# Integration of elementary functions

## Abstract

We extend a recent algorithm of Trager to a decision procedure for the indefinite integration of elementary functions. We can express the integral as an elementary function or prove that it is not elementary. We show that if the problem of integration in finite terms is solvable on a given elementary function field k, then it is solvable in any algebraic extension of k(δ), where δ is a logarithm or exponential of an element of k. Our proof considers an element of such an extension field to be an algebraic function of one variable over k. In his algorithm for the integration of algebraic functions, Trager describes a Hermitetypereduction to reduce the problem to an integrand with only simple finite poles on the associated Riemann surface. We generalize that technique to curves over liouvillian ground fields, and use it to simplify our integrands. Once the multiple finite poles have been removed, we use the Puiseux expansions of the integrand at infinity and n generalization of the residues to compute the integral. We also generalize a result of Rothstein that gives us a necessary condition for elementary integrability, and provide examples of its use. © 1990, Academic Press Limited. All rights reserved.