An interesting reference is

astro-ph/9912320

in arXiv: there he calculates the Kretschmann invariants of arbitrary black holes. His claim is that it is in fact very hard to do without symbolic algebra software.

The work referred to above by Richard Henry in http://arxiv.org/abs/astro-ph/9912320 (published in ApJ 535, 1, 2000) has been generalized to Kerr-Newman black holes (with spin and charge) by Henry, Overduin & Wilcomb in http://arxiv.org/abs/1512.02762. Symbolic algebraic software is indeed essential, not so much in calculating the invariants, but in simplifying the resulting expressions.

In the Maxima computer algebra system (see http://maxima.sourceforge.net/), you can calculate the Kretchmann invariant for the Schwarzschild metric using the following short script:

load(ctensor)$

ct_coordsys(exteriorschwarzschild)$

cmetric()$

lriemann(false)$

uriemann(false)$

factor(rinvariant());

yielding K = 48m2 / r6.

Full source code is available. The code that does Kretchmann is nothing magical, it is in fact a brute-force summation, in the form of the expression

sum(sum(sum(sum(lriem[i%,j%,k%,l%]*uriem[i%,j%,k%,l%],

i%,1,dim),j%,1,dim),k%,1,dim),l%,1,dim)$

It is possible to compute the same invariant for other metrics, but the algebra gets quite messy. Computer algebra software really helps, but the result may still easily cover half a page, e.g., for Kerr-Newman.