A sine or a cosine of a shifted argument is written in the other

[Rafael-SOWNet]

Part of #718.

sin(a + b x) sec(c + b x)^3 was declined in a few milliseconds, with twenty-nine more of Rubi's 4.7.1 that put a sine or a cosine of one linear argument beside functions of another of the same slope: SolveByWritingTwoFunctionsOfShiftedArgumentsApart reads two functions of one kind, and nothing read a sine beside a secant of a shifted argument. With d = A - B a constant,

sin(A) = sin(d) cos(B) + cos(d) sin(B)
cos(A) = cos(d) cos(B) - sin(d) sin(B)

so SolveByWritingASineOfAShiftedArgumentInTheOther writes the sines and cosines of the one argument that stands only in sines and cosines in the other, and asks the same question of one argument.

Both arguments must carry an offset. A first version did not ask it, and lost nine of 4.3.1.2's sec(c + d x)^4/(a + i a tan(c + d x))^2 and its like: another rule had written cos(c + d x) as cos(c) cos(d x) - sin(c) sin(d x) on the way, and this one wrote it back, round and round past thirty seconds.

integrand	2.5.0	master 136a7f51	this
sin(a + b x) sec(c + b x)^3	declined	declined	82 characters
cos(a + b x) sec(c + b x)^2	declined	declined	111 characters
sin(a + b x) csc(c + b x)^4	declined	declined	215 characters

Tests: ASineOfAShiftedArgumentIntegralTest, the three rows above, each differentiated back and compared at six points.

Measured on every corpus problem of 4.7 and every one with a secant or a cosecant, 7,605 of them, at the corpus's 5-second budget, against master 136a7f51:

	master	this
solved	7372	7402
unevaluated	163	133
wrong	0	0
past the budget	50	50

Thirty problems are answered here and not on master, all of 4.7.1, and none the other way; on the 7,372 both answer the time is 1,922 seconds and 1,922.

	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 thirty problems the builds disagreed on, run again one build at a time: master answers none, this all thirty.

The suite passes: 15,164 passed, 13 skipped, none failed, on the change before its test was added; with it, the related tests pass, 534. The allocation gate passes: every gated benchmark allocates what the baseline says. The library builds for every target.

Left in 4.7.1: csc(a + b x)^k sin(2a + 2b x)^n, whose arguments' slopes differ by a factor of two.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura