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




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

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