Interestingly, Mathematica does evaluate the integral successfully, but unless you add the constraint that alpha is real, it outputs a truly hideous looking result.

This expression:

    Integrate[Log[1 - 2 \[Alpha] Cos[x] + \[Alpha]^2], 
      {x, 0, \[Pi]}, 
      Assumptions -> Abs[\[Alpha]] >= 1 
         && \[Alpha] \[Element] Reals]

Required 33 seconds of CPU time @ 4 GHz and outputs the short and neat expression in the article. Slow, but successful.

This is a symptom of Mathematica's symbolic engine being written with the "everything is a complex number" assumption deeply ingrained in it. Simplifying assumptions are treated as an afterthought. Conversely, it's not able to generalise to number spaces such as quaternions, matrices, or noncommutative algebras in general.

It's a pity the Mathematica engine has not had a revamp, it would be great if they did a major update to bring its capabilities in line with modern group theory and category theory.

Interesting. Did you try MatLab?

AKA Leibniz’s rule: https://en.wikipedia.org/wiki/Leibniz_integral_rule

Is there a link that doesn’t require signing in with Facebook, etc?

Use a private window

Thanks!

If I recall he used power of two trapazoids to bound above and below? Paywalled article.

Medium -> incognito