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



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

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

Time to compute: 0.12 second

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