I can recommend reading "<a href="http://carma.newcastle.edu.au/jon/closed-form.pdf">Closed forms: what they are and why we care</a>" by Jon Borwein and Richard Crandall, the article is to appear in <i>Notices Amer. Math. Soc.</i> <b>60</b> (2013).

**Edit** (Dec 2012). The paper has just appeared in the Notices: <a href="http://www.ams.org/notices/201301/rnoti-p50.pdf">pdf</a>. Together with the sad news about Richard Crandall: he passed away.