The integral $\int \frac{{\sin }^{2}x.{\cos }^{2}x}{{({\sin }^{3}x+{\cos }^{3}x)}^2}dx$ is equal to

$\begin{array}{rcl} & & \int \frac{{\sin }^2\mathrm{x}.{\cos }^2\mathrm{x}}{{({\sin }^3\mathrm{x}+{\cos }^3\mathrm{x})}^2}\mathrm{dx}\\ & =& \int \frac{{\sin }^2\mathrm{x} {\cos }^2\mathrm{x}}{{\cos }^6{\left({\tan }^3\mathrm{x}+1\right)}^2}\mathrm{dx} \\ & =& \int \frac{{\tan }^2\mathrm{x} {\sec }^2\mathrm{x}}{{({\tan }^3\mathrm{x}+1)}^2}\mathrm{dx} \\ & =& \int \frac{{\mathrm{t}}^2\mathrm{dt}}{{({\mathrm{t}}^3+1)}^2} \mathrm{Put} \tan \mathrm{x}=\mathrm{t} {\sec }^2\mathrm{x} \mathrm{dx}=\mathrm{dt}\\ & =& \frac{1}{3}\int \frac{3{\mathrm{t}}^2\mathrm{dt}}{{({\mathrm{t}}^3+1)}^2}\\ & =& \frac{1}{3}\left(\frac{-1}{{\mathrm{t}}^3+1}\right)+\mathrm{c}\\ & =& \frac{-1}{3({\tan }^3\mathrm{x}+1)}\\ & & \\ & & \end{array}$