In this paper, we compute a family of infinite integrals involving hyperbolic and trigonometric functions, which we will call Berndt-type integrals. By contour integration, these integrals are first converted to some hyperbolic (infinite) sums, all of which can be calculated in closed forms by two different methods. The first method is indirect and we need to apply some previous results which are computational in nature, and therefore it does not provide general formulas and is inexplicit. For the second method, by comparing the Fourier series expansion and Maclaurin series expansion of one of the Jacobi elliptic functions we are able to directly express the same type of hyperbolic sums more clearly and concisely. When a parameter in these series is set to π, these sums can be expressed as rational polynomials of Γ4(1/4) and 1/π which give rise to the closed formulas of the Berndt-type integrals we are interested in. Moreover, we obtain a structural theorem on the original Berndt integral.