Few weeks ago I tried to get creative integrals for function compositions of the Lambert $W$ function with elementary functions $W(f(x))$, by using the known as Frullani's theorem, also exploiting formulas that I know from the Wikipedia Lambert $W$ function. In addition I've combined identities (which are natural in the theory of this function), changes of variables, and as a last strategy summation over integers to get the examples below, where $W(x)$ is the principal branch of the Lambert $W$. In this Wikipedia are added the literature justifying the identities that I've used, see please also the corresponding article from the encyclopedia Wolfram MathWorld.
The Wikipedia's page for Frullani integral is this Frullani integral, I know several articles from the literature with generalizations of Frullani's theorem.
Example 1. Denoting $W(1)=\Omega$ the omega constant, one has the definite integral $$\int_0^\infty \left(W(\Omega x)-W\left(\frac{\Omega x^{\Omega}}{W(x)^{\Omega -1}}\right)\right)\frac{dx}{x^2}=\Omega^2.$$
Example 2. Denoting the Möbius function as $\mu(n)$, and $W(1)=\Omega$, then for $\Re s>1$ $$\sum_{n=2}^\infty\frac{\mu(n)}{n^s}\int_0^\infty\frac{W(e^{-x})-W(e^{-nx})}{x}dx=-\Omega\,\frac{\zeta'(s)}{(\zeta(s))^2}$$ where $\zeta(s)$ is the Riemann zeta function (I've created similar expressions than this invoking the dominated convergence theorem).
Example 3. Other example that I found nice, if I'm right, is $$\int_0^\infty\left(\int_0^\infty\frac{W(e^{-x})-W(e^{-xF(z)})}{x}dx\right)dz=\Omega,$$ where $W(1)=\Omega$ and $F(z)$ denotes the cumulative distribution function of the standard Gumbel distribution.
Question. I would like to know what can be other examples of nice integrals (or identities build on those) arising from the evaluation of certain integrals that involve, in particular, a branch of the Lambert $W$ function, and that arise when we invoke the Frullani integral (again to emphasize for functions, in the integrand, related to the Lambert $W$ function*). Many thanks
*The only requirement is invoke Frullani for a function related to the Lambert $W$ function.
After there is some answer(s) providing one or several nice examples I should to accept an answer.