A rational function of the hyperbolic functions odd in the sine or the cosine is integrated by the other

[Rafael-SOWNet]

sinh(x)^3/(a + b cosh(x)^2) was a search past the budget: under u = e^x it is a rational function of a symbolic palindromic quartic, and under u = tanh(x/2) of a symbolic polynomial of degree eight. It is (u^2 - 1)/(a + b u^2) under u = cosh(x) — Bioche's first two rules, which the trigonometric functions already had (SolveByBiochesOddSubstitution): odd in the hyperbolic sine, u = cosh with sinh^2 = u^2 - 1; odd in the hyperbolic cosine, u = sinh with cosh^2 = u^2 + 1, both exact on the reals (cosh is positive, so u = sinh is a bijection of the line and u = cosh one on each side of zero).

What is new in SolveByBiochesOddHyperbolicSubstitution:

• The six hyperbolic functions are read back from their exponential spellings through the gathered exponentials (TryGatherExponentialsOfOneLinearForm), for any linear argument and either sign of it (sinh(-x), or y named from e^-y); tanh(c + d x) alone, written with e^(2(c + d x)), is read as the tangent of half the gathered form.
• A written power is handed on factor by factor as the power it is: (a + b sech(x)^2)^2 becomes (a u^2 + b)^2/u^4 and not the expanded symbolic quartic the rational integrator would have to factor again. A factor neither even nor odd (1 + sinh(x)) falls back to the expanded-and-mirrored form the trigonometric rule uses.
• A root of a polynomial in the two even in the function is admitted; a Laurent radicand (a + b sech(x) is a + b/u) is declined — measured: with it admitted coth(c + d x)^3/(a + b sech(c + d x))^(3/2) went from 8.9 s to past the budget, and the family run peaked at 2.7 GB; declined, both are back (8.6 s, 503 MB).
• ParityIn and InU are lifted out of the trigonometric rule and shared (fSquared is the parameter now: 1 - u^2, u^2 - 1 or u^2 + 1).
• Registered right after the rational-in-tanh route and before the substitution search, which with a symbolic coefficient spends the budget on u = e^x.

Measured (final build):

• Family 6 (familycap.sh 6 20): 344 → 357/417, 0 wrong, timeouts 34 → 24 — 6.1.5, 6.1.7, 6.2.5, 6.2.7, 6.3.7, 6.5.3, 6.5.7 rows, nothing lost.
• Family 7: 215/270, 0 wrong (one row flipped Unsolved → Timeout on the 5 s cap; it is 20 s+ on master solo too). 1774-problem suite: 1707, 0 wrong, 0 timeout, wall 85 s, unchanged.
• Unit suite: 12,610 tests, 0 failed. bench.sh: PASSED on all 19 gated benchmarks.
• Every new test row verified by quadrature (IP_QUAD=1) as well as by the pinned differentiate-back test.

BREAKING-CHANGES.md carries the entry with the 2.5.0 column measured on a 2.5.0 build (every row unevaluated there, including the rational-coefficient one).

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura