In the context of quantum field theory, I am facing the following $1$-dimensional integral over Feynman parameters:
$$I(a,b) = \int_0^\infty d\alpha \frac{1}{F G} \arctan \frac{F}{G} \tag{1}$$
with
$$F(\alpha, a , b) := \sqrt{\alpha(1+\alpha)(a-b)^2-\alpha a^2+(1+\alpha)b^2} \tag{2}$$
$$G(\alpha, a , b) := \sqrt{\alpha \left(a-b-\alpha(a-b)^2\right)} \tag{3}$$
Added details: $a$ and $b$ are $4$-vectors in Euclidean space with $a^2 = 1$. Thus $b^2 > 0$ and $(a-b)^2 >0$.
Is there a way I can express this integral with, I don't know, polylogarithms or elliptic functions? Usually, my (rather primitive) method to reach this goal is to put the indefinite integral into Mathematica, pray that it can solve it, and take the appropriate limits. But this time, Mathematica can't do it, so it makes me wish for a more robust method to do that.