Optimal. Leaf size=31 \[ -e^6+\frac {x^2}{3}+\frac {\log (x)}{\log \left (4 e^x\right )-\log (6+x)} \]
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Rubi [F] time = 1.21, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (12 x^2+2 x^3\right ) \log ^2\left (4 e^x\right )+\left (-15 x-3 x^2\right ) \log (x)+(-18-3 x) \log (6+x)+\left (12 x^2+2 x^3\right ) \log ^2(6+x)+\log \left (4 e^x\right ) \left (18+3 x+\left (-24 x^2-4 x^3\right ) \log (6+x)\right )}{\left (18 x+3 x^2\right ) \log ^2\left (4 e^x\right )+\left (-36 x-6 x^2\right ) \log \left (4 e^x\right ) \log (6+x)+\left (18 x+3 x^2\right ) \log ^2(6+x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x^2 (6+x) \log ^2\left (4 e^x\right )-3 x (5+x) \log (x)+(6+x) \log (6+x) \left (-3+2 x^2 \log (6+x)\right )-(6+x) \log \left (4 e^x\right ) \left (-3+4 x^2 \log (6+x)\right )}{3 x (6+x) \left (\log \left (4 e^x\right )-\log (6+x)\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {2 x^2 (6+x) \log ^2\left (4 e^x\right )-3 x (5+x) \log (x)+(6+x) \log (6+x) \left (-3+2 x^2 \log (6+x)\right )-(6+x) \log \left (4 e^x\right ) \left (-3+4 x^2 \log (6+x)\right )}{x (6+x) \left (\log \left (4 e^x\right )-\log (6+x)\right )^2} \, dx\\ &=\frac {1}{3} \int \left (2 x-\frac {3 (5+x) \log (x)}{(6+x) \left (\log \left (4 e^x\right )-\log (6+x)\right )^2}+\frac {3}{x \left (\log \left (4 e^x\right )-\log (6+x)\right )}\right ) \, dx\\ &=\frac {x^2}{3}-\int \frac {(5+x) \log (x)}{(6+x) \left (\log \left (4 e^x\right )-\log (6+x)\right )^2} \, dx+\int \frac {1}{x \left (\log \left (4 e^x\right )-\log (6+x)\right )} \, dx\\ &=\frac {x^2}{3}-\int \left (\frac {\log (x)}{\left (\log \left (4 e^x\right )-\log (6+x)\right )^2}-\frac {\log (x)}{(6+x) \left (\log \left (4 e^x\right )-\log (6+x)\right )^2}\right ) \, dx+\int \frac {1}{x \left (\log \left (4 e^x\right )-\log (6+x)\right )} \, dx\\ &=\frac {x^2}{3}-\int \frac {\log (x)}{\left (\log \left (4 e^x\right )-\log (6+x)\right )^2} \, dx+\int \frac {\log (x)}{(6+x) \left (\log \left (4 e^x\right )-\log (6+x)\right )^2} \, dx+\int \frac {1}{x \left (\log \left (4 e^x\right )-\log (6+x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{3} \left (x^2-\frac {3 \log (x)}{-\log \left (4 e^x\right )+\log (6+x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 40, normalized size = 1.29 \begin {gather*} \frac {x^{3} + 2 \, x^{2} \log \relax (2) - x^{2} \log \left (x + 6\right ) + 3 \, \log \relax (x)}{3 \, {\left (x + 2 \, \log \relax (2) - \log \left (x + 6\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, x^{2} + \frac {\log \relax (x)}{x + 2 \, \log \relax (2) - \log \left (x + 6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.00, size = 33, normalized size = 1.06
method | result | size |
risch | \(\frac {x^{2}}{3}+\frac {2 i \ln \relax (x )}{4 i \ln \relax (2)-2 i \ln \left (x +6\right )+2 i \ln \left ({\mathrm e}^{x}\right )}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 40, normalized size = 1.29 \begin {gather*} \frac {x^{3} + 2 \, x^{2} \log \relax (2) - x^{2} \log \left (x + 6\right ) + 3 \, \log \relax (x)}{3 \, {\left (x + 2 \, \log \relax (2) - \log \left (x + 6\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.64, size = 40, normalized size = 1.29 \begin {gather*} \frac {3\,\ln \relax (x)-x^2\,\ln \left (x+6\right )+x^2\,\ln \relax (4)+x^3}{3\,\left (x-\ln \left (x+6\right )+\ln \relax (4)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 19, normalized size = 0.61 \begin {gather*} \frac {x^{2}}{3} - \frac {\log {\relax (x )}}{- x + \log {\left (x + 6 \right )} - 2 \log {\relax (2 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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