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.

Two restrictions keep the rewrite from costing anything elsewhere:

• A single factor. The constant comes out only where the rest of the base is one factor, which is where it pays: the power it leaves can then meet another power of the same function beside it. Over a product it would rewrite the question into one no easier and pay a whole descent to find out — (cos(x)^11 sin(x)^13)^(-1/4) measured at ten seconds against forty-seven.
• Not an even power. (a sin(x)^2)^(5/2) is a^(5/2) sgn(sin x) sin(x)^5 for any real a, since an even power is never negative, and the rule that takes a function out of its even power with its sign says so unconditionally. Taking the constant out there would answer the same shape provided a > 0 and lose the negative half of the line by getting there first. The evenness is read through the nesting and past the sign, because below the tangent substitution the factor arrives as (u^2)^(-1).

Measured

	before	after
Rubi family 4	290/422	309/422, no row lost, 30 timeouts against 32
Rubi corpus (1774)	1707, 0 wrong, 0 error, 0 timeout	1707, 0 wrong, 0 error, 0 timeout
Rubi family 1	158/228	158/228
suite	—	12,644 passed, 0 failed, 14 skipped
allocation gate	—	PASSED on all 19 gated benchmarks

Four 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