Integrand size = 78, antiderivative size = 28 \[ \int \frac {78+26 x+e^x \left (-6-5 x-2 x^2\right )+(-12-4 x) \log (3)}{169 x^3+e^{2 x} x^3-52 x^3 \log (3)+4 x^3 \log ^2(3)+e^x \left (-26 x^3+4 x^3 \log (3)\right )} \, dx=\frac {3+2 x}{x^2 \left (-5+e^x+2 x-2 (4+x-\log (3))\right )} \]
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\[ \int \frac {78+26 x+e^x \left (-6-5 x-2 x^2\right )+(-12-4 x) \log (3)}{169 x^3+e^{2 x} x^3-52 x^3 \log (3)+4 x^3 \log ^2(3)+e^x \left (-26 x^3+4 x^3 \log (3)\right )} \, dx=\int \frac {78+26 x+e^x \left (-6-5 x-2 x^2\right )+(-12-4 x) \log (3)}{169 x^3+e^{2 x} x^3-52 x^3 \log (3)+4 x^3 \log ^2(3)+e^x \left (-26 x^3+4 x^3 \log (3)\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {78+26 x+e^x \left (-6-5 x-2 x^2\right )+(-12-4 x) \log (3)}{e^{2 x} x^3+x^3 (169-52 \log (3))+4 x^3 \log ^2(3)+e^x \left (-26 x^3+4 x^3 \log (3)\right )} \, dx \\ & = \int \frac {78+26 x+e^x \left (-6-5 x-2 x^2\right )+(-12-4 x) \log (3)}{e^{2 x} x^3+e^x \left (-26 x^3+4 x^3 \log (3)\right )+x^3 \left (169-52 \log (3)+4 \log ^2(3)\right )} \, dx \\ & = \int \frac {-e^x \left (6+5 x+2 x^2\right )-2 (3+x) (-13+\log (9))}{x^3 \left (e^x-13 \left (1-\frac {2 \log (3)}{13}\right )\right )^2} \, dx \\ & = \int \left (\frac {-6-5 x-2 x^2}{x^3 \left (e^x-13 \left (1-\frac {2 \log (3)}{13}\right )\right )}+\frac {-39-x (26-\log (81))+\log (729)}{x^2 \left (e^x-13 \left (1-\frac {2 \log (3)}{13}\right )\right )^2}\right ) \, dx \\ & = \int \frac {-6-5 x-2 x^2}{x^3 \left (e^x-13 \left (1-\frac {2 \log (3)}{13}\right )\right )} \, dx+\int \frac {-39-x (26-\log (81))+\log (729)}{x^2 \left (e^x-13 \left (1-\frac {2 \log (3)}{13}\right )\right )^2} \, dx \\ & = \int \left (\frac {6}{x^3 \left (-e^x+13 \left (1-\frac {2 \log (3)}{13}\right )\right )}+\frac {5}{x^2 \left (-e^x+13 \left (1-\frac {2 \log (3)}{13}\right )\right )}+\frac {2}{x \left (-e^x+13 \left (1-\frac {2 \log (3)}{13}\right )\right )}\right ) \, dx+\int \left (\frac {-26+\log (81)}{x \left (e^x-13 \left (1-\frac {2 \log (3)}{13}\right )\right )^2}+\frac {-39+\log (729)}{x^2 \left (e^x-13 \left (1-\frac {2 \log (3)}{13}\right )\right )^2}\right ) \, dx \\ & = 2 \int \frac {1}{x \left (-e^x+13 \left (1-\frac {2 \log (3)}{13}\right )\right )} \, dx+5 \int \frac {1}{x^2 \left (-e^x+13 \left (1-\frac {2 \log (3)}{13}\right )\right )} \, dx+6 \int \frac {1}{x^3 \left (-e^x+13 \left (1-\frac {2 \log (3)}{13}\right )\right )} \, dx+(-26+\log (81)) \int \frac {1}{x \left (e^x-13 \left (1-\frac {2 \log (3)}{13}\right )\right )^2} \, dx+(-39+\log (729)) \int \frac {1}{x^2 \left (e^x-13 \left (1-\frac {2 \log (3)}{13}\right )\right )^2} \, dx \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {78+26 x+e^x \left (-6-5 x-2 x^2\right )+(-12-4 x) \log (3)}{169 x^3+e^{2 x} x^3-52 x^3 \log (3)+4 x^3 \log ^2(3)+e^x \left (-26 x^3+4 x^3 \log (3)\right )} \, dx=-\frac {-3-2 x}{x^2 \left (-13+e^x+\log (9)\right )} \]
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Time = 2.72 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71
method | result | size |
norman | \(\frac {3+2 x}{\left ({\mathrm e}^{x}-13+2 \ln \left (3\right )\right ) x^{2}}\) | \(20\) |
risch | \(\frac {3+2 x}{\left ({\mathrm e}^{x}-13+2 \ln \left (3\right )\right ) x^{2}}\) | \(20\) |
parallelrisch | \(\frac {3+2 x}{\left ({\mathrm e}^{x}-13+2 \ln \left (3\right )\right ) x^{2}}\) | \(20\) |
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {78+26 x+e^x \left (-6-5 x-2 x^2\right )+(-12-4 x) \log (3)}{169 x^3+e^{2 x} x^3-52 x^3 \log (3)+4 x^3 \log ^2(3)+e^x \left (-26 x^3+4 x^3 \log (3)\right )} \, dx=\frac {2 \, x + 3}{x^{2} e^{x} + 2 \, x^{2} \log \left (3\right ) - 13 \, x^{2}} \]
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Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {78+26 x+e^x \left (-6-5 x-2 x^2\right )+(-12-4 x) \log (3)}{169 x^3+e^{2 x} x^3-52 x^3 \log (3)+4 x^3 \log ^2(3)+e^x \left (-26 x^3+4 x^3 \log (3)\right )} \, dx=\frac {2 x + 3}{x^{2} e^{x} - 13 x^{2} + 2 x^{2} \log {\left (3 \right )}} \]
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Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {78+26 x+e^x \left (-6-5 x-2 x^2\right )+(-12-4 x) \log (3)}{169 x^3+e^{2 x} x^3-52 x^3 \log (3)+4 x^3 \log ^2(3)+e^x \left (-26 x^3+4 x^3 \log (3)\right )} \, dx=\frac {2 \, x + 3}{x^{2} {\left (2 \, \log \left (3\right ) - 13\right )} + x^{2} e^{x}} \]
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {78+26 x+e^x \left (-6-5 x-2 x^2\right )+(-12-4 x) \log (3)}{169 x^3+e^{2 x} x^3-52 x^3 \log (3)+4 x^3 \log ^2(3)+e^x \left (-26 x^3+4 x^3 \log (3)\right )} \, dx=\frac {2 \, x + 3}{x^{2} e^{x} + 2 \, x^{2} \log \left (3\right ) - 13 \, x^{2}} \]
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {78+26 x+e^x \left (-6-5 x-2 x^2\right )+(-12-4 x) \log (3)}{169 x^3+e^{2 x} x^3-52 x^3 \log (3)+4 x^3 \log ^2(3)+e^x \left (-26 x^3+4 x^3 \log (3)\right )} \, dx=\frac {2\,x+3}{x^2\,{\mathrm {e}}^x+2\,x^2\,\ln \left (3\right )-13\,x^2} \]
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