$$ \int \frac{e^x}{\sqrt{-1+ e^{2x}}}\,\mathrm{d}x.$$
Not sure where I messed up the formatting, but the denominator is suppose to be under a square root. Sorry for any confusion. I am trying to integrate this by $u$-sub and am getting lost. I under stand I should choose $u$ to be the whole denominator, but its not working out, I keep getting that the derivative of the denominator to be the whole equation itself and get stuck here.
Let $u = e^x$. Then the integral becomes
\begin{align} \int \frac{du}{\sqrt{u^2-1}} &= \int \frac{\sec t \tan t \,dt}{\tan t} \tag{$u = \sec t$} \\ &= \int \sec t \, dt \\ &= \ln|\sec t + \tan t| + C \\ &= \ln|u + \sqrt{u^2-1}| + C \\ &= \ln|e^x + \sqrt{e^{2x}-1}| + C \end{align}