[Rafael-SOWNet]

Three seams of Rubi's family 4:

• (b sec(y))^m tan(y)^3 was declined only because m was not a number. In u = sec(y) it is (b u)^m (u^2 - 1)/u, and the rule for an odd power of the tangent beside a function of the secant now reads a power free of x whatever its exponent. A negative power of the tangent beside it leaves something hypergeometric, and is still declined.
• 1/(sqrt(a + a sin(y)) (c + d sin(y))^2) and its kin are answered by the half angle at which a + a sin(y) is a square. They never reached it, because the polynomial term asked the substitution search first, and the search spent the budget. A half-odd power of a ± a sin(y) beside another linear in the same sine or cosine is now asked that first.
• cos(y)^4 sin(y)^3/(a + b sin(y))^2: an even whole power of the cosine is a power of 1 - sin(y)^2, so beside a power of the sine above the bar the integrand is a polynomial in the sine over the power. Without a sine above the bar, the rules after this one answer as before. cos(y)^4/(a + b sin(y))^8 is 40K characters there and was 106K here, using 11.9 GB against 8.1 GB, so the rule declines it.

Measured on 8a1ef454, branch against master:

Run	Problems	Master	Branch
Pocket: Rubi's 4.x	2,061	1,967 solved	1,994 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	2,360

Re-run alone on both builds, 28 problems go from timed out or unevaluated to answered, in 13 ms to 6.2 s: Rubi's 4.1.2.1:404–406, 412–414, 475, 644, 653, 654, 659–661 and 666–668; 4.1.2.2:1444, 1452, 1605–1607, 1615 and 1616; 4.1.3.1:364; and 4.3.0:444, 445, 478 and 479. None goes the other way.

The two sampled problems master answered and the branch did not, 6.4.2:50 and 7.4.1:369, are the ones that take 21–25 s against a 5 s budget. Re-run alone, the branch answers both, and master answers 6.4.2:50.

The pocket ran before the sine-above-the-bar condition was added. That condition only narrows the rule. With it, the seven 4.1.2.2 gains are answered in 0.1–0.4 s (master runs past 60 s), and 4.1.1.2:514 is master's answer again, character for character. No run had a wrong answer.

The new and extended tests differentiate the answers back, with a negative b for the secant. BREAKING-CHANGES.md has the entry, with the 2.5.0 column measured on v2.5.0.

Part of #233.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura