\(\int e^{-\frac {12 e^x}{x}} (-2 e^{\frac {12 e^x}{x}} x-4 x^3+e^x (-12 x^2+12 x^3)) \, dx\) [4205]

Optimal result

Integrand size = 45, antiderivative size = 22 \[ \int e^{-\frac {12 e^x}{x}} \left (-2 e^{\frac {12 e^x}{x}} x-4 x^3+e^x \left (-12 x^2+12 x^3\right )\right ) \, dx=2-x^2-e^{-\frac {12 e^x}{x}} x^4 \]

[Out]

2-x^4/exp(6*exp(x)/x)^2-x^2

Rubi [F]

\[ \int e^{-\frac {12 e^x}{x}} \left (-2 e^{\frac {12 e^x}{x}} x-4 x^3+e^x \left (-12 x^2+12 x^3\right )\right ) \, dx=\int e^{-\frac {12 e^x}{x}} \left (-2 e^{\frac {12 e^x}{x}} x-4 x^3+e^x \left (-12 x^2+12 x^3\right )\right ) \, dx \]

[In]

Int[(-2*E^((12*E^x)/x)*x - 4*x^3 + E^x*(-12*x^2 + 12*x^3))/E^((12*E^x)/x),x]

[Out]

-x^2 - 12*Defer[Int][E^((-12*E^x)/x + x)*x^2, x] - 4*Defer[Int][x^3/E^((12*E^x)/x), x] + 12*Defer[Int][E^((-12
*E^x)/x + x)*x^3, x]

Rubi steps \begin{align*} \text {integral}& = \int 2 e^{-\frac {12 e^x}{x}} x \left (-e^{\frac {12 e^x}{x}}-6 e^x x-2 x^2+6 e^x x^2\right ) \, dx \\ & = 2 \int e^{-\frac {12 e^x}{x}} x \left (-e^{\frac {12 e^x}{x}}-6 e^x x-2 x^2+6 e^x x^2\right ) \, dx \\ & = 2 \int \left (6 e^{-\frac {12 e^x}{x}+x} (-1+x) x^2-e^{-\frac {12 e^x}{x}} x \left (e^{\frac {12 e^x}{x}}+2 x^2\right )\right ) \, dx \\ & = -\left (2 \int e^{-\frac {12 e^x}{x}} x \left (e^{\frac {12 e^x}{x}}+2 x^2\right ) \, dx\right )+12 \int e^{-\frac {12 e^x}{x}+x} (-1+x) x^2 \, dx \\ & = -\left (2 \int \left (x+2 e^{-\frac {12 e^x}{x}} x^3\right ) \, dx\right )+12 \int \left (-e^{-\frac {12 e^x}{x}+x} x^2+e^{-\frac {12 e^x}{x}+x} x^3\right ) \, dx \\ & = -x^2-4 \int e^{-\frac {12 e^x}{x}} x^3 \, dx-12 \int e^{-\frac {12 e^x}{x}+x} x^2 \, dx+12 \int e^{-\frac {12 e^x}{x}+x} x^3 \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int e^{-\frac {12 e^x}{x}} \left (-2 e^{\frac {12 e^x}{x}} x-4 x^3+e^x \left (-12 x^2+12 x^3\right )\right ) \, dx=-x^2 \left (1+e^{-\frac {12 e^x}{x}} x^2\right ) \]

[In]

Integrate[(-2*E^((12*E^x)/x)*x - 4*x^3 + E^x*(-12*x^2 + 12*x^3))/E^((12*E^x)/x),x]

[Out]

-(x^2*(1 + x^2/E^((12*E^x)/x)))

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

method	result	size
risch	\(-x^{2}-x^{4} {\mathrm e}^{-\frac {12 \,{\mathrm e}^{x}}{x}}\)	\(20\)
norman	\(\left (-x^{4}-x^{2} {\mathrm e}^{\frac {12 \,{\mathrm e}^{x}}{x}}\right ) {\mathrm e}^{-\frac {12 \,{\mathrm e}^{x}}{x}}\)	\(33\)
parallelrisch	\(\frac {\left (-x^{5}-{\mathrm e}^{\frac {12 \,{\mathrm e}^{x}}{x}} x^{3}\right ) {\mathrm e}^{-\frac {12 \,{\mathrm e}^{x}}{x}}}{x}\)	\(36\)

[In]

int((-2*x*exp(6*exp(x)/x)^2+(12*x^3-12*x^2)*exp(x)-4*x^3)/exp(6*exp(x)/x)^2,x,method=_RETURNVERBOSE)

[Out]

-x^2-x^4*exp(-12*exp(x)/x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int e^{-\frac {12 e^x}{x}} \left (-2 e^{\frac {12 e^x}{x}} x-4 x^3+e^x \left (-12 x^2+12 x^3\right )\right ) \, dx=-{\left (x^{4} + x^{2} e^{\left (\frac {12 \, e^{x}}{x}\right )}\right )} e^{\left (-\frac {12 \, e^{x}}{x}\right )} \]

[In]

integrate((-2*x*exp(6*exp(x)/x)^2+(12*x^3-12*x^2)*exp(x)-4*x^3)/exp(6*exp(x)/x)^2,x, algorithm="fricas")

[Out]

-(x^4 + x^2*e^(12*e^x/x))*e^(-12*e^x/x)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int e^{-\frac {12 e^x}{x}} \left (-2 e^{\frac {12 e^x}{x}} x-4 x^3+e^x \left (-12 x^2+12 x^3\right )\right ) \, dx=- x^{4} e^{- \frac {12 e^{x}}{x}} - x^{2} \]

[In]

integrate((-2*x*exp(6*exp(x)/x)**2+(12*x**3-12*x**2)*exp(x)-4*x**3)/exp(6*exp(x)/x)**2,x)

[Out]

-x**4*exp(-12*exp(x)/x) - x**2

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int e^{-\frac {12 e^x}{x}} \left (-2 e^{\frac {12 e^x}{x}} x-4 x^3+e^x \left (-12 x^2+12 x^3\right )\right ) \, dx=-x^{4} e^{\left (-\frac {12 \, e^{x}}{x}\right )} - x^{2} \]

[In]

integrate((-2*x*exp(6*exp(x)/x)^2+(12*x^3-12*x^2)*exp(x)-4*x^3)/exp(6*exp(x)/x)^2,x, algorithm="maxima")

[Out]

-x^4*e^(-12*e^x/x) - x^2

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int e^{-\frac {12 e^x}{x}} \left (-2 e^{\frac {12 e^x}{x}} x-4 x^3+e^x \left (-12 x^2+12 x^3\right )\right ) \, dx=-{\left (x^{4} e^{\left (\frac {x^{2} - 12 \, e^{x}}{x}\right )} + x^{2} e^{x}\right )} e^{\left (-x\right )} \]

[In]

integrate((-2*x*exp(6*exp(x)/x)^2+(12*x^3-12*x^2)*exp(x)-4*x^3)/exp(6*exp(x)/x)^2,x, algorithm="giac")

[Out]

-(x^4*e^((x^2 - 12*e^x)/x) + x^2*e^x)*e^(-x)

Mupad [B] (verification not implemented)

Time = 10.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int e^{-\frac {12 e^x}{x}} \left (-2 e^{\frac {12 e^x}{x}} x-4 x^3+e^x \left (-12 x^2+12 x^3\right )\right ) \, dx=-x^4\,{\mathrm {e}}^{-\frac {12\,{\mathrm {e}}^x}{x}}-x^2 \]

[In]

int(-exp(-(12*exp(x))/x)*(exp(x)*(12*x^2 - 12*x^3) + 2*x*exp((12*exp(x))/x) + 4*x^3),x)

[Out]

- x^4*exp(-(12*exp(x))/x) - x^2