3.28.71 \(\int e^{-2 x-e^{-x} (x^2-e^x x^2)} (18 e^{4 x+e^{-x} (x^2-e^x x^2)}+2 x^2-x^3+e^x (-1+x-2 x^2)) \, dx\)

Optimal. Leaf size=29 \[ 9 e^{2 x}-e^{-x-\left (-1+e^{-x}\right ) x^2} x \]

________________________________________________________________________________________

Rubi [F]  time = 1.76, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int e^{-2 x-e^{-x} \left (x^2-e^x x^2\right )} \left (18 e^{4 x+e^{-x} \left (x^2-e^x x^2\right )}+2 x^2-x^3+e^x \left (-1+x-2 x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-2*x - (x^2 - E^x*x^2)/E^x)*(18*E^(4*x + (x^2 - E^x*x^2)/E^x) + 2*x^2 - x^3 + E^x*(-1 + x - 2*x^2)),x]

[Out]

(18*E^(4*x - (1 - E^(-x))*x^2 - (x*(2*E^x + x - E^x*x))/E^x))/(4 - 2*(1 - E^(-x))*x - x^2/E^x - (x*(1 + E^x -
E^x*x))/E^x - (2*E^x + x - E^x*x)/E^x + (x*(2*E^x + x - E^x*x))/E^x) - Defer[Int][E^((x*(-E^x - x + E^x*x))/E^
x), x] + Defer[Int][E^((x*(-E^x - x + E^x*x))/E^x)*x, x] + 2*Defer[Int][E^((x*(-2*E^x - x + E^x*x))/E^x)*x^2,
x] - 2*Defer[Int][E^((x*(-E^x - x + E^x*x))/E^x)*x^2, x] - Defer[Int][E^((x*(-2*E^x - x + E^x*x))/E^x)*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{e^{-x} x \left (-2 e^x-x+e^x x\right )} \left (18 e^{4 x+e^{-x} \left (x^2-e^x x^2\right )}+2 x^2-x^3+e^x \left (-1+x-2 x^2\right )\right ) \, dx\\ &=\int \left (18 \exp \left (4 x+\left (-1+e^{-x}\right ) x^2+e^{-x} x \left (-2 e^x-x+e^x x\right )\right )+2 e^{e^{-x} x \left (-2 e^x-x+e^x x\right )} x^2-e^{e^{-x} x \left (-2 e^x-x+e^x x\right )} x^3-e^{x+e^{-x} x \left (-2 e^x-x+e^x x\right )} \left (1-x+2 x^2\right )\right ) \, dx\\ &=2 \int e^{e^{-x} x \left (-2 e^x-x+e^x x\right )} x^2 \, dx+18 \int \exp \left (4 x+\left (-1+e^{-x}\right ) x^2+e^{-x} x \left (-2 e^x-x+e^x x\right )\right ) \, dx-\int e^{e^{-x} x \left (-2 e^x-x+e^x x\right )} x^3 \, dx-\int e^{x+e^{-x} x \left (-2 e^x-x+e^x x\right )} \left (1-x+2 x^2\right ) \, dx\\ &=2 \int e^{e^{-x} x \left (-2 e^x-x+e^x x\right )} x^2 \, dx+\frac {18 \operatorname {Subst}\left (\int 1 \, dx,x,\exp \left (4 x+\left (-1+e^{-x}\right ) x^2+e^{-x} x \left (-2 e^x-x+e^x x\right )\right )\right )}{4+2 \left (-1+e^{-x}\right ) x-e^{-x} x^2+e^{-x} x \left (-1-e^x+e^x x\right )+e^{-x} \left (-2 e^x-x+e^x x\right )-e^{-x} x \left (-2 e^x-x+e^x x\right )}-\int e^{e^{-x} x \left (-2 e^x-x+e^x x\right )} x^3 \, dx-\int e^{e^{-x} x \left (-e^x-x+e^x x\right )} \left (1-x+2 x^2\right ) \, dx\\ &=\frac {18 \exp \left (4 x-\left (1-e^{-x}\right ) x^2-e^{-x} x \left (2 e^x+x-e^x x\right )\right )}{4-2 \left (1-e^{-x}\right ) x-e^{-x} x^2-e^{-x} x \left (1+e^x-e^x x\right )-e^{-x} \left (2 e^x+x-e^x x\right )+e^{-x} x \left (2 e^x+x-e^x x\right )}+2 \int e^{e^{-x} x \left (-2 e^x-x+e^x x\right )} x^2 \, dx-\int e^{e^{-x} x \left (-2 e^x-x+e^x x\right )} x^3 \, dx-\int \left (e^{e^{-x} x \left (-e^x-x+e^x x\right )}-e^{e^{-x} x \left (-e^x-x+e^x x\right )} x+2 e^{e^{-x} x \left (-e^x-x+e^x x\right )} x^2\right ) \, dx\\ &=\frac {18 \exp \left (4 x-\left (1-e^{-x}\right ) x^2-e^{-x} x \left (2 e^x+x-e^x x\right )\right )}{4-2 \left (1-e^{-x}\right ) x-e^{-x} x^2-e^{-x} x \left (1+e^x-e^x x\right )-e^{-x} \left (2 e^x+x-e^x x\right )+e^{-x} x \left (2 e^x+x-e^x x\right )}+2 \int e^{e^{-x} x \left (-2 e^x-x+e^x x\right )} x^2 \, dx-2 \int e^{e^{-x} x \left (-e^x-x+e^x x\right )} x^2 \, dx-\int e^{e^{-x} x \left (-e^x-x+e^x x\right )} \, dx+\int e^{e^{-x} x \left (-e^x-x+e^x x\right )} x \, dx-\int e^{e^{-x} x \left (-2 e^x-x+e^x x\right )} x^3 \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 1.94, size = 27, normalized size = 0.93 \begin {gather*} 9 e^{2 x}-e^{-x \left (1+\left (-1+e^{-x}\right ) x\right )} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-2*x - (x^2 - E^x*x^2)/E^x)*(18*E^(4*x + (x^2 - E^x*x^2)/E^x) + 2*x^2 - x^3 + E^x*(-1 + x - 2*x^2
)),x]

[Out]

9*E^(2*x) - x/E^(x*(1 + (-1 + E^(-x))*x))

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 34, normalized size = 1.17 \begin {gather*} -x e^{\left (-{\left (x^{2} - {\left (x^{2} - 2 \, x\right )} e^{x}\right )} e^{\left (-x\right )} + x\right )} + 9 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*exp(x)^4*exp((-exp(x)*x^2+x^2)/exp(x))+(-2*x^2+x-1)*exp(x)-x^3+2*x^2)/exp(x)^2/exp((-exp(x)*x^2+
x^2)/exp(x)),x, algorithm="fricas")

[Out]

-x*e^(-(x^2 - (x^2 - 2*x)*e^x)*e^(-x) + x) + 9*e^(2*x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -{\left (x^{3} - 2 \, x^{2} + {\left (2 \, x^{2} - x + 1\right )} e^{x} - 18 \, e^{\left (-{\left (x^{2} e^{x} - x^{2}\right )} e^{\left (-x\right )} + 4 \, x\right )}\right )} e^{\left ({\left (x^{2} e^{x} - x^{2}\right )} e^{\left (-x\right )} - 2 \, x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*exp(x)^4*exp((-exp(x)*x^2+x^2)/exp(x))+(-2*x^2+x-1)*exp(x)-x^3+2*x^2)/exp(x)^2/exp((-exp(x)*x^2+
x^2)/exp(x)),x, algorithm="giac")

[Out]

integrate(-(x^3 - 2*x^2 + (2*x^2 - x + 1)*e^x - 18*e^(-(x^2*e^x - x^2)*e^(-x) + 4*x))*e^((x^2*e^x - x^2)*e^(-x
) - 2*x), x)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 26, normalized size = 0.90

method	result	size
risch	\(9 \,{\mathrm e}^{2 x}-x \,{\mathrm e}^{-x \left (x \,{\mathrm e}^{-x}-x +1\right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*exp(x)^4*exp((-exp(x)*x^2+x^2)/exp(x))+(-2*x^2+x-1)*exp(x)-x^3+2*x^2)/exp(x)^2/exp((-exp(x)*x^2+x^2)/e
xp(x)),x,method=_RETURNVERBOSE)

[Out]

9*exp(2*x)-x*exp(-x*(x*exp(-x)-x+1))

________________________________________________________________________________________

maxima [A]  time = 0.87, size = 27, normalized size = 0.93 \begin {gather*} -x e^{\left (-x^{2} e^{\left (-x\right )} + x^{2} - x\right )} + 9 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*exp(x)^4*exp((-exp(x)*x^2+x^2)/exp(x))+(-2*x^2+x-1)*exp(x)-x^3+2*x^2)/exp(x)^2/exp((-exp(x)*x^2+
x^2)/exp(x)),x, algorithm="maxima")

[Out]

-x*e^(-x^2*e^(-x) + x^2 - x) + 9*e^(2*x)

________________________________________________________________________________________

mupad [B]  time = 1.91, size = 28, normalized size = 0.97 \begin {gather*} 9\,{\mathrm {e}}^{2\,x}-x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^{-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-2*x)*exp(exp(-x)*(x^2*exp(x) - x^2))*(18*exp(4*x)*exp(-exp(-x)*(x^2*exp(x) - x^2)) - exp(x)*(2*x^2 -
x + 1) + 2*x^2 - x^3),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.28, size = 26, normalized size = 0.90 \begin {gather*} - x e^{- x} e^{- \left (- x^{2} e^{x} + x^{2}\right ) e^{- x}} + 9 e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*exp(x)**4*exp((-exp(x)*x**2+x**2)/exp(x))+(-2*x**2+x-1)*exp(x)-x**3+2*x**2)/exp(x)**2/exp((-exp(
x)*x**2+x**2)/exp(x)),x)

[Out]

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

________________________________________________________________________________________