This is an integral which I'm unable to solve till now.
$$\int \sqrt{ x + \frac1x}\text{d}x $$
I thought if Wolfram|Alpha solved it I would get some idea of how to solve it, but it gave a really complicated answer for it: http://www.wolframalpha.com/input/?i=integral+(x%2B(1%2Fx))%5E0.5
Maybe a substitution so that the squareroot vanishes (i.e. a squared substitution), but that didn't help me either. Any hints on how to solve it? I'd like to try to solve it first before knowing the solution.
Thanks!
If we replace $x$ with $t^2$ we are left with the integral: $$ 2\int \sqrt{t^4+1}\,dt $$ that can be managed through integration by parts in order to get: $$ \int\frac{t^4}{\sqrt{1+t^4}}\,dt = \int\sqrt{1+t^4}\,dt-\int\frac{dt}{\sqrt{t^4+1}}$$ so that the problem boils down to computing: $$ \int\frac{dt}{\sqrt{(t^2+i)(t^2-i)}} $$ that is an incomplete elliptic integral of the first kind. We get a series expansion of some definite integrals by using the Taylor series of $\sqrt{1+z}$ and integrating it termwise; for instance: $$ \frac{1}{2}\int_{0}^{1}\sqrt{x+\frac{1}{x}}\,dx = \phantom{}_2 F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-1\right)$$ where the RHS is a (ordinary) hypergeometric function. In general, the outcome is non-elementary.