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

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

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

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