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

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

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

               1
x (-Sqrt[1 - -----] + a x ArcSec[a x])
              2  2
             a  x
--------------------------------------
                 2 a
Time to compute: 0.05 second
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