Suppose, we have a definite convergent integral (possibly improper) with an elementary function as an integrand.
Is there an algorithm deciding whether the value of the integral has a closed-form expression ?
Such an algorithm would be similar to the Risch-algorithm. The reason, why I have some hope that such an algorithm exists is that some definite integrals can be calculated (for example with the contour-integration-method), although no closed-form antiderivate exists.