# Integration of &#x222B;<!-- ∫ --> 1 x <mrow class="MJX

Integration of $\int \frac{1}{{x}^{4}+1}\mathrm{d}x$
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$\int \frac{1}{1+{x}^{4}}dx=\frac{1}{2}\int \frac{1+{x}^{2}+1-{x}^{2}}{1+{x}^{4}}dx$
$\int \frac{1+{x}^{2}}{1+{x}^{4}}dx=\int \frac{\frac{1}{{x}^{2}}+1}{{\left(x-\frac{1}{x}\right)}^{2}+2}dx$
Set $x-\frac{1}{x}=\sqrt{2}\mathrm{tan}\varphi$
$\int \frac{1-{x}^{2}}{1+{x}^{4}}dx=-\int \frac{1-\frac{1}{{x}^{2}}}{{\left(x+\frac{1}{x}\right)}^{2}-2}dx$
Set $x+\frac{1}{x}=\sqrt{2}\mathrm{sec}\psi$
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