An odd power of the tangent beside a function of the secant, in the secant

[Rafael-SOWNet]

Part of #718.

cot(c + d x)^3/(a + b sec(c + d x))^(3/2) ran past the corpus's budget. In u = sec(y), du = sec(y) tan(y) dy, and the even power of the tangent left over is a power of u^2 - 1. So an odd power of the tangent beside a function of the secant is, exactly,

tan(y)^n f(sec(y)) dy = (u^2 - 1)^((n - 1)/2) f(u)/u du

which here is a rational function beside the root of a linear. SolveAnOddPowerOfTheTangentBesideAFunctionOfTheSecant takes the cotangent as the tangent's reciprocal, and the cotangent beside a function of the cosecant in u = csc(y). It stands after the secant's other rules.

integrand	2.5.0	master 60b8111c	this
cot(x)^3/(a + b sec(x))^(3/2)	declined	past 60 seconds	39,235 characters, 4.5 s
cot(x)^3/(a + b sec(x)^2)^3	declined	1,218 characters, 17.8 s	680 characters, 0.04 s
cot(x)/(a + b sec(x))^(3/2)	declined	22,324 characters, 3.2 s	21,460 characters, 2.0 s

x stands for e + f x, which is what was probed. The first row's answer is long for the same reason #1831's are: a case for each sign of the roots of the quadratics the root of the linear makes.

Tests: AnOddPowerOfTheTangentBesideTheSecantIntegralTest, three rows (a shifted argument, the cotangent cubed, the cosecant), each differentiated back and compared at six points.

Measured on every corpus problem with a secant or a cosecant in its integrand, 6,489 of them, at the corpus's 5-second budget, against master e7ef0b75, before #1830 and #1831:

	master	this
solved	6372	6374
unevaluated	65	64
wrong	0	0
past the budget	39	38

Two problems of 4.5.1.4 are answered here and not on master, and none the other way. #1831 has since answered one of them on master. On the 6,372 both answer the time is 1,621 seconds on master and 1,593 here.

	master	this
family 0, independent suites (1814)	1782	1782
family 1, 40 a file (1381)	1339	1339
families 2 to 8, sampled (2410)	2330	2330

The harness counts no answer wrong in either. The two problems the builds disagreed on, run again one build at a time: master answers neither, this both.

The chain's own run of the suite was stopped by the machine's memory guard at 8 GB, with 9,745 passed and none failed. Merged with master, the whole suite was run twice on each tree. The test host peaked at 6.9 GB then 5.5 GB here, and at 3.7 GB then 6.9 GB on master. It passes: 15,187 passed, 13 skipped, none failed. The allocation gate passes: every gated benchmark allocates what the baseline says. The library builds for every target.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura