A rational function of the hyperbolic tangent with a symbol among its coefficients is integrated by u = tanh(y) with the written factors kept, and an answer assembled from a decomposition with symbols is checked before it is given

[Rafael-SOWNet]

Rubi's hyperbolic families (family 6) with a symbol among the coefficients: 278 → 301 of the 417-problem sample, 0 wrong, 0 error, timeouts 57 → 47. The independent suites stay at 1707/1774, 0 wrong, 0 timeouts (1706 on master; the same set, ±2 s over 85 s).

Carries #1374's commit until that merges — this branch is cut from complex-division, because the sample was measured with the division fix in (a right antiderivative's derivative read as wrong at one point without it). Once #1374 is in, I rebase and the diff is the one commit.

What it does

A rational function of the hyperbolic tangent is integrated by u = tanh(y) with the written factors kept. Under u = e^x, Rubi's 1/(a + b coth(c + d x)^2)^2 is a rational function of the symbolic palindromic quartic (a + b) u^4 + 2(b − a) u^2 + (a + b) squared, whose roots are what the budget went on: 57 of the 417 timed out, most of them the tangent, cotangent, secant and cosecant families with a symbolic a + b f^2. Under u = tanh(y) it is u^4/((b + a u^2)^2 (1 − u^2)), which the symbolic partial fractions answer. SolveARationalFunctionOfTheHyperbolicTangent sits before the substitution search, which with a symbolic slope spent the whole budget on u = c + d x and never came back to the chain.

The rewrite keeps the written factors rather than expanding: every exponential is e^(k y) for one linear form y (the gather is shared with the tangent substitution for radicands of both signs, which now takes a linear argument too — the two routes ask for y at a different grain, see the commit); each maximal rational function of v = e^y among the factors, powers and radicands is read as v^m R(s) with s = v^2, homogenised on its own as Σ p_i (1 + u)^i (1 − u)^(d − i) over the same for the denominator, the common powers of 1 ± u divided out synthetically, the sides collected as polynomials over the symbols with the numeric content in front, and the powers of 1 + u, 1 − u, 1 − u^2 gathered into one each at the end. An integrand odd in v takes the half-angle u = tanh(y/2) only where every exponent is written with an even factor and there is no root; otherwise it is not this route's.

The exponential substitution takes the whole linear form. csch(29/10 + 13/10 x)^3 (17/10 + 23/10 sech(...)^2)^3 put u = e^(13/10 x) and left e^(29/10) in every coefficient, a quadratic the rational rules do not factor over — 110 s and no answer. Where every exponent is a multiple of one x + shift (cross-multiplied, so a symbolic slope is never divided by), u = e^(13/10 x + 29/10) and the coefficients are the numbers they were written with: 516 ms.

Two wrong answers and a crash found on the way, each guarded, each with an issue for the cause

#1369	1/((a + b x^2)^3 (x^2 − 1))	the Hermite ansatz's logarithmic part, coefficients that are quotients of 45th-degree polynomials in the symbols, decomposed correctly and integrated wrongly as a sum where each term is right alone. The Hermite branch checks its rest against the logarithmic part, and the written-factors branch its term-by-term integral, when a symbol is involved. The mixing itself is still open.
#1370	sqrt(a + b sech(x)) tanh(x)^5	integrated through i, a root of a negative radicand on the way; the tangent substitution checks its answer at sampled points when a symbol is involved.
#1373	csch(29/10 + 13/10 x)^3 (17/10 + 23/10 sech(...)^2)^3	stack overflow: the power-of-a-quadratic reduction recursed on a remainder the division had not reduced, a level a call. It declines a non-linear remainder now, in both places it is asked, and the case is a test.

The check the guards share, PartialFractions.DerivativeHoldsAtSampledPoints, pins the symbols before differentiating (sgn(f)' and |f|' want a real-valued f) and skips a point where the derivative cannot be evaluated rather than throwing out of the integrator.

Measured

Family 6, 20 problems a file, 5 s budget, master 54d8a30e against this branch:

Source	Run	master	branch
6.1.1 (c+d x)^m (a+b sinh)^n	20	14	12
6.1.3 (e x)^m (a+b sinh(c+d x^n))^p	20	20	20
6.1.5 Hyperbolic sine functions	20	14	14
6.1.7 hyper^m (a+b sinh^n)^p	20	12	15
6.2.1 (c+d x)^m (a+b cosh)^n	20	13	13
6.2.2 (e x)^m (a+b x^n)^p cosh	20	20	20
6.2.3 (e x)^m (a+b cosh(c+d x^n))^p	19	19	19
6.2.5 Hyperbolic cosine functions	20	15	15
6.2.7 hyper^m (a+b cosh^n)^p	20	13	13
6.3.1 (c+d x)^m (a+b tanh)^n	10	10	10
6.3.2 Hyperbolic tangent functions	20	16	17
6.3.7 (d hyper)^m (a+b (c tanh)^n)^p	20	10	13
6.4.1 (c+d x)^m (a+b coth)^n	10	10	10
6.4.2 Hyperbolic cotangent functions	20	17	16
6.4.7 (d hyper)^m (a+b (c coth)^n)^p	20	4	6
6.5.1 (c+d x)^m (a+b sech)^n	4	1	1
6.5.2 (e x)^m (a+b sech(c+d x^n))^p	14	8	8
6.5.3 Hyperbolic secant functions	20	12	13
6.5.7 (d hyper)^m (a+b (c sech)^n)^p	20	9	15
6.6.1 (c+d x)^m (a+b csch)^n	7	3	3
6.6.2 (e x)^m (a+b csch(c+d x^n))^p	13	7	7
6.6.3 Hyperbolic cosecant functions	20	8	9
6.6.7 (d hyper)^m (a+b (c csch)^n)^p	20	8	15
6.7.1 Hyperbolic functions	20	15	17
total	417	278	301

26 gained, 4 lost: two at the edge of the budget (1/(b coth(c + d x)^4)^(2/3), csch(c + d x)^20 (a + b sinh(c + d x)^4)^3) which master answered in the corpus run from a warm memo at 5.2 s and 22 s of wall time and times out on alone at the same 5 s budget, exactly as this branch does — probed one process each on both builds; and two with i in the coefficients (cosh(c + d x)/(a + i a sinh(c + d x)), (c + d x)^2 (a + i a sinh(pe + f x))^2) that the harness now reports unverifiable on the reals rather than solved — a different answer form, not a wrong one. Every answer is verified by differentiating back with the parameters pinned, in the harness and in the library.

Four suites green (11366 / 134 / 18 / 41). No gate row is integration, and the change is confined to the integrator and one helper in PartialFractions.

Part of #718.

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