Risch's algorithm for symbolic integration and its variations

Question

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:

1. 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?
2. Am I right that if constants are trancendental this algorithm may not work?
3. 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?

Answer 1

1. Yes, Risch's algorithm, given an elementary univariate function, will compute an elementary anti-derivative, if one exists. "Failure" of the algorithm implies that no elementary anti-derivative exists.

2. No, the algorithm can handle transcendental constants just fine. The problem arises when you don't know the full transcendence relationships between your constants. For example, we've proven that $e$ and $\pi$ are both transcendental, but are they "bi-transcendental"? In other words, is $\mathbb{Q}[e,\pi]$ a transcendental extension of $\mathbb{Q}[e]$? I don't think anybody knows. So there might be a polynomial $p(x,y) \in \mathbb{Q}[x,y]$ such that $p(e,\pi)=0$.

   If your integrand involves both $e$ and $\pi$, Risch's algorithm will produce polynomials in these constants and expect to be able to test if they are zero. Non-zero implies that the anti-derivative is non-elementary. So, you could just plug in approximations for $e$ and $\pi$ to establish that the polynomial is not zero and the integrand is not elementary. On the other hand, if you keep using more and more accurate approximations of $e$ and $\pi$ and keep getting approximately zero, you can't prove that the integral is non-elementary... but you've just found a candidate polynomial relationship between $e$ and $\pi$!

3. I'm not sure about this one, though I know Risch's algorithm fairly well. Manuel Bronstein's book Symbolic Integration I (the best graduate-level reference on the transcendental case of Risch's algorithm) includes some examples with parameters, but I don't think I've ever seen an algorithm to handle parametric problems in general.

   For your example of $\ln(ae^x + x^a)$, you'd need several cases. First, you need to consider when $a$ is an integer, so that $x^a$ is a polynomial or rational function in the field $\mathbb{C}(x)$. Then, you need to consider the case where $a$ is a rational number, but not an integer, so that an algebraic extension is needed to construct $x^a$. Finally, you need to consider when $a$ is irrational, in which case I'd write $x^a = e^{a\ln x}$, and use two transcendental extensions (first $\ln x$, then $e^{a\ln x}$) to construct $x^a$. All three cases are handled differently by the algorithm.

   The more I think about it, the more unlikely I think it is that any extension to Risch's algorithm has been developed to handle integrands with arbitrary parameters as you desire.