Integrand size = 41, antiderivative size = 20 \[ \int \frac {-3-2 x+3 x^2+\log \left (\frac {3}{\log (5)}\right )}{-3 x-x^2+x^3+x \log \left (\frac {3}{\log (5)}\right )} \, dx=\log \left (x \left (-3-(1-x) x+\log \left (\frac {3}{\log (5)}\right )\right )\right ) \]
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Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {6, 1601} \[ \int \frac {-3-2 x+3 x^2+\log \left (\frac {3}{\log (5)}\right )}{-3 x-x^2+x^3+x \log \left (\frac {3}{\log (5)}\right )} \, dx=\log \left (-x^3+x^2+x \left (3-\log \left (\frac {3}{\log (5)}\right )\right )\right ) \]
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Rule 6
Rule 1601
Rubi steps \begin{align*} \text {integral}& = \int \frac {-3-2 x+3 x^2+\log \left (\frac {3}{\log (5)}\right )}{-x^2+x^3+x \left (-3+\log \left (\frac {3}{\log (5)}\right )\right )} \, dx \\ & = \log \left (x^2-x^3+x \left (3-\log \left (\frac {3}{\log (5)}\right )\right )\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {-3-2 x+3 x^2+\log \left (\frac {3}{\log (5)}\right )}{-3 x-x^2+x^3+x \log \left (\frac {3}{\log (5)}\right )} \, dx=\log (x)+\log \left (3+x-x^2-\log \left (\frac {3}{\log (5)}\right )\right ) \]
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Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
default | \(\ln \left (x \left (x^{2}+\ln \left (\frac {3}{\ln \left (5\right )}\right )-x -3\right )\right )\) | \(19\) |
norman | \(\ln \left (x \right )+\ln \left (x^{2}+\ln \left (\frac {3}{\ln \left (5\right )}\right )-x -3\right )\) | \(20\) |
parallelrisch | \(\ln \left (x \right )+\ln \left (x^{2}+\ln \left (\frac {3}{\ln \left (5\right )}\right )-x -3\right )\) | \(20\) |
risch | \(\ln \left (x^{3}-x^{2}+\left (\ln \left (3\right )-\ln \left (\ln \left (5\right )\right )-3\right ) x \right )\) | \(22\) |
derivativedivides | \(\ln \left (x \ln \left (\frac {3}{\ln \left (5\right )}\right )+x^{3}-x^{2}-3 x \right )\) | \(23\) |
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Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {-3-2 x+3 x^2+\log \left (\frac {3}{\log (5)}\right )}{-3 x-x^2+x^3+x \log \left (\frac {3}{\log (5)}\right )} \, dx=\log \left (x^{3} - x^{2} + x \log \left (\frac {3}{\log \left (5\right )}\right ) - 3 \, x\right ) \]
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {-3-2 x+3 x^2+\log \left (\frac {3}{\log (5)}\right )}{-3 x-x^2+x^3+x \log \left (\frac {3}{\log (5)}\right )} \, dx=\log {\left (x^{3} - x^{2} + x \left (-3 - \log {\left (\log {\left (5 \right )} \right )} + \log {\left (3 \right )}\right ) \right )} \]
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Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {-3-2 x+3 x^2+\log \left (\frac {3}{\log (5)}\right )}{-3 x-x^2+x^3+x \log \left (\frac {3}{\log (5)}\right )} \, dx=\log \left (x^{3} - x^{2} + x \log \left (\frac {3}{\log \left (5\right )}\right ) - 3 \, x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {-3-2 x+3 x^2+\log \left (\frac {3}{\log (5)}\right )}{-3 x-x^2+x^3+x \log \left (\frac {3}{\log (5)}\right )} \, dx=\log \left ({\left | x^{3} - x^{2} + x \log \left (\frac {3}{\log \left (5\right )}\right ) - 3 \, x \right |}\right ) \]
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Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {-3-2 x+3 x^2+\log \left (\frac {3}{\log (5)}\right )}{-3 x-x^2+x^3+x \log \left (\frac {3}{\log (5)}\right )} \, dx=\ln \left (x^3-x^2+\left (\ln \left (3\right )-\ln \left (\ln \left (5\right )\right )-3\right )\,x\right ) \]
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