\(\int \frac {4 \log (\frac {1}{x})+4 \log (\frac {1}{x}) \log (x)+2 \log (x) \log (e^2 \log (\frac {1}{x}))+(-\log (\frac {1}{x})-\log (\frac {1}{x}) \log (x)) \log ^2(e^2 \log (\frac {1}{x}))+(-4 \log (\frac {1}{x}) \log (x)+\log (\frac {1}{x}) \log (x) \log ^2(e^2 \log (\frac {1}{x}))) \log (4 x \log (x)-x \log (x) \log ^2(e^2 \log (\frac {1}{x})))}{(-4 \log (\frac {1}{x}) \log (x)+\log (\frac {1}{x}) \log (x) \log ^2(e^2 \log (\frac {1}{x}))) \log ^2(4 x \log (x)-x \log (x) \log ^2(e^2 \log (\frac {1}{x})))} \, dx\) [218]

Optimal result

Integrand size = 161, antiderivative size = 24 \[ \int \frac {4 \log \left (\frac {1}{x}\right )+4 \log \left (\frac {1}{x}\right ) \log (x)+2 \log (x) \log \left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-\log \left (\frac {1}{x}\right )-\log \left (\frac {1}{x}\right ) \log (x)\right ) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log \left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )}{\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log ^2\left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )} \, dx=\frac {x}{\log \left (x \log (x) \left (4-\log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {4 \log \left (\frac {1}{x}\right )+4 \log \left (\frac {1}{x}\right ) \log (x)+2 \log (x) \log \left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-\log \left (\frac {1}{x}\right )-\log \left (\frac {1}{x}\right ) \log (x)\right ) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log \left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )}{\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log ^2\left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )} \, dx=\frac {x}{\log \left (-x \log (x) \left (-4+\left (2+\log \left (\log \left (\frac {1}{x}\right )\right )\right )^2\right )\right )} \] Input:

Integrate[(4*Log[x^(-1)] + 4*Log[x^(-1)]*Log[x] + 2*Log[x]*Log[E^2*Log[x^( 
-1)]] + (-Log[x^(-1)] - Log[x^(-1)]*Log[x])*Log[E^2*Log[x^(-1)]]^2 + (-4*L 
og[x^(-1)]*Log[x] + Log[x^(-1)]*Log[x]*Log[E^2*Log[x^(-1)]]^2)*Log[4*x*Log 
[x] - x*Log[x]*Log[E^2*Log[x^(-1)]]^2])/((-4*Log[x^(-1)]*Log[x] + Log[x^(- 
1)]*Log[x]*Log[E^2*Log[x^(-1)]]^2)*Log[4*x*Log[x] - x*Log[x]*Log[E^2*Log[x 
^(-1)]]^2]^2),x]
 

Output:

x/Log[-(x*Log[x]*(-4 + (2 + Log[Log[x^(-1)]])^2))]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (-\log (x) \log \left (\frac {1}{x}\right )-\log \left (\frac {1}{x}\right )\right ) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )-4 \log \left (\frac {1}{x}\right ) \log (x)\right ) \log \left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )+2 \log (x) \log \left (e^2 \log \left (\frac {1}{x}\right )\right )+4 \log \left (\frac {1}{x}\right )+4 \log \left (\frac {1}{x}\right ) \log (x)}{\left (\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )-4 \log \left (\frac {1}{x}\right ) \log (x)\right ) \log ^2\left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-\log (x) \log \left (\frac {1}{x}\right )-\log \left (\frac {1}{x}\right )\right ) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )-4 \log \left (\frac {1}{x}\right ) \log (x)\right ) \log \left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )+2 \log (x) \log \left (e^2 \log \left (\frac {1}{x}\right )\right )+4 \log \left (\frac {1}{x}\right )+4 \log \left (\frac {1}{x}\right ) \log (x)}{\log \left (\frac {1}{x}\right ) \log (x) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right ) \log ^2\left (-x \log (x) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-\log \left (\frac {1}{x}\right ) \log ^2\left (\log \left (\frac {1}{x}\right )\right )-\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (\log \left (\frac {1}{x}\right )\right )-4 \log \left (\frac {1}{x}\right ) \log \left (\log \left (\frac {1}{x}\right )\right )-4 \log \left (\frac {1}{x}\right ) \log (x) \log \left (\log \left (\frac {1}{x}\right )\right )+2 \log (x) \log \left (\log \left (\frac {1}{x}\right )\right )+4 \log (x)}{\log \left (\frac {1}{x}\right ) \log (x) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right ) \log ^2\left (-x \log (x) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right )\right )}+\frac {1}{\log \left (-x \log (x) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right )\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -4 \int \frac {1}{\left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right ) \log ^2\left (-x \log (x) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right )\right )}dx+2 \int \frac {1}{\log \left (\frac {1}{x}\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right ) \log ^2\left (-x \log (x) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right )\right )}dx-4 \int \frac {1}{\log (x) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right ) \log ^2\left (-x \log (x) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right )\right )}dx+4 \int \frac {1}{\log \left (\frac {1}{x}\right ) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right ) \log ^2\left (-x \log (x) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right )\right )}dx-\int \frac {\log \left (\log \left (\frac {1}{x}\right )\right )}{\left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right ) \log ^2\left (-x \log (x) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right )\right )}dx-\int \frac {\log \left (\log \left (\frac {1}{x}\right )\right )}{\log (x) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right ) \log ^2\left (-x \log (x) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right )\right )}dx+\int \frac {1}{\log \left (-x \log (x) \log \left (\log \left (\frac {1}{x}\right )\right ) \left (\log \left (\log \left (\frac {1}{x}\right )\right )+4\right )\right )}dx\)

Input:

Int[(4*Log[x^(-1)] + 4*Log[x^(-1)]*Log[x] + 2*Log[x]*Log[E^2*Log[x^(-1)]] 
+ (-Log[x^(-1)] - Log[x^(-1)]*Log[x])*Log[E^2*Log[x^(-1)]]^2 + (-4*Log[x^( 
-1)]*Log[x] + Log[x^(-1)]*Log[x]*Log[E^2*Log[x^(-1)]]^2)*Log[4*x*Log[x] - 
x*Log[x]*Log[E^2*Log[x^(-1)]]^2])/((-4*Log[x^(-1)]*Log[x] + Log[x^(-1)]*Lo 
g[x]*Log[E^2*Log[x^(-1)]]^2)*Log[4*x*Log[x] - x*Log[x]*Log[E^2*Log[x^(-1)] 
]^2]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 229.60 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96

method	result	size
parallelrisch	\(\frac {x}{\ln \left (-x \ln \left (x \right ) \left (\ln \left ({\mathrm e}^{2} \ln \left (\frac {1}{x}\right )\right )^{2}-4\right )\right )}\)	\(23\)

Input:

int(((ln(1/x)*ln(x)*ln(exp(2)*ln(1/x))^2-4*ln(1/x)*ln(x))*ln(-x*ln(x)*ln(e 
xp(2)*ln(1/x))^2+4*x*ln(x))+(-ln(1/x)*ln(x)-ln(1/x))*ln(exp(2)*ln(1/x))^2+ 
2*ln(x)*ln(exp(2)*ln(1/x))+4*ln(1/x)*ln(x)+4*ln(1/x))/(ln(1/x)*ln(x)*ln(ex 
p(2)*ln(1/x))^2-4*ln(1/x)*ln(x))/ln(-x*ln(x)*ln(exp(2)*ln(1/x))^2+4*x*ln(x 
))^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {4 \log \left (\frac {1}{x}\right )+4 \log \left (\frac {1}{x}\right ) \log (x)+2 \log (x) \log \left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-\log \left (\frac {1}{x}\right )-\log \left (\frac {1}{x}\right ) \log (x)\right ) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log \left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )}{\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log ^2\left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )} \, dx=\frac {x}{\log \left (x \log \left (e^{2} \log \left (\frac {1}{x}\right )\right )^{2} \log \left (\frac {1}{x}\right ) - 4 \, x \log \left (\frac {1}{x}\right )\right )} \] Input:

integrate(((log(1/x)*log(x)*log(exp(2)*log(1/x))^2-4*log(1/x)*log(x))*log( 
-x*log(x)*log(exp(2)*log(1/x))^2+4*x*log(x))+(-log(1/x)*log(x)-log(1/x))*l 
og(exp(2)*log(1/x))^2+2*log(x)*log(exp(2)*log(1/x))+4*log(1/x)*log(x)+4*lo 
g(1/x))/(log(1/x)*log(x)*log(exp(2)*log(1/x))^2-4*log(1/x)*log(x))/log(-x* 
log(x)*log(exp(2)*log(1/x))^2+4*x*log(x))^2,x, algorithm="fricas")
 

Output:

x/log(x*log(e^2*log(1/x))^2*log(1/x) - 4*x*log(1/x))
 

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {4 \log \left (\frac {1}{x}\right )+4 \log \left (\frac {1}{x}\right ) \log (x)+2 \log (x) \log \left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-\log \left (\frac {1}{x}\right )-\log \left (\frac {1}{x}\right ) \log (x)\right ) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log \left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )}{\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log ^2\left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )} \, dx=\frac {x}{\log {\left (- x \log {\left (x \right )} \log {\left (- e^{2} \log {\left (x \right )} \right )}^{2} + 4 x \log {\left (x \right )} \right )}} \] Input:

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

Output:

x/log(-x*log(x)*log(-exp(2)*log(x))**2 + 4*x*log(x))
 

Maxima [F]

\[ \int \frac {4 \log \left (\frac {1}{x}\right )+4 \log \left (\frac {1}{x}\right ) \log (x)+2 \log (x) \log \left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-\log \left (\frac {1}{x}\right )-\log \left (\frac {1}{x}\right ) \log (x)\right ) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log \left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )}{\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log ^2\left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )} \, dx=\int { -\frac {{\left (\log \left (x\right ) \log \left (\frac {1}{x}\right ) + \log \left (\frac {1}{x}\right )\right )} \log \left (e^{2} \log \left (\frac {1}{x}\right )\right )^{2} - {\left (\log \left (e^{2} \log \left (\frac {1}{x}\right )\right )^{2} \log \left (x\right ) \log \left (\frac {1}{x}\right ) - 4 \, \log \left (x\right ) \log \left (\frac {1}{x}\right )\right )} \log \left (-x \log \left (e^{2} \log \left (\frac {1}{x}\right )\right )^{2} \log \left (x\right ) + 4 \, x \log \left (x\right )\right ) - 2 \, \log \left (e^{2} \log \left (\frac {1}{x}\right )\right ) \log \left (x\right ) - 4 \, \log \left (x\right ) \log \left (\frac {1}{x}\right ) - 4 \, \log \left (\frac {1}{x}\right )}{{\left (\log \left (e^{2} \log \left (\frac {1}{x}\right )\right )^{2} \log \left (x\right ) \log \left (\frac {1}{x}\right ) - 4 \, \log \left (x\right ) \log \left (\frac {1}{x}\right )\right )} \log \left (-x \log \left (e^{2} \log \left (\frac {1}{x}\right )\right )^{2} \log \left (x\right ) + 4 \, x \log \left (x\right )\right )^{2}} \,d x } \] Input:

integrate(((log(1/x)*log(x)*log(exp(2)*log(1/x))^2-4*log(1/x)*log(x))*log( 
-x*log(x)*log(exp(2)*log(1/x))^2+4*x*log(x))+(-log(1/x)*log(x)-log(1/x))*l 
og(exp(2)*log(1/x))^2+2*log(x)*log(exp(2)*log(1/x))+4*log(1/x)*log(x)+4*lo 
g(1/x))/(log(1/x)*log(x)*log(exp(2)*log(1/x))^2-4*log(1/x)*log(x))/log(-x* 
log(x)*log(exp(2)*log(1/x))^2+4*x*log(x))^2,x, algorithm="maxima")
 

Output:

(((4*I*pi - pi^2)*x*log(x) + (x*log(x) + x)*log(log(x))^2 + (6*I*pi - pi^2 
 + 4)*x - 2*((-I*pi - 2)*x*log(x) + (-I*pi - 3)*x)*log(log(x)))*log(-log(x 
))^2 - 4*((-4*I*pi + pi^2)*x*log(x) - (x*log(x) + x)*log(log(x))^2 + (-6*I 
*pi + pi^2 - 4)*x + 2*((-I*pi - 2)*x*log(x) + (-I*pi - 3)*x)*log(log(x)))* 
log(-log(x)))/(((log(x) + 1)*log(log(x))^3 + (4*I*pi - pi^2)*log(x)^2 + (2 
*I*pi + (2*I*pi + 5)*log(x) + log(x)^2 + 4)*log(log(x))^2 + (4*I*pi - pi^2 
)*log(x) + (4*I*pi - 2*(-I*pi - 2)*log(x)^2 - pi^2 + (6*I*pi - pi^2 + 4)*l 
og(x))*log(log(x)))*log(-log(x))^2 - 4*(-2*I*pi - log(x) - 4)*log(log(x))^ 
2 + 4*log(log(x))^3 - 4*(-4*I*pi + pi^2)*log(x) + 2*((2*log(x) + 3)*log(lo 
g(x))^3 - 2*(-4*I*pi + pi^2)*log(x)^2 - (-6*I*pi + (-4*I*pi - 11)*log(x) - 
 2*log(x)^2 - 12)*log(log(x))^2 - 3*(-4*I*pi + pi^2)*log(x) - (-12*I*pi + 
4*(-I*pi - 2)*log(x)^2 + 3*pi^2 + 2*(-7*I*pi + pi^2 - 6)*log(x))*log(log(x 
)))*log(-log(x)) + (16*I*pi + (4*I*pi + (log(x) + 1)*log(log(x))^2 - pi^2 
+ (4*I*pi - pi^2)*log(x) - 2*(-I*pi + (-I*pi - 2)*log(x) - 2)*log(log(x))) 
*log(-log(x))^2 - 4*pi^2 - 2*(-12*I*pi - (2*log(x) + 3)*log(log(x))^2 + 3* 
pi^2 + 2*(-4*I*pi + pi^2)*log(x) + 2*(-3*I*pi + 2*(-I*pi - 2)*log(x) - 6)* 
log(log(x)))*log(-log(x)) - 8*(-I*pi - 2)*log(log(x)) + 4*log(log(x))^2)*l 
og(-log(-log(x)) - 4) - 4*(-4*I*pi + pi^2 + 2*(-I*pi - 2)*log(x))*log(log( 
x)) + (16*I*pi + (4*I*pi + (log(x) + 1)*log(log(x))^2 - pi^2 + (4*I*pi - p 
i^2)*log(x) - 2*(-I*pi + (-I*pi - 2)*log(x) - 2)*log(log(x)))*log(-log(...
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 1664, normalized size of antiderivative = 69.33 \[ \int \frac {4 \log \left (\frac {1}{x}\right )+4 \log \left (\frac {1}{x}\right ) \log (x)+2 \log (x) \log \left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-\log \left (\frac {1}{x}\right )-\log \left (\frac {1}{x}\right ) \log (x)\right ) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log \left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )}{\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log ^2\left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )} \, dx=\text {Too large to display} \] Input:

integrate(((log(1/x)*log(x)*log(exp(2)*log(1/x))^2-4*log(1/x)*log(x))*log( 
-x*log(x)*log(exp(2)*log(1/x))^2+4*x*log(x))+(-log(1/x)*log(x)-log(1/x))*l 
og(exp(2)*log(1/x))^2+2*log(x)*log(exp(2)*log(1/x))+4*log(1/x)*log(x)+4*lo 
g(1/x))/(log(1/x)*log(x)*log(exp(2)*log(1/x))^2-4*log(1/x)*log(x))/log(-x* 
log(x)*log(exp(2)*log(1/x))^2+4*x*log(x))^2,x, algorithm="giac")
 

Output:

(pi^2*x*log(x)*log(-log(x))^2 - 2*I*pi*x*log(x)*log(-log(x))^2*log(log(x)) 
 - x*log(x)*log(-log(x))^2*log(log(x))^2 + 4*pi^2*x*log(x)*log(-log(x)) + 
pi^2*x*log(-log(x))^2 - 4*I*pi*x*log(x)*log(-log(x))^2 - 8*I*pi*x*log(x)*l 
og(-log(x))*log(log(x)) - 2*I*pi*x*log(-log(x))^2*log(log(x)) - 4*x*log(x) 
*log(-log(x))^2*log(log(x)) - 4*x*log(x)*log(-log(x))*log(log(x))^2 - x*lo 
g(-log(x))^2*log(log(x))^2 + 4*pi^2*x*log(-log(x)) - 16*I*pi*x*log(x)*log( 
-log(x)) - 6*I*pi*x*log(-log(x))^2 - 8*I*pi*x*log(-log(x))*log(log(x)) - 1 
6*x*log(x)*log(-log(x))*log(log(x)) - 6*x*log(-log(x))^2*log(log(x)) - 4*x 
*log(-log(x))*log(log(x))^2 - 24*I*pi*x*log(-log(x)) - 4*x*log(-log(x))^2 
- 24*x*log(-log(x))*log(log(x)) - 16*x*log(-log(x)))/(I*pi^3*log(x)*log(-l 
og(x))^2 + pi^2*log(-log(-log(x))^2 - 4*log(-log(x)))*log(x)*log(-log(x))^ 
2 + pi^2*log(x)^2*log(-log(x))^2 + 3*pi^2*log(x)*log(-log(x))^2*log(log(x) 
) - 2*I*pi*log(-log(-log(x))^2 - 4*log(-log(x)))*log(x)*log(-log(x))^2*log 
(log(x)) - 2*I*pi*log(x)^2*log(-log(x))^2*log(log(x)) - 3*I*pi*log(x)*log( 
-log(x))^2*log(log(x))^2 - log(-log(-log(x))^2 - 4*log(-log(x)))*log(x)*lo 
g(-log(x))^2*log(log(x))^2 - log(x)^2*log(-log(x))^2*log(log(x))^2 - log(x 
)*log(-log(x))^2*log(log(x))^3 + 4*I*pi^3*log(x)*log(-log(x)) + 4*pi^2*log 
(-log(-log(x))^2 - 4*log(-log(x)))*log(x)*log(-log(x)) + 4*pi^2*log(x)^2*l 
og(-log(x)) + I*pi^3*log(-log(x))^2 + pi^2*log(-log(-log(x))^2 - 4*log(-lo 
g(x)))*log(-log(x))^2 + 5*pi^2*log(x)*log(-log(x))^2 - 4*I*pi*log(-log(...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {4 \log \left (\frac {1}{x}\right )+4 \log \left (\frac {1}{x}\right ) \log (x)+2 \log (x) \log \left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-\log \left (\frac {1}{x}\right )-\log \left (\frac {1}{x}\right ) \log (x)\right ) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log \left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )}{\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log ^2\left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )} \, dx=-\int \frac {4\,\ln \left (\frac {1}{x}\right )-{\ln \left (\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^2\right )}^2\,\left (\ln \left (\frac {1}{x}\right )+\ln \left (\frac {1}{x}\right )\,\ln \left (x\right )\right )+4\,\ln \left (\frac {1}{x}\right )\,\ln \left (x\right )+2\,\ln \left (\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^2\right )\,\ln \left (x\right )-\ln \left (4\,x\,\ln \left (x\right )-x\,{\ln \left (\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^2\right )}^2\,\ln \left (x\right )\right )\,\left (4\,\ln \left (\frac {1}{x}\right )\,\ln \left (x\right )-\ln \left (\frac {1}{x}\right )\,{\ln \left (\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^2\right )}^2\,\ln \left (x\right )\right )}{{\ln \left (4\,x\,\ln \left (x\right )-x\,{\ln \left (\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^2\right )}^2\,\ln \left (x\right )\right )}^2\,\left (4\,\ln \left (\frac {1}{x}\right )\,\ln \left (x\right )-\ln \left (\frac {1}{x}\right )\,{\ln \left (\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^2\right )}^2\,\ln \left (x\right )\right )} \,d x \] Input:

int(-(4*log(1/x) - log(log(1/x)*exp(2))^2*(log(1/x) + log(1/x)*log(x)) + 4 
*log(1/x)*log(x) + 2*log(log(1/x)*exp(2))*log(x) - log(4*x*log(x) - x*log( 
log(1/x)*exp(2))^2*log(x))*(4*log(1/x)*log(x) - log(1/x)*log(log(1/x)*exp( 
2))^2*log(x)))/(log(4*x*log(x) - x*log(log(1/x)*exp(2))^2*log(x))^2*(4*log 
(1/x)*log(x) - log(1/x)*log(log(1/x)*exp(2))^2*log(x))),x)
 

Output:

-int((4*log(1/x) - log(log(1/x)*exp(2))^2*(log(1/x) + log(1/x)*log(x)) + 4 
*log(1/x)*log(x) + 2*log(log(1/x)*exp(2))*log(x) - log(4*x*log(x) - x*log( 
log(1/x)*exp(2))^2*log(x))*(4*log(1/x)*log(x) - log(1/x)*log(log(1/x)*exp( 
2))^2*log(x)))/(log(4*x*log(x) - x*log(log(1/x)*exp(2))^2*log(x))^2*(4*log 
(1/x)*log(x) - log(1/x)*log(log(1/x)*exp(2))^2*log(x))), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {4 \log \left (\frac {1}{x}\right )+4 \log \left (\frac {1}{x}\right ) \log (x)+2 \log (x) \log \left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-\log \left (\frac {1}{x}\right )-\log \left (\frac {1}{x}\right ) \log (x)\right ) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )+\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log \left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )}{\left (-4 \log \left (\frac {1}{x}\right ) \log (x)+\log \left (\frac {1}{x}\right ) \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right ) \log ^2\left (4 x \log (x)-x \log (x) \log ^2\left (e^2 \log \left (\frac {1}{x}\right )\right )\right )} \, dx=\frac {x}{\mathrm {log}\left (-\mathrm {log}\left (-\mathrm {log}\left (x \right ) e^{2}\right )^{2} \mathrm {log}\left (x \right ) x +4 \,\mathrm {log}\left (x \right ) x \right )} \] Input:

int(((log(1/x)*log(x)*log(exp(2)*log(1/x))^2-4*log(1/x)*log(x))*log(-x*log 
(x)*log(exp(2)*log(1/x))^2+4*x*log(x))+(-log(1/x)*log(x)-log(1/x))*log(exp 
(2)*log(1/x))^2+2*log(x)*log(exp(2)*log(1/x))+4*log(1/x)*log(x)+4*log(1/x) 
)/(log(1/x)*log(x)*log(exp(2)*log(1/x))^2-4*log(1/x)*log(x))/log(-x*log(x) 
*log(exp(2)*log(1/x))^2+4*x*log(x))^2,x)
 

Output:

x/log( - log( - log(x)*e**2)**2*log(x)*x + 4*log(x)*x)