Hyperbolic powers two apart whose coefficients kill the reduction's residual are integrated

[Rafael-SOWNet]

cosh(x)^(5/2) - 3 sqrt(cosh(x))/5 was left as written, and neither of its terms has an elementary antiderivative — both are elliptic. Their combination has one, and that is the whole rule: from

int cosh^p = sinh cosh^(p - 1)/p + (p - 1)/p int cosh^(p - 2)

a sum of powers two apart is elementary exactly when the walk from the highest exponent down carries nothing past the lowest, and the answer is what the walk accumulated. The hyperbolic sine's reduction subtracts where the cosine's adds (cosh^2 = sinh^2 + 1 against sinh^2 = cosh^2 - 1), and a gap in the exponents is simply a term with coefficient zero — which is what lets the chain run several steps: Rubi's x/sech(x)^(7/2) - 5 x sqrt(sech(x))/21 closes over two, its coefficient being (5/7)(1/3).

Three things the measurements decided:

• One linear factor in front is taken by parts inside the rule — x F - int F, where F is the accumulated antiderivative and int F is the next power over p^2 a^2. A second power of x would need int f^p itself, which is elliptic, so it is declined.
• Before the sum split. Registered after it, the csch rows took 55 s: the split hands each elliptic term to the whole chain and only then finds out. Before it, they are 1–20 s.
• Each term is read over its own bar. 1/sech(y)^(5/2) arrives as (c^(-5/2))^(-1) or 1/(1/c)^(5/2) depending on what rewrote it, and both are c^(5/2); reading spellings instead of exponents is what made the first three versions decline the Rubi rows while answering the same integrand written as cosh^(5/2).

The gathering of a common linear factor of a sum is shared with a small new route that hands such a sum on as a product (x cosh(x)^(3/2) - x sqrt(cosh(x))/3 → x (…)), for the shapes where parts, not this rule, does the work.

Measured (final build):

• Family 6 (familycap.sh 6 20): 366 → 375/417, 0 wrong, timeouts 21 → 17 — 6.1.1:124, 6.2.1:98/127, 6.5.1:24/25/26, 6.6.1:24/25/26; nothing lost.
• Family 7: 215/270, 0 wrong, unchanged. 1774-problem suite: 1707, 0 wrong, 0 timeout, unchanged.
• Unit suite: 12,627 tests, 0 failed. bench.sh: PASSED on all 19 gated benchmarks.
• A combination whose coefficient does not kill the residual (cosh^(5/2) - cosh^(1/2)/2) is still declined, and the test pins that.

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