\frac{y(x^4+4x^2y^2-y^4)}{(x^2+y^2)^2}=y\left(\left(\frac{x^2}{x^2+y^2}\right)^2+\left(\frac{2xy}{x^2+y^2}\right)^2-\left(\frac{y^2}{x^2+y^2}\right)^2\right)