Definite integral of the logarithm hyperbolic secant function in terms of the Hurwitz zeta function

Abstract

We evaluate definite integrals of the form given by $\int_{0}^{\infty}R(a, x)\log (\cos (\alpha) \text{sech}(x)+1)dx$. The function $R(a, x)$ is a rational function with general complex number parameters. Definite integrals of this form yield closed forms for famous integrals in the books of Bierens de Haan [4] and Gradshteyn and Ryzhik [5].

Keywords

• entries of gradshteyn and ryzhik
• hyperbolic integrals
• bierens de haan
• hurwitz zeta function