Optimal. Leaf size=460 \[ -\frac {\sqrt [4]{123-55 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{123-55 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{123+55 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123+55 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+x-\frac {\sqrt [4]{123-55 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{123-55 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{123+55 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123+55 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}} \]
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Rubi [A] time = 0.42, antiderivative size = 440, normalized size of antiderivative = 0.96, number of steps used = 20, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1367, 1422, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\sqrt [4]{984-440 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{8 \sqrt {10}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{8 \sqrt {10}}+\frac {\sqrt [4]{123+55 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123+55 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+x-\frac {\sqrt [4]{984-440 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{123+55 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123+55 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1367
Rule 1422
Rubi steps
\begin {align*} \int \frac {x^8}{1+3 x^4+x^8} \, dx &=x-\int \frac {1+3 x^4}{1+3 x^4+x^8} \, dx\\ &=x-\frac {1}{10} \left (15-7 \sqrt {5}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx\\ &=x+\frac {1}{2} \sqrt {\frac {1}{10} \left (9-4 \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}{2} \sqrt {\frac {1}{10} \left (9-4 \sqrt {5}\right )} \int \frac {\sqrt {3-\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx-\frac {\left (15+7 \sqrt {5}\right ) \int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{20 \sqrt {3+\sqrt {5}}}-\frac {\left (15+7 \sqrt {5}\right ) \int \frac {\sqrt {3+\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{20 \sqrt {3+\sqrt {5}}}\\ &=x+\frac {1}{4} \sqrt {\frac {1}{5} \left (9-4 \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}{5} \left (9-4 \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 {\left (\sqrt {\frac {1}{5} \left (9-4 \sqrt {5}\right )} \sqrt [4]{3+\sqrt {5}}\right ) \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}}-\frac {\left (\sqrt {\frac {1}{5} \left (9-4 \sqrt {5}\right )} \sqrt [4]{3+\sqrt {5}}\right ) \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}}-\frac {1}{4} \sqrt {\frac {1}{5} \left (9+4 \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}{5} \left (9+4 \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]{123+55 \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]{123+55 \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}}\\ &=x-\frac {1}{8} \sqrt [4]{\frac {246}{25}-\frac {22}{\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 )+\frac {1}{8} \sqrt [4]{\frac {246}{25}-\frac {22}{\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 )+\frac {\sqrt [4]{123+55 \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]{123+55 \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]{123-55 \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]{123-55 \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]{123+55 \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]{123+55 \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}}\\ &=x-\frac {\sqrt [4]{123-55 \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]{123-55 \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]{123+55 \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]{123+55 \sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {1}{8} \sqrt [4]{\frac {246}{25}-\frac {22}{\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 )+\frac {1}{8} \sqrt [4]{\frac {246}{25}-\frac {22}{\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 )+\frac {\sqrt [4]{123+55 \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]{123+55 \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 = 58, normalized size = 0.13 \[ x-\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8+3 \text {$\#$1}^4+1\& ,\frac {3 \text {$\#$1}^4 \log (x-\text {$\#$1})+\log (x-\text {$\#$1})}{2 \text {$\#$1}^7+3 \text {$\#$1}^3}\& \right ] \]
Antiderivative was successfully verified.
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fricas [B] time = 1.05, size = 1012, normalized size = 2.20 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.73, size = 240, normalized size = 0.52 \[ -\frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {25 \, \sqrt {5} + 55} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {25 \, \sqrt {5} + 55} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {25 \, \sqrt {5} - 55} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {25 \, \sqrt {5} - 55} - \frac {1}{40} \, \sqrt {25 \, \sqrt {5} + 55} \log \left (722500 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 722500 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {25 \, \sqrt {5} + 55} \log \left (722500 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 722500 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {25 \, \sqrt {5} - 55} \log \left (2992900 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 2992900 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {25 \, \sqrt {5} - 55} \log \left (2992900 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 2992900 \, x^{2}\right ) + x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 46, normalized size = 0.10 \[ x +\frac {\left (-3 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{4}-1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{7}+12 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{3}} \]
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 \[ x - \int \frac {3 \, x^{4} + 1}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 216, normalized size = 0.47 \[ x-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {3\,2^{1/4}\,x}{2\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {5}\,x}{2\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}\right )\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {3\,2^{1/4}\,x}{2\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {5}\,x}{2\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}\right )\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{1/4}\,x\,3{}\mathrm {i}}{2\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {5}\,x\,1{}\mathrm {i}}{2\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}\right )\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}\,1{}\mathrm {i}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{1/4}\,x\,3{}\mathrm {i}}{2\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {5}\,x\,1{}\mathrm {i}}{2\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}\right )\,{\left (55\,\sqrt {5}-123\right )}^{1/4}\,1{}\mathrm {i}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.59, size = 29, normalized size = 0.06 \[ x + \operatorname {RootSum} {\left (40960000 t^{8} + 787200 t^{4} + 1, \left (t \mapsto t \log {\left (\frac {15360 t^{5}}{11} + \frac {1288 t}{55} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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