Integrand size = 56, antiderivative size = 18 \[ \int \frac {\left (3+x+9 x^2+3 x^3\right ) \log (3+x)+\left (3+x+x^3\right ) \log \left (-9-3 x-3 x^3\right )}{9+6 x+x^2+3 x^3+x^4} \, dx=\log (3+x) \log \left (3 \left (-3-x-x^3\right )\right ) \]
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Time = 0.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {6874, 2465, 2604} \[ \int \frac {\left (3+x+9 x^2+3 x^3\right ) \log (3+x)+\left (3+x+x^3\right ) \log \left (-9-3 x-3 x^3\right )}{9+6 x+x^2+3 x^3+x^4} \, dx=\log (x+3) \log \left (-3 \left (x^3+x+3\right )\right ) \]
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Rule 2465
Rule 2604
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (1+3 x^2\right ) \log (3+x)}{3+x+x^3}+\frac {\log \left (-3 \left (3+x+x^3\right )\right )}{3+x}\right ) \, dx \\ & = \int \frac {\left (1+3 x^2\right ) \log (3+x)}{3+x+x^3} \, dx+\int \frac {\log \left (-3 \left (3+x+x^3\right )\right )}{3+x} \, dx \\ & = \log (3+x) \log \left (-3 \left (3+x+x^3\right )\right )-\int \frac {\left (1+3 x^2\right ) \log (3+x)}{3+x+x^3} \, dx+\int \left (\frac {\log (3+x)}{3+x+x^3}+\frac {3 x^2 \log (3+x)}{3+x+x^3}\right ) \, dx \\ & = \log (3+x) \log \left (-3 \left (3+x+x^3\right )\right )+3 \int \frac {x^2 \log (3+x)}{3+x+x^3} \, dx+\int \frac {\log (3+x)}{3+x+x^3} \, dx-\int \left (\frac {\log (3+x)}{3+x+x^3}+\frac {3 x^2 \log (3+x)}{3+x+x^3}\right ) \, dx \\ & = \log (3+x) \log \left (-3 \left (3+x+x^3\right )\right ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\left (3+x+9 x^2+3 x^3\right ) \log (3+x)+\left (3+x+x^3\right ) \log \left (-9-3 x-3 x^3\right )}{9+6 x+x^2+3 x^3+x^4} \, dx=\log (3+x) \log \left (-3 \left (3+x+x^3\right )\right ) \]
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Time = 0.32 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\ln \left (3+x \right ) \ln \left (-3 x^{3}-3 x -9\right )\) | \(17\) |
default | \(\ln \left (3\right ) \ln \left (3+x \right )+\ln \left (3+x \right ) \ln \left (-x^{3}-x -3\right )\) | \(25\) |
parts | \(\ln \left (3\right ) \ln \left (3+x \right )+\ln \left (3+x \right ) \ln \left (-x^{3}-x -3\right )\) | \(25\) |
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none
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {\left (3+x+9 x^2+3 x^3\right ) \log (3+x)+\left (3+x+x^3\right ) \log \left (-9-3 x-3 x^3\right )}{9+6 x+x^2+3 x^3+x^4} \, dx=\log \left (-3 \, x^{3} - 3 \, x - 9\right ) \log \left (x + 3\right ) \]
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Time = 0.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {\left (3+x+9 x^2+3 x^3\right ) \log (3+x)+\left (3+x+x^3\right ) \log \left (-9-3 x-3 x^3\right )}{9+6 x+x^2+3 x^3+x^4} \, dx=\log {\left (x + 3 \right )} \log {\left (- 3 x^{3} - 3 x - 9 \right )} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {\left (3+x+9 x^2+3 x^3\right ) \log (3+x)+\left (3+x+x^3\right ) \log \left (-9-3 x-3 x^3\right )}{9+6 x+x^2+3 x^3+x^4} \, dx={\left (i \, \pi + \log \left (3\right )\right )} \log \left (x + 3\right ) + \log \left (x^{3} + x + 3\right ) \log \left (x + 3\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {\left (3+x+9 x^2+3 x^3\right ) \log (3+x)+\left (3+x+x^3\right ) \log \left (-9-3 x-3 x^3\right )}{9+6 x+x^2+3 x^3+x^4} \, dx=\log \left (-3 \, x^{3} - 3 \, x - 9\right ) \log \left (x + 3\right ) \]
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Time = 12.62 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\left (3+x+9 x^2+3 x^3\right ) \log (3+x)+\left (3+x+x^3\right ) \log \left (-9-3 x-3 x^3\right )}{9+6 x+x^2+3 x^3+x^4} \, dx=\ln \left (x+3\right )\,\left (\ln \left (3\right )+\ln \left (-x^3-x-3\right )\right ) \]
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