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I have tried really hard to compute the following integrals, but got nowhere:

  • Does anyone know a method that could work here ?
  • The integrals are $$ \int_{1}^{x}\frac{\operatorname{Li}_{2}\left(t\right)\log\left(t + 1\right)}{t}\,{\rm d}t\quad \mbox{and}\quad \int_{1}^{x}\frac{\operatorname{Li}_{2}\left(t\right)\log\left(t\right)}{t + 1}\,{\rm d}t $$

I am well aware of Dilogarithm Identities and I have tried several integrations by parts and variable substitutions. I have also checked Lewin's Polylogarithms and associated functions, as well as both books by Cornel Ioan Vălean on impossible integrals, Abramowitz and Stegun... But none of the methods from these seem to work.

Thank you !

Quanto
  • 120,125
DSG
  • 37
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    After one integration by parts, one of the integrals you’d need to solve is this one. The result is extremely cumbersome and unsatisfying. So you gotta ask yourself, how bad do you want to solve this integral? – David H Nov 21 '24 at 19:01
  • Very badly! I have been stuck for weeks :( This integral (or very similar ones) keep appearing in the problem I want to solve, yet none of the books I have consulted have a method that works. Strangely, very similar integrals can be solved by Mathematica even, but I could not find a way to express mine in terms of these. – DSG Nov 21 '24 at 22:58
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    In that case, the question I linked to in my first comment as well as this one will help you through some of the integrals you’ll need. But again, I have my doubts a full antiderivative can be found. – David H Nov 22 '24 at 03:18

0 Answers0