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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?

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I'm not sure if this is exactly what you're looking for, but my go-to volume for these kinds of question is Symbolic Integration I by Manuel Bronstein. Risch's original treatment is sketchy in many places, and Bronstein did a lot of work to flesh out the details and actually implement Risch's methods. From the Foreword by B. F. Caviness:

With the advent of general computer algebra systems, some kind of symbolic integration facility was implemented in most. These integration capabilities opened the eyes of many early users of symbolic mathematical computation to the amazing potential of this form of computation. But yet none of the systems had a complete implementation of the full algorithm that Risch had announced in barest outline in 1970. There were a number of reasons for this. First and foremost, no one had worked out the many aspects of the problem that Risch's announcement left incomplete.

Starting with his Ph.D. dissertation and continuing in a series of beautiful and important papers, Bronstein set out to fill in the missing components of Risch's 1970 announcement. Meanwhile working at the IBM T. J. Watson Research Center, he carried out an almost complete implementation of the integration algorithm for elementary functions. It is the most complete implementation of symbolic integration algorithms to date.

The book is a very nice blend of practical algorithms and general theory. As the title indicates, Bronstein planned to write at least one more volume, about the integration of algebraic functions. Sadly, he passed away before he could complete this task. But I think you'll find a lot of relevant information in this first volume.

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J. H. Davenport, On the integration of algebraic functions. Lecture Notes in Computer Science, 102. Springer-Verlag, Berlin-New York, 1981.

J. H. Davenport, Integration in closed form. Computers in mathematical research (Cardiff, 1986), 119–134, Inst. Math. Appl. Conf. Ser. New Ser., 14, Oxford Sci. Publ., Oxford Univ. Press, New York, 1988.

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