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Integrate a power beside a Laurent polynomial in an exponential term by term - #2103
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…by term (c + d x)^m/(a + a tanh(e + f x)) was declined. tanh(u) is (e^(2u) - 1)/(e^(2u) + 1) here, so the factor is (1 + e^(-2u))/(2a), a Laurent polynomial in e^(2u), and each of its terms beside the power of the linear is a gamma function. A rational function of one exponential e^B is now read in w = e^B, every exponential a whole multiple of B, and where the quotient has a power of w below the bar and two terms or more, each term is integrated alone. Beside it only constants and powers of x or of one linear, the shapes the gamma rule reads: - for a linear B, at least one of them not whole. Whole powers are elementary or the sine and cosine integrals, sin(x)/x is Si(x), and are answered as such; - for any other B, powers of x alone: x sinh(a + b x^n)^2 is x (e^(2a + 2b x^n) - 2 + e^(-2a - 2b x^n))/4. Beside anything else, sin(c + d/x)/(e + f x) or (a + b x^n)^p sin(c + d x), asking the gamma rule of each term was a slow decline, and the first version of this answered sin(x)/x in gamma functions. The sine, cosine, tangent and cotangent of a linear A are written in e^(i A) first: 1/(1 + i tan(u)) is (1 + e^(-2 i u))/2. And a power of an exponential, (F^E)^n, is F^(n E) times (F^E)^n F^(-n E), which is constant between the cuts and is carried through as a constant: (a + b (F^(g (e + f x)))^n)^2 (c + d x)^m, which timed out, is three gamma functions. Part of #2075. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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October 24, 2026 20:26
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Part of #2075.
(c + d x)^m/(a + a tanh(e + f x))was declined. Sincetanh(u)is(e^(2u) - 1)/(e^(2u) + 1), the factor is(1 + e^(-2u))/(2a): a Laurent polynomial ine^(2u). Each of its terms beside the power of the linear is a gamma function.This reads a rational function of one exponential
e^Binw = e^B, with every exponential a whole multiple ofB. Where the quotient inwhas a power ofwbelow the bar and two terms or more, each term is integrated alone. The sine, cosine, tangent and cotangent of a linear are first written ine^(i A), so1/(1 + i tan(u))is(1 + e^(-2 i u))/2. A power of an exponential,(F^E)^n, isF^(n E)times a factor that is constant between the cuts, and that factor is carried through as a constant.Only constants and the powers the gamma rule reads may stand beside it: powers of
x, or of one linear with at least one exponent not whole. The first version took anything beside, and it failed 23 suite tests, lost 3 corpus rows, and allocated up to 20 GB on(a + b x^n)^p sin(c + d x). It also answeredsin(x)/xin gamma functions rather than asSi(x).Corpus (the full fair Rubi corpus, at base
2be40badagainst68211405): +26 solved, none lost, 0 wrong. Five of the 26 (6.1.1:205–207, 6.1.3:98/100) are already solved on master by #2096–#2100, so the gain on master is 21:(c+d x)^m (a+b tan)^n(c+d x)^m (a+b cot)^n(c+d x)^m (a+b tanh)^n(c+d x)^m (a+b coth)^n(e x)^m (a+b sinh(c+d x^n))^p(e x)^m (a+b cosh(c+d x^n))^p(c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^pu (F^(c (a+b x)))^nAllocation per stream is 1.0010 / 1.0023 / 0.9759. The one row flagged, 6.1.7:196, is memo warmth that comes back on its own run. The branch is rebased onto
e1fe08c2, and the full suite passes there.Tests:
APowerBesideALaurentPolynomialInAnExponential, nine integrands, each differentiated back.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura