\(\int \frac {1}{9} e^{\frac {1}{9} (-144-2 x^2-e^{e^x} (72+x^2))} (-8 x+e^{e^x} (-4 x+e^x (-144-2 x^2))) \, dx\) [6777]

Optimal result

Integrand size = 54, antiderivative size = 22 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2 e^{-\left (\left (2+e^{e^x}\right ) \left (8+\frac {x^2}{9}\right )\right )} \]

[Out]

2/exp((8+1/9*x^2)*(2+exp(exp(x))))

Rubi [F]

\[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=\int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx \]

[In]

Int[(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]

[Out]

-16*Defer[Int][E^(E^x + x - ((2 + E^E^x)*(72 + x^2))/9), x] - (8*Defer[Int][x/E^(((2 + E^E^x)*(72 + x^2))/9),
x])/9 - (4*Defer[Int][E^(E^x - ((2 + E^E^x)*(72 + x^2))/9)*x, x])/9 - (2*Defer[Int][E^(E^x + x - ((2 + E^E^x)*
(72 + x^2))/9)*x^2, x])/9

Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} \int 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 {1}{9} \int e^{-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx \\ & = \frac {1}{9} \int \left (-8 e^{-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} x-2 e^{e^x-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} \left (72 e^x+2 x+e^x x^2\right )\right ) \, dx \\ & = -\left (\frac {2}{9} \int e^{e^x-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} \left (72 e^x+2 x+e^x x^2\right ) \, dx\right )-\frac {8}{9} \int e^{-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} x \, dx \\ & = -\left (\frac {2}{9} \int \left (72 e^{e^x+x-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )}+2 e^{e^x-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} x+e^{e^x+x-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} x^2\right ) \, dx\right )-\frac {8}{9} \int e^{-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} x \, dx \\ & = -\left (\frac {2}{9} \int e^{e^x+x-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} x^2 \, dx\right )-\frac {4}{9} \int e^{e^x-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} x \, dx-\frac {8}{9} \int e^{-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} x \, dx-16 \int e^{e^x+x-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2 e^{-\frac {1}{9} \left (2+e^{e^x}\right ) \left (72+x^2\right )} \]

[In]

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]

[Out]

2/E^(((2 + E^E^x)*(72 + x^2))/9)

Maple [A] (verified)

Time = 0.21 (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	\(2 \,{\mathrm e}^{-\frac {\left (x^{2}+72\right ) {\mathrm e}^{{\mathrm e}^{x}}}{9}-\frac {2 x^{2}}{9}-16}\)	\(23\)

[In]

int(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,method=_RETURNV
ERBOSE)

[Out]

2*exp(-1/9*(x^2+72)*(2+exp(exp(x))))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2 \, e^{\left (-\frac {2}{9} \, x^{2} - \frac {1}{9} \, {\left (x^{2} + 72\right )} e^{\left (e^{x}\right )} - 16\right )} \]

[In]

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, algorith
m="fricas")

[Out]

2*e^(-2/9*x^2 - 1/9*(x^2 + 72)*e^(e^x) - 16)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2 e^{- \frac {2 x^{2}}{9} - \left (\frac {x^{2}}{9} + 8\right ) e^{e^{x}} - 16} \]

[In]

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)

[Out]

2*exp(-2*x**2/9 - (x**2/9 + 8)*exp(exp(x)) - 16)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2 \, e^{\left (-\frac {1}{9} \, x^{2} e^{\left (e^{x}\right )} - \frac {2}{9} \, x^{2} - 8 \, e^{\left (e^{x}\right )} - 16\right )} \]

[In]

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, algorith
m="maxima")

[Out]

2*e^(-1/9*x^2*e^(e^x) - 2/9*x^2 - 8*e^(e^x) - 16)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2 \, e^{\left (-\frac {1}{9} \, x^{2} e^{\left (e^{x}\right )} - \frac {2}{9} \, x^{2} - 8 \, e^{\left (e^{x}\right )} - 16\right )} \]

[In]

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, algorith
m="giac")

[Out]

2*e^(-1/9*x^2*e^(e^x) - 2/9*x^2 - 8*e^(e^x) - 16)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {1}{9} e^{\frac {1}{9} \left (-144-2 x^2-e^{e^x} \left (72+x^2\right )\right )} \left (-8 x+e^{e^x} \left (-4 x+e^x \left (-144-2 x^2\right )\right )\right ) \, dx=2\,{\mathrm {e}}^{-8\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^{-16}\,{\mathrm {e}}^{-\frac {2\,x^2}{9}}\,{\mathrm {e}}^{-\frac {x^2\,{\mathrm {e}}^{{\mathrm {e}}^x}}{9}} \]

[In]

int(-exp(- (2*x^2)/9 - (exp(exp(x))*(x^2 + 72))/9 - 16)*((8*x)/9 + (exp(exp(x))*(4*x + exp(x)*(2*x^2 + 144)))/
9),x)

[Out]

2*exp(-8*exp(exp(x)))*exp(-16)*exp(-(2*x^2)/9)*exp(-(x^2*exp(exp(x)))/9)