Optimal. Leaf size=25 \[ -20+\frac {\log \left (-6+\frac {\log (x)}{x}\right )}{-4-e^x \log (3 x)} \]
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Rubi [F]
time = 6.34, 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 {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4-4 \log (x)-\left (-e^x+e^x \log (x)\right ) \log (3 x)-\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{x (6 x-\log (x)) \left (4+e^x \log (3 x)\right )^2} \, dx\\ &=\int \frac {4-4 \log (x)-e^x (-1+\log (x)) \log (3 x)+e^x (6 x-\log (x)) (1+x \log (3 x)) \log \left (-6+\frac {\log (x)}{x}\right )}{x (6 x-\log (x)) \left (4+e^x \log (3 x)\right )^2} \, dx\\ &=\int \left (-\frac {4 (1+x \log (3 x)) \log \left (-6+\frac {\log (x)}{x}\right )}{x \log (3 x) \left (4+e^x \log (3 x)\right )^2}+\frac {\log (3 x)-\log (x) \log (3 x)+6 x \log \left (-6+\frac {\log (x)}{x}\right )-\log (x) \log \left (-6+\frac {\log (x)}{x}\right )+6 x^2 \log (3 x) \log \left (-6+\frac {\log (x)}{x}\right )-x \log (x) \log (3 x) \log \left (-6+\frac {\log (x)}{x}\right )}{x (6 x-\log (x)) \log (3 x) \left (4+e^x \log (3 x)\right )}\right ) \, dx\\ &=-\left (4 \int \frac {(1+x \log (3 x)) \log \left (-6+\frac {\log (x)}{x}\right )}{x \log (3 x) \left (4+e^x \log (3 x)\right )^2} \, dx\right )+\int \frac {\log (3 x)-\log (x) \log (3 x)+6 x \log \left (-6+\frac {\log (x)}{x}\right )-\log (x) \log \left (-6+\frac {\log (x)}{x}\right )+6 x^2 \log (3 x) \log \left (-6+\frac {\log (x)}{x}\right )-x \log (x) \log (3 x) \log \left (-6+\frac {\log (x)}{x}\right )}{x (6 x-\log (x)) \log (3 x) \left (4+e^x \log (3 x)\right )} \, dx\\ &=-\left (4 \int \left (\frac {\log \left (-6+\frac {\log (x)}{x}\right )}{\left (4+e^x \log (3 x)\right )^2}+\frac {\log \left (-6+\frac {\log (x)}{x}\right )}{x \log (3 x) \left (4+e^x \log (3 x)\right )^2}\right ) \, dx\right )+\int \left (\frac {1}{x (6 x-\log (x)) \left (4+e^x \log (3 x)\right )}-\frac {\log (x)}{x (6 x-\log (x)) \left (4+e^x \log (3 x)\right )}+\frac {6 x \log \left (-6+\frac {\log (x)}{x}\right )}{(6 x-\log (x)) \left (4+e^x \log (3 x)\right )}-\frac {\log (x) \log \left (-6+\frac {\log (x)}{x}\right )}{(6 x-\log (x)) \left (4+e^x \log (3 x)\right )}+\frac {6 \log \left (-6+\frac {\log (x)}{x}\right )}{(6 x-\log (x)) \log (3 x) \left (4+e^x \log (3 x)\right )}-\frac {\log (x) \log \left (-6+\frac {\log (x)}{x}\right )}{x (6 x-\log (x)) \log (3 x) \left (4+e^x \log (3 x)\right )}\right ) \, dx\\ &=-\left (4 \int \frac {\log \left (-6+\frac {\log (x)}{x}\right )}{\left (4+e^x \log (3 x)\right )^2} \, dx\right )-4 \int \frac {\log \left (-6+\frac {\log (x)}{x}\right )}{x \log (3 x) \left (4+e^x \log (3 x)\right )^2} \, dx+6 \int \frac {x \log \left (-6+\frac {\log (x)}{x}\right )}{(6 x-\log (x)) \left (4+e^x \log (3 x)\right )} \, dx+6 \int \frac {\log \left (-6+\frac {\log (x)}{x}\right )}{(6 x-\log (x)) \log (3 x) \left (4+e^x \log (3 x)\right )} \, dx+\int \frac {1}{x (6 x-\log (x)) \left (4+e^x \log (3 x)\right )} \, dx-\int \frac {\log (x)}{x (6 x-\log (x)) \left (4+e^x \log (3 x)\right )} \, dx-\int \frac {\log (x) \log \left (-6+\frac {\log (x)}{x}\right )}{(6 x-\log (x)) \left (4+e^x \log (3 x)\right )} \, dx-\int \frac {\log (x) \log \left (-6+\frac {\log (x)}{x}\right )}{x (6 x-\log (x)) \log (3 x) \left (4+e^x \log (3 x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.09, size = 23, normalized size = 0.92 \begin {gather*} -\frac {\log \left (-6+\frac {\log (x)}{x}\right )}{4+e^x \log (3 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.66, size = 207, normalized size = 8.28
method | result | size |
risch | \(-\frac {2 i \ln \left (-\frac {\ln \left (x \right )}{6}+x \right )}{2 i \ln \left (3\right ) {\mathrm e}^{x}+2 i {\mathrm e}^{x} \ln \left (x \right )+8 i}-\frac {-2 i \pi \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (\frac {\ln \left (x \right )}{6}-x \right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (\frac {\ln \left (x \right )}{6}-x \right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )-i \pi \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )^{3}+i \pi \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+2 i \pi +2 \ln \left (2\right )+2 \ln \left (3\right )-2 \ln \left (x \right )}{8+2 \ln \left (3\right ) {\mathrm e}^{x}+2 \,{\mathrm e}^{x} \ln \left (x \right )}\) | \(207\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 25, normalized size = 1.00 \begin {gather*} \frac {\log \left (x\right ) - \log \left (-6 \, x + \log \left (x\right )\right )}{{\left (\log \left (3\right ) + \log \left (x\right )\right )} e^{x} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 30, normalized size = 1.20 \begin {gather*} -\frac {\log \left (-\frac {6 \, x - \log \left (x\right )}{x}\right )}{e^{x} \log \left (3\right ) + e^{x} \log \left (x\right ) + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 22, normalized size = 0.88 \begin {gather*} - \frac {\log {\left (\frac {- 6 x + \log {\left (x \right )}}{x} \right )}}{\left (\log {\left (x \right )} + \log {\left (3 \right )}\right ) e^{x} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 27, normalized size = 1.08 \begin {gather*} \frac {\log \left (x\right ) - \log \left (-6 \, x + \log \left (x\right )\right )}{e^{x} \log \left (3\right ) + e^{x} \log \left (x\right ) + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.45, size = 27, normalized size = 1.08 \begin {gather*} -\frac {\ln \left (-\frac {6\,x-\ln \left (x\right )}{x}\right )}{\ln \left (3\,x\right )\,{\mathrm {e}}^x+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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