$$\int \frac{x^4 - 1}{x^2(x^4 + x^2 + 1)^{1/2}} \, dx$$

The key is to recognize that the numerator $x^4 - 1$ and denominator structure suggest a substitution that simplifies the radical. Notice that $x^4 + x^2 + 1$ can be related to $x^4 - 1$ through algebraic manipulation—specifically, if you divide both numerator and denominator by $x^2$, you get a form in $x^2 - x^{-2}$ whose derivative appears naturally in the radical. Try the substitution $u = x - \frac{1}{x}$ to convert this into an integral of a simpler function.

The key is to recognize that the numerator $x^4 - 1$ and denominator structure suggest a substitution that simplifies the radical. Notice that $x^4 + x^2 + 1$ can be related to $x^4 - 1$ through algebraic manipulation—specifically, if you divide both numerator and denominator by $x^2$, you get a form in $x^2 - x^{-2}$ whose derivative appears naturally in the radical. Try the substitution $u = x - \frac{1}{x}$ to convert this into an integral of a simpler function.