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x*ArcCsc[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[ArcCsc[a*x], x] ==
x*ArcCsc[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[ArcCsc[a x], x] == 

x ArcCsc[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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