Optimal. Leaf size=21 \[ x \left (-x^2+e^{-x} \log (x-\log (3))\right ) \]
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Rubi [A]
time = 0.90, antiderivative size = 38, normalized size of antiderivative = 1.81, number of steps
used = 14, number of rules used = 7, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6874, 6820,
2230, 2225, 2209, 2207, 2634} \begin {gather*} -x^3+e^{-x} \log (x-\log (3))-e^{-x} (1-x) \log (x-\log (3)) \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2209
Rule 2225
Rule 2230
Rule 2634
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-3 x^2+\frac {e^{-x} \left (x-x^2 \log (x-\log (3))-\log (3) \log (x-\log (3))+x (1+\log (3)) \log (x-\log (3))\right )}{x-\log (3)}\right ) \, dx\\ &=-x^3+\int \frac {e^{-x} \left (x-x^2 \log (x-\log (3))-\log (3) \log (x-\log (3))+x (1+\log (3)) \log (x-\log (3))\right )}{x-\log (3)} \, dx\\ &=-x^3+\int e^{-x} \left (\frac {x}{x-\log (3)}-(-1+x) \log (x-\log (3))\right ) \, dx\\ &=-x^3+\int \left (\frac {e^{-x} x}{x-\log (3)}-e^{-x} (-1+x) \log (x-\log (3))\right ) \, dx\\ &=-x^3+\int \frac {e^{-x} x}{x-\log (3)} \, dx-\int e^{-x} (-1+x) \log (x-\log (3)) \, dx\\ &=-x^3+e^{-x} \log (x-\log (3))-e^{-x} (1-x) \log (x-\log (3))+\int \frac {e^{-x} x}{-x+\log (3)} \, dx+\int \left (e^{-x}+\frac {e^{-x} \log (3)}{x-\log (3)}\right ) \, dx\\ &=-x^3+e^{-x} \log (x-\log (3))-e^{-x} (1-x) \log (x-\log (3))+\log (3) \int \frac {e^{-x}}{x-\log (3)} \, dx+\int e^{-x} \, dx+\int \left (-e^{-x}-\frac {e^{-x} \log (3)}{x-\log (3)}\right ) \, dx\\ &=-e^{-x}-x^3+\frac {1}{3} \text {Ei}(-x+\log (3)) \log (3)+e^{-x} \log (x-\log (3))-e^{-x} (1-x) \log (x-\log (3))-\log (3) \int \frac {e^{-x}}{x-\log (3)} \, dx-\int e^{-x} \, dx\\ &=-x^3+e^{-x} \log (x-\log (3))-e^{-x} (1-x) \log (x-\log (3))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.50, size = 24, normalized size = 1.14 \begin {gather*} -x^3+\log ^3(3)+e^{-x} x \log (x-\log (3)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.95, size = 20, normalized size = 0.95
method | result | size |
default | \(\ln \left (-\ln \left (3\right )+x \right ) x \,{\mathrm e}^{-x}-x^{3}\) | \(20\) |
risch | \(\ln \left (-\ln \left (3\right )+x \right ) x \,{\mathrm e}^{-x}-x^{3}\) | \(20\) |
norman | \(\left (\ln \left (-\ln \left (3\right )+x \right ) x -{\mathrm e}^{x} x^{3}\right ) {\mathrm e}^{-x}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 19, normalized size = 0.90 \begin {gather*} -x^{3} + x e^{\left (-x\right )} \log \left (x - \log \left (3\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 23, normalized size = 1.10 \begin {gather*} -{\left (x^{3} e^{x} - x \log \left (x - \log \left (3\right )\right )\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 14, normalized size = 0.67 \begin {gather*} - x^{3} + x e^{- x} \log {\left (x - \log {\left (3 \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 19, normalized size = 0.90 \begin {gather*} -x^{3} + x e^{\left (-x\right )} \log \left (x - \log \left (3\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\mathrm {e}}^{-x}\,\left (x+\ln \left (x-\ln \left (3\right )\right )\,\left (x+\ln \left (3\right )\,\left (x-1\right )-x^2\right )+{\mathrm {e}}^x\,\left (3\,x^2\,\ln \left (3\right )-3\,x^3\right )\right )}{x-\ln \left (3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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