Optimal. Leaf size=20 \[ x \left (-3+e^x-\frac {x^2}{2 x+\log (x)}\right ) \]
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Rubi [F] time = 1.01, 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 {-11 x^2-4 x^3+e^x \left (4 x^2+4 x^3\right )+\left (-12 x-3 x^2+e^x \left (4 x+4 x^2\right )\right ) \log (x)+\left (-3+e^x (1+x)\right ) \log ^2(x)}{4 x^2+4 x \log (x)+\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 {x^2 \left (-11-4 x+4 e^x (1+x)\right )+x \left (4 e^x (1+x)-3 (4+x)\right ) \log (x)+\left (-3+e^x (1+x)\right ) \log ^2(x)}{(2 x+\log (x))^2} \, dx\\ &=\int \left (e^x (1+x)-\frac {11 x^2}{(2 x+\log (x))^2}-\frac {4 x^3}{(2 x+\log (x))^2}-\frac {3 x (4+x) \log (x)}{(2 x+\log (x))^2}-\frac {3 \log ^2(x)}{(2 x+\log (x))^2}\right ) \, dx\\ &=-\left (3 \int \frac {x (4+x) \log (x)}{(2 x+\log (x))^2} \, dx\right )-3 \int \frac {\log ^2(x)}{(2 x+\log (x))^2} \, dx-4 \int \frac {x^3}{(2 x+\log (x))^2} \, dx-11 \int \frac {x^2}{(2 x+\log (x))^2} \, dx+\int e^x (1+x) \, dx\\ &=e^x (1+x)-3 \int \left (1+\frac {4 x^2}{(2 x+\log (x))^2}-\frac {4 x}{2 x+\log (x)}\right ) \, dx-3 \int \left (-\frac {2 x^2 (4+x)}{(2 x+\log (x))^2}+\frac {x (4+x)}{2 x+\log (x)}\right ) \, dx-4 \int \frac {x^3}{(2 x+\log (x))^2} \, dx-11 \int \frac {x^2}{(2 x+\log (x))^2} \, dx-\int e^x \, dx\\ &=-e^x-3 x+e^x (1+x)-3 \int \frac {x (4+x)}{2 x+\log (x)} \, dx-4 \int \frac {x^3}{(2 x+\log (x))^2} \, dx+6 \int \frac {x^2 (4+x)}{(2 x+\log (x))^2} \, dx-11 \int \frac {x^2}{(2 x+\log (x))^2} \, dx-12 \int \frac {x^2}{(2 x+\log (x))^2} \, dx+12 \int \frac {x}{2 x+\log (x)} \, dx\\ &=-e^x-3 x+e^x (1+x)-3 \int \left (\frac {4 x}{2 x+\log (x)}+\frac {x^2}{2 x+\log (x)}\right ) \, dx-4 \int \frac {x^3}{(2 x+\log (x))^2} \, dx+6 \int \left (\frac {4 x^2}{(2 x+\log (x))^2}+\frac {x^3}{(2 x+\log (x))^2}\right ) \, dx-11 \int \frac {x^2}{(2 x+\log (x))^2} \, dx-12 \int \frac {x^2}{(2 x+\log (x))^2} \, dx+12 \int \frac {x}{2 x+\log (x)} \, dx\\ &=-e^x-3 x+e^x (1+x)-3 \int \frac {x^2}{2 x+\log (x)} \, dx-4 \int \frac {x^3}{(2 x+\log (x))^2} \, dx+6 \int \frac {x^3}{(2 x+\log (x))^2} \, dx-11 \int \frac {x^2}{(2 x+\log (x))^2} \, dx-12 \int \frac {x^2}{(2 x+\log (x))^2} \, dx+24 \int \frac {x^2}{(2 x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.33, size = 22, normalized size = 1.10 \begin {gather*} -3 x+e^x x-\frac {x^3}{2 x+\log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 38, normalized size = 1.90 \begin {gather*} -\frac {x^{3} - 2 \, x^{2} e^{x} + 6 \, x^{2} - {\left (x e^{x} - 3 \, x\right )} \log \relax (x)}{2 \, x + \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 = 38, normalized size = 1.90 \begin {gather*} -\frac {x^{3} - 2 \, x^{2} e^{x} - x e^{x} \log \relax (x) + 6 \, x^{2} + 3 \, x \log \relax (x)}{2 \, x + \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 22, normalized size = 1.10
method | result | size |
risch | \({\mathrm e}^{x} x -3 x -\frac {x^{3}}{2 x +\ln \relax (x )}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 38, normalized size = 1.90 \begin {gather*} -\frac {x^{3} + 6 \, x^{2} - {\left (2 \, x^{2} + x \log \relax (x)\right )} e^{x} + 3 \, x \log \relax (x)}{2 \, x + \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.89, size = 65, normalized size = 3.25 \begin {gather*} \frac {3}{16\,\left (x+\frac {1}{2}\right )}-\frac {9\,x}{4}+x\,{\mathrm {e}}^x+\frac {\frac {3\,x^3\,\ln \relax (x)}{2\,x+1}-\frac {x\,\left (x^2-4\,x^3\right )}{2\,x+1}}{2\,x+\ln \relax (x)}-\frac {3\,x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 17, normalized size = 0.85 \begin {gather*} - \frac {x^{3}}{2 x + \log {\relax (x )}} + x e^{x} - 3 x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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