How to fastest approximate definite integrals I know that a definite integral is a limit of Riemann sums. So if one wanted to estimate a definite integral (because one might not be able to find an antiderivative), then one can just take enough subintervals.

I have recently learnt that one can also write the function considered as a power series (Taylor series or Maclaurin series). Then it is easy to find an antiderivative power series that one can then use to estimate the definite integral.

My question is: Which of these two methods is in general fastest?

As an example I considered $f(x) = x^3\arctan(x)$ and the integral
$$
\int_0^{1/2} f(x) dx.
$$
I get that
$$
\int_0^{1/2} f(x) dx = \sum_{n=0}^{\infty} (-1)^n\frac{(1/2)^{2n+5}}{(2n+1)(2n+5)}.
$$
The first three terms alone give an estimate of $0.0059$.

Using three left rectangles, I get $0.1211$ and the real answer is $0.00591592$. It looks like it is faster to use the Taylor series approach. Is this in general true?
Taylor series don't always converge.  The Taylor series method can be very good if the interval of integration is inside the radius of convergence of the series, otherwise it may not converge at all.

But as Yves Daoust mentioned, you should be comparing this to more sophisticated numerical methods, not to Riemann sums.  In practice, Riemann sums are almost never used for numerical integration.
Take a look at the wiki page for Monte-Carlo Integration:

http://en.wikipedia.org/wiki/Monte_Carlo_integration

This is how mathematica and other systems approximate definite integrals.

And it's devilishly clever.