Is $\int_E \frac{1}{(x^2+y^2)^2}dxdy$ convergent?

If $x<0$ we have that $x^a$ is not necessarily defined, so I am going to assume that the actual problem is to discuss the convergence of $$I(a)=\iint_E \frac{dx\,dy}{(x^2+y^2)^2},\qquad E=\{(x,y):x^2+y^2\leq 1, x> 0, 0<y<x^a\}.$$

With these assumptions we have $$ I(a) = \int_{0}^{1}\frac{L(\rho)}{\rho^4}\,d\rho $$ hence the problem boils down to estimating $L(\rho)$ for $\rho\to 0^+$. If $a\leq 1$ we have $L(\rho)\geq c\rho$ and the integral is clearly divergent. It follows that we may assume that $x^a$ is a convex function on $[0,1]$. This easily leads to $$ L(\rho)\sim \rho^a\quad\text{as }\rho\to 0^+ $$ and to the fact that the integral is convergent for $\color{red}{a>3}$.