A power of a + i a csch is integrated in the exponential

[Rafael-SOWNet]

Part of #718.

sqrt(a + i a csch(c + d x)) was declined, with nine more of Rubi's 6.6.3 that are a power of a ± i a csch not whole. The sum is the square of a complex function over a real one, which nothing read. In w = e^y, csch(y) = 2 w/(w^2 - 1) and

a + i a csch(y) = a (w + i)^2/(w^2 - 1)

so its power is a^p (w + i)^(2p) (w^2 - 1)^(-p) times a constant on every interval where both are continuous, and dy = dw/w: a root of a quadratic times a linear power over w, which is answered at once. SolveAPowerOfAnImaginaryHyperbolicCosecantSumInTheExponential reads the hyperbolic functions as their sine, takes one power of a sum in it, writes the sum in w as one quotient, and finds a constant q with numerator q (w ± i)^2. The constant is not written: the answer is the integrand times the antiderivative in w over what that differentiates back to, a quotient whose logarithmic derivative is zero. The antiderivative is checked in w against the integrand in w; checked in x, with the slope a symbol pinned at a sample, e^(c + d x) at the sampled points was past what the evaluation decides.

integrand	2.5.0	master d8dff2a5	this
sqrt(a + i a csch(c + d x))	declined	declined	454 characters, 0.5 s
(a + i a csch(c + d x))^(3/2)	declined	declined	834 characters, 0.2 s
1/sqrt(a - i a csch(c + d x))	declined	declined after 8 s	575 characters, 0.1 s

Every one of 6.6.3's eleven failing rows was also probed at points of both signs of the argument; ten are answered, and 1/(a + i a csch(c + d x))^(3/2) is still declined.

Tests: APowerOfAnImaginaryHyperbolicCosecantSumIntegralTest, the three rows above, each differentiated back and compared as a complex number at six real points of both signs.

Measured on every corpus problem with i in its integrand, 2,253 of them, at the corpus's 5-second budget, against master d8dff2a5:

	master	this
solved	2030	2040
unevaluated	50	40
wrong	1	1
past the budget	101	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. Ten problems are answered here and not on master, all of 6.6.3, and none the other way; on the 2,030 both answer the time is 1,663 seconds and 1,668.

	master	this
family 0, independent suites (1814)	1782	1782
family 1, 40 a file (1381)	1341	1341
families 2 to 8, sampled (2410)	2329	2330

The harness counts no answer wrong in either. The eleven problems the builds disagreed on, run again one build at a time: master answers none, this the ten of 6.6.3; the eleventh, 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.

The suite passes: 15,162 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