Let
$$I(R)=\iint_{x^2+y^2\leq R^2}\left(\frac{1+2x^2}{1+x^4+6x^2y^2+y^4}-\frac{1+y^2}{2+x^4+y^4}\right)dxdy.$$ Find
$$\lim_{R\to\infty}I(R),$$
or show that this limit does not exist. The original answer is in \frac{\sqrt{a}}{b}\pi\logc format, please provide the value of a+b+c.

Let's think step by step and output the final answer within \boxed{}.
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