An odd power of a cos + b sin is rotated into one cosine

[Rafael-SOWNet]

1/(a cos(x) + b sin(x))^3 was left as written. The tangent turns a quotient of homogeneous polynomials in the sine and cosine into a rational function only when the degrees differ by an even number; by an odd number it leaves a root of 1 + t^2, and every row of that kind was declined after the substitution search had spent its budget.

Harmonic addition. a cos(y) + b sin(y) is R cos(u) for R = sqrt(a^2 + b^2) and a shifted argument u, so the quotient becomes one over a power of one cosine, with cos(y) and sin(y) above it written as (a cos(u) - b sin(u))/R and (b cos(u) + a sin(u))/R — shapes the rules for powers of the sine and cosine answer in a fraction of a second. The way back goes through cos(u) = (a cos(y) + b sin(y))/R and sin(u) = (a sin(y) - b cos(y))/R rather than through the phase, whose quadrant would depend on the signs of a and b (the tests pin a = -1.7 as well as 1.7 for that reason); a bare u is y less a constant. R is integrated as a symbol and written out only in the answer, because a root of a symbolic sum in the integrand sent it down the rules for radicals.

Three restrictions, each found by measuring rather than reasoning:

• Only a sum below the bar. The rotation writes such sums above it itself (cos(y) becomes (a cos(u) - b sin(u))/R), and the first version, which rotated any sum it found, rotated its own output again — u_1, u_2, u_3 until the budget ran out.
• Nothing else below the bar holds x, so that one rotation leaves a power of one cosine.
• Only sines and cosines. A secant, cosecant, tangent or cotangent puts a cosine or sine below the bar that the rotation turns into one more sum: csc(x)^2/(a cos(x) + b sin(x)), which master declines in a quarter of a second, spent two minutes that way. Those rows want partial fractions over the two linear factors — a different rule.

Measured

	before	after
1/(a cos(x) + b sin(x))^3	declined after 3.6 s	0.3 s
cos(x)^3/(a cos(x) + b sin(x))^2	declined after 2 s	0.9 s
Rubi 4.7.2 trig^m (a trig + b trig)^n, all 290 problems	215 (after #1483)	237, no row lost, 20 timeouts → 16
Rubi families 1, 4, 5, 6, 7 and the 1774-problem suite	—	unchanged, row for row — their samples hold few 4.7.2 rows
unit suite	12,682	12,689 (seven cases added), 0 failed, run to completion
allocation gate	—	PASSED on all 19 gated benchmarks

With #1483, the 4.7.2 file goes from 199 to 237 of 290.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura