3.60.41 \(\int e^{-2 e^x} (e^{2 e^x+2 x+x^2} (40+40 x)+e^{2 x+x^2} (4 x+4 x^2-4 e^x x^2+4 x^3)) \, dx\)

Optimal. Leaf size=24 \[ 2 e^{2 x+x^2} \left (10+e^{-2 e^x} x^2\right ) \]

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Rubi [F]  time = 0.91, 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 e^x} \left (e^{2 e^x+2 x+x^2} (40+40 x)+e^{2 x+x^2} \left (4 x+4 x^2-4 e^x x^2+4 x^3\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*E^x + 2*x + x^2)*(40 + 40*x) + E^(2*x + x^2)*(4*x + 4*x^2 - 4*E^x*x^2 + 4*x^3))/E^(2*E^x),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-2 e^x+x (2+x)} \left (-4 e^x x^2+40 e^{2 e^x} (1+x)+4 x \left (1+x+x^2\right )\right ) \, dx\\ &=\int \left (-4 e^{-2 e^x+x+x (2+x)} x^2+40 e^{x (2+x)} (1+x)+4 e^{-2 e^x+x (2+x)} x \left (1+x+x^2\right )\right ) \, dx\\ &=-\left (4 \int e^{-2 e^x+x+x (2+x)} x^2 \, dx\right )+4 \int e^{-2 e^x+x (2+x)} x \left (1+x+x^2\right ) \, dx+40 \int e^{x (2+x)} (1+x) \, dx\\ &=-\left (4 \int e^{-2 e^x+x+x (2+x)} x^2 \, dx\right )+4 \int \left (e^{-2 e^x+x (2+x)} x+e^{-2 e^x+x (2+x)} x^2+e^{-2 e^x+x (2+x)} x^3\right ) \, dx+40 \int e^{2 x+x^2} (1+x) \, dx\\ &=20 e^{2 x+x^2}+4 \int e^{-2 e^x+x (2+x)} x \, dx+4 \int e^{-2 e^x+x (2+x)} x^2 \, dx-4 \int e^{-2 e^x+x+x (2+x)} x^2 \, dx+4 \int e^{-2 e^x+x (2+x)} x^3 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 28, normalized size = 1.17 \begin {gather*} 2 e^{-2 e^x+x (2+x)} \left (10 e^{2 e^x}+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*E^x + 2*x + x^2)*(40 + 40*x) + E^(2*x + x^2)*(4*x + 4*x^2 - 4*E^x*x^2 + 4*x^3))/E^(2*E^x),x]

[Out]

2*E^(-2*E^x + x*(2 + x))*(10*E^(2*E^x) + x^2)

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fricas [B]  time = 0.75, size = 47, normalized size = 1.96 \begin {gather*} 2 \, {\left (x^{2} e^{\left (2 \, x^{2} + 4 \, x\right )} + 10 \, e^{\left (2 \, x^{2} + 4 \, x + 2 \, e^{x}\right )}\right )} e^{\left (-x^{2} - 2 \, x - 2 \, e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*x+40)*exp(x^2+2*x)*exp(exp(x))^2+(-4*exp(x)*x^2+4*x^3+4*x^2+4*x)*exp(x^2+2*x))/exp(exp(x))^2,x,
algorithm="fricas")

[Out]

2*(x^2*e^(2*x^2 + 4*x) + 10*e^(2*x^2 + 4*x + 2*e^x))*e^(-x^2 - 2*x - 2*e^x)

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giac [A]  time = 0.13, size = 28, normalized size = 1.17 \begin {gather*} 2 \, x^{2} e^{\left (x^{2} + 2 \, x - 2 \, e^{x}\right )} + 20 \, e^{\left (x^{2} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*x+40)*exp(x^2+2*x)*exp(exp(x))^2+(-4*exp(x)*x^2+4*x^3+4*x^2+4*x)*exp(x^2+2*x))/exp(exp(x))^2,x,
algorithm="giac")

[Out]

2*x^2*e^(x^2 + 2*x - 2*e^x) + 20*e^(x^2 + 2*x)

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maple [A]  time = 0.06, size = 27, normalized size = 1.12

method	result	size
risch	\(20 \,{\mathrm e}^{x \left (2+x \right )}+2 x^{2} {\mathrm e}^{x^{2}-2 \,{\mathrm e}^{x}+2 x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((40*x+40)*exp(x^2+2*x)*exp(exp(x))^2+(-4*exp(x)*x^2+4*x^3+4*x^2+4*x)*exp(x^2+2*x))/exp(exp(x))^2,x,method
=_RETURNVERBOSE)

[Out]

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

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maxima [C]  time = 0.44, size = 72, normalized size = 3.00 \begin {gather*} 2 \, x^{2} e^{\left (x^{2} + 2 \, x - 2 \, e^{x}\right )} - 20 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x + i\right ) e^{\left (-1\right )} - 20 \, {\left (\frac {\sqrt {\pi } {\left (x + 1\right )} {\left (\operatorname {erf}\left (\sqrt {-{\left (x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x + 1\right )}^{2}}} - e^{\left ({\left (x + 1\right )}^{2}\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*x+40)*exp(x^2+2*x)*exp(exp(x))^2+(-4*exp(x)*x^2+4*x^3+4*x^2+4*x)*exp(x^2+2*x))/exp(exp(x))^2,x,
algorithm="maxima")

[Out]

2*x^2*e^(x^2 + 2*x - 2*e^x) - 20*I*sqrt(pi)*erf(I*x + I)*e^(-1) - 20*(sqrt(pi)*(x + 1)*(erf(sqrt(-(x + 1)^2))
- 1)/sqrt(-(x + 1)^2) - e^((x + 1)^2))*e^(-1)

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mupad [B]  time = 0.22, size = 21, normalized size = 0.88 \begin {gather*} {\mathrm {e}}^{x^2+2\,x}\,\left (2\,x^2\,{\mathrm {e}}^{-2\,{\mathrm {e}}^x}+20\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-2*exp(x))*(exp(2*x + x^2)*(4*x - 4*x^2*exp(x) + 4*x^2 + 4*x^3) + exp(2*x + x^2)*exp(2*exp(x))*(40*x +
40)),x)

[Out]

exp(2*x + x^2)*(2*x^2*exp(-2*exp(x)) + 20)

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sympy [A]  time = 45.09, size = 29, normalized size = 1.21 \begin {gather*} 2 x^{2} e^{x^{2} + 2 x} e^{- 2 e^{x}} + 20 e^{x^{2} + 2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*x+40)*exp(x**2+2*x)*exp(exp(x))**2+(-4*exp(x)*x**2+4*x**3+4*x**2+4*x)*exp(x**2+2*x))/exp(exp(x)
)**2,x)

[Out]

2*x**2*exp(x**2 + 2*x)*exp(-2*exp(x)) + 20*exp(x**2 + 2*x)

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