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calculus and analysis - How does Mathematica integrate?



Basically, this question can be considered to be an extenstion to my other question.


What I wanted to do was this integral as homework (it is indefinite BTW so no approximations using Simpson's Rule or Boole's Rule)


$$\int(x^{3m}+x^{2m}+x^{m})(2x^{2m}+3x^{m}+6)^{\frac1{m}}dx$$


So using Mathematica's Integrate function the answer was


Answer


Apparently, after rigorous substitutions and transformations the answer was found to be correct.


What I wanted to know was how Mathematica integrates these functions that require a human tons of intuition to compute, within seconds, and often in the most simple way and also presents them in the most humanly computable form.


(Even differentiation for that matter)



Answer



I can only direct you to Some Notes on Internal Implementation:




Differentiation and Integration


Differentiation uses caching to avoid recomputing partial results.


For indefinite integrals, an extended version of the Risch algorithm is used whenever both the integrand and integral can be expressed in terms of elementary functions, exponential integral functions, polylogarithms, and other related functions.


For other indefinite integrals, heuristic simplification followed by pattern matching is used.


The algorithms in Mathematica cover all of the indefinite integrals in standard reference books such as Gradshteyn-Ryzhik.


Definite integrals that involve no singularities are mostly done by taking limits of the indefinite integrals.


Many other definite integrals are done using Marichev-Adamchik Mellin transform methods. The results are often initially expressed in terms of Meijer G functions, which are converted into hypergeometric functions using Slater's theorem and then simplified.


Integrals over multidimensional regions defined by inequalities are computed by iterative decomposition into disjoint cylindrical or triangular cells.


Integrate uses about 500 pages of Mathematica code and 600 pages of C code.




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