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Finding $\displaystyle \int \frac{\cos^4 x-\sin^4 x}{\sqrt{1+\cos^4 x}}dx$

Attempt let $\displaystyle I = \int\frac{(\cos^2 x-\sin^2 x)(\cos^2 x+\sin^2 x)}{\sqrt{1+\cos^4 x}}dx = \int\frac{\cos 2x}{\sqrt{1+\cos^4 x}}dx$

wan,t be able to go further, could some help me, thanks

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    Where did you get this integral? It must relate to the incomplete elliptic integral. – 2017-01-21
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    this integral can not expressed in the known elementary functions – 2017-01-21
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    Even $\int \frac{\cos^4 x-\sin^4 x}{\sqrt{1+\cos^2 x}}dx$ leads to elliptic integrals. – 2017-01-21
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    If you consider instead $\displaystyle{\int\frac{\cos^4(x)-\sin^4(x)}{\sqrt{1-\cos^4(x)}}}$, you get an integral which can be explicitly calculated in terms of elementary functions. Are you sure there was no typo ? – 2017-01-21

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