a + i a tan below an odd power of the secant is written over its conjugate

[Rafael-SOWNet]

Part of #718.

sec(c + d x)^7/(a + i a tan(c + d x))^4 was declined, with ten more of Rubi's 4.3.1.2 that are an odd power of the secant over a whole power of a + i a tan. #1812's rule integrates these in u = a + i a tan(z), where whole powers on both are left to the rules for the sine and the cosine but for s + 2 n = 1; those rules decline them or run past five seconds, and in u the s + 2 n = 1 rows took two seconds for answers thousands of characters long.

The sum times its conjugate is a^2 sec(z)^2, so

sec(z)^s/(a + i a tan(z))^n = sec(z)^(s - 2 n) (a - i a tan(z))^n / a^(2 n)

a whole power of the conjugate beside one of the secant, which the rules for the sine and the cosine answer in a fraction of a second. A positive odd power of the secant is rewritten so; an even one, and an odd power of the cosine, are left as they were, the rules answering them shorter or sooner as written.

integrand	2.5.0	master c18738db	this
sec(x)^7/(a + i a tan(x))^4	declined	declined	371 characters
sec(x)^9/(a + i a tan(x))^8	declined	declined	745 characters
sec(x)^5/(a + i a tan(x))^2	declined	2,431 characters, 1.2 s	225 characters, 0.05 s
sec(x)^7/(a + i a tan(x))^3	declined	3,774 characters, 1.5 s	289 characters, 0.06 s

Tests: three rows in SecantBesideAnImaginaryTangentSumIntegralTest, one with a - i a tan, each differentiated back and compared as a complex number at six real points.

Measured first on every corpus problem with i in its integrand, 2,253 of them, at the corpus's 5-second budget, against master 8a310f7f (#1814 changes none of these rows):

	master	this
solved	1987	1998
unevaluated	64	59
wrong	1	1
past the budget	136	130

The one counted wrong on both is the known 6.1.5 1/(a + i a sinh(c + d x))^(1/2), the harness's own. Eleven problems are answered here and not on master, all of 4.3.1.2, in 57 to 338 ms, and none the other way; on the 1,987 both answer the time goes from 1,944 seconds to 1,903.

Measured then on the Rubi corpus:

	master	this
family 0, independent suites (1814)	1782	1782
family 1, 40 a file (1381)	1340	1341
families 2 to 8, sampled (2410)	2328	2328

The harness counts no answer wrong in either. The twelve problems the builds disagreed on, run again one build at a time: master answers one, the family 1 problem that ran past its budget in the shared run, and this answers all twelve.

The suite passes on 3b4ec9ca, this change on 8a310f7f: 15,134 passed, 13 skipped, none failed; rebased onto c18738db without a conflict, the tests of both 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