# Closed Form for  ∫10arctanh xtan(π2 x) dx~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx

Does possess a closed form expression ?

This recent post, in conjunction with my age-old interest in Gudermannian functions, have inspired me to ask this question. The reason I suspect that such a closed form might possibly exist is because the integration interval is “meaningful” for both functions used in the integrand. However, none of the various approaches that I can think of seem to be of any help. Perhaps I’m missing something ?

Here is an approach.

We give a preliminary result.

## A series of squares of logarithms

Let us consider the poly-Hurwitz zeta function initially defined by the series

The function $\displaystyle \zeta(\cdot,\cdot \mid a,b)$ extends to a meromorphic function on $\mathbb{C}^2$ with only singularities on the set $\displaystyle \left\{(s,t) \in \mathbb{C}^2, \,\Re (s+t)=1\right\}$. It clearly generalizes the classic Hurwitz zeta function initially defined by the series

We have the following new result.

Theorem. Let $a, b$ be complex numbers such that $\Re a>-1$ and $\Re b>-1$.

Then

where $\log (z)$ denotes the principal value of the logarithm defined for all $z \neq 0$ by

$\displaystyle \zeta(\cdot,a)$ and $\displaystyle \zeta(\cdot,\cdot \mid a,b)$ denoting the Hurwitz zeta function and the poly-Hurwitz zeta function respectively and where

Proof.
On the one hand, one has
using Theorem $1$ here.

On the other hand, one has

using Theorem $2$ here.

Observing that
where $\zeta(\cdot)$ is the Riemann zeta function, then
and both sides of $(3)$ vanish at $a=b=0$.

Thus $(3)$ holds true. $\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \Box$

## Lucian's integral

We prove that Lucian's integral is related to the preceding family of logarithmic series.

Proposition 1. We have

Proof. Let us proceed on Jack D'Aurizio's route which starts by using the standard expansion
then integrating termwise using

to get

We are left with two non trivial series to evaluate.

We prove that each series may be evaluated using the poly-Stieltjes constants.

One may write
using Theorem $2$ here.

To evaluate the last series on the right hand side of $(7)$, one may check with some algebra that, for any complex number $z$ satisfying $|z|<1$, the following identity holds true:

Then

Inserting $(10)$ and $(8)$ into $(7)$ gives the announced result $(4)$. $\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \Box$

We deduce the following closed form.

Proposition 2. We have

Proof. One may observe that
and recalling that $\zeta'(0)=-\frac12 \ln (2 \pi)$, one may obtain

From $(4)$ and $(3)$, we have
by appealing to $(14)$ and $(15)$, we get $(11)$. $\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \Box$

By combining Proposition $2$ and pisco125's derivation we obtain the following new closed forms.

Proposition 3. We have

where $\text{Ci} (\cdot)$ is the cosine integral and where $\displaystyle \zeta(\cdot,\cdot\mid a,b)$ is the poly-Hurwitz zeta function.