Any Mathematics high school student or undergrad knows that derivation is easy: there is a fixed set of rules which, if followed, lets you compute the derivative of any elementary function. And any Mathematics undergrad or high school student also knows that integration is hard: integrating an elementary function (like sin(ln(x))) is decidedly non-trivial, and often appears to require the use of a trick.

Hence statements like this node's title: any machine can derive, but only a human artiste can integrate.

Unfortunately for this claim and fortunately for future generations, in 1970 Robert Risch came up with an algorithm for symbolic integration of elementary functions. You do have to make some assumptions; the most important one is that you are computing over the reals and can solve all polynomial equations (otherwise some undecidable problems pop up, related to the solution of Hilbert's Tenth Problem; but they are truly unimportant compared to the usefulness of an integration algorithm).

The algorithm is by no means efficient (although some work has been done, symbolic algebra is hard to do quickly). Whether or not it is useful is debatable. But the fact remains: an algorithm exists which can give the antiderivative of any elementary function which has one (or prove that none exists).

So, at least as regards the art of integration, there isn't one. Any machine can integrate. Though it might take a while...

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