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Is there any painless way of calculating the Riemann tensor?

I have the metric, and finding the Christoffel symbols isn't that hard, especially if I'm given a diagonal metric.

Out of 40 components, most will be zero. But how do I know how to pick the indices of Riemann tensor, given the non vanishing Christoffel symbols?

Because I can't just go and put all the possible combinations :\

I have the metric:

[itex]ds^2= -B(r)\text{dt}^2 +A(r)\text{dr}^2+r^2 \left(d \theta

^2+\sin^2\theta d\phi^2\right)[/itex]

And I have 13 Christoffel symbols that are different from zero. Mathematica says that there will be 6 nonzero components.

I also know that there are possible 20 linearly independent components of Riemann tensor, but how do I figure out which combinations of [itex]t,r,\theta,\phi[/itex] will give me the correct answer?

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# Easy way of calculating Riemann tensor?

Can you offer guidance or do you also need help?

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