A symbolic parameter no longer stops a rational integrand being integrated, and 16 more Rubi integrals come out

[Rafael-SOWNet]

1/(8 + x^3) and 1/(16 - x^4) are answered at once. 1/(a^3 + x^3) and 1/(a^4 - x^4) were not.

The parameter is the whole difference. The rational rules read a denominator as a polynomial over
the rationals — TryGetRationalCoefficients is the gate — so a^3 is not a coefficient they can
work with, and the factoring that answers x^3 + 8 has nothing to hold on to for x^3 + a^3.

Measured rather than assumed, on master:

ok   1/(8 + x^3)        ok   1/(16 - x^4)
NO   1/(a^3 + x^3)      NO   1/(a^4 - x^4)

Two things were in the way

A constant factor inside the denominator stayed there. The two branches that take a factor out
of a quotient each want the whole of one side free of the variable, so a * (1/(1 + x^3)) was taken
apart and 1/(a*(1 + x^3)), the same number, was not. Taking the constant out now happens first,
before the branch that turns c/g(x) into c * g(x)^(-1) — a power being a shape the rational
rules do not read, that ordering is what decides these.

And the integrand is scaled by its parameter. With x = c t and dx = c dt a homogeneous
integrand becomes a constant times a function of t alone: 1/(a^3 + x^3) becomes
a^(-2)/(1 + t^3), whose denominator has integer coefficients again. The answer is read at
t = x/c.

Homogeneity is checked, and the check is what makes it terminate

The scaled integrand has to divide into a factor free of t times a function of t, and only the
second is handed on — so the sub-problem has one variable and cannot be scaled again. Without that,
the rule would hand on something still carrying c and scale it once more at every level.

x/((a^2 + x^2)*(b^2 + x^2)) is declined for exactly that reason: scaling by one parameter leaves
the other. It is in the tests as a decline, so the check is exercised rather than assumed.

Measured on Rubi's independent test suites

Same corpus and flags, both arms on this machine:

	solved	wrong	timeouts	wall clock
2dbeedf7 (master)	913	0	13	722 s
+ this	929	0	16	794 s

Sixteen more answers and no wrong ones, for about 10% more wall clock. Newly answered includes the
whole 1/(x^k (a^n ± x^n)) family, which is where this runs long in Timofeev's set. Each was also
checked by differentiating it back with the parameter pinned and comparing at five points.

Where the answer is not defined

c = 0 is not a scaling, and a^(-2) G(x/a) has no value there. That is the honest report for a
substitution that does not exist rather than a wrong answer — and 1/(a^3 + x^3) at a = 0 is a
different function, 1/x^3, which is answered on its own if asked that way. Recorded in
BREAKING-CHANGES.md rather than left to be discovered.

Ordering

It runs last of the rewrites, because unlike the others it fires on an integrand nothing is
wrong with — it clears a parameter rather than a shape. a*sin(x), a*x^2, a/x, sin(a*x),
x*e^(a*x) and 1/(1 + x^3) are all in the tests to hold that: they were answered before and are
answered by the same rules now.

Suites

UnitTests 8514 and 1557, FSharpWrapperUnitTests 134, InteractiveWrapperUnitTests 18,
TerminalUnitTests 41. New file ScaledVariableIntegralTest.cs, 28 cases.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

[Rafael-SOWNet]

Merge order against the other two open ones. This does not conflict with #1245 at all. It conflicts with #1248 in two places — the end of IndefiniteIntegralSolver.cs and Sources/.editorconfig — and both are additive: keep both blocks, in either order.

Composed on a throwaway branch to check they do more than merge, since #1248 changes how partial fractions reads a quotient and this changes the order in which a factor comes out of one:

UnitTests (Calculus)     1570 passed
FSharpWrapperUnitTests    134 passed

Whichever you take second I will rebase.