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Integrate[x^n*ArcCosh[x], x] ==
(x^n*(-((Sqrt[(-1 + x)/(1 + x)]*(1 + x)* (AppellF1[-1 - n, -n, -1/2, -n, (1 - x)^(-1), -2/(-1 + x)]/(1 + n) + (2*n*AppellF1[1 - n, -n, 1/2, 2 - n, (1 - x)^(-1), -2/(-1 + x)] + (-1 + n + x - n*x)*AppellF1[-n, -n, -1/2, 1 - n, (1 - x)^(-1), -2/(-1 + x)])/ ((-1 + n)*n*(-1 + x)^2)))/((x/(-1 + x))^n* Sqrt[(1 + x)/(-1 + x)])) + x*ArcCosh[x]))/ (1 + n)



           n
Integrate[x  ArcCosh[x], x] == 

  n          -1 + x
(x  (-((Sqrt[------] (1 + x) 
             1 + x
 
                                   1         1
           (AppellF1[-1 - n, -n, -(-), -n, -----, 
                                   2       1 - x
 
                 -2
               ------] / (1 + n) + 
               -1 + x
 
                                      1
             (2 n AppellF1[1 - n, -n, -, 2 - n, 
                                      2
 
                    1      -2
                  -----, ------] + 
                  1 - x  -1 + x
 
                (-1 + n + x - n x) 
 
                                    1            1
                 AppellF1[-n, -n, -(-), 1 - n, -----, 
                                    2          1 - x
 
                    -2                           2
                  ------]) / ((-1 + n) n (-1 + x) )))\
                  -1 + x
 
                x    n      1 + x
          / ((------)  Sqrt[------])) + x ArcCosh[x]))
              -1 + x        -1 + x
 
    / (1 + n)

Time to compute: 0.91 second

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