Problem: We are given that $f:\mathbb{R}^2\to \mathbb{R}$ is a function defined by$$ f(x,y)=\frac{1}{x^4+2x^2y^2+y^4+1}. $$ Solve the improper integral
$$I = \iint_{\mathbb R^2}f(x,y) \,\mathrm{d}x\mathrm{d}y=\int^\infty_{-\infty}\int^\infty_{-\infty}\frac{\mathrm{d}x\mathrm{d}y}{x^4+2x^2y^2+y^4+1}.$$
I am not sure on how I should approach this. Is there anyway to simplify this so it is easier to solve?
Based on the comments, this is what i've done so far:
$$f(z)dz=\int^R_{-R}f(z)dz +\int_{\gamma R}f(z)dz$$
where $$f(z)=\frac{1}{z^4+1}$$