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Integrate[x^(3/7)*ArcTanh[x], x] ==
(7*(14*x^(3/7) + 6*x^(10/7)*ArcTanh[x] + 6*ArcTan[Cot[Pi/7] - x^(1/7)*Csc[Pi/7]]* Cos[Pi/14] - 6*ArcTan[Cot[Pi/7] + x^(1/7)*Csc[Pi/7]]*Cos[Pi/14] + 6*ArcTan[x^(1/7)*Sec[Pi/14] - Tan[Pi/14]]* Cos[(3*Pi)/14] + 6*ArcTan[x^(1/7)*Sec[Pi/14] + Tan[Pi/14]]* Cos[(3*Pi)/14] + 3*Log[-1 + x^(1/7)] - 3*Log[1 + x^(1/7)] - 3*Cos[Pi/7]* Log[1 + x^(2/7) - 2*x^(1/7)*Sin[(3*Pi)/14]] + 3*Cos[Pi/7]*Log[1 + x^(2/7) + 2*x^(1/7)* Sin[(3*Pi)/14]] + 3*Log[1 + x^(2/7) - 2*x^(1/7)*Cos[Pi/7]]* Sin[Pi/14] - 3*Log[1 + x^(2/7) + 2*x^(1/7)*Cos[Pi/7]]* Sin[Pi/14] - 6*ArcTan[Sec[(3*Pi)/14]* (x^(1/7) - Sin[(3*Pi)/14])]*Sin[Pi/7] - 6*ArcTan[Sec[(3*Pi)/14]*(x^(1/7) + Sin[(3*Pi)/14])]*Sin[Pi/7] - 3*Log[1 + x^(2/7) - 2*x^(1/7)*Sin[Pi/14]]* Sin[(3*Pi)/14] + 3*Log[1 + x^(2/7) + 2*x^(1/7)*Sin[Pi/14]]* Sin[(3*Pi)/14]))/60



           3/7
Integrate[x    ArcTanh[x], x] == 

        3/7      10/7
(7 (14 x    + 6 x     ArcTanh[x] + 
 
                   Pi     1/7     Pi       Pi
      6 ArcTan[Cot[--] - x    Csc[--]] Cos[--] - 
                   7              7        14
 
                   Pi     1/7     Pi       Pi
      6 ArcTan[Cot[--] + x    Csc[--]] Cos[--] + 
                   7              7        14
 
                1/7     Pi        Pi       3 Pi
      6 ArcTan[x    Sec[--] - Tan[--]] Cos[----] + 
                        14        14        14
 
                1/7     Pi        Pi       3 Pi
      6 ArcTan[x    Sec[--] + Tan[--]] Cos[----] + 
                        14        14        14
 
                  1/7               1/7
      3 Log[-1 + x   ] - 3 Log[1 + x   ] - 
 
            Pi           2/7      1/7     3 Pi
      3 Cos[--] Log[1 + x    - 2 x    Sin[----]] + 
            7                              14
 
            Pi           2/7      1/7     3 Pi
      3 Cos[--] Log[1 + x    + 2 x    Sin[----]] + 
            7                              14
 
                 2/7      1/7     Pi       Pi
      3 Log[1 + x    - 2 x    Cos[--]] Sin[--] - 
                                  7        14
 
                 2/7      1/7     Pi       Pi
      3 Log[1 + x    + 2 x    Cos[--]] Sin[--] - 
                                  7        14
 
                   3 Pi    1/7       3 Pi
      6 ArcTan[Sec[----] (x    - Sin[----])] 
                    14                14
 
           Pi                 3 Pi
       Sin[--] - 6 ArcTan[Sec[----] 
           7                   14
 
           1/7       3 Pi        Pi
         (x    + Sin[----])] Sin[--] - 
                      14         7
 
                 2/7      1/7     Pi       3 Pi
      3 Log[1 + x    - 2 x    Sin[--]] Sin[----] + 
                                  14        14
 
                 2/7      1/7     Pi       3 Pi
      3 Log[1 + x    + 2 x    Sin[--]] Sin[----])) / 
                                  14        14
 
  60

Time to compute: 0.58 second

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