Integral of a tetration?

$\begin{aligned} \int_e^{^3e} \frac 1{\ln (^2x)} dx & = \int_e^{e^{e^e}} \frac 1{\ln (x^x)} dx \\ & = \int_e^{e^{e^e}} \frac 1{x\ln x} dx & \small \blue{\text{Let }x = e^u \implies dx = e^u\ du} \\ & = \int_1^{e^e} \frac 1u du & \small \blue{\implies u = \ln x \implies \ln e = 1, \ \ln e^{e^e} = e^e} \\ & = \ln u \ \bigg|_1^{e^e} \\ & = \ln e^e - \ln 1 \\ & = e \approx \boxed{2.72} \end{aligned}$

$\int_{e}^{^{3}e} \frac{1}{\ln(^{2}x)}dx = \ln|\ln(x)|\, \bigg|^{^{3}e}_{e} = \ln|\ln(e^{e^e})| - \ln|\ln(e)| = \ln|e^e| - \ln|1| = \boxed{e}$