3.8.13 \(\int \frac {e^{\frac {11-4 x^2 \log (x)}{22 \log (x)}} (-11-8 x^2 \log ^2(x))}{22 x \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ e^{-\frac {1}{22} \log ^2\left (e^{2 x}\right )+\frac {1}{2 \log (x)}} \]

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Rubi [F]  time = 0.55, 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 {e^{\frac {11-4 x^2 \log (x)}{22 \log (x)}} \left (-11-8 x^2 \log ^2(x)\right )}{22 x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((11 - 4*x^2*Log[x])/(22*Log[x]))*(-11 - 8*x^2*Log[x]^2))/(22*x*Log[x]^2),x]

[Out]

(-4*Defer[Int][E^((11 - 4*x^2*Log[x])/(22*Log[x]))*x, x])/11 - Defer[Int][E^((11 - 4*x^2*Log[x])/(22*Log[x]))/
(x*Log[x]^2), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{22} \int \frac {e^{\frac {11-4 x^2 \log (x)}{22 \log (x)}} \left (-11-8 x^2 \log ^2(x)\right )}{x \log ^2(x)} \, dx\\ &=\frac {1}{22} \int \left (-8 e^{\frac {11-4 x^2 \log (x)}{22 \log (x)}} x-\frac {11 e^{\frac {11-4 x^2 \log (x)}{22 \log (x)}}}{x \log ^2(x)}\right ) \, dx\\ &=-\left (\frac {4}{11} \int e^{\frac {11-4 x^2 \log (x)}{22 \log (x)}} x \, dx\right )-\frac {1}{2} \int \frac {e^{\frac {11-4 x^2 \log (x)}{22 \log (x)}}}{x \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 18, normalized size = 0.78 \begin {gather*} e^{-\frac {2 x^2}{11}+\frac {1}{2 \log (x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((11 - 4*x^2*Log[x])/(22*Log[x]))*(-11 - 8*x^2*Log[x]^2))/(22*x*Log[x]^2),x]

[Out]

E^((-2*x^2)/11 + 1/(2*Log[x]))

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fricas [A]  time = 0.55, size = 16, normalized size = 0.70 \begin {gather*} e^{\left (-\frac {4 \, x^{2} \log \relax (x) - 11}{22 \, \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/22*(-8*x^2*log(x)^2-11)*exp(1/22*(-4*x^2*log(x)+11)/log(x))/x/log(x)^2,x, algorithm="fricas")

[Out]

e^(-1/22*(4*x^2*log(x) - 11)/log(x))

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giac [A]  time = 0.36, size = 13, normalized size = 0.57 \begin {gather*} e^{\left (-\frac {2}{11} \, x^{2} + \frac {1}{2 \, \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/22*(-8*x^2*log(x)^2-11)*exp(1/22*(-4*x^2*log(x)+11)/log(x))/x/log(x)^2,x, algorithm="giac")

[Out]

e^(-2/11*x^2 + 1/2/log(x))

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maple [A]  time = 0.03, size = 17, normalized size = 0.74

method	result	size
norman	\({\mathrm e}^{\frac {-4 x^{2} \ln \relax (x )+11}{22 \ln \relax (x )}}\)	\(17\)
risch	\({\mathrm e}^{-\frac {4 x^{2} \ln \relax (x )-11}{22 \ln \relax (x )}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/22*(-8*x^2*ln(x)^2-11)*exp(1/22*(-4*x^2*ln(x)+11)/ln(x))/x/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

exp(1/22*(-4*x^2*ln(x)+11)/ln(x))

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maxima [A]  time = 0.58, size = 13, normalized size = 0.57 \begin {gather*} e^{\left (-\frac {2}{11} \, x^{2} + \frac {1}{2 \, \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/22*(-8*x^2*log(x)^2-11)*exp(1/22*(-4*x^2*log(x)+11)/log(x))/x/log(x)^2,x, algorithm="maxima")

[Out]

e^(-2/11*x^2 + 1/2/log(x))

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mupad [B]  time = 0.58, size = 14, normalized size = 0.61 \begin {gather*} {\mathrm {e}}^{\frac {1}{2\,\ln \relax (x)}}\,{\mathrm {e}}^{-\frac {2\,x^2}{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-((2*x^2*log(x))/11 - 1/2)/log(x))*(8*x^2*log(x)^2 + 11))/(22*x*log(x)^2),x)

[Out]

exp(1/(2*log(x)))*exp(-(2*x^2)/11)

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sympy [A]  time = 0.30, size = 17, normalized size = 0.74 \begin {gather*} e^{\frac {- \frac {2 x^{2} \log {\relax (x )}}{11} + \frac {1}{2}}{\log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/22*(-8*x**2*ln(x)**2-11)*exp(1/22*(-4*x**2*ln(x)+11)/ln(x))/x/ln(x)**2,x)

[Out]

exp((-2*x**2*log(x)/11 + 1/2)/log(x))

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