It's a problem that I can sovle only for $n=2$ let me try it :
$$\int_{0}^{e}n^{\operatorname{W^n(x)}}dx=?$$ Where $n\geq 2$ is a natural number and $\operatorname{W(x)}$ denotes the Lambert's function .
The case $n=2$:
$$\int_{}^{}2^{\operatorname{W^2(x)}}dx=\frac{1}{\log(256)}\Big(2^{-1 - \frac{8}{\log^2(16)}}\Big) \Big(e^\frac{1}{\log(16)}\Big) \Big(\sqrt{\frac{π}{\log(2)}} (\log(256) - 4) \operatorname{erfi}\Big(\frac{(\log(4) \operatorname{W(x)} + 1)}{(2 \sqrt{\log(2)})}\Big) + 8 \exp\Big( \operatorname{W(x)} + \log^2(4) \frac{\operatorname{W(x)^2}}{\log(16)} + \frac{1}{\log(16)}\Big)\Big) + \operatorname{constant} $$ Where $\operatorname{erfi}$ denotes the imaginary error function
So in this case we have an antiderivative wich is relatively simple .
Unfortunately I can't find others antiderivatives of this kind for the others values of $n$.
My question
Have you an idea to solve this ?
Thanks a lot and have fun .