Can someone explain me the Risch algorithm for solving integrals?

I have been studying integration for quite a while now, and as swiftly as I went through differential calculus, I got struck in integration when I learnt that there are as such no "rules of integration" as we have got in differentiation. There are indeed some standard methods known like the method of substitution, by parts, by partial fraction decomposition, etc. however, there exists no rule to determine which method we should use when and how. I always had to go over a function and using trial-and-error use various methods until I could find a suitable procedure. However, I discovered that online calculators like www.integral-calculator.com quickly and easily solves all integrals showing all the steps. I often wondered how they did, until one day I read below the calculator, that it uses something called as Risch algorithm, which in fact all such integral calculators use, to solve the problems. I now, badly want to know what is the algorithm, and using that how can we find out the integral of a function.

Thanks for the help!

An introduction ...

Rosenlicht, Maxwell, Integration in finite terms, Am. Math. Mon. 79, 963-972 (1972). ZBL0249.12106.

The complete algorithm takes a big book to describe... Bronstein's book describes "half" of the algorithm. Unfortunately, he never did part II.

Bronstein, Manuel, Symbolic integration. I: Transcendental functions, Algorithms and Computation in Mathematics. 1. Berlin: Springer. xiii, 299 pp. (1997). ZBL0880.12005.