Optimal. Leaf size=85 \[ -\frac{\log \left (x^2-\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2} x+1\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{2 \sqrt{2}} \]
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Rubi [A] time = 0.0454182, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {28, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\log \left (x^2-\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2} x+1\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 28
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1+x^4}{1+2 x^4+x^8} \, dx &=\int \frac{1}{1+x^4} \, dx\\ &=\frac{1}{2} \int \frac{1-x^2}{1+x^4} \, dx+\frac{1}{2} \int \frac{1+x^2}{1+x^4} \, dx\\ &=\frac{1}{4} \int \frac{1}{1-\sqrt{2} x+x^2} \, dx+\frac{1}{4} \int \frac{1}{1+\sqrt{2} x+x^2} \, dx-\frac{\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx}{4 \sqrt{2}}-\frac{\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx}{4 \sqrt{2}}\\ &=-\frac{\log \left (1-\sqrt{2} x+x^2\right )}{4 \sqrt{2}}+\frac{\log \left (1+\sqrt{2} x+x^2\right )}{4 \sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} x\right )}{2 \sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} x\right )}{2 \sqrt{2}}\\ &=-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (1+\sqrt{2} x\right )}{2 \sqrt{2}}-\frac{\log \left (1-\sqrt{2} x+x^2\right )}{4 \sqrt{2}}+\frac{\log \left (1+\sqrt{2} x+x^2\right )}{4 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0199455, size = 64, normalized size = 0.75 \[ \frac{-\log \left (x^2-\sqrt{2} x+1\right )+\log \left (x^2+\sqrt{2} x+1\right )-2 \tan ^{-1}\left (1-\sqrt{2} x\right )+2 \tan ^{-1}\left (\sqrt{2} x+1\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 58, normalized size = 0.7 \begin{align*}{\frac{\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{4}}+{\frac{\arctan \left ( -1+x\sqrt{2} \right ) \sqrt{2}}{4}}+{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}+x\sqrt{2}}{1+{x}^{2}-x\sqrt{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50123, size = 97, normalized size = 1.14 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{8} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{8} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35686, size = 302, normalized size = 3.55 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} - 1\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} + 1\right ) + \frac{1}{8} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{8} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.146687, size = 73, normalized size = 0.86 \begin{align*} - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{8} + \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12382, size = 97, normalized size = 1.14 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{8} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{8} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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