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?