Integrand size = 108, antiderivative size = 30 \[ \int \frac {e^x (-1+2 x) \log (3)+e^x (-2+x) \log (3) \log \left (9 e^{-x}\right )+e^x (2-x) \log (3) \log (2 x)}{x^3+x^3 \log ^2\left (9 e^{-x}\right )-2 x^3 \log (2 x)+x^3 \log ^2(2 x)+\log \left (9 e^{-x}\right ) \left (2 x^3-2 x^3 \log (2 x)\right )} \, dx=\frac {e^x \log (3)}{x \left (x+x \log \left (9 e^{-x}\right )-x \log (2 x)\right )} \]
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\[ \int \frac {e^x (-1+2 x) \log (3)+e^x (-2+x) \log (3) \log \left (9 e^{-x}\right )+e^x (2-x) \log (3) \log (2 x)}{x^3+x^3 \log ^2\left (9 e^{-x}\right )-2 x^3 \log (2 x)+x^3 \log ^2(2 x)+\log \left (9 e^{-x}\right ) \left (2 x^3-2 x^3 \log (2 x)\right )} \, dx=\int \frac {e^x (-1+2 x) \log (3)+e^x (-2+x) \log (3) \log \left (9 e^{-x}\right )+e^x (2-x) \log (3) \log (2 x)}{x^3+x^3 \log ^2\left (9 e^{-x}\right )-2 x^3 \log (2 x)+x^3 \log ^2(2 x)+\log \left (9 e^{-x}\right ) \left (2 x^3-2 x^3 \log (2 x)\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \log (3) \left (-1+2 x+(-2+x) \log \left (9 e^{-x}\right )-(-2+x) \log (2 x)\right )}{x^3 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )^2} \, dx \\ & = \log (3) \int \frac {e^x \left (-1+2 x+(-2+x) \log \left (9 e^{-x}\right )-(-2+x) \log (2 x)\right )}{x^3 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )^2} \, dx \\ & = \log (3) \int \left (\frac {e^x (1+x)}{x^3 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )^2}+\frac {e^x (-2+x)}{x^3 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )}\right ) \, dx \\ & = \log (3) \int \frac {e^x (1+x)}{x^3 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )^2} \, dx+\log (3) \int \frac {e^x (-2+x)}{x^3 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )} \, dx \\ & = \log (3) \int \left (\frac {e^x}{x^3 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )^2}+\frac {e^x}{x^2 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )^2}\right ) \, dx+\log (3) \int \left (-\frac {2 e^x}{x^3 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )}+\frac {e^x}{x^2 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )}\right ) \, dx \\ & = \log (3) \int \frac {e^x}{x^3 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )^2} \, dx+\log (3) \int \frac {e^x}{x^2 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )^2} \, dx+\log (3) \int \frac {e^x}{x^2 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )} \, dx-(2 \log (3)) \int \frac {e^x}{x^3 \left (1+\log \left (9 e^{-x}\right )-\log (2 x)\right )} \, dx \\ \end{align*}
Time = 0.84 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {e^x (-1+2 x) \log (3)+e^x (-2+x) \log (3) \log \left (9 e^{-x}\right )+e^x (2-x) \log (3) \log (2 x)}{x^3+x^3 \log ^2\left (9 e^{-x}\right )-2 x^3 \log (2 x)+x^3 \log ^2(2 x)+\log \left (9 e^{-x}\right ) \left (2 x^3-2 x^3 \log (2 x)\right )} \, dx=-\frac {e^x \log (3)}{x^2 \left (-1-\log \left (9 e^{-x}\right )+\log (2 x)\right )} \]
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Time = 1.65 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(-\frac {\ln \left (3\right ) {\mathrm e}^{x}}{x^{2} \left (-1+\ln \left (2 x \right )-\ln \left (9 \,{\mathrm e}^{-x}\right )\right )}\) | \(27\) |
risch | \(\frac {2 \ln \left (3\right ) {\mathrm e}^{x}}{x^{2} \left (2+4 \ln \left (3\right )-2 \ln \left (2 x \right )-2 \ln \left ({\mathrm e}^{x}\right )\right )}\) | \(29\) |
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Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^x (-1+2 x) \log (3)+e^x (-2+x) \log (3) \log \left (9 e^{-x}\right )+e^x (2-x) \log (3) \log (2 x)}{x^3+x^3 \log ^2\left (9 e^{-x}\right )-2 x^3 \log (2 x)+x^3 \log ^2(2 x)+\log \left (9 e^{-x}\right ) \left (2 x^3-2 x^3 \log (2 x)\right )} \, dx=-\frac {e^{x} \log \left (3\right )}{x^{3} - 2 \, x^{2} \log \left (3\right ) + x^{2} \log \left (2 \, x\right ) - x^{2}} \]
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Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {e^x (-1+2 x) \log (3)+e^x (-2+x) \log (3) \log \left (9 e^{-x}\right )+e^x (2-x) \log (3) \log (2 x)}{x^3+x^3 \log ^2\left (9 e^{-x}\right )-2 x^3 \log (2 x)+x^3 \log ^2(2 x)+\log \left (9 e^{-x}\right ) \left (2 x^3-2 x^3 \log (2 x)\right )} \, dx=- \frac {e^{x} \log {\left (3 \right )}}{x^{3} + x^{2} \log {\left (2 x \right )} - 2 x^{2} \log {\left (3 \right )} - x^{2}} \]
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Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {e^x (-1+2 x) \log (3)+e^x (-2+x) \log (3) \log \left (9 e^{-x}\right )+e^x (2-x) \log (3) \log (2 x)}{x^3+x^3 \log ^2\left (9 e^{-x}\right )-2 x^3 \log (2 x)+x^3 \log ^2(2 x)+\log \left (9 e^{-x}\right ) \left (2 x^3-2 x^3 \log (2 x)\right )} \, dx=-\frac {e^{x} \log \left (3\right )}{x^{3} - x^{2} {\left (2 \, \log \left (3\right ) - \log \left (2\right ) + 1\right )} + x^{2} \log \left (x\right )} \]
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Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {e^x (-1+2 x) \log (3)+e^x (-2+x) \log (3) \log \left (9 e^{-x}\right )+e^x (2-x) \log (3) \log (2 x)}{x^3+x^3 \log ^2\left (9 e^{-x}\right )-2 x^3 \log (2 x)+x^3 \log ^2(2 x)+\log \left (9 e^{-x}\right ) \left (2 x^3-2 x^3 \log (2 x)\right )} \, dx=-\frac {e^{x} \log \left (3\right )}{x^{3} - 2 \, x^{2} \log \left (3\right ) + x^{2} \log \left (2\right ) + x^{2} \log \left (x\right ) - x^{2}} \]
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Time = 8.47 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {e^x (-1+2 x) \log (3)+e^x (-2+x) \log (3) \log \left (9 e^{-x}\right )+e^x (2-x) \log (3) \log (2 x)}{x^3+x^3 \log ^2\left (9 e^{-x}\right )-2 x^3 \log (2 x)+x^3 \log ^2(2 x)+\log \left (9 e^{-x}\right ) \left (2 x^3-2 x^3 \log (2 x)\right )} \, dx=\frac {{\mathrm {e}}^x\,\left (\ln \left (3\right )-2\,\ln \left (2\,x\right )\,\ln \left (3\right )+\ln \left (2\,x\right )\,\ln \left (9\right )+x\,\ln \left (3\right )\right )}{x^2\,\left (x+1\right )\,\left (\ln \left (9\,{\mathrm {e}}^{-x}\right )-\ln \left (2\,x\right )+1\right )} \]
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