An Alternate proof for a case of a Malmsten integral

TL;DR

This paper provides a novel direct proof for a specific case of a Malmsten integral using a combination of integration techniques, including parts, Laplace transform, and special function representations.

Contribution

It introduces a unique, direct proof method for a Malmsten integral case not previously documented in literature.

Findings

New direct proof method for a Malmsten integral case

Utilizes integration by parts, Laplace transform, and special functions

Results in a simplified derivation of the integral

Abstract

In this paper, a direct proof is presented for a case of a Malmsten integral. The method used in solving the integral is a direct one that the author has not come accross in any old or recent publication. Integration by parts, Laplace transform, an integral representation for the hyperbolic secant function, and the digamma representation for an alternating series are employed to derive the result.