I know that we can determine whether an integral has closed form, that is, is a composition of elementary functions. That problem is (more or less) solved by Risch algorithm. For differential equation solutions we consider a bit weaker condition, a solution can contain integrals of elementary functions.
Is there an algorithm which decides if solution can be expressed like that?
Yes. see this Wikipedia link (and some more characters to be on the safe side): https://www.wikiwand.com/en/Picard%E2%80%93Vessiot_theory