Is there 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 to evaluate integrals of the form below $$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ The elementary function must be continuous in the interval of integration. The constant, and limits of integration should not be engineered to contain $W$; e.g. writing $1$ as $W(k)/W(k)$, and expressing the constant as $W(ke^k)$ for some elementary $k$ is not accepted.

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

• 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 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 (cross-posted from MSE):

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