A tough one

Integral calculus isn't easy to pick up. The vast array of tools and trick needed to solve the typical anti-differentiation problem is, at best, unwieldy. I'm reminded of this each time I teach the class and problems like this

\int e^{\cos(x)} \sin(x) \; dx

stump some of my students on the quiz. I benefit from years of experience, and the approach is clear to me. But when you've just been crapped on by methods of integration by parts, substitution, partial fraction decomposition, trig substitution, and, the techniquetoendalltechniques, table look-up, life is not so good.

I presented a colleague with the following challenge:

\int \sqrt{\tan(x)} \; dx

I think I gave him a headache. It spread through the faculty like wildfire at lunch. They appreciate a good challenge. I was presented with one "solution" before the end of the day, though it had an error in step 2 which destroyed the validity of it all.

I've done this integral before, and I know the solution can be presented in closed form. Mathematica can produce the answer in about 30 seconds, and Wolfram Alpha even shows you the steps, although I'm convinced the algorithm is not the most efficient for this particular problem. You can probably google for the solution as well. You'll just miss out on all the fun.