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Currently we have

  • Polynomial Functions
  • Trigonometric functions
  • Hyperbolic Functions
  • Exponential Functions

and their reciprocals and inverses as the "standard library of elementary functions".

could we define a new set of functions based on operations between these elementary functions, then work out the algebraic identities, derivatives and integrals of these new functions and use them to help ease symbolic integration?

i envision a world where i approach an integral, substitute various compositions, products, divisions, sums of elementary functions with a newly defined function, use the identities, derivatives and integrals developed for the newly defined functions to trivially evaluate the integral. then rewrite the antiderivative back in terms of the standard elementary functions, to avoid confusing everyone else.

I imagine such functions would have to be defined with the goal of easing symbolic integration in mind

other than that, is what i'm saying possible?

to illustrate what I mean if the above was too abstract, imagine hyperbolic functions weren't already part of the standard elementary functions. you may come across various exponentials in integrads. you write them in terms of hyperbolic functions, if it makes it easier to integrate. then evaluate the integral, and then rewrite the hyperbolic functions contained in the antiderivative in terms of exponentials.

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    $\begingroup$ By Liouville's theorem, adding such special functions would not expand the set of antiderivatives expressible in terms of elementary functions. Otherwise, this is kind of what an integral table is: a list of functional forms with known antiderivatives you can try to massage your integral into. $\endgroup$ Mar 1, 2023 at 20:28
  • $\begingroup$ oh. i never actually used an integral table. I just memorised the derivatives and integrals of the standard elementary functions, of their reciprocals and inverses but was wary of turning it into a memory game by keeping track of various other integral forms i come across. is that how integration is approached generally??. oh my god. I had always wondered why there so much emphasis on the teaching of the substitutions and integration of specific structures like rationals, trig rationals, quadratics composed in roots etc. Damn $\endgroup$
    – Hisham
    Mar 1, 2023 at 20:38
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    $\begingroup$ Yeah integration is like that. Differentiation can be done with a few nice rules, but integration is all about hoping you can get the integrand to match the output of one of those rules. And there are a hell of a lot of "nice" looking integrands that have no elementary antiderivatives, and quite a few of those have no known antiderivative elementary or otherwise. $\endgroup$ Mar 1, 2023 at 20:47

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