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

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

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

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