Powers of asech(a x) and acsch(a x) beside a power of x are integrated, by rounds of parts

[Rafael-SOWNet]

Rubi's 7.5.1 and 7.6.1 with a power of the secant or cosecant: x^3 asech(a x)^2, asech(a x)^3/x^2, (a + b acsch(c x))^2/x^4 and their kin. One step of parts leaves x^2 asech(a x)/sqrt(1/(a x)^2 − 1), and the step on that — two levels below the question — was never offered. Traced, five things, each fixed in its place:

• The by-parts regrouping around the differentiated factor is offered up to two levels below the question where everything beside the factor is algebraic with a root in it: the other factor integrates to an algebraic expression or declines at once, and the remainder is algebraic — none of the logarithms by the dozen the existing top-only scope is there to stop. With a root only: beside a rational function the rule for a rational function times a logarithm answers, and offering parts there sent (a + b atanh(c x^2))^2/x^5 into a search that did not return (measured, and why the guard says root).
• SolveByWritingAPowerOfAQuotientApart runs at any depth where every root stands over a monomial with a polynomial above it — sqrt(1/(a x)^2 − 1) as the derivative of asech writes it — and answers with sgn(x)^m in front rather than by parity: the parity extension wrote its answer in |x|, and handed on by the next step that was a remainder in |x| and (1 − (|x| a)^2) no rule read. The sign goes in front only where every power of |x| taken came from a root that is a factor of the integrand (a root inside a sum keeps the parity extension). A first version required the monomial only, without the polynomial above — (1 + ln(u^2)/u^2)^(−3/2) qualified, and Bronstein's sqrt(x + ln x) row went from 0.9 s to 10.
• SolveByTakingASignOut: the sign of a real-valued factor is constant between its zeros and goes in front of the antiderivative of the rest (the generic case every rule answers in; sgn(g) for a complex g is g/|g| and is not taken). Placed after every rule that reads the sign where it stands, so cos(x) sgn(sin(x)) is still |sin(x)| and the five pinned IntegrationTest rows keep their forms.
• A radicand written in two orders is one spelling where it stands in two places that differ in the sign of the power or the side of the bar — q^(−1/2) above beside sqrt(q) below, or sqrt(A) above beside sqrt(A′) below — extending asech(a + b x) and acsch(a + b x) beside a power of x are integrated #1457's rule, which read the written sign only and missed the second shape (sqrt(1 − x^2 a^2)/(sqrt(1 + −a^2 x^2) x) was a 49,000-try search).
• A whole power of a fractional power is normalised to the power it is: ((c^2 x^2 + 1)^(−1/2))^2 is (c^2 x^2 + 1)^(−1), and left as a square of a root it was handed to Euler's substitution as a radical beside (c^2 x^2 + 1)^(−1), which answered the remainder of acsch(c x)^2/x^4 wrongly — the first corpus run of this branch had it as its one wrong answer (derivative off by 0.02%), and the piece was found by differentiating the answer's two parts separately. A fractional inner power only: flattening ((1 + x^2)^(−1))^4 too sent Timofeev's x^3 arctan(x)^2/(1 + x^2)^3 from 0.1 s to 8.

Measured:

• x^3 asech(a x)^2 5 s decline → 0.3 s; asech(a x)^2/x^2 60 s runaway → 0.5 s; asech(a x)^3/x^2 3 s decline → 0.8 s; (a + b asech(c x))^2/x^4 8 s decline → 0.4 s; (a + b acsch(c x))^2/x^4 7 s decline → 0.5 s; x^3 (a + b acsch(c x))^2 20 s → 0.9 s.
• Family 7: 210 → 215 of 270, 0 wrong, 0 error (7.5.1:11/25/53, 7.6.1:26/33); timeouts 26 → 30 are rows over the 5 s budget either way.
• The 1774-problem suite: unchanged, 1707/1774, 0 wrong, 0 timeout, wall 85 s; per-problem, nothing moved by more than 0.2 s (Timofeev 950 is 2 s faster).
• Full suite 12552 passed, 0 failed; the kernel gate PASSED on all 19 gated benchmarks. BREAKING-CHANGES.md entry with the 2.5.0 column measured on a build of v2.5.0.

Tests: five rows added to InverseHyperbolicSecantIntegralTest, each differentiated back and compared numerically at five real points with the parameters pinned.

Part of #718.

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