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Why is $\ln( \cos x)$ a function that cannot be integrated symbolically? From what I have read, it's integral would produce a result that is not an elementary function, but I don't see why getting something like a polylogarithm is so bad. Further, $\int\ln( u)\, du$ can be solved, so why not use $u$-sub and evaluate it in this manner?
Thank you for your help in advance.
There is nothing wrong with getting the polylogarithm as the integral per se. We just don't include this in what we call elementary functions. This is also true of the closely related zeta function and also the gamma function.
Mathematicians have a list of functions that are deemed elementary, these include polynomials, rational functions, exponentials, logarithms, sine, cosine, square roots, etc. These functions are generally considered well understood. Though some cannot be computed exactly, like the exponential function, it is still included in this list.
The gamma function, polylogarithms, the Bessel functions and the zeta function are all what are called special functions. There are entire fields of study dedicated to their research. There is one particularly important book about special functions. This is the "Handbook of Mathematical Functions" by Abramowitz and Stegun.
