Optimal. Leaf size=117 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{\sqrt{2}-1}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{1+\sqrt{2}}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{\sqrt{2}-1}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{1+\sqrt{2}}} \]
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Rubi [A] time = 0.0574979, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1419, 1093, 203, 207} \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{\sqrt{2}-1}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{1+\sqrt{2}}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{\sqrt{2}-1}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{1+\sqrt{2}}} \]
Antiderivative was successfully verified.
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Rule 1419
Rule 1093
Rule 203
Rule 207
Rubi steps
\begin{align*} \int \frac{1+x^4}{1-6 x^4+x^8} \, dx &=\frac{1}{2} \int \frac{1}{1-2 \sqrt{2} x^2+x^4} \, dx+\frac{1}{2} \int \frac{1}{1+2 \sqrt{2} x^2+x^4} \, dx\\ &=\frac{1}{4} \int \frac{1}{-1-\sqrt{2}+x^2} \, dx-\frac{1}{4} \int \frac{1}{1-\sqrt{2}+x^2} \, dx+\frac{1}{4} \int \frac{1}{-1+\sqrt{2}+x^2} \, dx-\frac{1}{4} \int \frac{1}{1+\sqrt{2}+x^2} \, dx\\ &=\frac{\tan ^{-1}\left (\frac{x}{\sqrt{-1+\sqrt{2}}}\right )}{4 \sqrt{-1+\sqrt{2}}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{1+\sqrt{2}}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{-1+\sqrt{2}}}\right )}{4 \sqrt{-1+\sqrt{2}}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{1+\sqrt{2}}}\\ \end{align*}
Mathematica [A] time = 0.0452306, size = 111, normalized size = 0.95 \[ \frac{1}{4} \left (\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )-\sqrt{\sqrt{2}-1} \tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )-\sqrt{\sqrt{2}-1} \tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 78, normalized size = 0.7 \begin{align*}{\frac{1}{4\,\sqrt{\sqrt{2}-1}}\arctan \left ({\frac{x}{\sqrt{\sqrt{2}-1}}} \right ) }+{\frac{1}{4\,\sqrt{\sqrt{2}-1}}{\it Artanh} \left ({\frac{x}{\sqrt{\sqrt{2}-1}}} \right ) }-{\frac{1}{4\,\sqrt{1+\sqrt{2}}}\arctan \left ({\frac{x}{\sqrt{1+\sqrt{2}}}} \right ) }-{\frac{1}{4\,\sqrt{1+\sqrt{2}}}{\it Artanh} \left ({\frac{x}{\sqrt{1+\sqrt{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} + 1}{x^{8} - 6 \, x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.27806, size = 616, normalized size = 5.26 \begin{align*} -\frac{1}{2} \, \sqrt{\sqrt{2} + 1} \arctan \left (-x \sqrt{\sqrt{2} + 1} + \sqrt{x^{2} + \sqrt{2} - 1} \sqrt{\sqrt{2} + 1}\right ) + \frac{1}{2} \, \sqrt{\sqrt{2} - 1} \arctan \left (-x \sqrt{\sqrt{2} - 1} + \sqrt{x^{2} + \sqrt{2} + 1} \sqrt{\sqrt{2} - 1}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} - 1} \log \left ({\left (\sqrt{2} + 1\right )} \sqrt{\sqrt{2} - 1} + x\right ) + \frac{1}{8} \, \sqrt{\sqrt{2} - 1} \log \left (-{\left (\sqrt{2} + 1\right )} \sqrt{\sqrt{2} - 1} + x\right ) + \frac{1}{8} \, \sqrt{\sqrt{2} + 1} \log \left (\sqrt{\sqrt{2} + 1}{\left (\sqrt{2} - 1\right )} + x\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 1} \log \left (-\sqrt{\sqrt{2} + 1}{\left (\sqrt{2} - 1\right )} + x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.860665, size = 49, normalized size = 0.42 \begin{align*} \operatorname{RootSum}{\left (4096 t^{4} - 128 t^{2} - 1, \left ( t \mapsto t \log{\left (16384 t^{5} - 20 t + x \right )} \right )\right )} + \operatorname{RootSum}{\left (4096 t^{4} + 128 t^{2} - 1, \left ( t \mapsto t \log{\left (16384 t^{5} - 20 t + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21443, size = 166, normalized size = 1.42 \begin{align*} -\frac{1}{4} \, \sqrt{\sqrt{2} - 1} \arctan \left (\frac{x}{\sqrt{\sqrt{2} + 1}}\right ) + \frac{1}{4} \, \sqrt{\sqrt{2} + 1} \arctan \left (\frac{x}{\sqrt{\sqrt{2} - 1}}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} - 1} \log \left ({\left | x + \sqrt{\sqrt{2} + 1} \right |}\right ) + \frac{1}{8} \, \sqrt{\sqrt{2} - 1} \log \left ({\left | x - \sqrt{\sqrt{2} + 1} \right |}\right ) + \frac{1}{8} \, \sqrt{\sqrt{2} + 1} \log \left ({\left | x + \sqrt{\sqrt{2} - 1} \right |}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 1} \log \left ({\left | x - \sqrt{\sqrt{2} - 1} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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