A whole power of a quotient with a symbol in it is integrated as the quotient of the powers

[Rafael-SOWNet]

Part of #718.

(c/(a + c x^2))^2 was declined after a search of half a minute, while c^2/(a + c x^2)^2, the same function, is answered at once: the table reads the quotient of the powers, and nothing read the power of the quotient. And the substitution u = x^2 writes x/(a + c x^4)^2 in that spelling, so an integrand 2.5.0 answered in a twentieth of a second takes twenty-one on master:

integrand	2.5.0	master ac6e33fa	this
(c/(a + c x^2))^2	declined after 30 s	declined after 25 s	answered in 0.4 s
(1/(a + c x^2))^2	declined	declined	answered in 0.03 s
x/(a + c x^4)^2	answered in 0.05 s	answered in 21 s	answered in 0.15 s
x^2/(a + b x^6)^2	answered in 0.06 s	answered in 22 s	answered in 0.03 s

What changes. A whole power of a quotient with a symbol in it, where what is left above the bar is a monomial, is written as the quotient of the powers, a negative one with the quotient turned over, and asked as the same question. That is exact for a whole power. Only a monomial above, the table's x^m/(a + b x^n)^p: written for every quotient, the square of tanh(a + 2 ln x), a quotient of binomials once it is folded, went from five seconds to forty-five.

Tests: WholePowerOfAQuotientIntegralTest.IsTheQuotientOfThePowers, seven rows differentiated back with the symbols pinned, on both sides of zero. Four are declined on master, and two more take over twenty seconds there.

Measured first on 205 problems, at the corpus's 5-second budget, against master 8f3757cd, the branch's base: the 184 of 1.1.2.2 and 1.1.3.2 that a master build declined, ran past its budget on or took more than five seconds over, and those that two earlier, wider builds of this branch disagreed with master on.

	master	this
solved	19	17
wrong	0	0
past the budget	13	16

Each of the three problems the two builds differ on takes over twenty seconds on either build, measured alone, and comes out on either side of the corpus's patience. x/(a + c x^4)^2 and x^2/(a + b x^6)^2 are counted solved on both, in 23 s on master and in 41 and 13 ms here.

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	1296
families 2 to 8, sampled (2410)	2215	2219

No answer is wrong in the sample on either build. Of the seven problems the two builds disagreed on, run again one build at a time, four agree, and three come out on one side of the patience on one build and on the other side on the other; all seven take over nineteen seconds on either build.

The suite passes on the commit measured, fa61df92, 14,598 tests with 13 skipped, and the allocation gate with it. On the merge with master ac6e33fa, ce5547cb, the 4,059 calculus and corpus tests pass, and every row of the first table is as it says.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura