Integrand size = 8, antiderivative size = 31 \[ \int x \log \left (x+x^3\right ) \, dx=-\frac {3 x^2}{4}+\frac {1}{2} \log \left (1+x^2\right )+\frac {1}{2} x^2 \log \left (x+x^3\right ) \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2605, 455, 45} \[ \int x \log \left (x+x^3\right ) \, dx=-\frac {3 x^2}{4}+\frac {1}{2} \log \left (x^2+1\right )+\frac {1}{2} x^2 \log \left (x^3+x\right ) \]
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Rule 45
Rule 455
Rule 2605
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \log \left (x+x^3\right )-\frac {1}{2} \int \frac {x \left (1+3 x^2\right )}{1+x^2} \, dx \\ & = \frac {1}{2} x^2 \log \left (x+x^3\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1+3 x}{1+x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} x^2 \log \left (x+x^3\right )-\frac {1}{4} \text {Subst}\left (\int \left (3-\frac {2}{1+x}\right ) \, dx,x,x^2\right ) \\ & = -\frac {3 x^2}{4}+\frac {1}{2} \log \left (1+x^2\right )+\frac {1}{2} x^2 \log \left (x+x^3\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int x \log \left (x+x^3\right ) \, dx=-\frac {3 x^2}{4}+\frac {1}{2} \log \left (1+x^2\right )+\frac {1}{2} x^2 \log \left (x+x^3\right ) \]
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Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {3 x^{2}}{4}+\frac {\ln \left (x^{2}+1\right )}{2}+\frac {x^{2} \ln \left (x^{3}+x \right )}{2}\) | \(26\) |
risch | \(-\frac {3 x^{2}}{4}+\frac {\ln \left (x^{2}+1\right )}{2}+\frac {x^{2} \ln \left (x^{3}+x \right )}{2}\) | \(26\) |
parts | \(-\frac {3 x^{2}}{4}+\frac {\ln \left (x^{2}+1\right )}{2}+\frac {x^{2} \ln \left (x^{3}+x \right )}{2}\) | \(26\) |
parallelrisch | \(\frac {x^{2} \ln \left (x^{3}+x \right )}{2}+\frac {3}{4}-\frac {3 x^{2}}{4}-\frac {\ln \left (x \right )}{2}+\frac {\ln \left (x^{3}+x \right )}{2}\) | \(31\) |
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Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int x \log \left (x+x^3\right ) \, dx=\frac {1}{2} \, x^{2} \log \left (x^{3} + x\right ) - \frac {3}{4} \, x^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int x \log \left (x+x^3\right ) \, dx=\frac {x^{2} \log {\left (x^{3} + x \right )}}{2} - \frac {3 x^{2}}{4} + \frac {\log {\left (x^{2} + 1 \right )}}{2} \]
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Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int x \log \left (x+x^3\right ) \, dx=\frac {1}{2} \, x^{2} \log \left (x^{3} + x\right ) - \frac {3}{4} \, x^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int x \log \left (x+x^3\right ) \, dx=\frac {1}{2} \, x^{2} \log \left (x^{3} + x\right ) - \frac {3}{4} \, x^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
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Time = 1.58 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int x \log \left (x+x^3\right ) \, dx=\frac {\ln \left (x^2+1\right )}{2}+\frac {x^2\,\ln \left (x^3+x\right )}{2}-\frac {3\,x^2}{4} \]
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