Take a repeated rational root out of a denominator all at once

[Rafael-SOWNet]

TrySplitOffRationalRoot divided one factor of (x - r) out of the denominator and gave up if the quotient still vanished there — the code said so plainly:

var atRoot = Evaluate(q, root);
if (atRoot.IsZero)
    continue;                       // repeated root, see above

So a denominator whose only rational root is a repeated one was never split at all. 1/(x^4 + x^2) had no antiderivative, though x^4 + x^2 is x^2(x^2 + 1) and the decomposition is just 1/x^2 - 1/(x^2 + 1).

The fix

A root of multiplicity m contributes a term over the m-th power rather than the first. So divide the root out as many times as it goes, and use the power that came out. What was already there is this with m fixed at 1 — one degree still comes off the denominator each step, so it terminates for exactly the reason the single-root case did.

	before	after
1 / (x^4 + x^2)	no antiderivative	solved
1 / (x^3 + x^2)	no antiderivative	solved
1 / (x^2 * (x + 1))	no antiderivative	solved
1 / ((x - 1)^2 * (x + 2))	no antiderivative	solved
1 / ((x + 1)^3 * (x - 2))	no antiderivative	solved
x / ((x - 1)^2 * (x^2 + 1))	no antiderivative	solved
(x + 1) / (x^3 - x^2)	no antiderivative	solved

Every one is checked by differentiating the answer back and comparing at points, so a wrong antiderivative cannot pass as a solved one. That is the existing AssertIsAntiderivative helper in the file, reused rather than reinvented.

How it was found

Not by reading the code. Running the casbench corpus against master left x^2/(x^4 + 1) unsolved, and while probing what was and was not reachable around it, 1/(x^4 + x^2) turned up — a denominator that factors over Q and still had no answer. The corpus had no repeated-root case at all, which is why this went unnoticed; four have been added.

The boundary, asserted rather than implied

Denominators with no rational root stay out of reach: x^4 + 1 is irreducible over Q and only factors once real coefficients are allowed, which needs partial fractions with algebraic coefficients. There is now a test asserting those are declined, so the edge is visible in the suite instead of inferred from an absence.

Worth recording: the pieces past that edge already work. (x + 1)/(x^2 - sqrt(2)*x + 1) and 1/(x^2 - sqrt(2)*x + 1) both integrate in 2 ms, so for x^4 + 1 the only missing step is the factorisation itself.

Verification

• 6118 C# tests pass, 0 failed — 9 new
• 130 F# wrapper tests pass
• propcheck: 1340 property checks, 0 failures — unchanged
• casbench: 117/119 of the problems with an elementary answer, up from 113/115 with the four new cases

🤖 Generated with Claude Code