A symbolic power of a + i a tan is gathered with the secant's in the sum

[Rafael-SOWNet]

Part of #718.

(k sec(c + d x))^(4 - 2 n) (a + i a tan(c + d x))^n was declined, with three more of Rubi's 4.3.1.2 whose powers are symbols. #1812's rule integrates a power of the secant beside a power of a + i a tan in u = a + i a tan(z), where the integrand is u^n (u/a)^r ((2 a - u)/a)^r with r = (s - 2)/2; here u^(2 - n) (u/a)^(n - 1) is one power of u/a, and written apart no rule read it. u^n is a^n (u/a)^n up to a factor constant on the path, which the rule's quotient already carries, so the two are gathered into (u/a)^(n + r), the exponent simplified.

integrand	2.5.0	master c417571d	this
(k sec(x))^(4 - 2 n) (a + i a tan(x))^n	declined	declined	466 characters
(k sec(x))^(2 n) (a + i a tan(x))^(3 - n)	declined	declined	675 characters
sec(x)^3 sqrt(a + i a tan(x))	declined	495 characters	285 characters
cos(x)^9 (a + i a tan(x))^(7/2)	declined	1,067 characters	768 characters

Every answer this rule gives comes out of the gathered integrand: of the 88 rows of 4.3.1.2, 54 change, 50 of them shorter -- by a third or more for most -- and about twice as fast; four are up to 7% longer, and (d sec(x))^(2/3)/(a + i a tan(x))^(1/3) takes 448 ms where it took 259.

Tests: two rows in SecantBesideAnImaginaryTangentSumIntegralTest, n pinned to 2.3, 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 c417571d:

	master	this
solved	2002	2005
unevaluated	55	49
wrong	1	1
past the budget	130	127

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. On the 2,001 both answer the time goes from 2,015 seconds to 1,901.

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	2328

The harness counts no answer wrong in either. The fourteen problems the builds disagreed on, run again one build at a time: the four above are answered here and not on master; six powers of the cosine beside a + i a tan, past the budget on master, are answered here and are unverifiable to the harness, which simplifies an answer it cannot check on the reals within the same five seconds and whose integrand is nowhere real; the other four come out the same on both builds alone.

The suite passes on 958752cc, this change on c417571d: 15,138 passed, 13 skipped, none failed. The chain's own run of it was stopped by the machine's memory guard at 8 GB after 14,904 had passed and none failed; run whole twice more on each tree, the test host peaked at 5.4 and 5.5 GB here and at 4.1 and 6.5 GB on master. Rebased onto c21de3ef without a conflict, the tests of the tangent's 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