[Rafael-SOWNet]

The rule that integrates f^m (alpha + beta f^2) as the derivative of a product (f a sine, cosine, secant or cosecant, m symbolic) found its power by writing 1 for the first such power anywhere in the tree. A power below the bar, or inside one term of a sum, was taken for a factor of the integrand. Five such integrands came back wrong on master. 2.5.0 declined them, so these wrong answers never shipped:

• (-m + (1+m) sec(x)^2)/sec(x)^m
• -m sec(x)^m + (1+m) sec(x)^2
• sin(x)^m (1+m) - (2+m) sin(x)^2
• (1+m - (2+m) sin(x)^2)/sin(x)^m
• (-m + (1+m) csc(x)^2)/csc(x)^m

None of these is the derivative of a single product. The rule no longer takes them, and any answer they get now differentiates back.

The rule reads its power among the integrand's factors instead. A power under the bar counts as f^(-m), so (m + (1-m) sec^2)/sec^m is integrated as the derivative of tan sec^(-m). The rule also takes a power that is not whole: the identities hold for any m on the principal branch. That answers Rubi's 4.5.4.1 (3 + 3 sec(c + d x)^2)/sqrt(sec(c + d x)).

Measured on 8a1ef454, branch against master:

Run	Problems	Master	Branch
Pocket: the files the rule reads	1,791	1,784 solved	1,785 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	2,361

Two problems moved. 4.5.4.1:35 goes from declined to answered in 2 ms. 3.2.1:292 times out on both builds when re-run alone. No run had a wrong answer.

ASymbolicPowerOfATrigonometricFunctionAsADerivativeIntegralTest checks that the five are not answered wrongly, and checks the under-the-bar and non-whole cases as answered. The BREAKING-CHANGES.md entry for the rule is extended.

Part of #233.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura