Integrand size = 18, antiderivative size = 32 \[ \int \frac {-8+2 x+3 x^2}{8+x^3} \, dx=\frac {\arctan \left (\frac {1-x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {3}{2} \log \left (4-2 x+x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1886, 648, 632, 210, 642} \[ \int \frac {-8+2 x+3 x^2}{8+x^3} \, dx=\frac {\arctan \left (\frac {1-x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {3}{2} \log \left (x^2-2 x+4\right ) \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1886
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {-8+6 x}{4-2 x+x^2} \, dx \\ & = \frac {3}{2} \int \frac {-2+2 x}{4-2 x+x^2} \, dx-\int \frac {1}{4-2 x+x^2} \, dx \\ & = \frac {3}{2} \log \left (4-2 x+x^2\right )+2 \text {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,-2+2 x\right ) \\ & = \frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {3}{2} \log \left (4-2 x+x^2\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-8+2 x+3 x^2}{8+x^3} \, dx=-\frac {\arctan \left (\frac {-1+x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {3}{2} \log \left (4-2 x+x^2\right ) \]
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Time = 0.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {3 \ln \left (x^{2}-2 x +4\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (x -1\right ) \sqrt {3}}{3}\right )}{3}\) | \(27\) |
default | \(\frac {3 \ln \left (x^{2}-2 x +4\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -2\right ) \sqrt {3}}{6}\right )}{3}\) | \(29\) |
meijerg | \(-\frac {2 x \ln \left (1+\frac {\left (x^{3}\right )^{\frac {1}{3}}}{2}\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}+\frac {x \ln \left (1-\frac {\left (x^{3}\right )^{\frac {1}{3}}}{2}+\frac {\left (x^{3}\right )^{\frac {2}{3}}}{4}\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}-\frac {2 x \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{4-\left (x^{3}\right )^{\frac {1}{3}}}\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}+\ln \left (1+\frac {x^{3}}{8}\right )-\frac {x^{2} \ln \left (1+\frac {\left (x^{3}\right )^{\frac {1}{3}}}{2}\right )}{3 \left (x^{3}\right )^{\frac {2}{3}}}+\frac {x^{2} \ln \left (1-\frac {\left (x^{3}\right )^{\frac {1}{3}}}{2}+\frac {\left (x^{3}\right )^{\frac {2}{3}}}{4}\right )}{6 \left (x^{3}\right )^{\frac {2}{3}}}+\frac {x^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{4-\left (x^{3}\right )^{\frac {1}{3}}}\right )}{3 \left (x^{3}\right )^{\frac {2}{3}}}\) | \(168\) |
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Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {-8+2 x+3 x^2}{8+x^3} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x - 1\right )}\right ) + \frac {3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \]
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Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {-8+2 x+3 x^2}{8+x^3} \, dx=\frac {3 \log {\left (x^{2} - 2 x + 4 \right )}}{2} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]
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Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {-8+2 x+3 x^2}{8+x^3} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x - 1\right )}\right ) + \frac {3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \]
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Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {-8+2 x+3 x^2}{8+x^3} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x - 1\right )}\right ) + \frac {3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \]
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Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {-8+2 x+3 x^2}{8+x^3} \, dx=\frac {3\,\ln \left (x^2-2\,x+4\right )}{2}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,x}{3}-\frac {\sqrt {3}}{3}\right )}{3} \]
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