3.4.78 \(\int \frac {(1-9 \log (\frac {3}{x})) \log (\log (3))+\log (\frac {3}{x}) \log (\log (3)) \log (\log (\frac {3}{x}))}{(81-72 x+16 x^2) \log (\frac {3}{x})+(-18+8 x) \log (\frac {3}{x}) \log (\log (\frac {3}{x}))+\log (\frac {3}{x}) \log ^2(\log (\frac {3}{x}))} \, dx\)

Optimal. Leaf size=22 \[ \log (\log (3)) \left (16+\frac {x}{-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )}\right ) \]

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Rubi [F]  time = 0.50, 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 {\left (1-9 \log \left (\frac {3}{x}\right )\right ) \log (\log (3))+\log \left (\frac {3}{x}\right ) \log (\log (3)) \log \left (\log \left (\frac {3}{x}\right )\right )}{\left (81-72 x+16 x^2\right ) \log \left (\frac {3}{x}\right )+(-18+8 x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\log \left (\frac {3}{x}\right ) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 - 9*Log[3/x])*Log[Log[3]] + Log[3/x]*Log[Log[3]]*Log[Log[3/x]])/((81 - 72*x + 16*x^2)*Log[3/x] + (-18
+ 8*x)*Log[3/x]*Log[Log[3/x]] + Log[3/x]*Log[Log[3/x]]^2),x]

[Out]

-4*Log[Log[3]]*Defer[Int][x/(-9 + 4*x + Log[Log[3/x]])^2, x] + Log[Log[3]]*Defer[Int][1/(Log[3/x]*(-9 + 4*x +
Log[Log[3/x]])^2), x] + Log[Log[3]]*Defer[Int][(-9 + 4*x + Log[Log[3/x]])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\log (\log (3)) \left (1+\log \left (\frac {3}{x}\right ) \left (-9+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )}{\log \left (\frac {3}{x}\right ) \left (9-4 x-\log \left (\log \left (\frac {3}{x}\right )\right )\right )^2} \, dx\\ &=\log (\log (3)) \int \frac {1+\log \left (\frac {3}{x}\right ) \left (-9+\log \left (\log \left (\frac {3}{x}\right )\right )\right )}{\log \left (\frac {3}{x}\right ) \left (9-4 x-\log \left (\log \left (\frac {3}{x}\right )\right )\right )^2} \, dx\\ &=\log (\log (3)) \int \left (\frac {1-4 x \log \left (\frac {3}{x}\right )}{\log \left (\frac {3}{x}\right ) \left (-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )\right )^2}+\frac {1}{-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )}\right ) \, dx\\ &=\log (\log (3)) \int \frac {1-4 x \log \left (\frac {3}{x}\right )}{\log \left (\frac {3}{x}\right ) \left (-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )\right )^2} \, dx+\log (\log (3)) \int \frac {1}{-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )} \, dx\\ &=\log (\log (3)) \int \frac {1}{-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )} \, dx+\log (\log (3)) \int \left (-\frac {4 x}{\left (-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )\right )^2}+\frac {1}{\log \left (\frac {3}{x}\right ) \left (-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )\right )^2}\right ) \, dx\\ &=\log (\log (3)) \int \frac {1}{\log \left (\frac {3}{x}\right ) \left (-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )\right )^2} \, dx+\log (\log (3)) \int \frac {1}{-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )} \, dx-(4 \log (\log (3))) \int \frac {x}{\left (-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 19, normalized size = 0.86 \begin {gather*} \frac {x \log (\log (3))}{-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 9*Log[3/x])*Log[Log[3]] + Log[3/x]*Log[Log[3]]*Log[Log[3/x]])/((81 - 72*x + 16*x^2)*Log[3/x] +
(-18 + 8*x)*Log[3/x]*Log[Log[3/x]] + Log[3/x]*Log[Log[3/x]]^2),x]

[Out]

(x*Log[Log[3]])/(-9 + 4*x + Log[Log[3/x]])

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fricas [A]  time = 0.85, size = 19, normalized size = 0.86 \begin {gather*} \frac {x \log \left (\log \relax (3)\right )}{4 \, x + \log \left (\log \left (\frac {3}{x}\right )\right ) - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(3/x)*log(log(3))*log(log(3/x))+(-9*log(3/x)+1)*log(log(3)))/(log(3/x)*log(log(3/x))^2+(8*x-18)*
log(3/x)*log(log(3/x))+(16*x^2-72*x+81)*log(3/x)),x, algorithm="fricas")

[Out]

x*log(log(3))/(4*x + log(log(3/x)) - 9)

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giac [B]  time = 0.63, size = 172, normalized size = 7.82 \begin {gather*} \frac {4 \, x^{2} \log \relax (3) \log \left (\frac {3}{x}\right ) \log \left (\log \relax (3)\right ) - 4 \, x^{2} \log \relax (x) \log \left (\frac {3}{x}\right ) \log \left (\log \relax (3)\right ) - x \log \left (\frac {3}{x}\right ) \log \left (\log \relax (3)\right )}{16 \, x^{2} \log \relax (3) \log \left (\frac {3}{x}\right ) - 16 \, x^{2} \log \relax (x) \log \left (\frac {3}{x}\right ) + 4 \, x \log \relax (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - 4 \, x \log \relax (x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - 36 \, x \log \relax (3) \log \left (\frac {3}{x}\right ) + 36 \, x \log \relax (x) \log \left (\frac {3}{x}\right ) - 4 \, x \log \relax (3) + 4 \, x \log \relax (x) - \log \relax (3) \log \left (\log \left (\frac {3}{x}\right )\right ) + \log \relax (x) \log \left (\log \left (\frac {3}{x}\right )\right ) + 9 \, \log \relax (3) - 9 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(3/x)*log(log(3))*log(log(3/x))+(-9*log(3/x)+1)*log(log(3)))/(log(3/x)*log(log(3/x))^2+(8*x-18)*
log(3/x)*log(log(3/x))+(16*x^2-72*x+81)*log(3/x)),x, algorithm="giac")

[Out]

(4*x^2*log(3)*log(3/x)*log(log(3)) - 4*x^2*log(x)*log(3/x)*log(log(3)) - x*log(3/x)*log(log(3)))/(16*x^2*log(3
)*log(3/x) - 16*x^2*log(x)*log(3/x) + 4*x*log(3)*log(3/x)*log(log(3/x)) - 4*x*log(x)*log(3/x)*log(log(3/x)) -
36*x*log(3)*log(3/x) + 36*x*log(x)*log(3/x) - 4*x*log(3) + 4*x*log(x) - log(3)*log(log(3/x)) + log(x)*log(log(
3/x)) + 9*log(3) - 9*log(x))

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (\frac {3}{x}\right ) \ln \left (\ln \relax (3)\right ) \ln \left (\ln \left (\frac {3}{x}\right )\right )+\left (-9 \ln \left (\frac {3}{x}\right )+1\right ) \ln \left (\ln \relax (3)\right )}{\ln \left (\frac {3}{x}\right ) \ln \left (\ln \left (\frac {3}{x}\right )\right )^{2}+\left (8 x -18\right ) \ln \left (\frac {3}{x}\right ) \ln \left (\ln \left (\frac {3}{x}\right )\right )+\left (16 x^{2}-72 x +81\right ) \ln \left (\frac {3}{x}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(3/x)*ln(ln(3))*ln(ln(3/x))+(-9*ln(3/x)+1)*ln(ln(3)))/(ln(3/x)*ln(ln(3/x))^2+(8*x-18)*ln(3/x)*ln(ln(3/x
))+(16*x^2-72*x+81)*ln(3/x)),x)

[Out]

int((ln(3/x)*ln(ln(3))*ln(ln(3/x))+(-9*ln(3/x)+1)*ln(ln(3)))/(ln(3/x)*ln(ln(3/x))^2+(8*x-18)*ln(3/x)*ln(ln(3/x
))+(16*x^2-72*x+81)*ln(3/x)),x)

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maxima [A]  time = 0.73, size = 20, normalized size = 0.91 \begin {gather*} \frac {x \log \left (\log \relax (3)\right )}{4 \, x + \log \left (\log \relax (3) - \log \relax (x)\right ) - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(3/x)*log(log(3))*log(log(3/x))+(-9*log(3/x)+1)*log(log(3)))/(log(3/x)*log(log(3/x))^2+(8*x-18)*
log(3/x)*log(log(3/x))+(16*x^2-72*x+81)*log(3/x)),x, algorithm="maxima")

[Out]

x*log(log(3))/(4*x + log(log(3) - log(x)) - 9)

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mupad [B]  time = 0.98, size = 31, normalized size = 1.41 \begin {gather*} \frac {\frac {9\,\ln \left (\ln \relax (3)\right )}{4}+\frac {\ln \left (\ln \relax (3)\right )\,\left (4\,x-9\right )}{4}}{4\,x+\ln \left (\ln \left (\frac {3}{x}\right )\right )-9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(log(3))*(9*log(3/x) - 1) - log(log(3/x))*log(3/x)*log(log(3)))/(log(3/x)*(16*x^2 - 72*x + 81) + log(
log(3/x))^2*log(3/x) + log(log(3/x))*log(3/x)*(8*x - 18)),x)

[Out]

((9*log(log(3)))/4 + (log(log(3))*(4*x - 9))/4)/(4*x + log(log(3/x)) - 9)

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sympy [A]  time = 0.29, size = 17, normalized size = 0.77 \begin {gather*} \frac {x \log {\left (\log {\relax (3 )} \right )}}{4 x + \log {\left (\log {\left (\frac {3}{x} \right )} \right )} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(3/x)*ln(ln(3))*ln(ln(3/x))+(-9*ln(3/x)+1)*ln(ln(3)))/(ln(3/x)*ln(ln(3/x))**2+(8*x-18)*ln(3/x)*ln
(ln(3/x))+(16*x**2-72*x+81)*ln(3/x)),x)

[Out]

x*log(log(3))/(4*x + log(log(3/x)) - 9)

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