Compute $\int_{x=-\infty} ^\infty \frac{\tanh (x)}{\prod_{k=1}^N (x-(x_k+i))}dx$

Question

I am trying to compute the following complex integral $$ \int_{-\infty}^{\infty} \frac{\tanh\left(x\right)} {\prod_{k = 1}^{N}\, \left[\,{x - \left(\,{x_{k} + \mathrm{i}}\,\right)}\right]}\,\mathrm{d}x $$ where $\forall k, x_{k}\in \mathbb{R}$, $\mathrm{i}$ imaginary unit and $N \geq 2$.

Any suggestions are welcome.