Optimal. Leaf size=29 \[ -x+x \log \left (-3+\log \left (\frac {e^x x}{5-\frac {2-x}{x}}\right )\right ) \]
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Rubi [F] time = 0.72, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{(1-3 x) \left (3-\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )} \, dx\\ &=\int \left (\frac {-5+11 x+3 x^2+\log \left (\frac {e^x x^2}{-2+6 x}\right )-3 x \log \left (\frac {e^x x^2}{-2+6 x}\right )}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}+\log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )\right ) \, dx\\ &=\int \frac {-5+11 x+3 x^2+\log \left (\frac {e^x x^2}{-2+6 x}\right )-3 x \log \left (\frac {e^x x^2}{-2+6 x}\right )}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )} \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx\\ &=\int \left (-1+\frac {-2+2 x+3 x^2}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}\right ) \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx\\ &=-x+\int \frac {-2+2 x+3 x^2}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )} \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx\\ &=-x+\int \left (\frac {1}{-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )}+\frac {x}{-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )}-\frac {1}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}\right ) \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx\\ &=-x+\int \frac {1}{-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx+\int \frac {x}{-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx-\int \frac {1}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )} \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 24, normalized size = 0.83 \begin {gather*} -x+x \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 24, normalized size = 0.83 \begin {gather*} x \log \left (\log \left (\frac {x^{2} e^{x}}{2 \, {\left (3 \, x - 1\right )}}\right ) - 3\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 23, normalized size = 0.79 \begin {gather*} x \log \left (x + \log \left (\frac {x^{2}}{2 \, {\left (3 \, x - 1\right )}}\right ) - 3\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.47, size = 187, normalized size = 6.45
method | result | size |
risch | \(\ln \left (-\ln \relax (2)-\ln \relax (3)+2 \ln \relax (x )+\ln \left ({\mathrm e}^{x}\right )-\ln \left (x -\frac {1}{3}\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right ) \left (-\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\mathrm {csgn}\left (i {\mathrm e}^{x}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\mathrm {csgn}\left (\frac {i}{x -\frac {1}{3}}\right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i x^{2} {\mathrm e}^{x}}{x -\frac {1}{3}}\right ) \left (-\mathrm {csgn}\left (\frac {i x^{2} {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\mathrm {csgn}\left (i x^{2}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i x^{2} {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right )\right )}{2}-3\right ) x -x\) | \(187\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 26, normalized size = 0.90 \begin {gather*} x \log \left (x - \log \relax (2) - \log \left (3 \, x - 1\right ) + 2 \, \log \relax (x) - 3\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.42, size = 21, normalized size = 0.72 \begin {gather*} x\,\left (\ln \left (\ln \left (\frac {x^2\,{\mathrm {e}}^x}{6\,x-2}\right )-3\right )-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.45, size = 41, normalized size = 1.41 \begin {gather*} - x + \left (x - \frac {1}{18}\right ) \log {\left (\log {\left (\frac {x^{2} e^{x}}{6 x - 2} \right )} - 3 \right )} + \frac {\log {\left (\log {\left (\frac {x^{2} e^{x}}{6 x - 2} \right )} - 3 \right )}}{18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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