Already the early papers on supergravity were written using computer algebra software to do some calculations. What modern packages do people normally use for doing such calculations? Of course Mathematica and presumable a lot of other programs can be made to do such calculations, I am wondering if there are special packages or tutorials how to do such calculations in them.
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As a free and powerful software I can recommend you Cadabra. It is designed specially for the field theory calculations.
Cadabra is a computer algebra system (CAS) designed specifically for the solution of problems encountered in field theory. It has extensive functionality for tensor computer algebra, tensor polynomial simplification including multi-term symmetries, fermions and anti-commuting variables, Clifford algebras and Fierz transformations, implicit coordinate dependence, multiple index types and many more. The input format is a subset of TeX. Both a command-line and a graphical interface are available.