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Integrate[1/(2 + x^6), x] ==
(4*ArcTan[x/2^(1/6)] - 2*ArcTan[Sqrt[3] - 2^(5/6)*x] + 2*ArcTan[Sqrt[3] + 2^(5/6)*x] - Sqrt[3]*Log[2 - 2^(5/6)*Sqrt[3]*x + 2^(2/3)*x^2] + Sqrt[3]*Log[2 + 2^(5/6)*Sqrt[3]*x + 2^(2/3)*x^2])/ (12*2^(5/6))

            1
Integrate[------, x] == 
               6
          2 + x

           x                          5/6
(4 ArcTan[----] - 2 ArcTan[Sqrt[3] - 2    x] + 
           1/6
          2
 
                        5/6
    2 ArcTan[Sqrt[3] + 2    x] - 
 
                     5/6              2/3  2
    Sqrt[3] Log[2 - 2    Sqrt[3] x + 2    x ] + 
 
                     5/6              2/3  2
    Sqrt[3] Log[2 + 2    Sqrt[3] x + 2    x ]) / 
 
       5/6
  (12 2   )