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-(ArcSin[a*x]/x) + a*Log[a*x] - a*Log[2*(1 + Sqrt[1 - a^2*x^2])]

Integrate[ArcSin[a*x]/x^2, x] ==
-(ArcSin[a*x]/x) + a*Log[a*x] - a*Log[2*(1 + Sqrt[1 - a^2*x^2])] 

          ArcSin[a x]
Integrate[-----------, x] == 
               2
              x

  ArcSin[a x]
-(-----------) + a Log[a x] - 
       x
 
                         2  2
  a Log[2 (1 + Sqrt[1 - a  x ])]
Time to compute: 0.02 second
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