TITLE: The unbeatable $\int e^{1/\cos(x)} dx$ integral
QUESTION [1 upvotes]: Is there any way to express this in non-elementary functions?
$$ \int e^{1/\cos(x)} dx$$ 
And/or to calculate this definite integrals?
$$ \int_{-\pi/2}^{\pi/2} e^{1/\cos(x)} dx$$
$$ \int_{\pi/2}^{3\pi/ 2} e^{1/\cos(x)} dx$$
Added later:
First integral does not converges, but second on seems to be smooth function and WolframAlpha thinks it is approximately $0.656573$.  

Really, nothing is known about this function?
$$f(x)=\int_{\pi/2}^x e^{1/\cos(x)} dx; x \in ({\pi/2} ,{3\pi/ 2}]$$

REPLY [2 votes]: The indefinite integral is not an elementary function.  This can be proven using the Risch algorithm.
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