#int((x(tanx)^-1)dx)/(1+x^2)^(3/2)#
Put #x=tanz#
#dx=sec^2z#
#int (tanz.z.(sec^2z))/((sec^2z)^(3/2)).dz#
#int(tanz.z.dz)/secz.dz#
#int(sinz/cosz).cosz.z.dz#
#intsinz.z.dz#
Now I'm gonna use integration by parts
#intu.dv = u.v -intdu.v#
#u = z dz #
#du = 1#
#dv=sinz#
#v=-cosz#
Now put values in formula
#intz.sinz.dz=z(-cosz)-int(-cosz)#
#intz.sinz.dz=z(-cosz)+intcosz#
#intz.sinz.dz=z(-cosz)+sinz +C#.......(1)
As we know #cosz=1/secz#
#secz=sqrt(1+tan^2z)#
And is equal to #tanz =x# above.
then, #secz=sqrt(1+x^2)#
#cosz=1/sqrt(1+x^2)#
#sinz=sqrt(1-cos^2z)#
#sinz=sqrt(1-1/(1+x^2))#
#sinz=sqrt(x^2/(1+x^2))#
#sinz = x/sqrt(1+x^2)#
Now put value of z, cosz, sinz in eq 1
Here is Final answer#int((x(tanx)^-1)dx)/(1+x^2)^(3/2)=tan^-1x(-1/sqrt(1+x^2))+x/sqrt(1+x^2) +C#