Help solving complicated integral

[elkface]

This integral is not overly complicated, but I am having a little trouble with it - I haven't been in calc for a year almost, and it seems I have lost my integration abilities with it.

Here's the integral:
$$\int \sqrt{1 + (a \pi x^{a-1})^{2} \cos^{2}({\pi x^{a}})} dx$$

I figure the best place to start is to factor out the inside expression and get rid of the square root, but even doing this I am lost.

Any help?

 

[CFDFEAGURU]

Can you show your steps so far on your attempt to solve this integral?

Thanks
Matt

 

[elkface]

Well, to be honest, all I can seem to do is stare at it. I can't factor it, clearly, to get it separated into parts so that I could even attempt to do int. by parts or anything of that sort. I have tried Wolfram online integrator, but that doesn't work because it takes the server too long to process.

If anyone has software that could integrate this, or can recommend some (preferably freeware), I'd appreciate it. I don't need the work - I just need the answer.

 

[HallsofIvy]

If you replace $cos^2$ by $1- sin^2$ that becomes an elliptic integral which cannot be integrated in terms of elementary functions.

 

[elkface]

Ok, thanks for the tip. In that case, I will probably not be able to put this integral to any good use, as I am not familiar with most non-elementary functions. Thanks anyway.

 

[CFDFEAGURU]

Here is a link to the Numerical Recipe C routine for solving eliptical integrals. Not sure if if will help you. Take caution when using the NR routines they can be problematic. You have been warned.

http://www.nrbook.com/a/bookcpdf/c6-11.pdf

Thanks
Matt