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(x^n*((1 + n)*x*ArcCsch[x] + (Sqrt[1 + x^2]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, -x^2])/Sqrt[1 + x^(-2)]))/(1 + n)^2


Integrate[x^n*ArcCsch[x], x] ==
(x^n*((1 + n)*x*ArcCsch[x] + (Sqrt[1 + x^2]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, -x^2])/Sqrt[1 + x^(-2)]))/(1 + n)^2



           n
Integrate[x  ArcCsch[x], x] == 

  n
(x  ((1 + n) x ArcCsch[x] + 
 
                 2                    1  1 + n
      (Sqrt[1 + x ] Hypergeometric2F1[-, -----, 
                                      2    2
 
          3 + n    2               -2             2
          -----, -x ]) / Sqrt[1 + x  ])) / (1 + n)
            2

Time to compute: 0.29 second

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