Evaluate in close form $\sum_{{n}=\mathbf{1}}^{\infty} \frac{\cos (\boldsymbol{\eta} \boldsymbol{n})}{\boldsymbol{n}^{2}+\boldsymbol{b}^{2}}$

Question

Let Parameters $\boldsymbol{b} \in \mathbb{R} \backslash\{\boldsymbol{0}\}$ and $\boldsymbol{\eta} \in[\mathbf{0}, \boldsymbol{\pi}] .$ I want to evaluate in close form using hyperbolic functions the sum $$\sum_{\boldsymbol{n}=\mathbf{1}}^{\infty} \frac{\cos (\boldsymbol{\eta} \boldsymbol{n})}{\boldsymbol{n}^{2}+\boldsymbol{b}^{2}}$$ It seems clear that the way to go is to use Poisson Summation Formula (PSF) which I tried. The sum $\sum_{\boldsymbol{n}=\mathbf{1}}^{\infty} \frac{1}{\boldsymbol{n}^{2}+\boldsymbol{b}^{2}}$ is classic and I know how to derive it using PSF. I tried to convolve and derive the CTFT of the term under the sum but with no clear result.

Any suggestions on how to obtain it??