Evaluate the indefinite integral, $$\int\frac{x}{(x^4+1)^2 \sqrt{(x^2-x+1)}} \mathrm{d}x$$
Found this problem in a mathematics group site, but the solution was never posted. I suspect it cannot be solved with pen and paper. Online integration sites failed to complete it also. Is it possible? Tried integration by parts, trig substitution. Problem is the denominator: product of the square root of a quadratic and a 4th degree polynomial squared.
Yes, it is possible. And no, it isn't very nice.
We can deal with $\sqrt{x^2-x+1}$ by the substitution $2x=1+\sqrt{3}\tan\theta$. This converts the integral to an integral of a rational function of $\sin\theta$ and $\cos\theta$, so you can use the $t$-substitution to get it as the integral of a rational function of $t$ (i.e., $x=\frac12+\sqrt{3}t/(1+t^2)$). Note that you need to factor a high degree polynomial in $t$, and this is not very nice.