A power of x one less than a half-odd multiple of the power inside is substituted by half the power

[Rafael-SOWNet]

Part of #1501.

x^(m) g(x^n) with a symbolic n was integrated under u = x^n where (m + 1)/n is a whole k ≥ 1: x^(k n - 1) dx is u^(k - 1) du/n. Where k is half an odd number, of either sign, it is now the same substitution by half the power. Under u = x^(n/2), x^(k n - 1) dx is (2/n) u^(2k - 1) du, and g(x^n) is g(u^2). Beside an exponential of x^n, that is a moment of the Gaussian, which #1512 answers.

integral	master	this PR
f^(a + b x^n) x^(-1 + 5n/2), x^(-1 + 3n/2), x^(-1 + n/2)	unevaluated	answered
f^(a + b x^n) x^(-1 - n/2), x^(-1 - 3n/2)	unevaluated	answered
F^(c (a + b x)^n) (a + b x)^(-1 + n/2), and over it	unevaluated	answered
x^(-1 + n/2) sinh(a + b x^n), cosh	unevaluated	answered

A whole k below 1 is still declined, as it was. There, u^(k - 1) g(u) has a negative power in front of an exponential of u, which is the exponential integral where it is 1/u.

Measured

Measured with the change on 22021b1c, before the rebase onto #1520 and #1521. After the rebase, the library builds for every target, and the power-substitution, Gaussian and exponentials-of-quadratics test classes pass (86).

• Rubi's problems whose answer uses erf, erfc or erfi: 811 across families 0 to 7, at a 3 s budget. This PR solves 602, where master (22021b1c) solves 593, with 0 wrong and nothing lost. Both have the same 14 timeouts, and the total time goes from 574 s to 579 s. The 9 gained are the 9 problems of this shape: 7 in 2.3, 1 in 6.1.3 and 1 in 6.2.3.
• The textbook suites (family 0) are unchanged: 1723/1778, 0 wrong, 0 timeouts.
• The unit suite passes on net10.0: 12945 passed, 14 skipped, none failed, of 12959.
• The allocation gate passes on all 19 gated benchmarks, with no allocation moved.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura