Optimal. Leaf size=25 \[ \frac {2 e^{-x-2 x^2} x}{-3+e^{-4+e^x}} \]
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Rubi [F] time = 2.43, 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 {-6+6 x+24 x^2+e^{-4+e^x} \left (2-2 x-2 e^x x-8 x^2\right )}{9 e^{x+2 x^2}-6 e^{-4+e^x+x+2 x^2}+e^{-8+2 e^x+x+2 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^{8-x-2 x^2} \left (-6+6 x+24 x^2+e^{-4+e^x} \left (2-2 x-2 e^x x-8 x^2\right )\right )}{\left (3 e^4-e^{e^x}\right )^2} \, dx\\ &=\int \left (-\frac {6 e^{8-x-2 x^2}}{\left (-3 e^4+e^{e^x}\right )^2}+\frac {2 e^{4+e^x-x-2 x^2}}{\left (-3 e^4+e^{e^x}\right )^2}-\frac {2 e^{4+e^x-2 x^2} x}{\left (-3 e^4+e^{e^x}\right )^2}+\frac {6 e^{8-x-2 x^2} x}{\left (-3 e^4+e^{e^x}\right )^2}-\frac {2 e^{4+e^x-x-2 x^2} x}{\left (-3 e^4+e^{e^x}\right )^2}+\frac {24 e^{8-x-2 x^2} x^2}{\left (-3 e^4+e^{e^x}\right )^2}-\frac {8 e^{4+e^x-x-2 x^2} x^2}{\left (-3 e^4+e^{e^x}\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{4+e^x-x-2 x^2}}{\left (-3 e^4+e^{e^x}\right )^2} \, dx-2 \int \frac {e^{4+e^x-2 x^2} x}{\left (-3 e^4+e^{e^x}\right )^2} \, dx-2 \int \frac {e^{4+e^x-x-2 x^2} x}{\left (-3 e^4+e^{e^x}\right )^2} \, dx-6 \int \frac {e^{8-x-2 x^2}}{\left (-3 e^4+e^{e^x}\right )^2} \, dx+6 \int \frac {e^{8-x-2 x^2} x}{\left (-3 e^4+e^{e^x}\right )^2} \, dx-8 \int \frac {e^{4+e^x-x-2 x^2} x^2}{\left (-3 e^4+e^{e^x}\right )^2} \, dx+24 \int \frac {e^{8-x-2 x^2} x^2}{\left (-3 e^4+e^{e^x}\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.69, size = 28, normalized size = 1.12 \begin {gather*} \frac {2 e^{4-x-2 x^2} x}{-3 e^4+e^{e^x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 27, normalized size = 1.08 \begin {gather*} \frac {2 \, x}{e^{\left (2 \, x^{2} + x + e^{x} - 4\right )} - 3 \, e^{\left (2 \, x^{2} + x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 29, normalized size = 1.16 \begin {gather*} \frac {2 \, x e^{4}}{e^{\left (2 \, x^{2} + x + e^{x}\right )} - 3 \, e^{\left (2 \, x^{2} + x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 22, normalized size = 0.88
method | result | size |
risch | \(\frac {2 x \,{\mathrm e}^{-\left (2 x +1\right ) x}}{{\mathrm e}^{{\mathrm e}^{x}-4}-3}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 29, normalized size = 1.16 \begin {gather*} \frac {2 \, x e^{4}}{e^{\left (2 \, x^{2} + x + e^{x}\right )} - 3 \, e^{\left (2 \, x^{2} + x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.48, size = 22, normalized size = 0.88 \begin {gather*} \frac {2\,x\,{\mathrm {e}}^{-2\,x^2-x}}{{\mathrm {e}}^{{\mathrm {e}}^x-4}-3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 27, normalized size = 1.08 \begin {gather*} \frac {2 x}{e^{2 x^{2} + x} e^{e^{x} - 4} - 3 e^{2 x^{2} + x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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