Integrand size = 208, antiderivative size = 25 \[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=e^{4 \left (\frac {x}{2}+\frac {x \log (4)}{(1+x)^2 (x+\log (x))}\right )} \]
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\[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=\int \frac {\exp \left (\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}\right ) \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2^{1+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}} \left (3 x^3+3 x^4+x^5+x^2 (1-4 \log (4))-2 \log (4)-2 x \log (4)+2 \left (x+3 x^2+3 x^3+x^4+\log (4)-x \log (4)\right ) \log (x)+(1+x)^3 \log ^2(x)\right )}{(1+x)^3 (x+\log (x))^2} \, dx \\ & = \int \left (2^{1+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}-\frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}} \log (4)}{(1+x) (x+\log (x))^2}-\frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} (-1+x) x^{\frac {2 x}{x+\log (x)}} \log (4)}{(1+x)^3 (x+\log (x))}\right ) \, dx \\ & = -\left (\log (4) \int \frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x) (x+\log (x))^2} \, dx\right )-\log (4) \int \frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} (-1+x) x^{\frac {2 x}{x+\log (x)}}}{(1+x)^3 (x+\log (x))} \, dx+\int 2^{1+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}} \, dx \\ & = -\left (\log (4) \int \frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x) (x+\log (x))^2} \, dx\right )-\log (4) \int \left (-\frac {2^{3+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x)^3 (x+\log (x))}+\frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x)^2 (x+\log (x))}\right ) \, dx+\int 2^{1+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}} \, dx \\ & = -\left (\log (4) \int \frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x) (x+\log (x))^2} \, dx\right )+\log (4) \int \frac {2^{3+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x)^3 (x+\log (x))} \, dx-\log (4) \int \frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x)^2 (x+\log (x))} \, dx+\int 2^{1+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}} \, dx \\ \end{align*}
\[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=\int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx \]
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Time = 21.91 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72
method | result | size |
risch | \({\mathrm e}^{\frac {2 x \left (x^{2} \ln \left (x \right )+x^{3}+2 x \ln \left (x \right )+2 x^{2}+\ln \left (x \right )+4 \ln \left (2\right )+x \right )}{\left (1+x \right )^{2} \left (x +\ln \left (x \right )\right )}}\) | \(43\) |
parallelrisch | \({\mathrm e}^{\frac {\left (2 x^{3}+4 x^{2}+2 x \right ) \ln \left (x \right )+8 x \ln \left (2\right )+2 x^{4}+4 x^{3}+2 x^{2}}{x^{2} \ln \left (x \right )+x^{3}+2 x \ln \left (x \right )+2 x^{2}+\ln \left (x \right )+x}}\) | \(66\) |
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=e^{\left (\frac {2 \, {\left (x^{4} + 2 \, x^{3} + x^{2} + 4 \, x \log \left (2\right ) + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \left (x\right )\right )}}{x^{3} + 2 \, x^{2} + {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right ) + x}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.40 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.44 \[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=e^{\frac {2 x^{4} + 4 x^{3} + 2 x^{2} + 8 x \log {\left (2 \right )} + \left (2 x^{3} + 4 x^{2} + 2 x\right ) \log {\left (x \right )}}{x^{3} + 2 x^{2} + x + \left (x^{2} + 2 x + 1\right ) \log {\left (x \right )}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (21) = 42\).
Time = 0.76 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.52 \[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=e^{\left (2 \, x + \frac {8 \, \log \left (2\right ) \log \left (x\right )}{{\left (x + 1\right )} \log \left (x\right )^{2} - 2 \, {\left (x + 1\right )} \log \left (x\right ) + x + 1} - \frac {8 \, \log \left (2\right ) \log \left (x\right )}{{\left (x - 2\right )} \log \left (x\right )^{2} + \log \left (x\right )^{3} - {\left (2 \, x - 1\right )} \log \left (x\right ) + x} + \frac {8 \, \log \left (2\right )}{x^{2} - {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right ) + 2 \, x + 1}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (21) = 42\).
Time = 0.63 (sec) , antiderivative size = 216, normalized size of antiderivative = 8.64 \[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=e^{\left (\frac {2 \, x^{4}}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )} + \frac {2 \, x^{3} \log \left (x\right )}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )} + \frac {4 \, x^{3}}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )} + \frac {4 \, x^{2} \log \left (x\right )}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )} + \frac {2 \, x^{2}}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )} + \frac {8 \, x \log \left (2\right )}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )} + \frac {2 \, x \log \left (x\right )}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )}\right )} \]
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Time = 13.43 (sec) , antiderivative size = 134, normalized size of antiderivative = 5.36 \[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx={256}^{\frac {x}{x+\ln \left (x\right )+x^2\,\ln \left (x\right )+2\,x\,\ln \left (x\right )+2\,x^2+x^3}}\,x^{\frac {2\,x}{x+\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {2\,x^2}{x+\ln \left (x\right )+x^2\,\ln \left (x\right )+2\,x\,\ln \left (x\right )+2\,x^2+x^3}}\,{\mathrm {e}}^{\frac {2\,x^4}{x+\ln \left (x\right )+x^2\,\ln \left (x\right )+2\,x\,\ln \left (x\right )+2\,x^2+x^3}}\,{\mathrm {e}}^{\frac {4\,x^3}{x+\ln \left (x\right )+x^2\,\ln \left (x\right )+2\,x\,\ln \left (x\right )+2\,x^2+x^3}} \]
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