Optimal. Leaf size=167 \[ \frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{144-64 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{144-64 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt{5}} \]
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Rubi [A] time = 0.0783492, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1374, 298, 203, 206} \[ \frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{144-64 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{144-64 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 1374
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{x^6}{1-3 x^4+x^8} \, dx &=\frac{1}{10} \left (5-3 \sqrt{5}\right ) \int \frac{x^2}{-\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx+\frac{1}{10} \left (5+3 \sqrt{5}\right ) \int \frac{x^2}{-\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx\\ &=\frac{\left (3-\sqrt{5}\right ) \int \frac{1}{\sqrt{3-\sqrt{5}}-\sqrt{2} x^2} \, dx}{2 \sqrt{10}}-\frac{\left (3-\sqrt{5}\right ) \int \frac{1}{\sqrt{3-\sqrt{5}}+\sqrt{2} x^2} \, dx}{2 \sqrt{10}}-\frac{\left (3+\sqrt{5}\right ) \int \frac{1}{\sqrt{3+\sqrt{5}}-\sqrt{2} x^2} \, dx}{2 \sqrt{10}}+\frac{\left (3+\sqrt{5}\right ) \int \frac{1}{\sqrt{3+\sqrt{5}}+\sqrt{2} x^2} \, dx}{2 \sqrt{10}}\\ &=\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\left (3-\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\left (3-\sqrt{5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2\ 2^{3/4} \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.150366, size = 160, normalized size = 0.96 \[ \frac{\frac{\left (\sqrt{5}-3\right ) \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{\sqrt{5}-1}}+\frac{\left (3+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{1+\sqrt{5}}}-\frac{\left (\sqrt{5}-3\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{\sqrt{5}-1}}-\frac{\left (3+\sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{1+\sqrt{5}}}}{2 \sqrt{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 206, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{3\,\sqrt{5}}{10\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{2\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{3\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{2\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{3\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{2\,\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }+{\frac{3\,\sqrt{5}}{10\,\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \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^{6}}{x^{8} - 3 \, x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96073, size = 844, normalized size = 5.05 \begin{align*} \frac{1}{5} \, \sqrt{5} \sqrt{\sqrt{5} + 2} \arctan \left (\frac{1}{4} \, \sqrt{2 \, x^{2} + \sqrt{5} + 1}{\left (\sqrt{5} \sqrt{2} - 3 \, \sqrt{2}\right )} \sqrt{\sqrt{5} + 2} - \frac{1}{2} \,{\left (\sqrt{5} x - 3 \, x\right )} \sqrt{\sqrt{5} + 2}\right ) + \frac{1}{5} \, \sqrt{5} \sqrt{\sqrt{5} - 2} \arctan \left (\frac{1}{4} \, \sqrt{2 \, x^{2} + \sqrt{5} - 1}{\left (\sqrt{5} \sqrt{2} + 3 \, \sqrt{2}\right )} \sqrt{\sqrt{5} - 2} - \frac{1}{2} \,{\left (\sqrt{5} x + 3 \, x\right )} \sqrt{\sqrt{5} - 2}\right ) - \frac{1}{20} \, \sqrt{5} \sqrt{\sqrt{5} + 2} \log \left (\sqrt{\sqrt{5} + 2}{\left (\sqrt{5} - 1\right )} + 2 \, x\right ) + \frac{1}{20} \, \sqrt{5} \sqrt{\sqrt{5} + 2} \log \left (-\sqrt{\sqrt{5} + 2}{\left (\sqrt{5} - 1\right )} + 2 \, x\right ) + \frac{1}{20} \, \sqrt{5} \sqrt{\sqrt{5} - 2} \log \left ({\left (\sqrt{5} + 1\right )} \sqrt{\sqrt{5} - 2} + 2 \, x\right ) - \frac{1}{20} \, \sqrt{5} \sqrt{\sqrt{5} - 2} \log \left (-{\left (\sqrt{5} + 1\right )} \sqrt{\sqrt{5} - 2} + 2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.896756, size = 53, normalized size = 0.32 \begin{align*} \operatorname{RootSum}{\left (6400 t^{4} - 320 t^{2} - 1, \left ( t \mapsto t \log{\left (- 1792000 t^{7} + 4920 t^{3} + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 320 t^{2} - 1, \left ( t \mapsto t \log{\left (- 1792000 t^{7} + 4920 t^{3} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21738, size = 198, normalized size = 1.19 \begin{align*} \frac{1}{10} \, \sqrt{5 \, \sqrt{5} + 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) - \frac{1}{10} \, \sqrt{5 \, \sqrt{5} - 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{20} \, \sqrt{5 \, \sqrt{5} + 10} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{20} \, \sqrt{5 \, \sqrt{5} + 10} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{20} \, \sqrt{5 \, \sqrt{5} - 10} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{20} \, \sqrt{5 \, \sqrt{5} - 10} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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