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