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

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

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

                                       2  2
  ArcCot[a x]               a Log[1 + a  x ]
-(-----------) - a Log[x] + ----------------
       x                           2
Time to compute: < 0.01 second
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