The integral \(\int \frac{\sin ^2 x \cos ^2 x}{\left(\sin ^5 x+\cos ^3 x \sin ^2 x+\sin ^3 x \cos ^2 x+\cos ^5 x\right)^2} d x\) is equal to (Where \(C\) is a constant of integration).

\(\int \frac{\sin ^2 x \cos ^2 x}{\left(\sin ^5 x+\cos ^3 x \sin ^2 x+\sin ^3 x \cos ^2 x+\cos ^5 x\right)^2} d x \)
\( =\int \frac{\tan ^2 x \cdot \sec ^6 x d x}{\left(\tan ^5 x+\tan ^2 x+\tan ^3 x+1\right)^2} \text { [dividing } N^r\) \(\text { and } D^r \text { by } \cos ^{10} x \text { ] } \)
\( =\int \frac{\tan ^2 x \cdot\left(\tan ^2 x+1\right)^2 \sec ^2 x d x}{\left(\tan ^2 x+1\right)^2\left(\tan ^3 x+1\right)^2} \)
\( =\int \frac{\tan ^2 x \cdot \sec ^2 x d x}{\left(\tan ^3 x+1\right)^2}\)
\(=\frac{1}{3} \int \frac{\mathrm{d} t}{t^2}\)
\(\left\{\text {Let } \tan ^3 x+1=t \text { i.e., } 3 \tan ^2 x \cdot \sec ^2 x \mathrm{~d} x=\mathrm{d} t\right\}\)
\(=-\frac{1}{3 t}+c\)
\(=\frac{-1}{3\left(\tan ^3 x+1\right)}+c\)