3.59.24 \(\int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log (\log (\frac {3}{x}))}} ((6+3 x) \log (\frac {3}{x})+6 \log (\frac {3}{x}) \log (x)-3 x \log ^2(x))}{(4+4 x+x^2) \log (\frac {3}{x})+(-4 x-2 x^2) \log (\frac {3}{x}) \log (x) \log (\log (\frac {3}{x}))+x^2 \log (\frac {3}{x}) \log ^2(x) \log ^2(\log (\frac {3}{x}))} \, dx\)

Optimal. Leaf size=27 \[ e^{\frac {3}{\frac {2+x}{x \log (x)}-\log \left (\log \left (\frac {3}{x}\right )\right )}} \]

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Rubi [F]  time = 1.64, 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 {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((6 + 3*x)*Log[3/x] + 6*Log[3/x]*Log[x] - 3*x*Log[x]^2)/(x^((3*x)/(-2 - x + x*Log[x]*Log[Log[3/x]]))*((4 +
4*x + x^2)*Log[3/x] + (-4*x - 2*x^2)*Log[3/x]*Log[x]*Log[Log[3/x]] + x^2*Log[3/x]*Log[x]^2*Log[Log[3/x]]^2)),
x]

[Out]

6*Defer[Int][x^((3*x)/(2 + x - x*Log[x]*Log[Log[3/x]]))/(-2 - x + x*Log[x]*Log[Log[3/x]])^2, x] + 3*Defer[Int]
[x^(1 + (3*x)/(2 + x - x*Log[x]*Log[Log[3/x]]))/(-2 - x + x*Log[x]*Log[Log[3/x]])^2, x] + 6*Defer[Int][(x^((3*
x)/(2 + x - x*Log[x]*Log[Log[3/x]]))*Log[x])/(-2 - x + x*Log[x]*Log[Log[3/x]])^2, x] - 3*Defer[Int][(x^(1 + (3
*x)/(2 + x - x*Log[x]*Log[Log[3/x]]))*Log[x]^2)/(Log[3/x]*(-2 - x + x*Log[x]*Log[Log[3/x]])^2), x]

Rubi steps

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

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Mathematica [A]  time = 0.61, size = 23, normalized size = 0.85 \begin {gather*} x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((6 + 3*x)*Log[3/x] + 6*Log[3/x]*Log[x] - 3*x*Log[x]^2)/(x^((3*x)/(-2 - x + x*Log[x]*Log[Log[3/x]]))
*((4 + 4*x + x^2)*Log[3/x] + (-4*x - 2*x^2)*Log[3/x]*Log[x]*Log[Log[3/x]] + x^2*Log[3/x]*Log[x]^2*Log[Log[3/x]
]^2)),x]

[Out]

x^((-3*x)/(-2 - x + x*Log[x]*Log[Log[3/x]]))

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fricas [A]  time = 1.00, size = 46, normalized size = 1.70 \begin {gather*} e^{\left (-\frac {3 \, {\left (x \log \relax (3) - x \log \left (\frac {3}{x}\right )\right )}}{{\left (x \log \relax (3) - x \log \left (\frac {3}{x}\right )\right )} \log \left (\log \left (\frac {3}{x}\right )\right ) - x - 2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(x)^2+6*log(3/x)*log(x)+(6+3*x)*log(3/x))*exp(-3*x*log(x)/(x*log(x)*log(log(3/x))-x-2))/(x^
2*log(3/x)*log(x)^2*log(log(3/x))^2+(-2*x^2-4*x)*log(3/x)*log(x)*log(log(3/x))+(x^2+4*x+4)*log(3/x)),x, algori
thm="fricas")

[Out]

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

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giac [A]  time = 0.95, size = 25, normalized size = 0.93 \begin {gather*} \frac {1}{x^{\frac {3 \, x}{x \log \relax (x) \log \left (\log \left (\frac {3}{x}\right )\right ) - x - 2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(x)^2+6*log(3/x)*log(x)+(6+3*x)*log(3/x))*exp(-3*x*log(x)/(x*log(x)*log(log(3/x))-x-2))/(x^
2*log(3/x)*log(x)^2*log(log(3/x))^2+(-2*x^2-4*x)*log(3/x)*log(x)*log(log(3/x))+(x^2+4*x+4)*log(3/x)),x, algori
thm="giac")

[Out]

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

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maple [A]  time = 0.49, size = 25, normalized size = 0.93

method	result	size
risch	\(x^{-\frac {3 x}{x \ln \relax (x ) \ln \left (\ln \relax (3)-\ln \relax (x )\right )-x -2}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.72, size = 69, normalized size = 2.56 \begin {gather*} e^{\left (-\frac {6 \, \log \relax (x)}{x \log \relax (x)^{2} \log \left (\log \relax (3) - \log \relax (x)\right )^{2} - 2 \, {\left (x \log \left (\log \relax (3) - \log \relax (x)\right ) + \log \left (\log \relax (3) - \log \relax (x)\right )\right )} \log \relax (x) + x + 2} - \frac {3 \, \log \relax (x)}{\log \relax (x) \log \left (\log \relax (3) - \log \relax (x)\right ) - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(x)^2+6*log(3/x)*log(x)+(6+3*x)*log(3/x))*exp(-3*x*log(x)/(x*log(x)*log(log(3/x))-x-2))/(x^
2*log(3/x)*log(x)^2*log(log(3/x))^2+(-2*x^2-4*x)*log(3/x)*log(x)*log(log(3/x))+(x^2+4*x+4)*log(3/x)),x, algori
thm="maxima")

[Out]

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

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mupad [B]  time = 4.59, size = 24, normalized size = 0.89 \begin {gather*} {\mathrm {e}}^{\frac {3\,x\,\ln \relax (x)}{x-x\,\ln \left (\ln \left (\frac {1}{x}\right )+\ln \relax (3)\right )\,\ln \relax (x)+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 5.12, size = 26, normalized size = 0.96 \begin {gather*} e^{- \frac {3 x \log {\relax (x )}}{x \log {\relax (x )} \log {\left (- \log {\relax (x )} + \log {\relax (3 )} \right )} - x - 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*ln(x)**2+6*ln(3/x)*ln(x)+(6+3*x)*ln(3/x))*exp(-3*x*ln(x)/(x*ln(x)*ln(ln(3/x))-x-2))/(x**2*ln(3
/x)*ln(x)**2*ln(ln(3/x))**2+(-2*x**2-4*x)*ln(3/x)*ln(x)*ln(ln(3/x))+(x**2+4*x+4)*ln(3/x)),x)

[Out]

exp(-3*x*log(x)/(x*log(x)*log(-log(x) + log(3)) - x - 2))

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