A quadratic with rational roots counts as two linears before the substitution search

[Rafael-SOWNet]

Part of #718.

sec(c + d x)^2 sin(c + d x)/(a + b sin(c + d x))^3 ran past thirty seconds, with eight more of Rubi's 4.1 and 4.5. A trace of the routes showed the half-angle substitution writing it as a rational function over (1 - u^2)^2 (a u^2 + 2 b u + a)^3, and SolveBySubstitution spending eighty seconds on it, to decline, before the partial fractions answered it in a fraction of one.

SolveBySubstitution already declines a rational function over written factors with a symbol in their coefficients, linear and quadratic, at least one of them linear -- the partial fractions read those off their roots. 1 - u^2 is two linear factors written together, and IsAProductOfSymbolicLinearFactors counted it as a quadratic, so there was no linear among them. A quadratic with rational coefficients and a discriminant that is the square of a rational now counts as two linears. Two symbolic quadratics alone are still asked of the search, where u = x^2 is often the shorter answer.

integrand	2.5.0	master feb0e25f	this
sec(x)^2 sin(x)/(a + b sin(x))^3	declined	past 30 seconds	5,017 characters, 1.7 s
tan(x)^2/(a + b sin(x))^3	declined	past 30 seconds	4,816 characters, 0.7 s

Tests: AHalfAngleRationalFunctionOverOneMinusTheSquareIntegralTest, the two rows above, each differentiated back and compared at six points.

Measured on every corpus problem of 4.1 and every one with a secant or a cosecant, 8,962 of them, at the corpus's 5-second budget, against master feb0e25f:

	master	this
solved	8695	8703
unevaluated	128	127
wrong	0	0
past the budget	126	119

On the 8,694 both answer the time goes from 2,781 seconds to 2,647.

	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 eleven problems the builds disagreed on, run again one build at a time: nine past the budget on master are answered here -- one each of 4.1.1.2, 4.1.1.3, 4.5.3.1 and 4.5.4.2, three of 4.1.2.2 and two of 4.1.3.1 -- and none the other way; 4.3.7's csc(e + f x)^5 (a + b tan(e + f x)^2)^(3/2), past the budget here in the pocket, is answered by both alone, in 18 and 20 seconds; one of 4.5.4.2 neither answers.

The suite passes: 15,169 passed, 13 skipped, none failed, on the change before its test was added; with it, the related tests pass, 780. 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