3.90.77 \(\int \frac {e^{-4 x^2} (2 x^4+(12 x^3-4 x^4-24 x^5+8 x^6) \log (-3+x))}{(-3+x) \log ^3(-3+x)} \, dx\)

Optimal. Leaf size=18 \[ -\frac {e^{-4 x^2} x^4}{\log ^2(-3+x)} \]

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Rubi [F]  time = 2.13, 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 {e^{-4 x^2} \left (2 x^4+\left (12 x^3-4 x^4-24 x^5+8 x^6\right ) \log (-3+x)\right )}{(-3+x) \log ^3(-3+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x^4 + (12*x^3 - 4*x^4 - 24*x^5 + 8*x^6)*Log[-3 + x])/(E^(4*x^2)*(-3 + x)*Log[-3 + x]^3),x]

[Out]

216*Defer[Int][1/(E^(4*x^2)*Log[-3 + x]^3), x] + 162*Defer[Int][1/(E^(4*x^2)*(-3 + x)*Log[-3 + x]^3), x] + 108
*Defer[Int][(-3 + x)/(E^(4*x^2)*Log[-3 + x]^3), x] + 24*Defer[Int][(-3 + x)^2/(E^(4*x^2)*Log[-3 + x]^3), x] +
2*Defer[Int][(-3 + x)^3/(E^(4*x^2)*Log[-3 + x]^3), x] + 1836*Defer[Int][1/(E^(4*x^2)*Log[-3 + x]^2), x] + 3132
*Defer[Int][(-3 + x)/(E^(4*x^2)*Log[-3 + x]^2), x] + 2124*Defer[Int][(-3 + x)^2/(E^(4*x^2)*Log[-3 + x]^2), x]
+ 716*Defer[Int][(-3 + x)^3/(E^(4*x^2)*Log[-3 + x]^2), x] + 120*Defer[Int][(-3 + x)^4/(E^(4*x^2)*Log[-3 + x]^2
), x] + 8*Defer[Int][(-3 + x)^5/(E^(4*x^2)*Log[-3 + x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 e^{-4 x^2} x^4}{(-3+x) \log ^3(-3+x)}+\frac {4 e^{-4 x^2} x^3 \left (-1+2 x^2\right )}{\log ^2(-3+x)}\right ) \, dx\\ &=2 \int \frac {e^{-4 x^2} x^4}{(-3+x) \log ^3(-3+x)} \, dx+4 \int \frac {e^{-4 x^2} x^3 \left (-1+2 x^2\right )}{\log ^2(-3+x)} \, dx\\ &=2 \int \left (\frac {27 e^{-4 x^2}}{\log ^3(-3+x)}+\frac {81 e^{-4 x^2}}{(-3+x) \log ^3(-3+x)}+\frac {9 e^{-4 x^2} x}{\log ^3(-3+x)}+\frac {3 e^{-4 x^2} x^2}{\log ^3(-3+x)}+\frac {e^{-4 x^2} x^3}{\log ^3(-3+x)}\right ) \, dx+4 \int \left (\frac {459 e^{-4 x^2}}{\log ^2(-3+x)}+\frac {783 e^{-4 x^2} (-3+x)}{\log ^2(-3+x)}+\frac {531 e^{-4 x^2} (-3+x)^2}{\log ^2(-3+x)}+\frac {179 e^{-4 x^2} (-3+x)^3}{\log ^2(-3+x)}+\frac {30 e^{-4 x^2} (-3+x)^4}{\log ^2(-3+x)}+\frac {2 e^{-4 x^2} (-3+x)^5}{\log ^2(-3+x)}\right ) \, dx\\ &=2 \int \frac {e^{-4 x^2} x^3}{\log ^3(-3+x)} \, dx+6 \int \frac {e^{-4 x^2} x^2}{\log ^3(-3+x)} \, dx+8 \int \frac {e^{-4 x^2} (-3+x)^5}{\log ^2(-3+x)} \, dx+18 \int \frac {e^{-4 x^2} x}{\log ^3(-3+x)} \, dx+54 \int \frac {e^{-4 x^2}}{\log ^3(-3+x)} \, dx+120 \int \frac {e^{-4 x^2} (-3+x)^4}{\log ^2(-3+x)} \, dx+162 \int \frac {e^{-4 x^2}}{(-3+x) \log ^3(-3+x)} \, dx+716 \int \frac {e^{-4 x^2} (-3+x)^3}{\log ^2(-3+x)} \, dx+1836 \int \frac {e^{-4 x^2}}{\log ^2(-3+x)} \, dx+2124 \int \frac {e^{-4 x^2} (-3+x)^2}{\log ^2(-3+x)} \, dx+3132 \int \frac {e^{-4 x^2} (-3+x)}{\log ^2(-3+x)} \, dx\\ &=2 \int \left (\frac {27 e^{-4 x^2}}{\log ^3(-3+x)}+\frac {27 e^{-4 x^2} (-3+x)}{\log ^3(-3+x)}+\frac {9 e^{-4 x^2} (-3+x)^2}{\log ^3(-3+x)}+\frac {e^{-4 x^2} (-3+x)^3}{\log ^3(-3+x)}\right ) \, dx+6 \int \left (\frac {9 e^{-4 x^2}}{\log ^3(-3+x)}+\frac {6 e^{-4 x^2} (-3+x)}{\log ^3(-3+x)}+\frac {e^{-4 x^2} (-3+x)^2}{\log ^3(-3+x)}\right ) \, dx+8 \int \frac {e^{-4 x^2} (-3+x)^5}{\log ^2(-3+x)} \, dx+18 \int \left (\frac {3 e^{-4 x^2}}{\log ^3(-3+x)}+\frac {e^{-4 x^2} (-3+x)}{\log ^3(-3+x)}\right ) \, dx+54 \int \frac {e^{-4 x^2}}{\log ^3(-3+x)} \, dx+120 \int \frac {e^{-4 x^2} (-3+x)^4}{\log ^2(-3+x)} \, dx+162 \int \frac {e^{-4 x^2}}{(-3+x) \log ^3(-3+x)} \, dx+716 \int \frac {e^{-4 x^2} (-3+x)^3}{\log ^2(-3+x)} \, dx+1836 \int \frac {e^{-4 x^2}}{\log ^2(-3+x)} \, dx+2124 \int \frac {e^{-4 x^2} (-3+x)^2}{\log ^2(-3+x)} \, dx+3132 \int \frac {e^{-4 x^2} (-3+x)}{\log ^2(-3+x)} \, dx\\ &=2 \int \frac {e^{-4 x^2} (-3+x)^3}{\log ^3(-3+x)} \, dx+6 \int \frac {e^{-4 x^2} (-3+x)^2}{\log ^3(-3+x)} \, dx+8 \int \frac {e^{-4 x^2} (-3+x)^5}{\log ^2(-3+x)} \, dx+18 \int \frac {e^{-4 x^2} (-3+x)}{\log ^3(-3+x)} \, dx+18 \int \frac {e^{-4 x^2} (-3+x)^2}{\log ^3(-3+x)} \, dx+36 \int \frac {e^{-4 x^2} (-3+x)}{\log ^3(-3+x)} \, dx+4 \left (54 \int \frac {e^{-4 x^2}}{\log ^3(-3+x)} \, dx\right )+54 \int \frac {e^{-4 x^2} (-3+x)}{\log ^3(-3+x)} \, dx+120 \int \frac {e^{-4 x^2} (-3+x)^4}{\log ^2(-3+x)} \, dx+162 \int \frac {e^{-4 x^2}}{(-3+x) \log ^3(-3+x)} \, dx+716 \int \frac {e^{-4 x^2} (-3+x)^3}{\log ^2(-3+x)} \, dx+1836 \int \frac {e^{-4 x^2}}{\log ^2(-3+x)} \, dx+2124 \int \frac {e^{-4 x^2} (-3+x)^2}{\log ^2(-3+x)} \, dx+3132 \int \frac {e^{-4 x^2} (-3+x)}{\log ^2(-3+x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.78, size = 18, normalized size = 1.00 \begin {gather*} -\frac {e^{-4 x^2} x^4}{\log ^2(-3+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^4 + (12*x^3 - 4*x^4 - 24*x^5 + 8*x^6)*Log[-3 + x])/(E^(4*x^2)*(-3 + x)*Log[-3 + x]^3),x]

[Out]

-(x^4/(E^(4*x^2)*Log[-3 + x]^2))

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fricas [A]  time = 0.49, size = 17, normalized size = 0.94 \begin {gather*} -\frac {x^{4} e^{\left (-4 \, x^{2}\right )}}{\log \left (x - 3\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^6-24*x^5-4*x^4+12*x^3)*log(x-3)+2*x^4)/(x-3)/exp(2*x^2)^2/log(x-3)^3,x, algorithm="fricas")

[Out]

-x^4*e^(-4*x^2)/log(x - 3)^2

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giac [A]  time = 0.21, size = 17, normalized size = 0.94 \begin {gather*} -\frac {x^{4} e^{\left (-4 \, x^{2}\right )}}{\log \left (x - 3\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^6-24*x^5-4*x^4+12*x^3)*log(x-3)+2*x^4)/(x-3)/exp(2*x^2)^2/log(x-3)^3,x, algorithm="giac")

[Out]

-x^4*e^(-4*x^2)/log(x - 3)^2

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maple [A]  time = 0.46, size = 18, normalized size = 1.00

method	result	size
risch	\(-\frac {x^{4} {\mathrm e}^{-4 x^{2}}}{\ln \left (x -3\right )^{2}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^6-24*x^5-4*x^4+12*x^3)*ln(x-3)+2*x^4)/(x-3)/exp(2*x^2)^2/ln(x-3)^3,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 17, normalized size = 0.94 \begin {gather*} -\frac {x^{4} e^{\left (-4 \, x^{2}\right )}}{\log \left (x - 3\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^6-24*x^5-4*x^4+12*x^3)*log(x-3)+2*x^4)/(x-3)/exp(2*x^2)^2/log(x-3)^3,x, algorithm="maxima")

[Out]

-x^4*e^(-4*x^2)/log(x - 3)^2

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mupad [B]  time = 0.28, size = 17, normalized size = 0.94 \begin {gather*} -\frac {x^4\,{\mathrm {e}}^{-4\,x^2}}{{\ln \left (x-3\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-4*x^2)*(2*x^4 + log(x - 3)*(12*x^3 - 4*x^4 - 24*x^5 + 8*x^6)))/(log(x - 3)^3*(x - 3)),x)

[Out]

-(x^4*exp(-4*x^2))/log(x - 3)^2

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sympy [A]  time = 0.30, size = 17, normalized size = 0.94 \begin {gather*} - \frac {x^{4} e^{- 4 x^{2}}}{\log {\left (x - 3 \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**6-24*x**5-4*x**4+12*x**3)*ln(x-3)+2*x**4)/(x-3)/exp(2*x**2)**2/ln(x-3)**3,x)

[Out]

-x**4*exp(-4*x**2)/log(x - 3)**2

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