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what is the the answer of this indefinite integral? $$\int \frac{1}{x^{3}\cos^{2}(x)}dx$$ I have used all methods I knew, but all failed.some methods made it more complicated!please tell me your Ideas,that would be great.

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I do not believe a "nice" anti-derivative exists. See wolfram alpha query. –  Eric Thoma Dec 15 '13 at 15:45
    
thank you very much.so what should we do with this kind of integrals?I mean that do we have a kind of expression for example with series or continued fractions or we should put them away as they are? –  kpax Dec 15 '13 at 15:51
    
The integrand has lots of infinite discontinuities (when $\cos x = 0$), so the best one could probably hope for is an expression for the definite integral across some nicely behaving region -- someone more experienced than I may be able to find out more. –  Eric Thoma Dec 15 '13 at 16:09
    
(1) That some techniques make the problem more complicated, rather than easier, is usual in the integration biz. Get used to it. (2) What to do with the integral depends a lot on what you want to do with it. Sometimes, it is indeed best to leave it alone. –  Harald Hanche-Olsen Dec 15 '13 at 16:11
    
The integral $\int 1/(x^3(1+\cos^2 x))dx$ seems to behave more nicely. I do not know of a way to evaluate it though. –  Eric Thoma Dec 15 '13 at 16:13

1 Answer 1

Integrate by parts, using $\tan'(x)=\dfrac1{\cos^2x}$ . Then apply Liouville's theorem to $\displaystyle\int\frac{\tan x}{x^4}dx$ :-)

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OK and then ? Cheers. –  Claude Leibovici Dec 15 '13 at 16:43
    
Lucian,I don't think that it is working,can you explain more? –  kpax Dec 15 '13 at 16:55
    
Yes. The primitive of your little function is not expressible in terms of elementary functions. :-) Mind telling us where you found it ? –  Lucian Dec 15 '13 at 17:02
    
one of my friends ask me!I try your answer but no good result,how I should use Liouville's theorem? –  kpax Dec 15 '13 at 17:29
    
It was a joke ! Just tell him it's a dead end. But if you're serious about it, see the Risch algorithm. –  Lucian Dec 15 '13 at 17:43

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