I want to explore symbolic integration, but for this I initially need to imagine what are the algorithmic achievements in this area today, so I have some questions about Risch's algorithm and all its variations. The problem is that I don't want to explore Risch's algorithm by itself, I only need to understand what it can today. In this way of formulation of the problem the best decision (how I see it) is the little consultation of a knowledgeable person (not Internet). These questions are:
- Am I right that today there exists algorithm (maybe approved Risch's algorithm) to find (or to define that it doesn't exist) accurate elementary (elementary means composed of basic elementary functions) antiderivative of elementary single variable function with fixed algebraic constants for any case of such type in a finite number of iterations? Or maybe it can but it is not proved and disproved that it can?
- Am I right that if constants are trancendental this algorithm may not work?
- Is it true that there does not exist approved Risch's algorithm that can find accurate elementary antiderivative (expressed through one variable and all symbol constants, which can be called parameteres) of elementary single variable function with parameteres (for example: $\ln(ae^x+x^a)$ where $a$ is parameter) for any case of such type?