Optimal. Leaf size=451 \[ \frac {\sqrt [4]{3-\sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}} \]
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Rubi [A] time = 0.28, antiderivative size = 451, 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, 211, 1165, 628, 1162, 617, 204} \[ \frac {\sqrt [4]{3-\sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1374
Rubi steps
\begin {align*} \int \frac {x^4}{1+3 x^4+x^8} \, dx &=-\left (\frac {1}{10} \left (-5+3 \sqrt {5}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx\right )+\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx\\ &=-\left (\frac {1}{4} \sqrt {\frac {1}{5} \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\right )-\frac {1}{4} \sqrt {\frac {1}{5} \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+\frac {1}{4} \sqrt {\frac {1}{5} \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+\frac {1}{4} \sqrt {\frac {1}{5} \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\\ &=-\left (\frac {1}{4} \sqrt {\frac {1}{10} \left (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\right )-\frac {1}{4} \sqrt {\frac {1}{10} \left (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]{3-\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\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3-\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\ 2^{3/4} \sqrt {5}}+\frac {1}{4} \sqrt {\frac {1}{10} \left (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}{4} \sqrt {\frac {1}{10} \left (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]{3+\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\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\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\ 2^{3/4} \sqrt {5}}\\ &=\frac {\sqrt [4]{3-\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\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\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\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\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\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\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\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3-\sqrt {5}} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}\\ &=\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3-\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\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\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\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\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\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\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\ 2^{3/4} \sqrt {5}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 39, normalized size = 0.09 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8+3 \text {$\#$1}^4+1\& ,\frac {\text {$\#$1} \log (x-\text {$\#$1})}{2 \text {$\#$1}^4+3}\& \right ] \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 843, normalized size = 1.87 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.77, size = 239, normalized size = 0.53 \[ \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {5 \, \sqrt {5} + 5} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {5 \, \sqrt {5} + 5} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {5 \, \sqrt {5} - 5} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {5 \, \sqrt {5} - 5} + \frac {1}{40} \, \sqrt {5 \, \sqrt {5} + 5} \log \left (625 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 625 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {5 \, \sqrt {5} + 5} \log \left (625 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 625 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {5 \, \sqrt {5} - 5} \log \left (4225 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 4225 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {5 \, \sqrt {5} - 5} \log \left (4225 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 4225 \, x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 40, normalized size = 0.09 \[ \frac {\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{4} \ln \left (-\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{7}+12 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{3}} \]
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 \[ \int \frac {x^{4}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 454, normalized size = 1.01 \[ \frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {3\,2^{3/4}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}}{2}-\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}}{2}\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}}{2}-\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}}{2}\right )}\right )\,{\left (-\sqrt {5}-3\right )}^{1/4}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}\,3{}\mathrm {i}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}}{2}-\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}}{2}\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}}{2}-\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}}{2}\right )}\right )\,{\left (-\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {3\,2^{3/4}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-3}}{2}+\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}}{2}\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-3}}{2}+\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}}{2}\right )}\right )\,{\left (\sqrt {5}-3\right )}^{1/4}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}\,3{}\mathrm {i}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-3}}{2}+\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}}{2}\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-3}}{2}+\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}}{2}\right )}\right )\,{\left (\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.48, size = 24, normalized size = 0.05 \[ \operatorname {RootSum} {\left (40960000 t^{8} + 19200 t^{4} + 1, \left (t \mapsto t \log {\left (- 51200 t^{5} - 12 t + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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