Evaluate a Trigonometric Integral

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Hi All,

[itex] \int_{0}^{2\pi}d\varphi \int_{0}^{\pi}\sqrt{\frac{1}{1-sin^2\theta cos^2\varphi }}d\theta[/itex]
I tried very hard to evaluate this integral but failed. Please help me if you know how to solve it.
 

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Thanks a lot!
This integral cannot be expressed in terms of a finite number of elementary functions.
Analytic expression requires a kind of special functions called "Elliptic Integrals". More precisely In the present case : K(x) the complete elliptic integral of the first kind. Of course, the result can also be expressed on the form of infinite series. See :
http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html
http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html
 

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