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.

在现代计算机代数系统（CAS）中实现的代数函数积分算法并不总是能够解决代数或基本函数类别中的经典积分问题。描述代数函数积分的最通用方法是找到Abelian积分的标准表示形式，一方面，它不会太麻烦；另一方面，它使我们可以立即回答许多有关积分的问题。对于这种表示，我们建议通过Weierstrass讲座中介绍的三种积分的线性组合来使用Abelian积分的表示。

在本文中，证明了这种表示可用于解决代数函数符号积分的经典问题，即，决定这种给定的积分可以用代数函数还是基本函数表示。在积分可以用基本函数表示的情况下，将获得反导数的显式表达式，否则将积分简化为计算其属性已知的积分。