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I stumbled across the following integral:- $$I_{n}=\int\frac{\cos nx}{5-4\cos x}dx$$ where $n$ is a positive integer.

I have no idea how to proceed....I tried integration by parts and even writing $\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$

I couldn't make much headway..... Any ideas on how to proceed would be appreciated.

EDIT:

This question is different from @amWhy has marked..I want to evaluate indefinite integral ..Not the definite one.

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    What is $n$? An natural? An integer? – 2017-02-25
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    An positive integer..... – 2017-02-25
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    Possibly Related, but is a definite integral: [Here](http://math.stackexchange.com/questions/2159720/evaluate-int-0-pi-frac-cos-2017x5-4-cos-xdx#comment4441925_2159720) – 2017-02-25
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    See here: http://math.stackexchange.com/questions/125399/how-to-solve-int-0-pi-frac-cosnx5-4-cosxdx/125553#comment4419379_125553 – 2017-02-25
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    For general natural number $\;n\;$ there doesn't seem to be a solution in terms of elementary functions. For the $\;n=1,2,3,4\;$ it is a reasonably nasty-like integral, but for $\;n\ge5\;$ it gets more and more nigthmarish... Next time try no to stumble across ugly things lke that one. – 2017-02-25
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    http://math.stackexchange.com/questions/2159720/evaluate-int-0-pi-frac-cos-2017x5-4-cos-xdx – 2017-02-25
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    @DonAntonio I can't believe it is so impossible. I think with the right substitutions and expansions, this may be doable – 2017-02-25
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    @SimplyBeautifulArt Well, perhaps. I only repeat what WA said, though sometimes it makes ugly mistakes. Nevertheless, in this case, the results for the first naturals seem to point towards the plausibility that, if possible at all to solve by elementary functions, it is going to be very, very hard. But, hey: I'd love to see someone succeeding... – 2017-02-25
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    @labbhattacharjee That link you give is for a specific case and for a *Riemann* (i.e., definite) integral. – 2017-02-25
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    Did anyone notice that the $dx$ is missing? – 2017-02-25
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    @NilabroSaha - while in this case, the $dx$ probably should be there, there are many variations of the notation for integrals. Leaving off the differential when what it should be is understood is actually quite common. – 2017-02-25
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    Probably the original problem was about $\int_{0}^{\pi}(\ldots)\,dx$. That is simple to compute, please see here: http://math.stackexchange.com/questions/2159720/evaluate-int-0-pi-frac-cos-2017x5-4-cos-xdx?noredirect=1&lq=1 – 2017-02-25

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Hint:

It is just a piece of cake of creating the reduction formula:

$\int\dfrac{\cos nx}{5-4\cos x}~dx$

$=\int\dfrac{2\cos x\cos((n-1)x)}{5-4\cos x}~dx-\int\dfrac{\cos((n-2)x)}{5-4\cos x}~dx$ (according to http://mathworld.wolfram.com/Multiple-AngleFormulas.html)

$=\dfrac{1}{2}\int\dfrac{4\cos x\cos((n-1)x)}{5-4\cos x}~dx-\int\dfrac{\cos((n-2)x)}{5-4\cos x}~dx$

$=\dfrac{1}{2}\int\dfrac{(4\cos x-5+5)\cos((n-1)x)}{5-4\cos x}~dx-\int\dfrac{\cos((n-2)x)}{5-4\cos x}~dx$

$=-\dfrac{1}{2}\int\cos((n-1)x)~dx+\dfrac{5}{2}\int\dfrac{\cos((n-1)x)}{5-4\cos x}~dx-\int\dfrac{\cos((n-2)x)}{5-4\cos x}~dx$

$=-\dfrac{\sin((n-1)x)}{2(n-1)}+\dfrac{5}{2}\int\dfrac{\cos((n-1)x)}{5-4\cos x}~dx-\int\dfrac{\cos((n-2)x)}{5-4\cos x}~dx$