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In this paper, we evaluate in closed forms two families of infinite integralscontaining hyperbolic and trigonometric functions in their integrands. We callthem Berndt-type integrals since he initiated the study of similar integrals.We first establish explicit evaluations of four classes of hyperbolic sums byspecial values of the Gamma function, by two completely different approaches,which extend those sums considered by Ramanujan and Zucker previously. Wediscover the first by refining two results of Ramanujan concerning some$q$-series. For the second we compare both the Fourier series expansions andthe Maclaurin series expansions of a few Jacobi elliptic functions. Next, bycontour integrations we convert two families of Berndt-type integrals to theabove hyperbolic sums, all of which can be evaluated in closed forms. We thendiscover explicit formulas for one of the two families. Throughout the paper wepresent many examples which enable us to formulate a conjectural explicitformula for the other family of the Berndt-type integrals at the end.

Berndt-Type Integrals and Series Associated with Ramanujan and Jacobi Elliptic Functions