I need to solve $\phi (x,y) = \frac{2V}{\pi} \int_{0}^{\infty} \frac{\sin(kx)\cosh(ky) dk}{k\cosh(ka)}$ I start with a integral in complex plane $$\oint_c \frac{e^{izx} e^{zy} dz}{z\cosh(za)}$$ where $c$ is a countour starting in $z = -R$ along the real axis and jumping the pole at origin and continuing to $z = R$ and closing above the real axis with a semicircle circulating all the poles. I got $$\int_{0}^{\infty} \frac{e^{ky}\sin(kx)}{k\cosh(ky)}dk = \frac{\pi}{2} + 2\pi\sum_{n=1}^{\infty} (-1)^{n+1} \exp\left\{\frac{-xn\pi}{2a}\right\} \frac{\cos{\frac{yn\pi}{2a}}}{n}$$ and now i have no idea.

Assuming your summation is correct we have
$$-\sum_{n=1}^\infty (-1)^ne^{-\frac{\pi n x}{2a}}\frac{e^{\frac{\pi n y}{2a}i}+e^{-\frac{\pi n y}{2a}i}}{2n}=-\frac{1}{2}\sum_{n=1}^\infty (-1)^n\frac{e^{-\frac{\pi n x}{2a}+\frac{\pi n y}{2a}i}}{n}-\frac{1}{2}\sum_{n=1}^\infty (-1)^n\frac{e^{-\frac{\pi n x}{2a}-\frac{\pi n y}{2a}i}}{n}.$$
This can be rewritten
$$-\frac{1}{2}\sum_{n=1}^\infty (-1)^n\frac{\left(e^{-\frac{\pi x}{2a}+\frac{\pi y}{2a}i}\right)^n}{n}-\frac{1}{2}\sum_{n=1}^\infty (-1)^n\frac{\left(e^{-\frac{\pi x}{2a}-\frac{\pi y}{2a}i}\right)^n}{n}.$$
Using the series $-\log(1+z)=\sum_{n=1}^\infty (-1)^n\frac{z^n}{n}$, $|z|<1$, we obtain
$$\frac{1}{2}\log\left(1+e^{-\frac{\pi x}{2a}+\frac{\pi y}{2a}i}\right)+\frac{1}{2}\log\left(1+e^{-\frac{\pi x}{2a}-\frac{\pi y}{2a}i}\right),$$ for $\left|e^{-\frac{\pi x}{2a}\pm\frac{\pi y}{2a}i}\right|=e^{-\frac{\pi x}{2a}}<1\implies \frac{\pi x}{2a}>0$.
Converting to trigonometric functions,
$$\frac{1}{2}\log 2+\frac{1}{2}\log\left(\cosh \left(\frac{\pi  x}{2 a}\right)-\sinh \left(\frac{\pi  x}{2 a}\right)\right)+\frac{1}{2}\log\left( \cosh \left(\frac{\pi  x}{2 a}\right)+\cos \left(\frac{\pi  y}{2 a}\right)\right).$$
Hence, your sum is
$$\frac{\pi}{2}+\pi\log 2+\pi\log\left(\cosh \left(\frac{\pi  x}{2 a}\right)-\sinh \left(\frac{\pi  x}{2 a}\right)\right)+\pi\log\left( \cosh \left(\frac{\pi  x}{2 a}\right)+\cos \left(\frac{\pi  y}{2 a}\right)\right).$$
Simplifying gives
$$\frac{\pi}{2}+\pi\log 2-\frac{\pi^2 x}{2a}+\log\left(\cos\left(\frac{\pi x}{2a}\right)+\cosh\left(\frac{\pi x}{2a}\right)\right).$$
