Two different ways of solving $\int \frac{e^x - e^{-x}}{e^x + e^{-x} } dx$, yield different results. What's the mistake?
$ \int\frac{g(x)}{f(x)} \, dx $ where $f(x) = \frac{1}{2}(e^x+e^{-x})$ and $g(x) = \frac{1}{2}(e^x - e^{-x})$?