Integrand size = 187, antiderivative size = 24 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=\frac {4}{125 \left (2-e^x\right ) x^5 (4+\log (5+x))} \]
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\[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=\int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {4 \left (-200-42 x+e^x \left (100+41 x+4 x^2\right )+(5+x) \left (-10+e^x (5+x)\right ) \log (5+x)\right )}{125 \left (2-e^x\right )^2 x^6 (5+x) (4+\log (5+x))^2} \, dx \\ & = \frac {4}{125} \int \frac {-200-42 x+e^x \left (100+41 x+4 x^2\right )+(5+x) \left (-10+e^x (5+x)\right ) \log (5+x)}{\left (2-e^x\right )^2 x^6 (5+x) (4+\log (5+x))^2} \, dx \\ & = \frac {4}{125} \int \left (\frac {2}{\left (-2+e^x\right )^2 x^5 (4+\log (5+x))}+\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^6 (5+x) (4+\log (5+x))^2}\right ) \, dx \\ & = \frac {4}{125} \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^6 (5+x) (4+\log (5+x))^2} \, dx+\frac {8}{125} \int \frac {1}{\left (-2+e^x\right )^2 x^5 (4+\log (5+x))} \, dx \\ & = \frac {4}{125} \int \left (\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{5 \left (-2+e^x\right ) x^6 (4+\log (5+x))^2}-\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{25 \left (-2+e^x\right ) x^5 (4+\log (5+x))^2}+\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{125 \left (-2+e^x\right ) x^4 (4+\log (5+x))^2}-\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{625 \left (-2+e^x\right ) x^3 (4+\log (5+x))^2}+\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{3125 \left (-2+e^x\right ) x^2 (4+\log (5+x))^2}-\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{15625 \left (-2+e^x\right ) x (4+\log (5+x))^2}+\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{15625 \left (-2+e^x\right ) (5+x) (4+\log (5+x))^2}\right ) \, dx+\frac {8}{125} \int \frac {1}{\left (-2+e^x\right )^2 x^5 (4+\log (5+x))} \, dx \\ & = -\frac {4 \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x (4+\log (5+x))^2} \, dx}{1953125}+\frac {4 \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2} \, dx}{1953125}+\frac {4 \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^2 (4+\log (5+x))^2} \, dx}{390625}-\frac {4 \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^3 (4+\log (5+x))^2} \, dx}{78125}+\frac {4 \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^4 (4+\log (5+x))^2} \, dx}{15625}-\frac {4 \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^5 (4+\log (5+x))^2} \, dx}{3125}+\frac {4}{625} \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^6 (4+\log (5+x))^2} \, dx+\frac {8}{125} \int \frac {1}{\left (-2+e^x\right )^2 x^5 (4+\log (5+x))} \, dx \\ & = -\frac {4 \int \left (\frac {41}{\left (-2+e^x\right ) (4+\log (5+x))^2}+\frac {100}{\left (-2+e^x\right ) x (4+\log (5+x))^2}+\frac {4 x}{\left (-2+e^x\right ) (4+\log (5+x))^2}+\frac {10 \log (5+x)}{\left (-2+e^x\right ) (4+\log (5+x))^2}+\frac {25 \log (5+x)}{\left (-2+e^x\right ) x (4+\log (5+x))^2}+\frac {x \log (5+x)}{\left (-2+e^x\right ) (4+\log (5+x))^2}\right ) \, dx}{1953125}+\frac {4 \int \left (\frac {100}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2}+\frac {41 x}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2}+\frac {4 x^2}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2}+\frac {25 \log (5+x)}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2}+\frac {10 x \log (5+x)}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2}+\frac {x^2 \log (5+x)}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2}\right ) \, dx}{1953125}+\frac {4 \int \left (\frac {4}{\left (-2+e^x\right ) (4+\log (5+x))^2}+\frac {100}{\left (-2+e^x\right ) x^2 (4+\log (5+x))^2}+\frac {41}{\left (-2+e^x\right ) x (4+\log (5+x))^2}+\frac {\log (5+x)}{\left (-2+e^x\right ) (4+\log (5+x))^2}+\frac {25 \log (5+x)}{\left (-2+e^x\right ) x^2 (4+\log (5+x))^2}+\frac {10 \log (5+x)}{\left (-2+e^x\right ) x (4+\log (5+x))^2}\right ) \, dx}{390625}-\frac {4 \int \left (\frac {100}{\left (-2+e^x\right ) x^3 (4+\log (5+x))^2}+\frac {41}{\left (-2+e^x\right ) x^2 (4+\log (5+x))^2}+\frac {4}{\left (-2+e^x\right ) x (4+\log (5+x))^2}+\frac {25 \log (5+x)}{\left (-2+e^x\right ) x^3 (4+\log (5+x))^2}+\frac {10 \log (5+x)}{\left (-2+e^x\right ) x^2 (4+\log (5+x))^2}+\frac {\log (5+x)}{\left (-2+e^x\right ) x (4+\log (5+x))^2}\right ) \, dx}{78125}+\frac {4 \int \left (\frac {100}{\left (-2+e^x\right ) x^4 (4+\log (5+x))^2}+\frac {41}{\left (-2+e^x\right ) x^3 (4+\log (5+x))^2}+\frac {4}{\left (-2+e^x\right ) x^2 (4+\log (5+x))^2}+\frac {25 \log (5+x)}{\left (-2+e^x\right ) x^4 (4+\log (5+x))^2}+\frac {10 \log (5+x)}{\left (-2+e^x\right ) x^3 (4+\log (5+x))^2}+\frac {\log (5+x)}{\left (-2+e^x\right ) x^2 (4+\log (5+x))^2}\right ) \, dx}{15625}-\frac {4 \int \left (\frac {100}{\left (-2+e^x\right ) x^5 (4+\log (5+x))^2}+\frac {41}{\left (-2+e^x\right ) x^4 (4+\log (5+x))^2}+\frac {4}{\left (-2+e^x\right ) x^3 (4+\log (5+x))^2}+\frac {25 \log (5+x)}{\left (-2+e^x\right ) x^5 (4+\log (5+x))^2}+\frac {10 \log (5+x)}{\left (-2+e^x\right ) x^4 (4+\log (5+x))^2}+\frac {\log (5+x)}{\left (-2+e^x\right ) x^3 (4+\log (5+x))^2}\right ) \, dx}{3125}+\frac {4}{625} \int \left (\frac {100}{\left (-2+e^x\right ) x^6 (4+\log (5+x))^2}+\frac {41}{\left (-2+e^x\right ) x^5 (4+\log (5+x))^2}+\frac {4}{\left (-2+e^x\right ) x^4 (4+\log (5+x))^2}+\frac {25 \log (5+x)}{\left (-2+e^x\right ) x^6 (4+\log (5+x))^2}+\frac {10 \log (5+x)}{\left (-2+e^x\right ) x^5 (4+\log (5+x))^2}+\frac {\log (5+x)}{\left (-2+e^x\right ) x^4 (4+\log (5+x))^2}\right ) \, dx+\frac {8}{125} \int \frac {1}{\left (-2+e^x\right )^2 x^5 (4+\log (5+x))} \, dx \\ & = -\frac {4 \int \frac {x \log (5+x)}{\left (-2+e^x\right ) (4+\log (5+x))^2} \, dx}{1953125}+\frac {4 \int \frac {x^2 \log (5+x)}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2} \, dx}{1953125}-\frac {16 \int \frac {x}{\left (-2+e^x\right ) (4+\log (5+x))^2} \, dx}{1953125}+\frac {16 \int \frac {x^2}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2} \, dx}{1953125}+\frac {4 \int \frac {\log (5+x)}{\left (-2+e^x\right ) (4+\log (5+x))^2} \, dx}{390625}-\frac {8 \int \frac {\log (5+x)}{\left (-2+e^x\right ) (4+\log (5+x))^2} \, dx}{390625}+\frac {8 \int \frac {x \log (5+x)}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2} \, dx}{390625}+\frac {16 \int \frac {1}{\left (-2+e^x\right ) (4+\log (5+x))^2} \, dx}{390625}-2 \frac {4 \int \frac {\log (5+x)}{\left (-2+e^x\right ) x (4+\log (5+x))^2} \, dx}{78125}+\frac {4 \int \frac {\log (5+x)}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2} \, dx}{78125}-\frac {164 \int \frac {1}{\left (-2+e^x\right ) (4+\log (5+x))^2} \, dx}{1953125}+\frac {164 \int \frac {x}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2} \, dx}{1953125}+\frac {8 \int \frac {\log (5+x)}{\left (-2+e^x\right ) x (4+\log (5+x))^2} \, dx}{78125}-2 \frac {16 \int \frac {1}{\left (-2+e^x\right ) x (4+\log (5+x))^2} \, dx}{78125}+\frac {16 \int \frac {1}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2} \, dx}{78125}+2 \frac {4 \int \frac {\log (5+x)}{\left (-2+e^x\right ) x^2 (4+\log (5+x))^2} \, dx}{15625}+\frac {164 \int \frac {1}{\left (-2+e^x\right ) x (4+\log (5+x))^2} \, dx}{390625}-\frac {8 \int \frac {\log (5+x)}{\left (-2+e^x\right ) x^2 (4+\log (5+x))^2} \, dx}{15625}+2 \frac {16 \int \frac {1}{\left (-2+e^x\right ) x^2 (4+\log (5+x))^2} \, dx}{15625}-2 \frac {4 \int \frac {\log (5+x)}{\left (-2+e^x\right ) x^3 (4+\log (5+x))^2} \, dx}{3125}-\frac {164 \int \frac {1}{\left (-2+e^x\right ) x^2 (4+\log (5+x))^2} \, dx}{78125}+\frac {8 \int \frac {\log (5+x)}{\left (-2+e^x\right ) x^3 (4+\log (5+x))^2} \, dx}{3125}-2 \frac {16 \int \frac {1}{\left (-2+e^x\right ) x^3 (4+\log (5+x))^2} \, dx}{3125}+2 \left (\frac {4}{625} \int \frac {\log (5+x)}{\left (-2+e^x\right ) x^4 (4+\log (5+x))^2} \, dx\right )+\frac {164 \int \frac {1}{\left (-2+e^x\right ) x^3 (4+\log (5+x))^2} \, dx}{15625}-\frac {8}{625} \int \frac {\log (5+x)}{\left (-2+e^x\right ) x^4 (4+\log (5+x))^2} \, dx+2 \left (\frac {16}{625} \int \frac {1}{\left (-2+e^x\right ) x^4 (4+\log (5+x))^2} \, dx\right )-\frac {4}{125} \int \frac {\log (5+x)}{\left (-2+e^x\right ) x^5 (4+\log (5+x))^2} \, dx-\frac {164 \int \frac {1}{\left (-2+e^x\right ) x^4 (4+\log (5+x))^2} \, dx}{3125}+\frac {8}{125} \int \frac {\log (5+x)}{\left (-2+e^x\right ) x^5 (4+\log (5+x))^2} \, dx+\frac {8}{125} \int \frac {1}{\left (-2+e^x\right )^2 x^5 (4+\log (5+x))} \, dx-\frac {16}{125} \int \frac {1}{\left (-2+e^x\right ) x^5 (4+\log (5+x))^2} \, dx+\frac {4}{25} \int \frac {\log (5+x)}{\left (-2+e^x\right ) x^6 (4+\log (5+x))^2} \, dx+\frac {164}{625} \int \frac {1}{\left (-2+e^x\right ) x^5 (4+\log (5+x))^2} \, dx+\frac {16}{25} \int \frac {1}{\left (-2+e^x\right ) x^6 (4+\log (5+x))^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=-\frac {4}{125 \left (-2+e^x\right ) x^5 (4+\log (5+x))} \]
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Time = 3.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-\frac {4}{125 \left ({\mathrm e}^{x}-2\right ) x^{5} \left (\ln \left (5+x \right )+4\right )}\) | \(20\) |
parallelrisch | \(-\frac {4}{125 \left ({\mathrm e}^{x}-2\right ) x^{5} \left (\ln \left (5+x \right )+4\right )}\) | \(20\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=-\frac {4}{125 \, {\left (4 \, x^{5} e^{x} - 8 \, x^{5} + {\left (x^{5} e^{x} - 2 \, x^{5}\right )} \log \left (x + 5\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=- \frac {4}{- 250 x^{5} \log {\left (x + 5 \right )} - 1000 x^{5} + \left (125 x^{5} \log {\left (x + 5 \right )} + 500 x^{5}\right ) e^{x}} \]
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Time = 0.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=-\frac {4}{125 \, {\left (4 \, x^{5} e^{x} - 8 \, x^{5} + {\left (x^{5} e^{x} - 2 \, x^{5}\right )} \log \left (x + 5\right )\right )}} \]
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Time = 0.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=-\frac {4}{125 \, {\left (x^{5} e^{x} \log \left (x + 5\right ) + 4 \, x^{5} e^{x} - 2 \, x^{5} \log \left (x + 5\right ) - 8 \, x^{5}\right )}} \]
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Time = 12.75 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=-\frac {4}{125\,x^5\,\left ({\mathrm {e}}^x-2\right )\,\left (\ln \left (x+5\right )+4\right )} \]
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