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

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7  
What have you tried? Where did you get stuck? Where is this problem from? Did you notice that $1-x^4 = (1-x^2)(1+x^2)$? – Fixee Apr 10 '11 at 19:07
up vote 20 down vote accepted

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

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