Area under the infinite tetration curve

Question

What is the area under the curve where the infinite power tower converges?

$$\lim_{y \to \infty} = {}^y x.$$

The formula for this curve is given by various sources as:

$$\frac{\mathrm{W}(-\ln x)}{-\ln x}.$$

And the limits are from $\mathrm{e}^{-\mathrm{e}}$ to $\mathrm{e}^{\frac1{\mathrm{e}}}$. So we have:

$$\int_{\mathrm{e}^{-\mathrm{e}}}^{\mathrm{e}^{\frac1{\mathrm{e}}}} \frac{\mathrm{W}(-\ln x)}{-\ln x} \,\mathrm{d}x.$$

Numerically this value is approximately $1.244131300633398$.

Is there an exact value for this integral known?

Tetration wikipedia page

Lambert's W Function wikipedia page

Answer 1

From Micah:

Since we know that

$$y=x^y\implies x=y^{1/y}$$

the inverse of the function is given by $x^{1/x}$, hence we have

$$\int_{\exp(-e)}^{\exp(1/e)}x^{x^{x^{\dots}}}~dx=\exp\left(1+\frac1e\right)-\exp(-1-e)-\int_{1/e}^ex^{1/x}~dx$$

Note that there is no known closed form for the integral on the right, even though it is in a much more manageable form. Numerically, the integral is given by WolframAlpha as $\approx2.658607452339$, so

$$\int_{\exp(-e)}^{\exp(1/e)}x^{x^{x^{\dots}}}~dx\approx1.2441313006$$