<s>[INST] Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$ 
Show that :
  $$
\int_{0}^{\Large\frac\pi2}
{\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right)
\ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right)
\over
\cos\left(x\right)\sin\left(x\right)}\,{\rm d}x
={1 \over 4}\,
\bigg[2\,\zeta\left(5\right) - \zeta\left(2\right)\zeta\left(3\right) \bigg]
$$

I can only do non squared one. Anyone has a clue?
 [/INST] Let  $x=\sin ^2 \theta$, then
$$
\begin{aligned}
\int_0^1 \frac{\ln ^2(1-x) \ln ^2 x}{x} d x&=\int_0^1 \frac{\ln ^2\left(\cos ^2 \theta\right) \ln ^2\left(\sin ^2 \theta\right)}{\sin ^2 \theta} 2 \sin \theta d \theta\\
&=32 \int_0^{\frac{\pi}{2}} \frac{\ln ^2(\cos \theta) \ln ^2(\sin \theta)}{\tan \theta} d \theta\\
&=\left.32 \cdot \frac{1}{2} \frac{\partial^4}{\partial a^2 \partial b^2} B\left(\frac{a}{2}, \frac{b}{2}+1\right)\right| _{{a \rightarrow 0}, {b \rightarrow 0}}\\
&=\lim _{a \rightarrow 0} \lim _{b \rightarrow 1}\left[\frac{\partial^{4}}{\partial a^2\partial b^2} B(a, b)\right]
\end{aligned}
$$
</s>