Optimal. Leaf size=32 \[ \frac {x+\frac {x \left (-e^x+x\right )}{4 \left (-3+x^2-\frac {\log (x)}{x}\right )^2}}{x} \]
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Rubi [F] time = 1.28, 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 {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(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 {x \left (x \left (-2+3 x+3 x^3\right )+e^x \left (2-3 x^2-4 x^3+x^4\right )+3 x \log (x)-e^x (2+x) \log (x)\right )}{4 \left (3 x-x^3+\log (x)\right )^3} \, dx\\ &=\frac {1}{4} \int \frac {x \left (x \left (-2+3 x+3 x^3\right )+e^x \left (2-3 x^2-4 x^3+x^4\right )+3 x \log (x)-e^x (2+x) \log (x)\right )}{\left (3 x-x^3+\log (x)\right )^3} \, dx\\ &=\frac {1}{4} \int \left (-\frac {x^2 \left (-2+3 x+3 x^3+3 \log (x)\right )}{\left (-3 x+x^3-\log (x)\right )^3}-\frac {e^x x \left (2-3 x^2-4 x^3+x^4-2 \log (x)-x \log (x)\right )}{\left (-3 x+x^3-\log (x)\right )^3}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {x^2 \left (-2+3 x+3 x^3+3 \log (x)\right )}{\left (-3 x+x^3-\log (x)\right )^3} \, dx\right )-\frac {1}{4} \int \frac {e^x x \left (2-3 x^2-4 x^3+x^4-2 \log (x)-x \log (x)\right )}{\left (-3 x+x^3-\log (x)\right )^3} \, dx\\ &=-\frac {e^x x \left (3 x^2-x^4+x \log (x)\right )}{4 \left (3 x-x^3+\log (x)\right )^3}-\frac {1}{4} \int \left (\frac {2 x^2 \left (-1-3 x+3 x^3\right )}{\left (-3 x+x^3-\log (x)\right )^3}-\frac {3 x^2}{\left (-3 x+x^3-\log (x)\right )^2}\right ) \, dx\\ &=-\frac {e^x x \left (3 x^2-x^4+x \log (x)\right )}{4 \left (3 x-x^3+\log (x)\right )^3}-\frac {1}{2} \int \frac {x^2 \left (-1-3 x+3 x^3\right )}{\left (-3 x+x^3-\log (x)\right )^3} \, dx+\frac {3}{4} \int \frac {x^2}{\left (-3 x+x^3-\log (x)\right )^2} \, dx\\ &=-\frac {e^x x \left (3 x^2-x^4+x \log (x)\right )}{4 \left (3 x-x^3+\log (x)\right )^3}-\frac {1}{2} \int \left (-\frac {x^2}{\left (-3 x+x^3-\log (x)\right )^3}-\frac {3 x^3}{\left (-3 x+x^3-\log (x)\right )^3}+\frac {3 x^5}{\left (-3 x+x^3-\log (x)\right )^3}\right ) \, dx+\frac {3}{4} \int \frac {x^2}{\left (-3 x+x^3-\log (x)\right )^2} \, dx\\ &=-\frac {e^x x \left (3 x^2-x^4+x \log (x)\right )}{4 \left (3 x-x^3+\log (x)\right )^3}+\frac {1}{2} \int \frac {x^2}{\left (-3 x+x^3-\log (x)\right )^3} \, dx+\frac {3}{4} \int \frac {x^2}{\left (-3 x+x^3-\log (x)\right )^2} \, dx+\frac {3}{2} \int \frac {x^3}{\left (-3 x+x^3-\log (x)\right )^3} \, dx-\frac {3}{2} \int \frac {x^5}{\left (-3 x+x^3-\log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.36, size = 27, normalized size = 0.84 \begin {gather*} -\frac {\left (e^x-x\right ) x^2}{4 \left (3 x-x^3+\log (x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 44, normalized size = 1.38 \begin {gather*} \frac {x^{3} - x^{2} e^{x}}{4 \, {\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} - 2 \, {\left (x^{3} - 3 \, x\right )} \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 45, normalized size = 1.41 \begin {gather*} \frac {x^{3} - x^{2} e^{x}}{4 \, {\left (x^{6} - 6 \, x^{4} - 2 \, x^{3} \log \relax (x) + 9 \, x^{2} + 6 \, x \log \relax (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 = 0.03, size = 25, normalized size = 0.78
method | result | size |
risch | \(\frac {\left (x -{\mathrm e}^{x}\right ) x^{2}}{4 \left (x^{3}-3 x -\ln \relax (x )\right )^{2}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 44, normalized size = 1.38 \begin {gather*} \frac {x^{3} - x^{2} e^{x}}{4 \, {\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} - 2 \, {\left (x^{3} - 3 \, x\right )} \log \relax (x) + \log \relax (x)^{2}\right )}} \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 = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (x^5-4\,x^4-3\,x^3+2\,x\right )-\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (x^2+2\,x\right )-3\,x^2\right )-2\,x^2+3\,x^3+3\,x^5}{{\ln \relax (x)}^2\,\left (36\,x-12\,x^3\right )+4\,{\ln \relax (x)}^3+\ln \relax (x)\,\left (12\,x^6-72\,x^4+108\,x^2\right )+108\,x^3-108\,x^5+36\,x^7-4\,x^9} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.42, size = 80, normalized size = 2.50 \begin {gather*} \frac {x^{3}}{4 x^{6} - 24 x^{4} + 36 x^{2} + \left (- 8 x^{3} + 24 x\right ) \log {\relax (x )} + 4 \log {\relax (x )}^{2}} - \frac {x^{2} e^{x}}{4 x^{6} - 24 x^{4} - 8 x^{3} \log {\relax (x )} + 36 x^{2} + 24 x \log {\relax (x )} + 4 \log {\relax (x )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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