<s>[INST] A closed form for $\int_0^1\frac{\ln(-\ln x)\ \operatorname{li}^2x}{x}dx$ Let $\operatorname{li}x$ denote the logarithmic integral $^{[1]}$$^{[2]}$$^{[3]}$:
$$\operatorname{li}x=\int_0^x\frac{dt}{\ln t}$$
and
$$I=\int_0^1\frac{\ln(-\ln x)\ \operatorname{li}^2x}{x}dx\approx-4.311872263...$$
Is it possible to express the integral $I$ in a closed form (using algebraic numbers, known mathematical constants, elementary and known special functions)?
 [/INST] I will start from Vladimir's result and my calculation. Let
$$ F(s) = \int_{0}^{\infty} z^{s-1} \mathrm{Ei}(-z)^{2} \, dz = \frac{2\Gamma(s)}{s} \int_{0}^{\frac{1}{2}} \frac{u^{s-1}}{1 - u} \, du. $$
Then we have
\begin{align*}
\int_{0}^{\infty} z^{s-1} \mathrm{Ei}(-z)^{2} \log z \, dz
&= F'(s) \\
&= F(s) \left( \psi_{0}(s) - \frac{1}{s} \right) + \frac{2\Gamma(s)}{s} \int_{0}^{\frac{1}{2}} \frac{u^{s-1}\log u}{1-u} \, du.
\end{align*}
Plugging $s = 1$ to both sides,
\begin{align*}
I
&= F(1) \left( \psi_{0}(1) - 1 \right) + 2 \int_{0}^{\frac{1}{2}} \frac{\log u}{1-u} \, du \\
&= -2(\gamma+1)\log 2 + 2 \left[ \mathrm{Li}_{2}(1-u) \right]_{0}^{\frac{1}{2}} \\
&= -\zeta(2) - \log^{2} 2 - 2(\gamma + 1) \log 2
\end{align*}
as desired. Here,
\begin{align*}
\mathrm{Li}_{2}(z) = - \int_{0}^{z} \frac{\log(1 - t)}{t} \, dt
\end{align*}
is the dilogarithm and we exploited the special values
\begin{align*}
\mathrm{Li}_{2}(1) &= \zeta(2), \\
\mathrm{Li}_{2}\left(\tfrac{1}{2}\right) &= \frac{1}{2}\zeta(2) - \frac{1}{2}\log^{2} 2.
\end{align*}
</s>