Symbolic powers of a ± a sin beside a power of the cosine are integrated through the sine

[Rafael-SOWNet]

Part of #718.

(a + a sin(h + f x))^m sqrt(c - c sin(h + f x)) was declined, and so were the other powers of a ± a sin with a symbolic exponent beside the cosine or a function of the sine -- Rubi's (a + b sin)^m (c + d sin)^n files with a^2 = b^2 and c^2 = d^2, most of them after the whole budget: the half angle at which 1 ± sin is a square wants numeric powers. 2.5.0 declined them too:

integrand	2.5.0	master 4dbe73a4	this
(a + a sin(h + f x))^m sqrt(c - c sin(h + f x))	declined	searched past 8 s	in 1.3 s
cos(h + f x)^2 (a + a sin(h + f x))^m/sqrt(c - c sin(h + f x))	declined	searched past 8 s	in 0.2 s
(g cos(h + f x))^(1 - 2m) (a + a sin(h + f x))^m (c - c sin(h + f x))^(m - 1)	declined	declined after 6.3 s	in 0.2 s
(a + a sin(h + f x))^m (c - c sin(h + f x))^(-1 - m)	declined	searched past 8 s	in 0.7 s
(a + a cos(h + f x))^m sqrt(c - c cos(h + f x))	declined	declined after 7.1 s	in 0.1 s

Each answer is differentiated back with the symbols pinned and compared as a complex number at x = -2.3, -1.1, -0.4, 0.4, 1.1, 1.9, 2.6; the times include that.

What changes. (1 + sin(y))(1 - sin(y)) is cos(y)^2, so under u = sin(y) each such power, a power of g cos(y) beside it and dy = du/cos(y) itself are powers of 1 + u and 1 - u, up to a factor constant on each interval where it is defined, and the rest of the integrand, a function of the sine, comes along. (1 + u)^A (1 - u)^B R(u) is asked in u, and the answer is that antiderivative times the factor: the powers as written over the form they were rewritten to, a quotient whose logarithmic derivative is zero, so it holds whatever a, c, g and the exponents are. The cosine's the same way, by u = cos(y). Only where an exponent is symbolic.

Tests: PowersOfOnePlusAndMinusASineIntegralTest, seven rows differentiated back with the symbols pinned, at six points on both sides of zero; none is answered on master.

Measured first on the 1,279 problems of family 4 with a power of a ± a sin or a ± a cos, 797 of them run, at the corpus's 5-second budget, against master 8f3757cd, the branch's base:

	master	this
solved	733	777
wrong	0	0
past the budget	52	16

The 697 with numeric powers are answered alike, all but one on both builds. Of the 100 with a symbolic power, master answers 37 and this 81. Thirteen of the 19 left ask (1 + u)^A (1 - u)^B beside a polynomial, with A + B a whole number below -2 -- (a + a sin)^m (c - c sin)^(-3 - m) and its kin -- which t = (1 + u)/(1 - u) makes a power of t beside a polynomial. A rule for two linear powers whose exponents sum to such a number follows in a PR of its own, and with both the thirteen are answered in 0.1 to 0.5 seconds, where either alone runs past eight.

Measured then on the Rubi corpus against master 8f3757cd:

	master	this
family 0, independent suites (1814)	1766	1766
family 1, 40 a file (1381)	1296	1297
families 2 to 8, sampled (2410)	2215	2225

The harness counts no answer wrong in the pocket or the sample on either build. Of the 49 problems the two builds disagreed on, pocket and sample together, run again one build at a time, master answers 1 and this 42. The 42 are family 4's symbolic powers of a ± a sin, answered here in a second or less but one at sixteen, where master declines them or runs past the budget. The one master answers, 1/(x (a x + b x^3 + c x^5)^2), at twenty-two seconds, ran past the harness's patience here; run again, twice on each build, it runs past it on both. (-3 + 3 sin(h + f x))^(-1 - m) (a + a sin(h + f x))^m, past the budget on master, is answered here where the harness cannot check it, its integrand being complex on the whole real line, and five run past the budget on both.

The suite passes on the commit measured, c342aaea, 14,598 tests with 13 skipped, and the allocation gate with it. On the merge with master 4dbe73a4, 3a351431, the 4,148 calculus and corpus tests that run pass, with 2 skipped. Every row of the first table is as it says on the merge.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura