Such a formal comparison may exist somewhere, but I can summarize the key differences between Mathematica and Sage, two tools I'm familiar with.
Sage relies on Python, Mathematica relies on a proprietary language.
Mathematica can do a number of things that Sage can't do, but Sage has a wider array of mathematical tools and environments bundled with it (and Sage is a big package -- 4.4 GB installed). Sage includes a number of specialized packages for research and scientific mathematics beyond the most often used functions and packages.
Sage can perform the usual symbolic math operations -- solve equations, derive, integrate, produce many kinds of symbolic and numerical results. Mathematica can solve more cases in the same domains.
Sage is free. Mathematica presently costs $2,745 for the default, single-seat license.
Sage relies on Python, Mathematica relies on a proprietary language.
Mathematica can do a number of things that Sage can't do, but Sage has a wider array of mathematical tools and environments bundled with it (and Sage is a big package -- 4.4 GB installed). Sage includes a number of specialized packages for research and scientific mathematics beyond the most often used functions and packages.
Sage can perform the usual symbolic math operations -- solve equations, derive, integrate, produce many kinds of symbolic and numerical results. Mathematica can solve more cases in the same domains.
Sage is free. Mathematica presently costs $2,745 for the default, single-seat license.
Sage: http://www.sagemath.org/
My Sage tutorial: http://www.arachnoid.com/sage
Mathematica: http://www.wolfram.com/mathematica/pricing/industry-individu...