Integrand size = 26, antiderivative size = 377 \[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{9 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac {\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{27 \sqrt {3 \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {\left (9-2^{2/3} \sqrt [3]{3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{27 \sqrt {6 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}+\frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{6\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {\sqrt [3]{-\frac {1}{3}} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{18\ 2^{2/3}}-\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{18\ 2^{2/3} \sqrt [3]{3}} \]
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Time = 0.89 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2122, 648, 632, 210, 642, 212} \[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{9 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac {\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{27 \sqrt {3 \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {\left (9-2^{2/3} \sqrt [3]{3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{27 \sqrt {6 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}+\frac {\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{6\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {\sqrt [3]{-\frac {1}{3}} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{18\ 2^{2/3}}-\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{18\ 2^{2/3} \sqrt [3]{3}} \]
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Rule 210
Rule 212
Rule 632
Rule 642
Rule 648
Rule 2122
Rubi steps \begin{align*} \text {integral}& = 1259712 \int \left (\frac {(-1)^{2/3} \left (-2+\sqrt [3]{-3} 2^{2/3} x\right )}{7558272 \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 {(-1)^{2/3} \left (2+(-2)^{2/3} \sqrt [3]{3} x\right )}{7558272 \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 {\sqrt [3]{2}+\sqrt [3]{3} x}{11337408\ 6^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx \\ & = -\frac {(-1)^{2/3} \int \frac {2+(-2)^{2/3} \sqrt [3]{3} x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{18 \sqrt [3]{2} 3^{2/3}}-\frac {\int \frac {\sqrt [3]{2}+\sqrt [3]{3} x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{9\ 6^{2/3}}+\frac {(-1)^{2/3} \int \frac {-2+\sqrt [3]{-3} 2^{2/3} x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{6 \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}{18\ 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}{18\ 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}{6\ 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}{18 \left (1+\sqrt [3]{-1}\right )^2}-\frac {1}{54} \left (-9+(-2)^{2/3} \sqrt [3]{3}\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx-\frac {1}{54} \left (-9+2^{2/3} \sqrt [3]{3}\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx \\ & = \frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{6\ 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 )}{18\ 2^{2/3}}-\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{18\ 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 )}{9 \left (1+\sqrt [3]{-1}\right )^2}-\frac {1}{27} \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}{27} \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 )}{9 \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 )}{27 \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 )}{27 \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 )}{6\ 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 )}{18\ 2^{2/3}}-\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{18\ 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 = 61, normalized size of antiderivative = 0.16 \[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^3}{36+162 \text {$\#$1}+12 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.15
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(56\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(56\) |
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Timed out. \[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\text {Timed out} \]
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Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.17 \[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\operatorname {RootSum} {\left (15695178850368 t^{6} - 2066242608 t^{4} + 1845163152 t^{3} - 1180980 t^{2} - 1944 t - 1, \left ( t \mapsto t \log {\left (\frac {614714526178551746208 t^{5}}{57121295165} - \frac {1270857362386176 t^{4}}{57121295165} - \frac {80483053187684376 t^{3}}{57121295165} + \frac {72431318325103884 t^{2}}{57121295165} - \frac {45358602689088 t}{57121295165} + x - \frac {44532180783}{57121295165} \right )} \right )\right )} \]
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\[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{4}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \]
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\[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{4}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \]
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Time = 9.23 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.03 \[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\sum _{k=1}^6\ln \left (-\frac {5038848\,\left (1377495072\,x+17006112\,x\,\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )-104976\,x\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^2+158112\,x\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^3+1946\,x\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^4+3\,x\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^5-4251528\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^2+3927852\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^3-1188\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^4-{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^5+7558272\,\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )+33519046752\right )}{{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^5}\right )\,\mathrm {root}\left (z^6-\frac {z^4}{7596}+\frac {217\,z^3}{1845828}-\frac {5\,z^2}{66449808}-\frac {z}{8073651672}-\frac {1}{15695178850368},z,k\right ) \]
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