\(\int e^{-e^x x} (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} (3 x+3 x^2)) \, dx\) [6225]

Optimal result

Integrand size = 49, antiderivative size = 26 \[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=3 \left (2 x-\left (-1+e^{2 x-e^x x}-x\right ) x\right ) \]

[Out]

6*x-3*(exp(x)^2/exp(exp(x)*x)-x-1)*x

Rubi [F]

\[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=\int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx \]

[In]

Int[(E^(2*x)*(-3 - 6*x) + E^(E^x*x)*(9 + 6*x) + E^(3*x)*(3*x + 3*x^2))/E^(E^x*x),x]

[Out]

(3*(3 + 2*x)^2)/4 - 3*Defer[Int][E^(2*x - E^x*x), x] - 6*Defer[Int][E^(2*x - E^x*x)*x, x] + 3*Defer[Int][E^(3*
x - E^x*x)*x, x] + 3*Defer[Int][E^(3*x - E^x*x)*x^2, x]

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69 \[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=3 x \left (3-e^{-\left (\left (-2+e^x\right ) x\right )}+x\right ) \]

[In]

Integrate[(E^(2*x)*(-3 - 6*x) + E^(E^x*x)*(9 + 6*x) + E^(3*x)*(3*x + 3*x^2))/E^(E^x*x),x]

[Out]

3*x*(3 - E^(-((-2 + E^x)*x)) + x)

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81

method	result	size
risch	\(3 x^{2}+9 x -3 x \,{\mathrm e}^{-x \left ({\mathrm e}^{x}-2\right )}\)	\(21\)
norman	\(\left (-3 x \,{\mathrm e}^{2 x}+3 x^{2} {\mathrm e}^{{\mathrm e}^{x} x}+9 \,{\mathrm e}^{{\mathrm e}^{x} x} x \right ) {\mathrm e}^{-{\mathrm e}^{x} x}\)	\(35\)
parallelrisch	\(-\left (-3 x^{2} {\mathrm e}^{{\mathrm e}^{x} x}+3 x \,{\mathrm e}^{2 x}-9 \,{\mathrm e}^{{\mathrm e}^{x} x} x \right ) {\mathrm e}^{-{\mathrm e}^{x} x}\)	\(36\)

[In]

int(((6*x+9)*exp(exp(x)*x)+(3*x^2+3*x)*exp(x)^3+(-6*x-3)*exp(x)^2)/exp(exp(x)*x),x,method=_RETURNVERBOSE)

[Out]

3*x^2+9*x-3*x*exp(-x*(exp(x)-2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=3 \, {\left ({\left (x^{2} + 3 \, x\right )} e^{\left (x e^{x}\right )} - x e^{\left (2 \, x\right )}\right )} e^{\left (-x e^{x}\right )} \]

[In]

integrate(((6*x+9)*exp(exp(x)*x)+(3*x^2+3*x)*exp(x)^3+(-6*x-3)*exp(x)^2)/exp(exp(x)*x),x, algorithm="fricas")

[Out]

3*((x^2 + 3*x)*e^(x*e^x) - x*e^(2*x))*e^(-x*e^x)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=3 x^{2} - 3 x e^{2 x} e^{- x e^{x}} + 9 x \]

[In]

integrate(((6*x+9)*exp(exp(x)*x)+(3*x**2+3*x)*exp(x)**3+(-6*x-3)*exp(x)**2)/exp(exp(x)*x),x)

[Out]

3*x**2 - 3*x*exp(2*x)*exp(-x*exp(x)) + 9*x

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=3 \, x^{2} - 3 \, x e^{\left (-x e^{x} + 2 \, x\right )} + 9 \, x \]

[In]

integrate(((6*x+9)*exp(exp(x)*x)+(3*x^2+3*x)*exp(x)^3+(-6*x-3)*exp(x)^2)/exp(exp(x)*x),x, algorithm="maxima")

[Out]

3*x^2 - 3*x*e^(-x*e^x + 2*x) + 9*x

Giac [F]

\[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=\int { 3 \, {\left ({\left (2 \, x + 3\right )} e^{\left (x e^{x}\right )} + {\left (x^{2} + x\right )} e^{\left (3 \, x\right )} - {\left (2 \, x + 1\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-x e^{x}\right )} \,d x } \]

[In]

integrate(((6*x+9)*exp(exp(x)*x)+(3*x^2+3*x)*exp(x)^3+(-6*x-3)*exp(x)^2)/exp(exp(x)*x),x, algorithm="giac")

[Out]

integrate(3*((2*x + 3)*e^(x*e^x) + (x^2 + x)*e^(3*x) - (2*x + 1)*e^(2*x))*e^(-x*e^x), x)

Mupad [B] (verification not implemented)

Time = 12.65 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69 \[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=3\,x\,\left (x-{\mathrm {e}}^{2\,x-x\,{\mathrm {e}}^x}+3\right ) \]

[In]

int(exp(-x*exp(x))*(exp(3*x)*(3*x + 3*x^2) + exp(x*exp(x))*(6*x + 9) - exp(2*x)*(6*x + 3)),x)

[Out]

3*x*(x - exp(2*x - x*exp(x)) + 3)