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

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 from the generalized Stirling polynomials, and we give new expressions for the signed generalized Stirling polynomials in terms of the Stirling cycle numbers and complete Bell polynomials. We establish the role of the signed generalized Stirling polynomials 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, and we discover identities and a functional equation for the signed generalized Stirling polynomials. Several finite sums involving the signed generalized Stirling polynomials and Hurwitz zeta function are also expressed in terms of known mathematical constants.Comment: Keywords: gamma function, polygamma function, Riemann zeta function, Hurwitz zeta function, Stirling numbers, Stirling polynomials, Bell polynomials, Catalan's constan