Marginalizing out one of the variable of my Bayesian model I arrived at an integral $I(a,b,c)$ given by:
$$I(a,b,c) = \int_{-\infty}^{\infty} \frac{e^{ax-be^x}}{c^2+x^2} dx, \;\;a \in \Bbb{N}, \;\; (b, c) \in \Bbb{R^+}.$$
I wonder if there is a way to get an analytic solution but it seems that contour integration fail for this integral. Ultimately I'm interesting in the ratio $\frac{I(a,b,c_1)}{I(a,b,c_2)}$ but it doesn't seems to make the problem easier.