\arctan(x+1)'=\frac{1}{1+(x+1)^2}=\frac{1}{2+2x+x^2}=\frac{2-2x+x^2}{(2+2x+x^2)(2-2x+x^2)}=\frac{2-2x+x^2}{4+x^4}=\frac{2-2x+x^2}{4}\cdot\frac{1}{1+\frac{x^4}{4}}.