How does $$\beta=\theta-\int\frac{\alpha }{r^2\sqrt {2mE+\frac{2me^2z}{r}-\frac{\alpha^2}{r^2}}}dr\\\implies \frac{1}{r}=\frac{me^2z}{\alpha^2}-\sqrt{\frac{m^2e^4z^2}{\alpha^2}+\frac{2mE}{\alpha^2}}\cos(\theta-\beta-\pi/2)$$ ??
This implication occured in Kepler's problem solution by Hamilton-Jacobi method, i've tried hard to prove this implication but unable to reach to correct conclusion.
Here's the original problem-
