e^(i arctan(L)) beside a function of x is written as two powers of linears

[Rafael-SOWNet]

Part of #718.

1/(e^(i arctan(a + b x)) x^2) ran past thirty seconds, with three more of Rubi's 5.3.6. SolveByWritingAnExponentialOfAnInverseAlgebraically wrote e^(i arctan(L)) as (1 + i L)/sqrt(1 + L^2), a root of a quadratic, beside the pole. For a real L it is also

e^(i arctan(L)) = (1 + i L)^(1/2) (1 - i L)^(-1/2)

-- the two principal powers' arguments are arctan(L)/2 each and their moduli cancel -- powers of two linears, which the rules for a product of linear powers answer in milliseconds. It is written so for e^(+-i arctan(L)) beside a function of x, where no power of a quadratic in x stands for the other form to meet.

Measured on all 288 rows of 5.3.6 first. Written so for every power, it lost sixteen -- the third powers and the rows beside sqrt(1 + a^2 x^2) -- and made the answers for powers not whole up to seventeen times longer; and the bare exponential alone, e^(i arctan(a + b x)), comes out shorter as the root. Restricted as above: four answered that were not, none lost, and the 28 answers that change are 7,601 characters together where they were 63,347, in 25 seconds where they took 84. Six of the 28 are up to 16% longer.

integrand	2.5.0	master 05855f92	this
1/(e^(i arctan(a + b x)) x^2)	declined	past 30 seconds	340 characters, 0.6 s
x^3/e^(i arctan(a + b x))	declined	past 30 seconds	373 characters, 5.2 s
x^3/e^(i arctan(a x))	declined	25,944 characters, 4.2 s	218 characters, 0.1 s
e^(i arctan(a x)) x^2	declined	396 characters	198 characters

Tests: AnExponentialOfAnArctangentBesideAPowerIntegralTest, the four rows above, 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 05855f92:

	master	this
solved	2023	2027
unevaluated	50	50
wrong	1	1
past the budget	108	103

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 5.3.6 above, and none the other way; on the 2,023 both answer the time goes from 1,720 seconds to 1,664.

	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 five problems the builds disagreed on, run again one build at a time: master answers none, this the four of 5.3.6; the fifth, 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: 15,157 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