TITLE: Numerical Analysis: Why this function is giving Sage / WolframAlpha so much trouble to integrate?
QUESTION [6 upvotes]: I am trying to find the numerical value of $$\int_{-\pi}^\pi \sin(2\cos\theta)\cos(4\theta)\,d\theta$$.
The integrand looks like a nice, continuous function that is finite, bounded, differentiable, etc. Furthermore, I am just looking for a numerical value, not the exact antiderivative.
However, upon trying out first with WolframAlpha and Sage, none are able to give a satisfactory numerical value.
WolframAlpha ran out of computation time, and Sage gave an answer of order $10^{-17}$ with an error of order $10^{-15}$, i.e. the error is larger than the answer itself.
I am not an expert in numerical analysis, so I am puzzled at why this happened?
Any help will be greatly appreciated. Thanks.
Bonus: If anyone can tell me what is the numerical value (just accurate to 3 significant figures will be enough), I will upvote and accept your answer gratefully.

REPLY [17 votes]: The graph of the function makes it obvious: