Integrand size = 132, antiderivative size = 25 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=-20+\frac {\log \left (-6+\frac {\log (x)}{x}\right )}{-4-e^x \log (3 x)} \]
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\[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=\int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx \]
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Rubi steps \begin{align*} \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{align*}
Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=-\frac {\log \left (-6+\frac {\log (x)}{x}\right )}{4+e^x \log (3 x)} \]
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Time = 32.52 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(-\frac {\ln \left (\frac {\ln \left (x \right )-6 x}{x}\right )}{\ln \left (3 x \right ) {\mathrm e}^{x}+4}\) | \(25\) |
risch | \(-\frac {2 \ln \left (-\frac {\ln \left (x \right )}{6}+x \right )}{8+2 \ln \left (3\right ) {\mathrm e}^{x}+2 \,{\mathrm e}^{x} \ln \left (x \right )}-\frac {-2 i \pi \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )^{2}-i \pi \,\operatorname {csgn}\left (i \left (\frac {\ln \left (x \right )}{6}-x \right )\right ) \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )^{2}-i \pi \,\operatorname {csgn}\left (i \left (\frac {\ln \left (x \right )}{6}-x \right )\right ) \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )-i \pi \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )^{3}+i \pi \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+2 i \pi +2 \ln \left (3\right )+2 \ln \left (2\right )-2 \ln \left (x \right )}{8+2 \ln \left (3\right ) {\mathrm e}^{x}+2 \,{\mathrm e}^{x} \ln \left (x \right )}\) | \(203\) |
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Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=-\frac {\log \left (-\frac {6 \, x - \log \left (x\right )}{x}\right )}{e^{x} \log \left (3\right ) + e^{x} \log \left (x\right ) + 4} \]
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Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=- \frac {\log {\left (\frac {- 6 x + \log {\left (x \right )}}{x} \right )}}{\left (\log {\left (x \right )} + \log {\left (3 \right )}\right ) e^{x} + 4} \]
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Time = 0.36 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=\frac {\log \left (x\right ) - \log \left (-6 \, x + \log \left (x\right )\right )}{{\left (\log \left (3\right ) + \log \left (x\right )\right )} e^{x} + 4} \]
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Time = 0.58 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=\frac {\log \left (x\right ) - \log \left (-6 \, x + \log \left (x\right )\right )}{e^{x} \log \left (3\right ) + e^{x} \log \left (x\right ) + 4} \]
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Time = 12.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=-\frac {\ln \left (-\frac {6\,x-\ln \left (x\right )}{x}\right )}{\ln \left (3\,x\right )\,{\mathrm {e}}^x+4} \]
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