  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.
