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Currently there are a number of applications for numerical integration in applied mathematics and physics. Many of these are integral transforms (often Fourier or Laplace), or solving definite integrals with no closed form solution such as running the numbers on the Dirac equation for molecular modelling or doing lattice quantum chromodynamics to predict things like the mass of the proton.

Years ago I thought it might be interesting to try to implement the Risch algorithm as a library function for similar HPC applications, but it's my understanding that symbolic integration of indefinite integrals isn't of much use in these areas because most integrals don't have closed form solutions. Apparently there are heuristics for finding definite integrals for expressions that don't have closed form antiderivatives that often exploit special functions to make the numerical integration easier.

I'm curious if there's any high performance computing need to do symbolic integration or if it could help improve the performance of the applications where we currently do fairly dumb brute force numerical integration. Do we only need symbolic integration for doing the set up for packages that run numerical methods?

Are there applications for symbolic integration where you want to run the Risch algorithm (and other symbolic integration methods) over millions of expressions, or is the utility of symbolic integration in computer algebra systems only as a research tool that doesn't require large amounts of computational power the way numerical integration does? Are there any applications for high performance computing symbolic integration packages? If so, what are some examples?

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