I am calculating the Riemann tensor for the Schwarzschild solution. I've calculated all 9 non-vanishing Christoffel symbols already. Now I need to evaluate the Riemann tensor and I find no easy way to do it. I have $$ R^\alpha{}_{\beta\gamma\delta} = \partial_\gamma\Gamma^\alpha_{\beta\delta} - \partial_\delta\Gamma^\alpha_{\beta\gamma} + \Gamma^\mu_{\beta\delta} \Gamma^\alpha_{\mu\gamma} - \Gamma^\mu_{\beta\gamma} \Gamma^\alpha_{\mu\delta} $$ and I only have the following connections different from zero: $$ \Gamma^r_{tt}, \Gamma^r_{\theta\theta},\Gamma^r_{\phi\phi},\Gamma^r_{rr}, \Gamma^\theta_{r\theta},\Gamma^\phi_{r\phi}, \Gamma^\theta_{\phi\phi}, \Gamma^\phi_{\theta\phi}, \Gamma^t_{tr} $$
I'm thinking of putting in the first partial derivative $\partial_\gamma\Gamma^\alpha_{\beta\delta}$ only one of those symbols I already have, however I'm afraid it won't make me go trough every possible non-vanishing Riemann tensor component I need.
Will I get all the components? Are there other easy ways to do it?
PS1: Yes, I know the symmetries and that there are only 20 independent components
PS2: I also know that I have the answer in the book, but I want to do it myself to practice
PS3: I don't want a specifically method for Schwarzschild solution only, a more "general easy way out" of it.