Another big list with integrals involving Lambert's function Omega,$\pi$,$e$ and others constants... Hi I would like to create another big list always with Lambert's function , but this time with the constant Omega and pi (don't mind if there is others constants as $e$) .I have :
$$\int_{1}^{e} \frac{\operatorname{W(x)}^π}{x} dx = \frac{(-\operatorname{W(1)}^π (π \operatorname{W(1)} + 1 + π) + 1 + 2 π)}{(π (1 + π))}≈0.483124$$
Another one is :
$$\int_{1}^{e} \frac{\operatorname{W(x)}^π}{x^{\pi}} dx = \frac{(\operatorname{W(1)}^{(π - 1)} ((π - 1) \operatorname{W(1)} + π) + e^{(1 - π)} (1 - 2 π))}{(π - 1)^2}≈0.146618$$
As you can see the result must be sober (relatively) and elegant .
A last but not the least :
$$\int_{}^{} \frac{\operatorname{W(x)}^{π^n}}{x^{\pi^n}} dx =  -\frac{(x^{(1 - π^n)} \operatorname{W(x)}^{(π^n - 1)} ((π^n - 1) \operatorname{W(x)} + π^n))}{(π^n - 1)^2} + \operatorname{constant}$$
Where $n\geq 1$ is a natural number .
So if you want help me to achieve this it would be cool .
Thanks a lot .
 A: $\require{begingroup} \begingroup$
$\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$
This pair is pretty nice:
\begin{align} 
\int_0^1 t^{-\Omega}\,(-\Wp(-t\,\Omega^{\frac1\Omega}))^\Omega \,\rm dt
&=
\frac{1-\Omega^{2-\frac 1\Omega}}{(1-\Omega)^2}
\approx 0.67067
,\\
\int_0^1 t^{\Omega}\,(-\Wm(-t\,\Omega^{\frac1\Omega}))^{-\Omega} \,\rm dt
&=
\frac 1{(1+\Omega)^2}
\approx 0.407176
.
\end{align}
\begin{align} 
\int_0^1 t^{-1/\Omega}\,(-\Wp(-t\,\Omega^{\frac1\Omega}))^{1/\Omega} \,\rm dt
&=
\frac{\Omega^2-\Omega^{(\Omega^2-\Omega+1)/\Omega^2}}
{(\Omega-1)^2}
\approx 0.305665869
,\\
\int_0^1 t^{1/\Omega}\,(-\Wm(-t\,\Omega^{\frac1\Omega}))^{-1/\Omega} \,\rm dt
&=
\frac{\Omega^2}{(\Omega+1)^2}
\approx 0.1309689
\end{align}
$\endgroup$
A: $\require{begingroup} \begingroup$
$\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}\def\Catalan{\mathsf{Catalan}}$
One more example, this time with the Catalan's constant:
\begin{align}
\int_0^{\Wp(\tfrac1\e)}
\frac{\sqrt{-\Wm(-t\exp(t))}-\sqrt{-\Wp(-t\exp(t))}}{2\sqrt{t}}
\, dt
&=\Catalan
\approx 0.9159655941772190
.
\end{align}
Or, in other somewhat elegant form,
\begin{align}
\int_0^1
\frac{\sqrt{-\Wm(-t/\e)}-
\sqrt{-\Wp(-t/\e)}}{
2t\left( 
\frac{\sqrt{\Wp(t/\e)}}1+\frac 1{\sqrt{\Wp(t/\e)}} 
\right)
}
\, dt
&=\Catalan
\end{align}
$\endgroup$
