Optimal. Leaf size=28 \[ x+e^{-x} \log ^2\left (\frac {\left (e^x-\frac {3}{x}\right )^2}{e^{8/5}}\right ) \]
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Rubi [F] time = 4.65, 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 {-3 e^x x+e^{2 x} x^2+\left (12+4 e^x x^2\right ) \log \left (\frac {9-6 e^x x+e^{2 x} x^2}{e^{8/5} x^2}\right )+\left (3 x-e^x x^2\right ) \log ^2\left (\frac {9-6 e^x x+e^{2 x} x^2}{e^{8/5} x^2}\right )}{-3 e^x x+e^{2 x} x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (3 e^x x-e^{2 x} x^2-\left (12+4 e^x x^2\right ) \log \left (\frac {9-6 e^x x+e^{2 x} x^2}{e^{8/5} x^2}\right )-\left (3 x-e^x x^2\right ) \log ^2\left (\frac {9-6 e^x x+e^{2 x} x^2}{e^{8/5} x^2}\right )\right )}{x \left (3-e^x x\right )} \, dx\\ &=\int \left (1+\frac {12 e^{-x} (1+x) \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )}{5 x \left (-3+e^x x\right )}-\frac {1}{25} e^{-x} \left (-28+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )\right ) \, dx\\ &=x-\frac {1}{25} \int e^{-x} \left (-28+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \, dx+\frac {12}{5} \int \frac {e^{-x} (1+x) \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )}{x \left (-3+e^x x\right )} \, dx\\ &=x-\frac {1}{25} \int \left (224 e^{-x}-180 e^{-x} \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )+25 e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \, dx+\frac {12}{5} \int \left (\frac {e^{-x} \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )}{-3+e^x x}+\frac {e^{-x} \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )}{x \left (-3+e^x x\right )}\right ) \, dx\\ &=x+\frac {12}{5} \int \frac {e^{-x} \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )}{-3+e^x x} \, dx+\frac {12}{5} \int \frac {e^{-x} \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )}{x \left (-3+e^x x\right )} \, dx+\frac {36}{5} \int e^{-x} \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx-\frac {224}{25} \int e^{-x} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )+\frac {12}{5} \int \left (-\frac {8 e^{-x}}{-3+e^x x}+\frac {5 e^{-x} \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )}{-3+e^x x}\right ) \, dx+\frac {12}{5} \int \left (-\frac {8 e^{-x}}{x \left (-3+e^x x\right )}+\frac {5 e^{-x} \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )}{x \left (-3+e^x x\right )}\right ) \, dx-\frac {36}{5} \int \frac {6 e^{-x}+2 x^2}{3 x-e^x x^2} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )-\frac {36}{5} \int \left (\frac {2 e^{-x}}{x}-\frac {2 (1+x)}{-3+e^x x}\right ) \, dx+12 \int \frac {e^{-x} \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )}{-3+e^x x} \, dx+12 \int \frac {e^{-x} \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )}{x \left (-3+e^x x\right )} \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )-12 \int \frac {2 \left (-3-e^x x^2\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx}{x \left (3-e^x x\right )} \, dx-12 \int \frac {2 \left (-3-e^x x^2\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{x \left (3-e^x x\right )} \, dx-\frac {72}{5} \int \frac {e^{-x}}{x} \, dx+\frac {72}{5} \int \frac {1+x}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {72 \text {Ei}(-x)}{5}-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )+\frac {72}{5} \int \left (\frac {1}{-3+e^x x}+\frac {x}{-3+e^x x}\right ) \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-24 \int \frac {\left (-3-e^x x^2\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx}{x \left (3-e^x x\right )} \, dx-24 \int \frac {\left (-3-e^x x^2\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{x \left (3-e^x x\right )} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {72 \text {Ei}(-x)}{5}-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )+\frac {72}{5} \int \frac {1}{-3+e^x x} \, dx+\frac {72}{5} \int \frac {x}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-24 \int \left (\int \frac {e^{-x}}{-3+e^x x} \, dx+\frac {3 (1+x) \int \frac {e^{-x}}{-3+e^x x} \, dx}{x \left (-3+e^x x\right )}\right ) \, dx-24 \int \left (\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx+\frac {3 (1+x) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{x \left (-3+e^x x\right )}\right ) \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {72 \text {Ei}(-x)}{5}-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )+\frac {72}{5} \int \frac {1}{-3+e^x x} \, dx+\frac {72}{5} \int \frac {x}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-24 \int \left (\int \frac {e^{-x}}{-3+e^x x} \, dx\right ) \, dx-24 \int \left (\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx\right ) \, dx-72 \int \frac {(1+x) \int \frac {e^{-x}}{-3+e^x x} \, dx}{x \left (-3+e^x x\right )} \, dx-72 \int \frac {(1+x) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{x \left (-3+e^x x\right )} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {72 \text {Ei}(-x)}{5}-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )+\frac {72}{5} \int \frac {1}{-3+e^x x} \, dx+\frac {72}{5} \int \frac {x}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-24 \int \left (\int \frac {e^{-x}}{-3+e^x x} \, dx\right ) \, dx-24 \int \left (\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx\right ) \, dx-72 \int \left (\frac {\int \frac {e^{-x}}{-3+e^x x} \, dx}{-3+e^x x}+\frac {\int \frac {e^{-x}}{-3+e^x x} \, dx}{x \left (-3+e^x x\right )}\right ) \, dx-72 \int \left (\frac {\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{-3+e^x x}+\frac {\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{x \left (-3+e^x x\right )}\right ) \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {72 \text {Ei}(-x)}{5}-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )+\frac {72}{5} \int \frac {1}{-3+e^x x} \, dx+\frac {72}{5} \int \frac {x}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-24 \int \left (\int \frac {e^{-x}}{-3+e^x x} \, dx\right ) \, dx-24 \int \left (\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx\right ) \, dx-72 \int \frac {\int \frac {e^{-x}}{-3+e^x x} \, dx}{-3+e^x x} \, dx-72 \int \frac {\int \frac {e^{-x}}{-3+e^x x} \, dx}{x \left (-3+e^x x\right )} \, dx-72 \int \frac {\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{-3+e^x x} \, dx-72 \int \frac {\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{x \left (-3+e^x x\right )} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.19, size = 51, normalized size = 1.82 \begin {gather*} \frac {1}{25} e^{-x} \left (64+25 e^x x-80 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )+25 \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 34, normalized size = 1.21 \begin {gather*} {\left (x e^{x} + \log \left (\frac {{\left (x^{2} e^{\left (2 \, x\right )} - 6 \, x e^{x} + 9\right )} e^{\left (-\frac {8}{5}\right )}}{x^{2}}\right )^{2}\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.10, size = 59, normalized size = 2.11 \begin {gather*} \frac {1}{25} \, {\left (25 \, x e^{x} + 25 \, \log \left (\frac {x^{2} e^{\left (2 \, x\right )} - 6 \, x e^{x} + 9}{x^{2}}\right )^{2} - 80 \, \log \left (\frac {x^{2} e^{\left (2 \, x\right )} - 6 \, x e^{x} + 9}{x^{2}}\right ) + 64\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.50, size = 2783, normalized size = 99.39
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2783\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 49, normalized size = 1.75 \begin {gather*} \frac {1}{25} \, {\left (25 \, x e^{x} - 40 \, {\left (5 \, \log \relax (x) + 4\right )} \log \left (x e^{x} - 3\right ) + 100 \, \log \left (x e^{x} - 3\right )^{2} + 100 \, \log \relax (x)^{2} + 160 \, \log \relax (x) + 64\right )} e^{\left (-x\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 = -0.04 \begin {gather*} \int \frac {\ln \left (\frac {{\mathrm {e}}^{-\frac {8}{5}}\,\left (x^2\,{\mathrm {e}}^{2\,x}-6\,x\,{\mathrm {e}}^x+9\right )}{x^2}\right )\,\left (4\,x^2\,{\mathrm {e}}^x+12\right )+x^2\,{\mathrm {e}}^{2\,x}+{\ln \left (\frac {{\mathrm {e}}^{-\frac {8}{5}}\,\left (x^2\,{\mathrm {e}}^{2\,x}-6\,x\,{\mathrm {e}}^x+9\right )}{x^2}\right )}^2\,\left (3\,x-x^2\,{\mathrm {e}}^x\right )-3\,x\,{\mathrm {e}}^x}{x^2\,{\mathrm {e}}^{2\,x}-3\,x\,{\mathrm {e}}^x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 32, normalized size = 1.14 \begin {gather*} x + e^{- x} \log {\left (\frac {x^{2} e^{2 x} - 6 x e^{x} + 9}{x^{2} e^{\frac {8}{5}}} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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