I know that in the literature there are a lot of integrals involving the fractional part (and other floor and ceiling functions). For some of these integrals is provide its evaluation as a series or sum of series involving particular values of special functions when isn't possible to get the sum in closed-form.

I've considered the following proposal that I believe that isn't in the literature, I hope that the resulting series are interesting for this site.

**Problem.** We denote the inverse of the Gudermannian function, see this Wikipedia dedicated to this function, as $\operatorname{gd}^{-1}(x)$. Calculate an expression for the evaluation of $$I=\int_0^{\pi/2}\{ \operatorname{gd}^{-1}(x)\}dx,$$
where $\{u\}$ denotes the fractional part function.

Question.Is it possible to get an expression for $I$ as the sum of certain series involving particular values of special functions and presented, if possible, in a simplified expression?Many thanks.

I am asking about an expression of series because I think that isn't feasible to get the closed-form of $I$.

Then if there aren't mistakes in my change of variable $x=\operatorname{gd}(y)$ one has $$I=\sum_{k=0}^\infty\int_k^{k+1}\frac{y-k}{\cosh y}dy.$$ I know the indefinite integral using Wolfram Alpha online calculator

`int (y-k)/cosh(y)dy`

My main problem is to glimpse if will be possible to simplify some of those series, and how to group them in the result.

## References:

Also the encyclopedia MathWorld has an article dedicated to the function, that is the article with title *Inverse Gudermannian*.