  We address a class of definite integrals known as Berndt-type integrals,
highlighting their role as specialized instances within the integral
representation framework of the Barnes-zeta function. Building upon the
foundational insights of Xu and Zhao, who adeptly evaluate these integrals
using rational linear combinations of Lambert-type series and derive
closed-form expressions involving products of $\Gamma^4(1/4)$ and $\pi^{-1}$,
we uncover direct evaluations of the Barnes-zeta function. Moreover, our
inquiry leads us to establish connections between Berndt-type integrals and
Jacobi elliptic functions, as well as moment polynomials investigated by Lomont
and Brillhart, a relationship elucidated through the seminal contributions of
Kuznetsov. In this manner, we extend and integrate these diverse mathematical
threads, unveiling deeper insights into the intrinsic connections and broader
implications of Berndt-type integrals in special function and integration
theory.
