A formula for a quartic integral: a survey of old proofs and some new ones

TL;DR

This paper surveys various proofs of a specific quartic integral's value, introduces a new proof based on rational Landen transformations, and compares different methods for evaluating such integrals.

Contribution

It provides a comprehensive survey of old proofs and introduces a novel proof technique using rational Landen transformations.

Findings

Multiple proofs of the quartic integral are analyzed.

A new proof method based on rational Landen transformations is presented.

The paper enhances understanding of integral evaluation techniques.

Abstract

We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations.