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


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



           n
Integrate[x  ArcSinh[x], x] == 

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

Time to compute: 0.10 second

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