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I'm aware that there are an infinite number of integrals that do not have a closed-form solution. But how does one know if their particular integral is one of these? For example, a particular integral I have at the moment is

Integrate[ Cos[x]^2 Log[ a + b Cos[x] + Sqrt[ c + d Cos[x] ] ] , x]

or

$$ \int \cos^2 x \,\log\left(a+b\cos x+\sqrt{c+d \cos x}\right) \,\mathrm d x $$

To me this looks rather daunting but I'm constantly surprised how trivial my own maths problems are to others. What resources are available to me to investigate this sort of problem? Are there any good reference books for this sort of thing? (my library is regretfully bare) — electronic books (esp. for pay) even better.

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btw, it gives closed form for a=b=c=d=1, but it doesn't look as nice as the original integral –  Yaroslav Bulatov Jan 20 '11 at 8:20
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This is basically a question in differential Galois theory. The Risch algorithm can decide if an integrand has an antiderivative (with certain caveats), but it's too computationally intensive for practical use (so, apparently, Mathematica does not use it). –  Zhen Lin Jan 20 '11 at 16:03
    
@Zhen Lin Could you please explain more in detail your point about this side of the Rish algorithm and Mathematica? Do you know math software that has a complete implementation of the Rish algorithm? –  Alexey Popkov Mar 11 '11 at 14:13
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@Alexey: Wikipedia discusses these points, though in not much detail. The main problem is computing whether a given symbolic expression is zero or not, apparently. –  Zhen Lin Mar 11 '11 at 21:49
    
According to Maple, even $\int \cos^2 x \log(\cos x)\,dx$ is not elementary (involves a dilogarithm). –  GEdgar Jul 14 '11 at 15:06
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1 Answer

There is a modest list of integrals-without-closed-form at http://www.math.unt.edu/integration_bee/AwfulTruth.html which also links to a nice essay at http://www.math.niu.edu/~rusin/known-math/97/nonelem_integr2

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+1 for the links. –  Américo Tavares Jul 14 '11 at 6:55
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