Finding irrational function including $\sin$ and $\cos$

Asked Jan 21 '17 at 07:22

Active Jan 21 '17 at 07:22

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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

asked Jan 21 '17 at 07:22

DXT

1

Where did you get this integral? It must relate to the incomplete elliptic integral. – Renascence_5. Jan 21 '17 at 07:33
this integral can not expressed in the known elementary functions – Dr. Sonnhard Graubner Jan 21 '17 at 08:19
Even $\int \frac{\cos^4 x-\sin^4 x}{\sqrt{1+\cos^2 x}}dx$ leads to elliptic integrals. – Claude Leibovici Jan 21 '17 at 08:21
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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 ? – Adren Jan 21 '17 at 08:23

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