A trigonometric function of the reciprocal of a linear beside a power of it is integrated under the reciprocal

[Rafael-SOWNet]

Part of #718.

sin(a + b/x)/x^3 was declined, while under u = 1/x it is -u sin(a + b u), one step by parts; sin(a + b/x)/x^2, the derivative of its argument beside it, was answered. 2.5.0 declined them too:

integrand	2.5.0	master 0a13a636	this
sin(a + b/x)/x^3	declined	declined	cos and sin of a + b/x beside powers of 1/x, in 0.4 s
sin(a + b/x)^2/x^3	declined	declined	the same, in 0.03 s
cos(a + b/x)^3/x^4	declined	declined	the same, in 0.06 s
sin(a + b/(c + d x))/(c + d x)^3	declined	declined after 2 s	the same in 1/(c + d x), in 1.8 s

What changes. A trigonometric function of something of L with 1/L^k in it, L a linear, beside L^m with m <= -3, is integrated under u = 1/L: dx = -du/(d u^2), so it is -(1/d) u^(-m - 2) times the function of a + b u^k, a polynomial times the function, and it is written back with u = 1/L. Every factor with the variable in it has to be read so, or the rule declines. The other powers of L were answered as written -- the derivative of the argument at m = -2, the sine and cosine integrals above it -- and are left to the rules that answer them.

Tests: FunctionOfTheReciprocalIntegralTest.UnderTheReciprocal, six rows, each differentiated back with the symbols pinned, on both sides of zero; all six are declined on master.

Measured first on the 293 problems with a function of the reciprocal of x or of a linear in them, 191 run, at the corpus's 5-second budget, against master 8f3757cd, the branch's base:

	master	this
solved	142	155
wrong	0	0
past the budget	4	4

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	2218

No answer is wrong in the pocket or the sample on either build. Of the 18 problems the two builds disagreed on, run again one build at a time, master answers none and this 16: thirteen declined on master in a second and answered here in a fifth of one -- sin(a + b/x)/x^3 to x^5 and their squares, cos(a + b/x)/x^3 and x^4, cos(1/x)/x^5, and four of sin(a + b/(c + d x)^(1/3))/(c e + d e x)^(5/3) and its kind -- and three at about twenty seconds, past the budget on master and inside the patience here. The two left take longer than that on either build.

The suite passes on the commit measured, 3d5bf6ee, 14,597 tests with 13 skipped, and the allocation gate with it. On the merge with master 0a13a636, f93622d5, 4,066 of the 4,068 calculus and corpus tests pass; the other two time themselves, ran past their bound on a loaded machine, and pass run again. Every row of the first table is as it says on the merge.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura