# How to approach an integral over $g(\cos(t))$ from $0$ to $2\pi$, where $g(x)$ is nasty?

For notational convenience, let $f(t) = a^2 + 2 a b \cdot \cos(t) + b^2$, where $a,b$ are both positive real constants and $t$ will be the integrand of the integral, which is supposed to be carried out from $t=0$ to $t=2 \pi$. I want to find an expression (or approximations) for \begin{align} \int\limits_0^{2\pi} \frac{\exp(-c(24m^2 - 24m \sqrt{f(t)} + 7 f(t) - 4 a b \cdot \cos(t)))}{(m^2\cdot f(t))^{1/4}} dt \end{align} where $c,m$ are again positive real constants.

My first attempt was to use Mathematica, but without any result. Are there maybe any tricks how to approximate the above integral? Any comments or suggestions are most welcome! Thanks in Advance.

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