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1203 lines
50 KiB
1203 lines
50 KiB
"""
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This code is automatically generated. Never edit it manually.
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For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
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"""
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from sympy.external import import_module
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matchpy = import_module("matchpy")
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if matchpy:
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from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
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from sympy.integrals.rubi.utility_function import (
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Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
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ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
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PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
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ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
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NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
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FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
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SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
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Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
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IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
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ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
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ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
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Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
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PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
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ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
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Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
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SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
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NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
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InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
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EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
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PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
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TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
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NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
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ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
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NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
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ExpandExpression, Apart, SmartApart, MatchQ,
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PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
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RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
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ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
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AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
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RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
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MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
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GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
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PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
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RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
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RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
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SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
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CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
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GeneralizedBinomialParts, GeneralizedTrinomialDegree,
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GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
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MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
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PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
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TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
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QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
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NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
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PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
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AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
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QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
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Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
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NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
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CombineExponents, FactorInteger, FactorAbsurdNumber,
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SubstForInverseFunction, SubstForFractionalPower,
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SubstForFractionalPowerOfQuotientOfLinears,
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FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
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SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
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FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
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MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
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TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
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OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
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PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
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PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
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FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
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FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
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NormalizeIntegrandAux, NormalizeIntegrandFactor,
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NormalizeIntegrandFactorBase, NormalizeTogether,
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NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
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SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
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TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
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UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
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PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
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IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
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FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
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FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
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SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
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SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
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SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
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InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
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FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
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PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
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KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
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KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
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AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
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ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
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ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
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FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
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PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
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FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
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FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
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FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
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PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
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SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
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Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
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SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
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ConstantFactor, SameQ, ReplacePart, CommonFactors,
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MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
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FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
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FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
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rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
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RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
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SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
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FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
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_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
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_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
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_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
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TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
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PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
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CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
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Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
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Negative, Quotient
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)
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from sympy.core.add import Add
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from sympy.core.mod import Mod
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from sympy.core.mul import Mul
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from sympy.core import EulerGamma
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from sympy.core.numbers import (Float, I, Integer)
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from sympy.core.power import Pow
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from sympy.core.singleton import S
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from sympy.functions.elementary.complexes import (Abs, sign)
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.integrals.integrals import Integral
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from sympy.logic.boolalg import (And, Or)
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from sympy.simplify.simplify import simplify
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from sympy.integrals.rubi.symbol import WC
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from sympy.core.symbol import symbols, Symbol
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from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
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from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
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from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
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from sympy.core.numbers import pi as Pi
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A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
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a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
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i, ii, Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
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_UseGamma = False
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ShowSteps = False
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StepCounter = None
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def miscellaneous_integration():
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from sympy.integrals.rubi.constraints import cons149, cons2004, cons2, cons3, cons8, cons4, cons5, cons388, cons29, cons52, cons2005, cons2006, cons2007, cons2008, cons50, cons127, cons210, cons36, cons37, cons38, cons1101, cons2009, cons68, cons19, cons86, cons1039, cons1038, cons40, cons2010, cons10, cons2011, cons2012, cons2013, cons211, cons1833, cons1246, cons2014, cons48, cons2015, cons2016, cons2017, cons2018, cons54, cons2019, cons802, cons2020, cons20, cons2021, cons588, cons2022, cons2023, cons2024, cons2025, cons2026, cons2027, cons2028, cons2029, cons2030, cons669, cons198, cons2031, cons842, cons2032, cons21, cons2033, cons150, cons47, cons2034, cons1856, cons1249, cons263, cons2035, cons369, cons2036, cons69, cons1481, cons746, cons1484, cons167, cons2037, cons2038, cons1678, cons1257, cons2039, cons349
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pattern6934 = Pattern(Integral(u_*((x_*WC('b', S(1)) + WC('a', S(0)))**n_*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons4, cons5, cons149, cons2004)
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rule6934 = ReplacementRule(pattern6934, replacement6934)
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pattern6935 = Pattern(Integral(((d_*(x_*WC('b', S(1)) + WC('a', S(0))))**p_*WC('c', S(1)))**q_*WC('u', S(1)), x_), cons2, cons3, cons8, cons29, cons5, cons52, cons149, cons388)
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rule6935 = ReplacementRule(pattern6935, replacement6935)
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pattern6936 = Pattern(Integral((((x_*WC('b', S(1)) + WC('a', S(0)))**n_*WC('d', S(1)))**p_*WC('c', S(1)))**q_*WC('u', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons52, cons149, cons388)
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rule6936 = ReplacementRule(pattern6936, replacement6936)
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pattern6937 = Pattern(Integral((F_*sqrt(x_*WC('e', S(1)) + WC('d', S(0)))*WC('b', S(1))*WC('c', S(1))/sqrt(x_*WC('g', S(1)) + WC('f', S(0))) + WC('a', S(0)))**WC('n', S(1))/(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons36, cons37, cons38, cons1101, cons2005, cons2006, cons2007, cons2008)
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rule6937 = ReplacementRule(pattern6937, replacement6937)
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pattern6938 = Pattern(Integral((F_*sqrt(x_*WC('e', S(1)) + S(1))*WC('b', S(1))*WC('c', S(1))/sqrt(x_*WC('g', S(1)) + S(1)) + WC('a', S(0)))**WC('n', S(1))/(x_**S(2)*WC('C', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons8, cons50, cons210, cons36, cons38, cons1101, cons2005, cons2009)
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rule6938 = ReplacementRule(pattern6938, replacement6938)
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pattern6939 = Pattern(Integral((F_**(sqrt(x_*WC('e', S(1)) + WC('d', S(0)))*WC('c', S(1))/sqrt(x_*WC('g', S(1)) + WC('f', S(0))))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons36, cons37, cons38, cons1101, cons2005, cons2006, cons2007, cons2008)
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rule6939 = ReplacementRule(pattern6939, replacement6939)
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pattern6940 = Pattern(Integral((F_**(sqrt(x_*WC('e', S(1)) + S(1))*WC('c', S(1))/sqrt(x_*WC('g', S(1)) + S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(x_**S(2)*WC('C', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons8, cons50, cons210, cons36, cons38, cons1101, cons2005, cons2009)
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rule6940 = ReplacementRule(pattern6940, replacement6940)
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pattern6941 = Pattern(Integral(u_/y_, x_), CustomConstraint(With6941))
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rule6941 = ReplacementRule(pattern6941, replacement6941)
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pattern6942 = Pattern(Integral(u_/(w_*y_), x_), CustomConstraint(With6942))
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rule6942 = ReplacementRule(pattern6942, replacement6942)
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pattern6943 = Pattern(Integral(u_*y_**WC('m', S(1)), x_), cons19, cons68, CustomConstraint(With6943))
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rule6943 = ReplacementRule(pattern6943, replacement6943)
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pattern6944 = Pattern(Integral(u_*y_**WC('m', S(1))*z_**WC('n', S(1)), x_), cons19, cons4, cons68, CustomConstraint(With6944))
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rule6944 = ReplacementRule(pattern6944, replacement6944)
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pattern6945 = Pattern(Integral(u_, x_), CustomConstraint(With6945))
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rule6945 = ReplacementRule(pattern6945, replacement6945)
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pattern6946 = Pattern(Integral((sqrt(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1)) + sqrt(x_**WC('n', S(1))*WC('d', S(1)) + WC('c', S(0)))*WC('f', S(1)))**m_*WC('u', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons86, cons1039)
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rule6946 = ReplacementRule(pattern6946, replacement6946)
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pattern6947 = Pattern(Integral((sqrt(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1)) + sqrt(x_**WC('n', S(1))*WC('d', S(1)) + WC('c', S(0)))*WC('f', S(1)))**m_*WC('u', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons86, cons1038)
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rule6947 = ReplacementRule(pattern6947, replacement6947)
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pattern6948 = Pattern(Integral(u_**WC('m', S(1))*w_*(u_**n_*WC('a', S(1)) + v_)**WC('p', S(1)), x_), cons2, cons19, cons4, cons40, cons2010, cons10)
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rule6948 = ReplacementRule(pattern6948, replacement6948)
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pattern6949 = Pattern(Integral(u_*(v_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(y_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons2011, CustomConstraint(With6949))
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rule6949 = ReplacementRule(pattern6949, replacement6949)
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pattern6950 = Pattern(Integral(u_*(v_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(w_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*(y_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons2011, cons2012, CustomConstraint(With6950))
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rule6950 = ReplacementRule(pattern6950, replacement6950)
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pattern6951 = Pattern(Integral(u_*(v_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(w_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*(y_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(z_*WC('h', S(1)) + WC('g', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons211, cons19, cons4, cons5, cons52, cons2011, cons2012, cons2013, CustomConstraint(With6951))
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rule6951 = ReplacementRule(pattern6951, replacement6951)
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pattern6952 = Pattern(Integral((a_ + y_**n_*WC('b', S(1)))*WC('u', S(1)), x_), cons2, cons3, cons4, cons1833, CustomConstraint(With6952))
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rule6952 = ReplacementRule(pattern6952, replacement6952)
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pattern6953 = Pattern(Integral((y_**n_*WC('b', S(1)) + WC('a', S(0)))**p_*WC('u', S(1)), x_), cons2, cons3, cons4, cons5, cons1246, CustomConstraint(With6953))
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rule6953 = ReplacementRule(pattern6953, replacement6953)
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pattern6954 = Pattern(Integral(v_**WC('m', S(1))*(y_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons19, cons4, cons5, cons2014, CustomConstraint(With6954))
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rule6954 = ReplacementRule(pattern6954, replacement6954)
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pattern6955 = Pattern(Integral((v_**WC('n2', S(1))*WC('c', S(1)) + y_**n_*WC('b', S(1)) + WC('a', S(0)))**p_*WC('u', S(1)), x_), cons2, cons3, cons8, cons4, cons5, cons48, cons2011, CustomConstraint(With6955))
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rule6955 = ReplacementRule(pattern6955, replacement6955)
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pattern6956 = Pattern(Integral((A_ + y_**n_*WC('B', S(1)))*(v_**n_*WC('b', S(1)) + w_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons8, cons36, cons37, cons4, cons5, cons48, cons2011, cons2012, CustomConstraint(With6956))
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rule6956 = ReplacementRule(pattern6956, replacement6956)
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pattern6957 = Pattern(Integral((A_ + y_**n_*WC('B', S(1)))*(w_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons8, cons36, cons37, cons4, cons5, cons48, cons2012, CustomConstraint(With6957))
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rule6957 = ReplacementRule(pattern6957, replacement6957)
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pattern6958 = Pattern(Integral(v_**WC('m', S(1))*(w_**WC('n2', S(1))*WC('c', S(1)) + y_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons8, cons19, cons4, cons5, cons48, cons2012, CustomConstraint(With6958))
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rule6958 = ReplacementRule(pattern6958, replacement6958)
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pattern6959 = Pattern(Integral(z_**WC('m', S(1))*(A_ + y_**n_*WC('B', S(1)))*(v_**n_*WC('b', S(1)) + w_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons8, cons36, cons37, cons19, cons4, cons5, cons48, cons2011, cons2012, CustomConstraint(With6959))
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rule6959 = ReplacementRule(pattern6959, replacement6959)
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pattern6960 = Pattern(Integral(z_**WC('m', S(1))*(A_ + y_**n_*WC('B', S(1)))*(w_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons8, cons36, cons37, cons19, cons4, cons5, cons48, cons2012, CustomConstraint(With6960))
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rule6960 = ReplacementRule(pattern6960, replacement6960)
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|
|
pattern6961 = Pattern(Integral((v_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('p', S(1))*(y_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*WC('u', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons2011, CustomConstraint(With6961))
|
|
rule6961 = ReplacementRule(pattern6961, replacement6961)
|
|
|
|
pattern6962 = Pattern(Integral((v_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('p', S(1))*(w_**n_*WC('f', S(1)) + WC('e', S(0)))**WC('q', S(1))*(y_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*WC('u', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons52, cons2011, cons2012, CustomConstraint(With6962))
|
|
rule6962 = ReplacementRule(pattern6962, replacement6962)
|
|
|
|
pattern6963 = Pattern(Integral(F_**v_*u_, x_), cons1101, cons1101, CustomConstraint(With6963))
|
|
rule6963 = ReplacementRule(pattern6963, replacement6963)
|
|
|
|
pattern6964 = Pattern(Integral(F_**v_*u_*w_**WC('m', S(1)), x_), cons1101, cons19, cons2015, CustomConstraint(With6964))
|
|
rule6964 = ReplacementRule(pattern6964, replacement6964)
|
|
|
|
pattern6965 = Pattern(Integral(u_*(a_ + v_**WC('p', S(1))*w_**WC('p', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons19, cons5, cons40, CustomConstraint(With6965))
|
|
rule6965 = ReplacementRule(pattern6965, replacement6965)
|
|
|
|
pattern6966 = Pattern(Integral(u_*v_**WC('r', S(1))*(a_ + v_**WC('p', S(1))*w_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons19, cons5, cons52, cons54, cons2016, cons2017, cons2018, CustomConstraint(With6966))
|
|
rule6966 = ReplacementRule(pattern6966, replacement6966)
|
|
|
|
pattern6967 = Pattern(Integral(u_*v_**WC('r', S(1))*w_**WC('s', S(1))*(a_ + v_**WC('p', S(1))*w_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons19, cons5, cons52, cons54, cons802, cons2019, cons2017, cons2018, CustomConstraint(With6967))
|
|
rule6967 = ReplacementRule(pattern6967, replacement6967)
|
|
|
|
pattern6968 = Pattern(Integral(u_*(v_**WC('p', S(1))*WC('a', S(1)) + w_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons19, cons5, cons52, cons2020, cons40, cons20, CustomConstraint(With6968))
|
|
rule6968 = ReplacementRule(pattern6968, replacement6968)
|
|
|
|
pattern6969 = Pattern(Integral(u_*v_**WC('r', S(1))*(v_**WC('p', S(1))*WC('a', S(1)) + w_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons19, cons5, cons52, cons54, cons2021, cons588, cons20, CustomConstraint(With6969))
|
|
rule6969 = ReplacementRule(pattern6969, replacement6969)
|
|
|
|
pattern6970 = Pattern(Integral(u_*w_**WC('s', S(1))*(v_**WC('p', S(1))*WC('a', S(1)) + w_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons19, cons5, cons52, cons802, cons2022, cons2023, cons2024, cons20, CustomConstraint(With6970))
|
|
rule6970 = ReplacementRule(pattern6970, replacement6970)
|
|
|
|
pattern6971 = Pattern(Integral(u_*v_**WC('r', S(1))*w_**WC('s', S(1))*(v_**WC('p', S(1))*WC('a', S(1)) + w_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons19, cons5, cons52, cons54, cons802, cons2025, cons2023, cons2024, cons20, CustomConstraint(With6971))
|
|
rule6971 = ReplacementRule(pattern6971, replacement6971)
|
|
|
|
pattern6972 = Pattern(Integral(u_*x_**WC('m', S(1)), x_), cons19, cons68, cons2026)
|
|
rule6972 = ReplacementRule(pattern6972, replacement6972)
|
|
|
|
pattern6973 = Pattern(Integral(u_, x_), CustomConstraint(With6973))
|
|
rule6973 = ReplacementRule(pattern6973, replacement6973)
|
|
|
|
pattern6974 = Pattern(Integral(u_, x_), CustomConstraint(With6974))
|
|
rule6974 = ReplacementRule(pattern6974, replacement6974)
|
|
|
|
pattern6975 = Pattern(Integral((v_**WC('m', S(1))*w_**WC('n', S(1))*z_**WC('q', S(1))*WC('a', S(1)))**p_*WC('u', S(1)), x_), cons2, cons19, cons4, cons5, cons52, cons149, cons10, cons2027, cons2028)
|
|
rule6975 = ReplacementRule(pattern6975, replacement6975)
|
|
|
|
pattern6976 = Pattern(Integral((v_**WC('m', S(1))*w_**WC('n', S(1))*WC('a', S(1)))**p_*WC('u', S(1)), x_), cons2, cons19, cons4, cons5, cons149, cons10, cons2027)
|
|
rule6976 = ReplacementRule(pattern6976, replacement6976)
|
|
|
|
pattern6977 = Pattern(Integral((v_**WC('m', S(1))*WC('a', S(1)))**p_*WC('u', S(1)), x_), cons2, cons19, cons5, cons149, cons10, cons2029, cons2030)
|
|
rule6977 = ReplacementRule(pattern6977, replacement6977)
|
|
|
|
pattern6978 = Pattern(Integral((x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_*WC('u', S(1)), x_), cons2, cons3, cons5, cons669, cons198, cons2031)
|
|
rule6978 = ReplacementRule(pattern6978, replacement6978)
|
|
|
|
pattern6979 = Pattern(Integral((v_**n_*WC('b', S(1)) + WC('a', S(0)))**p_*WC('u', S(1)), x_), cons2, cons3, cons5, cons149, cons198, cons842, cons2032)
|
|
rule6979 = ReplacementRule(pattern6979, replacement6979)
|
|
|
|
pattern6980 = Pattern(Integral((v_**n_*x_**WC('m', S(1))*WC('b', S(1)) + WC('a', S(0)))**p_*WC('u', S(1)), x_), cons2, cons3, cons19, cons5, cons149, cons198, cons842)
|
|
rule6980 = ReplacementRule(pattern6980, replacement6980)
|
|
|
|
pattern6981 = Pattern(Integral((x_**WC('r', S(1))*WC('a', S(1)) + x_**WC('s', S(1))*WC('b', S(1)))**m_*WC('u', S(1)), x_), cons2, cons3, cons19, cons54, cons802, cons21, cons2033, CustomConstraint(With6981))
|
|
rule6981 = ReplacementRule(pattern6981, replacement6981)
|
|
|
|
pattern6982 = Pattern(Integral(u_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons150, CustomConstraint(With6982))
|
|
rule6982 = ReplacementRule(pattern6982, replacement6982)
|
|
|
|
pattern6983 = Pattern(Integral(u_*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons4, cons48, cons47, cons40, cons2034)
|
|
rule6983 = ReplacementRule(pattern6983, replacement6983)
|
|
|
|
pattern6984 = Pattern(Integral(u_*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons4, cons5, cons48, cons47, cons149, cons2034)
|
|
rule6984 = ReplacementRule(pattern6984, replacement6984)
|
|
|
|
pattern6985 = Pattern(Integral(u_/(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons48, cons150, CustomConstraint(With6985))
|
|
rule6985 = ReplacementRule(pattern6985, replacement6985)
|
|
|
|
pattern6986 = Pattern(Integral(WC('u', S(1))/(x_**WC('m', S(1))*WC('a', S(1)) + sqrt(x_**n_*WC('c', S(1)))*WC('b', S(1))), x_), cons2, cons3, cons8, cons19, cons4, cons1856)
|
|
rule6986 = ReplacementRule(pattern6986, replacement6986)
|
|
|
|
pattern6987 = Pattern(Integral(u_, x_), CustomConstraint(With6987))
|
|
rule6987 = ReplacementRule(pattern6987, replacement6987)
|
|
|
|
pattern6988 = Pattern(Integral(u_/x_, x_), cons1249, cons2031, CustomConstraint(With6988))
|
|
rule6988 = ReplacementRule(pattern6988, replacement6988)
|
|
|
|
pattern6989 = Pattern(Integral(u_*x_**WC('m', S(1)), x_), cons20, cons263, cons1249, cons2035, CustomConstraint(With6989))
|
|
rule6989 = ReplacementRule(pattern6989, replacement6989)
|
|
|
|
pattern6990 = Pattern(Integral(u_*x_**m_, x_), cons369)
|
|
rule6990 = ReplacementRule(pattern6990, With6990)
|
|
|
|
pattern6991 = Pattern(Integral(u_, x_), cons2036, CustomConstraint(With6991))
|
|
rule6991 = ReplacementRule(pattern6991, replacement6991)
|
|
|
|
pattern6992 = Pattern(Integral(S(1)/(a_ + v_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons69)
|
|
rule6992 = ReplacementRule(pattern6992, replacement6992)
|
|
|
|
pattern6993 = Pattern(Integral(S(1)/(a_ + v_**n_*WC('b', S(1))), x_), cons2, cons3, cons1481, cons746)
|
|
rule6993 = ReplacementRule(pattern6993, replacement6993)
|
|
|
|
pattern6994 = Pattern(Integral(S(1)/(a_ + v_**n_*WC('b', S(1))), x_), cons2, cons3, cons1484, cons167)
|
|
rule6994 = ReplacementRule(pattern6994, replacement6994)
|
|
|
|
pattern6995 = Pattern(Integral(v_/(a_ + u_**WC('n', S(1))*WC('b', S(1))), x_), cons2, cons3, cons150, cons2037)
|
|
rule6995 = ReplacementRule(pattern6995, replacement6995)
|
|
|
|
pattern6996 = Pattern(Integral(u_, x_), CustomConstraint(With6996))
|
|
rule6996 = ReplacementRule(pattern6996, replacement6996)
|
|
|
|
pattern6997 = Pattern(Integral(u_, x_), CustomConstraint(With6997))
|
|
rule6997 = ReplacementRule(pattern6997, replacement6997)
|
|
|
|
pattern6998 = Pattern(Integral((x_**WC('m', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(x_**WC('n', S(1))*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1))*WC('u', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons52, cons2038, cons1678, cons1257, cons2039)
|
|
rule6998 = ReplacementRule(pattern6998, replacement6998)
|
|
|
|
pattern6999 = Pattern(Integral(u_*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons4, cons5, cons48, cons47, cons349)
|
|
rule6999 = ReplacementRule(pattern6999, replacement6999)
|
|
|
|
pattern7000 = Pattern(Integral(u_, x_), CustomConstraint(With7000))
|
|
rule7000 = ReplacementRule(pattern7000, replacement7000)
|
|
|
|
pattern7001 = Pattern(Integral(u_, x_))
|
|
rule7001 = ReplacementRule(pattern7001, replacement7001)
|
|
return [rule6934, rule6935, rule6936, rule6937, rule6938, rule6939, rule6940, rule6941, rule6942, rule6943, rule6944, rule6945, rule6946, rule6947, rule6948, rule6949, rule6950, rule6951, rule6952, rule6953, rule6954, rule6955, rule6956, rule6957, rule6958, rule6959, rule6960, rule6961, rule6962, rule6963, rule6964, rule6965, rule6966, rule6967, rule6968, rule6969, rule6970, rule6971, rule6972, rule6973, rule6974, rule6975, rule6976, rule6977, rule6978, rule6979, rule6980, rule6981, rule6982, rule6983, rule6984, rule6985, rule6986, rule6987, rule6988, rule6989, rule6990, rule6991, rule6992, rule6993, rule6994, rule6995, rule6996, rule6997, rule6998, rule6999, rule7000, rule7001, ]
|
|
|
|
|
|
|
|
|
|
|
|
def replacement6934(a, b, c, n, p, u, x):
|
|
return Dist(c**IntPart(p)*(c*(a + b*x)**n)**FracPart(p)*(a + b*x)**(-n*FracPart(p)), Int(u*(a + b*x)**(n*p), x), x)
|
|
|
|
|
|
def replacement6935(a, b, c, d, p, q, u, x):
|
|
return Dist((c*(d*(a + b*x))**p)**q*(a + b*x)**(-p*q), Int(u*(a + b*x)**(p*q), x), x)
|
|
|
|
|
|
def replacement6936(a, b, c, d, n, p, q, u, x):
|
|
return Dist((c*(d*(a + b*x)**n)**p)**q*(a + b*x)**(-n*p*q), Int(u*(a + b*x)**(n*p*q), x), x)
|
|
|
|
|
|
def replacement6937(A, B, C, F, a, b, c, d, e, f, g, n, x):
|
|
return Dist(g/C, Subst(Int((a + b*F(c*x))**n/x, x), x, sqrt(d + e*x)/sqrt(f + g*x)), x)
|
|
|
|
|
|
def replacement6938(A, C, F, a, b, c, e, g, n, x):
|
|
return Dist(g/C, Subst(Int((a + b*F(c*x))**n/x, x), x, sqrt(e*x + S(1))/sqrt(g*x + S(1))), x)
|
|
|
|
|
|
def replacement6939(A, B, C, F, a, b, c, d, e, f, g, n, x):
|
|
return Dist(g/C, Subst(Int((F**(c*x)*b + a)**n/x, x), x, sqrt(d + e*x)/sqrt(f + g*x)), x)
|
|
|
|
|
|
def replacement6940(A, C, F, a, b, c, e, g, n, x):
|
|
return Dist(g/C, Subst(Int((F**(c*x)*b + a)**n/x, x), x, sqrt(e*x + S(1))/sqrt(g*x + S(1))), x)
|
|
|
|
|
|
def With6941(u, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(y, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6941(u, x, y):
|
|
|
|
q = DerivativeDivides(y, u, x)
|
|
return Simp(q*log(RemoveContent(y, x)), x)
|
|
|
|
|
|
def With6942(u, w, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(w*y, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6942(u, w, x, y):
|
|
|
|
q = DerivativeDivides(w*y, u, x)
|
|
return Simp(q*log(RemoveContent(w*y, x)), x)
|
|
|
|
|
|
def With6943(m, u, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(y, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6943(m, u, x, y):
|
|
|
|
q = DerivativeDivides(y, u, x)
|
|
return Simp(q*y**(m + S(1))/(m + S(1)), x)
|
|
|
|
|
|
def With6944(m, n, u, x, y, z):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(y*z, u*z**(-m + n), x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6944(m, n, u, x, y, z):
|
|
|
|
q = DerivativeDivides(y*z, u*z**(-m + n), x)
|
|
return Simp(q*y**(m + S(1))*z**(m + S(1))/(m + S(1)), x)
|
|
|
|
|
|
def With6945(u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
v = SimplifyIntegrand(u, x)
|
|
if SimplerIntegrandQ(v, u, x):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6945(u, x):
|
|
|
|
v = SimplifyIntegrand(u, x)
|
|
return Int(v, x)
|
|
|
|
|
|
def replacement6946(a, b, c, d, e, f, m, n, u, x):
|
|
return Dist((a*e**S(2) - c*f**S(2))**m, Int(ExpandIntegrand(u*(e*sqrt(a + b*x**n) - f*sqrt(c + d*x**n))**(-m), x), x), x)
|
|
|
|
|
|
def replacement6947(a, b, c, d, e, f, m, n, u, x):
|
|
return Dist((b*e**S(2) - d*f**S(2))**m, Int(ExpandIntegrand(u*x**(m*n)*(e*sqrt(a + b*x**n) - f*sqrt(c + d*x**n))**(-m), x), x), x)
|
|
|
|
|
|
def replacement6948(a, m, n, p, u, v, w, x):
|
|
return Int(u**(m + n*p)*w*(a + u**(-n)*v)**p, x)
|
|
|
|
|
|
def With6949(a, b, c, d, m, n, u, v, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(y, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6949(a, b, c, d, m, n, u, v, x, y):
|
|
|
|
q = DerivativeDivides(y, u, x)
|
|
return Dist(q, Subst(Int((a + b*x)**m*(c + d*x)**n, x), x, y), x)
|
|
|
|
|
|
def With6950(a, b, c, d, e, f, m, n, p, u, v, w, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(y, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6950(a, b, c, d, e, f, m, n, p, u, v, w, x, y):
|
|
|
|
q = DerivativeDivides(y, u, x)
|
|
return Dist(q, Subst(Int((a + b*x)**m*(c + d*x)**n*(e + f*x)**p, x), x, y), x)
|
|
|
|
|
|
def With6951(a, b, c, d, e, f, g, h, m, n, p, q, u, v, w, x, y, z):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
r = DerivativeDivides(y, u, x)
|
|
res = Not(FalseQ(r))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6951(a, b, c, d, e, f, g, h, m, n, p, q, u, v, w, x, y, z):
|
|
|
|
r = DerivativeDivides(y, u, x)
|
|
return Dist(r, Subst(Int((a + b*x)**m*(c + d*x)**n*(e + f*x)**p*(g + h*x)**q, x), x, y), x)
|
|
|
|
|
|
def With6952(a, b, n, u, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(y, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6952(a, b, n, u, x, y):
|
|
|
|
q = DerivativeDivides(y, u, x)
|
|
return Dist(a, Int(u, x), x) + Dist(b*q, Subst(Int(x**n, x), x, y), x)
|
|
|
|
|
|
def With6953(a, b, n, p, u, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(y, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6953(a, b, n, p, u, x, y):
|
|
|
|
q = DerivativeDivides(y, u, x)
|
|
return Dist(q, Subst(Int((a + b*x**n)**p, x), x, y), x)
|
|
|
|
|
|
def With6954(a, b, m, n, p, u, v, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = Symbol('q')
|
|
r = Symbol('r')
|
|
r = Divides(y**m, v**m, x)
|
|
q = DerivativeDivides(y, u, x)
|
|
res = And(Not(FalseQ(Set(r, Divides(y**m, v**m, x)))), Not(FalseQ(Set(q, DerivativeDivides(y, u, x)))))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6954(a, b, m, n, p, u, v, x, y):
|
|
|
|
q = Symbol('q')
|
|
r = Symbol('r')
|
|
r = Divides(y**m, v**m, x)
|
|
q = DerivativeDivides(y, u, x)
|
|
return Dist(q*r, Subst(Int(x**m*(a + b*x**n)**p, x), x, y), x)
|
|
|
|
|
|
def With6955(a, b, c, n, n2, p, u, v, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(y, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6955(a, b, c, n, n2, p, u, v, x, y):
|
|
|
|
q = DerivativeDivides(y, u, x)
|
|
return Dist(q, Subst(Int((a + b*x**n + c*x**(S(2)*n))**p, x), x, y), x)
|
|
|
|
|
|
def With6956(A, B, a, b, c, n, n2, p, u, v, w, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(y, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6956(A, B, a, b, c, n, n2, p, u, v, w, x, y):
|
|
|
|
q = DerivativeDivides(y, u, x)
|
|
return Dist(q, Subst(Int((A + B*x**n)*(a + b*x**n + c*x**(S(2)*n))**p, x), x, y), x)
|
|
|
|
|
|
def With6957(A, B, a, c, n, n2, p, u, w, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(y, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6957(A, B, a, c, n, n2, p, u, w, x, y):
|
|
|
|
q = DerivativeDivides(y, u, x)
|
|
return Dist(q, Subst(Int((A + B*x**n)*(a + c*x**(S(2)*n))**p, x), x, y), x)
|
|
|
|
|
|
def With6958(a, b, c, m, n, n2, p, u, v, w, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = Symbol('q')
|
|
r = Symbol('r')
|
|
r = Divides(y**m, v**m, x)
|
|
q = DerivativeDivides(y, u, x)
|
|
res = And(Not(FalseQ(Set(r, Divides(y**m, v**m, x)))), Not(FalseQ(Set(q, DerivativeDivides(y, u, x)))))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6958(a, b, c, m, n, n2, p, u, v, w, x, y):
|
|
|
|
q = Symbol('q')
|
|
r = Symbol('r')
|
|
r = Divides(y**m, v**m, x)
|
|
q = DerivativeDivides(y, u, x)
|
|
return Dist(q*r, Subst(Int(x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x, y), x)
|
|
|
|
|
|
def With6959(A, B, a, b, c, m, n, n2, p, u, v, w, x, y, z):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = Symbol('q')
|
|
r = Symbol('r')
|
|
r = Divides(y**m, z**m, x)
|
|
q = DerivativeDivides(y, u, x)
|
|
res = And(Not(FalseQ(Set(r, Divides(y**m, z**m, x)))), Not(FalseQ(Set(q, DerivativeDivides(y, u, x)))))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6959(A, B, a, b, c, m, n, n2, p, u, v, w, x, y, z):
|
|
|
|
q = Symbol('q')
|
|
r = Symbol('r')
|
|
r = Divides(y**m, z**m, x)
|
|
q = DerivativeDivides(y, u, x)
|
|
return Dist(q*r, Subst(Int(x**m*(A + B*x**n)*(a + b*x**n + c*x**(S(2)*n))**p, x), x, y), x)
|
|
|
|
|
|
def With6960(A, B, a, c, m, n, n2, p, u, w, x, y, z):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = Symbol('q')
|
|
r = Symbol('r')
|
|
r = Divides(y**m, z**m, x)
|
|
q = DerivativeDivides(y, u, x)
|
|
res = And(Not(FalseQ(Set(r, Divides(y**m, z**m, x)))), Not(FalseQ(Set(q, DerivativeDivides(y, u, x)))))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6960(A, B, a, c, m, n, n2, p, u, w, x, y, z):
|
|
|
|
q = Symbol('q')
|
|
r = Symbol('r')
|
|
r = Divides(y**m, z**m, x)
|
|
q = DerivativeDivides(y, u, x)
|
|
return Dist(q*r, Subst(Int(x**m*(A + B*x**n)*(a + c*x**(S(2)*n))**p, x), x, y), x)
|
|
|
|
|
|
def With6961(a, b, c, d, m, n, p, u, v, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(y, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6961(a, b, c, d, m, n, p, u, v, x, y):
|
|
|
|
q = DerivativeDivides(y, u, x)
|
|
return Dist(q, Subst(Int((a + b*x**n)**m*(c + d*x**n)**p, x), x, y), x)
|
|
|
|
|
|
def With6962(a, b, c, d, e, f, m, n, p, q, u, v, w, x, y):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
r = DerivativeDivides(y, u, x)
|
|
res = Not(FalseQ(r))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6962(a, b, c, d, e, f, m, n, p, q, u, v, w, x, y):
|
|
|
|
r = DerivativeDivides(y, u, x)
|
|
return Dist(r, Subst(Int((a + b*x**n)**m*(c + d*x**n)**p*(e + f*x**n)**q, x), x, y), x)
|
|
|
|
|
|
def With6963(F, u, v, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(v, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6963(F, u, v, x):
|
|
|
|
q = DerivativeDivides(v, u, x)
|
|
return Simp(F**v*q/log(F), x)
|
|
|
|
|
|
def With6964(F, m, u, v, w, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
q = DerivativeDivides(v, u, x)
|
|
res = Not(FalseQ(q))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6964(F, m, u, v, w, x):
|
|
|
|
q = DerivativeDivides(v, u, x)
|
|
return Dist(q, Subst(Int(F**x*x**m, x), x, v), x)
|
|
|
|
|
|
def With6965(a, b, m, p, u, v, w, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
c = u/(v*D(w, x) + w*D(v, x))
|
|
if FreeQ(c, x):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6965(a, b, m, p, u, v, w, x):
|
|
|
|
c = u/(v*D(w, x) + w*D(v, x))
|
|
return Dist(c, Subst(Int((a + b*x**p)**m, x), x, v*w), x)
|
|
|
|
|
|
def With6966(a, b, m, p, q, r, u, v, w, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
c = u/(p*w*D(v, x) + q*v*D(w, x))
|
|
if FreeQ(c, x):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6966(a, b, m, p, q, r, u, v, w, x):
|
|
|
|
c = u/(p*w*D(v, x) + q*v*D(w, x))
|
|
return Dist(c*p/(r + S(1)), Subst(Int((a + b*x**(p/(r + S(1))))**m, x), x, v**(r + S(1))*w), x)
|
|
|
|
|
|
def With6967(a, b, m, p, q, r, s, u, v, w, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
c = u/(p*w*D(v, x) + q*v*D(w, x))
|
|
if FreeQ(c, x):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6967(a, b, m, p, q, r, s, u, v, w, x):
|
|
|
|
c = u/(p*w*D(v, x) + q*v*D(w, x))
|
|
return Dist(c*p/(r + S(1)), Subst(Int((a + b*x**(p/(r + S(1))))**m, x), x, v**(r + S(1))*w**(s + S(1))), x)
|
|
|
|
|
|
def With6968(a, b, m, p, q, u, v, w, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
c = u/(p*w*D(v, x) - q*v*D(w, x))
|
|
if FreeQ(c, x):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6968(a, b, m, p, q, u, v, w, x):
|
|
|
|
c = u/(p*w*D(v, x) - q*v*D(w, x))
|
|
return Dist(c*p, Subst(Int((a*x**p + b)**m, x), x, v*w**(m*q + S(1))), x)
|
|
|
|
|
|
def With6969(a, b, m, p, q, r, u, v, w, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
c = u/(p*w*D(v, x) - q*v*D(w, x))
|
|
if FreeQ(c, x):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6969(a, b, m, p, q, r, u, v, w, x):
|
|
|
|
c = u/(p*w*D(v, x) - q*v*D(w, x))
|
|
return -Dist(c*q, Subst(Int((a + b*x**q)**m, x), x, v**(m*p + r + S(1))*w), x)
|
|
|
|
|
|
def With6970(a, b, m, p, q, s, u, v, w, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
c = u/(p*w*D(v, x) - q*v*D(w, x))
|
|
if FreeQ(c, x):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6970(a, b, m, p, q, s, u, v, w, x):
|
|
|
|
c = u/(p*w*D(v, x) - q*v*D(w, x))
|
|
return -Dist(c*q/(s + S(1)), Subst(Int((a + b*x**(q/(s + S(1))))**m, x), x, v**(m*p + S(1))*w**(s + S(1))), x)
|
|
|
|
|
|
def With6971(a, b, m, p, q, r, s, u, v, w, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
c = u/(p*w*D(v, x) - q*v*D(w, x))
|
|
if FreeQ(c, x):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6971(a, b, m, p, q, r, s, u, v, w, x):
|
|
|
|
c = u/(p*w*D(v, x) - q*v*D(w, x))
|
|
return -Dist(c*q/(s + S(1)), Subst(Int((a + b*x**(q/(s + S(1))))**m, x), x, v**(m*p + r + S(1))*w**(s + S(1))), x)
|
|
|
|
|
|
def replacement6972(m, u, x):
|
|
return Dist(S(1)/(m + S(1)), Subst(Int(SubstFor(x**(m + S(1)), u, x), x), x, x**(m + S(1))), x)
|
|
|
|
|
|
def With6973(u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
lst = SubstForFractionalPowerOfLinear(u, x)
|
|
res = And(Not(FalseQ(lst)), SubstForFractionalPowerQ(u, Part(lst, S(3)), x))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6973(u, x):
|
|
|
|
lst = SubstForFractionalPowerOfLinear(u, x)
|
|
return Dist(Part(lst, S(2))*Part(lst, S(4)), Subst(Int(Part(lst, S(1)), x), x, Part(lst, S(3))**(S(1)/Part(lst, S(2)))), x)
|
|
|
|
|
|
def With6974(u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
lst = SubstForFractionalPowerOfQuotientOfLinears(u, x)
|
|
res = Not(FalseQ(lst))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6974(u, x):
|
|
|
|
lst = SubstForFractionalPowerOfQuotientOfLinears(u, x)
|
|
return Dist(Part(lst, S(2))*Part(lst, S(4)), Subst(Int(Part(lst, S(1)), x), x, Part(lst, S(3))**(S(1)/Part(lst, S(2)))), x)
|
|
|
|
|
|
def replacement6975(a, m, n, p, q, u, v, w, x, z):
|
|
return Dist(a**IntPart(p)*v**(-m*FracPart(p))*w**(-n*FracPart(p))*z**(-q*FracPart(p))*(a*v**m*w**n*z**q)**FracPart(p), Int(u*v**(m*p)*w**(n*p)*z**(p*q), x), x)
|
|
|
|
|
|
def replacement6976(a, m, n, p, u, v, w, x):
|
|
return Dist(a**IntPart(p)*v**(-m*FracPart(p))*w**(-n*FracPart(p))*(a*v**m*w**n)**FracPart(p), Int(u*v**(m*p)*w**(n*p), x), x)
|
|
|
|
|
|
def replacement6977(a, m, p, u, v, x):
|
|
return Dist(a**IntPart(p)*v**(-m*FracPart(p))*(a*v**m)**FracPart(p), Int(u*v**(m*p), x), x)
|
|
|
|
|
|
def replacement6978(a, b, n, p, u, x):
|
|
return Dist(FullSimplify(x**(-n/S(2))*sqrt(a + b*x**n)/sqrt(a*x**(-n) + b)), Int(u*x**(n*p)*(a*x**(-n) + b)**p, x), x)
|
|
|
|
|
|
def replacement6979(a, b, n, p, u, v, x):
|
|
return Dist(v**(-n*FracPart(p))*(a + b*v**n)**FracPart(p)*(a*v**(-n) + b)**(-FracPart(p)), Int(u*v**(n*p)*(a*v**(-n) + b)**p, x), x)
|
|
|
|
|
|
def replacement6980(a, b, m, n, p, u, v, x):
|
|
return Dist(v**(-n*FracPart(p))*(a + b*v**n*x**m)**FracPart(p)*(a*v**(-n) + b*x**m)**(-FracPart(p)), Int(u*v**(n*p)*(a*v**(-n) + b*x**m)**p, x), x)
|
|
|
|
|
|
def With6981(a, b, m, r, s, u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
v = x**(-r*FracPart(m))*(a + b*x**(-r + s))**(-FracPart(m))*(a*x**r + b*x**s)**FracPart(m)
|
|
if Not(EqQ(v, S(1))):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6981(a, b, m, r, s, u, x):
|
|
|
|
v = x**(-r*FracPart(m))*(a + b*x**(-r + s))**(-FracPart(m))*(a*x**r + b*x**s)**FracPart(m)
|
|
return Dist(v, Int(u*x**(m*r)*(a + b*x**(-r + s))**m, x), x)
|
|
|
|
|
|
def With6982(a, b, n, u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
v = RationalFunctionExpand(u/(a + b*x**n), x)
|
|
if SumQ(v):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6982(a, b, n, u, x):
|
|
|
|
v = RationalFunctionExpand(u/(a + b*x**n), x)
|
|
return Int(v, x)
|
|
|
|
|
|
def replacement6983(a, b, c, n, n2, p, u, x):
|
|
return Dist(S(4)**(-p)*c**(-p), Int(u*(b + S(2)*c*x**n)**(S(2)*p), x), x)
|
|
|
|
|
|
def replacement6984(a, b, c, n, n2, p, u, x):
|
|
return Dist((b + S(2)*c*x**n)**(-S(2)*p)*(a + b*x**n + c*x**(S(2)*n))**p, Int(u*(b + S(2)*c*x**n)**(S(2)*p), x), x)
|
|
|
|
|
|
def With6985(a, b, c, n, n2, u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
v = RationalFunctionExpand(u/(a + b*x**n + c*x**(S(2)*n)), x)
|
|
if SumQ(v):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6985(a, b, c, n, n2, u, x):
|
|
|
|
v = RationalFunctionExpand(u/(a + b*x**n + c*x**(S(2)*n)), x)
|
|
return Int(v, x)
|
|
|
|
|
|
def replacement6986(a, b, c, m, n, u, x):
|
|
return Int(u*(a*x**m - b*sqrt(c*x**n))/(a**S(2)*x**(S(2)*m) - b**S(2)*c*x**n), x)
|
|
|
|
|
|
def With6987(u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
lst = FunctionOfLinear(u, x)
|
|
res = Not(FalseQ(lst))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6987(u, x):
|
|
|
|
lst = FunctionOfLinear(u, x)
|
|
return Dist(S(1)/Part(lst, S(3)), Subst(Int(Part(lst, S(1)), x), x, x*Part(lst, S(3)) + Part(lst, S(2))), x)
|
|
|
|
|
|
def With6988(u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
lst = PowerVariableExpn(u, S(0), x)
|
|
res = And(Not(FalseQ(lst)), NonzeroQ(Part(lst, S(2))))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6988(u, x):
|
|
|
|
lst = PowerVariableExpn(u, S(0), x)
|
|
return Dist(S(1)/Part(lst, S(2)), Subst(Int(NormalizeIntegrand(Part(lst, S(1))/x, x), x), x, (x*Part(lst, S(3)))**Part(lst, S(2))), x)
|
|
|
|
|
|
def With6989(m, u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
lst = PowerVariableExpn(u, m + S(1), x)
|
|
res = And(Not(FalseQ(lst)), NonzeroQ(-m + Part(lst, S(2)) + S(-1)))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6989(m, u, x):
|
|
|
|
lst = PowerVariableExpn(u, m + S(1), x)
|
|
return Dist(S(1)/Part(lst, S(2)), Subst(Int(NormalizeIntegrand(Part(lst, S(1))/x, x), x), x, (x*Part(lst, S(3)))**Part(lst, S(2))), x)
|
|
|
|
|
|
def With6990(m, u, x):
|
|
k = Denominator(m)
|
|
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*ReplaceAll(u, Rule(x, x**k)), x), x, x**(S(1)/k)), x)
|
|
|
|
|
|
def With6991(u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
lst = FunctionOfSquareRootOfQuadratic(u, x)
|
|
res = Not(FalseQ(lst))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6991(u, x):
|
|
|
|
lst = FunctionOfSquareRootOfQuadratic(u, x)
|
|
return Dist(S(2), Subst(Int(Part(lst, S(1)), x), x, Part(lst, S(2))), x)
|
|
|
|
|
|
def replacement6992(a, b, v, x):
|
|
return Dist(S(1)/(S(2)*a), Int(Together(S(1)/(-v/Rt(-a/b, S(2)) + S(1))), x), x) + Dist(S(1)/(S(2)*a), Int(Together(S(1)/(v/Rt(-a/b, S(2)) + S(1))), x), x)
|
|
|
|
|
|
def replacement6993(a, b, n, v, x):
|
|
return Dist(S(2)/(a*n), Sum_doit(Int(Together(S(1)/(S(1) - (S(-1))**(-S(4)*k/n)*v**S(2)/Rt(-a/b, n/S(2)))), x), List(k, S(1), n/S(2))), x)
|
|
|
|
|
|
def replacement6994(a, b, n, v, x):
|
|
return Dist(S(1)/(a*n), Sum_doit(Int(Together(S(1)/(S(1) - (S(-1))**(-S(2)*k/n)*v/Rt(-a/b, n))), x), List(k, S(1), n)), x)
|
|
|
|
|
|
def replacement6995(a, b, n, u, v, x):
|
|
return Int(ReplaceAll(ExpandIntegrand(PolynomialInSubst(v, u, x)/(a + b*x**n), x), Rule(x, u)), x)
|
|
|
|
|
|
def With6996(u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
v = NormalizeIntegrand(u, x)
|
|
if UnsameQ(v, u):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6996(u, x):
|
|
|
|
v = NormalizeIntegrand(u, x)
|
|
return Int(v, x)
|
|
|
|
|
|
def With6997(u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
v = ExpandIntegrand(u, x)
|
|
if SumQ(v):
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement6997(u, x):
|
|
|
|
v = ExpandIntegrand(u, x)
|
|
return Int(v, x)
|
|
|
|
|
|
def replacement6998(a, b, c, d, m, n, p, q, u, x):
|
|
return Dist(x**(-m*p)*(a + b*x**m)**p*(c + d*x**n)**q, Int(u*x**(m*p), x), x)
|
|
|
|
|
|
def replacement6999(a, b, c, n, n2, p, u, x):
|
|
return Dist((S(4)*c)**(S(1)/2 - p)*sqrt(a + b*x**n + c*x**(S(2)*n))/(b + S(2)*c*x**n), Int(u*(b + S(2)*c*x**n)**(S(2)*p), x), x)
|
|
|
|
|
|
def With7000(u, x):
|
|
if isinstance(x, (int, Integer, float, Float)):
|
|
return False
|
|
try:
|
|
lst = SubstForFractionalPowerOfLinear(u, x)
|
|
res = Not(FalseQ(lst))
|
|
except (TypeError, AttributeError):
|
|
return False
|
|
if res:
|
|
return True
|
|
return False
|
|
|
|
|
|
def replacement7000(u, x):
|
|
|
|
lst = SubstForFractionalPowerOfLinear(u, x)
|
|
return Dist(Part(lst, S(2))*Part(lst, S(4)), Subst(Int(Part(lst, S(1)), x), x, Part(lst, S(3))**(S(1)/Part(lst, S(2)))), x)
|
|
|
|
|
|
def replacement7001(u, x):
|
|
return Int(u, x)
|