The Hermite reduction's system is solved in one row order whatever the spelling

[Rafael-SOWNet]

Part of #718.

How the denominator of a rational integrand was written decided how long its Hermite reduction took: x/((1 + x^2)^3 (2 a x + b (x^2 + 1))) took 45 s, and the same with (x^2 + 1)^3 below the bar 1.2 s.

integrand	master 97c4b692	this
x/((1 + x^2)^3 (2 a x + b (x^2 + 1)))	45 s	0.6 s
x/((x^2 + 1)^3 (2 a x + b (x^2 + 1)))	1.2 s	0.6 s
sin(c + d x)^4/(a + b tan(c + d x))^3	past 90 s	1.1 s
sinh(c + d x)^6/(a - b sinh(c + d x)^4)^2	past 150 s	0.9 s
(A + B x + K x^2)/(x^3 (a + b x^2 + c x^4)^2)	past 90 s	5.1 s

What changes. The rational part of a Hermite reduction is one linear system, and its rows were the powers of x in the order the coefficient dictionaries met them, which follows how the factors are written. With symbols in the coefficients the elimination's pivots followed that order, and in one order the solution came out in coefficients of the thirty-sixth degree in the symbols that the final cancellation did not reduce; declining the logarithmic part they made was forty seconds before the answer. The rows are ordered by their power of x now. The value of an answer does not change; where the order mattered, its coefficients can come out reduced where they were not.

Tests: RationalFactorsBesideASymbolicQuadraticTest.EitherSpellingOfOnePlusTheSquare, three rows, each answered within the test's minute and differentiated back.

Measured first on every fourth problem of 4.1.1.2, 4.1.2.1 and 4.1.2.2, the half-angle tangent's rational functions with symbols, at the corpus's 5-second budget, against master 9a33298e, the branch's base:

	master	this
solved	498	498
wrong	0	0

Measured then on the Rubi corpus against master 9a33298e:

	master	this
family 0, independent suites (1814)	1766	1766
family 1, 40 a file (1381)	1288	1290
families 2 to 8, sampled (2410)	2200	2204

No answer is wrong on either build. The 7 problems the two builds disagreed on, run again one build at a time, timed out on master, all 7; here 6 are answered -- among them the third, fourth and fifth rows above, 1/((d + e x^2)(a + b x^2 + c x^4)^2) and 1/(1 + cosh(x)^5) -- and x^7/(216 + 108 x^2 + 324 x^3 + 18 x^4 + x^6)^2 is declined in 67 ms.

The suite passes on the commit measured, 13eb63d1, 14,572 tests, and so does the allocation gate. Master 97c4b692 is merged in since, with one conflict in BREAKING-CHANGES.md, two entries at one place, both kept, and master dbe5b382 after it without conflicts; the calculus tests pass on that merge, 3,951 of them.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

[Happypig375]

In case your comment polling loop failed, tell it to read #1649 (comment)

[Rafael-SOWNet]

Read, and answered on #1649. You've since opened #1716 and AngouriMathSite#40 for the revert, so I've dropped my own; new links use asc-community.