[Rafael-SOWNet]

The derivative of sgn(f) and of abs(f) was left unevaluated wherever f is not real on the whole line, and a root of anything that can be negative is not.

Where f is real wherever the bases of its roots are positive, the derivative is now a piecewise:

• where every such base is positive, the real-line rule: 0 for sgn, and sgn(f) f' for abs;
• elsewhere, the derivative left as written, since there a root is imaginary and sgn(z) = z/|z| turns with z.

RealValued.WhereRealValued gives the conjunction of base > 0 over the powers that are not whole. A derivative node no longer simplifies a result that contains itself; without that guard, this change looped.

The integrator's antiderivatives of Rubi's x^(-1 - 3n/2) (b x^n)^(3/2), sqrt(1 + x + sqrt(1 + x)), (x - 4)/((1 + x^(1/3)) sqrt(x)) and their kin carry such a sign. They are right, by quadrature on both sides of their roots, but could not be differentiated back anywhere, so they read as unverified. Three answers built under u = sqrt(1 + x) or u = sqrt(x) lose a provided 1 + x >= 0 that had been attached for want of that derivative. sqrt(1 + x + sqrt(1 + x)) keeps the condition of its answer in u instead, 26 characters longer.

Measured on 8270086f, branch against master:

Run	Problems	Master	Branch
Pocket: Rubi's 1.1.3.x, 1.2.4.2 and 1.3.2	3,549	3,502 solved	3,512 solved
Family 1, 40 a file	1,381	1,368	1,369
Family 0, whole	1,814	1,789	1,789
Families 2 to 8, sampled	2,410	2,380	2,382

Re-run alone, 12 problems go from unverified to verified: 1.1.3.2:211–214, 1.2.4.2:185, 1.3.2:935, 939, 966, 967 and 1228, 4.2.2.1:320 and 5.5.2:64. The 13th move, 7.4.1:369, a problem master answers in 24 s, is answered on both builds. Nothing goes the other way. Across the whole 1.3.2 file and 360 solved root integrands from families 0, 1 and 4–7, the answers are otherwise byte-identical. None of the problems answered on both builds allocates more on the branch. No run had a wrong answer.

ASignOfARootIsFlatWhereTheRootIsRealTest has 28 cases. The 5,665 calculus, derivative, limit and piecewise tests pass. One cache wall-clock test failed under load at 35 s against its 30 s bound, and passes alone in 3 s. BREAKING-CHANGES.md has the entry, with the 2.5.0 column measured on v2.5.0.

Part of #233.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura