Look at this one I just found on Google Books. It's fairly old, but the entire text is there for free.I posted this in the "calculus tutorial" thread at the top of this forum but I guess no one's observing that thread anymore.
Regarding elliptical integrals the author of that thread, TPH, says: "Now the problem is that this integral cannot be expressed in elementary functions (the standard functions) like I mentioned in the previous lecture that that sometimes happens. This problem was studied primarily by Abel and lead to something call Elliptic Functions. These are new types of functions that integrate this. I myself do not know what courses actually teach these functions, not because they are difficult but because it is unnecessary."
And elsewhere, regarding the integration of radicals (particularly the arclength formula):
"We did not discuss how to integrate that, it can be done. The types of functions used are hyperbolic functions, they are related to the exponental e^x function. But that is too advacned for us, and all we did was set up the integral."
I would like to look into these matters--elliptic integrals, using hyperbolic functions to integrate equations under a radical sign, etc.--a little more deeply. Can anyone recommend a text please?