In finite element method, especially when developing new elements, we often need integration techniques. Generally, numerical integration is used, however, in order to enhance the accuracy of our computations, we might be interested in performing integration over the continuum using symbols. In your opinion, what is the best programming language that can not only perform this kind of integration but also work with large symbolic arrays (where a single element is several lines of text long)? Of course, being in academia, cost effectiveness is also a factor.
I would go for Python (SymPy). Doing symbolic calculations, or in your case integration using symbols, on an array of 'n-dim' should be efficient.
SageMath provides an easy to use Expression class for symbolic calculations, it is also based on python (you can even use python packages in its interpreter) and is open-sourced, you might want to try it for prototyping usage. Note unlike differentials, there exist integrations cannot be solved analytically, like sin(x) /x. If you really want to solve techniqually hard symbolic integration problem, Mathematica or Maple is highly recommended. If you need to solve ODEs numerically, Matlab's ODE45 solver is always your friend (P.S. Sagemath's support for numerical ODE solver seems to be limited compared with matlab since I cannot see more accurate methods other than RK4).
MATLAB's Symbolic Math toolbox is a powerful tool for symbolic calculations, which is the successor of the MuPad. It's integration with MATLAB alleviates symbolic programming and conversion to and from double precision floating point numbers.
Now days this is not a matter of language. It is a matter of libraries. For example, you can use C++ with atermpp library, or Java with a similar one (all are free).
Fundamentally, symbolic computations stems from Mathematical elements hence MATLAB will be of great help.
Dear Gul Jokhio
I suggest you use the most powerful software "python".
SymPy is an open-source Python library for symbolic computation. It provides computer algebra capabilities either as a standalone application, as a library to other applications, or live on the web as SymPy Live or SymPy Gamma.
In my experience, Mathematica is the best for analytic integration. Besides, for your purpose, you can use the free version of Mathematica 12, for Raspberry Pi 4.
If your integrals are not so complicated, wxMaxima (free, open-source, and grandfather of Mathematica !) is as good as the Mathematica. Python is another good choice, but can not beat Mathematica and wxMaxima in symbolic integration.
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