A rational function of the secant with two sums below the bar is written in the cosine

[Rafael-SOWNet]

Part of #718.

sec(e + f x)/((a + b sec(e + f x)) (c + d sec(e + f x))^2) ran past thirty seconds, with two more of Rubi's 4.5.2.3, where cos(x)^2/((a cos(x) + b) (c cos(x) + d)^2), the same function, is answered in half of one. A sample of the stack showed SolveByHalfAngleSubstitution reading the secant's spelling into a rational function of the half-angle tangent, and a substitution search under it expanding that without end. SolveByWritingARationalFunctionOfTheSecantInTheCosine, ahead of Bioche's substitutions and the half angle, writes a rational function of the secant of one argument, or the cosecant, in the cosine as one quotient and asks the same question.

Only where two sums in the secant stand below the bar and none above. Written so wherever the secant stands alone, it was measured on all 6,489 corpus problems with a secant or a cosecant in them and lost 51 that master answers -- sec(c + d x)^5/(a + b sec(c + d x))^3 and its like, which the secant's own rules answer in half a second and the cosine's spelling sends past five -- for 6 gained.

integrand	2.5.0	master 1ea1f8ac	this
sec(x)/((a + b sec(x)) (c + d sec(x))^2)	declined	past 30 seconds	9,439 characters, 0.8 s
sec(x)/((a + b sec(x))^2 (c + d sec(x)))	declined	past 30 seconds	9,430 characters, 0.6 s

The answers carry a case for each sign of two discriminants, which is most of their length.

Tests: ARationalFunctionOfTheSecantInTheCosineIntegralTest, the two rows above, 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 1ea1f8ac:

	master	this
solved	6330	6336
unevaluated	99	99
wrong	0	0
past the budget	50	44

Six problems are answered here and not on master -- three of 4.5.2.3, two of 4.5.4.2 and one of 4.7.7 -- and none the other way; on the 6,330 both answer the time is 1,927 seconds and 1,921.

	master	this
family 0, independent suites (1814)	1782	1782
family 1, 40 a file (1381)	1341	1341
families 2 to 8, sampled (2410)	2330	2330

The harness counts no answer wrong in either. The six problems the builds disagreed on, run again one build at a time: master answers none, this all six.

The suite passes: 15,162 passed, 13 skipped, none failed, on the change before its test was added; with it, the secant, Bioche and half-angle tests pass, 354. 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