Is there any example of a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert W?

Question

I have never seen a real-analytic approach before to evaluate integrals of the form below $$\int_a^b\text{elementary function}(x)\,dx=\text{constant involving}\,W(\cdot)\,\text{in its simplest form}\tag1.$$

For instance, on MSE, all use the residue theorem:

• $\int_{-\infty}^{\infty}{e^x+1\over (e^x-x+1)^2+\pi^2}\mathrm dx=\int_{-\infty}^{\infty}{e^x+1\over (e^x+x+1)^2+\pi^2}\mathrm dx=1$

• Interesting integral related to the Omega Constant/Lambert W Function

• Prove that $\int_0^\infty \frac{1+2\cos x+x\sin x}{1+2x\sin x +x^2}dx=\frac{\pi}{1+\Omega}$ where $\Omega e^\Omega=1$

• Proof for integral representation of Lambert W function

• Evaluate $\int_{0}^{\infty} \ln(1+\frac{2\cos x}{x^2} +\frac{1}{x^4}) \, dx$

And the same applies to some of the wider literature I have come across:

• Stieltjes, Poisson and other integral representations for functions of Lambert W

• An Integral Representation of the Lambert $W$ Function Note: the proof is real-analytic, but the very first line assumes the validity of an integral identity which was only proven using complex analysis (Hankel contour).

So, my question is this:

Does anyone know of a proof of an identity of the form in $(1)$ that involves only real analysis (i.e. does not assume the existence of $\sqrt{-1}$)?