Official solution
\[\begin{aligned}
& \int_\limits0^{\pi / 4} \frac{\cos ^2 x \cdot \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x \\
& =\int_\limits0^{\pi / 4} \frac{\tan ^2 x \cdot \sec ^2 x}{\left(1+\tan ^3 x\right)^2} d x
\end{aligned}\]
Let \[\tan x=t\]
\[\int_\limits0^1 \frac{t^2 d t}{\left(1+t^3\right)^2}\]
Let \[1+t^3=\mathrm{z}\]
\[\begin{gathered}
3 t^2 d t=d z \\
\frac{1}{3} \int_\limits1^2 \frac{d z}{z^2}=\left.\frac{1}{3}\left(-\frac{1}{z}\right)\right|_1 ^2 \\
=-\frac{1}{3}\left(\frac{1}{2}-1\right)=\frac{1}{6}\end{gathered}\]