ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 15 May 2013 03:43:21 +0200Integrate with elliptic integral special function in resulthttps://ask.sagemath.org/question/10123/integrate-with-elliptic-integral-special-function-in-result/I'm trying to work with the following integral:
$$\int\sqrt{1-\frac14(\cosh x)^2}\mathrm dx$$
Feeding this to sage as `integrate(sqrt(1-1/4*cosh(x)^2),x)` leaves it pretty much as it stands. [Feeding the same to Wolfram Alpha](http://www.wolframalpha.com/input/?i=integrate%28sqrt%281-1%2F4*cosh%28x%29^2%29%2Cx%29), I get a solution which at least at first glance looks better:
$$\int\sqrt{1-\frac14(\cosh x)^2}\mathrm dx=-\frac12i\sqrt3E\left(ix\Big\vert-\frac13\right)$$
So I wonder:
* **Is there a way to obtain this kind of output using sage?** (This is my main question.)
* In particular, is there a way to manually indicate that a given integral can likely be expressed in terms of elliptic integral functions, and that these would be of interest?
* Are these [elliptic integral functions](http://en.wikipedia.org/wiki/Elliptic_integral) even available at all inside sage? If they are, under what name?
* Is there any benefit in using these special elliptic integral functions, as opposed to (a `numeric_integral` version of) the original integral, in terms of performance or accuracy when dealing with actual numeric data?MvGWed, 15 May 2013 03:43:21 +0200https://ask.sagemath.org/question/10123/