A secant beside powers of its conjugate sums, in the sum

[Rafael-SOWNet]

Part of #718.

sec(e + f x) (a + a sec(e + f x))^m sqrt(c - c sec(e + f x)) was declined, with two more of Rubi's 4.5.2.3 that differ only in the power of the second sum. The two sums multiply to a square,

(a + a sec(y)) (c - c sec(y)) = -a c tan(y)^2

so in u = a + a sec(y) the root of the second is tan(y) over sqrt(u) times a constant, sec(y) tan(y) dy is du over one, and c - c sec(y) is c (2 - u/a). What is left is u^(m - 1/2) (2 - u/a)^(n - 1/2), a whole power of a linear beside a symbolic power of u, which is answered at once. SolveASecantBesidePowersOfItsConjugateSums takes either sign and the cosecant's. The constants are not written: the answer is the integrand times the antiderivative in u over what that differentiates back to, a quotient whose square is one. It is kept only where that square is one at the sampled points.

integrand	2.5.0	master 61fb760b	this
sec(x) (a + a sec(x))^m (c - c sec(x))^(5/2)	declined	declined	469 characters, 0.5 s
sec(x) (a + a sec(x))^m (c - c sec(x))^(3/2)	declined	declined	385 characters, 0.1 s
sec(x) (a + a sec(x))^m sqrt(c - c sec(x))	declined	declined	268 characters, 0.1 s
sec(x) (a - a sec(x))^m sqrt(c + c sec(x))	declined	declined	193 characters, 0.1 s
csc(x) (a + a csc(x))^m sqrt(c - c csc(x))	declined	declined	195 characters, 0.1 s

x stands for e + f x in the first three rows, which is what was probed.

Tests: ASecantBesideConjugateSumsIntegralTest, four rows with m = 2.3, each differentiated back and compared as a complex number at six real points.

Measured on every corpus problem with a secant or a cosecant in its integrand, 6,489 of them, at the corpus's 5-second budget, against master e1a1a3fc, before #1828:

	master	this
solved	6363	6363
answered, not checkable	0	3
unevaluated	78	75
wrong	0	0
past the budget	38	38

The three of 4.5.2.3 are answered here and not on master, and none the other way. The harness cannot check them: with a and c of one sign, the integrand is -a c tan^2 under a root and is complex at every real point it samples. The test above compares them as complex numbers. On the 6,363 both answer the time is 1,691 seconds on master and 1,696 here.

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

The suite passes: 15,175 passed, 13 skipped, none failed. The allocation gate passes: every gated benchmark allocates what the baseline says. Merged with master after #1828: the library builds for every target, and the secant and half-angle tests pass, 348.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura