Integrand size = 75, antiderivative size = 26 \[ \int \frac {-4 e-8 x+e^x \left (-64-56 x+4 x^2\right )+32 e^x \log (x)+e^x (16+16 x) \log ^2(x)}{-e x-x^2+e^x \left (-16 x+x^2\right )+4 e^x x \log ^2(x)} \, dx=4 \log \left (x \left (-e-x+e^x \left (x+4 \left (-4+\log ^2(x)\right )\right )\right )\right ) \]
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\[ \int \frac {-4 e-8 x+e^x \left (-64-56 x+4 x^2\right )+32 e^x \log (x)+e^x (16+16 x) \log ^2(x)}{-e x-x^2+e^x \left (-16 x+x^2\right )+4 e^x x \log ^2(x)} \, dx=\int \frac {-4 e-8 x+e^x \left (-64-56 x+4 x^2\right )+32 e^x \log (x)+e^x (16+16 x) \log ^2(x)}{-e x-x^2+e^x \left (-16 x+x^2\right )+4 e^x x \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 \left (-16-14 x+x^2+8 \log (x)+4 \log ^2(x)+4 x \log ^2(x)\right )}{x \left (-16+x+4 \log ^2(x)\right )}+\frac {4 \left (16 \left (1-\frac {15 e}{16}\right ) x-16 \left (1-\frac {e}{16}\right ) x^2+x^3+8 e \log (x)+8 x \log (x)-4 (1-e) x \log ^2(x)+4 x^2 \log ^2(x)\right )}{x \left (16-x-4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )}\right ) \, dx \\ & = 4 \int \frac {-16-14 x+x^2+8 \log (x)+4 \log ^2(x)+4 x \log ^2(x)}{x \left (-16+x+4 \log ^2(x)\right )} \, dx+4 \int \frac {16 \left (1-\frac {15 e}{16}\right ) x-16 \left (1-\frac {e}{16}\right ) x^2+x^3+8 e \log (x)+8 x \log (x)-4 (1-e) x \log ^2(x)+4 x^2 \log ^2(x)}{x \left (16-x-4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )} \, dx \\ & = 4 \int \frac {x \left (16+e (-15+x)-16 x+x^2\right )+8 (e+x) \log (x)+4 x (-1+e+x) \log ^2(x)}{x \left (16-x-4 \log ^2(x)\right ) \left (e-e^x (-16+x)+x-4 e^x \log ^2(x)\right )} \, dx+4 \int \left (\frac {1+x}{x}+\frac {x+8 \log (x)}{x \left (-16+x+4 \log ^2(x)\right )}\right ) \, dx \\ & = 4 \int \frac {1+x}{x} \, dx+4 \int \frac {x+8 \log (x)}{x \left (-16+x+4 \log ^2(x)\right )} \, dx+4 \int \left (\frac {15 \left (1-\frac {16}{15 e}\right ) e}{\left (-16+x+4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )}-\frac {\left (1-\frac {16}{e}\right ) e x}{\left (-16+x+4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )}-\frac {8 e \log (x)}{x \left (-16+x+4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )}-\frac {4 \left (1-\frac {1}{e}\right ) e \log ^2(x)}{\left (-16+x+4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )}+\frac {x^2}{\left (-16+x+4 \log ^2(x)\right ) \left (-e-16 e^x-x+e^x x+4 e^x \log ^2(x)\right )}+\frac {8 \log (x)}{\left (-16+x+4 \log ^2(x)\right ) \left (-e-16 e^x-x+e^x x+4 e^x \log ^2(x)\right )}+\frac {4 x \log ^2(x)}{\left (-16+x+4 \log ^2(x)\right ) \left (-e-16 e^x-x+e^x x+4 e^x \log ^2(x)\right )}\right ) \, dx \\ & = 4 \log \left (16-x-4 \log ^2(x)\right )+4 \int \left (1+\frac {1}{x}\right ) \, dx+4 \int \frac {x^2}{\left (-16+x+4 \log ^2(x)\right ) \left (-e-16 e^x-x+e^x x+4 e^x \log ^2(x)\right )} \, dx+16 \int \frac {x \log ^2(x)}{\left (-16+x+4 \log ^2(x)\right ) \left (-e-16 e^x-x+e^x x+4 e^x \log ^2(x)\right )} \, dx+32 \int \frac {\log (x)}{\left (-16+x+4 \log ^2(x)\right ) \left (-e-16 e^x-x+e^x x+4 e^x \log ^2(x)\right )} \, dx-(4 (16-15 e)) \int \frac {1}{\left (-16+x+4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )} \, dx+(16 (1-e)) \int \frac {\log ^2(x)}{\left (-16+x+4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )} \, dx+(4 (16-e)) \int \frac {x}{\left (-16+x+4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )} \, dx-(32 e) \int \frac {\log (x)}{x \left (-16+x+4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )} \, dx \\ & = 4 x+4 \log (x)+4 \log \left (16-x-4 \log ^2(x)\right )+4 \int \frac {x^2}{\left (-16+x+4 \log ^2(x)\right ) \left (-e-16 e^x-x+e^x x+4 e^x \log ^2(x)\right )} \, dx+16 \int \frac {x \log ^2(x)}{\left (-16+x+4 \log ^2(x)\right ) \left (-e-16 e^x-x+e^x x+4 e^x \log ^2(x)\right )} \, dx+32 \int \frac {\log (x)}{\left (-16+x+4 \log ^2(x)\right ) \left (-e-16 e^x-x+e^x x+4 e^x \log ^2(x)\right )} \, dx-(4 (16-15 e)) \int \frac {1}{\left (-16+x+4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )} \, dx+(16 (1-e)) \int \frac {\log ^2(x)}{\left (-16+x+4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )} \, dx+(4 (16-e)) \int \frac {x}{\left (-16+x+4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )} \, dx-(32 e) \int \frac {\log (x)}{x \left (-16+x+4 \log ^2(x)\right ) \left (e+16 e^x+x-e^x x-4 e^x \log ^2(x)\right )} \, dx \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {-4 e-8 x+e^x \left (-64-56 x+4 x^2\right )+32 e^x \log (x)+e^x (16+16 x) \log ^2(x)}{-e x-x^2+e^x \left (-16 x+x^2\right )+4 e^x x \log ^2(x)} \, dx=-4 \left (-\log (x)-\log \left (-e-16 e^x-x+e^x x+4 e^x \log ^2(x)\right )\right ) \]
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Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15
method | result | size |
norman | \(4 \ln \left (x \right )+4 \ln \left (-4 \,{\mathrm e}^{x} \ln \left (x \right )^{2}-{\mathrm e}^{x} x +{\mathrm e}+x +16 \,{\mathrm e}^{x}\right )\) | \(30\) |
parallelrisch | \(4 \ln \left (4 \,{\mathrm e}^{x} \ln \left (x \right )^{2}+{\mathrm e}^{x} x -{\mathrm e}-16 \,{\mathrm e}^{x}-x \right )+4 \ln \left (x \right )\) | \(33\) |
risch | \(4 x +4 \ln \left (x \right )+4 \ln \left (\ln \left (x \right )^{2}+\frac {x}{4}-\frac {{\mathrm e}^{1-x}}{4}-4-\frac {x \,{\mathrm e}^{-x}}{4}\right )\) | \(36\) |
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {-4 e-8 x+e^x \left (-64-56 x+4 x^2\right )+32 e^x \log (x)+e^x (16+16 x) \log ^2(x)}{-e x-x^2+e^x \left (-16 x+x^2\right )+4 e^x x \log ^2(x)} \, dx=4 \, x + 4 \, \log \left ({\left (4 \, e^{x} \log \left (x\right )^{2} + {\left (x - 16\right )} e^{x} - x - e\right )} e^{\left (-x\right )}\right ) + 4 \, \log \left (x\right ) \]
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Time = 0.84 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {-4 e-8 x+e^x \left (-64-56 x+4 x^2\right )+32 e^x \log (x)+e^x (16+16 x) \log ^2(x)}{-e x-x^2+e^x \left (-16 x+x^2\right )+4 e^x x \log ^2(x)} \, dx=4 \log {\left (x \right )} + 4 \log {\left (\frac {- x - e}{x + 4 \log {\left (x \right )}^{2} - 16} + e^{x} \right )} + 4 \log {\left (\frac {x}{4} + \log {\left (x \right )}^{2} - 4 \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (25) = 50\).
Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int \frac {-4 e-8 x+e^x \left (-64-56 x+4 x^2\right )+32 e^x \log (x)+e^x (16+16 x) \log ^2(x)}{-e x-x^2+e^x \left (-16 x+x^2\right )+4 e^x x \log ^2(x)} \, dx=4 \, \log \left (\log \left (x\right )^{2} + \frac {1}{4} \, x - 4\right ) + 4 \, \log \left (x\right ) + 4 \, \log \left (\frac {{\left (4 \, \log \left (x\right )^{2} + x - 16\right )} e^{x} - x - e}{4 \, \log \left (x\right )^{2} + x - 16}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-4 e-8 x+e^x \left (-64-56 x+4 x^2\right )+32 e^x \log (x)+e^x (16+16 x) \log ^2(x)}{-e x-x^2+e^x \left (-16 x+x^2\right )+4 e^x x \log ^2(x)} \, dx=4 \, \log \left (-4 \, e^{x} \log \left (x\right )^{2} - x e^{x} + x + e + 16 \, e^{x}\right ) + 4 \, \log \left (x\right ) \]
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Time = 11.47 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-4 e-8 x+e^x \left (-64-56 x+4 x^2\right )+32 e^x \log (x)+e^x (16+16 x) \log ^2(x)}{-e x-x^2+e^x \left (-16 x+x^2\right )+4 e^x x \log ^2(x)} \, dx=4\,\ln \left (x\,\mathrm {e}+{\mathrm {e}}^x\,\left (16\,x-x^2\right )+x^2-4\,x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2\right ) \]
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