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(2*Sqrt[3]*ArcTan[(-c + 2*x)/(Sqrt[3]*c)] + 2*Log[c + x] - Log[c^2 - c*x + x^2])/(6*c^2)

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

             1
Integrate[-------, x] == 
           3    3
          c  + x

                  -c + 2 x
(2 Sqrt[3] ArcTan[---------] + 2 Log[c + x] - 
                  Sqrt[3] c
 
         2          2         2
    Log[c  - c x + x ]) / (6 c )
Time to compute: 0.02 second
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