Optimal. Leaf size=431 \[ -\frac {\sqrt [4]{9-4 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{9-4 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{8 \sqrt [4]{2} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{8 \sqrt [4]{2} \sqrt {5}}+\frac {\sqrt [4]{9-4 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {10}}-\frac {\sqrt [4]{9-4 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2 \sqrt {10}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{4 \sqrt [4]{2} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{4 \sqrt [4]{2} \sqrt {5}} \]
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Rubi [A] time = 0.29, antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1374, 297, 1162, 617, 204, 1165, 628} \[ -\frac {\sqrt [4]{9-4 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{9-4 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{8 \sqrt [4]{2} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{8 \sqrt [4]{2} \sqrt {5}}+\frac {\sqrt [4]{9-4 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {10}}-\frac {\sqrt [4]{9-4 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2 \sqrt {10}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{4 \sqrt [4]{2} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{4 \sqrt [4]{2} \sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1374
Rubi steps
\begin {align*} \int \frac {x^6}{1+3 x^4+x^8} \, dx &=-\left (\frac {1}{10} \left (-5+3 \sqrt {5}\right ) \int \frac {x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx\right )+\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 {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{4 \sqrt {10}}-\frac {\left (3-\sqrt {5}\right ) \int \frac {\sqrt {3-\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{4 \sqrt {10}}-\frac {\left (3+\sqrt {5}\right ) \int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{4 \sqrt {10}}+\frac {\left (3+\sqrt {5}\right ) \int \frac {\sqrt {3+\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{4 \sqrt {10}}\\ &=-\frac {\sqrt [4]{9-4 \sqrt {5}} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}+2 x}{-\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x-x^2} \, dx}{4 \sqrt {10}}-\frac {\sqrt [4]{9-4 \sqrt {5}} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{-\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x-x^2} \, dx}{4 \sqrt {10}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}+2 x}{-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x-x^2} \, dx}{8 \sqrt [4]{2} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x-x^2} \, dx}{8 \sqrt [4]{2} \sqrt {5}}-\frac {1}{40} \left (-5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+x^2} \, dx-\frac {1}{40} \left (-5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+x^2} \, dx+\frac {1}{40} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+x^2} \, dx+\frac {1}{40} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+x^2} \, dx\\ &=-\frac {\sqrt [4]{9-4 \sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{9-4 \sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {10}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{8 \sqrt [4]{2} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{8 \sqrt [4]{2} \sqrt {5}}-\frac {\left (3-\sqrt {5}\right )^{3/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4 \sqrt [4]{2} \sqrt {5}}-\frac {\left (5-3 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{20 \sqrt [4]{2 \left (3-\sqrt {5}\right )}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{4 \sqrt [4]{2} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{4 \sqrt [4]{2} \sqrt {5}}\\ &=\frac {\left (3-\sqrt {5}\right )^{3/4} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4 \sqrt [4]{2} \sqrt {5}}-\frac {\sqrt [4]{36-16 \sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4 \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{4 \sqrt [4]{2} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{4 \sqrt [4]{2} \sqrt {5}}-\frac {\sqrt [4]{9-4 \sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{9-4 \sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {10}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{8 \sqrt [4]{2} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{8 \sqrt [4]{2} \sqrt {5}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 41, normalized size = 0.10 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8+3 \text {$\#$1}^4+1\& ,\frac {\text {$\#$1}^3 \log (x-\text {$\#$1})}{2 \text {$\#$1}^4+3}\& \right ] \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 725, normalized size = 1.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 239, normalized size = 0.55 \[ \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {10 \, \sqrt {5} + 20} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {10 \, \sqrt {5} + 20} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {10 \, \sqrt {5} - 20} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {10 \, \sqrt {5} - 20} - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 20} \log \left (400 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 400 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 20} \log \left (400 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 400 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 20} \log \left (10000 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 10000 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 20} \log \left (10000 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 10000 \, x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 40, normalized size = 0.09 \[ \frac {\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{6} \ln \left (-\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{7}+12 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{3}} \]
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 = 1.46, size = 149, normalized size = 0.35 \[ \frac {\sqrt {5}\,\mathrm {atan}\left (\frac {16\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{8\,\sqrt {5}+24}\right )\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {16\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{8\,\sqrt {5}-24}\right )\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,16{}\mathrm {i}}{8\,\sqrt {5}+24}\right )\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,16{}\mathrm {i}}{8\,\sqrt {5}-24}\right )\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,1{}\mathrm {i}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.53, size = 26, normalized size = 0.06 \[ \operatorname {RootSum} {\left (40960000 t^{8} + 115200 t^{4} + 1, \left (t \mapsto t \log {\left (- 1792000 t^{7} - 4920 t^{3} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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