Can we find a closed form for the following integral:
$$\int_0^{\infty} \frac{e^{-x} \cos x}{1+x} \, {\rm d}x$$
No matter how hard I tried I cannot tackle it. I am pretty much afraid that if a closed form exists then it will involve hypergeometric functions. If that is the case, then I am not so interested in finding the closed form.
Notice that $$ \int_0^{\infty} \frac{e^{-x} \cos \left( x \right)}{1+x} = Re \int_0^{\infty} \frac{e^{-x} e^{ix}}{1+x}$$ Now, substitute $x \rightarrow x-1$ and then $x \rightarrow \frac{x}{i-1}$ now and use the fact that $$ Ei \left( x \right) = -\int_{-x}^\infty \frac{e^{-x}}{x} \mathrm{d}x$$