It may be possible to find an antiderivative symbolically, but it may be easier to compute a numerical approximation than to compute the antiderivative.
The importance of the partial fraction decomposition lies in the fact that it provides an algorithm for computing the antiderivative of a rational function.
Furthermore, holomorphy is a necessary condition for a function to have an antiderivative, since the derivative of any holomorphic function is holomorphic.
