A little game around Lambert's function and simple and beautiful integral I like very much to play with the Lambert's function so I was wondering if we can find integrals collecting famous constant and involving just logarithm and Lambert's function .
I have proposed an example  here and this integral :
$$\int_{0}^{e}\operatorname{W(x)}(\ln(x)+1)dx=-\operatorname{Ei(1)}+1+e+\gamma$$
Or this one :
$$\int_{1}^{e}\ln(\operatorname{W(x)})dx=e^{\operatorname{W(1)}}-\ln(\operatorname{W(1)})-e $$
Furthermore I want to find the simplest integrals with this esthetic result.
Any helps are welcome .
Thanks a lot for sharing time , knowledge and patience .
Another for the fun :
$$\int_{0}^{e}\operatorname{W(x)^i}+\operatorname{W(x)^{-i}}dx=-e^π (i e (!(i)) + Γ(i)) + (\sinh(π) - \cosh(π)) (Γ(-i) - i e(!(-i))) + 2 e$$
Where there is the gamma function ,cosine and sine hyperbolic function and subfactorial . this integral gives a real number .
 A: $\require{begingroup} \begingroup$
$\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}\def\Ei{\operatorname{Ei}}$
\begin{align}
\int_0^1\!-\Wm(-x\,\exp(-x))\,dx
&=\tfrac16\pi^2+\tfrac12
\tag{1}\label{1}
,\\
\int_0^1\!-\Wp(-\tfrac1x\,\exp(-\tfrac1x))\,dx
&=1-\gamma
\tag{2}\label{2}
,\\
\int_0^1\!-\left( \Wm (-x\,\exp(-x))  \right)^{-1}\,dx
&=\gamma
\tag{3}\label{3}
,\\
\int_0^1\!-\Wp(-\tfrac1x\,\exp(-\tfrac1x)
-\left( \Wm (-x\,\exp(-x))  \right)^{-1}\,dx
&=1
\tag{4}\label{4}
,\\
\int_0^1\!\frac{\Wp(-\tfrac1x\,\exp(-\tfrac1x))}
{ \Wm (-\tfrac1x\,\exp(-\tfrac1x))}\,dx
&=1-\ln2
\tag{5}\label{5}
.
\end{align}
\begin{align} 
\int_0^1 \Wp(-\tfrac t\e)\,\ln t \, dt
&=
5-\e\,(1+\gamma+\Ei(1,1))
\tag{6}\label{6}
.
\end{align} 
\begin{align} 
\int_0^1 
\left(\Big(-\Wp(-\tfrac t\e)\Big)^{-\tfrac1\e}
-\Big(-\Wm(-\tfrac t\e)\Big)^{-\tfrac1\e}\right)
\, dt
&=
-\tfrac1\e\Gamma(-\tfrac1\e)
\tag{7}\label{7}
,\\
\int_0^1 
\left(\Big(-\Wm(-\tfrac t\e)\Big)^{\tfrac1\e}
-\Big(-\Wp(-\tfrac t\e)\Big)^{\tfrac1\e}\right)
\, dt
&=
\tfrac1\e\Gamma(\tfrac1\e)
\tag{8}\label{8}
.
\end{align} 
\begin{align} 
\int_0^1 
\left(\sqrt{-\Wm(-\tfrac t\e)}-\sqrt{-\Wp(-\tfrac t\e)}\right)
\, dt
&=
\frac{\e\sqrt\pi}4
\tag{9}\label{9}
,\\
\int_0^1 \left(\frac 1{\sqrt{-\Wp(-\tfrac t\e)}}-\frac 1{\sqrt{-\Wm(-\tfrac t\e)}}\right) \,dt
&=
\frac{\e\sqrt\pi}2
\tag{10}\label{10}
.
\end{align}
And this one is also one of
integrals-invariant-to-the-choice-of-the-real-branch-of-the-labmert-W-function:
\begin{align} 
\int_0^1 \left(2\,\sqrt{-\W(-\tfrac t\e)}+\frac 1{\sqrt{-\W(-\tfrac t\e)}} \right)\, dt
&=\int_0^1 \left(2\,\sqrt{-\Wp(-\tfrac t\e)}+\frac 1{\sqrt{-\Wp(-\tfrac t\e)}} \right)\, dt
\\
&=\int_0^1 \left(2\,\sqrt{-\Wm(-\tfrac t\e)}+\frac 1{\sqrt{-\Wm(-\tfrac t\e)}} \right)\, dt
\\
&=4
\tag{11}\label{11}
.
\end{align}
$\endgroup$
