Optimal. Leaf size=21 \[ 4+\frac {2}{3} e^{5+x+\frac {1}{e^x+x}} x \log (3) \]
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Rubi [F] time = 6.44, 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 {\exp \left (\frac {1+5 x+x^2+e^x (5+x)+\left (e^x+x\right ) \log (x)}{e^x+x}\right ) \left (e^{2 x} (2+2 x) \log (3)+e^x \left (2 x+4 x^2\right ) \log (3)+\left (-2 x+2 x^2+2 x^3\right ) \log (3)\right )}{3 e^{2 x} x+6 e^x x^2+3 x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} \left (e^{2 x} (1+x)+e^x x (1+2 x)+x \left (-1+x+x^2\right )\right ) \log (3)}{3 \left (e^x+x\right )^2} \, dx\\ &=\frac {1}{3} (2 \log (3)) \int \frac {e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} \left (e^{2 x} (1+x)+e^x x (1+2 x)+x \left (-1+x+x^2\right )\right )}{\left (e^x+x\right )^2} \, dx\\ &=\frac {1}{3} (2 \log (3)) \int \left (e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}}+e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} x+\frac {e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} (-1+x) x}{\left (e^x+x\right )^2}-\frac {e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} x}{e^x+x}\right ) \, dx\\ &=\frac {1}{3} (2 \log (3)) \int e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} \, dx+\frac {1}{3} (2 \log (3)) \int e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} x \, dx+\frac {1}{3} (2 \log (3)) \int \frac {e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} (-1+x) x}{\left (e^x+x\right )^2} \, dx-\frac {1}{3} (2 \log (3)) \int \frac {e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} x}{e^x+x} \, dx\\ &=\frac {1}{3} (2 \log (3)) \int e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} \, dx+\frac {1}{3} (2 \log (3)) \int e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} x \, dx-\frac {1}{3} (2 \log (3)) \int \frac {e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} x}{e^x+x} \, dx+\frac {1}{3} (2 \log (3)) \int \left (-\frac {e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} x}{\left (e^x+x\right )^2}+\frac {e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} x^2}{\left (e^x+x\right )^2}\right ) \, dx\\ &=\frac {1}{3} (2 \log (3)) \int e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} \, dx+\frac {1}{3} (2 \log (3)) \int e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} x \, dx-\frac {1}{3} (2 \log (3)) \int \frac {e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} x}{\left (e^x+x\right )^2} \, dx+\frac {1}{3} (2 \log (3)) \int \frac {e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} x^2}{\left (e^x+x\right )^2} \, dx-\frac {1}{3} (2 \log (3)) \int \frac {e^{\frac {1+5 x+x^2+e^x (5+x)}{e^x+x}} x}{e^x+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.67, size = 19, normalized size = 0.90 \begin {gather*} \frac {2}{3} e^{5+x+\frac {1}{e^x+x}} x \log (3) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 33, normalized size = 1.57 \begin {gather*} \frac {2}{3} \, e^{\left (\frac {x^{2} + {\left (x + 5\right )} e^{x} + {\left (x + e^{x}\right )} \log \relax (x) + 5 \, x + 1}{x + e^{x}}\right )} \log \relax (3) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 29, normalized size = 1.38 \begin {gather*} \frac {2}{3} \, x e^{\left (\frac {x^{2} + x e^{x} + 5 \, x + 5 \, e^{x} + 1}{x + e^{x}}\right )} \log \relax (3) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 1.81
method | result | size |
risch | \(\frac {2 \ln \relax (3) {\mathrm e}^{\frac {{\mathrm e}^{x} \ln \relax (x )+x \ln \relax (x )+{\mathrm e}^{x} x +x^{2}+5 \,{\mathrm e}^{x}+5 x +1}{{\mathrm e}^{x}+x}}}{3}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2}{3} \, \int \frac {{\left ({\left (x + 1\right )} e^{\left (2 \, x\right )} \log \relax (3) + {\left (2 \, x^{2} + x\right )} e^{x} \log \relax (3) + {\left (x^{3} + x^{2} - x\right )} \log \relax (3)\right )} e^{\left (\frac {x^{2} + {\left (x + 5\right )} e^{x} + {\left (x + e^{x}\right )} \log \relax (x) + 5 \, x + 1}{x + e^{x}}\right )}}{x^{3} + 2 \, x^{2} e^{x} + x e^{\left (2 \, x\right )}}\,{d x} \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 = -0.05 \begin {gather*} \int \frac {{\mathrm {e}}^{\frac {5\,x+{\mathrm {e}}^x\,\left (x+5\right )+x^2+\ln \relax (x)\,\left (x+{\mathrm {e}}^x\right )+1}{x+{\mathrm {e}}^x}}\,\left (\ln \relax (3)\,\left (2\,x^3+2\,x^2-2\,x\right )+{\mathrm {e}}^x\,\ln \relax (3)\,\left (4\,x^2+2\,x\right )+{\mathrm {e}}^{2\,x}\,\ln \relax (3)\,\left (2\,x+2\right )\right )}{3\,x\,{\mathrm {e}}^{2\,x}+6\,x^2\,{\mathrm {e}}^x+3\,x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.70, size = 36, normalized size = 1.71 \begin {gather*} \frac {2 e^{\frac {x^{2} + 5 x + \left (x + 5\right ) e^{x} + \left (x + e^{x}\right ) \log {\relax (x )} + 1}{x + e^{x}}} \log {\relax (3 )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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