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 .
Best Answer
$\require{begingroup} \begingroup$
$\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}\def\Ei{\operatorname{Ei}}$
\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}
$\endgroup$