Integrand size = 16, antiderivative size = 57 \[ \int \frac {x^9}{1-x^4+x^8} \, dx=\frac {x^2}{2}+\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}} \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1373, 1136, 1178, 642} \[ \int \frac {x^9}{1-x^4+x^8} \, dx=\frac {x^2}{2}+\frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{4 \sqrt {3}}-\frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{4 \sqrt {3}} \]
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Rule 642
Rule 1136
Rule 1178
Rule 1373
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^4}{1-x^2+x^4} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2}-\frac {1}{2} \text {Subst}\left (\int \frac {1-x^2}{1-x^2+x^4} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2}+\frac {\text {Subst}\left (\int \frac {\sqrt {3}+2 x}{-1-\sqrt {3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt {3}}+\frac {\text {Subst}\left (\int \frac {\sqrt {3}-2 x}{-1+\sqrt {3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt {3}} \\ & = \frac {x^2}{2}+\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96 \[ \int \frac {x^9}{1-x^4+x^8} \, dx=\frac {1}{12} \left (6 x^2+\sqrt {3} \log \left (-1+\sqrt {3} x^2-x^4\right )-\sqrt {3} \log \left (1+\sqrt {3} x^2+x^4\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {x^{2}}{2}+\frac {\ln \left (1+x^{4}-x^{2} \sqrt {3}\right ) \sqrt {3}}{12}-\frac {\ln \left (1+x^{4}+x^{2} \sqrt {3}\right ) \sqrt {3}}{12}\) | \(44\) |
risch | \(\frac {x^{2}}{2}+\frac {\ln \left (1+x^{4}-x^{2} \sqrt {3}\right ) \sqrt {3}}{12}-\frac {\ln \left (1+x^{4}+x^{2} \sqrt {3}\right ) \sqrt {3}}{12}\) | \(44\) |
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none
Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82 \[ \int \frac {x^9}{1-x^4+x^8} \, dx=\frac {1}{2} \, x^{2} + \frac {1}{12} \, \sqrt {3} \log \left (\frac {x^{8} + 5 \, x^{4} - 2 \, \sqrt {3} {\left (x^{6} + x^{2}\right )} + 1}{x^{8} - x^{4} + 1}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84 \[ \int \frac {x^9}{1-x^4+x^8} \, dx=\frac {x^{2}}{2} + \frac {\sqrt {3} \log {\left (x^{4} - \sqrt {3} x^{2} + 1 \right )}}{12} - \frac {\sqrt {3} \log {\left (x^{4} + \sqrt {3} x^{2} + 1 \right )}}{12} \]
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\[ \int \frac {x^9}{1-x^4+x^8} \, dx=\int { \frac {x^{9}}{x^{8} - x^{4} + 1} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (43) = 86\).
Time = 0.34 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.74 \[ \int \frac {x^9}{1-x^4+x^8} \, dx=\frac {1}{2} \, x^{2} + \frac {1}{4} \, {\left (x^{4} - 1\right )} \arctan \left (2 \, x^{2} + \sqrt {3}\right ) + \frac {1}{4} \, {\left (x^{4} - 1\right )} \arctan \left (2 \, x^{2} - \sqrt {3}\right ) + \frac {1}{24} \, {\left (\sqrt {3} x^{4} - \sqrt {3}\right )} \log \left (x^{4} + \sqrt {3} x^{2} + 1\right ) - \frac {1}{24} \, {\left (\sqrt {3} x^{4} - \sqrt {3}\right )} \log \left (x^{4} - \sqrt {3} x^{2} + 1\right ) \]
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Time = 8.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.51 \[ \int \frac {x^9}{1-x^4+x^8} \, dx=\frac {x^2}{2}-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,x^2}{9\,\left (\frac {2\,x^4}{9}+\frac {2}{9}\right )}\right )}{6} \]
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