A derivative calculator seems a lot easier to build as it much more “rule” based and solutions are usually what you would expect them to be. But how on earth can integral calculators figure out absolutely brutal integrals like sqrt(tanx) is in less than 5 seconds?
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Not having coded a computer algebra system myself, I can’t tell you for sure what exactly the implementation is for any given system.
That said, for some simple functions I suspect it’s just a lookup table.
For more complicated general functions, there is a generalized way to determine whether you can compute an antiderivative using only elementary functions, and if so, what it is. It’s called the Risch algorithm. The algorithm itself is so complicated that it isn’t actually fully implemented in any computer algebra system, but the way it works is that it provides a method to turn the procedure of finding the antiderivative of a given function into a giant algebra problem.
It’s really not possible to go into more detail in anything approaching like ELI5, nor would I be competent to do so.
https://en.wikipedia.org/wiki/Risch_algorithm
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