A rational function of e^y with a symbolic denominator is integrated by the hyperbolic half angle

[Rafael-SOWNet]

csch(x)^2/(a + b sinh(x))^2 was a search past eighty seconds: under u = e^x its denominator is a symbolic quartic nothing factors. Under u = tanh(x/2) it is a rational function whose written factors the symbolic partial fractions answer — and SolveARationalFunctionOfTheHyperbolicTangent took that half only where every exponential was written with an even factor (coth(c + d x)'s e^(2(c + d x))). It is taken now for a root-free integrand of a plain e^y as well, where the denominator's shape depends on a symbol: read as a polynomial in v = e^y, a coefficient that is not a number times the first, and two distinct factors of v below the bar.

Both conditions come from measurements, and each cost a round:

• The content test. tanh(x)^6/(a + a sech(x)) has a for its content and the rest factors over the rationals, so u = e^x answers it a page shorter. The first version took the half there and produced 12,502 characters against the 8,000 the suite pins (SymbolicFactorsPartialFractionsTest.TheCoefficientsAreInLowestTerms) — the corpus was green and the unit suite caught it.
• Two factors. e^asinh(a + b x)/x^4 leaves e^u (e^u + e^-u)/(e^u - e^-u - 2a)^4 after the inverse hyperbolic substitution: root-free, symbolic, but a single quadratic in v to the fourth, whose roots the quadratic formula gives — one second under u = e^x against thirty under the half. Bare v is not counted, being where an odd power of it was gathered.
• The order. With the half taken for a symbolic denominator, SolveByBiochesOddHyperbolicSubstitution (A rational function of the hyperbolic functions odd in the sine or the cosine is integrated by the other #1464) must come before this route: cosh(x)^7/(a + b cosh(x)^2) and sech(x)^3/(a + b sinh(x))^2 are half a second by u = cosh(x) and past the budget by the half, which makes a polynomial of twice the degree. Measured: without the reorder family 6 fell to 359, below master.

Measured (final build):

• Family 6 (familycap.sh 6 20): 363 → 366/417, 0 wrong, timeouts 22 → 21; 6.1.5:116 and 6.6.3:112/157 gained, nothing lost.
• Family 7: 215/270, 0 wrong, unchanged. 1774-problem suite: 1707, 0 wrong, 0 timeout, unchanged.
• Unit suite: 12,619 tests, 0 failed. bench.sh: PASSED on all 19 gated benchmarks.

BREAKING-CHANGES.md carries the entry with the 2.5.0 column measured on a 2.5.0 build.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura