  In this paper, we prove two structural theorems on the general Berndt-type
integrals with the denominator having arbitrary positive degrees by contour
integrations involving hyperbolic and trigonometric functions, and hyperbolic
sums associated with Jacobi elliptic functions. We first establish explicit
relations between these integrals and four classes of hyperbolic sums. Then,
using our previous results on hyperbolic series and applying the matrix method
from linear algebra, we compute explicitly several general hyperbolic sums and
their higher derivatives. These enable us to express two families of general
Berndt-type integrals as polynomials in $\Gamma^4(1/4)$ and $\pi^{-1}$ with
rational coefficients, where $\Gamma$ is the Euler gamma function. At the end
of the paper, we provide some conjectures of general Berndt-type integrals.
