Integration problem [Solution]

The puzzle was to compute

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

Solution

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As usual, a clever substitution needs to be sought.

In this case, you divide the numerator and denominator by $x^2$ and then make the substitution $t = x - \frac{1}{x}$.

This results in

$$-\int \frac{dt}{t\sqrt{t^2+1}}$$

which is a standard integral.