Difficult integral?

Question

There's an integral I've no idea how to solve. Even Wolfram|Alpha gives a very odd result.
$$\int \frac{\sqrt{1 - x^2} + \sqrt{1 + x^2}}{\sqrt{1 - x^4}}dx$$

Answer 1

Have you tried to take apart the integrand into two fractions? This immediately gives $$\int \left( \frac{1}{\sqrt{1 + x^2}} + \frac{1}{\sqrt{1 - x^2}} \right) \, dx = \sinh^{-1} (x) + \sin^{-1} (x) + C.$$