$\int \frac{\left({\sin }^{2}x{\cos }^{2}x\right)dx}{{({\sin }^{5}x+{\cos }^{3}x{\sin }^{2}x+{\sin }^{3}x{\cos }^{2}x+{\cos }^{5}x)}^2}$

$I=\int \frac{{\sin }^{2}x{\cos }^{2}xdx}{{({\sin }^{5}x+{\cos }^{3}x{\sin }^{2}x+{\sin }^{3}x{\cos }^{2}x+{\cos }^{5}x)}^2}$

$=\int \frac{{\sin }^{2}x.{\cos }^{2}xdx}{{\left[({\sin }^{2}x+{\cos }^{2}x)({\sin }^3+{\cos }^{3}x)\right]}^2}$

$=\int \frac{{\sin }^{2}x.{\cos }^{2}xdx}{{({\sin }^{3}x+{\cos }^{3}x)}^2}=\int \frac{{\tan }^{2}x.{\sec }^{2}x}{{(1+{\tan }^{3}x)}^2}dx$

Put $1+{\tan }^{3}xdx=t\Rightarrow 3{\tan }^{2}x{\sec }^{2}xdx=dt$
$$\begin{gathered} \frac{1}{3}\int \frac{dt}{t^2} \\ =-\frac{1}{3t}+c \\ =-\frac{1}{3(1+{\tan }^{3}x)}+c \end{gathered}$$