An even rational function with symbols skips the substitution search

[Rafael-SOWNet]

Part of #718.

(d + e x^2 + f x^4)/(x^4 (a + b x^2 + c x^4)^2) ran past the corpus's budget, with four more of Rubi's. In each the time went to SolveBySubstitution declining an even rational function with symbols in it. The half-angle tangent makes one of 1/((c + d sec(x))^3 sqrt(a + a sec(x))), (1 + x^2)^2/((c + d + (c + 3d) x^2 + 2d x^4)^3 (1 + 2x^2)). The search spent 31 seconds simplifying its quotient by each candidate's derivative, and the partial fractions after it answered in under one. The derivative of anything even is odd, so no polynomial candidate is the substitution for an even rational function. The search now skips one that holds a symbol and whose denominator is past a sextic as written.

integrand	2.5.0	master e7ef0b75	this
(d + e x^2 + f x^4)/(x^4 (a + b x^2 + c x^4)^2)	declined	past 60 seconds	3,715 characters, 1.1 s
(d + e x^2 + f x^4 + g x^6)/(x^4 (a + b x^2 + c x^4)^2)	declined	past 60 seconds	3,727 characters, 0.9 s
cos(x)^5/(a + b sec(x)^2)^3	declined	past 60 seconds	27,944 characters, 0.7 s
1/((c + d sec(x))^3 sqrt(a + a sec(x)))	declined	55 s	the same answer, 4 s
1/((a + a sec(x))^(3/2) (c + d sec(x))^3)	declined	41 s	the same answer, 25 s

x stands for e + f x in the trigonometric rows, which is what was probed. Where master answers, the answer is the same. The last row's remaining time is the partial fractions over a symbolic quadratic to the third power.

Tests: AnEvenRationalFunctionIntegralTest, the first row and the half-angle tangent's rational function above, each differentiated back and compared at six points.

Measured on 14,109 corpus problems, every one whose integrand is a rational function and every one with a secant or a cosecant in it, at the corpus's 5-second budget, against master 61fb760b:

	master	this
solved	13889	13894
unevaluated	102	102
wrong	0	0
past the budget	108	103

Five problems are answered here and not on master, two of 1.2.2.6, two of 4.5.2.1 and one of 4.5.7, and none the other way. On the 13,889 both answer the time is 2,400 seconds on master and 2,371 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 harness counts no answer wrong in either. The five problems the builds disagreed on, run again one build at a time: master answers none within the budget, this all five.

The suite passes: 15,176 passed, 13 skipped, none failed. The allocation gate passes: every gated benchmark allocates what the baseline says. Merged with master: the library builds for every target.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura