∫(x^4-1)/(x^2 √(x^4+x^2+1)) dx
This one is long, but not overly complicated. I expect that u=x^2 is a good substitution to start with. Then I expect a trig or hyperbolic trig substitution will come in handy.
I suggest you enter your integrand at this URL and it will show all the steps involved.
http://www.integral-calculator.com/
yep, as I surmised. Note that at the end there are some complicated logs involved, but those are just arcsec and arctan expressed as logs.
Recall that
arctanh(z) = 1/2 (ln(1+z)-ln(1-z))