A shared symbol among a denominator sum's coefficients comes out in front of it

[Rafael-SOWNet]

tan(x)/(a + a csc(x)) came back unevaluated while tan(x)/(1 + csc(x)) was answered in a quarter of a second. Traced: under u = sin(x) the denominator is (a u + a)(1 - u^2), a symbolic linear beside a rational quadratic that shares its root, which the symbolic split declines (rightly — the root is not that block's alone); under the half-angle it is -a t^6 - 2a t^5 - a t^4 + a t^2 + 2a t + a, a sextic with a symbol in every coefficient, which nothing factors — where it is a times -(t - 1)(t + 1)^3 (t^2 + 1).

Change. In SolveByPartialFractions, before the denominator is refactored over the rationals: every written sum among its factors whose coefficients share a symbol is written as its content times its primitive part — the content by PolynomialGcd.ContentIn over the coefficients as polynomials in the symbols — so (a u + a) is a (u + 1), and the refactoring that follows finds the repeated (u + 1). The lowest power of the variable every term holds comes out the same way: a u^4 + b u^3 is u^3 (a u + b), which is what u = sin(x) makes of cot(x)^3/(a + b csc(x)), and neither the refactoring over the rationals nor the symbolic split reads a symbolic polynomial for the monomial it is a multiple of. Whole powers of the sum only: (c e + d e x)^(1/3) is e^(1/3) (c + d x)^(1/3) for a positive e and not for a negative one, and a symbol is neither — a first version read the fractional power too and answered #1386's sin(a + b (c + d x)^(1/3))/(c e + d e x)^(1/3) with e^(1/3) pulled out, right where e > 0 only, and that row is back to unevaluated. Once, since a primitive factor has no content to take out; the numerator is not read, since a factor there is not what stops anything.

Measured. Rubi's family 4 sample (422, six per file): 243 → 255, 0 wrong — tan(x)/(a + a csc(x)), tan(x)^4/…, sec(x)^2/…, sin(x)^4/(a + a cos(x)), cos(c + d x)^4/(a + a sin(c + d x))^2, (c + d x)/(a + a sin(e + f x))^2, cot^2/(a + a sec)^2 and two more powers of (a + a sin) from the symbol content, then cot(x)^3/(a + b csc(x)), 1/(a + b csc(c + d x)^2) and cot^4 sqrt(a + b tan^2) from the monomial (second commit); no row goes the other way (one sqrt(a + a sin)/(sin (c + d sin)) reads Unsolved at 9.4 s on one run and Timeout on the other, both past the budget). The 1,774-problem independent suite: 1706/1774, 0 wrong, 0 timeouts, verdicts identical to master. All four test suites green; DenominatorContentTest (6 rows, each at a = 2 and a = -3) with its .editorconfig section; BREAKING-CHANGES entry.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

[Rafael-SOWNet]

Merged master (#1401 and #1402; one same-anchor conflict in .editorconfig, both kept). On the merge: the new test classes and the corpus gate green, the 1,774-problem suite 1706/1774 with 0 wrong and 0 timeouts, verdicts identical to master, 89.6 s → 83.6 s.