Answer
$$\frac{1}{4}{\arctan ^2}\left( {\frac{x}{2}} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{\arctan \left( {x/2} \right)}}{{4 + {x^2}}}} dx \cr
& = \int {\arctan \left( {\frac{x}{2}} \right)\left( {\frac{1}{{4 + {x^2}}}} \right)} dx \cr
& {\text{Integrate by substitution}} \cr
& u = \arctan \left( {\frac{x}{2}} \right),{\text{ }}du = \frac{1}{{1 + {{\left( {x/2} \right)}^2}}}\left( {\frac{1}{2}} \right)dx \cr
& du = \frac{4}{{4 + {x^2}}}\left( {\frac{1}{2}} \right)dx,{\text{ }}du = \frac{2}{{4 + {x^2}}}dx \cr
& dx = \frac{{4 + {x^2}}}{2}du \cr
& {\text{Substituting}} \cr
& \int {\arctan \left( {\frac{x}{2}} \right)\left( {\frac{1}{{4 + {x^2}}}} \right)} dx = \int u \left( {\frac{1}{{4 + {x^2}}}} \right)\frac{{4 + {x^2}}}{2}du \cr
& = \frac{1}{2}\int u du \cr
& = \frac{1}{4}{u^2} + C \cr
& {\text{Write in terms of }}x \cr
& = \frac{1}{4}{\left( {\arctan \left( {\frac{x}{2}} \right)} \right)^2} + C \cr
& = \frac{1}{4}{\arctan ^2}\left( {\frac{x}{2}} \right) + C \cr} $$
