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I have to integrate a trigonometric expression that looks fairly straightforward, however, the expression I obtain after I evaluate the integral is massive and unusable. As the integrand is relatively straightforward I am wondering if it is possible to reduce the expression to a more manageable form, or if necessary, approximate it somehow. The integral is

Integrate[(Cos[x - y] Cos[5 y])/(C + Cos[m - y] + B Cos[x - y]), y]

The $5$ in the above expression is just a placeholder for a variable $n \in \mathbb{Z}$ as if I use $n$ it won't evaluate the integral at all.

The value $B$ will be a real number in the open interval $(-.5, 0)$ and $C > 1$, in case that is of any use, I am a newcomer to Mathematica.

So is it possible to get a usable expression somehow?

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    $\begingroup$ What problem are you solving with this? Or in other words, what's your application for the solution? Do you really need the indefinite integral or are you using it for definite integrals? What is the domain of m? As a general tip, you can use Assumptions -> {-1/2 < B < 0, C > 1} as an option to Integrate though in this case it won't solve your problem of a too big output. $\endgroup$
    – Thies Heidecke
    Commented Mar 6, 2017 at 19:51
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    $\begingroup$ Applying the Assuming-Option as suggested by Thies Heidecke and Simplifiyng afterwards with the Assumptions leads to a formula which is 'only' one page long (after about 15min evaluation time). $\endgroup$
    – Julien Kluge
    Commented Mar 6, 2017 at 20:49
  • $\begingroup$ @ThiesHeidecke This integral arose when analyzing a frequency bandgap problem in a 2D periodic crystal..it is part of a very complicated eigenvalue calculation. $m$ and $x$ are both in $[0, 2 \pi)$. I actually need to evaluate the definite integral from $y = 0$ to $2 \pi$. I posted it as an indefinite integral as I thought it might be simpler to obtain that. I need a usable closed form expression for this integral as it is needed to develop other components of the larger eigenvalue problem. $\endgroup$
    – sonicboom
    Commented Mar 6, 2017 at 22:13

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