Optimal. Leaf size=18 \[ e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \]
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Rubi [F] time = 127.75, 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 (9 x+3 x^2+\left (3 x+x^2\right ) \log \left (-1+e^{e^x}+x\right )\right ) \left (-9+6 x+7 x^2+e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )+\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+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 {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \left (9-6 x-7 x^2-e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )-\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{1-e^{e^x}-x} \, dx\\ &=\int \left (\frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x (3+x)}{-1+e^{e^x}+x}+\frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \left (-9+9 e^{e^x}+6 x+6 e^{e^x} x+7 x^2-3 \log \left (-1+e^{e^x}+x\right )+3 e^{e^x} \log \left (-1+e^{e^x}+x\right )+x \log \left (-1+e^{e^x}+x\right )+2 e^{e^x} x \log \left (-1+e^{e^x}+x\right )+2 x^2 \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x}\right ) \, dx\\ &=\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x (3+x)}{-1+e^{e^x}+x} \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \left (-9+9 e^{e^x}+6 x+6 e^{e^x} x+7 x^2-3 \log \left (-1+e^{e^x}+x\right )+3 e^{e^x} \log \left (-1+e^{e^x}+x\right )+x \log \left (-1+e^{e^x}+x\right )+2 e^{e^x} x \log \left (-1+e^{e^x}+x\right )+2 x^2 \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx\\ &=\int \left (\frac {3 e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x}+\frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x}\right ) \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \left (9-6 x-7 x^2-e^{e^x} (9+6 x)-\left (-1+e^{e^x}+x\right ) (3+2 x) \log \left (-1+e^{e^x}+x\right )\right )}{1-e^{e^x}-x} \, dx\\ &=3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \left (\frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x (3+x)}{-1+e^{e^x}+x}+e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )\right ) \, dx\\ &=3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x (3+x)}{-1+e^{e^x}+x} \, dx+\int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \left (3+\log \left (-1+e^{e^x}+x\right )\right ) \, dx\\ &=3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \left (\frac {3 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x}+\frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x}\right ) \, dx+\int \left (3 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x)+e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \log \left (-1+e^{e^x}+x\right )\right ) \, dx\\ &=3 \int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \log \left (-1+e^{e^x}+x\right ) \, dx\\ &=3 \int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int \left (3 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )}+2 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x\right ) \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \left (3 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \log \left (-1+e^{e^x}+x\right )+2 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x \log \left (-1+e^{e^x}+x\right )\right ) \, dx\\ &=2 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x \log \left (-1+e^{e^x}+x\right ) \, dx+3 \int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \log \left (-1+e^{e^x}+x\right ) \, dx+6 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x \, dx+9 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.16, size = 23, normalized size = 1.28 \begin {gather*} e^{3 x (3+x)} \left (-1+e^{e^x}+x\right )^{x (3+x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 25, normalized size = 1.39 \begin {gather*} e^{\left (3 \, x^{2} + {\left (x^{2} + 3 \, x\right )} \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 9 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.39, size = 31, normalized size = 1.72 \begin {gather*} e^{\left (x^{2} \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 3 \, x^{2} + 3 \, x \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 9 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 21, normalized size = 1.17
method | result | size |
risch | \(\left ({\mathrm e}^{{\mathrm e}^{x}}+x -1\right )^{\left (3+x \right ) x} {\mathrm e}^{3 \left (3+x \right ) x}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 31, normalized size = 1.72 \begin {gather*} e^{\left (x^{2} \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 3 \, x^{2} + 3 \, x \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 9 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 25, normalized size = 1.39 \begin {gather*} {\mathrm {e}}^{3\,x^2+9\,x}\,{\left (x+{\mathrm {e}}^{{\mathrm {e}}^x}-1\right )}^{x^2+3\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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