How to find the limit of this integral over the disk when its radius tends to infinity? – mathematica.stackexchange.com

I came across this problem in Facebook. What is the limit of the definite integral $\int\limits_{x^2+y^2\le R^2}\frac{2 x^2+1}{x^4+6 x^2 y^2+y^4+1}-\frac{y^2+1}{x^4+y^4+2} \,dx dy$ as $R\to\infty$? ...