Optimal. Leaf size=22 \[ e^{-\frac {1}{2} e^{-2 x} (1-x)-2 x} x \]
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Rubi [F] time = 0.65, 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 {1}{2} \exp \left (-2 x-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )\right ) \left (e^{2 x} (2-4 x)+3 x-2 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} &=\frac {1}{2} \int \exp \left (-2 x-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )\right ) \left (e^{2 x} (2-4 x)+3 x-2 x^2\right ) \, dx\\ &=\frac {1}{2} \int e^{-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} \left (e^{2 x} (2-4 x)+3 x-2 x^2\right ) \, dx\\ &=\frac {1}{2} \int \left (3 e^{-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} x-2 e^{-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} x^2-2 e^{2 x-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} (-1+2 x)\right ) \, dx\\ &=\frac {3}{2} \int e^{-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} x \, dx-\int e^{-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} x^2 \, dx-\int e^{2 x-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} (-1+2 x) \, dx\\ &=\frac {3}{2} \int e^{-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} x \, dx-\int e^{-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} x^2 \, dx-\int e^{-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )} (-1+2 x) \, dx\\ &=\frac {3}{2} \int e^{-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} x \, dx-\int e^{-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} x^2 \, dx-\int \left (-e^{-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )}+2 e^{-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )} x\right ) \, dx\\ &=\frac {3}{2} \int e^{-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} x \, dx-2 \int e^{-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )} x \, dx+\int e^{-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )} \, dx-\int e^{-\frac {1}{2} e^{-2 x} \left (1-x+8 e^{2 x} x\right )} x^2 \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.23, size = 24, normalized size = 1.09 \begin {gather*} e^{\frac {1}{2} e^{-2 x} \left (-1+x-4 e^{2 x} x\right )} x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 25, normalized size = 1.14 \begin {gather*} x e^{\left (-\frac {1}{2} \, {\left (8 \, x e^{\left (2 \, x\right )} - x + 1\right )} e^{\left (-2 \, 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 -\frac {1}{2} \, {\left (2 \, x^{2} + 2 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - 3 \, x\right )} e^{\left (-\frac {1}{2} \, {\left (4 \, x e^{\left (2 \, x\right )} - x + 1\right )} e^{\left (-2 \, 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 = 0.06, size = 22, normalized size = 1.00
method | result | size |
risch | \(x \,{\mathrm e}^{-\frac {\left (4 x \,{\mathrm e}^{2 x}-x +1\right ) {\mathrm e}^{-2 x}}{2}}\) | \(22\) |
norman | \(x \,{\mathrm e}^{-\frac {\left (4 x \,{\mathrm e}^{2 x}-x +1\right ) {\mathrm e}^{-2 x}}{2}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 20, normalized size = 0.91 \begin {gather*} x e^{\left (\frac {1}{2} \, x e^{\left (-2 \, x\right )} - 2 \, x - \frac {1}{2} \, e^{\left (-2 \, x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.18, size = 21, normalized size = 0.95 \begin {gather*} x\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{-2\,x}}{2}}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{-2\,x}}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 22, normalized size = 1.00 \begin {gather*} x e^{- 2 \left (x e^{2 x} - \frac {x}{4} + \frac {1}{4}\right ) e^{- 2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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