For what values of $\alpha$ does $\int_{(0, +\infty) \times (0, +\infty)} \frac{e^{-y} |\sin (x)|}{(1+xy)^{\alpha}}\: d(x, y)$ converge?

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

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

converge?

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?