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

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

Integrate[ArcSec[a x], x] == 

x ArcSec[a x] - 
 
              1                                2  2
   Sqrt[1 - -----] x Log[2 a (a x + Sqrt[-1 + a  x ])]
             2  2
            a  x
   ---------------------------------------------------
                               2  2
                    Sqrt[-1 + a  x ]
Time to compute: 0.18 second
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