In this paper, we focus on the calculation of a specific type of Berndt integral, which exclusively involves (hyperbolic) cosine functions. Initially, this integral is transformed into a Ramanujan-type hyperbolic (infinite) sum via contour integration. Subsequently, a function incorporating \(\theta \) is defined. By employing the residue theorem, the mixed Ramanujan-type hyperbolic (infinite) sum with both hyperbolic cosine and hyperbolic sine in the denominator is converted into a simpler Ramanujan-type hyperbolic (infinite) sum, which contains only hyperbolic cosine or hyperbolic sine in the denominator. The simpler Ramanujan-type hyperbolic (infinite) sum is then evaluated using Jacobi elliptic functions, Fourier series expansions, and Maclaurin series expansions. Ultimately, the result is expressed as a rational polynomial of \(\Gamma (1/4)\) and \(\pi ^{-1/2}\) . Additionally, the integral is related to the Barnes multiple zeta function, which provides an alternative method for its calculation.
