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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?

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?