3.17.42 \(\int \frac {1}{24} e^{-1-\frac {x^2+x^3}{2 e}} (12 e+48 e^{1+\frac {x^2+x^3}{2 e}}-12 x^2-18 x^3+(2 x+3 x^2) \log (3)) \, dx\)

Optimal. Leaf size=33 \[ 2 x-\frac {1}{4} e^{-\frac {x \left (x+x^2\right )}{2 e}} \left (-2 x+\frac {\log (3)}{3}\right ) \]

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Rubi [F]  time = 1.55, 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}{24} e^{-1-\frac {x^2+x^3}{2 e}} \left (12 e+48 e^{1+\frac {x^2+x^3}{2 e}}-12 x^2-18 x^3+\left (2 x+3 x^2\right ) \log (3)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-1 - (x^2 + x^3)/(2*E))*(12*E + 48*E^(1 + (x^2 + x^3)/(2*E)) - 12*x^2 - 18*x^3 + (2*x + 3*x^2)*Log[3])
)/24,x]

[Out]

2*x - (E^(-1/2*x^2/E - x^3/(2*E))*Log[3])/12 + Defer[Int][E^(-1/2*(x^2*(1 + x))/E), x]/2 - Defer[Int][E^(-1 -
x^2/(2*E) - x^3/(2*E))*x^2, x]/2 - (3*Defer[Int][E^(-1 - x^2/(2*E) - x^3/(2*E))*x^3, x])/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{24} \int e^{-1-\frac {x^2+x^3}{2 e}} \left (12 e+48 e^{1+\frac {x^2+x^3}{2 e}}-12 x^2-18 x^3+\left (2 x+3 x^2\right ) \log (3)\right ) \, dx\\ &=\frac {1}{24} \int e^{-1-\frac {x^2}{2 e}-\frac {x^3}{2 e}} \left (12 e+48 e^{1+\frac {x^2+x^3}{2 e}}-12 x^2-18 x^3+\left (2 x+3 x^2\right ) \log (3)\right ) \, dx\\ &=\frac {1}{24} \int \left (48+12 e^{-\frac {x^2}{2 e}-\frac {x^3}{2 e}}-12 e^{-1-\frac {x^2}{2 e}-\frac {x^3}{2 e}} x^2-18 e^{-1-\frac {x^2}{2 e}-\frac {x^3}{2 e}} x^3+e^{-1-\frac {x^2}{2 e}-\frac {x^3}{2 e}} x (2+3 x) \log (3)\right ) \, dx\\ &=2 x+\frac {1}{2} \int e^{-\frac {x^2}{2 e}-\frac {x^3}{2 e}} \, dx-\frac {1}{2} \int e^{-1-\frac {x^2}{2 e}-\frac {x^3}{2 e}} x^2 \, dx-\frac {3}{4} \int e^{-1-\frac {x^2}{2 e}-\frac {x^3}{2 e}} x^3 \, dx+\frac {1}{24} \log (3) \int e^{-1-\frac {x^2}{2 e}-\frac {x^3}{2 e}} x (2+3 x) \, dx\\ &=2 x-\frac {1}{12} e^{-\frac {x^2}{2 e}-\frac {x^3}{2 e}} \log (3)+\frac {1}{2} \int e^{-\frac {x^2 (1+x)}{2 e}} \, dx-\frac {1}{2} \int e^{-1-\frac {x^2}{2 e}-\frac {x^3}{2 e}} x^2 \, dx-\frac {3}{4} \int e^{-1-\frac {x^2}{2 e}-\frac {x^3}{2 e}} x^3 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 48, normalized size = 1.45 \begin {gather*} \frac {1}{24} \left (48 x+\frac {e^{-\frac {x^2 (1+x)}{2 e}} \left (36 x^2-2 \log (9)-2 x (-12+\log (27))\right )}{2+3 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-1 - (x^2 + x^3)/(2*E))*(12*E + 48*E^(1 + (x^2 + x^3)/(2*E)) - 12*x^2 - 18*x^3 + (2*x + 3*x^2)*L
og[3]))/24,x]

[Out]

(48*x + (36*x^2 - 2*Log[9] - 2*x*(-12 + Log[27]))/(E^((x^2*(1 + x))/(2*E))*(2 + 3*x)))/24

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fricas [A]  time = 0.76, size = 49, normalized size = 1.48 \begin {gather*} \frac {1}{12} \, {\left (6 \, x e + 24 \, x e^{\left (\frac {1}{2} \, {\left (x^{3} + x^{2} + 2 \, e\right )} e^{\left (-1\right )}\right )} - e \log \relax (3)\right )} e^{\left (-\frac {1}{2} \, {\left (x^{3} + x^{2} + 2 \, e\right )} e^{\left (-1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/24*(48*exp(1)*exp(1/2*(x^3+x^2)/exp(1))+(3*x^2+2*x)*log(3)+12*exp(1)-18*x^3-12*x^2)/exp(1)/exp(1/2
*(x^3+x^2)/exp(1)),x, algorithm="fricas")

[Out]

1/12*(6*x*e + 24*x*e^(1/2*(x^3 + x^2 + 2*e)*e^(-1)) - e*log(3))*e^(-1/2*(x^3 + x^2 + 2*e)*e^(-1))

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giac [A]  time = 0.35, size = 35, normalized size = 1.06 \begin {gather*} \frac {1}{2} \, x e^{\left (-\frac {1}{2} \, {\left (x^{3} + x^{2}\right )} e^{\left (-1\right )}\right )} - \frac {1}{12} \, e^{\left (-\frac {1}{2} \, {\left (x^{3} + x^{2}\right )} e^{\left (-1\right )}\right )} \log \relax (3) + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/24*(48*exp(1)*exp(1/2*(x^3+x^2)/exp(1))+(3*x^2+2*x)*log(3)+12*exp(1)-18*x^3-12*x^2)/exp(1)/exp(1/2
*(x^3+x^2)/exp(1)),x, algorithm="giac")

[Out]

1/2*x*e^(-1/2*(x^3 + x^2)*e^(-1)) - 1/12*e^(-1/2*(x^3 + x^2)*e^(-1))*log(3) + 2*x

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maple [A]  time = 0.10, size = 36, normalized size = 1.09

method	result	size
risch	\(2 x +\frac {\left (-2 \,{\mathrm e} \ln \relax (3)+12 x \,{\mathrm e}\right ) {\mathrm e}^{-1-\frac {x^{3} {\mathrm e}^{-1}}{2}-\frac {x^{2} {\mathrm e}^{-1}}{2}}}{24}\)	\(36\)
norman	\(\left (\frac {x}{2}+2 x \,{\mathrm e}^{\frac {\left (x^{3}+x^{2}\right ) {\mathrm e}^{-1}}{2}}-\frac {\ln \relax (3)}{12}\right ) {\mathrm e}^{-\frac {\left (x^{3}+x^{2}\right ) {\mathrm e}^{-1}}{2}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/24*(48*exp(1)*exp(1/2*(x^3+x^2)/exp(1))+(3*x^2+2*x)*ln(3)+12*exp(1)-18*x^3-12*x^2)/exp(1)/exp(1/2*(x^3+x
^2)/exp(1)),x,method=_RETURNVERBOSE)

[Out]

2*x+1/24*(-2*exp(1)*ln(3)+12*x*exp(1))*exp(-1-1/2*x^3*exp(-1)-1/2*x^2*exp(-1))

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maxima [A]  time = 0.83, size = 30, normalized size = 0.91 \begin {gather*} \frac {1}{12} \, {\left (6 \, x - \log \relax (3)\right )} e^{\left (-\frac {1}{2} \, x^{3} e^{\left (-1\right )} - \frac {1}{2} \, x^{2} e^{\left (-1\right )}\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/24*(48*exp(1)*exp(1/2*(x^3+x^2)/exp(1))+(3*x^2+2*x)*log(3)+12*exp(1)-18*x^3-12*x^2)/exp(1)/exp(1/2
*(x^3+x^2)/exp(1)),x, algorithm="maxima")

[Out]

1/12*(6*x - log(3))*e^(-1/2*x^3*e^(-1) - 1/2*x^2*e^(-1)) + 2*x

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mupad [B]  time = 1.11, size = 43, normalized size = 1.30 \begin {gather*} 2\,x-\frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^{-1}\,x^3}{2}-\frac {{\mathrm {e}}^{-1}\,x^2}{2}}\,\ln \relax (3)}{12}+\frac {x\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{-1}\,x^3}{2}-\frac {{\mathrm {e}}^{-1}\,x^2}{2}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-1)*exp(-exp(-1)*(x^2/2 + x^3/2))*(exp(1)/2 + (log(3)*(2*x + 3*x^2))/24 + 2*exp(1)*exp(exp(-1)*(x^2/2
+ x^3/2)) - x^2/2 - (3*x^3)/4),x)

[Out]

2*x - (exp(- (x^2*exp(-1))/2 - (x^3*exp(-1))/2)*log(3))/12 + (x*exp(- (x^2*exp(-1))/2 - (x^3*exp(-1))/2))/2

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sympy [A]  time = 0.18, size = 26, normalized size = 0.79 \begin {gather*} 2 x + \frac {\left (6 x - \log {\relax (3 )}\right ) e^{- \frac {\frac {x^{3}}{2} + \frac {x^{2}}{2}}{e}}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/24*(48*exp(1)*exp(1/2*(x**3+x**2)/exp(1))+(3*x**2+2*x)*ln(3)+12*exp(1)-18*x**3-12*x**2)/exp(1)/exp
(1/2*(x**3+x**2)/exp(1)),x)

[Out]

2*x + (6*x - log(3))*exp(-(x**3/2 + x**2/2)*exp(-1))/12

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