Numerical Analysis: Why this function is giving Sage / WolframAlpha so much trouble to integrate?

The integrand $f(\theta):=\sin(2\cos\theta)\cos(4\theta)$ is an even function: $f(-\theta)=f(\theta)$. So the integral of $f$ over the interval $[-\pi,\pi]$ is double its value over $[0,\pi]$. However, check also that $$f\left(\frac\pi2+t\right)=-f\left(\frac\pi2-t\right),$$ i.e. the integrand has odd symmetry around the point $(\pi/2,0)$. Hence the integral equals zero.

The graph of the function makes it obvious: