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

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

               1
Integrate[------------, x] == 
              3    3 2
          x (c  - x )

   3
  c                        3    3
------- + 3 Log[x] - Log[-c  + x ]
 3    3
c  - x
----------------------------------
                  6
               3 c
Time to compute: 0.04 second
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