[Rafael-SOWNet]

1/(1 + sin(x)^5) is asked one level down as (1 + sin^5)^(-1). The half-angle substitution then asks for 2 (t^2 + 1)^4/((t^2 + 1)^5 + 32 t^5), whose denominator is (t + 1)^2 times an irreducible octic. Its logarithms want residues of degree eight, which only the second pass writes, as a sum over roots. The first pass declined only after partial fractions had tried every split and spelling for 28 seconds; the second pass then answered in half a second.

In the first pass, the half-angle substitution now declines at once and sets ASumOverRootsWouldAnswer, when the question in t meets three conditions:

• its coefficients are rational;
• Hermite's reduction, the resultant and its factors show a residue field of degree past two;
• no product of the denominator's irreducible factors is a polynomial in (t + s)^k with k >= 2. That form covers the binomials, biquadratics and blocks that the closed-form rules read; x^5 + 1 is caught as a whole.

The test is RothsteinTrager.WantsASumOverRootsAlone. Its residue fields come from a helper, ResidueFields, that LogarithmicPart now shares. #1996's mod-p screen, which spares the factorization where a field past the second degree is a decline anyway, moved into that helper behind a parameter and is applied exactly where it was.

Measured on 20a4b391, branch against master:

Run	Problems	Master	Branch
Pocket: Rubi's 4.1, 4.2, 6.1.7, 6.2.7 and 6.7.1, sampled	1,886	1,826 solved	1,827 solved
Family 1, 40 a file	1,381	1,370	1,370
Family 0, whole	1,814	1,796	1,796
Families 2 to 8, sampled	2,410	2,391	2,390

Re-run alone, 4.1.7:326 goes from timed out to answered. Measured as one process each, it drops from 29.4 s and 56 GB to 2.3 s and 0.75 GB, with the same answer character for character. 4.1.7:329 and sin(x)/(1 + sin(x)^5) go from 12 s and 31 GB to under a second, also with identical answers.

The one-row sample difference is 6.6.1:25, which master answers in 23 s; both builds answer it alone. None of the following changed its answer, character for character: 394 solved rows of 4.1.7, 86 of 4.2.7, 244 of 6.7.1 and 302 reciprocals of radicals. 1/(1 + cos(x)^3), over a binomial the first pass writes at its roots, keeps its closed form. On the combined build, #1996's Hearn 365 and Bronstein 22 still answer. No run had a wrong answer.

ASumOverRootsLeftToTheSecondPassIntegralTest checks that these come back with a sum over roots, or in closed form for the even block, differentiated back. 190 related tests pass. No answer changes, so BREAKING-CHANGES.md has no entry.

Part of #233.

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