\(\int e^{-x} (3 e^x x^2+e^{e^{-x} (3 x^3+e^x (-3 x-3 x^3))} (9 x^4-3 x^5+e^x (2 x-3 x^2-9 x^4))) \, dx\) [10086]

Optimal result

Integrand size = 72, antiderivative size = 31 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=-4+x^2 \left (e^{3 \left (-x+x^2 \left (-x+e^{-x} x\right )\right )}+x\right ) \]

[Out]

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

Rubi [F]

\[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=\int e^{-x} \left (3 e^x x^2+\exp \left (e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )\right ) \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx \]

[In]

Int[(3*E^x*x^2 + E^((3*x^3 + E^x*(-3*x - 3*x^3))/E^x)*(9*x^4 - 3*x^5 + E^x*(2*x - 3*x^2 - 9*x^4)))/E^x,x]

[Out]

x^3 + 2*Defer[Int][E^(-3*x + (-3 + 3/E^x)*x^3)*x, x] - 3*Defer[Int][E^(-3*x + (-3 + 3/E^x)*x^3)*x^2, x] + 9*De
fer[Int][E^(-4*x + (-3 + 3/E^x)*x^3)*x^4, x] - 9*Defer[Int][E^(-3*x + (-3 + 3/E^x)*x^3)*x^4, x] - 3*Defer[Int]
[E^(-4*x + (-3 + 3/E^x)*x^3)*x^5, x]

Rubi steps \begin{align*} \text {integral}& = \int x \left (3 x-e^{-4 x+\left (-3+3 e^{-x}\right ) x^3} \left (3 (-3+x) x^3+e^x \left (-2+3 x+9 x^3\right )\right )\right ) \, dx \\ & = \int \left (3 x^2-e^{-4 x+\left (-3+3 e^{-x}\right ) x^3} x \left (-2 e^x+3 e^x x-9 x^3+9 e^x x^3+3 x^4\right )\right ) \, dx \\ & = x^3-\int e^{-4 x+\left (-3+3 e^{-x}\right ) x^3} x \left (-2 e^x+3 e^x x-9 x^3+9 e^x x^3+3 x^4\right ) \, dx \\ & = x^3-\int \left (3 e^{-4 x+\left (-3+3 e^{-x}\right ) x^3} (-3+x) x^4+e^{-3 x+\left (-3+3 e^{-x}\right ) x^3} x \left (-2+3 x+9 x^3\right )\right ) \, dx \\ & = x^3-3 \int e^{-4 x+\left (-3+3 e^{-x}\right ) x^3} (-3+x) x^4 \, dx-\int e^{-3 x+\left (-3+3 e^{-x}\right ) x^3} x \left (-2+3 x+9 x^3\right ) \, dx \\ & = x^3-3 \int \left (-3 e^{-4 x+\left (-3+3 e^{-x}\right ) x^3} x^4+e^{-4 x+\left (-3+3 e^{-x}\right ) x^3} x^5\right ) \, dx-\int \left (-2 e^{-3 x+\left (-3+3 e^{-x}\right ) x^3} x+3 e^{-3 x+\left (-3+3 e^{-x}\right ) x^3} x^2+9 e^{-3 x+\left (-3+3 e^{-x}\right ) x^3} x^4\right ) \, dx \\ & = x^3+2 \int e^{-3 x+\left (-3+3 e^{-x}\right ) x^3} x \, dx-3 \int e^{-3 x+\left (-3+3 e^{-x}\right ) x^3} x^2 \, dx-3 \int e^{-4 x+\left (-3+3 e^{-x}\right ) x^3} x^5 \, dx+9 \int e^{-4 x+\left (-3+3 e^{-x}\right ) x^3} x^4 \, dx-9 \int e^{-3 x+\left (-3+3 e^{-x}\right ) x^3} x^4 \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=x^2 \left (e^{-3 x+\left (-3+3 e^{-x}\right ) x^3}+x\right ) \]

[In]

Integrate[(3*E^x*x^2 + E^((3*x^3 + E^x*(-3*x - 3*x^3))/E^x)*(9*x^4 - 3*x^5 + E^x*(2*x - 3*x^2 - 9*x^4)))/E^x,x
]

[Out]

x^2*(E^(-3*x + (-3 + 3/E^x)*x^3) + x)

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00

method	result	size
risch	\(x^{3}+x^{2} {\mathrm e}^{-3 x \left ({\mathrm e}^{x} x^{2}-x^{2}+{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}\)	\(31\)
parallelrisch	\(x^{3}+x^{2} {\mathrm e}^{\left (\left (-3 x^{3}-3 x \right ) {\mathrm e}^{x}+3 x^{3}\right ) {\mathrm e}^{-x}}\)	\(33\)
norman	\(\left ({\mathrm e}^{x} x^{3}+{\mathrm e}^{x} x^{2} {\mathrm e}^{\left (\left (-3 x^{3}-3 x \right ) {\mathrm e}^{x}+3 x^{3}\right ) {\mathrm e}^{-x}}\right ) {\mathrm e}^{-x}\)	\(43\)

[In]

int((((-9*x^4-3*x^2+2*x)*exp(x)-3*x^5+9*x^4)*exp(((-3*x^3-3*x)*exp(x)+3*x^3)/exp(x))+3*exp(x)*x^2)/exp(x),x,me
thod=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=x^{3} + x^{2} e^{\left (3 \, {\left (x^{3} - {\left (x^{3} + x\right )} e^{x}\right )} e^{\left (-x\right )}\right )} \]

[In]

integrate((((-9*x^4-3*x^2+2*x)*exp(x)-3*x^5+9*x^4)*exp(((-3*x^3-3*x)*exp(x)+3*x^3)/exp(x))+3*exp(x)*x^2)/exp(x
),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=x^{3} + x^{2} e^{\left (3 x^{3} + \left (- 3 x^{3} - 3 x\right ) e^{x}\right ) e^{- x}} \]

[In]

integrate((((-9*x**4-3*x**2+2*x)*exp(x)-3*x**5+9*x**4)*exp(((-3*x**3-3*x)*exp(x)+3*x**3)/exp(x))+3*exp(x)*x**2
)/exp(x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=x^{3} + x^{2} e^{\left (3 \, x^{3} e^{\left (-x\right )} - 3 \, x^{3} - 3 \, x\right )} \]

[In]

integrate((((-9*x^4-3*x^2+2*x)*exp(x)-3*x^5+9*x^4)*exp(((-3*x^3-3*x)*exp(x)+3*x^3)/exp(x))+3*exp(x)*x^2)/exp(x
),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=\int { {\left (3 \, x^{2} e^{x} - {\left (3 \, x^{5} - 9 \, x^{4} + {\left (9 \, x^{4} + 3 \, x^{2} - 2 \, x\right )} e^{x}\right )} e^{\left (3 \, {\left (x^{3} - {\left (x^{3} + x\right )} e^{x}\right )} e^{\left (-x\right )}\right )}\right )} e^{\left (-x\right )} \,d x } \]

[In]

integrate((((-9*x^4-3*x^2+2*x)*exp(x)-3*x^5+9*x^4)*exp(((-3*x^3-3*x)*exp(x)+3*x^3)/exp(x))+3*exp(x)*x^2)/exp(x
),x, algorithm="giac")

[Out]

integrate((3*x^2*e^x - (3*x^5 - 9*x^4 + (9*x^4 + 3*x^2 - 2*x)*e^x)*e^(3*(x^3 - (x^3 + x)*e^x)*e^(-x)))*e^(-x),
x)

Mupad [B] (verification not implemented)

Time = 17.37 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=x^3+x^2\,{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^{-3\,x^3}\,{\mathrm {e}}^{3\,x^3\,{\mathrm {e}}^{-x}} \]

[In]

int(exp(-x)*(3*x^2*exp(x) - exp(-exp(-x)*(exp(x)*(3*x + 3*x^3) - 3*x^3))*(3*x^5 - 9*x^4 + exp(x)*(3*x^2 - 2*x
+ 9*x^4))),x)

[Out]

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