Integrand size = 123, antiderivative size = 28 \[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=e^{\frac {\left (-x+x^4\right ) \left (2+e^x+\log (\log (5))\right )}{7 (5+x)}} x \]
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\[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=\int \frac {\exp \left (\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}\right ) \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}\right ) \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{7 (5+x)^2} \, dx \\ & = \frac {1}{7} \int \frac {\exp \left (\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}\right ) \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{(5+x)^2} \, dx \\ & = \frac {1}{7} \int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{(5+x)^2} \, dx \\ & = \frac {1}{7} \int \left (\frac {175 \exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right )}{(5+x)^2}+\frac {60 \exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x}{(5+x)^2}+\frac {7 \exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x^2}{(5+x)^2}+\frac {40 \exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x^4}{(5+x)^2}+\frac {6 \exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x^5}{(5+x)^2}+\frac {\exp \left (x+\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x \left (-5-5 x-x^2+20 x^3+8 x^4+x^5\right )}{(5+x)^2}+\frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x \left (-5+20 x^3+3 x^4\right ) \log (\log (5))}{(5+x)^2}\right ) \, dx \\ & = \frac {1}{7} \int \frac {\exp \left (x+\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x \left (-5-5 x-x^2+20 x^3+8 x^4+x^5\right )}{(5+x)^2} \, dx+\frac {6}{7} \int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x^5}{(5+x)^2} \, dx+\frac {40}{7} \int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x^4}{(5+x)^2} \, dx+\frac {60}{7} \int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x}{(5+x)^2} \, dx+25 \int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right )}{(5+x)^2} \, dx+\frac {1}{7} \log (\log (5)) \int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x \left (-5+20 x^3+3 x^4\right )}{(5+x)^2} \, dx+\int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x^2}{(5+x)^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=e^{\frac {\left (2+e^x\right ) x \left (-1+x^3\right )}{7 (5+x)}} x \log ^{\frac {x \left (-1+x^3\right )}{7 (5+x)}}(5) \]
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Time = 4.85 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
method | result | size |
risch | \(x \,{\mathrm e}^{\frac {\left (2+{\mathrm e}^{x}+\ln \left (\ln \left (5\right )\right )\right ) \left (x^{2}+x +1\right ) x \left (-1+x \right )}{7 x +35}}\) | \(28\) |
parallelrisch | \(x \,{\mathrm e}^{\frac {\left (x^{4}-x \right ) \ln \left (\ln \left (5\right )\right )+\left (x^{4}-x \right ) {\mathrm e}^{x}+2 x^{4}-2 x}{7 x +35}}\) | \(41\) |
norman | \(\frac {x^{2} {\mathrm e}^{\frac {\left (x^{4}-x \right ) \ln \left (\ln \left (5\right )\right )+\left (x^{4}-x \right ) {\mathrm e}^{x}+2 x^{4}-2 x}{7 x +35}}+5 x \,{\mathrm e}^{\frac {\left (x^{4}-x \right ) \ln \left (\ln \left (5\right )\right )+\left (x^{4}-x \right ) {\mathrm e}^{x}+2 x^{4}-2 x}{7 x +35}}}{5+x}\) | \(93\) |
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Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=x e^{\left (\frac {2 \, x^{4} + {\left (x^{4} - x\right )} e^{x} + {\left (x^{4} - x\right )} \log \left (\log \left (5\right )\right ) - 2 \, x}{7 \, {\left (x + 5\right )}}\right )} \]
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Time = 6.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=x e^{\frac {2 x^{4} - 2 x + \left (x^{4} - x\right ) e^{x} + \left (x^{4} - x\right ) \log {\left (\log {\left (5 \right )} \right )}}{7 x + 35}} \]
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\[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=\int { \frac {{\left (6 \, x^{5} + 40 \, x^{4} + 7 \, x^{2} + {\left (x^{6} + 8 \, x^{5} + 20 \, x^{4} - x^{3} - 5 \, x^{2} - 5 \, x\right )} e^{x} + {\left (3 \, x^{5} + 20 \, x^{4} - 5 \, x\right )} \log \left (\log \left (5\right )\right ) + 60 \, x + 175\right )} e^{\left (\frac {2 \, x^{4} + {\left (x^{4} - x\right )} e^{x} + {\left (x^{4} - x\right )} \log \left (\log \left (5\right )\right ) - 2 \, x}{7 \, {\left (x + 5\right )}}\right )}}{7 \, {\left (x^{2} + 10 \, x + 25\right )}} \,d x } \]
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\[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=\int { \frac {{\left (6 \, x^{5} + 40 \, x^{4} + 7 \, x^{2} + {\left (x^{6} + 8 \, x^{5} + 20 \, x^{4} - x^{3} - 5 \, x^{2} - 5 \, x\right )} e^{x} + {\left (3 \, x^{5} + 20 \, x^{4} - 5 \, x\right )} \log \left (\log \left (5\right )\right ) + 60 \, x + 175\right )} e^{\left (\frac {2 \, x^{4} + {\left (x^{4} - x\right )} e^{x} + {\left (x^{4} - x\right )} \log \left (\log \left (5\right )\right ) - 2 \, x}{7 \, {\left (x + 5\right )}}\right )}}{7 \, {\left (x^{2} + 10 \, x + 25\right )}} \,d x } \]
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Time = 12.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61 \[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=\frac {x\,{\mathrm {e}}^{\frac {x^4\,{\mathrm {e}}^x}{7\,x+35}}\,{\mathrm {e}}^{\frac {2\,x^4}{7\,x+35}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^x}{7\,x+35}}\,{\mathrm {e}}^{-\frac {2\,x}{7\,x+35}}}{{\ln \left (5\right )}^{\frac {x-x^4}{7\,\left (x+5\right )}}} \]
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