I need to calculate the following: $\displaystyle \int \sqrt{2\cosh t(\cosh t-1)} dt$, assuming $\displaystyle t\ge0$. So far I have tried some things which, quite messily, arrive at a solution but I cannot simplify my result back to the solution in the book, so I think I have taken a more difficult approach.

Here's what I've done:

$\displaystyle \int \sqrt{2\cosh t(\cosh t-1)} dt=\int 2 \sinh(\tfrac t 2) \sqrt{\cosh t}dt=\tfrac 1 {\sqrt{2}} \int(e^{t/2}-e^{-t/2}) \sqrt{e^{-t}+e^t}dt$

This can then be solved using a u-substitution, such as $\displaystyle u=e^t$, but the result is really ugly, and I don't think this is the path the book would have intended for me to arrive at the simpler result.

I would appreciate a hint. I'm guessing that there's some hyperbolic identity I'm forgetting.