On symbolic integration of algebraic functions

Algorithms for integration of the algebraic functions implemented in modern computer algebra systems (CAS) are not always able to solve the classical problems of integration in the class of algebraic or elementary functions. The most general approach to describing the integral of an algebraic function is to find a standard representation for Abelian integrals, which, on the one hand, would not be too cumbersome, and on the other hand, would allow us to immediately answer a number of questions about the integral. For such a representation, we propose to use the representation of the Abelian integral by a linear combination of integrals of three kinds presented in the Lectures of Weierstrass. In this paper, it is proved that this representation can be used to solve the classical problems of symbolic integration of algebraic functions, that is, to decide whether such a given integral can be expressed in terms of algebraic or elementary functions. In cases when the integral can be expressed in elementary functions, an explicit expression for the antiderivative is obtained, otherwise the integration is reduced to the calculation the integrals whose properties are known. © 2020 Elsevier Ltd