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This work merges two key areas in computer algebra: symbolic integration and summation, alongside fast algorithmics. In algorithmically driven fields, the analysis of algorithms, highlighted by Don Knuth's influential talk at the 1970 ICM, serves as a clear success criterion. Researchers who develop algorithms that outperform previous methods (asymptotically, in the worst case) gain immediate recognition for their contributions. However, such breakthroughs are rare, despite ongoing efforts. An alternative evaluation method involves testing new algorithms on examples, which has its limitations but can be the best available approach. George Collins, a pioneer in computer algebra, noted in 1969 that simple analysis often provides more insight than extensive empirical data, although both are valuable. Within computer algebra, some areas, like polynomial algebra and linear algebra, have adhered to this analytical methodology, while others, such as polynomial system solving, have not fully embraced it. The conventional “input size” parameters used in computer science appear insufficient, and while some natural “geometric” parameters have been identified (like solution dimension and regularity), they do not encompass all potential major advancements. Symbolic integration and summation face similar challenges in this regard.

This book introduces the software package MuPAD, a computer algebra system developed at the University of Paderborn, Germany. It targets mathematicians, engineers, computer scientists, natural scientists, and anyone needing mathematical computations for education or professional purposes. The content is designed for anyone interested in leveraging the capabilities of a modern computer algebra package. There are two primary ways to utilize a computer algebra system. First, users can interactively access the system's mathematical knowledge by calling functions to perform tasks such as computing symbolic integrals or generating and inverting matrices. Chapters 2 through 15 focus on this interactive use of MuPAD. The second approach involves using MuPAD's programming language to enhance the system by creating custom algorithms as procedures, which is beneficial for specialized applications lacking appropriate system functions. Chapters 16 through 18 serve as an introduction to programming in MuPAD. While the book can be read linearly from start to finish, alternative approaches to navigation are encouraged for a more tailored learning experience.