Integrand size = 17, antiderivative size = 268 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=-\frac {\arctan \left (\frac {3-\left (1+\frac {4}{x}\right )^2}{6 \sqrt {7}}\right )}{12 \sqrt {7}}-\frac {1}{12} \sqrt {\frac {109+67 \sqrt {29}}{1218}} \arctan \left (\frac {2-\sqrt {6 \left (1+\sqrt {29}\right )}+\frac {8}{x}}{\sqrt {6 \left (-1+\sqrt {29}\right )}}\right )-\frac {1}{12} \sqrt {\frac {109+67 \sqrt {29}}{1218}} \arctan \left (\frac {2+\sqrt {6 \left (1+\sqrt {29}\right )}+\frac {8}{x}}{\sqrt {6 \left (-1+\sqrt {29}\right )}}\right )-\frac {1}{24} \sqrt {\frac {-109+67 \sqrt {29}}{1218}} \log \left (3 \sqrt {29}-\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )+\left (1+\frac {4}{x}\right )^2\right )+\frac {1}{24} \sqrt {\frac {-109+67 \sqrt {29}}{1218}} \log \left (3 \sqrt {29}+\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )+\left (1+\frac {4}{x}\right )^2\right ) \]
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Time = 0.31 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {2094, 12, 1687, 1183, 648, 632, 210, 642, 1121} \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=-\frac {\arctan \left (\frac {3-\left (\frac {4}{x}+1\right )^2}{6 \sqrt {7}}\right )}{12 \sqrt {7}}-\frac {1}{12} \sqrt {\frac {109+67 \sqrt {29}}{1218}} \arctan \left (\frac {\frac {8}{x}-\sqrt {6 \left (1+\sqrt {29}\right )}+2}{\sqrt {6 \left (\sqrt {29}-1\right )}}\right )-\frac {1}{12} \sqrt {\frac {109+67 \sqrt {29}}{1218}} \arctan \left (\frac {\frac {8}{x}+\sqrt {6 \left (1+\sqrt {29}\right )}+2}{\sqrt {6 \left (\sqrt {29}-1\right )}}\right )-\frac {1}{24} \sqrt {\frac {67 \sqrt {29}-109}{1218}} \log \left (\left (\frac {4}{x}+1\right )^2-\sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {4}{x}+1\right )+3 \sqrt {29}\right )+\frac {1}{24} \sqrt {\frac {67 \sqrt {29}-109}{1218}} \log \left (\left (\frac {4}{x}+1\right )^2+\sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {4}{x}+1\right )+3 \sqrt {29}\right ) \]
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Rule 12
Rule 210
Rule 632
Rule 642
Rule 648
Rule 1121
Rule 1183
Rule 1687
Rule 2094
Rubi steps \begin{align*} \text {integral}& = -\left (1024 \text {Subst}\left (\int \frac {(8-32 x)^2}{8 \left (1069056-393216 x^2+1048576 x^4\right )} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )\right ) \\ & = -\left (128 \text {Subst}\left (\int \frac {(8-32 x)^2}{1069056-393216 x^2+1048576 x^4} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )\right ) \\ & = -\left (128 \text {Subst}\left (\int -\frac {512 x}{1069056-393216 x^2+1048576 x^4} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )\right )-128 \text {Subst}\left (\int \frac {64+1024 x^2}{1069056-393216 x^2+1048576 x^4} \, dx,x,\frac {1}{4}+\frac {1}{x}\right ) \\ & = 65536 \text {Subst}\left (\int \frac {x}{1069056-393216 x^2+1048576 x^4} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )-\frac {\text {Subst}\left (\int \frac {16 \sqrt {6 \left (1+\sqrt {29}\right )}-\left (64-192 \sqrt {29}\right ) x}{\frac {3 \sqrt {29}}{16}-\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} x+x^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{768 \sqrt {174 \left (1+\sqrt {29}\right )}}-\frac {\text {Subst}\left (\int \frac {16 \sqrt {6 \left (1+\sqrt {29}\right )}+\left (64-192 \sqrt {29}\right ) x}{\frac {3 \sqrt {29}}{16}+\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} x+x^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{768 \sqrt {174 \left (1+\sqrt {29}\right )}} \\ & = 32768 \text {Subst}\left (\int \frac {1}{1069056-393216 x+1048576 x^2} \, dx,x,\left (\frac {1}{4}+\frac {1}{x}\right )^2\right )-\frac {\left (87+\sqrt {29}\right ) \text {Subst}\left (\int \frac {1}{\frac {3 \sqrt {29}}{16}-\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} x+x^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{2784}-\frac {\left (87+\sqrt {29}\right ) \text {Subst}\left (\int \frac {1}{\frac {3 \sqrt {29}}{16}+\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} x+x^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{2784}-\frac {1}{24} \sqrt {\frac {-109+67 \sqrt {29}}{1218}} \text {Subst}\left (\int \frac {-\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}+2 x}{\frac {3 \sqrt {29}}{16}-\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} x+x^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )+\frac {1}{24} \sqrt {\frac {-109+67 \sqrt {29}}{1218}} \text {Subst}\left (\int \frac {\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}+2 x}{\frac {3 \sqrt {29}}{16}+\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} x+x^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right ) \\ & = -\frac {1}{24} \sqrt {\frac {-109+67 \sqrt {29}}{1218}} \log \left (3 \sqrt {29}-\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )+\left (1+\frac {4}{x}\right )^2\right )+\frac {1}{24} \sqrt {\frac {-109+67 \sqrt {29}}{1218}} \log \left (3 \sqrt {29}+\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )+\left (1+\frac {4}{x}\right )^2\right )-65536 \text {Subst}\left (\int \frac {1}{-4329327034368-x^2} \, dx,x,-393216+2097152 \left (\frac {1}{4}+\frac {1}{x}\right )^2\right )+\frac {\left (87+\sqrt {29}\right ) \text {Subst}\left (\int \frac {1}{\frac {3}{8} \left (1-\sqrt {29}\right )-x^2} \, dx,x,-\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}+2 \left (\frac {1}{4}+\frac {1}{x}\right )\right )}{1392}+\frac {\left (87+\sqrt {29}\right ) \text {Subst}\left (\int \frac {1}{\frac {3}{8} \left (1-\sqrt {29}\right )-x^2} \, dx,x,\frac {1}{4} \left (2+\sqrt {6 \left (1+\sqrt {29}\right )}+\frac {8}{x}\right )\right )}{1392} \\ & = -\frac {\tan ^{-1}\left (\frac {3-\left (1+\frac {4}{x}\right )^2}{6 \sqrt {7}}\right )}{12 \sqrt {7}}-\frac {1}{12} \sqrt {\frac {109+67 \sqrt {29}}{1218}} \tan ^{-1}\left (\frac {2+\sqrt {6 \left (1+\sqrt {29}\right )}+\frac {8}{x}}{\sqrt {6 \left (-1+\sqrt {29}\right )}}\right )-\frac {1}{12} \sqrt {\frac {109+67 \sqrt {29}}{1218}} \tan ^{-1}\left (\frac {8+\left (2-\sqrt {6 \left (1+\sqrt {29}\right )}\right ) x}{\sqrt {6 \left (-1+\sqrt {29}\right )} x}\right )-\frac {1}{24} \sqrt {\frac {-109+67 \sqrt {29}}{1218}} \log \left (3 \sqrt {29}-\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )+\left (1+\frac {4}{x}\right )^2\right )+\frac {1}{24} \sqrt {\frac {-109+67 \sqrt {29}}{1218}} \log \left (3 \sqrt {29}+\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )+\left (1+\frac {4}{x}\right )^2\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.17 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\text {RootSum}\left [8+8 \text {$\#$1}-\text {$\#$1}^3+8 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{8-3 \text {$\#$1}^2+32 \text {$\#$1}^3}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.15
method | result | size |
default | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-\textit {\_Z}^{3}+8 \textit {\_Z} +8\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-3 \textit {\_R}^{2}+8}\) | \(41\) |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-\textit {\_Z}^{3}+8 \textit {\_Z} +8\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-3 \textit {\_R}^{2}+8}\) | \(41\) |
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Result contains complex when optimal does not.
Time = 1.09 (sec) , antiderivative size = 1015, normalized size of antiderivative = 3.79 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\text {Too large to display} \]
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Time = 0.54 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.15 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\operatorname {RootSum} {\left (66298176 t^{4} + 74088 t^{2} + 4095 t + 64, \left ( t \mapsto t \log {\left (\frac {35914274424 t^{3}}{2109763} - \frac {1504863360 t^{2}}{2109763} + \frac {102851343 t}{2109763} + x + \frac {6055613}{16878104} \right )} \right )\right )} \]
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\[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\int { \frac {1}{8 \, x^{4} - x^{3} + 8 \, x + 8} \,d x } \]
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\[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\int { \frac {1}{8 \, x^{4} - x^{3} + 8 \, x + 8} \,d x } \]
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Time = 10.58 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.46 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\sum _{k=1}^4\ln \left (-\frac {\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )\,\left (8064\,\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )+256\,x+\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )\,x\,12285+{\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )}^2\,x\,148176+198072\,{\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )}^2-8\right )}{4096}\right )\,\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right ) \]
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