I want to show that the Schwarz-Christoffel Integral
$$ I(z)=\int^z_0\dfrac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}} $$
is a biholomorphic mapping from the upper half complex plane to the rectangle. If $I$ is injective, then by using the fact that $I$ maps the boundary to the boundary I think I can show that it is surjective. However, I am stuck at how to the integral is injective. Any hints are appreciated.
Thanks!