[Rafael-SOWNet]

Part of #718.

• What was declined. Rubi's 6.7.1 sqrt(csch(x)) (x cosh(x) - 4 sech(x) tanh(x)) is the derivative of 2 x/sqrt(csch(x)) - 4 sech(x)/csch(x)^(3/2). By parts leaves -2/sqrt(csch(x)), which is elliptic, and so is the other term, so the split declined both.
• The rule. With t = e^(s x), a power that is not whole of a rational function of t, W = B(t)^p, beside a polynomial in x whose coefficients are rational in t, is W R, and W'/W = p s t B_t/B is rational in t. By Liouville's theorem an elementary antiderivative with no logarithmic part is W times a rational function of x and t. Its denominator is free of x, as R's is, and its degree in x is at most R's, since a higher coefficient would be a constant over W, which is not rational. So F = W P/(t^m E), with P of unknown coefficients and E read off R's denominator, makes F' = W R a linear system. Solved, the answer is checked by its derivative.
• Branches. The answer keeps the integrand's own power. Every other power of its base is written as W times a whole power of the base, B^r = B^p B^(r - p), which holds on every branch. Respelled in powers of the hyperbolic sine, sqrt(1/S) as 1/sqrt(S), the same integrand was right only where sinh(x) > 0. The base is compared by its value in lowest terms, since two spellings of csch(x) that print alike need not be one tree.
• Where it runs. Last in the dispatch, at the question asked only, with at most degree 3 in x, degree 12 in t and 60 unknowns.

Input	master b7977b2f	this branch
6.7.1:1243 sqrt(csch(x)) (x cosh(x) - 4 sech(x) tanh(x))	integral(...)	137 characters, 1.8 s
sqrt(csch(a + x)) ((a + x) cosh(a + x) - 4 sech(a + x) tanh(a + x))	integral(...)	191 characters
(sqrt(csch(x)) (x^2 sinh(x) + x cosh(x)))'	integral(...)	213 characters

APowerOfAFunctionOfAnExponentialBesideXIntegralTest checks each on both sides of 0. The full unit suite passes (16,689). The library builds for netstandard2.0. The BREAKING-CHANGES lengths were measured on the branch and the 2.5.0 column on a v2.5.0 build. Rebased onto be7e92ed.

Whole corpus, 48,305 rows at a 5 s budget, against master b7977b2f (the branch on it, 5c1d509f):

• 6.7.1:1243 moved from unsolved to solved, and no row moved from solved to anything else.
• Twelve of 5.1.5's inverse-sine rows land either side of the harness's hard limit on both builds, which is the harness's far-point recheck and is fixed there. They are not this change's.
• No row's allocation grew past 1.3x and 300 MB.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura