Mathematica gives $$g(a)=\frac{\pi}{2} \, \text{sgn}(a) \ln(\left| a\right| +1) $$ for real $a$.
This can be obtained as follows: $$g'(a)=\int_0^{\pi/2}\frac{dx}{1+a^2\tan^2 x}=\int_0^\infty\frac{dx}{(1+a^2u^2)(1+u^2)}, $$ and the latter integral can be taken by partial fraction decomposition.