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(x^(1 + n)*((2 + n)*ArcCos[a*x] + a*x*Hypergeometric2F1[1/2, 1 + n/2, 2 + n/2, a^2*x^2]))/(2 + 3*n + n^2)


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



           n
Integrate[x  ArcCos[a x], x] == 

  1 + n
(x      ((2 + n) ArcCos[a x] + 
 
                            1      n      n   2  2
      a x Hypergeometric2F1[-, 1 + -, 2 + -, a  x ]))\
                            2      2      2
 
                 2
   / (2 + 3 n + n )

Time to compute: 0.07 second

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