Here's a simple integral for you that has a nasty result.
If can be represented as
Details and Assumptions
1) are positive integers not necessarily distinct, are co-prime to each other, are co-prime to each other , are not perfect power of any integer (that it is not a perfect square, cube etc.)
2). It is commonly known as Polylogarithm.
I myself am trying to find the closed form of this integral for about a month, and finally found it.