∫arctanxdx/[x^2(1+x^2)]
=∫arctanxdx/x^2 -∫arctanxdx/(1+x^2)
=∫arctanxd(-1/x)-∫arctanxdarctanx
=-(arctanx)/x +∫(1/x)darctanx-(arctanx)^2/2
=-(arctanx)/x-(arctanx)^2/2+∫dx/[x(1+x^2)]
其中 ∫dx/[x(1+x^2)]=∫[(1+x^2)-x^2]dx/[x(1+x^2)]=∫dx/x-∫xdx/(1+x^2)=lnx-(1/2)ln(1+x^2)+C
原式=-(arctanx)/x-(arctanx)^2/2+lnx-(1/2)ln(1+x^2)+C