Integrand size = 152, antiderivative size = 29 \[ \int \frac {e^{-e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}}} \left (64+32 x+36 x^2+8 x^3+4 x^4+e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}} \left (-64 x-32 x^2-36 x^3-8 x^4-4 x^5+e^x \left (-3 x+x^2-x^3\right )\right )\right )}{16+8 x+9 x^2+2 x^3+x^4} \, dx=3+4 e^{-e^{5+x+\frac {e^x}{4 \left (4+x+x^2\right )}}} x \] Output:
3+4*x/exp(exp(x+exp(x)/(4*x^2+4*x+16)+5))
Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}}} \left (64+32 x+36 x^2+8 x^3+4 x^4+e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}} \left (-64 x-32 x^2-36 x^3-8 x^4-4 x^5+e^x \left (-3 x+x^2-x^3\right )\right )\right )}{16+8 x+9 x^2+2 x^3+x^4} \, dx=4 e^{-e^{5+x+\frac {e^x}{4 \left (4+x+x^2\right )}}} x \] Input:
Integrate[(64 + 32*x + 36*x^2 + 8*x^3 + 4*x^4 + E^((80 + E^x + 36*x + 24*x ^2 + 4*x^3)/(16 + 4*x + 4*x^2))*(-64*x - 32*x^2 - 36*x^3 - 8*x^4 - 4*x^5 + E^x*(-3*x + x^2 - x^3)))/(E^E^((80 + E^x + 36*x + 24*x^2 + 4*x^3)/(16 + 4 *x + 4*x^2))*(16 + 8*x + 9*x^2 + 2*x^3 + x^4)),x]
Output:
(4*x)/E^E^(5 + x + E^x/(4*(4 + x + x^2)))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (4 x^4+8 x^3+36 x^2+e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 x^2+4 x+16}} \left (-4 x^5-8 x^4-36 x^3-32 x^2+e^x \left (-x^3+x^2-3 x\right )-64 x\right )+32 x+64\right ) \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 x^2+4 x+16}}\right )}{x^4+2 x^3+9 x^2+8 x+16} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {4 i \left (4 x^4+8 x^3+36 x^2+e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 x^2+4 x+16}} \left (-4 x^5-8 x^4-36 x^3-32 x^2+e^x \left (-x^3+x^2-3 x\right )-64 x\right )+32 x+64\right ) \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 x^2+4 x+16}}\right )}{15 \sqrt {15} \left (-2 x+i \sqrt {15}-1\right )}+\frac {4 i \left (4 x^4+8 x^3+36 x^2+e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 x^2+4 x+16}} \left (-4 x^5-8 x^4-36 x^3-32 x^2+e^x \left (-x^3+x^2-3 x\right )-64 x\right )+32 x+64\right ) \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 x^2+4 x+16}}\right )}{15 \sqrt {15} \left (2 x+i \sqrt {15}+1\right )}-\frac {4 \left (4 x^4+8 x^3+36 x^2+e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 x^2+4 x+16}} \left (-4 x^5-8 x^4-36 x^3-32 x^2+e^x \left (-x^3+x^2-3 x\right )-64 x\right )+32 x+64\right ) \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 x^2+4 x+16}}\right )}{15 \left (-2 x+i \sqrt {15}-1\right )^2}-\frac {4 \left (4 x^4+8 x^3+36 x^2+e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 x^2+4 x+16}} \left (-4 x^5-8 x^4-36 x^3-32 x^2+e^x \left (-x^3+x^2-3 x\right )-64 x\right )+32 x+64\right ) \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 x^2+4 x+16}}\right )}{15 \left (2 x+i \sqrt {15}+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right ) \left (-x^3 \left (36 \exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )+e^{\frac {8 x^3+28 x^2+52 x+e^x+80}{4 \left (x^2+x+4\right )}}-8\right )+x^2 \left (-32 \exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )+e^{\frac {8 x^3+28 x^2+52 x+e^x+80}{4 \left (x^2+x+4\right )}}+36\right )+x \left (-64 \exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )-3 e^{\frac {8 x^3+28 x^2+52 x+e^x+80}{4 \left (x^2+x+4\right )}}+32\right )-4 x^5 \exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )+x^4 \left (4-8 \exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )+64\right )}{\left (x^2+x+4\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8 x^3 \exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )}{\left (x^2+x+4\right )^2}+\frac {36 x^2 \exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )}{\left (x^2+x+4\right )^2}-4 x \exp \left (\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )+\frac {32 x \exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )}{\left (x^2+x+4\right )^2}+\frac {\left (-x^2+x-3\right ) x \exp \left (\frac {8 x^3+28 x^2+52 x+e^x+80}{4 \left (x^2+x+4\right )}-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )}{\left (x^2+x+4\right )^2}+\frac {64 \exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )}{\left (x^2+x+4\right )^2}+\frac {4 x^4 \exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )}{\left (x^2+x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8 x^3 \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}+\frac {36 x^2 \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}-4 x \exp \left (-\exp \left (\frac {6 x^2}{x^2+x+4}+\frac {9 x}{x^2+x+4}+\frac {e^x}{4 \left (x^2+x+4\right )}+\frac {20}{x^2+x+4}+\frac {x^3}{x^2+x+4}\right )+\frac {e^x}{4 \left (x^2+x+4\right )}+x+5\right )+\frac {32 x \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}-\frac {\left (x^2-x+3\right ) x \exp \left (-\exp \left (\frac {6 x^2}{x^2+x+4}+\frac {9 x}{x^2+x+4}+\frac {e^x}{4 \left (x^2+x+4\right )}+\frac {20}{x^2+x+4}+\frac {x^3}{x^2+x+4}\right )+\frac {e^x}{4 \left (x^2+x+4\right )}+2 x+5\right )}{\left (x^2+x+4\right )^2}+\frac {64 \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}+\frac {4 x^4 \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (\left (-32 e^{\frac {e^x}{4 \left (x^2+x+4\right )}+x+5}+e^{\frac {e^x}{4 \left (x^2+x+4\right )}+2 x+5}+36\right ) x^2+\left (-64 e^{\frac {e^x}{4 \left (x^2+x+4\right )}+x+5}-3 e^{\frac {e^x}{4 \left (x^2+x+4\right )}+2 x+5}+32\right ) x-4 e^{\frac {e^x}{4 \left (x^2+x+4\right )}+x+5} x^5+\left (4-8 e^{\frac {e^x}{4 \left (x^2+x+4\right )}+x+5}\right ) x^4-\left (36 e^{\frac {e^x}{4 \left (x^2+x+4\right )}+x+5}+e^{\frac {e^x}{4 \left (x^2+x+4\right )}+2 x+5}-8\right ) x^3+64\right ) \exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )}{\left (x^2+x+4\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8 x^3 \exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )}{\left (x^2+x+4\right )^2}+\frac {36 x^2 \exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )}{\left (x^2+x+4\right )^2}+\frac {32 x \exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )}{\left (x^2+x+4\right )^2}+\frac {64 \exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )}{\left (x^2+x+4\right )^2}+\frac {4 x^4 \exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )\right )}{\left (x^2+x+4\right )^2}-\frac {\left (4 x^4+8 x^3+e^x x^2+36 x^2-e^x x+32 x+3 e^x+64\right ) x \exp \left (-\exp \left (\frac {4 \left (x^3+6 x^2+9 x+20\right )+e^x}{4 \left (x^2+x+4\right )}\right )+\frac {e^x}{4 \left (x^2+x+4\right )}+x+5\right )}{\left (x^2+x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8 x^3 \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}+\frac {36 x^2 \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}+\frac {32 x \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}+\frac {64 \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}+\frac {4 x^4 \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}-\frac {\left (4 x^4+8 x^3+e^x x^2+36 x^2-e^x x+32 x+3 e^x+64\right ) x \exp \left (-\exp \left (\frac {6 x^2}{x^2+x+4}+\frac {9 x}{x^2+x+4}+\frac {e^x}{4 \left (x^2+x+4\right )}+\frac {20}{x^2+x+4}+\frac {x^3}{x^2+x+4}\right )+\frac {e^x}{4 \left (x^2+x+4\right )}+x+5\right )}{\left (x^2+x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {8 x^3 \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}+\frac {36 x^2 \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}+\frac {32 x \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}+\frac {64 \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}+\frac {4 x^4 \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^2+x+4\right )^2}-\frac {\left (4 x^4+8 x^3+e^x x^2+36 x^2-e^x x+32 x+3 e^x+64\right ) x \exp \left (-\exp \left (\frac {6 x^2}{x^2+x+4}+\frac {9 x}{x^2+x+4}+\frac {e^x}{4 \left (x^2+x+4\right )}+\frac {20}{x^2+x+4}+\frac {x^3}{x^2+x+4}\right )+\frac {e^x}{4 \left (x^2+x+4\right )}+x+5\right )}{\left (x^2+x+4\right )^2}\right )dx\) |
Input:
Int[(64 + 32*x + 36*x^2 + 8*x^3 + 4*x^4 + E^((80 + E^x + 36*x + 24*x^2 + 4 *x^3)/(16 + 4*x + 4*x^2))*(-64*x - 32*x^2 - 36*x^3 - 8*x^4 - 4*x^5 + E^x*( -3*x + x^2 - x^3)))/(E^E^((80 + E^x + 36*x + 24*x^2 + 4*x^3)/(16 + 4*x + 4 *x^2))*(16 + 8*x + 9*x^2 + 2*x^3 + x^4)),x]
Output:
$Aborted
Time = 9.51 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21
| method | result | size |
| risch | \(4 x \,{\mathrm e}^{-{\mathrm e}^{\frac {{\mathrm e}^{x}+4 x^{3}+24 x^{2}+36 x +80}{4 x^{2}+4 x +16}}}\) | \(35\) |
| parallelrisch | \(4 x \,{\mathrm e}^{-{\mathrm e}^{\frac {{\mathrm e}^{x}+4 x^{3}+24 x^{2}+36 x +80}{4 x^{2}+4 x +16}}}\) | \(35\) |
Input:
int((((-x^3+x^2-3*x)*exp(x)-4*x^5-8*x^4-36*x^3-32*x^2-64*x)*exp((exp(x)+4* x^3+24*x^2+36*x+80)/(4*x^2+4*x+16))+4*x^4+8*x^3+36*x^2+32*x+64)/(x^4+2*x^3 +9*x^2+8*x+16)/exp(exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16))),x,me thod=_RETURNVERBOSE)
Output:
4*x*exp(-exp(1/4*(exp(x)+4*x^3+24*x^2+36*x+80)/(x^2+x+4)))
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}}} \left (64+32 x+36 x^2+8 x^3+4 x^4+e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}} \left (-64 x-32 x^2-36 x^3-8 x^4-4 x^5+e^x \left (-3 x+x^2-x^3\right )\right )\right )}{16+8 x+9 x^2+2 x^3+x^4} \, dx=4 \, x e^{\left (-e^{\left (\frac {4 \, x^{3} + 24 \, x^{2} + 36 \, x + e^{x} + 80}{4 \, {\left (x^{2} + x + 4\right )}}\right )}\right )} \] Input:
integrate((((-x^3+x^2-3*x)*exp(x)-4*x^5-8*x^4-36*x^3-32*x^2-64*x)*exp((exp (x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16))+4*x^4+8*x^3+36*x^2+32*x+64)/(x^4 +2*x^3+9*x^2+8*x+16)/exp(exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16)) ),x, algorithm="fricas")
Output:
4*x*e^(-e^(1/4*(4*x^3 + 24*x^2 + 36*x + e^x + 80)/(x^2 + x + 4)))
Timed out. \[ \int \frac {e^{-e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}}} \left (64+32 x+36 x^2+8 x^3+4 x^4+e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}} \left (-64 x-32 x^2-36 x^3-8 x^4-4 x^5+e^x \left (-3 x+x^2-x^3\right )\right )\right )}{16+8 x+9 x^2+2 x^3+x^4} \, dx=\text {Timed out} \] Input:
integrate((((-x**3+x**2-3*x)*exp(x)-4*x**5-8*x**4-36*x**3-32*x**2-64*x)*ex p((exp(x)+4*x**3+24*x**2+36*x+80)/(4*x**2+4*x+16))+4*x**4+8*x**3+36*x**2+3 2*x+64)/(x**4+2*x**3+9*x**2+8*x+16)/exp(exp((exp(x)+4*x**3+24*x**2+36*x+80 )/(4*x**2+4*x+16))),x)
Output:
Timed out
\[ \int \frac {e^{-e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}}} \left (64+32 x+36 x^2+8 x^3+4 x^4+e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}} \left (-64 x-32 x^2-36 x^3-8 x^4-4 x^5+e^x \left (-3 x+x^2-x^3\right )\right )\right )}{16+8 x+9 x^2+2 x^3+x^4} \, dx=\int { \frac {{\left (4 \, x^{4} + 8 \, x^{3} + 36 \, x^{2} - {\left (4 \, x^{5} + 8 \, x^{4} + 36 \, x^{3} + 32 \, x^{2} + {\left (x^{3} - x^{2} + 3 \, x\right )} e^{x} + 64 \, x\right )} e^{\left (\frac {4 \, x^{3} + 24 \, x^{2} + 36 \, x + e^{x} + 80}{4 \, {\left (x^{2} + x + 4\right )}}\right )} + 32 \, x + 64\right )} e^{\left (-e^{\left (\frac {4 \, x^{3} + 24 \, x^{2} + 36 \, x + e^{x} + 80}{4 \, {\left (x^{2} + x + 4\right )}}\right )}\right )}}{x^{4} + 2 \, x^{3} + 9 \, x^{2} + 8 \, x + 16} \,d x } \] Input:
integrate((((-x^3+x^2-3*x)*exp(x)-4*x^5-8*x^4-36*x^3-32*x^2-64*x)*exp((exp (x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16))+4*x^4+8*x^3+36*x^2+32*x+64)/(x^4 +2*x^3+9*x^2+8*x+16)/exp(exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16)) ),x, algorithm="maxima")
Output:
4*((x^3 - x^2 + 3*x)*e^(2*x) + 4*(x^5 + 2*x^4 + 9*x^3 + 8*x^2 + 16*x)*e^x) *e^(-e^(x + 1/4*e^x/(x^2 + x + 4) + 5))/((x^2 - x + 3)*e^(2*x) + 4*(x^4 + 2*x^3 + 9*x^2 + 8*x + 16)*e^x) - integrate(-4*((x^4 - 2*x^3 + 7*x^2 - 6*x + 9)*e^(3*x) - (4*x^7 + 48*x^5 + 28*x^4 + 156*x^3 + 84*x^2 + (x^4 - 2*x^3 + 7*x^2 - 6*x + 9)*e^x + 64*x + 192)*e^(2*x) + 8*(x^6 + x^5 + 10*x^4 + 5*x ^3 + 35*x^2 + 8*x + 48)*e^(2*x) - 4*(4*x^8 + 16*x^7 + 88*x^6 + 208*x^5 + 5 80*x^4 + 832*x^3 + 1408*x^2 - (x^7 - 2*x^6 + 10*x^5 - 13*x^4 + 29*x^3 - 49 *x^2 - 48)*e^x + 1024*x + 1024)*e^x + 16*(x^8 + 4*x^7 + 22*x^6 + 52*x^5 + 145*x^4 + 208*x^3 + 352*x^2 + 256*x + 256)*e^x)*e^(-e^(x + 1/4*e^x/(x^2 + x + 4) + 5))/((x^4 - 2*x^3 + 7*x^2 - 6*x + 9)*e^(3*x) + 8*(x^6 + x^5 + 10* x^4 + 5*x^3 + 35*x^2 + 8*x + 48)*e^(2*x) + 16*(x^8 + 4*x^7 + 22*x^6 + 52*x ^5 + 145*x^4 + 208*x^3 + 352*x^2 + 256*x + 256)*e^x), x)
\[ \int \frac {e^{-e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}}} \left (64+32 x+36 x^2+8 x^3+4 x^4+e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}} \left (-64 x-32 x^2-36 x^3-8 x^4-4 x^5+e^x \left (-3 x+x^2-x^3\right )\right )\right )}{16+8 x+9 x^2+2 x^3+x^4} \, dx=\int { \frac {{\left (4 \, x^{4} + 8 \, x^{3} + 36 \, x^{2} - {\left (4 \, x^{5} + 8 \, x^{4} + 36 \, x^{3} + 32 \, x^{2} + {\left (x^{3} - x^{2} + 3 \, x\right )} e^{x} + 64 \, x\right )} e^{\left (\frac {4 \, x^{3} + 24 \, x^{2} + 36 \, x + e^{x} + 80}{4 \, {\left (x^{2} + x + 4\right )}}\right )} + 32 \, x + 64\right )} e^{\left (-e^{\left (\frac {4 \, x^{3} + 24 \, x^{2} + 36 \, x + e^{x} + 80}{4 \, {\left (x^{2} + x + 4\right )}}\right )}\right )}}{x^{4} + 2 \, x^{3} + 9 \, x^{2} + 8 \, x + 16} \,d x } \] Input:
integrate((((-x^3+x^2-3*x)*exp(x)-4*x^5-8*x^4-36*x^3-32*x^2-64*x)*exp((exp (x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16))+4*x^4+8*x^3+36*x^2+32*x+64)/(x^4 +2*x^3+9*x^2+8*x+16)/exp(exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16)) ),x, algorithm="giac")
Output:
integrate((4*x^4 + 8*x^3 + 36*x^2 - (4*x^5 + 8*x^4 + 36*x^3 + 32*x^2 + (x^ 3 - x^2 + 3*x)*e^x + 64*x)*e^(1/4*(4*x^3 + 24*x^2 + 36*x + e^x + 80)/(x^2 + x + 4)) + 32*x + 64)*e^(-e^(1/4*(4*x^3 + 24*x^2 + 36*x + e^x + 80)/(x^2 + x + 4)))/(x^4 + 2*x^3 + 9*x^2 + 8*x + 16), x)
Time = 3.80 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.48 \[ \int \frac {e^{-e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}}} \left (64+32 x+36 x^2+8 x^3+4 x^4+e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}} \left (-64 x-32 x^2-36 x^3-8 x^4-4 x^5+e^x \left (-3 x+x^2-x^3\right )\right )\right )}{16+8 x+9 x^2+2 x^3+x^4} \, dx=4\,x\,{\mathrm {e}}^{-{\mathrm {e}}^{\frac {9\,x}{x^2+x+4}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{4\,x^2+4\,x+16}}\,{\mathrm {e}}^{\frac {x^3}{x^2+x+4}}\,{\mathrm {e}}^{\frac {6\,x^2}{x^2+x+4}}\,{\mathrm {e}}^{\frac {20}{x^2+x+4}}} \] Input:
int((exp(-exp((36*x + exp(x) + 24*x^2 + 4*x^3 + 80)/(4*x + 4*x^2 + 16)))*( 32*x - exp((36*x + exp(x) + 24*x^2 + 4*x^3 + 80)/(4*x + 4*x^2 + 16))*(64*x + exp(x)*(3*x - x^2 + x^3) + 32*x^2 + 36*x^3 + 8*x^4 + 4*x^5) + 36*x^2 + 8*x^3 + 4*x^4 + 64))/(8*x + 9*x^2 + 2*x^3 + x^4 + 16),x)
Output:
4*x*exp(-exp((9*x)/(x + x^2 + 4))*exp(exp(x)/(4*x + 4*x^2 + 16))*exp(x^3/( x + x^2 + 4))*exp((6*x^2)/(x + x^2 + 4))*exp(20/(x + x^2 + 4)))
Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {e^{-e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}}} \left (64+32 x+36 x^2+8 x^3+4 x^4+e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}} \left (-64 x-32 x^2-36 x^3-8 x^4-4 x^5+e^x \left (-3 x+x^2-x^3\right )\right )\right )}{16+8 x+9 x^2+2 x^3+x^4} \, dx=\frac {4 x}{e^{e^{\frac {e^{x}+4 x^{3}+4 x^{2}+16 x}{4 x^{2}+4 x +16}} e^{5}}} \] Input:
int((((-x^3+x^2-3*x)*exp(x)-4*x^5-8*x^4-36*x^3-32*x^2-64*x)*exp((exp(x)+4* x^3+24*x^2+36*x+80)/(4*x^2+4*x+16))+4*x^4+8*x^3+36*x^2+32*x+64)/(x^4+2*x^3 +9*x^2+8*x+16)/exp(exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16))),x)
Output:
(4*x)/e**(e**((e**x + 4*x**3 + 4*x**2 + 16*x)/(4*x**2 + 4*x + 16))*e**5)