A new approach to evaluating Malmsten's integral and related integrals

TL;DR

This paper introduces new mathematical tools and formulas to generalize and evaluate Malmsten's integral and related integrals, connecting them with Stirling and Bell polynomials.

Contribution

It presents a novel proof for a known integral, introduces signed generalized Stirling polynomials, and establishes their role in generalizing Malmsten's integral for hyperbolic secant powers.

Findings

New expressions for signed generalized Stirling polynomials in terms of Stirling cycle numbers and Bell polynomials

Generalized Malmsten's integral for all natural powers of hyperbolic secant

Derived reduction formulas and identities for related integral sequences

Abstract

This paper discusses generalizations of logarithmic and hyperbolic integrals. A new proof for an integral presented by Vardi and several other integrals in relation to known mathematical constants are discovered. We introduce the signed generalized Stirling polynomials of the first kind from the generalized Stirling polynomials of the first kind, and we give new expressions for the signed generalized Stirling polynomials of the first kind in terms of the Stirling cycle numbers and complete Bell polynomials. We establish the role of the signed generalized Stirling polynomials of the first kind and complete Bell polynomials in generalizing Malmsten's integral for all natural powers of the hyperbolic secant function, and we derive a reduction formula for the integral sequence. We give expressions for new integral sequences, which possess similar properties with Malmsten's integral, in terms of the signed generalized Stirling polynomials of the first kind, and we discover identities and a functional equation for the signed generalized Stirling polynomials of the first kind.