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

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

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