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This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
    )import_modulematchpy)PatternReplacementRuleCustomConstraintis_match(  IntSumSetWithModuleScanMapAndFalseQZeroQ	NegativeQNonzeroQFreeQNFreeQListLog	PositiveQPositiveIntegerQNegativeIntegerQIntegerQ	IntegersQComplexNumberQPureComplexNumberQRealNumericQPositiveOrZeroQNegativeOrZeroQFractionOrNegativeQNegQEqualUnequalIntPartFracPart	RationalQProductQSumQNonsumQSubstFirstRestSqrtNumberQSqrtNumberSumQLinearQSqrtArcCoshCoefficientDenominatorHypergeometric2F1NotSimplifyFractionalPartIntegerPartAppellF1
EllipticPi	EllipticE	EllipticFArcTanArcCotArcCothArcTanhArcSinArcSinhArcCosArcCscArcSecArcCschArcSechSinhTanhCoshSechCschCoth	LessEqualLessGreaterGreaterEqual	FractionQIntLinearcQExpandIndependentQPowerQIntegerPowerQPositiveIntegerPowerQFractionalPowerQAtomQExpQLogQHeadMemberQTrigQSinQCosQTanQCotQSecQCscQSinCosTanCotSecCscHyperbolicQSinhQCoshQTanhQCothQSechQCschQInverseTrigQSinCosQ	SinhCoshQ	LeafCount	NumeratorNumberQNumericQLengthListQImReInverseHyperbolicQInverseFunctionQTrigHyperbolicFreeQInverseFunctionFreeQRealQEqQFractionalPowerFreeQComplexFreeQPolynomialQFactorSquareFreePowerOfLinearQExponent
QuadraticQLinearPairQBinomialPartsTrinomialPartsPolyQEvenQOddQPerfectSquareQNiceSqrtAuxQ	NiceSqrtQTogetherPosAuxPosQCoefficientList
ReplaceAllExpandLinearProductGCDContentFactorNumericFactorNonnumericFactorsMakeAssocListGensymSubstKernelSubstExpandExpressionApart
SmartApartMatchQPolynomialQuotientRemainderFreeFactorsNonfreeFactorsRemoveContentAuxRemoveContent	FreeTermsNonfreeTermsExpandAlgebraicFunctionCollectReciprocalsExpandCleanupAlgebraicFunctionQCoeffLeadTermRemainingTerms
LeadFactorRemainingFactorsLeadBase
LeadDegreeNumerDenom	hypergeomExponMergeMonomialsPolynomialDivide	BinomialQ
TrinomialQGeneralizedBinomialQGeneralizedTrinomialQFactorSquareFreeListPerfectPowerTestSquareFreeFactorTestRationalFunctionQRationalFunctionFactorsNonrationalFunctionFactorsReverseRationalFunctionExponentsRationalFunctionExpandExpandIntegrandSimplerQSimplerSqrtQSumSimplerQBinomialDegreeTrinomialDegreeCancelCommonFactorsSimplerIntegrandQGeneralizedBinomialDegreeGeneralizedBinomialPartsGeneralizedTrinomialDegreeGeneralizedTrinomialParts	MonomialQMonomialSumQMinimumMonomialExponentMonomialExponentLinearMatchQPowerOfLinearMatchQQuadraticMatchQCubicMatchQBinomialMatchQTrinomialMatchQGeneralizedBinomialMatchQGeneralizedTrinomialMatchQQuotientOfLinearsMatchQPolynomialTermQPolynomialTermsNonpolynomialTermsPseudoBinomialPartsNormalizePseudoBinomialPseudoBinomialPairQPseudoBinomialQPolynomialGCDPolyGCDAlgebraicFunctionFactorsNonalgebraicFunctionFactorsQuotientOfLinearsPQuotientOfLinearsPartsQuotientOfLinearsQFlattenSortAbsurdNumberQAbsurdNumberFactorsNonabsurdNumberFactorsSumSimplerAuxQPrependDropCombineExponentsFactorIntegerFactorAbsurdNumberSubstForInverseFunctionSubstForFractionalPower*SubstForFractionalPowerOfQuotientOfLinears"FractionalPowerOfQuotientOfLinearsSubstForFractionalPowerQSubstForFractionalPowerAuxQFractionalPowerOfSquareQFractionalPowerSubexpressionQApplyFactorNumericGcdMergeableFactorQMergeFactorMergeFactorsTrigSimplifyQTrigSimplifyTrigSimplifyRecurOrderFactorOrderSmallestOrderedQMinimumDegreePositiveFactorsSignNonpositiveFactorsPolynomialInAuxQPolynomialInQExponentInAux
ExponentInPolynomialInSubstAuxPolynomialInSubstDistribDistributeDegreeFunctionOfPowerDivideDegreesOfFactorsMonomialFactorFullSimplifyFunctionOfLinearSubstFunctionOfLinearNormalizeIntegrandNormalizeIntegrandAuxNormalizeIntegrandFactorNormalizeIntegrandFactorBaseNormalizeTogetherNormalizeLeadTermSignsAbsorbMinusSignNormalizeSumFactorsSignOfFactorNormalizePowerOfLinearSimplifyIntegrandSimplifyTermTogetherSimplifySmartSimplifySubstForExpnExpandToSumUnifySum
UnifyTerms	UnifyTerm	CalculusQFunctionOfInverseLinearPureFunctionOfSinhQPureFunctionOfTanhQPureFunctionOfCoshQIntegerQuotientQOddQuotientQEvenQuotientQFindTrigFactorFunctionOfSinhQFunctionOfCoshQOddHyperbolicPowerQFunctionOfTanhQFunctionOfTanhWeightFunctionOfHyperbolicQSmartNumeratorSmartDenominatorSubstForAuxActivateTrig
ExpandTrig
TrigExpandSubstForTrigSubstForHyperbolicInertTrigFreeQLCMSubstForFractionalPowerOfLinearFractionalPowerOfLinearInverseFunctionOfLinear
InertTrigQInertReciprocalQDeactivateTrigFixInertTrigFunctionDeactivateTrigAuxPowerOfInertTrigSumQPiecewiseLinearQKnownTrigIntegrandQKnownSineIntegrandQKnownTangentIntegrandQKnownCotangentIntegrandQKnownSecantIntegrandQTryPureTanSubstTryTanhSubstTryPureTanhSubstAbsurdNumberGCDAbsurdNumberGCDListExpandTrigExpandExpandTrigReduceExpandTrigReduceAuxNormalizeTrig	TrigToExpExpandTrigToExp
TrigReduceFunctionOfTrigAlgebraicTrigFunctionQFunctionOfHyperbolicFunctionOfQFunctionOfExpnQPureFunctionOfSinQPureFunctionOfCosQPureFunctionOfTanQPureFunctionOfCotQFunctionOfCosQFunctionOfSinQOddTrigPowerQFunctionOfTanQFunctionOfTanWeightFunctionOfTrigQFunctionOfDensePolynomialsQFunctionOfLogPowerVariableExpnPowerVariableDegreePowerVariableSubstEulerIntegrandQFunctionOfSquareRootOfQuadraticSquareRootOfQuadraticSubstDividesEasyDQProductOfLinearPowersQRtNthRoot	AtomBaseQSumBaseQNegSumBaseQAllNegTermQSomeNegTermQTrigSquareQRtAux
TrigSquareIntSumIntTermMap2ConstantFactorSameQReplacePartCommonFactorsMostMainFactorPositionFunctionOfExponentialQFunctionOfExponentialFunctionOfExponentialFunction FunctionOfExponentialFunctionAuxFunctionOfExponentialTestFunctionOfExponentialTestAuxstdev	rubi_testIfIntQuadraticQIntBinomialQRectifyTangentRectifyCotangent
Inequality	ConditionSimpSimpHelpSplitProductSplitSumSubstForrG  FresnelSFresnelCErfcErfiGammaFunctionOfTrigOfLinearQElementaryFunctionQComplexUnsameQ_SimpFixFactorSimpFixFactor_FixSimplifyFixSimplify_SimplifyAntiderivativeSumSimplifyAntiderivativeSum_SimplifyAntiderivativeSimplifyAntiderivative_TrigSimplifyAuxTrigSimplifyAuxCancelPartPolyLogDDistSum_doitPolynomialQuotientFloorPolynomialRemainderFactorr  CosIntegralSinIntegralLogIntegralSinhIntegralCoshIntegralRuleErf	PolyGammaExpIntegralEiExpIntegralELogGammaUtilityOperator	FactorialZeta
ProductLogDerivativeDividesHypergeometricPFQIntHideOneQNullrubi_exprubi_logDiscriminantNegativeQuotient)Add)Mod)Mul)
EulerGamma)FloatIInteger)Pow)S)Abssign)sqrt)Integral)AndOr)simplifyWC)symbolsSymbol)sincostancotcscsecr  erf)acoshasinhatanhacothacschasechcoshsinhtanhcothsechcsch)atanacscasinacotacosasecatan2)pic                 C      g | ]}t |qS  r  .0ir  r  Q/tmp/pip-target-vg8gfxp4/lib/python/sympy/integrals/rubi/rules/linear_products.py
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||	}t|tԃ}tttttdtd  tdtd ttdtd tdtd tdtd  t| |||	||
|||
}t|tՃ}ttttdtd tdtd t ttdtd tdtd  t| |||	|
||}t|tփ}ttttdtd tdtd t ttdtd tdtd  t| |||	|
||}t|t׃}tttdttdtd tdtd ttdtd tdtd tdd	    t| |||	| }t|t؃}tttdttdtd tdtd ttdtd tdtd tdd	    t| |||	|!}t|tك}tttdttdtd tdtd ttdtd tdtd td
d	    t| |||	| }t|tڃ}tttdttdtd tdtd ttdtd tdtd td
d	    t| |||	|!}t|tۃ}ttttdtd tdtd t ttdtd tdtd  t| |||	|
||"}t|t܃}tttttdtd  t t t||	||#}t|t݃}ttttdtd tdtd t tttdtd   t| |||	||
|#}t|tރ}ttttdtd tdtd t ttdtd tdtd t  t| |||	|
|$|%||&|'|(}t|t߃}ttttdtd tdtd t ttdtd tdtd t  t| |||	|
|$|%|)|(
}t|t}ttttdtd tdtd t ttdtd tdtd t  t| |||	|
|$||*|+|,|(}t|t}tttdttttdtd  tttdtd tdtd   t| |||	|-|.}t|t}tttdtttdtd tdtd tttdtd tdtd   t| |||	|/|0}t|t}tttdttttdtd  tttdtd tdtd   t| |||	|
|1}t|t}tttdtttdtd tdtd tttdtd tdtd   t| |||	|
}t|t}tttttdtd  t ttdtd tdtd t  t| |||	|
|2|3|4	}t|t} tttdttdtd tdtd tdd	  ttdtd tdtd td
d	    t| |||	|
|5}t|t}tttdttdtd tdtd tdd	  ttdtd tdtd td
d	    t| |||	|
|6}t|t}ttttdtd tdtd t ttdtd tdtd t  t| |||	|
|$|3|7	}t|t}ttttdtd tdtd t ttdtd tdtd t  t| |||	|
|$|3|"|8|(}t|t}ttttdtd tdtd t ttdtd tdtd t  t| |||	|||
|9||:}	t|	t}
ttttdtd t tttdtd  t  t|||	|||;|<}t|t}ttttdtd t tttdtd  t  t|||	|||#|=}t|t}ttttdtd t tttdtd  t  t|||	|||;|#|>|?|@}t|t}ttttdtd t tttdtd  t  t|||	|||;|#|>|?
}t|t}tttttdtd  t tttdtd  t  t| |||	||
|;|	}t|t}tttttdtd  t tttdtd  t  t| |||	|||
|;|#|A|B}t|t}tttttdtd  t tttdtd  t  t| |||	|||
|;|#|C}t|t}ttttdtd tdtd tdtd ttdtd tdtd tdtd  t| |||	||||D	}t|t}tttttdtd  tdtd tttdtd  tdtd  ttdtd tdtd tdtd  t| |||	|G|H|||I||E|F}t|t}ttttdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|J|K}t|t}ttttdtd tdtd tttdtd   tttdtd  tdtd  t| ||	|G|H||L|M|N
}t|t} ttttdtd tdtd tttdtd   tttdtd  tdtd  t| ||	|G|H||L|O|P|Q}!t|!t}"tttttdtd  tdtd ttdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||
|R
}#t|#t}$ttttdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|S|T|U}%t|%t}&ttttdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||J|V|W|X}'t|'t}(ttttdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|J|Y|Z})t|)t}*ttttdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|J
}+t|+t},ttttdtd tdtd td
 ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|J|[|\}-t|-t}.ttttdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|H|||I||]|Y|^|_}/t|/t}0ttttdtd tdtd tdtd ttdtd tdtd ttdtd tdtd   t| |||	|G|H|`}1t|1t }2ttttdtd tdtd tdtd ttdtd tdtd ttdtd tdtd   t| |||	|G|H|V|a	}3t|3t}4ttttdtd tdtd t ttdtd tdtd ttdtd tdtd   t| |||	|G|H|V|b	}5t|5t}6ttttdtd tdtd t ttdtd tdtd ttdtd tdtd   t| |||	|G|H|V|W	}7t|7t}8ttttdtd tdtd t ttdtd tdtd ttdtd tdtd   t| |||	|G|H|I|c	}9t|9t}:ttttdtd tdtd tdtd ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H|d|e|W
};t|;t}<ttttdtd tdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H|I|f|g
}=t|=t}>ttttdtd tdtd td
 ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|h
}?t|?t}@ttttdtd tdtd td
 ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|[
}At|At	}Btttdttdtd tdtd tdd	  ttdtd tdtd td
d	   ttdtd tdtd   t| |||	|G|H|i}Ct|Ct
}Dtttdtttdtd tdtd tttdtd tdtd  ttdtd tdtd   t| |||	|G|H|j}Et|Et}Fttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H|k|$|3|l}Gt|Gt}Httttdtd tdtd t ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|m|||n}It|It}Jttttdtd tdtd t ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H|||I|o|p|}Kt|Kt}Lttttdtd tdtd t ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H|||I|o|q}Mt|Mt}Nttttdtd tdtd t ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H|r|%||s|t}Ot|Ot}Pttttdtd tdtd t ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H|I|r|%|u|t}Qt|Qt}Rttttdtd tdtd t ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H|I|r|%||t}St|St}Tttttdtd tdtd t ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|2|v|w|F}Ut|Ut}Vttttdtd tdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H|r|x||w|y}Wt|Wt}Xttttdtd tdtd t ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|2|v|w|z}Yt|Yt}Zttttdtd tdtd t ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|2|%|F|{}[t|[t}\ttttdtd tdtd t ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|2|%|z}]t|]t}^ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H|||||}}_t|_t}`tttdttdtd tdtd tttdtd tdtd  ttdtd tdtd tdd    t| |||	|G|H|~}at|at}btttdttdtd tdtd tttdtd tdtd  ttdtd tdtd tdd    t| |||	|G|H|}ct|ct}dtttdttdtd tdtd tttdtd tdtd  ttdtd tdtd td	d    t| |||	|G|H|~}et|et}ftttdttdtd tdtd tttdtd tdtd  ttdtd tdtd td	d    t| |||	|G|H|}gt|gt}httttttdtd  tttdtd ttttdtd    t|||	|G|H||||
}it|it}jttttttdtd  tttdtd ttttdtd    t|||	|G|H||||
}kt|kt}lttttttdtd  tttdtd ttttdtd    t|||	|G|H||}mt|mt}ntttttdtd tdtd ttttdtd  ttttdtd    t| |||	|G|H|A|||}ot|ot }ptttttdtd tdtd ttttdtd  ttttdtd    t| |||	|G|H||	}qt|qt!}rtttdtttdtd ttttdtd   ttttdtd    t|||	|G|H|||	}st|st"}ttttdtttdtd ttttdtd   ttttdtd    t|||	|G|H|||	}ut|ut#}vtttdtttdtd ttttdtd   ttttdtd    t|||	|G|H|}wt|wt$}xtttdttttdtd  ttttdtd   ttttdtd    t| |||	|G|H|A||l||}yt|yt%}ztttdttttdtd  ttttdtd   ttttdtd    t| |||	|G|H|A||l||}{t|{t&}|tttdttttdtd  ttttdtd   ttttdtd    t| |||	|G|H||l|
}}t|}t'}~tttdttdtd tdtd ttdtd tdtd tdd	   ttdtd tdtd tdd	    t| |||	|G|H|}t|t(}ttttdtd tdtd t ttdtd tdtd tdd	  ttdtd tdtd tdd	    t| |||	|G|H||F|%
}t|t)}ttttdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|H|||I||E||}t|t*}ttttdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|H|||I||E}t|t+}ttttdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|H|||I|||_}t|t,}ttttdtd tdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H||I|
}t|t-}ttttdtd tdtd t ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H|||I|||}t|t.}ttttdtd tdtd t ttdtd tdtd tdtd  ttdtd tdtd t  t| |||	|G|H||I|m|}t|t/}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  t| |||	|G|H|||I|m|#}t|t0}ttttdtd t tttdtd  t  tttdtd  t  t|||	|G|H|||I|;|#||}t|t1}ttttdtd t tttdtd  t  tttdtd  t  t|||	|G|H|||I|;|#||}t|t2}ttttdtd t tttdtd  t  tttdtd  t  t|||	|G|H|||I|;|#|>}t|t3}tttttdtd  t ttdtd tdtd t  ttdtd tdtd t  t| |||	|G|H|||;|#|`|A|}t|t4}tttttdtd  t ttdtd tdtd t  ttdtd tdtd t  t| |||	|G|H|||;|#|`||}t|t5}tttttdtd  t ttdtd tdtd t  ttdtd tdtd t  t| |||	|G|H|||I|;|#|c|A|||}t|t6}tttttdtd  t ttdtd tdtd t  ttdtd tdtd t  t| |||	|G|H|||I|;|#|c|A|}t|t7}tttttdtd  t ttdtd tdtd t  ttdtd tdtd t  t| |||	|G|H|||I|;|#|c|||}t|t8}tttttdtd  tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H|||I||}t|t9}tttttdtd  ttdtd tdtd tdtd  ttdtd tdtd tdtd  ttdtd tdtd  t| |||	|G|H|||
}t|t:}tttttdtd  ttdtd tdtd t  ttdtd tdtd tdtd  ttdtd tdtd  t| |||	|G|H|||||||}t|t;}tttttdtd  ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H|||$|%|}t|t<}tttttdtd  ttdtd tdtd t  ttdtd tdtd tdtd  ttdtd tdtd  t| |||	|G|H|||||}t|t=}tttttdtd  ttdtd tdtd t  ttdtd tdtd tdtd  ttdtd tdtd  t| |||	|G|H|||||||}t|t>}tttttdtd  ttdtd tdtd tdtd  ttdtd tdtd tdtd  ttdtd tdtd  t| |||	|G|H||||||}t|t?}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H||||}t|t@}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H|||I|$|%||F}t|tA}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H|||I|$|%||z}t|tB}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H||||I|2|%|F}t|tC}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H||||I|2|%|z}t|tD}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H||||I|2|x|_|F}t|tE}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H||||I|2|x|_|z}t|tF}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H||||I|||}t|tG}ttttdtd tdtd t ttdtd tdtd  ttdtd tdtd ttdtd tdtd   t| |||	|G|H|||
}t|tH}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd  ttdtd tdtd  t| |||	|G|H||||I|}t|tI}ttttdtd tdtd ttttdtd  ttttdtd   tttdtd tdtd   t| |||	|G|H||||}t|ÐtJ}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd  t| |||	|G|H|||||I|}t|ŐtK}ttttdtd tdtd t ttdtd tdtd tL  ttdtd tdtd ttdtd tdtd   t| |||	|G|H||||V|a}t|ǐtM}tttdttdtd tdtd tttdtd tdtd  tttdtd tdtd  tttdtd tdtd   t| |||	|G|H|||
}t|ɐtN}ttttdtd tdtd t ttdtd tdtd tttdtd tdtd  tttdtd tdtd   t| |||	|G|H|||
}t|ːtO}tttttdtd tdtd tttdtd tdtd  ttdtd tdtd tttdtd tdtd   t| |||	|G|H|||
}t|͐tP}tttdtttdtd tdtd tttdtd tdtd  tttdtd tdtd  tttdtd tdtd   t| |||	|G|H|||
}t|ϐtQ}tttttdtd tdtd ttdtd tdtd td	d
  tttdtd tdtd  tttdtd tdtd   t| |||	|G|H|||
}t|ѐtR}tttttdtd tdtd tttdtd tdtd tttdtd tdtd  tttdtd tdtd   t| |||	|G|H|||
}t|ӐtS}tttdttdtd tdtd td	d
  tttdtd tdtd  tttdtd tdtd  tttdtd tdtd   t| |||	|G|H|||
}t|ՐtT}tttttdtd tdtd tttdtd tdtd  tttdtd tdtd tttdtd tdtd   t| |||	|G|H|||
}t|אtU}ttttdtd tdtd td	d
  tttdtd tdtd tttdtd tdtd  tttdtd tdtd   t| |||	|G|H|||
}t|ِtV}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd tL  t| |||	|G|H|||||}t|ېtW}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd tL  t| |||	|G|H|||||I||}t|ݐtX}ttttdtd tdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H|||||I||}t|ߐtY}ttttdtd tdtd tdtd ttdtd tdtd tdtd  ttdtd tdtd tdtd  ttdtd tdtd tdtd  t| |||	|G|H|||||I|||}t|tZ}ttttdtd tdtd t ttdtd tdtd t  ttdtd tdtd t  ttdtd tdtd tL  tdtd t[ t| |||	|G|H||||||I|||}t|t\}g |||||||||đ|Ƒ|ȑ|ʑ|̑|Α|Б|ґ|ԑ|֑|ؑ|ڑ|ܑ|ޑ||||||||||||||||| |||||
||||||||||| |"|$|&|(|*|,|.|0|2|4|6|8|:|<|>|@|B|D|F|H|J|L|N|P|R|T|V|X|Z|\|^|`|b|d|f|h|j|l|n|p|r|t|v|x|z|||~|||||||||||||||||||||||||||||||||||đ|Ƒ|ȑ|ʑ|̑|Α|Б|ґ|ԑ|֑|ؑ|ڑ|ܑ|ޑ|||S )Nr   )cons2cons68cons19cons69cons3cons70cons71cons72cons8cons29cons73cons74cons4cons75cons76cons77cons45cons78cons79cons80cons81cons82cons83cons84cons64cons85cons86cons87cons88cons89cons90cons91cons92cons93cons94cons25cons95cons96cons97cons98cons99cons100cons101cons102cons103cons104cons105cons106cons107cons108cons33cons109cons110cons111cons112cons113cons114cons115cons116cons21cons117cons118cons119cons120cons121cons122cons123cons124cons125cons126cons20cons50cons127cons5cons128cons129cons130cons131cons132cons133cons134cons135cons136cons137cons56cons138cons13cons139cons140cons12cons141cons142cons143cons144cons145cons146cons40cons147cons148cons149cons150cons151cons152cons153cons154cons155cons156cons157cons158cons159cons160cons161cons162cons163cons164cons165cons166cons167cons168cons169cons170cons171cons172cons173cons174cons175cons176cons177cons178cons179cons180cons181cons182cons183cons184cons185cons186cons187cons188cons189cons190cons191cons192cons193cons194cons195cons196cons197cons198cons199cons200cons201cons202cons203cons204cons205cons206cons207cons208cons209cons210cons211cons212cons213cons214cons215cons216cons217cons218cons219cons220cons221cons52cons222cons223cons224cons225cons226cons54   mbadc            	   nfephgqr  (]   sympy.integrals.rubi.constraintsr4  r5  r6  r7  r8  r9  r:  r;  r<  r=  r>  r?  r@  rA  rB  rC  rD  rE  rF  rG  rH  rI  rJ  rK  rL  rM  rN  rO  rP  rQ  rR  rS  rT  rU  rV  rW  rX  rY  rZ  r[  r\  r]  r^  r_  r`  ra  rb  rc  rd  re  rf  rg  rh  ri  rj  rk  rl  rm  rn  ro  rp  rq  rr  rs  rt  ru  rv  rw  rx  ry  rz  r{  r|  r}  r~  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r   r  a_x_r   replacement39r  replacement40r  replacement41replacement42m_replacement43u_replacement44c_replacement45replacement46n_replacement47replacement48replacement49replacement50replacement51r  replacement52replacement53replacement54replacement55replacement56replacement57replacement58replacement59replacement60replacement61replacement62With63With64With65With66With67replacement68replacement69replacement70replacement71replacement72replacement73replacement74replacement75replacement76replacement77With78With79With80With81replacement82replacement83replacement84replacement85replacement86replacement87replacement88replacement89replacement90replacement91replacement92e_replacement93replacement94replacement95replacement96replacement97replacement98replacement99replacement100replacement101replacement102replacement103p_replacement104replacement105replacement106replacement107replacement108replacement109replacement110With111replacement112With113replacement114replacement115replacement116replacement117replacement118replacement119replacement120replacement121replacement122replacement123replacement124replacement125replacement126replacement127replacement128replacement129replacement130replacement131replacement132replacement133replacement134replacement135replacement136replacement137replacement138replacement139replacement140With141replacement142replacement143replacement144replacement145replacement146replacement147replacement148replacement149replacement150replacement151replacement152replacement153replacement154replacement155replacement156replacement157replacement158replacement159replacement160replacement161replacement162replacement163replacement164replacement165replacement166replacement167replacement168replacement169replacement170replacement171replacement172replacement173replacement174replacement175replacement176q_replacement177replacement178replacement179replacement180replacement181replacement182replacement183replacement184replacement185replacement186replacement187replacement188replacement189replacement190r_replacement191(  r4  r5  r6  r7  r8  r9  r:  r;  r<  r=  r>  r?  r@  rA  rB  rC  rD  rE  rF  rG  rH  rI  rJ  rK  rL  rM  rN  rO  rP  rQ  rR  rS  rT  rU  rV  rW  rX  rY  rZ  r[  r\  r]  r^  r_  r`  ra  rb  rc  rd  re  rf  rg  rh  ri  rj  rk  rl  rm  rn  ro  rp  rq  rr  rs  rt  ru  rv  rw  rx  ry  rz  r{  r|  r}  r~  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  	pattern39rule39	pattern40rule40	pattern41rule41	pattern42rule42	pattern43rule43	pattern44rule44	pattern45rule45	pattern46rule46	pattern47rule47	pattern48rule48	pattern49rule49	pattern50rule50	pattern51rule51	pattern52rule52	pattern53rule53	pattern54rule54	pattern55rule55	pattern56rule56	pattern57rule57	pattern58rule58	pattern59rule59	pattern60rule60	pattern61rule61	pattern62rule62	pattern63rule63	pattern64rule64	pattern65rule65	pattern66rule66	pattern67rule67	pattern68rule68	pattern69rule69	pattern70rule70	pattern71rule71	pattern72rule72	pattern73rule73	pattern74rule74	pattern75rule75	pattern76rule76	pattern77rule77	pattern78rule78	pattern79rule79	pattern80rule80	pattern81rule81	pattern82rule82	pattern83rule83	pattern84rule84	pattern85rule85	pattern86rule86	pattern87rule87	pattern88rule88	pattern89rule89	pattern90rule90	pattern91rule91	pattern92rule92	pattern93rule93	pattern94rule94	pattern95rule95	pattern96rule96	pattern97rule97	pattern98rule98	pattern99rule99
pattern100rule100
pattern101rule101
pattern102rule102
pattern103rule103
pattern104rule104
pattern105rule105
pattern106rule106
pattern107rule107
pattern108rule108
pattern109rule109
pattern110rule110
pattern111rule111
pattern112rule112
pattern113rule113
pattern114rule114
pattern115rule115
pattern116rule116
pattern117rule117
pattern118rule118
pattern119rule119
pattern120rule120
pattern121rule121
pattern122rule122
pattern123rule123
pattern124rule124
pattern125rule125
pattern126rule126
pattern127rule127
pattern128rule128
pattern129rule129
pattern130rule130
pattern131rule131
pattern132rule132
pattern133rule133
pattern134rule134
pattern135rule135
pattern136rule136
pattern137rule137
pattern138rule138
pattern139rule139
pattern140rule140
pattern141rule141
pattern142rule142
pattern143rule143
pattern144rule144
pattern145rule145
pattern146rule146
pattern147rule147
pattern148rule148
pattern149rule149
pattern150rule150
pattern151rule151
pattern152rule152
pattern153rule153
pattern154rule154
pattern155rule155
pattern156rule156
pattern157rule157
pattern158rule158
pattern159rule159
pattern160rule160
pattern161rule161
pattern162rule162
pattern163rule163
pattern164rule164
pattern165rule165
pattern166rule166
pattern167rule167
pattern168rule168
pattern169rule169
pattern170rule170
pattern171rule171
pattern172rule172
pattern173rule173
pattern174rule174
pattern175rule175
pattern176rule176
pattern177rule177
pattern178rule178
pattern179rule179
pattern180rule180
pattern181rule181
pattern182rule182
pattern183rule183
pattern184rule184
pattern185rule185
pattern186rule186
pattern187rule187
pattern188rule188
pattern189rule189
pattern190rule190
pattern191rule191r  r  r  linear_products   sr    

 
.
6
8
H
\
d
J
`
J
`
T
P
L
b
b
N
N
v
n
\
\
h
h
h
h
\
0
R
h
d
h
\
f
\
d
XxxdhhJJPNPVTxxzhhdxtnnjxxtppn  T r  c                 C   s   t t|  | S N)r  r  xr  r  r  r  h     r  c                 C   s   t t| | S r  )r  logr  r  r  r  r  l  r  r  c                 C   s"   t || td  | td  |S Nr  r  r  )r  r  r  r  r  r  p  s   "r  c                 C   s    t tt| ||  || |S r  )r  r  r   )r  r  r  r  r  r  r  t  s    r  c                 C   s.   t | ||  |td  ||td   |S r  r  )r  r  r  r  r  r  r  r  x  s   .r  c                 C   s:   t tdt||td tt| ||  | ||||S r  r  r  r3   r+   r   )r  r  r  ur  r  r  r  r  |  s   :r  c                 C   s*   t td| | || |td    |S Nr  r  r   r  r  r  r  r  r  r  r  r  r    s   *r  c                 C   sh   t ||  | ||   ttd| ||   ||t ||  | ||   ttd|||   || S r  r  r   r  r  r  r  r  r    s   hr  c                 C   sT   t | ||  |td  |||  |td   |td |  | ||    |S r  r  r  r  r  r  r  r  r  r  r  r  r    s   Tr  c                 C   s   t td|  | | td| td  t| ||  |td  |||  |td   ||t|| ||  |  |||  |  td| td  | S Nr  r  r  r  r   r  r  r  r  r  r  r  r  r  r  r	       r	  c                 C   s2   t || | t| ||   t|||    |S r  )r  r  r  r  r  r  r
       2r
  c                 C   s   t td| td td|  | |td   t| ||  |td  |||  |td   ||t|| ||  |td   |||  |td   td|  | |td   | S )Nr  r  r  r  r  r  r  r  r    s   r  c                 C   s&   t | | || |td   | |S Nr  r  r  r  r  r  r    s   &r  c                 C   s   t t|| |  | |S r  )r  r  r  r  r  r  r    s   r  c              
   C   R   t tdtttd|||td    ||t| ||  t|||   |S Nr  r  r  r  r+   r   r  r  r  r  r  r       Rr  c                 C   sx   t | ||  t| |||  t|  | | || |td   t|   t| | || |td   | ||S r  r  r&   r  r   r  r  r  r  r       xr  c                 C   s   t |  | ||  td|  ttd| ||  tdd  |||  tdd    ||ttd || ||  tdd   |||  tdd    | S )Nr  r  r  r  r  r  r  r  r  r       r  c                 C   s   t |td|  ttd| ||  tdd  |||  tdd    || ttd td| | ||  tdd   |||  tdd    | S )Nr  r  r  r  r  r  r  r  r    r  r  c                 C   s   t td| | || td  t| ||  | |||  |td   ||t| ||  |td  |||  |  ||| td   | S r  r  r  r  r  r  r       r  c                 C   s   t || td td|  |td   t| ||  |td  |||  |  ||t| ||  |td  |||  |td   td|  | |td   | S r  r  r  r  r  r  r       r  c                 C   ,   t t| ||  | |||  |  ||S r  r   r   r  r  r  r  r       ,r  c                 C   r  r  r  r  r  r  r  r    r  r  c                 C   sb   t |  | ||  | t|||  |td  | ||   ||t|||  | ||  | S Nr  r  r   r  r  r  r  r  r  r  r  r  r  r  r       br  c                 C   s   t ||  | ||   t|||  |td  | ||   ||t|||  |td  |td |  | ||    | S r  r  r  r  r  r  r       r  c              	   C   s   t |  | ||  | td}ttdtd|  tttd|td ||  |td   |||||  tdd  |ttdtd| |  tttd||  |||||  tdd  | ttt| ||  |td| |  | S Nr  r  r  r  r  r  r+   r   r  r  r   r  r  r  r  r  r  r  r  r  r    s    r  c              	   C   s   t |  | ||   | td}ttdtd|  tttd|td ||  |td   |||||  tdd  |ttdtd| |  tttd||  |||||  tdd  | ttt| ||  |td| |  | S r  r  r  r  r  r  r    s   "r  c              
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   C   sL   t td| tttdt|  | |td   ||t| ||  |S r  r  r  r  r  r  r$       Lr$  c              
   C   r  r  r  r  r  r  r  r%    r  r%  c                 C   s   t | ||  | |||  |  | | || |td   || | ||    |   t| | || |td   || | ||    | ||S r  )r  r  r   r  r  r  r  r&    r  r&  c              	   C   s  t || td}t|t|||   td|  | ttd| t|| ||  tdd   |||  tdd   td  td|  | tttd| ttdttd | | ||  tdd   td|||  tdd    ttdtd   | | S )Nr  r  r  r  r  r  r  r  r  r>   r  r  r  r  r'  	  s   
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 r(  c              	   C   sx   t |}t|tt|||td  td  ||||    ||| ||  td|  |||  td |   |S r  r  r  r  r  r  r  r  r  r  r  r  r  r)       pr)  c              	   C   sx   t |}t|| tt|||td  td  |  | | | |||  |  |  ||| ||  td|  |S r  r  r  r  r  r  r*    r  r*  c                 C   r  r  r  r  r  r  r  r+    r  r+  c                 C   s\   t || | | |td   t| |td |td | | |  | |td   |S r  r  r  r  r  r  r  r  r  r  r  r,  !  r  r,  c              	   C   sr   t | | |  |  |||  |td   t| |td |td td|| |   ||td   |S r  r  r  r  r  r  r-  %  s   rr-  c                 C   sl   t |t| td|| |  t|   |||  t|  t| | | td|| |  |  ||S r  r  r%   r  r&   r   r  r  r  r  r.  )     lr.  c                 C   sj   t | | t| |  | | t|  | | | t|   t| | | | |||  |  ||S r  )r  r&   r%   r   r  r  r  r  r/  -     jr/  c              
   C   s   t || td  | ||  |td   |  | ||  |  t| |td |td | | ||   |  | ||    |td  |S Nr  r  r  r  r  r  r  r  r0  1  r  r0  c              
   C   s   t ||  | ||   |  | ||  |td   t| |td |td | | ||   |  | ||    ||td   |S r  r  r  r  r  r  r1  5  s   r1  c              	   C   s   t ||  | ||   t|  ||||   |  | ||   t|   |||  t|  t| ||  | || |  | ||   || | |  | ||    |  ||S r  r  r%   r&   r   r  r  r  r  r2  9  s   r2  c                 C   sJ   t tdt||td tt| ||  | |||  |  ||||S r  r  )r  r  r  r  r  r  r  r  r  r  r  r3  =     Jr3  c
           
      C   s6   t |||	  | | | || |	td   |  |	S r  r  
r  r  r  r  r  r  r  r  r  r  r  r  r  r4  A  s   6r4  c	           	      C   sR   t ||||  |td   |||  |td   || || td   |S r  r  	r  r  r  r  r  r  r  r  r  r  r  r  r5  E  r  r5  c                 C   4   t t|| | | ||   |||  |  ||S r  r  r  r  r  r  r  r  r  r  r  r  r  r7  I     4r7  c                 C   r  r  r  r  r  r  r  r8  M  r  r8  c	           	      C   s8   t t| ||  |||  |  |||  |  ||S r  r  r  r  r  r  r9  Q  s   8r9  c	           	      C   s   t || t|||  | |||  |td   ||t|||  |td  |||  |td   |  | ||   ||td  || ||    | S r  r  r  r  r  r  r:  U  s   r:  c	           	      C     t | | | || td  ||| |td  || |td     ||td  || ||    t|||  | |||  |td   || t|||  |td  |||  |td   |  | ||   ||td  || ||    | S r  r  r  r  r  r  r;  Y      r;  c	           	      C   r  r  r  r  r  r  r  r<  ]  r  r<  c	           	      C   s   t | | | || td  ||| |td  || |td     || || td   t|||  | |||  |  ||t||||  |td   |||  |td   || || td   | S r  r  r  r  r  r  r=  a  s   r=  c	           	   	   C   s   t ||||  |td   |||  |td   td|  | | || td  || | | || td   ||| |td  || |td      |td |td  || td  || td   |S )Nr  r  r  r  r  r  r  r  r>  e  s   r>  c	           	      C   s   t | t|| | | ||  |  |||  |  ||t || t|| |td  | ||  |  |||  |  || S r  r  	r  r  r  r  r  r  r  r  r  r  r  r  r?  i  r  r?  c                 C   s4   t t|||  | | ||  |||    ||S r  r  r  r  r  r  r  r  r  r  r  r  r  r@  m  r  r@  c                 C   s   t |  | ||  |  | ||   t|||  |td  | ||   ||t | | ||  |  | ||   t|||  |td  |||   || S r  r  r  r  r  r  rA  q  r  rA  c              	   C   s   t td||  t|||  |td  |  | |td  || |td   || |  | | || |  td| | |     | ||  |||    ||t||||  |td   || |td   | S Nr  r  r  r  r  r  r  r  rC  u  s   rC  c                 C   s   t td|  | ||  | | ||    t|||  |td  |  | | || |  || |  || | |   | ||  |||    ||t||||  |td   |td |  | ||   | | ||    | S r  r  r  r  r  r  rD  y  s   rD  c                 C   sx   t ||  | ||   t|||  | | ||   ||t ||  | ||   t|||  | |||   || S r  r  r   r  r  r  r  rE  }  r  rE  c	           	      C   sN   t t|||  t| |||  | |||  t|  | ||   ||S r  )r   r   r8   r9   r  r  r  r  rF    s   NrF  c
           
      C   <   t t| ||	  | |||	  |  |||	  |  |	|	S r  r  r  r  r  r  rG       <rG  c	           	      C   s  t td|td |td  | | ||    t|||  |td  |||  |  t| td |td  | || td  td|  | | || |td  || |td     |td | || |td  || |td     |td | | |td  | | ||    | || t|||  |td  |||  |td   |  | ||  td  |td |td  | | ||    | S r  r  r  r  r  r  rH        rH  c	           	      C   s^  t td|| || td   t|||  | |||  |  t| td | | || td  || | | | || td  ||| |td  || |td       || || |td  || |td    || |    | ||t|| ||   |||  |td   |||  |td   || || td   | S )Nr  r  r  r  r  r  r  r  r  rI    s   ` rI  c              	   C   sf  t | | ||  |  | ||   td}t|t|||   td | | td| |   |ttd| t|| ||  tdd   |||  tdd    td | | td| |   | tttd| ttdttd | | ||  tdd   td|||  tdd    ttdtd   | | ||   | S r  r  r  r  r  r  r  r  r  r  r  r  r  rJ    s   .: rJ  c              
   C   sx   t || tttd||td  |td  ||  | ||  td    ||t| ||  t|||   |S r  )r  r+   r   r  r  r  r  r  r  r  r  r  r  r  r  rK    r  rK  c	           
   
   C   s   t |}	t|	tt||	|td  td  |  | ||  ||	 | | ||     ||| ||  td|	  |||  td |	   |S r  r  )
r  r  r  r  r  r  r  r  r  r  r  r  r  rL    s   rL  c
           
      C   s   t || | ||   |td |  | ||    t| ||	  |td  |||	  |td   |||	  |  |	|	 t| ||	  |td  |||	  |  |||	  |td   |td |  | ||    |	 S r  r  r  r  r  r  rM    s   rM  c
           
      C   s   t || ||	  |td   |||	  |td   |||	  |td   |td |  | ||   |  | ||    |	S r  r  r  r  r  r  rN    r  rN  c
           
      C   s<  t | | | |td  || | |td   || | |td   |td |  | ||   |  | ||    t| ||	  |td  |||	  |  |||	  |  |	|	t|| ||	  |td   |||	  |td   |||	  |td   |td |  | ||   |  | ||    |	 S r  r  r  r  r  r  rO    s   > rO  c
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   C   s   t td|td |  | ||    t| ||	  |td  |||	  |td   |||	  |  t|| || td  || |  || |	 || | td   |	 |	|	 t| ||	  |td  |||	  |  |||	  |td   |td |  | ||    |	 S Nr  r  r  r  r  r  r  r  rR    s   " rR  c
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   C   l   t tdttt| | t|t|  | td  || ||   t|  | td | t|  |S r  r  r  r=   r	  r  r  r)  r  r  r  rb    r	  rb  c              
   C   r,  r  r-  r)  r  r  r  rc    r	  rc  c              	   C   s   t ttd|| |  ttd|| |   t|||  t|||    ttdt| | ttd|| |   ttd|| |    ||S r  r+  r)  r  r  r  rd    r   rd  c              	   C      t tdt|td |  | ||  |  | ||     ttt| ||  t|  | ||   | td ||  | ||   ||  | ||     t|  | ||   | td | |S r  r  r  r  r=   r	  r  r  r  r  r  re       re  c              	   C   r.  r  r/  r  r  r  r  rf    r0  rf  c                 C   s  t t||||   |  | ||   t||||   |  | ||    t|||  t|||    ttdt| ||  t|| |  | ||   || | |  | ||     t|| |  | ||   || | |  | ||      ||S r  r(  r  r  r  r  rg    s    rg  c              	   C   sl  t ||  | ||   |  | ||  td  td}tt| ||  td| |  | ||    | ttdt||||  tdd   |||  tdd    td| |  | ||    | tttdttdttd | |||  tdd   td|||  tdd    ttdtd   td| |  | ||    | S )Nr  r  r  r  r  r  r  r  r  rh    s   :4 rh  c              	   C   sx  t |td|td  |  | ||   |  | ||    t| ||  |td  | | td| td  td| | td| td   td| | | td| td    |||  tdd  |||  tdd    ||t|| ||  |td   |||  tdd   |||  tdd   |td |  | ||   |  | ||    | S )N   r  r  r  r     r  )r  r  r  r  r  r  r  r  r  r  r  ri       z ri  c	           	      C   s2   t || | | | || |td   |  |S r  r  r  r  r  r  rj    r  rj  c	           	   	   C   s   t | ||  t| |||  t|  | | || |td   t|   t|| | | | || |td   |  ||S r  r  r  r  r  r  rk    r'  rk  c	           	      C   sJ   t t|| | | ||  |  |||  |  | ||  ||  ||S r  r  r  r  r  r  rl    r  rl  c
           
      C   r  r  r  r  r  r  r  rm     r  rm  c
           
      C   r#  r$  r  r  r  r  r  rn  $  r%  rn  c
           
      C   s   t | ||	  |td  |||	  | td   |  | ||  |  |  | ||  | td   t|td | |td | ||	   | | ||   |||	  |  | ||     |td  |	S r  r  r  r  r  r  ro  (     ro  c
           
      C   s   t |||	  |  | ||   |||	  |  | ||    |  | ||	  |td   |||	  |  |||	  |td   t|td | |td | ||	   | | ||   |||	  |  | ||     |td |  | ||    |	S r  r  r  r  r  r  rp  ,  s    rp  c	           	   
   C   st   t || ||  | | |td   t|td | | |td | | | | | |  | |td   |S r  r  r  r:   	r  r  r  r  r  r  r  r  r  r  r  r  rq  0  s   trq  c	           	      C   s   t || | ||   |  | | |  |   |||  |td   t|td | | |td td|| |  | |||   | | ||    ||td   |S r  r5  r6  r  r  r  rr  4  s   rr  c	           	      C   s|   t |t| td|| |  t|   |||  t|  t| | | td|| |  |  |||  |  ||S r  r  r6  r  r  r  rs  8  s   |rs  c
           
      C   s   t || td  ||  | ||   |   | ||	  |td   |  | ||  |  t|td | | |td | | ||	   |  | ||   | | ||	   |  | ||    |td  |	S r  r5  r  r  r  r  rt  <  r4  rt  c
           
   	   C      t ||  | ||   t|  ||||	   |  | ||   t|   |||	  t|  t| ||	  | |||	  |  || |  | ||   || |	 |  | ||    |  |	|	S r  r  r  r  r  r  ru  @     ru  c
           
      C   s   t ||  | ||   |  ||  | ||   |   | ||	  |td   t|td | | |td | | ||	   |  | ||   | | ||	   |  | ||    ||td   |	S r  r5  r  r  r  r  rv  D  r0  rv  c
           
   	   C   s   t ||  | ||   t|  ||||	   |  | ||   t|   |||	  t|  t| ||	  | |||	  |  || |  | ||   || |	 |  | ||    |  |	|	S r  r  r  r  r  r  rw  H  r8  rw  c
           
   	   C   r7  r  r  r  r  r  r  rx  L  r8  rx  c                 C   sZ   t tdt|	|
td tt| ||
  | |||
  |  |||
  |  |
|
|	|
S r  r  )r  r  r  r  r  r  r  r  r  r  r  r  r  r  ry  P  s   Zry  c                 C   sD   t t| ||
  | |||
  |	  |||
   |||
   |
|
S r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  rz  T  s   Drz  c              
   C   s  t | | | | | || | | |td  ||| ||      |td |  t| ||
  |td  |||
  |	  |
|
t| ||
  |td  |||
  |	td   | td  | | | | | | | | | |td  ||| ||      |td | | |  || | |
 |td  |  | ||     |td | |td  |  | ||    |
 S r  r  r9  r  r  r  r{  X  s    r{  c                 C   s  t | td |td  | | |	td td|	  td  | | | |	td  td| | | |td  ||| ||   ||	 td     |td |td | | |td td|  td  || |td  || ||   ||	 td   |td | | |td |td|	 td   |	td  td|	  td     || |td  |	td  |  | ||  td   t| ||
  |td  |||
  |	td   |
|
 t| ||
  |td  |||
  |	td   | td | | | | |	td  | | |td | | |td  || || ||   ||	 td   |td | | |td     |td | | | | |	td   |
| td |td  | | |	td  | | |td  |	td  || ||    |td |td | | |td  || |td  || ||    |td | | ||	 td        || |td  |	td  |  | ||  td   |
 S )Nr  r  r  r  r1  r  r9  r  r  r  r|  \  s      r|  c                 C   sL  t | ||	 td  | td | | | ||	  | | td| | | |td  ||	td  || ||      |td ||td  || ||   || | ||	 td      |td |td  |td  |  | ||  td   || |td   t| ||
  |td  |||
  |	  |
|
t| ||
  |td  |||
  |	td   | td  | | | |	td  | td | || | | ||| ||   ||	 td     | |td  ||| ||   || | td| |	 td     |td | | | |td   ||
 | td | | | ||	  | | td| | | |td  ||	td  || ||      |td ||td  || ||   || | ||	 td        |td |td  |td  |  | ||  td   |
 S )Nr  r  r  r  r  r9  r  r  r  r}  `  s     R r}  c              
   C   s  t | td |td  | | |	td  |	td  | | | |	td  td| | | |td  ||| ||   ||	 td     |td |td | | |td  |td  || |td  || ||   ||	 td   |td | | ||	 td  ||	 td     |td | |td  |  | ||   ||	 td   t| ||
  |td  |||
  |	  |
|
 t| ||
  |td  |||
  |	td   | td | | | |	td  | | || | |td  ||| ||   ||	 td     |td | | | ||	 td   || | |
 |td  |  | ||     |td | |td  |  | ||   ||	 td   |
 S Nr  r  r  r  r9  r  r  r  r~  d  s     r~  c                 C   s  t | td |td  | | |	td  |	td  | | | |	td  td| | | |td  ||| ||   ||	 td     |td |td | | |td  |td  || |td  || ||   ||	 td   |td | | ||	 td  ||	 td     |td |td  ||	 td  ||	 td   t| ||
  | |||
  |	  |
|
t| ||
  |td  |||
  |	td   | | | | |	td  || | | |td   || | | |
 ||	 td   || || ||   ||	 td    |td |td  ||	 td  ||	 td   |
 S r:  r  r9  r  r  r  r  h  s     r  c                 C   sH   t t| ||  | |||  |	  |||  |
  |||   ||S r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  l  s   Hr  c                 C     t td||td  |  | ||    t| ||  |td  |||  |	td   |||  |
  t|| |td  | | ||   || ||td  | | ||   ||  | ||   |	|
 td     |  | ||  || |
td  || |	    | || t| ||  |td  |||  |	  |||  |
td   |  | ||   ||td  |  | ||    | S r  r  r;  r  r  r  r  p      r  c                 C   r<  r  r  r;  r  r  r  r  t  r=  r  c                 C     t td|td |  | ||   |  | ||    t| ||  |td  |||  |	  |||  |
  t| | | |  | ||   ||	 |
 td  |td | | | | || | |  || || ||      |  | ||  || |
td  || |	td     | ||t| ||  |td  |||  |	td   |||  |
td   |  | ||   |td |  | ||   |  | ||    | S Nr  r  r  r;  r  r  r  r  x      r  c                 C   r>  r?  r  r;  r  r  r  r  |  r@  r  c                 C     t td|| ||	 |
 td   t| ||  |td  |||  |	  |||  |
  t| | | | ||	 |
 td  || || |
td  || |	td    || | |    ||| | | ||	 |
 td  || | | | ||| ||
 td  || ||	 td         | ||t|| ||  |  |||  |	td   |||  |
td   || ||	 |
 td   | S Nr  r  r  r  r;  r  r  r  r        r  c                 C   rA  rB  r  r;  r  r  r  r    rC  r  c                 C   r>  r?  r  r;  r  r  r  r    r@  r  c
           
      C   s   t |  | ||  |  | ||   t|||	  | | ||	   |	|	t | | ||  |  | ||   t|||	  | |||	   |	|	 S r  r  )
r  r  r  r  r  r  r  r  r  r  r  r  r  r    r  r  c                 C   s~   t || t|||
  | |||
  |	  |
|
t |  | ||  | t|||
  | |||
  |	  | ||
   |
|
 S r  r  )r  r  r  r  r  r  r  r  r  r  r  r  r  r  r    s   ~r  c	           	   
   C   s   t || tt|||  t| ||  t|||    ||t | | ||  | ttdt| ||  t|||   t|||    || S r  r  r   r  r  	r  r  r  r  r  r  r  r  r  r  r  r  r       r  c                 C   s   t || t| ||  |td  |||  |	  |||  |
  ||t |  | ||  | t| ||  | |||  |	  |||  |
  || S r  r  r;  r  r  r  r    rF  r  c                 C   s   t |  | ||  |  | ||   t|||
  |td  |||
  |	  | ||
   |
|
t | | ||  |  | ||   t|||
  |td  |||
  |	  |||
   |
|
 S r  r  )r  r  r  r  r  r  r  r  r  r  r  r  r  r  r    s   r  c	           	   
   C   s4  t td t||||   | | ||    t||||   | | ||    t| | | ||   ||  | ||    tt| | | ||   t|||   || | ||   || | ||     t| | | ||   t|||   t|||   |  | ||    |S r  )r  r  r  r;   r	  rE  r  r  r  r    r"  r  c
           
      C   sj   t ttdt|||	  t|||	   t|||	    |||	  |tdd   | ||	   |	|	S r  )r   r   r  r  )
r  r  r  r  r  r  r  r  r  r  r  r  r  r    r
  r  c	           	   
   C   s   t |td t|  | | || |  || |  || | |  t|||  t|||   t|||    ||t |  | ||  |  | ||   |td  ttd| ||  t|||   t|||   t|||    || S )Nr  r  r  )r  r  r   r  rE  r  r  r  r       r  c	           	      C   sv  t td t|||  |  | ||   | ||  | | ||     t|||  |  | ||   | ||  | | ||     | ||   t|||  t|||   |  | ||    tttdt|td |  | ||   | | ||   td t|td |  | ||   | | ||   td   ||t|||  t| ||   |S r  r  rE  r  r  r  r    s   x r  c	           	   
   C   s  t td t|||  |  | ||   | ||  | | ||     t|||  |  | ||   | ||  | | ||     | ||   | | ||   t|||  t|||   |  | ||  td   ttt|td |  | ||   | | ||   td t|td |  | ||   | | ||   td  ||t|||  t| ||   |S r  r  rE  r  r  r  r    s    r  c	           	      C   sx  t tdt|||  |  | ||   | ||  | | ||     t|||  |  | ||   | ||  | | ||     | ||   t|||  t|||    tttd| |td  | t|td |  | ||   | | ||   td  t|td |  | ||   | | ||   td   ||t|||  t| ||   |S r  r  rE  r  r  r  r    r3  r  c	           	   
   C   s   t ||  | ||   tt|||  | ||  tdd  t|||   t|||    ||t ||  | ||   ttdt| ||  t|||   t|||   t|||    || S r  rD  rE  r  r  r  r    s   r  c	           	   
   C   s  t | | | | || | | || |  || |    td|td  |  tt|||  t| ||  t|||   t|||    ||t | | ||  td |  | | || |  || |   td|td  |  ttdt| ||  t|||   t|||   t|||    || t | | ||  | | ||   td| |  tt| ||  t|||  |||  tdd   t|||    || tt| ||  t|||   t|||   |t|||    | S r:  )r  r  r   r  r  rE  r  r  r  r    s     r  c	           	   
   C   s   t || tt| ||  t|||   t|||  t|||    ||t |  | ||  | tt| ||  t|||  t|||   t|||    || S r  )r  r   r  rE  r  r  r  r    s   r  c                 C   sL   t t| ||  | |||  |	  |||  |
  |||  |  ||S r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r    r  r  c              	   C   s   t || t| ||  |td  |||  |	  |||  |
  |||  |td   ||t |  | ||  | t| ||  | |||  |	  |||  |
  |||  |td   || S r  r  rH  r  r  r  r    s   r  c                 C   sF   t | ||  | |||  |	  |||  |
  |||  |  |S r  )r   rH  r  r  r  r    s   Fr  c                 C   sj   t tdt||td tt| ||  | |||  |	  |||  |
  |||  |  ||||S r  r  )r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r    r
  r  c                 C   s   t || ||  |	  |||  |
  |||  |  |||  |  | | ||  |	 |   |||  |
 |   |||  | |   |||  | |   t| ||  |	|  |||  |
|   |||  ||   |||  ||   ||S r  r  )r  r  r  r  r  r  r  r  r  r  r  r  r  rr  r  r  r  r    rG  r  (  __doc__sympy.externalr   r   r   r   r   r   %sympy.integrals.rubi.utility_functionr   r	   r
   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r    r!   r"   r#   r$   r%   r&   r'   r(   r)   r*   r+   r,   r-   r.   r/   r0   r1   r2   r3   r4   r5   r6   r7   r8   r9   r:   r;   r<   r=   r>   r?   r@   rA   rB   rC   rD   rE   rF   rG   rH   rI   rJ   rK   rL   rM   rN   rO   rP   rQ   rR   rS   rT   rU   rV   rW   rX   rY   rZ   r[   r\   r]   r^   r_   r`   ra   rb   rc   rd   re   rf   rg   rh   ri   rj   rk   rl   rm   rn   ro   rp   rq   rr   rs   rt   ru   rv   rw   rx   ry   rz   r{   r|   r}   r~   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r  r  r  r  r  r  r  r  r	  r
  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r   r!  r"  r#  r$  r%  r&  r'  r(  r)  r*  r+  r,  r-  r.  r/  r0  r1  r2  r3  r4  r5  r6  r7  r8  r9  r:  r;  r<  r=  r>  r?  r@  rA  rB  rC  rD  rE  rF  rG  rH  rI  rJ  rK  rL  rM  rN  rO  rP  rQ  rR  rS  rT  rU  rV  rW  rX  rY  rZ  r[  r\  r]  r^  r_  r`  ra  rb  rc  rd  re  rf  rg  rh  ri  rj  rk  rl  rm  rn  ro  rp  rq  rr  rs  rt  ru  rv  rw  rx  ry  rz  r{  r|  r}  r~  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  expr  r  r  r  r  sympy.core.addr  sympy.core.modr  sympy.core.mulr  
sympy.corer  sympy.core.numbersr  r  r  sympy.core.powerr  sympy.core.singletonr  $sympy.functions.elementary.complexesr  r  (sympy.functions.elementary.miscellaneousr  sympy.integrals.integralsr  sympy.logic.boolalgr  r  sympy.simplify.simplifyr  sympy.integrals.rubi.symbolr  sympy.core.symbolr  r  sympy.functionsr  r  r  r  r  r  r  %sympy.functions.elementary.hyperbolicr  r  r  r  r  r   r  r  r  r  r  r  (sympy.functions.elementary.trigonometricr  r  r	  r
  r  r  r  r  PiA_B_C_F_G_H_r  b_r  d_r6  f_g_h_i_j_k_l_r   r  rB  r  r  t_r  v_s_w_r  y_z_a1_a2_b1_b2_c1_c2_d1_d2_n1_n2_e1_e2_f1_f2_g1_g2_n3_Pq_Pm_Px_Qm_Qr_Qx_jn_mn_non2_RFx_RGx_r  iiPqqQRrI  Ckr  	_UseGamma	ShowStepsStepCounterr  r  r  r  r  r  r  r  r  r  r	  r
  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r   r!  r"  r#  r$  r%  r&  r'  r(  r)  r*  r+  r,  r-  r.  r/  r0  r1  r2  r3  r4  r5  r7  r8  r9  r:  r;  r<  r=  r>  r?  r@  rA  rC  rD  rE  rF  rG  rH  rI  rJ  rK  rL  rM  rN  rO  rP  rQ  rR  rS  rT  rU  rV  rW  rX  rY  rZ  r[  r\  r]  r^  r_  r`  ra  rb  rc  rd  re  rf  rg  rh  ri  rj  rk  rl  rm  rn  ro  rp  rq  rr  rs  rt  ru  rv  rw  rx  ry  rz  r{  r|  r}  r~  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  <module>   s              n"JjB.
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