I wonder if anyone would find Einstein's field equation aesthetically pleasing.
As to the Einstein Field Equations (EFEs) themselves, the aesthetically pleasing quality is the terse notation brought about by the Einstein summation convention, abstraction into the Einstein tensor G_{\mu\nu}, the stress-energy-momentum tensor T_{\mu\nu} and other objects, and so forth (it's even shorter with geometrized units c = G = 1, ignoring the cosmological constant (say if one relies on thin shells and junctions[1]), \pi = 1, and/or 0 on the RHS).
This terseness can hide several pages of partial derivatives that look like http://4.bp.blogspot.com/-0e2Zl5QrRiA/UZfL29xVTZI/AAAAAAAAAF... (from NCSA's (offline) numerical relativity mathmine1.html originally) -- there will be sixteen of those for the Ricci tensor, plus some more pages for the other parts of the left-hand-side, although by introducing lots of symmetries and simplifications we can cut down by quite a bit.
One can get a feel for how these are generated with the example at https://www.maplesoft.com/support/help/Maple/view.aspx?path=... (eqn 12 for the metric at eqn 6)
Hartle's (anathe)Mathematica examples https://web.physics.ucsb.edu/~gravitybook/math/curvature.pdf are nice but don't take you to an interesting Ricci tensor.
Instead, perhaps see the answer https://mathematica.stackexchange.com/a/8908
Frankly, the mechanics of working with known solutions of the EFEs can be a pain. Perturbing against those can be even more of a pain. Junction conditions[1] between different exact solutions can be a super pain. Modern symbolic computing systems help a lot: https://en.wikipedia.org/wiki/Tensor_software (sigh, that needs updating, e.g. GRTensorII->GRTensorIII https://github.com/grtensor/grtensor/wiki )
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[1] https://arxiv.org/abs/gr-qc/9510052v3 since absorbed into GRTensorIII