Mathematica does not have a complete implementation of the Risch algorithm. It actually cannot do the integral in the mathoverflow question.

Macaulay2 has a focus on commutative algebra and algebraic geometry and doesn't have an implementation of the Risch algorithm either.

It's written in the mathoverflow question, did you read it?

> I have access to Maple 2018, and it doesn't seem to have a complete implementation either. A useful test case is the following integral, taken from the (apparently unpublished) paper Trager's algorithm for the integration of algebraic functions revisited by Daniel Schultz: ∫29x2+18x−3x6+4x5+6x4−12x3+33x2−16x−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√dx Schultz explicitly provides an elementary antiderivative in his paper, but Maple 2018 returns the integral unevaluated.

The question at hand is "how do you verify that an implementation is complete?", not "what is one test case whose failure would demonstrate that an implementation is incomplete?"

Yes, I read it.

What you responded does not address what i asked.

For example, you could have an implementation that handles the example you cited, but fails at others you did not mention. So would your comment claim "complete" for handling what you mentioned, when it is not?