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Let $$\text{I(a,b)}=\int _0^1\frac{\ln(1+x^a)}{1+x^b}$$ where $a=\text{irrational},b=\text{rational}$. Can this integral have a closed form ie definite value at some special $a's,b's$?

Archis Welankar
  • 15,910
  • I can't answer in general, but a fairly famous problem from an old Putnam exam asks you to evaluate $I(1,2)$. See https://math.stackexchange.com/questions/1875684/how-to-evaluate-the-following-integral-int-01-frac-log1x1x2dx/1875735#1875735 – User8128 Oct 04 '17 at 16:50
  • There are formulas using dilogarithms, which I would not consider a closed form solution. – MrYouMath Oct 04 '17 at 17:03
  • See Herglotz integrals, https://math.stackexchange.com/questions/522913/are-there-other-cases-similar-to-herglotzs-integral-int-01-frac-ln-left1t and there, https://math.stackexchange.com/questions/426325/evaluate-int-01-frac-log-left-1x2-sqrt3-right1x-mathrm-dx?rq=1 – FDP Oct 11 '17 at 09:08

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