I understand that what we need to do in this situation. But, I can not find the exact solution. I found the measure of the arc which is the image of the segment $[0, a]$ but still didn't describe its formula.
Problem: "The upper half-plane is mapped onto the unit disk so that the point $z = hi(h > 0)$ passes into the center of the circle. Find the length $\Gamma$ of the image of the segment $[0, a]$ of the real axis $(a > 0)$ and obtain linear approximate formulae for $\Gamma$ for small $a/h$ and for small $h/a$."