I have tried hard to estimate this definite integral $I:=\int_0^1 x^2 \arctan(e^{-x}) dx,$ but all in vain. This exercise is from Page 204, 4.115, of Giaquinta & Modica's Book Mathematical Analysis, Foundations of One Variable. Actually I have tried using substitution $t=arctan(e^{-x})$, or $s=e^{-x}$, and with the help of integration by parts, but can not get the result. Even by using Maple, what I got is just the following: $1/2\,i \left( -2\,{\it polylog} \left( 4,i \right) +2\,{\it polylog} \left( 4,-i \right) +{\it polylog} \left( 2,{\frac {i}{e}} \right) \left( \ln \left( e \right) \right) ^{2}+2\,{\it polylog} \left( 3, {\frac {i}{e}} \right) \ln \left( e \right) +2\,{\it polylog} \left( 4,{\frac {i}{e}} \right) -{\it polylog} \left( 2,{\frac {-i}{e}} \right) \left( \ln \left( e \right) \right) ^{2}-2\,{\it polylog} \left( 3,{\frac {-i}{e}} \right) \ln \left( e \right) -2\,{\it polylog} \left( 4,{\frac {-i}{e}} \right) \right) \left( \ln \left( e \right) \right) ^{-3}. $
Can anyone help me? Thanks in advance.