How would we evaluate the definite integral $ \int_0^{e^{\frac{1}{e}}}\frac {-W(-\ln(x))}{\ln(x)}dx $ How would we evaluate the definite integral: 
$$ \int_0^{e^{\frac{1}{e}}}\frac {-W(-\ln(x))}{\ln(x)}dx $$
Here $W$ is the Lambert W function. More information about this function can be found here.
Edit 1: Found a mistake in the limits so corrected it.
Edit 2: Here are some alternate forms that I derived that might help:
$$ \int_{-1/e}^{\infty} e^{(-u-W(u))}du $$
$$ -\int_{-1/e}^{\infty} \frac{W(x)}{Inv(W(x))}du $$
$$ \int_{-1/e}^{\infty} \frac { \sum_{n=0}^{ \infty } \frac {(-1)^{n-1}n^{n-1}x^{n}} {n!}} {x\sum_{n=0}^{ \infty }\frac{x^n}{n!} } dx$$
Edit 3: After a bit of more investigation, I have managed to further simplify the integral:
$$ \int_0^{e^{\frac{1}{e}}}\frac {-W(-\ln(x))}{\ln(x)}dx = -\int_{-1/e}^{\infty} \frac{W(u)}{u(e^u)}du = -\int_{-1/e}^{\infty} \sum_{n=0}^{\infty} \frac {(-1)^nT(n)u^n }{n!}du = \sum_{n=0}^{\infty}\left[ \frac {(-1)^nT(n)u^{(n+1)} }{n!(n+1)} \right]_{-1/e}^{\infty} $$
where 
T(n)
counts the number of forests of rooted labeled trees using labels in a subset of 
{1,⋯,n}
, which is also the hyperbinomial transform of the constant sequence of 
1's. (See A088957.) 
Now the problem is that I don't know how to evaluate this summation due to T(n), and even wolfram alpha cant do it. Any help on evaluating this summation?

Not an answer 
Change variables $y=-W(-\log x)$ to convert it to the integral
$$
\int_{-\infty}^{1}(1-y)\exp(ye^{-y})\;dy
$$
But that is still not elementary of course.
