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


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



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

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

Time to compute: 0.32 second

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