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Integrate a power of a quotient of linears over a half-odd power of their product - #1960
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…heir product The parser folds e^(n acoth(a x)) into ((a x + 1)/(a x - 1))^(n/2), a power T^alpha of a quotient of linears T = (p + q x)/(r + s x). Over a half-odd power of a multiple Q of their product, with a power of x beside it, the antiderivative has Liouville's shape T^alpha P(x) Q^(1/2 - K). Since T'/T is D/((p + q x)(r + s x)), asking F' to be the integrand is a three-term recurrence for P's coefficients, solved from the bottom up. The last equation fixes p_0 and the one past it is checked: where it fails, the antiderivative is hypergeometric and the rule declines. Rubi's 7.4.2 e^(n acoth(a x))/(c - a^2 c x^2)^(3/2) to ^(9/2), and the same beside x, x^2 and x^3, were declined. Part of #233. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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The parser folds
e^(n acoth(a x))into((a x + 1)/(a x - 1))^(n/2), a powerT^alphaof a quotient of linears. Over a half-odd power of a multipleQof their product, with a power ofxbeside it, the antiderivative has Liouville's shapeT^alpha P(x) Q^(1/2 - K). SinceT'/T = D/((p + q x)(r + s x)), askingF'to equal the integrand gives a three-term recurrence for the coefficients ofP. Where the equation past the last one fails, the antiderivative is hypergeometric and the rule declines.Rubi's 7.4.2
e^(n acoth(a x))/(c - a^2 c x^2)^(3/2)to^(9/2), and the same besidex,x^2andx^3, were declined. They are answered now.Measured on
8a1ef454, branch against master:Re-run alone on both builds, 10 of the 13 problems that moved go from declined to answered (7.4.2:858–861, 864, 865, 868–871). The other three are load:
No run had a wrong answer.
On this master the dispatch line sits after
SolveAFunctionOfTheHyperbolicTangent, the last rule of its group. The Moebius rule (#1953) and its tests pass beside it.APowerOfAQuotientOfLinearsOverAHalfOddPowerOfTheirProductIntegralTestdifferentiates the answers back.BREAKING-CHANGES.mdhas the entry, with the 2.5.0 column measured on v2.5.0.Part of #233.
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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura