Evaluating the integral $\int_{-\ln 2}^{\ln 2} e^{-x}(\sin x+x)^{1/3}dx$

As the title states, I'm having quite a lot of trouble with the integral:

$$\int_{-\ln 2}^{\ln 2} \frac{(\sin x+x)^{1/3}}{e^x}dx$$

The problem is that my standard (admittedly sparse) repertoire of tricks seems to have no effect at all. I can't think of any clever substitution that would work, and it looks like a nasty integral all-round.

I was hoping to apply some symmetry arguments (especially considering the limits of integration), but the exponential function is neither even nor odd, so I don't see how this would work.

I'm afraid I have absolutely no idea how to approach this.

Solution 1:

With that $(\sin(x)+x)^{1/3}$, it's very unlikely to have a closed form antiderivative, and I don't see any way to get help from a contour integration. Taking some advantage of symmetry, you can express your integral as $-2 \int_0^{\ln 2} (\sin(x)+x)^{1/3} \sinh(x)\; dx$. The numerical value is approximately $-.4758557412777$. The Inverse Symbolic Calculator doesn't find anything for this. So I would hazard to guess there is no closed form.