This week I'm trying to get integrals involving the Gudermannian function and other functions. My attempts are with Wolfram Alpha. I don't know if these integrals are in the literature are similar than $\int\operatorname{gd}(\log(\sin(x)))dx$ or $\int\operatorname{gd}(W(x))dx$, where $W(x)$ denotes the principal branch of Lambert W function. Wolfram Alpha online calculator knows how to compute the corresponding indefinite integrals.
I add below the reference [1] from Wikipedia for the Gudermannian function, that I've denoted in this post as $\operatorname{gd}(x)$.
Question. Explain how to get the indefinite integral for a novel integral (your example isn't in the literature: is your invention) involving the Gudermannian function, and get a definite integral from this (isn't required that this closed-form for the definite integral has mathematical beauty). Many thanks.
By application of trigonometric formulas, one can to consider more elaborated formulas as (if there aren't mistakes in my reasoning for real numbers $x>1$)$$\int\left(2\arctan\left(\frac{e^{W(x)}+W(x)}{1-x}\right)-\pi\right)\operatorname{mod }\pi\, dx.$$
Other available idea in this post is to use convergence theorems for series and limits to get more eleborated integrals than the previous.
Other interesting special function is the density and cumulative distribution of Gumbel distribution (see Wikipedia). Maybe one can to deduce other novel integrals involving the Gudermannian function, and pathological functions, or functions from mathematical physics (for example Hermite polynomials, see the online encyclopedias).
I think that the problem is interesting (the Gudermannian function and the Lambert W function are very interesting) as illustration of how to explore the creation of new integrals and the calculation of some definite integral as application/consequence.
References:
[1] The article Gudermannian function from Wikipedia.