Question Number 211399 by Spillover last updated on 08/Sep/24
$$\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{4}} \right)\sqrt{\mathrm{1}+{x}^{\mathrm{4}} −{x}^{\mathrm{2}} }} \\ $$$$ \\ $$
Commented by Frix last updated on 08/Sep/24
$${t}=\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}\:\Rightarrow \\ $$$$\mathrm{2}\sqrt{\mathrm{2}}\int\frac{{dt}}{\left({t}^{\mathrm{2}} +\mathrm{2}{t}+\mathrm{5}\right)\sqrt{\left({t}+\mathrm{1}\right)\left({t}^{\mathrm{2}} +\mathrm{3}\right)}} \\ $$$$\mathrm{We}\:\mathrm{need}\:\mathrm{elliptic}\:\mathrm{functions}\:\mathrm{for}\:\mathrm{this}… \\ $$