Need help with $\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$ Please help me to evaluate this integral:
$$\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$$

\begin{align*}
  \int_{0}^{\infty} \, \frac{e^{-x}}{2^{1/3}+\cos{x}}\, dx &= \int_{0}^{\infty} \, \sum_{n\ge 0} (-1)^n \frac{e^{-x}\, \left(\cos{x}\right)^n}{2^{(n+1)/3}} \, dx \tag 1
\end{align*}
Let $\displaystyle I_n=\int_{0}^{\infty} \, e^{-x}\, \left(\cos{x}\right)^n \, dx$, which has a reduction formula:
$\displaystyle I_{n+2}=\frac{\left(n^2+3\, n+2\right)I_n + 1}{n^2+4\, n+5}\; \; , I_0=1, I_1=\frac{1}{2}$
Hence, $(1)$ can be written as the sum:
\begin{align*}
  \int_{0}^{\infty} \, \frac{e^{-x}}{2^{1/3}+\cos{x}}\, dx &= \sum_{n\ge 0} (-1)^{n}\frac{I_n}{2^{(n+1)/3}} \approx 0.763521951811874989
\end{align*}
If there exists a closed form for the recurrence, then we can expect a closed form for this integral as well.
