Integrand size = 129, antiderivative size = 22 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{-x+x^4-\frac {4 x}{-3+\log (4+4 x)}} \]
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\[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{(1+x) (3-\log (4 (1+x)))^2} \, dx \\ & = \int \left (\frac {3 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2}+\frac {7 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x}{(1+x) (-3+\log (4 (1+x)))^2}+\frac {36 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3}{(1+x) (-3+\log (4 (1+x)))^2}+\frac {36 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^4}{(1+x) (-3+\log (4 (1+x)))^2}+\frac {2 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (1-12 x^3\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2}+\frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (-1+4 x^3\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2}\right ) \, dx \\ & = 2 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (1-12 x^3\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+3 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^4}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (-1+4 x^3\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx \\ & = 2 \int \left (\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \log (4+4 x)}{(3-\log (4 (1+x)))^2}-\frac {12 e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^3 \log (4+4 x)}{(3-\log (4 (1+x)))^2}\right ) \, dx+3 \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+7 \int \left (\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(-3+\log (4 (1+x)))^2}-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(1+x) (-3+\log (4 (1+x)))^2}\right ) \, dx+36 \int \left (\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(-3+\log (4 (1+x)))^2}-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x}{(-3+\log (4 (1+x)))^2}+\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^2}{(-3+\log (4 (1+x)))^2}-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(1+x) (-3+\log (4 (1+x)))^2}\right ) \, dx+36 \int \left (-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(-3+\log (4 (1+x)))^2}+\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x}{(-3+\log (4 (1+x)))^2}-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^2}{(-3+\log (4 (1+x)))^2}+\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^3}{(-3+\log (4 (1+x)))^2}+\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(1+x) (-3+\log (4 (1+x)))^2}\right ) \, dx+\int \left (-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2}+\frac {4 e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^3 \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2}\right ) \, dx \\ & = 2 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+3 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+4 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2} \, dx-7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx-24 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3}{(-3+\log (4 (1+x)))^2} \, dx-\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx \\ & = 2 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+3 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+4 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2} \, dx-7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx-24 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+36 \int \left (-\frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2}+\frac {3 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)}{(-3+\log (4 (1+x)))^2}-\frac {3 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)^2}{(-3+\log (4 (1+x)))^2}+\frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)^3}{(-3+\log (4 (1+x)))^2}\right ) \, dx-\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx \\ & = 2 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+3 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+4 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2} \, dx-7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx-24 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx-36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2} \, dx+36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)^3}{(-3+\log (4 (1+x)))^2} \, dx+108 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)}{(-3+\log (4 (1+x)))^2} \, dx-108 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)^2}{(-3+\log (4 (1+x)))^2} \, dx-\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\frac {x \left (-1-3 x^3+\left (-1+x^3\right ) \log (4 (1+x))\right )}{-3+\log (4 (1+x))}} \]
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Time = 0.81 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {\left (x^{4}-x \right ) \ln \left (4+4 x \right )-3 x^{4}-x}{\ln \left (4+4 x \right )-3}}\) | \(36\) |
risch | \({\mathrm e}^{\frac {x \left (\ln \left (4+4 x \right ) x^{3}-3 x^{3}-\ln \left (4+4 x \right )-1\right )}{\ln \left (4+4 x \right )-3}}\) | \(39\) |
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\left (-\frac {3 \, x^{4} - {\left (x^{4} - x\right )} \log \left (4 \, x + 4\right ) + x}{\log \left (4 \, x + 4\right ) - 3}\right )} \]
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\frac {- 3 x^{4} - x + \left (x^{4} - x\right ) \log {\left (4 x + 4 \right )}}{\log {\left (4 x + 4 \right )} - 3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (21) = 42\).
Time = 0.43 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.95 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\left (\frac {2 \, x^{4} \log \left (2\right )}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} + \frac {x^{4} \log \left (x + 1\right )}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} - \frac {3 \, x^{4}}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} - \frac {2 \, x \log \left (2\right )}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} - \frac {x \log \left (x + 1\right )}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} - \frac {x}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (21) = 42\).
Time = 0.83 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.14 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\left (\frac {x^{4} \log \left (4 \, x + 4\right )}{\log \left (4 \, x + 4\right ) - 3} - \frac {3 \, x^{4}}{\log \left (4 \, x + 4\right ) - 3} - \frac {x \log \left (4 \, x + 4\right )}{\log \left (4 \, x + 4\right ) - 3} - \frac {x}{\log \left (4 \, x + 4\right ) - 3}\right )} \]
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Time = 15.84 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.59 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=\frac {{\mathrm {e}}^{-\frac {x}{\ln \left (4\,x+4\right )-3}}\,{\mathrm {e}}^{-\frac {3\,x^4}{\ln \left (4\,x+4\right )-3}}}{{\left (4\,x+4\right )}^{\frac {x-x^4}{\ln \left (4\,x+4\right )-3}}} \]
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