We extend a recent algorithm of Tragcr 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 kt then it is solvable in any algebraic extension of k(O)t where O 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 Je.