Optimal. Leaf size=23 \[ e^{-1-3 x} x \left (-\frac {x^2}{4}+\log (2)-\log (16)\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 46, normalized size of antiderivative = 2.00, number of steps used = 12, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 2196, 2176, 2194} \begin {gather*} -\frac {1}{4} e^{-3 x-1} x^3-\frac {1}{3} e^{-3 x-1} \log (8)+\frac {1}{3} e^{-3 x-1} (1-3 x) \log (8) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int e^{-1-3 x} \left (-3 x^2+3 x^3+(4-12 x) \log (2)+(-4+12 x) \log (16)\right ) \, dx\\ &=\frac {1}{4} \int \left (-3 e^{-1-3 x} x^2+3 e^{-1-3 x} x^3-4 e^{-1-3 x} (-1+3 x) \log (2) \left (1-\frac {\log (16)}{\log (2)}\right )\right ) \, dx\\ &=-\left (\frac {3}{4} \int e^{-1-3 x} x^2 \, dx\right )+\frac {3}{4} \int e^{-1-3 x} x^3 \, dx+\log (8) \int e^{-1-3 x} (-1+3 x) \, dx\\ &=\frac {1}{4} e^{-1-3 x} x^2-\frac {1}{4} e^{-1-3 x} x^3+\frac {1}{3} e^{-1-3 x} (1-3 x) \log (8)-\frac {1}{2} \int e^{-1-3 x} x \, dx+\frac {3}{4} \int e^{-1-3 x} x^2 \, dx+\log (8) \int e^{-1-3 x} \, dx\\ &=\frac {1}{6} e^{-1-3 x} x-\frac {1}{4} e^{-1-3 x} x^3-\frac {1}{3} e^{-1-3 x} \log (8)+\frac {1}{3} e^{-1-3 x} (1-3 x) \log (8)-\frac {1}{6} \int e^{-1-3 x} \, dx+\frac {1}{2} \int e^{-1-3 x} x \, dx\\ &=\frac {1}{18} e^{-1-3 x}-\frac {1}{4} e^{-1-3 x} x^3-\frac {1}{3} e^{-1-3 x} \log (8)+\frac {1}{3} e^{-1-3 x} (1-3 x) \log (8)+\frac {1}{6} \int e^{-1-3 x} \, dx\\ &=-\frac {1}{4} e^{-1-3 x} x^3-\frac {1}{3} e^{-1-3 x} \log (8)+\frac {1}{3} e^{-1-3 x} (1-3 x) \log (8)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 0.96 \begin {gather*} \frac {1}{4} e^{-1-3 x} \left (-x^3-4 x \log (8)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 17, normalized size = 0.74 \begin {gather*} -\frac {1}{4} \, {\left (x^{3} + 12 \, x \log \relax (2)\right )} e^{\left (-3 \, x - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 17, normalized size = 0.74 \begin {gather*} -\frac {1}{4} \, {\left (x^{3} + 12 \, x \log \relax (2)\right )} e^{\left (-3 \, x - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 18, normalized size = 0.78
method | result | size |
gosper | \(-\frac {{\mathrm e}^{-3 x -1} x \left (x^{2}+12 \ln \relax (2)\right )}{4}\) | \(18\) |
risch | \(\frac {\left (-x^{3}-12 x \ln \relax (2)\right ) {\mathrm e}^{-3 x -1}}{4}\) | \(20\) |
norman | \(-\frac {x^{3} {\mathrm e}^{-3 x -1}}{4}-3 x \ln \relax (2) {\mathrm e}^{-3 x -1}\) | \(24\) |
derivativedivides | \(\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )}{36}+\frac {{\mathrm e}^{-3 x -1}}{108}+\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )^{2}}{36}+\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )^{3}}{108}+{\mathrm e}^{-3 x -1} \ln \relax (2)+{\mathrm e}^{-3 x -1} \ln \relax (2) \left (-3 x -1\right )\) | \(76\) |
default | \(\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )}{36}+\frac {{\mathrm e}^{-3 x -1}}{108}+\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )^{2}}{36}+\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )^{3}}{108}+{\mathrm e}^{-3 x -1} \ln \relax (2)+{\mathrm e}^{-3 x -1} \ln \relax (2) \left (-3 x -1\right )\) | \(76\) |
meijerg | \(-\ln \relax (2) {\mathrm e}^{-1} \left (1-{\mathrm e}^{-3 x}\right )+\ln \relax (2) {\mathrm e}^{-1} \left (1-\frac {\left (6 x +2\right ) {\mathrm e}^{-3 x}}{2}\right )+\frac {{\mathrm e}^{-1} \left (6-\frac {\left (108 x^{3}+108 x^{2}+72 x +24\right ) {\mathrm e}^{-3 x}}{4}\right )}{108}-\frac {{\mathrm e}^{-1} \left (2-\frac {\left (27 x^{2}+18 x +6\right ) {\mathrm e}^{-3 x}}{3}\right )}{36}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 66, normalized size = 2.87 \begin {gather*} -{\left (3 \, x + 1\right )} e^{\left (-3 \, x - 1\right )} \log \relax (2) - \frac {1}{36} \, {\left (9 \, x^{3} + 9 \, x^{2} + 6 \, x + 2\right )} e^{\left (-3 \, x - 1\right )} + \frac {1}{36} \, {\left (9 \, x^{2} + 6 \, x + 2\right )} e^{\left (-3 \, x - 1\right )} + e^{\left (-3 \, x - 1\right )} \log \relax (2) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 17, normalized size = 0.74 \begin {gather*} -\frac {x\,{\mathrm {e}}^{-3\,x-1}\,\left (x^2+12\,\ln \relax (2)\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 20, normalized size = 0.87 \begin {gather*} \frac {\left (- x^{3} - 12 x \log {\relax (2 )}\right ) e^{- 3 x - 1}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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