Integrand size = 26, antiderivative size = 448 \[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=-\frac {1}{216 x}-\frac {\left (27 \sqrt [3]{-6}-(-2)^{2/3}+12\ 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{5832 \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{1944 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{5832 \sqrt [6]{6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {\left (3 (-6)^{2/3}+2 \sqrt [3]{-2}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{7776 \sqrt [3]{3}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}} \]
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Time = 1.09 (sec) , antiderivative size = 448, 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} \[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=-\frac {\left (27 \sqrt [3]{-6}-(-2)^{2/3}+12\ 3^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{5832 \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{1944 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{5832 \sqrt [6]{6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}-\frac {(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {\left (3 (-6)^{2/3}+2 \sqrt [3]{-2}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{7776 \sqrt [3]{3}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{3888 \sqrt [3]{6}}-\frac {1}{216 x} \]
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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}{272097792 x^2}+\frac {(-1)^{2/3} \left (-1+9 (-3)^{2/3} \sqrt [3]{2}+27 \sqrt [3]{-3} 2^{2/3}-\left (9+\sqrt [3]{-3} 2^{2/3}\right ) x\right )}{816293376 \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 (1+27 (-2)^{2/3} \sqrt [3]{3}+9 \sqrt [3]{-2} 3^{2/3}+\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) x\right )}{816293376 \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 {-54+2^{2/3} \sqrt [3]{3}+54 \sqrt [3]{2} 3^{2/3}-6^{2/3} \left (2^{2/3}-3\ 3^{2/3}\right ) x}{14693280768 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx \\ & = -\frac {1}{216 x}+\frac {\int \frac {-54+2^{2/3} \sqrt [3]{3}+54 \sqrt [3]{2} 3^{2/3}-6^{2/3} \left (2^{2/3}-3\ 3^{2/3}\right ) x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{11664}+\frac {(-1)^{2/3} \int \frac {1+27 (-2)^{2/3} \sqrt [3]{3}+9 \sqrt [3]{-2} 3^{2/3}+\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1944 \sqrt [3]{2} 3^{2/3}}+\frac {(-1)^{2/3} \int \frac {-1+9 (-3)^{2/3} \sqrt [3]{2}+27 \sqrt [3]{-3} 2^{2/3}-\left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{648 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2} \\ & = -\frac {1}{216 x}-\frac {\left ((-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right )\right ) \int \frac {-3 \sqrt [3]{-3} 2^{2/3}+2 x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\left ((-1)^{2/3} \left ((-2)^{2/3}-3\ 3^{2/3}\right )\right ) \int \frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{3888 \sqrt [3]{6}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \int \frac {3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{3888 \sqrt [3]{6}}+\frac {\left ((-1)^{2/3} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right )\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{3888 \sqrt [3]{2} 3^{2/3}}+\frac {\left ((-1)^{2/3} \left (3 \sqrt [3]{-3} 2^{2/3} \left (-9-\sqrt [3]{-3} 2^{2/3}\right )+2 \left (-1+9 (-3)^{2/3} \sqrt [3]{2}+27 \sqrt [3]{-3} 2^{2/3}\right )\right )\right ) \int \frac {1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {\left (18 \sqrt [3]{2} \left (2^{2/3}-3\ 3^{2/3}\right )+2 \left (-54+2^{2/3} \sqrt [3]{3}+54 \sqrt [3]{2} 3^{2/3}\right )\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{23328} \\ & = -\frac {1}{216 x}-\frac {(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {(-1)^{2/3} \left ((-2)^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}}-\frac {\left ((-1)^{2/3} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right )\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 )}{1944 \sqrt [3]{2} 3^{2/3}}-\frac {\left ((-1)^{2/3} \left (3 \sqrt [3]{-3} 2^{2/3} \left (-9-\sqrt [3]{-3} 2^{2/3}\right )+2 \left (-1+9 (-3)^{2/3} \sqrt [3]{2}+27 \sqrt [3]{-3} 2^{2/3}\right )\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 )}{648 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\left (18 \sqrt [3]{2} \left (2^{2/3}-3\ 3^{2/3}\right )+2 \left (-54+2^{2/3} \sqrt [3]{3}+54 \sqrt [3]{2} 3^{2/3}\right )\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 )}{11664} \\ & = -\frac {1}{216 x}+\frac {(-1)^{2/3} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 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 )}{5832\ 2^{5/6} \sqrt [6]{3} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {(-1)^{2/3} \left (2-12 (-3)^{2/3} \sqrt [3]{2}-27 \sqrt [3]{-3} 2^{2/3}\right ) \tan ^{-1}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{1944\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\left (18\ 2^{2/3}-27\ 3^{2/3}-\sqrt [3]{6}\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 )}{5832 \sqrt [6]{2} \sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {(-1)^{2/3} \left ((-2)^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.24 \[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=-\frac {1}{216 x}-\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {108 \log (x-\text {$\#$1})+324 \log (x-\text {$\#$1}) \text {$\#$1}+18 \log (x-\text {$\#$1}) \text {$\#$1}^2+\log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{1296} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.15
method | result | size |
risch | \(-\frac {1}{216 x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (633 \textit {\_Z}^{6}+204849 \textit {\_Z}^{4}-5446980 \textit {\_Z}^{3}-80433 \textit {\_Z}^{2}-72 \textit {\_Z} -1\right )}{\sum }\textit {\_R} \ln \left (-462040439801351484393 \textit {\_R}^{5}+1364231865933925308 \textit {\_R}^{4}-149523740969574483417612 \textit {\_R}^{3}+3976310471903162636736042 \textit {\_R}^{2}+46967454543463546461111 \textit {\_R} +24700899569407983590 x -25597852658707816584\right )\right )}{11664}\) | \(69\) |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-\textit {\_R}^{4}-18 \textit {\_R}^{2}-324 \textit {\_R} -108\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{1296}-\frac {1}{216 x}\) | \(74\) |
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Timed out. \[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.16 \[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\operatorname {RootSum} {\left (1594001683946413330255577088 t^{6} + 3791612026460331638784 t^{4} - 8643672699589509120 t^{3} - 10942820851968 t^{2} - 839808 t - 1, \left ( t \mapsto t \log {\left (- \frac {49875532761902496003293561236914468028416 t^{5}}{12350449784703991795} + \frac {12625489872431620388005975200497664 t^{4}}{12350449784703991795} - \frac {118637692607573771238550798852644864 t^{3}}{12350449784703991795} + \frac {270486324927832147818193778754816 t^{2}}{12350449784703991795} + \frac {273914194897479402961199352 t}{12350449784703991795} + x - \frac {12798926329353908292}{12350449784703991795} \right )} \right )\right )} - \frac {1}{216 x} \]
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\[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} x^{2}} \,d x } \]
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\[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} x^{2}} \,d x } \]
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Time = 9.04 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\left (\sum _{k=1}^6\ln \left (\frac {5\,\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}{8}-\frac {\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )\,x}{216}-{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^2\,x\,396252-{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^3\,x\,598229670528+{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^4\,x\,82120746212352-{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^5\,x\,6940988288557056+2344464\,{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^2-210297580992\,{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^3-10535082310656\,{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^4-168897381688221696\,{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^5\right )\,\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )\right )-\frac {1}{216\,x} \]
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