A constant comes out of a fractional power beside another power of the same function

[Rafael-SOWNet]

sqrt(b sec(x))/sec(x)^(7/2) was left as written. The two powers of the secant are the same function to a total power of -3, which the rules for a power of a secant answer — and they cannot see it while one of them is written over b sec(x) rather than over sec(x).

(c f)^b is c^b f^b for a positive real c and any f: both sides pick up the same phase where f is negative, so the rewrite needs nothing assumed about f, and a symbolic c gives the answer provided c > 0.

That condition is not decoration. 2.5.0 answered sec(x)^(3/2)/(b sec(x))^(5/2) without it, and that answer is wrong wherever b and the secant are both negative — at b = -3 its derivative is +0.0267i against the integrand's -0.0267i at x = 2, while the two agree at x = 0.5. #1388 withdrew the unconditional reading for exactly that reason, and this brings the shape back with the condition it owes.

Where it is asked, and what it takes, keeps it from costing anything elsewhere:

• After the rules that answer the same shapes for any real constant. A function comes out of its even power with its sign — (a sin(x)^2)^(5/2) for every real a — and sqrt(a + a sin(x)) is integrated by the half angle, again for every real a. Asked before them this answered both provided a > 0; it is asked after them, and before the substitution search. The monomial split from A power of the variable times a sine or cosine of a logarithm, and a power of a monomial #1479 keeps its own, earlier place.
• A single factor in which x enters only through trigonometric functions, with a constant written in every term of such a sum read too (a - a sin(x)^2 is a (1 - sin(x)^2)). The rewritten question is asked again at the same depth, so a rewrite that brings nothing closer multiplies the search at every level: over a product of trigonometric powers it cost 10 s → 47 s, and over csch(x), which is written in exponentials, it hung the test run — see the comment below.
• Never the question it was asked. A rewrite that simplifies back to its input is declined rather than asked again.

Measured

	before	after
Rubi family 4	290/422	309/422, no row lost, 27 timeouts against 32
Rubi family 6	377/417	378/417
Rubi families 1, 5, 7	158, 212, 215	158, 212, 215, no row lost
Rubi corpus (1774)	1707, 0 wrong, 0 error, 0 timeout	1707, 0 wrong, 0 error, 0 timeout
suite	12,675	12,677, 0 failed, run to completion
allocation gate	—	PASSED on all 19 gated benchmarks

Six tests added, and BREAKING-CHANGES.md records both changed answers with their 2.5.0 values.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

[Rafael-SOWNet]

The first head of this PR hung the test run, and I cancelled its CI. The three C# Test legs sat in the Test step for 3½ hours; master's runs of that workflow finish in about 26 minutes. Fixed in 2de06de. What happened, for the record:

Where. HalfAngleSquareTest.ThePairOfPowersTwoApartIsElementary on x/csch(x)^(3/2) + x*sqrt(csch(x))/3 started and never finished — named by --blame-hang with the dump turned off, from the entry in the Sequence file that never reached Completed. Locally the run looked green at first: after Blame kills a hung host, vstest still prints Passed! Failed: 0 over the tests that finished, and its total was 12,658 against a complete run's 12,675.

Why. The library writes csch(x) as 1/((e^x - e^(-x))/2), and the rule re-asks through ComputeAsTheSameQuestion, which keeps the rewritten question at the same depth. Two things then went unbounded:

1. An identity rewrite. 1/s splits into the factors 1 and s^(-1); the constant accumulated as 1 * 1, which is not One as a tree, so the rule believed it had taken something out and asked the identical integrand again — 38 of the first 40 traced firings were expr ==> expr.
2. A real rewrite at every level. With that fixed, csch's genuine 2 was still taken out again under each exponential substitution below it: x/csch(x)^(3/2), declined in 7.4 s on master, did not return in 90.

What changed.

• A same-question rewrite must change the question: the rule declines when its rewrite, after InnerSimplified, is what it was asked. Products are built without a leading 1 *.
• The constant is taken out only of a single factor in which x enters only through trigonometric functions — the family it was written and measured for. Below a substitution the trigonometric functions are gone, so this also keeps it off the levels where it compounded. A constant written in every term of such a sum counts too (a - a sin(x)^2 is a (1 - sin(x)^2)), which was the difference between sqrt(a - a sin(pe + f x)^2) answering and sqrt(a - a sin(x)^2) not.
• Split into two registrations. The monomial split from A power of the variable times a sine or cosine of a logarithm, and a power of a monomial #1479 stays where it was. The constant pull now runs after the rules that answer the same shapes for any real constant — the even power with its sign, and the half angle of a ± a sin — because asked first it answered sqrt(a + a sin(x)) provided a > 0, which HalfAngleSquareTest pins at a = -3.

Measured on the new head:

	master	this PR
Rubi family 4	290/422	309/422, 27 timeouts against 32
Rubi family 6	377/417	378/417
families 1, 5, 7	158, 212, 215	158, 212, 215 — no row lost in any of the five
corpus (1774)	1707, 0 wrong	1707, 0 wrong, 0 error, 0 timeout
suite	12,675	12,677 (two cases added), 0 failed, run to completion — Blame: "All tests finished running"
allocation gate	—	PASSED on all 19 gated benchmarks

The shapes that hung now match master: x/csch(x)^(3/2) + x sqrt(csch(x))/3 16.96 s against 17.12 s, x/csch(x)^(3/2) 7.39 s against 7.42 s.

🤖 Generated with Claude Code