Integration in Finite Terms with Special Functions: The Logarithmic Integral

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Abstract
Since R. Risch published an algorithm for calculating symbolic integrals of elementary functions in 1969 (Traps. Amer. Math. Soc., 139 (1969), pp. 167–189), there has been an interest in extending his methods to include nonelementary functions. In this paper, we use the framework of differential algebra to make precise the notion of integration in terms of elementary functions and logarithmic integrals. Basing our work on a recent extension of Liouville’s theorem on integration in finite terms, we then describe a decision procedure for determining if a given element in a transcendental elementary field has an integral which can be written in terms of elementary functions and logarithmic integrals. This algorithm first examines the structure of the integrand in order to limit the logarithmic integrals which could appear in the integral to a finite number. This allows us to write a general expression for the integral and then use techniques similar to those employed by Risch to calculate the undetermined parts.

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