Optimal. Leaf size=29 \[ 2 \left (6-\frac {13 x}{16}+e^{-x} x+\frac {1}{2} \left (-x-x^2\right )\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 19, normalized size of antiderivative = 0.66, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {12, 6742, 2194, 2176} \begin {gather*} -x^2+2 e^{-x} x-\frac {21 x}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int e^{-x} \left (16+e^x (-21-16 x)-16 x\right ) \, dx\\ &=\frac {1}{8} \int \left (-21+16 e^{-x}-16 x-16 e^{-x} x\right ) \, dx\\ &=-\frac {21 x}{8}-x^2+2 \int e^{-x} \, dx-2 \int e^{-x} x \, dx\\ &=-2 e^{-x}-\frac {21 x}{8}+2 e^{-x} x-x^2-2 \int e^{-x} \, dx\\ &=-\frac {21 x}{8}+2 e^{-x} x-x^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 0.66 \begin {gather*} -\frac {21 x}{8}+2 e^{-x} x-x^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 22, normalized size = 0.76 \begin {gather*} -\frac {1}{8} \, {\left ({\left (8 \, x^{2} + 21 \, x\right )} e^{x} - 16 \, x\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 16, normalized size = 0.55 \begin {gather*} -x^{2} + 2 \, x e^{\left (-x\right )} - \frac {21}{8} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 17, normalized size = 0.59
method | result | size |
default | \(-x^{2}-\frac {21 x}{8}+2 x \,{\mathrm e}^{-x}\) | \(17\) |
risch | \(-x^{2}-\frac {21 x}{8}+2 x \,{\mathrm e}^{-x}\) | \(17\) |
norman | \(\left (2 x -\frac {21 \,{\mathrm e}^{x} x}{8}-{\mathrm e}^{x} x^{2}\right ) {\mathrm e}^{-x}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 24, normalized size = 0.83 \begin {gather*} -x^{2} + 2 \, {\left (x + 1\right )} e^{\left (-x\right )} - \frac {21}{8} \, x - 2 \, e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 14, normalized size = 0.48 \begin {gather*} -\frac {x\,\left (8\,x-16\,{\mathrm {e}}^{-x}+21\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 14, normalized size = 0.48 \begin {gather*} - x^{2} - \frac {21 x}{8} + 2 x e^{- x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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