The substitution search drops a trigonometric candidate its quotient cannot be a function of

[Rafael-SOWNet]

The first measurable result for #1486's question of whether a stated precondition can make the search cheaper.

Over the problems the integrator answers, most of its time goes to SolveBySubstitution declining candidates. Each candidate costs a few milliseconds: the integrand over du/dx is written in u and simplified, and the candidate is refused if x survives. A candidate that takes one value at two points, where that quotient takes two, can't be a substitution whatever the simplifier makes of it, because a function of u agrees wherever u does.

For a sine, cosine, tangent, cotangent, secant or cosecant of a linear, the second point is known. A sine repeats at pi - theta, a cosine at -theta, and a tangent a period on. So two evaluations with the symbols pinned now settle it before the simplification runs. The complement pass is the same function spelled differently, so it's skipped with it. Other candidates aren't screened: for x^2 the mirror point -x can sit off a branch the rule answers on today.

Measured

	master	this branch
Rubi 4.7.2, 4.3.10, 4.4.10: time on the 269 problems answered, two interleaved pairs	154.2 s, 153.4 s	120.5 s, 119.5 s (−22%)
the same, every answer	—	identical, verdict and text, on all 297 that finish
Rubi family 4 (not interleaved)	148.3 s	132.8 s
Rubi families 1, 4–7, the 7.1 files and the 1774-problem suite	—	every row unchanged
unit suite	12,718	12,718 run to completion, 0 failed; the six new cases pass on their own
allocation gate	—	PASSED on all 19 gated benchmarks

The new theory integrates six substitutions under a trigonometric function of a linear, which the screen has to keep, and checks each by differentiating back.

Part of #1486.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura