I'm trying to solve the integral
$$\int_a^b\int_a^b\frac{dxdy}{1+\left(x^2+y^2\right)^\alpha}$$
where the constant $\alpha$ is real-valued and in the range $\alpha\in[1/2,\infty)$. The bounds $a$ and $b$ lie along the real axis but we cannot make any arguments regarding symmetry about the origin.
My first idea was to do the integral over $x$ first, for instance, and use a contour integral using the poles
$$x_\pm=\pm\left[\left(-1\right)^\frac{1}{\alpha}-\beta\right]^\frac{1}{2}$$
Here I've taken $\beta\equiv y^2$ for convenience and we can say $\beta$ is real-valued and lies in the range $\beta\in[0,\infty)$.
My first question is this: How would one draw these poles in the complex plane? And how many are there?
Second, what would the contour look like? My initial thought was a keyhole contour, but the interval is not semi-infinite (unfortunately) - and my second thought was perhaps a dumbbell contour. It appears there might be branch cuts depending on the value of $\alpha$...
Does anyone dare take on this ugly problem? Thanks in advance for any suggestions.