A fractional power of a ± a cosh(y) is integrated by the half angle at which it is a square

[Rafael-SOWNet]

x^2 sqrt(a + a cosh(c + d x)) was left as written: no substitution rationalises a root of a hyperbolic function beside a power of x, and beside the sine and cosine of y the root is of a quotient under every one. 1 + cosh(y) is 2 cosh(y/2)^2, so the root is sqrt(2a) cosh(y/2) with no sign to write (the hyperbolic cosine is positive); 1 - cosh(y) is -2 sinh(y/2)^2, so (a - a cosh(y))^(3/2) is (-2a)^(3/2) sgn(sinh(y/2)) sinh(y/2)^3, the sign a constant between the zeros that comes out in front — SolveByTheHalfAngleWhereOnePlusASineIsASquare's identity for the hyperbolic cosine. cosh(y) and sinh(y) beside the power are written in the half angle (2 cosh(y/2)^2 - 1, 2 sinh(y/2) cosh(y/2)) and the question is asked again in x (ComputeAsTheSameQuestion), where by parts, the hyperbolic Bioche rule (#1464) and the exponential substitution answer it.

Exact for any a: cosh(y/2)^2 and sinh(y/2)^2 are not negative, so (a q)^p = a^p q^p for the principal powers whatever a is, with 2a and -2a kept as the one constant they are ((-2a)^p is not (-2)^p a^p for a symbolic a). Whole products 2p only; at the top only, as every rule that writes a sign for a function. The reading of the six hyperbolic functions from their exponential spellings is lifted out of SolveByBiochesOddHyperbolicSubstitution into ReadTheHyperbolicFunctions and shared.

Measured (final build):

• Family 6 (familycap.sh 6 20): 357 → 363/417, 0 wrong — 6.2.1:165/170/176/178 and 6.2.5:75/141 answered; 6.2.5:64 cosh(x)/sqrt(a - a cosh(x)) answered too, graded unverifiable on the reals by the harness (the integrand is complex for a positive a; the test pins a = -3 as well as 2 and checks both sides of zero).
• Family 7: 215/270, 0 wrong, unchanged. 1774-problem suite: 1707, 0 wrong, 0 timeout, unchanged.
• Unit suite: 12,616 tests, 0 failed. bench.sh: PASSED on all 19 gated benchmarks.
• Each new test row verified by quadrature (IP_QUAD=1) as well as by differentiating back with a = 2 and a = -3.

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