Integrand size = 54, antiderivative size = 22 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2 e^{-\left (\left (2+e^{e^x}\right ) \left (8+\frac {x^2}{9}\right )\right )} \] Output:
2/exp((8+1/9*x^2)*(2+exp(exp(x))))
Time = 0.50 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2 e^{-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} \] Input:
Integrate[(E^((-144 - 2*x^2 - E^E^x*(72 + x^2))/9)*(-8*x + E^E^x*(-4*x + E ^x*(-144 - 2*x^2))))/9,x]
Output:
2/E^(((2 + E^E^x)*(72 + x^2))/9)
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.
\(\displaystyle \int \frac {1}{9} e^{\frac {1}{9} \left (-2 x^2-e^{e^x} \left (x^2+72\right )-144\right )} \left (e^{e^x} \left (e^x \left (-2 x^2-144\right )-4 x\right )-8 x\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int -2 e^{\frac {1}{9} \left (-2 x^2-e^{e^x} \left (x^2+72\right )-144\right )} \left (4 x+e^{e^x} \left (2 x+e^x \left (x^2+72\right )\right )\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2}{9} \int e^{\frac {1}{9} \left (-2 x^2-e^{e^x} \left (x^2+72\right )-144\right )} \left (4 x+e^{e^x} \left (2 x+e^x \left (x^2+72\right )\right )\right )dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle -\frac {2}{9} \int e^{-\frac {1}{9} \left (2+e^{e^x}\right ) \left (x^2+72\right )} \left (4 x+e^{e^x} \left (2 x+e^x \left (x^2+72\right )\right )\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2}{9} \int \left (4 e^{-\frac {1}{9} \left (2+e^{e^x}\right ) \left (x^2+72\right )} x+e^{e^x-\frac {1}{9} \left (2+e^{e^x}\right ) \left (x^2+72\right )} \left (e^x x^2+2 x+72 e^x\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2}{9} \left (72 \int e^{x+e^x-\frac {1}{9} \left (2+e^{e^x}\right ) \left (x^2+72\right )}dx+4 \int e^{-\frac {1}{9} \left (2+e^{e^x}\right ) \left (x^2+72\right )} xdx+2 \int e^{e^x-\frac {1}{9} \left (2+e^{e^x}\right ) \left (x^2+72\right )} xdx+\int e^{x+e^x-\frac {1}{9} \left (2+e^{e^x}\right ) \left (x^2+72\right )} x^2dx\right )\) |
Input:
Int[(E^((-144 - 2*x^2 - E^E^x*(72 + x^2))/9)*(-8*x + E^E^x*(-4*x + E^x*(-1 44 - 2*x^2))))/9,x]
Output:
$Aborted
Time = 0.33 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73
method | result | size |
risch | \(2 \,{\mathrm e}^{-\frac {\left (x^{2}+72\right ) \left (2+{\mathrm e}^{{\mathrm e}^{x}}\right )}{9}}\) | \(16\) |
norman | \(2 \,{\mathrm e}^{-\frac {\left (x^{2}+72\right ) {\mathrm e}^{{\mathrm e}^{x}}}{9}-\frac {2 x^{2}}{9}-16}\) | \(23\) |
parallelrisch | \(2 \,{\mathrm e}^{-\frac {\left (x^{2}+72\right ) {\mathrm e}^{{\mathrm e}^{x}}}{9}-\frac {2 x^{2}}{9}-16}\) | \(23\) |
Input:
int(1/9*(((-2*x^2-144)*exp(x)-4*x)*exp(exp(x))-8*x)/exp(1/9*(x^2+72)*exp(e xp(x))+2/9*x^2+16),x,method=_RETURNVERBOSE)
Output:
2*exp(-1/9*(x^2+72)*(2+exp(exp(x))))
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2 \, e^{\left (-\frac {2}{9} \, x^{2} - \frac {1}{9} \, {\left (x^{2} + 72\right )} e^{\left (e^{x}\right )} - 16\right )} \] Input:
integrate(1/9*(((-2*x^2-144)*exp(x)-4*x)*exp(exp(x))-8*x)/exp(1/9*(x^2+72) *exp(exp(x))+2/9*x^2+16),x, algorithm="fricas")
Output:
2*e^(-2/9*x^2 - 1/9*(x^2 + 72)*e^(e^x) - 16)
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2 e^{- \frac {2 x^{2}}{9} - \left (\frac {x^{2}}{9} + 8\right ) e^{e^{x}} - 16} \] Input:
integrate(1/9*(((-2*x**2-144)*exp(x)-4*x)*exp(exp(x))-8*x)/exp(1/9*(x**2+7 2)*exp(exp(x))+2/9*x**2+16),x)
Output:
2*exp(-2*x**2/9 - (x**2/9 + 8)*exp(exp(x)) - 16)
Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2 \, e^{\left (-\frac {1}{9} \, x^{2} e^{\left (e^{x}\right )} - \frac {2}{9} \, x^{2} - 8 \, e^{\left (e^{x}\right )} - 16\right )} \] Input:
integrate(1/9*(((-2*x^2-144)*exp(x)-4*x)*exp(exp(x))-8*x)/exp(1/9*(x^2+72) *exp(exp(x))+2/9*x^2+16),x, algorithm="maxima")
Output:
2*e^(-1/9*x^2*e^(e^x) - 2/9*x^2 - 8*e^(e^x) - 16)
Time = 0.13 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2 \, e^{\left (-\frac {1}{9} \, x^{2} e^{\left (e^{x}\right )} - \frac {2}{9} \, x^{2} - 8 \, e^{\left (e^{x}\right )} - 16\right )} \] Input:
integrate(1/9*(((-2*x^2-144)*exp(x)-4*x)*exp(exp(x))-8*x)/exp(1/9*(x^2+72) *exp(exp(x))+2/9*x^2+16),x, algorithm="giac")
Output:
2*e^(-1/9*x^2*e^(e^x) - 2/9*x^2 - 8*e^(e^x) - 16)
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2\,{\mathrm {e}}^{-8\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^{-16}\,{\mathrm {e}}^{-\frac {2\,x^2}{9}}\,{\mathrm {e}}^{-\frac {x^2\,{\mathrm {e}}^{{\mathrm {e}}^x}}{9}} \] Input:
int(-exp(- (2*x^2)/9 - (exp(exp(x))*(x^2 + 72))/9 - 16)*((8*x)/9 + (exp(ex p(x))*(4*x + exp(x)*(2*x^2 + 144)))/9),x)
Output:
2*exp(-8*exp(exp(x)))*exp(-16)*exp(-(2*x^2)/9)*exp(-(x^2*exp(exp(x)))/9)
Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=\frac {2}{e^{\frac {e^{e^{x}} x^{2}}{9}+8 e^{e^{x}}+\frac {2 x^{2}}{9}} e^{16}} \] Input:
int(1/9*(((-2*x^2-144)*exp(x)-4*x)*exp(exp(x))-8*x)/exp(1/9*(x^2+72)*exp(e xp(x))+2/9*x^2+16),x)
Output:
2/(e**((e**(e**x)*x**2 + 72*e**(e**x) + 2*x**2)/9)*e**16)