Integrand size = 176, antiderivative size = 31 \[ \int \frac {40+e^{-5+x} (-20-10 x)+20 x+\left (-40-40 x+e^{-5+x} \left (20+20 x+10 x^2+5 x^3\right )\right ) \log (x)+\log \left (2-e^{-5+x}\right ) \left (-20 x-10 x^2+e^{-5+x} \left (10 x+5 x^2\right )+\left (10 x^2-5 e^{-5+x} x^2\right ) \log (x)\right )}{\left (-8+4 e^{-5+x}\right ) \log ^2(x)+\left (8 x-4 e^{-5+x} x\right ) \log \left (2-e^{-5+x}\right ) \log ^2(x)+\left (-2 x^2+e^{-5+x} x^2\right ) \log ^2\left (2-e^{-5+x}\right ) \log ^2(x)} \, dx=4+\frac {5 (2+x)}{\left (\frac {2}{x}-\log \left (2-e^{-5+x}\right )\right ) \log (x)} \]
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\[ \int \frac {40+e^{-5+x} (-20-10 x)+20 x+\left (-40-40 x+e^{-5+x} \left (20+20 x+10 x^2+5 x^3\right )\right ) \log (x)+\log \left (2-e^{-5+x}\right ) \left (-20 x-10 x^2+e^{-5+x} \left (10 x+5 x^2\right )+\left (10 x^2-5 e^{-5+x} x^2\right ) \log (x)\right )}{\left (-8+4 e^{-5+x}\right ) \log ^2(x)+\left (8 x-4 e^{-5+x} x\right ) \log \left (2-e^{-5+x}\right ) \log ^2(x)+\left (-2 x^2+e^{-5+x} x^2\right ) \log ^2\left (2-e^{-5+x}\right ) \log ^2(x)} \, dx=\int \frac {40+e^{-5+x} (-20-10 x)+20 x+\left (-40-40 x+e^{-5+x} \left (20+20 x+10 x^2+5 x^3\right )\right ) \log (x)+\log \left (2-e^{-5+x}\right ) \left (-20 x-10 x^2+e^{-5+x} \left (10 x+5 x^2\right )+\left (10 x^2-5 e^{-5+x} x^2\right ) \log (x)\right )}{\left (-8+4 e^{-5+x}\right ) \log ^2(x)+\left (8 x-4 e^{-5+x} x\right ) \log \left (2-e^{-5+x}\right ) \log ^2(x)+\left (-2 x^2+e^{-5+x} x^2\right ) \log ^2\left (2-e^{-5+x}\right ) \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {5 \left (2 \left (-2 e^5+e^x\right ) (2+x)-\left (-8 e^5 (1+x)+e^x \left (4+4 x+2 x^2+x^3\right )\right ) \log (x)+\left (-2 e^5+e^x\right ) x \log \left (2-e^{-5+x}\right ) (-2-x+x \log (x))\right )}{\left (2 e^5-e^x\right ) \left (2-x \log \left (2-e^{-5+x}\right )\right )^2 \log ^2(x)} \, dx \\ & = 5 \int \frac {2 \left (-2 e^5+e^x\right ) (2+x)-\left (-8 e^5 (1+x)+e^x \left (4+4 x+2 x^2+x^3\right )\right ) \log (x)+\left (-2 e^5+e^x\right ) x \log \left (2-e^{-5+x}\right ) (-2-x+x \log (x))}{\left (2 e^5-e^x\right ) \left (2-x \log \left (2-e^{-5+x}\right )\right )^2 \log ^2(x)} \, dx \\ & = 5 \int \left (-\frac {2 e^5 x^2 (2+x)}{\left (2 e^5-e^x\right ) \left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)}+\frac {-4-2 x+2 x \log \left (2-e^{-5+x}\right )+x^2 \log \left (2-e^{-5+x}\right )+4 \log (x)+4 x \log (x)+2 x^2 \log (x)+x^3 \log (x)-x^2 \log \left (2-e^{-5+x}\right ) \log (x)}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log ^2(x)}\right ) \, dx \\ & = 5 \int \frac {-4-2 x+2 x \log \left (2-e^{-5+x}\right )+x^2 \log \left (2-e^{-5+x}\right )+4 \log (x)+4 x \log (x)+2 x^2 \log (x)+x^3 \log (x)-x^2 \log \left (2-e^{-5+x}\right ) \log (x)}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log ^2(x)} \, dx-\left (10 e^5\right ) \int \frac {x^2 (2+x)}{\left (2 e^5-e^x\right ) \left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx \\ & = 5 \int \frac {-2 (2+x)+\left (4+4 x+2 x^2+x^3\right ) \log (x)+x \log \left (2-e^{-5+x}\right ) (2+x-x \log (x))}{\left (2-x \log \left (2-e^{-5+x}\right )\right )^2 \log ^2(x)} \, dx-\left (10 e^5\right ) \int \left (-\frac {2 x^2}{\left (-2 e^5+e^x\right ) \left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)}-\frac {x^3}{\left (-2 e^5+e^x\right ) \left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)}\right ) \, dx \\ & = 5 \int \left (\frac {2+x}{\left (-2+x \log \left (2-e^{-5+x}\right )\right ) \log ^2(x)}+\frac {4+4 x+2 x^2+x^3-x^2 \log \left (2-e^{-5+x}\right )}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)}\right ) \, dx+\left (10 e^5\right ) \int \frac {x^3}{\left (-2 e^5+e^x\right ) \left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx+\left (20 e^5\right ) \int \frac {x^2}{\left (-2 e^5+e^x\right ) \left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx \\ & = 5 \int \frac {2+x}{\left (-2+x \log \left (2-e^{-5+x}\right )\right ) \log ^2(x)} \, dx+5 \int \frac {4+4 x+2 x^2+x^3-x^2 \log \left (2-e^{-5+x}\right )}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx+\left (10 e^5\right ) \int \frac {x^3}{\left (-2 e^5+e^x\right ) \left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx+\left (20 e^5\right ) \int \frac {x^2}{\left (-2 e^5+e^x\right ) \left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx \\ & = 5 \int \left (\frac {2}{\left (-2+x \log \left (2-e^{-5+x}\right )\right ) \log ^2(x)}+\frac {x}{\left (-2+x \log \left (2-e^{-5+x}\right )\right ) \log ^2(x)}\right ) \, dx+5 \int \left (\frac {4}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)}+\frac {4 x}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)}+\frac {2 x^2}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)}+\frac {x^3}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)}-\frac {x^2 \log \left (2-e^{-5+x}\right )}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)}\right ) \, dx+\left (10 e^5\right ) \int \frac {x^3}{\left (-2 e^5+e^x\right ) \left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx+\left (20 e^5\right ) \int \frac {x^2}{\left (-2 e^5+e^x\right ) \left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx \\ & = 5 \int \frac {x}{\left (-2+x \log \left (2-e^{-5+x}\right )\right ) \log ^2(x)} \, dx+5 \int \frac {x^3}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx-5 \int \frac {x^2 \log \left (2-e^{-5+x}\right )}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx+10 \int \frac {1}{\left (-2+x \log \left (2-e^{-5+x}\right )\right ) \log ^2(x)} \, dx+10 \int \frac {x^2}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx+20 \int \frac {1}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx+20 \int \frac {x}{\left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx+\left (10 e^5\right ) \int \frac {x^3}{\left (-2 e^5+e^x\right ) \left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx+\left (20 e^5\right ) \int \frac {x^2}{\left (-2 e^5+e^x\right ) \left (-2+x \log \left (2-e^{-5+x}\right )\right )^2 \log (x)} \, dx \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {40+e^{-5+x} (-20-10 x)+20 x+\left (-40-40 x+e^{-5+x} \left (20+20 x+10 x^2+5 x^3\right )\right ) \log (x)+\log \left (2-e^{-5+x}\right ) \left (-20 x-10 x^2+e^{-5+x} \left (10 x+5 x^2\right )+\left (10 x^2-5 e^{-5+x} x^2\right ) \log (x)\right )}{\left (-8+4 e^{-5+x}\right ) \log ^2(x)+\left (8 x-4 e^{-5+x} x\right ) \log \left (2-e^{-5+x}\right ) \log ^2(x)+\left (-2 x^2+e^{-5+x} x^2\right ) \log ^2\left (2-e^{-5+x}\right ) \log ^2(x)} \, dx=-\frac {5 x (2+x)}{\left (-2+x \log \left (2-e^{-5+x}\right )\right ) \log (x)} \]
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Time = 5.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84
method | result | size |
risch | \(-\frac {5 x \left (2+x \right )}{\ln \left (x \right ) \left (x \ln \left (-{\mathrm e}^{-5+x}+2\right )-2\right )}\) | \(26\) |
parallelrisch | \(\frac {-40 x^{2}-80 x}{8 \left (x \ln \left (-{\mathrm e}^{-5+x}+2\right )-2\right ) \ln \left (x \right )}\) | \(31\) |
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {40+e^{-5+x} (-20-10 x)+20 x+\left (-40-40 x+e^{-5+x} \left (20+20 x+10 x^2+5 x^3\right )\right ) \log (x)+\log \left (2-e^{-5+x}\right ) \left (-20 x-10 x^2+e^{-5+x} \left (10 x+5 x^2\right )+\left (10 x^2-5 e^{-5+x} x^2\right ) \log (x)\right )}{\left (-8+4 e^{-5+x}\right ) \log ^2(x)+\left (8 x-4 e^{-5+x} x\right ) \log \left (2-e^{-5+x}\right ) \log ^2(x)+\left (-2 x^2+e^{-5+x} x^2\right ) \log ^2\left (2-e^{-5+x}\right ) \log ^2(x)} \, dx=-\frac {5 \, {\left (x^{2} + 2 \, x\right )}}{x \log \left (x\right ) \log \left (-e^{\left (x - 5\right )} + 2\right ) - 2 \, \log \left (x\right )} \]
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Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {40+e^{-5+x} (-20-10 x)+20 x+\left (-40-40 x+e^{-5+x} \left (20+20 x+10 x^2+5 x^3\right )\right ) \log (x)+\log \left (2-e^{-5+x}\right ) \left (-20 x-10 x^2+e^{-5+x} \left (10 x+5 x^2\right )+\left (10 x^2-5 e^{-5+x} x^2\right ) \log (x)\right )}{\left (-8+4 e^{-5+x}\right ) \log ^2(x)+\left (8 x-4 e^{-5+x} x\right ) \log \left (2-e^{-5+x}\right ) \log ^2(x)+\left (-2 x^2+e^{-5+x} x^2\right ) \log ^2\left (2-e^{-5+x}\right ) \log ^2(x)} \, dx=\frac {- 5 x^{2} - 10 x}{x \log {\left (x \right )} \log {\left (2 - e^{x - 5} \right )} - 2 \log {\left (x \right )}} \]
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {40+e^{-5+x} (-20-10 x)+20 x+\left (-40-40 x+e^{-5+x} \left (20+20 x+10 x^2+5 x^3\right )\right ) \log (x)+\log \left (2-e^{-5+x}\right ) \left (-20 x-10 x^2+e^{-5+x} \left (10 x+5 x^2\right )+\left (10 x^2-5 e^{-5+x} x^2\right ) \log (x)\right )}{\left (-8+4 e^{-5+x}\right ) \log ^2(x)+\left (8 x-4 e^{-5+x} x\right ) \log \left (2-e^{-5+x}\right ) \log ^2(x)+\left (-2 x^2+e^{-5+x} x^2\right ) \log ^2\left (2-e^{-5+x}\right ) \log ^2(x)} \, dx=-\frac {5 \, {\left (x^{2} + 2 \, x\right )}}{x \log \left (x\right ) \log \left (2 \, e^{5} - e^{x}\right ) - {\left (5 \, x + 2\right )} \log \left (x\right )} \]
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Time = 0.50 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {40+e^{-5+x} (-20-10 x)+20 x+\left (-40-40 x+e^{-5+x} \left (20+20 x+10 x^2+5 x^3\right )\right ) \log (x)+\log \left (2-e^{-5+x}\right ) \left (-20 x-10 x^2+e^{-5+x} \left (10 x+5 x^2\right )+\left (10 x^2-5 e^{-5+x} x^2\right ) \log (x)\right )}{\left (-8+4 e^{-5+x}\right ) \log ^2(x)+\left (8 x-4 e^{-5+x} x\right ) \log \left (2-e^{-5+x}\right ) \log ^2(x)+\left (-2 x^2+e^{-5+x} x^2\right ) \log ^2\left (2-e^{-5+x}\right ) \log ^2(x)} \, dx=-\frac {5 \, {\left (x^{2} + 2 \, x\right )}}{x \log \left (x\right ) \log \left (2 \, e^{5} - e^{x}\right ) - 5 \, x \log \left (x\right ) - 2 \, \log \left (x\right )} \]
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Time = 9.15 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.61 \[ \int \frac {40+e^{-5+x} (-20-10 x)+20 x+\left (-40-40 x+e^{-5+x} \left (20+20 x+10 x^2+5 x^3\right )\right ) \log (x)+\log \left (2-e^{-5+x}\right ) \left (-20 x-10 x^2+e^{-5+x} \left (10 x+5 x^2\right )+\left (10 x^2-5 e^{-5+x} x^2\right ) \log (x)\right )}{\left (-8+4 e^{-5+x}\right ) \log ^2(x)+\left (8 x-4 e^{-5+x} x\right ) \log \left (2-e^{-5+x}\right ) \log ^2(x)+\left (-2 x^2+e^{-5+x} x^2\right ) \log ^2\left (2-e^{-5+x}\right ) \log ^2(x)} \, dx=-\frac {80\,x-80\,x\,{\mathrm {e}}^{x-5}+20\,x\,{\mathrm {e}}^{2\,x-10}-40\,x^2\,{\mathrm {e}}^{x-5}-20\,x^3\,{\mathrm {e}}^{x-5}-10\,x^4\,{\mathrm {e}}^{x-5}+10\,x^2\,{\mathrm {e}}^{2\,x-10}+10\,x^3\,{\mathrm {e}}^{2\,x-10}+5\,x^4\,{\mathrm {e}}^{2\,x-10}+40\,x^2}{\ln \left (x\right )\,\left (x\,\ln \left (2-{\mathrm {e}}^{-5}\,{\mathrm {e}}^x\right )-2\right )\,\left (2\,{\mathrm {e}}^{2\,x-10}-8\,{\mathrm {e}}^{x-5}-2\,x^2\,{\mathrm {e}}^{x-5}+x^2\,{\mathrm {e}}^{2\,x-10}+8\right )} \]
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