Show that ∫π/20log2sinxlog2cosxcosxsinxdx=14(2ζ(5)−ζ(2)ζ(3))\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 :

I can only do non squared one. Anyone has a clue?

Related problems: (I), (II), (III), (IV), (V), (6). Use the change of variables $\ln(\cos(x))=t$ to transform the integral to

Follow it by another change of variables $1-e^{2t}=z$ gives

Getting the exact result: Integral (1) can be evaluated as

Other forms for the solution 1: Using integration by parts with $u=\ln^2(1-z)$, integral $(1)$ can be written as

You can use the identity $H_{n-1}=\psi(n)+\gamma$, where $H_n$ are the harmonic numbers, to write the result as

Other forms for the solution 2: We can have the following form for the solution

Note 1: we used the power series expansion of the function $\frac{\ln(1-z)}{1-z},$

Note 2: Try to tackle integral $(1)$ using the technique used in solving your previous question.