I am trying to solve complicated systems of polynomial equations. The first step is to determine maximal sets of independent variables for the solution manifold (ideal) or the number of isolated solutions using Gröbner bases. In some cases, Mathematica, Maple and SymPy do not seem to be able to determine the Gröbner basis (in reasonable time).

Do you recommend some tools for this task?

Let me give a test case for which Mathematica, Maple and SymPy did not finish in reasonable time (~<10h) on my computer. I used grevlex monomial order. Do you know a tool that can compute the Gröbner basis for this case? There are 21 polynomials and 8 variables u, q1, r1, q2, r2, q3, r3, q4 (using Gram-Schmidt, one can reduce to 8 polynomials): https://pastebin.com/mpqZUQqC