An alternative proof of an integral given by Vardi is presented, along with generalizations of logarithmic and hyperbolic integrals and related evaluations. The signed generalized Stirling polynomials of the first kind are introduced as a variant of the generalized Stirling polynomials of the first kind. Expressions for these polynomials are given in terms of the Stirling cycle numbers and complete Bell polynomials. We show how these polynomials arise naturally in a generalization of Malmsten’s integral to all natural powers of the hyperbolic secant function, and we derive a corresponding reduction formula. Furthermore, we present new integral sequences that exhibit properties similar to those of Malmsten’s integral and express them using the signed generalized Stirling polynomials of the first kind. Several identities and a functional equation involving these polynomials are also studied.