Abel-Plana formula for specific function

Question

Suppose we have the function $f(n) = \sqrt{n^{2}+a^{2}}$, where $a$ is real. I need to calculate the finite expression $$ \sum_{n = 0}^{\infty}f(n) - \int \limits_{0}^{\infty}f(x)dx $$ By using the Abel-Plana formula, I obtain $$ \sum_{n = 0}^{\infty}f(n) = \int \limits_{0}^{\infty}f(x)dx + \frac{a}{2} + i\int \limits_{0}^{\infty}\frac{f(ix) - f(-ix)}{e^{2\pi x}-1}dx $$ How to show that the last integral is $$ i\int\limits_{0}^{\infty}\frac{f(ix) - f(-ix)}{e^{2\pi x}-1}dx = -2\int\limits_{a}^{\infty} \frac{\sqrt{x^{2}-a^{2}}dx}{e^{2\pi x}-1}? $$ The problem is with the branching point $x = \pm ia$, which leads to appearance of the additional phase for $f(ix)$ for $x>a$, and for $f(ix)$ it is $+i$, while for $f(-ix)$ it is $-\pi$.