Calculation of a Coxeter integral

Questions
1

Mathematica does not formally calculate the following Coxeter integral:

Integrate[ArcCos[Cos[w]/(1 + 2 Cos[w])], {w, 0, Pi/3}]

Using some results from the article: “Some properties on generalized Ahmed´ integral, Sangchul Lee, December 1,2013” said Coxeter integral can be expressed as an Ahmed`s integral and in turn this integral can be expressed in terms of the LerchPhi function and integrals of the form

Integrate[ArcTan[t]/ ( t^2+ a)), {t, 0, t1}]

MATHEMATICA calculates these integrals but the result is complex. My problem and my question is if the

Open in Wolfram Cloud ↗

can be simplified further “result” until reaching the real value (2 [Pi]^2)/15.
The details are in the notebook I attached.
Thank you

Coxeter_integral.nb (30.4 KB)

2

I will say:

Mathematica version 14 is not a magic box that’ll spit out a solution to any
problem.

All computer algebra systems, including Mathematica, are limited in
their capabilities.

Regards M.I.

3

Not much, but it recognizes that it is a real number:

Im[resultado - (2 \[Pi]^2)/15] // FullSimplify
4

This doesn’t prove this is the solution, but it does show that the solution is probable according to Mathematica.
Here is how I approach those integrals:

Open in Wolfram Cloud ↗