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Integrate[1/(-1 + x^7), x] ==
(-2*ArcTan[Sec[Pi/14]*(x + Sin[Pi/14])]*Cos[Pi/14])/ 7 - (2*ArcTan[Sec[(3*Pi)/14]*(x - Sin[(3*Pi)/14])]* Cos[(3*Pi)/14])/7 + Log[-1 + x]/7 - (Cos[Pi/7]*Log[1 + x^2 + 2*x*Cos[Pi/7]])/7 - (Log[1 + x^2 + 2*x*Sin[Pi/14]]*Sin[Pi/14])/7 - (2*ArcTan[(x + Cos[Pi/7])*Csc[Pi/7]]*Sin[Pi/7])/7 + (Log[1 + x^2 - 2*x*Sin[(3*Pi)/14]]*Sin[(3*Pi)/14])/7



             1
Integrate[-------, x] == 
                7
          -1 + x

              Pi           Pi        Pi
-2 ArcTan[Sec[--] (x + Sin[--])] Cos[--]
              14           14        14
---------------------------------------- - 
                   7
 
               3 Pi           3 Pi        3 Pi
  2 ArcTan[Sec[----] (x - Sin[----])] Cos[----]
                14             14          14
  --------------------------------------------- + 
                        7
 
                    Pi           2           Pi
                Cos[--] Log[1 + x  + 2 x Cos[--]]
  Log[-1 + x]       7                        7
  ----------- - --------------------------------- - 
       7                        7
 
           2           Pi       Pi
  Log[1 + x  + 2 x Sin[--]] Sin[--]
                       14       14
  --------------------------------- - 
                  7
 
                    Pi       Pi       Pi
  2 ArcTan[(x + Cos[--]) Csc[--]] Sin[--]
                    7        7        7
  --------------------------------------- + 
                     7
 
           2           3 Pi       3 Pi
  Log[1 + x  - 2 x Sin[----]] Sin[----]
                        14         14
  -------------------------------------
                    7

Time to compute: < 0.01 second

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