Simple, yet evasive integral from zero to $\pi/2$

Change variable to $t = \sin^2 x$ and notice $$\begin{align}\frac{dx}{\sin x\cos x} &= \frac{\sin x\cos x dx}{\sin^2 x\cos^2 x} = \frac12\frac{dt}{t(1-t)}\\ \log\sin x \log\cos x &= \frac14\log\sin^2 x \log \cos^2 x = \frac14 \log t\log(1-t) \end{align} $$ The integral at hand can be rewritten as $$\mathcal{I} \stackrel{def}{=} \int_0^{\pi/2} \frac{(\log\sin x\log\cos x)^2}{\sin x\cos x} dx = \frac{1}{32}\int_0^1 \frac{\log^2 t \log^2(1-t)}{t(1-t)} dt$$ Since $\displaystyle\frac{1}{t(1-t)} = \frac1t + \frac{1}{1-t}$, by replacing $t$ by $1-t$ in part of the integral, we find

$$\begin{align} \mathcal{I} &= \frac{1}{16}\int_0^1 \frac{\log^2 t\log^2(1-t)}{t} dt = \frac{1}{48}\int_0^1 \log^2(1-t) d\log^3 t\\ &\stackrel{\text{I.by.P}}{=} \frac{1}{48}\left\{ \left[\log^2(1-t)\log^3 t\right]_0^1 + 2 \int_0^1 \log^3 t\frac{\log(1-t)}{1-t}dt\right\}\\ &= -\frac{1}{24}\int_0^1 \frac{\log^3 t}{1-t}\sum_{n=1}^\infty\frac{t^n}{n} dt = -\frac{1}{24}\int_0^1 \log^3 t\sum_{n=1}^\infty H_nt^n dt\\ &\stackrel{t = e^{-y}}{=} \frac{1}{24} \int_0^\infty y^3 \sum_{n=1}^\infty H_n e^{-(n+1)y} dy = \frac14 \sum_{n=1}^\infty \frac{H_n}{(n+1)^4} \end{align} $$ where $H_n$ are the $n^{th}$ harmonic number. By rearranging its terms, the last sum should be expressible in terms of zeta functions. I'm lazy, I just ask WA to evaluate the sum. As expected, last sum equals to $2\zeta(5) - \frac{\pi^2}{6}\zeta(3)$${}^\color{blue}{[1]}$.

As a consequence, the integral at hand equals to:

$$\mathcal{I} = \frac{12\zeta(5) - \pi^2\zeta(3)}{24} \approx 0.024137789997360933616411382857235691008...$$

Notes

• $\color{blue}{[1]}$ It turns out we can compute this sum using an identity by Euler. $$2\sum_{n=1}^\infty \frac{H_n}{n^m} = (m+2)\zeta(m+1) - \sum_{n=1}^{m-2}\zeta(m-n)\zeta(n+1),\quad\text{ for } m = 2, 3, \ldots$$ In particular, for $m = 4$, this identity becomes $$\sum_{n=1}^\infty \frac{\zeta(n)}{n^4} = 3\zeta(5) - \zeta(2)\zeta(3)$$ and we can evaluate our sum as $$\begin{align}\mathcal{I} &= \frac14\sum_{n=1}^{\infty}\frac{H_{n}}{(n+1)^4} = \frac14\sum_{n=1}^{\infty}\left(\frac{H_{n+1}}{(n+1)^4}-\frac{1}{(n+1)^5}\right) = \frac14\left(\sum_{n=1}^\infty \frac{H_n}{n^4} - \zeta(5)\right)\\ &= \frac14(2\zeta(5) - \zeta(2)\zeta(3)) = \frac{12\zeta(5) - \pi^2\zeta(3)}{24} \end{align} $$