Determination of Ricci tensor and Scalar curvature from vielbeins

Question

Consider the following metric:

$ds^2=h(r)\bigg(dr^2+r^2\big(d\theta^2+\sin^2\theta ~d\phi^2+(d\psi+\cos\theta ~d\phi)^2\big)\bigg)$

We can try to compute the Ricci scalar of this metric by using Mathematica packages like RGTC but the result is horrendous. Instead we could introduce the following one forms:

$e^{\theta}=r~d\theta\\ e^{\phi}=r\sin\theta~d\phi\\ e^{\psi}=r~d\psi+r\cos\theta~d\phi. $

Using these forms the metric now looks diagonal

$ds^2=h(r)\bigg(dr^2+e^{\theta}\otimes e^{\theta}+e^{\phi}\otimes e^{\phi}+ ~e^{\psi}\otimes e^{\psi}\bigg)$

However I cannot plug this metric as it is into Mathematica. It will give wrong results. My question is: Are there Mathematica packages online that compute Einstein tensor using the vielbeins in which the metric is diagonal?

This post imported from StackExchange Mathematica at 2014-11-29 15:08 (UTC), posted by SE-user Orbifold