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I am trying to get a closed form for the following definite integral:

$$f(\theta)= \int_\frac{\pi}{2}^\pi \frac{1}{\sqrt{1-\alpha^2 \cos^2\theta}}\exp\left(C_2\cos\theta-C_1\sqrt{1-\alpha^2 \cos^2\theta}\right) \,d\theta$$

where $0<\alpha<1$ and $C_1$ and $C_2$ are constants .

Can this function be related to some special function? Can anyone give me some advice or recommend some references?

Thanks in advance.

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    $\begingroup$ Why are you trying to get a closed form for this particular integral, and why are you confident that such form exists? $\endgroup$
    – Stefan Kohl ♦
    Commented Nov 6, 2014 at 23:44
  • $\begingroup$ @Stefan Kohl, Maybe I should write, ‘I am trying to solve’ as I am not sure whether the closed form for such problem exists or not. Actually, this function refers to a system in my work where I need the closed form to perform further analysis. I know that not all integrals can have the closed form and some integral solution may have non-elementary functions. I wonder if there is any special function that can be related to this function which can be used to solve this problem. Thank you. $\endgroup$
    – Nor
    Commented Nov 7, 2014 at 3:02
  • $\begingroup$ You mention that you want to perform some analysis on f. Most of the time, this can be done without getting a "closed form". So, your real question is about properties of this function, not about the closed form. $\endgroup$
    – Boris Bukh
    Commented Nov 10, 2014 at 19:02
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    $\begingroup$ The question shows no research effort. $\endgroup$
    – Boris Bukh
    Commented Nov 10, 2014 at 19:03

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