What is the runtime of the 'Risch Algorithm'?
The Risch algorithm is undecidable by Richardson Theorem if absolute value is allowed or semi-undecidable with $\log_2, \pi, e^x, \sin x$.
Akitoshi Kawamura in his dissertation Computational Complexity in Analysis and Geometry has proved that integration is $\#P{-}complete$ operation.

There are multiple modifications of the Risch algorithm, but still nobody have implemented the full algorithm (source: Wolfram, survey over CAS systems) in algebraic case.