How do I integrate this?

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How do I integrate this?

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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|dw:1326789570521:dw|
could you draw the diagram again, it's not clear
\[\int\limits \cos(\frac{1}{1+\cos(x)})dx\]

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Off the top of my head, I'd say that is transcendental and has no elementary antiderivative. Wolfram|Alpha backs me up on that.
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|dw:1326790075583:dw|
Perhaps just a rough area approximation near pi based on the width of the slice, something like the Taylor expansion around zero of the function given by: \[F(\epsilon)=\int\limits_{\pi-\epsilon}^{\pi} \cos(\frac{1}{1+\cos(t)})dt\]

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