Optimal. Leaf size=20 \[ \frac {1}{3} \log (x) \left (x^2-\log \left (\frac {1}{3}+4 x\right )\right ) \]
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Rubi [A] time = 0.24, antiderivative size = 30, normalized size of antiderivative = 1.50, number of steps used = 11, number of rules used = 8, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1593, 6688, 12, 2357, 2304, 2317, 2391, 2392} \begin {gather*} \frac {1}{3} x^2 \log (x)+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} \log (x) \log (12 x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1593
Rule 2304
Rule 2317
Rule 2357
Rule 2391
Rule 2392
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2+12 x^3+\left (-12 x+2 x^2+24 x^3\right ) \log (x)+(-1-12 x) \log \left (\frac {1}{3} (1+12 x)\right )}{x (3+36 x)} \, dx\\ &=\int \frac {1}{3} \left (x+\frac {2 \left (-6+x+12 x^2\right ) \log (x)}{1+12 x}-\frac {\log \left (\frac {1}{3}+4 x\right )}{x}\right ) \, dx\\ &=\frac {1}{3} \int \left (x+\frac {2 \left (-6+x+12 x^2\right ) \log (x)}{1+12 x}-\frac {\log \left (\frac {1}{3}+4 x\right )}{x}\right ) \, dx\\ &=\frac {x^2}{6}-\frac {1}{3} \int \frac {\log \left (\frac {1}{3}+4 x\right )}{x} \, dx+\frac {2}{3} \int \frac {\left (-6+x+12 x^2\right ) \log (x)}{1+12 x} \, dx\\ &=\frac {x^2}{6}+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} \int \frac {\log (1+12 x)}{x} \, dx+\frac {2}{3} \int \left (x \log (x)-\frac {6 \log (x)}{1+12 x}\right ) \, dx\\ &=\frac {x^2}{6}+\frac {1}{3} \log (3) \log (x)+\frac {\text {Li}_2(-12 x)}{3}+\frac {2}{3} \int x \log (x) \, dx-4 \int \frac {\log (x)}{1+12 x} \, dx\\ &=\frac {1}{3} x^2 \log (x)+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} \log (x) \log (1+12 x)+\frac {\text {Li}_2(-12 x)}{3}+\frac {1}{3} \int \frac {\log (1+12 x)}{x} \, dx\\ &=\frac {1}{3} x^2 \log (x)+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} \log (x) \log (1+12 x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 1.30 \begin {gather*} \frac {1}{3} \left (x^2 \log (x)+\log (3) \log (x)-\log (x) \log (1+12 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 18, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, x^{2} \log \relax (x) - \frac {1}{3} \, \log \left (4 \, x + \frac {1}{3}\right ) \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 24, normalized size = 1.20 \begin {gather*} \frac {1}{3} \, x^{2} \log \relax (x) + \frac {1}{3} \, \log \relax (3) \log \relax (x) - \frac {1}{3} \, \log \left (12 \, x + 1\right ) \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 19, normalized size = 0.95
method | result | size |
risch | \(-\frac {\ln \relax (x ) \ln \left (4 x +\frac {1}{3}\right )}{3}+\frac {x^{2} \ln \relax (x )}{3}\) | \(19\) |
default | \(\frac {\ln \relax (3) \ln \relax (x )}{3}+\frac {x^{2} \ln \relax (x )}{3}-\frac {\ln \relax (x ) \ln \left (12 x +1\right )}{3}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 21, normalized size = 1.05 \begin {gather*} \frac {1}{3} \, {\left (x^{2} + \log \relax (3)\right )} \log \relax (x) - \frac {1}{3} \, \log \left (12 \, x + 1\right ) \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.36, size = 16, normalized size = 0.80 \begin {gather*} -\frac {\ln \relax (x)\,\left (\ln \left (4\,x+\frac {1}{3}\right )-x^2\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 20, normalized size = 1.00 \begin {gather*} \frac {x^{2} \log {\relax (x )}}{3} - \frac {\log {\relax (x )} \log {\left (4 x + \frac {1}{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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