A power of the cosine beside a power of a + i a tan is read as one of the secant

[Rafael-SOWNet]

Part of #718.

sqrt(a + i a tan(c + d x))/(k cos(c + d x))^(3/2) was declined, with the three like it in Rubi's 4.3.1.2. #1796's rule integrates a power of a + i a tan(z) beside a power of the secant in e^(i z) where the powers add up to a whole number, and it read the secant only; a power of the cosine is the same power of the secant with its sign turned.

integrand	2.5.0	master 8a310f7f	this
sqrt(a + i a tan(x))/(k cos(x))^(3/2)	declined	declined	533 characters
(k cos(x))^(3/2) sqrt(a + i a tan(x))	declined	763 characters	172 characters

What changes. SolveAPowerOfAnImaginaryTangentBesideAPowerOfTheSecant reads cos(z)^p, and (k cos(z))^p, as a secant power -p, the factors kept as they are written. Rows that #1812's rule answered in u = a + i a tan(z) are now answered by this one first, all of them shorter: of the 88 rows of 4.3.1.2, 54 have the same answer as on master, 11 a shorter one (920 characters to 231 is typical), and 4 an answer where master has none, 579 to 766 characters long.

Tests: two rows added to ImaginaryTangentBesideTheSecantIntegralTest, one with the cosine below the bar and one above it, each differentiated back and compared as a complex number at 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:

	master	this
solved	1987	1991
unevaluated	63	60
wrong	1	1
past the budget	137	136

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. Four problems are answered here and not on master, the four of 4.3.1.2 named above, and none the other way; on the 1,987 both answer the time goes from 1,882 seconds to 1,852.

Measured then on the Rubi corpus against master 8a310f7f:

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

The harness counts no answer wrong in either. The five problems the builds disagreed on, run again one build at a time: master answers none, this answers the four; the fifth, a 4.3.2.1 quotient, neither answers.

The suite passes: 15,133 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