Integrand size = 237, antiderivative size = 20 \[ \int \frac {4-17 x+3 e x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx=\frac {4+x}{-2+e+e^x+x-\log (3+\log (x))} \]
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\[ \int \frac {4-17 x+3 e x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx=\int \frac {4-17 x+3 e x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {4+(-17+3 e) x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx \\ & = \int \frac {4+(-17+3 e) x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{3 e^{2 x} x+\left (12+3 e^2\right ) x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx \\ & = \int \frac {4-17 \left (1-\frac {3 e}{17}\right ) x-9 e^x x-3 e^x x^2-3 x \log (3+\log (x))-x \log (x) \left (6-e+e^x (3+x)+\log (3+\log (x))\right )}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx \\ & = \int \left (\frac {3+x}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))}+\frac {(4+x) \left (1-9 \left (1-\frac {e}{3}\right ) x+3 x^2-3 \left (1-\frac {e}{3}\right ) x \log (x)+x^2 \log (x)-3 x \log (3+\log (x))-x \log (x) \log (3+\log (x))\right )}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}\right ) \, dx \\ & = \int \frac {3+x}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx+\int \frac {(4+x) \left (1-9 \left (1-\frac {e}{3}\right ) x+3 x^2-3 \left (1-\frac {e}{3}\right ) x \log (x)+x^2 \log (x)-3 x \log (3+\log (x))-x \log (x) \log (3+\log (x))\right )}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx \\ & = \int \left (\frac {3}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))}+\frac {x}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))}\right ) \, dx+\int \left (\frac {1-9 \left (1-\frac {e}{3}\right ) x+3 x^2-3 \left (1-\frac {e}{3}\right ) x \log (x)+x^2 \log (x)-3 x \log (3+\log (x))-x \log (x) \log (3+\log (x))}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {4 \left (1-9 \left (1-\frac {e}{3}\right ) x+3 x^2-3 \left (1-\frac {e}{3}\right ) x \log (x)+x^2 \log (x)-3 x \log (3+\log (x))-x \log (x) \log (3+\log (x))\right )}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}\right ) \, dx \\ & = 3 \int \frac {1}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx+4 \int \frac {1-9 \left (1-\frac {e}{3}\right ) x+3 x^2-3 \left (1-\frac {e}{3}\right ) x \log (x)+x^2 \log (x)-3 x \log (3+\log (x))-x \log (x) \log (3+\log (x))}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+\int \frac {x}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx+\int \frac {1-9 \left (1-\frac {e}{3}\right ) x+3 x^2-3 \left (1-\frac {e}{3}\right ) x \log (x)+x^2 \log (x)-3 x \log (3+\log (x))-x \log (x) \log (3+\log (x))}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx \\ & = 3 \int \frac {1}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx+4 \int \left (\frac {3 (-3+e)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {1}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {3 x}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {(-3+e) \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {x \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {3 \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {\log (x) \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}\right ) \, dx+\int \frac {x}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx+\int \left (\frac {1}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {3 (-3+e) x}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {3 x^2}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {(-3+e) x \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {x^2 \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {3 x \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {x \log (x) \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}\right ) \, dx \\ & = 3 \int \frac {x^2}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+3 \int \frac {x \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+3 \int \frac {1}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx+4 \int \frac {1}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+4 \int \frac {x \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+4 \int \frac {\log (x) \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+12 \int \frac {x}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+12 \int \frac {\log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx-(3 (3-e)) \int \frac {x}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx-(4 (3-e)) \int \frac {\log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx-(12 (3-e)) \int \frac {1}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+(-3+e) \int \frac {x \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+\int \frac {1}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+\int \frac {x^2 \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+\int \frac {x \log (x) \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+\int \frac {x}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {4-17 x+3 e x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx=\frac {4+x}{-2+e+e^x+x-\log (3+\log (x))} \]
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Time = 4.71 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05
method | result | size |
risch | \(\frac {4+x}{x -2+{\mathrm e}^{x}+{\mathrm e}-\ln \left (3+\ln \left (x \right )\right )}\) | \(21\) |
parallelrisch | \(-\frac {-4-x}{x -2+{\mathrm e}^{x}+{\mathrm e}-\ln \left (3+\ln \left (x \right )\right )}\) | \(24\) |
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Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {4-17 x+3 e x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx=\frac {x + 4}{x + e + e^{x} - \log \left (\log \left (x\right ) + 3\right ) - 2} \]
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Time = 0.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {4-17 x+3 e x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx=\frac {x + 4}{x + e^{x} - \log {\left (\log {\left (x \right )} + 3 \right )} - 2 + e} \]
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Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {4-17 x+3 e x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx=\frac {x + 4}{x + e + e^{x} - \log \left (\log \left (x\right ) + 3\right ) - 2} \]
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Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {4-17 x+3 e x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx=\frac {x + 4}{x + e + e^{x} - \log \left (\log \left (x\right ) + 3\right ) - 2} \]
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Timed out. \[ \int \frac {4-17 x+3 e x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx=\int -\frac {17\,x+\ln \left (\ln \left (x\right )+3\right )\,\left (3\,x+x\,\ln \left (x\right )\right )+\ln \left (x\right )\,\left (6\,x-x\,\mathrm {e}+{\mathrm {e}}^x\,\left (x^2+3\,x\right )\right )-3\,x\,\mathrm {e}+{\mathrm {e}}^x\,\left (3\,x^2+9\,x\right )-4}{12\,x-\ln \left (\ln \left (x\right )+3\right )\,\left (6\,x\,\mathrm {e}-12\,x+\ln \left (x\right )\,\left (2\,x\,\mathrm {e}-4\,x+2\,x\,{\mathrm {e}}^x+2\,x^2\right )+6\,x\,{\mathrm {e}}^x+6\,x^2\right )+3\,x\,{\mathrm {e}}^{2\,x}-\mathrm {e}\,\left (12\,x-6\,x^2\right )+3\,x\,{\mathrm {e}}^2+{\ln \left (\ln \left (x\right )+3\right )}^2\,\left (3\,x+x\,\ln \left (x\right )\right )-12\,x^2+3\,x^3+{\mathrm {e}}^x\,\left (6\,x\,\mathrm {e}-12\,x+6\,x^2\right )+\ln \left (x\right )\,\left (4\,x+x\,{\mathrm {e}}^{2\,x}-\mathrm {e}\,\left (4\,x-2\,x^2\right )+x\,{\mathrm {e}}^2-4\,x^2+x^3+{\mathrm {e}}^x\,\left (2\,x\,\mathrm {e}-4\,x+2\,x^2\right )\right )} \,d x \]
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