Integrand size = 22, antiderivative size = 377 \[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{324 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac {\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{972 \sqrt {3 \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {\left (9-2^{2/3} \sqrt [3]{3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{972 \sqrt {6 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648\ 2^{2/3}}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{648\ 2^{2/3} \sqrt [3]{3}} \]
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Time = 0.82 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2095, 648, 632, 210, 642, 212} \[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{324 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac {\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{972 \sqrt {3 \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {\left (9-2^{2/3} \sqrt [3]{3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{972 \sqrt {6 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}-\frac {\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{648\ 2^{2/3}}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{648\ 2^{2/3} \sqrt [3]{3}} \]
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Rule 210
Rule 212
Rule 632
Rule 642
Rule 648
Rule 2095
Rubi steps \begin{align*} \text {integral}& = 1259712 \int \left (\frac {(-1)^{2/3} \left (-2+6 (-3)^{2/3} \sqrt [3]{2}-\sqrt [3]{-3} 2^{2/3} x\right )}{272097792 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )}+\frac {2 (-1)^{2/3}-6 \sqrt [3]{2} 3^{2/3}+\sqrt [3]{-3} 2^{2/3} x}{272097792 \sqrt [3]{2} 3^{2/3} \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {18-2^{2/3} \sqrt [3]{3}+\sqrt [3]{2} 3^{2/3} x}{2448880128 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {18-2^{2/3} \sqrt [3]{3}+\sqrt [3]{2} 3^{2/3} x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1944}-\frac {\int \frac {2 (-1)^{2/3}-6 \sqrt [3]{2} 3^{2/3}+\sqrt [3]{-3} 2^{2/3} x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{648 \sqrt [3]{2} 3^{2/3}}+\frac {(-1)^{2/3} \int \frac {-2+6 (-3)^{2/3} \sqrt [3]{2}-\sqrt [3]{-3} 2^{2/3} x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2} \\ & = -\frac {\sqrt [3]{-\frac {1}{3}} \int \frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{648\ 2^{2/3}}+\frac {\int \frac {3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{648\ 2^{2/3} \sqrt [3]{3}}-\frac {\int \frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\left (\sqrt [3]{-1} \left (9+\sqrt [3]{-3} 2^{2/3}\right )\right ) \int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{648 \left (1+\sqrt [3]{-1}\right )^2}-\frac {\left (-9+(-2)^{2/3} \sqrt [3]{3}\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1944}+\frac {\left (9-2^{2/3} \sqrt [3]{3}\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1944} \\ & = -\frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648\ 2^{2/3}}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{648\ 2^{2/3} \sqrt [3]{3}}+\frac {\left (\sqrt [3]{-1} \left (9+\sqrt [3]{-3} 2^{2/3}\right )\right ) \text {Subst}\left (\int \frac {1}{-6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )-x^2} \, dx,x,3 \sqrt [3]{-3} 2^{2/3}-2 x\right )}{324 \left (1+\sqrt [3]{-1}\right )^2}-\frac {1}{972} \left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \text {Subst}\left (\int \frac {1}{-6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )-x^2} \, dx,x,3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )+\frac {1}{972} \left (-9+2^{2/3} \sqrt [3]{3}\right ) \text {Subst}\left (\int \frac {1}{-6 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )-x^2} \, dx,x,3\ 2^{2/3} \sqrt [3]{3}+2 x\right ) \\ & = -\frac {\sqrt [3]{-1} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{324 \left (1+\sqrt [3]{-1}\right )^2 \sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\frac {\left ((-2)^{2/3}-3\ 3^{2/3}\right ) \tan ^{-1}\left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{972 \sqrt [6]{3} \sqrt {2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}+\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{972 \sqrt [6]{3} \sqrt {2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648\ 2^{2/3}}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{648\ 2^{2/3} \sqrt [3]{3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.16 \[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.14
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(53\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(53\) |
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Timed out. \[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\text {Timed out} \]
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Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.17 \[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\operatorname {RootSum} {\left (34164988081841849499648 t^{6} - 3470494144278528 t^{4} - 86087932019712 t^{3} - 1530550080 t^{2} + 69984 t - 1, \left ( t \mapsto t \log {\left (\frac {185904446699109611410573787136 t^{5}}{57121295165} + \frac {6377301253267917382766592 t^{4}}{57121295165} - \frac {18904636002388564311552 t^{3}}{57121295165} - \frac {469080552915181723968 t^{2}}{57121295165} - \frac {24358640509989936 t}{57121295165} + x + \frac {152427895956}{57121295165} \right )} \right )\right )} \]
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\[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {1}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \]
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\[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {1}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \]
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Time = 9.36 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.81 \[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\sum _{k=1}^6\ln \left (-\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )\,x\,6+{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^2\,x\,349920-{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^3\,x\,6122200320-{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^4\,x\,258263796059136-{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^5\,x\,6940988288557056+944784\,{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^2-16529940864\,{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^3-33192121254912\,{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^4-168897381688221696\,{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^5\right )\,\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right ) \]
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