Integration in Finite Terms: Dilogarithmic Integrals

TL;DR

This paper extends Liouville's theorem to include dilogarithmic integrals, providing criteria for antiderivatives in extended fields and exploring algebraic independence of these integrals.

Contribution

It introduces a necessary and sufficient condition for antiderivatives involving dilogarithmic integrals within elementary function extensions.

Findings

Extended Liouville's theorem to dilogarithmic integrals

Established criteria for antiderivatives in extended fields

Analyzed algebraic independence of dilogarithmic integrals

Abstract

We extend the theorem of Liouville on integration in finite terms to include dilogarithmic integrals. The results provide a necessary and sufficient condition for an element of the base field to have an antiderivative in a field extension generated by transcendental elementary functions and dilogarithmic integrals. We also study algebraic independence of certain dilogarithmic integrals.