\(\int \frac {(1-9 \log (\frac {3}{x})) \log (\log (3))+\log (\frac {3}{x}) \log (\log (3)) \log (\log (\frac {3}{x}))}{(81-72 x+16 x^2) \log (\frac {3}{x})+(-18+8 x) \log (\frac {3}{x}) \log (\log (\frac {3}{x}))+\log (\frac {3}{x}) \log ^2(\log (\frac {3}{x}))} \, dx\) [378]

Optimal result

Integrand size = 88, antiderivative size = 22 \[ \int \frac {\left (1-9 \log \left (\frac {3}{x}\right )\right ) \log (\log (3))+\log \left (\frac {3}{x}\right ) \log (\log (3)) \log \left (\log \left (\frac {3}{x}\right )\right )}{\left (81-72 x+16 x^2\right ) \log \left (\frac {3}{x}\right )+(-18+8 x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\log \left (\frac {3}{x}\right ) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\log (\log (3)) \left (16+\frac {x}{-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )}\right ) \]

[Out]

ln(ln(3))*(x/(4*x+ln(ln(3/x))-9)+16)

Rubi [F]

\[ \int \frac {\left (1-9 \log \left (\frac {3}{x}\right )\right ) \log (\log (3))+\log \left (\frac {3}{x}\right ) \log (\log (3)) \log \left (\log \left (\frac {3}{x}\right )\right )}{\left (81-72 x+16 x^2\right ) \log \left (\frac {3}{x}\right )+(-18+8 x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\log \left (\frac {3}{x}\right ) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\int \frac {\left (1-9 \log \left (\frac {3}{x}\right )\right ) \log (\log (3))+\log \left (\frac {3}{x}\right ) \log (\log (3)) \log \left (\log \left (\frac {3}{x}\right )\right )}{\left (81-72 x+16 x^2\right ) \log \left (\frac {3}{x}\right )+(-18+8 x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\log \left (\frac {3}{x}\right ) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx \]

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\left (1-9 \log \left (\frac {3}{x}\right )\right ) \log (\log (3))+\log \left (\frac {3}{x}\right ) \log (\log (3)) \log \left (\log \left (\frac {3}{x}\right )\right )}{\left (81-72 x+16 x^2\right ) \log \left (\frac {3}{x}\right )+(-18+8 x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\log \left (\frac {3}{x}\right ) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\frac {x \log (\log (3))}{-9+4 x+\log \left (\log \left (\frac {3}{x}\right )\right )} \]

[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]])

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

method	result	size
parallelrisch	\(\frac {\ln \left (\ln \left (3\right )\right ) x}{4 x +\ln \left (\ln \left (\frac {3}{x}\right )\right )-9}\)	\(20\)
default	\(\frac {\ln \left (\ln \left (3\right )\right )}{\frac {\ln \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )}{x}-\frac {9}{x}+4}\)	\(26\)

[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,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\left (1-9 \log \left (\frac {3}{x}\right )\right ) \log (\log (3))+\log \left (\frac {3}{x}\right ) \log (\log (3)) \log \left (\log \left (\frac {3}{x}\right )\right )}{\left (81-72 x+16 x^2\right ) \log \left (\frac {3}{x}\right )+(-18+8 x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\log \left (\frac {3}{x}\right ) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\frac {x \log \left (\log \left (3\right )\right )}{4 \, x + \log \left (\log \left (\frac {3}{x}\right )\right ) - 9} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {\left (1-9 \log \left (\frac {3}{x}\right )\right ) \log (\log (3))+\log \left (\frac {3}{x}\right ) \log (\log (3)) \log \left (\log \left (\frac {3}{x}\right )\right )}{\left (81-72 x+16 x^2\right ) \log \left (\frac {3}{x}\right )+(-18+8 x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\log \left (\frac {3}{x}\right ) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\frac {x \log {\left (\log {\left (3 \right )} \right )}}{4 x + \log {\left (\log {\left (\frac {3}{x} \right )} \right )} - 9} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\left (1-9 \log \left (\frac {3}{x}\right )\right ) \log (\log (3))+\log \left (\frac {3}{x}\right ) \log (\log (3)) \log \left (\log \left (\frac {3}{x}\right )\right )}{\left (81-72 x+16 x^2\right ) \log \left (\frac {3}{x}\right )+(-18+8 x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\log \left (\frac {3}{x}\right ) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\frac {x \log \left (\log \left (3\right )\right )}{4 \, x + \log \left (\log \left (3\right ) - \log \left (x\right )\right ) - 9} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (22) = 44\).

Time = 0.32 (sec) , antiderivative size = 172, normalized size of antiderivative = 7.82 \[ \int \frac {\left (1-9 \log \left (\frac {3}{x}\right )\right ) \log (\log (3))+\log \left (\frac {3}{x}\right ) \log (\log (3)) \log \left (\log \left (\frac {3}{x}\right )\right )}{\left (81-72 x+16 x^2\right ) \log \left (\frac {3}{x}\right )+(-18+8 x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\log \left (\frac {3}{x}\right ) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\frac {4 \, x^{2} \log \left (3\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (3\right )\right ) - 4 \, x^{2} \log \left (x\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (3\right )\right ) - x \log \left (\frac {3}{x}\right ) \log \left (\log \left (3\right )\right )}{16 \, x^{2} \log \left (3\right ) \log \left (\frac {3}{x}\right ) - 16 \, x^{2} \log \left (x\right ) \log \left (\frac {3}{x}\right ) + 4 \, x \log \left (3\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - 4 \, x \log \left (x\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - 36 \, x \log \left (3\right ) \log \left (\frac {3}{x}\right ) + 36 \, x \log \left (x\right ) \log \left (\frac {3}{x}\right ) - 4 \, x \log \left (3\right ) + 4 \, x \log \left (x\right ) - \log \left (3\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + \log \left (x\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + 9 \, \log \left (3\right ) - 9 \, \log \left (x\right )} \]

[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))

Mupad [B] (verification not implemented)

Time = 9.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {\left (1-9 \log \left (\frac {3}{x}\right )\right ) \log (\log (3))+\log \left (\frac {3}{x}\right ) \log (\log (3)) \log \left (\log \left (\frac {3}{x}\right )\right )}{\left (81-72 x+16 x^2\right ) \log \left (\frac {3}{x}\right )+(-18+8 x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\log \left (\frac {3}{x}\right ) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\frac {\frac {9\,\ln \left (\ln \left (3\right )\right )}{4}+\frac {\ln \left (\ln \left (3\right )\right )\,\left (4\,x-9\right )}{4}}{4\,x+\ln \left (\ln \left (\frac {3}{x}\right )\right )-9} \]

[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)