Integrand size = 123, antiderivative size = 25 \[ \int \frac {e^{\frac {1}{2} (2 x+3 \log (x-\log (x)))} \left (11 x-x^2-2 x^3+\left (4-19 x-9 x^2-2 x^3\right ) \log (x)+\left (8+8 x+2 x^2\right ) \log ^2(x)\right )}{-2 x^3+\left (-14 x^2-4 x^3\right ) \log (x)+\left (-16 x-12 x^2-2 x^3\right ) \log ^2(x)+\left (32+16 x+2 x^2\right ) \log ^3(x)} \, dx=\frac {e^x x (x-\log (x))^{3/2}}{x+(4+x) \log (x)} \]
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\[ \int \frac {e^{\frac {1}{2} (2 x+3 \log (x-\log (x)))} \left (11 x-x^2-2 x^3+\left (4-19 x-9 x^2-2 x^3\right ) \log (x)+\left (8+8 x+2 x^2\right ) \log ^2(x)\right )}{-2 x^3+\left (-14 x^2-4 x^3\right ) \log (x)+\left (-16 x-12 x^2-2 x^3\right ) \log ^2(x)+\left (32+16 x+2 x^2\right ) \log ^3(x)} \, dx=\int \frac {e^{\frac {1}{2} (2 x+3 \log (x-\log (x)))} \left (11 x-x^2-2 x^3+\left (4-19 x-9 x^2-2 x^3\right ) \log (x)+\left (8+8 x+2 x^2\right ) \log ^2(x)\right )}{-2 x^3+\left (-14 x^2-4 x^3\right ) \log (x)+\left (-16 x-12 x^2-2 x^3\right ) \log ^2(x)+\left (32+16 x+2 x^2\right ) \log ^3(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x (x-\log (x))^{3/2} \left (11 x-x^2-2 x^3+\left (4-19 x-9 x^2-2 x^3\right ) \log (x)+\left (8+8 x+2 x^2\right ) \log ^2(x)\right )}{-2 x^3+\left (-14 x^2-4 x^3\right ) \log (x)+\left (-16 x-12 x^2-2 x^3\right ) \log ^2(x)+\left (32+16 x+2 x^2\right ) \log ^3(x)} \, dx \\ & = \int \frac {e^x \sqrt {x-\log (x)} \left (x \left (-11+x+2 x^2\right )+\left (-4+19 x+9 x^2+2 x^3\right ) \log (x)-2 (2+x)^2 \log ^2(x)\right )}{2 (x+(4+x) \log (x))^2} \, dx \\ & = \frac {1}{2} \int \frac {e^x \sqrt {x-\log (x)} \left (x \left (-11+x+2 x^2\right )+\left (-4+19 x+9 x^2+2 x^3\right ) \log (x)-2 (2+x)^2 \log ^2(x)\right )}{(x+(4+x) \log (x))^2} \, dx \\ & = \frac {1}{2} \int \left (-\frac {2 e^x (2+x)^2 \sqrt {x-\log (x)}}{(4+x)^2}-\frac {2 e^x x \left (80+76 x+17 x^2+x^3\right ) \sqrt {x-\log (x)}}{(4+x)^2 (x+4 \log (x)+x \log (x))^2}+\frac {e^x \left (-16+88 x+71 x^2+21 x^3+2 x^4\right ) \sqrt {x-\log (x)}}{(4+x)^2 (x+4 \log (x)+x \log (x))}\right ) \, dx \\ & = \frac {1}{2} \int \frac {e^x \left (-16+88 x+71 x^2+21 x^3+2 x^4\right ) \sqrt {x-\log (x)}}{(4+x)^2 (x+4 \log (x)+x \log (x))} \, dx-\int \frac {e^x (2+x)^2 \sqrt {x-\log (x)}}{(4+x)^2} \, dx-\int \frac {e^x x \left (80+76 x+17 x^2+x^3\right ) \sqrt {x-\log (x)}}{(4+x)^2 (x+4 \log (x)+x \log (x))^2} \, dx \\ & = \frac {1}{2} \int \left (-\frac {e^x \sqrt {x-\log (x)}}{x+4 \log (x)+x \log (x)}+\frac {5 e^x x \sqrt {x-\log (x)}}{x+4 \log (x)+x \log (x)}+\frac {2 e^x x^2 \sqrt {x-\log (x)}}{x+4 \log (x)+x \log (x)}-\frac {64 e^x \sqrt {x-\log (x)}}{(4+x)^2 (x+4 \log (x)+x \log (x))}+\frac {16 e^x \sqrt {x-\log (x)}}{(4+x) (x+4 \log (x)+x \log (x))}\right ) \, dx-\int \left (e^x \sqrt {x-\log (x)}+\frac {4 e^x \sqrt {x-\log (x)}}{(4+x)^2}-\frac {4 e^x \sqrt {x-\log (x)}}{4+x}\right ) \, dx-\int \left (-\frac {12 e^x \sqrt {x-\log (x)}}{(x+4 \log (x)+x \log (x))^2}+\frac {9 e^x x \sqrt {x-\log (x)}}{(x+4 \log (x)+x \log (x))^2}+\frac {e^x x^2 \sqrt {x-\log (x)}}{(x+4 \log (x)+x \log (x))^2}+\frac {64 e^x \sqrt {x-\log (x)}}{(4+x)^2 (x+4 \log (x)+x \log (x))^2}+\frac {32 e^x \sqrt {x-\log (x)}}{(4+x) (x+4 \log (x)+x \log (x))^2}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {e^x \sqrt {x-\log (x)}}{x+4 \log (x)+x \log (x)} \, dx\right )+\frac {5}{2} \int \frac {e^x x \sqrt {x-\log (x)}}{x+4 \log (x)+x \log (x)} \, dx-4 \int \frac {e^x \sqrt {x-\log (x)}}{(4+x)^2} \, dx+4 \int \frac {e^x \sqrt {x-\log (x)}}{4+x} \, dx+8 \int \frac {e^x \sqrt {x-\log (x)}}{(4+x) (x+4 \log (x)+x \log (x))} \, dx-9 \int \frac {e^x x \sqrt {x-\log (x)}}{(x+4 \log (x)+x \log (x))^2} \, dx+12 \int \frac {e^x \sqrt {x-\log (x)}}{(x+4 \log (x)+x \log (x))^2} \, dx-32 \int \frac {e^x \sqrt {x-\log (x)}}{(4+x) (x+4 \log (x)+x \log (x))^2} \, dx-32 \int \frac {e^x \sqrt {x-\log (x)}}{(4+x)^2 (x+4 \log (x)+x \log (x))} \, dx-64 \int \frac {e^x \sqrt {x-\log (x)}}{(4+x)^2 (x+4 \log (x)+x \log (x))^2} \, dx-\int e^x \sqrt {x-\log (x)} \, dx-\int \frac {e^x x^2 \sqrt {x-\log (x)}}{(x+4 \log (x)+x \log (x))^2} \, dx+\int \frac {e^x x^2 \sqrt {x-\log (x)}}{x+4 \log (x)+x \log (x)} \, dx \\ \end{align*}
Time = 5.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {1}{2} (2 x+3 \log (x-\log (x)))} \left (11 x-x^2-2 x^3+\left (4-19 x-9 x^2-2 x^3\right ) \log (x)+\left (8+8 x+2 x^2\right ) \log ^2(x)\right )}{-2 x^3+\left (-14 x^2-4 x^3\right ) \log (x)+\left (-16 x-12 x^2-2 x^3\right ) \log ^2(x)+\left (32+16 x+2 x^2\right ) \log ^3(x)} \, dx=\frac {e^x x (x-\log (x))^{3/2}}{x+(4+x) \log (x)} \]
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Time = 2.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {x \left (x -\ln \left (x \right )\right )^{\frac {3}{2}} {\mathrm e}^{x}}{x \ln \left (x \right )+4 \ln \left (x \right )+x}\) | \(25\) |
parallelrisch | \(\frac {x \,{\mathrm e}^{\frac {3 \ln \left (x -\ln \left (x \right )\right )}{2}+x}}{x \ln \left (x \right )+4 \ln \left (x \right )+x}\) | \(27\) |
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {1}{2} (2 x+3 \log (x-\log (x)))} \left (11 x-x^2-2 x^3+\left (4-19 x-9 x^2-2 x^3\right ) \log (x)+\left (8+8 x+2 x^2\right ) \log ^2(x)\right )}{-2 x^3+\left (-14 x^2-4 x^3\right ) \log (x)+\left (-16 x-12 x^2-2 x^3\right ) \log ^2(x)+\left (32+16 x+2 x^2\right ) \log ^3(x)} \, dx=\frac {x e^{\left (x + \frac {3}{2} \, \log \left (x - \log \left (x\right )\right )\right )}}{{\left (x + 4\right )} \log \left (x\right ) + x} \]
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Timed out. \[ \int \frac {e^{\frac {1}{2} (2 x+3 \log (x-\log (x)))} \left (11 x-x^2-2 x^3+\left (4-19 x-9 x^2-2 x^3\right ) \log (x)+\left (8+8 x+2 x^2\right ) \log ^2(x)\right )}{-2 x^3+\left (-14 x^2-4 x^3\right ) \log (x)+\left (-16 x-12 x^2-2 x^3\right ) \log ^2(x)+\left (32+16 x+2 x^2\right ) \log ^3(x)} \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\frac {1}{2} (2 x+3 \log (x-\log (x)))} \left (11 x-x^2-2 x^3+\left (4-19 x-9 x^2-2 x^3\right ) \log (x)+\left (8+8 x+2 x^2\right ) \log ^2(x)\right )}{-2 x^3+\left (-14 x^2-4 x^3\right ) \log (x)+\left (-16 x-12 x^2-2 x^3\right ) \log ^2(x)+\left (32+16 x+2 x^2\right ) \log ^3(x)} \, dx=\frac {{\left (x^{2} - x \log \left (x\right )\right )} \sqrt {x - \log \left (x\right )} e^{x}}{{\left (x + 4\right )} \log \left (x\right ) + x} \]
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\[ \int \frac {e^{\frac {1}{2} (2 x+3 \log (x-\log (x)))} \left (11 x-x^2-2 x^3+\left (4-19 x-9 x^2-2 x^3\right ) \log (x)+\left (8+8 x+2 x^2\right ) \log ^2(x)\right )}{-2 x^3+\left (-14 x^2-4 x^3\right ) \log (x)+\left (-16 x-12 x^2-2 x^3\right ) \log ^2(x)+\left (32+16 x+2 x^2\right ) \log ^3(x)} \, dx=\int { -\frac {{\left (2 \, x^{3} - 2 \, {\left (x^{2} + 4 \, x + 4\right )} \log \left (x\right )^{2} + x^{2} + {\left (2 \, x^{3} + 9 \, x^{2} + 19 \, x - 4\right )} \log \left (x\right ) - 11 \, x\right )} e^{\left (x + \frac {3}{2} \, \log \left (x - \log \left (x\right )\right )\right )}}{2 \, {\left ({\left (x^{2} + 8 \, x + 16\right )} \log \left (x\right )^{3} - x^{3} - {\left (x^{3} + 6 \, x^{2} + 8 \, x\right )} \log \left (x\right )^{2} - {\left (2 \, x^{3} + 7 \, x^{2}\right )} \log \left (x\right )\right )}} \,d x } \]
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Time = 11.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {1}{2} (2 x+3 \log (x-\log (x)))} \left (11 x-x^2-2 x^3+\left (4-19 x-9 x^2-2 x^3\right ) \log (x)+\left (8+8 x+2 x^2\right ) \log ^2(x)\right )}{-2 x^3+\left (-14 x^2-4 x^3\right ) \log (x)+\left (-16 x-12 x^2-2 x^3\right ) \log ^2(x)+\left (32+16 x+2 x^2\right ) \log ^3(x)} \, dx=\frac {x\,{\mathrm {e}}^x\,{\left (x-\ln \left (x\right )\right )}^{3/2}}{x+4\,\ln \left (x\right )+x\,\ln \left (x\right )} \]
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