3.7.32 \(\int \frac {1}{3} (-6+3 e^{-e^{\frac {1}{3} (3 x-x^2)}+\frac {1}{3} (3 x-x^2)} (-3+2 x)) \, dx\)

Optimal. Leaf size=21 \[ 3 e^{-e^{x-\frac {x^2}{3}}}-2 x \]

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Rubi [F]  time = 0.30, 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 \frac {1}{3} \left (-6+3 e^{-e^{\frac {1}{3} \left (3 x-x^2\right )}+\frac {1}{3} \left (3 x-x^2\right )} (-3+2 x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 21, normalized size = 1.00 \begin {gather*} 3 e^{-e^{x-\frac {x^2}{3}}}-2 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3/E^E^(x - x^2/3) - 2*x

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fricas [B]  time = 0.67, size = 46, normalized size = 2.19 \begin {gather*} -{\left (2 \, x e^{\left (-\frac {1}{3} \, x^{2} + x\right )} - e^{\left (-\frac {1}{3} \, x^{2} + x - e^{\left (-\frac {1}{3} \, x^{2} + x\right )} + \log \relax (3)\right )}\right )} e^{\left (\frac {1}{3} \, x^{2} - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*x*e^(-1/3*x^2 + x) - e^(-1/3*x^2 + x - e^(-1/3*x^2 + x) + log(3)))*e^(1/3*x^2 - x)

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giac [A]  time = 0.56, size = 17, normalized size = 0.81 \begin {gather*} -2 \, x + 3 \, e^{\left (-e^{\left (-\frac {1}{3} \, x^{2} + x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x + 3*e^(-e^(-1/3*x^2 + x))

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maple [A]  time = 0.08, size = 17, normalized size = 0.81

method	result	size
risch	\(3 \,{\mathrm e}^{-{\mathrm e}^{-\frac {x \left (x -3\right )}{3}}}-2 x\)	\(17\)
default	\({\mathrm e}^{-{\mathrm e}^{-\frac {1}{3} x^{2}+x}+\ln \relax (3)}-2 x\)	\(19\)
norman	\({\mathrm e}^{-{\mathrm e}^{-\frac {1}{3} x^{2}+x}+\ln \relax (3)}-2 x\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(2*x-3)*exp(-1/3*x^2+x)*exp(-exp(-1/3*x^2+x)+ln(3))-2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.80, size = 17, normalized size = 0.81 \begin {gather*} -2 \, x + 3 \, e^{\left (-e^{\left (-\frac {1}{3} \, x^{2} + x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x + 3*e^(-e^(-1/3*x^2 + x))

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mupad [B]  time = 0.11, size = 17, normalized size = 0.81 \begin {gather*} 3\,{\mathrm {e}}^{-{\mathrm {e}}^{-\frac {x^2}{3}}\,{\mathrm {e}}^x}-2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(3) - exp(x - x^2/3))*exp(x - x^2/3)*(2*x - 3))/3 - 2,x)

[Out]

3*exp(-exp(-x^2/3)*exp(x)) - 2*x

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sympy [A]  time = 0.29, size = 14, normalized size = 0.67 \begin {gather*} - 2 x + 3 e^{- e^{- \frac {x^{2}}{3} + x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x + 3*exp(-exp(-x**2/3 + x))

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