Let's have equation $$ \cosh(2 \pi x) = \cos\left[\text{Re}\int \limits_{0}^{2 \pi}\sqrt{A - 2q\cos(2z)}dz \right]\times \cosh\left[ \text{Im}\int \limits_{0}^{2 \pi}\sqrt{A - 2q\cos(2z)}dz\right], $$ and $A, q$ - parameters. This equation, for example, has simple solution if $A = -2q$: $x = \frac{1}{4 \pi}\sqrt{q}$.
I need to find its solution for large positive $x$ in case $0 < A < 2q >> 1$. For large $x, q$ the equation is simplified to the form $$ e^{2 \pi x} = \cosh\left[ \sqrt{2q}\text{Re} \int \limits_{0}^{2 \pi}\sqrt{\cos(2z) - \frac{A}{2q}}dz\right]\times \cos\left[ \sqrt{2q}\text{Im} \int \limits_{0}^{2 \pi}\sqrt{\cos(2z) - \frac{A}{2q}}dz\right] $$ Is it possible to get approximate solution of this equation (note that I look for large positive x)? I only need terms up to $\frac{A}{\sqrt{2q}}$.