Integration itself is quite a bit of problem solving itself, and much more fundamentally understandable.

I'd rather students understand the fundamentals of that integration than have a computer apply an algorithm they don't understand. You can construct problems plenty complicated enough in intro calculus courses for students without resorting to functions they can't integrate with the methods they learn.

Have you looked at the syllabus of a numerical analysis course recently? There is plenty to learn at that level, even for students who had an exceptionally good calculus course beforehand.

Funny, I just forwarded this to my numerical analysis prof. we mostly studied solutions based on laplace in my course, but this is quite interesting (as it appeals to my inner hacker)