I want to compute the following integral:
$\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$
with $\alpha, \beta, c$ real constants, and $\alpha>0,\beta=0$. Mathematica was unable to give a solution.
Attempt: $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt = \frac{1}{2i} \int_{-\infty}^{\infty} \frac{e^{-\alpha t} (e^{i(t+y)} - e^{-i(t+y)})}{1+\beta e^{-2\alpha t} } dt = \frac{e^{iy}}{2i} \int_{-\infty}^{\infty} \frac{(e^{it} - e^{-it})}{e^{\alpha t} +\beta e^{-\alpha t} } dt = \frac{e^{iy}}{2i} \int_{-\infty}^{\infty} \frac{e^{it}}{e^{\alpha t} +\beta e^{-\alpha t} } dt - \frac{e^{iy}}{2i} \int_{-\infty}^{\infty} \frac{e^{-it}}{e^{\alpha t} -\beta e^{-\alpha t} } dt$