A half-angle integrand with symbols in it is put over one bar and not simplified

[Rafael-SOWNet]

Part of #718.

Under t = tan(x/2) a rational function of the sine and cosine was simplified before it was integrated, and with symbols in it the simplifier's search grew past any budget, where integrating what it returned takes a fraction of a second:

integrand	2.5.0	master ba8eb99b	this
sec(x)^2/(a + b cos(x))^3	declined	38 s	1.6 s
sec(x)^2/(a + b cos(x))^4	declined	past two minutes	1.4 s
sec(c + d x)^2/(a + b cos(c + d x))^4	declined	past two minutes	0.9 s
sec(x)/(a + b cos(x))^3	declined	7.5 s	0.25 s
1/(cos(x) (a + b cos(x))^2)	declined	1.6 s	0.3 s

Timed one integrand at a time, beside other work on the machine.

What changes. A rational function of t with a symbol in it is put over one bar, with its whole powers of products written as products of powers and its bases of a lower degree than they are written in lowered, and is not simplified further. One without a symbol, or with a root of t in it, is simplified as before: under a root the simplifier's writing of the radicand is what the radical rules read, and 1/(b cos(x) + c sin(x) - sqrt(b^2 + c^2))^(3/2) went from a decline in three seconds to a search past two minutes without it.

Tests: HalfAngleSubstitutionIntegralTest.APowerOfALinearInTheCosineBesideASecant, five rows, each differentiated back with the symbols pinned.

Measured first on every sixth of the 5,655 problems in Rubi's family 4 that are rational functions of trigonometric functions of one argument with a symbol in them, 938 run, at the corpus's 5-second budget, against master 8f3757cd, the branch's base:

	master	this
solved	871	902
wrong	0	0
past the budget	59	28

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	1295
families 2 to 8, sampled (2410)	2215	2232

No answer is wrong in the sample on either build. Of the 49 problems the two builds disagreed on, run again one build at a time, 43 ran out of time on master and are answered here, 34 of them in the secant files 4.5 -- 1/(a + b sec(c + d x))^2, sec(e + f x) (a + b sec(e + f x))/(c + d sec(e + f x))^3 and (A + B sec(c + d x))/(a + b sec(c + d x))^2 among them -- and the other nine rational functions of the trigonometric functions over a power of a linear or a quadratic in one of them. Five are answered on both: four take 21 to 23 s on either build, past the corpus's five seconds, and family 1's one difference in the sample, x^4/((a + b x^2)^2 (c + d x^2)^3), is one of them; the fifth, (A + B sin(e + f x))/((a + a sin(e + f x))^3 (c + d sin(e + f x))^3), takes 21 s on master and 7 s here. One, sqrt(g sin(e + f x))/((c + d sin(e + f x)) sqrt(a + a sin(e + f x))), runs out of time on both.

The suite passes on the merge with master ba8eb99b, 095cb42a, 14,673 tests with 13 skipped, run under a 4 GB heap limit, and the allocation gate on the commit measured, 489eb68b; every row above is as the table says on the merge.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura