Integrand size = 179, antiderivative size = 33 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4+4 e^{-\frac {-\frac {2}{5}+x}{\frac {12}{5+e^x}+x (-2+2 x)}} x \]
[Out]
\[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=\int \frac {\exp \left (-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}\right ) \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (360+2 \left (-350-59 e^x+e^{2 x}\right ) x+6 \left (125+25 e^x+e^{2 x}\right ) x^2-15 \left (5+e^x\right )^2 x^3+10 \left (5+e^x\right )^2 x^4\right )}{5 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )^2} \, dx \\ & = \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (360+2 \left (-350-59 e^x+e^{2 x}\right ) x+6 \left (125+25 e^x+e^{2 x}\right ) x^2-15 \left (5+e^x\right )^2 x^3+10 \left (5+e^x\right )^2 x^4\right )}{\left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )^2} \, dx \\ & = \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (4-11 x+8 x^2+5 x^3\right )}{(-1+x)^2 x \left (6-5 x-e^x x+5 x^2+e^x x^2\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (2+6 x-15 x^2+10 x^3\right )}{(-1+x)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (12-42 x+8 x^2+75 x^3-60 x^4+25 x^5\right )}{(-1+x)^2 x \left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2}\right ) \, dx \\ & = \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (2+6 x-15 x^2+10 x^3\right )}{(-1+x)^2 x} \, dx-\frac {12}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (4-11 x+8 x^2+5 x^3\right )}{(-1+x)^2 x \left (6-5 x-e^x x+5 x^2+e^x x^2\right )} \, dx+\frac {12}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (12-42 x+8 x^2+75 x^3-60 x^4+25 x^5\right )}{(-1+x)^2 x \left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2} \, dx \\ & = \frac {2}{5} \int \left (10 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}+\frac {3 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{(-1+x)^2}+\frac {3 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{-1+x}+\frac {2 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{x}\right ) \, dx+\frac {12}{5} \int \left (\frac {30 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{\left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2}+\frac {18 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{(-1+x)^2 \left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2}+\frac {66 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{(-1+x) \left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2}+\frac {12 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{x \left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2}-\frac {10 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}} x}{\left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2}+\frac {25 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}} x^2}{\left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2}\right ) \, dx-\frac {12}{5} \int \left (\frac {5 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{6-5 x-e^x x+5 x^2+e^x x^2}+\frac {6 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{(-1+x)^2 \left (6-5 x-e^x x+5 x^2+e^x x^2\right )}+\frac {14 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{(-1+x) \left (6-5 x-e^x x+5 x^2+e^x x^2\right )}+\frac {4 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{x \left (6-5 x-e^x x+5 x^2+e^x x^2\right )}\right ) \, dx \\ & = \frac {4}{5} \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{x} \, dx+\frac {6}{5} \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{(-1+x)^2} \, dx+\frac {6}{5} \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{-1+x} \, dx+4 \int e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}} \, dx-\frac {48}{5} \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{x \left (6-5 x-e^x x+5 x^2+e^x x^2\right )} \, dx-12 \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{6-5 x-e^x x+5 x^2+e^x x^2} \, dx-\frac {72}{5} \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{(-1+x)^2 \left (6-5 x-e^x x+5 x^2+e^x x^2\right )} \, dx-24 \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}} x}{\left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2} \, dx+\frac {144}{5} \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{x \left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2} \, dx-\frac {168}{5} \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{(-1+x) \left (6-5 x-e^x x+5 x^2+e^x x^2\right )} \, dx+\frac {216}{5} \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{(-1+x)^2 \left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2} \, dx+60 \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}} x^2}{\left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2} \, dx+72 \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{\left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2} \, dx+\frac {792}{5} \int \frac {e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}}}{(-1+x) \left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2} \, dx \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}} x \]
[In]
[Out]
Time = 2.71 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18
method | result | size |
risch | \(4 x \,{\mathrm e}^{-\frac {\left (5 x -2\right ) \left ({\mathrm e}^{x}+5\right )}{10 \left ({\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x +5 x^{2}-5 x +6\right )}}\) | \(39\) |
norman | \(\frac {\left (24 x -20 x^{2}+20 x^{3}-4 \,{\mathrm e}^{x} x^{2}+4 \,{\mathrm e}^{x} x^{3}\right ) {\mathrm e}^{-\frac {\left (5 x -2\right ) {\mathrm e}^{x}+25 x -10}{\left (10 x^{2}-10 x \right ) {\mathrm e}^{x}+50 x^{2}-50 x +60}}}{{\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x +5 x^{2}-5 x +6}\) | \(94\) |
parallelrisch | \(\frac {\left (10000 \,{\mathrm e}^{x} x^{3}+50000 x^{3}-10000 \,{\mathrm e}^{x} x^{2}-50000 x^{2}+60000 x \right ) {\mathrm e}^{-\frac {5 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{x}+25 x -10}{10 \left ({\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x +5 x^{2}-5 x +6\right )}}}{2500 \,{\mathrm e}^{x} x^{2}-2500 \,{\mathrm e}^{x} x +12500 x^{2}-12500 x +15000}\) | \(96\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4 \, x e^{\left (-\frac {{\left (5 \, x - 2\right )} e^{x} + 25 \, x - 10}{10 \, {\left (5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6\right )}}\right )} \]
[In]
[Out]
Time = 11.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4 x e^{- \frac {25 x + \left (5 x - 2\right ) e^{x} - 10}{50 x^{2} - 50 x + \left (10 x^{2} - 10 x\right ) e^{x} + 60}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (29) = 58\).
Time = 0.46 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.18 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4 \, x e^{\left (-\frac {x e^{x}}{2 \, {\left (5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6\right )}} - \frac {5 \, x}{2 \, {\left (5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6\right )}} + \frac {e^{x}}{5 \, {\left (5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6\right )}} + \frac {1}{5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6}\right )} \]
[In]
[Out]
none
Time = 1.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4 \, x e^{\left (-\frac {5 \, x^{2} e^{x} + 25 \, x^{2} + 10 \, x e^{x} + 50 \, x - 6 \, e^{x}}{30 \, {\left (x^{2} e^{x} + 5 \, x^{2} - x e^{x} - 5 \, x + 6\right )}} + \frac {1}{6}\right )} \]
[In]
[Out]
Time = 0.71 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.42 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4\,x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{5\,x^2\,{\mathrm {e}}^x-25\,x-5\,x\,{\mathrm {e}}^x+25\,x^2+30}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^x}{2\,x^2\,{\mathrm {e}}^x-10\,x-2\,x\,{\mathrm {e}}^x+10\,x^2+12}}\,{\mathrm {e}}^{-\frac {5\,x}{2\,x^2\,{\mathrm {e}}^x-10\,x-2\,x\,{\mathrm {e}}^x+10\,x^2+12}}\,{\mathrm {e}}^{\frac {1}{x^2\,{\mathrm {e}}^x-5\,x-x\,{\mathrm {e}}^x+5\,x^2+6}} \]
[In]
[Out]