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


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



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

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

Time to compute: 0.11 second

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