Integrand size = 35, antiderivative size = 10 \[ \int \frac {243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{729-64 x^6} \, dx=-\frac {1}{2} \log (3-2 x) \]
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Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1600, 31} \[ \int \frac {243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{729-64 x^6} \, dx=-\frac {1}{2} \log (3-2 x) \]
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Rule 31
Rule 1600
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{3-2 x} \, dx \\ & = -\frac {1}{2} \log (3-2 x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{729-64 x^6} \, dx=-\frac {1}{2} \log (3-2 x) \]
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Time = 1.54 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70
method | result | size |
parallelrisch | \(-\frac {\ln \left (x -\frac {3}{2}\right )}{2}\) | \(7\) |
default | \(-\frac {\ln \left (-3+2 x \right )}{2}\) | \(9\) |
norman | \(-\frac {\ln \left (-3+2 x \right )}{2}\) | \(9\) |
risch | \(-\frac {\ln \left (-3+2 x \right )}{2}\) | \(9\) |
meijerg | \(-\frac {x \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{12 \left (x^{6}\right )^{\frac {1}{6}}}-\frac {\ln \left (1-\frac {64 x^{6}}{729}\right )}{12}-\frac {x^{5} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{12 \left (x^{6}\right )^{\frac {5}{6}}}-\frac {x^{4} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{12 \left (x^{6}\right )^{\frac {2}{3}}}+\frac {\operatorname {arctanh}\left (\frac {8 x^{3}}{27}\right )}{6}-\frac {x^{2} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{12 \left (x^{6}\right )^{\frac {1}{3}}}\) | \(399\) |
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Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{729-64 x^6} \, dx=-\frac {1}{2} \, \log \left (2 \, x - 3\right ) \]
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Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{729-64 x^6} \, dx=- \frac {\log {\left (2 x - 3 \right )}}{2} \]
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Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{729-64 x^6} \, dx=-\frac {1}{2} \, \log \left (2 \, x - 3\right ) \]
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Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{729-64 x^6} \, dx=-\frac {1}{2} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \]
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Time = 8.89 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \frac {243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{729-64 x^6} \, dx=-\frac {\ln \left (x-\frac {3}{2}\right )}{2} \]
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