3.25.32 \(\int \frac {e^{-x^2+\frac {2 e^{-x^2} (6+2 \log (5))}{x^3}} (-36-24 x^2+(-12-8 x^2) \log (5))}{x^4} \, dx\)

Optimal. Leaf size=18 \[ e^{\frac {4 e^{-x^2} (3+\log (5))}{x^3}} \]

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Rubi [F]  time = 1.46, 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^{-x^2+\frac {2 e^{-x^2} (6+2 \log (5))}{x^3}} \left (-36-24 x^2+\left (-12-8 x^2\right ) \log (5)\right )}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-x^2 + (2*(6 + 2*Log[5]))/(E^x^2*x^3))*(-36 - 24*x^2 + (-12 - 8*x^2)*Log[5]))/x^4,x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x^2+\frac {2 e^{-x^2} (6+2 \log (5))}{x^3}} \left (-12 (3+\log (5))-8 x^2 (3+\log (5))\right )}{x^4} \, dx\\ &=\int \left (-\frac {12 e^{-x^2+\frac {2 e^{-x^2} (6+2 \log (5))}{x^3}} (3+\log (5))}{x^4}-\frac {8 e^{-x^2+\frac {2 e^{-x^2} (6+2 \log (5))}{x^3}} (3+\log (5))}{x^2}\right ) \, dx\\ &=-\left ((8 (3+\log (5))) \int \frac {e^{-x^2+\frac {2 e^{-x^2} (6+2 \log (5))}{x^3}}}{x^2} \, dx\right )-(12 (3+\log (5))) \int \frac {e^{-x^2+\frac {2 e^{-x^2} (6+2 \log (5))}{x^3}}}{x^4} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 29, normalized size = 1.61 \begin {gather*} 5^{\frac {4 e^{-x^2}}{x^3}} e^{\frac {12 e^{-x^2}}{x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-x^2 + (2*(6 + 2*Log[5]))/(E^x^2*x^3))*(-36 - 24*x^2 + (-12 - 8*x^2)*Log[5]))/x^4,x]

[Out]

5^(4/(E^x^2*x^3))*E^(12/(E^x^2*x^3))

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fricas [A]  time = 0.80, size = 26, normalized size = 1.44 \begin {gather*} e^{\left (x^{2} - \frac {x^{5} - 4 \, {\left (\log \relax (5) + 3\right )} e^{\left (-x^{2}\right )}}{x^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2-12)*log(5)-24*x^2-36)*exp((2*log(5)+6)/x^3/exp(x^2))^2/x^4/exp(x^2),x, algorithm="fricas")

[Out]

e^(x^2 - (x^5 - 4*(log(5) + 3)*e^(-x^2))/x^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {4 \, {\left (6 \, x^{2} + {\left (2 \, x^{2} + 3\right )} \log \relax (5) + 9\right )} e^{\left (-x^{2} + \frac {4 \, {\left (\log \relax (5) + 3\right )} e^{\left (-x^{2}\right )}}{x^{3}}\right )}}{x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2-12)*log(5)-24*x^2-36)*exp((2*log(5)+6)/x^3/exp(x^2))^2/x^4/exp(x^2),x, algorithm="giac")

[Out]

integrate(-4*(6*x^2 + (2*x^2 + 3)*log(5) + 9)*e^(-x^2 + 4*(log(5) + 3)*e^(-x^2)/x^3)/x^4, x)

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maple [A]  time = 0.11, size = 17, normalized size = 0.94

method	result	size
risch	\({\mathrm e}^{\frac {4 \left (\ln \relax (5)+3\right ) {\mathrm e}^{-x^{2}}}{x^{3}}}\)	\(17\)
norman	\({\mathrm e}^{\frac {2 \left (2 \ln \relax (5)+6\right ) {\mathrm e}^{-x^{2}}}{x^{3}}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^2-12)*ln(5)-24*x^2-36)*exp((2*ln(5)+6)/x^3/exp(x^2))^2/x^4/exp(x^2),x,method=_RETURNVERBOSE)

[Out]

exp(4*(ln(5)+3)*exp(-x^2)/x^3)

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maxima [A]  time = 0.84, size = 26, normalized size = 1.44 \begin {gather*} e^{\left (\frac {4 \, e^{\left (-x^{2}\right )} \log \relax (5)}{x^{3}} + \frac {12 \, e^{\left (-x^{2}\right )}}{x^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2-12)*log(5)-24*x^2-36)*exp((2*log(5)+6)/x^3/exp(x^2))^2/x^4/exp(x^2),x, algorithm="maxima")

[Out]

e^(4*e^(-x^2)*log(5)/x^3 + 12*e^(-x^2)/x^3)

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mupad [B]  time = 1.69, size = 26, normalized size = 1.44 \begin {gather*} 5^{\frac {4\,{\mathrm {e}}^{-x^2}}{x^3}}\,{\mathrm {e}}^{\frac {12\,{\mathrm {e}}^{-x^2}}{x^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((2*exp(-x^2)*(2*log(5) + 6))/x^3)*exp(-x^2)*(log(5)*(8*x^2 + 12) + 24*x^2 + 36))/x^4,x)

[Out]

5^((4*exp(-x^2))/x^3)*exp((12*exp(-x^2))/x^3)

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sympy [A]  time = 0.39, size = 15, normalized size = 0.83 \begin {gather*} e^{\frac {2 \left (2 \log {\relax (5 )} + 6\right ) e^{- x^{2}}}{x^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**2-12)*ln(5)-24*x**2-36)*exp((2*ln(5)+6)/x**3/exp(x**2))**2/x**4/exp(x**2),x)

[Out]

exp(2*(2*log(5) + 6)*exp(-x**2)/x**3)

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