TITLE: Is there a general approach to solve integrals of the form $\int\frac f{f'}$?
QUESTION [5 upvotes]: It is easy to solve integrals of the form $\int\frac{f'}f$ using the defintion of the natural logarithm: $\int \frac{f'(x)}{f(x)}\;\mathrm dx = \ln f(x).\ $ Is there a similar identity for the case $\int\frac f{f'}$?

REPLY [8 votes]: Writing $f = e^g$ we have $\int \frac{f}{f'} = \int \frac{1}{g'}$ and this can be a more or less arbitrary integrand so no. Already taking $g = x \ln x - x$ we get a non-elementary integral.