[Rafael-SOWNet]

cot(c + d x)^6/(a + b sin(c + d x))^3 is (1 - s^2)^3/(s^6 (a + b s)^3) in s = sin(x). That is a quotient over powers of linears in s: the sine, 1 ± s from an even power of the cosine, and a + b s. It is now split by the residues at their roots, each block a Taylor series at a root. Every piece is a constant over a power of one linear, which the rules for csc^n, 1/(1 ± sin)^n and 1/(a + b sin)^n answer.

Master took these under the half-angle tangent instead. There every linear but the sine becomes a quadratic, and the cube of a symbolic quadratic ran out of time. The split runs only where the half-angle is that slow: a linear with a symbolic root at the third power or more, with the half-angle's denominator past the eleventh degree. It is not used for the cosine, whose half-angle answer is the shorter.

A repeated quadratic with Gaussian rational roots is split by the residues at one root, with the other root's its conjugates. That answers sin(c + d x)^6/(a + b tan(c + d x))^2.

Measured on 8a1ef454, branch against master:

Run	Problems	Master	Branch
Pocket: Rubi's family 4	1,817	1,769 solved	1,783 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

Re-run alone, against 8a1ef45, 15 rows go from timed out to answered. Five of them, 4.1.2.2:1444, 1605–1607 and 1616, have since been answered by #1975.

Re-measured on the master this PR opens against (685093c), with both probes at a 30 s budget:

• 10 rows go from past the budget to answered in 0.2–1.5 s: 4.1.1.2:505, 4.1.1.3:235 and 239, 4.1.2.1:843, 4.1.2.2:1623, 1624 and 1878, and 4.3.1.3:84, 96 and 108. Their answers run to 1.8–15K characters.
• Integrate a symbolic secant power, a half-odd power of a ± a sin, and an even cosine power #1975's five rows, and the other moves, give the same answers on both builds, character for character, so the two rules do not compete.

The sample moves are 3.2.1:292, 7.3.5:50 and 7.4.1:369, rows that take 23 s against a 5 s budget. Re-run alone, the first times out on both builds and the other two are answered on both. No run had a wrong answer.

PowersOfLinearsByTheirResiduesIntegralTest differentiates the answers back, and 1,139 related tests pass. 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