I'm writing a physics simulation code, one portion of which involves the integration of the frequency response of an arbitrary $N^{th}$-order Butterworth bandpass filter. That is, I wish to solve:
$$\int \sqrt{\frac{1}{1+\bigg(\frac{\big(\frac{f}{f_0} - \frac{f_0}{f}\big)^2}{W^2}\bigg)^N}}\; df$$
...where $f,f_0$, and $W$ are arbitrary real-valued constants, and $N$ is some positive nonzero integer (see also).
It occurs to me that I could greatly simplify (and speed up) this portion of the code by instead finding an analytic solution (or a symbolic approximation). I've tried working this out myself, with no success, and have also tried passing this to Mathematica, SymPy, and various integral solvers, to no avail. Encouragingly, SymPy's Risch-based solver (while unable to return a solution) claims that the integral is not nonelementary.
Is there an analytic solution to this integral? If not, is there a relatively straight-forward symbolic approximation?