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.