Integrand size = 130, antiderivative size = 34 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\frac {(7-x) x \log ^2(x)}{256 \left (-2+\frac {x^2}{2 \left (5-e^x\right )}\right )} \]
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\[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\log (x) \left (2 \left (-5+e^x\right ) (-7+x) \left (-20+4 e^x+x^2\right )+\left (4 e^{2 x} (-7+2 x)-5 \left (140-40 x+7 x^2\right )+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log (x)\right )}{128 \left (20-4 e^x-x^2\right )^2} \, dx \\ & = \frac {1}{128} \int \frac {\log (x) \left (2 \left (-5+e^x\right ) (-7+x) \left (-20+4 e^x+x^2\right )+\left (4 e^{2 x} (-7+2 x)-5 \left (140-40 x+7 x^2\right )+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log (x)\right )}{\left (20-4 e^x-x^2\right )^2} \, dx \\ & = \frac {1}{128} \int \left (-\frac {x^3 \left (140-6 x-9 x^2+x^3\right ) \log ^2(x)}{4 \left (-20+4 e^x+x^2\right )^2}+\frac {1}{4} \log (x) (-14+2 x-7 \log (x)+2 x \log (x))+\frac {x^2 \log (x) \left (14-2 x+21 \log (x)-11 x \log (x)+x^2 \log (x)\right )}{4 \left (-20+4 e^x+x^2\right )}\right ) \, dx \\ & = -\left (\frac {1}{512} \int \frac {x^3 \left (140-6 x-9 x^2+x^3\right ) \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx\right )+\frac {1}{512} \int \log (x) (-14+2 x-7 \log (x)+2 x \log (x)) \, dx+\frac {1}{512} \int \frac {x^2 \log (x) \left (14-2 x+21 \log (x)-11 x \log (x)+x^2 \log (x)\right )}{-20+4 e^x+x^2} \, dx \\ & = \frac {1}{512} \int \left (2 (-7+x) \log (x)+(-7+2 x) \log ^2(x)\right ) \, dx-\frac {1}{512} \int \left (\frac {140 x^3 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2}-\frac {6 x^4 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2}-\frac {9 x^5 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2}+\frac {x^6 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2}\right ) \, dx+\frac {1}{512} \int \left (\frac {14 x^2 \log (x)}{-20+4 e^x+x^2}-\frac {2 x^3 \log (x)}{-20+4 e^x+x^2}+\frac {21 x^2 \log ^2(x)}{-20+4 e^x+x^2}-\frac {11 x^3 \log ^2(x)}{-20+4 e^x+x^2}+\frac {x^4 \log ^2(x)}{-20+4 e^x+x^2}\right ) \, dx \\ & = \frac {1}{512} \int (-7+2 x) \log ^2(x) \, dx-\frac {1}{512} \int \frac {x^6 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {1}{512} \int \frac {x^4 \log ^2(x)}{-20+4 e^x+x^2} \, dx+\frac {1}{256} \int (-7+x) \log (x) \, dx-\frac {1}{256} \int \frac {x^3 \log (x)}{-20+4 e^x+x^2} \, dx+\frac {3}{256} \int \frac {x^4 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {9}{512} \int \frac {x^5 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {11}{512} \int \frac {x^3 \log ^2(x)}{-20+4 e^x+x^2} \, dx+\frac {7}{256} \int \frac {x^2 \log (x)}{-20+4 e^x+x^2} \, dx+\frac {21}{512} \int \frac {x^2 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {35}{128} \int \frac {x^3 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx \\ & = -\frac {7}{256} x \log (x)+\frac {1}{512} x^2 \log (x)-\frac {1}{512} \int \frac {x^6 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {1}{512} \int \frac {x^4 \log ^2(x)}{-20+4 e^x+x^2} \, dx+\frac {1}{512} \int \left (-7 \log ^2(x)+2 x \log ^2(x)\right ) \, dx-\frac {1}{256} \int \left (-7+\frac {x}{2}\right ) \, dx+\frac {1}{256} \int \frac {\int \frac {x^3}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {3}{256} \int \frac {x^4 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {9}{512} \int \frac {x^5 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {11}{512} \int \frac {x^3 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {7}{256} \int \frac {\int \frac {x^2}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {21}{512} \int \frac {x^2 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {35}{128} \int \frac {x^3 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {1}{256} \log (x) \int \frac {x^3}{-20+4 e^x+x^2} \, dx+\frac {1}{256} (7 \log (x)) \int \frac {x^2}{-20+4 e^x+x^2} \, dx \\ & = \frac {7 x}{256}-\frac {x^2}{1024}-\frac {7}{256} x \log (x)+\frac {1}{512} x^2 \log (x)-\frac {1}{512} \int \frac {x^6 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {1}{512} \int \frac {x^4 \log ^2(x)}{-20+4 e^x+x^2} \, dx+\frac {1}{256} \int x \log ^2(x) \, dx+\frac {1}{256} \int \frac {\int \frac {x^3}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {3}{256} \int \frac {x^4 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {7}{512} \int \log ^2(x) \, dx+\frac {9}{512} \int \frac {x^5 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {11}{512} \int \frac {x^3 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {7}{256} \int \frac {\int \frac {x^2}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {21}{512} \int \frac {x^2 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {35}{128} \int \frac {x^3 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {1}{256} \log (x) \int \frac {x^3}{-20+4 e^x+x^2} \, dx+\frac {1}{256} (7 \log (x)) \int \frac {x^2}{-20+4 e^x+x^2} \, dx \\ & = \frac {7 x}{256}-\frac {x^2}{1024}-\frac {7}{256} x \log (x)+\frac {1}{512} x^2 \log (x)-\frac {7}{512} x \log ^2(x)+\frac {1}{512} x^2 \log ^2(x)-\frac {1}{512} \int \frac {x^6 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {1}{512} \int \frac {x^4 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {1}{256} \int x \log (x) \, dx+\frac {1}{256} \int \frac {\int \frac {x^3}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {3}{256} \int \frac {x^4 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {9}{512} \int \frac {x^5 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {11}{512} \int \frac {x^3 \log ^2(x)}{-20+4 e^x+x^2} \, dx+\frac {7}{256} \int \log (x) \, dx-\frac {7}{256} \int \frac {\int \frac {x^2}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {21}{512} \int \frac {x^2 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {35}{128} \int \frac {x^3 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {1}{256} \log (x) \int \frac {x^3}{-20+4 e^x+x^2} \, dx+\frac {1}{256} (7 \log (x)) \int \frac {x^2}{-20+4 e^x+x^2} \, dx \\ & = -\frac {7}{512} x \log ^2(x)+\frac {1}{512} x^2 \log ^2(x)-\frac {1}{512} \int \frac {x^6 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {1}{512} \int \frac {x^4 \log ^2(x)}{-20+4 e^x+x^2} \, dx+\frac {1}{256} \int \frac {\int \frac {x^3}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {3}{256} \int \frac {x^4 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {9}{512} \int \frac {x^5 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {11}{512} \int \frac {x^3 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {7}{256} \int \frac {\int \frac {x^2}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {21}{512} \int \frac {x^2 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {35}{128} \int \frac {x^3 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {1}{256} \log (x) \int \frac {x^3}{-20+4 e^x+x^2} \, dx+\frac {1}{256} (7 \log (x)) \int \frac {x^2}{-20+4 e^x+x^2} \, dx \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\frac {\left (-5+e^x\right ) (-7+x) x \log ^2(x)}{128 \left (-20+4 e^x+x^2\right )} \]
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Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {x \left ({\mathrm e}^{x} x -5 x -7 \,{\mathrm e}^{x}+35\right ) \ln \left (x \right )^{2}}{128 x^{2}+512 \,{\mathrm e}^{x}-2560}\) | \(32\) |
parallelrisch | \(-\frac {-4 x^{2} {\mathrm e}^{x} \ln \left (x \right )^{2}+20 x^{2} \ln \left (x \right )^{2}+28 x \,{\mathrm e}^{x} \ln \left (x \right )^{2}-140 x \ln \left (x \right )^{2}}{512 \left (x^{2}+4 \,{\mathrm e}^{x}-20\right )}\) | \(51\) |
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=-\frac {{\left (5 \, x^{2} - {\left (x^{2} - 7 \, x\right )} e^{x} - 35 \, x\right )} \log \left (x\right )^{2}}{128 \, {\left (x^{2} + 4 \, e^{x} - 20\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\left (\frac {x^{2}}{512} - \frac {7 x}{512}\right ) \log {\left (x \right )}^{2} + \frac {- x^{4} \log {\left (x \right )}^{2} + 7 x^{3} \log {\left (x \right )}^{2}}{512 x^{2} + 2048 e^{x} - 10240} \]
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\frac {{\left (x^{2} - 7 \, x\right )} e^{x} \log \left (x\right )^{2} - 5 \, {\left (x^{2} - 7 \, x\right )} \log \left (x\right )^{2}}{128 \, {\left (x^{2} + 4 \, e^{x} - 20\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\frac {x^{2} e^{x} \log \left (x\right )^{2} - 5 \, x^{2} \log \left (x\right )^{2} - 7 \, x e^{x} \log \left (x\right )^{2} + 35 \, x \log \left (x\right )^{2}}{128 \, {\left (x^{2} + 4 \, e^{x} - 20\right )}} \]
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Time = 14.80 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\frac {x\,{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^x-5\right )\,\left (x-7\right )}{128\,\left (4\,{\mathrm {e}}^x+x^2-20\right )} \]
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