Optimal. Leaf size=416 \[ -\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {1}{40} \sqrt [4]{6150-2750 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )-\frac {1}{40} \sqrt [4]{6150-2750 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )-\frac {1}{x}+\frac {\left (3+\sqrt {5}\right )^{5/4} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{5/4} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {1}{20} \sqrt [4]{6150-2750 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )+\frac {1}{20} \sqrt [4]{6150-2750 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right ) \]
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Rubi [A] time = 0.29, antiderivative size = 416, normalized size of antiderivative = 1.00, 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 = {1368, 1510, 297, 1162, 617, 204, 1165, 628} \[ -\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {1}{40} \sqrt [4]{6150-2750 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )-\frac {1}{40} \sqrt [4]{6150-2750 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )-\frac {1}{x}+\frac {\left (3+\sqrt {5}\right )^{5/4} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{5/4} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {1}{20} \sqrt [4]{6150-2750 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )+\frac {1}{20} \sqrt [4]{6150-2750 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1368
Rule 1510
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (1+3 x^4+x^8\right )} \, dx &=-\frac {1}{x}+\int \frac {x^2 \left (-3-x^4\right )}{1+3 x^4+x^8} \, dx\\ &=-\frac {1}{x}+\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 {1}{x}-\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 {1}{x}-\frac {\left (3+\sqrt {5}\right )^{5/4} \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\ 2^{3/4} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{5/4} \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\ 2^{3/4} \sqrt {5}}+\frac {\left (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}{8 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}+\frac {\left (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}{8 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}+\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 {1}{x}-\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {1}{40} \sqrt [4]{6150-2750 \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}{40} \sqrt [4]{6150-2750 \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 {\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 (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 (-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 (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 {1}{x}+\frac {\sqrt [4]{246+110 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{246+110 \sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4 \sqrt {5}}-\frac {1}{20} \sqrt [4]{6150-2750 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )+\frac {\sqrt [4]{246-110 \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 )^{5/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {1}{40} \sqrt [4]{6150-2750 \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}{40} \sqrt [4]{6150-2750 \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 )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 61, normalized size = 0.15 \[ -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8+3 \text {$\#$1}^4+1\& ,\frac {\text {$\#$1}^4 \log (x-\text {$\#$1})+3 \log (x-\text {$\#$1})}{2 \text {$\#$1}^5+3 \text {$\#$1}}\& \right ]-\frac {1}{x} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 1017, normalized size = 2.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 244, normalized size = 0.59 \[ -\frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {25 \, \sqrt {5} + 55} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {25 \, \sqrt {5} + 55} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {25 \, \sqrt {5} - 55} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {25 \, \sqrt {5} - 55} - \frac {1}{40} \, \sqrt {25 \, \sqrt {5} - 55} \log \left (748225 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 748225 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {25 \, \sqrt {5} - 55} \log \left (748225 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 748225 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {25 \, \sqrt {5} + 55} \log \left (180625 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 180625 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {25 \, \sqrt {5} + 55} \log \left (180625 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 180625 \, x^{2}\right ) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 52, normalized size = 0.12 \[ -\frac {\left (\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{6}+3 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{2}\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )+x \right )}{4 \left (2 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{7}+3 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{3}\right )}-\frac {1}{x} \]
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 \[ -\frac {1}{x} - \int \frac {x^{6} + 3 \, x^{2}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 292, normalized size = 0.70 \[ -\frac {1}{x}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2585\,2^{3/4}\,x\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}{2\,\left (3025\,\sqrt {5}+6765\right )}+\frac {1155\,2^{3/4}\,\sqrt {5}\,x\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}{2\,\left (3025\,\sqrt {5}+6765\right )}\right )\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2585\,2^{3/4}\,x\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}{2\,\left (3025\,\sqrt {5}-6765\right )}-\frac {1155\,2^{3/4}\,\sqrt {5}\,x\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}{2\,\left (3025\,\sqrt {5}-6765\right )}\right )\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}\,2585{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}+6765\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}\,1155{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}+6765\right )}\right )\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}\,1{}\mathrm {i}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (55\,\sqrt {5}-123\right )}^{1/4}\,2585{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}-6765\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (55\,\sqrt {5}-123\right )}^{1/4}\,1155{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}-6765\right )}\right )\,{\left (55\,\sqrt {5}-123\right )}^{1/4}\,1{}\mathrm {i}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.61, size = 32, normalized size = 0.08 \[ \operatorname {RootSum} {\left (40960000 t^{8} + 787200 t^{4} + 1, \left (t \mapsto t \log {\left (\frac {19251200 t^{7}}{11} + \frac {369792 t^{3}}{11} + x \right )} \right )\right )} - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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