Many academics and researchers get annoyed when students use computer algebra systems (CAS) such as Mathematica to evaluate simple integrals that they maintain should be done by hand. The question I ask is "At what point do you expect your students to switch over to using a computer?". Most mathematical examples are artificial in that closed-form expressions exist. However, in nearly any real problem, this is not the case.
Aside: see Closed-form expression, Closed forms: What they are and why they matter, and What is a Closed-Form Number?
I learnt mathematics using slide rules and tables, then calculators, then computers, then CAS. This is a useful and valid progression, but not one that everyone should have to go through. I feel that the only way true progress in teaching and learning can be made is if not all students have to learn the set of special rules (algorithms) for topics such as integration, that are better done by a CAS anyway. If we had to do calculus using Newton's geometrical constructs then progress would be very slow.
The real question is "what are the essential tools and lessons?". To me, deep understanding of the meaning of differentiation and integration ("mathematically" and "physically") is far more important than knowing how to compute a specific integral, or solving a particular differential equation.
Many people feel that reliance on a CAS means that students can't do calculus by hand and hence the really don't understand what's going on, just how to get the answer by computer. Calculus concepts are subtle. However, just knowing the mechanics of computing an integral or derivative does not imply understanding.
I believe that it is possible to have true understanding without knowing how to do algorithmic computation (see also Linear independence vs determinants), and so proper integration of CAS into mathematics courses is not only advisable, it is essential.
In a future world without computers, if all these human computational skills have been lost then it could take a long time to recover or re-discover them. In 1981, because of my physics background and experience using slide rules and tables, I was asked to navigate on a yacht sailing from Middle Harbour, Sydney, to Lord Howe Island. Learning how to use a sextant and reading GMT tables, with interpolation and hand computation, was straightforward. If there was no global GPS, I could survive again with these instruments, so the skills I learnt were not wasted. However, that still does not mean that everyone needs to have such skills.