 1 soln = list(linsolve((a, b)))[0] return ImmutableMatrix([[sol for sol in soln]]) def fixed_row_vector(self): """ A wrapper for ``stationary_distribution()``. """ return self.stationary_distribution() @property def limiting_distribution(self): """ The fixed row vector is the limiting distribution of a discrete Markov chain. """ return self.fixed_row_vector() def decompose(self) -> tTuple[tList[Basic], ImmutableMatrix, ImmutableMatrix, ImmutableMatrix]: """ Decomposes the transition matrix into submatrices with special properties. The transition matrix can be decomposed into 4 submatrices: - A - the submatrix from recurrent states to recurrent states. - B - the submatrix from transient to recurrent states. - C - the submatrix from transient to transient states. - O - the submatrix of zeros for recurrent to transient states. Returns ======= states, A, B, C ``states`` - a list of state names with the first being the recurrent states and the last being the transient states in the order of the row names of A and then the row names of C. ``A`` - the submatrix from recurrent states to recurrent states. ``B`` - the submatrix from transient to recurrent states. ``C`` - the submatrix from transient to transient states. Examples ======== >>> from sympy.stats import DiscreteMarkovChain >>> from sympy import Matrix, S One can decompose this chain for example: >>> T = Matrix([[S(1)/2, S(1)/2, 0, 0, 0], ... [S(2)/5, S(1)/5, S(2)/5, 0, 0], ... [0, 0, 1, 0, 0], ... [0, 0, S(1)/2, S(1)/2, 0], ... [S(1)/2, 0, 0, 0, S(1)/2]]) >>> X = DiscreteMarkovChain('X', trans_probs=T) >>> states, A, B, C = X.decompose() >>> states [2, 0, 1, 3, 4] >>> A # recurrent to recurrent Matrix([[1]]) >>> B # transient to recurrent Matrix([ [ 0], [2/5], [1/2], [ 0]]) >>> C # transient to transient Matrix([ [1/2, 1/2, 0, 0], [2/5, 1/5, 0, 0], [ 0, 0, 1/2, 0], [1/2, 0, 0, 1/2]]) This means that state 2 is the only absorbing state (since A is a 1x1 matrix). B is a 4x1 matrix since the 4 remaining transient states all merge into reccurent state 2. And C is the 4x4 matrix that shows how the transient states 0, 1, 3, 4 all interact. See Also ======== sympy.stats.DiscreteMarkovChain.communication_classes sympy.stats.DiscreteMarkovChain.canonical_form References ========== .. [1] https://en.wikipedia.org/wiki/Absorbing_Markov_chain .. [2] http://people.brandeis.edu/~igusa/Math56aS08/Math56a_S08_notes015.pdf """ trans_probs = self.transition_probabilities classes = self.communication_classes() r_states = [] t_states = [] for states, recurrent, period in classes: if recurrent: r_states += states else: t_states += states states = r_states + t_states indexes = [self.index_of[state] for state in states] # type: ignore A = Matrix(len(r_states), len(r_states), lambda i, j: trans_probs[indexes[i], indexes[j]]) B = Matrix(len(t_states), len(r_states), lambda i, j: trans_probs[indexes[len(r_states) + i], indexes[j]]) C = Matrix(len(t_states), len(t_states), lambda i, j: trans_probs[indexes[len(r_states) + i], indexes[len(r_states) + j]]) return states, A.as_immutable(), B.as_immutable(), C.as_immutable() def canonical_form(self) -> tTuple[tList[Basic], ImmutableMatrix]: """ Reorders the one-step transition matrix so that recurrent states appear first and transient states appear last. Other representations include inserting transient states first and recurrent states last. Returns ======= states, P_new ``states`` is the list that describes the order of the new states in the matrix so that the ith element in ``states`` is the state of the ith row of A. ``P_new`` is the new transition matrix in canonical form. Examples ======== >>> from sympy.stats import DiscreteMarkovChain >>> from sympy import Matrix, S You can convert your chain into canonical form: >>> T = Matrix([[S(1)/2, S(1)/2, 0, 0, 0], ... [S(2)/5, S(1)/5, S(2)/5, 0, 0], ... [0, 0, 1, 0, 0], ... [0, 0, S(1)/2, S(1)/2, 0], ... [S(1)/2, 0, 0, 0, S(1)/2]]) >>> X = DiscreteMarkovChain('X', list(range(1, 6)), trans_probs=T) >>> states, new_matrix = X.canonical_form() >>> states [3, 1, 2, 4, 5] >>> new_matrix Matrix([ [ 1, 0, 0, 0, 0], [ 0, 1/2, 1/2, 0, 0], [2/5, 2/5, 1/5, 0, 0], [1/2, 0, 0, 1/2, 0], [ 0, 1/2, 0, 0, 1/2]]) The new states are [3, 1, 2, 4, 5] and you can create a new chain with this and its canonical form will remain the same (since it is already in canonical form). >>> X = DiscreteMarkovChain('X', states, new_matrix) >>> states, new_matrix = X.canonical_form() >>> states [3, 1, 2, 4, 5] >>> new_matrix Matrix([ [ 1, 0, 0, 0, 0], [ 0, 1/2, 1/2, 0, 0], [2/5, 2/5, 1/5, 0, 0], [1/2, 0, 0, 1/2, 0], [ 0, 1/2, 0, 0, 1/2]]) This is not limited to absorbing chains: >>> T = Matrix([[0, 5, 5, 0, 0], ... [0, 0, 0, 10, 0], ... [5, 0, 5, 0, 0], ... [0, 10, 0, 0, 0], ... [0, 3, 0, 3, 4]])/10 >>> X = DiscreteMarkovChain('X', trans_probs=T) >>> states, new_matrix = X.canonical_form() >>> states [1, 3, 0, 2, 4] >>> new_matrix Matrix([ [ 0, 1, 0, 0, 0], [ 1, 0, 0, 0, 0], [ 1/2, 0, 0, 1/2, 0], [ 0, 0, 1/2, 1/2, 0], [3/10, 3/10, 0, 0, 2/5]]) See Also ======== sympy.stats.DiscreteMarkovChain.communication_classes sympy.stats.DiscreteMarkovChain.decompose References ========== .. [1] https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470316887.app1 .. [2] http://www.columbia.edu/~ww2040/6711F12/lect1023big.pdf """ states, A, B, C = self.decompose() O = zeros(A.shape[0], C.shape[1]) return states, BlockMatrix([[A, O], [B, C]]).as_explicit() def sample(self): """ Returns ======= sample: iterator object iterator object containing the sample """ if not isinstance(self.transition_probabilities, (Matrix, ImmutableMatrix)): raise ValueError("Transition Matrix must be provided for sampling") Tlist = self.transition_probabilities.tolist() samps = [random.choice(list(self.state_space))] yield samps[0] time = 1 densities = {} for state in self.state_space: states = list(self.state_space) densities[state] = {states[i]: Tlist[state][i] for i in range(len(states))} while time < S.Infinity: samps.append((next(sample_iter(FiniteRV("_", densities[samps[time - 1]]))))) yield samps[time] time += 1 class ContinuousMarkovChain(ContinuousTimeStochasticProcess, MarkovProcess): """ Represents continuous time Markov chain. Parameters ========== sym : Symbol/str state_space : Set Optional, by default, S.Reals gen_mat : Matrix/ImmutableMatrix/MatrixSymbol Optional, by default, None Examples ======== >>> from sympy.stats import ContinuousMarkovChain, P >>> from sympy import Matrix, S, Eq, Gt >>> G = Matrix([[-S(1), S(1)], [S(1), -S(1)]]) >>> C = ContinuousMarkovChain('C', state_space=[0, 1], gen_mat=G) >>> C.limiting_distribution() Matrix([[1/2, 1/2]]) >>> C.state_space {0, 1} >>> C.generator_matrix Matrix([ [-1, 1], [ 1, -1]]) Probability queries are supported >>> P(Eq(C(1.96), 0), Eq(C(0.78), 1)).round(5) 0.45279 >>> P(Gt(C(1.7), 0), Eq(C(0.82), 1)).round(5) 0.58602 Probability of expressions with multiple RandomIndexedSymbols can also be calculated provided there is only 1 RandomIndexedSymbol in the given condition. It is always better to use Rational instead of floating point numbers for the probabilities in the generator matrix to avoid errors. >>> from sympy import Gt, Le, Rational >>> G = Matrix([[-S(1), Rational(1, 10), Rational(9, 10)], [Rational(2, 5), -S(1), Rational(3, 5)], [Rational(1, 2), Rational(1, 2), -S(1)]]) >>> C = ContinuousMarkovChain('C', state_space=[0, 1, 2], gen_mat=G) >>> P(Eq(C(3.92), C(1.75)), Eq(C(0.46), 0)).round(5) 0.37933 >>> P(Gt(C(3.92), C(1.75)), Eq(C(0.46), 0)).round(5) 0.34211 >>> P(Le(C(1.57), C(3.14)), Eq(C(1.22), 1)).round(4) 0.7143 Symbolic probability queries are also supported >>> from sympy import symbols >>> a,b,c,d = symbols('a b c d') >>> G = Matrix([[-S(1), Rational(1, 10), Rational(9, 10)], [Rational(2, 5), -S(1), Rational(3, 5)], [Rational(1, 2), Rational(1, 2), -S(1)]]) >>> C = ContinuousMarkovChain('C', state_space=[0, 1, 2], gen_mat=G) >>> query = P(Eq(C(a), b), Eq(C(c), d)) >>> query.subs({a:3.65, b:2, c:1.78, d:1}).evalf().round(10) 0.4002723175 >>> P(Eq(C(3.65), 2), Eq(C(1.78), 1)).round(10) 0.4002723175 >>> query_gt = P(Gt(C(a), b), Eq(C(c), d)) >>> query_gt.subs({a:43.2, b:0, c:3.29, d:2}).evalf().round(10) 0.6832579186 >>> P(Gt(C(43.2), 0), Eq(C(3.29), 2)).round(10) 0.6832579186 References ========== .. [1] https://en.wikipedia.org/wiki/Markov_chain#Continuous-time_Markov_chain .. [2] http://u.math.biu.ac.il/~amirgi/CTMCnotes.pdf """ index_set = S.Reals def __new__(cls, sym, state_space=None, gen_mat=None): sym = _symbol_converter(sym) state_space, gen_mat = MarkovProcess._sanity_checks(state_space, gen_mat) obj = Basic.__new__(cls, sym, state_space, gen_mat) indices = dict() if isinstance(obj.number_of_states, Integer): for index, state in enumerate(obj.state_space): indices[state] = index obj.index_of = indices return obj @property def generator_matrix(self): return self.args[2] @cacheit def transition_probabilities(self, gen_mat=None): t = Dummy('t') if isinstance(gen_mat, (Matrix, ImmutableMatrix)) and \ gen_mat.is_diagonalizable(): # for faster computation use diagonalized generator matrix Q, D = gen_mat.diagonalize() return Lambda(t, Qexp(tD)Q.inv()) if gen_mat != None: return Lambda(t, exp(tgen_mat)) def limiting_distribution(self): gen_mat = self.generator_matrix if gen_mat is None: return None if isinstance(gen_mat, MatrixSymbol): wm = MatrixSymbol('wm', 1, gen_mat.shape[0]) return Lambda((wm, gen_mat), Eq(wmgen_mat, wm)) w = IndexedBase('w') wi = [w[i] for i in range(gen_mat.shape[0])] wm = Matrix([wi]) eqs = (wmgen_mat).tolist()[0] eqs.append(sum(wi) - 1) soln = list(linsolve(eqs, wi))[0] return ImmutableMatrix([[sol for sol in soln]]) class BernoulliProcess(DiscreteTimeStochasticProcess): """ The Bernoulli process consists of repeated independent Bernoulli process trials with the same parameter `p`. It's assumed that the probability `p` applies to every trial and that the outcomes of each trial are independent of all the rest. Therefore Bernoulli Processs is Discrete State and Discrete Time Stochastic Process. Parameters ========== sym : Symbol/str success : Integer/str The event which is considered to be success. Default: 1. failure: Integer/str The event which is considered to be failure. Default: 0. p : Real Number between 0 and 1 Represents the probability of getting success. Examples ======== >>> from sympy.stats import BernoulliProcess, P, E >>> from sympy import Eq, Gt >>> B = BernoulliProcess("B", p=0.7, success=1, failure=0) >>> B.state_space {0, 1} >>> (B.p).round(2) 0.70 >>> B.success 1 >>> B.failure 0 >>> X = B[1] + B[2] + B[3] >>> P(Eq(X, 0)).round(2) 0.03 >>> P(Eq(X, 2)).round(2) 0.44 >>> P(Eq(X, 4)).round(2) 0 >>> P(Gt(X, 1)).round(2) 0.78 >>> P(Eq(B[1], 0) & Eq(B[2], 1) & Eq(B[3], 0) & Eq(B[4], 1)).round(2) 0.04 >>> B.joint_distribution(B[1], B[2]) JointDistributionHandmade(Lambda((B[1], B[2]), Piecewise((0.7, Eq(B[1], 1)), (0.3, Eq(B[1], 0)), (0, True))Piecewise((0.7, Eq(B[2], 1)), (0.3, Eq(B[2], 0)), (0, True)))) >>> E(2B[1] + B[2]).round(2) 2.10 >>> P(B[1] < 1).round(2) 0.30 References ========== .. [1] https://en.wikipedia.org/wiki/Bernoulli_process .. [2] https://mathcs.clarku.edu/~djoyce/ma217/bernoulli.pdf """ index_set = S.Naturals0 def __new__(cls, sym, p, success=1, failure=0): _value_check(p >= 0 and p <= 1, 'Value of p must be between 0 and 1.') sym = _symbol_converter(sym) p = _sympify(p) success = _sym_sympify(success) failure = _sym_sympify(failure) return Basic.__new__(cls, sym, p, success, failure) @property def symbol(self): return self.args[0] @property def p(self): return self.args[1] @property def success(self): return self.args[2] @property def failure(self): return self.args[3] @property def state_space(self): return _set_converter([self.success, self.failure]) def distribution(self, key=None): if key is None: self._deprecation_warn_distribution() return BernoulliDistribution(self.p) return BernoulliDistribution(self.p, self.success, self.failure) def simple_rv(self, rv): return Bernoulli(rv.name, p=self.p, succ=self.success, fail=self.failure) def expectation(self, expr, condition=None, evaluate=True, kwargs): """ Computes expectation. Parameters ========== expr : RandomIndexedSymbol, Relational, Logic Condition for which expectation has to be computed. Must contain a RandomIndexedSymbol of the process. condition : Relational, Logic The given conditions under which computations should be done. Returns ======= Expectation of the RandomIndexedSymbol. """ return _SubstituteRV._expectation(expr, condition, evaluate, kwargs) def probability(self, condition, given_condition=None, evaluate=True, kwargs): """ Computes probability. Parameters ========== condition : Relational Condition for which probability has to be computed. Must contain a RandomIndexedSymbol of the process. given_condition : Relational, Logic The given conditions under which computations should be done. Returns ======= Probability of the condition. """ return _SubstituteRV._probability(condition, given_condition, evaluate, kwargs) def density(self, x): return Piecewise((self.p, Eq(x, self.success)), (1 - self.p, Eq(x, self.failure)), (S.Zero, True)) class _SubstituteRV: """ Internal class to handle the queries of expectation and probability by substitution. """ @staticmethod def _rvindexed_subs(expr, condition=None): """ Substitutes the RandomIndexedSymbol with the RandomSymbol with same name, distribution and probability as RandomIndexedSymbol. Parameters ========== expr: RandomIndexedSymbol, Relational, Logic Condition for which expectation has to be computed. Must contain a RandomIndexedSymbol of the process. condition: Relational, Logic The given conditions under which computations should be done. """ rvs_expr = random_symbols(expr) if len(rvs_expr) != 0: swapdict_expr = {} for rv in rvs_expr: if isinstance(rv, RandomIndexedSymbol): newrv = rv.pspace.process.simple_rv(rv) # substitute with equivalent simple rv swapdict_expr[rv] = newrv expr = expr.subs(swapdict_expr) rvs_cond = random_symbols(condition) if len(rvs_cond)!=0: swapdict_cond = {} for rv in rvs_cond: if isinstance(rv, RandomIndexedSymbol): newrv = rv.pspace.process.simple_rv(rv) swapdict_cond[rv] = newrv condition = condition.subs(swapdict_cond) return expr, condition @classmethod def _expectation(self, expr, condition=None, evaluate=True, *kwargs): """ Internal method for computing expectation of indexed RV. Parameters ========== expr: RandomIndexedSymbol, Relational, Logic Condition for which expectation has to be computed. Must contain a RandomIndexedSymbol of the process. condition: Relational, Logic The given conditions under which computations should be done. Returns ======= Expectation of the RandomIndexedSymbol. """ new_expr, new_condition = self._rvindexed_subs(expr, condition) if not is_random(new_expr): return new_expr new_pspace = pspace(new_expr) if new_condition is not None: new_expr = given(new_expr, new_condition) if new_expr.is_Add: # As E is Linear return Add([new_pspace.compute_expectation( expr=arg, evaluate=evaluate, kwargs) for arg in new_expr.args]) return new_pspace.compute_expectation( new_expr, evaluate=evaluate, kwargs) @classmethod def _probability(self, condition, given_condition=None, evaluate=True, kwargs): """ Internal method for computing probability of indexed RV Parameters ========== condition: Relational Condition for which probability has to be computed. Must contain a RandomIndexedSymbol of the process. given_condition: Relational/And The given conditions under which computations should be done. Returns ======= Probability of the condition. """ new_condition, new_givencondition = self._rvindexed_subs(condition, given_condition) if isinstance(new_givencondition, RandomSymbol): condrv = random_symbols(new_condition) if len(condrv) == 1 and condrv[0] == new_givencondition: return BernoulliDistribution(self._probability(new_condition), 0, 1) if any(dependent(rv, new_givencondition) for rv in condrv): return Probability(new_condition, new_givencondition) else: return self._probability(new_condition) if new_givencondition is not None and \ not isinstance(new_givencondition, (Relational, Boolean)): raise ValueError("%s is not a relational or combination of relationals" % (new_givencondition)) if new_givencondition == False or new_condition == False: return S.Zero if new_condition == True: return S.One if not isinstance(new_condition, (Relational, Boolean)): raise ValueError("%s is not a relational or combination of relationals" % (new_condition)) if new_givencondition is not None: # If there is a condition # Recompute on new conditional expr return self._probability(given(new_condition, new_givencondition, kwargs), kwargs) result = pspace(new_condition).probability(new_condition, kwargs) if evaluate and hasattr(result, 'doit'): return result.doit() else: return result def get_timerv_swaps(expr, condition): """ Finds the appropriate interval for each time stamp in expr by parsing the given condition and returns intervals for each timestamp and dictionary that maps variable time-stamped Random Indexed Symbol to its corresponding Random Indexed variable with fixed time stamp. Parameters ========== expr: SymPy Expression Expression containing Random Indexed Symbols with variable time stamps condition: Relational/Boolean Expression Expression containing time bounds of variable time stamps in expr Examples ======== >>> from sympy.stats.stochastic_process_types import get_timerv_swaps, PoissonProcess >>> from sympy import symbols, Contains, Interval >>> x, t, d = symbols('x t d', positive=True) >>> X = PoissonProcess("X", 3) >>> get_timerv_swaps(xX(t), Contains(t, Interval.Lopen(0, 1))) ([Interval.Lopen(0, 1)], {X(t): X(1)}) >>> get_timerv_swaps((X(t)2 + X(d)2), Contains(t, Interval.Lopen(0, 1)) ... & Contains(d, Interval.Ropen(1, 4))) # doctest: +SKIP ([Interval.Ropen(1, 4), Interval.Lopen(0, 1)], {X(d): X(3), X(t): X(1)}) Returns ======= intervals: list List of Intervals/FiniteSet on which each time stamp is defined rv_swap: dict Dictionary mapping variable time Random Indexed Symbol to constant time Random Indexed Variable """ if not isinstance(condition, (Relational, Boolean)): raise ValueError("%s is not a relational or combination of relationals" % (condition)) expr_syms = list(expr.atoms(RandomIndexedSymbol)) if isinstance(condition, (And, Or)): given_cond_args = condition.args else: # single condition given_cond_args = (condition, ) rv_swap = {} intervals = [] for expr_sym in expr_syms: for arg in given_cond_args: if arg.has(expr_sym.key) and isinstance(expr_sym.key, Symbol): intv = _set_converter(arg.args[1]) diff_key = intv._sup - intv._inf if diff_key == oo: raise ValueError("%s should have finite bounds" % str(expr_sym.name)) elif diff_key == S.Zero: # has singleton set diff_key = intv._sup rv_swap[expr_sym] = expr_sym.subs({expr_sym.key: diff_key}) intervals.append(intv) return intervals, rv_swap class CountingProcess(ContinuousTimeStochasticProcess): """ This class handles the common methods of the Counting Processes such as Poisson, Wiener and Gamma Processes """ index_set = _set_converter(Interval(0, oo)) @property def symbol(self): return self.args[0] def expectation(self, expr, condition=None, evaluate=True, kwargs): """ Computes expectation Parameters ========== expr: RandomIndexedSymbol, Relational, Logic Condition for which expectation has to be computed. Must contain a RandomIndexedSymbol of the process. condition: Relational, Boolean The given conditions under which computations should be done, i.e, the intervals on which each variable time stamp in expr is defined Returns ======= Expectation of the given expr """ if condition is not None: intervals, rv_swap = get_timerv_swaps(expr, condition) # they are independent when they have non-overlapping intervals if len(intervals) == 1 or all(Intersection(intv_comb) == EmptySet for intv_comb in itertools.combinations(intervals, 2)): if expr.is_Add: return Add.fromiter(self.expectation(arg, condition) for arg in expr.args) expr = expr.subs(rv_swap) else: return Expectation(expr, condition) return _SubstituteRV._expectation(expr, evaluate=evaluate, *kwargs) def _solve_argwith_tworvs(self, arg): if arg.args[0].key >= arg.args[1].key or isinstance(arg, Eq): diff_key = abs(arg.args[0].key - arg.args[1].key) rv = arg.args[0] arg = arg.__class__(rv.pspace.process(diff_key), 0) else: diff_key = arg.args[1].key - arg.args[0].key rv = arg.args[1] arg = arg.__class__(rv.pspace.process(diff_key), 0) return arg def _solve_numerical(self, condition, given_condition=None): if isinstance(condition, And): args_list = list(condition.args) else: args_list = [condition] if given_condition is not None: if isinstance(given_condition, And): args_list.extend(list(given_condition.args)) else: args_list.extend([given_condition]) # sort the args based on timestamp to get the independent increments in # each segment using all the condition args as well as given_condition args args_list = sorted(args_list, key=lambda x: x.args[0].key) result = [] cond_args = list(condition.args) if isinstance(condition, And) else [condition] if args_list[0] in cond_args and not (is_random(args_list[0].args[0]) and is_random(args_list[0].args[1])): result.append(_SubstituteRV._probability(args_list[0])) if is_random(args_list[0].args[0]) and is_random(args_list[0].args[1]): arg = self._solve_argwith_tworvs(args_list[0]) result.append(_SubstituteRV._probability(arg)) for i in range(len(args_list) - 1): curr, nex = args_list[i], args_list[i + 1] diff_key = nex.args[0].key - curr.args[0].key working_set = curr.args[0].pspace.process.state_space if curr.args[1] > nex.args[1]: #impossible condition so return 0 result.append(0) break if isinstance(curr, Eq): working_set = Intersection(working_set, Interval.Lopen(curr.args[1], oo)) else: working_set = Intersection(working_set, curr.as_set()) if isinstance(nex, Eq): working_set = Intersection(working_set, Interval(-oo, nex.args[1])) else: working_set = Intersection(working_set, nex.as_set()) if working_set == EmptySet: rv = Eq(curr.args[0].pspace.process(diff_key), 0) result.append(_SubstituteRV._probability(rv)) else: if working_set.is_finite_set: if isinstance(curr, Eq) and isinstance(nex, Eq): rv = Eq(curr.args[0].pspace.process(diff_key), len(working_set)) result.append(_SubstituteRV._probability(rv)) elif isinstance(curr, Eq) ^ isinstance(nex, Eq): result.append(Add.fromiter(_SubstituteRV._probability(Eq( curr.args[0].pspace.process(diff_key), x)) for x in range(len(working_set)))) else: n = len(working_set) result.append(Add.fromiter((n - x)_SubstituteRV._probability(Eq( curr.args[0].pspace.process(diff_key), x)) for x in range(n))) else: result.append(_SubstituteRV._probability( curr.args[0].pspace.process(diff_key) <= working_set._sup - working_set._inf)) return Mul.fromiter(result) def probability(self, condition, given_condition=None, evaluate=True, *kwargs): """ Computes probability. Parameters ========== condition: Relational Condition for which probability has to be computed. Must contain a RandomIndexedSymbol of the process. given_condition: Relational, Boolean The given conditions under which computations should be done, i.e, the intervals on which each variable time stamp in expr is defined Returns ======= Probability of the condition """ check_numeric = True if isinstance(condition, (And, Or)): cond_args = condition.args else: cond_args = (condition, ) # check that condition args are numeric or not if not all(arg.args[0].key.is_number for arg in cond_args): check_numeric = False if given_condition is not None: check_given_numeric = True if isinstance(given_condition, (And, Or)): given_cond_args = given_condition.args else: given_cond_args = (given_condition, ) # check that given condition args are numeric or not if given_condition.has(Contains): check_given_numeric = False # Handle numerical queries if check_numeric and check_given_numeric: res = [] if isinstance(condition, Or): res.append(Add.fromiter(self._solve_numerical(arg, given_condition) for arg in condition.args)) if isinstance(given_condition, Or): res.append(Add.fromiter(self._solve_numerical(condition, arg) for arg in given_condition.args)) if res: return Add.fromiter(res) return self._solve_numerical(condition, given_condition) # No numeric queries, go by Contains?... then check that all the # given condition are in form of `Contains` if not all(arg.has(Contains) for arg in given_cond_args): raise ValueError("If given condition is passed with `Contains`, then " "please pass the evaluated condition with its corresponding information " "in terms of intervals of each time stamp to be passed in given condition.") intervals, rv_swap = get_timerv_swaps(condition, given_condition) # they are independent when they have non-overlapping intervals if len(intervals) == 1 or all(Intersection(intv_comb) == EmptySet for intv_comb in itertools.combinations(intervals, 2)): if isinstance(condition, And): return Mul.fromiter(self.probability(arg, given_condition) for arg in condition.args) elif isinstance(condition, Or): return Add.fromiter(self.probability(arg, given_condition) for arg in condition.args) condition = condition.subs(rv_swap) else: return Probability(condition, given_condition) if check_numeric: return self._solve_numerical(condition) return _SubstituteRV._probability(condition, evaluate=evaluate, *kwargs) class PoissonProcess(CountingProcess): """ The Poisson process is a counting process. It is usually used in scenarios where we are counting the occurrences of certain events that appear to happen at a certain rate, but completely at random. Parameters ========== sym : Symbol/str lamda : Positive number Rate of the process, ``lambda > 0`` Examples ======== >>> from sympy.stats import PoissonProcess, P, E >>> from sympy import symbols, Eq, Ne, Contains, Interval >>> X = PoissonProcess("X", lamda=3) >>> X.state_space Naturals0 >>> X.lamda 3 >>> t1, t2 = symbols('t1 t2', positive=True) >>> P(X(t1) < 4) (9t1*3/2 + 9t1*2/2 + 3t1 + 1)exp(-3t1) >>> P(Eq(X(t1), 2) \| Ne(X(t1), 4), Contains(t1, Interval.Ropen(2, 4))) 1 - 36exp(-6) >>> P(Eq(X(t1), 2) & Eq(X(t2), 3), Contains(t1, Interval.Lopen(0, 2)) ... & Contains(t2, Interval.Lopen(2, 4))) 648exp(-12) >>> E(X(t1)) 3t1 >>> E(X(t1)2 + 2X(t2), Contains(t1, Interval.Lopen(0, 1)) ... & Contains(t2, Interval.Lopen(1, 2))) 18 >>> P(X(3) < 1, Eq(X(1), 0)) exp(-6) >>> P(Eq(X(4), 3), Eq(X(2), 3)) exp(-6) >>> P(X(2) <= 3, X(1) > 1) 5exp(-3) Merging two Poisson Processes >>> Y = PoissonProcess("Y", lamda=4) >>> Z = X + Y >>> Z.lamda 7 Splitting a Poisson Process into two independent Poisson Processes >>> N, M = Z.split(l1=2, l2=5) >>> N.lamda, M.lamda (2, 5) References ========== .. [1] https://www.probabilitycourse.com/chapter11/11_0_0_intro.php .. [2] https://en.wikipedia.org/wiki/Poisson_point_process """ def __new__(cls, sym, lamda): _value_check(lamda > 0, 'lamda should be a positive number.') sym = _symbol_converter(sym) lamda = _sympify(lamda) return Basic.__new__(cls, sym, lamda) @property def lamda(self): return self.args[1] @property def state_space(self): return S.Naturals0 def distribution(self, key): if isinstance(key, RandomIndexedSymbol): self._deprecation_warn_distribution() return PoissonDistribution(self.lamdakey.key) return PoissonDistribution(self.lamdakey) def density(self, x): return (self.lamdax.key)*x / factorial(x) exp(-(self.lamdax.key)) def simple_rv(self, rv): return Poisson(rv.name, lamda=self.lamdarv.key) def __add__(self, other): if not isinstance(other, PoissonProcess): raise ValueError("Only instances of Poisson Process can be merged") return PoissonProcess(Dummy(self.symbol.name + other.symbol.name), self.lamda + other.lamda) def split(self, l1, l2): if _sympify(l1 + l2) != self.lamda: raise ValueError("Sum of l1 and l2 should be %s" % str(self.lamda)) return PoissonProcess(Dummy("l1"), l1), PoissonProcess(Dummy("l2"), l2) class WienerProcess(CountingProcess): """ The Wiener process is a real valued continuous-time stochastic process. In physics it is used to study Brownian motion and it is often also called Brownian motion due to its historical connection with physical process of the same name originally observed by Scottish botanist Robert Brown. Parameters ========== sym : Symbol/str Examples ======== >>> from sympy.stats import WienerProcess, P, E >>> from sympy import symbols, Contains, Interval >>> X = WienerProcess("X") >>> X.state_space Reals >>> t1, t2 = symbols('t1 t2', positive=True) >>> P(X(t1) < 7).simplify() erf(7sqrt(2)/(2sqrt(t1)))/2 + 1/2 >>> P((X(t1) > 2) \| (X(t1) < 4), Contains(t1, Interval.Ropen(2, 4))).simplify() -erf(1)/2 + erf(2)/2 + 1 >>> E(X(t1)) 0 >>> E(X(t1) + 2X(t2), Contains(t1, Interval.Lopen(0, 1)) ... & Contains(t2, Interval.Lopen(1, 2))) 0 References ========== .. [1] https://www.probabilitycourse.com/chapter11/11_4_0_brownian_motion_wiener_process.php .. [2] https://en.wikipedia.org/wiki/Wiener_process """ def __new__(cls, sym): sym = _symbol_converter(sym) return Basic.__new__(cls, sym) @property def state_space(self): return S.Reals def distribution(self, key): if isinstance(key, RandomIndexedSymbol): self._deprecation_warn_distribution() return NormalDistribution(0, sqrt(key.key)) return NormalDistribution(0, sqrt(key)) def density(self, x): return exp(-x2/(2x.key)) / (sqrt(2pi)sqrt(x.key)) def simple_rv(self, rv): return Normal(rv.name, 0, sqrt(rv.key)) class GammaProcess(CountingProcess): r""" A Gamma process is a random process with independent gamma distributed increments. It is a pure-jump increasing Levy process. Parameters ========== sym : Symbol/str lamda : Positive number Jump size of the process, ``lamda > 0`` gamma : Positive number Rate of jump arrivals, `\gamma > 0` Examples ======== >>> from sympy.stats import GammaProcess, E, P, variance >>> from sympy import symbols, Contains, Interval, Not >>> t, d, x, l, g = symbols('t d x l g', positive=True) >>> X = GammaProcess("X", l, g) >>> E(X(t)) gt/l >>> variance(X(t)).simplify() gt/l*2 >>> X = GammaProcess('X', 1, 2) >>> P(X(t) < 1).simplify() lowergamma(2t, 1)/gamma(2t) >>> P(Not((X(t) < 5) & (X(d) > 3)), Contains(t, Interval.Ropen(2, 4)) & ... Contains(d, Interval.Lopen(7, 8))).simplify() -4exp(-3) + 472exp(-8)/3 + 1 >>> E(X(2) + xE(X(5))) 10x + 4 References ========== .. [1] https://en.wikipedia.org/wiki/Gamma_process """ def __new__(cls, sym, lamda, gamma): _value_check(lamda > 0, 'lamda should be a positive number') _value_check(gamma > 0, 'gamma should be a positive number') sym = _symbol_converter(sym) gamma = _sympify(gamma) lamda = _sympify(lamda) return Basic.__new__(cls, sym, lamda, gamma) @property def lamda(self): return self.args[1] @property def gamma(self): return self.args[2] @property def state_space(self): return _set_converter(Interval(0, oo)) def distribution(self, key): if isinstance(key, RandomIndexedSymbol): self._deprecation_warn_distribution() return GammaDistribution(self.gammakey.key, 1/self.lamda) return GammaDistribution(self.gammakey, 1/self.lamda) def density(self, x): k = self.gammax.key theta = 1/self.lamda return x*(k - 1) exp(-x/theta) / (gamma(k)thetak) def simple_rv(self, rv): return Gamma(rv.name, self.gammarv.key, 1/self.lamda)
68db3f7627e64f0eded8621f2c668f2e94be99796db6bac1dd35453c349d1b05	""" SymPy statistics module Introduces a random variable type into the SymPy language. Random variables may be declared using prebuilt functions such as Normal, Exponential, Coin, Die, etc... or built with functions like FiniteRV. Queries on random expressions can be made using the functions ========================= ============================= Expression Meaning ------------------------- ----------------------------- ``P(condition)`` Probability ``E(expression)`` Expected value ``H(expression)`` Entropy ``variance(expression)`` Variance ``density(expression)`` Probability Density Function ``sample(expression)`` Produce a realization ``where(condition)`` Where the condition is true ========================= ============================= Examples ======== >>> from sympy.stats import P, E, variance, Die, Normal >>> from sympy import simplify >>> X, Y = Die('X', 6), Die('Y', 6) # Define two six sided dice >>> Z = Normal('Z', 0, 1) # Declare a Normal random variable with mean 0, std 1 >>> P(X>3) # Probability X is greater than 3 1/2 >>> E(X+Y) # Expectation of the sum of two dice 7 >>> variance(X+Y) # Variance of the sum of two dice 35/6 >>> simplify(P(Z>1)) # Probability of Z being greater than 1 1/2 - erf(sqrt(2)/2)/2 One could also create custom distribution and define custom random variables as follows: 1. If you want to create a Continuous Random Variable: >>> from sympy.stats import ContinuousRV, P, E >>> from sympy import exp, Symbol, Interval, oo >>> x = Symbol('x') >>> pdf = exp(-x) # pdf of the Continuous Distribution >>> Z = ContinuousRV(x, pdf, set=Interval(0, oo)) >>> E(Z) 1 >>> P(Z > 5) exp(-5) 1.1 To create an instance of Continuous Distribution: >>> from sympy.stats import ContinuousDistributionHandmade >>> from sympy import Lambda >>> dist = ContinuousDistributionHandmade(Lambda(x, pdf), set=Interval(0, oo)) >>> dist.pdf(x) exp(-x) 2. If you want to create a Discrete Random Variable: >>> from sympy.stats import DiscreteRV, P, E >>> from sympy import Symbol, S >>> p = S(1)/2 >>> x = Symbol('x', integer=True, positive=True) >>> pdf = p(1 - p)(x - 1) >>> D = DiscreteRV(x, pdf, set=S.Naturals) >>> E(D) 2 >>> P(D > 3) 1/8 2.1 To create an instance of Discrete Distribution: >>> from sympy.stats import DiscreteDistributionHandmade >>> from sympy import Lambda >>> dist = DiscreteDistributionHandmade(Lambda(x, pdf), set=S.Naturals) >>> dist.pdf(x) 2*(1 - x)/2 3. If you want to create a Finite Random Variable: >>> from sympy.stats import FiniteRV, P, E >>> from sympy import Rational, Eq >>> pmf = {1: Rational(1, 3), 2: Rational(1, 6), 3: Rational(1, 4), 4: Rational(1, 4)} >>> X = FiniteRV('X', pmf) >>> E(X) 29/12 >>> P(X > 3) 1/4 3.1 To create an instance of Finite Distribution: >>> from sympy.stats import FiniteDistributionHandmade >>> dist = FiniteDistributionHandmade(pmf) >>> dist.pmf(x) Lambda(x, Piecewise((1/3, Eq(x, 1)), (1/6, Eq(x, 2)), (1/4, Eq(x, 3) \| Eq(x, 4)), (0, True))) """ __all__ = [ 'P', 'E', 'H', 'density', 'where', 'given', 'sample', 'cdf','median', 'characteristic_function', 'pspace', 'sample_iter', 'variance', 'std', 'skewness', 'kurtosis', 'covariance', 'dependent', 'entropy', 'independent', 'random_symbols', 'correlation', 'factorial_moment', 'moment', 'cmoment', 'sampling_density', 'moment_generating_function', 'smoment', 'quantile', 'coskewness', 'sample_stochastic_process', 'FiniteRV', 'DiscreteUniform', 'Die', 'Bernoulli', 'Coin', 'Binomial', 'BetaBinomial', 'Hypergeometric', 'Rademacher', 'IdealSoliton', 'RobustSoliton', 'FiniteDistributionHandmade', 'ContinuousRV', 'Arcsin', 'Benini', 'Beta', 'BetaNoncentral', 'BetaPrime', 'BoundedPareto', 'Cauchy', 'Chi', 'ChiNoncentral', 'ChiSquared', 'Dagum', 'Erlang', 'ExGaussian', 'Exponential', 'ExponentialPower', 'FDistribution', 'FisherZ', 'Frechet', 'Gamma', 'GammaInverse', 'Gompertz', 'Gumbel', 'Kumaraswamy', 'Laplace', 'Levy', 'Logistic','LogCauchy', 'LogLogistic', 'LogitNormal', 'LogNormal', 'Lomax', 'Moyal', 'Maxwell', 'Nakagami', 'Normal', 'GaussianInverse', 'Pareto', 'PowerFunction', 'QuadraticU', 'RaisedCosine', 'Rayleigh','Reciprocal', 'StudentT', 'ShiftedGompertz', 'Trapezoidal', 'Triangular', 'Uniform', 'UniformSum', 'VonMises', 'Wald', 'Weibull', 'WignerSemicircle', 'ContinuousDistributionHandmade', 'FlorySchulz', 'Geometric','Hermite', 'Logarithmic', 'NegativeBinomial', 'Poisson', 'Skellam', 'YuleSimon', 'Zeta', 'DiscreteRV', 'DiscreteDistributionHandmade', 'JointRV', 'Dirichlet', 'GeneralizedMultivariateLogGamma', 'GeneralizedMultivariateLogGammaOmega', 'Multinomial', 'MultivariateBeta', 'MultivariateEwens', 'MultivariateT', 'NegativeMultinomial', 'NormalGamma', 'MultivariateNormal', 'MultivariateLaplace', 'marginal_distribution', 'StochasticProcess', 'DiscreteTimeStochasticProcess', 'DiscreteMarkovChain', 'TransitionMatrixOf', 'StochasticStateSpaceOf', 'GeneratorMatrixOf', 'ContinuousMarkovChain', 'BernoulliProcess', 'PoissonProcess', 'WienerProcess', 'GammaProcess', 'CircularEnsemble', 'CircularUnitaryEnsemble', 'CircularOrthogonalEnsemble', 'CircularSymplecticEnsemble', 'GaussianEnsemble', 'GaussianUnitaryEnsemble', 'GaussianOrthogonalEnsemble', 'GaussianSymplecticEnsemble', 'joint_eigen_distribution', 'JointEigenDistribution', 'level_spacing_distribution', 'MatrixGamma', 'Wishart', 'MatrixNormal', 'MatrixStudentT', 'Probability', 'Expectation', 'Variance', 'Covariance', 'Moment', 'CentralMoment', 'ExpectationMatrix', 'VarianceMatrix', 'CrossCovarianceMatrix' ] from .rv_interface import (P, E, H, density, where, given, sample, cdf, median, characteristic_function, pspace, sample_iter, variance, std, skewness, kurtosis, covariance, dependent, entropy, independent, random_symbols, correlation, factorial_moment, moment, cmoment, sampling_density, moment_generating_function, smoment, quantile, coskewness, sample_stochastic_process) from .frv_types import (FiniteRV, DiscreteUniform, Die, Bernoulli, Coin, Binomial, BetaBinomial, Hypergeometric, Rademacher, FiniteDistributionHandmade, IdealSoliton, RobustSoliton) from .crv_types import (ContinuousRV, Arcsin, Benini, Beta, BetaNoncentral, BetaPrime, BoundedPareto, Cauchy, Chi, ChiNoncentral, ChiSquared, Dagum, Erlang, ExGaussian, Exponential, ExponentialPower, FDistribution, FisherZ, Frechet, Gamma, GammaInverse, GaussianInverse, Gompertz, Gumbel, Kumaraswamy, Laplace, Levy, Logistic, LogCauchy, LogLogistic, LogitNormal, LogNormal, Lomax, Maxwell, Moyal, Nakagami, Normal, Pareto, QuadraticU, RaisedCosine, Rayleigh, Reciprocal, StudentT, PowerFunction, ShiftedGompertz, Trapezoidal, Triangular, Uniform, UniformSum, VonMises, Wald, Weibull, WignerSemicircle, ContinuousDistributionHandmade) from .drv_types import (FlorySchulz, Geometric, Hermite, Logarithmic, NegativeBinomial, Poisson, Skellam, YuleSimon, Zeta, DiscreteRV, DiscreteDistributionHandmade) from .joint_rv_types import (JointRV, Dirichlet, GeneralizedMultivariateLogGamma, GeneralizedMultivariateLogGammaOmega, Multinomial, MultivariateBeta, MultivariateEwens, MultivariateT, NegativeMultinomial, NormalGamma, MultivariateNormal, MultivariateLaplace, marginal_distribution) from .stochastic_process_types import (StochasticProcess, DiscreteTimeStochasticProcess, DiscreteMarkovChain, TransitionMatrixOf, StochasticStateSpaceOf, GeneratorMatrixOf, ContinuousMarkovChain, BernoulliProcess, PoissonProcess, WienerProcess, GammaProcess) from .random_matrix_models import (CircularEnsemble, CircularUnitaryEnsemble, CircularOrthogonalEnsemble, CircularSymplecticEnsemble, GaussianEnsemble, GaussianUnitaryEnsemble, GaussianOrthogonalEnsemble, GaussianSymplecticEnsemble, joint_eigen_distribution, JointEigenDistribution, level_spacing_distribution) from .matrix_distributions import MatrixGamma, Wishart, MatrixNormal, MatrixStudentT from .symbolic_probability import (Probability, Expectation, Variance, Covariance, Moment, CentralMoment) from .symbolic_multivariate_probability import (ExpectationMatrix, VarianceMatrix, CrossCovarianceMatrix)
7f3a880c47198fac385bd59bb9e696bfe5f752bb51fa91bcf8295a1152f48390	from sympy.concrete.products import Product from sympy.concrete.summations import Sum from sympy.core.add import Add from sympy.core.function import Lambda from sympy.core.mul import Mul from sympy.core.numbers import (Integer, Rational, pi) from sympy.core.power import Pow from sympy.core.relational import Eq from sympy.core.singleton import S from sympy.core.symbol import (Symbol, symbols) from sympy.core.sympify import sympify from sympy.functions.combinatorial.factorials import (rf, factorial) from sympy.functions.elementary.exponential import exp from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.piecewise import Piecewise from sympy.functions.special.bessel import besselk from sympy.functions.special.gamma_functions import gamma from sympy.matrices.dense import (Matrix, ones) from sympy.sets.fancysets import Range from sympy.sets.sets import (Intersection, Interval) from sympy.tensor.indexed import (Indexed, IndexedBase) from sympy.matrices import ImmutableMatrix, MatrixSymbol from sympy.matrices.expressions.determinant import det from sympy.matrices.expressions.matexpr import MatrixElement from sympy.stats.joint_rv import JointDistribution, JointPSpace, MarginalDistribution from sympy.stats.rv import _value_check, random_symbols __all__ = ['JointRV', 'MultivariateNormal', 'MultivariateLaplace', 'Dirichlet', 'GeneralizedMultivariateLogGamma', 'GeneralizedMultivariateLogGammaOmega', 'Multinomial', 'MultivariateBeta', 'MultivariateEwens', 'MultivariateT', 'NegativeMultinomial', 'NormalGamma' ] def multivariate_rv(cls, sym, args): args = list(map(sympify, args)) dist = cls(args) args = dist.args dist.check(args) return JointPSpace(sym, dist).value def marginal_distribution(rv, indices): """ Marginal distribution function of a joint random variable. Parameters ========== rv : A random variable with a joint probability distribution. indices : Component indices or the indexed random symbol for which the joint distribution is to be calculated Returns ======= A Lambda expression in `sym`. Examples ======== >>> from sympy.stats import MultivariateNormal, marginal_distribution >>> m = MultivariateNormal('X', [1, 2], [[2, 1], [1, 2]]) >>> marginal_distribution(m, m[0])(1) 1/(2sqrt(pi)) """ indices = list(indices) for i in range(len(indices)): if isinstance(indices[i], Indexed): indices[i] = indices[i].args[1] prob_space = rv.pspace if not indices: raise ValueError( "At least one component for marginal density is needed.") if hasattr(prob_space.distribution, '_marginal_distribution'): return prob_space.distribution._marginal_distribution(indices, rv.symbol) return prob_space.marginal_distribution(indices) class JointDistributionHandmade(JointDistribution): _argnames = ('pdf',) is_Continuous = True @property def set(self): return self.args[1] def JointRV(symbol, pdf, _set=None): """ Create a Joint Random Variable where each of its component is continuous, given the following: Parameters ========== symbol : Symbol Represents name of the random variable. pdf : A PDF in terms of indexed symbols of the symbol given as the first argument NOTE ==== As of now, the set for each component for a ``JointRV`` is equal to the set of all integers, which cannot be changed. Examples ======== >>> from sympy import exp, pi, Indexed, S >>> from sympy.stats import density, JointRV >>> x1, x2 = (Indexed('x', i) for i in (1, 2)) >>> pdf = exp(-x12/2 + x1 - x22/2 - S(1)/2)/(2pi) >>> N1 = JointRV('x', pdf) #Multivariate Normal distribution >>> density(N1)(1, 2) exp(-2)/(2pi) Returns ======= RandomSymbol """ #TODO: Add support for sets provided by the user symbol = sympify(symbol) syms = list(i for i in pdf.free_symbols if isinstance(i, Indexed) and i.base == IndexedBase(symbol)) syms = tuple(sorted(syms, key = lambda index: index.args[1])) _set = S.Realslen(syms) pdf = Lambda(syms, pdf) dist = JointDistributionHandmade(pdf, _set) jrv = JointPSpace(symbol, dist).value rvs = random_symbols(pdf) if len(rvs) != 0: dist = MarginalDistribution(dist, (jrv,)) return JointPSpace(symbol, dist).value return jrv #------------------------------------------------------------------------------- # Multivariate Normal distribution --------------------------------------------- class MultivariateNormalDistribution(JointDistribution): _argnames = ('mu', 'sigma') is_Continuous=True @property def set(self): k = self.mu.shape[0] return S.Realsk @staticmethod def check(mu, sigma): _value_check(mu.shape[0] == sigma.shape[0], "Size of the mean vector and covariance matrix are incorrect.") #check if covariance matrix is positive semi definite or not. if not isinstance(sigma, MatrixSymbol): _value_check(sigma.is_positive_semidefinite, "The covariance matrix must be positive semi definite. ") def pdf(self, args): mu, sigma = self.mu, self.sigma k = mu.shape[0] if len(args) == 1 and args[0].is_Matrix: args = args[0] else: args = ImmutableMatrix(args) x = args - mu density = S.One/sqrt((2pi)*(k)det(sigma))exp( Rational(-1, 2)x.transpose()(sigma.inv()x)) return MatrixElement(density, 0, 0) def _marginal_distribution(self, indices, sym): sym = ImmutableMatrix([Indexed(sym, i) for i in indices]) _mu, _sigma = self.mu, self.sigma k = self.mu.shape[0] for i in range(k): if i not in indices: _mu = _mu.row_del(i) _sigma = _sigma.col_del(i) _sigma = _sigma.row_del(i) return Lambda(tuple(sym), S.One/sqrt((2pi)(len(_mu))det(_sigma))exp( Rational(-1, 2)(_mu - sym).transpose()(_sigma.inv()\ (_mu - sym)))[0]) def MultivariateNormal(name, mu, sigma): r""" Creates a continuous random variable with Multivariate Normal Distribution. The density of the multivariate normal distribution can be found at [1]. Parameters ========== mu : List representing the mean or the mean vector sigma : Positive semidefinite square matrix Represents covariance Matrix. If `\sigma` is noninvertible then only sampling is supported currently Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import MultivariateNormal, density, marginal_distribution >>> from sympy import symbols, MatrixSymbol >>> X = MultivariateNormal('X', [3, 4], [[2, 1], [1, 2]]) >>> y, z = symbols('y z') >>> density(X)(y, z) sqrt(3)exp(-y2/3 + yz/3 + 2y/3 - z2/3 + 5z/3 - 13/3)/(6pi) >>> density(X)(1, 2) sqrt(3)exp(-4/3)/(6pi) >>> marginal_distribution(X, X[1])(y) exp(-(y - 4)2/4)/(2sqrt(pi)) >>> marginal_distribution(X, X[0])(y) exp(-(y - 3)*2/4)/(2sqrt(pi)) The example below shows that it is also possible to use symbolic parameters to define the MultivariateNormal class. >>> n = symbols('n', integer=True, positive=True) >>> Sg = MatrixSymbol('Sg', n, n) >>> mu = MatrixSymbol('mu', n, 1) >>> obs = MatrixSymbol('obs', n, 1) >>> X = MultivariateNormal('X', mu, Sg) The density of a multivariate normal can be calculated using a matrix argument, as shown below. >>> density(X)(obs) (exp(((1/2)mu.T - (1/2)obs.T)Sg(-1)(-mu + obs))/sqrt((2pi)nDeterminant(Sg)))[0, 0] References ========== .. [1] https://en.wikipedia.org/wiki/Multivariate_normal_distribution """ return multivariate_rv(MultivariateNormalDistribution, name, mu, sigma) #------------------------------------------------------------------------------- # Multivariate Laplace distribution -------------------------------------------- class MultivariateLaplaceDistribution(JointDistribution): _argnames = ('mu', 'sigma') is_Continuous=True @property def set(self): k = self.mu.shape[0] return S.Reals*k @staticmethod def check(mu, sigma): _value_check(mu.shape[0] == sigma.shape[0], "Size of the mean vector and covariance matrix are incorrect.") # check if covariance matrix is positive definite or not. if not isinstance(sigma, MatrixSymbol): _value_check(sigma.is_positive_definite, "The covariance matrix must be positive definite. ") def pdf(self, args): mu, sigma = self.mu, self.sigma mu_T = mu.transpose() k = S(mu.shape[0]) sigma_inv = sigma.inv() args = ImmutableMatrix(args) args_T = args.transpose() x = (mu_Tsigma_invmu)[0] y = (args_Tsigma_invargs)[0] v = 1 - k/2 return (2 * (y/(2 + x))*(v/2) besselk(v, sqrt((2 + x)y)) exp((args_T * sigma_inv * mu)[0]) / ((2 * pi)*(k/2) sqrt(det(sigma)))) def MultivariateLaplace(name, mu, sigma): """ Creates a continuous random variable with Multivariate Laplace Distribution. The density of the multivariate Laplace distribution can be found at [1]. Parameters ========== mu : List representing the mean or the mean vector sigma : Positive definite square matrix Represents covariance Matrix Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import MultivariateLaplace, density >>> from sympy import symbols >>> y, z = symbols('y z') >>> X = MultivariateLaplace('X', [2, 4], [[3, 1], [1, 3]]) >>> density(X)(y, z) sqrt(2)exp(y/4 + 5z/4)besselk(0, sqrt(15y(3y/8 - z/8)/2 + 15z(-y/8 + 3z/8)/2))/(4pi) >>> density(X)(1, 2) sqrt(2)exp(11/4)besselk(0, sqrt(165)/4)/(4pi) References ========== .. [1] https://en.wikipedia.org/wiki/Multivariate_Laplace_distribution """ return multivariate_rv(MultivariateLaplaceDistribution, name, mu, sigma) #------------------------------------------------------------------------------- # Multivariate StudentT distribution ------------------------------------------- class MultivariateTDistribution(JointDistribution): _argnames = ('mu', 'shape_mat', 'dof') is_Continuous=True @property def set(self): k = self.mu.shape[0] return S.Realsk @staticmethod def check(mu, sigma, v): _value_check(mu.shape[0] == sigma.shape[0], "Size of the location vector and shape matrix are incorrect.") # check if covariance matrix is positive definite or not. if not isinstance(sigma, MatrixSymbol): _value_check(sigma.is_positive_definite, "The shape matrix must be positive definite. ") def pdf(self, args): mu, sigma = self.mu, self.shape_mat v = S(self.dof) k = S(mu.shape[0]) sigma_inv = sigma.inv() args = ImmutableMatrix(args) x = args - mu return gamma((k + v)/2)/(gamma(v/2)(vpi)*(k/2)sqrt(det(sigma)))\ (1 + 1/v(x.transpose()sigma_invx)[0])*((-v - k)/2) def MultivariateT(syms, mu, sigma, v): """ Creates a joint random variable with multivariate T-distribution. Parameters ========== syms : A symbol/str For identifying the random variable. mu : A list/matrix Representing the location vector sigma : The shape matrix for the distribution Examples ======== >>> from sympy.stats import density, MultivariateT >>> from sympy import Symbol >>> x = Symbol("x") >>> X = MultivariateT("x", [1, 1], [[1, 0], [0, 1]], 2) >>> density(X)(1, 2) 2/(9pi) Returns ======= RandomSymbol """ return multivariate_rv(MultivariateTDistribution, syms, mu, sigma, v) #------------------------------------------------------------------------------- # Multivariate Normal Gamma distribution --------------------------------------- class NormalGammaDistribution(JointDistribution): _argnames = ('mu', 'lamda', 'alpha', 'beta') is_Continuous=True @staticmethod def check(mu, lamda, alpha, beta): _value_check(mu.is_real, "Location must be real.") _value_check(lamda > 0, "Lambda must be positive") _value_check(alpha > 0, "alpha must be positive") _value_check(beta > 0, "beta must be positive") @property def set(self): return S.RealsInterval(0, S.Infinity) def pdf(self, x, tau): beta, alpha, lamda = self.beta, self.alpha, self.lamda mu = self.mu return betaalphasqrt(lamda)/(gamma(alpha)sqrt(2pi))\ tau(alpha - S.Half)exp(-1betatau)\ exp(-1(lamdatau(x - mu)*2)/S(2)) def _marginal_distribution(self, indices, sym): if len(indices) == 2: return self.pdf(sym) if indices[0] == 0: #For marginal over `x`, return non-standardized Student-T's #distribution x = sym[0] v, mu, sigma = self.alpha - S.Half, self.mu, \ S(self.beta)/(self.lamda self.alpha) return Lambda(sym, gamma((v + 1)/2)/(gamma(v/2)sqrt(piv)sigma)\ (1 + 1/v((x - mu)/sigma)2)((-v -1)/2)) #For marginal over `tau`, return Gamma distribution as per construction from sympy.stats.crv_types import GammaDistribution return Lambda(sym, GammaDistribution(self.alpha, self.beta)(sym[0])) def NormalGamma(sym, mu, lamda, alpha, beta): """ Creates a bivariate joint random variable with multivariate Normal gamma distribution. Parameters ========== sym : A symbol/str For identifying the random variable. mu : A real number The mean of the normal distribution lamda : A positive integer Parameter of joint distribution alpha : A positive integer Parameter of joint distribution beta : A positive integer Parameter of joint distribution Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import density, NormalGamma >>> from sympy import symbols >>> X = NormalGamma('x', 0, 1, 2, 3) >>> y, z = symbols('y z') >>> density(X)(y, z) 9sqrt(2)z(3/2)exp(-3z)exp(-y*2z/2)/(2sqrt(pi)) References ========== .. [1] https://en.wikipedia.org/wiki/Normal-gamma_distribution """ return multivariate_rv(NormalGammaDistribution, sym, mu, lamda, alpha, beta) #------------------------------------------------------------------------------- # Multivariate Beta/Dirichlet distribution ------------------------------------- class MultivariateBetaDistribution(JointDistribution): _argnames = ('alpha',) is_Continuous = True @staticmethod def check(alpha): _value_check(len(alpha) >= 2, "At least two categories should be passed.") for a_k in alpha: _value_check((a_k > 0) != False, "Each concentration parameter" " should be positive.") @property def set(self): k = len(self.alpha) return Interval(0, 1)k def pdf(self, syms): alpha = self.alpha B = Mul.fromiter(map(gamma, alpha))/gamma(Add(alpha)) return Mul.fromiter(sym(a_k - 1) for a_k, sym in zip(alpha, syms))/B def MultivariateBeta(syms, alpha): """ Creates a continuous random variable with Dirichlet/Multivariate Beta Distribution. The density of the Dirichlet distribution can be found at [1]. Parameters ========== alpha : Positive real numbers Signifies concentration numbers. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import density, MultivariateBeta, marginal_distribution >>> from sympy import Symbol >>> a1 = Symbol('a1', positive=True) >>> a2 = Symbol('a2', positive=True) >>> B = MultivariateBeta('B', [a1, a2]) >>> C = MultivariateBeta('C', a1, a2) >>> x = Symbol('x') >>> y = Symbol('y') >>> density(B)(x, y) x*(a1 - 1)y*(a2 - 1)gamma(a1 + a2)/(gamma(a1)gamma(a2)) >>> marginal_distribution(C, C[0])(x) x(a1 - 1)gamma(a1 + a2)/(a2gamma(a1)gamma(a2)) References ========== .. [1] https://en.wikipedia.org/wiki/Dirichlet_distribution .. [2] http://mathworld.wolfram.com/DirichletDistribution.html """ if not isinstance(alpha[0], list): alpha = (list(alpha),) return multivariate_rv(MultivariateBetaDistribution, syms, alpha[0]) Dirichlet = MultivariateBeta #------------------------------------------------------------------------------- # Multivariate Ewens distribution ---------------------------------------------- class MultivariateEwensDistribution(JointDistribution): _argnames = ('n', 'theta') is_Discrete = True is_Continuous = False @staticmethod def check(n, theta): _value_check((n > 0), "sample size should be positive integer.") _value_check(theta.is_positive, "mutation rate should be positive.") @property def set(self): if not isinstance(self.n, Integer): i = Symbol('i', integer=True, positive=True) return Product(Intersection(S.Naturals0, Interval(0, self.n//i)), (i, 1, self.n)) prod_set = Range(0, self.n + 1) for i in range(2, self.n + 1): prod_set = Range(0, self.n//i + 1) return prod_set.flatten() def pdf(self, syms): n, theta = self.n, self.theta condi = isinstance(self.n, Integer) if not (isinstance(syms[0], IndexedBase) or condi): raise ValueError("Please use IndexedBase object for syms as " "the dimension is symbolic") term_1 = factorial(n)/rf(theta, n) if condi: term_2 = Mul.fromiter(thetasyms[j]/((j+1)syms[j]factorial(syms[j])) for j in range(n)) cond = Eq(sum([(k + 1)syms[k] for k in range(n)]), n) return Piecewise((term_1 * term_2, cond), (0, True)) syms = syms[0] j, k = symbols('j, k', positive=True, integer=True) term_2 = Product(thetasyms[j]/((j+1)syms[j]factorial(syms[j])), (j, 0, n - 1)) cond = Eq(Sum((k + 1)syms[k], (k, 0, n - 1)), n) return Piecewise((term_1 * term_2, cond), (0, True)) def MultivariateEwens(syms, n, theta): """ Creates a discrete random variable with Multivariate Ewens Distribution. The density of the said distribution can be found at [1]. Parameters ========== n : Positive integer Size of the sample or the integer whose partitions are considered theta : Positive real number Denotes Mutation rate Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import density, marginal_distribution, MultivariateEwens >>> from sympy import Symbol >>> a1 = Symbol('a1', positive=True) >>> a2 = Symbol('a2', positive=True) >>> ed = MultivariateEwens('E', 2, 1) >>> density(ed)(a1, a2) Piecewise((1/(2*a2factorial(a1)factorial(a2)), Eq(a1 + 2a2, 2)), (0, True)) >>> marginal_distribution(ed, ed[0])(a1) Piecewise((1/factorial(a1), Eq(a1, 2)), (0, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Ewens%27s_sampling_formula .. [2] http://www.stat.rutgers.edu/home/hcrane/Papers/STS529.pdf """ return multivariate_rv(MultivariateEwensDistribution, syms, n, theta) #------------------------------------------------------------------------------- # Generalized Multivariate Log Gamma distribution ------------------------------ class GeneralizedMultivariateLogGammaDistribution(JointDistribution): _argnames = ('delta', 'v', 'lamda', 'mu') is_Continuous=True def check(self, delta, v, l, mu): _value_check((delta >= 0, delta <= 1), "delta must be in range [0, 1].") _value_check((v > 0), "v must be positive") for lk in l: _value_check((lk > 0), "lamda must be a positive vector.") for muk in mu: _value_check((muk > 0), "mu must be a positive vector.") _value_check(len(l) > 1,"the distribution should have at least" " two random variables.") @property def set(self): return S.Reals*len(self.lamda) def pdf(self, y): d, v, l, mu = self.delta, self.v, self.lamda, self.mu n = Symbol('n', negative=False, integer=True) k = len(l) sterm1 = Pow((1 - d), n)/\ ((gamma(v + n)*(k - 1))gamma(v)gamma(n + 1)) sterm2 = Mul.fromiter(muili*(-v - n) for mui, li in zip(mu, l)) term1 = sterm1 sterm2 sterm3 = (v + n) * sum([mui * yi for mui, yi in zip(mu, y)]) sterm4 = sum([exp(mui * yi)/li for (mui, yi, li) in zip(mu, y, l)]) term2 = exp(sterm3 - sterm4) return Pow(d, v) * Sum(term1 * term2, (n, 0, S.Infinity)) def GeneralizedMultivariateLogGamma(syms, delta, v, lamda, mu): """ Creates a joint random variable with generalized multivariate log gamma distribution. The joint pdf can be found at [1]. Parameters ========== syms : list/tuple/set of symbols for identifying each component delta : A constant in range $[0, 1]$ v : Positive real number lamda : List of positive real numbers mu : List of positive real numbers Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import density >>> from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGamma >>> from sympy import symbols, S >>> v = 1 >>> l, mu = [1, 1, 1], [1, 1, 1] >>> d = S.Half >>> y = symbols('y_1:4', positive=True) >>> Gd = GeneralizedMultivariateLogGamma('G', d, v, l, mu) >>> density(Gd)(y[0], y[1], y[2]) Sum(exp((n + 1)(y_1 + y_2 + y_3) - exp(y_1) - exp(y_2) - exp(y_3))/(2ngamma(n + 1)*3), (n, 0, oo))/2 References ========== .. [1] https://en.wikipedia.org/wiki/Generalized_multivariate_log-gamma_distribution .. [2] https://www.researchgate.net/publication/234137346_On_a_multivariate_log-gamma_distribution_and_the_use_of_the_distribution_in_the_Bayesian_analysis Note ==== If the GeneralizedMultivariateLogGamma is too long to type use, >>> from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGamma as GMVLG >>> Gd = GMVLG('G', d, v, l, mu) If you want to pass the matrix omega instead of the constant delta, then use ``GeneralizedMultivariateLogGammaOmega``. """ return multivariate_rv(GeneralizedMultivariateLogGammaDistribution, syms, delta, v, lamda, mu) def GeneralizedMultivariateLogGammaOmega(syms, omega, v, lamda, mu): """ Extends GeneralizedMultivariateLogGamma. Parameters ========== syms : list/tuple/set of symbols For identifying each component omega : A square matrix Every element of square matrix must be absolute value of square root of correlation coefficient v : Positive real number lamda : List of positive real numbers mu : List of positive real numbers Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import density >>> from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGammaOmega >>> from sympy import Matrix, symbols, S >>> omega = Matrix([[1, S.Half, S.Half], [S.Half, 1, S.Half], [S.Half, S.Half, 1]]) >>> v = 1 >>> l, mu = [1, 1, 1], [1, 1, 1] >>> G = GeneralizedMultivariateLogGammaOmega('G', omega, v, l, mu) >>> y = symbols('y_1:4', positive=True) >>> density(G)(y[0], y[1], y[2]) sqrt(2)Sum((1 - sqrt(2)/2)*nexp((n + 1)(y_1 + y_2 + y_3) - exp(y_1) - exp(y_2) - exp(y_3))/gamma(n + 1)3, (n, 0, oo))/2 References ========== .. [1] https://en.wikipedia.org/wiki/Generalized_multivariate_log-gamma_distribution .. [2] https://www.researchgate.net/publication/234137346_On_a_multivariate_log-gamma_distribution_and_the_use_of_the_distribution_in_the_Bayesian_analysis Notes ===== If the GeneralizedMultivariateLogGammaOmega is too long to type use, >>> from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGammaOmega as GMVLGO >>> G = GMVLGO('G', omega, v, l, mu) """ _value_check((omega.is_square, isinstance(omega, Matrix)), "omega must be a" " square matrix") for val in omega.values(): _value_check((val >= 0, val <= 1), "all values in matrix must be between 0 and 1(both inclusive).") _value_check(omega.diagonal().equals(ones(1, omega.shape[0])), "all the elements of diagonal should be 1.") _value_check((omega.shape[0] == len(lamda), len(lamda) == len(mu)), "lamda, mu should be of same length and omega should " " be of shape (length of lamda, length of mu)") _value_check(len(lamda) > 1,"the distribution should have at least" " two random variables.") delta = Pow(Rational(omega.det()), Rational(1, len(lamda) - 1)) return GeneralizedMultivariateLogGamma(syms, delta, v, lamda, mu) #------------------------------------------------------------------------------- # Multinomial distribution ----------------------------------------------------- class MultinomialDistribution(JointDistribution): _argnames = ('n', 'p') is_Continuous=False is_Discrete = True @staticmethod def check(n, p): _value_check(n > 0, "number of trials must be a positive integer") for p_k in p: _value_check((p_k >= 0, p_k <= 1), "probability must be in range [0, 1]") _value_check(Eq(sum(p), 1), "probabilities must sum to 1") @property def set(self): return Intersection(S.Naturals0, Interval(0, self.n))len(self.p) def pdf(self, x): n, p = self.n, self.p term_1 = factorial(n)/Mul.fromiter(factorial(x_k) for x_k in x) term_2 = Mul.fromiter(p_k*x_k for p_k, x_k in zip(p, x)) return Piecewise((term_1 term_2, Eq(sum(x), n)), (0, True)) def Multinomial(syms, n, p): """ Creates a discrete random variable with Multinomial Distribution. The density of the said distribution can be found at [1]. Parameters ========== n : Positive integer Represents number of trials p : List of event probabilites Must be in the range of $[0, 1]$. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import density, Multinomial, marginal_distribution >>> from sympy import symbols >>> x1, x2, x3 = symbols('x1, x2, x3', nonnegative=True, integer=True) >>> p1, p2, p3 = symbols('p1, p2, p3', positive=True) >>> M = Multinomial('M', 3, p1, p2, p3) >>> density(M)(x1, x2, x3) Piecewise((6p1*x1p2*x2p3*x3/(factorial(x1)factorial(x2)factorial(x3)), Eq(x1 + x2 + x3, 3)), (0, True)) >>> marginal_distribution(M, M[0])(x1).subs(x1, 1) 3p1p22 + 6p1p2p3 + 3p1p32 References ========== .. [1] https://en.wikipedia.org/wiki/Multinomial_distribution .. [2] http://mathworld.wolfram.com/MultinomialDistribution.html """ if not isinstance(p[0], list): p = (list(p), ) return multivariate_rv(MultinomialDistribution, syms, n, p[0]) #------------------------------------------------------------------------------- # Negative Multinomial Distribution -------------------------------------------- class NegativeMultinomialDistribution(JointDistribution): _argnames = ('k0', 'p') is_Continuous=False is_Discrete = True @staticmethod def check(k0, p): _value_check(k0 > 0, "number of failures must be a positive integer") for p_k in p: _value_check((p_k >= 0, p_k <= 1), "probability must be in range [0, 1].") _value_check(sum(p) <= 1, "success probabilities must not be greater than 1.") @property def set(self): return Range(0, S.Infinity)len(self.p) def pdf(self, k): k0, p = self.k0, self.p term_1 = (gamma(k0 + sum(k))(1 - sum(p))k0)/gamma(k0) term_2 = Mul.fromiter(piki/factorial(ki) for pi, ki in zip(p, k)) return term_1 * term_2 def NegativeMultinomial(syms, k0, p): """ Creates a discrete random variable with Negative Multinomial Distribution. The density of the said distribution can be found at [1]. Parameters ========== k0 : positive integer Represents number of failures before the experiment is stopped p : List of event probabilites Must be in the range of $[0, 1]$ Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import density, NegativeMultinomial, marginal_distribution >>> from sympy import symbols >>> x1, x2, x3 = symbols('x1, x2, x3', nonnegative=True, integer=True) >>> p1, p2, p3 = symbols('p1, p2, p3', positive=True) >>> N = NegativeMultinomial('M', 3, p1, p2, p3) >>> N_c = NegativeMultinomial('M', 3, 0.1, 0.1, 0.1) >>> density(N)(x1, x2, x3) p1x1p2*x2p3*x3(-p1 - p2 - p3 + 1)*3gamma(x1 + x2 + x3 + 3)/(2factorial(x1)factorial(x2)*factorial(x3)) >>> marginal_distribution(N_c, N_c[0])(1).evalf().round(2) 0.25 References ========== .. [1] https://en.wikipedia.org/wiki/Negative_multinomial_distribution .. [2] http://mathworld.wolfram.com/NegativeBinomialDistribution.html """ if not isinstance(p[0], list): p = (list(p), ) return multivariate_rv(NegativeMultinomialDistribution, syms, k0, p[0])
32db35bef9441e831d36d09f1f4fb31b3485f4028bc8f8076d3f364c4f99920f	""" Contains ======== FlorySchulz Geometric Hermite Logarithmic NegativeBinomial Poisson Skellam YuleSimon Zeta """ from sympy.concrete.summations import Sum from sympy.core.basic import Basic from sympy.core.function import Lambda from sympy.core.numbers import I from sympy.core.relational import Eq from sympy.core.singleton import S from sympy.core.symbol import Dummy from sympy.core.sympify import sympify from sympy.functions.combinatorial.factorials import (binomial, factorial) from sympy.functions.elementary.exponential import (exp, log) from sympy.functions.elementary.integers import floor from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.piecewise import Piecewise from sympy.functions.special.bessel import besseli from sympy.functions.special.beta_functions import beta from sympy.functions.special.hyper import hyper from sympy.functions.special.zeta_functions import (polylog, zeta) from sympy.stats.drv import SingleDiscreteDistribution, SingleDiscretePSpace from sympy.stats.rv import _value_check, is_random __all__ = ['FlorySchulz', 'Geometric', 'Hermite', 'Logarithmic', 'NegativeBinomial', 'Poisson', 'Skellam', 'YuleSimon', 'Zeta' ] def rv(symbol, cls, args, kwargs): args = list(map(sympify, args)) dist = cls(args) if kwargs.pop('check', True): dist.check(args) pspace = SingleDiscretePSpace(symbol, dist) if any(is_random(arg) for arg in args): from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution pspace = CompoundPSpace(symbol, CompoundDistribution(dist)) return pspace.value class DiscreteDistributionHandmade(SingleDiscreteDistribution): _argnames = ('pdf',) def __new__(cls, pdf, set=S.Integers): return Basic.__new__(cls, pdf, set) @property def set(self): return self.args[1] @staticmethod def check(pdf, set): x = Dummy('x') val = Sum(pdf(x), (x, set._inf, set._sup)).doit() _value_check(Eq(val, 1) != S.false, "The pdf is incorrect on the given set.") def DiscreteRV(symbol, density, set=S.Integers, kwargs): """ Create a Discrete Random Variable given the following: Parameters ========== symbol : Symbol Represents name of the random variable. density : Expression containing symbol Represents probability density function. set : set Represents the region where the pdf is valid, by default is real line. check : bool If True, it will check whether the given density integrates to 1 over the given set. If False, it will not perform this check. Default is False. Examples ======== >>> from sympy.stats import DiscreteRV, P, E >>> from sympy import Rational, Symbol >>> x = Symbol('x') >>> n = 10 >>> density = Rational(1, 10) >>> X = DiscreteRV(x, density, set=set(range(n))) >>> E(X) 9/2 >>> P(X>3) 3/5 Returns ======= RandomSymbol """ set = sympify(set) pdf = Piecewise((density, set.as_relational(symbol)), (0, True)) pdf = Lambda(symbol, pdf) # have a default of False while `rv` should have a default of True kwargs['check'] = kwargs.pop('check', False) return rv(symbol.name, DiscreteDistributionHandmade, pdf, set, kwargs) #------------------------------------------------------------------------------- # Flory-Schulz distribution ------------------------------------------------------------ class FlorySchulzDistribution(SingleDiscreteDistribution): _argnames = ('a',) set = S.Naturals @staticmethod def check(a): _value_check((0 < a, a < 1), "a must be between 0 and 1") def pdf(self, k): a = self.a return (a2 k * (1 - a)(k - 1)) def _characteristic_function(self, t): a = self.a return a2exp(It)/((1 + (a - 1)exp(It))2) def _moment_generating_function(self, t): a = self.a return a2exp(t)/((1 + (a - 1)exp(t))2) def FlorySchulz(name, a): r""" Create a discrete random variable with a FlorySchulz distribution. The density of the FlorySchulz distribution is given by .. math:: f(k) := (a^2) k (1 - a)^{k-1} Parameters ========== a : A real number between 0 and 1 Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import density, E, variance, FlorySchulz >>> from sympy import Symbol, S >>> a = S.One / 5 >>> z = Symbol("z") >>> X = FlorySchulz("x", a) >>> density(X)(z) (4/5)(z - 1)z/25 >>> E(X) 9 >>> variance(X) 40 References ========== https://en.wikipedia.org/wiki/Flory%E2%80%93Schulz_distribution """ return rv(name, FlorySchulzDistribution, a) #------------------------------------------------------------------------------- # Geometric distribution ------------------------------------------------------------ class GeometricDistribution(SingleDiscreteDistribution): _argnames = ('p',) set = S.Naturals @staticmethod def check(p): _value_check((0 < p, p <= 1), "p must be between 0 and 1") def pdf(self, k): return (1 - self.p)(k - 1) self.p def _characteristic_function(self, t): p = self.p return p * exp(It) / (1 - (1 - p)exp(It)) def _moment_generating_function(self, t): p = self.p return p exp(t) / (1 - (1 - p) * exp(t)) def Geometric(name, p): r""" Create a discrete random variable with a Geometric distribution. Explanation =========== The density of the Geometric distribution is given by .. math:: f(k) := p (1 - p)^{k - 1} Parameters ========== p : A probability between 0 and 1 Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Geometric, density, E, variance >>> from sympy import Symbol, S >>> p = S.One / 5 >>> z = Symbol("z") >>> X = Geometric("x", p) >>> density(X)(z) (4/5)(z - 1)/5 >>> E(X) 5 >>> variance(X) 20 References ========== .. [1] https://en.wikipedia.org/wiki/Geometric_distribution .. [2] http://mathworld.wolfram.com/GeometricDistribution.html """ return rv(name, GeometricDistribution, p) #------------------------------------------------------------------------------- # Hermite distribution --------------------------------------------------------- class HermiteDistribution(SingleDiscreteDistribution): _argnames = ('a1', 'a2') set = S.Naturals0 @staticmethod def check(a1, a2): _value_check(a1.is_nonnegative, 'Parameter a1 must be >= 0.') _value_check(a2.is_nonnegative, 'Parameter a2 must be >= 0.') def pdf(self, k): a1, a2 = self.a1, self.a2 term1 = exp(-(a1 + a2)) j = Dummy("j", integer=True) num = a1(k - 2j) a2*j den = factorial(k - 2j) * factorial(j) return term1 * Sum(num/den, (j, 0, k//2)).doit() def _moment_generating_function(self, t): a1, a2 = self.a1, self.a2 term1 = a1 * (exp(t) - 1) term2 = a2 * (exp(2t) - 1) return exp(term1 + term2) def _characteristic_function(self, t): a1, a2 = self.a1, self.a2 term1 = a1 (exp(It) - 1) term2 = a2 (exp(2It) - 1) return exp(term1 + term2) def Hermite(name, a1, a2): r""" Create a discrete random variable with a Hermite distribution. Explanation =========== The density of the Hermite distribution is given by .. math:: f(x):= e^{-a_1 -a_2}\sum_{j=0}^{\left \lfloor x/2 \right \rfloor} \frac{a_{1}^{x-2j}a_{2}^{j}}{(x-2j)!j!} Parameters ========== a1 : A Positive number greater than equal to 0. a2 : A Positive number greater than equal to 0. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Hermite, density, E, variance >>> from sympy import Symbol >>> a1 = Symbol("a1", positive=True) >>> a2 = Symbol("a2", positive=True) >>> x = Symbol("x") >>> H = Hermite("H", a1=5, a2=4) >>> density(H)(2) 33exp(-9)/2 >>> E(H) 13 >>> variance(H) 21 References ========== .. [1] https://en.wikipedia.org/wiki/Hermite_distribution """ return rv(name, HermiteDistribution, a1, a2) #------------------------------------------------------------------------------- # Logarithmic distribution ------------------------------------------------------------ class LogarithmicDistribution(SingleDiscreteDistribution): _argnames = ('p',) set = S.Naturals @staticmethod def check(p): _value_check((p > 0, p < 1), "p should be between 0 and 1") def pdf(self, k): p = self.p return (-1) p*k / (k log(1 - p)) def _characteristic_function(self, t): p = self.p return log(1 - p * exp(It)) / log(1 - p) def _moment_generating_function(self, t): p = self.p return log(1 - p exp(t)) / log(1 - p) def Logarithmic(name, p): r""" Create a discrete random variable with a Logarithmic distribution. Explanation =========== The density of the Logarithmic distribution is given by .. math:: f(k) := \frac{-p^k}{k \ln{(1 - p)}} Parameters ========== p : A value between 0 and 1 Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Logarithmic, density, E, variance >>> from sympy import Symbol, S >>> p = S.One / 5 >>> z = Symbol("z") >>> X = Logarithmic("x", p) >>> density(X)(z) -1/(5*zzlog(4/5)) >>> E(X) -1/(-4log(5) + 8log(2)) >>> variance(X) -1/((-4log(5) + 8log(2))(-2log(5) + 4log(2))) + 1/(-64log(2)log(5) + 64log(2)2 + 16log(5)*2) - 10/(-32log(5) + 64log(2)) References ========== .. [1] https://en.wikipedia.org/wiki/Logarithmic_distribution .. [2] http://mathworld.wolfram.com/LogarithmicDistribution.html """ return rv(name, LogarithmicDistribution, p) #------------------------------------------------------------------------------- # Negative binomial distribution ------------------------------------------------------------ class NegativeBinomialDistribution(SingleDiscreteDistribution): _argnames = ('r', 'p') set = S.Naturals0 @staticmethod def check(r, p): _value_check(r > 0, 'r should be positive') _value_check((p > 0, p < 1), 'p should be between 0 and 1') def pdf(self, k): r = self.r p = self.p return binomial(k + r - 1, k) (1 - p)*r p*k def _characteristic_function(self, t): r = self.r p = self.p return ((1 - p) / (1 - p exp(It)))r def _moment_generating_function(self, t): r = self.r p = self.p return ((1 - p) / (1 - p exp(t)))*r def NegativeBinomial(name, r, p): r""" Create a discrete random variable with a Negative Binomial distribution. Explanation =========== The density of the Negative Binomial distribution is given by .. math:: f(k) := \binom{k + r - 1}{k} (1 - p)^r p^k Parameters ========== r : A positive value p : A value between 0 and 1 Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import NegativeBinomial, density, E, variance >>> from sympy import Symbol, S >>> r = 5 >>> p = S.One / 5 >>> z = Symbol("z") >>> X = NegativeBinomial("x", r, p) >>> density(X)(z) 1024binomial(z + 4, z)/(31255z) >>> E(X) 5/4 >>> variance(X) 25/16 References ========== .. [1] https://en.wikipedia.org/wiki/Negative_binomial_distribution .. [2] http://mathworld.wolfram.com/NegativeBinomialDistribution.html """ return rv(name, NegativeBinomialDistribution, r, p) #------------------------------------------------------------------------------- # Poisson distribution ------------------------------------------------------------ class PoissonDistribution(SingleDiscreteDistribution): _argnames = ('lamda',) set = S.Naturals0 @staticmethod def check(lamda): _value_check(lamda > 0, "Lambda must be positive") def pdf(self, k): return self.lamdak / factorial(k) exp(-self.lamda) def _characteristic_function(self, t): return exp(self.lamda * (exp(It) - 1)) def _moment_generating_function(self, t): return exp(self.lamda (exp(t) - 1)) def Poisson(name, lamda): r""" Create a discrete random variable with a Poisson distribution. Explanation =========== The density of the Poisson distribution is given by .. math:: f(k) := \frac{\lambda^{k} e^{- \lambda}}{k!} Parameters ========== lamda : Positive number, a rate Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Poisson, density, E, variance >>> from sympy import Symbol, simplify >>> rate = Symbol("lambda", positive=True) >>> z = Symbol("z") >>> X = Poisson("x", rate) >>> density(X)(z) lambda*zexp(-lambda)/factorial(z) >>> E(X) lambda >>> simplify(variance(X)) lambda References ========== .. [1] https://en.wikipedia.org/wiki/Poisson_distribution .. [2] http://mathworld.wolfram.com/PoissonDistribution.html """ return rv(name, PoissonDistribution, lamda) # ----------------------------------------------------------------------------- # Skellam distribution -------------------------------------------------------- class SkellamDistribution(SingleDiscreteDistribution): _argnames = ('mu1', 'mu2') set = S.Integers @staticmethod def check(mu1, mu2): _value_check(mu1 >= 0, 'Parameter mu1 must be >= 0') _value_check(mu2 >= 0, 'Parameter mu2 must be >= 0') def pdf(self, k): (mu1, mu2) = (self.mu1, self.mu2) term1 = exp(-(mu1 + mu2)) * (mu1 / mu2) ** (k / 2) term2 = besseli(k, 2 * sqrt(mu1 * mu2)) return term1 * term2 def _cdf(self, x): raise NotImplementedError( "Skellam doesn't have closed form for the CDF.") def _characteristic_function(self, t): (mu1, mu2) = (self.mu1, self.mu2) return exp(-(mu1 + mu2) + mu1 * exp(I * t) + mu2 * exp(-I * t)) def _moment_generating_function(self, t): (mu1, mu2) = (self.mu1, self.mu2) return exp(-(mu1 + mu2) + mu1 * exp(t) + mu2 * exp(-t)) def Skellam(name, mu1, mu2): r""" Create a discrete random variable with a Skellam distribution. Explanation =========== The Skellam is the distribution of the difference N1 - N2 of two statistically independent random variables N1 and N2 each Poisson-distributed with respective expected values mu1 and mu2. The density of the Skellam distribution is given by .. math:: f(k) := e^{-(\mu_1+\mu_2)}(\frac{\mu_1}{\mu_2})^{k/2}I_k(2\sqrt{\mu_1\mu_2}) Parameters ========== mu1 : A non-negative value mu2 : A non-negative value Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Skellam, density, E, variance >>> from sympy import Symbol, pprint >>> z = Symbol("z", integer=True) >>> mu1 = Symbol("mu1", positive=True) >>> mu2 = Symbol("mu2", positive=True) >>> X = Skellam("x", mu1, mu2) >>> pprint(density(X)(z), use_unicode=False) z - 2 /mu1\ -mu1 - mu2 / _____ _____\ \|---\| e besseli\z, 2\/ mu1 \/ mu2 / \mu2/ >>> E(X) mu1 - mu2 >>> variance(X).expand() mu1 + mu2 References ========== .. [1] https://en.wikipedia.org/wiki/Skellam_distribution """ return rv(name, SkellamDistribution, mu1, mu2) #------------------------------------------------------------------------------- # Yule-Simon distribution ------------------------------------------------------------ class YuleSimonDistribution(SingleDiscreteDistribution): _argnames = ('rho',) set = S.Naturals @staticmethod def check(rho): _value_check(rho > 0, 'rho should be positive') def pdf(self, k): rho = self.rho return rho * beta(k, rho + 1) def _cdf(self, x): return Piecewise((1 - floor(x) * beta(floor(x), self.rho + 1), x >= 1), (0, True)) def _characteristic_function(self, t): rho = self.rho return rho * hyper((1, 1), (rho + 2,), exp(It)) exp(It) / (rho + 1) def _moment_generating_function(self, t): rho = self.rho return rho hyper((1, 1), (rho + 2,), exp(t)) * exp(t) / (rho + 1) def YuleSimon(name, rho): r""" Create a discrete random variable with a Yule-Simon distribution. Explanation =========== The density of the Yule-Simon distribution is given by .. math:: f(k) := \rho B(k, \rho + 1) Parameters ========== rho : A positive value Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import YuleSimon, density, E, variance >>> from sympy import Symbol, simplify >>> p = 5 >>> z = Symbol("z") >>> X = YuleSimon("x", p) >>> density(X)(z) 5beta(z, 6) >>> simplify(E(X)) 5/4 >>> simplify(variance(X)) 25/48 References ========== .. [1] https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution """ return rv(name, YuleSimonDistribution, rho) #------------------------------------------------------------------------------- # Zeta distribution ------------------------------------------------------------ class ZetaDistribution(SingleDiscreteDistribution): _argnames = ('s',) set = S.Naturals @staticmethod def check(s): _value_check(s > 1, 's should be greater than 1') def pdf(self, k): s = self.s return 1 / (ks zeta(s)) def _characteristic_function(self, t): return polylog(self.s, exp(It)) / zeta(self.s) def _moment_generating_function(self, t): return polylog(self.s, exp(t)) / zeta(self.s) def Zeta(name, s): r""" Create a discrete random variable with a Zeta distribution. Explanation =========== The density of the Zeta distribution is given by .. math:: f(k) := \frac{1}{k^s \zeta{(s)}} Parameters ========== s : A value greater than 1 Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Zeta, density, E, variance >>> from sympy import Symbol >>> s = 5 >>> z = Symbol("z") >>> X = Zeta("x", s) >>> density(X)(z) 1/(z5zeta(5)) >>> E(X) pi*4/(90zeta(5)) >>> variance(X) -pi*8/(8100zeta(5)**2) + zeta(3)/zeta(5) References ========== .. [1] https://en.wikipedia.org/wiki/Zeta_distribution """ return rv(name, ZetaDistribution, s)
196774a055ab8f6030cfbf22675e9120f5be531d74cf47bfd81d4b94c12ae521	""" Main Random Variables Module Defines abstract random variable type. Contains interfaces for probability space object (PSpace) as well as standard operators, P, E, sample, density, where, quantile See Also ======== sympy.stats.crv sympy.stats.frv sympy.stats.rv_interface """ from functools import singledispatch from typing import Tuple as tTuple from sympy.core.add import Add from sympy.core.basic import Basic from sympy.core.containers import Tuple from sympy.core.expr import Expr from sympy.core.function import (Function, Lambda) from sympy.core.logic import fuzzy_and from sympy.core.mul import (Mul, prod) from sympy.core.relational import (Eq, Ne) from sympy.core.singleton import S from sympy.core.symbol import (Dummy, Symbol) from sympy.core.sympify import sympify from sympy.functions.special.delta_functions import DiracDelta from sympy.functions.special.tensor_functions import KroneckerDelta from sympy.logic.boolalg import (And, Or) from sympy.matrices.expressions.matexpr import MatrixSymbol from sympy.tensor.indexed import Indexed from sympy.utilities.lambdify import lambdify from sympy.core.relational import Relational from sympy.core.sympify import _sympify from sympy.sets.sets import FiniteSet, ProductSet, Intersection from sympy.solvers.solveset import solveset from sympy.external import import_module from sympy.utilities.decorator import doctest_depends_on from sympy.utilities.exceptions import sympy_deprecation_warning from sympy.utilities.iterables import iterable x = Symbol('x') @singledispatch def is_random(x): return False @is_random.register(Basic) def _(x): atoms = x.free_symbols return any(is_random(i) for i in atoms) class RandomDomain(Basic): """ Represents a set of variables and the values which they can take. See Also ======== sympy.stats.crv.ContinuousDomain sympy.stats.frv.FiniteDomain """ is_ProductDomain = False is_Finite = False is_Continuous = False is_Discrete = False def __new__(cls, symbols, args): symbols = FiniteSet(symbols) return Basic.__new__(cls, symbols, args) @property def symbols(self): return self.args[0] @property def set(self): return self.args[1] def __contains__(self, other): raise NotImplementedError() def compute_expectation(self, expr): raise NotImplementedError() class SingleDomain(RandomDomain): """ A single variable and its domain. See Also ======== sympy.stats.crv.SingleContinuousDomain sympy.stats.frv.SingleFiniteDomain """ def __new__(cls, symbol, set): assert symbol.is_Symbol return Basic.__new__(cls, symbol, set) @property def symbol(self): return self.args[0] @property def symbols(self): return FiniteSet(self.symbol) def __contains__(self, other): if len(other) != 1: return False sym, val = tuple(other)[0] return self.symbol == sym and val in self.set class MatrixDomain(RandomDomain): """ A Random Matrix variable and its domain. """ def __new__(cls, symbol, set): symbol, set = _symbol_converter(symbol), _sympify(set) return Basic.__new__(cls, symbol, set) @property def symbol(self): return self.args[0] @property def symbols(self): return FiniteSet(self.symbol) class ConditionalDomain(RandomDomain): """ A RandomDomain with an attached condition. See Also ======== sympy.stats.crv.ConditionalContinuousDomain sympy.stats.frv.ConditionalFiniteDomain """ def __new__(cls, fulldomain, condition): condition = condition.xreplace({rs: rs.symbol for rs in random_symbols(condition)}) return Basic.__new__(cls, fulldomain, condition) @property def symbols(self): return self.fulldomain.symbols @property def fulldomain(self): return self.args[0] @property def condition(self): return self.args[1] @property def set(self): raise NotImplementedError("Set of Conditional Domain not Implemented") def as_boolean(self): return And(self.fulldomain.as_boolean(), self.condition) class PSpace(Basic): """ A Probability Space. Explanation =========== Probability Spaces encode processes that equal different values probabilistically. These underly Random Symbols which occur in SymPy expressions and contain the mechanics to evaluate statistical statements. See Also ======== sympy.stats.crv.ContinuousPSpace sympy.stats.frv.FinitePSpace """ is_Finite = None # type: bool is_Continuous = None # type: bool is_Discrete = None # type: bool is_real = None # type: bool @property def domain(self): return self.args[0] @property def density(self): return self.args[1] @property def values(self): return frozenset(RandomSymbol(sym, self) for sym in self.symbols) @property def symbols(self): return self.domain.symbols def where(self, condition): raise NotImplementedError() def compute_density(self, expr): raise NotImplementedError() def sample(self, size=(), library='scipy', seed=None): raise NotImplementedError() def probability(self, condition): raise NotImplementedError() def compute_expectation(self, expr): raise NotImplementedError() class SinglePSpace(PSpace): """ Represents the probabilities of a set of random events that can be attributed to a single variable/symbol. """ def __new__(cls, s, distribution): s = _symbol_converter(s) return Basic.__new__(cls, s, distribution) @property def value(self): return RandomSymbol(self.symbol, self) @property def symbol(self): return self.args[0] @property def distribution(self): return self.args[1] @property def pdf(self): return self.distribution.pdf(self.symbol) class RandomSymbol(Expr): """ Random Symbols represent ProbabilitySpaces in SymPy Expressions. In principle they can take on any value that their symbol can take on within the associated PSpace with probability determined by the PSpace Density. Explanation =========== Random Symbols contain pspace and symbol properties. The pspace property points to the represented Probability Space The symbol is a standard SymPy Symbol that is used in that probability space for example in defining a density. You can form normal SymPy expressions using RandomSymbols and operate on those expressions with the Functions E - Expectation of a random expression P - Probability of a condition density - Probability Density of an expression given - A new random expression (with new random symbols) given a condition An object of the RandomSymbol type should almost never be created by the user. They tend to be created instead by the PSpace class's value method. Traditionally a user does not even do this but instead calls one of the convenience functions Normal, Exponential, Coin, Die, FiniteRV, etc.... """ def __new__(cls, symbol, pspace=None): from sympy.stats.joint_rv import JointRandomSymbol if pspace is None: # Allow single arg, representing pspace == PSpace() pspace = PSpace() symbol = _symbol_converter(symbol) if not isinstance(pspace, PSpace): raise TypeError("pspace variable should be of type PSpace") if cls == JointRandomSymbol and isinstance(pspace, SinglePSpace): cls = RandomSymbol return Basic.__new__(cls, symbol, pspace) is_finite = True is_symbol = True is_Atom = True _diff_wrt = True pspace = property(lambda self: self.args[1]) symbol = property(lambda self: self.args[0]) name = property(lambda self: self.symbol.name) def _eval_is_positive(self): return self.symbol.is_positive def _eval_is_integer(self): return self.symbol.is_integer def _eval_is_real(self): return self.symbol.is_real or self.pspace.is_real @property def is_commutative(self): return self.symbol.is_commutative @property def free_symbols(self): return {self} class RandomIndexedSymbol(RandomSymbol): def __new__(cls, idx_obj, pspace=None): if pspace is None: # Allow single arg, representing pspace == PSpace() pspace = PSpace() if not isinstance(idx_obj, (Indexed, Function)): raise TypeError("An Function or Indexed object is expected not %s"%(idx_obj)) return Basic.__new__(cls, idx_obj, pspace) symbol = property(lambda self: self.args[0]) name = property(lambda self: str(self.args[0])) @property def key(self): if isinstance(self.symbol, Indexed): return self.symbol.args[1] elif isinstance(self.symbol, Function): return self.symbol.args[0] @property def free_symbols(self): if self.key.free_symbols: free_syms = self.key.free_symbols free_syms.add(self) return free_syms return {self} @property def pspace(self): return self.args[1] class RandomMatrixSymbol(RandomSymbol, MatrixSymbol): # type: ignore def __new__(cls, symbol, n, m, pspace=None): n, m = _sympify(n), _sympify(m) symbol = _symbol_converter(symbol) if pspace is None: # Allow single arg, representing pspace == PSpace() pspace = PSpace() return Basic.__new__(cls, symbol, n, m, pspace) symbol = property(lambda self: self.args[0]) pspace = property(lambda self: self.args[3]) class ProductPSpace(PSpace): """ Abstract class for representing probability spaces with multiple random variables. See Also ======== sympy.stats.rv.IndependentProductPSpace sympy.stats.joint_rv.JointPSpace """ pass class IndependentProductPSpace(ProductPSpace): """ A probability space resulting from the merger of two independent probability spaces. Often created using the function, pspace. """ def __new__(cls, spaces): rs_space_dict = {} for space in spaces: for value in space.values: rs_space_dict[value] = space symbols = FiniteSet([val.symbol for val in rs_space_dict.keys()]) # Overlapping symbols from sympy.stats.joint_rv import MarginalDistribution from sympy.stats.compound_rv import CompoundDistribution if len(symbols) < sum(len(space.symbols) for space in spaces if not isinstance(space.distribution, ( CompoundDistribution, MarginalDistribution))): raise ValueError("Overlapping Random Variables") if all(space.is_Finite for space in spaces): from sympy.stats.frv import ProductFinitePSpace cls = ProductFinitePSpace obj = Basic.__new__(cls, FiniteSet(spaces)) return obj @property def pdf(self): p = Mul([space.pdf for space in self.spaces]) return p.subs({rv: rv.symbol for rv in self.values}) @property def rs_space_dict(self): d = {} for space in self.spaces: for value in space.values: d[value] = space return d @property def symbols(self): return FiniteSet([val.symbol for val in self.rs_space_dict.keys()]) @property def spaces(self): return FiniteSet(self.args) @property def values(self): return sumsets(space.values for space in self.spaces) def compute_expectation(self, expr, rvs=None, evaluate=False, kwargs): rvs = rvs or self.values rvs = frozenset(rvs) for space in self.spaces: expr = space.compute_expectation(expr, rvs & space.values, evaluate=False, kwargs) if evaluate and hasattr(expr, 'doit'): return expr.doit(*kwargs) return expr @property def domain(self): return ProductDomain([space.domain for space in self.spaces]) @property def density(self): raise NotImplementedError("Density not available for ProductSpaces") def sample(self, size=(), library='scipy', seed=None): return {k: v for space in self.spaces for k, v in space.sample(size=size, library=library, seed=seed).items()} def probability(self, condition, *kwargs): cond_inv = False if isinstance(condition, Ne): condition = Eq(condition.args[0], condition.args[1]) cond_inv = True elif isinstance(condition, And): # they are independent return Mul([self.probability(arg) for arg in condition.args]) elif isinstance(condition, Or): # they are independent return Add([self.probability(arg) for arg in condition.args]) expr = condition.lhs - condition.rhs rvs = random_symbols(expr) dens = self.compute_density(expr) if any(pspace(rv).is_Continuous for rv in rvs): from sympy.stats.crv import SingleContinuousPSpace from sympy.stats.crv_types import ContinuousDistributionHandmade if expr in self.values: # Marginalize all other random symbols out of the density randomsymbols = tuple(set(self.values) - frozenset([expr])) symbols = tuple(rs.symbol for rs in randomsymbols) pdf = self.domain.integrate(self.pdf, symbols, kwargs) return Lambda(expr.symbol, pdf) dens = ContinuousDistributionHandmade(dens) z = Dummy('z', real=True) space = SingleContinuousPSpace(z, dens) result = space.probability(condition.__class__(space.value, 0)) else: from sympy.stats.drv import SingleDiscretePSpace from sympy.stats.drv_types import DiscreteDistributionHandmade dens = DiscreteDistributionHandmade(dens) z = Dummy('z', integer=True) space = SingleDiscretePSpace(z, dens) result = space.probability(condition.__class__(space.value, 0)) return result if not cond_inv else S.One - result def compute_density(self, expr, kwargs): rvs = random_symbols(expr) if any(pspace(rv).is_Continuous for rv in rvs): z = Dummy('z', real=True) expr = self.compute_expectation(DiracDelta(expr - z), kwargs) else: z = Dummy('z', integer=True) expr = self.compute_expectation(KroneckerDelta(expr, z), kwargs) return Lambda(z, expr) def compute_cdf(self, expr, kwargs): raise ValueError("CDF not well defined on multivariate expressions") def conditional_space(self, condition, normalize=True, kwargs): rvs = random_symbols(condition) condition = condition.xreplace({rv: rv.symbol for rv in self.values}) pspaces = [pspace(rv) for rv in rvs] if any(ps.is_Continuous for ps in pspaces): from sympy.stats.crv import (ConditionalContinuousDomain, ContinuousPSpace) space = ContinuousPSpace domain = ConditionalContinuousDomain(self.domain, condition) elif any(ps.is_Discrete for ps in pspaces): from sympy.stats.drv import (ConditionalDiscreteDomain, DiscretePSpace) space = DiscretePSpace domain = ConditionalDiscreteDomain(self.domain, condition) elif all(ps.is_Finite for ps in pspaces): from sympy.stats.frv import FinitePSpace return FinitePSpace.conditional_space(self, condition) if normalize: replacement = {rv: Dummy(str(rv)) for rv in self.symbols} norm = domain.compute_expectation(self.pdf, kwargs) pdf = self.pdf / norm.xreplace(replacement) # XXX: Converting symbols from set to tuple. The order matters to # Lambda though so we shouldn't be starting with a set here... density = Lambda(tuple(domain.symbols), pdf) return space(domain, density) class ProductDomain(RandomDomain): """ A domain resulting from the merger of two independent domains. See Also ======== sympy.stats.crv.ProductContinuousDomain sympy.stats.frv.ProductFiniteDomain """ is_ProductDomain = True def __new__(cls, domains): # Flatten any product of products domains2 = [] for domain in domains: if not domain.is_ProductDomain: domains2.append(domain) else: domains2.extend(domain.domains) domains2 = FiniteSet(domains2) if all(domain.is_Finite for domain in domains2): from sympy.stats.frv import ProductFiniteDomain cls = ProductFiniteDomain if all(domain.is_Continuous for domain in domains2): from sympy.stats.crv import ProductContinuousDomain cls = ProductContinuousDomain if all(domain.is_Discrete for domain in domains2): from sympy.stats.drv import ProductDiscreteDomain cls = ProductDiscreteDomain return Basic.__new__(cls, domains2) @property def sym_domain_dict(self): return {symbol: domain for domain in self.domains for symbol in domain.symbols} @property def symbols(self): return FiniteSet([sym for domain in self.domains for sym in domain.symbols]) @property def domains(self): return self.args @property def set(self): return ProductSet((domain.set for domain in self.domains)) def __contains__(self, other): # Split event into each subdomain for domain in self.domains: # Collect the parts of this event which associate to this domain elem = frozenset([item for item in other if sympify(domain.symbols.contains(item[0])) is S.true]) # Test this sub-event if elem not in domain: return False # All subevents passed return True def as_boolean(self): return And([domain.as_boolean() for domain in self.domains]) def random_symbols(expr): """ Returns all RandomSymbols within a SymPy Expression. """ atoms = getattr(expr, 'atoms', None) if atoms is not None: comp = lambda rv: rv.symbol.name l = list(atoms(RandomSymbol)) return sorted(l, key=comp) else: return [] def pspace(expr): """ Returns the underlying Probability Space of a random expression. For internal use. Examples ======== >>> from sympy.stats import pspace, Normal >>> X = Normal('X', 0, 1) >>> pspace(2X + 1) == X.pspace True """ expr = sympify(expr) if isinstance(expr, RandomSymbol) and expr.pspace is not None: return expr.pspace if expr.has(RandomMatrixSymbol): rm = list(expr.atoms(RandomMatrixSymbol))[0] return rm.pspace rvs = random_symbols(expr) if not rvs: raise ValueError("Expression containing Random Variable expected, not %s" % (expr)) # If only one space present if all(rv.pspace == rvs[0].pspace for rv in rvs): return rvs[0].pspace from sympy.stats.compound_rv import CompoundPSpace from sympy.stats.stochastic_process import StochasticPSpace for rv in rvs: if isinstance(rv.pspace, (CompoundPSpace, StochasticPSpace)): return rv.pspace # Otherwise make a product space return IndependentProductPSpace([rv.pspace for rv in rvs]) def sumsets(sets): """ Union of sets """ return frozenset().union(sets) def rs_swap(a, b): """ Build a dictionary to swap RandomSymbols based on their underlying symbol. i.e. if ``X = ('x', pspace1)`` and ``Y = ('x', pspace2)`` then ``X`` and ``Y`` match and the key, value pair ``{X:Y}`` will appear in the result Inputs: collections a and b of random variables which share common symbols Output: dict mapping RVs in a to RVs in b """ d = {} for rsa in a: d[rsa] = [rsb for rsb in b if rsa.symbol == rsb.symbol][0] return d def given(expr, condition=None, *kwargs): r""" Conditional Random Expression. Explanation =========== From a random expression and a condition on that expression creates a new probability space from the condition and returns the same expression on that conditional probability space. Examples ======== >>> from sympy.stats import given, density, Die >>> X = Die('X', 6) >>> Y = given(X, X > 3) >>> density(Y).dict {4: 1/3, 5: 1/3, 6: 1/3} Following convention, if the condition is a random symbol then that symbol is considered fixed. >>> from sympy.stats import Normal >>> from sympy import pprint >>> from sympy.abc import z >>> X = Normal('X', 0, 1) >>> Y = Normal('Y', 0, 1) >>> pprint(density(X + Y, Y)(z), use_unicode=False) 2 -(-Y + z) ----------- ___ 2 \/ 2 e ------------------ ____ 2\/ pi """ if not is_random(condition) or pspace_independent(expr, condition): return expr if isinstance(condition, RandomSymbol): condition = Eq(condition, condition.symbol) condsymbols = random_symbols(condition) if (isinstance(condition, Eq) and len(condsymbols) == 1 and not isinstance(pspace(expr).domain, ConditionalDomain)): rv = tuple(condsymbols)[0] results = solveset(condition, rv) if isinstance(results, Intersection) and S.Reals in results.args: results = list(results.args[1]) sums = 0 for res in results: temp = expr.subs(rv, res) if temp == True: return True if temp != False: # XXX: This seems nonsensical but preserves existing behaviour # after the change that Relational is no longer a subclass of # Expr. Here expr is sometimes Relational and sometimes Expr # but we are trying to add them with +=. This needs to be # fixed somehow. if sums == 0 and isinstance(expr, Relational): sums = expr.subs(rv, res) else: sums += expr.subs(rv, res) if sums == 0: return False return sums # Get full probability space of both the expression and the condition fullspace = pspace(Tuple(expr, condition)) # Build new space given the condition space = fullspace.conditional_space(condition, kwargs) # Dictionary to swap out RandomSymbols in expr with new RandomSymbols # That point to the new conditional space swapdict = rs_swap(fullspace.values, space.values) # Swap random variables in the expression expr = expr.xreplace(swapdict) return expr def expectation(expr, condition=None, numsamples=None, evaluate=True, kwargs): """ Returns the expected value of a random expression. Parameters ========== expr : Expr containing RandomSymbols The expression of which you want to compute the expectation value given : Expr containing RandomSymbols A conditional expression. E(X, X>0) is expectation of X given X > 0 numsamples : int Enables sampling and approximates the expectation with this many samples evalf : Bool (defaults to True) If sampling return a number rather than a complex expression evaluate : Bool (defaults to True) In case of continuous systems return unevaluated integral Examples ======== >>> from sympy.stats import E, Die >>> X = Die('X', 6) >>> E(X) 7/2 >>> E(2X + 1) 8 >>> E(X, X > 3) # Expectation of X given that it is above 3 5 """ if not is_random(expr): # expr isn't random? return expr kwargs['numsamples'] = numsamples from sympy.stats.symbolic_probability import Expectation if evaluate: return Expectation(expr, condition).doit(kwargs) return Expectation(expr, condition) def probability(condition, given_condition=None, numsamples=None, evaluate=True, kwargs): """ Probability that a condition is true, optionally given a second condition. Parameters ========== condition : Combination of Relationals containing RandomSymbols The condition of which you want to compute the probability given_condition : Combination of Relationals containing RandomSymbols A conditional expression. P(X > 1, X > 0) is expectation of X > 1 given X > 0 numsamples : int Enables sampling and approximates the probability with this many samples evaluate : Bool (defaults to True) In case of continuous systems return unevaluated integral Examples ======== >>> from sympy.stats import P, Die >>> from sympy import Eq >>> X, Y = Die('X', 6), Die('Y', 6) >>> P(X > 3) 1/2 >>> P(Eq(X, 5), X > 2) # Probability that X == 5 given that X > 2 1/4 >>> P(X > Y) 5/12 """ kwargs['numsamples'] = numsamples from sympy.stats.symbolic_probability import Probability if evaluate: return Probability(condition, given_condition).doit(kwargs) return Probability(condition, given_condition) class Density(Basic): expr = property(lambda self: self.args[0]) def __new__(cls, expr, condition = None): expr = _sympify(expr) if condition is None: obj = Basic.__new__(cls, expr) else: condition = _sympify(condition) obj = Basic.__new__(cls, expr, condition) return obj @property def condition(self): if len(self.args) > 1: return self.args[1] else: return None def doit(self, evaluate=True, kwargs): from sympy.stats.random_matrix import RandomMatrixPSpace from sympy.stats.joint_rv import JointPSpace from sympy.stats.matrix_distributions import MatrixPSpace from sympy.stats.compound_rv import CompoundPSpace from sympy.stats.frv import SingleFiniteDistribution expr, condition = self.expr, self.condition if isinstance(expr, SingleFiniteDistribution): return expr.dict if condition is not None: # Recompute on new conditional expr expr = given(expr, condition, kwargs) if not random_symbols(expr): return Lambda(x, DiracDelta(x - expr)) if isinstance(expr, RandomSymbol): if isinstance(expr.pspace, (SinglePSpace, JointPSpace, MatrixPSpace)) and \ hasattr(expr.pspace, 'distribution'): return expr.pspace.distribution elif isinstance(expr.pspace, RandomMatrixPSpace): return expr.pspace.model if isinstance(pspace(expr), CompoundPSpace): kwargs['compound_evaluate'] = evaluate result = pspace(expr).compute_density(expr, kwargs) if evaluate and hasattr(result, 'doit'): return result.doit() else: return result def density(expr, condition=None, evaluate=True, numsamples=None, *kwargs): """ Probability density of a random expression, optionally given a second condition. Explanation =========== This density will take on different forms for different types of probability spaces. Discrete variables produce Dicts. Continuous variables produce Lambdas. Parameters ========== expr : Expr containing RandomSymbols The expression of which you want to compute the density value condition : Relational containing RandomSymbols A conditional expression. density(X > 1, X > 0) is density of X > 1 given X > 0 numsamples : int Enables sampling and approximates the density with this many samples Examples ======== >>> from sympy.stats import density, Die, Normal >>> from sympy import Symbol >>> x = Symbol('x') >>> D = Die('D', 6) >>> X = Normal(x, 0, 1) >>> density(D).dict {1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6} >>> density(2D).dict {2: 1/6, 4: 1/6, 6: 1/6, 8: 1/6, 10: 1/6, 12: 1/6} >>> density(X)(x) sqrt(2)exp(-x2/2)/(2sqrt(pi)) """ if numsamples: return sampling_density(expr, condition, numsamples=numsamples, kwargs) return Density(expr, condition).doit(evaluate=evaluate, kwargs) def cdf(expr, condition=None, evaluate=True, *kwargs): """ Cumulative Distribution Function of a random expression. optionally given a second condition. Explanation =========== This density will take on different forms for different types of probability spaces. Discrete variables produce Dicts. Continuous variables produce Lambdas. Examples ======== >>> from sympy.stats import density, Die, Normal, cdf >>> D = Die('D', 6) >>> X = Normal('X', 0, 1) >>> density(D).dict {1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6} >>> cdf(D) {1: 1/6, 2: 1/3, 3: 1/2, 4: 2/3, 5: 5/6, 6: 1} >>> cdf(3D, D > 2) {9: 1/4, 12: 1/2, 15: 3/4, 18: 1} >>> cdf(X) Lambda(_z, erf(sqrt(2)_z/2)/2 + 1/2) """ if condition is not None: # If there is a condition # Recompute on new conditional expr return cdf(given(expr, condition, kwargs), kwargs) # Otherwise pass work off to the ProbabilitySpace result = pspace(expr).compute_cdf(expr, kwargs) if evaluate and hasattr(result, 'doit'): return result.doit() else: return result def characteristic_function(expr, condition=None, evaluate=True, kwargs): """ Characteristic function of a random expression, optionally given a second condition. Returns a Lambda. Examples ======== >>> from sympy.stats import Normal, DiscreteUniform, Poisson, characteristic_function >>> X = Normal('X', 0, 1) >>> characteristic_function(X) Lambda(_t, exp(-_t2/2)) >>> Y = DiscreteUniform('Y', [1, 2, 7]) >>> characteristic_function(Y) Lambda(_t, exp(7_tI)/3 + exp(2_tI)/3 + exp(_tI)/3) >>> Z = Poisson('Z', 2) >>> characteristic_function(Z) Lambda(_t, exp(2exp(_tI) - 2)) """ if condition is not None: return characteristic_function(given(expr, condition, kwargs), kwargs) result = pspace(expr).compute_characteristic_function(expr, kwargs) if evaluate and hasattr(result, 'doit'): return result.doit() else: return result def moment_generating_function(expr, condition=None, evaluate=True, kwargs): if condition is not None: return moment_generating_function(given(expr, condition, kwargs), kwargs) result = pspace(expr).compute_moment_generating_function(expr, kwargs) if evaluate and hasattr(result, 'doit'): return result.doit() else: return result def where(condition, given_condition=None, kwargs): """ Returns the domain where a condition is True. Examples ======== >>> from sympy.stats import where, Die, Normal >>> from sympy import And >>> D1, D2 = Die('a', 6), Die('b', 6) >>> a, b = D1.symbol, D2.symbol >>> X = Normal('x', 0, 1) >>> where(X2<1) Domain: (-1 < x) & (x < 1) >>> where(X2<1).set Interval.open(-1, 1) >>> where(And(D1<=D2, D2<3)) Domain: (Eq(a, 1) & Eq(b, 1)) \| (Eq(a, 1) & Eq(b, 2)) \| (Eq(a, 2) & Eq(b, 2)) """ if given_condition is not None: # If there is a condition # Recompute on new conditional expr return where(given(condition, given_condition, kwargs), kwargs) # Otherwise pass work off to the ProbabilitySpace return pspace(condition).where(condition, kwargs) @doctest_depends_on(modules=('scipy',)) def sample(expr, condition=None, size=(), library='scipy', numsamples=1, seed=None, kwargs): """ A realization of the random expression. Parameters ========== expr : Expression of random variables Expression from which sample is extracted condition : Expr containing RandomSymbols A conditional expression size : int, tuple Represents size of each sample in numsamples library : str - 'scipy' : Sample using scipy - 'numpy' : Sample using numpy - 'pymc3' : Sample using PyMC3 Choose any of the available options to sample from as string, by default is 'scipy' numsamples : int Number of samples, each with size as ``size``. .. deprecated:: 1.9 The ``numsamples`` parameter is deprecated and is only provided for compatibility with v1.8. Use a list comprehension or an additional dimension in ``size`` instead. See :ref:`deprecated-sympy-stats-numsamples` for details. seed : An object to be used as seed by the given external library for sampling `expr`. Following is the list of possible types of object for the supported libraries, - 'scipy': int, numpy.random.RandomState, numpy.random.Generator - 'numpy': int, numpy.random.RandomState, numpy.random.Generator - 'pymc3': int Optional, by default None, in which case seed settings related to the given library will be used. No modifications to environment's global seed settings are done by this argument. Returns ======= sample: float/list/numpy.ndarray one sample or a collection of samples of the random expression. - sample(X) returns float/numpy.float64/numpy.int64 object. - sample(X, size=int/tuple) returns numpy.ndarray object. Examples ======== >>> from sympy.stats import Die, sample, Normal, Geometric >>> X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6) # Finite Random Variable >>> die_roll = sample(X + Y + Z) >>> die_roll # doctest: +SKIP 3 >>> N = Normal('N', 3, 4) # Continuous Random Variable >>> samp = sample(N) >>> samp in N.pspace.domain.set True >>> samp = sample(N, N>0) >>> samp > 0 True >>> samp_list = sample(N, size=4) >>> [sam in N.pspace.domain.set for sam in samp_list] [True, True, True, True] >>> sample(N, size = (2,3)) # doctest: +SKIP array([[5.42519758, 6.40207856, 4.94991743], [1.85819627, 6.83403519, 1.9412172 ]]) >>> G = Geometric('G', 0.5) # Discrete Random Variable >>> samp_list = sample(G, size=3) >>> samp_list # doctest: +SKIP [1, 3, 2] >>> [sam in G.pspace.domain.set for sam in samp_list] [True, True, True] >>> MN = Normal("MN", [3, 4], [[2, 1], [1, 2]]) # Joint Random Variable >>> samp_list = sample(MN, size=4) >>> samp_list # doctest: +SKIP [array([2.85768055, 3.38954165]), array([4.11163337, 4.3176591 ]), array([0.79115232, 1.63232916]), array([4.01747268, 3.96716083])] >>> [tuple(sam) in MN.pspace.domain.set for sam in samp_list] [True, True, True, True] .. versionchanged:: 1.7.0 sample used to return an iterator containing the samples instead of value. .. versionchanged:: 1.9.0 sample returns values or array of values instead of an iterator and numsamples is deprecated. """ iterator = sample_iter(expr, condition, size=size, library=library, numsamples=numsamples, seed=seed) if numsamples != 1: sympy_deprecation_warning( f""" The numsamples parameter to sympy.stats.sample() is deprecated. Either use a list comprehension, like [sample(...) for i in range({numsamples})] or add a dimension to size, like sample(..., size={(numsamples,) + size}) """, deprecated_since_version="1.9", active_deprecations_target="deprecated-sympy-stats-numsamples", ) return [next(iterator) for i in range(numsamples)] return next(iterator) def quantile(expr, evaluate=True, kwargs): r""" Return the :math:`p^{th}` order quantile of a probability distribution. Explanation =========== Quantile is defined as the value at which the probability of the random variable is less than or equal to the given probability. .. math:: Q(p) = \inf\{x \in (-\infty, \infty) : p \le F(x)\} Examples ======== >>> from sympy.stats import quantile, Die, Exponential >>> from sympy import Symbol, pprint >>> p = Symbol("p") >>> l = Symbol("lambda", positive=True) >>> X = Exponential("x", l) >>> quantile(X)(p) -log(1 - p)/lambda >>> D = Die("d", 6) >>> pprint(quantile(D)(p), use_unicode=False) /nan for Or(p > 1, p < 0) \| \| 1 for p <= 1/6 \| \| 2 for p <= 1/3 \| < 3 for p <= 1/2 \| \| 4 for p <= 2/3 \| \| 5 for p <= 5/6 \| \ 6 for p <= 1 """ result = pspace(expr).compute_quantile(expr, kwargs) if evaluate and hasattr(result, 'doit'): return result.doit() else: return result def sample_iter(expr, condition=None, size=(), library='scipy', numsamples=S.Infinity, seed=None, *kwargs): """ Returns an iterator of realizations from the expression given a condition. Parameters ========== expr: Expr Random expression to be realized condition: Expr, optional A conditional expression size : int, tuple Represents size of each sample in numsamples numsamples: integer, optional Length of the iterator (defaults to infinity) seed : An object to be used as seed by the given external library for sampling `expr`. Following is the list of possible types of object for the supported libraries, - 'scipy': int, numpy.random.RandomState, numpy.random.Generator - 'numpy': int, numpy.random.RandomState, numpy.random.Generator - 'pymc3': int Optional, by default None, in which case seed settings related to the given library will be used. No modifications to environment's global seed settings are done by this argument. Examples ======== >>> from sympy.stats import Normal, sample_iter >>> X = Normal('X', 0, 1) >>> expr = XX + 3 >>> iterator = sample_iter(expr, numsamples=3) # doctest: +SKIP >>> list(iterator) # doctest: +SKIP [12, 4, 7] Returns ======= sample_iter: iterator object iterator object containing the sample/samples of given expr See Also ======== sample sampling_P sampling_E """ from sympy.stats.joint_rv import JointRandomSymbol if not import_module(library): raise ValueError("Failed to import %s" % library) if condition is not None: ps = pspace(Tuple(expr, condition)) else: ps = pspace(expr) rvs = list(ps.values) if isinstance(expr, JointRandomSymbol): expr = expr.subs({expr: RandomSymbol(expr.symbol, expr.pspace)}) else: sub = {} for arg in expr.args: if isinstance(arg, JointRandomSymbol): sub[arg] = RandomSymbol(arg.symbol, arg.pspace) expr = expr.subs(sub) def fn_subs(args): return expr.subs({rv: arg for rv, arg in zip(rvs, args)}) def given_fn_subs(args): if condition is not None: return condition.subs({rv: arg for rv, arg in zip(rvs, args)}) return False if library == 'pymc3': # Currently unable to lambdify in pymc3 # TODO : Remove 'pymc3' when lambdify accepts 'pymc3' as module fn = lambdify(rvs, expr, kwargs) else: fn = lambdify(rvs, expr, modules=library, kwargs) if condition is not None: given_fn = lambdify(rvs, condition, *kwargs) def return_generator_infinite(): count = 0 _size = (1,)+((size,) if isinstance(size, int) else size) while count < numsamples: d = ps.sample(size=_size, library=library, seed=seed) # a dictionary that maps RVs to values args = [d[rv][0] for rv in rvs] if condition is not None: # Check that these values satisfy the condition # TODO: Replace the try-except block with only given_fn(args) # once lambdify works with unevaluated SymPy objects. try: gd = given_fn(args) except (NameError, TypeError): gd = given_fn_subs(args) if gd != True and gd != False: raise ValueError( "Conditions must not contain free symbols") if not gd: # If the values don't satisfy then try again continue yield fn(args) count += 1 def return_generator_finite(): faulty = True while faulty: d = ps.sample(size=(numsamples,) + ((size,) if isinstance(size, int) else size), library=library, seed=seed) # a dictionary that maps RVs to values faulty = False count = 0 while count < numsamples and not faulty: args = [d[rv][count] for rv in rvs] if condition is not None: # Check that these values satisfy the condition # TODO: Replace the try-except block with only given_fn(args) # once lambdify works with unevaluated SymPy objects. try: gd = given_fn(args) except (NameError, TypeError): gd = given_fn_subs(args) if gd != True and gd != False: raise ValueError( "Conditions must not contain free symbols") if not gd: # If the values don't satisfy then try again faulty = True count += 1 count = 0 while count < numsamples: args = [d[rv][count] for rv in rvs] # TODO: Replace the try-except block with only fn(args) # once lambdify works with unevaluated SymPy objects. try: yield fn(args) except (NameError, TypeError): yield fn_subs(args) count += 1 if numsamples is S.Infinity: return return_generator_infinite() return return_generator_finite() def sample_iter_lambdify(expr, condition=None, size=(), numsamples=S.Infinity, seed=None, kwargs): return sample_iter(expr, condition=condition, size=size, numsamples=numsamples, seed=seed, kwargs) def sample_iter_subs(expr, condition=None, size=(), numsamples=S.Infinity, seed=None, kwargs): return sample_iter(expr, condition=condition, size=size, numsamples=numsamples, seed=seed, kwargs) def sampling_P(condition, given_condition=None, library='scipy', numsamples=1, evalf=True, seed=None, kwargs): """ Sampling version of P. See Also ======== P sampling_E sampling_density """ count_true = 0 count_false = 0 samples = sample_iter(condition, given_condition, library=library, numsamples=numsamples, seed=seed, kwargs) for sample in samples: if sample: count_true += 1 else: count_false += 1 result = S(count_true) / numsamples if evalf: return result.evalf() else: return result def sampling_E(expr, given_condition=None, library='scipy', numsamples=1, evalf=True, seed=None, kwargs): """ Sampling version of E. See Also ======== P sampling_P sampling_density """ samples = list(sample_iter(expr, given_condition, library=library, numsamples=numsamples, seed=seed, kwargs)) result = Add([samp for samp in samples]) / numsamples if evalf: return result.evalf() else: return result def sampling_density(expr, given_condition=None, library='scipy', numsamples=1, seed=None, kwargs): """ Sampling version of density. See Also ======== density sampling_P sampling_E """ results = {} for result in sample_iter(expr, given_condition, library=library, numsamples=numsamples, seed=seed, kwargs): results[result] = results.get(result, 0) + 1 return results def dependent(a, b): """ Dependence of two random expressions. Two expressions are independent if knowledge of one does not change computations on the other. Examples ======== >>> from sympy.stats import Normal, dependent, given >>> from sympy import Tuple, Eq >>> X, Y = Normal('X', 0, 1), Normal('Y', 0, 1) >>> dependent(X, Y) False >>> dependent(2X + Y, -Y) True >>> X, Y = given(Tuple(X, Y), Eq(X + Y, 3)) >>> dependent(X, Y) True See Also ======== independent """ if pspace_independent(a, b): return False z = Symbol('z', real=True) # Dependent if density is unchanged when one is given information about # the other return (density(a, Eq(b, z)) != density(a) or density(b, Eq(a, z)) != density(b)) def independent(a, b): """ Independence of two random expressions. Two expressions are independent if knowledge of one does not change computations on the other. Examples ======== >>> from sympy.stats import Normal, independent, given >>> from sympy import Tuple, Eq >>> X, Y = Normal('X', 0, 1), Normal('Y', 0, 1) >>> independent(X, Y) True >>> independent(2X + Y, -Y) False >>> X, Y = given(Tuple(X, Y), Eq(X + Y, 3)) >>> independent(X, Y) False See Also ======== dependent """ return not dependent(a, b) def pspace_independent(a, b): """ Tests for independence between a and b by checking if their PSpaces have overlapping symbols. This is a sufficient but not necessary condition for independence and is intended to be used internally. Notes ===== pspace_independent(a, b) implies independent(a, b) independent(a, b) does not imply pspace_independent(a, b) """ a_symbols = set(pspace(b).symbols) b_symbols = set(pspace(a).symbols) if len(set(random_symbols(a)).intersection(random_symbols(b))) != 0: return False if len(a_symbols.intersection(b_symbols)) == 0: return True return None def rv_subs(expr, symbols=None): """ Given a random expression replace all random variables with their symbols. If symbols keyword is given restrict the swap to only the symbols listed. """ if symbols is None: symbols = random_symbols(expr) if not symbols: return expr swapdict = {rv: rv.symbol for rv in symbols} return expr.subs(swapdict) class NamedArgsMixin: _argnames = () # type: tTuple[str, ...] def __getattr__(self, attr): try: return self.args[self._argnames.index(attr)] except ValueError: raise AttributeError("'%s' object has no attribute '%s'" % ( type(self).__name__, attr)) class Distribution(Basic): def sample(self, size=(), library='scipy', seed=None): """ A random realization from the distribution """ module = import_module(library) if library in {'scipy', 'numpy', 'pymc3'} and module is None: raise ValueError("Failed to import %s" % library) if library == 'scipy': # scipy does not require map as it can handle using custom distributions. # However, we will still use a map where we can. # TODO: do this for drv.py and frv.py if necessary. # TODO: add more distributions here if there are more # See links below referring to sections beginning with "A common parametrization..." # I will remove all these comments if everything is ok. from sympy.stats.sampling.sample_scipy import do_sample_scipy import numpy if seed is None or isinstance(seed, int): rand_state = numpy.random.default_rng(seed=seed) else: rand_state = seed samps = do_sample_scipy(self, size, rand_state) elif library == 'numpy': from sympy.stats.sampling.sample_numpy import do_sample_numpy import numpy if seed is None or isinstance(seed, int): rand_state = numpy.random.default_rng(seed=seed) else: rand_state = seed _size = None if size == () else size samps = do_sample_numpy(self, _size, rand_state) elif library == 'pymc3': from sympy.stats.sampling.sample_pymc3 import do_sample_pymc3 import logging logging.getLogger("pymc3").setLevel(logging.ERROR) import pymc3 with pymc3.Model(): if do_sample_pymc3(self): samps = pymc3.sample(draws=prod(size), chains=1, compute_convergence_checks=False, progressbar=False, random_seed=seed, return_inferencedata=False)[:]['X'] samps = samps.reshape(size) else: samps = None else: raise NotImplementedError("Sampling from %s is not supported yet." % str(library)) if samps is not None: return samps raise NotImplementedError( "Sampling for %s is not currently implemented from %s" % (self, library)) def _value_check(condition, message): """ Raise a ValueError with message if condition is False, else return True if all conditions were True, else False. Examples ======== >>> from sympy.stats.rv import _value_check >>> from sympy.abc import a, b, c >>> from sympy import And, Dummy >>> _value_check(2 < 3, '') True Here, the condition is not False, but it does not evaluate to True so False is returned (but no error is raised). So checking if the return value is True or False will tell you if all conditions were evaluated. >>> _value_check(a < b, '') False In this case the condition is False so an error is raised: >>> r = Dummy(real=True) >>> _value_check(r < r - 1, 'condition is not true') Traceback (most recent call last): ... ValueError: condition is not true If no condition of many conditions must be False, they can be checked by passing them as an iterable: >>> _value_check((a < 0, b < 0, c < 0), '') False The iterable can be a generator, too: >>> _value_check((i < 0 for i in (a, b, c)), '') False The following are equivalent to the above but do not pass an iterable: >>> all(_value_check(i < 0, '') for i in (a, b, c)) False >>> _value_check(And(a < 0, b < 0, c < 0), '') False """ if not iterable(condition): condition = [condition] truth = fuzzy_and(condition) if truth == False: raise ValueError(message) return truth == True def _symbol_converter(sym): """ Casts the parameter to Symbol if it is 'str' otherwise no operation is performed on it. Parameters ========== sym The parameter to be converted. Returns ======= Symbol the parameter converted to Symbol. Raises ====== TypeError If the parameter is not an instance of both str and Symbol. Examples ======== >>> from sympy import Symbol >>> from sympy.stats.rv import _symbol_converter >>> s = _symbol_converter('s') >>> isinstance(s, Symbol) True >>> _symbol_converter(1) Traceback (most recent call last): ... TypeError: 1 is neither a Symbol nor a string >>> r = Symbol('r') >>> isinstance(r, Symbol) True """ if isinstance(sym, str): sym = Symbol(sym) if not isinstance(sym, Symbol): raise TypeError("%s is neither a Symbol nor a string"%(sym)) return sym def sample_stochastic_process(process): """ This function is used to sample from stochastic process. Parameters ========== process: StochasticProcess Process used to extract the samples. It must be an instance of StochasticProcess Examples ======== >>> from sympy.stats import sample_stochastic_process, DiscreteMarkovChain >>> from sympy import Matrix >>> T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]]) >>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T) >>> next(sample_stochastic_process(Y)) in Y.state_space # doctest: +SKIP True >>> next(sample_stochastic_process(Y)) # doctest: +SKIP 0 >>> next(sample_stochastic_process(Y)) # doctest: +SKIP 2 Returns ======= sample: iterator object iterator object containing the sample of given process """ from sympy.stats.stochastic_process_types import StochasticProcess if not isinstance(process, StochasticProcess): raise ValueError("Process must be an instance of Stochastic Process") return process.sample()
32e2cdb7ff64f0768996b7ec3010a03efc38a7ff200ee8772f33fb26ae884e7c	""" Generating and counting primes. """ import random from bisect import bisect from itertools import count # Using arrays for sieving instead of lists greatly reduces # memory consumption from array import array as _array from sympy.core.function import Function from sympy.core.singleton import S from .primetest import isprime from sympy.utilities.misc import as_int def _azeros(n): return _array('l', [0]n) def _aset(v): return _array('l', v) def _arange(a, b): return _array('l', range(a, b)) def _as_int_ceiling(a): """ Wrapping ceiling in as_int will raise an error if there was a problem determining whether the expression was exactly an integer or not.""" from sympy.functions.elementary.integers import ceiling return as_int(ceiling(a)) class Sieve: """An infinite list of prime numbers, implemented as a dynamically growing sieve of Eratosthenes. When a lookup is requested involving an odd number that has not been sieved, the sieve is automatically extended up to that number. Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> 25 in sieve False >>> sieve._list array('l', [2, 3, 5, 7, 11, 13, 17, 19, 23]) """ # data shared (and updated) by all Sieve instances def __init__(self): self._n = 6 self._list = _aset(2, 3, 5, 7, 11, 13) # primes self._tlist = _aset(0, 1, 1, 2, 2, 4) # totient self._mlist = _aset(0, 1, -1, -1, 0, -1) # mobius assert all(len(i) == self._n for i in (self._list, self._tlist, self._mlist)) def __repr__(self): return ("<%s sieve (%i): %i, %i, %i, ... %i, %i\n" "%s sieve (%i): %i, %i, %i, ... %i, %i\n" "%s sieve (%i): %i, %i, %i, ... %i, %i>") % ( 'prime', len(self._list), self._list[0], self._list[1], self._list[2], self._list[-2], self._list[-1], 'totient', len(self._tlist), self._tlist[0], self._tlist[1], self._tlist[2], self._tlist[-2], self._tlist[-1], 'mobius', len(self._mlist), self._mlist[0], self._mlist[1], self._mlist[2], self._mlist[-2], self._mlist[-1]) def _reset(self, prime=None, totient=None, mobius=None): """Reset all caches (default). To reset one or more set the desired keyword to True.""" if all(i is None for i in (prime, totient, mobius)): prime = totient = mobius = True if prime: self._list = self._list[:self._n] if totient: self._tlist = self._tlist[:self._n] if mobius: self._mlist = self._mlist[:self._n] def extend(self, n): """Grow the sieve to cover all primes <= n (a real number). Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend(30) >>> sieve[10] == 29 True """ n = int(n) if n <= self._list[-1]: return # We need to sieve against all bases up to sqrt(n). # This is a recursive call that will do nothing if there are enough # known bases already. maxbase = int(n*0.5) + 1 self.extend(maxbase) # Create a new sieve starting from sqrt(n) begin = self._list[-1] + 1 newsieve = _arange(begin, n + 1) # Now eliminate all multiples of primes in [2, sqrt(n)] for p in self.primerange(maxbase): # Start counting at a multiple of p, offsetting # the index to account for the new sieve's base index startindex = (-begin) % p for i in range(startindex, len(newsieve), p): newsieve[i] = 0 # Merge the sieves self._list += _array('l', [x for x in newsieve if x]) def extend_to_no(self, i): """Extend to include the ith prime number. Parameters ========== i : integer Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend_to_no(9) >>> sieve._list array('l', [2, 3, 5, 7, 11, 13, 17, 19, 23]) Notes ===== The list is extended by 50% if it is too short, so it is likely that it will be longer than requested. """ i = as_int(i) while len(self._list) < i: self.extend(int(self._list[-1] 1.5)) def primerange(self, a, b=None): """Generate all prime numbers in the range [2, a) or [a, b). Examples ======== >>> from sympy import sieve, prime All primes less than 19: >>> print([i for i in sieve.primerange(19)]) [2, 3, 5, 7, 11, 13, 17] All primes greater than or equal to 7 and less than 19: >>> print([i for i in sieve.primerange(7, 19)]) [7, 11, 13, 17] All primes through the 10th prime >>> list(sieve.primerange(prime(10) + 1)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] """ if b is None: b = _as_int_ceiling(a) a = 2 else: a = max(2, _as_int_ceiling(a)) b = _as_int_ceiling(b) if a >= b: return self.extend(b) i = self.search(a)[1] maxi = len(self._list) + 1 while i < maxi: p = self._list[i - 1] if p < b: yield p i += 1 else: return def totientrange(self, a, b): """Generate all totient numbers for the range [a, b). Examples ======== >>> from sympy import sieve >>> print([i for i in sieve.totientrange(7, 18)]) [6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16] """ a = max(1, _as_int_ceiling(a)) b = _as_int_ceiling(b) n = len(self._tlist) if a >= b: return elif b <= n: for i in range(a, b): yield self._tlist[i] else: self._tlist += _arange(n, b) for i in range(1, n): ti = self._tlist[i] startindex = (n + i - 1) // i * i for j in range(startindex, b, i): self._tlist[j] -= ti if i >= a: yield ti for i in range(n, b): ti = self._tlist[i] for j in range(2 * i, b, i): self._tlist[j] -= ti if i >= a: yield ti def mobiusrange(self, a, b): """Generate all mobius numbers for the range [a, b). Parameters ========== a : integer First number in range b : integer First number outside of range Examples ======== >>> from sympy import sieve >>> print([i for i in sieve.mobiusrange(7, 18)]) [-1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1] """ a = max(1, _as_int_ceiling(a)) b = _as_int_ceiling(b) n = len(self._mlist) if a >= b: return elif b <= n: for i in range(a, b): yield self._mlist[i] else: self._mlist += _azeros(b - n) for i in range(1, n): mi = self._mlist[i] startindex = (n + i - 1) // i * i for j in range(startindex, b, i): self._mlist[j] -= mi if i >= a: yield mi for i in range(n, b): mi = self._mlist[i] for j in range(2 * i, b, i): self._mlist[j] -= mi if i >= a: yield mi def search(self, n): """Return the indices i, j of the primes that bound n. If n is prime then i == j. Although n can be an expression, if ceiling cannot convert it to an integer then an n error will be raised. Examples ======== >>> from sympy import sieve >>> sieve.search(25) (9, 10) >>> sieve.search(23) (9, 9) """ test = _as_int_ceiling(n) n = as_int(n) if n < 2: raise ValueError("n should be >= 2 but got: %s" % n) if n > self._list[-1]: self.extend(n) b = bisect(self._list, n) if self._list[b - 1] == test: return b, b else: return b, b + 1 def __contains__(self, n): try: n = as_int(n) assert n >= 2 except (ValueError, AssertionError): return False if n % 2 == 0: return n == 2 a, b = self.search(n) return a == b def __iter__(self): for n in count(1): yield self[n] def __getitem__(self, n): """Return the nth prime number""" if isinstance(n, slice): self.extend_to_no(n.stop) # Python 2.7 slices have 0 instead of None for start, so # we can't default to 1. start = n.start if n.start is not None else 0 if start < 1: # sieve[:5] would be empty (starting at -1), let's # just be explicit and raise. raise IndexError("Sieve indices start at 1.") return self._list[start - 1:n.stop - 1:n.step] else: if n < 1: # offset is one, so forbid explicit access to sieve[0] # (would surprisingly return the last one). raise IndexError("Sieve indices start at 1.") n = as_int(n) self.extend_to_no(n) return self._list[n - 1] # Generate a global object for repeated use in trial division etc sieve = Sieve() def prime(nth): r""" Return the nth prime, with the primes indexed as prime(1) = 2, prime(2) = 3, etc.... The nth prime is approximately $n\log(n)$. Logarithmic integral of $x$ is a pretty nice approximation for number of primes $\le x$, i.e. li(x) ~ pi(x) In fact, for the numbers we are concerned about( x<1e11 ), li(x) - pi(x) < 50000 Also, li(x) > pi(x) can be safely assumed for the numbers which can be evaluated by this function. Here, we find the least integer m such that li(m) > n using binary search. Now pi(m-1) < li(m-1) <= n, We find pi(m - 1) using primepi function. Starting from m, we have to find n - pi(m-1) more primes. For the inputs this implementation can handle, we will have to test primality for at max about 10*5 numbers, to get our answer. Examples ======== >>> from sympy import prime >>> prime(10) 29 >>> prime(1) 2 >>> prime(100000) 1299709 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range primepi : Return the number of primes less than or equal to n References ========== .. [1] https://en.wikipedia.org/wiki/Prime_number_theorem#Table_of_.CF.80.28x.29.2C_x_.2F_log_x.2C_and_li.28x.29 .. [2] https://en.wikipedia.org/wiki/Prime_number_theorem#Approximations_for_the_nth_prime_number .. [3] https://en.wikipedia.org/wiki/Skewes%27_number """ n = as_int(nth) if n < 1: raise ValueError("nth must be a positive integer; prime(1) == 2") if n <= len(sieve._list): return sieve[n] from sympy.functions.elementary.exponential import log from sympy.functions.special.error_functions import li a = 2 # Lower bound for binary search b = int(n(log(n) + log(log(n)))) # Upper bound for the search. while a < b: mid = (a + b) >> 1 if li(mid) > n: b = mid else: a = mid + 1 n_primes = primepi(a - 1) while n_primes < n: if isprime(a): n_primes += 1 a += 1 return a - 1 class primepi(Function): r""" Represents the prime counting function pi(n) = the number of prime numbers less than or equal to n. Algorithm Description: In sieve method, we remove all multiples of prime p except p itself. Let phi(i,j) be the number of integers 2 <= k <= i which remain after sieving from primes less than or equal to j. Clearly, pi(n) = phi(n, sqrt(n)) If j is not a prime, phi(i,j) = phi(i, j - 1) if j is a prime, We remove all numbers(except j) whose smallest prime factor is j. Let $x= j \times a$ be such a number, where $2 \le a \le i / j$ Now, after sieving from primes $\le j - 1$, a must remain (because x, and hence a has no prime factor $\le j - 1$) Clearly, there are phi(i / j, j - 1) such a which remain on sieving from primes $\le j - 1$ Now, if a is a prime less than equal to j - 1, $x= j \times a$ has smallest prime factor = a, and has already been removed(by sieving from a). So, we do not need to remove it again. (Note: there will be pi(j - 1) such x) Thus, number of x, that will be removed are: phi(i / j, j - 1) - phi(j - 1, j - 1) (Note that pi(j - 1) = phi(j - 1, j - 1)) $\Rightarrow$ phi(i,j) = phi(i, j - 1) - phi(i / j, j - 1) + phi(j - 1, j - 1) So,following recursion is used and implemented as dp: phi(a, b) = phi(a, b - 1), if b is not a prime phi(a, b) = phi(a, b-1)-phi(a / b, b-1) + phi(b-1, b-1), if b is prime Clearly a is always of the form floor(n / k), which can take at most $2\sqrt{n}$ values. Two arrays arr1,arr2 are maintained arr1[i] = phi(i, j), arr2[i] = phi(n // i, j) Finally the answer is arr2[1] Examples ======== >>> from sympy import primepi, prime, prevprime, isprime >>> primepi(25) 9 So there are 9 primes less than or equal to 25. Is 25 prime? >>> isprime(25) False It is not. So the first prime less than 25 must be the 9th prime: >>> prevprime(25) == prime(9) True See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range prime : Return the nth prime """ @classmethod def eval(cls, n): if n is S.Infinity: return S.Infinity if n is S.NegativeInfinity: return S.Zero try: n = int(n) except TypeError: if n.is_real == False or n is S.NaN: raise ValueError("n must be real") return if n < 2: return S.Zero if n <= sieve._list[-1]: return S(sieve.search(n)[0]) lim = int(n ** 0.5) lim -= 1 lim = max(lim, 0) while lim * lim <= n: lim += 1 lim -= 1 arr1 = [0] * (lim + 1) arr2 = [0] * (lim + 1) for i in range(1, lim + 1): arr1[i] = i - 1 arr2[i] = n // i - 1 for i in range(2, lim + 1): # Presently, arr1[k]=phi(k,i - 1), # arr2[k] = phi(n // k,i - 1) if arr1[i] == arr1[i - 1]: continue p = arr1[i - 1] for j in range(1, min(n // (i * i), lim) + 1): st = i * j if st <= lim: arr2[j] -= arr2[st] - p else: arr2[j] -= arr1[n // st] - p lim2 = min(lim, i * i - 1) for j in range(lim, lim2, -1): arr1[j] -= arr1[j // i] - p return S(arr2[1]) def nextprime(n, ith=1): """ Return the ith prime greater than n. i must be an integer. Notes ===== Potential primes are located at 6j +/- 1. This property is used during searching. >>> from sympy import nextprime >>> [(i, nextprime(i)) for i in range(10, 15)] [(10, 11), (11, 13), (12, 13), (13, 17), (14, 17)] >>> nextprime(2, ith=2) # the 2nd prime after 2 5 See Also ======== prevprime : Return the largest prime smaller than n primerange : Generate all primes in a given range """ n = int(n) i = as_int(ith) if i > 1: pr = n j = 1 while 1: pr = nextprime(pr) j += 1 if j > i: break return pr if n < 2: return 2 if n < 7: return {2: 3, 3: 5, 4: 5, 5: 7, 6: 7}[n] if n <= sieve._list[-2]: l, u = sieve.search(n) if l == u: return sieve[u + 1] else: return sieve[u] nn = 6(n//6) if nn == n: n += 1 if isprime(n): return n n += 4 elif n - nn == 5: n += 2 if isprime(n): return n n += 4 else: n = nn + 5 while 1: if isprime(n): return n n += 2 if isprime(n): return n n += 4 def prevprime(n): """ Return the largest prime smaller than n. Notes ===== Potential primes are located at 6j +/- 1. This property is used during searching. >>> from sympy import prevprime >>> [(i, prevprime(i)) for i in range(10, 15)] [(10, 7), (11, 7), (12, 11), (13, 11), (14, 13)] See Also ======== nextprime : Return the ith prime greater than n primerange : Generates all primes in a given range """ n = _as_int_ceiling(n) if n < 3: raise ValueError("no preceding primes") if n < 8: return {3: 2, 4: 3, 5: 3, 6: 5, 7: 5}[n] if n <= sieve._list[-1]: l, u = sieve.search(n) if l == u: return sieve[l-1] else: return sieve[l] nn = 6(n//6) if n - nn <= 1: n = nn - 1 if isprime(n): return n n -= 4 else: n = nn + 1 while 1: if isprime(n): return n n -= 2 if isprime(n): return n n -= 4 def primerange(a, b=None): """ Generate a list of all prime numbers in the range [2, a), or [a, b). If the range exists in the default sieve, the values will be returned from there; otherwise values will be returned but will not modify the sieve. Examples ======== >>> from sympy import primerange, prime All primes less than 19: >>> list(primerange(19)) [2, 3, 5, 7, 11, 13, 17] All primes greater than or equal to 7 and less than 19: >>> list(primerange(7, 19)) [7, 11, 13, 17] All primes through the 10th prime >>> list(primerange(prime(10) + 1)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] The Sieve method, primerange, is generally faster but it will occupy more memory as the sieve stores values. The default instance of Sieve, named sieve, can be used: >>> from sympy import sieve >>> list(sieve.primerange(1, 30)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] Notes ===== Some famous conjectures about the occurrence of primes in a given range are [1]: - Twin primes: though often not, the following will give 2 primes an infinite number of times: primerange(6n - 1, 6n + 2) - Legendre's: the following always yields at least one prime primerange(n2, (n+1)2+1) - Bertrand's (proven): there is always a prime in the range primerange(n, 2n) - Brocard's: there are at least four primes in the range primerange(prime(n)2, prime(n+1)2) The average gap between primes is log(n) [2]; the gap between primes can be arbitrarily large since sequences of composite numbers are arbitrarily large, e.g. the numbers in the sequence n! + 2, n! + 3 ... n! + n are all composite. See Also ======== prime : Return the nth prime nextprime : Return the ith prime greater than n prevprime : Return the largest prime smaller than n randprime : Returns a random prime in a given range primorial : Returns the product of primes based on condition Sieve.primerange : return range from already computed primes or extend the sieve to contain the requested range. References ========== .. [1] https://en.wikipedia.org/wiki/Prime_number .. [2] http://primes.utm.edu/notes/gaps.html """ if b is None: a, b = 2, a if a >= b: return # if we already have the range, return it if b <= sieve._list[-1]: yield from sieve.primerange(a, b) return # otherwise compute, without storing, the desired range. a = _as_int_ceiling(a) - 1 b = _as_int_ceiling(b) while 1: a = nextprime(a) if a < b: yield a else: return def randprime(a, b): """ Return a random prime number in the range [a, b). Bertrand's postulate assures that randprime(a, 2a) will always succeed for a > 1. Examples ======== >>> from sympy import randprime, isprime >>> randprime(1, 30) #doctest: +SKIP 13 >>> isprime(randprime(1, 30)) True See Also ======== primerange : Generate all primes in a given range References ========== .. [1] https://en.wikipedia.org/wiki/Bertrand's_postulate """ if a >= b: return a, b = map(int, (a, b)) n = random.randint(a - 1, b) p = nextprime(n) if p >= b: p = prevprime(b) if p < a: raise ValueError("no primes exist in the specified range") return p def primorial(n, nth=True): """ Returns the product of the first n primes (default) or the primes less than or equal to n (when ``nth=False``). Examples ======== >>> from sympy.ntheory.generate import primorial, primerange >>> from sympy import factorint, Mul, primefactors, sqrt >>> primorial(4) # the first 4 primes are 2, 3, 5, 7 210 >>> primorial(4, nth=False) # primes <= 4 are 2 and 3 6 >>> primorial(1) 2 >>> primorial(1, nth=False) 1 >>> primorial(sqrt(101), nth=False) 210 One can argue that the primes are infinite since if you take a set of primes and multiply them together (e.g. the primorial) and then add or subtract 1, the result cannot be divided by any of the original factors, hence either 1 or more new primes must divide this product of primes. In this case, the number itself is a new prime: >>> factorint(primorial(4) + 1) {211: 1} In this case two new primes are the factors: >>> factorint(primorial(4) - 1) {11: 1, 19: 1} Here, some primes smaller and larger than the primes multiplied together are obtained: >>> p = list(primerange(10, 20)) >>> sorted(set(primefactors(Mul(p) + 1)).difference(set(p))) [2, 5, 31, 149] See Also ======== primerange : Generate all primes in a given range """ if nth: n = as_int(n) else: n = int(n) if n < 1: raise ValueError("primorial argument must be >= 1") p = 1 if nth: for i in range(1, n + 1): p = prime(i) else: for i in primerange(2, n + 1): p = i return p def cycle_length(f, x0, nmax=None, values=False): """For a given iterated sequence, return a generator that gives the length of the iterated cycle (lambda) and the length of terms before the cycle begins (mu); if ``values`` is True then the terms of the sequence will be returned instead. The sequence is started with value ``x0``. Note: more than the first lambda + mu terms may be returned and this is the cost of cycle detection with Brent's method; there are, however, generally less terms calculated than would have been calculated if the proper ending point were determined, e.g. by using Floyd's method. >>> from sympy.ntheory.generate import cycle_length This will yield successive values of i <-- func(i): >>> def iter(func, i): ... while 1: ... ii = func(i) ... yield ii ... i = ii ... A function is defined: >>> func = lambda i: (i2 + 1) % 51 and given a seed of 4 and the mu and lambda terms calculated: >>> next(cycle_length(func, 4)) (6, 2) We can see what is meant by looking at the output: >>> n = cycle_length(func, 4, values=True) >>> list(ni for ni in n) [17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14] There are 6 repeating values after the first 2. If a sequence is suspected of being longer than you might wish, ``nmax`` can be used to exit early (and mu will be returned as None): >>> next(cycle_length(func, 4, nmax = 4)) (4, None) >>> [ni for ni in cycle_length(func, 4, nmax = 4, values=True)] [17, 35, 2, 5] Code modified from: https://en.wikipedia.org/wiki/Cycle_detection. """ nmax = int(nmax or 0) # main phase: search successive powers of two power = lam = 1 tortoise, hare = x0, f(x0) # f(x0) is the element/node next to x0. i = 0 while tortoise != hare and (not nmax or i < nmax): i += 1 if power == lam: # time to start a new power of two? tortoise = hare power = 2 lam = 0 if values: yield hare hare = f(hare) lam += 1 if nmax and i == nmax: if values: return else: yield nmax, None return if not values: # Find the position of the first repetition of length lambda mu = 0 tortoise = hare = x0 for i in range(lam): hare = f(hare) while tortoise != hare: tortoise = f(tortoise) hare = f(hare) mu += 1 if mu: mu -= 1 yield lam, mu def composite(nth): """ Return the nth composite number, with the composite numbers indexed as composite(1) = 4, composite(2) = 6, etc.... Examples ======== >>> from sympy import composite >>> composite(36) 52 >>> composite(1) 4 >>> composite(17737) 20000 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range primepi : Return the number of primes less than or equal to n prime : Return the nth prime compositepi : Return the number of positive composite numbers less than or equal to n """ n = as_int(nth) if n < 1: raise ValueError("nth must be a positive integer; composite(1) == 4") composite_arr = [4, 6, 8, 9, 10, 12, 14, 15, 16, 18] if n <= 10: return composite_arr[n - 1] a, b = 4, sieve._list[-1] if n <= b - primepi(b) - 1: while a < b - 1: mid = (a + b) >> 1 if mid - primepi(mid) - 1 > n: b = mid else: a = mid if isprime(a): a -= 1 return a from sympy.functions.elementary.exponential import log from sympy.functions.special.error_functions import li a = 4 # Lower bound for binary search b = int(n*(log(n) + log(log(n)))) # Upper bound for the search. while a < b: mid = (a + b) >> 1 if mid - li(mid) - 1 > n: b = mid else: a = mid + 1 n_composites = a - primepi(a) - 1 while n_composites > n: if not isprime(a): n_composites -= 1 a -= 1 if isprime(a): a -= 1 return a def compositepi(n): """ Return the number of positive composite numbers less than or equal to n. The first positive composite is 4, i.e. compositepi(4) = 1. Examples ======== >>> from sympy import compositepi >>> compositepi(25) 15 >>> compositepi(1000) 831 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range prime : Return the nth prime primepi : Return the number of primes less than or equal to n composite : Return the nth composite number """ n = int(n) if n < 4: return 0 return n - primepi(n) - 1
a4ae3be21cae07bd1ff5cddecffc73d1b9cc6252db8a7ec88c989c400c9b706c	from sympy.ntheory import sieve, isprime from sympy.core.numbers import mod_inverse from sympy.core.power import integer_log from sympy.utilities.misc import as_int import random rgen = random.Random() #----------------------------------------------------------------------------# # # # Lenstra's Elliptic Curve Factorization # # # #----------------------------------------------------------------------------# class Point: """Montgomery form of Points in an elliptic curve. In this form, the addition and doubling of points does not need any y-coordinate information thus decreasing the number of operations. Using Montgomery form we try to perform point addition and doubling in least amount of multiplications. The elliptic curve used here is of the form (E : by2z = x*3 + ax*2z + xz2). The a_24 parameter is equal to (a + 2)/4. References ========== .. [1] http://www.hyperelliptic.org/tanja/SHARCS/talks06/Gaj.pdf """ def __init__(self, x_cord, z_cord, a_24, mod): """ Initial parameters for the Point class. Parameters ========== x_cord : X coordinate of the Point z_cord : Z coordinate of the Point a_24 : Parameter of the elliptic curve in Montgomery form mod : modulus """ self.x_cord = x_cord self.z_cord = z_cord self.a_24 = a_24 self.mod = mod def __eq__(self, other): """Two points are equal if X/Z of both points are equal """ if self.a_24 != other.a_24 or self.mod != other.mod: return False return self.x_cord mod_inverse(self.z_cord, self.mod) % self.mod ==\ other.x_cord * mod_inverse(other.z_cord, self.mod) % self.mod def add(self, Q, diff): """ Add two points self and Q where diff = self - Q. Moreover the assumption is self.x_cordQ.x_cord(self.x_cord - Q.x_cord) != 0. This algorithm requires 6 multiplications. Here the difference between the points is already known and using this algorihtm speeds up the addition by reducing the number of multiplication required. Also in the mont_ladder algorithm is constructed in a way so that the difference between intermediate points is always equal to the initial point. So, we always know what the difference between the point is. Parameters ========== Q : point on the curve in Montgomery form diff : self - Q Examples ======== >>> from sympy.ntheory.ecm import Point >>> p1 = Point(11, 16, 7, 29) >>> p2 = Point(13, 10, 7, 29) >>> p3 = p2.add(p1, p1) >>> p3.x_cord 23 >>> p3.z_cord 17 """ u = (self.x_cord - self.z_cord)(Q.x_cord + Q.z_cord) v = (self.x_cord + self.z_cord)(Q.x_cord - Q.z_cord) add, subt = u + v, u - v x_cord = diff.z_cord * add * add % self.mod z_cord = diff.x_cord * subt * subt % self.mod return Point(x_cord, z_cord, self.a_24, self.mod) def double(self): """ Doubles a point in an elliptic curve in Montgomery form. This algorithm requires 5 multiplications. Examples ======== >>> from sympy.ntheory.ecm import Point >>> p1 = Point(11, 16, 7, 29) >>> p2 = p1.double() >>> p2.x_cord 13 >>> p2.z_cord 10 """ u, v = self.x_cord + self.z_cord, self.x_cord - self.z_cord u, v = uu, vv diff = u - v x_cord = uv % self.mod z_cord = diff(v + self.a_24diff) % self.mod return Point(x_cord, z_cord, self.a_24, self.mod) def mont_ladder(self, k): """ Scalar multiplication of a point in Montgomery form using Montgomery Ladder Algorithm. A total of 11 multiplications are required in each step of this algorithm. Parameters ========== k : The positive integer multiplier Examples ======== >>> from sympy.ntheory.ecm import Point >>> p1 = Point(11, 16, 7, 29) >>> p3 = p1.mont_ladder(3) >>> p3.x_cord 23 >>> p3.z_cord 17 """ Q = self R = self.double() for i in bin(k)[3:]: if i == '1': Q = R.add(Q, self) R = R.double() else: R = Q.add(R, self) Q = Q.double() return Q def _ecm_one_factor(n, B1=10000, B2=100000, max_curve=200): """Returns one factor of n using Lenstra's 2 Stage Elliptic curve Factorization with Suyama's Parameterization. Here Montgomery arithmetic is used for fast computation of addition and doubling of points in elliptic curve. This ECM method considers elliptic curves in Montgomery form (E : by*2z = x*3 + ax*2z + xz2) and involves elliptic curve operations (mod N), where the elements in Z are reduced (mod N). Since N is not a prime, E over FF(N) is not really an elliptic curve but we can still do point additions and doubling as if FF(N) was a field. Stage 1 : The basic algorithm involves taking a random point (P) on an elliptic curve in FF(N). The compute kP using Montgomery ladder algorithm. Let q be an unknown factor of N. Then the order of the curve E, \|E(FF(q))\|, might be a smooth number that divides k. Then we have k = l * \|E(FF(q))\| for some l. For any point belonging to the curve E, \|E(FF(q))\|P = O, hence kP = l\|E(FF(q))\|P. Thus kP.z_cord = 0 (mod q), and the unknownn factor of N (q) can be recovered by taking gcd(kP.z_cord, N). Stage 2 : This is a continuation of Stage 1 if kP != O. The idea utilize the fact that even if kP != 0, the value of k might miss just one large prime divisor of \|E(FF(q))\|. In this case we only need to compute the scalar multiplication by p to get pkP = O. Here a second bound B2 restrict the size of possible values of p. Parameters ========== n : Number to be Factored B1 : Stage 1 Bound B2 : Stage 2 Bound max_curve : Maximum number of curves generated References ========== .. [1] Carl Pomerance and Richard Crandall "Prime Numbers: A Computational Perspective" (2nd Ed.), page 344 """ n = as_int(n) if B1 % 2 != 0 or B2 % 2 != 0: raise ValueError("The Bounds should be an even integer") sieve.extend(B2) if isprime(n): return n from sympy.functions.elementary.miscellaneous import sqrt from sympy.polys.polytools import gcd curve = 0 D = int(sqrt(B2)) beta = [0](D + 1) S = [0](D + 1) k = 1 for p in sieve.primerange(1, B1 + 1): k = pow(p, integer_log(B1, p)[0]) while(curve <= max_curve): curve += 1 #Suyama's Paramatrization sigma = rgen.randint(6, n - 1) u = (sigmasigma - 5) % n v = (4sigma) % n diff = v - u u_3 = pow(u, 3, n) try: C = (pow(diff, 3, n)(3u + v)mod_inverse(4u_3v, n) - 2) % n except ValueError: #If the mod_inverse(4u_3v, n) doesn't exist return gcd(4u_3v, n) a24 = (C + 2)mod_inverse(4, n) % n Q = Point(u_3, pow(v, 3, n), a24, n) Q = Q.mont_ladder(k) g = gcd(Q.z_cord, n) #Stage 1 factor if g != 1 and g != n: return g #Stage 1 failure. Q.z = 0, Try another curve elif g == n: continue #Stage 2 - Improved Standard Continuation S[1] = Q.double() S[2] = S[1].double() beta[1] = (S[1].x_cordS[1].z_cord) % n beta[2] = (S[2].x_cordS[2].z_cord) % n for d in range(3, D + 1): S[d] = S[d - 1].add(S[1], S[d - 2]) beta[d] = (S[d].x_cordS[d].z_cord) % n g = 1 B = B1 - 1 T = Q.mont_ladder(B - 2D) R = Q.mont_ladder(B) for r in range(B, B2, 2D): alpha = (R.x_cordR.z_cord) % n for q in sieve.primerange(r + 2, r + 2D + 1): delta = (q - r) // 2 f = (R.x_cord - S[d].x_cord)(R.z_cord + S[d].z_cord) -\ alpha + beta[delta] g = (g*f) % n #Swap T, R = R, R.add(S[D], T) g = gcd(n, g) #Stage 2 Factor found if g != 1 and g != n: return g #ECM failed, Increase the bounds raise ValueError("Increase the bounds") def ecm(n, B1=10000, B2=100000, max_curve=200, seed=1234): """Performs factorization using Lenstra's Elliptic curve method. This function repeatedly calls `ecm_one_factor` to compute the factors of n. First all the small factors are taken out using trial division. Then `ecm_one_factor` is used to compute one factor at a time. Parameters ========== n : Number to be Factored B1 : Stage 1 Bound B2 : Stage 2 Bound max_curve : Maximum number of curves generated seed : Initialize pseudorandom generator Examples ======== >>> from sympy.ntheory import ecm >>> ecm(25645121643901801) {5394769, 4753701529} >>> ecm(9804659461513846513) {4641991, 2112166839943} """ _factors = set() for prime in sieve.primerange(1, 100000): if n % prime == 0: _factors.add(prime) while(n % prime == 0): n //= prime rgen.seed(seed) while(n > 1): try: factor = _ecm_one_factor(n, B1, B2, max_curve) except ValueError: raise ValueError("Increase the bounds") _factors.add(factor) n //= factor factors = set() for factor in _factors: if isprime(factor): factors.add(factor) continue factors \|= ecm(factor) return factors
956676ab7960bd6a8aaeaefd0fcd4f72fb82e42b3a9061d20cb257c30441d9b3	from sympy.core.function import Function from sympy.core.numbers import igcd, igcdex, mod_inverse from sympy.core.power import isqrt from sympy.core.singleton import S from sympy.polys import Poly from sympy.polys.domains import ZZ from sympy.polys.galoistools import gf_crt1, gf_crt2, linear_congruence from .primetest import isprime from .factor_ import factorint, trailing, totient, multiplicity from sympy.utilities.misc import as_int from sympy.core.random import _randint, randint from itertools import cycle, product def n_order(a, n): """Returns the order of ``a`` modulo ``n``. The order of ``a`` modulo ``n`` is the smallest integer ``k`` such that ``a*k`` leaves a remainder of 1 with ``n``. Examples ======== >>> from sympy.ntheory import n_order >>> n_order(3, 7) 6 >>> n_order(4, 7) 3 """ from collections import defaultdict a, n = as_int(a), as_int(n) if igcd(a, n) != 1: raise ValueError("The two numbers should be relatively prime") factors = defaultdict(int) f = factorint(n) for px, kx in f.items(): if kx > 1: factors[px] += kx - 1 fpx = factorint(px - 1) for py, ky in fpx.items(): factors[py] += ky group_order = 1 for px, kx in factors.items(): group_order = px*kx order = 1 if a > n: a = a % n for p, e in factors.items(): exponent = group_order for f in range(e + 1): if pow(a, exponent, n) != 1: order = p ** (e - f + 1) break exponent = exponent // p return order def _primitive_root_prime_iter(p): """ Generates the primitive roots for a prime ``p`` Examples ======== >>> from sympy.ntheory.residue_ntheory import _primitive_root_prime_iter >>> list(_primitive_root_prime_iter(19)) [2, 3, 10, 13, 14, 15] References ========== .. [1] W. Stein "Elementary Number Theory" (2011), page 44 """ # it is assumed that p is an int v = [(p - 1) // i for i in factorint(p - 1).keys()] a = 2 while a < p: for pw in v: # a TypeError below may indicate that p was not an int if pow(a, pw, p) == 1: break else: yield a a += 1 def primitive_root(p): """ Returns the smallest primitive root or None Parameters ========== p : positive integer Examples ======== >>> from sympy.ntheory.residue_ntheory import primitive_root >>> primitive_root(19) 2 References ========== .. [1] W. Stein "Elementary Number Theory" (2011), page 44 .. [2] P. Hackman "Elementary Number Theory" (2009), Chapter C """ p = as_int(p) if p < 1: raise ValueError('p is required to be positive') if p <= 2: return 1 f = factorint(p) if len(f) > 2: return None if len(f) == 2: if 2 not in f or f[2] > 1: return None # case p = 2p1k, p1 prime for p1, e1 in f.items(): if p1 != 2: break i = 1 while i < p: i += 2 if i % p1 == 0: continue if is_primitive_root(i, p): return i else: if 2 in f: if p == 4: return 3 return None p1, n = list(f.items())[0] if n > 1: # see Ref [2], page 81 g = primitive_root(p1) if is_primitive_root(g, p12): return g else: for i in range(2, g + p1 + 1): if igcd(i, p) == 1 and is_primitive_root(i, p): return i return next(_primitive_root_prime_iter(p)) def is_primitive_root(a, p): """ Returns True if ``a`` is a primitive root of ``p`` ``a`` is said to be the primitive root of ``p`` if gcd(a, p) == 1 and totient(p) is the smallest positive number s.t. atotient(p) cong 1 mod(p) Examples ======== >>> from sympy.ntheory import is_primitive_root, n_order, totient >>> is_primitive_root(3, 10) True >>> is_primitive_root(9, 10) False >>> n_order(3, 10) == totient(10) True >>> n_order(9, 10) == totient(10) False """ a, p = as_int(a), as_int(p) if igcd(a, p) != 1: raise ValueError("The two numbers should be relatively prime") if a > p: a = a % p return n_order(a, p) == totient(p) def _sqrt_mod_tonelli_shanks(a, p): """ Returns the square root in the case of ``p`` prime with ``p == 1 (mod 8)`` References ========== .. [1] R. Crandall and C. Pomerance "Prime Numbers", 2nt Ed., page 101 """ s = trailing(p - 1) t = p >> s # find a non-quadratic residue while 1: d = randint(2, p - 1) r = legendre_symbol(d, p) if r == -1: break #assert legendre_symbol(d, p) == -1 A = pow(a, t, p) D = pow(d, t, p) m = 0 for i in range(s): adm = Apow(D, m, p) % p adm = pow(adm, 2(s - 1 - i), p) if adm % p == p - 1: m += 2i #assert Apow(D, m, p) % p == 1 x = pow(a, (t + 1)//2, p)pow(D, m//2, p) % p return x def sqrt_mod(a, p, all_roots=False): """ Find a root of ``x*2 = a mod p`` Parameters ========== a : integer p : positive integer all_roots : if True the list of roots is returned or None Notes ===== If there is no root it is returned None; else the returned root is less or equal to ``p // 2``; in general is not the smallest one. It is returned ``p // 2`` only if it is the only root. Use ``all_roots`` only when it is expected that all the roots fit in memory; otherwise use ``sqrt_mod_iter``. Examples ======== >>> from sympy.ntheory import sqrt_mod >>> sqrt_mod(11, 43) 21 >>> sqrt_mod(17, 32, True) [7, 9, 23, 25] """ if all_roots: return sorted(list(sqrt_mod_iter(a, p))) try: p = abs(as_int(p)) it = sqrt_mod_iter(a, p) r = next(it) if r > p // 2: return p - r elif r < p // 2: return r else: try: r = next(it) if r > p // 2: return p - r except StopIteration: pass return r except StopIteration: return None def _product(iters): """ Cartesian product generator Notes ===== Unlike itertools.product, it works also with iterables which do not fit in memory. See http://bugs.python.org/issue10109 Author: Fernando Sumudu with small changes """ inf_iters = tuple(cycle(enumerate(it)) for it in iters) num_iters = len(inf_iters) cur_val = [None]num_iters first_v = True while True: i, p = 0, num_iters while p and not i: p -= 1 i, cur_val[p] = next(inf_iters[p]) if not p and not i: if first_v: first_v = False else: break yield cur_val def sqrt_mod_iter(a, p, domain=int): """ Iterate over solutions to ``x2 = a mod p`` Parameters ========== a : integer p : positive integer domain : integer domain, ``int``, ``ZZ`` or ``Integer`` Examples ======== >>> from sympy.ntheory.residue_ntheory import sqrt_mod_iter >>> list(sqrt_mod_iter(11, 43)) [21, 22] """ a, p = as_int(a), abs(as_int(p)) if isprime(p): a = a % p if a == 0: res = _sqrt_mod1(a, p, 1) else: res = _sqrt_mod_prime_power(a, p, 1) if res: if domain is ZZ: yield from res else: for x in res: yield domain(x) else: f = factorint(p) v = [] pv = [] for px, ex in f.items(): if a % px == 0: rx = _sqrt_mod1(a, px, ex) if not rx: return else: rx = _sqrt_mod_prime_power(a, px, ex) if not rx: return v.append(rx) pv.append(pxex) mm, e, s = gf_crt1(pv, ZZ) if domain is ZZ: for vx in _product(v): r = gf_crt2(vx, pv, mm, e, s, ZZ) yield r else: for vx in _product(v): r = gf_crt2(vx, pv, mm, e, s, ZZ) yield domain(r) def _sqrt_mod_prime_power(a, p, k): """ Find the solutions to ``x2 = a mod pk`` when ``a % p != 0`` Parameters ========== a : integer p : prime number k : positive integer Examples ======== >>> from sympy.ntheory.residue_ntheory import _sqrt_mod_prime_power >>> _sqrt_mod_prime_power(11, 43, 1) [21, 22] References ========== .. [1] P. Hackman "Elementary Number Theory" (2009), page 160 .. [2] http://www.numbertheory.org/php/squareroot.html .. [3] [Gathen99]_ """ pk = pk a = a % pk if k == 1: if p == 2: return [ZZ(a)] if not (a % p < 2 or pow(a, (p - 1) // 2, p) == 1): return None if p % 4 == 3: res = pow(a, (p + 1) // 4, p) elif p % 8 == 5: sign = pow(a, (p - 1) // 4, p) if sign == 1: res = pow(a, (p + 3) // 8, p) else: b = pow(4a, (p - 5) // 8, p) x = (2ab) % p if pow(x, 2, p) == a: res = x else: res = _sqrt_mod_tonelli_shanks(a, p) # ``_sqrt_mod_tonelli_shanks(a, p)`` is not deterministic; # sort to get always the same result return sorted([ZZ(res), ZZ(p - res)]) if k > 1: # see Ref.[2] if p == 2: if a % 8 != 1: return None if k <= 3: s = set() for i in range(0, pk, 4): s.add(1 + i) s.add(-1 + i) return list(s) # according to Ref.[2] for k > 2 there are two solutions # (mod 2k-1), that is four solutions (mod 2k), which can be # obtained from the roots of x2 = 0 (mod 8) rv = [ZZ(1), ZZ(3), ZZ(5), ZZ(7)] # hensel lift them to solutions of x2 = 0 (mod 2k) # if r2 - a = 0 mod 2nx but not mod 2(nx+1) # then r + 2(nx - 1) is a root mod 2(nx+1) n = 3 res = [] for r in rv: nx = n while nx < k: r1 = (r2 - a) >> nx if r1 % 2: r = r + (1 << (nx - 1)) #assert (r2 - a)% (1 << (nx + 1)) == 0 nx += 1 if r not in res: res.append(r) x = r + (1 << (k - 1)) #assert (x2 - a) % pk == 0 if x < (1 << nx) and x not in res: if (x2 - a) % pk == 0: res.append(x) return res rv = _sqrt_mod_prime_power(a, p, 1) if not rv: return None r = rv[0] fr = r2 - a # hensel lifting with Newton iteration, see Ref.[3] chapter 9 # with f(x) = x2 - a; one has f'(a) != 0 (mod p) for p != 2 n = 1 px = p while 1: n1 = n n1 = 2 if n1 > k: break n = n1 px = px2 frinv = igcdex(2r, px)[0] r = (r - frfrinv) % px fr = r2 - a if n < k: px = pk frinv = igcdex(2r, px)[0] r = (r - frfrinv) % px return [r, px - r] def _sqrt_mod1(a, p, n): """ Find solution to ``x2 == a mod pn`` when ``a % p == 0`` see http://www.numbertheory.org/php/squareroot.html """ pn = pn a = a % pn if a == 0: # case gcd(a, pk) = pn m = n // 2 if n % 2 == 1: pm1 = p(m + 1) def _iter0a(): i = 0 while i < pn: yield i i += pm1 return _iter0a() else: pm = pm def _iter0b(): i = 0 while i < pn: yield i i += pm return _iter0b() # case gcd(a, pk) = pr, r < n f = factorint(a) r = f[p] if r % 2 == 1: return None m = r // 2 a1 = a >> r if p == 2: if n - r == 1: pnm1 = 1 << (n - m + 1) pm1 = 1 << (m + 1) def _iter1(): k = 1 << (m + 2) i = 1 << m while i < pnm1: j = i while j < pn: yield j j += k i += pm1 return _iter1() if n - r == 2: res = _sqrt_mod_prime_power(a1, p, n - r) if res is None: return None pnm = 1 << (n - m) def _iter2(): s = set() for r in res: i = 0 while i < pn: x = (r << m) + i if x not in s: s.add(x) yield x i += pnm return _iter2() if n - r > 2: res = _sqrt_mod_prime_power(a1, p, n - r) if res is None: return None pnm1 = 1 << (n - m - 1) def _iter3(): s = set() for r in res: i = 0 while i < pn: x = ((r << m) + i) % pn if x not in s: s.add(x) yield x i += pnm1 return _iter3() else: m = r // 2 a1 = a // pr res1 = _sqrt_mod_prime_power(a1, p, n - r) if res1 is None: return None pm = pm pnr = p(n-r) pnm = p(n-m) def _iter4(): s = set() pm = pm for rx in res1: i = 0 while i < pnm: x = ((rx + i) % pn) if x not in s: s.add(x) yield xpm i += pnr return _iter4() def is_quad_residue(a, p): """ Returns True if ``a`` (mod ``p``) is in the set of squares mod ``p``, i.e a % p in set([i2 % p for i in range(p)]). If ``p`` is an odd prime, an iterative method is used to make the determination: >>> from sympy.ntheory import is_quad_residue >>> sorted(set([i2 % 7 for i in range(7)])) [0, 1, 2, 4] >>> [j for j in range(7) if is_quad_residue(j, 7)] [0, 1, 2, 4] See Also ======== legendre_symbol, jacobi_symbol """ a, p = as_int(a), as_int(p) if p < 1: raise ValueError('p must be > 0') if a >= p or a < 0: a = a % p if a < 2 or p < 3: return True if not isprime(p): if p % 2 and jacobi_symbol(a, p) == -1: return False r = sqrt_mod(a, p) if r is None: return False else: return True return pow(a, (p - 1) // 2, p) == 1 def is_nthpow_residue(a, n, m): """ Returns True if ``x*n == a (mod m)`` has solutions. References ========== .. [1] P. Hackman "Elementary Number Theory" (2009), page 76 """ a = a % m a, n, m = as_int(a), as_int(n), as_int(m) if m <= 0: raise ValueError('m must be > 0') if n < 0: raise ValueError('n must be >= 0') if n == 0: if m == 1: return False return a == 1 if a == 0: return True if n == 1: return True if n == 2: return is_quad_residue(a, m) return _is_nthpow_residue_bign(a, n, m) def _is_nthpow_residue_bign(a, n, m): r"""Returns True if `x^n = a \pmod{n}` has solutions for `n > 2`.""" # assert n > 2 # assert a > 0 and m > 0 if primitive_root(m) is None or igcd(a, m) != 1: # assert m >= 8 for prime, power in factorint(m).items(): if not _is_nthpow_residue_bign_prime_power(a, n, prime, power): return False return True f = totient(m) k = f // igcd(f, n) return pow(a, k, m) == 1 def _is_nthpow_residue_bign_prime_power(a, n, p, k): r"""Returns True/False if a solution for `x^n = a \pmod{p^k}` does/does not exist.""" # assert a > 0 # assert n > 2 # assert p is prime # assert k > 0 if a % p: if p != 2: return _is_nthpow_residue_bign(a, n, pow(p, k)) if n & 1: return True c = trailing(n) return a % pow(2, min(c + 2, k)) == 1 else: a %= pow(p, k) if not a: return True mu = multiplicity(p, a) if mu % n: return False pm = pow(p, mu) return _is_nthpow_residue_bign_prime_power(a//pm, n, p, k - mu) def _nthroot_mod2(s, q, p): f = factorint(q) v = [] for b, e in f.items(): v.extend([b]e) for qx in v: s = _nthroot_mod1(s, qx, p, False) return s def _nthroot_mod1(s, q, p, all_roots): """ Root of ``x*q = s mod p``, ``p`` prime and ``q`` divides ``p - 1`` References ========== .. [1] A. M. Johnston "A Generalized qth Root Algorithm" """ g = primitive_root(p) if not isprime(q): r = _nthroot_mod2(s, q, p) else: f = p - 1 assert (p - 1) % q == 0 # determine k k = 0 while f % q == 0: k += 1 f = f // q # find z, x, r1 f1 = igcdex(-f, q)[0] % q z = ff1 x = (1 + z) // q r1 = pow(s, x, p) s1 = pow(s, f, p) h = pow(g, fq, p) t = discrete_log(p, s1, h) g2 = pow(g, zt, p) g3 = igcdex(g2, p)[0] r = r1g3 % p #assert pow(r, q, p) == s res = [r] h = pow(g, (p - 1) // q, p) #assert pow(h, q, p) == 1 hx = r for i in range(q - 1): hx = (hxh) % p res.append(hx) if all_roots: res.sort() return res return min(res) def _help(m, prime_modulo_method, diff_method, expr_val): """ Helper function for _nthroot_mod_composite and polynomial_congruence. Parameters ========== m : positive integer prime_modulo_method : function to calculate the root of the congruence equation for the prime divisors of m diff_method : function to calculate derivative of expression at any given point expr_val : function to calculate value of the expression at any given point """ from sympy.ntheory.modular import crt f = factorint(m) dd = {} for p, e in f.items(): tot_roots = set() if e == 1: tot_roots.update(prime_modulo_method(p)) else: for root in prime_modulo_method(p): diff = diff_method(root, p) if diff != 0: ppow = p m_inv = mod_inverse(diff, p) for j in range(1, e): ppow = p root = (root - expr_val(root, ppow) m_inv) % ppow tot_roots.add(root) else: new_base = p roots_in_base = {root} while new_base < pow(p, e): new_base = p new_roots = set() for k in roots_in_base: if expr_val(k, new_base)!= 0: continue while k not in new_roots: new_roots.add(k) k = (k + (new_base // p)) % new_base roots_in_base = new_roots tot_roots = tot_roots \| roots_in_base if tot_roots == set(): return [] dd[pow(p, e)] = tot_roots a = [] m = [] for x, y in dd.items(): m.append(x) a.append(list(y)) return sorted({crt(m, list(i))[0] for i in product(a)}) def _nthroot_mod_composite(a, n, m): """ Find the solutions to ``x*n = a mod m`` when m is not prime. """ return _help(m, lambda p: nthroot_mod(a, n, p, True), lambda root, p: (pow(root, n - 1, p) (n % p)) % p, lambda root, p: (pow(root, n, p) - a) % p) def nthroot_mod(a, n, p, all_roots=False): """ Find the solutions to ``xn = a mod p`` Parameters ========== a : integer n : positive integer p : positive integer all_roots : if False returns the smallest root, else the list of roots Examples ======== >>> from sympy.ntheory.residue_ntheory import nthroot_mod >>> nthroot_mod(11, 4, 19) 8 >>> nthroot_mod(11, 4, 19, True) [8, 11] >>> nthroot_mod(68, 3, 109) 23 """ a = a % p a, n, p = as_int(a), as_int(n), as_int(p) if n == 2: return sqrt_mod(a, p, all_roots) # see Hackman "Elementary Number Theory" (2009), page 76 if not isprime(p): return _nthroot_mod_composite(a, n, p) if a % p == 0: return [0] if not is_nthpow_residue(a, n, p): return [] if all_roots else None if (p - 1) % n == 0: return _nthroot_mod1(a, n, p, all_roots) # The roots of ``xn - a = 0 (mod p)`` are roots of # ``gcd(xn - a, x(p - 1) - 1) = 0 (mod p)`` pa = n pb = p - 1 b = 1 if pa < pb: a, pa, b, pb = b, pb, a, pa while pb: # xpa - a = 0; xpb - b = 0 # xpa - a = x(qpb + r) - a = (xpb)q xr - a = # bq * xr - a; xr - c = 0; c = b*-q a mod p q, r = divmod(pa, pb) c = pow(b, q, p) c = igcdex(c, p)[0] c = (c * a) % p pa, pb = pb, r a, b = b, c if pa == 1: if all_roots: res = [a] else: res = a elif pa == 2: return sqrt_mod(a, p, all_roots) else: res = _nthroot_mod1(a, pa, p, all_roots) return res def quadratic_residues(p): """ Returns the list of quadratic residues. Examples ======== >>> from sympy.ntheory.residue_ntheory import quadratic_residues >>> quadratic_residues(7) [0, 1, 2, 4] """ p = as_int(p) r = set() for i in range(p // 2 + 1): r.add(pow(i, 2, p)) return sorted(list(r)) def legendre_symbol(a, p): r""" Returns the Legendre symbol `(a / p)`. For an integer ``a`` and an odd prime ``p``, the Legendre symbol is defined as .. math :: \genfrac(){}{}{a}{p} = \begin{cases} 0 & \text{if } p \text{ divides } a\\ 1 & \text{if } a \text{ is a quadratic residue modulo } p\\ -1 & \text{if } a \text{ is a quadratic nonresidue modulo } p \end{cases} Parameters ========== a : integer p : odd prime Examples ======== >>> from sympy.ntheory import legendre_symbol >>> [legendre_symbol(i, 7) for i in range(7)] [0, 1, 1, -1, 1, -1, -1] >>> sorted(set([i2 % 7 for i in range(7)])) [0, 1, 2, 4] See Also ======== is_quad_residue, jacobi_symbol """ a, p = as_int(a), as_int(p) if not isprime(p) or p == 2: raise ValueError("p should be an odd prime") a = a % p if not a: return 0 if pow(a, (p - 1) // 2, p) == 1: return 1 return -1 def jacobi_symbol(m, n): r""" Returns the Jacobi symbol `(m / n)`. For any integer ``m`` and any positive odd integer ``n`` the Jacobi symbol is defined as the product of the Legendre symbols corresponding to the prime factors of ``n``: .. math :: \genfrac(){}{}{m}{n} = \genfrac(){}{}{m}{p^{1}}^{\alpha_1} \genfrac(){}{}{m}{p^{2}}^{\alpha_2} ... \genfrac(){}{}{m}{p^{k}}^{\alpha_k} \text{ where } n = p_1^{\alpha_1} p_2^{\alpha_2} ... p_k^{\alpha_k} Like the Legendre symbol, if the Jacobi symbol `\genfrac(){}{}{m}{n} = -1` then ``m`` is a quadratic nonresidue modulo ``n``. But, unlike the Legendre symbol, if the Jacobi symbol `\genfrac(){}{}{m}{n} = 1` then ``m`` may or may not be a quadratic residue modulo ``n``. Parameters ========== m : integer n : odd positive integer Examples ======== >>> from sympy.ntheory import jacobi_symbol, legendre_symbol >>> from sympy import S >>> jacobi_symbol(45, 77) -1 >>> jacobi_symbol(60, 121) 1 The relationship between the ``jacobi_symbol`` and ``legendre_symbol`` can be demonstrated as follows: >>> L = legendre_symbol >>> S(45).factors() {3: 2, 5: 1} >>> jacobi_symbol(7, 45) == L(7, 3)2 * L(7, 5)*1 True See Also ======== is_quad_residue, legendre_symbol """ m, n = as_int(m), as_int(n) if n < 0 or not n % 2: raise ValueError("n should be an odd positive integer") if m < 0 or m > n: m %= n if not m: return int(n == 1) if n == 1 or m == 1: return 1 if igcd(m, n) != 1: return 0 j = 1 if m < 0: m = -m if n % 4 == 3: j = -j while m != 0: while m % 2 == 0 and m > 0: m >>= 1 if n % 8 in [3, 5]: j = -j m, n = n, m if m % 4 == n % 4 == 3: j = -j m %= n if n != 1: j = 0 return j class mobius(Function): """ Mobius function maps natural number to {-1, 0, 1} It is defined as follows: 1) `1` if `n = 1`. 2) `0` if `n` has a squared prime factor. 3) `(-1)^k` if `n` is a square-free positive integer with `k` number of prime factors. It is an important multiplicative function in number theory and combinatorics. It has applications in mathematical series, algebraic number theory and also physics (Fermion operator has very concrete realization with Mobius Function model). Parameters ========== n : positive integer Examples ======== >>> from sympy.ntheory import mobius >>> mobius(137) 1 >>> mobius(1) 1 >>> mobius(1375) -1 >>> mobius(132) 0 References ========== .. [1] https://en.wikipedia.org/wiki/M%C3%B6bius_function .. [2] Thomas Koshy "Elementary Number Theory with Applications" """ @classmethod def eval(cls, n): if n.is_integer: if n.is_positive is not True: raise ValueError("n should be a positive integer") else: raise TypeError("n should be an integer") if n.is_prime: return S.NegativeOne elif n is S.One: return S.One elif n.is_Integer: a = factorint(n) if any(i > 1 for i in a.values()): return S.Zero return S.NegativeOnelen(a) def _discrete_log_trial_mul(n, a, b, order=None): """ Trial multiplication algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. The algorithm finds the discrete logarithm using exhaustive search. This naive method is used as fallback algorithm of ``discrete_log`` when the group order is very small. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_trial_mul >>> _discrete_log_trial_mul(41, 15, 7) 3 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). """ a %= n b %= n if order is None: order = n x = 1 for i in range(order): if x == a: return i x = x * b % n raise ValueError("Log does not exist") def _discrete_log_shanks_steps(n, a, b, order=None): """ Baby-step giant-step algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. The algorithm is a time-memory trade-off of the method of exhaustive search. It uses `O(sqrt(m))` memory, where `m` is the group order. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_shanks_steps >>> _discrete_log_shanks_steps(41, 15, 7) 3 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). """ a %= n b %= n if order is None: order = n_order(b, n) m = isqrt(order) + 1 T = dict() x = 1 for i in range(m): T[x] = i x = x * b % n z = mod_inverse(b, n) z = pow(z, m, n) x = a for i in range(m): if x in T: return i * m + T[x] x = x * z % n raise ValueError("Log does not exist") def _discrete_log_pollard_rho(n, a, b, order=None, retries=10, rseed=None): """ Pollard's Rho algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. It is a randomized algorithm with the same expected running time as ``_discrete_log_shanks_steps``, but requires a negligible amount of memory. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_pollard_rho >>> _discrete_log_pollard_rho(227, 3*7, 3) 7 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). """ a %= n b %= n if order is None: order = n_order(b, n) randint = _randint(rseed) for i in range(retries): aa = randint(1, order - 1) ba = randint(1, order - 1) xa = pow(b, aa, n) pow(a, ba, n) % n c = xa % 3 if c == 0: xb = a * xa % n ab = aa bb = (ba + 1) % order elif c == 1: xb = xa * xa % n ab = (aa + aa) % order bb = (ba + ba) % order else: xb = b * xa % n ab = (aa + 1) % order bb = ba for j in range(order): c = xa % 3 if c == 0: xa = a * xa % n ba = (ba + 1) % order elif c == 1: xa = xa * xa % n aa = (aa + aa) % order ba = (ba + ba) % order else: xa = b * xa % n aa = (aa + 1) % order c = xb % 3 if c == 0: xb = a * xb % n bb = (bb + 1) % order elif c == 1: xb = xb * xb % n ab = (ab + ab) % order bb = (bb + bb) % order else: xb = b * xb % n ab = (ab + 1) % order c = xb % 3 if c == 0: xb = a * xb % n bb = (bb + 1) % order elif c == 1: xb = xb * xb % n ab = (ab + ab) % order bb = (bb + bb) % order else: xb = b * xb % n ab = (ab + 1) % order if xa == xb: r = (ba - bb) % order try: e = mod_inverse(r, order) * (ab - aa) % order if (pow(b, e, n) - a) % n == 0: return e except ValueError: pass break raise ValueError("Pollard's Rho failed to find logarithm") def _discrete_log_pohlig_hellman(n, a, b, order=None): """ Pohlig-Hellman algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. In order to compute the discrete logarithm, the algorithm takes advantage of the factorization of the group order. It is more efficient when the group order factors into many small primes. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_pohlig_hellman >>> _discrete_log_pohlig_hellman(251, 210, 71) 197 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). """ from .modular import crt a %= n b %= n if order is None: order = n_order(b, n) f = factorint(order) l = [0] * len(f) for i, (pi, ri) in enumerate(f.items()): for j in range(ri): gj = pow(b, l[i], n) aj = pow(a * mod_inverse(gj, n), order // pi*(j + 1), n) bj = pow(b, order // pi, n) cj = discrete_log(n, aj, bj, pi, True) l[i] += cj pij d, _ = crt([piri for pi, ri in f.items()], l) return d def discrete_log(n, a, b, order=None, prime_order=None): """ Compute the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. This is a recursive function to reduce the discrete logarithm problem in cyclic groups of composite order to the problem in cyclic groups of prime order. It employs different algorithms depending on the problem (subgroup order size, prime order or not): * Trial multiplication * Baby-step giant-step * Pollard's Rho * Pohlig-Hellman Examples ======== >>> from sympy.ntheory import discrete_log >>> discrete_log(41, 15, 7) 3 References ========== .. [1] http://mathworld.wolfram.com/DiscreteLogarithm.html .. [2] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). """ n, a, b = as_int(n), as_int(a), as_int(b) if order is None: order = n_order(b, n) if prime_order is None: prime_order = isprime(order) if order < 1000: return _discrete_log_trial_mul(n, a, b, order) elif prime_order: if order < 1000000000000: return _discrete_log_shanks_steps(n, a, b, order) return _discrete_log_pollard_rho(n, a, b, order) return _discrete_log_pohlig_hellman(n, a, b, order) def quadratic_congruence(a, b, c, p): """ Find the solutions to ``a x*2 + b x + c = 0 mod p a : integer b : integer c : integer p : positive integer """ a = as_int(a) b = as_int(b) c = as_int(c) p = as_int(p) a = a % p b = b % p c = c % p if a == 0: return linear_congruence(b, -c, p) if p == 2: roots = [] if c % 2 == 0: roots.append(0) if (a + b + c) % 2 == 0: roots.append(1) return roots if isprime(p): inv_a = mod_inverse(a, p) b = inv_a c = inv_a if b % 2 == 1: b = b + p d = ((b b) // 4 - c) % p y = sqrt_mod(d, p, all_roots=True) res = set() for i in y: res.add((i - b // 2) % p) return sorted(res) y = sqrt_mod(b * b - 4 * a * c, 4 * a * p, all_roots=True) res = set() for i in y: root = linear_congruence(2 * a, i - b, 4 * a * p) for j in root: res.add(j % p) return sorted(res) def _polynomial_congruence_prime(coefficients, p): """A helper function used by polynomial_congruence. It returns the root of a polynomial modulo prime number by naive search from [0, p). Parameters ========== coefficients : list of integers p : prime number """ roots = [] rank = len(coefficients) for i in range(0, p): f_val = 0 for coeff in range(0,rank - 1): f_val = (f_val + pow(i, int(rank - coeff - 1), p) * coefficients[coeff]) % p f_val = f_val + coefficients[-1] if f_val % p == 0: roots.append(i) return roots def _diff_poly(root, coefficients, p): """A helper function used by polynomial_congruence. It returns the derivative of the polynomial evaluated at the root (mod p). Parameters ========== coefficients : list of integers p : prime number root : integer """ diff = 0 rank = len(coefficients) for coeff in range(0, rank - 1): if not coefficients[coeff]: continue diff = (diff + pow(root, rank - coeff - 2, p)(rank - coeff - 1) coefficients[coeff]) % p return diff % p def _val_poly(root, coefficients, p): """A helper function used by polynomial_congruence. It returns value of the polynomial at root (mod p). Parameters ========== coefficients : list of integers p : prime number root : integer """ rank = len(coefficients) f_val = 0 for coeff in range(0, rank - 1): f_val = (f_val + pow(root, rank - coeff - 1, p)* coefficients[coeff]) % p f_val = f_val + coefficients[-1] return f_val % p def _valid_expr(expr): """ return coefficients of expr if it is a univariate polynomial with integer coefficients else raise a ValueError. """ if not expr.is_polynomial(): raise ValueError("The expression should be a polynomial") polynomial = Poly(expr) if not polynomial.is_univariate: raise ValueError("The expression should be univariate") if not polynomial.domain == ZZ: raise ValueError("The expression should should have integer coefficients") return polynomial.all_coeffs() def polynomial_congruence(expr, m): """ Find the solutions to a polynomial congruence equation modulo m. Parameters ========== coefficients : Coefficients of the Polynomial m : positive integer Examples ======== >>> from sympy.ntheory import polynomial_congruence >>> from sympy.abc import x >>> expr = x*6 - 2x*5 -35 >>> polynomial_congruence(expr, 6125) [3257] """ coefficients = _valid_expr(expr) coefficients = [num % m for num in coefficients] rank = len(coefficients) if rank == 3: return quadratic_congruence(coefficients, m) if rank == 2: return quadratic_congruence(0, *coefficients, m) if coefficients[0] == 1 and 1 + coefficients[-1] == sum(coefficients): return nthroot_mod(-coefficients[-1], rank - 1, m, True) if isprime(m): return _polynomial_congruence_prime(coefficients, m) return _help(m, lambda p: _polynomial_congruence_prime(coefficients, p), lambda root, p: _diff_poly(root, coefficients, p), lambda root, p: _val_poly(root, coefficients, p))
2747af33c9f9652b2611fd8caf0ca9e27a40476164fe8b4e75f2c62ad8ac107a	""" Primality testing """ from sympy.core.numbers import igcd from sympy.core.power import integer_nthroot from sympy.core.sympify import sympify from sympy.external.gmpy import HAS_GMPY from sympy.utilities.misc import as_int from mpmath.libmp import bitcount as _bitlength def _int_tuple(i): return tuple(int(_) for _ in i) def is_euler_pseudoprime(n, b): """Returns True if n is prime or an Euler pseudoprime to base b, else False. Euler Pseudoprime : In arithmetic, an odd composite integer n is called an euler pseudoprime to base a, if a and n are coprime and satisfy the modular arithmetic congruence relation : a ^ (n-1)/2 = + 1(mod n) or a ^ (n-1)/2 = - 1(mod n) (where mod refers to the modulo operation). Examples ======== >>> from sympy.ntheory.primetest import is_euler_pseudoprime >>> is_euler_pseudoprime(2, 5) True References ========== .. [1] https://en.wikipedia.org/wiki/Euler_pseudoprime """ from sympy.ntheory.factor_ import trailing if not mr(n, [b]): return False n = as_int(n) r = n - 1 c = pow(b, r >> trailing(r), n) if c == 1: return True while True: if c == n - 1: return True c = pow(c, 2, n) if c == 1: return False def is_square(n, prep=True): """Return True if n == a a for some integer a, else False. If n is suspected of not being a square then this is a quick method of confirming that it is not. Examples ======== >>> from sympy.ntheory.primetest import is_square >>> is_square(25) True >>> is_square(2) False References ========== .. [1] http://mersenneforum.org/showpost.php?p=110896 See Also ======== sympy.core.power.integer_nthroot """ if prep: n = as_int(n) if n < 0: return False if n in (0, 1): return True # def magic(n): # s = {x*2 % n for x in range(n)} # return sum(1 << bit for bit in s) # >>> print(hex(magic(128))) # 0x2020212020202130202021202030213 # >>> print(hex(magic(99))) # 0x209060049048220348a410213 # >>> print(hex(magic(91))) # 0x102e403012a0c9862c14213 # >>> print(hex(magic(85))) # 0x121065188e001c46298213 if not 0x2020212020202130202021202030213 & (1 << (n & 127)): return False # e.g. 2, 3 m = n % (99 91 * 85) if not 0x209060049048220348a410213 & (1 << (m % 99)): return False # e.g. 17, 68 if not 0x102e403012a0c9862c14213 & (1 << (m % 91)): return False # e.g. 97, 388 if not 0x121065188e001c46298213 & (1 << (m % 85)): return False # e.g. 793, 1408 # n is either: # a) odd = 4even + 1 (and square if even = k(k + 1)) # b) even with # odd multiplicity of 2 --> not square, e.g. 39040 # even multiplicity of 2, e.g. 4, 16, 36, ..., 16324 # removal of factors of 2 to give an odd, and rejection if # any(i%2 for i in divmod(odd - 1, 4)) # will give an odd number in form 4even + 1. # Use of `trailing` to check the power of 2 is not done since it # does not apply to a large percentage of arbitrary numbers # and the integer_nthroot is able to quickly resolve these cases. return integer_nthroot(n, 2)[1] def _test(n, base, s, t): """Miller-Rabin strong pseudoprime test for one base. Return False if n is definitely composite, True if n is probably prime, with a probability greater than 3/4. """ # do the Fermat test b = pow(base, t, n) if b == 1 or b == n - 1: return True else: for j in range(1, s): b = pow(b, 2, n) if b == n - 1: return True # see I. Niven et al. "An Introduction to Theory of Numbers", page 78 if b == 1: return False return False def mr(n, bases): """Perform a Miller-Rabin strong pseudoprime test on n using a given list of bases/witnesses. References ========== .. [1] Richard Crandall & Carl Pomerance (2005), "Prime Numbers: A Computational Perspective", Springer, 2nd edition, 135-138 A list of thresholds and the bases they require are here: https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Deterministic_variants Examples ======== >>> from sympy.ntheory.primetest import mr >>> mr(1373651, [2, 3]) False >>> mr(479001599, [31, 73]) True """ from sympy.ntheory.factor_ import trailing from sympy.polys.domains import ZZ n = as_int(n) if n < 2: return False # remove powers of 2 from n-1 (= t 2s) s = trailing(n - 1) t = n >> s for base in bases: # Bases >= n are wrapped, bases < 2 are invalid if base >= n: base %= n if base >= 2: base = ZZ(base) if not _test(n, base, s, t): return False return True def _lucas_sequence(n, P, Q, k): """Return the modular Lucas sequence (U_k, V_k, Q_k). Given a Lucas sequence defined by P, Q, returns the kth values for U and V, along with Q^k, all modulo n. This is intended for use with possibly very large values of n and k, where the combinatorial functions would be completely unusable. The modular Lucas sequences are used in numerous places in number theory, especially in the Lucas compositeness tests and the various n + 1 proofs. Examples ======== >>> from sympy.ntheory.primetest import _lucas_sequence >>> N = 102000 + 4561 >>> sol = U, V, Qk = _lucas_sequence(N, 3, 1, N//2); sol (0, 2, 1) """ D = PP - 4Q if n < 2: raise ValueError("n must be >= 2") if k < 0: raise ValueError("k must be >= 0") if D == 0: raise ValueError("D must not be zero") if k == 0: return _int_tuple(0, 2, Q) U = 1 V = P Qk = Q b = _bitlength(k) if Q == 1: # Optimization for extra strong tests. while b > 1: U = (UV) % n V = (VV - 2) % n b -= 1 if (k >> (b - 1)) & 1: U, V = UP + V, VP + UD if U & 1: U += n if V & 1: V += n U, V = U >> 1, V >> 1 elif P == 1 and Q == -1: # Small optimization for 50% of Selfridge parameters. while b > 1: U = (UV) % n if Qk == 1: V = (VV - 2) % n else: V = (VV + 2) % n Qk = 1 b -= 1 if (k >> (b-1)) & 1: U, V = U + V, V + UD if U & 1: U += n if V & 1: V += n U, V = U >> 1, V >> 1 Qk = -1 else: # The general case with any P and Q. while b > 1: U = (UV) % n V = (VV - 2Qk) % n Qk = Qk b -= 1 if (k >> (b - 1)) & 1: U, V = UP + V, VP + UD if U & 1: U += n if V & 1: V += n U, V = U >> 1, V >> 1 Qk = Q Qk %= n return _int_tuple(U % n, V % n, Qk) def _lucas_selfridge_params(n): """Calculates the Selfridge parameters (D, P, Q) for n. This is method A from page 1401 of Baillie and Wagstaff. References ========== .. [1] "Lucas Pseudoprimes", Baillie and Wagstaff, 1980. http://mpqs.free.fr/LucasPseudoprimes.pdf """ from sympy.ntheory.residue_ntheory import jacobi_symbol D = 5 while True: g = igcd(abs(D), n) if g > 1 and g != n: return (0, 0, 0) if jacobi_symbol(D, n) == -1: break if D > 0: D = -D - 2 else: D = -D + 2 return _int_tuple(D, 1, (1 - D)/4) def _lucas_extrastrong_params(n): """Calculates the "extra strong" parameters (D, P, Q) for n. References ========== .. [1] OEIS A217719: Extra Strong Lucas Pseudoprimes https://oeis.org/A217719 .. [1] https://en.wikipedia.org/wiki/Lucas_pseudoprime """ from sympy.ntheory.residue_ntheory import jacobi_symbol P, Q, D = 3, 1, 5 while True: g = igcd(D, n) if g > 1 and g != n: return (0, 0, 0) if jacobi_symbol(D, n) == -1: break P += 1 D = PP - 4 return _int_tuple(D, P, Q) def is_lucas_prp(n): """Standard Lucas compositeness test with Selfridge parameters. Returns False if n is definitely composite, and True if n is a Lucas probable prime. This is typically used in combination with the Miller-Rabin test. References ========== - "Lucas Pseudoprimes", Baillie and Wagstaff, 1980. http://mpqs.free.fr/LucasPseudoprimes.pdf - OEIS A217120: Lucas Pseudoprimes https://oeis.org/A217120 - https://en.wikipedia.org/wiki/Lucas_pseudoprime Examples ======== >>> from sympy.ntheory.primetest import isprime, is_lucas_prp >>> for i in range(10000): ... if is_lucas_prp(i) and not isprime(i): ... print(i) 323 377 1159 1829 3827 5459 5777 9071 9179 """ n = as_int(n) if n == 2: return True if n < 2 or (n % 2) == 0: return False if is_square(n, False): return False D, P, Q = _lucas_selfridge_params(n) if D == 0: return False U, V, Qk = _lucas_sequence(n, P, Q, n+1) return U == 0 def is_strong_lucas_prp(n): """Strong Lucas compositeness test with Selfridge parameters. Returns False if n is definitely composite, and True if n is a strong Lucas probable prime. This is often used in combination with the Miller-Rabin test, and in particular, when combined with M-R base 2 creates the strong BPSW test. References ========== - "Lucas Pseudoprimes", Baillie and Wagstaff, 1980. http://mpqs.free.fr/LucasPseudoprimes.pdf - OEIS A217255: Strong Lucas Pseudoprimes https://oeis.org/A217255 - https://en.wikipedia.org/wiki/Lucas_pseudoprime - https://en.wikipedia.org/wiki/Baillie-PSW_primality_test Examples ======== >>> from sympy.ntheory.primetest import isprime, is_strong_lucas_prp >>> for i in range(20000): ... if is_strong_lucas_prp(i) and not isprime(i): ... print(i) 5459 5777 10877 16109 18971 """ from sympy.ntheory.factor_ import trailing n = as_int(n) if n == 2: return True if n < 2 or (n % 2) == 0: return False if is_square(n, False): return False D, P, Q = _lucas_selfridge_params(n) if D == 0: return False # remove powers of 2 from n+1 (= k * 2*s) s = trailing(n + 1) k = (n+1) >> s U, V, Qk = _lucas_sequence(n, P, Q, k) if U == 0 or V == 0: return True for r in range(1, s): V = (VV - 2Qk) % n if V == 0: return True Qk = pow(Qk, 2, n) return False def is_extra_strong_lucas_prp(n): """Extra Strong Lucas compositeness test. Returns False if n is definitely composite, and True if n is a "extra strong" Lucas probable prime. The parameters are selected using P = 3, Q = 1, then incrementing P until (D\|n) == -1. The test itself is as defined in Grantham 2000, from the Mo and Jones preprint. The parameter selection and test are the same as used in OEIS A217719, Perl's Math::Prime::Util, and the Lucas pseudoprime page on Wikipedia. With these parameters, there are no counterexamples below 2^64 nor any known above that range. It is 20-50% faster than the strong test. Because of the different parameters selected, there is no relationship between the strong Lucas pseudoprimes and extra strong Lucas pseudoprimes. In particular, one is not a subset of the other. References ========== - "Frobenius Pseudoprimes", Jon Grantham, 2000. http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01197-2/ - OEIS A217719: Extra Strong Lucas Pseudoprimes https://oeis.org/A217719 - https://en.wikipedia.org/wiki/Lucas_pseudoprime Examples ======== >>> from sympy.ntheory.primetest import isprime, is_extra_strong_lucas_prp >>> for i in range(20000): ... if is_extra_strong_lucas_prp(i) and not isprime(i): ... print(i) 989 3239 5777 10877 """ # Implementation notes: # 1) the parameters differ from Thomas R. Nicely's. His parameter # selection leads to pseudoprimes that overlap M-R tests, and # contradict Baillie and Wagstaff's suggestion of (D\|n) = -1. # 2) The MathWorld page as of June 2013 specifies Q=-1. The Lucas # sequence must have Q=1. See Grantham theorem 2.3, any of the # references on the MathWorld page, or run it and see Q=-1 is wrong. from sympy.ntheory.factor_ import trailing n = as_int(n) if n == 2: return True if n < 2 or (n % 2) == 0: return False if is_square(n, False): return False D, P, Q = _lucas_extrastrong_params(n) if D == 0: return False # remove powers of 2 from n+1 (= k 2*s) s = trailing(n + 1) k = (n+1) >> s U, V, Qk = _lucas_sequence(n, P, Q, k) if U == 0 and (V == 2 or V == n - 2): return True for r in range(1, s): if V == 0: return True V = (VV - 2) % n return False def isprime(n): """ Test if n is a prime number (True) or not (False). For n < 2^64 the answer is definitive; larger n values have a small probability of actually being pseudoprimes. Negative numbers (e.g. -2) are not considered prime. The first step is looking for trivial factors, which if found enables a quick return. Next, if the sieve is large enough, use bisection search on the sieve. For small numbers, a set of deterministic Miller-Rabin tests are performed with bases that are known to have no counterexamples in their range. Finally if the number is larger than 2^64, a strong BPSW test is performed. While this is a probable prime test and we believe counterexamples exist, there are no known counterexamples. Examples ======== >>> from sympy.ntheory import isprime >>> isprime(13) True >>> isprime(13.0) # limited precision False >>> isprime(15) False Notes ===== This routine is intended only for integer input, not numerical expressions which may represent numbers. Floats are also rejected as input because they represent numbers of limited precision. While it is tempting to permit 7.0 to represent an integer there are errors that may "pass silently" if this is allowed: >>> from sympy import Float, S >>> int(1e3) == 1e3 == 103 True >>> int(1e23) == 1e23 True >>> int(1e23) == 1023 False >>> near_int = 1 + S(1)/1019 >>> near_int == int(near_int) False >>> n = Float(near_int, 10) # truncated by precision >>> n == int(n) True >>> n = Float(near_int, 20) >>> n == int(n) False See Also ======== sympy.ntheory.generate.primerange : Generates all primes in a given range sympy.ntheory.generate.primepi : Return the number of primes less than or equal to n sympy.ntheory.generate.prime : Return the nth prime References ========== - https://en.wikipedia.org/wiki/Strong_pseudoprime - "Lucas Pseudoprimes", Baillie and Wagstaff, 1980. http://mpqs.free.fr/LucasPseudoprimes.pdf - https://en.wikipedia.org/wiki/Baillie-PSW_primality_test """ try: n = as_int(n) except ValueError: return False # Step 1, do quick composite testing via trial division. The individual # modulo tests benchmark faster than one or two primorial igcds for me. # The point here is just to speedily handle small numbers and many # composites. Step 2 only requires that n <= 2 get handled here. if n in [2, 3, 5]: return True if n < 2 or (n % 2) == 0 or (n % 3) == 0 or (n % 5) == 0: return False if n < 49: return True if (n % 7) == 0 or (n % 11) == 0 or (n % 13) == 0 or (n % 17) == 0 or \ (n % 19) == 0 or (n % 23) == 0 or (n % 29) == 0 or (n % 31) == 0 or \ (n % 37) == 0 or (n % 41) == 0 or (n % 43) == 0 or (n % 47) == 0: return False if n < 2809: return True if n <= 23001: return pow(2, n, n) == 2 and n not in [7957, 8321, 13747, 18721, 19951] # bisection search on the sieve if the sieve is large enough from sympy.ntheory.generate import sieve as s if n <= s._list[-1]: l, u = s.search(n) return l == u # If we have GMPY2, skip straight to step 3 and do a strong BPSW test. # This should be a bit faster than our step 2, and for large values will # be a lot faster than our step 3 (C+GMP vs. Python). if HAS_GMPY == 2: from gmpy2 import is_strong_prp, is_strong_selfridge_prp return is_strong_prp(n, 2) and is_strong_selfridge_prp(n) # Step 2: deterministic Miller-Rabin testing for numbers < 2^64. See: # https://miller-rabin.appspot.com/ # for lists. We have made sure the M-R routine will successfully handle # bases larger than n, so we can use the minimal set. if n < 341531: return mr(n, [9345883071009581737]) if n < 885594169: return mr(n, [725270293939359937, 3569819667048198375]) if n < 350269456337: return mr(n, [4230279247111683200, 14694767155120705706, 16641139526367750375]) if n < 55245642489451: return mr(n, [2, 141889084524735, 1199124725622454117, 11096072698276303650]) if n < 7999252175582851: return mr(n, [2, 4130806001517, 149795463772692060, 186635894390467037, 3967304179347715805]) if n < 585226005592931977: return mr(n, [2, 123635709730000, 9233062284813009, 43835965440333360, 761179012939631437, 1263739024124850375]) if n < 18446744073709551616: return mr(n, [2, 325, 9375, 28178, 450775, 9780504, 1795265022]) # We could do this instead at any point: #if n < 18446744073709551616: # return mr(n, [2]) and is_extra_strong_lucas_prp(n) # Here are tests that are safe for MR routines that don't understand # large bases. #if n < 9080191: # return mr(n, [31, 73]) #if n < 19471033: # return mr(n, [2, 299417]) #if n < 38010307: # return mr(n, [2, 9332593]) #if n < 316349281: # return mr(n, [11000544, 31481107]) #if n < 4759123141: # return mr(n, [2, 7, 61]) #if n < 105936894253: # return mr(n, [2, 1005905886, 1340600841]) #if n < 31858317218647: # return mr(n, [2, 642735, 553174392, 3046413974]) #if n < 3071837692357849: # return mr(n, [2, 75088, 642735, 203659041, 3613982119]) #if n < 18446744073709551616: # return mr(n, [2, 325, 9375, 28178, 450775, 9780504, 1795265022]) # Step 3: BPSW. # # Time for isprime(102000 + 4561), no gmpy or gmpy2 installed # 44.0s old isprime using 46 bases # 5.3s strong BPSW + one random base # 4.3s extra strong BPSW + one random base # 4.1s strong BPSW # 3.2s extra strong BPSW # Classic BPSW from page 1401 of the paper. See alternate ideas below. return mr(n, [2]) and is_strong_lucas_prp(n) # Using extra strong test, which is somewhat faster #return mr(n, [2]) and is_extra_strong_lucas_prp(n) # Add a random M-R base #import random #return mr(n, [2, random.randint(3, n-1)]) and is_strong_lucas_prp(n) def is_gaussian_prime(num): r"""Test if num is a Gaussian prime number. References ========== .. [1] https://oeis.org/wiki/Gaussian_primes """ num = sympify(num) a, b = num.as_real_imag() a = as_int(a, strict=False) b = as_int(b, strict=False) if a == 0: b = abs(b) return isprime(b) and b % 4 == 3 elif b == 0: a = abs(a) return isprime(a) and a % 4 == 3 return isprime(a2 + b2)
3d525836e098b8263459094af6bc807511a3bfcd73ea320c22309bfff1345deb	from sympy.core.exprtools import factor_terms from sympy.core.numbers import Integer, Rational from sympy.core.singleton import S from sympy.core.symbol import Dummy from sympy.core.sympify import _sympify from sympy.utilities.misc import as_int def continued_fraction(a): """Return the continued fraction representation of a Rational or quadratic irrational. Examples ======== >>> from sympy.ntheory.continued_fraction import continued_fraction >>> from sympy import sqrt >>> continued_fraction((1 + 2sqrt(3))/5) [0, 1, [8, 3, 34, 3]] See Also ======== continued_fraction_periodic, continued_fraction_reduce, continued_fraction_convergents """ e = _sympify(a) if all(i.is_Rational for i in e.atoms()): if e.is_Integer: return continued_fraction_periodic(e, 1, 0) elif e.is_Rational: return continued_fraction_periodic(e.p, e.q, 0) elif e.is_Pow and e.exp is S.Half and e.base.is_Integer: return continued_fraction_periodic(0, 1, e.base) elif e.is_Mul and len(e.args) == 2 and ( e.args[0].is_Rational and e.args[1].is_Pow and e.args[1].base.is_Integer and e.args[1].exp is S.Half): a, b = e.args return continued_fraction_periodic(0, a.q, b.base, a.p) else: # this should not have to work very hard- no # simplification, cancel, etc... which should be # done by the user. e.g. This is a fancy 1 but # the user should simplify it first: # sqrt(2)(1 + sqrt(2))/(sqrt(2) + 2) p, d = e.expand().as_numer_denom() if d.is_Integer: if p.is_Rational: return continued_fraction_periodic(p, d) # look for a + bc # with c = sqrt(s) if p.is_Add and len(p.args) == 2: a, bc = p.args else: a = S.Zero bc = p if a.is_Integer: b = S.NaN if bc.is_Mul and len(bc.args) == 2: b, c = bc.args elif bc.is_Pow: b = Integer(1) c = bc if b.is_Integer and ( c.is_Pow and c.exp is S.Half and c.base.is_Integer): # (a + bsqrt(c))/d c = c.base return continued_fraction_periodic(a, d, c, b) raise ValueError( 'expecting a rational or quadratic irrational, not %s' % e) def continued_fraction_periodic(p, q, d=0, s=1): r""" Find the periodic continued fraction expansion of a quadratic irrational. Compute the continued fraction expansion of a rational or a quadratic irrational number, i.e. `\frac{p + s\sqrt{d}}{q}`, where `p`, `q \ne 0` and `d \ge 0` are integers. Returns the continued fraction representation (canonical form) as a list of integers, optionally ending (for quadratic irrationals) with list of integers representing the repeating digits. Parameters ========== p : int the rational part of the number's numerator q : int the denominator of the number d : int, optional the irrational part (discriminator) of the number's numerator s : int, optional the coefficient of the irrational part Examples ======== >>> from sympy.ntheory.continued_fraction import continued_fraction_periodic >>> continued_fraction_periodic(3, 2, 7) [2, [1, 4, 1, 1]] Golden ratio has the simplest continued fraction expansion: >>> continued_fraction_periodic(1, 2, 5) [[1]] If the discriminator is zero or a perfect square then the number will be a rational number: >>> continued_fraction_periodic(4, 3, 0) [1, 3] >>> continued_fraction_periodic(4, 3, 49) [3, 1, 2] See Also ======== continued_fraction_iterator, continued_fraction_reduce References ========== .. [1] https://en.wikipedia.org/wiki/Periodic_continued_fraction .. [2] K. Rosen. Elementary Number theory and its applications. Addison-Wesley, 3 Sub edition, pages 379-381, January 1992. """ from sympy.functions import sqrt, floor p, q, d, s = list(map(as_int, [p, q, d, s])) if d < 0: raise ValueError("expected non-negative for `d` but got %s" % d) if q == 0: raise ValueError("The denominator cannot be 0.") if not s: d = 0 # check for rational case sd = sqrt(d) if sd.is_Integer: return list(continued_fraction_iterator(Rational(p + ssd, q))) # irrational case with sd != Integer if q < 0: p, q, s = -p, -q, -s n = (p + ssd)/q if n < 0: w = floor(-n) f = -n - w one_f = continued_fraction(1 - f) # 1-f < 1 so cf is [0 ... [...]] one_f[0] -= w + 1 return one_f d = s2 sd = s if (d - p*2)%q: d = q*2 sd = q p = q q = q terms = [] pq = {} while (p, q) not in pq: pq[(p, q)] = len(terms) terms.append((p + sd)//q) p = terms[-1]q - p q = (d - p2)//q i = pq[(p, q)] return terms[:i] + [terms[i:]] def continued_fraction_reduce(cf): """ Reduce a continued fraction to a rational or quadratic irrational. Compute the rational or quadratic irrational number from its terminating or periodic continued fraction expansion. The continued fraction expansion (cf) should be supplied as a terminating iterator supplying the terms of the expansion. For terminating continued fractions, this is equivalent to ``list(continued_fraction_convergents(cf))[-1]``, only a little more efficient. If the expansion has a repeating part, a list of the repeating terms should be returned as the last element from the iterator. This is the format returned by continued_fraction_periodic. For quadratic irrationals, returns the largest solution found, which is generally the one sought, if the fraction is in canonical form (all terms positive except possibly the first). Examples ======== >>> from sympy.ntheory.continued_fraction import continued_fraction_reduce >>> continued_fraction_reduce([1, 2, 3, 4, 5]) 225/157 >>> continued_fraction_reduce([-2, 1, 9, 7, 1, 2]) -256/233 >>> continued_fraction_reduce([2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8]).n(10) 2.718281835 >>> continued_fraction_reduce([1, 4, 2, [3, 1]]) (sqrt(21) + 287)/238 >>> continued_fraction_reduce([[1]]) (1 + sqrt(5))/2 >>> from sympy.ntheory.continued_fraction import continued_fraction_periodic >>> continued_fraction_reduce(continued_fraction_periodic(8, 5, 13)) (sqrt(13) + 8)/5 See Also ======== continued_fraction_periodic """ from sympy.solvers import solve period = [] x = Dummy('x') def untillist(cf): for nxt in cf: if isinstance(nxt, list): period.extend(nxt) yield x break yield nxt a = S.Zero for a in continued_fraction_convergents(untillist(cf)): pass if period: y = Dummy('y') solns = solve(continued_fraction_reduce(period + [y]) - y, y) solns.sort() pure = solns[-1] rv = a.subs(x, pure).radsimp() else: rv = a if rv.is_Add: rv = factor_terms(rv) if rv.is_Mul and rv.args[0] == -1: rv = rv.func(rv.args) return rv def continued_fraction_iterator(x): """ Return continued fraction expansion of x as iterator. Examples ======== >>> from sympy import Rational, pi >>> from sympy.ntheory.continued_fraction import continued_fraction_iterator >>> list(continued_fraction_iterator(Rational(3, 8))) [0, 2, 1, 2] >>> list(continued_fraction_iterator(Rational(-3, 8))) [-1, 1, 1, 1, 2] >>> for i, v in enumerate(continued_fraction_iterator(pi)): ... if i > 7: ... break ... print(v) 3 7 15 1 292 1 1 1 References ========== .. [1] https://en.wikipedia.org/wiki/Continued_fraction """ from sympy.functions import floor while True: i = floor(x) yield i x -= i if not x: break x = 1/x def continued_fraction_convergents(cf): """ Return an iterator over the convergents of a continued fraction (cf). The parameter should be an iterable returning successive partial quotients of the continued fraction, such as might be returned by continued_fraction_iterator. In computing the convergents, the continued fraction need not be strictly in canonical form (all integers, all but the first positive). Rational and negative elements may be present in the expansion. Examples ======== >>> from sympy.core import pi >>> from sympy import S >>> from sympy.ntheory.continued_fraction import \ continued_fraction_convergents, continued_fraction_iterator >>> list(continued_fraction_convergents([0, 2, 1, 2])) [0, 1/2, 1/3, 3/8] >>> list(continued_fraction_convergents([1, S('1/2'), -7, S('1/4')])) [1, 3, 19/5, 7] >>> it = continued_fraction_convergents(continued_fraction_iterator(pi)) >>> for n in range(7): ... print(next(it)) 3 22/7 333/106 355/113 103993/33102 104348/33215 208341/66317 See Also ======== continued_fraction_iterator """ p_2, q_2 = S.Zero, S.One p_1, q_1 = S.One, S.Zero for a in cf: p, q = ap_1 + p_2, aq_1 + q_2 p_2, q_2 = p_1, q_1 p_1, q_1 = p, q yield p/q
2b9933797ef12e7c74af6d1d7b4511f6a293701789c108275fb99ccd1cd0fb04	from sympy.core.numbers import igcd, mod_inverse from sympy.core.power import integer_nthroot from sympy.ntheory.residue_ntheory import _sqrt_mod_prime_power from sympy.ntheory import isprime from math import log, sqrt import random rgen = random.Random() class SievePolynomial: def __init__(self, modified_coeff=(), a=None, b=None): """This class denotes the seive polynomial. If ``g(x) = (ax + b)2 - N``. `g(x)` can be expanded to ``ax*2 + 2abx + b2 - N``, so the coefficient is stored in the form `[a2, 2ab, b2 - N]`. This ensures faster `eval` method because we dont have to perform `a2, 2ab, b*2` every time we call the `eval` method. As multiplication is more expensive than addition, by using modified_coefficient we get a faster seiving process. Parameters ========== modified_coeff : modified_coefficient of sieve polynomial a : parameter of the sieve polynomial b : parameter of the sieve polynomial """ self.modified_coeff = modified_coeff self.a = a self.b = b def eval(self, x): """ Compute the value of the sieve polynomial at point x. Parameters ========== x : Integer parameter for sieve polynomial """ ans = 0 for coeff in self.modified_coeff: ans = x ans += coeff return ans class FactorBaseElem: """This class stores an element of the `factor_base`. """ def __init__(self, prime, tmem_p, log_p): """ Initialization of factor_base_elem. Parameters ========== prime : prime number of the factor_base tmem_p : Integer square root of x*2 = n mod prime log_p : Compute Natural Logarithm of the prime """ self.prime = prime self.tmem_p = tmem_p self.log_p = log_p self.soln1 = None self.soln2 = None self.a_inv = None self.b_ainv = None def _generate_factor_base(prime_bound, n): """Generate `factor_base` for Quadratic Sieve. The `factor_base` consists of all the points whose ``legendre_symbol(n, p) == 1`` and ``p < num_primes``. Along with the prime `factor_base` also stores natural logarithm of prime and the residue n modulo p. It also returns the of primes numbers in the `factor_base` which are close to 1000 and 5000. Parameters ========== prime_bound : upper prime bound of the factor_base n : integer to be factored """ from sympy.ntheory.generate import sieve factor_base = [] idx_1000, idx_5000 = None, None for prime in sieve.primerange(1, prime_bound): if pow(n, (prime - 1) // 2, prime) == 1: if prime > 1000 and idx_1000 is None: idx_1000 = len(factor_base) - 1 if prime > 5000 and idx_5000 is None: idx_5000 = len(factor_base) - 1 residue = _sqrt_mod_prime_power(n, prime, 1)[0] log_p = round(log(prime)2*10) factor_base.append(FactorBaseElem(prime, residue, log_p)) return idx_1000, idx_5000, factor_base def _initialize_first_polynomial(N, M, factor_base, idx_1000, idx_5000, seed=None): """This step is the initialization of the 1st sieve polynomial. Here `a` is selected as a product of several primes of the factor_base such that `a` is about to ``sqrt(2N) / M``. Other initial values of factor_base elem are also intialized which includes a_inv, b_ainv, soln1, soln2 which are used when the sieve polynomial is changed. The b_ainv is required for fast polynomial change as we do not have to calculate `2bmod_inverse(a, prime)` every time. We also ensure that the `factor_base` primes which make `a` are between 1000 and 5000. Parameters ========== N : Number to be factored M : sieve interval factor_base : factor_base primes idx_1000 : index of prime numbe in the factor_base near 1000 idx_5000 : index of primenumber in the factor_base near to 5000 seed : Generate pseudoprime numbers """ if seed is not None: rgen.seed(seed) approx_val = sqrt(2N) / M # `a` is a parameter of the sieve polynomial and `q` is the prime factors of `a` # randomly search for a combination of primes whose multiplication is close to approx_val # This multiplication of primes will be `a` and the primes will be `q` # `best_a` denotes that `a` is close to approx_val in the random search of combination best_a, best_q, best_ratio = None, None, None start = 0 if idx_1000 is None else idx_1000 end = len(factor_base) - 1 if idx_5000 is None else idx_5000 for _ in range(50): a = 1 q = [] while(a < approx_val): rand_p = 0 while(rand_p == 0 or rand_p in q): rand_p = rgen.randint(start, end) p = factor_base[rand_p].prime a = p q.append(rand_p) ratio = a / approx_val if best_ratio is None or abs(ratio - 1) < abs(best_ratio - 1): best_q = q best_a = a best_ratio = ratio a = best_a q = best_q B = [] for idx, val in enumerate(q): q_l = factor_base[val].prime gamma = factor_base[val].tmem_p * mod_inverse(a // q_l, q_l) % q_l if gamma > q_l / 2: gamma = q_l - gamma B.append(a//q_lgamma) b = sum(B) g = SievePolynomial([aa, 2ab, bb - N], a, b) for fb in factor_base: if a % fb.prime == 0: continue fb.a_inv = mod_inverse(a, fb.prime) fb.b_ainv = [2b_elemfb.a_inv % fb.prime for b_elem in B] fb.soln1 = (fb.a_inv(fb.tmem_p - b)) % fb.prime fb.soln2 = (fb.a_inv(-fb.tmem_p - b)) % fb.prime return g, B def _initialize_ith_poly(N, factor_base, i, g, B): """Initialization stage of ith poly. After we finish sieving 1`st polynomial here we quickly change to the next polynomial from which we will again start sieving. Suppose we generated ith sieve polynomial and now we want to generate (i + 1)th polynomial, where ``1 <= i <= 2(j - 1) - 1`` where `j` is the number of prime factors of the coefficient `a` then this function can be used to go to the next polynomial. If ``i = 2(j - 1) - 1`` then go to _initialize_first_polynomial stage. Parameters ========== N : number to be factored factor_base : factor_base primes i : integer denoting ith polynomial g : (i - 1)th polynomial B : array that stores a//q_lgamma """ from sympy.functions.elementary.integers import ceiling v = 1 j = i while(j % 2 == 0): v += 1 j //= 2 if ceiling(i / (2*v)) % 2 == 1: neg_pow = -1 else: neg_pow = 1 b = g.b + 2neg_powB[v - 1] a = g.a g = SievePolynomial([aa, 2ab, bb - N], a, b) for fb in factor_base: if a % fb.prime == 0: continue fb.soln1 = (fb.soln1 - neg_powfb.b_ainv[v - 1]) % fb.prime fb.soln2 = (fb.soln2 - neg_powfb.b_ainv[v - 1]) % fb.prime return g def _gen_sieve_array(M, factor_base): """Sieve Stage of the Quadratic Sieve. For every prime in the factor_base that does not divide the coefficient `a` we add log_p over the sieve_array such that ``-M <= soln1 + ip <= M`` and ``-M <= soln2 + ip <= M`` where `i` is an integer. When p = 2 then log_p is only added using ``-M <= soln1 + ip <= M``. Parameters ========== M : sieve interval factor_base : factor_base primes """ sieve_array = [0](2M + 1) for factor in factor_base: if factor.soln1 is None: #The prime does not divides a continue for idx in range((M + factor.soln1) % factor.prime, 2M, factor.prime): sieve_array[idx] += factor.log_p if factor.prime == 2: continue #if prime is 2 then sieve only with soln_1_p for idx in range((M + factor.soln2) % factor.prime, 2M, factor.prime): sieve_array[idx] += factor.log_p return sieve_array def _check_smoothness(num, factor_base): """Here we check that if `num` is a smooth number or not. If `a` is a smooth number then it returns a vector of prime exponents modulo 2. For example if a = 2 * 5*2 7*3 and the factor base contains {2, 3, 5, 7} then `a` is a smooth number and this function returns ([1, 0, 0, 1], True). If `a` is a partial relation which means that `a` a has one prime factor greater than the `factor_base` then it returns `(a, False)` which denotes `a` is a partial relation. Parameters ========== a : integer whose smootheness is to be checked factor_base : factor_base primes """ vec = [] if num < 0: vec.append(1) num = -1 else: vec.append(0) #-1 is not included in factor_base add -1 in vector for factor in factor_base: if num % factor.prime != 0: vec.append(0) continue factor_exp = 0 while num % factor.prime == 0: factor_exp += 1 num //= factor.prime vec.append(factor_exp % 2) if num == 1: return vec, True if isprime(num): return num, False return None, None def _trial_division_stage(N, M, factor_base, sieve_array, sieve_poly, partial_relations, ERROR_TERM): """Trial division stage. Here we trial divide the values generetated by sieve_poly in the sieve interval and if it is a smooth number then it is stored in `smooth_relations`. Moreover, if we find two partial relations with same large prime then they are combined to form a smooth relation. First we iterate over sieve array and look for values which are greater than accumulated_val, as these values have a high chance of being smooth number. Then using these values we find smooth relations. In general, let ``t*2 = up modN`` and ``r*2 = vp modN`` be two partial relations with the same large prime p. Then they can be combined ``(tr/p)2 = uv modN`` to form a smooth relation. Parameters ========== N : Number to be factored M : sieve interval factor_base : factor_base primes sieve_array : stores log_p values sieve_poly : polynomial from which we find smooth relations partial_relations : stores partial relations with one large prime ERROR_TERM : error term for accumulated_val """ sqrt_n = sqrt(float(N)) accumulated_val = log(M * sqrt_n)210 - ERROR_TERM smooth_relations = [] proper_factor = set() partial_relation_upper_bound = 128factor_base[-1].prime for idx, val in enumerate(sieve_array): if val < accumulated_val: continue x = idx - M v = sieve_poly.eval(x) vec, is_smooth = _check_smoothness(v, factor_base) if is_smooth is None:#Neither smooth nor partial continue u = sieve_poly.ax + sieve_poly.b # Update the partial relation # If 2 partial relation with same large prime is found then generate smooth relation if is_smooth is False:#partial relation found large_prime = vec #Consider the large_primes under 128F if large_prime > partial_relation_upper_bound: continue if large_prime not in partial_relations: partial_relations[large_prime] = (u, v) continue else: u_prev, v_prev = partial_relations[large_prime] partial_relations.pop(large_prime) try: large_prime_inv = mod_inverse(large_prime, N) except ValueError:#if large_prine divides N proper_factor.add(large_prime) continue u = uu_prevlarge_prime_inv v = vv_prev // (large_primelarge_prime) vec, is_smooth = _check_smoothness(v, factor_base) #assert uu % N == v % N smooth_relations.append((u, v, vec)) return smooth_relations, proper_factor #LINEAR ALGEBRA STAGE def _build_matrix(smooth_relations): """Build a 2D matrix from smooth relations. Parameters ========== smooth_relations : Stores smooth relations """ matrix = [] for s_relation in smooth_relations: matrix.append(s_relation[2]) return matrix def _gauss_mod_2(A): """Fast gaussian reduction for modulo 2 matrix. Parameters ========== A : Matrix Examples ======== >>> from sympy.ntheory.qs import _gauss_mod_2 >>> _gauss_mod_2([[0, 1, 1], [1, 0, 1], [0, 1, 0], [1, 1, 1]]) ([[[1, 0, 1], 3]], [True, True, True, False], [[0, 1, 0], [1, 0, 0], [0, 0, 1], [1, 0, 1]]) Reference ========== .. [1] A fast algorithm for gaussian elimination over GF(2) and its implementation on the GAPP. Cetin K.Koc, Sarath N.Arachchige""" import copy matrix = copy.deepcopy(A) row = len(matrix) col = len(matrix[0]) mark = [False]row for c in range(col): for r in range(row): if matrix[r][c] == 1: break mark[r] = True for c1 in range(col): if c1 == c: continue if matrix[r][c1] == 1: for r2 in range(row): matrix[r2][c1] = (matrix[r2][c1] + matrix[r2][c]) % 2 dependent_row = [] for idx, val in enumerate(mark): if val == False: dependent_row.append([matrix[idx], idx]) return dependent_row, mark, matrix def _find_factor(dependent_rows, mark, gauss_matrix, index, smooth_relations, N): """Finds proper factor of N. Here, transform the dependent rows as a combination of independent rows of the gauss_matrix to form the desired relation of the form ``X2 = Y2 modN``. After obtaining the desired relation we obtain a proper factor of N by `gcd(X - Y, N)`. Parameters ========== dependent_rows : denoted dependent rows in the reduced matrix form mark : boolean array to denoted dependent and independent rows gauss_matrix : Reduced form of the smooth relations matrix index : denoted the index of the dependent_rows smooth_relations : Smooth relations vectors matrix N : Number to be factored """ idx_in_smooth = dependent_rows[index][1] independent_u = [smooth_relations[idx_in_smooth][0]] independent_v = [smooth_relations[idx_in_smooth][1]] dept_row = dependent_rows[index][0] for idx, val in enumerate(dept_row): if val == 1: for row in range(len(gauss_matrix)): if gauss_matrix[row][idx] == 1 and mark[row] == True: independent_u.append(smooth_relations[row][0]) independent_v.append(smooth_relations[row][1]) break u = 1 v = 1 for i in independent_u: u = i for i in independent_v: v = i #assert u2 % N == v % N v = integer_nthroot(v, 2)[0] return igcd(u - v, N) def qs(N, prime_bound, M, ERROR_TERM=25, seed=1234): """Performs factorization using Self-Initializing Quadratic Sieve. In SIQS, let N be a number to be factored, and this N should not be a perfect power. If we find two integers such that ``X2 = Y2 modN`` and ``X != +-Y modN``, then `gcd(X + Y, N)` will reveal a proper factor of N. In order to find these integers X and Y we try to find relations of form t2 = u modN where u is a product of small primes. If we have enough of these relations then we can form ``(t1t2...ti)2 = u1u2...ui modN`` such that the right hand side is a square, thus we found a relation of ``X2 = Y2 modN``. Here, several optimizations are done like using muliple polynomials for sieving, fast changing between polynomials and using partial relations. The use of partial relations can speeds up the factoring by 2 times. Parameters ========== N : Number to be Factored prime_bound : upper bound for primes in the factor base M : Sieve Interval ERROR_TERM : Error term for checking smoothness threshold : Extra smooth relations for factorization seed : generate pseudo prime numbers Examples ======== >>> from sympy.ntheory import qs >>> qs(25645121643901801, 2000, 10000) {5394769, 4753701529} >>> qs(9804659461513846513, 2000, 10000) {4641991, 2112166839943} References ========== .. [1] https://pdfs.semanticscholar.org/5c52/8a975c1405bd35c65993abf5a4edb667c1db.pdf .. [2] https://www.rieselprime.de/ziki/Self-initializing_quadratic_sieve """ ERROR_TERM=210 rgen.seed(seed) idx_1000, idx_5000, factor_base = _generate_factor_base(prime_bound, N) smooth_relations = [] ith_poly = 0 partial_relations = {} proper_factor = set() threshold = 5len(factor_base) // 100 while True: if ith_poly == 0: ith_sieve_poly, B_array = _initialize_first_polynomial(N, M, factor_base, idx_1000, idx_5000) else: ith_sieve_poly = _initialize_ith_poly(N, factor_base, ith_poly, ith_sieve_poly, B_array) ith_poly += 1 if ith_poly >= 2**(len(B_array) - 1): # time to start with a new sieve polynomial ith_poly = 0 sieve_array = _gen_sieve_array(M, factor_base) s_rel, p_f = _trial_division_stage(N, M, factor_base, sieve_array, ith_sieve_poly, partial_relations, ERROR_TERM) smooth_relations += s_rel proper_factor \|= p_f if len(smooth_relations) >= len(factor_base) + threshold: break matrix = _build_matrix(smooth_relations) dependent_row, mark, gauss_matrix = _gauss_mod_2(matrix) N_copy = N for index in range(len(dependent_row)): factor = _find_factor(dependent_row, mark, gauss_matrix, index, smooth_relations, N) if factor > 1 and factor < N: proper_factor.add(factor) while(N_copy % factor == 0): N_copy //= factor if isprime(N_copy): proper_factor.add(N_copy) break if(N_copy == 1): break return proper_factor
d8a85eeb2707ed4889f3e0266cbc87a384564b1c3e52fcbd7d730555d8449681	""" Integer factorization """ from collections import defaultdict from functools import reduce import random import math from sympy.core import sympify from sympy.core.containers import Dict from sympy.core.evalf import bitcount from sympy.core.expr import Expr from sympy.core.function import Function from sympy.core.logic import fuzzy_and from sympy.core.mul import Mul, prod from sympy.core.numbers import igcd, ilcm, Rational, Integer from sympy.core.power import integer_nthroot, Pow, integer_log from sympy.core.singleton import S from sympy.external.gmpy import SYMPY_INTS from .primetest import isprime from .generate import sieve, primerange, nextprime from .digits import digits from sympy.utilities.iterables import flatten from sympy.utilities.misc import as_int, filldedent from .ecm import _ecm_one_factor # Note: This list should be updated whenever new Mersenne primes are found. # Refer: https://www.mersenne.org/ MERSENNE_PRIME_EXPONENTS = (2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933) # compute more when needed for i in Mersenne prime exponents PERFECT = [6] # 2*(i-1)(2i-1) MERSENNES = [3] # 2i - 1 def _ismersenneprime(n): global MERSENNES j = len(MERSENNES) while n > MERSENNES[-1] and j < len(MERSENNE_PRIME_EXPONENTS): # conservatively grow the list MERSENNES.append(2MERSENNE_PRIME_EXPONENTS[j] - 1) j += 1 return n in MERSENNES def _isperfect(n): global PERFECT if n % 2 == 0: j = len(PERFECT) while n > PERFECT[-1] and j < len(MERSENNE_PRIME_EXPONENTS): # conservatively grow the list t = 2(MERSENNE_PRIME_EXPONENTS[j] - 1) PERFECT.append(t(2t - 1)) j += 1 return n in PERFECT small_trailing = [0] * 256 for j in range(1,8): small_trailing[1<<j::1<<(j+1)] = [j] * (1<<(7-j)) def smoothness(n): """ Return the B-smooth and B-power smooth values of n. The smoothness of n is the largest prime factor of n; the power- smoothness is the largest divisor raised to its multiplicity. Examples ======== >>> from sympy.ntheory.factor_ import smoothness >>> smoothness(2*732) (3, 128) >>> smoothness(2413) (13, 16) >>> smoothness(2) (2, 2) See Also ======== factorint, smoothness_p """ if n == 1: return (1, 1) # not prime, but otherwise this causes headaches facs = factorint(n) return max(facs), max(mfacs[m] for m in facs) def smoothness_p(n, m=-1, power=0, visual=None): """ Return a list of [m, (p, (M, sm(p + m), psm(p + m)))...] where: 1. pM is the base-p divisor of n 2. sm(p + m) is the smoothness of p + m (m = -1 by default) 3. psm(p + m) is the power smoothness of p + m The list is sorted according to smoothness (default) or by power smoothness if power=1. The smoothness of the numbers to the left (m = -1) or right (m = 1) of a factor govern the results that are obtained from the p +/- 1 type factoring methods. >>> from sympy.ntheory.factor_ import smoothness_p, factorint >>> smoothness_p(10431, m=1) (1, [(3, (2, 2, 4)), (19, (1, 5, 5)), (61, (1, 31, 31))]) >>> smoothness_p(10431) (-1, [(3, (2, 2, 2)), (19, (1, 3, 9)), (61, (1, 5, 5))]) >>> smoothness_p(10431, power=1) (-1, [(3, (2, 2, 2)), (61, (1, 5, 5)), (19, (1, 3, 9))]) If visual=True then an annotated string will be returned: >>> print(smoothness_p(21477639576571, visual=1)) pi=44103171 has p-1 B=1787, B-pow=1787 pi=48698631 has p-1 B=2434931, B-pow=2434931 This string can also be generated directly from a factorization dictionary and vice versa: >>> factorint(179) {3: 2, 17: 1} >>> smoothness_p(_) 'pi=32 has p-1 B=2, B-pow=2\\npi=171 has p-1 B=2, B-pow=16' >>> smoothness_p(_) {3: 2, 17: 1} The table of the output logic is: ====== ====== ======= ======= \| Visual ------ ---------------------- Input True False other ====== ====== ======= ======= dict str tuple str str str tuple dict tuple str tuple str n str tuple tuple mul str tuple tuple ====== ====== ======= ======= See Also ======== factorint, smoothness """ # visual must be True, False or other (stored as None) if visual in (1, 0): visual = bool(visual) elif visual not in (True, False): visual = None if isinstance(n, str): if visual: return n d = {} for li in n.splitlines(): k, v = [int(i) for i in li.split('has')[0].split('=')[1].split('')] d[k] = v if visual is not True and visual is not False: return d return smoothness_p(d, visual=False) elif not isinstance(n, tuple): facs = factorint(n, visual=False) if power: k = -1 else: k = 1 if isinstance(n, tuple): rv = n else: rv = (m, sorted([(f, tuple([M] + list(smoothness(f + m)))) for f, M in [i for i in facs.items()]], key=lambda x: (x[1][k], x[0]))) if visual is False or (visual is not True) and (type(n) in [int, Mul]): return rv lines = [] for dat in rv[1]: dat = flatten(dat) dat.insert(2, m) lines.append('pi=%i%i has p%+i B=%i, B-pow=%i' % tuple(dat)) return '\n'.join(lines) def trailing(n): """Count the number of trailing zero digits in the binary representation of n, i.e. determine the largest power of 2 that divides n. Examples ======== >>> from sympy import trailing >>> trailing(128) 7 >>> trailing(63) 0 """ n = abs(int(n)) if not n: return 0 low_byte = n & 0xff if low_byte: return small_trailing[low_byte] # 2m is quick for z up through 2*30 z = bitcount(n) - 1 if isinstance(z, SYMPY_INTS): if n == 1 << z: return z if z < 300: # fixed 8-byte reduction t = 8 n >>= 8 while not n & 0xff: n >>= 8 t += 8 return t + small_trailing[n & 0xff] # binary reduction important when there might be a large # number of trailing 0s t = 0 p = 8 while not n & 1: while not n & ((1 << p) - 1): n >>= p t += p p = 2 p //= 2 return t def multiplicity(p, n): """ Find the greatest integer m such that pm divides n. Examples ======== >>> from sympy import multiplicity, Rational >>> [multiplicity(5, n) for n in [8, 5, 25, 125, 250]] [0, 1, 2, 3, 3] >>> multiplicity(3, Rational(1, 9)) -2 Note: when checking for the multiplicity of a number in a large factorial it is most efficient to send it as an unevaluated factorial or to call ``multiplicity_in_factorial`` directly: >>> from sympy.ntheory import multiplicity_in_factorial >>> from sympy import factorial >>> p = factorial(25) >>> n = 2100 >>> nfac = factorial(n, evaluate=False) >>> multiplicity(p, nfac) 52818775009509558395695966887 >>> _ == multiplicity_in_factorial(p, n) True """ try: p, n = as_int(p), as_int(n) except ValueError: from sympy.functions.combinatorial.factorials import factorial if all(isinstance(i, (SYMPY_INTS, Rational)) for i in (p, n)): p = Rational(p) n = Rational(n) if p.q == 1: if n.p == 1: return -multiplicity(p.p, n.q) return multiplicity(p.p, n.p) - multiplicity(p.p, n.q) elif p.p == 1: return multiplicity(p.q, n.q) else: like = min( multiplicity(p.p, n.p), multiplicity(p.q, n.q)) cross = min( multiplicity(p.q, n.p), multiplicity(p.p, n.q)) return like - cross elif (isinstance(p, (SYMPY_INTS, Integer)) and isinstance(n, factorial) and isinstance(n.args[0], Integer) and n.args[0] >= 0): return multiplicity_in_factorial(p, n.args[0]) raise ValueError('expecting ints or fractions, got %s and %s' % (p, n)) if n == 0: raise ValueError('no such integer exists: multiplicity of %s is not-defined' %(n)) if p == 2: return trailing(n) if p < 2: raise ValueError('p must be an integer, 2 or larger, but got %s' % p) if p == n: return 1 m = 0 n, rem = divmod(n, p) while not rem: m += 1 if m > 5: # The multiplicity could be very large. Better # to increment in powers of two e = 2 while 1: ppow = p*e if ppow < n: nnew, rem = divmod(n, ppow) if not rem: m += e e = 2 n = nnew continue return m + multiplicity(p, n) n, rem = divmod(n, p) return m def multiplicity_in_factorial(p, n): """return the largest integer ``m`` such that ``pm`` divides ``n!`` without calculating the factorial of ``n``. Examples ======== >>> from sympy.ntheory import multiplicity_in_factorial >>> from sympy import factorial >>> multiplicity_in_factorial(2, 3) 1 An instructive use of this is to tell how many trailing zeros a given factorial has. For example, there are 6 in 25!: >>> factorial(25) 15511210043330985984000000 >>> multiplicity_in_factorial(10, 25) 6 For large factorials, it is much faster/feasible to use this function rather than computing the actual factorial: >>> multiplicity_in_factorial(factorial(25), 2100) 52818775009509558395695966887 """ p, n = as_int(p), as_int(n) if p <= 0: raise ValueError('expecting positive integer got %s' % p ) if n < 0: raise ValueError('expecting non-negative integer got %s' % n ) factors = factorint(p) # keep only the largest of a given multiplicity since those # of a given multiplicity will be goverened by the behavior # of the largest factor test = defaultdict(int) for k, v in factors.items(): test[v] = max(k, test[v]) keep = set(test.values()) # remove others from factors for k in list(factors.keys()): if k not in keep: factors.pop(k) mp = S.Infinity for i in factors: # multiplicity of i in n! is mi = (n - (sum(digits(n, i)) - i))//(i - 1) # multiplicity of p in n! depends on multiplicity # of prime `i` in p, so we floor divide by factors[i] # and keep it if smaller than the multiplicity of p # seen so far mp = min(mp, mi//factors[i]) return mp def perfect_power(n, candidates=None, big=True, factor=True): """ Return ``(b, e)`` such that ``n`` == ``be`` if ``n`` is a unique perfect power with ``e > 1``, else ``False`` (e.g. 1 is not a perfect power). A ValueError is raised if ``n`` is not Rational. By default, the base is recursively decomposed and the exponents collected so the largest possible ``e`` is sought. If ``big=False`` then the smallest possible ``e`` (thus prime) will be chosen. If ``factor=True`` then simultaneous factorization of ``n`` is attempted since finding a factor indicates the only possible root for ``n``. This is True by default since only a few small factors will be tested in the course of searching for the perfect power. The use of ``candidates`` is primarily for internal use; if provided, False will be returned if ``n`` cannot be written as a power with one of the candidates as an exponent and factoring (beyond testing for a factor of 2) will not be attempted. Examples ======== >>> from sympy import perfect_power, Rational >>> perfect_power(16) (2, 4) >>> perfect_power(16, big=False) (4, 2) Negative numbers can only have odd perfect powers: >>> perfect_power(-4) False >>> perfect_power(-8) (-2, 3) Rationals are also recognized: >>> perfect_power(Rational(1, 2)3) (1/2, 3) >>> perfect_power(Rational(-3, 2)3) (-3/2, 3) Notes ===== To know whether an integer is a perfect power of 2 use >>> is2pow = lambda n: bool(n and not n & (n - 1)) >>> [(i, is2pow(i)) for i in range(5)] [(0, False), (1, True), (2, True), (3, False), (4, True)] It is not necessary to provide ``candidates``. When provided it will be assumed that they are ints. The first one that is larger than the computed maximum possible exponent will signal failure for the routine. >>> perfect_power(38, [9]) False >>> perfect_power(38, [2, 4, 8]) (3, 8) >>> perfect_power(38, [4, 8], big=False) (9, 4) See Also ======== sympy.core.power.integer_nthroot sympy.ntheory.primetest.is_square """ if isinstance(n, Rational) and not n.is_Integer: p, q = n.as_numer_denom() if p is S.One: pp = perfect_power(q) if pp: pp = (n.func(1, pp[0]), pp[1]) else: pp = perfect_power(p) if pp: num, e = pp pq = perfect_power(q, [e]) if pq: den, _ = pq pp = n.func(num, den), e return pp n = as_int(n) if n < 0: pp = perfect_power(-n) if pp: b, e = pp if e % 2: return -b, e return False if n <= 3: # no unique exponent for 0, 1 # 2 and 3 have exponents of 1 return False logn = math.log(n, 2) max_possible = int(logn) + 2 # only check values less than this not_square = n % 10 in [2, 3, 7, 8] # squares cannot end in 2, 3, 7, 8 min_possible = 2 + not_square if not candidates: candidates = primerange(min_possible, max_possible) else: candidates = sorted([i for i in candidates if min_possible <= i < max_possible]) if n%2 == 0: e = trailing(n) candidates = [i for i in candidates if e%i == 0] if big: candidates = reversed(candidates) for e in candidates: r, ok = integer_nthroot(n, e) if ok: return (r, e) return False def _factors(): rv = 2 + n % 2 while True: yield rv rv = nextprime(rv) for fac, e in zip(_factors(), candidates): # see if there is a factor present if factor and n % fac == 0: # find what the potential power is if fac == 2: e = trailing(n) else: e = multiplicity(fac, n) # if it's a trivial power we are done if e == 1: return False # maybe the e-th root of n is exact r, exact = integer_nthroot(n, e) if not exact: # Having a factor, we know that e is the maximal # possible value for a root of n. # If n = fac*em can be written as a perfect # power then see if m can be written as rE where # gcd(e, E) != 1 so n = (fac(e//E)r)E m = n//face rE = perfect_power(m, candidates=divisors(e, generator=True)) if not rE: return False else: r, E = rE r, e = fac(e//E)r, E if not big: e0 = primefactors(e) if e0[0] != e: r, e = r(e//e0[0]), e0[0] return r, e # Weed out downright impossible candidates if logn/e < 40: b = 2.0(logn/e) if abs(int(b + 0.5) - b) > 0.01: continue # now see if the plausible e makes a perfect power r, exact = integer_nthroot(n, e) if exact: if big: m = perfect_power(r, big=big, factor=factor) if m: r, e = m[0], em[1] return int(r), e return False def pollard_rho(n, s=2, a=1, retries=5, seed=1234, max_steps=None, F=None): r""" Use Pollard's rho method to try to extract a nontrivial factor of ``n``. The returned factor may be a composite number. If no factor is found, ``None`` is returned. The algorithm generates pseudo-random values of x with a generator function, replacing x with F(x). If F is not supplied then the function x2 + ``a`` is used. The first value supplied to F(x) is ``s``. Upon failure (if ``retries`` is > 0) a new ``a`` and ``s`` will be supplied; the ``a`` will be ignored if F was supplied. The sequence of numbers generated by such functions generally have a a lead-up to some number and then loop around back to that number and begin to repeat the sequence, e.g. 1, 2, 3, 4, 5, 3, 4, 5 -- this leader and loop look a bit like the Greek letter rho, and thus the name, 'rho'. For a given function, very different leader-loop values can be obtained so it is a good idea to allow for retries: >>> from sympy.ntheory.generate import cycle_length >>> n = 16843009 >>> F = lambda x:(2048pow(x, 2, n) + 32767) % n >>> for s in range(5): ... print('loop length = %4i; leader length = %3i' % next(cycle_length(F, s))) ... loop length = 2489; leader length = 42 loop length = 78; leader length = 120 loop length = 1482; leader length = 99 loop length = 1482; leader length = 285 loop length = 1482; leader length = 100 Here is an explicit example where there is a two element leadup to a sequence of 3 numbers (11, 14, 4) that then repeat: >>> x=2 >>> for i in range(9): ... x=(x2+12)%17 ... print(x) ... 16 13 11 14 4 11 14 4 11 >>> next(cycle_length(lambda x: (x2+12)%17, 2)) (3, 2) >>> list(cycle_length(lambda x: (x*2+12)%17, 2, values=True)) [16, 13, 11, 14, 4] Instead of checking the differences of all generated values for a gcd with n, only the kth and 2kth numbers are checked, e.g. 1st and 2nd, 2nd and 4th, 3rd and 6th until it has been detected that the loop has been traversed. Loops may be many thousands of steps long before rho finds a factor or reports failure. If ``max_steps`` is specified, the iteration is cancelled with a failure after the specified number of steps. Examples ======== >>> from sympy import pollard_rho >>> n=16843009 >>> F=lambda x:(2048pow(x,2,n) + 32767) % n >>> pollard_rho(n, F=F) 257 Use the default setting with a bad value of ``a`` and no retries: >>> pollard_rho(n, a=n-2, retries=0) If retries is > 0 then perhaps the problem will correct itself when new values are generated for a: >>> pollard_rho(n, a=n-2, retries=1) 257 References ========== .. [1] Richard Crandall & Carl Pomerance (2005), "Prime Numbers: A Computational Perspective", Springer, 2nd edition, 229-231 """ n = int(n) if n < 5: raise ValueError('pollard_rho should receive n > 4') prng = random.Random(seed + retries) V = s for i in range(retries + 1): U = V if not F: F = lambda x: (pow(x, 2, n) + a) % n j = 0 while 1: if max_steps and (j > max_steps): break j += 1 U = F(U) V = F(F(V)) # V is 2x further along than U g = igcd(U - V, n) if g == 1: continue if g == n: break return int(g) V = prng.randint(0, n - 1) a = prng.randint(1, n - 3) # for x2 + a, a%n should not be 0 or -2 F = None return None def pollard_pm1(n, B=10, a=2, retries=0, seed=1234): """ Use Pollard's p-1 method to try to extract a nontrivial factor of ``n``. Either a divisor (perhaps composite) or ``None`` is returned. The value of ``a`` is the base that is used in the test gcd(aM - 1, n). The default is 2. If ``retries`` > 0 then if no factor is found after the first attempt, a new ``a`` will be generated randomly (using the ``seed``) and the process repeated. Note: the value of M is lcm(1..B) = reduce(ilcm, range(2, B + 1)). A search is made for factors next to even numbers having a power smoothness less than ``B``. Choosing a larger B increases the likelihood of finding a larger factor but takes longer. Whether a factor of n is found or not depends on ``a`` and the power smoothness of the even number just less than the factor p (hence the name p - 1). Although some discussion of what constitutes a good ``a`` some descriptions are hard to interpret. At the modular.math site referenced below it is stated that if gcd(aM - 1, n) = N then aM % qr is 1 for every prime power divisor of N. But consider the following: >>> from sympy.ntheory.factor_ import smoothness_p, pollard_pm1 >>> n=2571009 >>> smoothness_p(n) (-1, [(257, (1, 2, 256)), (1009, (1, 7, 16))]) So we should (and can) find a root with B=16: >>> pollard_pm1(n, B=16, a=3) 1009 If we attempt to increase B to 256 we find that it does not work: >>> pollard_pm1(n, B=256) >>> But if the value of ``a`` is changed we find that only multiples of 257 work, e.g.: >>> pollard_pm1(n, B=256, a=257) 1009 Checking different ``a`` values shows that all the ones that did not work had a gcd value not equal to ``n`` but equal to one of the factors: >>> from sympy import ilcm, igcd, factorint, Pow >>> M = 1 >>> for i in range(2, 256): ... M = ilcm(M, i) ... >>> set([igcd(pow(a, M, n) - 1, n) for a in range(2, 256) if ... igcd(pow(a, M, n) - 1, n) != n]) {1009} But does aM % d for every divisor of n give 1? >>> aM = pow(255, M, n) >>> [(d, aM%Pow(d.args)) for d in factorint(n, visual=True).args] [(2571, 1), (10091, 1)] No, only one of them. So perhaps the principle is that a root will be found for a given value of B provided that: 1) the power smoothness of the p - 1 value next to the root does not exceed B 2) aM % p != 1 for any of the divisors of n. By trying more than one ``a`` it is possible that one of them will yield a factor. Examples ======== With the default smoothness bound, this number cannot be cracked: >>> from sympy.ntheory import pollard_pm1 >>> pollard_pm1(21477639576571) Increasing the smoothness bound helps: >>> pollard_pm1(21477639576571, B=2000) 4410317 Looking at the smoothness of the factors of this number we find: >>> from sympy.ntheory.factor_ import smoothness_p, factorint >>> print(smoothness_p(21477639576571, visual=1)) pi=44103171 has p-1 B=1787, B-pow=1787 pi=48698631 has p-1 B=2434931, B-pow=2434931 The B and B-pow are the same for the p - 1 factorizations of the divisors because those factorizations had a very large prime factor: >>> factorint(4410317 - 1) {2: 2, 617: 1, 1787: 1} >>> factorint(4869863-1) {2: 1, 2434931: 1} Note that until B reaches the B-pow value of 1787, the number is not cracked; >>> pollard_pm1(21477639576571, B=1786) >>> pollard_pm1(21477639576571, B=1787) 4410317 The B value has to do with the factors of the number next to the divisor, not the divisors themselves. A worst case scenario is that the number next to the factor p has a large prime divisisor or is a perfect power. If these conditions apply then the power-smoothness will be about p/2 or p. The more realistic is that there will be a large prime factor next to p requiring a B value on the order of p/2. Although primes may have been searched for up to this level, the p/2 is a factor of p - 1, something that we do not know. The modular.math reference below states that 15% of numbers in the range of 1015 to 1515 + 104 are 106 power smooth so a B of 106 will fail 85% of the time in that range. From 108 to 108 + 103 the percentages are nearly reversed...but in that range the simple trial division is quite fast. References ========== .. [1] Richard Crandall & Carl Pomerance (2005), "Prime Numbers: A Computational Perspective", Springer, 2nd edition, 236-238 .. [2] http://modular.math.washington.edu/edu/2007/spring/ent/ent-html/node81.html .. [3] https://www.cs.toronto.edu/~yuvalf/Factorization.pdf """ n = int(n) if n < 4 or B < 3: raise ValueError('pollard_pm1 should receive n > 3 and B > 2') prng = random.Random(seed + B) # computing alcm(1,2,3,..B) % n for B > 2 # it looks weird, but it's right: primes run [2, B] # and the answer's not right until the loop is done. for i in range(retries + 1): aM = a for p in sieve.primerange(2, B + 1): e = int(math.log(B, p)) aM = pow(aM, pow(p, e), n) g = igcd(aM - 1, n) if 1 < g < n: return int(g) # get a new a: # since the exponent, lcm(1..B), is even, if we allow 'a' to be 'n-1' # then (n - 1)even % n will be 1 which will give a g of 0 and 1 will # give a zero, too, so we set the range as [2, n-2]. Some references # say 'a' should be coprime to n, but either will detect factors. a = prng.randint(2, n - 2) def _trial(factors, n, candidates, verbose=False): """ Helper function for integer factorization. Trial factors ``n` against all integers given in the sequence ``candidates`` and updates the dict ``factors`` in-place. Returns the reduced value of ``n`` and a flag indicating whether any factors were found. """ if verbose: factors0 = list(factors.keys()) nfactors = len(factors) for d in candidates: if n % d == 0: m = multiplicity(d, n) n //= dm factors[d] = m if verbose: for k in sorted(set(factors).difference(set(factors0))): print(factor_msg % (k, factors[k])) return int(n), len(factors) != nfactors def _check_termination(factors, n, limitp1, use_trial, use_rho, use_pm1, verbose): """ Helper function for integer factorization. Checks if ``n`` is a prime or a perfect power, and in those cases updates the factorization and raises ``StopIteration``. """ if verbose: print('Check for termination') # since we've already been factoring there is no need to do # simultaneous factoring with the power check p = perfect_power(n, factor=False) if p is not False: base, exp = p if limitp1: limit = limitp1 - 1 else: limit = limitp1 facs = factorint(base, limit, use_trial, use_rho, use_pm1, verbose=False) for b, e in facs.items(): if verbose: print(factor_msg % (b, e)) factors[b] = expe raise StopIteration if isprime(n): factors[int(n)] = 1 raise StopIteration if n == 1: raise StopIteration trial_int_msg = "Trial division with ints [%i ... %i] and fail_max=%i" trial_msg = "Trial division with primes [%i ... %i]" rho_msg = "Pollard's rho with retries %i, max_steps %i and seed %i" pm1_msg = "Pollard's p-1 with smoothness bound %i and seed %i" ecm_msg = "Elliptic Curve with B1 bound %i, B2 bound %i, num_curves %i" factor_msg = '\t%i ** %i' fermat_msg = 'Close factors satisying Fermat condition found.' complete_msg = 'Factorization is complete.' def _factorint_small(factors, n, limit, fail_max): """ Return the value of n and either a 0 (indicating that factorization up to the limit was complete) or else the next near-prime that would have been tested. Factoring stops if there are fail_max unsuccessful tests in a row. If factors of n were found they will be in the factors dictionary as {factor: multiplicity} and the returned value of n will have had those factors removed. The factors dictionary is modified in-place. """ def done(n, d): """return n, d if the sqrt(n) was not reached yet, else n, 0 indicating that factoring is done. """ if dd <= n: return n, d return n, 0 d = 2 m = trailing(n) if m: factors[d] = m n >>= m d = 3 if limit < d: if n > 1: factors[n] = 1 return done(n, d) # reduce m = 0 while n % d == 0: n //= d m += 1 if m == 20: mm = multiplicity(d, n) m += mm n //= dmm break if m: factors[d] = m # when dd exceeds maxx or n we are done; if limit*2 is greater # than n then maxx is set to zero so the value of n will flag the finish if limitlimit > n: maxx = 0 else: maxx = limitlimit dd = maxx or n d = 5 fails = 0 while fails < fail_max: if dd > dd: break # d = 6i - 1 # reduce m = 0 while n % d == 0: n //= d m += 1 if m == 20: mm = multiplicity(d, n) m += mm n //= dmm break if m: factors[d] = m dd = maxx or n fails = 0 else: fails += 1 d += 2 if dd > dd: break # d = 6i - 1 # reduce m = 0 while n % d == 0: n //= d m += 1 if m == 20: mm = multiplicity(d, n) m += mm n //= dmm break if m: factors[d] = m dd = maxx or n fails = 0 else: fails += 1 # d = 6(i + 1) - 1 d += 4 return done(n, d) def factorint(n, limit=None, use_trial=True, use_rho=True, use_pm1=True, use_ecm=True, verbose=False, visual=None, multiple=False): r""" Given a positive integer ``n``, ``factorint(n)`` returns a dict containing the prime factors of ``n`` as keys and their respective multiplicities as values. For example: >>> from sympy.ntheory import factorint >>> factorint(2000) # 2000 = (2*4) (5*3) {2: 4, 5: 3} >>> factorint(65537) # This number is prime {65537: 1} For input less than 2, factorint behaves as follows: - ``factorint(1)`` returns the empty factorization, ``{}`` - ``factorint(0)`` returns ``{0:1}`` - ``factorint(-n)`` adds ``-1:1`` to the factors and then factors ``n`` Partial Factorization: If ``limit`` (> 3) is specified, the search is stopped after performing trial division up to (and including) the limit (or taking a corresponding number of rho/p-1 steps). This is useful if one has a large number and only is interested in finding small factors (if any). Note that setting a limit does not prevent larger factors from being found early; it simply means that the largest factor may be composite. Since checking for perfect power is relatively cheap, it is done regardless of the limit setting. This number, for example, has two small factors and a huge semi-prime factor that cannot be reduced easily: >>> from sympy.ntheory import isprime >>> a = 1407633717262338957430697921446883 >>> f = factorint(a, limit=10000) >>> f == {991: 1, int(202916782076162456022877024859): 1, 7: 1} True >>> isprime(max(f)) False This number has a small factor and a residual perfect power whose base is greater than the limit: >>> factorint(3101*7, limit=5) {3: 1, 101: 7} List of Factors: If ``multiple`` is set to ``True`` then a list containing the prime factors including multiplicities is returned. >>> factorint(24, multiple=True) [2, 2, 2, 3] Visual Factorization: If ``visual`` is set to ``True``, then it will return a visual factorization of the integer. For example: >>> from sympy import pprint >>> pprint(factorint(4200, visual=True)) 3 1 2 1 2 3 5 7 Note that this is achieved by using the evaluate=False flag in Mul and Pow. If you do other manipulations with an expression where evaluate=False, it may evaluate. Therefore, you should use the visual option only for visualization, and use the normal dictionary returned by visual=False if you want to perform operations on the factors. You can easily switch between the two forms by sending them back to factorint: >>> from sympy import Mul >>> regular = factorint(1764); regular {2: 2, 3: 2, 7: 2} >>> pprint(factorint(regular)) 2 2 2 2 3 7 >>> visual = factorint(1764, visual=True); pprint(visual) 2 2 2 2 3 7 >>> print(factorint(visual)) {2: 2, 3: 2, 7: 2} If you want to send a number to be factored in a partially factored form you can do so with a dictionary or unevaluated expression: >>> factorint(factorint({4: 2, 12: 3})) # twice to toggle to dict form {2: 10, 3: 3} >>> factorint(Mul(4, 12, evaluate=False)) {2: 4, 3: 1} The table of the output logic is: ====== ====== ======= ======= Visual ------ ---------------------- Input True False other ====== ====== ======= ======= dict mul dict mul n mul dict dict mul mul dict dict ====== ====== ======= ======= Notes ===== Algorithm: The function switches between multiple algorithms. Trial division quickly finds small factors (of the order 1-5 digits), and finds all large factors if given enough time. The Pollard rho and p-1 algorithms are used to find large factors ahead of time; they will often find factors of the order of 10 digits within a few seconds: >>> factors = factorint(12345678910111213141516) >>> for base, exp in sorted(factors.items()): ... print('%s %s' % (base, exp)) ... 2 2 2507191691 1 1231026625769 1 Any of these methods can optionally be disabled with the following boolean parameters: - ``use_trial``: Toggle use of trial division - ``use_rho``: Toggle use of Pollard's rho method - ``use_pm1``: Toggle use of Pollard's p-1 method ``factorint`` also periodically checks if the remaining part is a prime number or a perfect power, and in those cases stops. For unevaluated factorial, it uses Legendre's formula(theorem). If ``verbose`` is set to ``True``, detailed progress is printed. See Also ======== smoothness, smoothness_p, divisors """ if isinstance(n, Dict): n = dict(n) if multiple: fac = factorint(n, limit=limit, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, verbose=verbose, visual=False, multiple=False) factorlist = sum(([p] * fac[p] if fac[p] > 0 else [S.One/p](-fac[p]) for p in sorted(fac)), []) return factorlist factordict = {} if visual and not isinstance(n, (Mul, dict)): factordict = factorint(n, limit=limit, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, verbose=verbose, visual=False) elif isinstance(n, Mul): factordict = {int(k): int(v) for k, v in n.as_powers_dict().items()} elif isinstance(n, dict): factordict = n if factordict and isinstance(n, (Mul, dict)): # check it for key in list(factordict.keys()): if isprime(key): continue e = factordict.pop(key) d = factorint(key, limit=limit, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, verbose=verbose, visual=False) for k, v in d.items(): if k in factordict: factordict[k] += ve else: factordict[k] = ve if visual or (type(n) is dict and visual is not True and visual is not False): if factordict == {}: return S.One if -1 in factordict: factordict.pop(-1) args = [S.NegativeOne] else: args = [] args.extend([Pow(i, evaluate=False) for i in sorted(factordict.items())]) return Mul(args, evaluate=False) elif isinstance(n, (dict, Mul)): return factordict assert use_trial or use_rho or use_pm1 or use_ecm from sympy.functions.combinatorial.factorials import factorial if isinstance(n, factorial): x = as_int(n.args[0]) if x >= 20: factors = {} m = 2 # to initialize the if condition below for p in sieve.primerange(2, x + 1): if m > 1: m, q = 0, x // p while q != 0: m += q q //= p factors[p] = m if factors and verbose: for k in sorted(factors): print(factor_msg % (k, factors[k])) if verbose: print(complete_msg) return factors else: # if n < 20!, direct computation is faster # since it uses a lookup table n = n.func(x) n = as_int(n) if limit: limit = int(limit) use_ecm = False # special cases if n < 0: factors = factorint( -n, limit=limit, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, verbose=verbose, visual=False) factors[-1] = 1 return factors if limit and limit < 2: if n == 1: return {} return {n: 1} elif n < 10: # doing this we are assured of getting a limit > 2 # when we have to compute it later return [{0: 1}, {}, {2: 1}, {3: 1}, {2: 2}, {5: 1}, {2: 1, 3: 1}, {7: 1}, {2: 3}, {3: 2}][n] factors = {} # do simplistic factorization if verbose: sn = str(n) if len(sn) > 50: print('Factoring %s' % sn[:5] + \ '..(%i other digits)..' % (len(sn) - 10) + sn[-5:]) else: print('Factoring', n) if use_trial: # this is the preliminary factorization for small factors small = 215 fail_max = 600 small = min(small, limit or small) if verbose: print(trial_int_msg % (2, small, fail_max)) n, next_p = _factorint_small(factors, n, small, fail_max) else: next_p = 2 if factors and verbose: for k in sorted(factors): print(factor_msg % (k, factors[k])) if next_p == 0: if n > 1: factors[int(n)] = 1 if verbose: print(complete_msg) return factors # continue with more advanced factorization methods # first check if the simplistic run didn't finish # because of the limit and check for a perfect # power before exiting try: if limit and next_p > limit: if verbose: print('Exceeded limit:', limit) _check_termination(factors, n, limit, use_trial, use_rho, use_pm1, verbose) if n > 1: factors[int(n)] = 1 return factors else: # Before quitting (or continuing on)... # ...do a Fermat test since it's so easy and we need the # square root anyway. Finding 2 factors is easy if they are # "close enough." This is the big root equivalent of dividing by # 2, 3, 5. sqrt_n = integer_nthroot(n, 2)[0] a = sqrt_n + 1 a2 = a2 b2 = a2 - n for i in range(3): b, fermat = integer_nthroot(b2, 2) if fermat: break b2 += 2a + 1 # equiv to (a + 1)*2 - n a += 1 if fermat: if verbose: print(fermat_msg) if limit: limit -= 1 for r in [a - b, a + b]: facs = factorint(r, limit=limit, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, verbose=verbose) for k, v in facs.items(): factors[k] = factors.get(k, 0) + v raise StopIteration # ...see if factorization can be terminated _check_termination(factors, n, limit, use_trial, use_rho, use_pm1, verbose) except StopIteration: if verbose: print(complete_msg) return factors # these are the limits for trial division which will # be attempted in parallel with pollard methods low, high = next_p, 2next_p limit = limit or sqrt_n # add 1 to make sure limit is reached in primerange calls limit += 1 iteration = 0 while 1: try: high_ = high if limit < high_: high_ = limit # Trial division if use_trial: if verbose: print(trial_msg % (low, high_)) ps = sieve.primerange(low, high_) n, found_trial = _trial(factors, n, ps, verbose) if found_trial: _check_termination(factors, n, limit, use_trial, use_rho, use_pm1, verbose) else: found_trial = False if high > limit: if verbose: print('Exceeded limit:', limit) if n > 1: factors[int(n)] = 1 raise StopIteration # Only used advanced methods when no small factors were found if not found_trial: if (use_pm1 or use_rho): high_root = max(int(math.log(high_*0.7)), low, 3) # Pollard p-1 if use_pm1: if verbose: print(pm1_msg % (high_root, high_)) c = pollard_pm1(n, B=high_root, seed=high_) if c: # factor it and let _trial do the update ps = factorint(c, limit=limit - 1, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, use_ecm=use_ecm, verbose=verbose) n, _ = _trial(factors, n, ps, verbose=False) _check_termination(factors, n, limit, use_trial, use_rho, use_pm1, verbose) # Pollard rho if use_rho: max_steps = high_root if verbose: print(rho_msg % (1, max_steps, high_)) c = pollard_rho(n, retries=1, max_steps=max_steps, seed=high_) if c: # factor it and let _trial do the update ps = factorint(c, limit=limit - 1, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, use_ecm=use_ecm, verbose=verbose) n, _ = _trial(factors, n, ps, verbose=False) _check_termination(factors, n, limit, use_trial, use_rho, use_pm1, verbose) except StopIteration: if verbose: print(complete_msg) return factors #Use subexponential algorithms if use_ecm #Use pollard algorithms for finding small factors for 3 iterations #if after small factors the number of digits of n is >= 20 then use ecm iteration += 1 if use_ecm and iteration >= 3 and len(str(n)) >= 25: break low, high = high, high2 B1 = 10000 B2 = 100B1 num_curves = 50 while(1): if verbose: print(ecm_msg % (B1, B2, num_curves)) while(1): try: factor = _ecm_one_factor(n, B1, B2, num_curves) ps = factorint(factor, limit=limit - 1, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, use_ecm=use_ecm, verbose=verbose) n, _ = _trial(factors, n, ps, verbose=False) _check_termination(factors, n, limit, use_trial, use_rho, use_pm1, verbose) except ValueError: break except StopIteration: if verbose: print(complete_msg) return factors B1 = 5 B2 = 100B1 num_curves = 4 def factorrat(rat, limit=None, use_trial=True, use_rho=True, use_pm1=True, verbose=False, visual=None, multiple=False): r""" Given a Rational ``r``, ``factorrat(r)`` returns a dict containing the prime factors of ``r`` as keys and their respective multiplicities as values. For example: >>> from sympy import factorrat, S >>> factorrat(S(8)/9) # 8/9 = (2*3) (3*-2) {2: 3, 3: -2} >>> factorrat(S(-1)/987) # -1/789 = -1 (3*-1) (7*-1) (47*-1) {-1: 1, 3: -1, 7: -1, 47: -1} Please see the docstring for ``factorint`` for detailed explanations and examples of the following keywords: - ``limit``: Integer limit up to which trial division is done - ``use_trial``: Toggle use of trial division - ``use_rho``: Toggle use of Pollard's rho method - ``use_pm1``: Toggle use of Pollard's p-1 method - ``verbose``: Toggle detailed printing of progress - ``multiple``: Toggle returning a list of factors or dict - ``visual``: Toggle product form of output """ if multiple: fac = factorrat(rat, limit=limit, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, verbose=verbose, visual=False, multiple=False) factorlist = sum(([p] fac[p] if fac[p] > 0 else [S.One/p](-fac[p]) for p, _ in sorted(fac.items(), key=lambda elem: elem[0] if elem[1] > 0 else 1/elem[0])), []) return factorlist f = factorint(rat.p, limit=limit, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, verbose=verbose).copy() f = defaultdict(int, f) for p, e in factorint(rat.q, limit=limit, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, verbose=verbose).items(): f[p] += -e if len(f) > 1 and 1 in f: del f[1] if not visual: return dict(f) else: if -1 in f: f.pop(-1) args = [S.NegativeOne] else: args = [] args.extend([Pow(i, evaluate=False) for i in sorted(f.items())]) return Mul(args, evaluate=False) def primefactors(n, limit=None, verbose=False): """Return a sorted list of n's prime factors, ignoring multiplicity and any composite factor that remains if the limit was set too low for complete factorization. Unlike factorint(), primefactors() does not return -1 or 0. Examples ======== >>> from sympy.ntheory import primefactors, factorint, isprime >>> primefactors(6) [2, 3] >>> primefactors(-5) [5] >>> sorted(factorint(123456).items()) [(2, 6), (3, 1), (643, 1)] >>> primefactors(123456) [2, 3, 643] >>> sorted(factorint(10000000001, limit=200).items()) [(101, 1), (99009901, 1)] >>> isprime(99009901) False >>> primefactors(10000000001, limit=300) [101] See Also ======== divisors """ n = int(n) factors = sorted(factorint(n, limit=limit, verbose=verbose).keys()) s = [f for f in factors[:-1:] if f not in [-1, 0, 1]] if factors and isprime(factors[-1]): s += [factors[-1]] return s def _divisors(n, proper=False): """Helper function for divisors which generates the divisors.""" factordict = factorint(n) ps = sorted(factordict.keys()) def rec_gen(n=0): if n == len(ps): yield 1 else: pows = [1] for j in range(factordict[ps[n]]): pows.append(pows[-1] ps[n]) for q in rec_gen(n + 1): for p in pows: yield p * q if proper: for p in rec_gen(): if p != n: yield p else: yield from rec_gen() def divisors(n, generator=False, proper=False): r""" Return all divisors of n sorted from 1..n by default. If generator is ``True`` an unordered generator is returned. The number of divisors of n can be quite large if there are many prime factors (counting repeated factors). If only the number of factors is desired use divisor_count(n). Examples ======== >>> from sympy import divisors, divisor_count >>> divisors(24) [1, 2, 3, 4, 6, 8, 12, 24] >>> divisor_count(24) 8 >>> list(divisors(120, generator=True)) [1, 2, 4, 8, 3, 6, 12, 24, 5, 10, 20, 40, 15, 30, 60, 120] Notes ===== This is a slightly modified version of Tim Peters referenced at: https://stackoverflow.com/questions/1010381/python-factorization See Also ======== primefactors, factorint, divisor_count """ n = as_int(abs(n)) if isprime(n): if proper: return [1] return [1, n] if n == 1: if proper: return [] return [1] if n == 0: return [] rv = _divisors(n, proper) if not generator: return sorted(rv) return rv def divisor_count(n, modulus=1, proper=False): """ Return the number of divisors of ``n``. If ``modulus`` is not 1 then only those that are divisible by ``modulus`` are counted. If ``proper`` is True then the divisor of ``n`` will not be counted. Examples ======== >>> from sympy import divisor_count >>> divisor_count(6) 4 >>> divisor_count(6, 2) 2 >>> divisor_count(6, proper=True) 3 See Also ======== factorint, divisors, totient, proper_divisor_count """ if not modulus: return 0 elif modulus != 1: n, r = divmod(n, modulus) if r: return 0 if n == 0: return 0 n = Mul([v + 1 for k, v in factorint(n).items() if k > 1]) if n and proper: n -= 1 return n def proper_divisors(n, generator=False): """ Return all divisors of n except n, sorted by default. If generator is ``True`` an unordered generator is returned. Examples ======== >>> from sympy import proper_divisors, proper_divisor_count >>> proper_divisors(24) [1, 2, 3, 4, 6, 8, 12] >>> proper_divisor_count(24) 7 >>> list(proper_divisors(120, generator=True)) [1, 2, 4, 8, 3, 6, 12, 24, 5, 10, 20, 40, 15, 30, 60] See Also ======== factorint, divisors, proper_divisor_count """ return divisors(n, generator=generator, proper=True) def proper_divisor_count(n, modulus=1): """ Return the number of proper divisors of ``n``. Examples ======== >>> from sympy import proper_divisor_count >>> proper_divisor_count(6) 3 >>> proper_divisor_count(6, modulus=2) 1 See Also ======== divisors, proper_divisors, divisor_count """ return divisor_count(n, modulus=modulus, proper=True) def _udivisors(n): """Helper function for udivisors which generates the unitary divisors.""" factorpows = [pe for p, e in factorint(n).items()] for i in range(2len(factorpows)): d, j, k = 1, i, 0 while j: if (j & 1): d = factorpows[k] j >>= 1 k += 1 yield d def udivisors(n, generator=False): r""" Return all unitary divisors of n sorted from 1..n by default. If generator is ``True`` an unordered generator is returned. The number of unitary divisors of n can be quite large if there are many prime factors. If only the number of unitary divisors is desired use udivisor_count(n). Examples ======== >>> from sympy.ntheory.factor_ import udivisors, udivisor_count >>> udivisors(15) [1, 3, 5, 15] >>> udivisor_count(15) 4 >>> sorted(udivisors(120, generator=True)) [1, 3, 5, 8, 15, 24, 40, 120] See Also ======== primefactors, factorint, divisors, divisor_count, udivisor_count References ========== .. [1] https://en.wikipedia.org/wiki/Unitary_divisor .. [2] http://mathworld.wolfram.com/UnitaryDivisor.html """ n = as_int(abs(n)) if isprime(n): return [1, n] if n == 1: return [1] if n == 0: return [] rv = _udivisors(n) if not generator: return sorted(rv) return rv def udivisor_count(n): """ Return the number of unitary divisors of ``n``. Parameters ========== n : integer Examples ======== >>> from sympy.ntheory.factor_ import udivisor_count >>> udivisor_count(120) 8 See Also ======== factorint, divisors, udivisors, divisor_count, totient References ========== .. [1] http://mathworld.wolfram.com/UnitaryDivisorFunction.html """ if n == 0: return 0 return 2*len([p for p in factorint(n) if p > 1]) def _antidivisors(n): """Helper function for antidivisors which generates the antidivisors.""" for d in _divisors(n): y = 2d if n > y and n % y: yield y for d in _divisors(2n-1): if n > d >= 2 and n % d: yield d for d in _divisors(2n+1): if n > d >= 2 and n % d: yield d def antidivisors(n, generator=False): r""" Return all antidivisors of n sorted from 1..n by default. Antidivisors [1]_ of n are numbers that do not divide n by the largest possible margin. If generator is True an unordered generator is returned. Examples ======== >>> from sympy.ntheory.factor_ import antidivisors >>> antidivisors(24) [7, 16] >>> sorted(antidivisors(128, generator=True)) [3, 5, 15, 17, 51, 85] See Also ======== primefactors, factorint, divisors, divisor_count, antidivisor_count References ========== .. [1] definition is described in https://oeis.org/A066272/a066272a.html """ n = as_int(abs(n)) if n <= 2: return [] rv = _antidivisors(n) if not generator: return sorted(rv) return rv def antidivisor_count(n): """ Return the number of antidivisors [1]_ of ``n``. Parameters ========== n : integer Examples ======== >>> from sympy.ntheory.factor_ import antidivisor_count >>> antidivisor_count(13) 4 >>> antidivisor_count(27) 5 See Also ======== factorint, divisors, antidivisors, divisor_count, totient References ========== .. [1] formula from https://oeis.org/A066272 """ n = as_int(abs(n)) if n <= 2: return 0 return divisor_count(2n - 1) + divisor_count(2n + 1) + \ divisor_count(n) - divisor_count(n, 2) - 5 class totient(Function): r""" Calculate the Euler totient function phi(n) ``totient(n)`` or `\phi(n)` is the number of positive integers `\leq` n that are relatively prime to n. Parameters ========== n : integer Examples ======== >>> from sympy.ntheory import totient >>> totient(1) 1 >>> totient(25) 20 >>> totient(45) == totient(5)totient(9) True See Also ======== divisor_count References ========== .. [1] https://en.wikipedia.org/wiki/Euler%27s_totient_function .. [2] http://mathworld.wolfram.com/TotientFunction.html """ @classmethod def eval(cls, n): n = sympify(n) if n.is_Integer: if n < 1: raise ValueError("n must be a positive integer") factors = factorint(n) return cls._from_factors(factors) elif not isinstance(n, Expr) or (n.is_integer is False) or (n.is_positive is False): raise ValueError("n must be a positive integer") def _eval_is_integer(self): return fuzzy_and([self.args[0].is_integer, self.args[0].is_positive]) @classmethod def _from_distinct_primes(self, args): """Subroutine to compute totient from the list of assumed distinct primes Examples ======== >>> from sympy.ntheory.factor_ import totient >>> totient._from_distinct_primes(5, 7) 24 """ return reduce(lambda i, j: i * (j-1), args, 1) @classmethod def _from_factors(self, factors): """Subroutine to compute totient from already-computed factors Examples ======== >>> from sympy.ntheory.factor_ import totient >>> totient._from_factors({5: 2}) 20 """ t = 1 for p, k in factors.items(): t = (p - 1) p(k - 1) return t class reduced_totient(Function): r""" Calculate the Carmichael reduced totient function lambda(n) ``reduced_totient(n)`` or `\lambda(n)` is the smallest m > 0 such that `k^m \equiv 1 \mod n` for all k relatively prime to n. Examples ======== >>> from sympy.ntheory import reduced_totient >>> reduced_totient(1) 1 >>> reduced_totient(8) 2 >>> reduced_totient(30) 4 See Also ======== totient References ========== .. [1] https://en.wikipedia.org/wiki/Carmichael_function .. [2] http://mathworld.wolfram.com/CarmichaelFunction.html """ @classmethod def eval(cls, n): n = sympify(n) if n.is_Integer: if n < 1: raise ValueError("n must be a positive integer") factors = factorint(n) return cls._from_factors(factors) @classmethod def _from_factors(self, factors): """Subroutine to compute totient from already-computed factors """ t = 1 for p, k in factors.items(): if p == 2 and k > 2: t = ilcm(t, 2(k - 2)) else: t = ilcm(t, (p - 1) * p*(k - 1)) return t @classmethod def _from_distinct_primes(self, args): """Subroutine to compute totient from the list of assumed distinct primes """ args = [p - 1 for p in args] return ilcm(args) def _eval_is_integer(self): return fuzzy_and([self.args[0].is_integer, self.args[0].is_positive]) class divisor_sigma(Function): r""" Calculate the divisor function `\sigma_k(n)` for positive integer n ``divisor_sigma(n, k)`` is equal to ``sum([xk for x in divisors(n)])`` If n's prime factorization is: .. math :: n = \prod_{i=1}^\omega p_i^{m_i}, then .. math :: \sigma_k(n) = \prod_{i=1}^\omega (1+p_i^k+p_i^{2k}+\cdots + p_i^{m_ik}). Parameters ========== n : integer k : integer, optional power of divisors in the sum for k = 0, 1: ``divisor_sigma(n, 0)`` is equal to ``divisor_count(n)`` ``divisor_sigma(n, 1)`` is equal to ``sum(divisors(n))`` Default for k is 1. Examples ======== >>> from sympy.ntheory import divisor_sigma >>> divisor_sigma(18, 0) 6 >>> divisor_sigma(39, 1) 56 >>> divisor_sigma(12, 2) 210 >>> divisor_sigma(37) 38 See Also ======== divisor_count, totient, divisors, factorint References ========== .. [1] https://en.wikipedia.org/wiki/Divisor_function """ @classmethod def eval(cls, n, k=1): n = sympify(n) k = sympify(k) if n.is_prime: return 1 + nk if n.is_Integer: if n <= 0: raise ValueError("n must be a positive integer") elif k.is_Integer: k = int(k) return Integer(prod( (p(k(e + 1)) - 1)//(p*k - 1) if k != 0 else e + 1 for p, e in factorint(n).items())) else: return Mul([(p*(k(e + 1)) - 1)/(pk - 1) if k != 0 else e + 1 for p, e in factorint(n).items()]) if n.is_integer: # symbolic case args = [] for p, e in (_.as_base_exp() for _ in Mul.make_args(n)): if p.is_prime and e.is_positive: args.append((p(k(e + 1)) - 1)/(pk - 1) if k != 0 else e + 1) else: return return Mul(args) def core(n, t=2): r""" Calculate core(n, t) = `core_t(n)` of a positive integer n ``core_2(n)`` is equal to the squarefree part of n If n's prime factorization is: .. math :: n = \prod_{i=1}^\omega p_i^{m_i}, then .. math :: core_t(n) = \prod_{i=1}^\omega p_i^{m_i \mod t}. Parameters ========== n : integer t : integer core(n, t) calculates the t-th power free part of n ``core(n, 2)`` is the squarefree part of ``n`` ``core(n, 3)`` is the cubefree part of ``n`` Default for t is 2. Examples ======== >>> from sympy.ntheory.factor_ import core >>> core(24, 2) 6 >>> core(9424, 3) 1178 >>> core(379238) 379238 >>> core(15*11, 10) 15 See Also ======== factorint, sympy.solvers.diophantine.diophantine.square_factor References ========== .. [1] https://en.wikipedia.org/wiki/Square-free_integer#Squarefree_core """ n = as_int(n) t = as_int(t) if n <= 0: raise ValueError("n must be a positive integer") elif t <= 1: raise ValueError("t must be >= 2") else: y = 1 for p, e in factorint(n).items(): y = p*(e % t) return y class udivisor_sigma(Function): r""" Calculate the unitary divisor function `\sigma_k^(n)` for positive integer n ``udivisor_sigma(n, k)`` is equal to ``sum([x*k for x in udivisors(n)])`` If n's prime factorization is: .. math :: n = \prod_{i=1}^\omega p_i^{m_i}, then .. math :: \sigma_k^(n) = \prod_{i=1}^\omega (1+ p_i^{m_ik}). Parameters ========== k : power of divisors in the sum for k = 0, 1: ``udivisor_sigma(n, 0)`` is equal to ``udivisor_count(n)`` ``udivisor_sigma(n, 1)`` is equal to ``sum(udivisors(n))`` Default for k is 1. Examples ======== >>> from sympy.ntheory.factor_ import udivisor_sigma >>> udivisor_sigma(18, 0) 4 >>> udivisor_sigma(74, 1) 114 >>> udivisor_sigma(36, 3) 47450 >>> udivisor_sigma(111) 152 See Also ======== divisor_count, totient, divisors, udivisors, udivisor_count, divisor_sigma, factorint References ========== .. [1] http://mathworld.wolfram.com/UnitaryDivisorFunction.html """ @classmethod def eval(cls, n, k=1): n = sympify(n) k = sympify(k) if n.is_prime: return 1 + n*k if n.is_Integer: if n <= 0: raise ValueError("n must be a positive integer") else: return Mul([1+p*(ke) for p, e in factorint(n).items()]) class primenu(Function): r""" Calculate the number of distinct prime factors for a positive integer n. If n's prime factorization is: .. math :: n = \prod_{i=1}^k p_i^{m_i}, then ``primenu(n)`` or `\nu(n)` is: .. math :: \nu(n) = k. Examples ======== >>> from sympy.ntheory.factor_ import primenu >>> primenu(1) 0 >>> primenu(30) 3 See Also ======== factorint References ========== .. [1] http://mathworld.wolfram.com/PrimeFactor.html """ @classmethod def eval(cls, n): n = sympify(n) if n.is_Integer: if n <= 0: raise ValueError("n must be a positive integer") else: return len(factorint(n).keys()) class primeomega(Function): r""" Calculate the number of prime factors counting multiplicities for a positive integer n. If n's prime factorization is: .. math :: n = \prod_{i=1}^k p_i^{m_i}, then ``primeomega(n)`` or `\Omega(n)` is: .. math :: \Omega(n) = \sum_{i=1}^k m_i. Examples ======== >>> from sympy.ntheory.factor_ import primeomega >>> primeomega(1) 0 >>> primeomega(20) 3 See Also ======== factorint References ========== .. [1] http://mathworld.wolfram.com/PrimeFactor.html """ @classmethod def eval(cls, n): n = sympify(n) if n.is_Integer: if n <= 0: raise ValueError("n must be a positive integer") else: return sum(factorint(n).values()) def mersenne_prime_exponent(nth): """Returns the exponent ``i`` for the nth Mersenne prime (which has the form `2^i - 1`). Examples ======== >>> from sympy.ntheory.factor_ import mersenne_prime_exponent >>> mersenne_prime_exponent(1) 2 >>> mersenne_prime_exponent(20) 4423 """ n = as_int(nth) if n < 1: raise ValueError("nth must be a positive integer; mersenne_prime_exponent(1) == 2") if n > 51: raise ValueError("There are only 51 perfect numbers; nth must be less than or equal to 51") return MERSENNE_PRIME_EXPONENTS[n - 1] def is_perfect(n): """Returns True if ``n`` is a perfect number, else False. A perfect number is equal to the sum of its positive, proper divisors. Examples ======== >>> from sympy.ntheory.factor_ import is_perfect, divisors, divisor_sigma >>> is_perfect(20) False >>> is_perfect(6) True >>> 6 == divisor_sigma(6) - 6 == sum(divisors(6)[:-1]) True References ========== .. [1] http://mathworld.wolfram.com/PerfectNumber.html .. [2] https://en.wikipedia.org/wiki/Perfect_number """ n = as_int(n) if _isperfect(n): return True # all perfect numbers for Mersenne primes with exponents # less than or equal to 43112609 are known iknow = MERSENNE_PRIME_EXPONENTS.index(43112609) if iknow <= len(PERFECT) - 1 and n <= PERFECT[iknow]: # there may be gaps between this and larger known values # so only conclude in the range for which all values # are known return False if n%2 == 0: last2 = n % 100 if last2 != 28 and last2 % 10 != 6: return False r, b = integer_nthroot(1 + 8n, 2) if not b: return False m, x = divmod(1 + r, 4) if x: return False e, b = integer_log(m, 2) if not b: return False else: if n < 102000: # http://www.lirmm.fr/~ochem/opn/ return False if n % 105 == 0: # not divis by 105 return False if not any(n%m == r for m, r in [(12, 1), (468, 117), (324, 81)]): return False # there are many criteria that the factor structure of n # must meet; since we will have to factor it to test the # structure we will have the factors and can then check # to see whether it is a perfect number or not. So we # skip the structure checks and go straight to the final # test below. rv = divisor_sigma(n) - n if rv == n: if n%2 == 0: raise ValueError(filldedent(''' This even number is perfect and is associated with a Mersenne Prime, 2^%s - 1. It should be added to SymPy.''' % (e + 1))) else: raise ValueError(filldedent('''In 1888, Sylvester stated: " ...a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] -- its escape, so to say, from the complex web of conditions which hem it in on all sides -- would be little short of a miracle." I guess SymPy just found that miracle and it factors like this: %s''' % factorint(n))) def is_mersenne_prime(n): """Returns True if ``n`` is a Mersenne prime, else False. A Mersenne prime is a prime number having the form `2^i - 1`. Examples ======== >>> from sympy.ntheory.factor_ import is_mersenne_prime >>> is_mersenne_prime(6) False >>> is_mersenne_prime(127) True References ========== .. [1] http://mathworld.wolfram.com/MersennePrime.html """ n = as_int(n) if _ismersenneprime(n): return True if not isprime(n): return False r, b = integer_log(n + 1, 2) if not b: return False raise ValueError(filldedent(''' This Mersenne Prime, 2^%s - 1, should be added to SymPy's known values.''' % r)) def abundance(n): """Returns the difference between the sum of the positive proper divisors of a number and the number. Examples ======== >>> from sympy.ntheory import abundance, is_perfect, is_abundant >>> abundance(6) 0 >>> is_perfect(6) True >>> abundance(10) -2 >>> is_abundant(10) False """ return divisor_sigma(n, 1) - 2 n def is_abundant(n): """Returns True if ``n`` is an abundant number, else False. A abundant number is smaller than the sum of its positive proper divisors. Examples ======== >>> from sympy.ntheory.factor_ import is_abundant >>> is_abundant(20) True >>> is_abundant(15) False References ========== .. [1] http://mathworld.wolfram.com/AbundantNumber.html """ n = as_int(n) if is_perfect(n): return False return n % 6 == 0 or bool(abundance(n) > 0) def is_deficient(n): """Returns True if ``n`` is a deficient number, else False. A deficient number is greater than the sum of its positive proper divisors. Examples ======== >>> from sympy.ntheory.factor_ import is_deficient >>> is_deficient(20) False >>> is_deficient(15) True References ========== .. [1] http://mathworld.wolfram.com/DeficientNumber.html """ n = as_int(n) if is_perfect(n): return False return bool(abundance(n) < 0) def is_amicable(m, n): """Returns True if the numbers `m` and `n` are "amicable", else False. Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to that of the other. Examples ======== >>> from sympy.ntheory.factor_ import is_amicable, divisor_sigma >>> is_amicable(220, 284) True >>> divisor_sigma(220) == divisor_sigma(284) True References ========== .. [1] https://en.wikipedia.org/wiki/Amicable_numbers """ if m == n: return False a, b = map(lambda i: divisor_sigma(i), (m, n)) return a == b == (m + n) def dra(n, b): """ Returns the additive digital root of a natural number ``n`` in base ``b`` which is a single digit value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. Examples ======== >>> from sympy.ntheory.factor_ import dra >>> dra(3110, 12) 8 References ========== .. [1] https://en.wikipedia.org/wiki/Digital_root """ num = abs(as_int(n)) b = as_int(b) if b <= 1: raise ValueError("Base should be an integer greater than 1") if num == 0: return 0 return (1 + (num - 1) % (b - 1)) def drm(n, b): """ Returns the multiplicative digital root of a natural number ``n`` in a given base ``b`` which is a single digit value obtained by an iterative process of multiplying digits, on each iteration using the result from the previous iteration to compute the digit multiplication. Examples ======== >>> from sympy.ntheory.factor_ import drm >>> drm(9876, 10) 0 >>> drm(49, 10) 8 References ========== .. [1] http://mathworld.wolfram.com/MultiplicativeDigitalRoot.html """ n = abs(as_int(n)) b = as_int(b) if b <= 1: raise ValueError("Base should be an integer greater than 1") while n > b: mul = 1 while n > 1: n, r = divmod(n, b) if r == 0: return 0 mul *= r n = mul return n
e4bb392f4955302b2d42482b377b34585c0e5829586b2f642a7936d9758cb42c	from math import log from itertools import chain, islice, product from sympy.combinatorics import Permutation from sympy.combinatorics.permutations import (_af_commutes_with, _af_invert, _af_rmul, _af_rmuln, _af_pow, Cycle) from sympy.combinatorics.util import (_check_cycles_alt_sym, _distribute_gens_by_base, _orbits_transversals_from_bsgs, _handle_precomputed_bsgs, _base_ordering, _strong_gens_from_distr, _strip, _strip_af) from sympy.core import Basic from sympy.core.random import _randrange, randrange, choice from sympy.core.symbol import Symbol from sympy.core.sympify import _sympify from sympy.functions.combinatorial.factorials import factorial from sympy.ntheory import primefactors, sieve from sympy.ntheory.factor_ import (factorint, multiplicity) from sympy.ntheory.primetest import isprime from sympy.utilities.iterables import has_variety, is_sequence, uniq rmul = Permutation.rmul_with_af _af_new = Permutation._af_new class PermutationGroup(Basic): r"""The class defining a Permutation group. Explanation =========== ``PermutationGroup([p1, p2, ..., pn])`` returns the permutation group generated by the list of permutations. This group can be supplied to Polyhedron if one desires to decorate the elements to which the indices of the permutation refer. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> from sympy.combinatorics import Polyhedron The permutations corresponding to motion of the front, right and bottom face of a $2 \times 2$ Rubik's cube are defined: >>> F = Permutation(2, 19, 21, 8)(3, 17, 20, 10)(4, 6, 7, 5) >>> R = Permutation(1, 5, 21, 14)(3, 7, 23, 12)(8, 10, 11, 9) >>> D = Permutation(6, 18, 14, 10)(7, 19, 15, 11)(20, 22, 23, 21) These are passed as permutations to PermutationGroup: >>> G = PermutationGroup(F, R, D) >>> G.order() 3674160 The group can be supplied to a Polyhedron in order to track the objects being moved. An example involving the $2 \times 2$ Rubik's cube is given there, but here is a simple demonstration: >>> a = Permutation(2, 1) >>> b = Permutation(1, 0) >>> G = PermutationGroup(a, b) >>> P = Polyhedron(list('ABC'), pgroup=G) >>> P.corners (A, B, C) >>> P.rotate(0) # apply permutation 0 >>> P.corners (A, C, B) >>> P.reset() >>> P.corners (A, B, C) Or one can make a permutation as a product of selected permutations and apply them to an iterable directly: >>> P10 = G.make_perm([0, 1]) >>> P10('ABC') ['C', 'A', 'B'] See Also ======== sympy.combinatorics.polyhedron.Polyhedron, sympy.combinatorics.permutations.Permutation References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" .. [2] Seress, A. "Permutation Group Algorithms" .. [3] https://en.wikipedia.org/wiki/Schreier_vector .. [4] https://en.wikipedia.org/wiki/Nielsen_transformation#Product_replacement_algorithm .. [5] Frank Celler, Charles R.Leedham-Green, Scott H.Murray, Alice C.Niemeyer, and E.A.O'Brien. "Generating Random Elements of a Finite Group" .. [6] https://en.wikipedia.org/wiki/Block_%28permutation_group_theory%29 .. [7] http://www.algorithmist.com/index.php/Union_Find .. [8] https://en.wikipedia.org/wiki/Multiply_transitive_group#Multiply_transitive_groups .. [9] https://en.wikipedia.org/wiki/Center_%28group_theory%29 .. [10] https://en.wikipedia.org/wiki/Centralizer_and_normalizer .. [11] http://groupprops.subwiki.org/wiki/Derived_subgroup .. [12] https://en.wikipedia.org/wiki/Nilpotent_group .. [13] http://www.math.colostate.edu/~hulpke/CGT/cgtnotes.pdf .. [14] https://www.gap-system.org/Manuals/doc/ref/manual.pdf """ is_group = True def __new__(cls, args, dups=True, kwargs): """The default constructor. Accepts Cycle and Permutation forms. Removes duplicates unless ``dups`` keyword is ``False``. """ if not args: args = [Permutation()] else: args = list(args[0] if is_sequence(args[0]) else args) if not args: args = [Permutation()] if any(isinstance(a, Cycle) for a in args): args = [Permutation(a) for a in args] if has_variety(a.size for a in args): degree = kwargs.pop('degree', None) if degree is None: degree = max(a.size for a in args) for i in range(len(args)): if args[i].size != degree: args[i] = Permutation(args[i], size=degree) if dups: args = list(uniq([_af_new(list(a)) for a in args])) if len(args) > 1: args = [g for g in args if not g.is_identity] return Basic.__new__(cls, args, *kwargs) def __init__(self, args, *kwargs): self._generators = list(self.args) self._order = None self._center = [] self._is_abelian = None self._is_transitive = None self._is_sym = None self._is_alt = None self._is_primitive = None self._is_nilpotent = None self._is_solvable = None self._is_trivial = None self._transitivity_degree = None self._max_div = None self._is_perfect = None self._is_cyclic = None self._r = len(self._generators) self._degree = self._generators[0].size # these attributes are assigned after running schreier_sims self._base = [] self._strong_gens = [] self._strong_gens_slp = [] self._basic_orbits = [] self._transversals = [] self._transversal_slp = [] # these attributes are assigned after running _random_pr_init self._random_gens = [] # finite presentation of the group as an instance of `FpGroup` self._fp_presentation = None def __getitem__(self, i): return self._generators[i] def __contains__(self, i): """Return ``True`` if i* is contained in PermutationGroup. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> p = Permutation(1, 2, 3) >>> Permutation(3) in PermutationGroup(p) True """ if not isinstance(i, Permutation): raise TypeError("A PermutationGroup contains only Permutations as " "elements, not elements of type %s" % type(i)) return self.contains(i) def __len__(self): return len(self._generators) def equals(self, other): """Return ``True`` if PermutationGroup generated by elements in the group are same i.e they represent the same PermutationGroup. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> p = Permutation(0, 1, 2, 3, 4, 5) >>> G = PermutationGroup([p, p2]) >>> H = PermutationGroup([p2, p]) >>> G.generators == H.generators False >>> G.equals(H) True """ if not isinstance(other, PermutationGroup): return False set_self_gens = set(self.generators) set_other_gens = set(other.generators) # before reaching the general case there are also certain # optimisation and obvious cases requiring less or no actual # computation. if set_self_gens == set_other_gens: return True # in the most general case it will check that each generator of # one group belongs to the other PermutationGroup and vice-versa for gen1 in set_self_gens: if not other.contains(gen1): return False for gen2 in set_other_gens: if not self.contains(gen2): return False return True def __mul__(self, other): """ Return the direct product of two permutation groups as a permutation group. Explanation =========== This implementation realizes the direct product by shifting the index set for the generators of the second group: so if we have ``G`` acting on ``n1`` points and ``H`` acting on ``n2`` points, ``GH`` acts on ``n1 + n2`` points. Examples ======== >>> from sympy.combinatorics.named_groups import CyclicGroup >>> G = CyclicGroup(5) >>> H = GG >>> H PermutationGroup([ (9)(0 1 2 3 4), (5 6 7 8 9)]) >>> H.order() 25 """ if isinstance(other, Permutation): return Coset(other, self, dir='+') gens1 = [perm._array_form for perm in self.generators] gens2 = [perm._array_form for perm in other.generators] n1 = self._degree n2 = other._degree start = list(range(n1)) end = list(range(n1, n1 + n2)) for i in range(len(gens2)): gens2[i] = [x + n1 for x in gens2[i]] gens2 = [start + gen for gen in gens2] gens1 = [gen + end for gen in gens1] together = gens1 + gens2 gens = [_af_new(x) for x in together] return PermutationGroup(gens) def _random_pr_init(self, r, n, _random_prec_n=None): r"""Initialize random generators for the product replacement algorithm. Explanation =========== The implementation uses a modification of the original product replacement algorithm due to Leedham-Green, as described in [1], pp. 69-71; also, see [2], pp. 27-29 for a detailed theoretical analysis of the original product replacement algorithm, and [4]. The product replacement algorithm is used for producing random, uniformly distributed elements of a group `G` with a set of generators `S`. For the initialization ``_random_pr_init``, a list ``R`` of `\max\{r, \|S\|\}` group generators is created as the attribute ``G._random_gens``, repeating elements of `S` if necessary, and the identity element of `G` is appended to ``R`` - we shall refer to this last element as the accumulator. Then the function ``random_pr()`` is called ``n`` times, randomizing the list ``R`` while preserving the generation of `G` by ``R``. The function ``random_pr()`` itself takes two random elements ``g, h`` among all elements of ``R`` but the accumulator and replaces ``g`` with a randomly chosen element from `\{gh, g(~h), hg, (~h)g\}`. Then the accumulator is multiplied by whatever ``g`` was replaced by. The new value of the accumulator is then returned by ``random_pr()``. The elements returned will eventually (for ``n`` large enough) become uniformly distributed across `G` ([5]). For practical purposes however, the values ``n = 50, r = 11`` are suggested in [1]. Notes ===== THIS FUNCTION HAS SIDE EFFECTS: it changes the attribute self._random_gens See Also ======== random_pr """ deg = self.degree random_gens = [x._array_form for x in self.generators] k = len(random_gens) if k < r: for i in range(k, r): random_gens.append(random_gens[i - k]) acc = list(range(deg)) random_gens.append(acc) self._random_gens = random_gens # handle randomized input for testing purposes if _random_prec_n is None: for i in range(n): self.random_pr() else: for i in range(n): self.random_pr(_random_prec=_random_prec_n[i]) def _union_find_merge(self, first, second, ranks, parents, not_rep): """Merges two classes in a union-find data structure. Explanation =========== Used in the implementation of Atkinson's algorithm as suggested in [1], pp. 83-87. The class merging process uses union by rank as an optimization. ([7]) Notes ===== THIS FUNCTION HAS SIDE EFFECTS: the list of class representatives, ``parents``, the list of class sizes, ``ranks``, and the list of elements that are not representatives, ``not_rep``, are changed due to class merging. See Also ======== minimal_block, _union_find_rep References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of computational group theory" .. [7] http://www.algorithmist.com/index.php/Union_Find """ rep_first = self._union_find_rep(first, parents) rep_second = self._union_find_rep(second, parents) if rep_first != rep_second: # union by rank if ranks[rep_first] >= ranks[rep_second]: new_1, new_2 = rep_first, rep_second else: new_1, new_2 = rep_second, rep_first total_rank = ranks[new_1] + ranks[new_2] if total_rank > self.max_div: return -1 parents[new_2] = new_1 ranks[new_1] = total_rank not_rep.append(new_2) return 1 return 0 def _union_find_rep(self, num, parents): """Find representative of a class in a union-find data structure. Explanation =========== Used in the implementation of Atkinson's algorithm as suggested in [1], pp. 83-87. After the representative of the class to which ``num`` belongs is found, path compression is performed as an optimization ([7]). Notes ===== THIS FUNCTION HAS SIDE EFFECTS: the list of class representatives, ``parents``, is altered due to path compression. See Also ======== minimal_block, _union_find_merge References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of computational group theory" .. [7] http://www.algorithmist.com/index.php/Union_Find """ rep, parent = num, parents[num] while parent != rep: rep = parent parent = parents[rep] # path compression temp, parent = num, parents[num] while parent != rep: parents[temp] = rep temp = parent parent = parents[temp] return rep @property def base(self): r"""Return a base from the Schreier-Sims algorithm. Explanation =========== For a permutation group `G`, a base is a sequence of points `B = (b_1, b_2, \dots, b_k)` such that no element of `G` apart from the identity fixes all the points in `B`. The concepts of a base and strong generating set and their applications are discussed in depth in [1], pp. 87-89 and [2], pp. 55-57. An alternative way to think of `B` is that it gives the indices of the stabilizer cosets that contain more than the identity permutation. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> G = PermutationGroup([Permutation(0, 1, 3)(2, 4)]) >>> G.base [0, 2] See Also ======== strong_gens, basic_transversals, basic_orbits, basic_stabilizers """ if self._base == []: self.schreier_sims() return self._base def baseswap(self, base, strong_gens, pos, randomized=False, transversals=None, basic_orbits=None, strong_gens_distr=None): r"""Swap two consecutive base points in base and strong generating set. Explanation =========== If a base for a group `G` is given by `(b_1, b_2, \dots, b_k)`, this function returns a base `(b_1, b_2, \dots, b_{i+1}, b_i, \dots, b_k)`, where `i` is given by ``pos``, and a strong generating set relative to that base. The original base and strong generating set are not modified. The randomized version (default) is of Las Vegas type. Parameters ========== base, strong_gens The base and strong generating set. pos The position at which swapping is performed. randomized A switch between randomized and deterministic version. transversals The transversals for the basic orbits, if known. basic_orbits The basic orbits, if known. strong_gens_distr The strong generators distributed by basic stabilizers, if known. Returns ======= (base, strong_gens) ``base`` is the new base, and ``strong_gens`` is a generating set relative to it. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.testutil import _verify_bsgs >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> S = SymmetricGroup(4) >>> S.schreier_sims() >>> S.base [0, 1, 2] >>> base, gens = S.baseswap(S.base, S.strong_gens, 1, randomized=False) >>> base, gens ([0, 2, 1], [(0 1 2 3), (3)(0 1), (1 3 2), (2 3), (1 3)]) check that base, gens is a BSGS >>> S1 = PermutationGroup(gens) >>> _verify_bsgs(S1, base, gens) True See Also ======== schreier_sims Notes ===== The deterministic version of the algorithm is discussed in [1], pp. 102-103; the randomized version is discussed in [1], p.103, and [2], p.98. It is of Las Vegas type. Notice that [1] contains a mistake in the pseudocode and discussion of BASESWAP: on line 3 of the pseudocode, `\|\beta_{i+1}^{\left\langle T\right\rangle}\|` should be replaced by `\|\beta_{i}^{\left\langle T\right\rangle}\|`, and the same for the discussion of the algorithm. """ # construct the basic orbits, generators for the stabilizer chain # and transversal elements from whatever was provided transversals, basic_orbits, strong_gens_distr = \ _handle_precomputed_bsgs(base, strong_gens, transversals, basic_orbits, strong_gens_distr) base_len = len(base) degree = self.degree # size of orbit of base[pos] under the stabilizer we seek to insert # in the stabilizer chain at position pos + 1 size = len(basic_orbits[pos])len(basic_orbits[pos + 1]) \ //len(_orbit(degree, strong_gens_distr[pos], base[pos + 1])) # initialize the wanted stabilizer by a subgroup if pos + 2 > base_len - 1: T = [] else: T = strong_gens_distr[pos + 2][:] # randomized version if randomized is True: stab_pos = PermutationGroup(strong_gens_distr[pos]) schreier_vector = stab_pos.schreier_vector(base[pos + 1]) # add random elements of the stabilizer until they generate it while len(_orbit(degree, T, base[pos])) != size: new = stab_pos.random_stab(base[pos + 1], schreier_vector=schreier_vector) T.append(new) # deterministic version else: Gamma = set(basic_orbits[pos]) Gamma.remove(base[pos]) if base[pos + 1] in Gamma: Gamma.remove(base[pos + 1]) # add elements of the stabilizer until they generate it by # ruling out member of the basic orbit of base[pos] along the way while len(_orbit(degree, T, base[pos])) != size: gamma = next(iter(Gamma)) x = transversals[pos][gamma] temp = x._array_form.index(base[pos + 1]) # (~x)(base[pos + 1]) if temp not in basic_orbits[pos + 1]: Gamma = Gamma - _orbit(degree, T, gamma) else: y = transversals[pos + 1][temp] el = rmul(x, y) if el(base[pos]) not in _orbit(degree, T, base[pos]): T.append(el) Gamma = Gamma - _orbit(degree, T, base[pos]) # build the new base and strong generating set strong_gens_new_distr = strong_gens_distr[:] strong_gens_new_distr[pos + 1] = T base_new = base[:] base_new[pos], base_new[pos + 1] = base_new[pos + 1], base_new[pos] strong_gens_new = _strong_gens_from_distr(strong_gens_new_distr) for gen in T: if gen not in strong_gens_new: strong_gens_new.append(gen) return base_new, strong_gens_new @property def basic_orbits(self): r""" Return the basic orbits relative to a base and strong generating set. Explanation =========== If `(b_1, b_2, \dots, b_k)` is a base for a group `G`, and `G^{(i)} = G_{b_1, b_2, \dots, b_{i-1}}` is the ``i``-th basic stabilizer (so that `G^{(1)} = G`), the ``i``-th basic orbit relative to this base is the orbit of `b_i` under `G^{(i)}`. See [1], pp. 87-89 for more information. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> S = SymmetricGroup(4) >>> S.basic_orbits [[0, 1, 2, 3], [1, 2, 3], [2, 3]] See Also ======== base, strong_gens, basic_transversals, basic_stabilizers """ if self._basic_orbits == []: self.schreier_sims() return self._basic_orbits @property def basic_stabilizers(self): r""" Return a chain of stabilizers relative to a base and strong generating set. Explanation =========== The ``i``-th basic stabilizer `G^{(i)}` relative to a base `(b_1, b_2, \dots, b_k)` is `G_{b_1, b_2, \dots, b_{i-1}}`. For more information, see [1], pp. 87-89. Examples ======== >>> from sympy.combinatorics.named_groups import AlternatingGroup >>> A = AlternatingGroup(4) >>> A.schreier_sims() >>> A.base [0, 1] >>> for g in A.basic_stabilizers: ... print(g) ... PermutationGroup([ (3)(0 1 2), (1 2 3)]) PermutationGroup([ (1 2 3)]) See Also ======== base, strong_gens, basic_orbits, basic_transversals """ if self._transversals == []: self.schreier_sims() strong_gens = self._strong_gens base = self._base if not base: # e.g. if self is trivial return [] strong_gens_distr = _distribute_gens_by_base(base, strong_gens) basic_stabilizers = [] for gens in strong_gens_distr: basic_stabilizers.append(PermutationGroup(gens)) return basic_stabilizers @property def basic_transversals(self): """ Return basic transversals relative to a base and strong generating set. Explanation =========== The basic transversals are transversals of the basic orbits. They are provided as a list of dictionaries, each dictionary having keys - the elements of one of the basic orbits, and values - the corresponding transversal elements. See [1], pp. 87-89 for more information. Examples ======== >>> from sympy.combinatorics.named_groups import AlternatingGroup >>> A = AlternatingGroup(4) >>> A.basic_transversals [{0: (3), 1: (3)(0 1 2), 2: (3)(0 2 1), 3: (0 3 1)}, {1: (3), 2: (1 2 3), 3: (1 3 2)}] See Also ======== strong_gens, base, basic_orbits, basic_stabilizers """ if self._transversals == []: self.schreier_sims() return self._transversals def composition_series(self): r""" Return the composition series for a group as a list of permutation groups. Explanation =========== The composition series for a group `G` is defined as a subnormal series `G = H_0 > H_1 > H_2 \ldots` A composition series is a subnormal series such that each factor group `H(i+1) / H(i)` is simple. A subnormal series is a composition series only if it is of maximum length. The algorithm works as follows: Starting with the derived series the idea is to fill the gap between `G = der[i]` and `H = der[i+1]` for each `i` independently. Since, all subgroups of the abelian group `G/H` are normal so, first step is to take the generators `g` of `G` and add them to generators of `H` one by one. The factor groups formed are not simple in general. Each group is obtained from the previous one by adding one generator `g`, if the previous group is denoted by `H` then the next group `K` is generated by `g` and `H`. The factor group `K/H` is cyclic and it's order is `K.order()//G.order()`. The series is then extended between `K` and `H` by groups generated by powers of `g` and `H`. The series formed is then prepended to the already existing series. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.named_groups import CyclicGroup >>> S = SymmetricGroup(12) >>> G = S.sylow_subgroup(2) >>> C = G.composition_series() >>> [H.order() for H in C] [1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1] >>> G = S.sylow_subgroup(3) >>> C = G.composition_series() >>> [H.order() for H in C] [243, 81, 27, 9, 3, 1] >>> G = CyclicGroup(12) >>> C = G.composition_series() >>> [H.order() for H in C] [12, 6, 3, 1] """ der = self.derived_series() if not all(g.is_identity for g in der[-1].generators): raise NotImplementedError('Group should be solvable') series = [] for i in range(len(der)-1): H = der[i+1] up_seg = [] for g in der[i].generators: K = PermutationGroup([g] + H.generators) order = K.order() // H.order() down_seg = [] for p, e in factorint(order).items(): for _ in range(e): down_seg.append(PermutationGroup([g] + H.generators)) g = gp up_seg = down_seg + up_seg H = K up_seg[0] = der[i] series.extend(up_seg) series.append(der[-1]) return series def coset_transversal(self, H): """Return a transversal of the right cosets of self by its subgroup H using the second method described in [1], Subsection 4.6.7 """ if not H.is_subgroup(self): raise ValueError("The argument must be a subgroup") if H.order() == 1: return self._elements self._schreier_sims(base=H.base) # make G.base an extension of H.base base = self.base base_ordering = _base_ordering(base, self.degree) identity = Permutation(self.degree - 1) transversals = self.basic_transversals[:] # transversals is a list of dictionaries. Get rid of the keys # so that it is a list of lists and sort each list in # the increasing order of base[l]^x for l, t in enumerate(transversals): transversals[l] = sorted(t.values(), key = lambda x: base_ordering[base[l]^x]) orbits = H.basic_orbits h_stabs = H.basic_stabilizers g_stabs = self.basic_stabilizers indices = [x.order()//y.order() for x, y in zip(g_stabs, h_stabs)] # T^(l) should be a right transversal of H^(l) in G^(l) for # 1<=l<=len(base). While H^(l) is the trivial group, T^(l) # contains all the elements of G^(l) so we might just as well # start with l = len(h_stabs)-1 if len(g_stabs) > len(h_stabs): T = g_stabs[len(h_stabs)]._elements else: T = [identity] l = len(h_stabs)-1 t_len = len(T) while l > -1: T_next = [] for u in transversals[l]: if u == identity: continue b = base_ordering[base[l]^u] for t in T: p = tu if all(base_ordering[h^p] >= b for h in orbits[l]): T_next.append(p) if t_len + len(T_next) == indices[l]: break if t_len + len(T_next) == indices[l]: break T += T_next t_len += len(T_next) l -= 1 T.remove(identity) T = [identity] + T return T def _coset_representative(self, g, H): """Return the representative of Hg from the transversal that would be computed by ``self.coset_transversal(H)``. """ if H.order() == 1: return g # The base of self must be an extension of H.base. if not(self.base[:len(H.base)] == H.base): self._schreier_sims(base=H.base) orbits = H.basic_orbits[:] h_transversals = [list(_.values()) for _ in H.basic_transversals] transversals = [list(_.values()) for _ in self.basic_transversals] base = self.base base_ordering = _base_ordering(base, self.degree) def step(l, x): gamma = sorted(orbits[l], key = lambda y: base_ordering[y^x])[0] i = [base[l]^h for h in h_transversals[l]].index(gamma) x = h_transversals[l][i]x if l < len(orbits)-1: for u in transversals[l]: if base[l]^u == base[l]^x: break x = step(l+1, xu*-1)u return x return step(0, g) def coset_table(self, H): """Return the standardised (right) coset table of self in H as a list of lists. """ # Maybe this should be made to return an instance of CosetTable # from fp_groups.py but the class would need to be changed first # to be compatible with PermutationGroups if not H.is_subgroup(self): raise ValueError("The argument must be a subgroup") T = self.coset_transversal(H) n = len(T) A = list(chain.from_iterable((gen, gen*-1) for gen in self.generators)) table = [] for i in range(n): row = [self._coset_representative(T[i]x, H) for x in A] row = [T.index(r) for r in row] table.append(row) # standardize (this is the same as the algorithm used in coset_table) # If CosetTable is made compatible with PermutationGroups, this # should be replaced by table.standardize() A = range(len(A)) gamma = 1 for alpha, a in product(range(n), A): beta = table[alpha][a] if beta >= gamma: if beta > gamma: for x in A: z = table[gamma][x] table[gamma][x] = table[beta][x] table[beta][x] = z for i in range(n): if table[i][x] == beta: table[i][x] = gamma elif table[i][x] == gamma: table[i][x] = beta gamma += 1 if gamma >= n-1: return table def center(self): r""" Return the center of a permutation group. Explanation =========== The center for a group `G` is defined as `Z(G) = \{z\in G \| \forall g\in G, zg = gz \}`, the set of elements of `G` that commute with all elements of `G`. It is equal to the centralizer of `G` inside `G`, and is naturally a subgroup of `G` ([9]). Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> D = DihedralGroup(4) >>> G = D.center() >>> G.order() 2 See Also ======== centralizer Notes ===== This is a naive implementation that is a straightforward application of ``.centralizer()`` """ return self.centralizer(self) def centralizer(self, other): r""" Return the centralizer of a group/set/element. Explanation =========== The centralizer of a set of permutations ``S`` inside a group ``G`` is the set of elements of ``G`` that commute with all elements of ``S``:: `C_G(S) = \{ g \in G \| gs = sg \forall s \in S\}` ([10]) Usually, ``S`` is a subset of ``G``, but if ``G`` is a proper subgroup of the full symmetric group, we allow for ``S`` to have elements outside ``G``. It is naturally a subgroup of ``G``; the centralizer of a permutation group is equal to the centralizer of any set of generators for that group, since any element commuting with the generators commutes with any product of the generators. Parameters ========== other a permutation group/list of permutations/single permutation Examples ======== >>> from sympy.combinatorics.named_groups import (SymmetricGroup, ... CyclicGroup) >>> S = SymmetricGroup(6) >>> C = CyclicGroup(6) >>> H = S.centralizer(C) >>> H.is_subgroup(C) True See Also ======== subgroup_search Notes ===== The implementation is an application of ``.subgroup_search()`` with tests using a specific base for the group ``G``. """ if hasattr(other, 'generators'): if other.is_trivial or self.is_trivial: return self degree = self.degree identity = _af_new(list(range(degree))) orbits = other.orbits() num_orbits = len(orbits) orbits.sort(key=lambda x: -len(x)) long_base = [] orbit_reps = [None]num_orbits orbit_reps_indices = [None]num_orbits orbit_descr = [None]degree for i in range(num_orbits): orbit = list(orbits[i]) orbit_reps[i] = orbit[0] orbit_reps_indices[i] = len(long_base) for point in orbit: orbit_descr[point] = i long_base = long_base + orbit base, strong_gens = self.schreier_sims_incremental(base=long_base) strong_gens_distr = _distribute_gens_by_base(base, strong_gens) i = 0 for i in range(len(base)): if strong_gens_distr[i] == [identity]: break base = base[:i] base_len = i for j in range(num_orbits): if base[base_len - 1] in orbits[j]: break rel_orbits = orbits[: j + 1] num_rel_orbits = len(rel_orbits) transversals = [None]num_rel_orbits for j in range(num_rel_orbits): rep = orbit_reps[j] transversals[j] = dict( other.orbit_transversal(rep, pairs=True)) trivial_test = lambda x: True tests = [None]base_len for l in range(base_len): if base[l] in orbit_reps: tests[l] = trivial_test else: def test(computed_words, l=l): g = computed_words[l] rep_orb_index = orbit_descr[base[l]] rep = orbit_reps[rep_orb_index] im = g._array_form[base[l]] im_rep = g._array_form[rep] tr_el = transversals[rep_orb_index][base[l]] # using the definition of transversal, # base[l]^g = rep^(tr_elg); # if g belongs to the centralizer, then # base[l]^g = (rep^g)^tr_el return im == tr_el._array_form[im_rep] tests[l] = test def prop(g): return [rmul(g, gen) for gen in other.generators] == \ [rmul(gen, g) for gen in other.generators] return self.subgroup_search(prop, base=base, strong_gens=strong_gens, tests=tests) elif hasattr(other, '__getitem__'): gens = list(other) return self.centralizer(PermutationGroup(gens)) elif hasattr(other, 'array_form'): return self.centralizer(PermutationGroup([other])) def commutator(self, G, H): """ Return the commutator of two subgroups. Explanation =========== For a permutation group ``K`` and subgroups ``G``, ``H``, the commutator of ``G`` and ``H`` is defined as the group generated by all the commutators `[g, h] = hgh^{-1}g^{-1}` for ``g`` in ``G`` and ``h`` in ``H``. It is naturally a subgroup of ``K`` ([1], p.27). Examples ======== >>> from sympy.combinatorics.named_groups import (SymmetricGroup, ... AlternatingGroup) >>> S = SymmetricGroup(5) >>> A = AlternatingGroup(5) >>> G = S.commutator(S, A) >>> G.is_subgroup(A) True See Also ======== derived_subgroup Notes ===== The commutator of two subgroups `H, G` is equal to the normal closure of the commutators of all the generators, i.e. `hgh^{-1}g^{-1}` for `h` a generator of `H` and `g` a generator of `G` ([1], p.28) """ ggens = G.generators hgens = H.generators commutators = [] for ggen in ggens: for hgen in hgens: commutator = rmul(hgen, ggen, ~hgen, ~ggen) if commutator not in commutators: commutators.append(commutator) res = self.normal_closure(commutators) return res def coset_factor(self, g, factor_index=False): """Return ``G``'s (self's) coset factorization of ``g`` Explanation =========== If ``g`` is an element of ``G`` then it can be written as the product of permutations drawn from the Schreier-Sims coset decomposition, The permutations returned in ``f`` are those for which the product gives ``g``: ``g = f[n]...f[1]f[0]`` where ``n = len(B)`` and ``B = G.base``. f[i] is one of the permutations in ``self._basic_orbits[i]``. If factor_index==True, returns a tuple ``[b[0],..,b[n]]``, where ``b[i]`` belongs to ``self._basic_orbits[i]`` Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation(0, 1, 3, 7, 6, 4)(2, 5) >>> b = Permutation(0, 1, 3, 2)(4, 5, 7, 6) >>> G = PermutationGroup([a, b]) Define g: >>> g = Permutation(7)(1, 2, 4)(3, 6, 5) Confirm that it is an element of G: >>> G.contains(g) True Thus, it can be written as a product of factors (up to 3) drawn from u. See below that a factor from u1 and u2 and the Identity permutation have been used: >>> f = G.coset_factor(g) >>> f[2]f[1]f[0] == g True >>> f1 = G.coset_factor(g, True); f1 [0, 4, 4] >>> tr = G.basic_transversals >>> f[0] == tr[0][f1[0]] True If g is not an element of G then [] is returned: >>> c = Permutation(5, 6, 7) >>> G.coset_factor(c) [] See Also ======== sympy.combinatorics.util._strip """ if isinstance(g, (Cycle, Permutation)): g = g.list() if len(g) != self._degree: # this could either adjust the size or return [] immediately # but we don't choose between the two and just signal a possible # error raise ValueError('g should be the same size as permutations of G') I = list(range(self._degree)) basic_orbits = self.basic_orbits transversals = self._transversals factors = [] base = self.base h = g for i in range(len(base)): beta = h[base[i]] if beta == base[i]: factors.append(beta) continue if beta not in basic_orbits[i]: return [] u = transversals[i][beta]._array_form h = _af_rmul(_af_invert(u), h) factors.append(beta) if h != I: return [] if factor_index: return factors tr = self.basic_transversals factors = [tr[i][factors[i]] for i in range(len(base))] return factors def generator_product(self, g, original=False): r''' Return a list of strong generators `[s1, \dots, sn]` s.t `g = sn \times \dots \times s1`. If ``original=True``, make the list contain only the original group generators ''' product = [] if g.is_identity: return [] if g in self.strong_gens: if not original or g in self.generators: return [g] else: slp = self._strong_gens_slp[g] for s in slp: product.extend(self.generator_product(s, original=True)) return product elif g-1 in self.strong_gens: g = g-1 if not original or g in self.generators: return [g-1] else: slp = self._strong_gens_slp[g] for s in slp: product.extend(self.generator_product(s, original=True)) l = len(product) product = [product[l-i-1]-1 for i in range(l)] return product f = self.coset_factor(g, True) for i, j in enumerate(f): slp = self._transversal_slp[i][j] for s in slp: if not original: product.append(self.strong_gens[s]) else: s = self.strong_gens[s] product.extend(self.generator_product(s, original=True)) return product def coset_rank(self, g): """rank using Schreier-Sims representation. Explanation =========== The coset rank of ``g`` is the ordering number in which it appears in the lexicographic listing according to the coset decomposition The ordering is the same as in G.generate(method='coset'). If ``g`` does not belong to the group it returns None. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation(0, 1, 3, 7, 6, 4)(2, 5) >>> b = Permutation(0, 1, 3, 2)(4, 5, 7, 6) >>> G = PermutationGroup([a, b]) >>> c = Permutation(7)(2, 4)(3, 5) >>> G.coset_rank(c) 16 >>> G.coset_unrank(16) (7)(2 4)(3 5) See Also ======== coset_factor """ factors = self.coset_factor(g, True) if not factors: return None rank = 0 b = 1 transversals = self._transversals base = self._base basic_orbits = self._basic_orbits for i in range(len(base)): k = factors[i] j = basic_orbits[i].index(k) rank += bj b = blen(transversals[i]) return rank def coset_unrank(self, rank, af=False): """unrank using Schreier-Sims representation coset_unrank is the inverse operation of coset_rank if 0 <= rank < order; otherwise it returns None. """ if rank < 0 or rank >= self.order(): return None base = self.base transversals = self.basic_transversals basic_orbits = self.basic_orbits m = len(base) v = [0]m for i in range(m): rank, c = divmod(rank, len(transversals[i])) v[i] = basic_orbits[i][c] a = [transversals[i][v[i]]._array_form for i in range(m)] h = _af_rmuln(a) if af: return h else: return _af_new(h) @property def degree(self): """Returns the size of the permutations in the group. Explanation =========== The number of permutations comprising the group is given by ``len(group)``; the number of permutations that can be generated by the group is given by ``group.order()``. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([1, 0, 2]) >>> G = PermutationGroup([a]) >>> G.degree 3 >>> len(G) 1 >>> G.order() 2 >>> list(G.generate()) [(2), (2)(0 1)] See Also ======== order """ return self._degree @property def identity(self): ''' Return the identity element of the permutation group. ''' return _af_new(list(range(self.degree))) @property def elements(self): """Returns all the elements of the permutation group as a set Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> p = PermutationGroup(Permutation(1, 3), Permutation(1, 2)) >>> p.elements {(1 2 3), (1 3 2), (1 3), (2 3), (3), (3)(1 2)} """ return set(self._elements) @property def _elements(self): """Returns all the elements of the permutation group as a list Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> p = PermutationGroup(Permutation(1, 3), Permutation(1, 2)) >>> p._elements [(3), (3)(1 2), (1 3), (2 3), (1 2 3), (1 3 2)] """ return list(islice(self.generate(), None)) def derived_series(self): r"""Return the derived series for the group. Explanation =========== The derived series for a group `G` is defined as `G = G_0 > G_1 > G_2 > \ldots` where `G_i = [G_{i-1}, G_{i-1}]`, i.e. `G_i` is the derived subgroup of `G_{i-1}`, for `i\in\mathbb{N}`. When we have `G_k = G_{k-1}` for some `k\in\mathbb{N}`, the series terminates. Returns ======= A list of permutation groups containing the members of the derived series in the order `G = G_0, G_1, G_2, \ldots`. Examples ======== >>> from sympy.combinatorics.named_groups import (SymmetricGroup, ... AlternatingGroup, DihedralGroup) >>> A = AlternatingGroup(5) >>> len(A.derived_series()) 1 >>> S = SymmetricGroup(4) >>> len(S.derived_series()) 4 >>> S.derived_series()[1].is_subgroup(AlternatingGroup(4)) True >>> S.derived_series()[2].is_subgroup(DihedralGroup(2)) True See Also ======== derived_subgroup """ res = [self] current = self nxt = self.derived_subgroup() while not current.is_subgroup(nxt): res.append(nxt) current = nxt nxt = nxt.derived_subgroup() return res def derived_subgroup(self): r"""Compute the derived subgroup. Explanation =========== The derived subgroup, or commutator subgroup is the subgroup generated by all commutators `[g, h] = hgh^{-1}g^{-1}` for `g, h\in G` ; it is equal to the normal closure of the set of commutators of the generators ([1], p.28, [11]). Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([1, 0, 2, 4, 3]) >>> b = Permutation([0, 1, 3, 2, 4]) >>> G = PermutationGroup([a, b]) >>> C = G.derived_subgroup() >>> list(C.generate(af=True)) [[0, 1, 2, 3, 4], [0, 1, 3, 4, 2], [0, 1, 4, 2, 3]] See Also ======== derived_series """ r = self._r gens = [p._array_form for p in self.generators] set_commutators = set() degree = self._degree rng = list(range(degree)) for i in range(r): for j in range(r): p1 = gens[i] p2 = gens[j] c = list(range(degree)) for k in rng: c[p2[p1[k]]] = p1[p2[k]] ct = tuple(c) if ct not in set_commutators: set_commutators.add(ct) cms = [_af_new(p) for p in set_commutators] G2 = self.normal_closure(cms) return G2 def generate(self, method="coset", af=False): """Return iterator to generate the elements of the group. Explanation =========== Iteration is done with one of these methods:: method='coset' using the Schreier-Sims coset representation method='dimino' using the Dimino method If ``af = True`` it yields the array form of the permutations Examples ======== >>> from sympy.combinatorics import PermutationGroup >>> from sympy.combinatorics.polyhedron import tetrahedron The permutation group given in the tetrahedron object is also true groups: >>> G = tetrahedron.pgroup >>> G.is_group True Also the group generated by the permutations in the tetrahedron pgroup -- even the first two -- is a proper group: >>> H = PermutationGroup(G[0], G[1]) >>> J = PermutationGroup(list(H.generate())); J PermutationGroup([ (0 1)(2 3), (1 2 3), (1 3 2), (0 3 1), (0 2 3), (0 3)(1 2), (0 1 3), (3)(0 2 1), (0 3 2), (3)(0 1 2), (0 2)(1 3)]) >>> _.is_group True """ if method == "coset": return self.generate_schreier_sims(af) elif method == "dimino": return self.generate_dimino(af) else: raise NotImplementedError('No generation defined for %s' % method) def generate_dimino(self, af=False): """Yield group elements using Dimino's algorithm. If ``af == True`` it yields the array form of the permutations. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([0, 2, 1, 3]) >>> b = Permutation([0, 2, 3, 1]) >>> g = PermutationGroup([a, b]) >>> list(g.generate_dimino(af=True)) [[0, 1, 2, 3], [0, 2, 1, 3], [0, 2, 3, 1], [0, 1, 3, 2], [0, 3, 2, 1], [0, 3, 1, 2]] References ========== .. [1] The Implementation of Various Algorithms for Permutation Groups in the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis """ idn = list(range(self.degree)) order = 0 element_list = [idn] set_element_list = {tuple(idn)} if af: yield idn else: yield _af_new(idn) gens = [p._array_form for p in self.generators] for i in range(len(gens)): # D elements of the subgroup G_i generated by gens[:i] D = element_list[:] N = [idn] while N: A = N N = [] for a in A: for g in gens[:i + 1]: ag = _af_rmul(a, g) if tuple(ag) not in set_element_list: # produce G_ig for d in D: order += 1 ap = _af_rmul(d, ag) if af: yield ap else: p = _af_new(ap) yield p element_list.append(ap) set_element_list.add(tuple(ap)) N.append(ap) self._order = len(element_list) def generate_schreier_sims(self, af=False): """Yield group elements using the Schreier-Sims representation in coset_rank order If ``af = True`` it yields the array form of the permutations Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([0, 2, 1, 3]) >>> b = Permutation([0, 2, 3, 1]) >>> g = PermutationGroup([a, b]) >>> list(g.generate_schreier_sims(af=True)) [[0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 2, 1], [0, 1, 3, 2], [0, 2, 3, 1], [0, 3, 1, 2]] """ n = self._degree u = self.basic_transversals basic_orbits = self._basic_orbits if len(u) == 0: for x in self.generators: if af: yield x._array_form else: yield x return if len(u) == 1: for i in basic_orbits[0]: if af: yield u[0][i]._array_form else: yield u[0][i] return u = list(reversed(u)) basic_orbits = basic_orbits[::-1] # stg stack of group elements stg = [list(range(n))] posmax = [len(x) for x in u] n1 = len(posmax) - 1 pos = [0]n1 h = 0 while 1: # backtrack when finished iterating over coset if pos[h] >= posmax[h]: if h == 0: return pos[h] = 0 h -= 1 stg.pop() continue p = _af_rmul(u[h][basic_orbits[h][pos[h]]]._array_form, stg[-1]) pos[h] += 1 stg.append(p) h += 1 if h == n1: if af: for i in basic_orbits[-1]: p = _af_rmul(u[-1][i]._array_form, stg[-1]) yield p else: for i in basic_orbits[-1]: p = _af_rmul(u[-1][i]._array_form, stg[-1]) p1 = _af_new(p) yield p1 stg.pop() h -= 1 @property def generators(self): """Returns the generators of the group. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([0, 2, 1]) >>> b = Permutation([1, 0, 2]) >>> G = PermutationGroup([a, b]) >>> G.generators [(1 2), (2)(0 1)] """ return self._generators def contains(self, g, strict=True): """Test if permutation ``g`` belong to self, ``G``. Explanation =========== If ``g`` is an element of ``G`` it can be written as a product of factors drawn from the cosets of ``G``'s stabilizers. To see if ``g`` is one of the actual generators defining the group use ``G.has(g)``. If ``strict`` is not ``True``, ``g`` will be resized, if necessary, to match the size of permutations in ``self``. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation(1, 2) >>> b = Permutation(2, 3, 1) >>> G = PermutationGroup(a, b, degree=5) >>> G.contains(G[0]) # trivial check True >>> elem = Permutation([[2, 3]], size=5) >>> G.contains(elem) True >>> G.contains(Permutation(4)(0, 1, 2, 3)) False If strict is False, a permutation will be resized, if necessary: >>> H = PermutationGroup(Permutation(5)) >>> H.contains(Permutation(3)) False >>> H.contains(Permutation(3), strict=False) True To test if a given permutation is present in the group: >>> elem in G.generators False >>> G.has(elem) False See Also ======== coset_factor, sympy.core.basic.Basic.has, __contains__ """ if not isinstance(g, Permutation): return False if g.size != self.degree: if strict: return False g = Permutation(g, size=self.degree) if g in self.generators: return True return bool(self.coset_factor(g.array_form, True)) @property def is_perfect(self): """Return ``True`` if the group is perfect. A group is perfect if it equals to its derived subgroup. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation(1,2,3)(4,5) >>> b = Permutation(1,2,3,4,5) >>> G = PermutationGroup([a, b]) >>> G.is_perfect False """ if self._is_perfect is None: self._is_perfect = self.equals(self.derived_subgroup()) return self._is_perfect @property def is_abelian(self): """Test if the group is Abelian. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([0, 2, 1]) >>> b = Permutation([1, 0, 2]) >>> G = PermutationGroup([a, b]) >>> G.is_abelian False >>> a = Permutation([0, 2, 1]) >>> G = PermutationGroup([a]) >>> G.is_abelian True """ if self._is_abelian is not None: return self._is_abelian self._is_abelian = True gens = [p._array_form for p in self.generators] for x in gens: for y in gens: if y <= x: continue if not _af_commutes_with(x, y): self._is_abelian = False return False return True def abelian_invariants(self): """ Returns the abelian invariants for the given group. Let ``G`` be a nontrivial finite abelian group. Then G is isomorphic to the direct product of finitely many nontrivial cyclic groups of prime-power order. Explanation =========== The prime-powers that occur as the orders of the factors are uniquely determined by G. More precisely, the primes that occur in the orders of the factors in any such decomposition of ``G`` are exactly the primes that divide ``\|G\|`` and for any such prime ``p``, if the orders of the factors that are p-groups in one such decomposition of ``G`` are ``p^{t_1} >= p^{t_2} >= ... p^{t_r}``, then the orders of the factors that are p-groups in any such decomposition of ``G`` are ``p^{t_1} >= p^{t_2} >= ... p^{t_r}``. The uniquely determined integers ``p^{t_1} >= p^{t_2} >= ... p^{t_r}``, taken for all primes that divide ``\|G\|`` are called the invariants of the nontrivial group ``G`` as suggested in ([14], p. 542). Notes ===== We adopt the convention that the invariants of a trivial group are []. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([0, 2, 1]) >>> b = Permutation([1, 0, 2]) >>> G = PermutationGroup([a, b]) >>> G.abelian_invariants() [2] >>> from sympy.combinatorics import CyclicGroup >>> G = CyclicGroup(7) >>> G.abelian_invariants() [7] """ if self.is_trivial: return [] gns = self.generators inv = [] G = self H = G.derived_subgroup() Hgens = H.generators for p in primefactors(G.order()): ranks = [] while True: pows = [] for g in gns: elm = g*p if not H.contains(elm): pows.append(elm) K = PermutationGroup(Hgens + pows) if pows else H r = G.order()//K.order() G = K gns = pows if r == 1: break ranks.append(multiplicity(p, r)) if ranks: pows = [1]ranks[0] for i in ranks: for j in range(0, i): pows[j] = pows[j]p inv.extend(pows) inv.sort() return inv def is_elementary(self, p): """Return ``True`` if the group is elementary abelian. An elementary abelian group is a finite abelian group, where every nontrivial element has order `p`, where `p` is a prime. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([0, 2, 1]) >>> G = PermutationGroup([a]) >>> G.is_elementary(2) True >>> a = Permutation([0, 2, 1, 3]) >>> b = Permutation([3, 1, 2, 0]) >>> G = PermutationGroup([a, b]) >>> G.is_elementary(2) True >>> G.is_elementary(3) False """ return self.is_abelian and all(g.order() == p for g in self.generators) def _eval_is_alt_sym_naive(self, only_sym=False, only_alt=False): """A naive test using the group order.""" if only_sym and only_alt: raise ValueError( "Both {} and {} cannot be set to True" .format(only_sym, only_alt)) n = self.degree sym_order = 1 for i in range(2, n+1): sym_order = i order = self.order() if order == sym_order: self._is_sym = True self._is_alt = False if only_alt: return False return True elif 2order == sym_order: self._is_sym = False self._is_alt = True if only_sym: return False return True return False def _eval_is_alt_sym_monte_carlo(self, eps=0.05, perms=None): """A test using monte-carlo algorithm. Parameters ========== eps : float, optional The criterion for the incorrect ``False`` return. perms : list[Permutation], optional If explicitly given, it tests over the given candidats for testing. If ``None``, it randomly computes ``N_eps`` and chooses ``N_eps`` sample of the permutation from the group. See Also ======== _check_cycles_alt_sym """ if perms is None: n = self.degree if n < 17: c_n = 0.34 else: c_n = 0.57 d_n = (c_nlog(2))/log(n) N_eps = int(-log(eps)/d_n) perms = (self.random_pr() for i in range(N_eps)) return self._eval_is_alt_sym_monte_carlo(perms=perms) for perm in perms: if _check_cycles_alt_sym(perm): return True return False def is_alt_sym(self, eps=0.05, _random_prec=None): r"""Monte Carlo test for the symmetric/alternating group for degrees >= 8. Explanation =========== More specifically, it is one-sided Monte Carlo with the answer True (i.e., G is symmetric/alternating) guaranteed to be correct, and the answer False being incorrect with probability eps. For degree < 8, the order of the group is checked so the test is deterministic. Notes ===== The algorithm itself uses some nontrivial results from group theory and number theory: 1) If a transitive group ``G`` of degree ``n`` contains an element with a cycle of length ``n/2 < p < n-2`` for ``p`` a prime, ``G`` is the symmetric or alternating group ([1], pp. 81-82) 2) The proportion of elements in the symmetric/alternating group having the property described in 1) is approximately `\log(2)/\log(n)` ([1], p.82; [2], pp. 226-227). The helper function ``_check_cycles_alt_sym`` is used to go over the cycles in a permutation and look for ones satisfying 1). Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> D = DihedralGroup(10) >>> D.is_alt_sym() False See Also ======== _check_cycles_alt_sym """ if _random_prec is not None: N_eps = _random_prec['N_eps'] perms= (_random_prec[i] for i in range(N_eps)) return self._eval_is_alt_sym_monte_carlo(perms=perms) if self._is_sym or self._is_alt: return True if self._is_sym is False and self._is_alt is False: return False n = self.degree if n < 8: return self._eval_is_alt_sym_naive() elif self.is_transitive(): return self._eval_is_alt_sym_monte_carlo(eps=eps) self._is_sym, self._is_alt = False, False return False @property def is_nilpotent(self): """Test if the group is nilpotent. Explanation =========== A group `G` is nilpotent if it has a central series of finite length. Alternatively, `G` is nilpotent if its lower central series terminates with the trivial group. Every nilpotent group is also solvable ([1], p.29, [12]). Examples ======== >>> from sympy.combinatorics.named_groups import (SymmetricGroup, ... CyclicGroup) >>> C = CyclicGroup(6) >>> C.is_nilpotent True >>> S = SymmetricGroup(5) >>> S.is_nilpotent False See Also ======== lower_central_series, is_solvable """ if self._is_nilpotent is None: lcs = self.lower_central_series() terminator = lcs[len(lcs) - 1] gens = terminator.generators degree = self.degree identity = _af_new(list(range(degree))) if all(g == identity for g in gens): self._is_solvable = True self._is_nilpotent = True return True else: self._is_nilpotent = False return False else: return self._is_nilpotent def is_normal(self, gr, strict=True): """Test if ``G=self`` is a normal subgroup of ``gr``. Explanation =========== G is normal in gr if for each g2 in G, g1 in gr, ``g = g1g2g1*-1`` belongs to G It is sufficient to check this for each g1 in gr.generators and g2 in G.generators. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([1, 2, 0]) >>> b = Permutation([1, 0, 2]) >>> G = PermutationGroup([a, b]) >>> G1 = PermutationGroup([a, Permutation([2, 0, 1])]) >>> G1.is_normal(G) True """ if not self.is_subgroup(gr, strict=strict): return False d_self = self.degree d_gr = gr.degree if self.is_trivial and (d_self == d_gr or not strict): return True if self._is_abelian: return True new_self = self.copy() if not strict and d_self != d_gr: if d_self < d_gr: new_self = PermGroup(new_self.generators + [Permutation(d_gr - 1)]) else: gr = PermGroup(gr.generators + [Permutation(d_self - 1)]) gens2 = [p._array_form for p in new_self.generators] gens1 = [p._array_form for p in gr.generators] for g1 in gens1: for g2 in gens2: p = _af_rmuln(g1, g2, _af_invert(g1)) if not new_self.coset_factor(p, True): return False return True def is_primitive(self, randomized=True): r"""Test if a group is primitive. Explanation =========== A permutation group ``G`` acting on a set ``S`` is called primitive if ``S`` contains no nontrivial block under the action of ``G`` (a block is nontrivial if its cardinality is more than ``1``). Notes ===== The algorithm is described in [1], p.83, and uses the function minimal_block to search for blocks of the form `\{0, k\}` for ``k`` ranging over representatives for the orbits of `G_0`, the stabilizer of ``0``. This algorithm has complexity `O(n^2)` where ``n`` is the degree of the group, and will perform badly if `G_0` is small. There are two implementations offered: one finds `G_0` deterministically using the function ``stabilizer``, and the other (default) produces random elements of `G_0` using ``random_stab``, hoping that they generate a subgroup of `G_0` with not too many more orbits than `G_0` (this is suggested in [1], p.83). Behavior is changed by the ``randomized`` flag. Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> D = DihedralGroup(10) >>> D.is_primitive() False See Also ======== minimal_block, random_stab """ if self._is_primitive is not None: return self._is_primitive if self.is_transitive() is False: return False if randomized: random_stab_gens = [] v = self.schreier_vector(0) for _ in range(len(self)): random_stab_gens.append(self.random_stab(0, v)) stab = PermutationGroup(random_stab_gens) else: stab = self.stabilizer(0) orbits = stab.orbits() for orb in orbits: x = orb.pop() if x != 0 and any(e != 0 for e in self.minimal_block([0, x])): self._is_primitive = False return False self._is_primitive = True return True def minimal_blocks(self, randomized=True): ''' For a transitive group, return the list of all minimal block systems. If a group is intransitive, return `False`. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> from sympy.combinatorics.named_groups import DihedralGroup >>> DihedralGroup(6).minimal_blocks() [[0, 1, 0, 1, 0, 1], [0, 1, 2, 0, 1, 2]] >>> G = PermutationGroup(Permutation(1,2,5)) >>> G.minimal_blocks() False See Also ======== minimal_block, is_transitive, is_primitive ''' def _number_blocks(blocks): # number the blocks of a block system # in order and return the number of # blocks and the tuple with the # reordering n = len(blocks) appeared = {} m = 0 b = [None]n for i in range(n): if blocks[i] not in appeared: appeared[blocks[i]] = m b[i] = m m += 1 else: b[i] = appeared[blocks[i]] return tuple(b), m if not self.is_transitive(): return False blocks = [] num_blocks = [] rep_blocks = [] if randomized: random_stab_gens = [] v = self.schreier_vector(0) for i in range(len(self)): random_stab_gens.append(self.random_stab(0, v)) stab = PermutationGroup(random_stab_gens) else: stab = self.stabilizer(0) orbits = stab.orbits() for orb in orbits: x = orb.pop() if x != 0: block = self.minimal_block([0, x]) num_block, _ = _number_blocks(block) # a representative block (containing 0) rep = {j for j in range(self.degree) if num_block[j] == 0} # check if the system is minimal with # respect to the already discovere ones minimal = True blocks_remove_mask = [False] * len(blocks) for i, r in enumerate(rep_blocks): if len(r) > len(rep) and rep.issubset(r): # i-th block system is not minimal blocks_remove_mask[i] = True elif len(r) < len(rep) and r.issubset(rep): # the system being checked is not minimal minimal = False break # remove non-minimal representative blocks blocks = [b for i, b in enumerate(blocks) if not blocks_remove_mask[i]] num_blocks = [n for i, n in enumerate(num_blocks) if not blocks_remove_mask[i]] rep_blocks = [r for i, r in enumerate(rep_blocks) if not blocks_remove_mask[i]] if minimal and num_block not in num_blocks: blocks.append(block) num_blocks.append(num_block) rep_blocks.append(rep) return blocks @property def is_solvable(self): """Test if the group is solvable. ``G`` is solvable if its derived series terminates with the trivial group ([1], p.29). Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> S = SymmetricGroup(3) >>> S.is_solvable True See Also ======== is_nilpotent, derived_series """ if self._is_solvable is None: if self.order() % 2 != 0: return True ds = self.derived_series() terminator = ds[len(ds) - 1] gens = terminator.generators degree = self.degree identity = _af_new(list(range(degree))) if all(g == identity for g in gens): self._is_solvable = True return True else: self._is_solvable = False return False else: return self._is_solvable def is_subgroup(self, G, strict=True): """Return ``True`` if all elements of ``self`` belong to ``G``. If ``strict`` is ``False`` then if ``self``'s degree is smaller than ``G``'s, the elements will be resized to have the same degree. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> from sympy.combinatorics import SymmetricGroup, CyclicGroup Testing is strict by default: the degree of each group must be the same: >>> p = Permutation(0, 1, 2, 3, 4, 5) >>> G1 = PermutationGroup([Permutation(0, 1, 2), Permutation(0, 1)]) >>> G2 = PermutationGroup([Permutation(0, 2), Permutation(0, 1, 2)]) >>> G3 = PermutationGroup([p, p*2]) >>> assert G1.order() == G2.order() == G3.order() == 6 >>> G1.is_subgroup(G2) True >>> G1.is_subgroup(G3) False >>> G3.is_subgroup(PermutationGroup(G3[1])) False >>> G3.is_subgroup(PermutationGroup(G3[0])) True To ignore the size, set ``strict`` to ``False``: >>> S3 = SymmetricGroup(3) >>> S5 = SymmetricGroup(5) >>> S3.is_subgroup(S5, strict=False) True >>> C7 = CyclicGroup(7) >>> G = S5C7 >>> S5.is_subgroup(G, False) True >>> C7.is_subgroup(G, 0) False """ if isinstance(G, SymmetricPermutationGroup): if self.degree != G.degree: return False return True if not isinstance(G, PermutationGroup): return False if self == G or self.generators[0]==Permutation(): return True if G.order() % self.order() != 0: return False if self.degree == G.degree or \ (self.degree < G.degree and not strict): gens = self.generators else: return False return all(G.contains(g, strict=strict) for g in gens) @property def is_polycyclic(self): """Return ``True`` if a group is polycyclic. A group is polycyclic if it has a subnormal series with cyclic factors. For finite groups, this is the same as if the group is solvable. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([0, 2, 1, 3]) >>> b = Permutation([2, 0, 1, 3]) >>> G = PermutationGroup([a, b]) >>> G.is_polycyclic True """ return self.is_solvable def is_transitive(self, strict=True): """Test if the group is transitive. Explanation =========== A group is transitive if it has a single orbit. If ``strict`` is ``False`` the group is transitive if it has a single orbit of length different from 1. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([0, 2, 1, 3]) >>> b = Permutation([2, 0, 1, 3]) >>> G1 = PermutationGroup([a, b]) >>> G1.is_transitive() False >>> G1.is_transitive(strict=False) True >>> c = Permutation([2, 3, 0, 1]) >>> G2 = PermutationGroup([a, c]) >>> G2.is_transitive() True >>> d = Permutation([1, 0, 2, 3]) >>> e = Permutation([0, 1, 3, 2]) >>> G3 = PermutationGroup([d, e]) >>> G3.is_transitive() or G3.is_transitive(strict=False) False """ if self._is_transitive: # strict or not, if True then True return self._is_transitive if strict: if self._is_transitive is not None: # we only store strict=True return self._is_transitive ans = len(self.orbit(0)) == self.degree self._is_transitive = ans return ans got_orb = False for x in self.orbits(): if len(x) > 1: if got_orb: return False got_orb = True return got_orb @property def is_trivial(self): """Test if the group is the trivial group. This is true if the group contains only the identity permutation. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> G = PermutationGroup([Permutation([0, 1, 2])]) >>> G.is_trivial True """ if self._is_trivial is None: self._is_trivial = len(self) == 1 and self[0].is_Identity return self._is_trivial def lower_central_series(self): r"""Return the lower central series for the group. The lower central series for a group `G` is the series `G = G_0 > G_1 > G_2 > \ldots` where `G_k = [G, G_{k-1}]`, i.e. every term after the first is equal to the commutator of `G` and the previous term in `G1` ([1], p.29). Returns ======= A list of permutation groups in the order `G = G_0, G_1, G_2, \ldots` Examples ======== >>> from sympy.combinatorics.named_groups import (AlternatingGroup, ... DihedralGroup) >>> A = AlternatingGroup(4) >>> len(A.lower_central_series()) 2 >>> A.lower_central_series()[1].is_subgroup(DihedralGroup(2)) True See Also ======== commutator, derived_series """ res = [self] current = self nxt = self.commutator(self, current) while not current.is_subgroup(nxt): res.append(nxt) current = nxt nxt = self.commutator(self, current) return res @property def max_div(self): """Maximum proper divisor of the degree of a permutation group. Explanation =========== Obviously, this is the degree divided by its minimal proper divisor (larger than ``1``, if one exists). As it is guaranteed to be prime, the ``sieve`` from ``sympy.ntheory`` is used. This function is also used as an optimization tool for the functions ``minimal_block`` and ``_union_find_merge``. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> G = PermutationGroup([Permutation([0, 2, 1, 3])]) >>> G.max_div 2 See Also ======== minimal_block, _union_find_merge """ if self._max_div is not None: return self._max_div n = self.degree if n == 1: return 1 for x in sieve: if n % x == 0: d = n//x self._max_div = d return d def minimal_block(self, points): r"""For a transitive group, finds the block system generated by ``points``. Explanation =========== If a group ``G`` acts on a set ``S``, a nonempty subset ``B`` of ``S`` is called a block under the action of ``G`` if for all ``g`` in ``G`` we have ``gB = B`` (``g`` fixes ``B``) or ``gB`` and ``B`` have no common points (``g`` moves ``B`` entirely). ([1], p.23; [6]). The distinct translates ``gB`` of a block ``B`` for ``g`` in ``G`` partition the set ``S`` and this set of translates is known as a block system. Moreover, we obviously have that all blocks in the partition have the same size, hence the block size divides ``\|S\|`` ([1], p.23). A ``G``-congruence is an equivalence relation ``~`` on the set ``S`` such that ``a ~ b`` implies ``g(a) ~ g(b)`` for all ``g`` in ``G``. For a transitive group, the equivalence classes of a ``G``-congruence and the blocks of a block system are the same thing ([1], p.23). The algorithm below checks the group for transitivity, and then finds the ``G``-congruence generated by the pairs ``(p_0, p_1), (p_0, p_2), ..., (p_0,p_{k-1})`` which is the same as finding the maximal block system (i.e., the one with minimum block size) such that ``p_0, ..., p_{k-1}`` are in the same block ([1], p.83). It is an implementation of Atkinson's algorithm, as suggested in [1], and manipulates an equivalence relation on the set ``S`` using a union-find data structure. The running time is just above `O(\|points\|\|S\|)`. ([1], pp. 83-87; [7]). Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> D = DihedralGroup(10) >>> D.minimal_block([0, 5]) [0, 1, 2, 3, 4, 0, 1, 2, 3, 4] >>> D.minimal_block([0, 1]) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] See Also ======== _union_find_rep, _union_find_merge, is_transitive, is_primitive """ if not self.is_transitive(): return False n = self.degree gens = self.generators # initialize the list of equivalence class representatives parents = list(range(n)) ranks = [1]n not_rep = [] k = len(points) # the block size must divide the degree of the group if k > self.max_div: return [0]n for i in range(k - 1): parents[points[i + 1]] = points[0] not_rep.append(points[i + 1]) ranks[points[0]] = k i = 0 len_not_rep = k - 1 while i < len_not_rep: gamma = not_rep[i] i += 1 for gen in gens: # find has side effects: performs path compression on the list # of representatives delta = self._union_find_rep(gamma, parents) # union has side effects: performs union by rank on the list # of representatives temp = self._union_find_merge(gen(gamma), gen(delta), ranks, parents, not_rep) if temp == -1: return [0]n len_not_rep += temp for i in range(n): # force path compression to get the final state of the equivalence # relation self._union_find_rep(i, parents) # rewrite result so that block representatives are minimal new_reps = {} return [new_reps.setdefault(r, i) for i, r in enumerate(parents)] def conjugacy_class(self, x): r"""Return the conjugacy class of an element in the group. Explanation =========== The conjugacy class of an element ``g`` in a group ``G`` is the set of elements ``x`` in ``G`` that are conjugate with ``g``, i.e. for which ``g = xax^{-1}`` for some ``a`` in ``G``. Note that conjugacy is an equivalence relation, and therefore that conjugacy classes are partitions of ``G``. For a list of all the conjugacy classes of the group, use the conjugacy_classes() method. In a permutation group, each conjugacy class corresponds to a particular `cycle structure': for example, in ``S_3``, the conjugacy classes are: the identity class, ``{()}`` * all transpositions, ``{(1 2), (1 3), (2 3)}`` * all 3-cycles, ``{(1 2 3), (1 3 2)}`` Examples ======== >>> from sympy.combinatorics import Permutation, SymmetricGroup >>> S3 = SymmetricGroup(3) >>> S3.conjugacy_class(Permutation(0, 1, 2)) {(0 1 2), (0 2 1)} Notes ===== This procedure computes the conjugacy class directly by finding the orbit of the element under conjugation in G. This algorithm is only feasible for permutation groups of relatively small order, but is like the orbit() function itself in that respect. """ # Ref: "Computing the conjugacy classes of finite groups"; Butler, G. # Groups '93 Galway/St Andrews; edited by Campbell, C. M. new_class = {x} last_iteration = new_class while len(last_iteration) > 0: this_iteration = set() for y in last_iteration: for s in self.generators: conjugated = s * y * (~s) if conjugated not in new_class: this_iteration.add(conjugated) new_class.update(last_iteration) last_iteration = this_iteration return new_class def conjugacy_classes(self): r"""Return the conjugacy classes of the group. Explanation =========== As described in the documentation for the .conjugacy_class() function, conjugacy is an equivalence relation on a group G which partitions the set of elements. This method returns a list of all these conjugacy classes of G. Examples ======== >>> from sympy.combinatorics import SymmetricGroup >>> SymmetricGroup(3).conjugacy_classes() [{(2)}, {(0 1 2), (0 2 1)}, {(0 2), (1 2), (2)(0 1)}] """ identity = _af_new(list(range(self.degree))) known_elements = {identity} classes = [known_elements.copy()] for x in self.generate(): if x not in known_elements: new_class = self.conjugacy_class(x) classes.append(new_class) known_elements.update(new_class) return classes def normal_closure(self, other, k=10): r"""Return the normal closure of a subgroup/set of permutations. Explanation =========== If ``S`` is a subset of a group ``G``, the normal closure of ``A`` in ``G`` is defined as the intersection of all normal subgroups of ``G`` that contain ``A`` ([1], p.14). Alternatively, it is the group generated by the conjugates ``x^{-1}yx`` for ``x`` a generator of ``G`` and ``y`` a generator of the subgroup ``\left\langle S\right\rangle`` generated by ``S`` (for some chosen generating set for ``\left\langle S\right\rangle``) ([1], p.73). Parameters ========== other a subgroup/list of permutations/single permutation k an implementation-specific parameter that determines the number of conjugates that are adjoined to ``other`` at once Examples ======== >>> from sympy.combinatorics.named_groups import (SymmetricGroup, ... CyclicGroup, AlternatingGroup) >>> S = SymmetricGroup(5) >>> C = CyclicGroup(5) >>> G = S.normal_closure(C) >>> G.order() 60 >>> G.is_subgroup(AlternatingGroup(5)) True See Also ======== commutator, derived_subgroup, random_pr Notes ===== The algorithm is described in [1], pp. 73-74; it makes use of the generation of random elements for permutation groups by the product replacement algorithm. """ if hasattr(other, 'generators'): degree = self.degree identity = _af_new(list(range(degree))) if all(g == identity for g in other.generators): return other Z = PermutationGroup(other.generators[:]) base, strong_gens = Z.schreier_sims_incremental() strong_gens_distr = _distribute_gens_by_base(base, strong_gens) basic_orbits, basic_transversals = \ _orbits_transversals_from_bsgs(base, strong_gens_distr) self._random_pr_init(r=10, n=20) _loop = True while _loop: Z._random_pr_init(r=10, n=10) for _ in range(k): g = self.random_pr() h = Z.random_pr() conj = h^g res = _strip(conj, base, basic_orbits, basic_transversals) if res[0] != identity or res[1] != len(base) + 1: gens = Z.generators gens.append(conj) Z = PermutationGroup(gens) strong_gens.append(conj) temp_base, temp_strong_gens = \ Z.schreier_sims_incremental(base, strong_gens) base, strong_gens = temp_base, temp_strong_gens strong_gens_distr = \ _distribute_gens_by_base(base, strong_gens) basic_orbits, basic_transversals = \ _orbits_transversals_from_bsgs(base, strong_gens_distr) _loop = False for g in self.generators: for h in Z.generators: conj = h^g res = _strip(conj, base, basic_orbits, basic_transversals) if res[0] != identity or res[1] != len(base) + 1: _loop = True break if _loop: break return Z elif hasattr(other, '__getitem__'): return self.normal_closure(PermutationGroup(other)) elif hasattr(other, 'array_form'): return self.normal_closure(PermutationGroup([other])) def orbit(self, alpha, action='tuples'): r"""Compute the orbit of alpha `\{g(\alpha) \| g \in G\}` as a set. Explanation =========== The time complexity of the algorithm used here is `O(\|Orb\|r)` where `\|Orb\|` is the size of the orbit and ``r`` is the number of generators of the group. For a more detailed analysis, see [1], p.78, [2], pp. 19-21. Here alpha can be a single point, or a list of points. If alpha is a single point, the ordinary orbit is computed. if alpha is a list of points, there are three available options: 'union' - computes the union of the orbits of the points in the list 'tuples' - computes the orbit of the list interpreted as an ordered tuple under the group action ( i.e., g((1,2,3)) = (g(1), g(2), g(3)) ) 'sets' - computes the orbit of the list interpreted as a sets Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([1, 2, 0, 4, 5, 6, 3]) >>> G = PermutationGroup([a]) >>> G.orbit(0) {0, 1, 2} >>> G.orbit([0, 4], 'union') {0, 1, 2, 3, 4, 5, 6} See Also ======== orbit_transversal """ return _orbit(self.degree, self.generators, alpha, action) def orbit_rep(self, alpha, beta, schreier_vector=None): """Return a group element which sends ``alpha`` to ``beta``. Explanation =========== If ``beta`` is not in the orbit of ``alpha``, the function returns ``False``. This implementation makes use of the schreier vector. For a proof of correctness, see [1], p.80 Examples ======== >>> from sympy.combinatorics.named_groups import AlternatingGroup >>> G = AlternatingGroup(5) >>> G.orbit_rep(0, 4) (0 4 1 2 3) See Also ======== schreier_vector """ if schreier_vector is None: schreier_vector = self.schreier_vector(alpha) if schreier_vector[beta] is None: return False k = schreier_vector[beta] gens = [x._array_form for x in self.generators] a = [] while k != -1: a.append(gens[k]) beta = gens[k].index(beta) # beta = (~gens[k])(beta) k = schreier_vector[beta] if a: return _af_new(_af_rmuln(a)) else: return _af_new(list(range(self._degree))) def orbit_transversal(self, alpha, pairs=False): r"""Computes a transversal for the orbit of ``alpha`` as a set. Explanation =========== For a permutation group `G`, a transversal for the orbit `Orb = \{g(\alpha) \| g \in G\}` is a set `\{g_\beta \| g_\beta(\alpha) = \beta\}` for `\beta \in Orb`. Note that there may be more than one possible transversal. If ``pairs`` is set to ``True``, it returns the list of pairs `(\beta, g_\beta)`. For a proof of correctness, see [1], p.79 Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> G = DihedralGroup(6) >>> G.orbit_transversal(0) [(5), (0 1 2 3 4 5), (0 5)(1 4)(2 3), (0 2 4)(1 3 5), (5)(0 4)(1 3), (0 3)(1 4)(2 5)] See Also ======== orbit """ return _orbit_transversal(self._degree, self.generators, alpha, pairs) def orbits(self, rep=False): """Return the orbits of ``self``, ordered according to lowest element in each orbit. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation(1, 5)(2, 3)(4, 0, 6) >>> b = Permutation(1, 5)(3, 4)(2, 6, 0) >>> G = PermutationGroup([a, b]) >>> G.orbits() [{0, 2, 3, 4, 6}, {1, 5}] """ return _orbits(self._degree, self._generators) def order(self): """Return the order of the group: the number of permutations that can be generated from elements of the group. The number of permutations comprising the group is given by ``len(group)``; the length of each permutation in the group is given by ``group.size``. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([1, 0, 2]) >>> G = PermutationGroup([a]) >>> G.degree 3 >>> len(G) 1 >>> G.order() 2 >>> list(G.generate()) [(2), (2)(0 1)] >>> a = Permutation([0, 2, 1]) >>> b = Permutation([1, 0, 2]) >>> G = PermutationGroup([a, b]) >>> G.order() 6 See Also ======== degree """ if self._order is not None: return self._order if self._is_sym: n = self._degree self._order = factorial(n) return self._order if self._is_alt: n = self._degree self._order = factorial(n)/2 return self._order basic_transversals = self.basic_transversals m = 1 for x in basic_transversals: m = len(x) self._order = m return m def index(self, H): """ Returns the index of a permutation group. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation(1,2,3) >>> b =Permutation(3) >>> G = PermutationGroup([a]) >>> H = PermutationGroup([b]) >>> G.index(H) 3 """ if H.is_subgroup(self): return self.order()//H.order() @property def is_symmetric(self): """Return ``True`` if the group is symmetric. Examples ======== >>> from sympy.combinatorics import SymmetricGroup >>> g = SymmetricGroup(5) >>> g.is_symmetric True >>> from sympy.combinatorics import Permutation, PermutationGroup >>> g = PermutationGroup( ... Permutation(0, 1, 2, 3, 4), ... Permutation(2, 3)) >>> g.is_symmetric True Notes ===== This uses a naive test involving the computation of the full group order. If you need more quicker taxonomy for large groups, you can use :meth:`PermutationGroup.is_alt_sym`. However, :meth:`PermutationGroup.is_alt_sym` may not be accurate and is not able to distinguish between an alternating group and a symmetric group. See Also ======== is_alt_sym """ _is_sym = self._is_sym if _is_sym is not None: return _is_sym n = self.degree if n >= 8: if self.is_transitive(): _is_alt_sym = self._eval_is_alt_sym_monte_carlo() if _is_alt_sym: if any(g.is_odd for g in self.generators): self._is_sym, self._is_alt = True, False return True self._is_sym, self._is_alt = False, True return False return self._eval_is_alt_sym_naive(only_sym=True) self._is_sym, self._is_alt = False, False return False return self._eval_is_alt_sym_naive(only_sym=True) @property def is_alternating(self): """Return ``True`` if the group is alternating. Examples ======== >>> from sympy.combinatorics import AlternatingGroup >>> g = AlternatingGroup(5) >>> g.is_alternating True >>> from sympy.combinatorics import Permutation, PermutationGroup >>> g = PermutationGroup( ... Permutation(0, 1, 2, 3, 4), ... Permutation(2, 3, 4)) >>> g.is_alternating True Notes ===== This uses a naive test involving the computation of the full group order. If you need more quicker taxonomy for large groups, you can use :meth:`PermutationGroup.is_alt_sym`. However, :meth:`PermutationGroup.is_alt_sym` may not be accurate and is not able to distinguish between an alternating group and a symmetric group. See Also ======== is_alt_sym """ _is_alt = self._is_alt if _is_alt is not None: return _is_alt n = self.degree if n >= 8: if self.is_transitive(): _is_alt_sym = self._eval_is_alt_sym_monte_carlo() if _is_alt_sym: if all(g.is_even for g in self.generators): self._is_sym, self._is_alt = False, True return True self._is_sym, self._is_alt = True, False return False return self._eval_is_alt_sym_naive(only_alt=True) self._is_sym, self._is_alt = False, False return False return self._eval_is_alt_sym_naive(only_alt=True) @classmethod def _distinct_primes_lemma(cls, primes): """Subroutine to test if there is only one cyclic group for the order.""" primes = sorted(primes) l = len(primes) for i in range(l): for j in range(i+1, l): if primes[j] % primes[i] == 1: return None return True @property def is_cyclic(self): r""" Return ``True`` if the group is Cyclic. Examples ======== >>> from sympy.combinatorics.named_groups import AbelianGroup >>> G = AbelianGroup(3, 4) >>> G.is_cyclic True >>> G = AbelianGroup(4, 4) >>> G.is_cyclic False Notes ===== If the order of a group $n$ can be factored into the distinct primes $p_1, p_2, \dots , p_s$ and if .. math:: \forall i, j \in \{1, 2, \dots, s \}: p_i \not \equiv 1 \pmod {p_j} holds true, there is only one group of the order $n$ which is a cyclic group [1]_. This is a generalization of the lemma that the group of order $15, 35, \dots$ are cyclic. And also, these additional lemmas can be used to test if a group is cyclic if the order of the group is already found. - If the group is abelian and the order of the group is square-free, the group is cyclic. - If the order of the group is less than $6$ and is not $4$, the group is cyclic. - If the order of the group is prime, the group is cyclic. References ========== .. [1] 1978: John S. Rose: A Course on Group Theory, Introduction to Finite Group Theory: 1.4 """ if self._is_cyclic is not None: return self._is_cyclic if len(self.generators) == 1: self._is_cyclic = True self._is_abelian = True return True if self._is_abelian is False: self._is_cyclic = False return False order = self.order() if order < 6: self._is_abelian = True if order != 4: self._is_cyclic = True return True factors = factorint(order) if all(v == 1 for v in factors.values()): if self._is_abelian: self._is_cyclic = True return True primes = list(factors.keys()) if PermutationGroup._distinct_primes_lemma(primes) is True: self._is_cyclic = True self._is_abelian = True return True for p in factors: pgens = [] for g in self.generators: pgens.append(gp) if self.index(self.subgroup(pgens)) != p: self._is_cyclic = False return False self._is_cyclic = True self._is_abelian = True return True def pointwise_stabilizer(self, points, incremental=True): r"""Return the pointwise stabilizer for a set of points. Explanation =========== For a permutation group `G` and a set of points `\{p_1, p_2,\ldots, p_k\}`, the pointwise stabilizer of `p_1, p_2, \ldots, p_k` is defined as `G_{p_1,\ldots, p_k} = \{g\in G \| g(p_i) = p_i \forall i\in\{1, 2,\ldots,k\}\}` ([1],p20). It is a subgroup of `G`. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> S = SymmetricGroup(7) >>> Stab = S.pointwise_stabilizer([2, 3, 5]) >>> Stab.is_subgroup(S.stabilizer(2).stabilizer(3).stabilizer(5)) True See Also ======== stabilizer, schreier_sims_incremental Notes ===== When incremental == True, rather than the obvious implementation using successive calls to ``.stabilizer()``, this uses the incremental Schreier-Sims algorithm to obtain a base with starting segment - the given points. """ if incremental: base, strong_gens = self.schreier_sims_incremental(base=points) stab_gens = [] degree = self.degree for gen in strong_gens: if [gen(point) for point in points] == points: stab_gens.append(gen) if not stab_gens: stab_gens = _af_new(list(range(degree))) return PermutationGroup(stab_gens) else: gens = self._generators degree = self.degree for x in points: gens = _stabilizer(degree, gens, x) return PermutationGroup(gens) def make_perm(self, n, seed=None): """ Multiply ``n`` randomly selected permutations from pgroup together, starting with the identity permutation. If ``n`` is a list of integers, those integers will be used to select the permutations and they will be applied in L to R order: make_perm((A, B, C)) will give CBA(I) where I is the identity permutation. ``seed`` is used to set the seed for the random selection of permutations from pgroup. If this is a list of integers, the corresponding permutations from pgroup will be selected in the order give. This is mainly used for testing purposes. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a, b = [Permutation([1, 0, 3, 2]), Permutation([1, 3, 0, 2])] >>> G = PermutationGroup([a, b]) >>> G.make_perm(1, [0]) (0 1)(2 3) >>> G.make_perm(3, [0, 1, 0]) (0 2 3 1) >>> G.make_perm([0, 1, 0]) (0 2 3 1) See Also ======== random """ if is_sequence(n): if seed is not None: raise ValueError('If n is a sequence, seed should be None') n, seed = len(n), n else: try: n = int(n) except TypeError: raise ValueError('n must be an integer or a sequence.') randomrange = _randrange(seed) # start with the identity permutation result = Permutation(list(range(self.degree))) m = len(self) for _ in range(n): p = self[randomrange(m)] result = rmul(result, p) return result def random(self, af=False): """Return a random group element """ rank = randrange(self.order()) return self.coset_unrank(rank, af) def random_pr(self, gen_count=11, iterations=50, _random_prec=None): """Return a random group element using product replacement. Explanation =========== For the details of the product replacement algorithm, see ``_random_pr_init`` In ``random_pr`` the actual 'product replacement' is performed. Notice that if the attribute ``_random_gens`` is empty, it needs to be initialized by ``_random_pr_init``. See Also ======== _random_pr_init """ if self._random_gens == []: self._random_pr_init(gen_count, iterations) random_gens = self._random_gens r = len(random_gens) - 1 # handle randomized input for testing purposes if _random_prec is None: s = randrange(r) t = randrange(r - 1) if t == s: t = r - 1 x = choice([1, 2]) e = choice([-1, 1]) else: s = _random_prec['s'] t = _random_prec['t'] if t == s: t = r - 1 x = _random_prec['x'] e = _random_prec['e'] if x == 1: random_gens[s] = _af_rmul(random_gens[s], _af_pow(random_gens[t], e)) random_gens[r] = _af_rmul(random_gens[r], random_gens[s]) else: random_gens[s] = _af_rmul(_af_pow(random_gens[t], e), random_gens[s]) random_gens[r] = _af_rmul(random_gens[s], random_gens[r]) return _af_new(random_gens[r]) def random_stab(self, alpha, schreier_vector=None, _random_prec=None): """Random element from the stabilizer of ``alpha``. The schreier vector for ``alpha`` is an optional argument used for speeding up repeated calls. The algorithm is described in [1], p.81 See Also ======== random_pr, orbit_rep """ if schreier_vector is None: schreier_vector = self.schreier_vector(alpha) if _random_prec is None: rand = self.random_pr() else: rand = _random_prec['rand'] beta = rand(alpha) h = self.orbit_rep(alpha, beta, schreier_vector) return rmul(~h, rand) def schreier_sims(self): """Schreier-Sims algorithm. Explanation =========== It computes the generators of the chain of stabilizers `G > G_{b_1} > .. > G_{b1,..,b_r} > 1` in which `G_{b_1,..,b_i}` stabilizes `b_1,..,b_i`, and the corresponding ``s`` cosets. An element of the group can be written as the product `h_1..h_s`. We use the incremental Schreier-Sims algorithm. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([0, 2, 1]) >>> b = Permutation([1, 0, 2]) >>> G = PermutationGroup([a, b]) >>> G.schreier_sims() >>> G.basic_transversals [{0: (2)(0 1), 1: (2), 2: (1 2)}, {0: (2), 2: (0 2)}] """ if self._transversals: return self._schreier_sims() return def _schreier_sims(self, base=None): schreier = self.schreier_sims_incremental(base=base, slp_dict=True) base, strong_gens = schreier[:2] self._base = base self._strong_gens = strong_gens self._strong_gens_slp = schreier[2] if not base: self._transversals = [] self._basic_orbits = [] return strong_gens_distr = _distribute_gens_by_base(base, strong_gens) basic_orbits, transversals, slps = _orbits_transversals_from_bsgs(base,\ strong_gens_distr, slp=True) # rewrite the indices stored in slps in terms of strong_gens for i, slp in enumerate(slps): gens = strong_gens_distr[i] for k in slp: slp[k] = [strong_gens.index(gens[s]) for s in slp[k]] self._transversals = transversals self._basic_orbits = [sorted(x) for x in basic_orbits] self._transversal_slp = slps def schreier_sims_incremental(self, base=None, gens=None, slp_dict=False): """Extend a sequence of points and generating set to a base and strong generating set. Parameters ========== base The sequence of points to be extended to a base. Optional parameter with default value ``[]``. gens The generating set to be extended to a strong generating set relative to the base obtained. Optional parameter with default value ``self.generators``. slp_dict If `True`, return a dictionary `{g: gens}` for each strong generator `g` where `gens` is a list of strong generators coming before `g` in `strong_gens`, such that the product of the elements of `gens` is equal to `g`. Returns ======= (base, strong_gens) ``base`` is the base obtained, and ``strong_gens`` is the strong generating set relative to it. The original parameters ``base``, ``gens`` remain unchanged. Examples ======== >>> from sympy.combinatorics.named_groups import AlternatingGroup >>> from sympy.combinatorics.testutil import _verify_bsgs >>> A = AlternatingGroup(7) >>> base = [2, 3] >>> seq = [2, 3] >>> base, strong_gens = A.schreier_sims_incremental(base=seq) >>> _verify_bsgs(A, base, strong_gens) True >>> base[:2] [2, 3] Notes ===== This version of the Schreier-Sims algorithm runs in polynomial time. There are certain assumptions in the implementation - if the trivial group is provided, ``base`` and ``gens`` are returned immediately, as any sequence of points is a base for the trivial group. If the identity is present in the generators ``gens``, it is removed as it is a redundant generator. The implementation is described in [1], pp. 90-93. See Also ======== schreier_sims, schreier_sims_random """ if base is None: base = [] if gens is None: gens = self.generators[:] degree = self.degree id_af = list(range(degree)) # handle the trivial group if len(gens) == 1 and gens[0].is_Identity: if slp_dict: return base, gens, {gens[0]: [gens[0]]} return base, gens # prevent side effects _base, _gens = base[:], gens[:] # remove the identity as a generator _gens = [x for x in _gens if not x.is_Identity] # make sure no generator fixes all base points for gen in _gens: if all(x == gen._array_form[x] for x in _base): for new in id_af: if gen._array_form[new] != new: break else: assert None # can this ever happen? _base.append(new) # distribute generators according to basic stabilizers strong_gens_distr = _distribute_gens_by_base(_base, _gens) strong_gens_slp = [] # initialize the basic stabilizers, basic orbits and basic transversals orbs = {} transversals = {} slps = {} base_len = len(_base) for i in range(base_len): transversals[i], slps[i] = _orbit_transversal(degree, strong_gens_distr[i], _base[i], pairs=True, af=True, slp=True) transversals[i] = dict(transversals[i]) orbs[i] = list(transversals[i].keys()) # main loop: amend the stabilizer chain until we have generators # for all stabilizers i = base_len - 1 while i >= 0: # this flag is used to continue with the main loop from inside # a nested loop continue_i = False # test the generators for being a strong generating set db = {} for beta, u_beta in list(transversals[i].items()): for j, gen in enumerate(strong_gens_distr[i]): gb = gen._array_form[beta] u1 = transversals[i][gb] g1 = _af_rmul(gen._array_form, u_beta) slp = [(i, g) for g in slps[i][beta]] slp = [(i, j)] + slp if g1 != u1: # test if the schreier generator is in the i+1-th # would-be basic stabilizer y = True try: u1_inv = db[gb] except KeyError: u1_inv = db[gb] = _af_invert(u1) schreier_gen = _af_rmul(u1_inv, g1) u1_inv_slp = slps[i][gb][:] u1_inv_slp.reverse() u1_inv_slp = [(i, (g,)) for g in u1_inv_slp] slp = u1_inv_slp + slp h, j, slp = _strip_af(schreier_gen, _base, orbs, transversals, i, slp=slp, slps=slps) if j <= base_len: # new strong generator h at level j y = False elif h: # h fixes all base points y = False moved = 0 while h[moved] == moved: moved += 1 _base.append(moved) base_len += 1 strong_gens_distr.append([]) if y is False: # if a new strong generator is found, update the # data structures and start over h = _af_new(h) strong_gens_slp.append((h, slp)) for l in range(i + 1, j): strong_gens_distr[l].append(h) transversals[l], slps[l] =\ _orbit_transversal(degree, strong_gens_distr[l], _base[l], pairs=True, af=True, slp=True) transversals[l] = dict(transversals[l]) orbs[l] = list(transversals[l].keys()) i = j - 1 # continue main loop using the flag continue_i = True if continue_i is True: break if continue_i is True: break if continue_i is True: continue i -= 1 strong_gens = _gens[:] if slp_dict: # create the list of the strong generators strong_gens and # rewrite the indices of strong_gens_slp in terms of the # elements of strong_gens for k, slp in strong_gens_slp: strong_gens.append(k) for i in range(len(slp)): s = slp[i] if isinstance(s[1], tuple): slp[i] = strong_gens_distr[s[0]][s[1][0]]-1 else: slp[i] = strong_gens_distr[s[0]][s[1]] strong_gens_slp = dict(strong_gens_slp) # add the original generators for g in _gens: strong_gens_slp[g] = [g] return (_base, strong_gens, strong_gens_slp) strong_gens.extend([k for k, _ in strong_gens_slp]) return _base, strong_gens def schreier_sims_random(self, base=None, gens=None, consec_succ=10, _random_prec=None): r"""Randomized Schreier-Sims algorithm. Explanation =========== The randomized Schreier-Sims algorithm takes the sequence ``base`` and the generating set ``gens``, and extends ``base`` to a base, and ``gens`` to a strong generating set relative to that base with probability of a wrong answer at most `2^{-consec\_succ}`, provided the random generators are sufficiently random. Parameters ========== base The sequence to be extended to a base. gens The generating set to be extended to a strong generating set. consec_succ The parameter defining the probability of a wrong answer. _random_prec An internal parameter used for testing purposes. Returns ======= (base, strong_gens) ``base`` is the base and ``strong_gens`` is the strong generating set relative to it. Examples ======== >>> from sympy.combinatorics.testutil import _verify_bsgs >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> S = SymmetricGroup(5) >>> base, strong_gens = S.schreier_sims_random(consec_succ=5) >>> _verify_bsgs(S, base, strong_gens) #doctest: +SKIP True Notes ===== The algorithm is described in detail in [1], pp. 97-98. It extends the orbits ``orbs`` and the permutation groups ``stabs`` to basic orbits and basic stabilizers for the base and strong generating set produced in the end. The idea of the extension process is to "sift" random group elements through the stabilizer chain and amend the stabilizers/orbits along the way when a sift is not successful. The helper function ``_strip`` is used to attempt to decompose a random group element according to the current state of the stabilizer chain and report whether the element was fully decomposed (successful sift) or not (unsuccessful sift). In the latter case, the level at which the sift failed is reported and used to amend ``stabs``, ``base``, ``gens`` and ``orbs`` accordingly. The halting condition is for ``consec_succ`` consecutive successful sifts to pass. This makes sure that the current ``base`` and ``gens`` form a BSGS with probability at least `1 - 1/\text{consec\_succ}`. See Also ======== schreier_sims """ if base is None: base = [] if gens is None: gens = self.generators base_len = len(base) n = self.degree # make sure no generator fixes all base points for gen in gens: if all(gen(x) == x for x in base): new = 0 while gen._array_form[new] == new: new += 1 base.append(new) base_len += 1 # distribute generators according to basic stabilizers strong_gens_distr = _distribute_gens_by_base(base, gens) # initialize the basic stabilizers, basic transversals and basic orbits transversals = {} orbs = {} for i in range(base_len): transversals[i] = dict(_orbit_transversal(n, strong_gens_distr[i], base[i], pairs=True)) orbs[i] = list(transversals[i].keys()) # initialize the number of consecutive elements sifted c = 0 # start sifting random elements while the number of consecutive sifts # is less than consec_succ while c < consec_succ: if _random_prec is None: g = self.random_pr() else: g = _random_prec['g'].pop() h, j = _strip(g, base, orbs, transversals) y = True # determine whether a new base point is needed if j <= base_len: y = False elif not h.is_Identity: y = False moved = 0 while h(moved) == moved: moved += 1 base.append(moved) base_len += 1 strong_gens_distr.append([]) # if the element doesn't sift, amend the strong generators and # associated stabilizers and orbits if y is False: for l in range(1, j): strong_gens_distr[l].append(h) transversals[l] = dict(_orbit_transversal(n, strong_gens_distr[l], base[l], pairs=True)) orbs[l] = list(transversals[l].keys()) c = 0 else: c += 1 # build the strong generating set strong_gens = strong_gens_distr[0][:] for gen in strong_gens_distr[1]: if gen not in strong_gens: strong_gens.append(gen) return base, strong_gens def schreier_vector(self, alpha): """Computes the schreier vector for ``alpha``. Explanation =========== The Schreier vector efficiently stores information about the orbit of ``alpha``. It can later be used to quickly obtain elements of the group that send ``alpha`` to a particular element in the orbit. Notice that the Schreier vector depends on the order in which the group generators are listed. For a definition, see [3]. Since list indices start from zero, we adopt the convention to use "None" instead of 0 to signify that an element does not belong to the orbit. For the algorithm and its correctness, see [2], pp.78-80. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([2, 4, 6, 3, 1, 5, 0]) >>> b = Permutation([0, 1, 3, 5, 4, 6, 2]) >>> G = PermutationGroup([a, b]) >>> G.schreier_vector(0) [-1, None, 0, 1, None, 1, 0] See Also ======== orbit """ n = self.degree v = [None]n v[alpha] = -1 orb = [alpha] used = [False]n used[alpha] = True gens = self.generators r = len(gens) for b in orb: for i in range(r): temp = gens[i]._array_form[b] if used[temp] is False: orb.append(temp) used[temp] = True v[temp] = i return v def stabilizer(self, alpha): r"""Return the stabilizer subgroup of ``alpha``. Explanation =========== The stabilizer of `\alpha` is the group `G_\alpha = \{g \in G \| g(\alpha) = \alpha\}`. For a proof of correctness, see [1], p.79. Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> G = DihedralGroup(6) >>> G.stabilizer(5) PermutationGroup([ (5)(0 4)(1 3)]) See Also ======== orbit """ return PermGroup(_stabilizer(self._degree, self._generators, alpha)) @property def strong_gens(self): r"""Return a strong generating set from the Schreier-Sims algorithm. Explanation =========== A generating set `S = \{g_1, g_2, \dots, g_t\}` for a permutation group `G` is a strong generating set relative to the sequence of points (referred to as a "base") `(b_1, b_2, \dots, b_k)` if, for `1 \leq i \leq k` we have that the intersection of the pointwise stabilizer `G^{(i+1)} := G_{b_1, b_2, \dots, b_i}` with `S` generates the pointwise stabilizer `G^{(i+1)}`. The concepts of a base and strong generating set and their applications are discussed in depth in [1], pp. 87-89 and [2], pp. 55-57. Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> D = DihedralGroup(4) >>> D.strong_gens [(0 1 2 3), (0 3)(1 2), (1 3)] >>> D.base [0, 1] See Also ======== base, basic_transversals, basic_orbits, basic_stabilizers """ if self._strong_gens == []: self.schreier_sims() return self._strong_gens def subgroup(self, gens): """ Return the subgroup generated by `gens` which is a list of elements of the group """ if not all(g in self for g in gens): raise ValueError("The group does not contain the supplied generators") G = PermutationGroup(gens) return G def subgroup_search(self, prop, base=None, strong_gens=None, tests=None, init_subgroup=None): """Find the subgroup of all elements satisfying the property ``prop``. Explanation =========== This is done by a depth-first search with respect to base images that uses several tests to prune the search tree. Parameters ========== prop The property to be used. Has to be callable on group elements and always return ``True`` or ``False``. It is assumed that all group elements satisfying ``prop`` indeed form a subgroup. base A base for the supergroup. strong_gens A strong generating set for the supergroup. tests A list of callables of length equal to the length of ``base``. These are used to rule out group elements by partial base images, so that ``tests[l](g)`` returns False if the element ``g`` is known not to satisfy prop base on where g sends the first ``l + 1`` base points. init_subgroup if a subgroup of the sought group is known in advance, it can be passed to the function as this parameter. Returns ======= res The subgroup of all elements satisfying ``prop``. The generating set for this group is guaranteed to be a strong generating set relative to the base ``base``. Examples ======== >>> from sympy.combinatorics.named_groups import (SymmetricGroup, ... AlternatingGroup) >>> from sympy.combinatorics.testutil import _verify_bsgs >>> S = SymmetricGroup(7) >>> prop_even = lambda x: x.is_even >>> base, strong_gens = S.schreier_sims_incremental() >>> G = S.subgroup_search(prop_even, base=base, strong_gens=strong_gens) >>> G.is_subgroup(AlternatingGroup(7)) True >>> _verify_bsgs(G, base, G.generators) True Notes ===== This function is extremely lengthy and complicated and will require some careful attention. The implementation is described in [1], pp. 114-117, and the comments for the code here follow the lines of the pseudocode in the book for clarity. The complexity is exponential in general, since the search process by itself visits all members of the supergroup. However, there are a lot of tests which are used to prune the search tree, and users can define their own tests via the ``tests`` parameter, so in practice, and for some computations, it's not terrible. A crucial part in the procedure is the frequent base change performed (this is line 11 in the pseudocode) in order to obtain a new basic stabilizer. The book mentiones that this can be done by using ``.baseswap(...)``, however the current implementation uses a more straightforward way to find the next basic stabilizer - calling the function ``.stabilizer(...)`` on the previous basic stabilizer. """ # initialize BSGS and basic group properties def get_reps(orbits): # get the minimal element in the base ordering return [min(orbit, key = lambda x: base_ordering[x]) \ for orbit in orbits] def update_nu(l): temp_index = len(basic_orbits[l]) + 1 -\ len(res_basic_orbits_init_base[l]) # this corresponds to the element larger than all points if temp_index >= len(sorted_orbits[l]): nu[l] = base_ordering[degree] else: nu[l] = sorted_orbits[l][temp_index] if base is None: base, strong_gens = self.schreier_sims_incremental() base_len = len(base) degree = self.degree identity = _af_new(list(range(degree))) base_ordering = _base_ordering(base, degree) # add an element larger than all points base_ordering.append(degree) # add an element smaller than all points base_ordering.append(-1) # compute BSGS-related structures strong_gens_distr = _distribute_gens_by_base(base, strong_gens) basic_orbits, transversals = _orbits_transversals_from_bsgs(base, strong_gens_distr) # handle subgroup initialization and tests if init_subgroup is None: init_subgroup = PermutationGroup([identity]) if tests is None: trivial_test = lambda x: True tests = [] for i in range(base_len): tests.append(trivial_test) # line 1: more initializations. res = init_subgroup f = base_len - 1 l = base_len - 1 # line 2: set the base for K to the base for G res_base = base[:] # line 3: compute BSGS and related structures for K res_base, res_strong_gens = res.schreier_sims_incremental( base=res_base) res_strong_gens_distr = _distribute_gens_by_base(res_base, res_strong_gens) res_generators = res.generators res_basic_orbits_init_base = \ [_orbit(degree, res_strong_gens_distr[i], res_base[i])\ for i in range(base_len)] # initialize orbit representatives orbit_reps = [None]base_len # line 4: orbit representatives for f-th basic stabilizer of K orbits = _orbits(degree, res_strong_gens_distr[f]) orbit_reps[f] = get_reps(orbits) # line 5: remove the base point from the representatives to avoid # getting the identity element as a generator for K orbit_reps[f].remove(base[f]) # line 6: more initializations c = [0]base_len u = [identity]base_len sorted_orbits = [None]base_len for i in range(base_len): sorted_orbits[i] = basic_orbits[i][:] sorted_orbits[i].sort(key=lambda point: base_ordering[point]) # line 7: initializations mu = [None]base_len nu = [None]base_len # this corresponds to the element smaller than all points mu[l] = degree + 1 update_nu(l) # initialize computed words computed_words = [identity]base_len # line 8: main loop while True: # apply all the tests while l < base_len - 1 and \ computed_words[l](base[l]) in orbit_reps[l] and \ base_ordering[mu[l]] < \ base_ordering[computed_words[l](base[l])] < \ base_ordering[nu[l]] and \ tests[l](computed_words): # line 11: change the (partial) base of K new_point = computed_words[l](base[l]) res_base[l] = new_point new_stab_gens = _stabilizer(degree, res_strong_gens_distr[l], new_point) res_strong_gens_distr[l + 1] = new_stab_gens # line 12: calculate minimal orbit representatives for the # l+1-th basic stabilizer orbits = _orbits(degree, new_stab_gens) orbit_reps[l + 1] = get_reps(orbits) # line 13: amend sorted orbits l += 1 temp_orbit = [computed_words[l - 1](point) for point in basic_orbits[l]] temp_orbit.sort(key=lambda point: base_ordering[point]) sorted_orbits[l] = temp_orbit # lines 14 and 15: update variables used minimality tests new_mu = degree + 1 for i in range(l): if base[l] in res_basic_orbits_init_base[i]: candidate = computed_words[i](base[i]) if base_ordering[candidate] > base_ordering[new_mu]: new_mu = candidate mu[l] = new_mu update_nu(l) # line 16: determine the new transversal element c[l] = 0 temp_point = sorted_orbits[l][c[l]] gamma = computed_words[l - 1]._array_form.index(temp_point) u[l] = transversals[l][gamma] # update computed words computed_words[l] = rmul(computed_words[l - 1], u[l]) # lines 17 & 18: apply the tests to the group element found g = computed_words[l] temp_point = g(base[l]) if l == base_len - 1 and \ base_ordering[mu[l]] < \ base_ordering[temp_point] < base_ordering[nu[l]] and \ temp_point in orbit_reps[l] and \ tests[l](computed_words) and \ prop(g): # line 19: reset the base of K res_generators.append(g) res_base = base[:] # line 20: recalculate basic orbits (and transversals) res_strong_gens.append(g) res_strong_gens_distr = _distribute_gens_by_base(res_base, res_strong_gens) res_basic_orbits_init_base = \ [_orbit(degree, res_strong_gens_distr[i], res_base[i]) \ for i in range(base_len)] # line 21: recalculate orbit representatives # line 22: reset the search depth orbit_reps[f] = get_reps(orbits) l = f # line 23: go up the tree until in the first branch not fully # searched while l >= 0 and c[l] == len(basic_orbits[l]) - 1: l = l - 1 # line 24: if the entire tree is traversed, return K if l == -1: return PermutationGroup(res_generators) # lines 25-27: update orbit representatives if l < f: # line 26 f = l c[l] = 0 # line 27 temp_orbits = _orbits(degree, res_strong_gens_distr[f]) orbit_reps[f] = get_reps(temp_orbits) # line 28: update variables used for minimality testing mu[l] = degree + 1 temp_index = len(basic_orbits[l]) + 1 - \ len(res_basic_orbits_init_base[l]) if temp_index >= len(sorted_orbits[l]): nu[l] = base_ordering[degree] else: nu[l] = sorted_orbits[l][temp_index] # line 29: set the next element from the current branch and update # accordingly c[l] += 1 if l == 0: gamma = sorted_orbits[l][c[l]] else: gamma = computed_words[l - 1]._array_form.index(sorted_orbits[l][c[l]]) u[l] = transversals[l][gamma] if l == 0: computed_words[l] = u[l] else: computed_words[l] = rmul(computed_words[l - 1], u[l]) @property def transitivity_degree(self): r"""Compute the degree of transitivity of the group. Explanation =========== A permutation group `G` acting on `\Omega = \{0, 1, \dots, n-1\}` is ``k``-fold transitive, if, for any `k` points `(a_1, a_2, \dots, a_k) \in \Omega` and any `k` points `(b_1, b_2, \dots, b_k) \in \Omega` there exists `g \in G` such that `g(a_1) = b_1, g(a_2) = b_2, \dots, g(a_k) = b_k` The degree of transitivity of `G` is the maximum ``k`` such that `G` is ``k``-fold transitive. ([8]) Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> a = Permutation([1, 2, 0]) >>> b = Permutation([1, 0, 2]) >>> G = PermutationGroup([a, b]) >>> G.transitivity_degree 3 See Also ======== is_transitive, orbit """ if self._transitivity_degree is None: n = self.degree G = self # if G is k-transitive, a tuple (a_0,..,a_k) # can be brought to (b_0,...,b_(k-1), b_k) # where b_0,...,b_(k-1) are fixed points; # consider the group G_k which stabilizes b_0,...,b_(k-1) # if G_k is transitive on the subset excluding b_0,...,b_(k-1) # then G is (k+1)-transitive for i in range(n): orb = G.orbit(i) if len(orb) != n - i: self._transitivity_degree = i return i G = G.stabilizer(i) self._transitivity_degree = n return n else: return self._transitivity_degree def _p_elements_group(self, p): ''' For an abelian p-group, return the subgroup consisting of all elements of order p (and the identity) ''' gens = self.generators[:] gens = sorted(gens, key=lambda x: x.order(), reverse=True) gens_p = [g(g.order()/p) for g in gens] gens_r = [] for i in range(len(gens)): x = gens[i] x_order = x.order() # x_p has order p x_p = x(x_order/p) if i > 0: P = PermutationGroup(gens_p[:i]) else: P = PermutationGroup(self.identity) if x(x_order/p) not in P: gens_r.append(x(x_order/p)) else: # replace x by an element of order (x.order()/p) # so that gens still generates G g = P.generator_product(x_p, original=True) for s in g: x = xs-1 x_order = x_order/p # insert x to gens so that the sorting is preserved del gens[i] del gens_p[i] j = i - 1 while j < len(gens) and gens[j].order() >= x_order: j += 1 gens = gens[:j] + [x] + gens[j:] gens_p = gens_p[:j] + [x] + gens_p[j:] return PermutationGroup(gens_r) def _sylow_alt_sym(self, p): ''' Return a p-Sylow subgroup of a symmetric or an alternating group. Explanation =========== The algorithm for this is hinted at in [1], Chapter 4, Exercise 4. For Sym(n) with n = p^i, the idea is as follows. Partition the interval [0..n-1] into p equal parts, each of length p^(i-1): [0..p^(i-1)-1], [p^(i-1)..2p^(i-1)-1]...[(p-1)p^(i-1)..p^i-1]. Find a p-Sylow subgroup of Sym(p^(i-1)) (treated as a subgroup of ``self``) acting on each of the parts. Call the subgroups P_1, P_2...P_p. The generators for the subgroups P_2...P_p can be obtained from those of P_1 by applying a "shifting" permutation to them, that is, a permutation mapping [0..p^(i-1)-1] to the second part (the other parts are obtained by using the shift multiple times). The union of this permutation and the generators of P_1 is a p-Sylow subgroup of ``self``. For n not equal to a power of p, partition [0..n-1] in accordance with how n would be written in base p. E.g. for p=2 and n=11, 11 = 2^3 + 2^2 + 1 so the partition is [[0..7], [8..9], {10}]. To generate a p-Sylow subgroup, take the union of the generators for each of the parts. For the above example, {(0 1), (0 2)(1 3), (0 4), (1 5)(2 7)} from the first part, {(8 9)} from the second part and nothing from the third. This gives 4 generators in total, and the subgroup they generate is p-Sylow. Alternating groups are treated the same except when p=2. In this case, (0 1)(s s+1) should be added for an appropriate s (the start of a part) for each part in the partitions. See Also ======== sylow_subgroup, is_alt_sym ''' n = self.degree gens = [] identity = Permutation(n-1) # the case of 2-sylow subgroups of alternating groups # needs special treatment alt = p == 2 and all(g.is_even for g in self.generators) # find the presentation of n in base p coeffs = [] m = n while m > 0: coeffs.append(m % p) m = m // p power = len(coeffs)-1 # for a symmetric group, gens[:i] is the generating # set for a p-Sylow subgroup on [0..p(i-1)-1]. For # alternating groups, the same is given by gens[:2(i-1)] for i in range(1, power+1): if i == 1 and alt: # (0 1) shouldn't be added for alternating groups continue gen = Permutation([(j + p(i-1)) % pi for j in range(p*i)]) gens.append(identitygen) if alt: gen = Permutation(0, 1)genPermutation(0, 1)gen gens.append(gen) # the first point in the current part (see the algorithm # description in the docstring) start = 0 while power > 0: a = coeffs[power] # make the permutation shifting the start of the first # part ([0..p^i-1] for some i) to the current one for _ in range(a): shift = Permutation() if start > 0: for i in range(ppower): shift = shift(i, start + i) if alt: gen = Permutation(0, 1)shiftPermutation(0, 1)shift gens.append(gen) j = 2(power - 1) else: j = power for i, gen in enumerate(gens[:j]): if alt and i % 2 == 1: continue # shift the generator to the start of the # partition part gen = shiftgenshift gens.append(gen) start += ppower power = power-1 return gens def sylow_subgroup(self, p): ''' Return a p-Sylow subgroup of the group. The algorithm is described in [1], Chapter 4, Section 7 Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.named_groups import AlternatingGroup >>> D = DihedralGroup(6) >>> S = D.sylow_subgroup(2) >>> S.order() 4 >>> G = SymmetricGroup(6) >>> S = G.sylow_subgroup(5) >>> S.order() 5 >>> G1 = AlternatingGroup(3) >>> G2 = AlternatingGroup(5) >>> G3 = AlternatingGroup(9) >>> S1 = G1.sylow_subgroup(3) >>> S2 = G2.sylow_subgroup(3) >>> S3 = G3.sylow_subgroup(3) >>> len1 = len(S1.lower_central_series()) >>> len2 = len(S2.lower_central_series()) >>> len3 = len(S3.lower_central_series()) >>> len1 == len2 True >>> len1 < len3 True ''' from sympy.combinatorics.homomorphisms import ( orbit_homomorphism, block_homomorphism) if not isprime(p): raise ValueError("p must be a prime") def is_p_group(G): # check if the order of G is a power of p # and return the power m = G.order() n = 0 while m % p == 0: m = m/p n += 1 if m == 1: return True, n return False, n def _sylow_reduce(mu, nu): # reduction based on two homomorphisms # mu and nu with trivially intersecting # kernels Q = mu.image().sylow_subgroup(p) Q = mu.invert_subgroup(Q) nu = nu.restrict_to(Q) R = nu.image().sylow_subgroup(p) return nu.invert_subgroup(R) order = self.order() if order % p != 0: return PermutationGroup([self.identity]) p_group, n = is_p_group(self) if p_group: return self if self.is_alt_sym(): return PermutationGroup(self._sylow_alt_sym(p)) # if there is a non-trivial orbit with size not divisible # by p, the sylow subgroup is contained in its stabilizer # (by orbit-stabilizer theorem) orbits = self.orbits() non_p_orbits = [o for o in orbits if len(o) % p != 0 and len(o) != 1] if non_p_orbits: G = self.stabilizer(list(non_p_orbits[0]).pop()) return G.sylow_subgroup(p) if not self.is_transitive(): # apply _sylow_reduce to orbit actions orbits = sorted(orbits, key=len) omega1 = orbits.pop() omega2 = orbits[0].union(orbits) mu = orbit_homomorphism(self, omega1) nu = orbit_homomorphism(self, omega2) return _sylow_reduce(mu, nu) blocks = self.minimal_blocks() if len(blocks) > 1: # apply _sylow_reduce to block system actions mu = block_homomorphism(self, blocks[0]) nu = block_homomorphism(self, blocks[1]) return _sylow_reduce(mu, nu) elif len(blocks) == 1: block = list(blocks)[0] if any(e != 0 for e in block): # self is imprimitive mu = block_homomorphism(self, block) if not is_p_group(mu.image())[0]: S = mu.image().sylow_subgroup(p) return mu.invert_subgroup(S).sylow_subgroup(p) # find an element of order p g = self.random() g_order = g.order() while g_order % p != 0 or g_order == 0: g = self.random() g_order = g.order() g = g(g_order // p) if order % p2 != 0: return PermutationGroup(g) C = self.centralizer(g) while C.order() % p*n != 0: S = C.sylow_subgroup(p) s_order = S.order() Z = S.center() P = Z._p_elements_group(p) h = P.random() C_h = self.centralizer(h) while C_h.order() % ps_order != 0: h = P.random() C_h = self.centralizer(h) C = C_h return C.sylow_subgroup(p) def _block_verify(self, L, alpha): delta = sorted(list(self.orbit(alpha))) # p[i] will be the number of the block # delta[i] belongs to p = [-1]len(delta) blocks = [-1]len(delta) B = [[]] # future list of blocks u = [0]len(delta) # u[i] in L s.t. alpha^u[i] = B[0][i] t = L.orbit_transversal(alpha, pairs=True) for a, beta in t: B[0].append(a) i_a = delta.index(a) p[i_a] = 0 blocks[i_a] = alpha u[i_a] = beta rho = 0 m = 0 # number of blocks - 1 while rho <= m: beta = B[rho][0] for g in self.generators: d = beta^g i_d = delta.index(d) sigma = p[i_d] if sigma < 0: # define a new block m += 1 sigma = m u[i_d] = u[delta.index(beta)]g p[i_d] = sigma rep = d blocks[i_d] = rep newb = [rep] for gamma in B[rho][1:]: i_gamma = delta.index(gamma) d = gamma^g i_d = delta.index(d) if p[i_d] < 0: u[i_d] = u[i_gamma]g p[i_d] = sigma blocks[i_d] = rep newb.append(d) else: # B[rho] is not a block s = u[i_gamma]gu[i_d](-1) return False, s B.append(newb) else: for h in B[rho][1:]: if h^g not in B[sigma]: # B[rho] is not a block s = u[delta.index(beta)]gu[i_d](-1) return False, s rho += 1 return True, blocks def _verify(H, K, phi, z, alpha): ''' Return a list of relators ``rels`` in generators ``gens`_h` that are mapped to ``H.generators`` by ``phi`` so that given a finite presentation <gens_k \| rels_k> of ``K`` on a subset of ``gens_h`` <gens_h \| rels_k + rels> is a finite presentation of ``H``. Explanation =========== ``H`` should be generated by the union of ``K.generators`` and ``z`` (a single generator), and ``H.stabilizer(alpha) == K``; ``phi`` is a canonical injection from a free group into a permutation group containing ``H``. The algorithm is described in [1], Chapter 6. Examples ======== >>> from sympy.combinatorics import free_group, Permutation, PermutationGroup >>> from sympy.combinatorics.homomorphisms import homomorphism >>> from sympy.combinatorics.fp_groups import FpGroup >>> H = PermutationGroup(Permutation(0, 2), Permutation (1, 5)) >>> K = PermutationGroup(Permutation(5)(0, 2)) >>> F = free_group("x_0 x_1")[0] >>> gens = F.generators >>> phi = homomorphism(F, H, F.generators, H.generators) >>> rels_k = [gens[0]2] # relators for presentation of K >>> z= Permutation(1, 5) >>> check, rels_h = H._verify(K, phi, z, 1) >>> check True >>> rels = rels_k + rels_h >>> G = FpGroup(F, rels) # presentation of H >>> G.order() == H.order() True See also ======== strong_presentation, presentation, stabilizer ''' orbit = H.orbit(alpha) beta = alpha^(z-1) K_beta = K.stabilizer(beta) # orbit representatives of K_beta gammas = [alpha, beta] orbits = list({tuple(K_beta.orbit(o)) for o in orbit}) orbit_reps = [orb[0] for orb in orbits] for rep in orbit_reps: if rep not in gammas: gammas.append(rep) # orbit transversal of K betas = [alpha, beta] transversal = {alpha: phi.invert(H.identity), beta: phi.invert(z-1)} for s, g in K.orbit_transversal(beta, pairs=True): if s not in transversal: transversal[s] = transversal[beta]phi.invert(g) union = K.orbit(alpha).union(K.orbit(beta)) while (len(union) < len(orbit)): for gamma in gammas: if gamma in union: r = gamma^z if r not in union: betas.append(r) transversal[r] = transversal[gamma]phi.invert(z) for s, g in K.orbit_transversal(r, pairs=True): if s not in transversal: transversal[s] = transversal[r]phi.invert(g) union = union.union(K.orbit(r)) break # compute relators rels = [] for b in betas: k_gens = K.stabilizer(b).generators for y in k_gens: new_rel = transversal[b] gens = K.generator_product(y, original=True) for g in gens[::-1]: new_rel = new_relphi.invert(g) new_rel = new_reltransversal[b]*-1 perm = phi(new_rel) try: gens = K.generator_product(perm, original=True) except ValueError: return False, perm for g in gens: new_rel = new_relphi.invert(g)*-1 if new_rel not in rels: rels.append(new_rel) for gamma in gammas: new_rel = transversal[gamma]phi.invert(z)transversal[gamma^z]-1 perm = phi(new_rel) try: gens = K.generator_product(perm, original=True) except ValueError: return False, perm for g in gens: new_rel = new_relphi.invert(g)-1 if new_rel not in rels: rels.append(new_rel) return True, rels def strong_presentation(self): ''' Return a strong finite presentation of group. The generators of the returned group are in the same order as the strong generators of group. The algorithm is based on Sims' Verify algorithm described in [1], Chapter 6. Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> P = DihedralGroup(4) >>> G = P.strong_presentation() >>> P.order() == G.order() True See Also ======== presentation, _verify ''' from sympy.combinatorics.fp_groups import (FpGroup, simplify_presentation) from sympy.combinatorics.free_groups import free_group from sympy.combinatorics.homomorphisms import (block_homomorphism, homomorphism, GroupHomomorphism) strong_gens = self.strong_gens[:] stabs = self.basic_stabilizers[:] base = self.base[:] # injection from a free group on len(strong_gens) # generators into G gen_syms = [('x_%d'%i) for i in range(len(strong_gens))] F = free_group(', '.join(gen_syms))[0] phi = homomorphism(F, self, F.generators, strong_gens) H = PermutationGroup(self.identity) while stabs: alpha = base.pop() K = H H = stabs.pop() new_gens = [g for g in H.generators if g not in K] if K.order() == 1: z = new_gens.pop() rels = [F.generators[-1]z.order()] intermediate_gens = [z] K = PermutationGroup(intermediate_gens) # add generators one at a time building up from K to H while new_gens: z = new_gens.pop() intermediate_gens = [z] + intermediate_gens K_s = PermutationGroup(intermediate_gens) orbit = K_s.orbit(alpha) orbit_k = K.orbit(alpha) # split into cases based on the orbit of K_s if orbit_k == orbit: if z in K: rel = phi.invert(z) perm = z else: t = K.orbit_rep(alpha, alpha^z) rel = phi.invert(z)phi.invert(t)-1 perm = zt*-1 for g in K.generator_product(perm, original=True): rel = relphi.invert(g)*-1 new_rels = [rel] elif len(orbit_k) == 1: # `success` is always true because `strong_gens` # and `base` are already a verified BSGS. Later # this could be changed to start with a randomly # generated (potential) BSGS, and then new elements # would have to be appended to it when `success` # is false. success, new_rels = K_s._verify(K, phi, z, alpha) else: # K.orbit(alpha) should be a block # under the action of K_s on K_s.orbit(alpha) check, block = K_s._block_verify(K, alpha) if check: # apply _verify to the action of K_s # on the block system; for convenience, # add the blocks as additional points # that K_s should act on t = block_homomorphism(K_s, block) m = t.codomain.degree # number of blocks d = K_s.degree # conjugating with p will shift # permutations in t.image() to # higher numbers, e.g. # p(0 1)p = (m m+1) p = Permutation() for i in range(m): p = Permutation(i, i+d) t_img = t.images # combine generators of K_s with their # action on the block system images = {g: gpt_img[g]p for g in t_img} for g in self.strong_gens[:-len(K_s.generators)]: images[g] = g K_s_act = PermutationGroup(list(images.values())) f = GroupHomomorphism(self, K_s_act, images) K_act = PermutationGroup([f(g) for g in K.generators]) success, new_rels = K_s_act._verify(K_act, f.compose(phi), f(z), d) for n in new_rels: if n not in rels: rels.append(n) K = K_s group = FpGroup(F, rels) return simplify_presentation(group) def presentation(self, eliminate_gens=True): ''' Return an `FpGroup` presentation of the group. The algorithm is described in [1], Chapter 6.1. ''' from sympy.combinatorics.fp_groups import (FpGroup, simplify_presentation) from sympy.combinatorics.coset_table import CosetTable from sympy.combinatorics.free_groups import free_group from sympy.combinatorics.homomorphisms import homomorphism if self._fp_presentation: return self._fp_presentation def _factor_group_by_rels(G, rels): if isinstance(G, FpGroup): rels.extend(G.relators) return FpGroup(G.free_group, list(set(rels))) return FpGroup(G, rels) gens = self.generators len_g = len(gens) if len_g == 1: order = gens[0].order() # handle the trivial group if order == 1: return free_group([])[0] F, x = free_group('x') return FpGroup(F, [xorder]) if self.order() > 20: half_gens = self.generators[0:(len_g+1)//2] else: half_gens = [] H = PermutationGroup(half_gens) H_p = H.presentation() len_h = len(H_p.generators) C = self.coset_table(H) n = len(C) # subgroup index gen_syms = [('x_%d'%i) for i in range(len(gens))] F = free_group(', '.join(gen_syms))[0] # mapping generators of H_p to those of F images = [F.generators[i] for i in range(len_h)] R = homomorphism(H_p, F, H_p.generators, images, check=False) # rewrite relators rels = R(H_p.relators) G_p = FpGroup(F, rels) # injective homomorphism from G_p into self T = homomorphism(G_p, self, G_p.generators, gens) C_p = CosetTable(G_p, []) C_p.table = [[None](2len_g) for i in range(n)] # initiate the coset transversal transversal = [None]n transversal[0] = G_p.identity # fill in the coset table as much as possible for i in range(2len_h): C_p.table[0][i] = 0 gamma = 1 for alpha, x in product(range(0, n), range(2len_g)): beta = C[alpha][x] if beta == gamma: gen = G_p.generators[x//2]((-1)(x % 2)) transversal[beta] = transversal[alpha]gen C_p.table[alpha][x] = beta C_p.table[beta][x + (-1)(x % 2)] = alpha gamma += 1 if gamma == n: break C_p.p = list(range(n)) beta = x = 0 while not C_p.is_complete(): # find the first undefined entry while C_p.table[beta][x] == C[beta][x]: x = (x + 1) % (2len_g) if x == 0: beta = (beta + 1) % n # define a new relator gen = G_p.generators[x//2]((-1)(x % 2)) new_rel = transversal[beta]gentransversal[C[beta][x]]*-1 perm = T(new_rel) nxt = G_p.identity for s in H.generator_product(perm, original=True): nxt = nxtT.invert(s)*-1 new_rel = new_relnxt # continue coset enumeration G_p = _factor_group_by_rels(G_p, [new_rel]) C_p.scan_and_fill(0, new_rel) C_p = G_p.coset_enumeration([], strategy="coset_table", draft=C_p, max_cosets=n, incomplete=True) self._fp_presentation = simplify_presentation(G_p) return self._fp_presentation def polycyclic_group(self): """ Return the PolycyclicGroup instance with below parameters: Explanation =========== * ``pc_sequence`` : Polycyclic sequence is formed by collecting all the missing generators between the adjacent groups in the derived series of given permutation group. * ``pc_series`` : Polycyclic series is formed by adding all the missing generators of ``der[i+1]`` in ``der[i]``, where ``der`` represents the derived series. * ``relative_order`` : A list, computed by the ratio of adjacent groups in pc_series. """ from sympy.combinatorics.pc_groups import PolycyclicGroup if not self.is_polycyclic: raise ValueError("The group must be solvable") der = self.derived_series() pc_series = [] pc_sequence = [] relative_order = [] pc_series.append(der[-1]) der.reverse() for i in range(len(der)-1): H = der[i] for g in der[i+1].generators: if g not in H: H = PermutationGroup([g] + H.generators) pc_series.insert(0, H) pc_sequence.insert(0, g) G1 = pc_series[0].order() G2 = pc_series[1].order() relative_order.insert(0, G1 // G2) return PolycyclicGroup(pc_sequence, pc_series, relative_order, collector=None) def _orbit(degree, generators, alpha, action='tuples'): r"""Compute the orbit of alpha `\{g(\alpha) \| g \in G\}` as a set. Explanation =========== The time complexity of the algorithm used here is `O(\|Orb\|r)` where `\|Orb\|` is the size of the orbit and ``r`` is the number of generators of the group. For a more detailed analysis, see [1], p.78, [2], pp. 19-21. Here alpha can be a single point, or a list of points. If alpha is a single point, the ordinary orbit is computed. if alpha is a list of points, there are three available options: 'union' - computes the union of the orbits of the points in the list 'tuples' - computes the orbit of the list interpreted as an ordered tuple under the group action ( i.e., g((1, 2, 3)) = (g(1), g(2), g(3)) ) 'sets' - computes the orbit of the list interpreted as a sets Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup >>> from sympy.combinatorics.perm_groups import _orbit >>> a = Permutation([1, 2, 0, 4, 5, 6, 3]) >>> G = PermutationGroup([a]) >>> _orbit(G.degree, G.generators, 0) {0, 1, 2} >>> _orbit(G.degree, G.generators, [0, 4], 'union') {0, 1, 2, 3, 4, 5, 6} See Also ======== orbit, orbit_transversal """ if not hasattr(alpha, '__getitem__'): alpha = [alpha] gens = [x._array_form for x in generators] if len(alpha) == 1 or action == 'union': orb = alpha used = [False]degree for el in alpha: used[el] = True for b in orb: for gen in gens: temp = gen[b] if used[temp] == False: orb.append(temp) used[temp] = True return set(orb) elif action == 'tuples': alpha = tuple(alpha) orb = [alpha] used = {alpha} for b in orb: for gen in gens: temp = tuple([gen[x] for x in b]) if temp not in used: orb.append(temp) used.add(temp) return set(orb) elif action == 'sets': alpha = frozenset(alpha) orb = [alpha] used = {alpha} for b in orb: for gen in gens: temp = frozenset([gen[x] for x in b]) if temp not in used: orb.append(temp) used.add(temp) return {tuple(x) for x in orb} def _orbits(degree, generators): """Compute the orbits of G. If ``rep=False`` it returns a list of sets else it returns a list of representatives of the orbits Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy.combinatorics.perm_groups import _orbits >>> a = Permutation([0, 2, 1]) >>> b = Permutation([1, 0, 2]) >>> _orbits(a.size, [a, b]) [{0, 1, 2}] """ orbs = [] sorted_I = list(range(degree)) I = set(sorted_I) while I: i = sorted_I[0] orb = _orbit(degree, generators, i) orbs.append(orb) # remove all indices that are in this orbit I -= orb sorted_I = [i for i in sorted_I if i not in orb] return orbs def _orbit_transversal(degree, generators, alpha, pairs, af=False, slp=False): r"""Computes a transversal for the orbit of ``alpha`` as a set. Explanation =========== generators generators of the group ``G`` For a permutation group ``G``, a transversal for the orbit `Orb = \{g(\alpha) \| g \in G\}` is a set `\{g_\beta \| g_\beta(\alpha) = \beta\}` for `\beta \in Orb`. Note that there may be more than one possible transversal. If ``pairs`` is set to ``True``, it returns the list of pairs `(\beta, g_\beta)`. For a proof of correctness, see [1], p.79 if ``af`` is ``True``, the transversal elements are given in array form. If `slp` is `True`, a dictionary `{beta: slp_beta}` is returned for `\beta \in Orb` where `slp_beta` is a list of indices of the generators in `generators` s.t. if `slp_beta = [i_1 \dots i_n]` `g_\beta = generators[i_n] \times \dots \times generators[i_1]`. Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> from sympy.combinatorics.perm_groups import _orbit_transversal >>> G = DihedralGroup(6) >>> _orbit_transversal(G.degree, G.generators, 0, False) [(5), (0 1 2 3 4 5), (0 5)(1 4)(2 3), (0 2 4)(1 3 5), (5)(0 4)(1 3), (0 3)(1 4)(2 5)] """ tr = [(alpha, list(range(degree)))] slp_dict = {alpha: []} used = [False]degree used[alpha] = True gens = [x._array_form for x in generators] for x, px in tr: px_slp = slp_dict[x] for gen in gens: temp = gen[x] if used[temp] == False: slp_dict[temp] = [gens.index(gen)] + px_slp tr.append((temp, _af_rmul(gen, px))) used[temp] = True if pairs: if not af: tr = [(x, _af_new(y)) for x, y in tr] if not slp: return tr return tr, slp_dict if af: tr = [y for _, y in tr] if not slp: return tr return tr, slp_dict tr = [_af_new(y) for _, y in tr] if not slp: return tr return tr, slp_dict def _stabilizer(degree, generators, alpha): r"""Return the stabilizer subgroup of ``alpha``. Explanation =========== The stabilizer of `\alpha` is the group `G_\alpha = \{g \in G \| g(\alpha) = \alpha\}`. For a proof of correctness, see [1], p.79. degree : degree of G generators : generators of G Examples ======== >>> from sympy.combinatorics.perm_groups import _stabilizer >>> from sympy.combinatorics.named_groups import DihedralGroup >>> G = DihedralGroup(6) >>> _stabilizer(G.degree, G.generators, 5) [(5)(0 4)(1 3), (5)] See Also ======== orbit """ orb = [alpha] table = {alpha: list(range(degree))} table_inv = {alpha: list(range(degree))} used = [False]degree used[alpha] = True gens = [x._array_form for x in generators] stab_gens = [] for b in orb: for gen in gens: temp = gen[b] if used[temp] is False: gen_temp = _af_rmul(gen, table[b]) orb.append(temp) table[temp] = gen_temp table_inv[temp] = _af_invert(gen_temp) used[temp] = True else: schreier_gen = _af_rmuln(table_inv[temp], gen, table[b]) if schreier_gen not in stab_gens: stab_gens.append(schreier_gen) return [_af_new(x) for x in stab_gens] PermGroup = PermutationGroup class SymmetricPermutationGroup(Basic): """ The class defining the lazy form of SymmetricGroup. deg : int """ def __new__(cls, deg): deg = _sympify(deg) obj = Basic.__new__(cls, deg) return obj def __init__(self, args, kwargs): self._deg = self.args[0] self._order = None def __contains__(self, i): """Return ``True`` if i* is contained in SymmetricPermutationGroup. Examples ======== >>> from sympy.combinatorics import Permutation, SymmetricPermutationGroup >>> G = SymmetricPermutationGroup(4) >>> Permutation(1, 2, 3) in G True """ if not isinstance(i, Permutation): raise TypeError("A SymmetricPermutationGroup contains only Permutations as " "elements, not elements of type %s" % type(i)) return i.size == self.degree def order(self): """ Return the order of the SymmetricPermutationGroup. Examples ======== >>> from sympy.combinatorics import SymmetricPermutationGroup >>> G = SymmetricPermutationGroup(4) >>> G.order() 24 """ if self._order is not None: return self._order n = self._deg self._order = factorial(n) return self._order @property def degree(self): """ Return the degree of the SymmetricPermutationGroup. Examples ======== >>> from sympy.combinatorics import SymmetricPermutationGroup >>> G = SymmetricPermutationGroup(4) >>> G.degree 4 """ return self._deg @property def identity(self): ''' Return the identity element of the SymmetricPermutationGroup. Examples ======== >>> from sympy.combinatorics import SymmetricPermutationGroup >>> G = SymmetricPermutationGroup(4) >>> G.identity() (3) ''' return _af_new(list(range(self._deg))) class Coset(Basic): """A left coset of a permutation group with respect to an element. Parameters ========== g : Permutation H : PermutationGroup dir : "+" or "-", If not specified by default it will be "+" here ``dir`` specified the type of coset "+" represent the right coset and "-" represent the left coset. G : PermutationGroup, optional The group which contains H as its subgroup and g as its element. If not specified, it would automatically become a symmetric group ``SymmetricPermutationGroup(g.size)`` and ``SymmetricPermutationGroup(H.degree)`` if ``g.size`` and ``H.degree`` are matching.``SymmetricPermutationGroup`` is a lazy form of SymmetricGroup used for representation purpose. """ def __new__(cls, g, H, G=None, dir="+"): g = _sympify(g) if not isinstance(g, Permutation): raise NotImplementedError H = _sympify(H) if not isinstance(H, PermutationGroup): raise NotImplementedError if G is not None: G = _sympify(G) if not isinstance(G, (PermutationGroup, SymmetricPermutationGroup)): raise NotImplementedError if not H.is_subgroup(G): raise ValueError("{} must be a subgroup of {}.".format(H, G)) if g not in G: raise ValueError("{} must be an element of {}.".format(g, G)) else: g_size = g.size h_degree = H.degree if g_size != h_degree: raise ValueError( "The size of the permutation {} and the degree of " "the permutation group {} should be matching " .format(g, H)) G = SymmetricPermutationGroup(g.size) if isinstance(dir, str): dir = Symbol(dir) elif not isinstance(dir, Symbol): raise TypeError("dir must be of type basestring or " "Symbol, not %s" % type(dir)) if str(dir) not in ('+', '-'): raise ValueError("dir must be one of '+' or '-' not %s" % dir) obj = Basic.__new__(cls, g, H, G, dir) return obj def __init__(self, args, kwargs): self._dir = self.args[3] @property def is_left_coset(self): """ Check if the coset is left coset that is ``gH``. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup, Coset >>> a = Permutation(1, 2) >>> b = Permutation(0, 1) >>> G = PermutationGroup([a, b]) >>> cst = Coset(a, G, dir="-") >>> cst.is_left_coset True """ return str(self._dir) == '-' @property def is_right_coset(self): """ Check if the coset is right coset that is ``Hg``. Examples ======== >>> from sympy.combinatorics import Permutation, PermutationGroup, Coset >>> a = Permutation(1, 2) >>> b = Permutation(0, 1) >>> G = PermutationGroup([a, b]) >>> cst = Coset(a, G, dir="+") >>> cst.is_right_coset True """ return str(self._dir) == '+' def as_list(self): """ Return all the elements of coset in the form of list. """ g = self.args[0] H = self.args[1] cst = [] if str(self._dir) == '+': for h in H.elements: cst.append(hg) else: for h in H.elements: cst.append(g*h) return cst
7db9bfe4e49cc383211afde7ac40eec593cb638dd9f994750ed2a47bf65c18a7	from sympy.core import Basic, Integer import random class GrayCode(Basic): """ A Gray code is essentially a Hamiltonian walk on a n-dimensional cube with edge length of one. The vertices of the cube are represented by vectors whose values are binary. The Hamilton walk visits each vertex exactly once. The Gray code for a 3d cube is ['000','100','110','010','011','111','101', '001']. A Gray code solves the problem of sequentially generating all possible subsets of n objects in such a way that each subset is obtained from the previous one by either deleting or adding a single object. In the above example, 1 indicates that the object is present, and 0 indicates that its absent. Gray codes have applications in statistics as well when we want to compute various statistics related to subsets in an efficient manner. Examples ======== >>> from sympy.combinatorics import GrayCode >>> a = GrayCode(3) >>> list(a.generate_gray()) ['000', '001', '011', '010', '110', '111', '101', '100'] >>> a = GrayCode(4) >>> list(a.generate_gray()) ['0000', '0001', '0011', '0010', '0110', '0111', '0101', '0100', \ '1100', '1101', '1111', '1110', '1010', '1011', '1001', '1000'] References ========== .. [1] Nijenhuis,A. and Wilf,H.S.(1978). Combinatorial Algorithms. Academic Press. .. [2] Knuth, D. (2011). The Art of Computer Programming, Vol 4 Addison Wesley """ _skip = False _current = 0 _rank = None def __new__(cls, n, args, kw_args): """ Default constructor. It takes a single argument ``n`` which gives the dimension of the Gray code. The starting Gray code string (``start``) or the starting ``rank`` may also be given; the default is to start at rank = 0 ('0...0'). Examples ======== >>> from sympy.combinatorics import GrayCode >>> a = GrayCode(3) >>> a GrayCode(3) >>> a.n 3 >>> a = GrayCode(3, start='100') >>> a.current '100' >>> a = GrayCode(4, rank=4) >>> a.current '0110' >>> a.rank 4 """ if n < 1 or int(n) != n: raise ValueError( 'Gray code dimension must be a positive integer, not %i' % n) n = Integer(n) args = (n,) + args obj = Basic.__new__(cls, args) if 'start' in kw_args: obj._current = kw_args["start"] if len(obj._current) > n: raise ValueError('Gray code start has length %i but ' 'should not be greater than %i' % (len(obj._current), n)) elif 'rank' in kw_args: if int(kw_args["rank"]) != kw_args["rank"]: raise ValueError('Gray code rank must be a positive integer, ' 'not %i' % kw_args["rank"]) obj._rank = int(kw_args["rank"]) % obj.selections obj._current = obj.unrank(n, obj._rank) return obj def next(self, delta=1): """ Returns the Gray code a distance ``delta`` (default = 1) from the current value in canonical order. Examples ======== >>> from sympy.combinatorics import GrayCode >>> a = GrayCode(3, start='110') >>> a.next().current '111' >>> a.next(-1).current '010' """ return GrayCode(self.n, rank=(self.rank + delta) % self.selections) @property def selections(self): """ Returns the number of bit vectors in the Gray code. Examples ======== >>> from sympy.combinatorics import GrayCode >>> a = GrayCode(3) >>> a.selections 8 """ return 2self.n @property def n(self): """ Returns the dimension of the Gray code. Examples ======== >>> from sympy.combinatorics import GrayCode >>> a = GrayCode(5) >>> a.n 5 """ return self.args[0] def generate_gray(self, hints): """ Generates the sequence of bit vectors of a Gray Code. Examples ======== >>> from sympy.combinatorics import GrayCode >>> a = GrayCode(3) >>> list(a.generate_gray()) ['000', '001', '011', '010', '110', '111', '101', '100'] >>> list(a.generate_gray(start='011')) ['011', '010', '110', '111', '101', '100'] >>> list(a.generate_gray(rank=4)) ['110', '111', '101', '100'] See Also ======== skip References ========== .. [1] Knuth, D. (2011). The Art of Computer Programming, Vol 4, Addison Wesley """ bits = self.n start = None if "start" in hints: start = hints["start"] elif "rank" in hints: start = GrayCode.unrank(self.n, hints["rank"]) if start is not None: self._current = start current = self.current graycode_bin = gray_to_bin(current) if len(graycode_bin) > self.n: raise ValueError('Gray code start has length %i but should ' 'not be greater than %i' % (len(graycode_bin), bits)) self._current = int(current, 2) graycode_int = int(''.join(graycode_bin), 2) for i in range(graycode_int, 1 << bits): if self._skip: self._skip = False else: yield self.current bbtc = (i ^ (i + 1)) gbtc = (bbtc ^ (bbtc >> 1)) self._current = (self._current ^ gbtc) self._current = 0 def skip(self): """ Skips the bit generation. Examples ======== >>> from sympy.combinatorics import GrayCode >>> a = GrayCode(3) >>> for i in a.generate_gray(): ... if i == '010': ... a.skip() ... print(i) ... 000 001 011 010 111 101 100 See Also ======== generate_gray """ self._skip = True @property def rank(self): """ Ranks the Gray code. A ranking algorithm determines the position (or rank) of a combinatorial object among all the objects w.r.t. a given order. For example, the 4 bit binary reflected Gray code (BRGC) '0101' has a rank of 6 as it appears in the 6th position in the canonical ordering of the family of 4 bit Gray codes. Examples ======== >>> from sympy.combinatorics import GrayCode >>> a = GrayCode(3) >>> list(a.generate_gray()) ['000', '001', '011', '010', '110', '111', '101', '100'] >>> GrayCode(3, start='100').rank 7 >>> GrayCode(3, rank=7).current '100' See Also ======== unrank References ========== .. [1] http://statweb.stanford.edu/~susan/courses/s208/node12.html """ if self._rank is None: self._rank = int(gray_to_bin(self.current), 2) return self._rank @property def current(self): """ Returns the currently referenced Gray code as a bit string. Examples ======== >>> from sympy.combinatorics import GrayCode >>> GrayCode(3, start='100').current '100' """ rv = self._current or '0' if not isinstance(rv, str): rv = bin(rv)[2:] return rv.rjust(self.n, '0') @classmethod def unrank(self, n, rank): """ Unranks an n-bit sized Gray code of rank k. This method exists so that a derivative GrayCode class can define its own code of a given rank. The string here is generated in reverse order to allow for tail-call optimization. Examples ======== >>> from sympy.combinatorics import GrayCode >>> GrayCode(5, rank=3).current '00010' >>> GrayCode.unrank(5, 3) '00010' See Also ======== rank """ def _unrank(k, n): if n == 1: return str(k % 2) m = 2**(n - 1) if k < m: return '0' + _unrank(k, n - 1) return '1' + _unrank(m - (k % m) - 1, n - 1) return _unrank(rank, n) def random_bitstring(n): """ Generates a random bitlist of length n. Examples ======== >>> from sympy.combinatorics.graycode import random_bitstring >>> random_bitstring(3) # doctest: +SKIP 100 """ return ''.join([random.choice('01') for i in range(n)]) def gray_to_bin(bin_list): """ Convert from Gray coding to binary coding. We assume big endian encoding. Examples ======== >>> from sympy.combinatorics.graycode import gray_to_bin >>> gray_to_bin('100') '111' See Also ======== bin_to_gray """ b = [bin_list[0]] for i in range(1, len(bin_list)): b += str(int(b[i - 1] != bin_list[i])) return ''.join(b) def bin_to_gray(bin_list): """ Convert from binary coding to gray coding. We assume big endian encoding. Examples ======== >>> from sympy.combinatorics.graycode import bin_to_gray >>> bin_to_gray('111') '100' See Also ======== gray_to_bin """ b = [bin_list[0]] for i in range(1, len(bin_list)): b += str(int(bin_list[i]) ^ int(bin_list[i - 1])) return ''.join(b) def get_subset_from_bitstring(super_set, bitstring): """ Gets the subset defined by the bitstring. Examples ======== >>> from sympy.combinatorics.graycode import get_subset_from_bitstring >>> get_subset_from_bitstring(['a', 'b', 'c', 'd'], '0011') ['c', 'd'] >>> get_subset_from_bitstring(['c', 'a', 'c', 'c'], '1100') ['c', 'a'] See Also ======== graycode_subsets """ if len(super_set) != len(bitstring): raise ValueError("The sizes of the lists are not equal") return [super_set[i] for i, j in enumerate(bitstring) if bitstring[i] == '1'] def graycode_subsets(gray_code_set): """ Generates the subsets as enumerated by a Gray code. Examples ======== >>> from sympy.combinatorics.graycode import graycode_subsets >>> list(graycode_subsets(['a', 'b', 'c'])) [[], ['c'], ['b', 'c'], ['b'], ['a', 'b'], ['a', 'b', 'c'], \ ['a', 'c'], ['a']] >>> list(graycode_subsets(['a', 'b', 'c', 'c'])) [[], ['c'], ['c', 'c'], ['c'], ['b', 'c'], ['b', 'c', 'c'], \ ['b', 'c'], ['b'], ['a', 'b'], ['a', 'b', 'c'], ['a', 'b', 'c', 'c'], \ ['a', 'b', 'c'], ['a', 'c'], ['a', 'c', 'c'], ['a', 'c'], ['a']] See Also ======== get_subset_from_bitstring """ for bitstring in list(GrayCode(len(gray_code_set)).generate_gray()): yield get_subset_from_bitstring(gray_code_set, bitstring)
3e8f0b1dd20d0e8d19989441bc655ec9ff07147a86c4d7b645b8993cce202848	from collections import deque from sympy.combinatorics.rewritingsystem_fsm import StateMachine class RewritingSystem: ''' A class implementing rewriting systems for `FpGroup`s. References ========== .. [1] Epstein, D., Holt, D. and Rees, S. (1991). The use of Knuth-Bendix methods to solve the word problem in automatic groups. Journal of Symbolic Computation, 12(4-5), pp.397-414. .. [2] GAP's Manual on its KBMAG package https://www.gap-system.org/Manuals/pkg/kbmag-1.5.3/doc/manual.pdf ''' def __init__(self, group): self.group = group self.alphabet = group.generators self._is_confluent = None # these values are taken from [2] self.maxeqns = 32767 # max rules self.tidyint = 100 # rules before tidying # _max_exceeded is True if maxeqns is exceeded # at any point self._max_exceeded = False # Reduction automaton self.reduction_automaton = None self._new_rules = {} # dictionary of reductions self.rules = {} self.rules_cache = deque([], 50) self._init_rules() # All the transition symbols in the automaton generators = list(self.alphabet) generators += [gen*-1 for gen in generators] # Create a finite state machine as an instance of the StateMachine object self.reduction_automaton = StateMachine('Reduction automaton for '+ repr(self.group), generators) self.construct_automaton() def set_max(self, n): ''' Set the maximum number of rules that can be defined ''' if n > self.maxeqns: self._max_exceeded = False self.maxeqns = n return @property def is_confluent(self): ''' Return `True` if the system is confluent ''' if self._is_confluent is None: self._is_confluent = self._check_confluence() return self._is_confluent def _init_rules(self): identity = self.group.free_group.identity for r in self.group.relators: self.add_rule(r, identity) self._remove_redundancies() return def _add_rule(self, r1, r2): ''' Add the rule r1 -> r2 with no checking or further deductions ''' if len(self.rules) + 1 > self.maxeqns: self._is_confluent = self._check_confluence() self._max_exceeded = True raise RuntimeError("Too many rules were defined.") self.rules[r1] = r2 # Add the newly added rule to the `new_rules` dictionary. if self.reduction_automaton: self._new_rules[r1] = r2 def add_rule(self, w1, w2, check=False): new_keys = set() if w1 == w2: return new_keys if w1 < w2: w1, w2 = w2, w1 if (w1, w2) in self.rules_cache: return new_keys self.rules_cache.append((w1, w2)) s1, s2 = w1, w2 # The following is the equivalent of checking # s1 for overlaps with the implicit reductions # {gg-1 -> <identity>} and {g-1g -> <identity>} # for any generator g without installing the # redundant rules that would result from processing # the overlaps. See [1], Section 3 for details. if len(s1) - len(s2) < 3: if s1 not in self.rules: new_keys.add(s1) if not check: self._add_rule(s1, s2) if s2-1 > s1-1 and s2-1 not in self.rules: new_keys.add(s2-1) if not check: self._add_rule(s2-1, s1-1) # overlaps on the right while len(s1) - len(s2) > -1: g = s1[len(s1)-1] s1 = s1.subword(0, len(s1)-1) s2 = s2g-1 if len(s1) - len(s2) < 0: if s2 not in self.rules: if not check: self._add_rule(s2, s1) new_keys.add(s2) elif len(s1) - len(s2) < 3: new = self.add_rule(s1, s2, check) new_keys.update(new) # overlaps on the left while len(w1) - len(w2) > -1: g = w1[0] w1 = w1.subword(1, len(w1)) w2 = g-1w2 if len(w1) - len(w2) < 0: if w2 not in self.rules: if not check: self._add_rule(w2, w1) new_keys.add(w2) elif len(w1) - len(w2) < 3: new = self.add_rule(w1, w2, check) new_keys.update(new) return new_keys def _remove_redundancies(self, changes=False): ''' Reduce left- and right-hand sides of reduction rules and remove redundant equations (i.e. those for which lhs == rhs). If `changes` is `True`, return a set containing the removed keys and a set containing the added keys ''' removed = set() added = set() rules = self.rules.copy() for r in rules: v = self.reduce(r, exclude=r) w = self.reduce(rules[r]) if v != r: del self.rules[r] removed.add(r) if v > w: added.add(v) self.rules[v] = w elif v < w: added.add(w) self.rules[w] = v else: self.rules[v] = w if changes: return removed, added return def make_confluent(self, check=False): ''' Try to make the system confluent using the Knuth-Bendix completion algorithm ''' if self._max_exceeded: return self._is_confluent lhs = list(self.rules.keys()) def _overlaps(r1, r2): len1 = len(r1) len2 = len(r2) result = [] for j in range(1, len1 + len2): if (r1.subword(len1 - j, len1 + len2 - j, strict=False) == r2.subword(j - len1, j, strict=False)): a = r1.subword(0, len1-j, strict=False) a = ar2.subword(0, j-len1, strict=False) b = r2.subword(j-len1, j, strict=False) c = r2.subword(j, len2, strict=False) c = cr1.subword(len1 + len2 - j, len1, strict=False) result.append(abc) return result def _process_overlap(w, r1, r2, check): s = w.eliminate_word(r1, self.rules[r1]) s = self.reduce(s) t = w.eliminate_word(r2, self.rules[r2]) t = self.reduce(t) if s != t: if check: # system not confluent return [0] try: new_keys = self.add_rule(t, s, check) return new_keys except RuntimeError: return False return added = 0 i = 0 while i < len(lhs): r1 = lhs[i] i += 1 # j could be i+1 to not # check each pair twice but lhs # is extended in the loop and the new # elements have to be checked with the # preceding ones. there is probably a better way # to handle this j = 0 while j < len(lhs): r2 = lhs[j] j += 1 if r1 == r2: continue overlaps = _overlaps(r1, r2) overlaps.extend(_overlaps(r1-1, r2)) if not overlaps: continue for w in overlaps: new_keys = _process_overlap(w, r1, r2, check) if new_keys: if check: return False lhs.extend(new_keys) added += len(new_keys) elif new_keys == False: # too many rules were added so the process # couldn't complete return self._is_confluent if added > self.tidyint and not check: # tidy up r, a = self._remove_redundancies(changes=True) added = 0 if r: # reset i since some elements were removed i = min([lhs.index(s) for s in r]) lhs = [l for l in lhs if l not in r] lhs.extend(a) if r1 in r: # r1 was removed as redundant break self._is_confluent = True if not check: self._remove_redundancies() return True def _check_confluence(self): return self.make_confluent(check=True) def reduce(self, word, exclude=None): ''' Apply reduction rules to `word` excluding the reduction rule for the lhs equal to `exclude` ''' rules = {r: self.rules[r] for r in self.rules if r != exclude} # the following is essentially `eliminate_words()` code from the # `FreeGroupElement` class, the only difference being the first # "if" statement again = True new = word while again: again = False for r in rules: prev = new if rules[r]-1 > r-1: new = new.eliminate_word(r, rules[r], _all=True, inverse=False) else: new = new.eliminate_word(r, rules[r], _all=True) if new != prev: again = True return new def _compute_inverse_rules(self, rules): ''' Compute the inverse rules for a given set of rules. The inverse rules are used in the automaton for word reduction. Arguments: rules (dictionary): Rules for which the inverse rules are to computed. Returns: Dictionary of inverse_rules. ''' inverse_rules = {} for r in rules: rule_key_inverse = r-1 rule_value_inverse = (rules[r])-1 if (rule_value_inverse < rule_key_inverse): inverse_rules[rule_key_inverse] = rule_value_inverse else: inverse_rules[rule_value_inverse] = rule_key_inverse return inverse_rules def construct_automaton(self): ''' Construct the automaton based on the set of reduction rules of the system. Automata Design: The accept states of the automaton are the proper prefixes of the left hand side of the rules. The complete left hand side of the rules are the dead states of the automaton. ''' self._add_to_automaton(self.rules) def _add_to_automaton(self, rules): ''' Add new states and transitions to the automaton. Summary: States corresponding to the new rules added to the system are computed and added to the automaton. Transitions in the previously added states are also modified if necessary. Arguments: rules (dictionary) -- Dictionary of the newly added rules. ''' # Automaton variables automaton_alphabet = [] proper_prefixes = {} # compute the inverses of all the new rules added all_rules = rules inverse_rules = self._compute_inverse_rules(all_rules) all_rules.update(inverse_rules) # Keep track of the accept_states. accept_states = [] for rule in all_rules: # The symbols present in the new rules are the symbols to be verified at each state. # computes the automaton_alphabet, as the transitions solely depend upon the new states. automaton_alphabet += rule.letter_form_elm # Compute the proper prefixes for every rule. proper_prefixes[rule] = [] letter_word_array = [s for s in rule.letter_form_elm] len_letter_word_array = len(letter_word_array) for i in range (1, len_letter_word_array): letter_word_array[i] = letter_word_array[i-1]letter_word_array[i] # Add accept states. elem = letter_word_array[i-1] if elem not in self.reduction_automaton.states: self.reduction_automaton.add_state(elem, state_type='a') accept_states.append(elem) proper_prefixes[rule] = letter_word_array # Check for overlaps between dead and accept states. if rule in accept_states: self.reduction_automaton.states[rule].state_type = 'd' self.reduction_automaton.states[rule].rh_rule = all_rules[rule] accept_states.remove(rule) # Add dead states if rule not in self.reduction_automaton.states: self.reduction_automaton.add_state(rule, state_type='d', rh_rule=all_rules[rule]) automaton_alphabet = set(automaton_alphabet) # Add new transitions for every state. for state in self.reduction_automaton.states: current_state_name = state current_state_type = self.reduction_automaton.states[state].state_type # Transitions will be modified only when suffixes of the current_state # belongs to the proper_prefixes of the new rules. # The rest are ignored if they cannot lead to a dead state after a finite number of transisitons. if current_state_type == 's': for letter in automaton_alphabet: if letter in self.reduction_automaton.states: self.reduction_automaton.states[state].add_transition(letter, letter) else: self.reduction_automaton.states[state].add_transition(letter, current_state_name) elif current_state_type == 'a': # Check if the transition to any new state in possible. for letter in automaton_alphabet: _next = current_state_nameletter while len(_next) and _next not in self.reduction_automaton.states: _next = _next.subword(1, len(_next)) if not len(_next): _next = 'start' self.reduction_automaton.states[state].add_transition(letter, _next) # Add transitions for new states. All symbols used in the automaton are considered here. # Ignore this if `reduction_automaton.automaton_alphabet` = `automaton_alphabet`. if len(self.reduction_automaton.automaton_alphabet) != len(automaton_alphabet): for state in accept_states: current_state_name = state for letter in self.reduction_automaton.automaton_alphabet: _next = current_state_nameletter while len(_next) and _next not in self.reduction_automaton.states: _next = _next.subword(1, len(_next)) if not len(_next): _next = 'start' self.reduction_automaton.states[state].add_transition(letter, _next) def reduce_using_automaton(self, word): ''' Reduce a word using an automaton. Summary: All the symbols of the word are stored in an array and are given as the input to the automaton. If the automaton reaches a dead state that subword is replaced and the automaton is run from the beginning. The complete word has to be replaced when the word is read and the automaton reaches a dead state. So, this process is repeated until the word is read completely and the automaton reaches the accept state. Arguments: word (instance of FreeGroupElement) -- Word that needs to be reduced. ''' # Modify the automaton if new rules are found. if self._new_rules: self._add_to_automaton(self._new_rules) self._new_rules = {} flag = 1 while flag: flag = 0 current_state = self.reduction_automaton.states['start'] word_array = [s for s in word.letter_form_elm] for i in range (0, len(word_array)): next_state_name = current_state.transitions[word_array[i]] next_state = self.reduction_automaton.states[next_state_name] if next_state.state_type == 'd': subst = next_state.rh_rule word = word.substituted_word(i - len(next_state_name) + 1, i+1, subst) flag = 1 break current_state = next_state return word
3dfad67593dbf7c44df1e02d6bffe1a2b19a328fa7b2b4a5652fd685f1aa031c	import random from collections import defaultdict from collections.abc import Iterable from functools import reduce from sympy.core.parameters import global_parameters from sympy.core.basic import Atom from sympy.core.expr import Expr from sympy.core.numbers import Integer from sympy.core.sympify import _sympify from sympy.matrices import zeros from sympy.polys.polytools import lcm from sympy.printing.repr import srepr from sympy.utilities.iterables import (flatten, has_variety, minlex, has_dups, runs, is_sequence) from sympy.utilities.misc import as_int from mpmath.libmp.libintmath import ifac from sympy.multipledispatch import dispatch def _af_rmul(a, b): """ Return the product ba; input and output are array forms. The ith value is a[b[i]]. Examples ======== >>> from sympy.combinatorics.permutations import _af_rmul, Permutation >>> a, b = [1, 0, 2], [0, 2, 1] >>> _af_rmul(a, b) [1, 2, 0] >>> [a[b[i]] for i in range(3)] [1, 2, 0] This handles the operands in reverse order compared to the ```` operator: >>> a = Permutation(a) >>> b = Permutation(b) >>> list(ab) [2, 0, 1] >>> [b(a(i)) for i in range(3)] [2, 0, 1] See Also ======== rmul, _af_rmuln """ return [a[i] for i in b] def _af_rmuln(abc): """ Given [a, b, c, ...] return the product of ...cba using array forms. The ith value is a[b[c[i]]]. Examples ======== >>> from sympy.combinatorics.permutations import _af_rmul, Permutation >>> a, b = [1, 0, 2], [0, 2, 1] >>> _af_rmul(a, b) [1, 2, 0] >>> [a[b[i]] for i in range(3)] [1, 2, 0] This handles the operands in reverse order compared to the ```` operator: >>> a = Permutation(a); b = Permutation(b) >>> list(ab) [2, 0, 1] >>> [b(a(i)) for i in range(3)] [2, 0, 1] See Also ======== rmul, _af_rmul """ a = abc m = len(a) if m == 3: p0, p1, p2 = a return [p0[p1[i]] for i in p2] if m == 4: p0, p1, p2, p3 = a return [p0[p1[p2[i]]] for i in p3] if m == 5: p0, p1, p2, p3, p4 = a return [p0[p1[p2[p3[i]]]] for i in p4] if m == 6: p0, p1, p2, p3, p4, p5 = a return [p0[p1[p2[p3[p4[i]]]]] for i in p5] if m == 7: p0, p1, p2, p3, p4, p5, p6 = a return [p0[p1[p2[p3[p4[p5[i]]]]]] for i in p6] if m == 8: p0, p1, p2, p3, p4, p5, p6, p7 = a return [p0[p1[p2[p3[p4[p5[p6[i]]]]]]] for i in p7] if m == 1: return a[0][:] if m == 2: a, b = a return [a[i] for i in b] if m == 0: raise ValueError("String must not be empty") p0 = _af_rmuln(a[:m//2]) p1 = _af_rmuln(a[m//2:]) return [p0[i] for i in p1] def _af_parity(pi): """ Computes the parity of a permutation in array form. Explanation =========== The parity of a permutation reflects the parity of the number of inversions in the permutation, i.e., the number of pairs of x and y such that x > y but p[x] < p[y]. Examples ======== >>> from sympy.combinatorics.permutations import _af_parity >>> _af_parity([0, 1, 2, 3]) 0 >>> _af_parity([3, 2, 0, 1]) 1 See Also ======== Permutation """ n = len(pi) a = [0] n c = 0 for j in range(n): if a[j] == 0: c += 1 a[j] = 1 i = j while pi[i] != j: i = pi[i] a[i] = 1 return (n - c) % 2 def _af_invert(a): """ Finds the inverse, ~A, of a permutation, A, given in array form. Examples ======== >>> from sympy.combinatorics.permutations import _af_invert, _af_rmul >>> A = [1, 2, 0, 3] >>> _af_invert(A) [2, 0, 1, 3] >>> _af_rmul(_, A) [0, 1, 2, 3] See Also ======== Permutation, __invert__ """ inv_form = [0] * len(a) for i, ai in enumerate(a): inv_form[ai] = i return inv_form def _af_pow(a, n): """ Routine for finding powers of a permutation. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy.combinatorics.permutations import _af_pow >>> p = Permutation([2, 0, 3, 1]) >>> p.order() 4 >>> _af_pow(p._array_form, 4) [0, 1, 2, 3] """ if n == 0: return list(range(len(a))) if n < 0: return _af_pow(_af_invert(a), -n) if n == 1: return a[:] elif n == 2: b = [a[i] for i in a] elif n == 3: b = [a[a[i]] for i in a] elif n == 4: b = [a[a[a[i]]] for i in a] else: # use binary multiplication b = list(range(len(a))) while 1: if n & 1: b = [b[i] for i in a] n -= 1 if not n: break if n % 4 == 0: a = [a[a[a[i]]] for i in a] n = n // 4 elif n % 2 == 0: a = [a[i] for i in a] n = n // 2 return b def _af_commutes_with(a, b): """ Checks if the two permutations with array forms given by ``a`` and ``b`` commute. Examples ======== >>> from sympy.combinatorics.permutations import _af_commutes_with >>> _af_commutes_with([1, 2, 0], [0, 2, 1]) False See Also ======== Permutation, commutes_with """ return not any(a[b[i]] != b[a[i]] for i in range(len(a) - 1)) class Cycle(dict): """ Wrapper around dict which provides the functionality of a disjoint cycle. Explanation =========== A cycle shows the rule to use to move subsets of elements to obtain a permutation. The Cycle class is more flexible than Permutation in that 1) all elements need not be present in order to investigate how multiple cycles act in sequence and 2) it can contain singletons: >>> from sympy.combinatorics.permutations import Perm, Cycle A Cycle will automatically parse a cycle given as a tuple on the rhs: >>> Cycle(1, 2)(2, 3) (1 3 2) The identity cycle, Cycle(), can be used to start a product: >>> Cycle()(1, 2)(2, 3) (1 3 2) The array form of a Cycle can be obtained by calling the list method (or passing it to the list function) and all elements from 0 will be shown: >>> a = Cycle(1, 2) >>> a.list() [0, 2, 1] >>> list(a) [0, 2, 1] If a larger (or smaller) range is desired use the list method and provide the desired size -- but the Cycle cannot be truncated to a size smaller than the largest element that is out of place: >>> b = Cycle(2, 4)(1, 2)(3, 1, 4)(1, 3) >>> b.list() [0, 2, 1, 3, 4] >>> b.list(b.size + 1) [0, 2, 1, 3, 4, 5] >>> b.list(-1) [0, 2, 1] Singletons are not shown when printing with one exception: the largest element is always shown -- as a singleton if necessary: >>> Cycle(1, 4, 10)(4, 5) (1 5 4 10) >>> Cycle(1, 2)(4)(5)(10) (1 2)(10) The array form can be used to instantiate a Permutation so other properties of the permutation can be investigated: >>> Perm(Cycle(1, 2)(3, 4).list()).transpositions() [(1, 2), (3, 4)] Notes ===== The underlying structure of the Cycle is a dictionary and although the __iter__ method has been redefined to give the array form of the cycle, the underlying dictionary items are still available with the such methods as items(): >>> list(Cycle(1, 2).items()) [(1, 2), (2, 1)] See Also ======== Permutation """ def __missing__(self, arg): """Enter arg into dictionary and return arg.""" return as_int(arg) def __iter__(self): yield from self.list() def __call__(self, other): """Return product of cycles processed from R to L. Examples ======== >>> from sympy.combinatorics import Cycle >>> Cycle(1, 2)(2, 3) (1 3 2) An instance of a Cycle will automatically parse list-like objects and Permutations that are on the right. It is more flexible than the Permutation in that all elements need not be present: >>> a = Cycle(1, 2) >>> a(2, 3) (1 3 2) >>> a(2, 3)(4, 5) (1 3 2)(4 5) """ rv = Cycle(other) for k, v in zip(list(self.keys()), [rv[self[k]] for k in self.keys()]): rv[k] = v return rv def list(self, size=None): """Return the cycles as an explicit list starting from 0 up to the greater of the largest value in the cycles and size. Truncation of trailing unmoved items will occur when size is less than the maximum element in the cycle; if this is desired, setting ``size=-1`` will guarantee such trimming. Examples ======== >>> from sympy.combinatorics import Cycle >>> p = Cycle(2, 3)(4, 5) >>> p.list() [0, 1, 3, 2, 5, 4] >>> p.list(10) [0, 1, 3, 2, 5, 4, 6, 7, 8, 9] Passing a length too small will trim trailing, unchanged elements in the permutation: >>> Cycle(2, 4)(1, 2, 4).list(-1) [0, 2, 1] """ if not self and size is None: raise ValueError('must give size for empty Cycle') if size is not None: big = max([i for i in self.keys() if self[i] != i] + [0]) size = max(size, big + 1) else: size = self.size return [self[i] for i in range(size)] def __repr__(self): """We want it to print as a Cycle, not as a dict. Examples ======== >>> from sympy.combinatorics import Cycle >>> Cycle(1, 2) (1 2) >>> print(_) (1 2) >>> list(Cycle(1, 2).items()) [(1, 2), (2, 1)] """ if not self: return 'Cycle()' cycles = Permutation(self).cyclic_form s = ''.join(str(tuple(c)) for c in cycles) big = self.size - 1 if not any(i == big for c in cycles for i in c): s += '(%s)' % big return 'Cycle%s' % s def __str__(self): """We want it to be printed in a Cycle notation with no comma in-between. Examples ======== >>> from sympy.combinatorics import Cycle >>> Cycle(1, 2) (1 2) >>> Cycle(1, 2, 4)(5, 6) (1 2 4)(5 6) """ if not self: return '()' cycles = Permutation(self).cyclic_form s = ''.join(str(tuple(c)) for c in cycles) big = self.size - 1 if not any(i == big for c in cycles for i in c): s += '(%s)' % big s = s.replace(',', '') return s def __init__(self, args): """Load up a Cycle instance with the values for the cycle. Examples ======== >>> from sympy.combinatorics import Cycle >>> Cycle(1, 2, 6) (1 2 6) """ if not args: return if len(args) == 1: if isinstance(args[0], Permutation): for c in args[0].cyclic_form: self.update(self(c)) return elif isinstance(args[0], Cycle): for k, v in args[0].items(): self[k] = v return args = [as_int(a) for a in args] if any(i < 0 for i in args): raise ValueError('negative integers are not allowed in a cycle.') if has_dups(args): raise ValueError('All elements must be unique in a cycle.') for i in range(-len(args), 0): self[args[i]] = args[i + 1] @property def size(self): if not self: return 0 return max(self.keys()) + 1 def copy(self): return Cycle(self) class Permutation(Atom): r""" A permutation, alternatively known as an 'arrangement number' or 'ordering' is an arrangement of the elements of an ordered list into a one-to-one mapping with itself. The permutation of a given arrangement is given by indicating the positions of the elements after re-arrangement [2]_. For example, if one started with elements ``[x, y, a, b]`` (in that order) and they were reordered as ``[x, y, b, a]`` then the permutation would be ``[0, 1, 3, 2]``. Notice that (in SymPy) the first element is always referred to as 0 and the permutation uses the indices of the elements in the original ordering, not the elements ``(a, b, ...)`` themselves. >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) Permutations Notation ===================== Permutations are commonly represented in disjoint cycle or array forms. Array Notation and 2-line Form ------------------------------------ In the 2-line form, the elements and their final positions are shown as a matrix with 2 rows: [0 1 2 ... n-1] [p(0) p(1) p(2) ... p(n-1)] Since the first line is always ``range(n)``, where n is the size of p, it is sufficient to represent the permutation by the second line, referred to as the "array form" of the permutation. This is entered in brackets as the argument to the Permutation class: >>> p = Permutation([0, 2, 1]); p Permutation([0, 2, 1]) Given i in range(p.size), the permutation maps i to i^p >>> [i^p for i in range(p.size)] [0, 2, 1] The composite of two permutations pq means first apply p, then q, so i^(pq) = (i^p)^q which is i^p^q according to Python precedence rules: >>> q = Permutation([2, 1, 0]) >>> [i^p^q for i in range(3)] [2, 0, 1] >>> [i^(pq) for i in range(3)] [2, 0, 1] One can use also the notation p(i) = i^p, but then the composition rule is (pq)(i) = q(p(i)), not p(q(i)): >>> [(pq)(i) for i in range(p.size)] [2, 0, 1] >>> [q(p(i)) for i in range(p.size)] [2, 0, 1] >>> [p(q(i)) for i in range(p.size)] [1, 2, 0] Disjoint Cycle Notation ----------------------- In disjoint cycle notation, only the elements that have shifted are indicated. For example, [1, 3, 2, 0] can be represented as (0, 1, 3)(2). This can be understood from the 2 line format of the given permutation. In the 2-line form, [0 1 2 3] [1 3 2 0] The element in the 0th position is 1, so 0 -> 1. The element in the 1st position is three, so 1 -> 3. And the element in the third position is again 0, so 3 -> 0. Thus, 0 -> 1 -> 3 -> 0, and 2 -> 2. Thus, this can be represented as 2 cycles: (0, 1, 3)(2). In common notation, singular cycles are not explicitly written as they can be inferred implicitly. Only the relative ordering of elements in a cycle matter: >>> Permutation(1,2,3) == Permutation(2,3,1) == Permutation(3,1,2) True The disjoint cycle notation is convenient when representing permutations that have several cycles in them: >>> Permutation(1, 2)(3, 5) == Permutation([[1, 2], [3, 5]]) True It also provides some economy in entry when computing products of permutations that are written in disjoint cycle notation: >>> Permutation(1, 2)(1, 3)(2, 3) Permutation([0, 3, 2, 1]) >>> _ == Permutation([[1, 2]])Permutation([[1, 3]])Permutation([[2, 3]]) True Caution: when the cycles have common elements between them then the order in which the permutations are applied matters. This module applies the permutations from left to right. >>> Permutation(1, 2)(2, 3) == Permutation([(1, 2), (2, 3)]) True >>> Permutation(1, 2)(2, 3).list() [0, 3, 1, 2] In the above case, (1,2) is computed before (2,3). As 0 -> 0, 0 -> 0, element in position 0 is 0. As 1 -> 2, 2 -> 3, element in position 1 is 3. As 2 -> 1, 1 -> 1, element in position 2 is 1. As 3 -> 3, 3 -> 2, element in position 3 is 2. If the first and second elements had been swapped first, followed by the swapping of the second and third, the result would have been [0, 2, 3, 1]. If, you want to apply the cycles in the conventional right to left order, call the function with arguments in reverse order as demonstrated below: >>> Permutation([(1, 2), (2, 3)][::-1]).list() [0, 2, 3, 1] Entering a singleton in a permutation is a way to indicate the size of the permutation. The ``size`` keyword can also be used. Array-form entry: >>> Permutation([[1, 2], [9]]) Permutation([0, 2, 1], size=10) >>> Permutation([[1, 2]], size=10) Permutation([0, 2, 1], size=10) Cyclic-form entry: >>> Permutation(1, 2, size=10) Permutation([0, 2, 1], size=10) >>> Permutation(9)(1, 2) Permutation([0, 2, 1], size=10) Caution: no singleton containing an element larger than the largest in any previous cycle can be entered. This is an important difference in how Permutation and Cycle handle the ``__call__`` syntax. A singleton argument at the start of a Permutation performs instantiation of the Permutation and is permitted: >>> Permutation(5) Permutation([], size=6) A singleton entered after instantiation is a call to the permutation -- a function call -- and if the argument is out of range it will trigger an error. For this reason, it is better to start the cycle with the singleton: The following fails because there is no element 3: >>> Permutation(1, 2)(3) Traceback (most recent call last): ... IndexError: list index out of range This is ok: only the call to an out of range singleton is prohibited; otherwise the permutation autosizes: >>> Permutation(3)(1, 2) Permutation([0, 2, 1, 3]) >>> Permutation(1, 2)(3, 4) == Permutation(3, 4)(1, 2) True Equality testing ---------------- The array forms must be the same in order for permutations to be equal: >>> Permutation([1, 0, 2, 3]) == Permutation([1, 0]) False Identity Permutation -------------------- The identity permutation is a permutation in which no element is out of place. It can be entered in a variety of ways. All the following create an identity permutation of size 4: >>> I = Permutation([0, 1, 2, 3]) >>> all(p == I for p in [ ... Permutation(3), ... Permutation(range(4)), ... Permutation([], size=4), ... Permutation(size=4)]) True Watch out for entering the range inside* a set of brackets (which is cycle notation): >>> I == Permutation([range(4)]) False Permutation Printing ==================== There are a few things to note about how Permutations are printed. .. deprecated:: 1.6 Configuring Permutation printing by setting ``Permutation.print_cyclic`` is deprecated. Users should use the ``perm_cyclic`` flag to the printers, as described below. 1) If you prefer one form (array or cycle) over another, you can set ``init_printing`` with the ``perm_cyclic`` flag. >>> from sympy import init_printing >>> p = Permutation(1, 2)(4, 5)(3, 4) >>> p Permutation([0, 2, 1, 4, 5, 3]) >>> init_printing(perm_cyclic=True, pretty_print=False) >>> p (1 2)(3 4 5) 2) Regardless of the setting, a list of elements in the array for cyclic form can be obtained and either of those can be copied and supplied as the argument to Permutation: >>> p.array_form [0, 2, 1, 4, 5, 3] >>> p.cyclic_form [[1, 2], [3, 4, 5]] >>> Permutation(_) == p True 3) Printing is economical in that as little as possible is printed while retaining all information about the size of the permutation: >>> init_printing(perm_cyclic=False, pretty_print=False) >>> Permutation([1, 0, 2, 3]) Permutation([1, 0, 2, 3]) >>> Permutation([1, 0, 2, 3], size=20) Permutation([1, 0], size=20) >>> Permutation([1, 0, 2, 4, 3, 5, 6], size=20) Permutation([1, 0, 2, 4, 3], size=20) >>> p = Permutation([1, 0, 2, 3]) >>> init_printing(perm_cyclic=True, pretty_print=False) >>> p (3)(0 1) >>> init_printing(perm_cyclic=False, pretty_print=False) The 2 was not printed but it is still there as can be seen with the array_form and size methods: >>> p.array_form [1, 0, 2, 3] >>> p.size 4 Short introduction to other methods =================================== The permutation can act as a bijective function, telling what element is located at a given position >>> q = Permutation([5, 2, 3, 4, 1, 0]) >>> q.array_form[1] # the hard way 2 >>> q(1) # the easy way 2 >>> {i: q(i) for i in range(q.size)} # showing the bijection {0: 5, 1: 2, 2: 3, 3: 4, 4: 1, 5: 0} The full cyclic form (including singletons) can be obtained: >>> p.full_cyclic_form [[0, 1], [2], [3]] Any permutation can be factored into transpositions of pairs of elements: >>> Permutation([[1, 2], [3, 4, 5]]).transpositions() [(1, 2), (3, 5), (3, 4)] >>> Permutation.rmul([Permutation([ti], size=6) for ti in _]).cyclic_form [[1, 2], [3, 4, 5]] The number of permutations on a set of n elements is given by n! and is called the cardinality. >>> p.size 4 >>> p.cardinality 24 A given permutation has a rank among all the possible permutations of the same elements, but what that rank is depends on how the permutations are enumerated. (There are a number of different methods of doing so.) The lexicographic rank is given by the rank method and this rank is used to increment a permutation with addition/subtraction: >>> p.rank() 6 >>> p + 1 Permutation([1, 0, 3, 2]) >>> p.next_lex() Permutation([1, 0, 3, 2]) >>> _.rank() 7 >>> p.unrank_lex(p.size, rank=7) Permutation([1, 0, 3, 2]) The product of two permutations p and q is defined as their composition as functions, (pq)(i) = q(p(i)) [6]_. >>> p = Permutation([1, 0, 2, 3]) >>> q = Permutation([2, 3, 1, 0]) >>> list(qp) [2, 3, 0, 1] >>> list(pq) [3, 2, 1, 0] >>> [q(p(i)) for i in range(p.size)] [3, 2, 1, 0] The permutation can be 'applied' to any list-like object, not only Permutations: >>> p(['zero', 'one', 'four', 'two']) ['one', 'zero', 'four', 'two'] >>> p('zo42') ['o', 'z', '4', '2'] If you have a list of arbitrary elements, the corresponding permutation can be found with the from_sequence method: >>> Permutation.from_sequence('SymPy') Permutation([1, 3, 2, 0, 4]) Checking if a Permutation is contained in a Group ================================================= Generally if you have a group of permutations G on n symbols, and you're checking if a permutation on less than n symbols is part of that group, the check will fail. Here is an example for n=5 and we check if the cycle (1,2,3) is in G: >>> from sympy import init_printing >>> init_printing(perm_cyclic=True, pretty_print=False) >>> from sympy.combinatorics import Cycle, Permutation >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> G = PermutationGroup(Cycle(2, 3)(4, 5), Cycle(1, 2, 3, 4, 5)) >>> p1 = Permutation(Cycle(2, 5, 3)) >>> p2 = Permutation(Cycle(1, 2, 3)) >>> a1 = Permutation(Cycle(1, 2, 3).list(6)) >>> a2 = Permutation(Cycle(1, 2, 3)(5)) >>> a3 = Permutation(Cycle(1, 2, 3),size=6) >>> for p in [p1,p2,a1,a2,a3]: p, G.contains(p) ((2 5 3), True) ((1 2 3), False) ((5)(1 2 3), True) ((5)(1 2 3), True) ((5)(1 2 3), True) The check for p2 above will fail. Checking if p1 is in G works because SymPy knows G is a group on 5 symbols, and p1 is also on 5 symbols (its largest element is 5). For ``a1``, the ``.list(6)`` call will extend the permutation to 5 symbols, so the test will work as well. In the case of ``a2`` the permutation is being extended to 5 symbols by using a singleton, and in the case of ``a3`` it's extended through the constructor argument ``size=6``. There is another way to do this, which is to tell the ``contains`` method that the number of symbols the group is on does not need to match perfectly the number of symbols for the permutation: >>> G.contains(p2,strict=False) True This can be via the ``strict`` argument to the ``contains`` method, and SymPy will try to extend the permutation on its own and then perform the containment check. See Also ======== Cycle References ========== .. [1] Skiena, S. 'Permutations.' 1.1 in Implementing Discrete Mathematics Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 3-16, 1990. .. [2] Knuth, D. E. The Art of Computer Programming, Vol. 4: Combinatorial Algorithms, 1st ed. Reading, MA: Addison-Wesley, 2011. .. [3] Wendy Myrvold and Frank Ruskey. 2001. Ranking and unranking permutations in linear time. Inf. Process. Lett. 79, 6 (September 2001), 281-284. DOI=10.1016/S0020-0190(01)00141-7 .. [4] D. L. Kreher, D. R. Stinson 'Combinatorial Algorithms' CRC Press, 1999 .. [5] Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. .. [6] https://en.wikipedia.org/wiki/Permutation#Product_and_inverse .. [7] https://en.wikipedia.org/wiki/Lehmer_code """ is_Permutation = True _array_form = None _cyclic_form = None _cycle_structure = None _size = None _rank = None def __new__(cls, args, size=None, kwargs): """ Constructor for the Permutation object from a list or a list of lists in which all elements of the permutation may appear only once. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) Permutations entered in array-form are left unaltered: >>> Permutation([0, 2, 1]) Permutation([0, 2, 1]) Permutations entered in cyclic form are converted to array form; singletons need not be entered, but can be entered to indicate the largest element: >>> Permutation([[4, 5, 6], [0, 1]]) Permutation([1, 0, 2, 3, 5, 6, 4]) >>> Permutation([[4, 5, 6], [0, 1], [19]]) Permutation([1, 0, 2, 3, 5, 6, 4], size=20) All manipulation of permutations assumes that the smallest element is 0 (in keeping with 0-based indexing in Python) so if the 0 is missing when entering a permutation in array form, an error will be raised: >>> Permutation([2, 1]) Traceback (most recent call last): ... ValueError: Integers 0 through 2 must be present. If a permutation is entered in cyclic form, it can be entered without singletons and the ``size`` specified so those values can be filled in, otherwise the array form will only extend to the maximum value in the cycles: >>> Permutation([[1, 4], [3, 5, 2]], size=10) Permutation([0, 4, 3, 5, 1, 2], size=10) >>> _.array_form [0, 4, 3, 5, 1, 2, 6, 7, 8, 9] """ if size is not None: size = int(size) #a) () #b) (1) = identity #c) (1, 2) = cycle #d) ([1, 2, 3]) = array form #e) ([[1, 2]]) = cyclic form #f) (Cycle) = conversion to permutation #g) (Permutation) = adjust size or return copy ok = True if not args: # a return cls._af_new(list(range(size or 0))) elif len(args) > 1: # c return cls._af_new(Cycle(args).list(size)) if len(args) == 1: a = args[0] if isinstance(a, cls): # g if size is None or size == a.size: return a return cls(a.array_form, size=size) if isinstance(a, Cycle): # f return cls._af_new(a.list(size)) if not is_sequence(a): # b if size is not None and a + 1 > size: raise ValueError('size is too small when max is %s' % a) return cls._af_new(list(range(a + 1))) if has_variety(is_sequence(ai) for ai in a): ok = False else: ok = False if not ok: raise ValueError("Permutation argument must be a list of ints, " "a list of lists, Permutation or Cycle.") # safe to assume args are valid; this also makes a copy # of the args args = list(args[0]) is_cycle = args and is_sequence(args[0]) if is_cycle: # e args = [[int(i) for i in c] for c in args] else: # d args = [int(i) for i in args] # if there are n elements present, 0, 1, ..., n-1 should be present # unless a cycle notation has been provided. A 0 will be added # for convenience in case one wants to enter permutations where # counting starts from 1. temp = flatten(args) if has_dups(temp) and not is_cycle: raise ValueError('there were repeated elements.') temp = set(temp) if not is_cycle: if temp != set(range(len(temp))): raise ValueError('Integers 0 through %s must be present.' % max(temp)) if size is not None and temp and max(temp) + 1 > size: raise ValueError('max element should not exceed %s' % (size - 1)) if is_cycle: # it's not necessarily canonical so we won't store # it -- use the array form instead c = Cycle() for ci in args: c = c(ci) aform = c.list() else: aform = list(args) if size and size > len(aform): # don't allow for truncation of permutation which # might split a cycle and lead to an invalid aform # but do allow the permutation size to be increased aform.extend(list(range(len(aform), size))) return cls._af_new(aform) @classmethod def _af_new(cls, perm): """A method to produce a Permutation object from a list; the list is bound to the _array_form attribute, so it must not be modified; this method is meant for internal use only; the list ``a`` is supposed to be generated as a temporary value in a method, so p = Perm._af_new(a) is the only object to hold a reference to ``a``:: Examples ======== >>> from sympy.combinatorics.permutations import Perm >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> a = [2, 1, 3, 0] >>> p = Perm._af_new(a) >>> p Permutation([2, 1, 3, 0]) """ p = super().__new__(cls) p._array_form = perm p._size = len(perm) return p def _hashable_content(self): # the array_form (a list) is the Permutation arg, so we need to # return a tuple, instead return tuple(self.array_form) @property def array_form(self): """ Return a copy of the attribute _array_form Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([[2, 0], [3, 1]]) >>> p.array_form [2, 3, 0, 1] >>> Permutation([[2, 0, 3, 1]]).array_form [3, 2, 0, 1] >>> Permutation([2, 0, 3, 1]).array_form [2, 0, 3, 1] >>> Permutation([[1, 2], [4, 5]]).array_form [0, 2, 1, 3, 5, 4] """ return self._array_form[:] def list(self, size=None): """Return the permutation as an explicit list, possibly trimming unmoved elements if size is less than the maximum element in the permutation; if this is desired, setting ``size=-1`` will guarantee such trimming. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation(2, 3)(4, 5) >>> p.list() [0, 1, 3, 2, 5, 4] >>> p.list(10) [0, 1, 3, 2, 5, 4, 6, 7, 8, 9] Passing a length too small will trim trailing, unchanged elements in the permutation: >>> Permutation(2, 4)(1, 2, 4).list(-1) [0, 2, 1] >>> Permutation(3).list(-1) [] """ if not self and size is None: raise ValueError('must give size for empty Cycle') rv = self.array_form if size is not None: if size > self.size: rv.extend(list(range(self.size, size))) else: # find first value from rhs where rv[i] != i i = self.size - 1 while rv: if rv[-1] != i: break rv.pop() i -= 1 return rv @property def cyclic_form(self): """ This is used to convert to the cyclic notation from the canonical notation. Singletons are omitted. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 3, 1, 2]) >>> p.cyclic_form [[1, 3, 2]] >>> Permutation([1, 0, 2, 4, 3, 5]).cyclic_form [[0, 1], [3, 4]] See Also ======== array_form, full_cyclic_form """ if self._cyclic_form is not None: return list(self._cyclic_form) array_form = self.array_form unchecked = [True] len(array_form) cyclic_form = [] for i in range(len(array_form)): if unchecked[i]: cycle = [] cycle.append(i) unchecked[i] = False j = i while unchecked[array_form[j]]: j = array_form[j] cycle.append(j) unchecked[j] = False if len(cycle) > 1: cyclic_form.append(cycle) assert cycle == list(minlex(cycle)) cyclic_form.sort() self._cyclic_form = cyclic_form[:] return cyclic_form @property def full_cyclic_form(self): """Return permutation in cyclic form including singletons. Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation([0, 2, 1]).full_cyclic_form [[0], [1, 2]] """ need = set(range(self.size)) - set(flatten(self.cyclic_form)) rv = self.cyclic_form + [[i] for i in need] rv.sort() return rv @property def size(self): """ Returns the number of elements in the permutation. Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation([[3, 2], [0, 1]]).size 4 See Also ======== cardinality, length, order, rank """ return self._size def support(self): """Return the elements in permutation, P, for which P[i] != i. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([[3, 2], [0, 1], [4]]) >>> p.array_form [1, 0, 3, 2, 4] >>> p.support() [0, 1, 2, 3] """ a = self.array_form return [i for i, e in enumerate(a) if a[i] != i] def __add__(self, other): """Return permutation that is other higher in rank than self. The rank is the lexicographical rank, with the identity permutation having rank of 0. Examples ======== >>> from sympy.combinatorics import Permutation >>> I = Permutation([0, 1, 2, 3]) >>> a = Permutation([2, 1, 3, 0]) >>> I + a.rank() == a True See Also ======== __sub__, inversion_vector """ rank = (self.rank() + other) % self.cardinality rv = self.unrank_lex(self.size, rank) rv._rank = rank return rv def __sub__(self, other): """Return the permutation that is other lower in rank than self. See Also ======== __add__ """ return self.__add__(-other) @staticmethod def rmul(args): """ Return product of Permutations [a, b, c, ...] as the Permutation whose ith value is a(b(c(i))). a, b, c, ... can be Permutation objects or tuples. Examples ======== >>> from sympy.combinatorics import Permutation >>> a, b = [1, 0, 2], [0, 2, 1] >>> a = Permutation(a); b = Permutation(b) >>> list(Permutation.rmul(a, b)) [1, 2, 0] >>> [a(b(i)) for i in range(3)] [1, 2, 0] This handles the operands in reverse order compared to the ```` operator: >>> a = Permutation(a); b = Permutation(b) >>> list(ab) [2, 0, 1] >>> [b(a(i)) for i in range(3)] [2, 0, 1] Notes ===== All items in the sequence will be parsed by Permutation as necessary as long as the first item is a Permutation: >>> Permutation.rmul(a, [0, 2, 1]) == Permutation.rmul(a, b) True The reverse order of arguments will raise a TypeError. """ rv = args[0] for i in range(1, len(args)): rv = args[i]rv return rv @classmethod def rmul_with_af(cls, args): """ same as rmul, but the elements of args are Permutation objects which have _array_form """ a = [x._array_form for x in args] rv = cls._af_new(_af_rmuln(a)) return rv def mul_inv(self, other): """ other~self, self and other have _array_form """ a = _af_invert(self._array_form) b = other._array_form return self._af_new(_af_rmul(a, b)) def __rmul__(self, other): """This is needed to coerce other to Permutation in rmul.""" cls = type(self) return cls(other)self def __mul__(self, other): """ Return the product ab as a Permutation; the ith value is b(a(i)). Examples ======== >>> from sympy.combinatorics.permutations import _af_rmul, Permutation >>> a, b = [1, 0, 2], [0, 2, 1] >>> a = Permutation(a); b = Permutation(b) >>> list(ab) [2, 0, 1] >>> [b(a(i)) for i in range(3)] [2, 0, 1] This handles operands in reverse order compared to _af_rmul and rmul: >>> al = list(a); bl = list(b) >>> _af_rmul(al, bl) [1, 2, 0] >>> [al[bl[i]] for i in range(3)] [1, 2, 0] It is acceptable for the arrays to have different lengths; the shorter one will be padded to match the longer one: >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> bPermutation([1, 0]) Permutation([1, 2, 0]) >>> Permutation([1, 0])b Permutation([2, 0, 1]) It is also acceptable to allow coercion to handle conversion of a single list to the left of a Permutation: >>> [0, 1]a # no change: 2-element identity Permutation([1, 0, 2]) >>> [[0, 1]]a # exchange first two elements Permutation([0, 1, 2]) You cannot use more than 1 cycle notation in a product of cycles since coercion can only handle one argument to the left. To handle multiple cycles it is convenient to use Cycle instead of Permutation: >>> [[1, 2]][[2, 3]]Permutation([]) # doctest: +SKIP >>> from sympy.combinatorics.permutations import Cycle >>> Cycle(1, 2)(2, 3) (1 3 2) """ from sympy.combinatorics.perm_groups import PermutationGroup, Coset if isinstance(other, PermutationGroup): return Coset(self, other, dir='-') a = self.array_form # __rmul__ makes sure the other is a Permutation b = other.array_form if not b: perm = a else: b.extend(list(range(len(b), len(a)))) perm = [b[i] for i in a] + b[len(a):] return self._af_new(perm) def commutes_with(self, other): """ Checks if the elements are commuting. Examples ======== >>> from sympy.combinatorics import Permutation >>> a = Permutation([1, 4, 3, 0, 2, 5]) >>> b = Permutation([0, 1, 2, 3, 4, 5]) >>> a.commutes_with(b) True >>> b = Permutation([2, 3, 5, 4, 1, 0]) >>> a.commutes_with(b) False """ a = self.array_form b = other.array_form return _af_commutes_with(a, b) def __pow__(self, n): """ Routine for finding powers of a permutation. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> p = Permutation([2, 0, 3, 1]) >>> p.order() 4 >>> p4 Permutation([0, 1, 2, 3]) """ if isinstance(n, Permutation): raise NotImplementedError( 'pp is not defined; do you mean p^p (conjugate)?') n = int(n) return self._af_new(_af_pow(self.array_form, n)) def __rxor__(self, i): """Return self(i) when ``i`` is an int. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation(1, 2, 9) >>> 2^p == p(2) == 9 True """ if int(i) == i: return self(i) else: raise NotImplementedError( "i^p = p(i) when i is an integer, not %s." % i) def __xor__(self, h): """Return the conjugate permutation ``~hselfh` `. Explanation =========== If ``a`` and ``b`` are conjugates, ``a = hb~h`` and ``b = ~hah`` and both have the same cycle structure. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation(1, 2, 9) >>> q = Permutation(6, 9, 8) >>> pq != qp True Calculate and check properties of the conjugate: >>> c = p^q >>> c == ~qpq and p == qc~q True The expression q^p^r is equivalent to q^(pr): >>> r = Permutation(9)(4, 6, 8) >>> q^p^r == q^(pr) True If the term to the left of the conjugate operator, i, is an integer then this is interpreted as selecting the ith element from the permutation to the right: >>> all(i^p == p(i) for i in range(p.size)) True Note that the * operator as higher precedence than the ^ operator: >>> q^rp^r == q^(rp)^r == Permutation(9)(1, 6, 4) True Notes ===== In Python the precedence rule is p^q^r = (p^q)^r which differs in general from p^(q^r) >>> q^p^r (9)(1 4 8) >>> q^(p^r) (9)(1 8 6) For a given r and p, both of the following are conjugates of p: ~rpr and rp~r. But these are not necessarily the same: >>> ~rpr == rp~r True >>> p = Permutation(1, 2, 9)(5, 6) >>> ~rpr == rp~r False The conjugate ~rpr was chosen so that ``p^q^r`` would be equivalent to ``p^(qr)`` rather than ``p^(rq)``. To obtain rp~r, pass ~r to this method: >>> p^~r == rp~r True """ if self.size != h.size: raise ValueError("The permutations must be of equal size.") a = [None]self.size h = h._array_form p = self._array_form for i in range(self.size): a[h[i]] = h[p[i]] return self._af_new(a) def transpositions(self): """ Return the permutation decomposed into a list of transpositions. Explanation =========== It is always possible to express a permutation as the product of transpositions, see [1] Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([[1, 2, 3], [0, 4, 5, 6, 7]]) >>> t = p.transpositions() >>> t [(0, 7), (0, 6), (0, 5), (0, 4), (1, 3), (1, 2)] >>> print(''.join(str(c) for c in t)) (0, 7)(0, 6)(0, 5)(0, 4)(1, 3)(1, 2) >>> Permutation.rmul([Permutation([ti], size=p.size) for ti in t]) == p True References ========== .. [1] https://en.wikipedia.org/wiki/Transposition_%28mathematics%29#Properties """ a = self.cyclic_form res = [] for x in a: nx = len(x) if nx == 2: res.append(tuple(x)) elif nx > 2: first = x[0] for y in x[nx - 1:0:-1]: res.append((first, y)) return res @classmethod def from_sequence(self, i, key=None): """Return the permutation needed to obtain ``i`` from the sorted elements of ``i``. If custom sorting is desired, a key can be given. Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.from_sequence('SymPy') (4)(0 1 3) >>> _(sorted("SymPy")) ['S', 'y', 'm', 'P', 'y'] >>> Permutation.from_sequence('SymPy', key=lambda x: x.lower()) (4)(0 2)(1 3) """ ic = list(zip(i, list(range(len(i))))) if key: ic.sort(key=lambda x: key(x[0])) else: ic.sort() return ~Permutation([i[1] for i in ic]) def __invert__(self): """ Return the inverse of the permutation. A permutation multiplied by its inverse is the identity permutation. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> p = Permutation([[2, 0], [3, 1]]) >>> ~p Permutation([2, 3, 0, 1]) >>> _ == p*-1 True >>> p~p == ~pp == Permutation([0, 1, 2, 3]) True """ return self._af_new(_af_invert(self._array_form)) def __iter__(self): """Yield elements from array form. Examples ======== >>> from sympy.combinatorics import Permutation >>> list(Permutation(range(3))) [0, 1, 2] """ yield from self.array_form def __repr__(self): return srepr(self) def __call__(self, i): """ Allows applying a permutation instance as a bijective function. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([[2, 0], [3, 1]]) >>> p.array_form [2, 3, 0, 1] >>> [p(i) for i in range(4)] [2, 3, 0, 1] If an array is given then the permutation selects the items from the array (i.e. the permutation is applied to the array): >>> from sympy.abc import x >>> p([x, 1, 0, x2]) [0, x2, x, 1] """ # list indices can be Integer or int; leave this # as it is (don't test or convert it) because this # gets called a lot and should be fast if len(i) == 1: i = i[0] if not isinstance(i, Iterable): i = as_int(i) if i < 0 or i > self.size: raise TypeError( "{} should be an integer between 0 and {}" .format(i, self.size-1)) return self._array_form[i] # P([a, b, c]) if len(i) != self.size: raise TypeError( "{} should have the length {}.".format(i, self.size)) return [i[j] for j in self._array_form] # P(1, 2, 3) return selfPermutation(Cycle(i), size=self.size) def atoms(self): """ Returns all the elements of a permutation Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation([0, 1, 2, 3, 4, 5]).atoms() {0, 1, 2, 3, 4, 5} >>> Permutation([[0, 1], [2, 3], [4, 5]]).atoms() {0, 1, 2, 3, 4, 5} """ return set(self.array_form) def apply(self, i): r"""Apply the permutation to an expression. Parameters ========== i : Expr It should be an integer between $0$ and $n-1$ where $n$ is the size of the permutation. If it is a symbol or a symbolic expression that can have integer values, an ``AppliedPermutation`` object will be returned which can represent an unevaluated function. Notes ===== Any permutation can be defined as a bijective function $\sigma : \{ 0, 1, \dots, n-1 \} \rightarrow \{ 0, 1, \dots, n-1 \}$ where $n$ denotes the size of the permutation. The definition may even be extended for any set with distinctive elements, such that the permutation can even be applied for real numbers or such, however, it is not implemented for now for computational reasons and the integrity with the group theory module. This function is similar to the ``__call__`` magic, however, ``__call__`` magic already has some other applications like permuting an array or attatching new cycles, which would not always be mathematically consistent. This also guarantees that the return type is a SymPy integer, which guarantees the safety to use assumptions. """ i = _sympify(i) if i.is_integer is False: raise NotImplementedError("{} should be an integer.".format(i)) n = self.size if (i < 0) == True or (i >= n) == True: raise NotImplementedError( "{} should be an integer between 0 and {}".format(i, n-1)) if i.is_Integer: return Integer(self._array_form[i]) return AppliedPermutation(self, i) def next_lex(self): """ Returns the next permutation in lexicographical order. If self is the last permutation in lexicographical order it returns None. See [4] section 2.4. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([2, 3, 1, 0]) >>> p = Permutation([2, 3, 1, 0]); p.rank() 17 >>> p = p.next_lex(); p.rank() 18 See Also ======== rank, unrank_lex """ perm = self.array_form[:] n = len(perm) i = n - 2 while perm[i + 1] < perm[i]: i -= 1 if i == -1: return None else: j = n - 1 while perm[j] < perm[i]: j -= 1 perm[j], perm[i] = perm[i], perm[j] i += 1 j = n - 1 while i < j: perm[j], perm[i] = perm[i], perm[j] i += 1 j -= 1 return self._af_new(perm) @classmethod def unrank_nonlex(self, n, r): """ This is a linear time unranking algorithm that does not respect lexicographic order [3]. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> Permutation.unrank_nonlex(4, 5) Permutation([2, 0, 3, 1]) >>> Permutation.unrank_nonlex(4, -1) Permutation([0, 1, 2, 3]) See Also ======== next_nonlex, rank_nonlex """ def _unrank1(n, r, a): if n > 0: a[n - 1], a[r % n] = a[r % n], a[n - 1] _unrank1(n - 1, r//n, a) id_perm = list(range(n)) n = int(n) r = r % ifac(n) _unrank1(n, r, id_perm) return self._af_new(id_perm) def rank_nonlex(self, inv_perm=None): """ This is a linear time ranking algorithm that does not enforce lexicographic order [3]. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 1, 2, 3]) >>> p.rank_nonlex() 23 See Also ======== next_nonlex, unrank_nonlex """ def _rank1(n, perm, inv_perm): if n == 1: return 0 s = perm[n - 1] t = inv_perm[n - 1] perm[n - 1], perm[t] = perm[t], s inv_perm[n - 1], inv_perm[s] = inv_perm[s], t return s + n_rank1(n - 1, perm, inv_perm) if inv_perm is None: inv_perm = (~self).array_form if not inv_perm: return 0 perm = self.array_form[:] r = _rank1(len(perm), perm, inv_perm) return r def next_nonlex(self): """ Returns the next permutation in nonlex order [3]. If self is the last permutation in this order it returns None. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> p = Permutation([2, 0, 3, 1]); p.rank_nonlex() 5 >>> p = p.next_nonlex(); p Permutation([3, 0, 1, 2]) >>> p.rank_nonlex() 6 See Also ======== rank_nonlex, unrank_nonlex """ r = self.rank_nonlex() if r == ifac(self.size) - 1: return None return self.unrank_nonlex(self.size, r + 1) def rank(self): """ Returns the lexicographic rank of the permutation. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 1, 2, 3]) >>> p.rank() 0 >>> p = Permutation([3, 2, 1, 0]) >>> p.rank() 23 See Also ======== next_lex, unrank_lex, cardinality, length, order, size """ if self._rank is not None: return self._rank rank = 0 rho = self.array_form[:] n = self.size - 1 size = n + 1 psize = int(ifac(n)) for j in range(size - 1): rank += rho[j]psize for i in range(j + 1, size): if rho[i] > rho[j]: rho[i] -= 1 psize //= n n -= 1 self._rank = rank return rank @property def cardinality(self): """ Returns the number of all possible permutations. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 1, 2, 3]) >>> p.cardinality 24 See Also ======== length, order, rank, size """ return int(ifac(self.size)) def parity(self): """ Computes the parity of a permutation. Explanation =========== The parity of a permutation reflects the parity of the number of inversions in the permutation, i.e., the number of pairs of x and y such that ``x > y`` but ``p[x] < p[y]``. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 1, 2, 3]) >>> p.parity() 0 >>> p = Permutation([3, 2, 0, 1]) >>> p.parity() 1 See Also ======== _af_parity """ if self._cyclic_form is not None: return (self.size - self.cycles) % 2 return _af_parity(self.array_form) @property def is_even(self): """ Checks if a permutation is even. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 1, 2, 3]) >>> p.is_even True >>> p = Permutation([3, 2, 1, 0]) >>> p.is_even True See Also ======== is_odd """ return not self.is_odd @property def is_odd(self): """ Checks if a permutation is odd. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 1, 2, 3]) >>> p.is_odd False >>> p = Permutation([3, 2, 0, 1]) >>> p.is_odd True See Also ======== is_even """ return bool(self.parity() % 2) @property def is_Singleton(self): """ Checks to see if the permutation contains only one number and is thus the only possible permutation of this set of numbers Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation([0]).is_Singleton True >>> Permutation([0, 1]).is_Singleton False See Also ======== is_Empty """ return self.size == 1 @property def is_Empty(self): """ Checks to see if the permutation is a set with zero elements Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation([]).is_Empty True >>> Permutation([0]).is_Empty False See Also ======== is_Singleton """ return self.size == 0 @property def is_identity(self): return self.is_Identity @property def is_Identity(self): """ Returns True if the Permutation is an identity permutation. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([]) >>> p.is_Identity True >>> p = Permutation([[0], [1], [2]]) >>> p.is_Identity True >>> p = Permutation([0, 1, 2]) >>> p.is_Identity True >>> p = Permutation([0, 2, 1]) >>> p.is_Identity False See Also ======== order """ af = self.array_form return not af or all(i == af[i] for i in range(self.size)) def ascents(self): """ Returns the positions of ascents in a permutation, ie, the location where p[i] < p[i+1] Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([4, 0, 1, 3, 2]) >>> p.ascents() [1, 2] See Also ======== descents, inversions, min, max """ a = self.array_form pos = [i for i in range(len(a) - 1) if a[i] < a[i + 1]] return pos def descents(self): """ Returns the positions of descents in a permutation, ie, the location where p[i] > p[i+1] Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([4, 0, 1, 3, 2]) >>> p.descents() [0, 3] See Also ======== ascents, inversions, min, max """ a = self.array_form pos = [i for i in range(len(a) - 1) if a[i] > a[i + 1]] return pos def max(self): """ The maximum element moved by the permutation. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([1, 0, 2, 3, 4]) >>> p.max() 1 See Also ======== min, descents, ascents, inversions """ max = 0 a = self.array_form for i in range(len(a)): if a[i] != i and a[i] > max: max = a[i] return max def min(self): """ The minimum element moved by the permutation. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 1, 4, 3, 2]) >>> p.min() 2 See Also ======== max, descents, ascents, inversions """ a = self.array_form min = len(a) for i in range(len(a)): if a[i] != i and a[i] < min: min = a[i] return min def inversions(self): """ Computes the number of inversions of a permutation. Explanation =========== An inversion is where i > j but p[i] < p[j]. For small length of p, it iterates over all i and j values and calculates the number of inversions. For large length of p, it uses a variation of merge sort to calculate the number of inversions. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 1, 2, 3, 4, 5]) >>> p.inversions() 0 >>> Permutation([3, 2, 1, 0]).inversions() 6 See Also ======== descents, ascents, min, max References ========== .. [1] http://www.cp.eng.chula.ac.th/~piak/teaching/algo/algo2008/count-inv.htm """ inversions = 0 a = self.array_form n = len(a) if n < 130: for i in range(n - 1): b = a[i] for c in a[i + 1:]: if b > c: inversions += 1 else: k = 1 right = 0 arr = a[:] temp = a[:] while k < n: i = 0 while i + k < n: right = i + k * 2 - 1 if right >= n: right = n - 1 inversions += _merge(arr, temp, i, i + k, right) i = i + k * 2 k = k * 2 return inversions def commutator(self, x): """Return the commutator of ``self`` and ``x``: ``~x~selfxself`` If f and g are part of a group, G, then the commutator of f and g is the group identity iff f and g commute, i.e. fg == gf. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> p = Permutation([0, 2, 3, 1]) >>> x = Permutation([2, 0, 3, 1]) >>> c = p.commutator(x); c Permutation([2, 1, 3, 0]) >>> c == ~x~pxp True >>> I = Permutation(3) >>> p = [I + i for i in range(6)] >>> for i in range(len(p)): ... for j in range(len(p)): ... c = p[i].commutator(p[j]) ... if p[i]p[j] == p[j]p[i]: ... assert c == I ... else: ... assert c != I ... References ========== .. [1] https://en.wikipedia.org/wiki/Commutator """ a = self.array_form b = x.array_form n = len(a) if len(b) != n: raise ValueError("The permutations must be of equal size.") inva = [None]n for i in range(n): inva[a[i]] = i invb = [None]n for i in range(n): invb[b[i]] = i return self._af_new([a[b[inva[i]]] for i in invb]) def signature(self): """ Gives the signature of the permutation needed to place the elements of the permutation in canonical order. The signature is calculated as (-1)^<number of inversions> Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 1, 2]) >>> p.inversions() 0 >>> p.signature() 1 >>> q = Permutation([0,2,1]) >>> q.inversions() 1 >>> q.signature() -1 See Also ======== inversions """ if self.is_even: return 1 return -1 def order(self): """ Computes the order of a permutation. When the permutation is raised to the power of its order it equals the identity permutation. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> p = Permutation([3, 1, 5, 2, 4, 0]) >>> p.order() 4 >>> (p*(p.order())) Permutation([], size=6) See Also ======== identity, cardinality, length, rank, size """ return reduce(lcm, [len(cycle) for cycle in self.cyclic_form], 1) def length(self): """ Returns the number of integers moved by a permutation. Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation([0, 3, 2, 1]).length() 2 >>> Permutation([[0, 1], [2, 3]]).length() 4 See Also ======== min, max, support, cardinality, order, rank, size """ return len(self.support()) @property def cycle_structure(self): """Return the cycle structure of the permutation as a dictionary indicating the multiplicity of each cycle length. Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation(3).cycle_structure {1: 4} >>> Permutation(0, 4, 3)(1, 2)(5, 6).cycle_structure {2: 2, 3: 1} """ if self._cycle_structure: rv = self._cycle_structure else: rv = defaultdict(int) singletons = self.size for c in self.cyclic_form: rv[len(c)] += 1 singletons -= len(c) if singletons: rv[1] = singletons self._cycle_structure = rv return dict(rv) # make a copy @property def cycles(self): """ Returns the number of cycles contained in the permutation (including singletons). Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation([0, 1, 2]).cycles 3 >>> Permutation([0, 1, 2]).full_cyclic_form [[0], [1], [2]] >>> Permutation(0, 1)(2, 3).cycles 2 See Also ======== sympy.functions.combinatorial.numbers.stirling """ return len(self.full_cyclic_form) def index(self): """ Returns the index of a permutation. The index of a permutation is the sum of all subscripts j such that p[j] is greater than p[j+1]. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([3, 0, 2, 1, 4]) >>> p.index() 2 """ a = self.array_form return sum([j for j in range(len(a) - 1) if a[j] > a[j + 1]]) def runs(self): """ Returns the runs of a permutation. An ascending sequence in a permutation is called a run [5]. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([2, 5, 7, 3, 6, 0, 1, 4, 8]) >>> p.runs() [[2, 5, 7], [3, 6], [0, 1, 4, 8]] >>> q = Permutation([1,3,2,0]) >>> q.runs() [[1, 3], [2], [0]] """ return runs(self.array_form) def inversion_vector(self): """Return the inversion vector of the permutation. The inversion vector consists of elements whose value indicates the number of elements in the permutation that are lesser than it and lie on its right hand side. The inversion vector is the same as the Lehmer encoding of a permutation. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([4, 8, 0, 7, 1, 5, 3, 6, 2]) >>> p.inversion_vector() [4, 7, 0, 5, 0, 2, 1, 1] >>> p = Permutation([3, 2, 1, 0]) >>> p.inversion_vector() [3, 2, 1] The inversion vector increases lexicographically with the rank of the permutation, the -ith element cycling through 0..i. >>> p = Permutation(2) >>> while p: ... print('%s %s %s' % (p, p.inversion_vector(), p.rank())) ... p = p.next_lex() (2) [0, 0] 0 (1 2) [0, 1] 1 (2)(0 1) [1, 0] 2 (0 1 2) [1, 1] 3 (0 2 1) [2, 0] 4 (0 2) [2, 1] 5 See Also ======== from_inversion_vector """ self_array_form = self.array_form n = len(self_array_form) inversion_vector = [0] (n - 1) for i in range(n - 1): val = 0 for j in range(i + 1, n): if self_array_form[j] < self_array_form[i]: val += 1 inversion_vector[i] = val return inversion_vector def rank_trotterjohnson(self): """ Returns the Trotter Johnson rank, which we get from the minimal change algorithm. See [4] section 2.4. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 1, 2, 3]) >>> p.rank_trotterjohnson() 0 >>> p = Permutation([0, 2, 1, 3]) >>> p.rank_trotterjohnson() 7 See Also ======== unrank_trotterjohnson, next_trotterjohnson """ if self.array_form == [] or self.is_Identity: return 0 if self.array_form == [1, 0]: return 1 perm = self.array_form n = self.size rank = 0 for j in range(1, n): k = 1 i = 0 while perm[i] != j: if perm[i] < j: k += 1 i += 1 j1 = j + 1 if rank % 2 == 0: rank = j1rank + j1 - k else: rank = j1rank + k - 1 return rank @classmethod def unrank_trotterjohnson(cls, size, rank): """ Trotter Johnson permutation unranking. See [4] section 2.4. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> Permutation.unrank_trotterjohnson(5, 10) Permutation([0, 3, 1, 2, 4]) See Also ======== rank_trotterjohnson, next_trotterjohnson """ perm = [0]size r2 = 0 n = ifac(size) pj = 1 for j in range(2, size + 1): pj = j r1 = (rank * pj) // n k = r1 - jr2 if r2 % 2 == 0: for i in range(j - 1, j - k - 1, -1): perm[i] = perm[i - 1] perm[j - k - 1] = j - 1 else: for i in range(j - 1, k, -1): perm[i] = perm[i - 1] perm[k] = j - 1 r2 = r1 return cls._af_new(perm) def next_trotterjohnson(self): """ Returns the next permutation in Trotter-Johnson order. If self is the last permutation it returns None. See [4] section 2.4. If it is desired to generate all such permutations, they can be generated in order more quickly with the ``generate_bell`` function. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> p = Permutation([3, 0, 2, 1]) >>> p.rank_trotterjohnson() 4 >>> p = p.next_trotterjohnson(); p Permutation([0, 3, 2, 1]) >>> p.rank_trotterjohnson() 5 See Also ======== rank_trotterjohnson, unrank_trotterjohnson, sympy.utilities.iterables.generate_bell """ pi = self.array_form[:] n = len(pi) st = 0 rho = pi[:] done = False m = n-1 while m > 0 and not done: d = rho.index(m) for i in range(d, m): rho[i] = rho[i + 1] par = _af_parity(rho[:m]) if par == 1: if d == m: m -= 1 else: pi[st + d], pi[st + d + 1] = pi[st + d + 1], pi[st + d] done = True else: if d == 0: m -= 1 st += 1 else: pi[st + d], pi[st + d - 1] = pi[st + d - 1], pi[st + d] done = True if m == 0: return None return self._af_new(pi) def get_precedence_matrix(self): """ Gets the precedence matrix. This is used for computing the distance between two permutations. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> p = Permutation.josephus(3, 6, 1) >>> p Permutation([2, 5, 3, 1, 4, 0]) >>> p.get_precedence_matrix() Matrix([ [0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 0], [1, 1, 0, 1, 1, 1], [1, 1, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0], [1, 1, 0, 1, 1, 0]]) See Also ======== get_precedence_distance, get_adjacency_matrix, get_adjacency_distance """ m = zeros(self.size) perm = self.array_form for i in range(m.rows): for j in range(i + 1, m.cols): m[perm[i], perm[j]] = 1 return m def get_precedence_distance(self, other): """ Computes the precedence distance between two permutations. Explanation =========== Suppose p and p' represent n jobs. The precedence metric counts the number of times a job j is preceded by job i in both p and p'. This metric is commutative. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([2, 0, 4, 3, 1]) >>> q = Permutation([3, 1, 2, 4, 0]) >>> p.get_precedence_distance(q) 7 >>> q.get_precedence_distance(p) 7 See Also ======== get_precedence_matrix, get_adjacency_matrix, get_adjacency_distance """ if self.size != other.size: raise ValueError("The permutations must be of equal size.") self_prec_mat = self.get_precedence_matrix() other_prec_mat = other.get_precedence_matrix() n_prec = 0 for i in range(self.size): for j in range(self.size): if i == j: continue if self_prec_mat[i, j] other_prec_mat[i, j] == 1: n_prec += 1 d = self.size * (self.size - 1)//2 - n_prec return d def get_adjacency_matrix(self): """ Computes the adjacency matrix of a permutation. Explanation =========== If job i is adjacent to job j in a permutation p then we set m[i, j] = 1 where m is the adjacency matrix of p. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation.josephus(3, 6, 1) >>> p.get_adjacency_matrix() Matrix([ [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]]) >>> q = Permutation([0, 1, 2, 3]) >>> q.get_adjacency_matrix() Matrix([ [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [0, 0, 0, 0]]) See Also ======== get_precedence_matrix, get_precedence_distance, get_adjacency_distance """ m = zeros(self.size) perm = self.array_form for i in range(self.size - 1): m[perm[i], perm[i + 1]] = 1 return m def get_adjacency_distance(self, other): """ Computes the adjacency distance between two permutations. Explanation =========== This metric counts the number of times a pair i,j of jobs is adjacent in both p and p'. If n_adj is this quantity then the adjacency distance is n - n_adj - 1 [1] [1] Reeves, Colin R. Landscapes, Operators and Heuristic search, Annals of Operational Research, 86, pp 473-490. (1999) Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 3, 1, 2, 4]) >>> q = Permutation.josephus(4, 5, 2) >>> p.get_adjacency_distance(q) 3 >>> r = Permutation([0, 2, 1, 4, 3]) >>> p.get_adjacency_distance(r) 4 See Also ======== get_precedence_matrix, get_precedence_distance, get_adjacency_matrix """ if self.size != other.size: raise ValueError("The permutations must be of the same size.") self_adj_mat = self.get_adjacency_matrix() other_adj_mat = other.get_adjacency_matrix() n_adj = 0 for i in range(self.size): for j in range(self.size): if i == j: continue if self_adj_mat[i, j] * other_adj_mat[i, j] == 1: n_adj += 1 d = self.size - n_adj - 1 return d def get_positional_distance(self, other): """ Computes the positional distance between two permutations. Examples ======== >>> from sympy.combinatorics import Permutation >>> p = Permutation([0, 3, 1, 2, 4]) >>> q = Permutation.josephus(4, 5, 2) >>> r = Permutation([3, 1, 4, 0, 2]) >>> p.get_positional_distance(q) 12 >>> p.get_positional_distance(r) 12 See Also ======== get_precedence_distance, get_adjacency_distance """ a = self.array_form b = other.array_form if len(a) != len(b): raise ValueError("The permutations must be of the same size.") return sum([abs(a[i] - b[i]) for i in range(len(a))]) @classmethod def josephus(cls, m, n, s=1): """Return as a permutation the shuffling of range(n) using the Josephus scheme in which every m-th item is selected until all have been chosen. The returned permutation has elements listed by the order in which they were selected. The parameter ``s`` stops the selection process when there are ``s`` items remaining and these are selected by continuing the selection, counting by 1 rather than by ``m``. Consider selecting every 3rd item from 6 until only 2 remain:: choices chosen ======== ====== 012345 01 345 2 01 34 25 01 4 253 0 4 2531 0 25314 253140 Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.josephus(3, 6, 2).array_form [2, 5, 3, 1, 4, 0] References ========== .. [1] https://en.wikipedia.org/wiki/Flavius_Josephus .. [2] https://en.wikipedia.org/wiki/Josephus_problem .. [3] http://www.wou.edu/~burtonl/josephus.html """ from collections import deque m -= 1 Q = deque(list(range(n))) perm = [] while len(Q) > max(s, 1): for dp in range(m): Q.append(Q.popleft()) perm.append(Q.popleft()) perm.extend(list(Q)) return cls(perm) @classmethod def from_inversion_vector(cls, inversion): """ Calculates the permutation from the inversion vector. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> Permutation.from_inversion_vector([3, 2, 1, 0, 0]) Permutation([3, 2, 1, 0, 4, 5]) """ size = len(inversion) N = list(range(size + 1)) perm = [] try: for k in range(size): val = N[inversion[k]] perm.append(val) N.remove(val) except IndexError: raise ValueError("The inversion vector is not valid.") perm.extend(N) return cls._af_new(perm) @classmethod def random(cls, n): """ Generates a random permutation of length ``n``. Uses the underlying Python pseudo-random number generator. Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.random(2) in (Permutation([1, 0]), Permutation([0, 1])) True """ perm_array = list(range(n)) random.shuffle(perm_array) return cls._af_new(perm_array) @classmethod def unrank_lex(cls, size, rank): """ Lexicographic permutation unranking. Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy import init_printing >>> init_printing(perm_cyclic=False, pretty_print=False) >>> a = Permutation.unrank_lex(5, 10) >>> a.rank() 10 >>> a Permutation([0, 2, 4, 1, 3]) See Also ======== rank, next_lex """ perm_array = [0] * size psize = 1 for i in range(size): new_psize = psize(i + 1) d = (rank % new_psize) // psize rank -= dpsize perm_array[size - i - 1] = d for j in range(size - i, size): if perm_array[j] > d - 1: perm_array[j] += 1 psize = new_psize return cls._af_new(perm_array) def resize(self, n): """Resize the permutation to the new size ``n``. Parameters ========== n : int The new size of the permutation. Raises ====== ValueError If the permutation cannot be resized to the given size. This may only happen when resized to a smaller size than the original. Examples ======== >>> from sympy.combinatorics import Permutation Increasing the size of a permutation: >>> p = Permutation(0, 1, 2) >>> p = p.resize(5) >>> p (4)(0 1 2) Decreasing the size of the permutation: >>> p = p.resize(4) >>> p (3)(0 1 2) If resizing to the specific size breaks the cycles: >>> p.resize(2) Traceback (most recent call last): ... ValueError: The permutation cannot be resized to 2 because the cycle (0, 1, 2) may break. """ aform = self.array_form l = len(aform) if n > l: aform += list(range(l, n)) return Permutation._af_new(aform) elif n < l: cyclic_form = self.full_cyclic_form new_cyclic_form = [] for cycle in cyclic_form: cycle_min = min(cycle) cycle_max = max(cycle) if cycle_min <= n-1: if cycle_max > n-1: raise ValueError( "The permutation cannot be resized to {} " "because the cycle {} may break." .format(n, tuple(cycle))) new_cyclic_form.append(cycle) return Permutation(new_cyclic_form) return self # XXX Deprecated flag print_cyclic = None def _merge(arr, temp, left, mid, right): """ Merges two sorted arrays and calculates the inversion count. Helper function for calculating inversions. This method is for internal use only. """ i = k = left j = mid inv_count = 0 while i < mid and j <= right: if arr[i] < arr[j]: temp[k] = arr[i] k += 1 i += 1 else: temp[k] = arr[j] k += 1 j += 1 inv_count += (mid -i) while i < mid: temp[k] = arr[i] k += 1 i += 1 if j <= right: k += right - j + 1 j += right - j + 1 arr[left:k + 1] = temp[left:k + 1] else: arr[left:right + 1] = temp[left:right + 1] return inv_count Perm = Permutation _af_new = Perm._af_new class AppliedPermutation(Expr): """A permutation applied to a symbolic variable. Parameters ========== perm : Permutation x : Expr Examples ======== >>> from sympy import Symbol >>> from sympy.combinatorics import Permutation Creating a symbolic permutation function application: >>> x = Symbol('x') >>> p = Permutation(0, 1, 2) >>> p.apply(x) AppliedPermutation((0 1 2), x) >>> _.subs(x, 1) 2 """ def __new__(cls, perm, x, evaluate=None): if evaluate is None: evaluate = global_parameters.evaluate perm = _sympify(perm) x = _sympify(x) if not isinstance(perm, Permutation): raise ValueError("{} must be a Permutation instance." .format(perm)) if evaluate: if x.is_Integer: return perm.apply(x) obj = super().__new__(cls, perm, x) return obj @dispatch(Permutation, Permutation) def _eval_is_eq(lhs, rhs): if lhs._size != rhs._size: return None return lhs._array_form == rhs._array_form
f2c0f32791e2dbdfe54f432876b51d25715e010471a9a30d6dd1bb47ac0cb821	from itertools import combinations from sympy.combinatorics.graycode import GrayCode class Subset(): """ Represents a basic subset object. Explanation =========== We generate subsets using essentially two techniques, binary enumeration and lexicographic enumeration. The Subset class takes two arguments, the first one describes the initial subset to consider and the second describes the superset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.next_binary().subset ['b'] >>> a.prev_binary().subset ['c'] """ _rank_binary = None _rank_lex = None _rank_graycode = None _subset = None _superset = None def __new__(cls, subset, superset): """ Default constructor. It takes the ``subset`` and its ``superset`` as its parameters. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.subset ['c', 'd'] >>> a.superset ['a', 'b', 'c', 'd'] >>> a.size 2 """ if len(subset) > len(superset): raise ValueError('Invalid arguments have been provided. The ' 'superset must be larger than the subset.') for elem in subset: if elem not in superset: raise ValueError('The superset provided is invalid as it does ' 'not contain the element {}'.format(elem)) obj = object.__new__(cls) obj._subset = subset obj._superset = superset return obj def __eq__(self, other): """Return a boolean indicating whether a == b on the basis of whether both objects are of the class Subset and if the values of the subset and superset attributes are the same. """ if not isinstance(other, Subset): return NotImplemented return self.subset == other.subset and self.superset == other.superset def iterate_binary(self, k): """ This is a helper function. It iterates over the binary subsets by ``k`` steps. This variable can be both positive or negative. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.iterate_binary(-2).subset ['d'] >>> a = Subset(['a', 'b', 'c'], ['a', 'b', 'c', 'd']) >>> a.iterate_binary(2).subset [] See Also ======== next_binary, prev_binary """ bin_list = Subset.bitlist_from_subset(self.subset, self.superset) n = (int(''.join(bin_list), 2) + k) % 2self.superset_size bits = bin(n)[2:].rjust(self.superset_size, '0') return Subset.subset_from_bitlist(self.superset, bits) def next_binary(self): """ Generates the next binary ordered subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.next_binary().subset ['b'] >>> a = Subset(['a', 'b', 'c', 'd'], ['a', 'b', 'c', 'd']) >>> a.next_binary().subset [] See Also ======== prev_binary, iterate_binary """ return self.iterate_binary(1) def prev_binary(self): """ Generates the previous binary ordered subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset([], ['a', 'b', 'c', 'd']) >>> a.prev_binary().subset ['a', 'b', 'c', 'd'] >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.prev_binary().subset ['c'] See Also ======== next_binary, iterate_binary """ return self.iterate_binary(-1) def next_lexicographic(self): """ Generates the next lexicographically ordered subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.next_lexicographic().subset ['d'] >>> a = Subset(['d'], ['a', 'b', 'c', 'd']) >>> a.next_lexicographic().subset [] See Also ======== prev_lexicographic """ i = self.superset_size - 1 indices = Subset.subset_indices(self.subset, self.superset) if i in indices: if i - 1 in indices: indices.remove(i - 1) else: indices.remove(i) i = i - 1 while i >= 0 and i not in indices: i = i - 1 if i >= 0: indices.remove(i) indices.append(i+1) else: while i not in indices and i >= 0: i = i - 1 indices.append(i + 1) ret_set = [] super_set = self.superset for i in indices: ret_set.append(super_set[i]) return Subset(ret_set, super_set) def prev_lexicographic(self): """ Generates the previous lexicographically ordered subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset([], ['a', 'b', 'c', 'd']) >>> a.prev_lexicographic().subset ['d'] >>> a = Subset(['c','d'], ['a', 'b', 'c', 'd']) >>> a.prev_lexicographic().subset ['c'] See Also ======== next_lexicographic """ i = self.superset_size - 1 indices = Subset.subset_indices(self.subset, self.superset) while i >= 0 and i not in indices: i = i - 1 if i == 0 or i - 1 in indices: indices.remove(i) else: if i >= 0: indices.remove(i) indices.append(i - 1) indices.append(self.superset_size - 1) ret_set = [] super_set = self.superset for i in indices: ret_set.append(super_set[i]) return Subset(ret_set, super_set) def iterate_graycode(self, k): """ Helper function used for prev_gray and next_gray. It performs ``k`` step overs to get the respective Gray codes. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset([1, 2, 3], [1, 2, 3, 4]) >>> a.iterate_graycode(3).subset [1, 4] >>> a.iterate_graycode(-2).subset [1, 2, 4] See Also ======== next_gray, prev_gray """ unranked_code = GrayCode.unrank(self.superset_size, (self.rank_gray + k) % self.cardinality) return Subset.subset_from_bitlist(self.superset, unranked_code) def next_gray(self): """ Generates the next Gray code ordered subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset([1, 2, 3], [1, 2, 3, 4]) >>> a.next_gray().subset [1, 3] See Also ======== iterate_graycode, prev_gray """ return self.iterate_graycode(1) def prev_gray(self): """ Generates the previous Gray code ordered subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset([2, 3, 4], [1, 2, 3, 4, 5]) >>> a.prev_gray().subset [2, 3, 4, 5] See Also ======== iterate_graycode, next_gray """ return self.iterate_graycode(-1) @property def rank_binary(self): """ Computes the binary ordered rank. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset([], ['a','b','c','d']) >>> a.rank_binary 0 >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.rank_binary 3 See Also ======== iterate_binary, unrank_binary """ if self._rank_binary is None: self._rank_binary = int("".join( Subset.bitlist_from_subset(self.subset, self.superset)), 2) return self._rank_binary @property def rank_lexicographic(self): """ Computes the lexicographic ranking of the subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.rank_lexicographic 14 >>> a = Subset([2, 4, 5], [1, 2, 3, 4, 5, 6]) >>> a.rank_lexicographic 43 """ if self._rank_lex is None: def _ranklex(self, subset_index, i, n): if subset_index == [] or i > n: return 0 if i in subset_index: subset_index.remove(i) return 1 + _ranklex(self, subset_index, i + 1, n) return 2(n - i - 1) + _ranklex(self, subset_index, i + 1, n) indices = Subset.subset_indices(self.subset, self.superset) self._rank_lex = _ranklex(self, indices, 0, self.superset_size) return self._rank_lex @property def rank_gray(self): """ Computes the Gray code ranking of the subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c','d'], ['a','b','c','d']) >>> a.rank_gray 2 >>> a = Subset([2, 4, 5], [1, 2, 3, 4, 5, 6]) >>> a.rank_gray 27 See Also ======== iterate_graycode, unrank_gray """ if self._rank_graycode is None: bits = Subset.bitlist_from_subset(self.subset, self.superset) self._rank_graycode = GrayCode(len(bits), start=bits).rank return self._rank_graycode @property def subset(self): """ Gets the subset represented by the current instance. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.subset ['c', 'd'] See Also ======== superset, size, superset_size, cardinality """ return self._subset @property def size(self): """ Gets the size of the subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.size 2 See Also ======== subset, superset, superset_size, cardinality """ return len(self.subset) @property def superset(self): """ Gets the superset of the subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.superset ['a', 'b', 'c', 'd'] See Also ======== subset, size, superset_size, cardinality """ return self._superset @property def superset_size(self): """ Returns the size of the superset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.superset_size 4 See Also ======== subset, superset, size, cardinality """ return len(self.superset) @property def cardinality(self): """ Returns the number of all possible subsets. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.cardinality 16 See Also ======== subset, superset, size, superset_size """ return 2*(self.superset_size) @classmethod def subset_from_bitlist(self, super_set, bitlist): """ Gets the subset defined by the bitlist. Examples ======== >>> from sympy.combinatorics import Subset >>> Subset.subset_from_bitlist(['a', 'b', 'c', 'd'], '0011').subset ['c', 'd'] See Also ======== bitlist_from_subset """ if len(super_set) != len(bitlist): raise ValueError("The sizes of the lists are not equal") ret_set = [] for i in range(len(bitlist)): if bitlist[i] == '1': ret_set.append(super_set[i]) return Subset(ret_set, super_set) @classmethod def bitlist_from_subset(self, subset, superset): """ Gets the bitlist corresponding to a subset. Examples ======== >>> from sympy.combinatorics import Subset >>> Subset.bitlist_from_subset(['c', 'd'], ['a', 'b', 'c', 'd']) '0011' See Also ======== subset_from_bitlist """ bitlist = ['0'] len(superset) if isinstance(subset, Subset): subset = subset.subset for i in Subset.subset_indices(subset, superset): bitlist[i] = '1' return ''.join(bitlist) @classmethod def unrank_binary(self, rank, superset): """ Gets the binary ordered subset of the specified rank. Examples ======== >>> from sympy.combinatorics import Subset >>> Subset.unrank_binary(4, ['a', 'b', 'c', 'd']).subset ['b'] See Also ======== iterate_binary, rank_binary """ bits = bin(rank)[2:].rjust(len(superset), '0') return Subset.subset_from_bitlist(superset, bits) @classmethod def unrank_gray(self, rank, superset): """ Gets the Gray code ordered subset of the specified rank. Examples ======== >>> from sympy.combinatorics import Subset >>> Subset.unrank_gray(4, ['a', 'b', 'c']).subset ['a', 'b'] >>> Subset.unrank_gray(0, ['a', 'b', 'c']).subset [] See Also ======== iterate_graycode, rank_gray """ graycode_bitlist = GrayCode.unrank(len(superset), rank) return Subset.subset_from_bitlist(superset, graycode_bitlist) @classmethod def subset_indices(self, subset, superset): """Return indices of subset in superset in a list; the list is empty if all elements of ``subset`` are not in ``superset``. Examples ======== >>> from sympy.combinatorics import Subset >>> superset = [1, 3, 2, 5, 4] >>> Subset.subset_indices([3, 2, 1], superset) [1, 2, 0] >>> Subset.subset_indices([1, 6], superset) [] >>> Subset.subset_indices([], superset) [] """ a, b = superset, subset sb = set(b) d = {} for i, ai in enumerate(a): if ai in sb: d[ai] = i sb.remove(ai) if not sb: break else: return list() return [d[bi] for bi in b] def ksubsets(superset, k): """ Finds the subsets of size ``k`` in lexicographic order. This uses the itertools generator. Examples ======== >>> from sympy.combinatorics.subsets import ksubsets >>> list(ksubsets([1, 2, 3], 2)) [(1, 2), (1, 3), (2, 3)] >>> list(ksubsets([1, 2, 3, 4, 5], 2)) [(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), \ (2, 5), (3, 4), (3, 5), (4, 5)] See Also ======== Subset """ return combinations(superset, k)
0f1f716a4a47b93220cb81bccd58c063aa53bbc27dfcadddfe328f8b344c755a	from sympy.combinatorics.permutations import Permutation, Cycle from sympy.combinatorics.prufer import Prufer from sympy.combinatorics.generators import cyclic, alternating, symmetric, dihedral from sympy.combinatorics.subsets import Subset from sympy.combinatorics.partitions import (Partition, IntegerPartition, RGS_rank, RGS_unrank, RGS_enum) from sympy.combinatorics.polyhedron import (Polyhedron, tetrahedron, cube, octahedron, dodecahedron, icosahedron) from sympy.combinatorics.perm_groups import PermutationGroup, Coset, SymmetricPermutationGroup from sympy.combinatorics.group_constructs import DirectProduct from sympy.combinatorics.graycode import GrayCode from sympy.combinatorics.named_groups import (SymmetricGroup, DihedralGroup, CyclicGroup, AlternatingGroup, AbelianGroup, RubikGroup) from sympy.combinatorics.pc_groups import PolycyclicGroup, Collector from sympy.combinatorics.free_groups import free_group __all__ = [ 'Permutation', 'Cycle', 'Prufer', 'cyclic', 'alternating', 'symmetric', 'dihedral', 'Subset', 'Partition', 'IntegerPartition', 'RGS_rank', 'RGS_unrank', 'RGS_enum', 'Polyhedron', 'tetrahedron', 'cube', 'octahedron', 'dodecahedron', 'icosahedron', 'PermutationGroup', 'Coset', 'SymmetricPermutationGroup', 'DirectProduct', 'GrayCode', 'SymmetricGroup', 'DihedralGroup', 'CyclicGroup', 'AlternatingGroup', 'AbelianGroup', 'RubikGroup', 'PolycyclicGroup', 'Collector', 'free_group', ]
0c2d56263f8ea646160611bcc44211ce86d8a968f36a15ee09b275b736f547c5	from typing import Dict as tDict, List from sympy.core import S from sympy.core.expr import Expr from sympy.core.symbol import Symbol, symbols as _symbols from sympy.core.sympify import CantSympify from sympy.printing.defaults import DefaultPrinting from sympy.utilities import public from sympy.utilities.iterables import flatten, is_sequence from sympy.utilities.magic import pollute from sympy.utilities.misc import as_int @public def free_group(symbols): """Construct a free group returning ``(FreeGroup, (f_0, f_1, ..., f_(n-1))``. Parameters ========== symbols : str, Symbol/Expr or sequence of str, Symbol/Expr (may be empty) Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y, z = free_group("x, y, z") >>> F <free group on the generators (x, y, z)> >>> x*2y-1 x2y-1 >>> type(_) <class 'sympy.combinatorics.free_groups.FreeGroupElement'> """ _free_group = FreeGroup(symbols) return (_free_group,) + tuple(_free_group.generators) @public def xfree_group(symbols): """Construct a free group returning ``(FreeGroup, (f_0, f_1, ..., f_(n-1)))``. Parameters ========== symbols : str, Symbol/Expr or sequence of str, Symbol/Expr (may be empty) Examples ======== >>> from sympy.combinatorics.free_groups import xfree_group >>> F, (x, y, z) = xfree_group("x, y, z") >>> F <free group on the generators (x, y, z)> >>> y2x*-2z-1 y2x-2z-1 >>> type(_) <class 'sympy.combinatorics.free_groups.FreeGroupElement'> """ _free_group = FreeGroup(symbols) return (_free_group, _free_group.generators) @public def vfree_group(symbols): """Construct a free group and inject ``f_0, f_1, ..., f_(n-1)`` as symbols into the global namespace. Parameters ========== symbols : str, Symbol/Expr or sequence of str, Symbol/Expr (may be empty) Examples ======== >>> from sympy.combinatorics.free_groups import vfree_group >>> vfree_group("x, y, z") <free group on the generators (x, y, z)> >>> x2y-2z # noqa: F821 x*2y*-2z >>> type(_) <class 'sympy.combinatorics.free_groups.FreeGroupElement'> """ _free_group = FreeGroup(symbols) pollute([sym.name for sym in _free_group.symbols], _free_group.generators) return _free_group def _parse_symbols(symbols): if not symbols: return tuple() if isinstance(symbols, str): return _symbols(symbols, seq=True) elif isinstance(symbols, Expr or FreeGroupElement): return (symbols,) elif is_sequence(symbols): if all(isinstance(s, str) for s in symbols): return _symbols(symbols) elif all(isinstance(s, Expr) for s in symbols): return symbols raise ValueError("The type of `symbols` must be one of the following: " "a str, Symbol/Expr or a sequence of " "one of these types") ############################################################################## # FREE GROUP # ############################################################################## _free_group_cache = {} # type: tDict[int, FreeGroup] class FreeGroup(DefaultPrinting): """ Free group with finite or infinite number of generators. Its input API is that of a str, Symbol/Expr or a sequence of one of these types (which may be empty) See Also ======== sympy.polys.rings.PolyRing References ========== .. [1] http://www.gap-system.org/Manuals/doc/ref/chap37.html .. [2] https://en.wikipedia.org/wiki/Free_group """ is_associative = True is_group = True is_FreeGroup = True is_PermutationGroup = False relators = [] # type: List[Expr] def __new__(cls, symbols): symbols = tuple(_parse_symbols(symbols)) rank = len(symbols) _hash = hash((cls.__name__, symbols, rank)) obj = _free_group_cache.get(_hash) if obj is None: obj = object.__new__(cls) obj._hash = _hash obj._rank = rank # dtype method is used to create new instances of FreeGroupElement obj.dtype = type("FreeGroupElement", (FreeGroupElement,), {"group": obj}) obj.symbols = symbols obj.generators = obj._generators() obj._gens_set = set(obj.generators) for symbol, generator in zip(obj.symbols, obj.generators): if isinstance(symbol, Symbol): name = symbol.name if hasattr(obj, name): setattr(obj, name, generator) _free_group_cache[_hash] = obj return obj def _generators(group): """Returns the generators of the FreeGroup. Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y, z = free_group("x, y, z") >>> F.generators (x, y, z) """ gens = [] for sym in group.symbols: elm = ((sym, 1),) gens.append(group.dtype(elm)) return tuple(gens) def clone(self, symbols=None): return self.__class__(symbols or self.symbols) def __contains__(self, i): """Return True if ``i`` is contained in FreeGroup.""" if not isinstance(i, FreeGroupElement): return False group = i.group return self == group def __hash__(self): return self._hash def __len__(self): return self.rank def __str__(self): if self.rank > 30: str_form = "<free group with %s generators>" % self.rank else: str_form = "<free group on the generators " gens = self.generators str_form += str(gens) + ">" return str_form __repr__ = __str__ def __getitem__(self, index): symbols = self.symbols[index] return self.clone(symbols=symbols) def __eq__(self, other): """No ``FreeGroup`` is equal to any "other" ``FreeGroup``. """ return self is other def index(self, gen): """Return the index of the generator `gen` from ``(f_0, ..., f_(n-1))``. Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> F.index(y) 1 >>> F.index(x) 0 """ if isinstance(gen, self.dtype): return self.generators.index(gen) else: raise ValueError("expected a generator of Free Group %s, got %s" % (self, gen)) def order(self): """Return the order of the free group. Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> F.order() oo >>> free_group("")[0].order() 1 """ if self.rank == 0: return S.One else: return S.Infinity @property def elements(self): """ Return the elements of the free group. Examples ======== >>> from sympy.combinatorics import free_group >>> (z,) = free_group("") >>> z.elements {<identity>} """ if self.rank == 0: # A set containing Identity element of `FreeGroup` self is returned return {self.identity} else: raise ValueError("Group contains infinitely many elements" ", hence cannot be represented") @property def rank(self): r""" In group theory, the `rank` of a group `G`, denoted `G.rank`, can refer to the smallest cardinality of a generating set for G, that is \operatorname{rank}(G)=\min\{ \|X\|: X\subseteq G, \left\langle X\right\rangle =G\}. """ return self._rank @property def is_abelian(self): """Returns if the group is Abelian. Examples ======== >>> from sympy.combinatorics import free_group >>> f, x, y, z = free_group("x y z") >>> f.is_abelian False """ return self.rank in (0, 1) @property def identity(self): """Returns the identity element of free group.""" return self.dtype() def contains(self, g): """Tests if Free Group element ``g`` belong to self, ``G``. In mathematical terms any linear combination of generators of a Free Group is contained in it. Examples ======== >>> from sympy.combinatorics import free_group >>> f, x, y, z = free_group("x y z") >>> f.contains(x*3y*2) True """ if not isinstance(g, FreeGroupElement): return False elif self != g.group: return False else: return True def center(self): """Returns the center of the free group `self`.""" return {self.identity} ############################################################################ # FreeGroupElement # ############################################################################ class FreeGroupElement(CantSympify, DefaultPrinting, tuple): """Used to create elements of FreeGroup. It cannot be used directly to create a free group element. It is called by the `dtype` method of the `FreeGroup` class. """ is_assoc_word = True def new(self, init): return self.__class__(init) _hash = None def __hash__(self): _hash = self._hash if _hash is None: self._hash = _hash = hash((self.group, frozenset(tuple(self)))) return _hash def copy(self): return self.new(self) @property def is_identity(self): if self.array_form == tuple(): return True else: return False @property def array_form(self): """ SymPy provides two different internal kinds of representation of associative words. The first one is called the `array_form` which is a tuple containing `tuples` as its elements, where the size of each tuple is two. At the first position the tuple contains the `symbol-generator`, while at the second position of tuple contains the exponent of that generator at the position. Since elements (i.e. words) do not commute, the indexing of tuple makes that property to stay. The structure in ``array_form`` of ``FreeGroupElement`` is of form: ``( ( symbol_of_gen, exponent ), ( , ), ... ( , ) )`` Examples ======== >>> from sympy.combinatorics import free_group >>> f, x, y, z = free_group("x y z") >>> (xz).array_form ((x, 1), (z, 1)) >>> (x*2zyx2).array_form ((x, 2), (z, 1), (y, 1), (x, 2)) See Also ======== letter_repr """ return tuple(self) @property def letter_form(self): """ The letter representation of a ``FreeGroupElement`` is a tuple of generator symbols, with each entry corresponding to a group generator. Inverses of the generators are represented by negative generator symbols. Examples ======== >>> from sympy.combinatorics import free_group >>> f, a, b, c, d = free_group("a b c d") >>> (a3).letter_form (a, a, a) >>> (a*2d*-2ab-4).letter_form (a, a, -d, -d, a, -b, -b, -b, -b) >>> (a-2b*3d).letter_form (-a, -a, b, b, b, d) See Also ======== array_form """ return tuple(flatten([(i,)j if j > 0 else (-i,)(-j) for i, j in self.array_form])) def __getitem__(self, i): group = self.group r = self.letter_form[i] if r.is_Symbol: return group.dtype(((r, 1),)) else: return group.dtype(((-r, -1),)) def index(self, gen): if len(gen) != 1: raise ValueError() return (self.letter_form).index(gen.letter_form[0]) @property def letter_form_elm(self): """ """ group = self.group r = self.letter_form return [group.dtype(((elm,1),)) if elm.is_Symbol \ else group.dtype(((-elm,-1),)) for elm in r] @property def ext_rep(self): """This is called the External Representation of ``FreeGroupElement`` """ return tuple(flatten(self.array_form)) def __contains__(self, gen): return gen.array_form[0][0] in tuple([r[0] for r in self.array_form]) def __str__(self): if self.is_identity: return "<identity>" str_form = "" array_form = self.array_form for i in range(len(array_form)): if i == len(array_form) - 1: if array_form[i][1] == 1: str_form += str(array_form[i][0]) else: str_form += str(array_form[i][0]) + \ "*" + str(array_form[i][1]) else: if array_form[i][1] == 1: str_form += str(array_form[i][0]) + "" else: str_form += str(array_form[i][0]) + \ "*" + str(array_form[i][1]) + "" return str_form __repr__ = __str__ def __pow__(self, n): n = as_int(n) group = self.group if n == 0: return group.identity if n < 0: n = -n return (self.inverse())*n result = self for i in range(n - 1): result = resultself # this method can be improved instead of just returning the # multiplication of elements return result def __mul__(self, other): """Returns the product of elements belonging to the same ``FreeGroup``. Examples ======== >>> from sympy.combinatorics import free_group >>> f, x, y, z = free_group("x y z") >>> xy2y*-4 xy*-2 >>> zy*-2 zy-2 >>> x2yy*-1x*-2 <identity> """ group = self.group if not isinstance(other, group.dtype): raise TypeError("only FreeGroup elements of same FreeGroup can " "be multiplied") if self.is_identity: return other if other.is_identity: return self r = list(self.array_form + other.array_form) zero_mul_simp(r, len(self.array_form) - 1) return group.dtype(tuple(r)) def __truediv__(self, other): group = self.group if not isinstance(other, group.dtype): raise TypeError("only FreeGroup elements of same FreeGroup can " "be multiplied") return self(other.inverse()) def __rtruediv__(self, other): group = self.group if not isinstance(other, group.dtype): raise TypeError("only FreeGroup elements of same FreeGroup can " "be multiplied") return other(self.inverse()) def __add__(self, other): return NotImplemented def inverse(self): """ Returns the inverse of a ``FreeGroupElement`` element Examples ======== >>> from sympy.combinatorics import free_group >>> f, x, y, z = free_group("x y z") >>> x.inverse() x-1 >>> (xy).inverse() y*-1x-1 """ group = self.group r = tuple([(i, -j) for i, j in self.array_form[::-1]]) return group.dtype(r) def order(self): """Find the order of a ``FreeGroupElement``. Examples ======== >>> from sympy.combinatorics import free_group >>> f, x, y = free_group("x y") >>> (x2yy*-1x*-2).order() 1 """ if self.is_identity: return S.One else: return S.Infinity def commutator(self, other): """ Return the commutator of `self` and `x`: ``~x~selfxself`` """ group = self.group if not isinstance(other, group.dtype): raise ValueError("commutator of only FreeGroupElement of the same " "FreeGroup exists") else: return self.inverse()other.inverse()selfother def eliminate_words(self, words, _all=False, inverse=True): ''' Replace each subword from the dictionary `words` by words[subword]. If words is a list, replace the words by the identity. ''' again = True new = self if isinstance(words, dict): while again: again = False for sub in words: prev = new new = new.eliminate_word(sub, words[sub], _all=_all, inverse=inverse) if new != prev: again = True else: while again: again = False for sub in words: prev = new new = new.eliminate_word(sub, _all=_all, inverse=inverse) if new != prev: again = True return new def eliminate_word(self, gen, by=None, _all=False, inverse=True): """ For an associative word `self`, a subword `gen`, and an associative word `by` (identity by default), return the associative word obtained by replacing each occurrence of `gen` in `self` by `by`. If `_all = True`, the occurrences of `gen` that may appear after the first substitution will also be replaced and so on until no occurrences are found. This might not always terminate (e.g. `(x).eliminate_word(x, x2, _all=True)`). Examples ======== >>> from sympy.combinatorics import free_group >>> f, x, y = free_group("x y") >>> w = x5yx2y*-4x >>> w.eliminate_word( x, x2 ) x10yx*4y*-4x2 >>> w.eliminate_word( x, y-1 ) y-11 >>> w.eliminate_word(x5) yx2y*-4x >>> w.eliminate_word(xy, y) x4yx2y*-4x See Also ======== substituted_word """ if by is None: by = self.group.identity if self.is_independent(gen) or gen == by: return self if gen == self: return by if gen-1 == by: _all = False word = self l = len(gen) try: i = word.subword_index(gen) k = 1 except ValueError: if not inverse: return word try: i = word.subword_index(gen-1) k = -1 except ValueError: return word word = word.subword(0, i)bykword.subword(i+l, len(word)).eliminate_word(gen, by) if _all: return word.eliminate_word(gen, by, _all=True, inverse=inverse) else: return word def __len__(self): """ For an associative word `self`, returns the number of letters in it. Examples ======== >>> from sympy.combinatorics import free_group >>> f, a, b = free_group("a b") >>> w = a*5ba2b*-4a >>> len(w) 13 >>> len(a17) 17 >>> len(w0) 0 """ return sum(abs(j) for (i, j) in self) def __eq__(self, other): """ Two associative words are equal if they are words over the same alphabet and if they are sequences of the same letters. This is equivalent to saying that the external representations of the words are equal. There is no "universal" empty word, every alphabet has its own empty word. Examples ======== >>> from sympy.combinatorics import free_group >>> f, swapnil0, swapnil1 = free_group("swapnil0 swapnil1") >>> f <free group on the generators (swapnil0, swapnil1)> >>> g, swap0, swap1 = free_group("swap0 swap1") >>> g <free group on the generators (swap0, swap1)> >>> swapnil0 == swapnil1 False >>> swapnil0swapnil1 == swapnil1/swapnil1swapnil0swapnil1 True >>> swapnil0swapnil1 == swapnil1swapnil0 False >>> swapnil10 == swap00 False """ group = self.group if not isinstance(other, group.dtype): return False return tuple.__eq__(self, other) def __lt__(self, other): """ The ordering of associative words is defined by length and lexicography (this ordering is called short-lex ordering), that is, shorter words are smaller than longer words, and words of the same length are compared w.r.t. the lexicographical ordering induced by the ordering of generators. Generators are sorted according to the order in which they were created. If the generators are invertible then each generator `g` is larger than its inverse `g^{-1}`, and `g^{-1}` is larger than every generator that is smaller than `g`. Examples ======== >>> from sympy.combinatorics import free_group >>> f, a, b = free_group("a b") >>> b < a False >>> a < a.inverse() False """ group = self.group if not isinstance(other, group.dtype): raise TypeError("only FreeGroup elements of same FreeGroup can " "be compared") l = len(self) m = len(other) # implement lenlex order if l < m: return True elif l > m: return False for i in range(l): a = self[i].array_form[0] b = other[i].array_form[0] p = group.symbols.index(a[0]) q = group.symbols.index(b[0]) if p < q: return True elif p > q: return False elif a[1] < b[1]: return True elif a[1] > b[1]: return False return False def __le__(self, other): return (self == other or self < other) def __gt__(self, other): """ Examples ======== >>> from sympy.combinatorics import free_group >>> f, x, y, z = free_group("x y z") >>> y2 > x2 True >>> yz > zy False >>> x > x.inverse() True """ group = self.group if not isinstance(other, group.dtype): raise TypeError("only FreeGroup elements of same FreeGroup can " "be compared") return not self <= other def __ge__(self, other): return not self < other def exponent_sum(self, gen): """ For an associative word `self` and a generator or inverse of generator `gen`, ``exponent_sum`` returns the number of times `gen` appears in `self` minus the number of times its inverse appears in `self`. If neither `gen` nor its inverse occur in `self` then 0 is returned. Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> w = x2y3 >>> w.exponent_sum(x) 2 >>> w.exponent_sum(x-1) -2 >>> w = x*2y*4x*-3 >>> w.exponent_sum(x) -1 See Also ======== generator_count """ if len(gen) != 1: raise ValueError("gen must be a generator or inverse of a generator") s = gen.array_form[0] return s[1]sum([i[1] for i in self.array_form if i[0] == s[0]]) def generator_count(self, gen): """ For an associative word `self` and a generator `gen`, ``generator_count`` returns the multiplicity of generator `gen` in `self`. Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> w = x*2y3 >>> w.generator_count(x) 2 >>> w = x2y4x*-3 >>> w.generator_count(x) 5 See Also ======== exponent_sum """ if len(gen) != 1 or gen.array_form[0][1] < 0: raise ValueError("gen must be a generator") s = gen.array_form[0] return s[1]sum([abs(i[1]) for i in self.array_form if i[0] == s[0]]) def subword(self, from_i, to_j, strict=True): """ For an associative word `self` and two positive integers `from_i` and `to_j`, `subword` returns the subword of `self` that begins at position `from_i` and ends at `to_j - 1`, indexing is done with origin 0. Examples ======== >>> from sympy.combinatorics import free_group >>> f, a, b = free_group("a b") >>> w = a*5ba2b*-4a >>> w.subword(2, 6) a*3b """ group = self.group if not strict: from_i = max(from_i, 0) to_j = min(len(self), to_j) if from_i < 0 or to_j > len(self): raise ValueError("`from_i`, `to_j` must be positive and no greater than " "the length of associative word") if to_j <= from_i: return group.identity else: letter_form = self.letter_form[from_i: to_j] array_form = letter_form_to_array_form(letter_form, group) return group.dtype(array_form) def subword_index(self, word, start = 0): ''' Find the index of `word` in `self`. Examples ======== >>> from sympy.combinatorics import free_group >>> f, a, b = free_group("a b") >>> w = a*2bab*3 >>> w.subword_index(abab) 1 ''' l = len(word) self_lf = self.letter_form word_lf = word.letter_form index = None for i in range(start,len(self_lf)-l+1): if self_lf[i:i+l] == word_lf: index = i break if index is not None: return index else: raise ValueError("The given word is not a subword of self") def is_dependent(self, word): """ Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> (x*4y-3).is_dependent(x4y-2) True >>> (x2y*-1).is_dependent(xy) False >>> (xy2xy2).is_dependent(xy2) True >>> (x12).is_dependent(x-4) True See Also ======== is_independent """ try: return self.subword_index(word) is not None except ValueError: pass try: return self.subword_index(word-1) is not None except ValueError: return False def is_independent(self, word): """ See Also ======== is_dependent """ return not self.is_dependent(word) def contains_generators(self): """ Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y, z = free_group("x, y, z") >>> (x*2y-1).contains_generators() {x, y} >>> (x3z).contains_generators() {x, z} """ group = self.group gens = set() for syllable in self.array_form: gens.add(group.dtype(((syllable[0], 1),))) return set(gens) def cyclic_subword(self, from_i, to_j): group = self.group l = len(self) letter_form = self.letter_form period1 = int(from_i/l) if from_i >= l: from_i -= lperiod1 to_j -= lperiod1 diff = to_j - from_i word = letter_form[from_i: to_j] period2 = int(to_j/l) - 1 word += letter_formperiod2 + letter_form[:diff-l+from_i-lperiod2] word = letter_form_to_array_form(word, group) return group.dtype(word) def cyclic_conjugates(self): """Returns a words which are cyclic to the word `self`. Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> w = xyxyx >>> w.cyclic_conjugates() {xyx2y, x*2yxy, yxyx2, yx*2yx, xyxyx} >>> s = xyx2yx >>> s.cyclic_conjugates() {x2yx2y, yx2yx2, xyx2yx} References ========== .. [1] http://planetmath.org/cyclicpermutation """ return {self.cyclic_subword(i, i+len(self)) for i in range(len(self))} def is_cyclic_conjugate(self, w): """ Checks whether words ``self``, ``w`` are cyclic conjugates. Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> w1 = x2y*5 >>> w2 = xy*5x >>> w1.is_cyclic_conjugate(w2) True >>> w3 = x*-1y*5x-1 >>> w3.is_cyclic_conjugate(w2) False """ l1 = len(self) l2 = len(w) if l1 != l2: return False w1 = self.identity_cyclic_reduction() w2 = w.identity_cyclic_reduction() letter1 = w1.letter_form letter2 = w2.letter_form str1 = ' '.join(map(str, letter1)) str2 = ' '.join(map(str, letter2)) if len(str1) != len(str2): return False return str1 in str2 + ' ' + str2 def number_syllables(self): """Returns the number of syllables of the associative word `self`. Examples ======== >>> from sympy.combinatorics import free_group >>> f, swapnil0, swapnil1 = free_group("swapnil0 swapnil1") >>> (swapnil13swapnil0swapnil1-1).number_syllables() 3 """ return len(self.array_form) def exponent_syllable(self, i): """ Returns the exponent of the `i`-th syllable of the associative word `self`. Examples ======== >>> from sympy.combinatorics import free_group >>> f, a, b = free_group("a b") >>> w = a5ba*2b*-4a >>> w.exponent_syllable( 2 ) 2 """ return self.array_form[i][1] def generator_syllable(self, i): """ Returns the symbol of the generator that is involved in the i-th syllable of the associative word `self`. Examples ======== >>> from sympy.combinatorics import free_group >>> f, a, b = free_group("a b") >>> w = a*5ba2b*-4a >>> w.generator_syllable( 3 ) b """ return self.array_form[i][0] def sub_syllables(self, from_i, to_j): """ `sub_syllables` returns the subword of the associative word `self` that consists of syllables from positions `from_to` to `to_j`, where `from_to` and `to_j` must be positive integers and indexing is done with origin 0. Examples ======== >>> from sympy.combinatorics import free_group >>> f, a, b = free_group("a, b") >>> w = a*5ba2b*-4a >>> w.sub_syllables(1, 2) b >>> w.sub_syllables(3, 3) <identity> """ if not isinstance(from_i, int) or not isinstance(to_j, int): raise ValueError("both arguments should be integers") group = self.group if to_j <= from_i: return group.identity else: r = tuple(self.array_form[from_i: to_j]) return group.dtype(r) def substituted_word(self, from_i, to_j, by): """ Returns the associative word obtained by replacing the subword of `self` that begins at position `from_i` and ends at position `to_j - 1` by the associative word `by`. `from_i` and `to_j` must be positive integers, indexing is done with origin 0. In other words, `w.substituted_word(w, from_i, to_j, by)` is the product of the three words: `w.subword(0, from_i)`, `by`, and `w.subword(to_j len(w))`. See Also ======== eliminate_word """ lw = len(self) if from_i >= to_j or from_i > lw or to_j > lw: raise ValueError("values should be within bounds") # otherwise there are four possibilities # first if from=1 and to=lw then if from_i == 0 and to_j == lw: return by elif from_i == 0: # second if from_i=1 (and to_j < lw) then return byself.subword(to_j, lw) elif to_j == lw: # third if to_j=1 (and from_i > 1) then return self.subword(0, from_i)by else: # finally return self.subword(0, from_i)byself.subword(to_j, lw) def is_cyclically_reduced(self): r"""Returns whether the word is cyclically reduced or not. A word is cyclically reduced if by forming the cycle of the word, the word is not reduced, i.e a word w = `a_1 ... a_n` is called cyclically reduced if `a_1 \ne a_n^{-1}`. Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> (x*2y*-1x*-1).is_cyclically_reduced() False >>> (yx*2y2).is_cyclically_reduced() True """ if not self: return True return self[0] != self[-1]-1 def identity_cyclic_reduction(self): """Return a unique cyclically reduced version of the word. Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> (x*2y*2x*-1).identity_cyclic_reduction() xy2 >>> (x-3y-1x5).identity_cyclic_reduction() x2y-1 References ========== .. [1] http://planetmath.org/cyclicallyreduced """ word = self.copy() group = self.group while not word.is_cyclically_reduced(): exp1 = word.exponent_syllable(0) exp2 = word.exponent_syllable(-1) r = exp1 + exp2 if r == 0: rep = word.array_form[1: word.number_syllables() - 1] else: rep = ((word.generator_syllable(0), exp1 + exp2),) + \ word.array_form[1: word.number_syllables() - 1] word = group.dtype(rep) return word def cyclic_reduction(self, removed=False): """Return a cyclically reduced version of the word. Unlike `identity_cyclic_reduction`, this will not cyclically permute the reduced word - just remove the "unreduced" bits on either side of it. Compare the examples with those of `identity_cyclic_reduction`. When `removed` is `True`, return a tuple `(word, r)` where self `r` is such that before the reduction the word was either `rwordr-1`. Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> (x2y*2x*-1).cyclic_reduction() xy2 >>> (x-3y-1x5).cyclic_reduction() y-1x2 >>> (x-3y*-1x5).cyclic_reduction(removed=True) (y-1x2, x-3) """ word = self.copy() g = self.group.identity while not word.is_cyclically_reduced(): exp1 = abs(word.exponent_syllable(0)) exp2 = abs(word.exponent_syllable(-1)) exp = min(exp1, exp2) start = word[0]abs(exp) end = word[-1]abs(exp) word = start-1wordend-1 g = gstart if removed: return word, g return word def power_of(self, other): ''' Check if `self == other*n` for some integer n. Examples ======== >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> ((xy)*2).power_of(xy) True >>> (x*-3y*-2x3).power_of(x-3yx3) True ''' if self.is_identity: return True l = len(other) if l == 1: # self has to be a power of one generator gens = self.contains_generators() s = other in gens or other-1 in gens return len(gens) == 1 and s # if self is not cyclically reduced and it is a power of other, # other isn't cyclically reduced and the parts removed during # their reduction must be equal reduced, r1 = self.cyclic_reduction(removed=True) if not r1.is_identity: other, r2 = other.cyclic_reduction(removed=True) if r1 == r2: return reduced.power_of(other) return False if len(self) < l or len(self) % l: return False prefix = self.subword(0, l) if prefix == other or prefix**-1 == other: rest = self.subword(l, len(self)) return rest.power_of(other) return False def letter_form_to_array_form(array_form, group): """ This method converts a list given with possible repetitions of elements in it. It returns a new list such that repetitions of consecutive elements is removed and replace with a tuple element of size two such that the first index contains `value` and the second index contains the number of consecutive repetitions of `value`. """ a = list(array_form[:]) new_array = [] n = 1 symbols = group.symbols for i in range(len(a)): if i == len(a) - 1: if a[i] == a[i - 1]: if (-a[i]) in symbols: new_array.append((-a[i], -n)) else: new_array.append((a[i], n)) else: if (-a[i]) in symbols: new_array.append((-a[i], -1)) else: new_array.append((a[i], 1)) return new_array elif a[i] == a[i + 1]: n += 1 else: if (-a[i]) in symbols: new_array.append((-a[i], -n)) else: new_array.append((a[i], n)) n = 1 def zero_mul_simp(l, index): """Used to combine two reduced words.""" while index >=0 and index < len(l) - 1 and l[index][0] == l[index + 1][0]: exp = l[index][1] + l[index + 1][1] base = l[index][0] l[index] = (base, exp) del l[index + 1] if l[index][1] == 0: del l[index] index -= 1
4e591ab16e0df719b9ec4f70bcfac3498df803ce1d4de791fd602219fdd1efd4	from sympy.ntheory.primetest import isprime from sympy.combinatorics.perm_groups import PermutationGroup from sympy.printing.defaults import DefaultPrinting from sympy.combinatorics.free_groups import free_group class PolycyclicGroup(DefaultPrinting): is_group = True is_solvable = True def __init__(self, pc_sequence, pc_series, relative_order, collector=None): """ Parameters ========== pc_sequence : list A sequence of elements whose classes generate the cyclic factor groups of pc_series. pc_series : list A subnormal sequence of subgroups where each factor group is cyclic. relative_order : list The orders of factor groups of pc_series. collector : Collector By default, it is None. Collector class provides the polycyclic presentation with various other functionalities. """ self.pcgs = pc_sequence self.pc_series = pc_series self.relative_order = relative_order self.collector = Collector(self.pcgs, pc_series, relative_order) if not collector else collector def is_prime_order(self): return all(isprime(order) for order in self.relative_order) def length(self): return len(self.pcgs) class Collector(DefaultPrinting): """ References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" Section 8.1.3 """ def __init__(self, pcgs, pc_series, relative_order, free_group_=None, pc_presentation=None): """ Most of the parameters for the Collector class are the same as for PolycyclicGroup. Others are described below. Parameters ========== free_group_ : tuple free_group_ provides the mapping of polycyclic generating sequence with the free group elements. pc_presentation : dict Provides the presentation of polycyclic groups with the help of power and conjugate relators. See Also ======== PolycyclicGroup """ self.pcgs = pcgs self.pc_series = pc_series self.relative_order = relative_order self.free_group = free_group('x:{}'.format(len(pcgs)))[0] if not free_group_ else free_group_ self.index = {s: i for i, s in enumerate(self.free_group.symbols)} self.pc_presentation = self.pc_relators() def minimal_uncollected_subword(self, word): r""" Returns the minimal uncollected subwords. Explanation =========== A word ``v`` defined on generators in ``X`` is a minimal uncollected subword of the word ``w`` if ``v`` is a subword of ``w`` and it has one of the following form * `v = {x_{i+1}}^{a_j}x_i` * `v = {x_{i+1}}^{a_j}{x_i}^{-1}` * `v = {x_i}^{a_j}` for `a_j` not in `\{1, \ldots, s-1\}`. Where, ``s`` is the power exponent of the corresponding generator. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics import free_group >>> G = SymmetricGroup(4) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> F, x1, x2 = free_group("x1, x2") >>> word = x2*2x17 >>> collector.minimal_uncollected_subword(word) ((x2, 2),) """ # To handle the case word = <identity> if not word: return None array = word.array_form re = self.relative_order index = self.index for i in range(len(array)): s1, e1 = array[i] if re[index[s1]] and (e1 < 0 or e1 > re[index[s1]]-1): return ((s1, e1), ) for i in range(len(array)-1): s1, e1 = array[i] s2, e2 = array[i+1] if index[s1] > index[s2]: e = 1 if e2 > 0 else -1 return ((s1, e1), (s2, e)) return None def relations(self): """ Separates the given relators of pc presentation in power and conjugate relations. Returns ======= (power_rel, conj_rel) Separates pc presentation into power and conjugate relations. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> G = SymmetricGroup(3) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> power_rel, conj_rel = collector.relations() >>> power_rel {x02: (), x13: ()} >>> conj_rel {x0-1x1x0: x12} See Also ======== pc_relators """ power_relators = {} conjugate_relators = {} for key, value in self.pc_presentation.items(): if len(key.array_form) == 1: power_relators[key] = value else: conjugate_relators[key] = value return power_relators, conjugate_relators def subword_index(self, word, w): """ Returns the start and ending index of a given subword in a word. Parameters ========== word : FreeGroupElement word defined on free group elements for a polycyclic group. w : FreeGroupElement subword of a given word, whose starting and ending index to be computed. Returns ======= (i, j) A tuple containing starting and ending index of ``w`` in the given word. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics import free_group >>> G = SymmetricGroup(4) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> F, x1, x2 = free_group("x1, x2") >>> word = x22x17 >>> w = x22x1 >>> collector.subword_index(word, w) (0, 3) >>> w = x1*7 >>> collector.subword_index(word, w) (2, 9) """ low = -1 high = -1 for i in range(len(word)-len(w)+1): if word.subword(i, i+len(w)) == w: low = i high = i+len(w) break if low == high == -1: return -1, -1 return low, high def map_relation(self, w): """ Return a conjugate relation. Explanation =========== Given a word formed by two free group elements, the corresponding conjugate relation with those free group elements is formed and mapped with the collected word in the polycyclic presentation. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics import free_group >>> G = SymmetricGroup(3) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> F, x0, x1 = free_group("x0, x1") >>> w = x1x0 >>> collector.map_relation(w) x1*2 See Also ======== pc_presentation """ array = w.array_form s1 = array[0][0] s2 = array[1][0] key = ((s2, -1), (s1, 1), (s2, 1)) key = self.free_group.dtype(key) return self.pc_presentation[key] def collected_word(self, word): r""" Return the collected form of a word. Explanation =========== A word ``w`` is called collected, if `w = {x_{i_1}}^{a_1} \ldots * {x_{i_r}}^{a_r}` with `i_1 < i_2< \ldots < i_r` and `a_j` is in `\{1, \ldots, {s_j}-1\}`. Otherwise w is uncollected. Parameters ========== word : FreeGroupElement An uncollected word. Returns ======= word A collected word of form `w = {x_{i_1}}^{a_1}, \ldots, {x_{i_r}}^{a_r}` with `i_1, i_2, \ldots, i_r` and `a_j \in \{1, \ldots, {s_j}-1\}`. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics import free_group >>> G = SymmetricGroup(4) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> F, x0, x1, x2, x3 = free_group("x0, x1, x2, x3") >>> word = x3x2x1x0 >>> collected_word = collector.collected_word(word) >>> free_to_perm = {} >>> free_group = collector.free_group >>> for sym, gen in zip(free_group.symbols, collector.pcgs): ... free_to_perm[sym] = gen >>> G1 = PermutationGroup() >>> for w in word: ... sym = w[0] ... perm = free_to_perm[sym] ... G1 = PermutationGroup([perm] + G1.generators) >>> G2 = PermutationGroup() >>> for w in collected_word: ... sym = w[0] ... perm = free_to_perm[sym] ... G2 = PermutationGroup([perm] + G2.generators) The two are not identical, but they are equivalent: >>> G1.equals(G2), G1 == G2 (True, False) See Also ======== minimal_uncollected_subword """ free_group = self.free_group while True: w = self.minimal_uncollected_subword(word) if not w: break low, high = self.subword_index(word, free_group.dtype(w)) if low == -1: continue s1, e1 = w[0] if len(w) == 1: re = self.relative_order[self.index[s1]] q = e1 // re r = e1-qre key = ((w[0][0], re), ) key = free_group.dtype(key) if self.pc_presentation[key]: presentation = self.pc_presentation[key].array_form sym, exp = presentation[0] word_ = ((w[0][0], r), (sym, qexp)) word_ = free_group.dtype(word_) else: if r != 0: word_ = ((w[0][0], r), ) word_ = free_group.dtype(word_) else: word_ = None word = word.eliminate_word(free_group.dtype(w), word_) if len(w) == 2 and w[1][1] > 0: s2, e2 = w[1] s2 = ((s2, 1), ) s2 = free_group.dtype(s2) word_ = self.map_relation(free_group.dtype(w)) word_ = s2word_e1 word_ = free_group.dtype(word_) word = word.substituted_word(low, high, word_) elif len(w) == 2 and w[1][1] < 0: s2, e2 = w[1] s2 = ((s2, 1), ) s2 = free_group.dtype(s2) word_ = self.map_relation(free_group.dtype(w)) word_ = s2-1word_e1 word_ = free_group.dtype(word_) word = word.substituted_word(low, high, word_) return word def pc_relators(self): r""" Return the polycyclic presentation. Explanation =========== There are two types of relations used in polycyclic presentation. ``Power relations`` : Power relators are of the form `x_i^{re_i}`, where `i \in \{0, \ldots, \mathrm{len(pcgs)}\}`, ``x`` represents polycyclic generator and ``re`` is the corresponding relative order. * ``Conjugate relations`` : Conjugate relators are of the form `x_j^-1x_ix_j`, where `j < i \in \{0, \ldots, \mathrm{len(pcgs)}\}`. Returns ======= A dictionary with power and conjugate relations as key and their collected form as corresponding values. Notes ===== Identity Permutation is mapped with empty ``()``. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.permutations import Permutation >>> S = SymmetricGroup(49).sylow_subgroup(7) >>> der = S.derived_series() >>> G = der[len(der)-2] >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> pcgs = PcGroup.pcgs >>> len(pcgs) 6 >>> free_group = collector.free_group >>> pc_resentation = collector.pc_presentation >>> free_to_perm = {} >>> for s, g in zip(free_group.symbols, pcgs): ... free_to_perm[s] = g >>> for k, v in pc_resentation.items(): ... k_array = k.array_form ... if v != (): ... v_array = v.array_form ... lhs = Permutation() ... for gen in k_array: ... s = gen[0] ... e = gen[1] ... lhs = lhsfree_to_perm[s]e ... if v == (): ... assert lhs.is_identity ... continue ... rhs = Permutation() ... for gen in v_array: ... s = gen[0] ... e = gen[1] ... rhs = rhsfree_to_perm[s]e ... assert lhs == rhs """ free_group = self.free_group rel_order = self.relative_order pc_relators = {} perm_to_free = {} pcgs = self.pcgs for gen, s in zip(pcgs, free_group.generators): perm_to_free[gen-1] = s-1 perm_to_free[gen] = s pcgs = pcgs[::-1] series = self.pc_series[::-1] rel_order = rel_order[::-1] collected_gens = [] for i, gen in enumerate(pcgs): re = rel_order[i] relation = perm_to_free[gen]re G = series[i] l = G.generator_product(gen*re, original = True) l.reverse() word = free_group.identity for g in l: word = wordperm_to_free[g] word = self.collected_word(word) pc_relators[relation] = word if word else () self.pc_presentation = pc_relators collected_gens.append(gen) if len(collected_gens) > 1: conj = collected_gens[len(collected_gens)-1] conjugator = perm_to_free[conj] for j in range(len(collected_gens)-1): conjugated = perm_to_free[collected_gens[j]] relation = conjugator*-1conjugatedconjugator gens = conj-1collected_gens[j]conj l = G.generator_product(gens, original = True) l.reverse() word = free_group.identity for g in l: word = wordperm_to_free[g] word = self.collected_word(word) pc_relators[relation] = word if word else () self.pc_presentation = pc_relators return pc_relators def exponent_vector(self, element): r""" Return the exponent vector of length equal to the length of polycyclic generating sequence. Explanation =========== For a given generator/element ``g`` of the polycyclic group, it can be represented as `g = {x_1}^{e_1}, \ldots, {x_n}^{e_n}`, where `x_i` represents polycyclic generators and ``n`` is the number of generators in the free_group equal to the length of pcgs. Parameters ========== element : Permutation Generator of a polycyclic group. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.permutations import Permutation >>> G = SymmetricGroup(4) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> pcgs = PcGroup.pcgs >>> collector.exponent_vector(G[0]) [1, 0, 0, 0] >>> exp = collector.exponent_vector(G[1]) >>> g = Permutation() >>> for i in range(len(exp)): ... g = gpcgs[i]exp[i] if exp[i] else g >>> assert g == G[1] References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" Section 8.1.1, Definition 8.4 """ free_group = self.free_group G = PermutationGroup() for g in self.pcgs: G = PermutationGroup([g] + G.generators) gens = G.generator_product(element, original = True) gens.reverse() perm_to_free = {} for sym, g in zip(free_group.generators, self.pcgs): perm_to_free[g-1] = sym-1 perm_to_free[g] = sym w = free_group.identity for g in gens: w = wperm_to_free[g] word = self.collected_word(w) index = self.index exp_vector = [0]len(free_group) word = word.array_form for t in word: exp_vector[index[t[0]]] = t[1] return exp_vector def depth(self, element): r""" Return the depth of a given element. Explanation =========== The depth of a given element ``g`` is defined by `\mathrm{dep}[g] = i` if `e_1 = e_2 = \ldots = e_{i-1} = 0` and `e_i != 0`, where ``e`` represents the exponent-vector. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> G = SymmetricGroup(3) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> collector.depth(G[0]) 2 >>> collector.depth(G[1]) 1 References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" Section 8.1.1, Definition 8.5 """ exp_vector = self.exponent_vector(element) return next((i+1 for i, x in enumerate(exp_vector) if x), len(self.pcgs)+1) def leading_exponent(self, element): r""" Return the leading non-zero exponent. Explanation =========== The leading exponent for a given element `g` is defined by `\mathrm{leading\_exponent}[g]` `= e_i`, if `\mathrm{depth}[g] = i`. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> G = SymmetricGroup(3) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> collector.leading_exponent(G[1]) 1 """ exp_vector = self.exponent_vector(element) depth = self.depth(element) if depth != len(self.pcgs)+1: return exp_vector[depth-1] return None def _sift(self, z, g): h = g d = self.depth(h) while d < len(self.pcgs) and z[d-1] != 1: k = z[d-1] e = self.leading_exponent(h)(self.leading_exponent(k))-1 e = e % self.relative_order[d-1] h = k-eh d = self.depth(h) return h def induced_pcgs(self, gens): """ Parameters ========== gens : list A list of generators on which polycyclic subgroup is to be defined. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> S = SymmetricGroup(8) >>> G = S.sylow_subgroup(2) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> gens = [G[0], G[1]] >>> ipcgs = collector.induced_pcgs(gens) >>> [gen.order() for gen in ipcgs] [2, 2, 2] >>> G = S.sylow_subgroup(3) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> gens = [G[0], G[1]] >>> ipcgs = collector.induced_pcgs(gens) >>> [gen.order() for gen in ipcgs] [3] """ z = [1]len(self.pcgs) G = gens while G: g = G.pop(0) h = self._sift(z, g) d = self.depth(h) if d < len(self.pcgs): for gen in z: if gen != 1: G.append(h*-1gen*-1hgen) z[d-1] = h; z = [gen for gen in z if gen != 1] return z def constructive_membership_test(self, ipcgs, g): """ Return the exponent vector for induced pcgs. """ e = [0]len(ipcgs) h = g d = self.depth(h) for i, gen in enumerate(ipcgs): while self.depth(gen) == d: f = self.leading_exponent(h)self.leading_exponent(gen) f = f % self.relative_order[d-1] h = gen(-f)h e[i] = f d = self.depth(h) if h == 1: return e return False
18fcd5ebc0640eb5f4f513bae674ab95d6c88ec68fc1621fa1e23d162d6f4961	from sympy.combinatorics import Permutation as Perm from sympy.combinatorics.perm_groups import PermutationGroup from sympy.core import Basic, Tuple, default_sort_key from sympy.sets import FiniteSet from sympy.utilities.iterables import (minlex, unflatten, flatten) from sympy.utilities.misc import as_int rmul = Perm.rmul class Polyhedron(Basic): """ Represents the polyhedral symmetry group (PSG). Explanation =========== The PSG is one of the symmetry groups of the Platonic solids. There are three polyhedral groups: the tetrahedral group of order 12, the octahedral group of order 24, and the icosahedral group of order 60. All doctests have been given in the docstring of the constructor of the object. References ========== .. [1] http://mathworld.wolfram.com/PolyhedralGroup.html """ _edges = None def __new__(cls, corners, faces=(), pgroup=()): """ The constructor of the Polyhedron group object. Explanation =========== It takes up to three parameters: the corners, faces, and allowed transformations. The corners/vertices are entered as a list of arbitrary expressions that are used to identify each vertex. The faces are entered as a list of tuples of indices; a tuple of indices identifies the vertices which define the face. They should be entered in a cw or ccw order; they will be standardized by reversal and rotation to be give the lowest lexical ordering. If no faces are given then no edges will be computed. >>> from sympy.combinatorics.polyhedron import Polyhedron >>> Polyhedron(list('abc'), [(1, 2, 0)]).faces {(0, 1, 2)} >>> Polyhedron(list('abc'), [(1, 0, 2)]).faces {(0, 1, 2)} The allowed transformations are entered as allowable permutations of the vertices for the polyhedron. Instance of Permutations (as with faces) should refer to the supplied vertices by index. These permutation are stored as a PermutationGroup. Examples ======== >>> from sympy.combinatorics.permutations import Permutation >>> from sympy import init_printing >>> from sympy.abc import w, x, y, z >>> init_printing(pretty_print=False, perm_cyclic=False) Here we construct the Polyhedron object for a tetrahedron. >>> corners = [w, x, y, z] >>> faces = [(0, 1, 2), (0, 2, 3), (0, 3, 1), (1, 2, 3)] Next, allowed transformations of the polyhedron must be given. This is given as permutations of vertices. Although the vertices of a tetrahedron can be numbered in 24 (4!) different ways, there are only 12 different orientations for a physical tetrahedron. The following permutations, applied once or twice, will generate all 12 of the orientations. (The identity permutation, Permutation(range(4)), is not included since it does not change the orientation of the vertices.) >>> pgroup = [Permutation([[0, 1, 2], [3]]), \ Permutation([[0, 1, 3], [2]]), \ Permutation([[0, 2, 3], [1]]), \ Permutation([[1, 2, 3], [0]]), \ Permutation([[0, 1], [2, 3]]), \ Permutation([[0, 2], [1, 3]]), \ Permutation([[0, 3], [1, 2]])] The Polyhedron is now constructed and demonstrated: >>> tetra = Polyhedron(corners, faces, pgroup) >>> tetra.size 4 >>> tetra.edges {(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)} >>> tetra.corners (w, x, y, z) It can be rotated with an arbitrary permutation of vertices, e.g. the following permutation is not in the pgroup: >>> tetra.rotate(Permutation([0, 1, 3, 2])) >>> tetra.corners (w, x, z, y) An allowed permutation of the vertices can be constructed by repeatedly applying permutations from the pgroup to the vertices. Here is a demonstration that applying p and p2 for every p in pgroup generates all the orientations of a tetrahedron and no others: >>> all = ( (w, x, y, z), \ (x, y, w, z), \ (y, w, x, z), \ (w, z, x, y), \ (z, w, y, x), \ (w, y, z, x), \ (y, z, w, x), \ (x, z, y, w), \ (z, y, x, w), \ (y, x, z, w), \ (x, w, z, y), \ (z, x, w, y) ) >>> got = [] >>> for p in (pgroup + [p2 for p in pgroup]): ... h = Polyhedron(corners) ... h.rotate(p) ... got.append(h.corners) ... >>> set(got) == set(all) True The make_perm method of a PermutationGroup will randomly pick permutations, multiply them together, and return the permutation that can be applied to the polyhedron to give the orientation produced by those individual permutations. Here, 3 permutations are used: >>> tetra.pgroup.make_perm(3) # doctest: +SKIP Permutation([0, 3, 1, 2]) To select the permutations that should be used, supply a list of indices to the permutations in pgroup in the order they should be applied: >>> use = [0, 0, 2] >>> p002 = tetra.pgroup.make_perm(3, use) >>> p002 Permutation([1, 0, 3, 2]) Apply them one at a time: >>> tetra.reset() >>> for i in use: ... tetra.rotate(pgroup[i]) ... >>> tetra.vertices (x, w, z, y) >>> sequentially = tetra.vertices Apply the composite permutation: >>> tetra.reset() >>> tetra.rotate(p002) >>> tetra.corners (x, w, z, y) >>> tetra.corners in all and tetra.corners == sequentially True Notes ===== Defining permutation groups --------------------------- It is not necessary to enter any permutations, nor is necessary to enter a complete set of transformations. In fact, for a polyhedron, all configurations can be constructed from just two permutations. For example, the orientations of a tetrahedron can be generated from an axis passing through a vertex and face and another axis passing through a different vertex or from an axis passing through the midpoints of two edges opposite of each other. For simplicity of presentation, consider a square -- not a cube -- with vertices 1, 2, 3, and 4: 1-----2 We could think of axes of rotation being: \| \| 1) through the face \| \| 2) from midpoint 1-2 to 3-4 or 1-3 to 2-4 3-----4 3) lines 1-4 or 2-3 To determine how to write the permutations, imagine 4 cameras, one at each corner, labeled A-D: A B A B 1-----2 1-----3 vertex index: \| \| \| \| 1 0 \| \| \| \| 2 1 3-----4 2-----4 3 2 C D C D 4 3 original after rotation along 1-4 A diagonal and a face axis will be chosen for the "permutation group" from which any orientation can be constructed. >>> pgroup = [] Imagine a clockwise rotation when viewing 1-4 from camera A. The new orientation is (in camera-order): 1, 3, 2, 4 so the permutation is given using the indices of the vertices as: >>> pgroup.append(Permutation((0, 2, 1, 3))) Now imagine rotating clockwise when looking down an axis entering the center of the square as viewed. The new camera-order would be 3, 1, 4, 2 so the permutation is (using indices): >>> pgroup.append(Permutation((2, 0, 3, 1))) The square can now be constructed: use real-world labels for the vertices, entering them in camera order for the faces we use zero-based indices of the vertices in edge-order as the face is traversed; neither the direction nor the starting point matter -- the faces are only used to define edges (if so desired). >>> square = Polyhedron((1, 2, 3, 4), [(0, 1, 3, 2)], pgroup) To rotate the square with a single permutation we can do: >>> square.rotate(square.pgroup[0]) >>> square.corners (1, 3, 2, 4) To use more than one permutation (or to use one permutation more than once) it is more convenient to use the make_perm method: >>> p011 = square.pgroup.make_perm([0, 1, 1]) # diag flip + 2 rotations >>> square.reset() # return to initial orientation >>> square.rotate(p011) >>> square.corners (4, 2, 3, 1) Thinking outside the box ------------------------ Although the Polyhedron object has a direct physical meaning, it actually has broader application. In the most general sense it is just a decorated PermutationGroup, allowing one to connect the permutations to something physical. For example, a Rubik's cube is not a proper polyhedron, but the Polyhedron class can be used to represent it in a way that helps to visualize the Rubik's cube. >>> from sympy import flatten, unflatten, symbols >>> from sympy.combinatorics import RubikGroup >>> facelets = flatten([symbols(s+'1:5') for s in 'UFRBLD']) >>> def show(): ... pairs = unflatten(r2.corners, 2) ... print(pairs[::2]) ... print(pairs[1::2]) ... >>> r2 = Polyhedron(facelets, pgroup=RubikGroup(2)) >>> show() [(U1, U2), (F1, F2), (R1, R2), (B1, B2), (L1, L2), (D1, D2)] [(U3, U4), (F3, F4), (R3, R4), (B3, B4), (L3, L4), (D3, D4)] >>> r2.rotate(0) # cw rotation of F >>> show() [(U1, U2), (F3, F1), (U3, R2), (B1, B2), (L1, D1), (R3, R1)] [(L4, L2), (F4, F2), (U4, R4), (B3, B4), (L3, D2), (D3, D4)] Predefined Polyhedra ==================== For convenience, the vertices and faces are defined for the following standard solids along with a permutation group for transformations. When the polyhedron is oriented as indicated below, the vertices in a given horizontal plane are numbered in ccw direction, starting from the vertex that will give the lowest indices in a given face. (In the net of the vertices, indices preceded by "-" indicate replication of the lhs index in the net.) tetrahedron, tetrahedron_faces ------------------------------ 4 vertices (vertex up) net: 0 0-0 1 2 3-1 4 faces: (0, 1, 2) (0, 2, 3) (0, 3, 1) (1, 2, 3) cube, cube_faces ---------------- 8 vertices (face up) net: 0 1 2 3-0 4 5 6 7-4 6 faces: (0, 1, 2, 3) (0, 1, 5, 4) (1, 2, 6, 5) (2, 3, 7, 6) (0, 3, 7, 4) (4, 5, 6, 7) octahedron, octahedron_faces ---------------------------- 6 vertices (vertex up) net: 0 0 0-0 1 2 3 4-1 5 5 5-5 8 faces: (0, 1, 2) (0, 2, 3) (0, 3, 4) (0, 1, 4) (1, 2, 5) (2, 3, 5) (3, 4, 5) (1, 4, 5) dodecahedron, dodecahedron_faces -------------------------------- 20 vertices (vertex up) net: 0 1 2 3 4 -0 5 6 7 8 9 -5 14 10 11 12 13-14 15 16 17 18 19-15 12 faces: (0, 1, 2, 3, 4) (0, 1, 6, 10, 5) (1, 2, 7, 11, 6) (2, 3, 8, 12, 7) (3, 4, 9, 13, 8) (0, 4, 9, 14, 5) (5, 10, 16, 15, 14) (6, 10, 16, 17, 11) (7, 11, 17, 18, 12) (8, 12, 18, 19, 13) (9, 13, 19, 15, 14)(15, 16, 17, 18, 19) icosahedron, icosahedron_faces ------------------------------ 12 vertices (face up) net: 0 0 0 0 -0 1 2 3 4 5 -1 6 7 8 9 10 -6 11 11 11 11 -11 20 faces: (0, 1, 2) (0, 2, 3) (0, 3, 4) (0, 4, 5) (0, 1, 5) (1, 2, 6) (2, 3, 7) (3, 4, 8) (4, 5, 9) (1, 5, 10) (2, 6, 7) (3, 7, 8) (4, 8, 9) (5, 9, 10) (1, 6, 10) (6, 7, 11) (7, 8, 11) (8, 9, 11) (9, 10, 11) (6, 10, 11) >>> from sympy.combinatorics.polyhedron import cube >>> cube.edges {(0, 1), (0, 3), (0, 4), (1, 2), (1, 5), (2, 3), (2, 6), (3, 7), (4, 5), (4, 7), (5, 6), (6, 7)} If you want to use letters or other names for the corners you can still use the pre-calculated faces: >>> corners = list('abcdefgh') >>> Polyhedron(corners, cube.faces).corners (a, b, c, d, e, f, g, h) References ========== .. [1] www.ocf.berkeley.edu/~wwu/articles/platonicsolids.pdf """ faces = [minlex(f, directed=False, key=default_sort_key) for f in faces] corners, faces, pgroup = args = \ [Tuple(a) for a in (corners, faces, pgroup)] obj = Basic.__new__(cls, args) obj._corners = tuple(corners) # in order given obj._faces = FiniteSet(faces) if pgroup and pgroup[0].size != len(corners): raise ValueError("Permutation size unequal to number of corners.") # use the identity permutation if none are given obj._pgroup = PermutationGroup( pgroup or [Perm(range(len(corners)))] ) return obj @property def corners(self): """ Get the corners of the Polyhedron. The method ``vertices`` is an alias for ``corners``. Examples ======== >>> from sympy.combinatorics import Polyhedron >>> from sympy.abc import a, b, c, d >>> p = Polyhedron(list('abcd')) >>> p.corners == p.vertices == (a, b, c, d) True See Also ======== array_form, cyclic_form """ return self._corners vertices = corners @property def array_form(self): """Return the indices of the corners. The indices are given relative to the original position of corners. Examples ======== >>> from sympy.combinatorics.polyhedron import tetrahedron >>> tetrahedron = tetrahedron.copy() >>> tetrahedron.array_form [0, 1, 2, 3] >>> tetrahedron.rotate(0) >>> tetrahedron.array_form [0, 2, 3, 1] >>> tetrahedron.pgroup[0].array_form [0, 2, 3, 1] See Also ======== corners, cyclic_form """ corners = list(self.args[0]) return [corners.index(c) for c in self.corners] @property def cyclic_form(self): """Return the indices of the corners in cyclic notation. The indices are given relative to the original position of corners. See Also ======== corners, array_form """ return Perm._af_new(self.array_form).cyclic_form @property def size(self): """ Get the number of corners of the Polyhedron. """ return len(self._corners) @property def faces(self): """ Get the faces of the Polyhedron. """ return self._faces @property def pgroup(self): """ Get the permutations of the Polyhedron. """ return self._pgroup @property def edges(self): """ Given the faces of the polyhedra we can get the edges. Examples ======== >>> from sympy.combinatorics import Polyhedron >>> from sympy.abc import a, b, c >>> corners = (a, b, c) >>> faces = [(0, 1, 2)] >>> Polyhedron(corners, faces).edges {(0, 1), (0, 2), (1, 2)} """ if self._edges is None: output = set() for face in self.faces: for i in range(len(face)): edge = tuple(sorted([face[i], face[i - 1]])) output.add(edge) self._edges = FiniteSet(output) return self._edges def rotate(self, perm): """ Apply a permutation to the polyhedron in place. The permutation may be given as a Permutation instance or an integer indicating which permutation from pgroup of the Polyhedron should be applied. This is an operation that is analogous to rotation about an axis by a fixed increment. Notes ===== When a Permutation is applied, no check is done to see if that is a valid permutation for the Polyhedron. For example, a cube could be given a permutation which effectively swaps only 2 vertices. A valid permutation (that rotates the object in a physical way) will be obtained if one only uses permutations from the ``pgroup`` of the Polyhedron. On the other hand, allowing arbitrary rotations (applications of permutations) gives a way to follow named elements rather than indices since Polyhedron allows vertices to be named while Permutation works only with indices. Examples ======== >>> from sympy.combinatorics import Polyhedron, Permutation >>> from sympy.combinatorics.polyhedron import cube >>> cube = cube.copy() >>> cube.corners (0, 1, 2, 3, 4, 5, 6, 7) >>> cube.rotate(0) >>> cube.corners (1, 2, 3, 0, 5, 6, 7, 4) A non-physical "rotation" that is not prohibited by this method: >>> cube.reset() >>> cube.rotate(Permutation([[1, 2]], size=8)) >>> cube.corners (0, 2, 1, 3, 4, 5, 6, 7) Polyhedron can be used to follow elements of set that are identified by letters instead of integers: >>> shadow = h5 = Polyhedron(list('abcde')) >>> p = Permutation([3, 0, 1, 2, 4]) >>> h5.rotate(p) >>> h5.corners (d, a, b, c, e) >>> _ == shadow.corners True >>> copy = h5.copy() >>> h5.rotate(p) >>> h5.corners == copy.corners False """ if not isinstance(perm, Perm): perm = self.pgroup[perm] # and we know it's valid else: if perm.size != self.size: raise ValueError('Polyhedron and Permutation sizes differ.') a = perm.array_form corners = [self.corners[a[i]] for i in range(len(self.corners))] self._corners = tuple(corners) def reset(self): """Return corners to their original positions. Examples ======== >>> from sympy.combinatorics.polyhedron import tetrahedron as T >>> T = T.copy() >>> T.corners (0, 1, 2, 3) >>> T.rotate(0) >>> T.corners (0, 2, 3, 1) >>> T.reset() >>> T.corners (0, 1, 2, 3) """ self._corners = self.args[0] def _pgroup_calcs(): """Return the permutation groups for each of the polyhedra and the face definitions: tetrahedron, cube, octahedron, dodecahedron, icosahedron, tetrahedron_faces, cube_faces, octahedron_faces, dodecahedron_faces, icosahedron_faces Explanation =========== (This author did not find and did not know of a better way to do it though there likely is such a way.) Although only 2 permutations are needed for a polyhedron in order to generate all the possible orientations, a group of permutations is provided instead. A set of permutations is called a "group" if:: ab = c (for any pair of permutations in the group, a and b, their product, c, is in the group) a(bc) = (ab)c (for any 3 permutations in the group associativity holds) there is an identity permutation, I, such that Ia = aI for all elements in the group ab = I (the inverse of each permutation is also in the group) None of the polyhedron groups defined follow these definitions of a group. Instead, they are selected to contain those permutations whose powers alone will construct all orientations of the polyhedron, i.e. for permutations ``a``, ``b``, etc... in the group, ``a, a2, ..., ao_a``, ``b, b2, ..., bo_b``, etc... (where ``o_i`` is the order of permutation ``i``) generate all permutations of the polyhedron instead of mixed products like ``ab``, ``ab2``, etc.... Note that for a polyhedron with n vertices, the valid permutations of the vertices exclude those that do not maintain its faces. e.g. the permutation BCDE of a square's four corners, ABCD, is a valid permutation while CBDE is not (because this would twist the square). Examples ======== The is_group checks for: closure, the presence of the Identity permutation, and the presence of the inverse for each of the elements in the group. This confirms that none of the polyhedra are true groups: >>> from sympy.combinatorics.polyhedron import ( ... tetrahedron, cube, octahedron, dodecahedron, icosahedron) ... >>> polyhedra = (tetrahedron, cube, octahedron, dodecahedron, icosahedron) >>> [h.pgroup.is_group for h in polyhedra] ... [True, True, True, True, True] Although tests in polyhedron's test suite check that powers of the permutations in the groups generate all permutations of the vertices of the polyhedron, here we also demonstrate the powers of the given permutations create a complete group for the tetrahedron: >>> from sympy.combinatorics import Permutation, PermutationGroup >>> for h in polyhedra[:1]: ... G = h.pgroup ... perms = set() ... for g in G: ... for e in range(g.order()): ... p = tuple((ge).array_form) ... perms.add(p) ... ... perms = [Permutation(p) for p in perms] ... assert PermutationGroup(perms).is_group In addition to doing the above, the tests in the suite confirm that the faces are all present after the application of each permutation. References ========== .. [1] http://dogschool.tripod.com/trianglegroup.html """ def _pgroup_of_double(polyh, ordered_faces, pgroup): n = len(ordered_faces[0]) # the vertices of the double which sits inside a give polyhedron # can be found by tracking the faces of the outer polyhedron. # A map between face and the vertex of the double is made so that # after rotation the position of the vertices can be located fmap = dict(zip(ordered_faces, range(len(ordered_faces)))) flat_faces = flatten(ordered_faces) new_pgroup = [] for i, p in enumerate(pgroup): h = polyh.copy() h.rotate(p) c = h.corners # reorder corners in the order they should appear when # enumerating the faces reorder = unflatten([c[j] for j in flat_faces], n) # make them canonical reorder = [tuple(map(as_int, minlex(f, directed=False))) for f in reorder] # map face to vertex: the resulting list of vertices are the # permutation that we seek for the double new_pgroup.append(Perm([fmap[f] for f in reorder])) return new_pgroup tetrahedron_faces = [ (0, 1, 2), (0, 2, 3), (0, 3, 1), # upper 3 (1, 2, 3), # bottom ] # cw from top # _t_pgroup = [ Perm([[1, 2, 3], [0]]), # cw from top Perm([[0, 1, 2], [3]]), # cw from front face Perm([[0, 3, 2], [1]]), # cw from back right face Perm([[0, 3, 1], [2]]), # cw from back left face Perm([[0, 1], [2, 3]]), # through front left edge Perm([[0, 2], [1, 3]]), # through front right edge Perm([[0, 3], [1, 2]]), # through back edge ] tetrahedron = Polyhedron( range(4), tetrahedron_faces, _t_pgroup) cube_faces = [ (0, 1, 2, 3), # upper (0, 1, 5, 4), (1, 2, 6, 5), (2, 3, 7, 6), (0, 3, 7, 4), # middle 4 (4, 5, 6, 7), # lower ] # U, D, F, B, L, R = up, down, front, back, left, right _c_pgroup = [Perm(p) for p in [ [1, 2, 3, 0, 5, 6, 7, 4], # cw from top, U [4, 0, 3, 7, 5, 1, 2, 6], # cw from F face [4, 5, 1, 0, 7, 6, 2, 3], # cw from R face [1, 0, 4, 5, 2, 3, 7, 6], # cw through UF edge [6, 2, 1, 5, 7, 3, 0, 4], # cw through UR edge [6, 7, 3, 2, 5, 4, 0, 1], # cw through UB edge [3, 7, 4, 0, 2, 6, 5, 1], # cw through UL edge [4, 7, 6, 5, 0, 3, 2, 1], # cw through FL edge [6, 5, 4, 7, 2, 1, 0, 3], # cw through FR edge [0, 3, 7, 4, 1, 2, 6, 5], # cw through UFL vertex [5, 1, 0, 4, 6, 2, 3, 7], # cw through UFR vertex [5, 6, 2, 1, 4, 7, 3, 0], # cw through UBR vertex [7, 4, 0, 3, 6, 5, 1, 2], # cw through UBL ]] cube = Polyhedron( range(8), cube_faces, _c_pgroup) octahedron_faces = [ (0, 1, 2), (0, 2, 3), (0, 3, 4), (0, 1, 4), # top 4 (1, 2, 5), (2, 3, 5), (3, 4, 5), (1, 4, 5), # bottom 4 ] octahedron = Polyhedron( range(6), octahedron_faces, _pgroup_of_double(cube, cube_faces, _c_pgroup)) dodecahedron_faces = [ (0, 1, 2, 3, 4), # top (0, 1, 6, 10, 5), (1, 2, 7, 11, 6), (2, 3, 8, 12, 7), # upper 5 (3, 4, 9, 13, 8), (0, 4, 9, 14, 5), (5, 10, 16, 15, 14), (6, 10, 16, 17, 11), (7, 11, 17, 18, 12), # lower 5 (8, 12, 18, 19, 13), (9, 13, 19, 15, 14), (15, 16, 17, 18, 19) # bottom ] def _string_to_perm(s): rv = [Perm(range(20))] p = None for si in s: if si not in '01': count = int(si) - 1 else: count = 1 if si == '0': p = _f0 elif si == '1': p = _f1 rv.extend([p]count) return Perm.rmul(rv) # top face cw _f0 = Perm([ 1, 2, 3, 4, 0, 6, 7, 8, 9, 5, 11, 12, 13, 14, 10, 16, 17, 18, 19, 15]) # front face cw _f1 = Perm([ 5, 0, 4, 9, 14, 10, 1, 3, 13, 15, 6, 2, 8, 19, 16, 17, 11, 7, 12, 18]) # the strings below, like 0104 are shorthand for F0F1F0**4 and are # the remaining 4 face rotations, 15 edge permutations, and the # 10 vertex rotations. _dodeca_pgroup = [_f0, _f1] + [_string_to_perm(s) for s in ''' 0104 140 014 0410 010 1403 03104 04103 102 120 1304 01303 021302 03130 0412041 041204103 04120410 041204104 041204102 10 01 1402 0140 04102 0412 1204 1302 0130 03120'''.strip().split()] dodecahedron = Polyhedron( range(20), dodecahedron_faces, _dodeca_pgroup) icosahedron_faces = [ (0, 1, 2), (0, 2, 3), (0, 3, 4), (0, 4, 5), (0, 1, 5), (1, 6, 7), (1, 2, 7), (2, 7, 8), (2, 3, 8), (3, 8, 9), (3, 4, 9), (4, 9, 10), (4, 5, 10), (5, 6, 10), (1, 5, 6), (6, 7, 11), (7, 8, 11), (8, 9, 11), (9, 10, 11), (6, 10, 11)] icosahedron = Polyhedron( range(12), icosahedron_faces, _pgroup_of_double( dodecahedron, dodecahedron_faces, _dodeca_pgroup)) return (tetrahedron, cube, octahedron, dodecahedron, icosahedron, tetrahedron_faces, cube_faces, octahedron_faces, dodecahedron_faces, icosahedron_faces) # ----------------------------------------------------------------------- # Standard Polyhedron groups # # These are generated using _pgroup_calcs() above. However to save # import time we encode them explicitly here. # ----------------------------------------------------------------------- tetrahedron = Polyhedron( Tuple(0, 1, 2, 3), Tuple( Tuple(0, 1, 2), Tuple(0, 2, 3), Tuple(0, 1, 3), Tuple(1, 2, 3)), Tuple( Perm(1, 2, 3), Perm(3)(0, 1, 2), Perm(0, 3, 2), Perm(0, 3, 1), Perm(0, 1)(2, 3), Perm(0, 2)(1, 3), Perm(0, 3)(1, 2) )) cube = Polyhedron( Tuple(0, 1, 2, 3, 4, 5, 6, 7), Tuple( Tuple(0, 1, 2, 3), Tuple(0, 1, 5, 4), Tuple(1, 2, 6, 5), Tuple(2, 3, 7, 6), Tuple(0, 3, 7, 4), Tuple(4, 5, 6, 7)), Tuple( Perm(0, 1, 2, 3)(4, 5, 6, 7), Perm(0, 4, 5, 1)(2, 3, 7, 6), Perm(0, 4, 7, 3)(1, 5, 6, 2), Perm(0, 1)(2, 4)(3, 5)(6, 7), Perm(0, 6)(1, 2)(3, 5)(4, 7), Perm(0, 6)(1, 7)(2, 3)(4, 5), Perm(0, 3)(1, 7)(2, 4)(5, 6), Perm(0, 4)(1, 7)(2, 6)(3, 5), Perm(0, 6)(1, 5)(2, 4)(3, 7), Perm(1, 3, 4)(2, 7, 5), Perm(7)(0, 5, 2)(3, 4, 6), Perm(0, 5, 7)(1, 6, 3), Perm(0, 7, 2)(1, 4, 6))) octahedron = Polyhedron( Tuple(0, 1, 2, 3, 4, 5), Tuple( Tuple(0, 1, 2), Tuple(0, 2, 3), Tuple(0, 3, 4), Tuple(0, 1, 4), Tuple(1, 2, 5), Tuple(2, 3, 5), Tuple(3, 4, 5), Tuple(1, 4, 5)), Tuple( Perm(5)(1, 2, 3, 4), Perm(0, 4, 5, 2), Perm(0, 1, 5, 3), Perm(0, 1)(2, 4)(3, 5), Perm(0, 2)(1, 3)(4, 5), Perm(0, 3)(1, 5)(2, 4), Perm(0, 4)(1, 3)(2, 5), Perm(0, 5)(1, 4)(2, 3), Perm(0, 5)(1, 2)(3, 4), Perm(0, 4, 1)(2, 3, 5), Perm(0, 1, 2)(3, 4, 5), Perm(0, 2, 3)(1, 5, 4), Perm(0, 4, 3)(1, 5, 2))) dodecahedron = Polyhedron( Tuple(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19), Tuple( Tuple(0, 1, 2, 3, 4), Tuple(0, 1, 6, 10, 5), Tuple(1, 2, 7, 11, 6), Tuple(2, 3, 8, 12, 7), Tuple(3, 4, 9, 13, 8), Tuple(0, 4, 9, 14, 5), Tuple(5, 10, 16, 15, 14), Tuple(6, 10, 16, 17, 11), Tuple(7, 11, 17, 18, 12), Tuple(8, 12, 18, 19, 13), Tuple(9, 13, 19, 15, 14), Tuple(15, 16, 17, 18, 19)), Tuple( Perm(0, 1, 2, 3, 4)(5, 6, 7, 8, 9)(10, 11, 12, 13, 14)(15, 16, 17, 18, 19), Perm(0, 5, 10, 6, 1)(2, 4, 14, 16, 11)(3, 9, 15, 17, 7)(8, 13, 19, 18, 12), Perm(0, 10, 17, 12, 3)(1, 6, 11, 7, 2)(4, 5, 16, 18, 8)(9, 14, 15, 19, 13), Perm(0, 6, 17, 19, 9)(1, 11, 18, 13, 4)(2, 7, 12, 8, 3)(5, 10, 16, 15, 14), Perm(0, 2, 12, 19, 14)(1, 7, 18, 15, 5)(3, 8, 13, 9, 4)(6, 11, 17, 16, 10), Perm(0, 4, 9, 14, 5)(1, 3, 13, 15, 10)(2, 8, 19, 16, 6)(7, 12, 18, 17, 11), Perm(0, 1)(2, 5)(3, 10)(4, 6)(7, 14)(8, 16)(9, 11)(12, 15)(13, 17)(18, 19), Perm(0, 7)(1, 2)(3, 6)(4, 11)(5, 12)(8, 10)(9, 17)(13, 16)(14, 18)(15, 19), Perm(0, 12)(1, 8)(2, 3)(4, 7)(5, 18)(6, 13)(9, 11)(10, 19)(14, 17)(15, 16), Perm(0, 8)(1, 13)(2, 9)(3, 4)(5, 12)(6, 19)(7, 14)(10, 18)(11, 15)(16, 17), Perm(0, 4)(1, 9)(2, 14)(3, 5)(6, 13)(7, 15)(8, 10)(11, 19)(12, 16)(17, 18), Perm(0, 5)(1, 14)(2, 15)(3, 16)(4, 10)(6, 9)(7, 19)(8, 17)(11, 13)(12, 18), Perm(0, 11)(1, 6)(2, 10)(3, 16)(4, 17)(5, 7)(8, 15)(9, 18)(12, 14)(13, 19), Perm(0, 18)(1, 12)(2, 7)(3, 11)(4, 17)(5, 19)(6, 8)(9, 16)(10, 13)(14, 15), Perm(0, 18)(1, 19)(2, 13)(3, 8)(4, 12)(5, 17)(6, 15)(7, 9)(10, 16)(11, 14), Perm(0, 13)(1, 19)(2, 15)(3, 14)(4, 9)(5, 8)(6, 18)(7, 16)(10, 12)(11, 17), Perm(0, 16)(1, 15)(2, 19)(3, 18)(4, 17)(5, 10)(6, 14)(7, 13)(8, 12)(9, 11), Perm(0, 18)(1, 17)(2, 16)(3, 15)(4, 19)(5, 12)(6, 11)(7, 10)(8, 14)(9, 13), Perm(0, 15)(1, 19)(2, 18)(3, 17)(4, 16)(5, 14)(6, 13)(7, 12)(8, 11)(9, 10), Perm(0, 17)(1, 16)(2, 15)(3, 19)(4, 18)(5, 11)(6, 10)(7, 14)(8, 13)(9, 12), Perm(0, 19)(1, 18)(2, 17)(3, 16)(4, 15)(5, 13)(6, 12)(7, 11)(8, 10)(9, 14), Perm(1, 4, 5)(2, 9, 10)(3, 14, 6)(7, 13, 16)(8, 15, 11)(12, 19, 17), Perm(19)(0, 6, 2)(3, 5, 11)(4, 10, 7)(8, 14, 17)(9, 16, 12)(13, 15, 18), Perm(0, 11, 8)(1, 7, 3)(4, 6, 12)(5, 17, 13)(9, 10, 18)(14, 16, 19), Perm(0, 7, 13)(1, 12, 9)(2, 8, 4)(5, 11, 19)(6, 18, 14)(10, 17, 15), Perm(0, 3, 9)(1, 8, 14)(2, 13, 5)(6, 12, 15)(7, 19, 10)(11, 18, 16), Perm(0, 14, 10)(1, 9, 16)(2, 13, 17)(3, 19, 11)(4, 15, 6)(7, 8, 18), Perm(0, 16, 7)(1, 10, 11)(2, 5, 17)(3, 14, 18)(4, 15, 12)(8, 9, 19), Perm(0, 16, 13)(1, 17, 8)(2, 11, 12)(3, 6, 18)(4, 10, 19)(5, 15, 9), Perm(0, 11, 15)(1, 17, 14)(2, 18, 9)(3, 12, 13)(4, 7, 19)(5, 6, 16), Perm(0, 8, 15)(1, 12, 16)(2, 18, 10)(3, 19, 5)(4, 13, 14)(6, 7, 17))) icosahedron = Polyhedron( Tuple(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11), Tuple( Tuple(0, 1, 2), Tuple(0, 2, 3), Tuple(0, 3, 4), Tuple(0, 4, 5), Tuple(0, 1, 5), Tuple(1, 6, 7), Tuple(1, 2, 7), Tuple(2, 7, 8), Tuple(2, 3, 8), Tuple(3, 8, 9), Tuple(3, 4, 9), Tuple(4, 9, 10), Tuple(4, 5, 10), Tuple(5, 6, 10), Tuple(1, 5, 6), Tuple(6, 7, 11), Tuple(7, 8, 11), Tuple(8, 9, 11), Tuple(9, 10, 11), Tuple(6, 10, 11)), Tuple( Perm(11)(1, 2, 3, 4, 5)(6, 7, 8, 9, 10), Perm(0, 5, 6, 7, 2)(3, 4, 10, 11, 8), Perm(0, 1, 7, 8, 3)(4, 5, 6, 11, 9), Perm(0, 2, 8, 9, 4)(1, 7, 11, 10, 5), Perm(0, 3, 9, 10, 5)(1, 2, 8, 11, 6), Perm(0, 4, 10, 6, 1)(2, 3, 9, 11, 7), Perm(0, 1)(2, 5)(3, 6)(4, 7)(8, 10)(9, 11), Perm(0, 2)(1, 3)(4, 7)(5, 8)(6, 9)(10, 11), Perm(0, 3)(1, 9)(2, 4)(5, 8)(6, 11)(7, 10), Perm(0, 4)(1, 9)(2, 10)(3, 5)(6, 8)(7, 11), Perm(0, 5)(1, 4)(2, 10)(3, 6)(7, 9)(8, 11), Perm(0, 6)(1, 5)(2, 10)(3, 11)(4, 7)(8, 9), Perm(0, 7)(1, 2)(3, 6)(4, 11)(5, 8)(9, 10), Perm(0, 8)(1, 9)(2, 3)(4, 7)(5, 11)(6, 10), Perm(0, 9)(1, 11)(2, 10)(3, 4)(5, 8)(6, 7), Perm(0, 10)(1, 9)(2, 11)(3, 6)(4, 5)(7, 8), Perm(0, 11)(1, 6)(2, 10)(3, 9)(4, 8)(5, 7), Perm(0, 11)(1, 8)(2, 7)(3, 6)(4, 10)(5, 9), Perm(0, 11)(1, 10)(2, 9)(3, 8)(4, 7)(5, 6), Perm(0, 11)(1, 7)(2, 6)(3, 10)(4, 9)(5, 8), Perm(0, 11)(1, 9)(2, 8)(3, 7)(4, 6)(5, 10), Perm(0, 5, 1)(2, 4, 6)(3, 10, 7)(8, 9, 11), Perm(0, 1, 2)(3, 5, 7)(4, 6, 8)(9, 10, 11), Perm(0, 2, 3)(1, 8, 4)(5, 7, 9)(6, 11, 10), Perm(0, 3, 4)(1, 8, 10)(2, 9, 5)(6, 7, 11), Perm(0, 4, 5)(1, 3, 10)(2, 9, 6)(7, 8, 11), Perm(0, 10, 7)(1, 5, 6)(2, 4, 11)(3, 9, 8), Perm(0, 6, 8)(1, 7, 2)(3, 5, 11)(4, 10, 9), Perm(0, 7, 9)(1, 11, 4)(2, 8, 3)(5, 6, 10), Perm(0, 8, 10)(1, 7, 6)(2, 11, 5)(3, 9, 4), Perm(0, 9, 6)(1, 3, 11)(2, 8, 7)(4, 10, 5))) tetrahedron_faces = list(tuple(arg) for arg in tetrahedron.faces) cube_faces = list(tuple(arg) for arg in cube.faces) octahedron_faces = list(tuple(arg) for arg in octahedron.faces) dodecahedron_faces = list(tuple(arg) for arg in dodecahedron.faces) icosahedron_faces = list(tuple(arg) for arg in icosahedron.faces)
6e59628e93527814b3b5c2fddcd7a12ff0a0abe020bfbfafeee7b0772f36f1f6	import itertools from sympy.combinatorics.fp_groups import FpGroup, FpSubgroup, simplify_presentation from sympy.combinatorics.free_groups import FreeGroup from sympy.combinatorics.perm_groups import PermutationGroup from sympy.core.numbers import igcd from sympy.ntheory.factor_ import totient from sympy.core.singleton import S class GroupHomomorphism: ''' A class representing group homomorphisms. Instantiate using `homomorphism()`. References ========== .. [1] Holt, D., Eick, B. and O'Brien, E. (2005). Handbook of computational group theory. ''' def __init__(self, domain, codomain, images): self.domain = domain self.codomain = codomain self.images = images self._inverses = None self._kernel = None self._image = None def _invs(self): ''' Return a dictionary with `{gen: inverse}` where `gen` is a rewriting generator of `codomain` (e.g. strong generator for permutation groups) and `inverse` is an element of its preimage ''' image = self.image() inverses = {} for k in list(self.images.keys()): v = self.images[k] if not (v in inverses or v.is_identity): inverses[v] = k if isinstance(self.codomain, PermutationGroup): gens = image.strong_gens else: gens = image.generators for g in gens: if g in inverses or g.is_identity: continue w = self.domain.identity if isinstance(self.codomain, PermutationGroup): parts = image._strong_gens_slp[g][::-1] else: parts = g for s in parts: if s in inverses: w = winverses[s] else: w = winverses[s-1]-1 inverses[g] = w return inverses def invert(self, g): ''' Return an element of the preimage of ``g`` or of each element of ``g`` if ``g`` is a list. Explanation =========== If the codomain is an FpGroup, the inverse for equal elements might not always be the same unless the FpGroup's rewriting system is confluent. However, making a system confluent can be time-consuming. If it's important, try `self.codomain.make_confluent()` first. ''' from sympy.combinatorics import Permutation from sympy.combinatorics.free_groups import FreeGroupElement if isinstance(g, (Permutation, FreeGroupElement)): if isinstance(self.codomain, FpGroup): g = self.codomain.reduce(g) if self._inverses is None: self._inverses = self._invs() image = self.image() w = self.domain.identity if isinstance(self.codomain, PermutationGroup): gens = image.generator_product(g)[::-1] else: gens = g # the following can't be "for s in gens:" # because that would be equivalent to # "for s in gens.array_form:" when g is # a FreeGroupElement. On the other hand, # when you call gens by index, the generator # (or inverse) at position i is returned. for i in range(len(gens)): s = gens[i] if s.is_identity: continue if s in self._inverses: w = wself._inverses[s] else: w = wself._inverses[s-1]-1 return w elif isinstance(g, list): return [self.invert(e) for e in g] def kernel(self): ''' Compute the kernel of `self`. ''' if self._kernel is None: self._kernel = self._compute_kernel() return self._kernel def _compute_kernel(self): G = self.domain G_order = G.order() if G_order is S.Infinity: raise NotImplementedError( "Kernel computation is not implemented for infinite groups") gens = [] if isinstance(G, PermutationGroup): K = PermutationGroup(G.identity) else: K = FpSubgroup(G, gens, normal=True) i = self.image().order() while K.order()i != G_order: r = G.random() k = rself.invert(self(r))*-1 if k not in K: gens.append(k) if isinstance(G, PermutationGroup): K = PermutationGroup(gens) else: K = FpSubgroup(G, gens, normal=True) return K def image(self): ''' Compute the image of `self`. ''' if self._image is None: values = list(set(self.images.values())) if isinstance(self.codomain, PermutationGroup): self._image = self.codomain.subgroup(values) else: self._image = FpSubgroup(self.codomain, values) return self._image def _apply(self, elem): ''' Apply `self` to `elem`. ''' if elem not in self.domain: if isinstance(elem, (list, tuple)): return [self._apply(e) for e in elem] raise ValueError("The supplied element does not belong to the domain") if elem.is_identity: return self.codomain.identity else: images = self.images value = self.codomain.identity if isinstance(self.domain, PermutationGroup): gens = self.domain.generator_product(elem, original=True) for g in gens: if g in self.images: value = images[g]value else: value = images[g-1]-1value else: i = 0 for _, p in elem.array_form: if p < 0: g = elem[i]-1 else: g = elem[i] value = valueimages[g]*p i += abs(p) return value def __call__(self, elem): return self._apply(elem) def is_injective(self): ''' Check if the homomorphism is injective ''' return self.kernel().order() == 1 def is_surjective(self): ''' Check if the homomorphism is surjective ''' im = self.image().order() oth = self.codomain.order() if im is S.Infinity and oth is S.Infinity: return None else: return im == oth def is_isomorphism(self): ''' Check if `self` is an isomorphism. ''' return self.is_injective() and self.is_surjective() def is_trivial(self): ''' Check is `self` is a trivial homomorphism, i.e. all elements are mapped to the identity. ''' return self.image().order() == 1 def compose(self, other): ''' Return the composition of `self` and `other`, i.e. the homomorphism phi such that for all g in the domain of `other`, phi(g) = self(other(g)) ''' if not other.image().is_subgroup(self.domain): raise ValueError("The image of `other` must be a subgroup of " "the domain of `self`") images = {g: self(other(g)) for g in other.images} return GroupHomomorphism(other.domain, self.codomain, images) def restrict_to(self, H): ''' Return the restriction of the homomorphism to the subgroup `H` of the domain. ''' if not isinstance(H, PermutationGroup) or not H.is_subgroup(self.domain): raise ValueError("Given H is not a subgroup of the domain") domain = H images = {g: self(g) for g in H.generators} return GroupHomomorphism(domain, self.codomain, images) def invert_subgroup(self, H): ''' Return the subgroup of the domain that is the inverse image of the subgroup ``H`` of the homomorphism image ''' if not H.is_subgroup(self.image()): raise ValueError("Given H is not a subgroup of the image") gens = [] P = PermutationGroup(self.image().identity) for h in H.generators: h_i = self.invert(h) if h_i not in P: gens.append(h_i) P = PermutationGroup(gens) for k in self.kernel().generators: if kh_i not in P: gens.append(kh_i) P = PermutationGroup(gens) return P def homomorphism(domain, codomain, gens, images=(), check=True): ''' Create (if possible) a group homomorphism from the group ``domain`` to the group ``codomain`` defined by the images of the domain's generators ``gens``. ``gens`` and ``images`` can be either lists or tuples of equal sizes. If ``gens`` is a proper subset of the group's generators, the unspecified generators will be mapped to the identity. If the images are not specified, a trivial homomorphism will be created. If the given images of the generators do not define a homomorphism, an exception is raised. If ``check`` is ``False``, do not check whether the given images actually define a homomorphism. ''' if not isinstance(domain, (PermutationGroup, FpGroup, FreeGroup)): raise TypeError("The domain must be a group") if not isinstance(codomain, (PermutationGroup, FpGroup, FreeGroup)): raise TypeError("The codomain must be a group") generators = domain.generators if not all(g in generators for g in gens): raise ValueError("The supplied generators must be a subset of the domain's generators") if not all(g in codomain for g in images): raise ValueError("The images must be elements of the codomain") if images and len(images) != len(gens): raise ValueError("The number of images must be equal to the number of generators") gens = list(gens) images = list(images) images.extend([codomain.identity](len(generators)-len(images))) gens.extend([g for g in generators if g not in gens]) images = dict(zip(gens,images)) if check and not _check_homomorphism(domain, codomain, images): raise ValueError("The given images do not define a homomorphism") return GroupHomomorphism(domain, codomain, images) def _check_homomorphism(domain, codomain, images): if hasattr(domain, 'relators'): rels = domain.relators else: gens = domain.presentation().generators rels = domain.presentation().relators identity = codomain.identity def _image(r): if r.is_identity: return identity else: w = identity r_arr = r.array_form i = 0 j = 0 # i is the index for r and j is for # r_arr. r_arr[j] is the tuple (sym, p) # where sym is the generator symbol # and p is the power to which it is # raised while r[i] is a generator # (not just its symbol) or the inverse of # a generator - hence the need for # both indices while i < len(r): power = r_arr[j][1] if isinstance(domain, PermutationGroup) and r[i] in gens: s = domain.generators[gens.index(r[i])] else: s = r[i] if s in images: w = wimages[s]power elif s-1 in images: w = wimages[s-1]power i += abs(power) j += 1 return w for r in rels: if isinstance(codomain, FpGroup): s = codomain.equals(_image(r), identity) if s is None: # only try to make the rewriting system # confluent when it can't determine the # truth of equality otherwise success = codomain.make_confluent() s = codomain.equals(_image(r), identity) if s is None and not success: raise RuntimeError("Can't determine if the images " "define a homomorphism. Try increasing " "the maximum number of rewriting rules " "(group._rewriting_system.set_max(new_value); " "the current value is stored in group._rewriting" "_system.maxeqns)") else: s = _image(r).is_identity if not s: return False return True def orbit_homomorphism(group, omega): ''' Return the homomorphism induced by the action of the permutation group ``group`` on the set ``omega`` that is closed under the action. ''' from sympy.combinatorics import Permutation from sympy.combinatorics.named_groups import SymmetricGroup codomain = SymmetricGroup(len(omega)) identity = codomain.identity omega = list(omega) images = {g: identityPermutation([omega.index(o^g) for o in omega]) for g in group.generators} group._schreier_sims(base=omega) H = GroupHomomorphism(group, codomain, images) if len(group.basic_stabilizers) > len(omega): H._kernel = group.basic_stabilizers[len(omega)] else: H._kernel = PermutationGroup([group.identity]) return H def block_homomorphism(group, blocks): ''' Return the homomorphism induced by the action of the permutation group ``group`` on the block system ``blocks``. The latter should be of the same form as returned by the ``minimal_block`` method for permutation groups, namely a list of length ``group.degree`` where the i-th entry is a representative of the block i belongs to. ''' from sympy.combinatorics import Permutation from sympy.combinatorics.named_groups import SymmetricGroup n = len(blocks) # number the blocks; m is the total number, # b is such that b[i] is the number of the block i belongs to, # p is the list of length m such that p[i] is the representative # of the i-th block m = 0 p = [] b = [None]n for i in range(n): if blocks[i] == i: p.append(i) b[i] = m m += 1 for i in range(n): b[i] = b[blocks[i]] codomain = SymmetricGroup(m) # the list corresponding to the identity permutation in codomain identity = range(m) images = {g: Permutation([b[p[i]^g] for i in identity]) for g in group.generators} H = GroupHomomorphism(group, codomain, images) return H def group_isomorphism(G, H, isomorphism=True): ''' Compute an isomorphism between 2 given groups. Parameters ========== G : A finite ``FpGroup`` or a ``PermutationGroup``. First group. H : A finite ``FpGroup`` or a ``PermutationGroup`` Second group. isomorphism : bool This is used to avoid the computation of homomorphism when the user only wants to check if there exists an isomorphism between the groups. Returns ======= If isomorphism = False -- Returns a boolean. If isomorphism = True -- Returns a boolean and an isomorphism between `G` and `H`. Examples ======== >>> from sympy.combinatorics import free_group, Permutation >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.fp_groups import FpGroup >>> from sympy.combinatorics.homomorphisms import group_isomorphism >>> from sympy.combinatorics.named_groups import DihedralGroup, AlternatingGroup >>> D = DihedralGroup(8) >>> p = Permutation(0, 1, 2, 3, 4, 5, 6, 7) >>> P = PermutationGroup(p) >>> group_isomorphism(D, P) (False, None) >>> F, a, b = free_group("a, b") >>> G = FpGroup(F, [a3, b3, (ab)2]) >>> H = AlternatingGroup(4) >>> (check, T) = group_isomorphism(G, H) >>> check True >>> T(bab-1a*-1b*-1) (0 2 3) Notes ===== Uses the approach suggested by Robert Tarjan to compute the isomorphism between two groups. First, the generators of ``G`` are mapped to the elements of ``H`` and we check if the mapping induces an isomorphism. ''' if not isinstance(G, (PermutationGroup, FpGroup)): raise TypeError("The group must be a PermutationGroup or an FpGroup") if not isinstance(H, (PermutationGroup, FpGroup)): raise TypeError("The group must be a PermutationGroup or an FpGroup") if isinstance(G, FpGroup) and isinstance(H, FpGroup): G = simplify_presentation(G) H = simplify_presentation(H) # Two infinite FpGroups with the same generators are isomorphic # when the relators are same but are ordered differently. if G.generators == H.generators and (G.relators).sort() == (H.relators).sort(): if not isomorphism: return True return (True, homomorphism(G, H, G.generators, H.generators)) # `_H` is the permutation group isomorphic to `H`. _H = H g_order = G.order() h_order = H.order() if g_order is S.Infinity: raise NotImplementedError("Isomorphism methods are not implemented for infinite groups.") if isinstance(H, FpGroup): if h_order is S.Infinity: raise NotImplementedError("Isomorphism methods are not implemented for infinite groups.") _H, h_isomorphism = H._to_perm_group() if (g_order != h_order) or (G.is_abelian != H.is_abelian): if not isomorphism: return False return (False, None) if not isomorphism: # Two groups of the same cyclic numbered order # are isomorphic to each other. n = g_order if (igcd(n, totient(n))) == 1: return True # Match the generators of `G` with subsets of `_H` gens = list(G.generators) for subset in itertools.permutations(_H, len(gens)): images = list(subset) images.extend([_H.identity](len(G.generators)-len(images))) _images = dict(zip(gens,images)) if _check_homomorphism(G, _H, _images): if isinstance(H, FpGroup): images = h_isomorphism.invert(images) T = homomorphism(G, H, G.generators, images, check=False) if T.is_isomorphism(): # It is a valid isomorphism if not isomorphism: return True return (True, T) if not isomorphism: return False return (False, None) def is_isomorphic(G, H): ''' Check if the groups are isomorphic to each other Parameters ========== G : A finite ``FpGroup`` or a ``PermutationGroup`` First group. H : A finite ``FpGroup`` or a ``PermutationGroup`` Second group. Returns ======= boolean ''' return group_isomorphism(G, H, isomorphism=False)
cf31d9c79e4c5f8122d87cdd2e13f884e695ad97f61439d1aff1a8a3b6618332	from sympy.core import Basic, Dict, sympify, Tuple from sympy.core.numbers import Integer from sympy.core.sorting import default_sort_key from sympy.core.sympify import _sympify from sympy.functions.combinatorial.numbers import bell from sympy.matrices import zeros from sympy.sets.sets import FiniteSet, Union from sympy.utilities.iterables import flatten, group from sympy.utilities.misc import as_int from collections import defaultdict class Partition(FiniteSet): """ This class represents an abstract partition. A partition is a set of disjoint sets whose union equals a given set. See Also ======== sympy.utilities.iterables.partitions, sympy.utilities.iterables.multiset_partitions """ _rank = None _partition = None def __new__(cls, partition): """ Generates a new partition object. This method also verifies if the arguments passed are valid and raises a ValueError if they are not. Examples ======== Creating Partition from Python lists: >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3]) >>> a Partition({3}, {1, 2}) >>> a.partition [[1, 2], [3]] >>> len(a) 2 >>> a.members (1, 2, 3) Creating Partition from Python sets: >>> Partition({1, 2, 3}, {4, 5}) Partition({4, 5}, {1, 2, 3}) Creating Partition from SymPy finite sets: >>> from sympy import FiniteSet >>> a = FiniteSet(1, 2, 3) >>> b = FiniteSet(4, 5) >>> Partition(a, b) Partition({4, 5}, {1, 2, 3}) """ args = [] dups = False for arg in partition: if isinstance(arg, list): as_set = set(arg) if len(as_set) < len(arg): dups = True break # error below arg = as_set args.append(_sympify(arg)) if not all(isinstance(part, FiniteSet) for part in args): raise ValueError( "Each argument to Partition should be " \ "a list, set, or a FiniteSet") # sort so we have a canonical reference for RGS U = Union(args) if dups or len(U) < sum(len(arg) for arg in args): raise ValueError("Partition contained duplicate elements.") obj = FiniteSet.__new__(cls, args) obj.members = tuple(U) obj.size = len(U) return obj def sort_key(self, order=None): """Return a canonical key that can be used for sorting. Ordering is based on the size and sorted elements of the partition and ties are broken with the rank. Examples ======== >>> from sympy import default_sort_key >>> from sympy.combinatorics import Partition >>> from sympy.abc import x >>> a = Partition([1, 2]) >>> b = Partition([3, 4]) >>> c = Partition([1, x]) >>> d = Partition(list(range(4))) >>> l = [d, b, a + 1, a, c] >>> l.sort(key=default_sort_key); l [Partition({1, 2}), Partition({1}, {2}), Partition({1, x}), Partition({3, 4}), Partition({0, 1, 2, 3})] """ if order is None: members = self.members else: members = tuple(sorted(self.members, key=lambda w: default_sort_key(w, order))) return tuple(map(default_sort_key, (self.size, members, self.rank))) @property def partition(self): """Return partition as a sorted list of lists. Examples ======== >>> from sympy.combinatorics import Partition >>> Partition([1], [2, 3]).partition [[1], [2, 3]] """ if self._partition is None: self._partition = sorted([sorted(p, key=default_sort_key) for p in self.args]) return self._partition def __add__(self, other): """ Return permutation whose rank is ``other`` greater than current rank, (mod the maximum rank for the set). Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3]) >>> a.rank 1 >>> (a + 1).rank 2 >>> (a + 100).rank 1 """ other = as_int(other) offset = self.rank + other result = RGS_unrank((offset) % RGS_enum(self.size), self.size) return Partition.from_rgs(result, self.members) def __sub__(self, other): """ Return permutation whose rank is ``other`` less than current rank, (mod the maximum rank for the set). Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3]) >>> a.rank 1 >>> (a - 1).rank 0 >>> (a - 100).rank 1 """ return self.__add__(-other) def __le__(self, other): """ Checks if a partition is less than or equal to the other based on rank. Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3, 4, 5]) >>> b = Partition([1], [2, 3], [4], [5]) >>> a.rank, b.rank (9, 34) >>> a <= a True >>> a <= b True """ return self.sort_key() <= sympify(other).sort_key() def __lt__(self, other): """ Checks if a partition is less than the other. Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3, 4, 5]) >>> b = Partition([1], [2, 3], [4], [5]) >>> a.rank, b.rank (9, 34) >>> a < b True """ return self.sort_key() < sympify(other).sort_key() @property def rank(self): """ Gets the rank of a partition. Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3], [4, 5]) >>> a.rank 13 """ if self._rank is not None: return self._rank self._rank = RGS_rank(self.RGS) return self._rank @property def RGS(self): """ Returns the "restricted growth string" of the partition. Explanation =========== The RGS is returned as a list of indices, L, where L[i] indicates the block in which element i appears. For example, in a partition of 3 elements (a, b, c) into 2 blocks ([c], [a, b]) the RGS is [1, 1, 0]: "a" is in block 1, "b" is in block 1 and "c" is in block 0. Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3], [4, 5]) >>> a.members (1, 2, 3, 4, 5) >>> a.RGS (0, 0, 1, 2, 2) >>> a + 1 Partition({3}, {4}, {5}, {1, 2}) >>> _.RGS (0, 0, 1, 2, 3) """ rgs = {} partition = self.partition for i, part in enumerate(partition): for j in part: rgs[j] = i return tuple([rgs[i] for i in sorted( [i for p in partition for i in p], key=default_sort_key)]) @classmethod def from_rgs(self, rgs, elements): """ Creates a set partition from a restricted growth string. Explanation =========== The indices given in rgs are assumed to be the index of the element as given in elements as provided* (the elements are not sorted by this routine). Block numbering starts from 0. If any block was not referenced in ``rgs`` an error will be raised. Examples ======== >>> from sympy.combinatorics import Partition >>> Partition.from_rgs([0, 1, 2, 0, 1], list('abcde')) Partition({c}, {a, d}, {b, e}) >>> Partition.from_rgs([0, 1, 2, 0, 1], list('cbead')) Partition({e}, {a, c}, {b, d}) >>> a = Partition([1, 4], [2], [3, 5]) >>> Partition.from_rgs(a.RGS, a.members) Partition({2}, {1, 4}, {3, 5}) """ if len(rgs) != len(elements): raise ValueError('mismatch in rgs and element lengths') max_elem = max(rgs) + 1 partition = [[] for i in range(max_elem)] j = 0 for i in rgs: partition[i].append(elements[j]) j += 1 if not all(p for p in partition): raise ValueError('some blocks of the partition were empty.') return Partition(partition) class IntegerPartition(Basic): """ This class represents an integer partition. Explanation =========== In number theory and combinatorics, a partition of a positive integer, ``n``, also called an integer partition, is a way of writing ``n`` as a list of positive integers that sum to n. Two partitions that differ only in the order of summands are considered to be the same partition; if order matters then the partitions are referred to as compositions. For example, 4 has five partitions: [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1]; the compositions [1, 2, 1] and [1, 1, 2] are the same as partition [2, 1, 1]. See Also ======== sympy.utilities.iterables.partitions, sympy.utilities.iterables.multiset_partitions References ========== .. [1] https://en.wikipedia.org/wiki/Partition_%28number_theory%29 """ _dict = None _keys = None def __new__(cls, partition, integer=None): """ Generates a new IntegerPartition object from a list or dictionary. Explantion ========== The partition can be given as a list of positive integers or a dictionary of (integer, multiplicity) items. If the partition is preceded by an integer an error will be raised if the partition does not sum to that given integer. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> a = IntegerPartition([5, 4, 3, 1, 1]) >>> a IntegerPartition(14, (5, 4, 3, 1, 1)) >>> print(a) [5, 4, 3, 1, 1] >>> IntegerPartition({1:3, 2:1}) IntegerPartition(5, (2, 1, 1, 1)) If the value that the partition should sum to is given first, a check will be made to see n error will be raised if there is a discrepancy: >>> IntegerPartition(10, [5, 4, 3, 1]) Traceback (most recent call last): ... ValueError: The partition is not valid """ if integer is not None: integer, partition = partition, integer if isinstance(partition, (dict, Dict)): _ = [] for k, v in sorted(list(partition.items()), reverse=True): if not v: continue k, v = as_int(k), as_int(v) _.extend([k]v) partition = tuple(_) else: partition = tuple(sorted(map(as_int, partition), reverse=True)) sum_ok = False if integer is None: integer = sum(partition) sum_ok = True else: integer = as_int(integer) if not sum_ok and sum(partition) != integer: raise ValueError("Partition did not add to %s" % integer) if any(i < 1 for i in partition): raise ValueError("All integer summands must be greater than one") obj = Basic.__new__(cls, Integer(integer), Tuple(partition)) obj.partition = list(partition) obj.integer = integer return obj def prev_lex(self): """Return the previous partition of the integer, n, in lexical order, wrapping around to [1, ..., 1] if the partition is [n]. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> p = IntegerPartition([4]) >>> print(p.prev_lex()) [3, 1] >>> p.partition > p.prev_lex().partition True """ d = defaultdict(int) d.update(self.as_dict()) keys = self._keys if keys == [1]: return IntegerPartition({self.integer: 1}) if keys[-1] != 1: d[keys[-1]] -= 1 if keys[-1] == 2: d[1] = 2 else: d[keys[-1] - 1] = d[1] = 1 else: d[keys[-2]] -= 1 left = d[1] + keys[-2] new = keys[-2] d[1] = 0 while left: new -= 1 if left - new >= 0: d[new] += left//new left -= d[new]new return IntegerPartition(self.integer, d) def next_lex(self): """Return the next partition of the integer, n, in lexical order, wrapping around to [n] if the partition is [1, ..., 1]. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> p = IntegerPartition([3, 1]) >>> print(p.next_lex()) [4] >>> p.partition < p.next_lex().partition True """ d = defaultdict(int) d.update(self.as_dict()) key = self._keys a = key[-1] if a == self.integer: d.clear() d[1] = self.integer elif a == 1: if d[a] > 1: d[a + 1] += 1 d[a] -= 2 else: b = key[-2] d[b + 1] += 1 d[1] = (d[b] - 1)b d[b] = 0 else: if d[a] > 1: if len(key) == 1: d.clear() d[a + 1] = 1 d[1] = self.integer - a - 1 else: a1 = a + 1 d[a1] += 1 d[1] = d[a]a - a1 d[a] = 0 else: b = key[-2] b1 = b + 1 d[b1] += 1 need = d[b]b + d[a]a - b1 d[a] = d[b] = 0 d[1] = need return IntegerPartition(self.integer, d) def as_dict(self): """Return the partition as a dictionary whose keys are the partition integers and the values are the multiplicity of that integer. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> IntegerPartition([1]3 + [2] + [3]4).as_dict() {1: 3, 2: 1, 3: 4} """ if self._dict is None: groups = group(self.partition, multiple=False) self._keys = [g[0] for g in groups] self._dict = dict(groups) return self._dict @property def conjugate(self): """ Computes the conjugate partition of itself. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> a = IntegerPartition([6, 3, 3, 2, 1]) >>> a.conjugate [5, 4, 3, 1, 1, 1] """ j = 1 temp_arr = list(self.partition) + [0] k = temp_arr[0] b = [0]k while k > 0: while k > temp_arr[j]: b[k - 1] = j k -= 1 j += 1 return b def __lt__(self, other): """Return True if self is less than other when the partition is listed from smallest to biggest. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> a = IntegerPartition([3, 1]) >>> a < a False >>> b = a.next_lex() >>> a < b True >>> a == b False """ return list(reversed(self.partition)) < list(reversed(other.partition)) def __le__(self, other): """Return True if self is less than other when the partition is listed from smallest to biggest. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> a = IntegerPartition([4]) >>> a <= a True """ return list(reversed(self.partition)) <= list(reversed(other.partition)) def as_ferrers(self, char='#'): """ Prints the ferrer diagram of a partition. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> print(IntegerPartition([1, 1, 5]).as_ferrers()) ##### # # """ return "\n".join([chari for i in self.partition]) def __str__(self): return str(list(self.partition)) def random_integer_partition(n, seed=None): """ Generates a random integer partition summing to ``n`` as a list of reverse-sorted integers. Examples ======== >>> from sympy.combinatorics.partitions import random_integer_partition For the following, a seed is given so a known value can be shown; in practice, the seed would not be given. >>> random_integer_partition(100, seed=[1, 1, 12, 1, 2, 1, 85, 1]) [85, 12, 2, 1] >>> random_integer_partition(10, seed=[1, 2, 3, 1, 5, 1]) [5, 3, 1, 1] >>> random_integer_partition(1) [1] """ from sympy.core.random import _randint n = as_int(n) if n < 1: raise ValueError('n must be a positive integer') randint = _randint(seed) partition = [] while (n > 0): k = randint(1, n) mult = randint(1, n//k) partition.append((k, mult)) n -= kmult partition.sort(reverse=True) partition = flatten([[k]m for k, m in partition]) return partition def RGS_generalized(m): """ Computes the m + 1 generalized unrestricted growth strings and returns them as rows in matrix. Examples ======== >>> from sympy.combinatorics.partitions import RGS_generalized >>> RGS_generalized(6) Matrix([ [ 1, 1, 1, 1, 1, 1, 1], [ 1, 2, 3, 4, 5, 6, 0], [ 2, 5, 10, 17, 26, 0, 0], [ 5, 15, 37, 77, 0, 0, 0], [ 15, 52, 151, 0, 0, 0, 0], [ 52, 203, 0, 0, 0, 0, 0], [203, 0, 0, 0, 0, 0, 0]]) """ d = zeros(m + 1) for i in range(0, m + 1): d[0, i] = 1 for i in range(1, m + 1): for j in range(m): if j <= m - i: d[i, j] = j * d[i - 1, j] + d[i - 1, j + 1] else: d[i, j] = 0 return d def RGS_enum(m): """ RGS_enum computes the total number of restricted growth strings possible for a superset of size m. Examples ======== >>> from sympy.combinatorics.partitions import RGS_enum >>> from sympy.combinatorics import Partition >>> RGS_enum(4) 15 >>> RGS_enum(5) 52 >>> RGS_enum(6) 203 We can check that the enumeration is correct by actually generating the partitions. Here, the 15 partitions of 4 items are generated: >>> a = Partition(list(range(4))) >>> s = set() >>> for i in range(20): ... s.add(a) ... a += 1 ... >>> assert len(s) == 15 """ if (m < 1): return 0 elif (m == 1): return 1 else: return bell(m) def RGS_unrank(rank, m): """ Gives the unranked restricted growth string for a given superset size. Examples ======== >>> from sympy.combinatorics.partitions import RGS_unrank >>> RGS_unrank(14, 4) [0, 1, 2, 3] >>> RGS_unrank(0, 4) [0, 0, 0, 0] """ if m < 1: raise ValueError("The superset size must be >= 1") if rank < 0 or RGS_enum(m) <= rank: raise ValueError("Invalid arguments") L = [1] * (m + 1) j = 1 D = RGS_generalized(m) for i in range(2, m + 1): v = D[m - i, j] cr = jv if cr <= rank: L[i] = j + 1 rank -= cr j += 1 else: L[i] = int(rank / v + 1) rank %= v return [x - 1 for x in L[1:]] def RGS_rank(rgs): """ Computes the rank of a restricted growth string. Examples ======== >>> from sympy.combinatorics.partitions import RGS_rank, RGS_unrank >>> RGS_rank([0, 1, 2, 1, 3]) 42 >>> RGS_rank(RGS_unrank(4, 7)) 4 """ rgs_size = len(rgs) rank = 0 D = RGS_generalized(rgs_size) for i in range(1, rgs_size): n = len(rgs[(i + 1):]) m = max(rgs[0:i]) rank += D[n, m + 1] rgs[i] return rank
16a4b16ad16738ab3206f9f9d35ef60124e0899b3b18005d2a5736eeb5116fc4	from sympy.combinatorics.permutations import Permutation, _af_invert, _af_rmul from sympy.ntheory import isprime rmul = Permutation.rmul _af_new = Permutation._af_new ############################################ # # Utilities for computational group theory # ############################################ def _base_ordering(base, degree): r""" Order `\{0, 1, \dots, n-1\}` so that base points come first and in order. Parameters ========== ``base`` : the base ``degree`` : the degree of the associated permutation group Returns ======= A list ``base_ordering`` such that ``base_ordering[point]`` is the number of ``point`` in the ordering. Examples ======== >>> from sympy.combinatorics import SymmetricGroup >>> from sympy.combinatorics.util import _base_ordering >>> S = SymmetricGroup(4) >>> S.schreier_sims() >>> _base_ordering(S.base, S.degree) [0, 1, 2, 3] Notes ===== This is used in backtrack searches, when we define a relation `\ll` on the underlying set for a permutation group of degree `n`, `\{0, 1, \dots, n-1\}`, so that if `(b_1, b_2, \dots, b_k)` is a base we have `b_i \ll b_j` whenever `i<j` and `b_i \ll a` for all `i\in\{1,2, \dots, k\}` and `a` is not in the base. The idea is developed and applied to backtracking algorithms in [1], pp.108-132. The points that are not in the base are taken in increasing order. References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of computational group theory" """ base_len = len(base) ordering = [0]degree for i in range(base_len): ordering[base[i]] = i current = base_len for i in range(degree): if i not in base: ordering[i] = current current += 1 return ordering def _check_cycles_alt_sym(perm): """ Checks for cycles of prime length p with n/2 < p < n-2. Explanation =========== Here `n` is the degree of the permutation. This is a helper function for the function is_alt_sym from sympy.combinatorics.perm_groups. Examples ======== >>> from sympy.combinatorics.util import _check_cycles_alt_sym >>> from sympy.combinatorics import Permutation >>> a = Permutation([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [11, 12]]) >>> _check_cycles_alt_sym(a) False >>> b = Permutation([[0, 1, 2, 3, 4, 5, 6], [7, 8, 9, 10]]) >>> _check_cycles_alt_sym(b) True See Also ======== sympy.combinatorics.perm_groups.PermutationGroup.is_alt_sym """ n = perm.size af = perm.array_form current_len = 0 total_len = 0 used = set() for i in range(n//2): if i not in used and i < n//2 - total_len: current_len = 1 used.add(i) j = i while af[j] != i: current_len += 1 j = af[j] used.add(j) total_len += current_len if current_len > n//2 and current_len < n - 2 and isprime(current_len): return True return False def _distribute_gens_by_base(base, gens): r""" Distribute the group elements ``gens`` by membership in basic stabilizers. Explanation =========== Notice that for a base `(b_1, b_2, \dots, b_k)`, the basic stabilizers are defined as `G^{(i)} = G_{b_1, \dots, b_{i-1}}` for `i \in\{1, 2, \dots, k\}`. Parameters ========== ``base`` : a sequence of points in `\{0, 1, \dots, n-1\}` ``gens`` : a list of elements of a permutation group of degree `n`. Returns ======= List of length `k`, where `k` is the length of ``base``. The `i`-th entry contains those elements in ``gens`` which fix the first `i` elements of ``base`` (so that the `0`-th entry is equal to ``gens`` itself). If no element fixes the first `i` elements of ``base``, the `i`-th element is set to a list containing the identity element. Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> from sympy.combinatorics.util import _distribute_gens_by_base >>> D = DihedralGroup(3) >>> D.schreier_sims() >>> D.strong_gens [(0 1 2), (0 2), (1 2)] >>> D.base [0, 1] >>> _distribute_gens_by_base(D.base, D.strong_gens) [[(0 1 2), (0 2), (1 2)], [(1 2)]] See Also ======== _strong_gens_from_distr, _orbits_transversals_from_bsgs, _handle_precomputed_bsgs """ base_len = len(base) degree = gens[0].size stabs = [[] for _ in range(base_len)] max_stab_index = 0 for gen in gens: j = 0 while j < base_len - 1 and gen._array_form[base[j]] == base[j]: j += 1 if j > max_stab_index: max_stab_index = j for k in range(j + 1): stabs[k].append(gen) for i in range(max_stab_index + 1, base_len): stabs[i].append(_af_new(list(range(degree)))) return stabs def _handle_precomputed_bsgs(base, strong_gens, transversals=None, basic_orbits=None, strong_gens_distr=None): """ Calculate BSGS-related structures from those present. Explanation =========== The base and strong generating set must be provided; if any of the transversals, basic orbits or distributed strong generators are not provided, they will be calculated from the base and strong generating set. Parameters ========== ``base`` - the base ``strong_gens`` - the strong generators ``transversals`` - basic transversals ``basic_orbits`` - basic orbits ``strong_gens_distr`` - strong generators distributed by membership in basic stabilizers Returns ======= ``(transversals, basic_orbits, strong_gens_distr)`` where ``transversals`` are the basic transversals, ``basic_orbits`` are the basic orbits, and ``strong_gens_distr`` are the strong generators distributed by membership in basic stabilizers. Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> from sympy.combinatorics.util import _handle_precomputed_bsgs >>> D = DihedralGroup(3) >>> D.schreier_sims() >>> _handle_precomputed_bsgs(D.base, D.strong_gens, ... basic_orbits=D.basic_orbits) ([{0: (2), 1: (0 1 2), 2: (0 2)}, {1: (2), 2: (1 2)}], [[0, 1, 2], [1, 2]], [[(0 1 2), (0 2), (1 2)], [(1 2)]]) See Also ======== _orbits_transversals_from_bsgs, _distribute_gens_by_base """ if strong_gens_distr is None: strong_gens_distr = _distribute_gens_by_base(base, strong_gens) if transversals is None: if basic_orbits is None: basic_orbits, transversals = \ _orbits_transversals_from_bsgs(base, strong_gens_distr) else: transversals = \ _orbits_transversals_from_bsgs(base, strong_gens_distr, transversals_only=True) else: if basic_orbits is None: base_len = len(base) basic_orbits = [None]base_len for i in range(base_len): basic_orbits[i] = list(transversals[i].keys()) return transversals, basic_orbits, strong_gens_distr def _orbits_transversals_from_bsgs(base, strong_gens_distr, transversals_only=False, slp=False): """ Compute basic orbits and transversals from a base and strong generating set. Explanation =========== The generators are provided as distributed across the basic stabilizers. If the optional argument ``transversals_only`` is set to True, only the transversals are returned. Parameters ========== ``base`` - The base. ``strong_gens_distr`` - Strong generators distributed by membership in basic stabilizers. ``transversals_only`` - bool A flag switching between returning only the transversals and both orbits and transversals. ``slp`` - If ``True``, return a list of dictionaries containing the generator presentations of the elements of the transversals, i.e. the list of indices of generators from ``strong_gens_distr[i]`` such that their product is the relevant transversal element. Examples ======== >>> from sympy.combinatorics import SymmetricGroup >>> from sympy.combinatorics.util import _distribute_gens_by_base >>> S = SymmetricGroup(3) >>> S.schreier_sims() >>> strong_gens_distr = _distribute_gens_by_base(S.base, S.strong_gens) >>> (S.base, strong_gens_distr) ([0, 1], [[(0 1 2), (2)(0 1), (1 2)], [(1 2)]]) See Also ======== _distribute_gens_by_base, _handle_precomputed_bsgs """ from sympy.combinatorics.perm_groups import _orbit_transversal base_len = len(base) degree = strong_gens_distr[0][0].size transversals = [None]base_len slps = [None]base_len if transversals_only is False: basic_orbits = [None]*base_len for i in range(base_len): transversals[i], slps[i] = _orbit_transversal(degree, strong_gens_distr[i], base[i], pairs=True, slp=True) transversals[i] = dict(transversals[i]) if transversals_only is False: basic_orbits[i] = list(transversals[i].keys()) if transversals_only: return transversals else: if not slp: return basic_orbits, transversals return basic_orbits, transversals, slps def _remove_gens(base, strong_gens, basic_orbits=None, strong_gens_distr=None): """ Remove redundant generators from a strong generating set. Parameters ========== ``base`` - a base ``strong_gens`` - a strong generating set relative to ``base`` ``basic_orbits`` - basic orbits ``strong_gens_distr`` - strong generators distributed by membership in basic stabilizers Returns ======= A strong generating set with respect to ``base`` which is a subset of ``strong_gens``. Examples ======== >>> from sympy.combinatorics import SymmetricGroup >>> from sympy.combinatorics.util import _remove_gens >>> from sympy.combinatorics.testutil import _verify_bsgs >>> S = SymmetricGroup(15) >>> base, strong_gens = S.schreier_sims_incremental() >>> new_gens = _remove_gens(base, strong_gens) >>> len(new_gens) 14 >>> _verify_bsgs(S, base, new_gens) True Notes ===== This procedure is outlined in [1],p.95. References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of computational group theory" """ from sympy.combinatorics.perm_groups import _orbit base_len = len(base) degree = strong_gens[0].size if strong_gens_distr is None: strong_gens_distr = _distribute_gens_by_base(base, strong_gens) if basic_orbits is None: basic_orbits = [] for i in range(base_len): basic_orbit = _orbit(degree, strong_gens_distr[i], base[i]) basic_orbits.append(basic_orbit) strong_gens_distr.append([]) res = strong_gens[:] for i in range(base_len - 1, -1, -1): gens_copy = strong_gens_distr[i][:] for gen in strong_gens_distr[i]: if gen not in strong_gens_distr[i + 1]: temp_gens = gens_copy[:] temp_gens.remove(gen) if temp_gens == []: continue temp_orbit = _orbit(degree, temp_gens, base[i]) if temp_orbit == basic_orbits[i]: gens_copy.remove(gen) res.remove(gen) return res def _strip(g, base, orbits, transversals): """ Attempt to decompose a permutation using a (possibly partial) BSGS structure. Explanation =========== This is done by treating the sequence ``base`` as an actual base, and the orbits ``orbits`` and transversals ``transversals`` as basic orbits and transversals relative to it. This process is called "sifting". A sift is unsuccessful when a certain orbit element is not found or when after the sift the decomposition does not end with the identity element. The argument ``transversals`` is a list of dictionaries that provides transversal elements for the orbits ``orbits``. Parameters ========== ``g`` - permutation to be decomposed ``base`` - sequence of points ``orbits`` - a list in which the ``i``-th entry is an orbit of ``base[i]`` under some subgroup of the pointwise stabilizer of ` `base[0], base[1], ..., base[i - 1]``. The groups themselves are implicit in this function since the only information we need is encoded in the orbits and transversals ``transversals`` - a list of orbit transversals associated with the orbits ``orbits``. Examples ======== >>> from sympy.combinatorics import Permutation, SymmetricGroup >>> from sympy.combinatorics.util import _strip >>> S = SymmetricGroup(5) >>> S.schreier_sims() >>> g = Permutation([0, 2, 3, 1, 4]) >>> _strip(g, S.base, S.basic_orbits, S.basic_transversals) ((4), 5) Notes ===== The algorithm is described in [1],pp.89-90. The reason for returning both the current state of the element being decomposed and the level at which the sifting ends is that they provide important information for the randomized version of the Schreier-Sims algorithm. References ========== .. [1] Holt, D., Eick, B., O'Brien, E."Handbook of computational group theory" See Also ======== sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random """ h = g._array_form base_len = len(base) for i in range(base_len): beta = h[base[i]] if beta == base[i]: continue if beta not in orbits[i]: return _af_new(h), i + 1 u = transversals[i][beta]._array_form h = _af_rmul(_af_invert(u), h) return _af_new(h), base_len + 1 def _strip_af(h, base, orbits, transversals, j, slp=[], slps={}): """ optimized _strip, with h, transversals and result in array form if the stripped elements is the identity, it returns False, base_len + 1 j h[base[i]] == base[i] for i <= j """ base_len = len(base) for i in range(j+1, base_len): beta = h[base[i]] if beta == base[i]: continue if beta not in orbits[i]: if not slp: return h, i + 1 return h, i + 1, slp u = transversals[i][beta] if h == u: if not slp: return False, base_len + 1 return False, base_len + 1, slp h = _af_rmul(_af_invert(u), h) if slp: u_slp = slps[i][beta][:] u_slp.reverse() u_slp = [(i, (g,)) for g in u_slp] slp = u_slp + slp if not slp: return h, base_len + 1 return h, base_len + 1, slp def _strong_gens_from_distr(strong_gens_distr): """ Retrieve strong generating set from generators of basic stabilizers. This is just the union of the generators of the first and second basic stabilizers. Parameters ========== ``strong_gens_distr`` - strong generators distributed by membership in basic stabilizers Examples ======== >>> from sympy.combinatorics import SymmetricGroup >>> from sympy.combinatorics.util import (_strong_gens_from_distr, ... _distribute_gens_by_base) >>> S = SymmetricGroup(3) >>> S.schreier_sims() >>> S.strong_gens [(0 1 2), (2)(0 1), (1 2)] >>> strong_gens_distr = _distribute_gens_by_base(S.base, S.strong_gens) >>> _strong_gens_from_distr(strong_gens_distr) [(0 1 2), (2)(0 1), (1 2)] See Also ======== _distribute_gens_by_base """ if len(strong_gens_distr) == 1: return strong_gens_distr[0][:] else: result = strong_gens_distr[0] for gen in strong_gens_distr[1]: if gen not in result: result.append(gen) return result
4612f7cabb197c10271025ff29526a99a3b339c4c92c43a8e354f613f5d116b5	from sympy.combinatorics.free_groups import free_group from sympy.printing.defaults import DefaultPrinting from itertools import chain, product from bisect import bisect_left ############################################################################### # COSET TABLE # ############################################################################### class CosetTable(DefaultPrinting): # coset_table: Mathematically a coset table # represented using a list of lists # alpha: Mathematically a coset (precisely, a live coset) # represented by an integer between i with 1 <= i <= n # alpha in c # x: Mathematically an element of "A" (set of generators and # their inverses), represented using "FpGroupElement" # fp_grp: Finitely Presented Group with < X\|R > as presentation. # H: subgroup of fp_grp. # NOTE: We start with H as being only a list of words in generators # of "fp_grp". Since `.subgroup` method has not been implemented. r""" Properties ========== [1] `0 \in \Omega` and `\tau(1) = \epsilon` [2] `\alpha^x = \beta \Leftrightarrow \beta^{x^{-1}} = \alpha` [3] If `\alpha^x = \beta`, then `H \tau(\alpha)x = H \tau(\beta)` [4] `\forall \alpha \in \Omega, 1^{\tau(\alpha)} = \alpha` References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" .. [2] John J. Cannon; Lucien A. Dimino; George Havas; Jane M. Watson Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 463-490. "Implementation and Analysis of the Todd-Coxeter Algorithm" """ # default limit for the number of cosets allowed in a # coset enumeration. coset_table_max_limit = 4096000 # limit for the current instance coset_table_limit = None # maximum size of deduction stack above or equal to # which it is emptied max_stack_size = 100 def __init__(self, fp_grp, subgroup, max_cosets=None): if not max_cosets: max_cosets = CosetTable.coset_table_max_limit self.fp_group = fp_grp self.subgroup = subgroup self.coset_table_limit = max_cosets # "p" is setup independent of Omega and n self.p = [0] # a list of the form `[gen_1, gen_1^{-1}, ... , gen_k, gen_k^{-1}]` self.A = list(chain.from_iterable((gen, gen*-1) \ for gen in self.fp_group.generators)) #P[alpha, x] Only defined when alpha^x is defined. self.P = [[None]len(self.A)] # the mathematical coset table which is a list of lists self.table = [[None]len(self.A)] self.A_dict = {x: self.A.index(x) for x in self.A} self.A_dict_inv = {} for x, index in self.A_dict.items(): if index % 2 == 0: self.A_dict_inv[x] = self.A_dict[x] + 1 else: self.A_dict_inv[x] = self.A_dict[x] - 1 # used in the coset-table based method of coset enumeration. Each of # the element is called a "deduction" which is the form (alpha, x) whenever # a value is assigned to alpha^x during a definition or "deduction process" self.deduction_stack = [] # Attributes for modified methods. H = self.subgroup self._grp = free_group(', ' .join(["a_%d" % i for i in range(len(H))]))[0] self.P = [[None]len(self.A)] self.p_p = {} @property def omega(self): """Set of live cosets. """ return [coset for coset in range(len(self.p)) if self.p[coset] == coset] def copy(self): """ Return a shallow copy of Coset Table instance ``self``. """ self_copy = self.__class__(self.fp_group, self.subgroup) self_copy.table = [list(perm_rep) for perm_rep in self.table] self_copy.p = list(self.p) self_copy.deduction_stack = list(self.deduction_stack) return self_copy def __str__(self): return "Coset Table on %s with %s as subgroup generators" \ % (self.fp_group, self.subgroup) __repr__ = __str__ @property def n(self): """The number `n` represents the length of the sublist containing the live cosets. """ if not self.table: return 0 return max(self.omega) + 1 # Pg. 152 [1] def is_complete(self): r""" The coset table is called complete if it has no undefined entries on the live cosets; that is, `\alpha^x` is defined for all `\alpha \in \Omega` and `x \in A`. """ return not any(None in self.table[coset] for coset in self.omega) # Pg. 153 [1] def define(self, alpha, x, modified=False): r""" This routine is used in the relator-based strategy of Todd-Coxeter algorithm if some `\alpha^x` is undefined. We check whether there is space available for defining a new coset. If there is enough space then we remedy this by adjoining a new coset `\beta` to `\Omega` (i.e to set of live cosets) and put that equal to `\alpha^x`, then make an assignment satisfying Property[1]. If there is not enough space then we halt the Coset Table creation. The maximum amount of space that can be used by Coset Table can be manipulated using the class variable ``CosetTable.coset_table_max_limit``. See Also ======== define_c """ A = self.A table = self.table len_table = len(table) if len_table >= self.coset_table_limit: # abort the further generation of cosets raise ValueError("the coset enumeration has defined more than " "%s cosets. Try with a greater value max number of cosets " % self.coset_table_limit) table.append([None]len(A)) self.P.append([None]len(self.A)) # beta is the new coset generated beta = len_table self.p.append(beta) table[alpha][self.A_dict[x]] = beta table[beta][self.A_dict_inv[x]] = alpha # P[alpha][x] = epsilon, P[beta][x*-1] = epsilon if modified: self.P[alpha][self.A_dict[x]] = self._grp.identity self.P[beta][self.A_dict_inv[x]] = self._grp.identity self.p_p[beta] = self._grp.identity def define_c(self, alpha, x): r""" A variation of ``define`` routine, described on Pg. 165 [1], used in the coset table-based strategy of Todd-Coxeter algorithm. It differs from ``define`` routine in that for each definition it also adds the tuple `(\alpha, x)` to the deduction stack. See Also ======== define """ A = self.A table = self.table len_table = len(table) if len_table >= self.coset_table_limit: # abort the further generation of cosets raise ValueError("the coset enumeration has defined more than " "%s cosets. Try with a greater value max number of cosets " % self.coset_table_limit) table.append([None]len(A)) # beta is the new coset generated beta = len_table self.p.append(beta) table[alpha][self.A_dict[x]] = beta table[beta][self.A_dict_inv[x]] = alpha # append to deduction stack self.deduction_stack.append((alpha, x)) def scan_c(self, alpha, word): """ A variation of ``scan`` routine, described on pg. 165 of [1], which puts at tuple, whenever a deduction occurs, to deduction stack. See Also ======== scan, scan_check, scan_and_fill, scan_and_fill_c """ # alpha is an integer representing a "coset" # since scanning can be in two cases # 1. for alpha=0 and w in Y (i.e generating set of H) # 2. alpha in Omega (set of live cosets), w in R (relators) A_dict = self.A_dict A_dict_inv = self.A_dict_inv table = self.table f = alpha i = 0 r = len(word) b = alpha j = r - 1 # list of union of generators and their inverses while i <= j and table[f][A_dict[word[i]]] is not None: f = table[f][A_dict[word[i]]] i += 1 if i > j: if f != b: self.coincidence_c(f, b) return while j >= i and table[b][A_dict_inv[word[j]]] is not None: b = table[b][A_dict_inv[word[j]]] j -= 1 if j < i: # we have an incorrect completed scan with coincidence f ~ b # run the "coincidence" routine self.coincidence_c(f, b) elif j == i: # deduction process table[f][A_dict[word[i]]] = b table[b][A_dict_inv[word[i]]] = f self.deduction_stack.append((f, word[i])) # otherwise scan is incomplete and yields no information # alpha, beta coincide, i.e. alpha, beta represent the pair of cosets where # coincidence occurs def coincidence_c(self, alpha, beta): """ A variation of ``coincidence`` routine used in the coset-table based method of coset enumeration. The only difference being on addition of a new coset in coset table(i.e new coset introduction), then it is appended to ``deduction_stack``. See Also ======== coincidence """ A_dict = self.A_dict A_dict_inv = self.A_dict_inv table = self.table # behaves as a queue q = [] self.merge(alpha, beta, q) while len(q) > 0: gamma = q.pop(0) for x in A_dict: delta = table[gamma][A_dict[x]] if delta is not None: table[delta][A_dict_inv[x]] = None # only line of difference from ``coincidence`` routine self.deduction_stack.append((delta, x*-1)) mu = self.rep(gamma) nu = self.rep(delta) if table[mu][A_dict[x]] is not None: self.merge(nu, table[mu][A_dict[x]], q) elif table[nu][A_dict_inv[x]] is not None: self.merge(mu, table[nu][A_dict_inv[x]], q) else: table[mu][A_dict[x]] = nu table[nu][A_dict_inv[x]] = mu def scan(self, alpha, word, y=None, fill=False, modified=False): r""" ``scan`` performs a scanning process on the input ``word``. It first locates the largest prefix ``s`` of ``word`` for which `\alpha^s` is defined (i.e is not ``None``), ``s`` may be empty. Let ``word=sv``, let ``t`` be the longest suffix of ``v`` for which `\alpha^{t^{-1}}` is defined, and let ``v=ut``. Then three possibilities are there: 1. If ``t=v``, then we say that the scan completes, and if, in addition `\alpha^s = \alpha^{t^{-1}}`, then we say that the scan completes correctly. 2. It can also happen that scan does not complete, but `\|u\|=1`; that is, the word ``u`` consists of a single generator `x \in A`. In that case, if `\alpha^s = \beta` and `\alpha^{t^{-1}} = \gamma`, then we can set `\beta^x = \gamma` and `\gamma^{x^{-1}} = \beta`. These assignments are known as deductions and enable the scan to complete correctly. 3. See ``coicidence`` routine for explanation of third condition. Notes ===== The code for the procedure of scanning `\alpha \in \Omega` under `w \in A` is defined on pg. 155 [1] See Also ======== scan_c, scan_check, scan_and_fill, scan_and_fill_c Scan and Fill ============= Performed when the default argument fill=True. Modified Scan ============= Performed when the default argument modified=True """ # alpha is an integer representing a "coset" # since scanning can be in two cases # 1. for alpha=0 and w in Y (i.e generating set of H) # 2. alpha in Omega (set of live cosets), w in R (relators) A_dict = self.A_dict A_dict_inv = self.A_dict_inv table = self.table f = alpha i = 0 r = len(word) b = alpha j = r - 1 b_p = y if modified: f_p = self._grp.identity flag = 0 while fill or flag == 0: flag = 1 while i <= j and table[f][A_dict[word[i]]] is not None: if modified: f_p = f_pself.P[f][A_dict[word[i]]] f = table[f][A_dict[word[i]]] i += 1 if i > j: if f != b: if modified: self.modified_coincidence(f, b, f_p-1y) else: self.coincidence(f, b) return while j >= i and table[b][A_dict_inv[word[j]]] is not None: if modified: b_p = b_pself.P[b][self.A_dict_inv[word[j]]] b = table[b][A_dict_inv[word[j]]] j -= 1 if j < i: # we have an incorrect completed scan with coincidence f ~ b # run the "coincidence" routine if modified: self.modified_coincidence(f, b, f_p-1b_p) else: self.coincidence(f, b) elif j == i: # deduction process table[f][A_dict[word[i]]] = b table[b][A_dict_inv[word[i]]] = f if modified: self.P[f][self.A_dict[word[i]]] = f_p*-1b_p self.P[b][self.A_dict_inv[word[i]]] = b_p*-1f_p return elif fill: self.define(f, word[i], modified=modified) # otherwise scan is incomplete and yields no information # used in the low-index subgroups algorithm def scan_check(self, alpha, word): r""" Another version of ``scan`` routine, described on, it checks whether `\alpha` scans correctly under `word`, it is a straightforward modification of ``scan``. ``scan_check`` returns ``False`` (rather than calling ``coincidence``) if the scan completes incorrectly; otherwise it returns ``True``. See Also ======== scan, scan_c, scan_and_fill, scan_and_fill_c """ # alpha is an integer representing a "coset" # since scanning can be in two cases # 1. for alpha=0 and w in Y (i.e generating set of H) # 2. alpha in Omega (set of live cosets), w in R (relators) A_dict = self.A_dict A_dict_inv = self.A_dict_inv table = self.table f = alpha i = 0 r = len(word) b = alpha j = r - 1 while i <= j and table[f][A_dict[word[i]]] is not None: f = table[f][A_dict[word[i]]] i += 1 if i > j: return f == b while j >= i and table[b][A_dict_inv[word[j]]] is not None: b = table[b][A_dict_inv[word[j]]] j -= 1 if j < i: # we have an incorrect completed scan with coincidence f ~ b # return False, instead of calling coincidence routine return False elif j == i: # deduction process table[f][A_dict[word[i]]] = b table[b][A_dict_inv[word[i]]] = f return True def merge(self, k, lamda, q, w=None, modified=False): """ Merge two classes with representatives ``k`` and ``lamda``, described on Pg. 157 [1] (for pseudocode), start by putting ``p[k] = lamda``. It is more efficient to choose the new representative from the larger of the two classes being merged, i.e larger among ``k`` and ``lamda``. procedure ``merge`` performs the merging operation, adds the deleted class representative to the queue ``q``. Parameters ========== 'k', 'lamda' being the two class representatives to be merged. Notes ===== Pg. 86-87 [1] contains a description of this method. See Also ======== coincidence, rep """ p = self.p rep = self.rep phi = rep(k, modified=modified) psi = rep(lamda, modified=modified) if phi != psi: mu = min(phi, psi) v = max(phi, psi) p[v] = mu if modified: if v == phi: self.p_p[phi] = self.p_p[k]*-1wself.p_p[lamda] else: self.p_p[psi] = self.p_p[lamda]-1w*-1self.p_p[k] q.append(v) def rep(self, k, modified=False): r""" Parameters ========== `k \in [0 \ldots n-1]`, as for ``self`` only array ``p`` is used Returns ======= Representative of the class containing ``k``. Returns the representative of `\sim` class containing ``k``, it also makes some modification to array ``p`` of ``self`` to ease further computations, described on Pg. 157 [1]. The information on classes under `\sim` is stored in array `p` of ``self`` argument, which will always satisfy the property: `p[\alpha] \sim \alpha` and `p[\alpha]=\alpha \iff \alpha=rep(\alpha)` `\forall \in [0 \ldots n-1]`. So, for `\alpha \in [0 \ldots n-1]`, we find `rep(self, \alpha)` by continually replacing `\alpha` by `p[\alpha]` until it becomes constant (i.e satisfies `p[\alpha] = \alpha`):w To increase the efficiency of later ``rep`` calculations, whenever we find `rep(self, \alpha)=\beta`, we set `p[\gamma] = \beta \forall \gamma \in p-chain` from `\alpha` to `\beta` Notes ===== ``rep`` routine is also described on Pg. 85-87 [1] in Atkinson's algorithm, this results from the fact that ``coincidence`` routine introduces functionality similar to that introduced by the ``minimal_block`` routine on Pg. 85-87 [1]. See Also ======== coincidence, merge """ p = self.p lamda = k rho = p[lamda] if modified: s = p[:] while rho != lamda: if modified: s[rho] = lamda lamda = rho rho = p[lamda] if modified: rho = s[lamda] while rho != k: mu = rho rho = s[mu] p[rho] = lamda self.p_p[rho] = self.p_p[rho]self.p_p[mu] else: mu = k rho = p[mu] while rho != lamda: p[mu] = lamda mu = rho rho = p[mu] return lamda # alpha, beta coincide, i.e. alpha, beta represent the pair of cosets # where coincidence occurs def coincidence(self, alpha, beta, w=None, modified=False): r""" The third situation described in ``scan`` routine is handled by this routine, described on Pg. 156-161 [1]. The unfortunate situation when the scan completes but not correctly, then ``coincidence`` routine is run. i.e when for some `i` with `1 \le i \le r+1`, we have `w=st` with `s = x_1 x_2 \dots x_{i-1}`, `t = x_i x_{i+1} \dots x_r`, and `\beta = \alpha^s` and `\gamma = \alpha^{t-1}` are defined but unequal. This means that `\beta` and `\gamma` represent the same coset of `H` in `G`. Described on Pg. 156 [1]. ``rep`` See Also ======== scan """ A_dict = self.A_dict A_dict_inv = self.A_dict_inv table = self.table # behaves as a queue q = [] if modified: self.modified_merge(alpha, beta, w, q) else: self.merge(alpha, beta, q) while len(q) > 0: gamma = q.pop(0) for x in A_dict: delta = table[gamma][A_dict[x]] if delta is not None: table[delta][A_dict_inv[x]] = None mu = self.rep(gamma, modified=modified) nu = self.rep(delta, modified=modified) if table[mu][A_dict[x]] is not None: if modified: v = self.p_p[delta]-1self.P[gamma][self.A_dict[x]]*-1 v = vself.p_p[gamma]self.P[mu][self.A_dict[x]] self.modified_merge(nu, table[mu][self.A_dict[x]], v, q) else: self.merge(nu, table[mu][A_dict[x]], q) elif table[nu][A_dict_inv[x]] is not None: if modified: v = self.p_p[gamma]-1self.P[gamma][self.A_dict[x]] v = vself.p_p[delta]self.P[mu][self.A_dict_inv[x]] self.modified_merge(mu, table[nu][self.A_dict_inv[x]], v, q) else: self.merge(mu, table[nu][A_dict_inv[x]], q) else: table[mu][A_dict[x]] = nu table[nu][A_dict_inv[x]] = mu if modified: v = self.p_p[gamma]*-1self.P[gamma][self.A_dict[x]]self.p_p[delta] self.P[mu][self.A_dict[x]] = v self.P[nu][self.A_dict_inv[x]] = v-1 # method used in the HLT strategy def scan_and_fill(self, alpha, word): """ A modified version of ``scan`` routine used in the relator-based method of coset enumeration, described on pg. 162-163 [1], which follows the idea that whenever the procedure is called and the scan is incomplete then it makes new definitions to enable the scan to complete; i.e it fills in the gaps in the scan of the relator or subgroup generator. """ self.scan(alpha, word, fill=True) def scan_and_fill_c(self, alpha, word): """ A modified version of ``scan`` routine, described on Pg. 165 second para. [1], with modification similar to that of ``scan_anf_fill`` the only difference being it calls the coincidence procedure used in the coset-table based method i.e. the routine ``coincidence_c`` is used. See Also ======== scan, scan_and_fill """ A_dict = self.A_dict A_dict_inv = self.A_dict_inv table = self.table r = len(word) f = alpha i = 0 b = alpha j = r - 1 # loop until it has filled the alpha row in the table. while True: # do the forward scanning while i <= j and table[f][A_dict[word[i]]] is not None: f = table[f][A_dict[word[i]]] i += 1 if i > j: if f != b: self.coincidence_c(f, b) return # forward scan was incomplete, scan backwards while j >= i and table[b][A_dict_inv[word[j]]] is not None: b = table[b][A_dict_inv[word[j]]] j -= 1 if j < i: self.coincidence_c(f, b) elif j == i: table[f][A_dict[word[i]]] = b table[b][A_dict_inv[word[i]]] = f self.deduction_stack.append((f, word[i])) else: self.define_c(f, word[i]) # method used in the HLT strategy def look_ahead(self): """ When combined with the HLT method this is known as HLT+Lookahead method of coset enumeration, described on pg. 164 [1]. Whenever ``define`` aborts due to lack of space available this procedure is executed. This routine helps in recovering space resulting from "coincidence" of cosets. """ R = self.fp_group.relators p = self.p # complete scan all relators under all cosets(obviously live) # without making new definitions for beta in self.omega: for w in R: self.scan(beta, w) if p[beta] < beta: break # Pg. 166 def process_deductions(self, R_c_x, R_c_x_inv): """ Processes the deductions that have been pushed onto ``deduction_stack``, described on Pg. 166 [1] and is used in coset-table based enumeration. See Also ======== deduction_stack """ p = self.p table = self.table while len(self.deduction_stack) > 0: if len(self.deduction_stack) >= CosetTable.max_stack_size: self.look_ahead() del self.deduction_stack[:] continue else: alpha, x = self.deduction_stack.pop() if p[alpha] == alpha: for w in R_c_x: self.scan_c(alpha, w) if p[alpha] < alpha: break beta = table[alpha][self.A_dict[x]] if beta is not None and p[beta] == beta: for w in R_c_x_inv: self.scan_c(beta, w) if p[beta] < beta: break def process_deductions_check(self, R_c_x, R_c_x_inv): """ A variation of ``process_deductions``, this calls ``scan_check`` wherever ``process_deductions`` calls ``scan``, described on Pg. [1]. See Also ======== process_deductions """ table = self.table while len(self.deduction_stack) > 0: alpha, x = self.deduction_stack.pop() for w in R_c_x: if not self.scan_check(alpha, w): return False beta = table[alpha][self.A_dict[x]] if beta is not None: for w in R_c_x_inv: if not self.scan_check(beta, w): return False return True def switch(self, beta, gamma): r"""Switch the elements `\beta, \gamma \in \Omega` of ``self``, used by the ``standardize`` procedure, described on Pg. 167 [1]. See Also ======== standardize """ A = self.A A_dict = self.A_dict table = self.table for x in A: z = table[gamma][A_dict[x]] table[gamma][A_dict[x]] = table[beta][A_dict[x]] table[beta][A_dict[x]] = z for alpha in range(len(self.p)): if self.p[alpha] == alpha: if table[alpha][A_dict[x]] == beta: table[alpha][A_dict[x]] = gamma elif table[alpha][A_dict[x]] == gamma: table[alpha][A_dict[x]] = beta def standardize(self): r""" A coset table is standardized if when running through the cosets and within each coset through the generator images (ignoring generator inverses), the cosets appear in order of the integers `0, 1, \dots, n`. "Standardize" reorders the elements of `\Omega` such that, if we scan the coset table first by elements of `\Omega` and then by elements of A, then the cosets occur in ascending order. ``standardize()`` is used at the end of an enumeration to permute the cosets so that they occur in some sort of standard order. Notes ===== procedure is described on pg. 167-168 [1], it also makes use of the ``switch`` routine to replace by smaller integer value. Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_r >>> F, x, y = free_group("x, y") # Example 5.3 from [1] >>> f = FpGroup(F, [x2y2, x3y5]) >>> C = coset_enumeration_r(f, []) >>> C.compress() >>> C.table [[1, 3, 1, 3], [2, 0, 2, 0], [3, 1, 3, 1], [0, 2, 0, 2]] >>> C.standardize() >>> C.table [[1, 2, 1, 2], [3, 0, 3, 0], [0, 3, 0, 3], [2, 1, 2, 1]] """ A = self.A A_dict = self.A_dict gamma = 1 for alpha, x in product(range(self.n), A): beta = self.table[alpha][A_dict[x]] if beta >= gamma: if beta > gamma: self.switch(gamma, beta) gamma += 1 if gamma == self.n: return # Compression of a Coset Table def compress(self): """Removes the non-live cosets from the coset table, described on pg. 167 [1]. """ gamma = -1 A = self.A A_dict = self.A_dict A_dict_inv = self.A_dict_inv table = self.table chi = tuple([i for i in range(len(self.p)) if self.p[i] != i]) for alpha in self.omega: gamma += 1 if gamma != alpha: # replace alpha by gamma in coset table for x in A: beta = table[alpha][A_dict[x]] table[gamma][A_dict[x]] = beta table[beta][A_dict_inv[x]] == gamma # all the cosets in the table are live cosets self.p = list(range(gamma + 1)) # delete the useless columns del table[len(self.p):] # re-define values for row in table: for j in range(len(self.A)): row[j] -= bisect_left(chi, row[j]) def conjugates(self, R): R_c = list(chain.from_iterable((rel.cyclic_conjugates(), \ (rel-1).cyclic_conjugates()) for rel in R)) R_set = set() for conjugate in R_c: R_set = R_set.union(conjugate) R_c_list = [] for x in self.A: r = {word for word in R_set if word[0] == x} R_c_list.append(r) R_set.difference_update(r) return R_c_list def coset_representative(self, coset): ''' Compute the coset representative of a given coset. Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_r >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x3, y3, x-1y*-1xy]) >>> C = coset_enumeration_r(f, [x]) >>> C.compress() >>> C.table [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1]] >>> C.coset_representative(0) <identity> >>> C.coset_representative(1) y >>> C.coset_representative(2) y-1 ''' for x in self.A: gamma = self.table[coset][self.A_dict[x]] if coset == 0: return self.fp_group.identity if gamma < coset: return self.coset_representative(gamma)x*-1 ############################## # Modified Methods # ############################## def modified_define(self, alpha, x): r""" Define a function p_p from from [1..n] to A as an additional component of the modified coset table. Parameters ========== \alpha \in \Omega x \in A* See Also ======== define """ self.define(alpha, x, modified=True) def modified_scan(self, alpha, w, y, fill=False): r""" Parameters ========== \alpha \in \Omega w \in A* y \in (YUY^-1) fill -- `modified_scan_and_fill` when set to True. See Also ======== scan """ self.scan(alpha, w, y=y, fill=fill, modified=True) def modified_scan_and_fill(self, alpha, w, y): self.modified_scan(alpha, w, y, fill=True) def modified_merge(self, k, lamda, w, q): r""" Parameters ========== 'k', 'lamda' -- the two class representatives to be merged. q -- queue of length l of elements to be deleted from `\Omega` . w -- Word in (YUY^-1) See Also ======== merge """ self.merge(k, lamda, q, w=w, modified=True) def modified_rep(self, k): r""" Parameters ========== `k \in [0 \ldots n-1]` See Also ======== rep """ self.rep(k, modified=True) def modified_coincidence(self, alpha, beta, w): r""" Parameters ========== A coincident pair `\alpha, \beta \in \Omega, w \in Y \cup Y^{-1}` See Also ======== coincidence """ self.coincidence(alpha, beta, w=w, modified=True) ############################################################################### # COSET ENUMERATION # ############################################################################### # relator-based method def coset_enumeration_r(fp_grp, Y, max_cosets=None, draft=None, incomplete=False, modified=False): """ This is easier of the two implemented methods of coset enumeration. and is often called the HLT method, after Hazelgrove, Leech, Trotter The idea is that we make use of ``scan_and_fill`` makes new definitions whenever the scan is incomplete to enable the scan to complete; this way we fill in the gaps in the scan of the relator or subgroup generator, that's why the name relator-based method. An instance of `CosetTable` for `fp_grp` can be passed as the keyword argument `draft` in which case the coset enumeration will start with that instance and attempt to complete it. When `incomplete` is `True` and the function is unable to complete for some reason, the partially complete table will be returned. # TODO: complete the docstring See Also ======== scan_and_fill, Examples ======== >>> from sympy.combinatorics.free_groups import free_group >>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_r >>> F, x, y = free_group("x, y") # Example 5.1 from [1] >>> f = FpGroup(F, [x3, y3, x-1y*-1xy]) >>> C = coset_enumeration_r(f, [x]) >>> for i in range(len(C.p)): ... if C.p[i] == i: ... print(C.table[i]) [0, 0, 1, 2] [1, 1, 2, 0] [2, 2, 0, 1] >>> C.p [0, 1, 2, 1, 1] # Example from exercises Q2 [1] >>> f = FpGroup(F, [x2y2, y-1xyx-3]) >>> C = coset_enumeration_r(f, []) >>> C.compress(); C.standardize() >>> C.table [[1, 2, 3, 4], [5, 0, 6, 7], [0, 5, 7, 6], [7, 6, 5, 0], [6, 7, 0, 5], [2, 1, 4, 3], [3, 4, 2, 1], [4, 3, 1, 2]] # Example 5.2 >>> f = FpGroup(F, [x2, y3, (xy)*3]) >>> Y = [xy] >>> C = coset_enumeration_r(f, Y) >>> for i in range(len(C.p)): ... if C.p[i] == i: ... print(C.table[i]) [1, 1, 2, 1] [0, 0, 0, 2] [3, 3, 1, 0] [2, 2, 3, 3] # Example 5.3 >>> f = FpGroup(F, [x*2y2, x3y5]) >>> Y = [] >>> C = coset_enumeration_r(f, Y) >>> for i in range(len(C.p)): ... if C.p[i] == i: ... print(C.table[i]) [1, 3, 1, 3] [2, 0, 2, 0] [3, 1, 3, 1] [0, 2, 0, 2] # Example 5.4 >>> F, a, b, c, d, e = free_group("a, b, c, d, e") >>> f = FpGroup(F, [abc-1, bcd-1, cde-1, dea-1, eab-1]) >>> Y = [a] >>> C = coset_enumeration_r(f, Y) >>> for i in range(len(C.p)): ... if C.p[i] == i: ... print(C.table[i]) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] # example of "compress" method >>> C.compress() >>> C.table [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]] # Exercises Pg. 161, Q2. >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x2y2, y-1xyx-3]) >>> Y = [] >>> C = coset_enumeration_r(f, Y) >>> C.compress() >>> C.standardize() >>> C.table [[1, 2, 3, 4], [5, 0, 6, 7], [0, 5, 7, 6], [7, 6, 5, 0], [6, 7, 0, 5], [2, 1, 4, 3], [3, 4, 2, 1], [4, 3, 1, 2]] # John J. Cannon; Lucien A. Dimino; George Havas; Jane M. Watson # Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 463-490 # from 1973chwd.pdf # Table 1. Ex. 1 >>> F, r, s, t = free_group("r, s, t") >>> E1 = FpGroup(F, [t-1rtr-2, r-1srs-2, s-1tst-2]) >>> C = coset_enumeration_r(E1, [r]) >>> for i in range(len(C.p)): ... if C.p[i] == i: ... print(C.table[i]) [0, 0, 0, 0, 0, 0] Ex. 2 >>> F, a, b = free_group("a, b") >>> Cox = FpGroup(F, [a6, b*6, (ab)2, (a2b2)2, (a3b3)5]) >>> C = coset_enumeration_r(Cox, [a]) >>> index = 0 >>> for i in range(len(C.p)): ... if C.p[i] == i: ... index += 1 >>> index 500 # Ex. 3 >>> F, a, b = free_group("a, b") >>> B_2_4 = FpGroup(F, [a4, b4, (ab)4, (a-1b)4, (a2b)4, \ (ab2)4, (a*2b2)4, (a*-1bab)*4, (ab*-1ab)4]) >>> C = coset_enumeration_r(B_2_4, [a]) >>> index = 0 >>> for i in range(len(C.p)): ... if C.p[i] == i: ... index += 1 >>> index 1024 References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of computational group theory" """ # 1. Initialize a coset table C for < X\|R > C = CosetTable(fp_grp, Y, max_cosets=max_cosets) # Define coset table methods. if modified: _scan_and_fill = C.modified_scan_and_fill _define = C.modified_define else: _scan_and_fill = C.scan_and_fill _define = C.define if draft: C.table = draft.table[:] C.p = draft.p[:] R = fp_grp.relators A_dict = C.A_dict p = C.p for i in range(0, len(Y)): if modified: _scan_and_fill(0, Y[i], C._grp.generators[i]) else: _scan_and_fill(0, Y[i]) alpha = 0 while alpha < C.n: if p[alpha] == alpha: try: for w in R: if modified: _scan_and_fill(alpha, w, C._grp.identity) else: _scan_and_fill(alpha, w) # if alpha was eliminated during the scan then break if p[alpha] < alpha: break if p[alpha] == alpha: for x in A_dict: if C.table[alpha][A_dict[x]] is None: _define(alpha, x) except ValueError as e: if incomplete: return C raise e alpha += 1 return C def modified_coset_enumeration_r(fp_grp, Y, max_cosets=None, draft=None, incomplete=False): r""" Introduce a new set of symbols y \in Y that correspond to the generators of the subgroup. Store the elements of Y as a word P[\alpha, x] and compute the coset table similar to that of the regular coset enumeration methods. Examples ======== >>> from sympy.combinatorics.free_groups import free_group >>> from sympy.combinatorics.fp_groups import FpGroup >>> from sympy.combinatorics.coset_table import modified_coset_enumeration_r >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x3, y3, x-1y*-1xy]) >>> C = modified_coset_enumeration_r(f, [x]) >>> C.table [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1], [None, 1, None, None], [1, 3, None, None]] See Also ======== coset_enumertation_r References ========== .. [1] Holt, D., Eick, B., O'Brien, E., "Handbook of Computational Group Theory", Section 5.3.2 """ return coset_enumeration_r(fp_grp, Y, max_cosets=max_cosets, draft=draft, incomplete=incomplete, modified=True) # Pg. 166 # coset-table based method def coset_enumeration_c(fp_grp, Y, max_cosets=None, draft=None, incomplete=False): """ >>> from sympy.combinatorics.free_groups import free_group >>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_c >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x3, y3, x-1y*-1xy]) >>> C = coset_enumeration_c(f, [x]) >>> C.table [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1]] """ # Initialize a coset table C for < X\|R > X = fp_grp.generators R = fp_grp.relators C = CosetTable(fp_grp, Y, max_cosets=max_cosets) if draft: C.table = draft.table[:] C.p = draft.p[:] C.deduction_stack = draft.deduction_stack for alpha, x in product(range(len(C.table)), X): if C.table[alpha][C.A_dict[x]] is not None: C.deduction_stack.append((alpha, x)) A = C.A # replace all the elements by cyclic reductions R_cyc_red = [rel.identity_cyclic_reduction() for rel in R] R_c = list(chain.from_iterable((rel.cyclic_conjugates(), (rel*-1).cyclic_conjugates()) \ for rel in R_cyc_red)) R_set = set() for conjugate in R_c: R_set = R_set.union(conjugate) # a list of subsets of R_c whose words start with "x". R_c_list = [] for x in C.A: r = {word for word in R_set if word[0] == x} R_c_list.append(r) R_set.difference_update(r) for w in Y: C.scan_and_fill_c(0, w) for x in A: C.process_deductions(R_c_list[C.A_dict[x]], R_c_list[C.A_dict_inv[x]]) alpha = 0 while alpha < len(C.table): if C.p[alpha] == alpha: try: for x in C.A: if C.p[alpha] != alpha: break if C.table[alpha][C.A_dict[x]] is None: C.define_c(alpha, x) C.process_deductions(R_c_list[C.A_dict[x]], R_c_list[C.A_dict_inv[x]]) except ValueError as e: if incomplete: return C raise e alpha += 1 return C
42378210605f4c16cc65df96a42398e62114cadf0eede2576b74432df302ef09	"""Finitely Presented Groups and its algorithms. """ from sympy.core.singleton import S from sympy.core.symbol import symbols from sympy.combinatorics.free_groups import (FreeGroup, FreeGroupElement, free_group) from sympy.combinatorics.rewritingsystem import RewritingSystem from sympy.combinatorics.coset_table import (CosetTable, coset_enumeration_r, coset_enumeration_c) from sympy.combinatorics import PermutationGroup from sympy.matrices.normalforms import invariant_factors from sympy.matrices import Matrix from sympy.polys.polytools import gcd from sympy.printing.defaults import DefaultPrinting from sympy.utilities import public from sympy.utilities.magic import pollute from itertools import product @public def fp_group(fr_grp, relators=()): _fp_group = FpGroup(fr_grp, relators) return (_fp_group,) + tuple(_fp_group._generators) @public def xfp_group(fr_grp, relators=()): _fp_group = FpGroup(fr_grp, relators) return (_fp_group, _fp_group._generators) # Does not work. Both symbols and pollute are undefined. Never tested. @public def vfp_group(fr_grpm, relators): _fp_group = FpGroup(symbols, relators) pollute([sym.name for sym in _fp_group.symbols], _fp_group.generators) return _fp_group def _parse_relators(rels): """Parse the passed relators.""" return rels ############################################################################### # FINITELY PRESENTED GROUPS # ############################################################################### class FpGroup(DefaultPrinting): """ The FpGroup would take a FreeGroup and a list/tuple of relators, the relators would be specified in such a way that each of them be equal to the identity of the provided free group. """ is_group = True is_FpGroup = True is_PermutationGroup = False def __init__(self, fr_grp, relators): relators = _parse_relators(relators) self.free_group = fr_grp self.relators = relators self.generators = self._generators() self.dtype = type("FpGroupElement", (FpGroupElement,), {"group": self}) # CosetTable instance on identity subgroup self._coset_table = None # returns whether coset table on identity subgroup # has been standardized self._is_standardized = False self._order = None self._center = None self._rewriting_system = RewritingSystem(self) self._perm_isomorphism = None return def _generators(self): return self.free_group.generators def make_confluent(self): ''' Try to make the group's rewriting system confluent ''' self._rewriting_system.make_confluent() return def reduce(self, word): ''' Return the reduced form of `word` in `self` according to the group's rewriting system. If it's confluent, the reduced form is the unique normal form of the word in the group. ''' return self._rewriting_system.reduce(word) def equals(self, word1, word2): ''' Compare `word1` and `word2` for equality in the group using the group's rewriting system. If the system is confluent, the returned answer is necessarily correct. (If it is not, `False` could be returned in some cases where in fact `word1 == word2`) ''' if self.reduce(word1word2-1) == self.identity: return True elif self._rewriting_system.is_confluent: return False return None @property def identity(self): return self.free_group.identity def __contains__(self, g): return g in self.free_group def subgroup(self, gens, C=None, homomorphism=False): ''' Return the subgroup generated by `gens` using the Reidemeister-Schreier algorithm homomorphism -- When set to True, return a dictionary containing the images of the presentation generators in the original group. Examples ======== >>> from sympy.combinatorics.fp_groups import FpGroup >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x3, y5, (xy)*2]) >>> H = [xy, x*-1y*-1xyx] >>> K, T = f.subgroup(H, homomorphism=True) >>> T(K.generators) [xy, x-1y*2x-1] ''' if not all(isinstance(g, FreeGroupElement) for g in gens): raise ValueError("Generators must be `FreeGroupElement`s") if not all(g.group == self.free_group for g in gens): raise ValueError("Given generators are not members of the group") if homomorphism: g, rels, _gens = reidemeister_presentation(self, gens, C=C, homomorphism=True) else: g, rels = reidemeister_presentation(self, gens, C=C) if g: g = FpGroup(g[0].group, rels) else: g = FpGroup(free_group('')[0], []) if homomorphism: from sympy.combinatorics.homomorphisms import homomorphism return g, homomorphism(g, self, g.generators, _gens, check=False) return g def coset_enumeration(self, H, strategy="relator_based", max_cosets=None, draft=None, incomplete=False): """ Return an instance of ``coset table``, when Todd-Coxeter algorithm is run over the ``self`` with ``H`` as subgroup, using ``strategy`` argument as strategy. The returned coset table is compressed but not standardized. An instance of `CosetTable` for `fp_grp` can be passed as the keyword argument `draft` in which case the coset enumeration will start with that instance and attempt to complete it. When `incomplete` is `True` and the function is unable to complete for some reason, the partially complete table will be returned. """ if not max_cosets: max_cosets = CosetTable.coset_table_max_limit if strategy == 'relator_based': C = coset_enumeration_r(self, H, max_cosets=max_cosets, draft=draft, incomplete=incomplete) else: C = coset_enumeration_c(self, H, max_cosets=max_cosets, draft=draft, incomplete=incomplete) if C.is_complete(): C.compress() return C def standardize_coset_table(self): """ Standardized the coset table ``self`` and makes the internal variable ``_is_standardized`` equal to ``True``. """ self._coset_table.standardize() self._is_standardized = True def coset_table(self, H, strategy="relator_based", max_cosets=None, draft=None, incomplete=False): """ Return the mathematical coset table of ``self`` in ``H``. """ if not H: if self._coset_table is not None: if not self._is_standardized: self.standardize_coset_table() else: C = self.coset_enumeration([], strategy, max_cosets=max_cosets, draft=draft, incomplete=incomplete) self._coset_table = C self.standardize_coset_table() return self._coset_table.table else: C = self.coset_enumeration(H, strategy, max_cosets=max_cosets, draft=draft, incomplete=incomplete) C.standardize() return C.table def order(self, strategy="relator_based"): """ Returns the order of the finitely presented group ``self``. It uses the coset enumeration with identity group as subgroup, i.e ``H=[]``. Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x, y2]) >>> f.order(strategy="coset_table_based") 2 """ if self._order is not None: return self._order if self._coset_table is not None: self._order = len(self._coset_table.table) elif len(self.relators) == 0: self._order = self.free_group.order() elif len(self.generators) == 1: self._order = abs(gcd([r.array_form[0][1] for r in self.relators])) elif self._is_infinite(): self._order = S.Infinity else: gens, C = self._finite_index_subgroup() if C: ind = len(C.table) self._order = indself.subgroup(gens, C=C).order() else: self._order = self.index([]) return self._order def _is_infinite(self): ''' Test if the group is infinite. Return `True` if the test succeeds and `None` otherwise ''' used_gens = set() for r in self.relators: used_gens.update(r.contains_generators()) if not set(self.generators) <= used_gens: return True # Abelianisation test: check is the abelianisation is infinite abelian_rels = [] for rel in self.relators: abelian_rels.append([rel.exponent_sum(g) for g in self.generators]) m = Matrix(Matrix(abelian_rels)) if 0 in invariant_factors(m): return True else: return None def _finite_index_subgroup(self, s=None): ''' Find the elements of `self` that generate a finite index subgroup and, if found, return the list of elements and the coset table of `self` by the subgroup, otherwise return `(None, None)` ''' gen = self.most_frequent_generator() rels = list(self.generators) rels.extend(self.relators) if not s: if len(self.generators) == 2: s = [gen] + [g for g in self.generators if g != gen] else: rand = self.free_group.identity i = 0 while ((rand in rels or rand-1 in rels or rand.is_identity) and i<10): rand = self.random() i += 1 s = [gen, rand] + [g for g in self.generators if g != gen] mid = (len(s)+1)//2 half1 = s[:mid] half2 = s[mid:] draft1 = None draft2 = None m = 200 C = None while not C and (m/2 < CosetTable.coset_table_max_limit): m = min(m, CosetTable.coset_table_max_limit) draft1 = self.coset_enumeration(half1, max_cosets=m, draft=draft1, incomplete=True) if draft1.is_complete(): C = draft1 half = half1 else: draft2 = self.coset_enumeration(half2, max_cosets=m, draft=draft2, incomplete=True) if draft2.is_complete(): C = draft2 half = half2 if not C: m = 2 if not C: return None, None C.compress() return half, C def most_frequent_generator(self): gens = self.generators rels = self.relators freqs = [sum([r.generator_count(g) for r in rels]) for g in gens] return gens[freqs.index(max(freqs))] def random(self): import random r = self.free_group.identity for i in range(random.randint(2,3)): r = rrandom.choice(self.generators)random.choice([1,-1]) return r def index(self, H, strategy="relator_based"): """ Return the index of subgroup ``H`` in group ``self``. Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x5, y4, yxy3x*3]) >>> f.index([x]) 4 """ # TODO: use \|G:H\| = \|G\|/\|H\| (currently H can't be made into a group) # when we know \|G\| and \|H\| if H == []: return self.order() else: C = self.coset_enumeration(H, strategy) return len(C.table) def __str__(self): if self.free_group.rank > 30: str_form = "<fp group with %s generators>" % self.free_group.rank else: str_form = "<fp group on the generators %s>" % str(self.generators) return str_form __repr__ = __str__ #============================================================================== # PERMUTATION GROUP METHODS #============================================================================== def _to_perm_group(self): ''' Return an isomorphic permutation group and the isomorphism. The implementation is dependent on coset enumeration so will only terminate for finite groups. ''' from sympy.combinatorics import Permutation from sympy.combinatorics.homomorphisms import homomorphism if self.order() is S.Infinity: raise NotImplementedError("Permutation presentation of infinite " "groups is not implemented") if self._perm_isomorphism: T = self._perm_isomorphism P = T.image() else: C = self.coset_table([]) gens = self.generators images = [[C[i][2gens.index(g)] for i in range(len(C))] for g in gens] images = [Permutation(i) for i in images] P = PermutationGroup(images) T = homomorphism(self, P, gens, images, check=False) self._perm_isomorphism = T return P, T def _perm_group_list(self, method_name, args): ''' Given the name of a `PermutationGroup` method (returning a subgroup or a list of subgroups) and (optionally) additional arguments it takes, return a list or a list of lists containing the generators of this (or these) subgroups in terms of the generators of `self`. ''' P, T = self._to_perm_group() perm_result = getattr(P, method_name)(args) single = False if isinstance(perm_result, PermutationGroup): perm_result, single = [perm_result], True result = [] for group in perm_result: gens = group.generators result.append(T.invert(gens)) return result[0] if single else result def derived_series(self): ''' Return the list of lists containing the generators of the subgroups in the derived series of `self`. ''' return self._perm_group_list('derived_series') def lower_central_series(self): ''' Return the list of lists containing the generators of the subgroups in the lower central series of `self`. ''' return self._perm_group_list('lower_central_series') def center(self): ''' Return the list of generators of the center of `self`. ''' return self._perm_group_list('center') def derived_subgroup(self): ''' Return the list of generators of the derived subgroup of `self`. ''' return self._perm_group_list('derived_subgroup') def centralizer(self, other): ''' Return the list of generators of the centralizer of `other` (a list of elements of `self`) in `self`. ''' T = self._to_perm_group()[1] other = T(other) return self._perm_group_list('centralizer', other) def normal_closure(self, other): ''' Return the list of generators of the normal closure of `other` (a list of elements of `self`) in `self`. ''' T = self._to_perm_group()[1] other = T(other) return self._perm_group_list('normal_closure', other) def _perm_property(self, attr): ''' Given an attribute of a `PermutationGroup`, return its value for a permutation group isomorphic to `self`. ''' P = self._to_perm_group()[0] return getattr(P, attr) @property def is_abelian(self): ''' Check if `self` is abelian. ''' return self._perm_property("is_abelian") @property def is_nilpotent(self): ''' Check if `self` is nilpotent. ''' return self._perm_property("is_nilpotent") @property def is_solvable(self): ''' Check if `self` is solvable. ''' return self._perm_property("is_solvable") @property def elements(self): ''' List the elements of `self`. ''' P, T = self._to_perm_group() return T.invert(P._elements) @property def is_cyclic(self): """ Return ``True`` if group is Cyclic. """ if len(self.generators) <= 1: return True try: P, T = self._to_perm_group() except NotImplementedError: raise NotImplementedError("Check for infinite Cyclic group " "is not implemented") return P.is_cyclic def abelian_invariants(self): """ Return Abelian Invariants of a group. """ try: P, T = self._to_perm_group() except NotImplementedError: raise NotImplementedError("abelian invariants is not implemented" "for infinite group") return P.abelian_invariants() def composition_series(self): """ Return subnormal series of maximum length for a group. """ try: P, T = self._to_perm_group() except NotImplementedError: raise NotImplementedError("composition series is not implemented" "for infinite group") return P.composition_series() class FpSubgroup(DefaultPrinting): ''' The class implementing a subgroup of an FpGroup or a FreeGroup (only finite index subgroups are supported at this point). This is to be used if one wishes to check if an element of the original group belongs to the subgroup ''' def __init__(self, G, gens, normal=False): super().__init__() self.parent = G self.generators = list({g for g in gens if g != G.identity}) self._min_words = None #for use in __contains__ self.C = None self.normal = normal def __contains__(self, g): if isinstance(self.parent, FreeGroup): if self._min_words is None: # make _min_words - a list of subwords such that # g is in the subgroup if and only if it can be # partitioned into these subwords. Infinite families of # subwords are presented by tuples, e.g. (r, w) # stands for the family of subwords rwnr-1 def _process(w): # this is to be used before adding new words # into _min_words; if the word w is not cyclically # reduced, it will generate an infinite family of # subwords so should be written as a tuple; # if it is, w-1 should be added to the list # as well p, r = w.cyclic_reduction(removed=True) if not r.is_identity: return [(r, p)] else: return [w, w-1] # make the initial list gens = [] for w in self.generators: if self.normal: w = w.cyclic_reduction() gens.extend(_process(w)) for w1 in gens: for w2 in gens: # if w1 and w2 are equal or are inverses, continue if w1 == w2 or (not isinstance(w1, tuple) and w1-1 == w2): continue # if the start of one word is the inverse of the # end of the other, their multiple should be added # to _min_words because of cancellation if isinstance(w1, tuple): # start, end s1, s2 = w1[0][0], w1[0][0]-1 else: s1, s2 = w1[0], w1[len(w1)-1] if isinstance(w2, tuple): # start, end r1, r2 = w2[0][0], w2[0][0]-1 else: r1, r2 = w2[0], w2[len(w1)-1] # p1 and p2 are w1 and w2 or, in case when # w1 or w2 is an infinite family, a representative p1, p2 = w1, w2 if isinstance(w1, tuple): p1 = w1[0]w1[1]w1[0]*-1 if isinstance(w2, tuple): p2 = w2[0]w2[1]w2[0]-1 # add the product of the words to the list is necessary if r1-1 == s2 and not (p1p2).is_identity: new = _process(p1p2) if new not in gens: gens.extend(new) if r2-1 == s1 and not (p2p1).is_identity: new = _process(p2p1) if new not in gens: gens.extend(new) self._min_words = gens min_words = self._min_words def _is_subword(w): # check if w is a word in _min_words or one of # the infinite families in it w, r = w.cyclic_reduction(removed=True) if r.is_identity or self.normal: return w in min_words else: t = [s[1] for s in min_words if isinstance(s, tuple) and s[0] == r] return [s for s in t if w.power_of(s)] != [] # store the solution of words for which the result of # _word_break (below) is known known = {} def _word_break(w): # check if w can be written as a product of words # in min_words if len(w) == 0: return True i = 0 while i < len(w): i += 1 prefix = w.subword(0, i) if not _is_subword(prefix): continue rest = w.subword(i, len(w)) if rest not in known: known[rest] = _word_break(rest) if known[rest]: return True return False if self.normal: g = g.cyclic_reduction() return _word_break(g) else: if self.C is None: C = self.parent.coset_enumeration(self.generators) self.C = C i = 0 C = self.C for j in range(len(g)): i = C.table[i][C.A_dict[g[j]]] return i == 0 def order(self): if not self.generators: return S.One if isinstance(self.parent, FreeGroup): return S.Infinity if self.C is None: C = self.parent.coset_enumeration(self.generators) self.C = C # This is valid because `len(self.C.table)` (the index of the subgroup) # will always be finite - otherwise coset enumeration doesn't terminate return self.parent.order()/len(self.C.table) def to_FpGroup(self): if isinstance(self.parent, FreeGroup): gen_syms = [('x_%d'%i) for i in range(len(self.generators))] return free_group(', '.join(gen_syms))[0] return self.parent.subgroup(C=self.C) def __str__(self): if len(self.generators) > 30: str_form = "<fp subgroup with %s generators>" % len(self.generators) else: str_form = "<fp subgroup on the generators %s>" % str(self.generators) return str_form __repr__ = __str__ ############################################################################### # LOW INDEX SUBGROUPS # ############################################################################### def low_index_subgroups(G, N, Y=()): """ Implements the Low Index Subgroups algorithm, i.e find all subgroups of ``G`` upto a given index ``N``. This implements the method described in [Sim94]. This procedure involves a backtrack search over incomplete Coset Tables, rather than over forced coincidences. Parameters ========== G: An FpGroup < X\|R > N: positive integer, representing the maximum index value for subgroups Y: (an optional argument) specifying a list of subgroup generators, such that each of the resulting subgroup contains the subgroup generated by Y. Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup, low_index_subgroups >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x2, y3, (xy)*4]) >>> L = low_index_subgroups(f, 4) >>> for coset_table in L: ... print(coset_table.table) [[0, 0, 0, 0]] [[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 3, 3]] [[0, 0, 1, 2], [2, 2, 2, 0], [1, 1, 0, 1]] [[1, 1, 0, 0], [0, 0, 1, 1]] References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" Section 5.4 .. [2] Marston Conder and Peter Dobcsanyi "Applications and Adaptions of the Low Index Subgroups Procedure" """ C = CosetTable(G, []) R = G.relators # length chosen for the length of the short relators len_short_rel = 5 # elements of R2 only checked at the last step for complete # coset tables R2 = {rel for rel in R if len(rel) > len_short_rel} # elements of R1 are used in inner parts of the process to prune # branches of the search tree, R1 = {rel.identity_cyclic_reduction() for rel in set(R) - R2} R1_c_list = C.conjugates(R1) S = [] descendant_subgroups(S, C, R1_c_list, C.A[0], R2, N, Y) return S def descendant_subgroups(S, C, R1_c_list, x, R2, N, Y): A_dict = C.A_dict A_dict_inv = C.A_dict_inv if C.is_complete(): # if C is complete then it only needs to test # whether the relators in R2 are satisfied for w, alpha in product(R2, C.omega): if not C.scan_check(alpha, w): return # relators in R2 are satisfied, append the table to list S.append(C) else: # find the first undefined entry in Coset Table for alpha, x in product(range(len(C.table)), C.A): if C.table[alpha][A_dict[x]] is None: # this is "x" in pseudo-code (using "y" makes it clear) undefined_coset, undefined_gen = alpha, x break # for filling up the undefine entry we try all possible values # of beta in Omega or beta = n where beta^(undefined_gen^-1) is undefined reach = C.omega + [C.n] for beta in reach: if beta < N: if beta == C.n or C.table[beta][A_dict_inv[undefined_gen]] is None: try_descendant(S, C, R1_c_list, R2, N, undefined_coset, \ undefined_gen, beta, Y) def try_descendant(S, C, R1_c_list, R2, N, alpha, x, beta, Y): r""" Solves the problem of trying out each individual possibility for `\alpha^x. """ D = C.copy() if beta == D.n and beta < N: D.table.append([None]len(D.A)) D.p.append(beta) D.table[alpha][D.A_dict[x]] = beta D.table[beta][D.A_dict_inv[x]] = alpha D.deduction_stack.append((alpha, x)) if not D.process_deductions_check(R1_c_list[D.A_dict[x]], \ R1_c_list[D.A_dict_inv[x]]): return for w in Y: if not D.scan_check(0, w): return if first_in_class(D, Y): descendant_subgroups(S, D, R1_c_list, x, R2, N, Y) def first_in_class(C, Y=()): """ Checks whether the subgroup ``H=G1`` corresponding to the Coset Table could possibly be the canonical representative of its conjugacy class. Parameters ========== C: CosetTable Returns ======= bool: True/False If this returns False, then no descendant of C can have that property, and so we can abandon C. If it returns True, then we need to process further the node of the search tree corresponding to C, and so we call ``descendant_subgroups`` recursively on C. Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup, CosetTable, first_in_class >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x2, y3, (xy)4]) >>> C = CosetTable(f, []) >>> C.table = [[0, 0, None, None]] >>> first_in_class(C) True >>> C.table = [[1, 1, 1, None], [0, 0, None, 1]]; C.p = [0, 1] >>> first_in_class(C) True >>> C.table = [[1, 1, 2, 1], [0, 0, 0, None], [None, None, None, 0]] >>> C.p = [0, 1, 2] >>> first_in_class(C) False >>> C.table = [[1, 1, 1, 2], [0, 0, 2, 0], [2, None, 0, 1]] >>> first_in_class(C) False # TODO:: Sims points out in [Sim94] that performance can be improved by # remembering some of the information computed by ``first_in_class``. If # the ``continue alpha`` statement is executed at line 14, then the same thing # will happen for that value of alpha in any descendant of the table C, and so # the values the values of alpha for which this occurs could profitably be # stored and passed through to the descendants of C. Of course this would # make the code more complicated. # The code below is taken directly from the function on page 208 of [Sim94] # nu[alpha] """ n = C.n # lamda is the largest numbered point in Omega_c_alpha which is currently defined lamda = -1 # for alpha in Omega_c, nu[alpha] is the point in Omega_c_alpha corresponding to alpha nu = [None]n # for alpha in Omega_c_alpha, mu[alpha] is the point in Omega_c corresponding to alpha mu = [None]n # mutually nu and mu are the mutually-inverse equivalence maps between # Omega_c_alpha and Omega_c next_alpha = False # For each 0!=alpha in [0 .. nc-1], we start by constructing the equivalent # standardized coset table C_alpha corresponding to H_alpha for alpha in range(1, n): # reset nu to "None" after previous value of alpha for beta in range(lamda+1): nu[mu[beta]] = None # we only want to reject our current table in favour of a preceding # table in the ordering in which 1 is replaced by alpha, if the subgroup # G_alpha corresponding to this preceding table definitely contains the # given subgroup for w in Y: # TODO: this should support input of a list of general words # not just the words which are in "A" (i.e gen and gen^-1) if C.table[alpha][C.A_dict[w]] != alpha: # continue with alpha next_alpha = True break if next_alpha: next_alpha = False continue # try alpha as the new point 0 in Omega_C_alpha mu[0] = alpha nu[alpha] = 0 # compare corresponding entries in C and C_alpha lamda = 0 for beta in range(n): for x in C.A: gamma = C.table[beta][C.A_dict[x]] delta = C.table[mu[beta]][C.A_dict[x]] # if either of the entries is undefined, # we move with next alpha if gamma is None or delta is None: # continue with alpha next_alpha = True break if nu[delta] is None: # delta becomes the next point in Omega_C_alpha lamda += 1 nu[delta] = lamda mu[lamda] = delta if nu[delta] < gamma: return False if nu[delta] > gamma: # continue with alpha next_alpha = True break if next_alpha: next_alpha = False break return True #======================================================================== # Simplifying Presentation #======================================================================== def simplify_presentation(args, change_gens=False): ''' For an instance of `FpGroup`, return a simplified isomorphic copy of the group (e.g. remove redundant generators or relators). Alternatively, a list of generators and relators can be passed in which case the simplified lists will be returned. By default, the generators of the group are unchanged. If you would like to remove redundant generators, set the keyword argument `change_gens = True`. ''' if len(args) == 1: if not isinstance(args[0], FpGroup): raise TypeError("The argument must be an instance of FpGroup") G = args[0] gens, rels = simplify_presentation(G.generators, G.relators, change_gens=change_gens) if gens: return FpGroup(gens[0].group, rels) return FpGroup(FreeGroup([]), []) elif len(args) == 2: gens, rels = args[0][:], args[1][:] if not gens: return gens, rels identity = gens[0].group.identity else: if len(args) == 0: m = "Not enough arguments" else: m = "Too many arguments" raise RuntimeError(m) prev_gens = [] prev_rels = [] while not set(prev_rels) == set(rels): prev_rels = rels while change_gens and not set(prev_gens) == set(gens): prev_gens = gens gens, rels = elimination_technique_1(gens, rels, identity) rels = _simplify_relators(rels, identity) if change_gens: syms = [g.array_form[0][0] for g in gens] F = free_group(syms)[0] identity = F.identity gens = F.generators subs = dict(zip(syms, gens)) for j, r in enumerate(rels): a = r.array_form rel = identity for sym, p in a: rel = relsubs[sym]p rels[j] = rel return gens, rels def _simplify_relators(rels, identity): """Relies upon ``_simplification_technique_1`` for its functioning. """ rels = rels[:] rels = list(set(_simplification_technique_1(rels))) rels.sort() rels = [r.identity_cyclic_reduction() for r in rels] try: rels.remove(identity) except ValueError: pass return rels # Pg 350, section 2.5.1 from [2] def elimination_technique_1(gens, rels, identity): rels = rels[:] # the shorter relators are examined first so that generators selected for # elimination will have shorter strings as equivalent rels.sort() gens = gens[:] redundant_gens = {} redundant_rels = [] used_gens = set() # examine each relator in relator list for any generator occurring exactly # once for rel in rels: # don't look for a redundant generator in a relator which # depends on previously found ones contained_gens = rel.contains_generators() if any(g in contained_gens for g in redundant_gens): continue contained_gens = list(contained_gens) contained_gens.sort(reverse = True) for gen in contained_gens: if rel.generator_count(gen) == 1 and gen not in used_gens: k = rel.exponent_sum(gen) gen_index = rel.index(genk) bk = rel.subword(gen_index + 1, len(rel)) fw = rel.subword(0, gen_index) chi = bkfw redundant_gens[gen] = chi*(-1k) used_gens.update(chi.contains_generators()) redundant_rels.append(rel) break rels = [r for r in rels if r not in redundant_rels] # eliminate the redundant generators from remaining relators rels = [r.eliminate_words(redundant_gens, _all = True).identity_cyclic_reduction() for r in rels] rels = list(set(rels)) try: rels.remove(identity) except ValueError: pass gens = [g for g in gens if g not in redundant_gens] return gens, rels def _simplification_technique_1(rels): """ All relators are checked to see if they are of the form `gen^n`. If any such relators are found then all other relators are processed for strings in the `gen` known order. Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import _simplification_technique_1 >>> F, x, y = free_group("x, y") >>> w1 = [x*2y4, x3] >>> _simplification_technique_1(w1) [x*-1y4, x3] >>> w2 = [x*2y*-4x5, x3, x*2y8, y5] >>> _simplification_technique_1(w2) [x*-1yx-1, x3, x-1y-2, y5] >>> w3 = [x*6y4, x4] >>> _simplification_technique_1(w3) [x*2y4, x4] """ rels = rels[:] # dictionary with "gen: n" where gen^n is one of the relators exps = {} for i in range(len(rels)): rel = rels[i] if rel.number_syllables() == 1: g = rel[0] exp = abs(rel.array_form[0][1]) if rel.array_form[0][1] < 0: rels[i] = rels[i]-1 g = g-1 if g in exps: exp = gcd(exp, exps[g].array_form[0][1]) exps[g] = gexp one_syllables_words = exps.values() # decrease some of the exponents in relators, making use of the single # syllable relators for i in range(len(rels)): rel = rels[i] if rel in one_syllables_words: continue rel = rel.eliminate_words(one_syllables_words, _all = True) # if rels[i] contains gn where abs(n) is greater than half of the power p # of g in exps, gn can be replaced by g(n-p) (or g(p-n) if n<0) for g in rel.contains_generators(): if g in exps: exp = exps[g].array_form[0][1] max_exp = (exp + 1)//2 rel = rel.eliminate_word(g(max_exp), g(max_exp-exp), _all = True) rel = rel.eliminate_word(g(-max_exp), g*(-(max_exp-exp)), _all = True) rels[i] = rel rels = [r.identity_cyclic_reduction() for r in rels] return rels ############################################################################### # SUBGROUP PRESENTATIONS # ############################################################################### # Pg 175 [1] def define_schreier_generators(C, homomorphism=False): ''' Parameters ========== C -- Coset table. homomorphism -- When set to True, return a dictionary containing the images of the presentation generators in the original group. ''' y = [] gamma = 1 f = C.fp_group X = f.generators if homomorphism: # `_gens` stores the elements of the parent group to # to which the schreier generators correspond to. _gens = {} # compute the schreier Traversal tau = {} tau[0] = f.identity C.P = [[None]len(C.A) for i in range(C.n)] for alpha, x in product(C.omega, C.A): beta = C.table[alpha][C.A_dict[x]] if beta == gamma: C.P[alpha][C.A_dict[x]] = "<identity>" C.P[beta][C.A_dict_inv[x]] = "<identity>" gamma += 1 if homomorphism: tau[beta] = tau[alpha]x elif x in X and C.P[alpha][C.A_dict[x]] is None: y_alpha_x = '%s_%s' % (x, alpha) y.append(y_alpha_x) C.P[alpha][C.A_dict[x]] = y_alpha_x if homomorphism: _gens[y_alpha_x] = tau[alpha]xtau[beta]-1 grp_gens = list(free_group(', '.join(y))) C._schreier_free_group = grp_gens.pop(0) C._schreier_generators = grp_gens if homomorphism: C._schreier_gen_elem = _gens # replace all elements of P by, free group elements for i, j in product(range(len(C.P)), range(len(C.A))): # if equals "<identity>", replace by identity element if C.P[i][j] == "<identity>": C.P[i][j] = C._schreier_free_group.identity elif isinstance(C.P[i][j], str): r = C._schreier_generators[y.index(C.P[i][j])] C.P[i][j] = r beta = C.table[i][j] C.P[beta][j + 1] = r-1 def reidemeister_relators(C): R = C.fp_group.relators rels = [rewrite(C, coset, word) for word in R for coset in range(C.n)] order_1_gens = {i for i in rels if len(i) == 1} # remove all the order 1 generators from relators rels = list(filter(lambda rel: rel not in order_1_gens, rels)) # replace order 1 generators by identity element in reidemeister relators for i in range(len(rels)): w = rels[i] w = w.eliminate_words(order_1_gens, _all=True) rels[i] = w C._schreier_generators = [i for i in C._schreier_generators if not (i in order_1_gens or i-1 in order_1_gens)] # Tietze transformation 1 i.e TT_1 # remove cyclic conjugate elements from relators i = 0 while i < len(rels): w = rels[i] j = i + 1 while j < len(rels): if w.is_cyclic_conjugate(rels[j]): del rels[j] else: j += 1 i += 1 C._reidemeister_relators = rels def rewrite(C, alpha, w): """ Parameters ========== C: CosetTable alpha: A live coset w: A word in `A` Returns ======= rho(tau(alpha), w) Examples ======== >>> from sympy.combinatorics.fp_groups import FpGroup, CosetTable, define_schreier_generators, rewrite >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x2, y3, (xy)6]) >>> C = CosetTable(f, []) >>> C.table = [[1, 1, 2, 3], [0, 0, 4, 5], [4, 4, 3, 0], [5, 5, 0, 2], [2, 2, 5, 1], [3, 3, 1, 4]] >>> C.p = [0, 1, 2, 3, 4, 5] >>> define_schreier_generators(C) >>> rewrite(C, 0, (xy)*6) x_4y_2x_3x_1x_2y_4x_5 """ v = C._schreier_free_group.identity for i in range(len(w)): x_i = w[i] v = vC.P[alpha][C.A_dict[x_i]] alpha = C.table[alpha][C.A_dict[x_i]] return v # Pg 350, section 2.5.2 from [2] def elimination_technique_2(C): """ This technique eliminates one generator at a time. Heuristically this seems superior in that we may select for elimination the generator with shortest equivalent string at each stage. >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_r, \ reidemeister_relators, define_schreier_generators, elimination_technique_2 >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x3, y5, (xy)2]); H = [xy, x*-1y*-1xyx] >>> C = coset_enumeration_r(f, H) >>> C.compress(); C.standardize() >>> define_schreier_generators(C) >>> reidemeister_relators(C) >>> elimination_technique_2(C) ([y_1, y_2], [y_2*-3, y_2y_1y_2y_1y_2y_1, y_12]) """ rels = C._reidemeister_relators rels.sort(reverse=True) gens = C._schreier_generators for i in range(len(gens) - 1, -1, -1): rel = rels[i] for j in range(len(gens) - 1, -1, -1): gen = gens[j] if rel.generator_count(gen) == 1: k = rel.exponent_sum(gen) gen_index = rel.index(genk) bk = rel.subword(gen_index + 1, len(rel)) fw = rel.subword(0, gen_index) rep_by = (bkfw)(-1k) del rels[i]; del gens[j] for l in range(len(rels)): rels[l] = rels[l].eliminate_word(gen, rep_by) break C._reidemeister_relators = rels C._schreier_generators = gens return C._schreier_generators, C._reidemeister_relators def reidemeister_presentation(fp_grp, H, C=None, homomorphism=False): """ Parameters ========== fp_group: A finitely presented group, an instance of FpGroup H: A subgroup whose presentation is to be found, given as a list of words in generators of `fp_grp` homomorphism: When set to True, return a homomorphism from the subgroup to the parent group Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup, reidemeister_presentation >>> F, x, y = free_group("x, y") Example 5.6 Pg. 177 from [1] >>> f = FpGroup(F, [x3, y5, (xy)2]) >>> H = [xy, x*-1y*-1xyx] >>> reidemeister_presentation(f, H) ((y_1, y_2), (y_12, y_23, y_2y_1y_2y_1y_2y_1)) Example 5.8 Pg. 183 from [1] >>> f = FpGroup(F, [x3, y3, (xy)*3]) >>> H = [xy, xy-1] >>> reidemeister_presentation(f, H) ((x_0, y_0), (x_03, y_03, x_0y_0x_0y_0x_0y_0)) Exercises Q2. Pg 187 from [1] >>> f = FpGroup(F, [x*2y2, y-1xyx-3]) >>> H = [x] >>> reidemeister_presentation(f, H) ((x_0,), (x_04,)) Example 5.9 Pg. 183 from [1] >>> f = FpGroup(F, [x3y*-3, (xy)*3, (xy-1)2]) >>> H = [x] >>> reidemeister_presentation(f, H) ((x_0,), (x_0**6,)) """ if not C: C = coset_enumeration_r(fp_grp, H) C.compress(); C.standardize() define_schreier_generators(C, homomorphism=homomorphism) reidemeister_relators(C) gens, rels = C._schreier_generators, C._reidemeister_relators gens, rels = simplify_presentation(gens, rels, change_gens=True) C.schreier_generators = tuple(gens) C.reidemeister_relators = tuple(rels) if homomorphism: _gens = [] for gen in gens: _gens.append(C._schreier_gen_elem[str(gen)]) return C.schreier_generators, C.reidemeister_relators, _gens return C.schreier_generators, C.reidemeister_relators FpGroupElement = FreeGroupElement
20ef9aba38058354b1b5b04ff9d32315c05bc68e17451459c00b5a3a6e0fee6b	from typing import Tuple as tTuple from .expr_with_intlimits import ExprWithIntLimits from .summations import Sum, summation, _dummy_with_inherited_properties_concrete from sympy.core.expr import Expr from sympy.core.exprtools import factor_terms from sympy.core.function import Derivative from sympy.core.mul import Mul from sympy.core.singleton import S from sympy.core.symbol import Dummy, Symbol from sympy.functions.combinatorial.factorials import RisingFactorial from sympy.functions.elementary.exponential import exp, log from sympy.functions.special.tensor_functions import KroneckerDelta from sympy.polys import quo, roots class Product(ExprWithIntLimits): r""" Represents unevaluated products. Explanation =========== ``Product`` represents a finite or infinite product, with the first argument being the general form of terms in the series, and the second argument being ``(dummy_variable, start, end)``, with ``dummy_variable`` taking all integer values from ``start`` through ``end``. In accordance with long-standing mathematical convention, the end term is included in the product. Finite products =============== For finite products (and products with symbolic limits assumed to be finite) we follow the analogue of the summation convention described by Karr [1], especially definition 3 of section 1.4. The product: .. math:: \prod_{m \leq i < n} f(i) has the obvious meaning for `m < n`, namely: .. math:: \prod_{m \leq i < n} f(i) = f(m) f(m+1) \cdot \ldots \cdot f(n-2) f(n-1) with the upper limit value `f(n)` excluded. The product over an empty set is one if and only if `m = n`: .. math:: \prod_{m \leq i < n} f(i) = 1 \quad \mathrm{for} \quad m = n Finally, for all other products over empty sets we assume the following definition: .. math:: \prod_{m \leq i < n} f(i) = \frac{1}{\prod_{n \leq i < m} f(i)} \quad \mathrm{for} \quad m > n It is important to note that above we define all products with the upper limit being exclusive. This is in contrast to the usual mathematical notation, but does not affect the product convention. Indeed we have: .. math:: \prod_{m \leq i < n} f(i) = \prod_{i = m}^{n - 1} f(i) where the difference in notation is intentional to emphasize the meaning, with limits typeset on the top being inclusive. Examples ======== >>> from sympy.abc import a, b, i, k, m, n, x >>> from sympy import Product, oo >>> Product(k, (k, 1, m)) Product(k, (k, 1, m)) >>> Product(k, (k, 1, m)).doit() factorial(m) >>> Product(k2,(k, 1, m)) Product(k2, (k, 1, m)) >>> Product(k2,(k, 1, m)).doit() factorial(m)2 Wallis' product for pi: >>> W = Product(2i/(2i-1) * 2i/(2i+1), (i, 1, oo)) >>> W Product(4i2/((2i - 1)(2i + 1)), (i, 1, oo)) Direct computation currently fails: >>> W.doit() Product(4i2/((2i - 1)(2i + 1)), (i, 1, oo)) But we can approach the infinite product by a limit of finite products: >>> from sympy import limit >>> W2 = Product(2i/(2i-1)2i/(2i+1), (i, 1, n)) >>> W2 Product(4i*2/((2i - 1)(2i + 1)), (i, 1, n)) >>> W2e = W2.doit() >>> W2e 4*nfactorial(n)2/(2(2n)RisingFactorial(1/2, n)RisingFactorial(3/2, n)) >>> limit(W2e, n, oo) pi/2 By the same formula we can compute sin(pi/2): >>> from sympy import combsimp, pi, gamma, simplify >>> P = pi x * Product(1 - x2/k2, (k, 1, n)) >>> P = P.subs(x, pi/2) >>> P pi*2Product(1 - pi*2/(4k2), (k, 1, n))/2 >>> Pe = P.doit() >>> Pe pi2RisingFactorial(1 - pi/2, n)RisingFactorial(1 + pi/2, n)/(2factorial(n)2) >>> limit(Pe, n, oo).gammasimp() sin(pi2/2) >>> Pe.rewrite(gamma) (-1)npi*2gamma(pi/2)gamma(n + 1 + pi/2)/(2gamma(1 + pi/2)gamma(-n + pi/2)gamma(n + 1)2) Products with the lower limit being larger than the upper one: >>> Product(1/i, (i, 6, 1)).doit() 120 >>> Product(i, (i, 2, 5)).doit() 120 The empty product: >>> Product(i, (i, n, n-1)).doit() 1 An example showing that the symbolic result of a product is still valid for seemingly nonsensical values of the limits. Then the Karr convention allows us to give a perfectly valid interpretation to those products by interchanging the limits according to the above rules: >>> P = Product(2, (i, 10, n)).doit() >>> P 2(n - 9) >>> P.subs(n, 5) 1/16 >>> Product(2, (i, 10, 5)).doit() 1/16 >>> 1/Product(2, (i, 6, 9)).doit() 1/16 An explicit example of the Karr summation convention applied to products: >>> P1 = Product(x, (i, a, b)).doit() >>> P1 x(-a + b + 1) >>> P2 = Product(x, (i, b+1, a-1)).doit() >>> P2 x(a - b - 1) >>> simplify(P1 * P2) 1 And another one: >>> P1 = Product(i, (i, b, a)).doit() >>> P1 RisingFactorial(b, a - b + 1) >>> P2 = Product(i, (i, a+1, b-1)).doit() >>> P2 RisingFactorial(a + 1, -a + b - 1) >>> P1 * P2 RisingFactorial(b, a - b + 1)RisingFactorial(a + 1, -a + b - 1) >>> combsimp(P1 P2) 1 See Also ======== Sum, summation product References ========== .. [1] Michael Karr, "Summation in Finite Terms", Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 http://dl.acm.org/citation.cfm?doid=322248.322255 .. [2] https://en.wikipedia.org/wiki/Multiplication#Capital_Pi_notation .. [3] https://en.wikipedia.org/wiki/Empty_product """ __slots__ = () limits: tTuple[tTuple[Symbol, Expr, Expr]] def __new__(cls, function, symbols, assumptions): obj = ExprWithIntLimits.__new__(cls, function, symbols, *assumptions) return obj def _eval_rewrite_as_Sum(self, args, *kwargs): return exp(Sum(log(self.function), self.limits)) @property def term(self): return self._args[0] function = term def _eval_is_zero(self): if self.has_empty_sequence: return False z = self.term.is_zero if z is True: return True if self.has_finite_limits: # A Product is zero only if its term is zero assuming finite limits. return z def _eval_is_extended_real(self): if self.has_empty_sequence: return True return self.function.is_extended_real def _eval_is_positive(self): if self.has_empty_sequence: return True if self.function.is_positive and self.has_finite_limits: return True def _eval_is_nonnegative(self): if self.has_empty_sequence: return True if self.function.is_nonnegative and self.has_finite_limits: return True def _eval_is_extended_nonnegative(self): if self.has_empty_sequence: return True if self.function.is_extended_nonnegative: return True def _eval_is_extended_nonpositive(self): if self.has_empty_sequence: return True def _eval_is_finite(self): if self.has_finite_limits and self.function.is_finite: return True def doit(self, hints): # first make sure any definite limits have product # variables with matching assumptions reps = {} for xab in self.limits: d = _dummy_with_inherited_properties_concrete(xab) if d: reps[xab[0]] = d if reps: undo = {v: k for k, v in reps.items()} did = self.xreplace(reps).doit(hints) if isinstance(did, tuple): # when separate=True did = tuple([i.xreplace(undo) for i in did]) else: did = did.xreplace(undo) return did from sympy.simplify.powsimp import powsimp f = self.function for index, limit in enumerate(self.limits): i, a, b = limit dif = b - a if dif.is_integer and dif.is_negative: a, b = b + 1, a - 1 f = 1 / f g = self._eval_product(f, (i, a, b)) if g in (None, S.NaN): return self.func(powsimp(f), self.limits[index:]) else: f = g if hints.get('deep', True): return f.doit(hints) else: return powsimp(f) def _eval_adjoint(self): if self.is_commutative: return self.func(self.function.adjoint(), self.limits) return None def _eval_conjugate(self): return self.func(self.function.conjugate(), self.limits) def _eval_product(self, term, limits): (k, a, n) = limits if k not in term.free_symbols: if (term - 1).is_zero: return S.One return term(n - a + 1) if a == n: return term.subs(k, a) from .delta import deltaproduct, _has_simple_delta if term.has(KroneckerDelta) and _has_simple_delta(term, limits[0]): return deltaproduct(term, limits) dif = n - a definite = dif.is_Integer if definite and (dif < 100): return self._eval_product_direct(term, limits) elif term.is_polynomial(k): poly = term.as_poly(k) A = B = Q = S.One all_roots = roots(poly) M = 0 for r, m in all_roots.items(): M += m A = RisingFactorial(a - r, n - a + 1)*m Q = (n - r)m if M < poly.degree(): arg = quo(poly, Q.as_poly(k)) B = self.func(arg, (k, a, n)).doit() return poly.LC()(n - a + 1) * A * B elif term.is_Add: factored = factor_terms(term, fraction=True) if factored.is_Mul: return self._eval_product(factored, (k, a, n)) elif term.is_Mul: # Factor in part without the summation variable and part with without_k, with_k = term.as_coeff_mul(k) if len(with_k) >= 2: # More than one term including k, so still a multiplication exclude, include = [], [] for t in with_k: p = self._eval_product(t, (k, a, n)) if p is not None: exclude.append(p) else: include.append(t) if not exclude: return None else: arg = term._new_rawargs(include) A = Mul(exclude) B = self.func(arg, (k, a, n)).doit() return without_k*(n - a + 1)A * B else: # Just a single term p = self._eval_product(with_k[0], (k, a, n)) if p is None: p = self.func(with_k[0], (k, a, n)).doit() return without_k*(n - a + 1)p elif term.is_Pow: if not term.base.has(k): s = summation(term.exp, (k, a, n)) return term.bases elif not term.exp.has(k): p = self._eval_product(term.base, (k, a, n)) if p is not None: return pterm.exp elif isinstance(term, Product): evaluated = term.doit() f = self._eval_product(evaluated, limits) if f is None: return self.func(evaluated, limits) else: return f if definite: return self._eval_product_direct(term, limits) def _eval_simplify(self, *kwargs): from sympy.simplify.simplify import product_simplify rv = product_simplify(self) return rv.doit() if kwargs['doit'] else rv def _eval_transpose(self): if self.is_commutative: return self.func(self.function.transpose(), self.limits) return None def _eval_product_direct(self, term, limits): (k, a, n) = limits return Mul([term.subs(k, a + i) for i in range(n - a + 1)]) def _eval_derivative(self, x): if isinstance(x, Symbol) and x not in self.free_symbols: return S.Zero f, limits = self.function, list(self.limits) limit = limits.pop(-1) if limits: f = self.func(f, limits) i, a, b = limit if x in a.free_symbols or x in b.free_symbols: return None h = Dummy() rv = Sum( Product(f, (i, a, h - 1)) * Product(f, (i, h + 1, b)) * Derivative(f, x, evaluate=True).subs(i, h), (h, a, b)) return rv def is_convergent(self): r""" See docs of :obj:`.Sum.is_convergent()` for explanation of convergence in SymPy. Explanation =========== The infinite product: .. math:: \prod_{1 \leq i < \infty} f(i) is defined by the sequence of partial products: .. math:: \prod_{i=1}^{n} f(i) = f(1) f(2) \cdots f(n) as n increases without bound. The product converges to a non-zero value if and only if the sum: .. math:: \sum_{1 \leq i < \infty} \log{f(n)} converges. Examples ======== >>> from sympy import Product, Symbol, cos, pi, exp, oo >>> n = Symbol('n', integer=True) >>> Product(n/(n + 1), (n, 1, oo)).is_convergent() False >>> Product(1/n2, (n, 1, oo)).is_convergent() False >>> Product(cos(pi/n), (n, 1, oo)).is_convergent() True >>> Product(exp(-n2), (n, 1, oo)).is_convergent() False References ========== .. [1] https://en.wikipedia.org/wiki/Infinite_product """ sequence_term = self.function log_sum = log(sequence_term) lim = self.limits try: is_conv = Sum(log_sum, lim).is_convergent() except NotImplementedError: if Sum(sequence_term - 1, lim).is_absolutely_convergent() is S.true: return S.true raise NotImplementedError("The algorithm to find the product convergence of %s " "is not yet implemented" % (sequence_term)) return is_conv def reverse_order(expr, indices): """ Reverse the order of a limit in a Product. Explanation =========== ``reverse_order(expr, indices)`` reverses some limits in the expression ``expr`` which can be either a ``Sum`` or a ``Product``. The selectors in the argument ``indices`` specify some indices whose limits get reversed. These selectors are either variable names or numerical indices counted starting from the inner-most limit tuple. Examples ======== >>> from sympy import gamma, Product, simplify, Sum >>> from sympy.abc import x, y, a, b, c, d >>> P = Product(x, (x, a, b)) >>> Pr = P.reverse_order(x) >>> Pr Product(1/x, (x, b + 1, a - 1)) >>> Pr = Pr.doit() >>> Pr 1/RisingFactorial(b + 1, a - b - 1) >>> simplify(Pr.rewrite(gamma)) Piecewise((gamma(b + 1)/gamma(a), b > -1), ((-1)*(-a + b + 1)gamma(1 - a)/gamma(-b), True)) >>> P = P.doit() >>> P RisingFactorial(a, -a + b + 1) >>> simplify(P.rewrite(gamma)) Piecewise((gamma(b + 1)/gamma(a), a > 0), ((-1)*(-a + b + 1)gamma(1 - a)/gamma(-b), True)) While one should prefer variable names when specifying which limits to reverse, the index counting notation comes in handy in case there are several symbols with the same name. >>> S = Sum(xy, (x, a, b), (y, c, d)) >>> S Sum(xy, (x, a, b), (y, c, d)) >>> S0 = S.reverse_order(0) >>> S0 Sum(-xy, (x, b + 1, a - 1), (y, c, d)) >>> S1 = S0.reverse_order(1) >>> S1 Sum(xy, (x, b + 1, a - 1), (y, d + 1, c - 1)) Of course we can mix both notations: >>> Sum(xy, (x, a, b), (y, 2, 5)).reverse_order(x, 1) Sum(xy, (x, b + 1, a - 1), (y, 6, 1)) >>> Sum(xy, (x, a, b), (y, 2, 5)).reverse_order(y, x) Sum(xy, (x, b + 1, a - 1), (y, 6, 1)) See Also ======== sympy.concrete.expr_with_intlimits.ExprWithIntLimits.index, reorder_limit, sympy.concrete.expr_with_intlimits.ExprWithIntLimits.reorder References ========== .. [1] Michael Karr, "Summation in Finite Terms", Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 http://dl.acm.org/citation.cfm?doid=322248.322255 """ l_indices = list(indices) for i, indx in enumerate(l_indices): if not isinstance(indx, int): l_indices[i] = expr.index(indx) e = 1 limits = [] for i, limit in enumerate(expr.limits): l = limit if i in l_indices: e = -e l = (limit[0], limit[2] + 1, limit[1] - 1) limits.append(l) return Product(expr.function ** e, limits) def product(args, kwargs): r""" Compute the product. Explanation =========== The notation for symbols is similar to the notation used in Sum or Integral. product(f, (i, a, b)) computes the product of f with respect to i from a to b, i.e., :: b _____ product(f(n), (i, a, b)) = \| \| f(n) \| \| i = a If it cannot compute the product, it returns an unevaluated Product object. Repeated products can be computed by introducing additional symbols tuples:: Examples ======== >>> from sympy import product, symbols >>> i, n, m, k = symbols('i n m k', integer=True) >>> product(i, (i, 1, k)) factorial(k) >>> product(m, (i, 1, k)) mk >>> product(i, (i, 1, k), (k, 1, n)) Product(factorial(k), (k, 1, n)) """ prod = Product(args, *kwargs) if isinstance(prod, Product): return prod.doit(deep=False) else: return prod
be3e0f8edb41f7113969462edfb475038fcab4683170f6bae63ce3bfcfa4aec7	from sympy.concrete.expr_with_limits import ExprWithLimits from sympy.core.singleton import S from sympy.core.relational import Eq class ReorderError(NotImplementedError): """ Exception raised when trying to reorder dependent limits. """ def __init__(self, expr, msg): super().__init__( "%s could not be reordered: %s." % (expr, msg)) class ExprWithIntLimits(ExprWithLimits): """ Superclass for Product and Sum. See Also ======== sympy.concrete.expr_with_limits.ExprWithLimits sympy.concrete.products.Product sympy.concrete.summations.Sum """ __slots__ = () def change_index(self, var, trafo, newvar=None): r""" Change index of a Sum or Product. Perform a linear transformation `x \mapsto a x + b` on the index variable `x`. For `a` the only values allowed are `\pm 1`. A new variable to be used after the change of index can also be specified. Explanation =========== ``change_index(expr, var, trafo, newvar=None)`` where ``var`` specifies the index variable `x` to transform. The transformation ``trafo`` must be linear and given in terms of ``var``. If the optional argument ``newvar`` is provided then ``var`` gets replaced by ``newvar`` in the final expression. Examples ======== >>> from sympy import Sum, Product, simplify >>> from sympy.abc import x, y, a, b, c, d, u, v, i, j, k, l >>> S = Sum(x, (x, a, b)) >>> S.doit() -a2/2 + a/2 + b2/2 + b/2 >>> Sn = S.change_index(x, x + 1, y) >>> Sn Sum(y - 1, (y, a + 1, b + 1)) >>> Sn.doit() -a2/2 + a/2 + b2/2 + b/2 >>> Sn = S.change_index(x, -x, y) >>> Sn Sum(-y, (y, -b, -a)) >>> Sn.doit() -a2/2 + a/2 + b2/2 + b/2 >>> Sn = S.change_index(x, x+u) >>> Sn Sum(-u + x, (x, a + u, b + u)) >>> Sn.doit() -a*2/2 - au + a/2 + b*2/2 + bu + b/2 - u(-a + b + 1) + u >>> simplify(Sn.doit()) -a2/2 + a/2 + b2/2 + b/2 >>> Sn = S.change_index(x, -x - u, y) >>> Sn Sum(-u - y, (y, -b - u, -a - u)) >>> Sn.doit() -a2/2 - au + a/2 + b*2/2 + bu + b/2 - u(-a + b + 1) + u >>> simplify(Sn.doit()) -a2/2 + a/2 + b2/2 + b/2 >>> P = Product(ij*2, (i, a, b), (j, c, d)) >>> P Product(ij2, (i, a, b), (j, c, d)) >>> P2 = P.change_index(i, i+3, k) >>> P2 Product(j2(k - 3), (k, a + 3, b + 3), (j, c, d)) >>> P3 = P2.change_index(j, -j, l) >>> P3 Product(l2(k - 3), (k, a + 3, b + 3), (l, -d, -c)) When dealing with symbols only, we can make a general linear transformation: >>> Sn = S.change_index(x, ux+v, y) >>> Sn Sum((-v + y)/u, (y, bu + v, au + v)) >>> Sn.doit() -v(au - bu + 1)/u + (a*2u*2/2 + auv + au/2 - b*2u*2/2 - buv + bu/2 + v)/u >>> simplify(Sn.doit()) a*2u/2 + a/2 - b*2u/2 + b/2 However, the last result can be inconsistent with usual summation where the index increment is always 1. This is obvious as we get back the original value only for ``u`` equal +1 or -1. See Also ======== sympy.concrete.expr_with_intlimits.ExprWithIntLimits.index, reorder_limit, sympy.concrete.expr_with_intlimits.ExprWithIntLimits.reorder, sympy.concrete.summations.Sum.reverse_order, sympy.concrete.products.Product.reverse_order """ if newvar is None: newvar = var limits = [] for limit in self.limits: if limit[0] == var: p = trafo.as_poly(var) if p.degree() != 1: raise ValueError("Index transformation is not linear") alpha = p.coeff_monomial(var) beta = p.coeff_monomial(S.One) if alpha.is_number: if alpha == S.One: limits.append((newvar, alphalimit[1] + beta, alphalimit[2] + beta)) elif alpha == S.NegativeOne: limits.append((newvar, alphalimit[2] + beta, alphalimit[1] + beta)) else: raise ValueError("Linear transformation results in non-linear summation stepsize") else: # Note that the case of alpha being symbolic can give issues if alpha < 0. limits.append((newvar, alphalimit[2] + beta, alphalimit[1] + beta)) else: limits.append(limit) function = self.function.subs(var, (var - beta)/alpha) function = function.subs(var, newvar) return self.func(function, limits) def index(expr, x): """ Return the index of a dummy variable in the list of limits. Explanation =========== ``index(expr, x)`` returns the index of the dummy variable ``x`` in the limits of ``expr``. Note that we start counting with 0 at the inner-most limits tuple. Examples ======== >>> from sympy.abc import x, y, a, b, c, d >>> from sympy import Sum, Product >>> Sum(xy, (x, a, b), (y, c, d)).index(x) 0 >>> Sum(xy, (x, a, b), (y, c, d)).index(y) 1 >>> Product(xy, (x, a, b), (y, c, d)).index(x) 0 >>> Product(xy, (x, a, b), (y, c, d)).index(y) 1 See Also ======== reorder_limit, reorder, sympy.concrete.summations.Sum.reverse_order, sympy.concrete.products.Product.reverse_order """ variables = [limit[0] for limit in expr.limits] if variables.count(x) != 1: raise ValueError(expr, "Number of instances of variable not equal to one") else: return variables.index(x) def reorder(expr, arg): """ Reorder limits in a expression containing a Sum or a Product. Explanation =========== ``expr.reorder(arg)`` reorders the limits in the expression ``expr`` according to the list of tuples given by ``arg``. These tuples can contain numerical indices or index variable names or involve both. Examples ======== >>> from sympy import Sum, Product >>> from sympy.abc import x, y, z, a, b, c, d, e, f >>> Sum(xy, (x, a, b), (y, c, d)).reorder((x, y)) Sum(xy, (y, c, d), (x, a, b)) >>> Sum(xyz, (x, a, b), (y, c, d), (z, e, f)).reorder((x, y), (x, z), (y, z)) Sum(xyz, (z, e, f), (y, c, d), (x, a, b)) >>> P = Product(xyz, (x, a, b), (y, c, d), (z, e, f)) >>> P.reorder((x, y), (x, z), (y, z)) Product(xyz, (z, e, f), (y, c, d), (x, a, b)) We can also select the index variables by counting them, starting with the inner-most one: >>> Sum(x2, (x, a, b), (x, c, d)).reorder((0, 1)) Sum(x2, (x, c, d), (x, a, b)) And of course we can mix both schemes: >>> Sum(xy, (x, a, b), (y, c, d)).reorder((y, x)) Sum(xy, (y, c, d), (x, a, b)) >>> Sum(xy, (x, a, b), (y, c, d)).reorder((y, 0)) Sum(xy, (y, c, d), (x, a, b)) See Also ======== reorder_limit, index, sympy.concrete.summations.Sum.reverse_order, sympy.concrete.products.Product.reverse_order """ new_expr = expr for r in arg: if len(r) != 2: raise ValueError(r, "Invalid number of arguments") index1 = r[0] index2 = r[1] if not isinstance(r[0], int): index1 = expr.index(r[0]) if not isinstance(r[1], int): index2 = expr.index(r[1]) new_expr = new_expr.reorder_limit(index1, index2) return new_expr def reorder_limit(expr, x, y): """ Interchange two limit tuples of a Sum or Product expression. Explanation =========== ``expr.reorder_limit(x, y)`` interchanges two limit tuples. The arguments ``x`` and ``y`` are integers corresponding to the index variables of the two limits which are to be interchanged. The expression ``expr`` has to be either a Sum or a Product. Examples ======== >>> from sympy.abc import x, y, z, a, b, c, d, e, f >>> from sympy import Sum, Product >>> Sum(xyz, (x, a, b), (y, c, d), (z, e, f)).reorder_limit(0, 2) Sum(xyz, (z, e, f), (y, c, d), (x, a, b)) >>> Sum(x2, (x, a, b), (x, c, d)).reorder_limit(1, 0) Sum(x2, (x, c, d), (x, a, b)) >>> Product(xyz, (x, a, b), (y, c, d), (z, e, f)).reorder_limit(0, 2) Product(xyz, (z, e, f), (y, c, d), (x, a, b)) See Also ======== index, reorder, sympy.concrete.summations.Sum.reverse_order, sympy.concrete.products.Product.reverse_order """ var = {limit[0] for limit in expr.limits} limit_x = expr.limits[x] limit_y = expr.limits[y] if (len(set(limit_x[1].free_symbols).intersection(var)) == 0 and len(set(limit_x[2].free_symbols).intersection(var)) == 0 and len(set(limit_y[1].free_symbols).intersection(var)) == 0 and len(set(limit_y[2].free_symbols).intersection(var)) == 0): limits = [] for i, limit in enumerate(expr.limits): if i == x: limits.append(limit_y) elif i == y: limits.append(limit_x) else: limits.append(limit) return type(expr)(expr.function, limits) else: raise ReorderError(expr, "could not interchange the two limits specified") @property def has_empty_sequence(self): """ Returns True if the Sum or Product is computed for an empty sequence. Examples ======== >>> from sympy import Sum, Product, Symbol >>> m = Symbol('m') >>> Sum(m, (m, 1, 0)).has_empty_sequence True >>> Sum(m, (m, 1, 1)).has_empty_sequence False >>> M = Symbol('M', integer=True, positive=True) >>> Product(m, (m, 1, M)).has_empty_sequence False >>> Product(m, (m, 2, M)).has_empty_sequence >>> Product(m, (m, M + 1, M)).has_empty_sequence True >>> N = Symbol('N', integer=True, positive=True) >>> Sum(m, (m, N, M)).has_empty_sequence >>> N = Symbol('N', integer=True, negative=True) >>> Sum(m, (m, N, M)).has_empty_sequence False See Also ======== has_reversed_limits has_finite_limits """ ret_None = False for lim in self.limits: dif = lim[1] - lim[2] eq = Eq(dif, 1) if eq == True: return True elif eq == False: continue else: ret_None = True if ret_None: return None return False
c24ea80a374b2909a53ebb525ff5e68c2938a56f11b75ce4f94b4d6011b4b822	""" This module implements sums and products containing the Kronecker Delta function. References ========== .. [1] http://mathworld.wolfram.com/KroneckerDelta.html """ from .products import product from .summations import Sum, summation from sympy.core import Add, Mul, S, Dummy from sympy.core.cache import cacheit from sympy.core.sorting import default_sort_key from sympy.functions import KroneckerDelta, Piecewise, piecewise_fold from sympy.polys.polytools import factor from sympy.sets.sets import Interval from sympy.solvers.solvers import solve @cacheit def _expand_delta(expr, index): """ Expand the first Add containing a simple KroneckerDelta. """ if not expr.is_Mul: return expr delta = None func = Add terms = [S.One] for h in expr.args: if delta is None and h.is_Add and _has_simple_delta(h, index): delta = True func = h.func terms = [terms[0]t for t in h.args] else: terms = [th for t in terms] return func(terms) @cacheit def _extract_delta(expr, index): """ Extract a simple KroneckerDelta from the expression. Explanation =========== Returns the tuple ``(delta, newexpr)`` where: - ``delta`` is a simple KroneckerDelta expression if one was found, or ``None`` if no simple KroneckerDelta expression was found. - ``newexpr`` is a Mul containing the remaining terms; ``expr`` is returned unchanged if no simple KroneckerDelta expression was found. Examples ======== >>> from sympy import KroneckerDelta >>> from sympy.concrete.delta import _extract_delta >>> from sympy.abc import x, y, i, j, k >>> _extract_delta(4xyKroneckerDelta(i, j), i) (KroneckerDelta(i, j), 4xy) >>> _extract_delta(4xyKroneckerDelta(i, j), k) (None, 4xyKroneckerDelta(i, j)) See Also ======== sympy.functions.special.tensor_functions.KroneckerDelta deltaproduct deltasummation """ if not _has_simple_delta(expr, index): return (None, expr) if isinstance(expr, KroneckerDelta): return (expr, S.One) if not expr.is_Mul: raise ValueError("Incorrect expr") delta = None terms = [] for arg in expr.args: if delta is None and _is_simple_delta(arg, index): delta = arg else: terms.append(arg) return (delta, expr.func(terms)) @cacheit def _has_simple_delta(expr, index): """ Returns True if ``expr`` is an expression that contains a KroneckerDelta that is simple in the index ``index``, meaning that this KroneckerDelta is nonzero for a single value of the index ``index``. """ if expr.has(KroneckerDelta): if _is_simple_delta(expr, index): return True if expr.is_Add or expr.is_Mul: for arg in expr.args: if _has_simple_delta(arg, index): return True return False @cacheit def _is_simple_delta(delta, index): """ Returns True if ``delta`` is a KroneckerDelta and is nonzero for a single value of the index ``index``. """ if isinstance(delta, KroneckerDelta) and delta.has(index): p = (delta.args[0] - delta.args[1]).as_poly(index) if p: return p.degree() == 1 return False @cacheit def _remove_multiple_delta(expr): """ Evaluate products of KroneckerDelta's. """ if expr.is_Add: return expr.func(list(map(_remove_multiple_delta, expr.args))) if not expr.is_Mul: return expr eqs = [] newargs = [] for arg in expr.args: if isinstance(arg, KroneckerDelta): eqs.append(arg.args[0] - arg.args[1]) else: newargs.append(arg) if not eqs: return expr solns = solve(eqs, dict=True) if len(solns) == 0: return S.Zero elif len(solns) == 1: for key in solns[0].keys(): newargs.append(KroneckerDelta(key, solns[0][key])) expr2 = expr.func(newargs) if expr != expr2: return _remove_multiple_delta(expr2) return expr @cacheit def _simplify_delta(expr): """ Rewrite a KroneckerDelta's indices in its simplest form. """ if isinstance(expr, KroneckerDelta): try: slns = solve(expr.args[0] - expr.args[1], dict=True) if slns and len(slns) == 1: return Mul([KroneckerDelta((key, value)) for key, value in slns[0].items()]) except NotImplementedError: pass return expr @cacheit def deltaproduct(f, limit): """ Handle products containing a KroneckerDelta. See Also ======== deltasummation sympy.functions.special.tensor_functions.KroneckerDelta sympy.concrete.products.product """ if ((limit[2] - limit[1]) < 0) == True: return S.One if not f.has(KroneckerDelta): return product(f, limit) if f.is_Add: # Identify the term in the Add that has a simple KroneckerDelta delta = None terms = [] for arg in sorted(f.args, key=default_sort_key): if delta is None and _has_simple_delta(arg, limit[0]): delta = arg else: terms.append(arg) newexpr = f.func(terms) k = Dummy("kprime", integer=True) if isinstance(limit[1], int) and isinstance(limit[2], int): result = deltaproduct(newexpr, limit) + sum([ deltaproduct(newexpr, (limit[0], limit[1], ik - 1)) * delta.subs(limit[0], ik) * deltaproduct(newexpr, (limit[0], ik + 1, limit[2])) for ik in range(int(limit[1]), int(limit[2] + 1))] ) else: result = deltaproduct(newexpr, limit) + deltasummation( deltaproduct(newexpr, (limit[0], limit[1], k - 1)) * delta.subs(limit[0], k) * deltaproduct(newexpr, (limit[0], k + 1, limit[2])), (k, limit[1], limit[2]), no_piecewise=_has_simple_delta(newexpr, limit[0]) ) return _remove_multiple_delta(result) delta, _ = _extract_delta(f, limit[0]) if not delta: g = _expand_delta(f, limit[0]) if f != g: try: return factor(deltaproduct(g, limit)) except AssertionError: return deltaproduct(g, limit) return product(f, limit) return _remove_multiple_delta(f.subs(limit[0], limit[1])KroneckerDelta(limit[2], limit[1])) + \ S.One_simplify_delta(KroneckerDelta(limit[2], limit[1] - 1)) @cacheit def deltasummation(f, limit, no_piecewise=False): """ Handle summations containing a KroneckerDelta. Explanation =========== The idea for summation is the following: - If we are dealing with a KroneckerDelta expression, i.e. KroneckerDelta(g(x), j), we try to simplify it. If we could simplify it, then we sum the resulting expression. We already know we can sum a simplified expression, because only simple KroneckerDelta expressions are involved. If we could not simplify it, there are two cases: 1) The expression is a simple expression: we return the summation, taking care if we are dealing with a Derivative or with a proper KroneckerDelta. 2) The expression is not simple (i.e. KroneckerDelta(cos(x))): we can do nothing at all. - If the expr is a multiplication expr having a KroneckerDelta term: First we expand it. If the expansion did work, then we try to sum the expansion. If not, we try to extract a simple KroneckerDelta term, then we have two cases: 1) We have a simple KroneckerDelta term, so we return the summation. 2) We did not have a simple term, but we do have an expression with simplified KroneckerDelta terms, so we sum this expression. Examples ======== >>> from sympy import oo, symbols >>> from sympy.abc import k >>> i, j = symbols('i, j', integer=True, finite=True) >>> from sympy.concrete.delta import deltasummation >>> from sympy import KroneckerDelta >>> deltasummation(KroneckerDelta(i, k), (k, -oo, oo)) 1 >>> deltasummation(KroneckerDelta(i, k), (k, 0, oo)) Piecewise((1, i >= 0), (0, True)) >>> deltasummation(KroneckerDelta(i, k), (k, 1, 3)) Piecewise((1, (i >= 1) & (i <= 3)), (0, True)) >>> deltasummation(kKroneckerDelta(i, j)KroneckerDelta(j, k), (k, -oo, oo)) jKroneckerDelta(i, j) >>> deltasummation(jKroneckerDelta(i, j), (j, -oo, oo)) i >>> deltasummation(iKroneckerDelta(i, j), (i, -oo, oo)) j See Also ======== deltaproduct sympy.functions.special.tensor_functions.KroneckerDelta sympy.concrete.sums.summation """ if ((limit[2] - limit[1]) < 0) == True: return S.Zero if not f.has(KroneckerDelta): return summation(f, limit) x = limit[0] g = _expand_delta(f, x) if g.is_Add: return piecewise_fold( g.func([deltasummation(h, limit, no_piecewise) for h in g.args])) # try to extract a simple KroneckerDelta term delta, expr = _extract_delta(g, x) if (delta is not None) and (delta.delta_range is not None): dinf, dsup = delta.delta_range if (limit[1] - dinf <= 0) == True and (limit[2] - dsup >= 0) == True: no_piecewise = True if not delta: return summation(f, limit) solns = solve(delta.args[0] - delta.args[1], x) if len(solns) == 0: return S.Zero elif len(solns) != 1: return Sum(f, limit) value = solns[0] if no_piecewise: return expr.subs(x, value) return Piecewise( (expr.subs(x, value), Interval(*limit[1:3]).as_relational(value)), (S.Zero, True) )
deaf20919cffa0078138151f39b32fc00ac190129845e1d1d29ad55cc22aacd1	"""Gosper's algorithm for hypergeometric summation. """ from sympy.core import S, Dummy, symbols from sympy.polys import Poly, parallel_poly_from_expr, factor from sympy.utilities.iterables import is_sequence def gosper_normal(f, g, n, polys=True): r""" Compute the Gosper's normal form of ``f`` and ``g``. Explanation =========== Given relatively prime univariate polynomials ``f`` and ``g``, rewrite their quotient to a normal form defined as follows: .. math:: \frac{f(n)}{g(n)} = Z \cdot \frac{A(n) C(n+1)}{B(n) C(n)} where ``Z`` is an arbitrary constant and ``A``, ``B``, ``C`` are monic polynomials in ``n`` with the following properties: 1. `\gcd(A(n), B(n+h)) = 1 \forall h \in \mathbb{N}` 2. `\gcd(B(n), C(n+1)) = 1` 3. `\gcd(A(n), C(n)) = 1` This normal form, or rational factorization in other words, is a crucial step in Gosper's algorithm and in solving of difference equations. It can be also used to decide if two hypergeometric terms are similar or not. This procedure will return a tuple containing elements of this factorization in the form ``(ZA, B, C)``. Examples ======== >>> from sympy.concrete.gosper import gosper_normal >>> from sympy.abc import n >>> gosper_normal(4n+5, 2(4n+1)(2n+3), n, polys=False) (1/4, n + 3/2, n + 1/4) """ (p, q), opt = parallel_poly_from_expr( (f, g), n, field=True, extension=True) a, A = p.LC(), p.monic() b, B = q.LC(), q.monic() C, Z = A.one, a/b h = Dummy('h') D = Poly(n + h, n, h, domain=opt.domain) R = A.resultant(B.compose(D)) roots = set(R.ground_roots().keys()) for r in set(roots): if not r.is_Integer or r < 0: roots.remove(r) for i in sorted(roots): d = A.gcd(B.shift(+i)) A = A.quo(d) B = B.quo(d.shift(-i)) for j in range(1, i + 1): C = d.shift(-j) A = A.mul_ground(Z) if not polys: A = A.as_expr() B = B.as_expr() C = C.as_expr() return A, B, C def gosper_term(f, n): r""" Compute Gosper's hypergeometric term for ``f``. Explanation =========== Suppose ``f`` is a hypergeometric term such that: .. math:: s_n = \sum_{k=0}^{n-1} f_k and `f_k` does not depend on `n`. Returns a hypergeometric term `g_n` such that `g_{n+1} - g_n = f_n`. Examples ======== >>> from sympy.concrete.gosper import gosper_term >>> from sympy import factorial >>> from sympy.abc import n >>> gosper_term((4n + 1)factorial(n)/factorial(2n + 1), n) (-n - 1/2)/(n + 1/4) """ from sympy.simplify import hypersimp r = hypersimp(f, n) if r is None: return None # 'f' is not a hypergeometric term p, q = r.as_numer_denom() A, B, C = gosper_normal(p, q, n) B = B.shift(-1) N = S(A.degree()) M = S(B.degree()) K = S(C.degree()) if (N != M) or (A.LC() != B.LC()): D = {K - max(N, M)} elif not N: D = {K - N + 1, S.Zero} else: D = {K - N + 1, (B.nth(N - 1) - A.nth(N - 1))/A.LC()} for d in set(D): if not d.is_Integer or d < 0: D.remove(d) if not D: return None # 'f(n)' is not Gosper-summable d = max(D) coeffs = symbols('c:%s' % (d + 1), cls=Dummy) domain = A.get_domain().inject(coeffs) x = Poly(coeffs, n, domain=domain) H = Ax.shift(1) - Bx - C from sympy.solvers.solvers import solve solution = solve(H.coeffs(), coeffs) if solution is None: return None # 'f(n)' is not* Gosper-summable x = x.as_expr().subs(solution) for coeff in coeffs: if coeff not in solution: x = x.subs(coeff, 0) if x.is_zero: return None # 'f(n)' is not Gosper-summable else: return B.as_expr()x/C.as_expr() def gosper_sum(f, k): r""" Gosper's hypergeometric summation algorithm. Explanation =========== Given a hypergeometric term ``f`` such that: .. math :: s_n = \sum_{k=0}^{n-1} f_k and `f(n)` does not depend on `n`, returns `g_{n} - g(0)` where `g_{n+1} - g_n = f_n`, or ``None`` if `s_n` cannot be expressed in closed form as a sum of hypergeometric terms. Examples ======== >>> from sympy.concrete.gosper import gosper_sum >>> from sympy import factorial >>> from sympy.abc import n, k >>> f = (4k + 1)factorial(k)/factorial(2k + 1) >>> gosper_sum(f, (k, 0, n)) (-factorial(n) + 2factorial(2n + 1))/factorial(2n + 1) >>> _.subs(n, 2) == sum(f.subs(k, i) for i in [0, 1, 2]) True >>> gosper_sum(f, (k, 3, n)) (-60factorial(n) + factorial(2n + 1))/(60factorial(2n + 1)) >>> _.subs(n, 5) == sum(f.subs(k, i) for i in [3, 4, 5]) True References ========== .. [1] Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger, A = B, AK Peters, Ltd., Wellesley, MA, USA, 1997, pp. 73--100 """ indefinite = False if is_sequence(k): k, a, b = k else: indefinite = True g = gosper_term(f, k) if g is None: return None if indefinite: result = fg else: result = (f(g + 1)).subs(k, b) - (fg).subs(k, a) if result is S.NaN: try: result = (f(g + 1)).limit(k, b) - (fg).limit(k, a) except NotImplementedError: result = None return factor(result)
fe9125a246ec3177db19be19e3fe6fc1a4bac8a49e5f845b706802d78136d56b	"""Various algorithms for helping identifying numbers and sequences.""" from sympy.concrete.products import (Product, product) from sympy.core import Function, S from sympy.core.numbers import (Zero, Integer, Rational) from sympy.core.symbol import Symbol, symbols from sympy.core.sympify import sympify from sympy.functions.elementary.exponential import exp from sympy.functions.elementary.integers import floor from sympy.integrals.integrals import integrate from sympy.polys.polyfuncs import rational_interpolate as rinterp from sympy.polys.polytools import lcm from sympy.simplify.radsimp import denom from sympy.utilities import public @public def find_simple_recurrence_vector(l): """ This function is used internally by other functions from the sympy.concrete.guess module. While most users may want to rather use the function find_simple_recurrence when looking for recurrence relations among rational numbers, the current function may still be useful when some post-processing has to be done. Explanation =========== The function returns a vector of length n when a recurrence relation of order n is detected in the sequence of rational numbers v. If the returned vector has a length 1, then the returned value is always the list [0], which means that no relation has been found. While the functions is intended to be used with rational numbers, it should work for other kinds of real numbers except for some cases involving quadratic numbers; for that reason it should be used with some caution when the argument is not a list of rational numbers. Examples ======== >>> from sympy.concrete.guess import find_simple_recurrence_vector >>> from sympy import fibonacci >>> find_simple_recurrence_vector([fibonacci(k) for k in range(12)]) [1, -1, -1] See Also ======== See the function sympy.concrete.guess.find_simple_recurrence which is more user-friendly. """ q1 = [0] q2 = [Integer(1)] b, z = 0, len(l) >> 1 while len(q2) <= z: while l[b]==0: b += 1 if b == len(l): c = 1 for x in q2: c = lcm(c, denom(x)) if q2[0]c < 0: c = -c for k in range(len(q2)): q2[k] = int(q2[k]c) return q2 a = Integer(1)/l[b] m = [a] for k in range(b+1, len(l)): m.append(-sum(l[j+1]m[b-j-1] for j in range(b, k))a) l, m = m, [0] * max(len(q2), b+len(q1)) for k in range(len(q2)): m[k] = aq2[k] for k in range(b, b+len(q1)): m[k] += q1[k-b] while m[-1]==0: m.pop() # because trailing zeros can occur q1, q2, b = q2, m, 1 return [0] @public def find_simple_recurrence(v, A=Function('a'), N=Symbol('n')): """ Detects and returns a recurrence relation from a sequence of several integer (or rational) terms. The name of the function in the returned expression is 'a' by default; the main variable is 'n' by default. The smallest index in the returned expression is always n (and never n-1, n-2, etc.). Examples ======== >>> from sympy.concrete.guess import find_simple_recurrence >>> from sympy import fibonacci >>> find_simple_recurrence([fibonacci(k) for k in range(12)]) -a(n) - a(n + 1) + a(n + 2) >>> from sympy import Function, Symbol >>> a = [1, 1, 1] >>> for k in range(15): a.append(5a[-1]-3a[-2]+8a[-3]) >>> find_simple_recurrence(a, A=Function('f'), N=Symbol('i')) -8f(i) + 3f(i + 1) - 5f(i + 2) + f(i + 3) """ p = find_simple_recurrence_vector(v) n = len(p) if n <= 1: return Zero() rel = Zero() for k in range(n): rel += A(N+n-1-k)p[k] return rel @public def rationalize(x, maxcoeff=10000): """ Helps identifying a rational number from a float (or mpmath.mpf) value by using a continued fraction. The algorithm stops as soon as a large partial quotient is detected (greater than 10000 by default). Examples ======== >>> from sympy.concrete.guess import rationalize >>> from mpmath import cos, pi >>> rationalize(cos(pi/3)) 1/2 >>> from mpmath import mpf >>> rationalize(mpf("0.333333333333333")) 1/3 While the function is rather intended to help 'identifying' rational values, it may be used in some cases for approximating real numbers. (Though other functions may be more relevant in that case.) >>> rationalize(pi, maxcoeff = 250) 355/113 See Also ======== Several other methods can approximate a real number as a rational, like: * fractions.Fraction.from_decimal * fractions.Fraction.from_float * mpmath.identify * mpmath.pslq by using the following syntax: mpmath.pslq([x, 1]) * mpmath.findpoly by using the following syntax: mpmath.findpoly(x, 1) * sympy.simplify.nsimplify (which is a more general function) The main difference between the current function and all these variants is that control focuses on magnitude of partial quotients here rather than on global precision of the approximation. If the real is "known to be" a rational number, the current function should be able to detect it correctly with the default settings even when denominator is great (unless its expansion contains unusually big partial quotients) which may occur when studying sequences of increasing numbers. If the user cares more on getting simple fractions, other methods may be more convenient. """ p0, p1 = 0, 1 q0, q1 = 1, 0 a = floor(x) while a < maxcoeff or q1==0: p = ap1 + p0 q = aq1 + q0 p0, p1 = p1, p q0, q1 = q1, q if x==a: break x = 1/(x-a) a = floor(x) return sympify(p) / q @public def guess_generating_function_rational(v, X=Symbol('x')): """ Tries to "guess" a rational generating function for a sequence of rational numbers v. Examples ======== >>> from sympy.concrete.guess import guess_generating_function_rational >>> from sympy import fibonacci >>> l = [fibonacci(k) for k in range(5,15)] >>> guess_generating_function_rational(l) (3x + 5)/(-x2 - x + 1) See Also ======== sympy.series.approximants mpmath.pade """ # a) compute the denominator as q q = find_simple_recurrence_vector(v) n = len(q) if n <= 1: return None # b) compute the numerator as p p = [sum(v[i-k]q[k] for k in range(min(i+1, n))) for i in range(len(v)>>1)] return (sum(p[k]Xk for k in range(len(p))) / sum(q[k]X*k for k in range(n))) @public def guess_generating_function(v, X=Symbol('x'), types=['all'], maxsqrtn=2): """ Tries to "guess" a generating function for a sequence of rational numbers v. Only a few patterns are implemented yet. Explanation =========== The function returns a dictionary where keys are the name of a given type of generating function. Six types are currently implemented: type \| formal definition -------+---------------------------------------------------------------- ogf \| f(x) = Sum( a_k x^k , k: 0..infinity ) egf \| f(x) = Sum( a_k * x^k / k! , k: 0..infinity ) lgf \| f(x) = Sum( (-1)^(k+1) a_k * x^k / k , k: 1..infinity ) \| (with initial index being hold as 1 rather than 0) hlgf \| f(x) = Sum( a_k * x^k / k , k: 1..infinity ) \| (with initial index being hold as 1 rather than 0) lgdogf \| f(x) = derivate( log(Sum( a_k * x^k, k: 0..infinity )), x) lgdegf \| f(x) = derivate( log(Sum( a_k * x^k / k!, k: 0..infinity )), x) In order to spare time, the user can select only some types of generating functions (default being ['all']). While forgetting to use a list in the case of a single type may seem to work most of the time as in: types='ogf' this (convenient) syntax may lead to unexpected extra results in some cases. Discarding a type when calling the function does not mean that the type will not be present in the returned dictionary; it only means that no extra computation will be performed for that type, but the function may still add it in the result when it can be easily converted from another type. Two generating functions (lgdogf and lgdegf) are not even computed if the initial term of the sequence is 0; it may be useful in that case to try again after having removed the leading zeros. Examples ======== >>> from sympy.concrete.guess import guess_generating_function as ggf >>> ggf([k+1 for k in range(12)], types=['ogf', 'lgf', 'hlgf']) {'hlgf': 1/(1 - x), 'lgf': 1/(x + 1), 'ogf': 1/(x*2 - 2x + 1)} >>> from sympy import sympify >>> l = sympify("[3/2, 11/2, 0, -121/2, -363/2, 121]") >>> ggf(l) {'ogf': (x + 3/2)/(11x2 - 3x + 1)} >>> from sympy import fibonacci >>> ggf([fibonacci(k) for k in range(5, 15)], types=['ogf']) {'ogf': (3x + 5)/(-x2 - x + 1)} >>> from sympy import factorial >>> ggf([factorial(k) for k in range(12)], types=['ogf', 'egf', 'lgf']) {'egf': 1/(1 - x)} >>> ggf([k+1 for k in range(12)], types=['egf']) {'egf': (x + 1)exp(x), 'lgdegf': (x + 2)/(x + 1)} N-th root of a rational function can also be detected (below is an example coming from the sequence A108626 from http://oeis.org). The greatest n-th root to be tested is specified as maxsqrtn (default 2). >>> ggf([1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122, 38919])['ogf'] sqrt(1/(x*4 + 2x*2 - 4x + 1)) References ========== .. [1] "Concrete Mathematics", R.L. Graham, D.E. Knuth, O. Patashnik .. [2] https://oeis.org/wiki/Generating_functions """ # List of all types of all g.f. known by the algorithm if 'all' in types: types = ['ogf', 'egf', 'lgf', 'hlgf', 'lgdogf', 'lgdegf'] result = {} # Ordinary Generating Function (ogf) if 'ogf' in types: # Perform some convolutions of the sequence with itself t = [1 if k==0 else 0 for k in range(len(v))] for d in range(max(1, maxsqrtn)): t = [sum(t[n-i]v[i] for i in range(n+1)) for n in range(len(v))] g = guess_generating_function_rational(t, X=X) if g: result['ogf'] = gRational(1, d+1) break # Exponential Generating Function (egf) if 'egf' in types: # Transform sequence (division by factorial) w, f = [], S.One for i, k in enumerate(v): f = i if i else 1 w.append(k/f) # Perform some convolutions of the sequence with itself t = [1 if k==0 else 0 for k in range(len(w))] for d in range(max(1, maxsqrtn)): t = [sum(t[n-i]w[i] for i in range(n+1)) for n in range(len(w))] g = guess_generating_function_rational(t, X=X) if g: result['egf'] = gRational(1, d+1) break # Logarithmic Generating Function (lgf) if 'lgf' in types: # Transform sequence (multiplication by (-1)^(n+1) / n) w, f = [], S.NegativeOne for i, k in enumerate(v): f = -f w.append(fk/Integer(i+1)) # Perform some convolutions of the sequence with itself t = [1 if k==0 else 0 for k in range(len(w))] for d in range(max(1, maxsqrtn)): t = [sum(t[n-i]w[i] for i in range(n+1)) for n in range(len(w))] g = guess_generating_function_rational(t, X=X) if g: result['lgf'] = gRational(1, d+1) break # Hyperbolic logarithmic Generating Function (hlgf) if 'hlgf' in types: # Transform sequence (division by n+1) w = [] for i, k in enumerate(v): w.append(k/Integer(i+1)) # Perform some convolutions of the sequence with itself t = [1 if k==0 else 0 for k in range(len(w))] for d in range(max(1, maxsqrtn)): t = [sum(t[n-i]w[i] for i in range(n+1)) for n in range(len(w))] g = guess_generating_function_rational(t, X=X) if g: result['hlgf'] = g*Rational(1, d+1) break # Logarithmic derivative of ordinary generating Function (lgdogf) if v[0] != 0 and ('lgdogf' in types or ('ogf' in types and 'ogf' not in result)): # Transform sequence by computing f'(x)/f(x) # because log(f(x)) = integrate( f'(x)/f(x) ) a, w = sympify(v[0]), [] for n in range(len(v)-1): w.append( (v[n+1](n+1) - sum(w[-i-1]v[i+1] for i in range(n)))/a) # Perform some convolutions of the sequence with itself t = [1 if k==0 else 0 for k in range(len(w))] for d in range(max(1, maxsqrtn)): t = [sum(t[n-i]w[i] for i in range(n+1)) for n in range(len(w))] g = guess_generating_function_rational(t, X=X) if g: result['lgdogf'] = g*Rational(1, d+1) if 'ogf' not in result: result['ogf'] = exp(integrate(result['lgdogf'], X)) break # Logarithmic derivative of exponential generating Function (lgdegf) if v[0] != 0 and ('lgdegf' in types or ('egf' in types and 'egf' not in result)): # Transform sequence / step 1 (division by factorial) z, f = [], Integer(1) for i, k in enumerate(v): f = i if i else 1 z.append(k/f) # Transform sequence / step 2 by computing f'(x)/f(x) # because log(f(x)) = integrate( f'(x)/f(x) ) a, w = z[0], [] for n in range(len(z)-1): w.append( (z[n+1](n+1) - sum(w[-i-1]z[i+1] for i in range(n)))/a) # Perform some convolutions of the sequence with itself t = [1 if k==0 else 0 for k in range(len(w))] for d in range(max(1, maxsqrtn)): t = [sum(t[n-i]w[i] for i in range(n+1)) for n in range(len(w))] g = guess_generating_function_rational(t, X=X) if g: result['lgdegf'] = gRational(1, d+1) if 'egf' not in result: result['egf'] = exp(integrate(result['lgdegf'], X)) break return result @public def guess(l, all=False, evaluate=True, niter=2, variables=None): """ This function is adapted from the Rate.m package for Mathematica written by Christian Krattenthaler. It tries to guess a formula from a given sequence of rational numbers. Explanation =========== In order to speed up the process, the 'all' variable is set to False by default, stopping the computation as some results are returned during an iteration; the variable can be set to True if more iterations are needed (other formulas may be found; however they may be equivalent to the first ones). Another option is the 'evaluate' variable (default is True); setting it to False will leave the involved products unevaluated. By default, the number of iterations is set to 2 but a greater value (up to len(l)-1) can be specified with the optional 'niter' variable. More and more convoluted results are found when the order of the iteration gets higher: first iteration returns polynomial or rational functions; * second iteration returns products of rising factorials and their inverses; * third iteration returns products of products of rising factorials and their inverses; * etc. The returned formulas contain symbols i0, i1, i2, ... where the main variables is i0 (and auxiliary variables are i1, i2, ...). A list of other symbols can be provided in the 'variables' option; the length of the least should be the value of 'niter' (more is acceptable but only the first symbols will be used); in this case, the main variable will be the first symbol in the list. Examples ======== >>> from sympy.concrete.guess import guess >>> guess([1,2,6,24,120], evaluate=False) [Product(i1 + 1, (i1, 1, i0 - 1))] >>> from sympy import symbols >>> r = guess([1,2,7,42,429,7436,218348,10850216], niter=4) >>> i0 = symbols("i0") >>> [r[0].subs(i0,n).doit() for n in range(1,10)] [1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460] """ if any(a==0 for a in l[:-1]): return [] N = len(l) niter = min(N-1, niter) myprod = product if evaluate else Product g = [] res = [] if variables is None: symb = symbols('i:'+str(niter)) else: symb = variables for k, s in enumerate(symb): g.append(l) n, r = len(l), [] for i in range(n-2-1, -1, -1): ri = rinterp(enumerate(g[k][:-1], start=1), i, X=s) if ((denom(ri).subs({s:n}) != 0) and (ri.subs({s:n}) - g[k][-1] == 0) and ri not in r): r.append(ri) if r: for i in range(k-1, -1, -1): r = list(map(lambda v: g[i][0] * myprod(v, (symb[i+1], 1, symb[i]-1)), r)) if not all: return r res += r l = [Rational(l[i+1], l[i]) for i in range(N-k-1)] return res

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