The infinite power tower or infinite tetration ${\displaystyle z^{z^{z^{\cdot ^{\cdot ^{\cdot }}}}}\!}$ is generally expressed in terms of the Lambert W function in the following form:
${\displaystyle {\frac {W(-\ln(z))}{-\ln(z)}}}$
Considering the integral
$\displaystyle \int -\frac{W(-\ln (z))}{\ln (z)} \, dz,$
Does this integral have a closed form solution?
Is there a series expansion for this integral?
