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Is there any way of evaluating this little monster of an integral...or its approximation?

$$\displaystyle\int_{\alpha}^{\alpha+4\pi}\dfrac{e^{-A\cos(\theta)}}{B+C\cos(\theta)}d\theta\qquad,\,A,B,C>0$$

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  • $\begingroup$ It's begging for a change of variable $x=cos(\alpha-2\theta)$. $\endgroup$
    – Paul
    Apr 20 '21 at 20:07
  • $\begingroup$ Why do you say little ? $\endgroup$ Apr 21 '21 at 5:10
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    $\begingroup$ The integrand is $2\pi$ periodic, hence its integral over any interval of length $2n\pi$ is the same. WLOG take $\alpha=0$. $\endgroup$
    – K.defaoite
    Apr 21 '21 at 18:30
  • $\begingroup$ An identity perhaps useful for $s>1$ is $$\int_{0}^{2\pi}\frac{1}{s+\cos x}\mathrm{d}x=\frac{2\pi}{\sqrt{s^2-1}}$$ $\endgroup$
    – K.defaoite
    Apr 21 '21 at 18:33

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