I recently came across the following claim on this Wikipedia article:
I assumed that this is due to a small change in one of the coefficients of the polynomial yielding a large change in the roots, but that cannot be the case here, as only the constant term changes. The number of real roots (two) also does not change. Is there a way to understand why this occurs? Is it something about the residues?
For a more open-ended follow-up question, are there other examples of this, that is, other functions with elementary antiderivatives that become non-elementary after some "small" change?