Evaluation of an improper integral with complex exponential

Question

Are there any convenient ways to calculate an integral of the form $$ \int_{-\infty}^\infty\frac{a_1 e^{j\omega\alpha}+a_2e^{j\omega\beta}}{1 + a_1a_2e^{j\omega\gamma}}d\omega$$ where $a_1,a_2,\alpha$, $\beta$ and $\gamma$ are real constants and $j=\sqrt{-1}$? The solution probably contains Dirac delta functions since the integral can be viewed as an inverse Fourier transform of a periodic function. Constants do not matter much, we can use specific numbers, say $a_1=2, a_2=3, \alpha=3, \beta=4$ and $\gamma=5$; the good method is what is of general interest..