Integrand size = 266, antiderivative size = 27 \[ \int \frac {-4+4 x+7 x^2-8 x^3-2 x^4-e^{2 x} x^4+4 x^5-x^6+e^x \left (-4 x^2+2 x^3+4 x^4-2 x^5\right )+\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log (x)+\left (6-4 x-5 x^2+6 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (3 x^2-3 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)}{\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)} \, dx=x+\frac {x}{2-x+x^2 \left (-2+e^x+x\right )}+\frac {x}{\log (x)} \]
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\[ \int \frac {-4+4 x+7 x^2-8 x^3-2 x^4-e^{2 x} x^4+4 x^5-x^6+e^x \left (-4 x^2+2 x^3+4 x^4-2 x^5\right )+\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log (x)+\left (6-4 x-5 x^2+6 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (3 x^2-3 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)}{\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)} \, dx=\int \frac {-4+4 x+7 x^2-8 x^3-2 x^4-e^{2 x} x^4+4 x^5-x^6+e^x \left (-4 x^2+2 x^3+4 x^4-2 x^5\right )+\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log (x)+\left (6-4 x-5 x^2+6 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (3 x^2-3 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)}{\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {6-4 x+\left (-5+3 e^x\right ) x^2-3 \left (-2+e^x\right ) x^3+\left (2-4 e^x+e^{2 x}\right ) x^4+2 \left (-2+e^x\right ) x^5+x^6}{\left (2-x+\left (-2+e^x\right ) x^2+x^3\right )^2}-\frac {1}{\log ^2(x)}+\frac {1}{\log (x)}\right ) \, dx \\ & = \int \frac {6-4 x+\left (-5+3 e^x\right ) x^2-3 \left (-2+e^x\right ) x^3+\left (2-4 e^x+e^{2 x}\right ) x^4+2 \left (-2+e^x\right ) x^5+x^6}{\left (2-x+\left (-2+e^x\right ) x^2+x^3\right )^2} \, dx-\int \frac {1}{\log ^2(x)} \, dx+\int \frac {1}{\log (x)} \, dx \\ & = \frac {x}{\log (x)}+\operatorname {LogIntegral}(x)+\int \left (1-\frac {1+x}{2-x-2 x^2+e^x x^2+x^3}+\frac {4+x-x^2-3 x^3+x^4}{\left (2-x-2 x^2+e^x x^2+x^3\right )^2}\right ) \, dx-\int \frac {1}{\log (x)} \, dx \\ & = x+\frac {x}{\log (x)}-\int \frac {1+x}{2-x-2 x^2+e^x x^2+x^3} \, dx+\int \frac {4+x-x^2-3 x^3+x^4}{\left (2-x-2 x^2+e^x x^2+x^3\right )^2} \, dx \\ & = x+\frac {x}{\log (x)}+\int \left (\frac {4}{\left (2-x-2 x^2+e^x x^2+x^3\right )^2}+\frac {x}{\left (2-x-2 x^2+e^x x^2+x^3\right )^2}-\frac {x^2}{\left (2-x-2 x^2+e^x x^2+x^3\right )^2}-\frac {3 x^3}{\left (2-x-2 x^2+e^x x^2+x^3\right )^2}+\frac {x^4}{\left (2-x-2 x^2+e^x x^2+x^3\right )^2}\right ) \, dx-\int \left (\frac {1}{2-x-2 x^2+e^x x^2+x^3}+\frac {x}{2-x-2 x^2+e^x x^2+x^3}\right ) \, dx \\ & = x+\frac {x}{\log (x)}-3 \int \frac {x^3}{\left (2-x-2 x^2+e^x x^2+x^3\right )^2} \, dx+4 \int \frac {1}{\left (2-x-2 x^2+e^x x^2+x^3\right )^2} \, dx+\int \frac {x}{\left (2-x-2 x^2+e^x x^2+x^3\right )^2} \, dx-\int \frac {x^2}{\left (2-x-2 x^2+e^x x^2+x^3\right )^2} \, dx+\int \frac {x^4}{\left (2-x-2 x^2+e^x x^2+x^3\right )^2} \, dx-\int \frac {1}{2-x-2 x^2+e^x x^2+x^3} \, dx-\int \frac {x}{2-x-2 x^2+e^x x^2+x^3} \, dx \\ \end{align*}
Time = 6.70 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {-4+4 x+7 x^2-8 x^3-2 x^4-e^{2 x} x^4+4 x^5-x^6+e^x \left (-4 x^2+2 x^3+4 x^4-2 x^5\right )+\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log (x)+\left (6-4 x-5 x^2+6 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (3 x^2-3 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)}{\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)} \, dx=x+\frac {x}{2-x-2 x^2+e^x x^2+x^3}+\frac {x}{\log (x)} \]
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Time = 0.80 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85
method | result | size |
risch | \(\frac {x \left (x^{3}+{\mathrm e}^{x} x^{2}-2 x^{2}-x +3\right )}{{\mathrm e}^{x} x^{2}+x^{3}-2 x^{2}-x +2}+\frac {x}{\ln \left (x \right )}\) | \(50\) |
parallelrisch | \(\frac {2 x +x^{4} \ln \left (x \right )+{\mathrm e}^{x} x^{3}+3 x \ln \left (x \right )+x^{3} {\mathrm e}^{x} \ln \left (x \right )+x^{4}-2 x^{3}-x^{2}-2 x^{3} \ln \left (x \right )-x^{2} \ln \left (x \right )}{\ln \left (x \right ) \left ({\mathrm e}^{x} x^{2}+x^{3}-2 x^{2}-x +2\right )}\) | \(83\) |
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78 \[ \int \frac {-4+4 x+7 x^2-8 x^3-2 x^4-e^{2 x} x^4+4 x^5-x^6+e^x \left (-4 x^2+2 x^3+4 x^4-2 x^5\right )+\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log (x)+\left (6-4 x-5 x^2+6 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (3 x^2-3 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)}{\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)} \, dx=\frac {x^{4} + x^{3} e^{x} - 2 \, x^{3} - x^{2} + {\left (x^{4} + x^{3} e^{x} - 2 \, x^{3} - x^{2} + 3 \, x\right )} \log \left (x\right ) + 2 \, x}{{\left (x^{3} + x^{2} e^{x} - 2 \, x^{2} - x + 2\right )} \log \left (x\right )} \]
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Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {-4+4 x+7 x^2-8 x^3-2 x^4-e^{2 x} x^4+4 x^5-x^6+e^x \left (-4 x^2+2 x^3+4 x^4-2 x^5\right )+\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log (x)+\left (6-4 x-5 x^2+6 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (3 x^2-3 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)}{\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)} \, dx=x + \frac {x}{\log {\left (x \right )}} + \frac {x}{x^{3} + x^{2} e^{x} - 2 x^{2} - x + 2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (26) = 52\).
Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.89 \[ \int \frac {-4+4 x+7 x^2-8 x^3-2 x^4-e^{2 x} x^4+4 x^5-x^6+e^x \left (-4 x^2+2 x^3+4 x^4-2 x^5\right )+\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log (x)+\left (6-4 x-5 x^2+6 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (3 x^2-3 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)}{\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)} \, dx=\frac {x^{4} - 2 \, x^{3} - x^{2} + {\left (x^{3} \log \left (x\right ) + x^{3}\right )} e^{x} + {\left (x^{4} - 2 \, x^{3} - x^{2} + 3 \, x\right )} \log \left (x\right ) + 2 \, x}{x^{2} e^{x} \log \left (x\right ) + {\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (26) = 52\).
Time = 0.38 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.33 \[ \int \frac {-4+4 x+7 x^2-8 x^3-2 x^4-e^{2 x} x^4+4 x^5-x^6+e^x \left (-4 x^2+2 x^3+4 x^4-2 x^5\right )+\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log (x)+\left (6-4 x-5 x^2+6 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (3 x^2-3 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)}{\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)} \, dx=\frac {x^{4} \log \left (x\right ) + x^{3} e^{x} \log \left (x\right ) + x^{4} + x^{3} e^{x} - 2 \, x^{3} \log \left (x\right ) - 2 \, x^{3} - x^{2} \log \left (x\right ) - x^{2} + 3 \, x \log \left (x\right ) + 2 \, x}{x^{3} \log \left (x\right ) + x^{2} e^{x} \log \left (x\right ) - 2 \, x^{2} \log \left (x\right ) - x \log \left (x\right ) + 2 \, \log \left (x\right )} \]
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Time = 14.94 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81 \[ \int \frac {-4+4 x+7 x^2-8 x^3-2 x^4-e^{2 x} x^4+4 x^5-x^6+e^x \left (-4 x^2+2 x^3+4 x^4-2 x^5\right )+\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log (x)+\left (6-4 x-5 x^2+6 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (3 x^2-3 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)}{\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)} \, dx=\frac {x\,\left (3\,\ln \left (x\right )-x+x^2\,{\mathrm {e}}^x-2\,x^2\,\ln \left (x\right )+x^3\,\ln \left (x\right )-x\,\ln \left (x\right )-2\,x^2+x^3+x^2\,{\mathrm {e}}^x\,\ln \left (x\right )+2\right )}{\ln \left (x\right )\,\left (x^2\,{\mathrm {e}}^x-x-2\,x^2+x^3+2\right )} \]
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