I vaguely remember a book/some lecture notes which introduce integration algorithms such as Risch algorithm by first giving a list of quasi-algorithmic way of evaluating symbolic integrals. (For example, when integrating a rational trignometric function, use one of the substitutions $u=\sin \theta$, $u=\cos \theta, u=\tan \frac12\theta$ and so on; when dealing with a rational function, express it in partial fractions.) Then it goes on to develop the theory of differential algebra.
I could not find the text now. After searching for quite a few sources, I have not found any books which contrast those easy procedures with Risch algorithm.
Where could I find a book which is similar to the one I describe above?