From the reviews of the second edition: "It discusses ... the integration of transcendental functions ... . Indeed, one of the most remarkable characteristics of this book is that it requires from its readers very little beyond a basic knowledge of calculus and algebra. ... This is an extremely well-written book on a beautiful topic that deserves to be better known to practising mathematicians and teachers of mathematics alike." S. C. Coutinho, The Mathematical Gazette, Vol. 90 (5l9), 2006 "[...] The book contains many examples and exercises for the reader to get a feeling not only of the theoretical but also the practical aspects of the algorithms. It is designed as a textbook and well suited for courses about these constructive aspects of calculus. A "must have" for the bookshelf of every library." H.-G.Grabe, Zeitschrift fur Angewandte Analysis und Ihre Anwendungen, Vol. 24, Issue 4, 2005 "The second edition of Bronstein's excellent book [...] contains a new chapter on parallel integration (also called the new Risch algorithm or the Risch-Norman algorithm) [...] The first nine chapters of the book contain some improvements and a couple of additional exercises. Bronstein's book still presents the state of the art in the domain of integration of transcendental functions. Its unique blend of detailed mathematics and clear description of algorithms make it both a standard reference and a handbook for researchers and designers of computer algebra systems and a useful, easy to read textbook for teachers and students." Friedrich Schwarz (Paderborn), Zentralblatt MATH 1059 From the reviews of the first edition: "The goal of this well-written book is to present the state of the art in the domain of "integration in finite terms". ... Both aspects of the work, mathematics and implementation, are present in the book. Moreover, Bronstein has chosen a good level of detail, and in such a way that he only deals with the case of transcendental functions. ..." J.M.Ollagnier, Mathematical Reviews 2002 "... It makes an excellent textbook for courses in computer algebra. It contains many exercises and the algorithms are presented in pseudocode, which is easy to implement in any computer algebra system. For those interested in symbolic integration it will become the standard reference." N.A.van Arkel, Medelingen van het wiskundig genootschap 1998 "... The writing is excellent, and the author provides a clear and coherent treatment of the problem of symbolic integration of transcendental functions. Each chapter includes several worked examples and a list of additional exercises. Every researcher and teacher in symbolic computation should have access to this book." F.Winkler, Computing Reviews 1997 "... In fact, this book is an excellent blend of mathematics and algorithms. ... This well-written book serves as a good foundation to the topic of symbolic integration. ... Highly recommendable." P.Kravanja, Journal of Computational and Applied Mathematics / Newsletter on Computational and Applied Mathematics 1997 "My first thought on seeing this book was "I wish I had written it" - it resembles my lecture notes on the subject, but is better and more complete. ... In sum, the book does what it sets out to do, does it well, and should be on the bookshelf of every implementer or teacher. For the latter, the author indicates various ways through the material, depending on the aims and background of the course being taught." J.Davenport, The SAC Newsletter 2, 1997 "Symbolic Integration I is the second edition of an extremely thorough account of the problem of integration in finite terms for transcendental functions. ... This book was written by the world's leading expert in the area. ... it does what it sets out to do and does it extremely well." (Sam Blake, SIAM Review, Vol. 50 (1), 2008)

• Algebraic Preliminaries.- Integration of Rational Functions.- Differential Fields.- The Order Function.- Integration of Transcendental Functions.- The Risch Differential Equation.- Parametric Problems.- The Coupled Differential System.- Structure Theorems.- Parallel Integration.

From the reviews of the second edition: "It discusses ... the integration of transcendental functions ... . Indeed, one of the most remarkable characteristics of this book is that it requires from its readers very little beyond a basic knowledge of calculus and algebra. ... This is an extremely well-written book on a beautiful topic that deserves to be better known to practising mathematicians and teachers of mathematics alike." S. C. Coutinho, The Mathematical Gazette, Vol. 90 (5l9), 2006 "[...] The book contains many examples and exercises for the reader to get a feeling not only of the theoretical but also the practical aspects of the algorithms. It is designed as a textbook and well suited for courses about these constructive aspects of calculus. A "must have" for the bookshelf of every library." H.-G.Grabe, Zeitschrift fur Angewandte Analysis und Ihre Anwendungen, Vol. 24, Issue 4, 2005 "The second edition of Bronstein's excellent book [...] contains a new chapter on parallel integration (also called the new Risch algorithm or the Risch-Norman algorithm) [...] The first nine chapters of the book contain some improvements and a couple of additional exercises. Bronstein's book still presents the state of the art in the domain of integration of transcendental functions. Its unique blend of detailed mathematics and clear description of algorithms make it both a standard reference and a handbook for researchers and designers of computer algebra systems and a useful, easy to read textbook for teachers and students." Friedrich Schwarz (Paderborn), Zentralblatt MATH 1059 From the reviews of the first edition: "The goal of this well-written book is to present the state of the art in the domain of "integration in finite terms". ... Both aspects of the work, mathematics and implementation, are present in the book. Moreover, Bronstein has chosen a good level of detail, and in such a way that he only deals with the case of transcendental functions. ..." J.M.Ollagnier, Mathematical Reviews 2002 "... It makes an excellent textbook for courses in computer algebra. It contains many exercises and the algorithms are presented in pseudocode, which is easy to implement in any computer algebra system. For those interested in symbolic integration it will become the standard reference." N.A.van Arkel, Medelingen van het wiskundig genootschap 1998 "... The writing is excellent, and the author provides a clear and coherent treatment of the problem of symbolic integration of transcendental functions. Each chapter includes several worked examples and a list of additional exercises. Every researcher and teacher in symbolic computation should have access to this book." F.Winkler, Computing Reviews 1997 "... In fact, this book is an excellent blend of mathematics and algorithms. ... This well-written book serves as a good foundation to the topic of symbolic integration. ... Highly recommendable." P.Kravanja, Journal of Computational and Applied Mathematics / Newsletter on Computational and Applied Mathematics 1997 "My first thought on seeing this book was "I wish I had written it" - it resembles my lecture notes on the subject, but is better and more complete. ... In sum, the book does what it sets out to do, does it well, and should be on the bookshelf of every implementer or teacher. For the latter, the author indicates various ways through the material, depending on the aims and background of the course being taught." J.Davenport, The SAC Newsletter 2, 1997 "Symbolic Integration I is the second edition of an extremely thorough account of the problem of integration in finite terms for transcendental functions. ... This book was written by the world's leading expert in the area. ... it does what it sets out to do and does it extremely well." (Sam Blake, SIAM Review, Vol. 50 (1), 2008)

• Algebraic Preliminaries.- Integration of Rational Functions.- Differential Fields.- The Order Function.- Integration of Transcendental Functions.- The Risch Differential Equation.- Parametric Problems.- The Coupled Differential System.- Structure Theorems.- Parallel Integration.

From the reviews of the second edition: "It discusses ... the integration of transcendental functions ... . Indeed, one of the most remarkable characteristics of this book is that it requires from its readers very little beyond a basic knowledge of calculus and algebra. ... This is an extremely well-written book on a beautiful topic that deserves to be better known to practising mathematicians and teachers of mathematics alike." S. C. Coutinho, The Mathematical Gazette, Vol. 90 (5l9), 2006 "[...] The book contains many examples and exercises for the reader to get a feeling not only of the theoretical but also the practical aspects of the algorithms. It is designed as a textbook and well suited for courses about these constructive aspects of calculus. A "must have" for the bookshelf of every library." H.-G.Grabe, Zeitschrift fur Angewandte Analysis und Ihre Anwendungen, Vol. 24, Issue 4, 2005 "The second edition of Bronstein's excellent book [...] contains a new chapter on parallel integration (also called the new Risch algorithm or the Risch-Norman algorithm) [...] The first nine chapters of the book contain some improvements and a couple of additional exercises. Bronstein's book still presents the state of the art in the domain of integration of transcendental functions. Its unique blend of detailed mathematics and clear description of algorithms make it both a standard reference and a handbook for researchers and designers of computer algebra systems and a useful, easy to read textbook for teachers and students." Friedrich Schwarz (Paderborn), Zentralblatt MATH 1059 From the reviews of the first edition: "The goal of this well-written book is to present the state of the art in the domain of "integration in finite terms". ... Both aspects of the work, mathematics and implementation, are present in the book. Moreover, Bronstein has chosen a good level of detail, and in such a way that he only deals with the case of transcendental functions. ..." J.M.Ollagnier, Mathematical Reviews 2002 "... It makes an excellent textbook for courses in computer algebra. It contains many exercises and the algorithms are presented in pseudocode, which is easy to implement in any computer algebra system. For those interested in symbolic integration it will become the standard reference." N.A.van Arkel, Medelingen van het wiskundig genootschap 1998 "... The writing is excellent, and the author provides a clear and coherent treatment of the problem of symbolic integration of transcendental functions. Each chapter includes several worked examples and a list of additional exercises. Every researcher and teacher in symbolic computation should have access to this book." F.Winkler, Computing Reviews 1997 "... In fact, this book is an excellent blend of mathematics and algorithms. ... This well-written book serves as a good foundation to the topic of symbolic integration. ... Highly recommendable." P.Kravanja, Journal of Computational and Applied Mathematics / Newsletter on Computational and Applied Mathematics 1997 "My first thought on seeing this book was "I wish I had written it" - it resembles my lecture notes on the subject, but is better and more complete. ... In sum, the book does what it sets out to do, does it well, and should be on the bookshelf of every implementer or teacher. For the latter, the author indicates various ways through the material, depending on the aims and background of the course being taught." J.Davenport, The SAC Newsletter 2, 1997 "Symbolic Integration I is the second edition of an extremely thorough account of the problem of integration in finite terms for transcendental functions. ... This book was written by the world's leading expert in the area. ... it does what it sets out to do and does it extremely well." (Sam Blake, SIAM Review, Vol. 50 (1), 2008)

• Algebraic Preliminaries.- Integration of Rational Functions.- Differential Fields.- The Order Function.- Integration of Transcendental Functions.- The Risch Differential Equation.- Parametric Problems.- The Coupled Differential System.- Structure Theorems.- Parallel Integration.