Optimal. Leaf size=23 \[ 5+x-e^{-x} x+\frac {2}{3} e^{-x} x \log (3) \]
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Rubi [A]
time = 0.08, antiderivative size = 37, normalized size of antiderivative = 1.61, number of steps
used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {12, 6820, 2207,
2225} \begin {gather*} x+\frac {1}{3} e^{-x} (1-x) (3-\log (9))-\frac {1}{3} e^{-x} (3-\log (9)) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2207
Rule 2225
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int e^{-x} \left (-3+3 e^x+3 x+(2-2 x) \log (3)\right ) \, dx\\ &=\frac {1}{3} \int \left (3-e^{-x} (-1+x) (-3+\log (9))\right ) \, dx\\ &=x+\frac {1}{3} (3-\log (9)) \int e^{-x} (-1+x) \, dx\\ &=x+\frac {1}{3} e^{-x} (1-x) (3-\log (9))+\frac {1}{3} (3-\log (9)) \int e^{-x} \, dx\\ &=x-\frac {1}{3} e^{-x} (3-\log (9))+\frac {1}{3} e^{-x} (1-x) (3-\log (9))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 16, normalized size = 0.70 \begin {gather*} x+\frac {1}{3} e^{-x} x (-3+\log (9)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 19, normalized size = 0.83
method | result | size |
risch | \(x +\frac {\left (2 \ln \left (3\right )-3\right ) x \,{\mathrm e}^{-x}}{3}\) | \(16\) |
default | \(x -x \,{\mathrm e}^{-x}+\frac {2 x \ln \left (3\right ) {\mathrm e}^{-x}}{3}\) | \(19\) |
norman | \(\left (\left (\frac {2 \ln \left (3\right )}{3}-1\right ) x +{\mathrm e}^{x} x \right ) {\mathrm e}^{-x}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 34, normalized size = 1.48 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )} e^{\left (-x\right )} \log \left (3\right ) - {\left (x + 1\right )} e^{\left (-x\right )} - \frac {2}{3} \, e^{\left (-x\right )} \log \left (3\right ) + x + e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 20, normalized size = 0.87 \begin {gather*} \frac {1}{3} \, {\left (3 \, x e^{x} + 2 \, x \log \left (3\right ) - 3 \, x\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 15, normalized size = 0.65 \begin {gather*} x + \frac {\left (- 3 x + 2 x \log {\left (3 \right )}\right ) e^{- x}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, {\left (2 \, x \log \left (3\right ) - 3 \, x\right )} e^{\left (-x\right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 19, normalized size = 0.83 \begin {gather*} \frac {x\,\left (2\,{\mathrm {e}}^{-x}\,\ln \left (3\right )-3\,{\mathrm {e}}^{-x}+3\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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