Radial Null Geodesic!

Hi,



I've found geodesic equations for the metric:



\begin{equation}



ds^{2} = -c^{2} \alpha dt^{2} + \frac{1}{ \alpha } dr^{2} + d \omega ^{2}



\end{equation}



where



\begin{equation}



\alpha = 1 - \frac{r^{2}}{r_{s}^{2}}



\end{equation}



I have found that for a light ray:



\begin{equation}



\alpha \frac{dt}{d \lambda} = 1 ;

\frac{dr}{d \lambda} = c



\end{equation}



Where lambda is an affine parameter and i have appliedthe conditions:



\begin{equation}





\frac{dt}{d \lambda} = 1 , r = 0



\end{equation}



I am then told to integrate these equations to get:



\begin{equation}



r = r_{S}tanh(\frac{ct}{r_{s}})



\end{equation}



However when I try to integrate I get:



\begin{equation}



\int_{0}^{r} \frac{1}{1-\frac{r^2}{r_{s}^{2}}}dr = ct



\end{equation}



goes to:



\begin{equation}



\frac{r_{s}}{2} ln \frac{r+r_{s}}{r-r_{s}} = ct



\end{equation}



Which rearranges to:



\begin{equation}



r = r_{s}coth( \frac{ct}{r_s})



\end{equation}



Any help would be appreciated, thanks!