An odd power of the secant is read in the tangent, with the cosine's sign

[Rafael-SOWNet]

Part of #718.

sec(c + d x)^5/(a + b tan(c + d x))^2 was declined, and sec^7/(a + b tan)^3 and two like it ran past the corpus's five seconds: Rubi's 4.3.1.2 with a general b. Under u = tan(x), SolveByTangentSubstitution reads an odd power of the sine or the cosine as its sign times a function of the tangent, cos(x) = sgn(cos(x))/sqrt(1 + u^2), and not the secant, which is the same the other way up. It reads the secant too now:

sec(x)^5/(a + b tan(x))^2 = sgn(cos(x)) (1 + u^2)^(5/2)/(a + b u)^2,  dx = du/(1 + u^2)

integrand	2.5.0	master c417571d	this
sec(x)^5/(a + b tan(x))^2	declined	declined	901 characters
sec(x)^7/(a + b tan(x))^3	declined	declined	1,253 characters
sec(x)/(a + b cot(x))	declined	declined	571 characters

Each answer carries sgn(cos(x)), constant between the zeros of the cosine.

Tests: three rows in TangentSubstitutionIntegralTest, each differentiated back with the symbols pinned and compared at six points on both signs of the cosine.

Measured first on every corpus problem with a secant in its integrand, 5,093 of them, at the corpus's 5-second budget, against master c18738db:

	master	this
solved	4965	4971
unevaluated	71	68
wrong	0	0
past the budget	51	48

Six problems are answered here and not on master, and none the other way: the five of 4.3.1.2 and 4.4.1.3's sec(x)/(a + b cot(x)). On the 4,965 both answer the time goes from 1,617 seconds to 1,629.

Measured then on the Rubi corpus:

	master	this
family 0, independent suites (1814)	1782	1782
family 1, 40 a file (1381)	1341	1341
families 2 to 8, sampled (2410)	2328	2327

The harness counts no answer wrong in either. The seven problems the builds disagreed on, run again one build at a time: master answers none, this answers all seven. The seventh, 3.2.3's log(pe (f (a + b x)^p (c + d x)^q)^r)/(a + b x)^5, has no trigonometric function in it; master answered it in 25 seconds in the sample and is past the budget alone, this the other way round.

The suite passes on a867b351, this change on c18738db: 15,136 passed, 13 skipped, none failed. Rebased onto c417571d without a conflict, the tangent substitution's tests and both imaginary-tangent rules' pass again. 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