Sorry for the confusing title but I'm having a problem and I can phrase the question in multiple different ways.
I was calculating with WolframAlpha
$$\int \text{atanh}(\cos(x))\mathrm{d}x= i \text{Li}_2(-e^{i x}) - i \text{Li}_2(e^{i x}) + x (\ln(1 - e^{i x}) - \ln(1 + e^{i x}) + \text{atanh}(\cos(x))) + C$$
In order to calculate the integral I consider only the real part of this result. As for the final part, it's not a problem.
I wanted to know if it was possible to express the first part with the polylogarithm in terms of known real functions.
I originally thought about Clausen functions because they have a very similar shape.
In fact, the sum can be expressed as:
$$\frac{\text{Li}_2(e^{ix})-\text{Li}_2(-e^{ix})}{2i}=i\cdot \text{Ti}_2(- ie^{ix})$$
but still takes a complex number as input
However, trying to calculate by hand using the definition of polylogarithm, I arrived at the following formula:
$$\Re\left[\frac{\text{Li}_2(e^{ix})-\text{Li}_2(-e^{ix})}{2i}\right]=\sum_{k=1}^{\infty}\frac{\sin((2k-1)x)}{(2k-1)^2}$$
I tried to have Wolfram evaluate this series too but the result always depends on complex numbers.
I also tried to see if by doing the series for $x=\dfrac{\pi}{2}$ a more interesting result was obtained and the Catalan constant actually comes out.
$$\Re\left[\frac{\text{Li}_2(e^{ix})-\text{Li}_2(-e^{ix})}{2i}\right]\approx C+\frac{x'}{2}-\frac{x'^2}{4}-\frac{x'^4}{48}-\frac{x'^6}{288}-\frac{61x'^8}{80640}-\frac{277x'^{10}}{1451520}-...$$
Where $x':=x-\dfrac{\pi}{2}$
In this case I tried searching on OEIS if the denominators were a known sequence but nothing.\
So my question is this
Is it possible to express that series in terms of known real functions?