We shall recall in short a history of the problem of integration by elementary functions (i.e. "in finite terms"). Then we shall discuss a problem of specialization concerning integration in finite terms along a pencil of algebraic differentials on a curve, an issue posed by James Davenport. We shall briefly discuss joint work with David Masser, giving counterexamples to the question whether a non-elementary generic integral remains non elementary for almost all specializations, and giving also many positive results. We shall present several explicit examples constructed out of  special functions.