Does this integral converge $\int_{\mathbb{R}^2}\frac{1}{x^4y^4+1}\ dxdy $?

Question

I want to find out whether this integral is convergent or not. $$\int_{\mathbb{R}^2}\frac{1}{x^4y^4+1}\ dxdy $$

I've tried to calculate it using the following variable changement, but it does'nt work i guess.$(x,y)=(r\cdot \cos(\theta),r\cdot \sin(\theta))$.

I also though of comparing the general term to another one that converge but i couldn't find.

Answer 1

This answer is wrong, as the integral diverges.

Since I missed the case of $|x|>1$ and $|y|<1$ and vice versa.

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Hint:

$$\int\frac1{x^4y^4+1}\ dx\ dy<\int\frac1{x^4y^4}\ dx\ dy$$

and use it to show convergence for when $|x|>1$ and $|y|>1$.

For $|\cdot|\le1$, show that it is finite.