Integrand size = 16, antiderivative size = 25 \[ \int \frac {x^2}{9-10 x^3+x^6} \, dx=-\frac {1}{24} \log \left (1-x^3\right )+\frac {1}{24} \log \left (9-x^3\right ) \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1366, 630, 31} \[ \int \frac {x^2}{9-10 x^3+x^6} \, dx=\frac {1}{24} \log \left (9-x^3\right )-\frac {1}{24} \log \left (1-x^3\right ) \]
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Rule 31
Rule 630
Rule 1366
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{9-10 x+x^2} \, dx,x,x^3\right ) \\ & = \frac {1}{24} \text {Subst}\left (\int \frac {1}{-9+x} \, dx,x,x^3\right )-\frac {1}{24} \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^3\right ) \\ & = -\frac {1}{24} \log \left (1-x^3\right )+\frac {1}{24} \log \left (9-x^3\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{9-10 x^3+x^6} \, dx=-\frac {1}{24} \log \left (1-x^3\right )+\frac {1}{24} \log \left (9-x^3\right ) \]
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Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72
method | result | size |
default | \(-\frac {\ln \left (x^{3}-1\right )}{24}+\frac {\ln \left (x^{3}-9\right )}{24}\) | \(18\) |
risch | \(-\frac {\ln \left (x^{3}-1\right )}{24}+\frac {\ln \left (x^{3}-9\right )}{24}\) | \(18\) |
norman | \(-\frac {\ln \left (-1+x \right )}{24}+\frac {\ln \left (x^{3}-9\right )}{24}-\frac {\ln \left (x^{2}+x +1\right )}{24}\) | \(25\) |
parallelrisch | \(-\frac {\ln \left (-1+x \right )}{24}+\frac {\ln \left (x^{3}-9\right )}{24}-\frac {\ln \left (x^{2}+x +1\right )}{24}\) | \(25\) |
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Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {x^2}{9-10 x^3+x^6} \, dx=-\frac {1}{24} \, \log \left (x^{3} - 1\right ) + \frac {1}{24} \, \log \left (x^{3} - 9\right ) \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60 \[ \int \frac {x^2}{9-10 x^3+x^6} \, dx=\frac {\log {\left (x^{3} - 9 \right )}}{24} - \frac {\log {\left (x^{3} - 1 \right )}}{24} \]
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Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {x^2}{9-10 x^3+x^6} \, dx=-\frac {1}{24} \, \log \left (x^{3} - 1\right ) + \frac {1}{24} \, \log \left (x^{3} - 9\right ) \]
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Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {x^2}{9-10 x^3+x^6} \, dx=-\frac {1}{24} \, \log \left ({\left | x^{3} - 1 \right |}\right ) + \frac {1}{24} \, \log \left ({\left | x^{3} - 9 \right |}\right ) \]
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Time = 0.61 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64 \[ \int \frac {x^2}{9-10 x^3+x^6} \, dx=\frac {\mathrm {atanh}\left (\frac {81}{320\,\left (\frac {5\,x^3}{4}-\frac {9}{8}\right )}-\frac {41}{40}\right )}{12} \]
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