Question Number 85488 by john santu last updated on 22/Mar/20
$$\int\:\frac{{dx}}{\left({x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{4}}} } \\ $$
Answered by MJS last updated on 22/Mar/20
$$\mathrm{trying}\:\mathrm{around}\:\mathrm{I}\:\mathrm{landed}\:\mathrm{on} \\ $$$${t}=\mathrm{arctan}\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{1}}\:\rightarrow\:{dx}=−\frac{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}{\sqrt{\mathrm{3}}{x}} \\ $$$$\Rightarrow\:\int\frac{{dx}}{\left({x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{4}}} }=−\frac{\mathrm{1}}{\sqrt[{\mathrm{4}}]{\mathrm{12}}}\int\frac{{dt}}{\sqrt{\mathrm{cos}\:\left({t}+\frac{\pi}{\mathrm{6}}\right)}} \\ $$$$\mathrm{and}\:\mathrm{this}\:\mathrm{leads}\:\mathrm{to}\:\mathrm{an}\:\mathrm{elliptic}\:\mathrm{integral}\:\mathrm{like}\:\mathrm{in} \\ $$$$\mathrm{qu}.\:\mathrm{85456} \\ $$
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
$${elliptic}\:{integral}\:?\:{it}\:{same} \\ $$$${to}\:{hiperbolic}\:{function}\: \\ $$