$$\begin{align} A&:=\int{\arctan(x)\over 1+x^2}\,\mathrm{d}x\tag{1} \\ &=\int\arctan(x){\mathrm{d}\over\mathrm{d}x}\arctan(x)\,\mathrm{d}x\tag{2} \\ &=\arctan(x)\arctan(x)-\int{\arctan(x)\over 1+x^2}\,\mathrm{d}x\tag{3}\\ &=\arctan(x)^2-A\tag{4}\\ \end{align}$$
$$\begin{align} A &= \arctan(x)^2-A\tag{5}\\ 2A &= \arctan(x)^2\tag{6}\\ A &= {1\over 2}\arctan(x)^2+\mathrm{const}\tag{7}\\ \end{align}$$
$$\int{\arctan(x)\over 1+x^2}\,\mathrm{d}x={1\over 2}\arctan(x)^2+\mathrm{const}\tag{8}$$