\(\int \frac {2 x^2 \log (\frac {2}{x})+e^{\frac {1}{x+\log (\frac {3}{\log (\frac {2}{x})})}} (-1-x \log (\frac {2}{x})) \log (x)+4 x \log (\frac {2}{x}) \log (\frac {3}{\log (\frac {2}{x})})+2 \log (\frac {2}{x}) \log ^2(\frac {3}{\log (\frac {2}{x})})}{x^3 \log (\frac {2}{x}) \log (x)+2 x^2 \log (\frac {2}{x}) \log (x) \log (\frac {3}{\log (\frac {2}{x})})+x \log (\frac {2}{x}) \log (x) \log ^2(\frac {3}{\log (\frac {2}{x})})} \, dx\) [1899]

Optimal result

Integrand size = 147, antiderivative size = 23 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}}+\log \left (\log ^2(x)\right ) \] Output:

ln(ln(x)^2)+exp(1/(ln(3/ln(2/x))+x))
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}}+2 \log (\log (x)) \] Input:

Integrate[(2*x^2*Log[2/x] + E^(x + Log[3/Log[2/x]])^(-1)*(-1 - x*Log[2/x]) 
*Log[x] + 4*x*Log[2/x]*Log[3/Log[2/x]] + 2*Log[2/x]*Log[3/Log[2/x]]^2)/(x^ 
3*Log[2/x]*Log[x] + 2*x^2*Log[2/x]*Log[x]*Log[3/Log[2/x]] + x*Log[2/x]*Log 
[x]*Log[3/Log[2/x]]^2),x]
 

Output:

E^(x + Log[3/Log[2/x]])^(-1) + 2*Log[Log[x]]
 

Rubi [A] (verified)

Time = 2.46 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {7292, 7293, 2009}

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.

\(\displaystyle \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-x \log \left (\frac {2}{x}\right )-1\right ) \log (x)}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-x \log \left (\frac {2}{x}\right )-1\right ) \log (x)}{x \log \left (\frac {2}{x}\right ) \log (x) \left (x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2}{x \log (x)}-\frac {e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (x \log \left (\frac {2}{x}\right )+1\right )}{x \log \left (\frac {2}{x}\right ) \left (x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \log (\log (x))+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}}\)

Input:

Int[(2*x^2*Log[2/x] + E^(x + Log[3/Log[2/x]])^(-1)*(-1 - x*Log[2/x])*Log[x 
] + 4*x*Log[2/x]*Log[3/Log[2/x]] + 2*Log[2/x]*Log[3/Log[2/x]]^2)/(x^3*Log[ 
2/x]*Log[x] + 2*x^2*Log[2/x]*Log[x]*Log[3/Log[2/x]] + x*Log[2/x]*Log[x]*Lo 
g[3/Log[2/x]]^2),x]
 

Output:

E^(x + Log[3/Log[2/x]])^(-1) + 2*Log[Log[x]]
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 178.94 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(2 \ln \left (\ln \left (x \right )\right )+{\mathrm e}^{\frac {1}{\ln \left (\frac {3}{\ln \left (\frac {2}{x}\right )}\right )+x}}\)	\(23\)
risch	\(2 \ln \left (\ln \left (x \right )\right )+{\mathrm e}^{\frac {1}{\ln \left (3\right )+x -\ln \left (\ln \left (2\right )-\ln \left (x \right )\right )}}\)	\(24\)

Input:

int(((-x*ln(2/x)-1)*ln(x)*exp(1/(ln(3/ln(2/x))+x))+2*ln(2/x)*ln(3/ln(2/x)) 
^2+4*x*ln(2/x)*ln(3/ln(2/x))+2*x^2*ln(2/x))/(x*ln(2/x)*ln(x)*ln(3/ln(2/x)) 
^2+2*x^2*ln(2/x)*ln(x)*ln(3/ln(2/x))+x^3*ln(2/x)*ln(x)),x,method=_RETURNVE 
RBOSE)
 

Output:

2*ln(ln(x))+exp(1/(ln(3/ln(2/x))+x))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=e^{\left (\frac {1}{x + \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}\right )} + 2 \, \log \left (-\log \left (2\right ) + \log \left (\frac {2}{x}\right )\right ) \] Input:

integrate(((-x*log(2/x)-1)*log(x)*exp(1/(log(3/log(2/x))+x))+2*log(2/x)*lo 
g(3/log(2/x))^2+4*x*log(2/x)*log(3/log(2/x))+2*x^2*log(2/x))/(x*log(2/x)*l 
og(x)*log(3/log(2/x))^2+2*x^2*log(2/x)*log(x)*log(3/log(2/x))+x^3*log(2/x) 
*log(x)),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.95 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=e^{\frac {1}{x + \log {\left (\frac {3}{- \log {\left (x \right )} + \log {\left (2 \right )}} \right )}}} + 2 \log {\left (\log {\left (x \right )} \right )} \] Input:

integrate(((-x*ln(2/x)-1)*ln(x)*exp(1/(ln(3/ln(2/x))+x))+2*ln(2/x)*ln(3/ln 
(2/x))**2+4*x*ln(2/x)*ln(3/ln(2/x))+2*x**2*ln(2/x))/(x*ln(2/x)*ln(x)*ln(3/ 
ln(2/x))**2+2*x**2*ln(2/x)*ln(x)*ln(3/ln(2/x))+x**3*ln(2/x)*ln(x)),x)
 

Output:

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

Maxima [F]

\[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=\int { -\frac {{\left (x \log \left (\frac {2}{x}\right ) + 1\right )} e^{\left (\frac {1}{x + \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}\right )} \log \left (x\right ) - 2 \, x^{2} \log \left (\frac {2}{x}\right ) - 4 \, x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right ) - 2 \, \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )^{2}}{x^{3} \log \left (x\right ) \log \left (\frac {2}{x}\right ) + 2 \, x^{2} \log \left (x\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right ) + x \log \left (x\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )^{2}} \,d x } \] Input:

integrate(((-x*log(2/x)-1)*log(x)*exp(1/(log(3/log(2/x))+x))+2*log(2/x)*lo 
g(3/log(2/x))^2+4*x*log(2/x)*log(3/log(2/x))+2*x^2*log(2/x))/(x*log(2/x)*l 
og(x)*log(3/log(2/x))^2+2*x^2*log(2/x)*log(x)*log(3/log(2/x))+x^3*log(2/x) 
*log(x)),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=e^{\left (\frac {1}{x + \log \left (3\right ) - \log \left (\log \left (2\right ) - \log \left (x\right )\right )}\right )} + 2 \, \log \left (\log \left (x\right )\right ) \] Input:

integrate(((-x*log(2/x)-1)*log(x)*exp(1/(log(3/log(2/x))+x))+2*log(2/x)*lo 
g(3/log(2/x))^2+4*x*log(2/x)*log(3/log(2/x))+2*x^2*log(2/x))/(x*log(2/x)*l 
og(x)*log(3/log(2/x))^2+2*x^2*log(2/x)*log(x)*log(3/log(2/x))+x^3*log(2/x) 
*log(x)),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 2.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=2\,\ln \left (\ln \left (x\right )\right )+{\mathrm {e}}^{\frac {1}{x+\ln \left (\frac {3}{\ln \left (\frac {2}{x}\right )}\right )}} \] Input:

int((2*log(2/x)*log(3/log(2/x))^2 + 2*x^2*log(2/x) + 4*x*log(2/x)*log(3/lo 
g(2/x)) - exp(1/(x + log(3/log(2/x))))*log(x)*(x*log(2/x) + 1))/(x^3*log(2 
/x)*log(x) + x*log(2/x)*log(3/log(2/x))^2*log(x) + 2*x^2*log(2/x)*log(3/lo 
g(2/x))*log(x)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=e^{\frac {1}{\mathrm {log}\left (\frac {3}{\mathrm {log}\left (\frac {2}{x}\right )}\right )+x}}+2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \] Input:

int(((-x*log(2/x)-1)*log(x)*exp(1/(log(3/log(2/x))+x))+2*log(2/x)*log(3/lo 
g(2/x))^2+4*x*log(2/x)*log(3/log(2/x))+2*x^2*log(2/x))/(x*log(2/x)*log(x)* 
log(3/log(2/x))^2+2*x^2*log(2/x)*log(x)*log(3/log(2/x))+x^3*log(2/x)*log(x 
)),x)
 

Output:

e**(1/(log(3/log(2/x)) + x)) + 2*log(log(x))