a + i a tan of a shifted linear below the bar is integrated in the linear

[Rafael-SOWNet]

Part of #718.

sqrt(a + i a tan(e + f x)) (A + B tan(e + f x))/sqrt(c - i c tan(e + f x)) ran past thirty seconds, with two more of Rubi's 4.3.3.1, where with x for the argument it is answered in a second. SolveByWritingAnImaginaryTangentAsAnExponential writes a + i a tan(z) as a e^(i z)/cos(z); a sample of the stack showed the time going to the phase of e^(i (g + f x)) expanded beside the rest and a substitution search simplifying it. Where every trigonometric function in the integrand is of one linear other than x, and a root stands in it, the rule now integrates in u = g + f x first, dx = du/f. Whole powers are left as they were: (a + i a tan(z))^3 (A + B tan(z))/(c - i c tan(z))^3 is answered in 0.6 s of the linear as written and in 5 s of x, and the first version of this lost it.

integrand	2.5.0	master ea2cb5e3	this
sqrt(a + i a tan(g + f x)) (A + B tan(g + f x))/sqrt(c - i c tan(g + f x))	declined	past 30 seconds	448 characters, 1.3 s
(A + B tan(g + f x))/(sqrt(a + i a tan(g + f x)) (c - i c tan(g + f x))^(3/2))	declined	past 30 seconds	597 characters, 0.9 s

Tests: AnImaginaryTangentSumOfAShiftedLinearIntegralTest, the two rows above with every symbol pinned, each differentiated back and compared as a complex number at six real points.

Measured on every corpus problem with i in its integrand, 2,253 of them, at the corpus's 5-second budget, against master ea2cb5e3:

	master	this
solved	2027	2030
unevaluated	50	50
wrong	1	1
past the budget	103	100

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. Three problems are answered here and not on master, all of 4.3.3.1, and none the other way; on the 2,027 both answer the time goes from 1,668 seconds to 1,615.

	master	this
family 0, independent suites (1814)	1782	1782
family 1, 40 a file (1381)	1341	1341
families 2 to 8, sampled (2410)	2329	2329

The harness counts no answer wrong in either. The four problems the builds disagreed on, run again one build at a time: master answers none, this the three of 4.3.3.1; the fourth, 4.3.1.2's cos(c + d x)^11 (a + i a tan(c + d x))^(7/2), is past the budget on both. A first version called the rewrite before the rule had read its sum, and moved 4.1.2.3's 1/((c - c sin(e + f x)) sqrt(g sin(e + f x)) sqrt(a + a sin(e + f x))), which has no imaginary tangent in it; it is called only once the sum is read.

The suite passes: 15,159 passed, 13 skipped, none failed. The chain's own run of it was stopped by the machine's memory guard at 8 GB after 11,271 had passed and none failed; run whole again on each tree, the test host peaked at 5.4 GB here and 5.7 GB on master. 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