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

$I=\int \frac{{\sin }^{2}x{\cos }^{2}x}{{({\sin }^{3}x+{\cos }^{3}x)}^2}dx$

$$\begin{gathered} =\int \frac{{\sin }^{2}x{\cos }^{2}x}{{\left({\cos }^{3}x\right)}^2\left[{\displaystyle \frac{{\sin }^{3}x}{{\cos }^{3}x}}+1\right]}dx \\ =\int \frac{{\sin }^{2}x}{Co{s}^{4}x}\frac{dx}{{\left[{\tan }^{3}x+1\right]}^2} \\ Put \tan x=t \\ se{c}^{2}x dx=dt \\ =\int \frac{t^2 dt}{{\left(t^3+1\right)}^2} \\ =\frac{1}{3}\int \frac{dy}{y^2} \\ =\frac{1}{3}\left(\frac{y^{-1}}{-1}\right)+C \\ =\frac{1}{3}\left(\frac{-1}{t^3+1}\right)+C \\ =\left(\frac{-1}{3}\right)\frac{1}{{\tan }^{3}x+1}+C \end{gathered}$$