A power of a homogeneous trigonometric factor is read as that power, and a cos + b sin with symbols reaches the tangent

[Rafael-SOWNet]

sec(x)^2/(a cos(x) + b sin(x))^4 was left as written after ten minutes, where sec(x)^2/(2 cos(x) + 3 sin(x))^4 took a tenth of a second.

Both are quotients of homogeneous polynomials in the sine and cosine, which t = tan(x) turns into a rational function (Bioche's third rule), and SolveByHomogeneousTrigonometricSubstitution answers the numeric one at the top. It read each side by expanding it first. With numbers, the expanded quartic in t is (2 + 3t)^4 again as soon as the rational integrator factors it. With symbols it is a^4 + 4a^3 b t + 6a^2 b^2 t^2 + 4a b^3 t^3 + b^4 t^4, a quartic with a symbol in every coefficient that nothing factors back — so the search went on past its budget. Timing each step of the rule settled it: read in 5 ms, substituted and simplified by 99 ms, and then the expanded quartic.

A side is now read a factor at a time, with a whole power of a homogeneous factor read as that power of it: the degrees add and the rewritten factors are raised as written, so the substitution hands on (1 + t^2)^2/(a + b t)^4, which takes half a second. A side whose factors are not homogeneous one by one is still read whole, as before.

Measured

	before	after
sec(x)^2/(a cos(x) + b sin(x))^4	> 10 min, unevaluated	88 ms
Rubi 4.7.2 trig^m (a trig + b trig)^n, all 290 problems	199	215, no row lost, 27 timeouts → 20
Rubi family 4	330/422	332/422
families 1, 5, 6, 7 and the 1774-problem suite	—	no row lost; the suite stays at 1719, 0 wrong
unit suite	12,677	12,682 (five cases added), 0 failed, run to completion
allocation gate	—	PASSED on all 19 gated benchmarks

The sixteen rows are all trig^m/(a cos + b sin)^n with symbolic a and b, sin, cos, sec and csc above — two of them the a cos + i a sin kind. The existing tests all had numeric coefficients, which is why this survived them; the new ones pin a = 1.7, b = 0.6 after integrating symbolically.

The family 1/5/6/7 counts are not quoted as gains here: their baselines predate the harness change that checks complex-valued integrands as complex numbers, so a raised count would mix the two. "No row lost" holds either way, since that change only ever adds answered rows.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura