`int(xtan^-1x)/(1+x^2)^(3/2)dx`

Explanation

Put x=tan t so that dx `=sec^(2)tdt`.
`:.int(xtan^(-1)x)/((1+x^(2))^(3//2))dx=int((tant)t)/((1+tan^(2)t)^(3//2))*sec^(2)tdt`
`int((tant)t)/(sect)dt=inttsintdt`
`=t(-cost)-int1*(-cost)-int1*(1-cost)dt" "` [integrating by parts]
`=-tcost+sint+C=(-tan^(-1)x)/(sqrt(1+x^(2)))+(x)/(sqrt(1+x^(2)))+(x)/(sqrt(1+x^(2)))+C`
`[becausesint=(x)/(sqrt(1+x^(2)))andcost=(1)/(sqrt(1+x^(2)))]`.