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](https://math.stackexchange.com/questions/45745/interesting-integral-related-to-the-omega-constant-lambert-w-function?noredirect=1&lq=1)

- [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 $\Omega e^\Omega=1$](https://math.stackexchange.com/questions/2331600/prove-that-int-0-infty-frac12-cos-xx-sin-x12x-sin-x-x2dx-frac-pi?noredirect=1&lq=1)

- [Proof for integral representation of Lambert W function](https://math.stackexchange.com/questions/3347447/proof-for-integral-representation-of-lambert-w-function)

- [Evaluate $\int_{0}^{\infty} \ln(1+\frac{2\cos x}{x^2} +\frac{1}{x^4}) \, dx$](https://math.stackexchange.com/questions/4433252/evaluate-int-0-infty-ln1-frac2-cos-xx2-frac1x4-dx)

And the same applies to literature I have come across:

- [Stieltjes, Poisson and other integral representations for functions of Lambert W](https://arxiv.org/pdf/1103.5640.pdf)

- [An Integral Representation of the Lambert $W$ Function](https://arxiv.org/pdf/2012.02480.pdf) **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](https://math.stackexchange.com/questions/4501736)): 

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 $\sqrt{-1}$)?