This leads to a similar integral: \begin{equation} \phi = \int d\phi = \pm \frac{l}{\sqrt{2m}}\int \frac{dr}{r \sqrt{Er^2 + GM m r - \frac{l^2}{2 m}}} \end{equation}

Let: \begin{equation} X = cx^2 + bx + a = E r^2 + GM\mu r - \frac{l^2}{2 \mu} \end{equation}

From the integral tables: \begin{equation} \int \frac{dx}{x \sqrt{X}} = \frac{1}{\sqrt{-a}} \sin^{-1} \left(\frac{b x + 2 a}{x \sqrt{-(4ac-b^2)}} \right) \end{equation}

Carry this through as before, and define: \begin{equation} \epsilon \equiv \sqrt{1 + \frac{2 E l^2}{G^2 M^2 m^3}} \end{equation}

1. Find $r(\phi)$