Given an algebraic function $f_0(x)$, when can a primitive of it be constructed by means of algebraic operations and taking primitives of some other given algebraic functions $f_1(x),\dots,f_m(x)$? When the only primitive allowed is the logarithm, this is the question of elementary integrability and a decision procedure has been given by Risch. We will present a general decision procedure for deciding this question, based on a new result of Ax-Schanuel type for primitives of differential forms on curves.

This is joint work with Jonathan Pila.