It sounds like what you're looking for is what used to be covered in advanced calculus courses. Typically, these were 2-semester length courses that followed the elementary calculus sequence and, in the U.S., taken by 3rd and 4th year undergraduate mathematics majors. There are quite a few standard texts for these courses (which in many universities have been "replaced" by an expansion of the undergraduate real analysis course), many of which have the title Advanced Calculus, such as the texts by Angus E. Taylor and W. Robert Mann, by David V. Widder, by Robert Creighton Buck, and by Wilfred Kaplan. For more extensive coverage of classical analysis topics such as the gamma function, Laplace transform, differentiating under the integral sign, etc., look for an older text, one from before the early 1950s (when things rapidly started to get more abstract and less applied in the undergraduate mathematics curriculum), such as Advanced Calculus by Frederick S. Woods, the book that many here will recognize as the book that Richard Feynman worked through on his own while in high school.
If there is a university within driving distance of where you live, the simplest way of finding what you want is to simply look on the library shelves. According to the on-line card catalog for the university near where I live, their library has over 30 books with the title Advanced Calculus. If I include the likely number of very similar books without this exact same title that are shelved in roughly the same location as these books, the total is probably over 100 books.