# 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, 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 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?