∫ (dx/((x^4 +x^2 +1)^(3/4) ))


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Question Number 85488 by john santu last updated on 22/Mar/20

$\int \frac{d x}{{(x^{4} + x^{2} + 1)}^{\frac{3}{4}}}$
	Answered by MJS last updated on 22/Mar/20

	$trying around I landed on$ $t = \arctan \frac{\sqrt{3}}{2 x^{2} + 1} \to d x = - \frac{x^{4} + x^{2} + 1}{\sqrt{3} x}$ $\Rightarrow \int \frac{d x}{{(x^{4} + x^{2} + 1)}^{\frac{3}{4}}} = - \frac{1}{\sqrt[4]{12}} \int \frac{d t}{\sqrt{\cos (t + \frac{π}{6})}}$ $and this leads to an elliptic integral like in$ $qu . 85456$
	Commented by MJS last updated on 22/Mar/20
	https://en.wikipedia.org/wiki/Elliptic_integral
	Commented by john santu last updated on 22/Mar/20

	$e l l i p t i c i n t e g r a l ? i t s a m e$ $t o h i p e r b o l i c f u n c t i o n$
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