[Rafael-SOWNet]

The parser writes each inverse hyperbolic function as a logarithm, ln(A + sqrt(A^2 + k)) or ln((1 + u)/(1 - u)). Differentiated as written, that is a quotient with the root in both parts, which the search never brought down to the root alone. The rule writes the derivative as the algebraic function it is, A'/sqrt(A^2 + k) or 2 u'/(1 - u^2), and integrates P + Q L as ∫P + G L - ∫G L', with G the antiderivative of Q.

It declines where:

• Q is a polynomial, which is already answered;
• Q is not rational: beside a root, the rules for the inverse function answer, and the root's integral asked here first took their time;
• G holds L itself;
• the answer would hold on part of the line only;
• a factor below Q's bar divides the function's own quadratic, as in atanh(a x)/(1 - a^2 x^2)^3, which the substitution under the function answers;
• an artanh's argument is not linear.

Each limit came from a measured loss. Validated by chain four times, on 8a1ef454, branch against master:

1. Early, unrestricted: 116 problems master answers timed out.
2. With Q rational: one wrong answer, the master defect fixed in Take an interval's size without squaring, so a size past the doubles does not agree #1966, and two losses.
3. With the quadratic exclusion: clean.

Run	Problems	Master	Branch
Pocket: Rubi's 7.x, every third solved problem	1,502	1,384 solved	1,395 solved
Family 1, 40 a file	1,381	1,362	1,362
Family 0, whole	1,814	1,789	1,789
Families 2 to 8, sampled	2,410	2,362	2,362

Re-run alone on both builds, 10 problems go from timed out to answered: Rubi's 7.1.4:20, 7.2.4:621/629/630, 7.5.1:165, 7.5.2:14/15, 7.6.1:151/152 and 7.6.2:13. None of master's answers is lost, and no run had a wrong answer.

In that chain, five 7.3.7 problems that master declines in 1–11 s timed out instead, all atanh(x sqrt(e)/sqrt(d + e x^2))/x^n. The last limit, a linear argument for the artanh, was added for them after the chain. Probed on this rebased branch, those problems answer, through #1952's rule, and so do all ten gains and both earlier losses: 13 of 13.

A sweep of the 55 rows with one linear inverse hyperbolic function and a whole power of a quadratic below the bar: master answers 49, the branch 55.

ByPartsAgainstAnInverseHyperbolicFunctionAsItsLogarithmIntegralTest includes 7.6.1:130 and both losses. BREAKING-CHANGES.md has the entry.

Part of #233.

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