What is the branch point of $f(z)=\tanh^{-1}(\frac{z}{\sqrt{(1+a^2)(z^2+1)}})$

Question

What is the branch point of $f(z)=\tanh^{-1}(\frac{z}{\sqrt{(1+a^2)(z^2+1)}})$, where $a$ is small real number $a\ll1$.

I know the branch point of $g(z)=\tanh^{-1}(\frac{z}{\sqrt{(z^2+1)}})=\sinh^{-1}z$ is $\pm i$, how can I deduce that for $f(z)$?