I haven't much experience with hyperbolic functions, having never learned them, but the result of this integral seems to involve one:

$\displaystyle \int{\sqrt{1+u^2} du}$

Their solution is:

$\displaystyle \frac{1}{2}u\sqrt{1+u^2}+\frac{1}{2}\text{arsinh}{ u}$

(What's the proper latex command for arsinh?)

I'm not sure how they derived that.