Optimal. Leaf size=22 \[ e^{\frac {e^x-x-3 x (x+\log (x))}{e^2}}+x \]
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Rubi [F] time = 1.74, 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 {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+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 \left (1+e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \left (-4+e^x-6 x-3 \log (x)\right )\right ) \, dx\\ &=x+\int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \left (-4+e^x-6 x-3 \log (x)\right ) \, dx\\ &=x+\int \left (-4 e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}}+e^{-2+e^{-2+x}+x-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}}-6 e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{1-\frac {3 x}{e^2}}-3 e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \log (x)\right ) \, dx\\ &=x-3 \int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \log (x) \, dx-4 \int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \, dx-6 \int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{1-\frac {3 x}{e^2}} \, dx+\int e^{-2+e^{-2+x}+x-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \, dx\\ &=x+3 \int \frac {\int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \, dx}{x} \, dx-4 \int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \, dx-6 \int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{1-\frac {3 x}{e^2}} \, dx-(3 \log (x)) \int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \, dx+\int e^{-2+e^{-2+x}+x-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 29, normalized size = 1.32 \begin {gather*} x+e^{\frac {e^x-x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 90, normalized size = 4.09 \begin {gather*} \frac {3 \, x^{3} + 3 \, x^{2} \log \relax (x) + x^{2} - x e^{x} - e^{\left (-{\left (3 \, x^{2} - e^{2} \log \left (-3 \, x^{2} - 3 \, x \log \relax (x) - x + e^{x}\right ) + 3 \, x \log \relax (x) + x + 2 \, e^{2} - e^{x}\right )} e^{\left (-2\right )} + 2\right )}}{3 \, x^{2} + 3 \, x \log \relax (x) + x - e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{2} + {\left (6 \, x - e^{x} + 3 \, \log \relax (x) + 4\right )} e^{\left (e^{\left (\log \left (-3 \, x^{2} - 3 \, x \log \relax (x) - x + e^{x}\right ) - 2\right )} + \log \left (-3 \, x^{2} - 3 \, x \log \relax (x) - x + e^{x}\right ) - 2\right )} + 3 \, x \log \relax (x) + x - e^{x}}{3 \, x^{2} + 3 \, x \log \relax (x) + x - e^{x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 23, normalized size = 1.05
method | result | size |
risch | \(x +{\mathrm e}^{\left (-3 x \ln \relax (x )+{\mathrm e}^{x}-3 x^{2}-x \right ) {\mathrm e}^{-2}}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 44, normalized size = 2.00 \begin {gather*} {\left (x e^{\left (3 \, x^{2} e^{\left (-2\right )} + x e^{\left (-2\right )}\right )} + e^{\left (-3 \, x e^{\left (-2\right )} \log \relax (x) + e^{\left (x - 2\right )}\right )}\right )} e^{\left (-3 \, x^{2} e^{\left (-2\right )} - x e^{\left (-2\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.19, size = 32, normalized size = 1.45 \begin {gather*} x+\frac {{\mathrm {e}}^{-3\,x^2\,{\mathrm {e}}^{-2}}\,{\mathrm {e}}^{{\mathrm {e}}^{-2}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-2}}}{x^{3\,x\,{\mathrm {e}}^{-2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 22, normalized size = 1.00 \begin {gather*} x + e^{\frac {- 3 x^{2} - 3 x \log {\relax (x )} - x + e^{x}}{e^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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