Optimal. Leaf size=28 \[ 5-\frac {x^2}{3-x-\frac {1}{3} \left (2-4 e^x\right ) x \log (x)} \]
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Rubi [F] time = 2.92, 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 {-54 x+3 x^2+12 e^x x^2+\left (6 x^2+e^x \left (-12 x^2+12 x^3\right )\right ) \log (x)}{81-54 x+9 x^2+\left (-36 x+12 x^2+e^x \left (72 x-24 x^2\right )\right ) \log (x)+\left (4 x^2-16 e^x x^2+16 e^{2 x} x^2\right ) \log ^2(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 {3 x \left (-18+x+4 e^x x+2 \left (1+2 e^x (-1+x)\right ) x \log (x)\right )}{\left (9-3 x+2 \left (-1+2 e^x\right ) x \log (x)\right )^2} \, dx\\ &=3 \int \frac {x \left (-18+x+4 e^x x+2 \left (1+2 e^x (-1+x)\right ) x \log (x)\right )}{\left (9-3 x+2 \left (-1+2 e^x\right ) x \log (x)\right )^2} \, dx\\ &=3 \int \left (\frac {x (1-\log (x)+x \log (x))}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )}+\frac {x \left (-9+3 x-9 \log (x)-9 x \log (x)+3 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}\right ) \, dx\\ &=3 \int \frac {x (1-\log (x)+x \log (x))}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )} \, dx+3 \int \frac {x \left (-9+3 x-9 \log (x)-9 x \log (x)+3 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx\\ &=3 \int \left (-\frac {9 x}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}-\frac {9 x^2}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}+\frac {3 x^3}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}-\frac {9 x}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}+\frac {3 x^2}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}+\frac {2 x^3 \log (x)}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}\right ) \, dx+3 \int \left (-\frac {x}{9-3 x-2 x \log (x)+4 e^x x \log (x)}+\frac {x^2}{9-3 x-2 x \log (x)+4 e^x x \log (x)}+\frac {x}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )}\right ) \, dx\\ &=-\left (3 \int \frac {x}{9-3 x-2 x \log (x)+4 e^x x \log (x)} \, dx\right )+3 \int \frac {x^2}{9-3 x-2 x \log (x)+4 e^x x \log (x)} \, dx+3 \int \frac {x}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )} \, dx+6 \int \frac {x^3 \log (x)}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx+9 \int \frac {x^3}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx+9 \int \frac {x^2}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx-27 \int \frac {x}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx-27 \int \frac {x^2}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx-27 \int \frac {x}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.52, size = 25, normalized size = 0.89 \begin {gather*} -\frac {3 x^2}{9-3 x-2 x \log (x)+4 e^x x \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 25, normalized size = 0.89 \begin {gather*} -\frac {3 \, x^{2}}{2 \, {\left (2 \, x e^{x} - x\right )} \log \relax (x) - 3 \, x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 24, normalized size = 0.86 \begin {gather*} -\frac {3 \, x^{2}}{4 \, x e^{x} \log \relax (x) - 2 \, x \log \relax (x) - 3 \, x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 25, normalized size = 0.89
method | result | size |
risch | \(-\frac {3 x^{2}}{4 x \,{\mathrm e}^{x} \ln \relax (x )-2 x \ln \relax (x )-3 x +9}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 24, normalized size = 0.86 \begin {gather*} -\frac {3 \, x^{2}}{4 \, x e^{x} \log \relax (x) - 2 \, x \log \relax (x) - 3 \, x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {54\,x-12\,x^2\,{\mathrm {e}}^x-3\,x^2+\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (12\,x^2-12\,x^3\right )-6\,x^2\right )}{\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (72\,x-24\,x^2\right )-36\,x+12\,x^2\right )-54\,x+{\ln \relax (x)}^2\,\left (16\,x^2\,{\mathrm {e}}^{2\,x}-16\,x^2\,{\mathrm {e}}^x+4\,x^2\right )+9\,x^2+81} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 27, normalized size = 0.96 \begin {gather*} - \frac {3 x^{2}}{4 x e^{x} \log {\relax (x )} - 2 x \log {\relax (x )} - 3 x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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