Optimal. Leaf size=35 \[ e^{-\frac {x}{-e^2+\frac {e^{2 e^x}}{x}+2 x}} x-3 x^3 \]
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Rubi [F]
time = 32.83, 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^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (e^{4 e^x}+e^4 x^2-3 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+2 x^2+2 e^x x^3\right )+e^{\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (-9 e^{4 e^x} x^2-9 e^4 x^4+36 e^2 x^5-36 x^6+e^{2 e^x} \left (18 e^2 x^3-36 x^4\right )\right )\right )}{e^{4 e^x}+e^4 x^2-4 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+4 x^2\right )} \, 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^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (e^{4 e^x}+e^4 x^2-3 e^2 x^3+4 x^4-9 e^{\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2 \left (e^{2 e^x}-e^2 x+2 x^2\right )^2+2 e^{2 e^x} x \left (-e^2+x+e^x x^2\right )\right )}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx\\ &=\int \left (-9 x^2+\frac {e^{4-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2}-\frac {3 e^{2-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2}+\frac {e^{4 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}}}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2}+\frac {4 e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^4}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2}+\frac {2 e^{2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x \left (-e^2+x+e^x x^2\right )}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2}\right ) \, dx\\ &=-3 x^3+2 \int \frac {e^{2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x \left (-e^2+x+e^x x^2\right )}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx-3 \int \frac {e^{2-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+4 \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^4}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+\int \frac {e^{4-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+\int \frac {e^{4 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}}}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx\\ &=-3 x^3+2 \int \left (\frac {e^{2 e^x+x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2}+\frac {e^{2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x \left (-e^2+x\right )}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2}\right ) \, dx-3 \int \frac {e^{2-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+4 \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^4}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+\int \frac {e^{4-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+\int \frac {e^{4 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}}}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx\\ &=-3 x^3+2 \int \frac {e^{2 e^x+x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+2 \int \frac {e^{2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x \left (-e^2+x\right )}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx-3 \int \frac {e^{2-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+4 \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^4}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+\int \frac {e^{4-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+\int \frac {e^{4 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}}}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx\\ &=-3 x^3+2 \int \frac {e^{2 e^x+x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+2 \int \left (-\frac {e^{2+2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2}+\frac {e^{2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2}\right ) \, dx-3 \int \frac {e^{2-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+4 \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^4}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+\int \frac {e^{4-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+\int \frac {e^{4 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}}}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx\\ &=-3 x^3-2 \int \frac {e^{2+2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+2 \int \frac {e^{2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+2 \int \frac {e^{2 e^x+x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx-3 \int \frac {e^{2-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+4 \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^4}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+\int \frac {e^{4-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+\int \frac {e^{4 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}}}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.29, size = 36, normalized size = 1.03 \begin {gather*} e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x-3 x^3 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 33, normalized size = 0.94
method | result | size |
risch | \(-3 x^{3}+x \,{\mathrm e}^{\frac {x^{2}}{-{\mathrm e}^{2 \,{\mathrm e}^{x}}+{\mathrm e}^{2} x -2 x^{2}}}\) | \(33\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 58, normalized size = 1.66 \begin {gather*} -{\left (3 \, x^{3} e^{\left (\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} - x\right )} e^{\left (-\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.72, size = 27, normalized size = 0.77 \begin {gather*} - 3 x^{3} + x e^{- \frac {x^{2}}{2 x^{2} - x e^{2} + e^{2 e^{x}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 58, normalized size = 1.66 \begin {gather*} -{\left (3 \, x^{3} e^{\left (\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} - x\right )} e^{\left (-\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.22, size = 32, normalized size = 0.91 \begin {gather*} x\,{\mathrm {e}}^{-\frac {x^2}{{\mathrm {e}}^{2\,{\mathrm {e}}^x}-x\,{\mathrm {e}}^2+2\,x^2}}-3\,x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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