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-(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x] - 2*a^2*x^2*ArcCosh[a*x] + Log[2*a*x + 2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]])/(4*a^2)


Integrate[x*ArcCosh[a*x], x] ==
-(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x] - 2*a^2*x^2*ArcCosh[a*x] + Log[2*a*x + 2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]])/ (4*a^2)



Integrate[x ArcCosh[a x], x] == 

-(a x Sqrt[-1 + a x] Sqrt[1 + a x] - 
 
        2  2
     2 a  x  ArcCosh[a x] + 
 
     Log[2 a x + 2 Sqrt[-1 + a x] Sqrt[1 + a x]]) / 
 
      2
  (4 a )

Time to compute: 0.16 second

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