For what values of $\alpha \in \mathbb R$ does

$$\int_{(0, +\infty) \times (0, +\infty)} \frac{e^{-y} |\sin (x)|}{(1+xy)^{\alpha}}\: d(x, y) $$


What I've tried:

Using the substitution $(x, y) = (u, \frac{v}{u})$, we get

$$ \int_{(0, +\infty) \times (0, +\infty)} \frac{e^{-\frac{v}{u}} |\sin u|}{u (1+v)^{\alpha} }\: d(u, v). $$

Using the substitution $(x, y) = (\frac{u}{v}, v)$, we get

$$ \int_{(0, +\infty) \times (0, +\infty)} \frac{e^{-v} |\sin \frac{u}{v}|}{v (1+u)^{\alpha} }\: d(u, v). $$

How do I proceed?


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