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x/(2*a) + (x^2*ArcCot[a*x])/2 - ArcTan[a*x]/(2*a^2)

Integrate[x*ArcCot[a*x], x] ==
x/(2*a) + (x^2*ArcCot[a*x])/2 - ArcTan[a*x]/(2*a^2) 

Integrate[x ArcCot[a x], x] == 

       2
 x    x  ArcCot[a x]   ArcTan[a x]
--- + -------------- - -----------
2 a         2                2
                          2 a
Time to compute: < 0.01 second
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