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 = 0.08, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 203
Rule 206
Rule 298
Rule 1374
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*}
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Mathematica [A] time = 0.15, 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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fricas [B] time = 0.75, size = 255, normalized size = 1.53 \[ \frac {1}{5} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \arctan \left (\frac {1}{4} \, \sqrt {2 \, x^{2} + \sqrt {5} + 1} {\left (\sqrt {5} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {\sqrt {5} + 2} - \frac {1}{2} \, {\left (\sqrt {5} x - 3 \, x\right )} \sqrt {\sqrt {5} + 2}\right ) + \frac {1}{5} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \arctan \left (\frac {1}{4} \, \sqrt {2 \, x^{2} + \sqrt {5} - 1} {\left (\sqrt {5} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {\sqrt {5} - 2} - \frac {1}{2} \, {\left (\sqrt {5} x + 3 \, x\right )} \sqrt {\sqrt {5} - 2}\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \log \left (\sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 1\right )} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \log \left (-\sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 1\right )} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left ({\left (\sqrt {5} + 1\right )} \sqrt {\sqrt {5} - 2} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left (-{\left (\sqrt {5} + 1\right )} \sqrt {\sqrt {5} - 2} + 2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.75, size = 147, normalized size = 0.88 \[ \frac {1}{10} \, \sqrt {5 \, \sqrt {5} + 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{10} \, \sqrt {5 \, \sqrt {5} - 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 206, normalized size = 1.23 \[ -\frac {\arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{2 \sqrt {-2+2 \sqrt {5}}}+\frac {3 \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}-\frac {\arctanh \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{2 \sqrt {2+2 \sqrt {5}}}-\frac {3 \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}+\frac {\arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{2 \sqrt {-2+2 \sqrt {5}}}-\frac {3 \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}+\frac {\arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{2 \sqrt {2+2 \sqrt {5}}}+\frac {3 \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{x^{8} - 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 147, normalized size = 0.88 \[ \frac {\sqrt {5}\,\mathrm {atan}\left (\frac {16\,x\,\sqrt {2-\sqrt {5}}}{8\,\sqrt {5}-24}\right )\,\sqrt {\sqrt {5}-2}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {16\,x\,\sqrt {-\sqrt {5}-2}}{8\,\sqrt {5}+24}\right )\,\sqrt {\sqrt {5}+2}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {2-\sqrt {5}}\,16{}\mathrm {i}}{8\,\sqrt {5}-24}\right )\,\sqrt {2-\sqrt {5}}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {5}-2}\,16{}\mathrm {i}}{8\,\sqrt {5}+24}\right )\,\sqrt {-\sqrt {5}-2}\,1{}\mathrm {i}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 53, normalized size = 0.32 \[ \operatorname {RootSum} {\left (6400 t^{4} - 320 t^{2} - 1, \left (t \mapsto t \log {\left (- 1792000 t^{7} + 4920 t^{3} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 320 t^{2} - 1, \left (t \mapsto t \log {\left (- 1792000 t^{7} + 4920 t^{3} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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