Integration vs differentiation

Integration is hard! Integration is generally much harder than differentiation. This little demo allows you to enter a function and then ask for the derivative or integral. You can also generate random functions of varying complexity. Differentiation is typically quite easy, taking a fraction of a second. Integration typically takes much longer, if the process completes at all!
The point? If integration seems hard - that's because it really is! This demo shows that to be the case from a purely algorithmic perspective.

computation time:

$$\int f(x) dx = ...$$
computation time:


This webpage implements a simple form that interfaces a program that lives on my webserver written using Python and SymPy. Sympy, it should be said, is not a particularly powerful computer algebra system when compared to a commercial system like Mathematica or even an open source system like Maxima. Those systems are typically much faster and can integrate more functions. The main advantage of sympy in the context here is that it is very easy to use as a library. Furthermore, it amply illustrates the main point that integration is much harder than differentiation in a computationally quantifiable way.