Another big list with integrals involving Lambert's function Omega,$\pi$,$e$ and others constants...

Question

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 .

Answer 1

$\require{begingroup} \begingroup$

$\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$

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}

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Answer 2

$\require{begingroup} \begingroup$ $\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}\def\Catalan{\mathsf{Catalan}}$

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}

$\endgroup$