Optimal. Leaf size=26 \[ 2 x+\frac {4}{3} e^x (2+x) \log \left (3-x-\log \left (\frac {3}{2}\right )\right ) \]
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Rubi [A]
time = 0.79, antiderivative size = 45, normalized size of antiderivative = 1.73, number of steps
used = 15, number of rules used = 8, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6873, 6874,
6820, 2230, 2225, 2209, 2207, 2634} \begin {gather*} 2 x-\frac {4}{3} e^x \log \left (-x+3-\log \left (\frac {3}{2}\right )\right )+\frac {4}{3} e^x (x+3) \log \left (-x+3-\log \left (\frac {3}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2209
Rule 2225
Rule 2230
Rule 2634
Rule 6820
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x (-8-4 x)-6 x+18 \left (1-\frac {1}{3} \log \left (\frac {3}{2}\right )\right )+e^x \left (36-4 x^2-(12+4 x) \log \left (\frac {3}{2}\right )\right ) \log \left (3-x-\log \left (\frac {3}{2}\right )\right )}{9-3 x-3 \log \left (\frac {3}{2}\right )} \, dx\\ &=\int \left (2+\frac {4 e^x \left (-2-x-x^2 \log \left (3-x-\log \left (\frac {3}{2}\right )\right )+9 \left (1-\frac {1}{3} \log \left (\frac {3}{2}\right )\right ) \log \left (3-x-\log \left (\frac {3}{2}\right )\right )-x \log \left (\frac {3}{2}\right ) \log \left (3-x-\log \left (\frac {3}{2}\right )\right )\right )}{3 \left (3-x-\log \left (\frac {3}{2}\right )\right )}\right ) \, dx\\ &=2 x+\frac {4}{3} \int \frac {e^x \left (-2-x-x^2 \log \left (3-x-\log \left (\frac {3}{2}\right )\right )+9 \left (1-\frac {1}{3} \log \left (\frac {3}{2}\right )\right ) \log \left (3-x-\log \left (\frac {3}{2}\right )\right )-x \log \left (\frac {3}{2}\right ) \log \left (3-x-\log \left (\frac {3}{2}\right )\right )\right )}{3-x-\log \left (\frac {3}{2}\right )} \, dx\\ &=2 x+\frac {4}{3} \int \frac {e^x \left (-2-x-(3+x) \left (-3+x+\log \left (\frac {3}{2}\right )\right ) \log \left (3-x-\log \left (\frac {3}{2}\right )\right )\right )}{3-x-\log \left (\frac {3}{2}\right )} \, dx\\ &=2 x+\frac {4}{3} \int \left (\frac {e^x (2+x)}{-3+x+\log \left (\frac {3}{2}\right )}+e^x (3+x) \log \left (3-x-\log \left (\frac {3}{2}\right )\right )\right ) \, dx\\ &=2 x+\frac {4}{3} \int \frac {e^x (2+x)}{-3+x+\log \left (\frac {3}{2}\right )} \, dx+\frac {4}{3} \int e^x (3+x) \log \left (3-x-\log \left (\frac {3}{2}\right )\right ) \, dx\\ &=2 x-\frac {4}{3} e^x \log \left (3-x-\log \left (\frac {3}{2}\right )\right )+\frac {4}{3} e^x (3+x) \log \left (3-x-\log \left (\frac {3}{2}\right )\right )-\frac {4}{3} \int \frac {e^x (-2-x)}{3-x-\log \left (\frac {3}{2}\right )} \, dx+\frac {4}{3} \int \left (e^x+\frac {e^x \left (5-\log \left (\frac {3}{2}\right )\right )}{-3+x+\log \left (\frac {3}{2}\right )}\right ) \, dx\\ &=2 x-\frac {4}{3} e^x \log \left (3-x-\log \left (\frac {3}{2}\right )\right )+\frac {4}{3} e^x (3+x) \log \left (3-x-\log \left (\frac {3}{2}\right )\right )+\frac {4 \int e^x \, dx}{3}-\frac {4}{3} \int \left (e^x+\frac {e^x \left (5-\log \left (\frac {3}{2}\right )\right )}{-3+x+\log \left (\frac {3}{2}\right )}\right ) \, dx+\frac {1}{3} \left (4 \left (5-\log \left (\frac {3}{2}\right )\right )\right ) \int \frac {e^x}{-3+x+\log \left (\frac {3}{2}\right )} \, dx\\ &=\frac {4 e^x}{3}+2 x+\frac {8}{9} e^3 \text {Ei}\left (-3+x+\log \left (\frac {3}{2}\right )\right ) \left (5-\log \left (\frac {3}{2}\right )\right )-\frac {4}{3} e^x \log \left (3-x-\log \left (\frac {3}{2}\right )\right )+\frac {4}{3} e^x (3+x) \log \left (3-x-\log \left (\frac {3}{2}\right )\right )-\frac {4 \int e^x \, dx}{3}-\frac {1}{3} \left (4 \left (5-\log \left (\frac {3}{2}\right )\right )\right ) \int \frac {e^x}{-3+x+\log \left (\frac {3}{2}\right )} \, dx\\ &=2 x-\frac {4}{3} e^x \log \left (3-x-\log \left (\frac {3}{2}\right )\right )+\frac {4}{3} e^x (3+x) \log \left (3-x-\log \left (\frac {3}{2}\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 1.87, size = 28, normalized size = 1.08 \begin {gather*} \frac {2}{3} \left (3 x+2 e^x (2+x) \log \left (3-x-\log \left (\frac {3}{2}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.25, size = 30, normalized size = 1.15
method | result | size |
risch | \(\frac {4 \left (2+x \right ) {\mathrm e}^{x} \ln \left (-\ln \left (3\right )+\ln \left (2\right )+3-x \right )}{3}+2 x\) | \(24\) |
default | \(2 x +\frac {8 \,{\mathrm e}^{x} \ln \left (\ln \left (\frac {2}{3}\right )+3-x \right )}{3}+\frac {4 \,{\mathrm e}^{x} x \ln \left (\ln \left (\frac {2}{3}\right )+3-x \right )}{3}\) | \(30\) |
norman | \(2 x +\frac {8 \,{\mathrm e}^{x} \ln \left (\ln \left (\frac {2}{3}\right )+3-x \right )}{3}+\frac {4 \,{\mathrm e}^{x} x \ln \left (\ln \left (\frac {2}{3}\right )+3-x \right )}{3}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 19, normalized size = 0.73 \begin {gather*} \frac {4}{3} \, {\left (x + 2\right )} e^{x} \log \left (-x + \log \left (\frac {2}{3}\right ) + 3\right ) + 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 32, normalized size = 1.23 \begin {gather*} 2 x + \frac {\left (4 x \log {\left (- x + \log {\left (\frac {2}{3} \right )} + 3 \right )} + 8 \log {\left (- x + \log {\left (\frac {2}{3} \right )} + 3 \right )}\right ) e^{x}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.43, size = 128, normalized size = 4.92 \begin {gather*} -\frac {4}{3} \, {\rm Ei}\left (x + \log \left (3\right ) - \log \left (2\right ) - 3\right ) e^{\left (-\log \left (3\right ) + \log \left (2\right ) + 3\right )} \log \left (3\right ) + \frac {4}{3} \, {\rm Ei}\left (x + \log \left (3\right ) - \log \left (2\right ) - 3\right ) e^{\left (-\log \left (3\right ) + \log \left (2\right ) + 3\right )} \log \left (2\right ) - \frac {4}{3} \, {\rm Ei}\left (x - \log \left (\frac {2}{3}\right ) - 3\right ) e^{\left (\log \left (\frac {2}{3}\right ) + 3\right )} \log \left (\frac {2}{3}\right ) + \frac {4}{3} \, x e^{x} \log \left (-x + \log \left (\frac {2}{3}\right ) + 3\right ) + \frac {20}{3} \, {\rm Ei}\left (x + \log \left (3\right ) - \log \left (2\right ) - 3\right ) e^{\left (-\log \left (3\right ) + \log \left (2\right ) + 3\right )} - \frac {20}{3} \, {\rm Ei}\left (x - \log \left (\frac {2}{3}\right ) - 3\right ) e^{\left (\log \left (\frac {2}{3}\right ) + 3\right )} + \frac {8}{3} \, e^{x} \log \left (-x + \log \left (\frac {2}{3}\right ) + 3\right ) + 2 \, x \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 = -0.04 \begin {gather*} \int \frac {6\,\ln \left (\frac {2}{3}\right )-6\,x-{\mathrm {e}}^x\,\left (4\,x+8\right )+\ln \left (\ln \left (\frac {2}{3}\right )-x+3\right )\,{\mathrm {e}}^x\,\left (\ln \left (\frac {2}{3}\right )\,\left (4\,x+12\right )-4\,x^2+36\right )+18}{3\,\ln \left (\frac {2}{3}\right )-3\,x+9} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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