[Rafael-SOWNet]

With u = a + b ln(c x^n), B = b n and k = m + 1, the derivative of x^k sec(u)^(p - 2) (B (p - 2) tan(u) - k) is x^m (D sec(u)^p - E sec(u)^(p - 2)), where D = B^2 (p - 1)(p - 2) and E = B^2 (p - 2)^2 + k^2. So x^m (alpha sec(u)^p + beta sec(u)^(p - 2)) is a multiple of that product wherever beta D + alpha E = 0. A lone power is the case where E is zero, which is Rubi's sec(a + i ln(c x^n)/(n (p - 2)))^p. The cosecant's product is -x^k csc(u)^(p - 2) (B (p - 2) cot(u) + k). The rule does not apply at k = 0, where the logarithm's derivative is beside the power and the substitution answers.

Measured on 8a1ef454, branch against master:

Run	Problems	Master	Branch
Pocket: Rubi's 4.x	1,525	1,480 solved	1,487 solved
Family 1, 40 a file	1,381	1,362	1,362
Family 0, whole	1,814	1,789	1,789
Families 2 to 8, sampled	2,410	2,362	2,363

Re-run alone on both builds, nine problems of Rubi's 4.7.5 that were unevaluated or timed out are answered, and none goes the other way. Seven are verified at sampled points, in 18–362 ms: 4.7.5:334, 341, 342, 392, 398, 399 and 400. The integrands of 4.7.5:337 and 395 are complex at every real point, through x^(sqrt(-(1+m)^2)/2), so the harness leaves them unverified. Under the probe's quadrature, both answers agree with Simpson's rule over [0.5, 1.5] to 1e-13, in the real part and in the imaginary part. None of the problems answered on both builds allocates more on the branch. No run had a wrong answer.

SecantPowersOfALogarithmTwoApartAsTheDerivativeOfAProductIntegralTest differentiates the answers back. BREAKING-CHANGES.md has the entry, with the 2.5.0 column measured on v2.5.0.

Part of #233.

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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura