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

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

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

  ArcCos[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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