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My initial thoughts are no, since all you have to do is code all known integral techniques, and have the computer brute force the result.

But would there be any integrals, or any integral techniques which would take less time to do by hand than by a program?

Apologies if this sounds like a silly question but I'm just now getting back into maths.

AdronVenelucio
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    If it can be done by hand, it can be implemented as an algorithm. As to whether or not a given method is implemented in a given program, that depends on the algorithmic strategy chosen by the programmers involved in the implementation. As regards efficiency or simplest form, humans can sometimes do better than a given program, but once again, it depends on which program, and future versions might change the results in favor of the program. – quasi Jun 23 '18 at 00:29
  • In terms of antiderivatives, the answer is more or less no, due to the Risch algorithm. The Risch algorithm is a semi-algorithm, so a computer may choke on some of the expressions that it needs to determine are zero or not, but these will generally be hard to prove to be zero or not anyway (and in some cases it will simply not be possible one way or the other). – Ian Jun 23 '18 at 01:01
  • In terms of definite integrals, the situation is a bit complicated. In principle you could tell a computer how to do anything after you did it, but in terms of training a computer to recognize what to do with a definite integral, that's a hard (and as far as I know not really solved) problem. A lot of it also depends on what kind of answer you want to get; for example, a computer may be OK with spitting out the answer to a definite integral in terms of some extremely general special function, about which a human may have no intuition. I'm grappling with this myself now. – Ian Jun 23 '18 at 01:02
  • No. ${}{}{}{}{}$ – copper.hat Jun 23 '18 at 04:29
  • May be many of Ramanujan's results are verified by machine but.. some of his work on integrals is still clueless? He used number theory for integration? – Narasimham Feb 07 '24 at 00:06

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Yes, there are. An example is integrating Dirichlet's function over a closed interval. The value of this function depends on whether the input to the function is rational or irrational. This type of function can be evaluated exactly. But, for a computer, everything is a rational number due to finite precision.

D.B.
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  • That's a matter of numerical integration, though. You can circumvent this with a clever enough program. – Ian Jun 23 '18 at 00:58
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    CAS programs such as Maple and Mathematica can use rational numbers in exact rational form, and some irrational numbers can also be expressed symbolically, so in principle, if the integral simplifies to an exact symbolic value, a CAS program could potentially be able to compute it symbolically. If you have a sample test integral, I'll try it in Maple. – quasi Jun 23 '18 at 00:59
  • Ok. My answer was not very thorough. – D.B. Jun 23 '18 at 03:22