Optimal. Leaf size=21 \[ \frac {8}{3} x \left (-\frac {e}{x}+x\right ) \log (4+2 x \log (x)) \]
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Rubi [F] time = 0.73, 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 {-8 e+8 x^2+\left (-8 e+8 x^2\right ) \log (x)+\left (32 x+16 x^2 \log (x)\right ) \log (4+2 x \log (x))}{6+3 x \log (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 {-8 e+8 x^2+\left (-8 e+8 x^2\right ) \log (x)+\left (32 x+16 x^2 \log (x)\right ) \log (4+2 x \log (x))}{3 (2+x \log (x))} \, dx\\ &=\frac {1}{3} \int \frac {-8 e+8 x^2+\left (-8 e+8 x^2\right ) \log (x)+\left (32 x+16 x^2 \log (x)\right ) \log (4+2 x \log (x))}{2+x \log (x)} \, dx\\ &=\frac {1}{3} \int \left (-\frac {8 \left (e-x^2\right ) (1+\log (x))}{2+x \log (x)}+16 x \log (2 (2+x \log (x)))\right ) \, dx\\ &=-\left (\frac {8}{3} \int \frac {\left (e-x^2\right ) (1+\log (x))}{2+x \log (x)} \, dx\right )+\frac {16}{3} \int x \log (2 (2+x \log (x))) \, dx\\ &=\frac {8}{3} x^2 \log (2 (2+x \log (x)))-\frac {8}{3} \int \left (\frac {e-x^2}{x}-\frac {(-2+x) \left (-e+x^2\right )}{x (2+x \log (x))}\right ) \, dx-\frac {16}{3} \int \frac {x^2 (1+\log (x))}{2 (2+x \log (x))} \, dx\\ &=\frac {8}{3} x^2 \log (2 (2+x \log (x)))-\frac {8}{3} \int \frac {e-x^2}{x} \, dx+\frac {8}{3} \int \frac {(-2+x) \left (-e+x^2\right )}{x (2+x \log (x))} \, dx-\frac {8}{3} \int \frac {x^2 (1+\log (x))}{2+x \log (x)} \, dx\\ &=\frac {8}{3} x^2 \log (2 (2+x \log (x)))-\frac {8}{3} \int \left (\frac {e}{x}-x\right ) \, dx-\frac {8}{3} \int \left (x+\frac {(-2+x) x}{2+x \log (x)}\right ) \, dx+\frac {8}{3} \int \left (-\frac {e}{2+x \log (x)}+\frac {2 e}{x (2+x \log (x))}-\frac {2 x}{2+x \log (x)}+\frac {x^2}{2+x \log (x)}\right ) \, dx\\ &=-\frac {8}{3} e \log (x)+\frac {8}{3} x^2 \log (2 (2+x \log (x)))-\frac {8}{3} \int \frac {(-2+x) x}{2+x \log (x)} \, dx+\frac {8}{3} \int \frac {x^2}{2+x \log (x)} \, dx-\frac {16}{3} \int \frac {x}{2+x \log (x)} \, dx-\frac {1}{3} (8 e) \int \frac {1}{2+x \log (x)} \, dx+\frac {1}{3} (16 e) \int \frac {1}{x (2+x \log (x))} \, dx\\ &=-\frac {8}{3} e \log (x)+\frac {8}{3} x^2 \log (2 (2+x \log (x)))+\frac {8}{3} \int \frac {x^2}{2+x \log (x)} \, dx-\frac {8}{3} \int \left (-\frac {2 x}{2+x \log (x)}+\frac {x^2}{2+x \log (x)}\right ) \, dx-\frac {16}{3} \int \frac {x}{2+x \log (x)} \, dx-\frac {1}{3} (8 e) \int \frac {1}{2+x \log (x)} \, dx+\frac {1}{3} (16 e) \int \frac {1}{x (2+x \log (x))} \, dx\\ &=-\frac {8}{3} e \log (x)+\frac {8}{3} x^2 \log (2 (2+x \log (x)))-\frac {1}{3} (8 e) \int \frac {1}{2+x \log (x)} \, dx+\frac {1}{3} (16 e) \int \frac {1}{x (2+x \log (x))} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.16, size = 30, normalized size = 1.43 \begin {gather*} 8 \left (-\frac {1}{3} e \log (2+x \log (x))+\frac {1}{3} x^2 \log (4+2 x \log (x))\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 18, normalized size = 0.86 \begin {gather*} \frac {8}{3} \, {\left (x^{2} - e\right )} \log \left (2 \, x \log \relax (x) + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 25, normalized size = 1.19 \begin {gather*} \frac {8}{3} \, x^{2} \log \left (2 \, x \log \relax (x) + 4\right ) - \frac {8}{3} \, e \log \left (x \log \relax (x) + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 27, normalized size = 1.29
method | result | size |
norman | \(-\frac {8 \,{\mathrm e} \ln \left (2 x \ln \relax (x )+4\right )}{3}+\frac {8 x^{2} \ln \left (2 x \ln \relax (x )+4\right )}{3}\) | \(27\) |
risch | \(\frac {8 x^{2} \ln \left (2 x \ln \relax (x )+4\right )}{3}-\frac {8 \,{\mathrm e} \ln \relax (x )}{3}-\frac {8 \,{\mathrm e} \ln \left (\frac {2}{x}+\ln \relax (x )\right )}{3}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.74, size = 41, normalized size = 1.95 \begin {gather*} \frac {8}{3} \, x^{2} \log \relax (2) + \frac {8}{3} \, x^{2} \log \left (x \log \relax (x) + 2\right ) - \frac {8}{3} \, e \log \relax (x) - \frac {8}{3} \, e \log \left (\frac {x \log \relax (x) + 2}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.87, size = 35, normalized size = 1.67 \begin {gather*} \frac {8\,x^2\,\ln \left (2\,x\,\ln \relax (x)+4\right )}{3}-\frac {8\,\mathrm {e}\,\ln \relax (x)}{3}-\frac {8\,\mathrm {e}\,\ln \left (\frac {x\,\ln \relax (x)+2}{x}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.41, size = 41, normalized size = 1.95 \begin {gather*} \frac {8 x^{2} \log {\left (2 x \log {\relax (x )} + 4 \right )}}{3} - \frac {8 e \log {\relax (x )}}{3} - \frac {8 e \log {\left (\log {\relax (x )} + \frac {2}{x} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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