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Integrate[ArcSin[a*x]^n, x] ==
((I/2)*ArcSin[a*x]^n* (-((I*ArcSin[a*x])^n*Gamma[1 + n, (-I)*ArcSin[a*x]]) + ((-I)*ArcSin[a*x])^n* Gamma[1 + n, I*ArcSin[a*x]]))/ (a*(ArcSin[a*x]^2)^n)

Integrate[ArcSin[a x] , x] == 

 I            n                   n
(- ArcSin[a x]  (-((I ArcSin[a x])  
         Gamma[1 + n, -I ArcSin[a x]]) + 
      (-I ArcSin[a x])  Gamma[1 + n, I ArcSin[a x]]))\
                    2 n
   / (a (ArcSin[a x] ) )

Time to compute: 0.12 second

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