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Why is it that differentiation of a function that is a composition of elementary functions (such as $\sin \:2^x$ or $\ln(\mathrm{arcsec}\: x^3)$ or $x^{1/x}$) always produces a composition of elementary functions, while integration sometimes does not? For example, the derivative of $\sin \:x^2$ is $2x\cos \:x^2$, but the integral of $\sin \:x^2$ produces the Fresnel sine function, the definition of which is simply the integral? To put it loosely, why does differentiation make functions simpler and integration make functions more complex? I guess what I'm trying to get at is what fundamental aspect of integration sometimes produces a non-elementary function from an elementary function?

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marked as duplicate by J. M. is a poor mathematician, JonMark Perry, Claude Leibovici calculus May 14 '16 at 6:01

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  • $\begingroup$ The derivative of $\sin x^2$ is $2x\cos x^2$. $\endgroup$ – choco_addicted May 14 '16 at 1:35
  • $\begingroup$ Oops, I've corrected it. $\endgroup$ – Yunfei Ma May 14 '16 at 1:36
  • $\begingroup$ But there also some functions that differentiating them is harder than integrating. For example $2x\cos x^2$, its anti-derivative is $\sin x^2$ and its derivative is $2\cos x^2-4x^2 \sin x^2 $ $\endgroup$ – Ruzayqat May 14 '16 at 1:44
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    $\begingroup$ One actually does not need to go further than the natural logarithm. Who'd expect that $\int \frac1{u}\mathrm du$ results in a transcendental function? $\endgroup$ – J. M. is a poor mathematician May 14 '16 at 1:59
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The phrase you are looking for is "integration in finite terms".

Here is one of the key papers in the field:

http://www.ams.org/journals/tran/1969-139-00/S0002-9947-1969-0237477-8/S0002-9947-1969-0237477-8.pdf

A useful book which I read many years ago is J. F. Ritt, Integration in finite terms, Liouville's theory of elementary methods, Columbia Univ. Press, New York, 1948.

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