dx-x-4-x-2-1-3-4-

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