Given the series expansion
$$1+b_1\arctan x+b_2 \arctan^2 x=1+\sum_{k=1}^\infty a_k x^k$$
one has the relationships
$$a_{2k+1}=(-1)^k\frac{b_1}{2k+1},$$
$$a_{2k}=(-1)^k b_2\sum_{i=0}^k \frac{1}{(2i+1)(2k-2i+1)}$$
$$\qquad=(-1)^k \frac{b_2}{4k+4}\left(H_{-k-\frac{3}{2}}+H_{k+\frac{1}{2}}+4\ln 2\right),$$
with $H_n$ the harmonic number.