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

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

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

       2         x     2         x     2         x
      x  ArcTanh[-]   a  Log[1 - -]   a  Log[1 + -]
a x              a               a               a
--- + ------------- + ------------- - -------------
 2          2               4               4
Time to compute: < 0.01 second
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