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I can not find any method to solve this, why do the typical methods fail? I have tried subtituion, parts, strange subitution, etc. How does one solve this integral?

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    @Java, the ugliness is only because Wolfram is rather crappy with the business of evaluating elliptic integrals.2013-04-03

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This is an elliptic integral. The elliptic integrals represent solutions for the arc length of an elliptic arc and they cannot be expressed in terms of elementary functions.

I do not know if the above statement is proved, or closed forms in terms of elementary functions exist but not yet discovered. Anyways, among the people who tried to find a solution was Euler and Legendre.

The elliptic integrals can be expressed (reduced) to Legendre canonical forms (there are three of them).

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    @Mao, the canonical forms are listed [here](http://dlmf.nist.gov/19.2.ii).2013-04-03