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(-(a*x*Sqrt[1 - a^2*x^2]) + 2*a^2*x^2*ArcCos[a*x] + ArcSin[a*x])/(4*a^2)


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



Integrate[x ArcCos[a x], x] == 

                 2  2        2  2
(-(a x Sqrt[1 - a  x ]) + 2 a  x  ArcCos[a x] + 
 
                       2
    ArcSin[a x]) / (4 a )

Time to compute: 0.04 second

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