I am currently studying the geodesics of different type of spacetimes and I'm not sure if I'm doing it in the correct way for Schwarzschild de Sitter space (SdS).

The metric in SdS is given by: $$ ds^2=-(1-\frac{2M}{r}-\frac{\Lambda r^2}{3})dt^2+(1-\frac{2M}{r}-\frac{\Lambda r^2}{3})^{-1}dr^2+r^2d\Omega^2 $$ with the usual definitions.

Solving for radial null geodesics directly from the metric gives $$ \frac{dr}{(1-\frac{2M}{r}-\frac{\Lambda r^2}{3})}=\pm dt $$ Integrating both sides will give a very nasty integral and I'm not sure this is what I should be doing.

Is there another way to calculate radial null geodesics for SdS?

The only way I can analytically solve this problem is by requiring an extremal black hole with $\Lambda=\frac{1}{9M^2}$ but that is not what I'm looking for.

The problem gets even worse when you look at the Reissner-Nordström in de Sitter space, because an extra term $-\frac{Q^2}{r^2}$ gets added inside the brackets.