Optimal. Leaf size=18 \[ 8-2 x+\frac {e^x x}{3}-x^2 \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {12, 2176, 2194} \begin {gather*} -x^2-2 x-\frac {e^x}{3}+\frac {1}{3} e^x (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (-6-6 x+e^x (1+x)\right ) \, dx\\ &=-2 x-x^2+\frac {1}{3} \int e^x (1+x) \, dx\\ &=-2 x-x^2+\frac {1}{3} e^x (1+x)-\frac {\int e^x \, dx}{3}\\ &=-\frac {e^x}{3}-2 x-x^2+\frac {1}{3} e^x (1+x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 0.94 \begin {gather*} \frac {1}{3} \left (e^x x-3 (1+x)^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 14, normalized size = 0.78 \begin {gather*} -x^{2} + \frac {1}{3} \, x e^{x} - 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 14, normalized size = 0.78 \begin {gather*} -x^{2} + \frac {1}{3} \, x e^{x} - 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 15, normalized size = 0.83
method | result | size |
default | \(-x^{2}-2 x +\frac {{\mathrm e}^{x} x}{3}\) | \(15\) |
norman | \(-x^{2}-2 x +\frac {{\mathrm e}^{x} x}{3}\) | \(15\) |
risch | \(-x^{2}-2 x +\frac {{\mathrm e}^{x} x}{3}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 14, normalized size = 0.78 \begin {gather*} -x^{2} + \frac {1}{3} \, x e^{x} - 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 12, normalized size = 0.67 \begin {gather*} -\frac {x\,\left (3\,x-{\mathrm {e}}^x+6\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 12, normalized size = 0.67 \begin {gather*} - x^{2} + \frac {x e^{x}}{3} - 2 x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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