Numerical evaluation of a Berndt-type integral of order three

Let $\Gamma=\Gamma(1/4)$. The Berndt-type integral

$$\displaystyle\!\int_{0}^{\infty}\displaystyle\!\frac{x\,dx}{(\cos x+\cosh x)^3}$$

Numerical evaluation. We believe that

$$\displaystyle\!\int_{0}^{\infty}\displaystyle\!\frac{x\,dx}{(\cos x+\cosh x)^3}=-\displaystyle\!\frac{\Gamma^4}{2^7\pi^2}+\displaystyle\!\frac{\Gamma^4}{2^9\pi}+\displaystyle\!\frac{\Gamma^{12}}{2^{13}\pi^7}.$$

This evaluation is supported by numerical computation and belongs to the family of Berndt-type integrals associated with Jacobi elliptic functions. The supplied text does not report a proof or a resolution.