$\require{begingroup} \begingroup$ $\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$
Consider definite integral \begin{align} I_n&= \int_0^1 \frac{\W_{n}(-t/\e)\ln(-\W_{n}(-t/\e))^2} {t\,(1+\W_{n}(-t/\e)^2)(1+\W_{n}(-t/\e))} \, dt \tag{1}\label{1} , \end{align}
where $n=0,-1$ and $\Wp,\Wm$ are the two real branches of the Lambert W function (also known as ProductLog).
Surprisingly, for $n=-1,0$
\begin{align} (-1)^{n+1}I_n= I_{-1}=-I_{0} &= \frac{\pi^3}{16} \approx 1.9378922925 \tag{2}\label{2} . \end{align}
Question: Are there any other known definite integrals in terms of $\Wp,\Wm$ with the same property?
Somewhat related is Integrals invariant to the choice of the real branch of the Lambert W..., but there the sign is the same.
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