[Rafael-SOWNet]

Part of #233.

• Under the half tangent, a factor the parts share is written out. Rubi's 6.7.1 sech(x)^2/(a + c sech(x) + b tanh(x)) ran past the corpus's budget. Master answered it in 1,383,216 characters after 22 s and 3.7 GB.
• Why it grew. SolveARationalFunctionOfTheHyperbolicTangent homogenises each maximal rational function of e^x apart, which keeps the written factors. A factor two parts share then stays inside the expanded polynomial of one of them. Under u = tanh(x/2) the integrand was (1 + u^2)^2 over (1 + u^2)^2, times a quartic that holds 1 + u^2 once more. The rational integrator then wrote that quartic's partial fractions with symbols in it.
• The change. The factors the homogenising makes, 1 + u^2, 1 - u^2, 1 - u, 1 + u and u, are divided out of every written factor of the denominator as often as they go exactly, and written as powers of their own. Then they are divided out of the numerator. The integrand becomes 1/((1 + u^2) Q(u)) for a quadratic Q. The helper is PartialFractions.WithTheCandidateFactorsWrittenOut.
• Where it applies. Only where a candidate then divides the numerator too. 1 - u^2 comes before its two linears, so that an integrand even in u stays written so.
• A polynomial gcd over the whole quotient was tried first. It cancels the same factor but expands the denominator, and the rational integrator declined the quartic it left.
• What the chains caught. Written out with nothing cancelled, 6.3.1's (c + d x)^3/(a + a tanh(e + f x))^3 and two like it ran past the budget they had been answered in. With 1 - u^2 split into its two linears, 6.1.7's sinh(u)^8/(a - b sinh(u)^4)^2 took 5.4 GB where it takes 0.5, confirmed alone. Both guards answer those as master does.

Input	master ef4150e	this branch
6.7.1:910 sech(x)^2/(a + c sech(x) + b tanh(x))	1,383,216 characters, 22 s, 3.7 GB	2,931 characters, 0.1 s, 0.04 GB
csch(x)^2/(a + c csch(x) + b coth(x))	1,690 characters	1,766 characters
6.3.1:68/69, 6.4.1:46, 6.1.7:286	answered	the same answers

Third chain, against master ef4150e (the branch rebased onto 7d1081e):

• The family-6 pocket of 3,690 rows went from 3,676 to 3,681 solved.
• Of the five rows that moved, run alone, 6.7.1:910 is this change's. The other four, 6.5.3:251/255 and 6.6.3:213/217, are Read x times a rational function of an exponential of a logarithm off the integrand #2047's, merged since the master arm was cut.
• Every family's sample is unchanged.
• 9,281 rows were compared for allocation, and none grew past 1.3x and 300 MB.

The library builds for netstandard2.0, net8.0 and net10.0. The suite is left to CI. The BREAKING-CHANGES lengths were measured on the branch, and the 2.5.0 column on a v2.5.0 build.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

[Rafael-SOWNet]

Rebased onto 1566841 (#2054). There was one conflict, in BREAKING-CHANGES.md: two entries at one anchor, sharing their citation line and table header. Both entries are rebuilt whole, #2054's first. The library builds for every target, and the hyperbolic rules' tests pass together, 6 of 6.