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Integrate[1/(1 + x^8), x] ==
(ArcTan[Sec[Pi/8]*(x - Sin[Pi/8])]*Cos[Pi/8])/4 + (ArcTan[Sec[Pi/8]*(x + Sin[Pi/8])]*Cos[Pi/8])/4 - (Cos[Pi/8]*Log[1 + x^2 - 2*x*Cos[Pi/8]])/8 + (Cos[Pi/8]*Log[1 + x^2 + 2*x*Cos[Pi/8]])/8 + (ArcTan[(x - Cos[Pi/8])*Csc[Pi/8]]*Sin[Pi/8])/4 + (ArcTan[(x + Cos[Pi/8])*Csc[Pi/8]]*Sin[Pi/8])/4 - (Log[1 + x^2 - 2*x*Sin[Pi/8]]*Sin[Pi/8])/8 + (Log[1 + x^2 + 2*x*Sin[Pi/8]]*Sin[Pi/8])/8



            1
Integrate[------, x] == 
               8
          1 + x

           Pi           Pi        Pi
ArcTan[Sec[--] (x - Sin[--])] Cos[--]
           8            8         8
------------------------------------- + 
                  4
 
             Pi           Pi        Pi
  ArcTan[Sec[--] (x + Sin[--])] Cos[--]
             8            8         8
  ------------------------------------- - 
                    4
 
      Pi           2           Pi
  Cos[--] Log[1 + x  - 2 x Cos[--]]
      8                        8
  --------------------------------- + 
                  8
 
      Pi           2           Pi
  Cos[--] Log[1 + x  + 2 x Cos[--]]
      8                        8
  --------------------------------- + 
                  8
 
                  Pi       Pi       Pi
  ArcTan[(x - Cos[--]) Csc[--]] Sin[--]
                  8        8        8
  ------------------------------------- + 
                    4
 
                  Pi       Pi       Pi
  ArcTan[(x + Cos[--]) Csc[--]] Sin[--]
                  8        8        8
  ------------------------------------- - 
                    4
 
           2           Pi       Pi
  Log[1 + x  - 2 x Sin[--]] Sin[--]
                       8        8
  --------------------------------- + 
                  8
 
           2           Pi       Pi
  Log[1 + x  + 2 x Sin[--]] Sin[--]
                       8        8
  ---------------------------------
                  8

Time to compute: 0.02 second

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