Integrand size = 246, antiderivative size = 28 \[ \int \frac {-5 e^7+e^6 \left (-5 e^4+20 e^2 x\right )+e^6 \left (20 e^2 x+10 e^4 x\right ) \log (x)+\left (-e^4 x+e^3 \left (-2 e^4 x+8 e^2 x^2\right )+e^6 \left (4 e^4 x+8 e^2 x^2-16 x^3\right )\right ) \log ^2(x)+\left (2 e^7 x^2+e^6 \left (2 e^4 x^2-8 e^2 x^3\right )\right ) \log ^3(x)-e^{10} x^3 \log ^4(x)}{\left (e^4 x+e^3 \left (2 e^4 x-8 e^2 x^2\right )+e^6 \left (e^4 x-8 e^2 x^2+16 x^3\right )\right ) \log ^2(x)+\left (-2 e^7 x^2+e^6 \left (-2 e^4 x^2+8 e^2 x^3\right )\right ) \log ^3(x)+e^{10} x^3 \log ^4(x)} \, dx=-x+\frac {5}{\log (x) \left (1+\frac {1}{e^3}-x \left (\frac {4}{e^2}+\log (x)\right )\right )} \]
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\[ \int \frac {-5 e^7+e^6 \left (-5 e^4+20 e^2 x\right )+e^6 \left (20 e^2 x+10 e^4 x\right ) \log (x)+\left (-e^4 x+e^3 \left (-2 e^4 x+8 e^2 x^2\right )+e^6 \left (4 e^4 x+8 e^2 x^2-16 x^3\right )\right ) \log ^2(x)+\left (2 e^7 x^2+e^6 \left (2 e^4 x^2-8 e^2 x^3\right )\right ) \log ^3(x)-e^{10} x^3 \log ^4(x)}{\left (e^4 x+e^3 \left (2 e^4 x-8 e^2 x^2\right )+e^6 \left (e^4 x-8 e^2 x^2+16 x^3\right )\right ) \log ^2(x)+\left (-2 e^7 x^2+e^6 \left (-2 e^4 x^2+8 e^2 x^3\right )\right ) \log ^3(x)+e^{10} x^3 \log ^4(x)} \, dx=\int \frac {-5 e^7+e^6 \left (-5 e^4+20 e^2 x\right )+e^6 \left (20 e^2 x+10 e^4 x\right ) \log (x)+\left (-e^4 x+e^3 \left (-2 e^4 x+8 e^2 x^2\right )+e^6 \left (4 e^4 x+8 e^2 x^2-16 x^3\right )\right ) \log ^2(x)+\left (2 e^7 x^2+e^6 \left (2 e^4 x^2-8 e^2 x^3\right )\right ) \log ^3(x)-e^{10} x^3 \log ^4(x)}{\left (e^4 x+e^3 \left (2 e^4 x-8 e^2 x^2\right )+e^6 \left (e^4 x-8 e^2 x^2+16 x^3\right )\right ) \log ^2(x)+\left (-2 e^7 x^2+e^6 \left (-2 e^4 x^2+8 e^2 x^3\right )\right ) \log ^3(x)+e^{10} x^3 \log ^4(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-5 e^3 \left (1+e^3-4 e x\right )+10 e^4 \left (2+e^2\right ) x \log (x)-x \left (1+2 e^3-4 e^6-8 e x-8 e^4 x+16 e^2 x^2\right ) \log ^2(x)+2 e^3 x^2 \left (1+e^3-4 e x\right ) \log ^3(x)-e^6 x^3 \log ^4(x)}{x \log ^2(x) \left (1+e^3-4 e x-e^3 x \log (x)\right )^2} \, dx \\ & = \int \left (-1+\frac {5 e^3}{x \left (-1-e^3+4 e x\right ) \log ^2(x)}+\frac {20 e^4}{\left (1+e^3-4 e x\right )^2 \log (x)}+\frac {5 \left (e^6+e^9+e^9 x\right )}{\left (1+e^3-4 e x\right ) \left (-1-e^3+4 e x+e^3 x \log (x)\right )^2}-\frac {20 e^7 x}{\left (1+e^3-4 e x\right )^2 \left (-1-e^3+4 e x+e^3 x \log (x)\right )}\right ) \, dx \\ & = -x+5 \int \frac {e^6+e^9+e^9 x}{\left (1+e^3-4 e x\right ) \left (-1-e^3+4 e x+e^3 x \log (x)\right )^2} \, dx+\left (5 e^3\right ) \int \frac {1}{x \left (-1-e^3+4 e x\right ) \log ^2(x)} \, dx+\left (20 e^4\right ) \int \frac {1}{\left (1+e^3-4 e x\right )^2 \log (x)} \, dx-\left (20 e^7\right ) \int \frac {x}{\left (1+e^3-4 e x\right )^2 \left (-1-e^3+4 e x+e^3 x \log (x)\right )} \, dx \\ & = -x+5 \int \left (-\frac {e^8}{4 \left (-1-e^3+4 e x+e^3 x \log (x)\right )^2}+\frac {e^6 \left (4+e^2+4 e^3+e^5\right )}{4 \left (1+e^3-4 e x\right ) \left (-1-e^3+4 e x+e^3 x \log (x)\right )^2}\right ) \, dx+\left (5 e^3\right ) \int \frac {1}{x \left (-1-e^3+4 e x\right ) \log ^2(x)} \, dx+\left (20 e^4\right ) \int \frac {1}{\left (1+e^3-4 e x\right )^2 \log (x)} \, dx-\left (20 e^7\right ) \int \left (\frac {1+e^3}{4 e \left (1+e^3-4 e x\right )^2 \left (-1-e^3+4 e x+e^3 x \log (x)\right )}-\frac {1}{4 e \left (1+e^3-4 e x\right ) \left (-1-e^3+4 e x+e^3 x \log (x)\right )}\right ) \, dx \\ & = -x+\left (5 e^3\right ) \int \frac {1}{x \left (-1-e^3+4 e x\right ) \log ^2(x)} \, dx+\left (20 e^4\right ) \int \frac {1}{\left (1+e^3-4 e x\right )^2 \log (x)} \, dx+\left (5 e^6\right ) \int \frac {1}{\left (1+e^3-4 e x\right ) \left (-1-e^3+4 e x+e^3 x \log (x)\right )} \, dx-\frac {1}{4} \left (5 e^8\right ) \int \frac {1}{\left (-1-e^3+4 e x+e^3 x \log (x)\right )^2} \, dx-\left (5 e^6 \left (1+e^3\right )\right ) \int \frac {1}{\left (1+e^3-4 e x\right )^2 \left (-1-e^3+4 e x+e^3 x \log (x)\right )} \, dx+\frac {1}{4} \left (5 e^6 \left (4+e^2+4 e^3+e^5\right )\right ) \int \frac {1}{\left (1+e^3-4 e x\right ) \left (-1-e^3+4 e x+e^3 x \log (x)\right )^2} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(28)=56\).
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.18 \[ \int \frac {-5 e^7+e^6 \left (-5 e^4+20 e^2 x\right )+e^6 \left (20 e^2 x+10 e^4 x\right ) \log (x)+\left (-e^4 x+e^3 \left (-2 e^4 x+8 e^2 x^2\right )+e^6 \left (4 e^4 x+8 e^2 x^2-16 x^3\right )\right ) \log ^2(x)+\left (2 e^7 x^2+e^6 \left (2 e^4 x^2-8 e^2 x^3\right )\right ) \log ^3(x)-e^{10} x^3 \log ^4(x)}{\left (e^4 x+e^3 \left (2 e^4 x-8 e^2 x^2\right )+e^6 \left (e^4 x-8 e^2 x^2+16 x^3\right )\right ) \log ^2(x)+\left (-2 e^7 x^2+e^6 \left (-2 e^4 x^2+8 e^2 x^3\right )\right ) \log ^3(x)+e^{10} x^3 \log ^4(x)} \, dx=-x+\frac {5 e^3}{\left (1+e^3-4 e x\right ) \log (x)}-\frac {5 e^6 x}{\left (1+e^3-4 e x\right ) \left (-1-e^3+4 e x+e^3 x \log (x)\right )} \]
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Time = 0.92 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14
method | result | size |
risch | \(-x -\frac {5 \,{\mathrm e}^{3}}{\left (x \,{\mathrm e}^{3} \ln \left (x \right )-{\mathrm e}^{3}+4 x \,{\mathrm e}-1\right ) \ln \left (x \right )}\) | \(32\) |
default | \(-\frac {-\frac {{\mathrm e}^{-3} \left ({\mathrm e}^{3}+1\right )^{2} {\mathrm e}^{4} \ln \left (x \right )}{4}+\frac {{\mathrm e}^{4} \left ({\mathrm e}^{3}+1\right ) \ln \left (x \right )^{2} x}{4}+{\mathrm e}^{2} \ln \left (x \right )^{2} {\mathrm e}^{3} x^{2}+5 \,{\mathrm e}^{2} {\mathrm e}^{3}+4 \,{\mathrm e}^{3} \ln \left (x \right ) x^{2}}{\ln \left (x \right ) \left ({\mathrm e}^{3} {\mathrm e}^{2} x \ln \left (x \right )-{\mathrm e}^{2} {\mathrm e}^{3}+4 x \,{\mathrm e}^{3}-{\mathrm e}^{2}\right )}\) | \(94\) |
norman | \(\frac {\frac {{\mathrm e}^{4} {\mathrm e}^{-3} \left ({\mathrm e}^{6}+2 \,{\mathrm e}^{3}+1\right ) \ln \left (x \right )}{4}+\left (-\frac {{\mathrm e}^{3} {\mathrm e}^{4}}{4}-\frac {{\mathrm e}^{4}}{4}\right ) x \ln \left (x \right )^{2}-5 \,{\mathrm e}^{2} {\mathrm e}^{3}-4 \,{\mathrm e}^{3} \ln \left (x \right ) x^{2}-{\mathrm e}^{2} \ln \left (x \right )^{2} {\mathrm e}^{3} x^{2}}{\ln \left (x \right ) \left ({\mathrm e}^{3} {\mathrm e}^{2} x \ln \left (x \right )-{\mathrm e}^{2} {\mathrm e}^{3}+4 x \,{\mathrm e}^{3}-{\mathrm e}^{2}\right )}\) | \(104\) |
parallelrisch | \(-\frac {\left ({\mathrm e}^{4} \ln \left (x \right )^{2} {\mathrm e}^{6} x +4 \,{\mathrm e}^{2} \ln \left (x \right )^{2} {\mathrm e}^{6} x^{2}+{\mathrm e}^{4} \ln \left (x \right )^{2} {\mathrm e}^{3} x -{\mathrm e}^{4} {\mathrm e}^{6} \ln \left (x \right )+16 \ln \left (x \right ) {\mathrm e}^{6} x^{2}-2 \,{\mathrm e}^{3} \ln \left (x \right ) {\mathrm e}^{4}-{\mathrm e}^{4} \ln \left (x \right )+20 \,{\mathrm e}^{2} {\mathrm e}^{6}\right ) {\mathrm e}^{-3}}{4 \ln \left (x \right ) \left ({\mathrm e}^{3} {\mathrm e}^{2} x \ln \left (x \right )-{\mathrm e}^{2} {\mathrm e}^{3}+4 x \,{\mathrm e}^{3}-{\mathrm e}^{2}\right )}\) | \(128\) |
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (27) = 54\).
Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.18 \[ \int \frac {-5 e^7+e^6 \left (-5 e^4+20 e^2 x\right )+e^6 \left (20 e^2 x+10 e^4 x\right ) \log (x)+\left (-e^4 x+e^3 \left (-2 e^4 x+8 e^2 x^2\right )+e^6 \left (4 e^4 x+8 e^2 x^2-16 x^3\right )\right ) \log ^2(x)+\left (2 e^7 x^2+e^6 \left (2 e^4 x^2-8 e^2 x^3\right )\right ) \log ^3(x)-e^{10} x^3 \log ^4(x)}{\left (e^4 x+e^3 \left (2 e^4 x-8 e^2 x^2\right )+e^6 \left (e^4 x-8 e^2 x^2+16 x^3\right )\right ) \log ^2(x)+\left (-2 e^7 x^2+e^6 \left (-2 e^4 x^2+8 e^2 x^3\right )\right ) \log ^3(x)+e^{10} x^3 \log ^4(x)} \, dx=-\frac {x^{2} e^{3} \log \left (x\right )^{2} + {\left (4 \, x^{2} e - x e^{3} - x\right )} \log \left (x\right ) + 5 \, e^{3}}{x e^{3} \log \left (x\right )^{2} + {\left (4 \, x e - e^{3} - 1\right )} \log \left (x\right )} \]
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Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-5 e^7+e^6 \left (-5 e^4+20 e^2 x\right )+e^6 \left (20 e^2 x+10 e^4 x\right ) \log (x)+\left (-e^4 x+e^3 \left (-2 e^4 x+8 e^2 x^2\right )+e^6 \left (4 e^4 x+8 e^2 x^2-16 x^3\right )\right ) \log ^2(x)+\left (2 e^7 x^2+e^6 \left (2 e^4 x^2-8 e^2 x^3\right )\right ) \log ^3(x)-e^{10} x^3 \log ^4(x)}{\left (e^4 x+e^3 \left (2 e^4 x-8 e^2 x^2\right )+e^6 \left (e^4 x-8 e^2 x^2+16 x^3\right )\right ) \log ^2(x)+\left (-2 e^7 x^2+e^6 \left (-2 e^4 x^2+8 e^2 x^3\right )\right ) \log ^3(x)+e^{10} x^3 \log ^4(x)} \, dx=- x - \frac {5 e^{3}}{x e^{3} \log {\left (x \right )}^{2} + \left (4 e x - e^{3} - 1\right ) \log {\left (x \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (27) = 54\).
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {-5 e^7+e^6 \left (-5 e^4+20 e^2 x\right )+e^6 \left (20 e^2 x+10 e^4 x\right ) \log (x)+\left (-e^4 x+e^3 \left (-2 e^4 x+8 e^2 x^2\right )+e^6 \left (4 e^4 x+8 e^2 x^2-16 x^3\right )\right ) \log ^2(x)+\left (2 e^7 x^2+e^6 \left (2 e^4 x^2-8 e^2 x^3\right )\right ) \log ^3(x)-e^{10} x^3 \log ^4(x)}{\left (e^4 x+e^3 \left (2 e^4 x-8 e^2 x^2\right )+e^6 \left (e^4 x-8 e^2 x^2+16 x^3\right )\right ) \log ^2(x)+\left (-2 e^7 x^2+e^6 \left (-2 e^4 x^2+8 e^2 x^3\right )\right ) \log ^3(x)+e^{10} x^3 \log ^4(x)} \, dx=-\frac {x^{2} e^{3} \log \left (x\right )^{2} + {\left (4 \, x^{2} e - x {\left (e^{3} + 1\right )}\right )} \log \left (x\right ) + 5 \, e^{3}}{x e^{3} \log \left (x\right )^{2} + {\left (4 \, x e - e^{3} - 1\right )} \log \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (27) = 54\).
Time = 0.44 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int \frac {-5 e^7+e^6 \left (-5 e^4+20 e^2 x\right )+e^6 \left (20 e^2 x+10 e^4 x\right ) \log (x)+\left (-e^4 x+e^3 \left (-2 e^4 x+8 e^2 x^2\right )+e^6 \left (4 e^4 x+8 e^2 x^2-16 x^3\right )\right ) \log ^2(x)+\left (2 e^7 x^2+e^6 \left (2 e^4 x^2-8 e^2 x^3\right )\right ) \log ^3(x)-e^{10} x^3 \log ^4(x)}{\left (e^4 x+e^3 \left (2 e^4 x-8 e^2 x^2\right )+e^6 \left (e^4 x-8 e^2 x^2+16 x^3\right )\right ) \log ^2(x)+\left (-2 e^7 x^2+e^6 \left (-2 e^4 x^2+8 e^2 x^3\right )\right ) \log ^3(x)+e^{10} x^3 \log ^4(x)} \, dx=-\frac {x^{2} e^{3} \log \left (x\right )^{2} + 4 \, x^{2} e \log \left (x\right ) - x e^{3} \log \left (x\right ) - x \log \left (x\right ) + 5 \, e^{3}}{x e^{3} \log \left (x\right )^{2} + 4 \, x e \log \left (x\right ) - e^{3} \log \left (x\right ) - \log \left (x\right )} \]
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Time = 14.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-5 e^7+e^6 \left (-5 e^4+20 e^2 x\right )+e^6 \left (20 e^2 x+10 e^4 x\right ) \log (x)+\left (-e^4 x+e^3 \left (-2 e^4 x+8 e^2 x^2\right )+e^6 \left (4 e^4 x+8 e^2 x^2-16 x^3\right )\right ) \log ^2(x)+\left (2 e^7 x^2+e^6 \left (2 e^4 x^2-8 e^2 x^3\right )\right ) \log ^3(x)-e^{10} x^3 \log ^4(x)}{\left (e^4 x+e^3 \left (2 e^4 x-8 e^2 x^2\right )+e^6 \left (e^4 x-8 e^2 x^2+16 x^3\right )\right ) \log ^2(x)+\left (-2 e^7 x^2+e^6 \left (-2 e^4 x^2+8 e^2 x^3\right )\right ) \log ^3(x)+e^{10} x^3 \log ^4(x)} \, dx=-x-\frac {5}{x\,\left ({\ln \left (x\right )}^2-\frac {{\mathrm {e}}^{-3}\,\ln \left (x\right )\,\left ({\mathrm {e}}^3-4\,x\,\mathrm {e}+1\right )}{x}\right )} \]
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