Optimal. Leaf size=18 \[ x+e^{-2 x-4 e^{e^x} x} x \]
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Rubi [F] time = 2.02, 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 e^{-2 x-4 e^{e^x} x} \left (1+e^{2 x+4 e^{e^x} x}-2 x+e^{e^x} \left (-4 x-4 e^x x^2\right )\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-2 \left (1+2 e^{e^x}\right ) x} \left (1+e^{2 x+4 e^{e^x} x}-2 x+e^{e^x} \left (-4 x-4 e^x x^2\right )\right ) \, dx\\ &=\int \left (1+e^{-2 \left (1+2 e^{e^x}\right ) x}-2 e^{-2 \left (1+2 e^{e^x}\right ) x} x-4 e^{e^x-2 \left (1+2 e^{e^x}\right ) x} x \left (1+e^x x\right )\right ) \, dx\\ &=x-2 \int e^{-2 \left (1+2 e^{e^x}\right ) x} x \, dx-4 \int e^{e^x-2 \left (1+2 e^{e^x}\right ) x} x \left (1+e^x x\right ) \, dx+\int e^{-2 \left (1+2 e^{e^x}\right ) x} \, dx\\ &=x-2 \int e^{-2 \left (1+2 e^{e^x}\right ) x} x \, dx-4 \int \left (e^{e^x-2 \left (1+2 e^{e^x}\right ) x} x+e^{e^x+x-2 \left (1+2 e^{e^x}\right ) x} x^2\right ) \, dx+\int e^{-2 \left (1+2 e^{e^x}\right ) x} \, dx\\ &=x-2 \int e^{-2 \left (1+2 e^{e^x}\right ) x} x \, dx-4 \int e^{e^x-2 \left (1+2 e^{e^x}\right ) x} x \, dx-4 \int e^{e^x+x-2 \left (1+2 e^{e^x}\right ) x} x^2 \, dx+\int e^{-2 \left (1+2 e^{e^x}\right ) x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.65, size = 18, normalized size = 1.00 \begin {gather*} x+e^{-2 x-4 e^{e^x} x} x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 27, normalized size = 1.50 \begin {gather*} {\left (x e^{\left (4 \, x e^{\left (e^{x}\right )} + 2 \, x\right )} + x\right )} e^{\left (-4 \, x e^{\left (e^{x}\right )} - 2 \, x\right )} \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 -{\left (4 \, {\left (x^{2} e^{x} + x\right )} e^{\left (e^{x}\right )} + 2 \, x - e^{\left (4 \, x e^{\left (e^{x}\right )} + 2 \, x\right )} - 1\right )} e^{\left (-4 \, x e^{\left (e^{x}\right )} - 2 \, x\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 16, normalized size = 0.89
method | result | size |
risch | \(x +x \,{\mathrm e}^{-2 x \left (2 \,{\mathrm e}^{{\mathrm e}^{x}}+1\right )}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 15, normalized size = 0.83 \begin {gather*} x e^{\left (-4 \, x e^{\left (e^{x}\right )} - 2 \, x\right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.09, size = 15, normalized size = 0.83 \begin {gather*} x+x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-4\,x\,{\mathrm {e}}^{{\mathrm {e}}^x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.09, size = 17, normalized size = 0.94 \begin {gather*} x e^{- 4 x e^{e^{x}} - 2 x} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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