[misan]

[JRav]

1/72 \[ImaginaryI] (48 \[ImaginaryI] Catalan +
3 \[Pi]^2 + \[ImaginaryI] \[Pi] (8 ArcCsch[Sqrt[2]] + Log[512] -
6 (Log[(1 - 2 \[ImaginaryI]) + \[ImaginaryI] Sqrt[3]] -
Log[2 + Sqrt[3]] +
Log[-\[ImaginaryI] ((2 + \[ImaginaryI]) + Sqrt[3])])) -
12 (2 ArcCsch[Sqrt[
2]] (Log[(1 - 2 \[ImaginaryI]) + \[ImaginaryI] Sqrt[3]] -
Log[-1 + Sqrt[3]] + Log[1 + Sqrt[3]] -
Log[-\[ImaginaryI] ((2 + \[ImaginaryI]) + Sqrt[3])]) -
PolyLog[2, 2 - Sqrt[3]] +
PolyLog[2, -\[ImaginaryI] (-2 + Sqrt[3])] -
PolyLog[2, -(\[ImaginaryI]/(2 + Sqrt[3]))] +
PolyLog[2, 1/(2 + Sqrt[3])]))

Whatever this is.

[shawtoos]

Hey JRav . . . whenever you want to post Mathematica code, you need to first convert it to Raw Input Form: Select the cell, then choose Cell/Convert To/Raw Input form. For the record, I realize that's not what Misan is asking for; he wants the manual calculations -- just trying to help JRav guys:

Code:

In[154]:=
Integrate[ArcTan[x]/(x*(1 + x^3)), {x, 0, 1}]

Out[154]=
(-(1/72))*I*(48*I*Catalan + 3*Pi^2 +
   I*Pi*(8*ArcCsch[Sqrt[2]] + Log[512] -
     6*(Log[(1 - 2*I) + I*Sqrt[3]] -
       Log[2 + Sqrt[3]] +
       Log[(-I)*((2 + I) + Sqrt[3])])) -
   12*(2*ArcCsch[Sqrt[2]]*
      (Log[(1 - 2*I) + I*Sqrt[3]] -
       Log[-1 + Sqrt[3]] +
       Log[1 + Sqrt[3]] -
       Log[(-I)*((2 + I) + Sqrt[3])]) -
     PolyLog[2, 2 - Sqrt[3]] +
     PolyLog[2, (-I)*(-2 + Sqrt[3])] -
     PolyLog[2, -(I/(2 + Sqrt[3]))] +
     PolyLog[2, 1/(2 + Sqrt[3])]))