Optimal. Leaf size=125 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{2 \left (\sqrt{2}-1\right )}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{2 \left (\sqrt{2}-1\right )}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}} \]
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Rubi [A] time = 0.0674187, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1419, 1093, 207, 203} \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{2 \left (\sqrt{2}-1\right )}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{2 \left (\sqrt{2}-1\right )}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}} \]
Antiderivative was successfully verified.
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Rule 1419
Rule 1093
Rule 207
Rule 203
Rubi steps
\begin{align*} \int \frac{1-x^4}{1-6 x^4+x^8} \, dx &=-\left (\frac{1}{2} \int \frac{1}{-1-2 x^2+x^4} \, dx\right )-\frac{1}{2} \int \frac{1}{-1+2 x^2+x^4} \, dx\\ &=-\frac{\int \frac{1}{-1-\sqrt{2}+x^2} \, dx}{4 \sqrt{2}}-\frac{\int \frac{1}{1-\sqrt{2}+x^2} \, dx}{4 \sqrt{2}}+\frac{\int \frac{1}{-1+\sqrt{2}+x^2} \, dx}{4 \sqrt{2}}+\frac{\int \frac{1}{1+\sqrt{2}+x^2} \, dx}{4 \sqrt{2}}\\ &=\frac{\tan ^{-1}\left (\frac{x}{\sqrt{-1+\sqrt{2}}}\right )}{4 \sqrt{2 \left (-1+\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{-1+\sqrt{2}}}\right )}{4 \sqrt{2 \left (-1+\sqrt{2}\right )}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}}\\ \end{align*}
Mathematica [A] time = 0.0541279, size = 114, normalized size = 0.91 \[ \frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )+\sqrt{\sqrt{2}-1} \tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )+\sqrt{\sqrt{2}-1} \tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 90, normalized size = 0.7 \begin{align*}{\frac{\sqrt{2}}{8\,\sqrt{\sqrt{2}-1}}{\it Artanh} \left ({\frac{x}{\sqrt{\sqrt{2}-1}}} \right ) }+{\frac{\sqrt{2}}{8\,\sqrt{1+\sqrt{2}}}\arctan \left ({\frac{x}{\sqrt{1+\sqrt{2}}}} \right ) }+{\frac{\sqrt{2}}{8\,\sqrt{1+\sqrt{2}}}{\it Artanh} \left ({\frac{x}{\sqrt{1+\sqrt{2}}}} \right ) }+{\frac{\sqrt{2}}{8\,\sqrt{\sqrt{2}-1}}\arctan \left ({\frac{x}{\sqrt{\sqrt{2}-1}}} \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 = 1.3468, size = 686, normalized size = 5.49 \begin{align*} -\frac{1}{4} \, \sqrt{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}{4} \, \sqrt{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}{16} \, \sqrt{2} \sqrt{\sqrt{2} - 1} \log \left ({\left (\sqrt{2} + 1\right )} \sqrt{\sqrt{2} - 1} + x\right ) - \frac{1}{16} \, \sqrt{2} \sqrt{\sqrt{2} - 1} \log \left (-{\left (\sqrt{2} + 1\right )} \sqrt{\sqrt{2} - 1} + x\right ) + \frac{1}{16} \, \sqrt{2} \sqrt{\sqrt{2} + 1} \log \left (\sqrt{\sqrt{2} + 1}{\left (\sqrt{2} - 1\right )} + x\right ) - \frac{1}{16} \, \sqrt{2} \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 = 0.868287, size = 51, normalized size = 0.41 \begin{align*} - \operatorname{RootSum}{\left (16384 t^{4} - 256 t^{2} - 1, \left ( t \mapsto t \log{\left (65536 t^{5} - 28 t + x \right )} \right )\right )} - \operatorname{RootSum}{\left (16384 t^{4} + 256 t^{2} - 1, \left ( t \mapsto t \log{\left (65536 t^{5} - 28 t + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2046, size = 182, normalized size = 1.46 \begin{align*} \frac{1}{8} \, \sqrt{2 \, \sqrt{2} - 2} \arctan \left (\frac{x}{\sqrt{\sqrt{2} + 1}}\right ) + \frac{1}{8} \, \sqrt{2 \, \sqrt{2} + 2} \arctan \left (\frac{x}{\sqrt{\sqrt{2} - 1}}\right ) + \frac{1}{16} \, \sqrt{2 \, \sqrt{2} - 2} \log \left ({\left | x + \sqrt{\sqrt{2} + 1} \right |}\right ) - \frac{1}{16} \, \sqrt{2 \, \sqrt{2} - 2} \log \left ({\left | x - \sqrt{\sqrt{2} + 1} \right |}\right ) + \frac{1}{16} \, \sqrt{2 \, \sqrt{2} + 2} \log \left ({\left | x + \sqrt{\sqrt{2} - 1} \right |}\right ) - \frac{1}{16} \, \sqrt{2 \, \sqrt{2} + 2} \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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