\(\int \frac {e^{\frac {-x^3+\log (\frac {x^4}{\log ^2(x)})+x \log (\frac {\log (3 x)}{\log (x)})}{-x^2+\log (\frac {\log (3 x)}{\log (x)})}} ((2 x^2+(-4 x^2+x^5) \log (x)) \log (3 x)+(-\log (x)+(1+2 x^2 \log (x)) \log (3 x)) \log (\frac {x^4}{\log ^2(x)})+(-2+(4-2 x^3) \log (x)) \log (3 x) \log (\frac {\log (3 x)}{\log (x)})+x \log (x) \log (3 x) \log ^2(\frac {\log (3 x)}{\log (x)}))}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log (\frac {\log (3 x)}{\log (x)})+x \log (x) \log (3 x) \log ^2(\frac {\log (3 x)}{\log (x)})} \, dx\) [876]

Optimal result

Integrand size = 203, antiderivative size = 32 \[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=e^{x+\frac {\log \left (\frac {x^4}{\log ^2(x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \] Output:

exp(x+ln(x^4/ln(x)^2)/(ln(ln(3*x)/ln(x))-x^2))
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=e^x \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \] Input:

Integrate[(E^((-x^3 + Log[x^4/Log[x]^2] + x*Log[Log[3*x]/Log[x]])/(-x^2 + 
Log[Log[3*x]/Log[x]]))*((2*x^2 + (-4*x^2 + x^5)*Log[x])*Log[3*x] + (-Log[x 
] + (1 + 2*x^2*Log[x])*Log[3*x])*Log[x^4/Log[x]^2] + (-2 + (4 - 2*x^3)*Log 
[x])*Log[3*x]*Log[Log[3*x]/Log[x]] + x*Log[x]*Log[3*x]*Log[Log[3*x]/Log[x] 
]^2))/(x^5*Log[x]*Log[3*x] - 2*x^3*Log[x]*Log[3*x]*Log[Log[3*x]/Log[x]] + 
x*Log[x]*Log[3*x]*Log[Log[3*x]/Log[x]]^2),x]
 

Output:

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

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.

\(\displaystyle \int \frac {\left (\left (\left (4-2 x^3\right ) \log (x)-2\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+\left (2 x^2+\left (x^5-4 x^2\right ) \log (x)\right ) \log (3 x)+\left (\left (2 x^2 \log (x)+1\right ) \log (3 x)-\log (x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right ) \exp \left (\frac {\log \left (\frac {x^4}{\log ^2(x)}\right )-x^3+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (\left (\left (4-2 x^3\right ) \log (x)-2\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+\left (2 x^2+\left (x^5-4 x^2\right ) \log (x)\right ) \log (3 x)+\left (\left (2 x^2 \log (x)+1\right ) \log (3 x)-\log (x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right ) \exp \left (\frac {\log \left (\frac {x^4}{\log ^2(x)}\right )-x^3+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}\right )}{x \log (x) \log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {2 (2 \log (x)-1) \exp \left (\frac {\log \left (\frac {x^4}{\log ^2(x)}\right )-x^3+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}\right )}{x \log (x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\exp \left (\frac {\log \left (\frac {x^4}{\log ^2(x)}\right )-x^3+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}\right )+\frac {\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right ) \exp \left (\frac {\log \left (\frac {x^4}{\log ^2(x)}\right )-x^3+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}\right )}{x \log (x) \log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (e^{-\frac {x^3}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}}-\frac {2 e^{-\frac {x^3}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}} (2 \log (x)-1) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}}}{x \log (x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{-\frac {x^3}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}}}{x \log (x) \log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \log (x) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^4}-\frac {2 e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} (2 \log (x)-1) \log (3 x) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}+\frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \log \left (\frac {x^4}{\log ^2(x)}\right ) \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1}}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {x^3}{x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}+1} \left (\frac {\log (3 x)}{\log (x)}\right )^{\frac {x}{\log \left (\frac {\log (3 x)}{\log (x)}\right )-x^2}-1} \left (\left (2 x^2 \log ^2(x)+x^2 \log (9) \log (x)+\log (3)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right ) \left (\log (x) \left (x^3-x \log \left (\frac {\log (3 x)}{\log (x)}\right )-4\right )+2\right )\right )}{x^5 \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}dx\)

Input:

Int[(E^((-x^3 + Log[x^4/Log[x]^2] + x*Log[Log[3*x]/Log[x]])/(-x^2 + Log[Lo 
g[3*x]/Log[x]]))*((2*x^2 + (-4*x^2 + x^5)*Log[x])*Log[3*x] + (-Log[x] + (1 
 + 2*x^2*Log[x])*Log[3*x])*Log[x^4/Log[x]^2] + (-2 + (4 - 2*x^3)*Log[x])*L 
og[3*x]*Log[Log[3*x]/Log[x]] + x*Log[x]*Log[3*x]*Log[Log[3*x]/Log[x]]^2))/ 
(x^5*Log[x]*Log[3*x] - 2*x^3*Log[x]*Log[3*x]*Log[Log[3*x]/Log[x]] + x*Log[ 
x]*Log[3*x]*Log[Log[3*x]/Log[x]]^2),x]
 

Output:

$Aborted
 

Maple [F(-1)]

Timed out.

\[\int \frac {\left (x \ln \left (x \right ) \ln \left (3 x \right ) \ln \left (\frac {\ln \left (3 x \right )}{\ln \left (x \right )}\right )^{2}+\left (\left (-2 x^{3}+4\right ) \ln \left (x \right )-2\right ) \ln \left (3 x \right ) \ln \left (\frac {\ln \left (3 x \right )}{\ln \left (x \right )}\right )+\left (\left (2 x^{2} \ln \left (x \right )+1\right ) \ln \left (3 x \right )-\ln \left (x \right )\right ) \ln \left (\frac {x^{4}}{\ln \left (x \right )^{2}}\right )+\left (\left (x^{5}-4 x^{2}\right ) \ln \left (x \right )+2 x^{2}\right ) \ln \left (3 x \right )\right ) {\mathrm e}^{\frac {x \ln \left (\frac {\ln \left (3 x \right )}{\ln \left (x \right )}\right )+\ln \left (\frac {x^{4}}{\ln \left (x \right )^{2}}\right )-x^{3}}{\ln \left (\frac {\ln \left (3 x \right )}{\ln \left (x \right )}\right )-x^{2}}}}{x \ln \left (x \right ) \ln \left (3 x \right ) \ln \left (\frac {\ln \left (3 x \right )}{\ln \left (x \right )}\right )^{2}-2 x^{3} \ln \left (x \right ) \ln \left (3 x \right ) \ln \left (\frac {\ln \left (3 x \right )}{\ln \left (x \right )}\right )+x^{5} \ln \left (x \right ) \ln \left (3 x \right )}d x\]

Input:

int((x*ln(x)*ln(3*x)*ln(ln(3*x)/ln(x))^2+((-2*x^3+4)*ln(x)-2)*ln(3*x)*ln(l 
n(3*x)/ln(x))+((2*x^2*ln(x)+1)*ln(3*x)-ln(x))*ln(x^4/ln(x)^2)+((x^5-4*x^2) 
*ln(x)+2*x^2)*ln(3*x))*exp((x*ln(ln(3*x)/ln(x))+ln(x^4/ln(x)^2)-x^3)/(ln(l 
n(3*x)/ln(x))-x^2))/(x*ln(x)*ln(3*x)*ln(ln(3*x)/ln(x))^2-2*x^3*ln(x)*ln(3* 
x)*ln(ln(3*x)/ln(x))+x^5*ln(x)*ln(3*x)),x)
 

Output:

int((x*ln(x)*ln(3*x)*ln(ln(3*x)/ln(x))^2+((-2*x^3+4)*ln(x)-2)*ln(3*x)*ln(l 
n(3*x)/ln(x))+((2*x^2*ln(x)+1)*ln(3*x)-ln(x))*ln(x^4/ln(x)^2)+((x^5-4*x^2) 
*ln(x)+2*x^2)*ln(3*x))*exp((x*ln(ln(3*x)/ln(x))+ln(x^4/ln(x)^2)-x^3)/(ln(l 
n(3*x)/ln(x))-x^2))/(x*ln(x)*ln(3*x)*ln(ln(3*x)/ln(x))^2-2*x^3*ln(x)*ln(3* 
x)*ln(ln(3*x)/ln(x))+x^5*ln(x)*ln(3*x)),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=e^{\left (\frac {x^{3} - x \log \left (\frac {\log \left (3\right ) + \log \left (x\right )}{\log \left (x\right )}\right ) - \log \left (\frac {x^{4}}{\log \left (x\right )^{2}}\right )}{x^{2} - \log \left (\frac {\log \left (3\right ) + \log \left (x\right )}{\log \left (x\right )}\right )}\right )} \] Input:

integrate((x*log(x)*log(3*x)*log(log(3*x)/log(x))^2+((-2*x^3+4)*log(x)-2)* 
log(3*x)*log(log(3*x)/log(x))+((2*x^2*log(x)+1)*log(3*x)-log(x))*log(x^4/l 
og(x)^2)+((x^5-4*x^2)*log(x)+2*x^2)*log(3*x))*exp((x*log(log(3*x)/log(x))+ 
log(x^4/log(x)^2)-x^3)/(log(log(3*x)/log(x))-x^2))/(x*log(x)*log(3*x)*log( 
log(3*x)/log(x))^2-2*x^3*log(x)*log(3*x)*log(log(3*x)/log(x))+x^5*log(x)*l 
og(3*x)),x, algorithm="fricas")
 

Output:

e^((x^3 - x*log((log(3) + log(x))/log(x)) - log(x^4/log(x)^2))/(x^2 - log( 
(log(3) + log(x))/log(x))))
 

Sympy [A] (verification not implemented)

Time = 28.73 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=e^{\frac {- x^{3} + x \log {\left (\frac {\log {\left (x \right )} + \log {\left (3 \right )}}{\log {\left (x \right )}} \right )} + \log {\left (\frac {x^{4}}{\log {\left (x \right )}^{2}} \right )}}{- x^{2} + \log {\left (\frac {\log {\left (x \right )} + \log {\left (3 \right )}}{\log {\left (x \right )}} \right )}}} \] Input:

integrate((x*ln(x)*ln(3*x)*ln(ln(3*x)/ln(x))**2+((-2*x**3+4)*ln(x)-2)*ln(3 
*x)*ln(ln(3*x)/ln(x))+((2*x**2*ln(x)+1)*ln(3*x)-ln(x))*ln(x**4/ln(x)**2)+( 
(x**5-4*x**2)*ln(x)+2*x**2)*ln(3*x))*exp((x*ln(ln(3*x)/ln(x))+ln(x**4/ln(x 
)**2)-x**3)/(ln(ln(3*x)/ln(x))-x**2))/(x*ln(x)*ln(3*x)*ln(ln(3*x)/ln(x))** 
2-2*x**3*ln(x)*ln(3*x)*ln(ln(3*x)/ln(x))+x**5*ln(x)*ln(3*x)),x)
 

Output:

exp((-x**3 + x*log((log(x) + log(3))/log(x)) + log(x**4/log(x)**2))/(-x**2 
 + log((log(x) + log(3))/log(x))))
 

Maxima [F]

\[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=\int { \frac {{\left (x \log \left (3 \, x\right ) \log \left (x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right )^{2} - 2 \, {\left ({\left (x^{3} - 2\right )} \log \left (x\right ) + 1\right )} \log \left (3 \, x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right ) + {\left ({\left (2 \, x^{2} \log \left (x\right ) + 1\right )} \log \left (3 \, x\right ) - \log \left (x\right )\right )} \log \left (\frac {x^{4}}{\log \left (x\right )^{2}}\right ) + {\left (2 \, x^{2} + {\left (x^{5} - 4 \, x^{2}\right )} \log \left (x\right )\right )} \log \left (3 \, x\right )\right )} e^{\left (\frac {x^{3} - x \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right ) - \log \left (\frac {x^{4}}{\log \left (x\right )^{2}}\right )}{x^{2} - \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right )}\right )}}{x^{5} \log \left (3 \, x\right ) \log \left (x\right ) - 2 \, x^{3} \log \left (3 \, x\right ) \log \left (x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right ) + x \log \left (3 \, x\right ) \log \left (x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right )^{2}} \,d x } \] Input:

integrate((x*log(x)*log(3*x)*log(log(3*x)/log(x))^2+((-2*x^3+4)*log(x)-2)* 
log(3*x)*log(log(3*x)/log(x))+((2*x^2*log(x)+1)*log(3*x)-log(x))*log(x^4/l 
og(x)^2)+((x^5-4*x^2)*log(x)+2*x^2)*log(3*x))*exp((x*log(log(3*x)/log(x))+ 
log(x^4/log(x)^2)-x^3)/(log(log(3*x)/log(x))-x^2))/(x*log(x)*log(3*x)*log( 
log(3*x)/log(x))^2-2*x^3*log(x)*log(3*x)*log(log(3*x)/log(x))+x^5*log(x)*l 
og(3*x)),x, algorithm="maxima")
 

Output:

integrate((x*log(3*x)*log(x)*log(log(3*x)/log(x))^2 - 2*((x^3 - 2)*log(x) 
+ 1)*log(3*x)*log(log(3*x)/log(x)) + ((2*x^2*log(x) + 1)*log(3*x) - log(x) 
)*log(x^4/log(x)^2) + (2*x^2 + (x^5 - 4*x^2)*log(x))*log(3*x))*e^((x^3 - x 
*log(log(3*x)/log(x)) - log(x^4/log(x)^2))/(x^2 - log(log(3*x)/log(x))))/( 
x^5*log(3*x)*log(x) - 2*x^3*log(3*x)*log(x)*log(log(3*x)/log(x)) + x*log(3 
*x)*log(x)*log(log(3*x)/log(x))^2), x)
 

Giac [F]

\[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=\int { \frac {{\left (x \log \left (3 \, x\right ) \log \left (x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right )^{2} - 2 \, {\left ({\left (x^{3} - 2\right )} \log \left (x\right ) + 1\right )} \log \left (3 \, x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right ) + {\left ({\left (2 \, x^{2} \log \left (x\right ) + 1\right )} \log \left (3 \, x\right ) - \log \left (x\right )\right )} \log \left (\frac {x^{4}}{\log \left (x\right )^{2}}\right ) + {\left (2 \, x^{2} + {\left (x^{5} - 4 \, x^{2}\right )} \log \left (x\right )\right )} \log \left (3 \, x\right )\right )} e^{\left (\frac {x^{3} - x \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right ) - \log \left (\frac {x^{4}}{\log \left (x\right )^{2}}\right )}{x^{2} - \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right )}\right )}}{x^{5} \log \left (3 \, x\right ) \log \left (x\right ) - 2 \, x^{3} \log \left (3 \, x\right ) \log \left (x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right ) + x \log \left (3 \, x\right ) \log \left (x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right )^{2}} \,d x } \] Input:

integrate((x*log(x)*log(3*x)*log(log(3*x)/log(x))^2+((-2*x^3+4)*log(x)-2)* 
log(3*x)*log(log(3*x)/log(x))+((2*x^2*log(x)+1)*log(3*x)-log(x))*log(x^4/l 
og(x)^2)+((x^5-4*x^2)*log(x)+2*x^2)*log(3*x))*exp((x*log(log(3*x)/log(x))+ 
log(x^4/log(x)^2)-x^3)/(log(log(3*x)/log(x))-x^2))/(x*log(x)*log(3*x)*log( 
log(3*x)/log(x))^2-2*x^3*log(x)*log(3*x)*log(log(3*x)/log(x))+x^5*log(x)*l 
og(3*x)),x, algorithm="giac")
 

Output:

undef
 

Mupad [B] (verification not implemented)

Time = 4.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.12 \[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx={\mathrm {e}}^{-\frac {x^3}{\ln \left (\frac {\ln \left (3\,x\right )}{\ln \left (x\right )}\right )-x^2}}\,{\left (\frac {1}{{\ln \left (x\right )}^2}\right )}^{\frac {1}{\ln \left (\frac {\ln \left (3\,x\right )}{\ln \left (x\right )}\right )-x^2}}\,{\left (x^4\right )}^{\frac {1}{\ln \left (\frac {\ln \left (3\,x\right )}{\ln \left (x\right )}\right )-x^2}}\,{\left (\frac {\ln \left (3\,x\right )}{\ln \left (x\right )}\right )}^{\frac {x}{\ln \left (\frac {\ln \left (3\,x\right )}{\ln \left (x\right )}\right )-x^2}} \] Input:

int(-(exp((log(x^4/log(x)^2) + x*log(log(3*x)/log(x)) - x^3)/(log(log(3*x) 
/log(x)) - x^2))*(log(3*x)*(log(x)*(4*x^2 - x^5) - 2*x^2) + log(x^4/log(x) 
^2)*(log(x) - log(3*x)*(2*x^2*log(x) + 1)) + log(3*x)*log(log(3*x)/log(x)) 
*(log(x)*(2*x^3 - 4) + 2) - x*log(3*x)*log(log(3*x)/log(x))^2*log(x)))/(x^ 
5*log(3*x)*log(x) + x*log(3*x)*log(log(3*x)/log(x))^2*log(x) - 2*x^3*log(3 
*x)*log(log(3*x)/log(x))*log(x)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=e^{\frac {\mathrm {log}\left (\frac {x^{4}}{\mathrm {log}\left (x \right )^{2}}\right )+\mathrm {log}\left (\frac {\mathrm {log}\left (3 x \right )}{\mathrm {log}\left (x \right )}\right ) x -x^{3}}{\mathrm {log}\left (\frac {\mathrm {log}\left (3 x \right )}{\mathrm {log}\left (x \right )}\right )-x^{2}}} \] Input:

int((x*log(x)*log(3*x)*log(log(3*x)/log(x))^2+((-2*x^3+4)*log(x)-2)*log(3* 
x)*log(log(3*x)/log(x))+((2*x^2*log(x)+1)*log(3*x)-log(x))*log(x^4/log(x)^ 
2)+((x^5-4*x^2)*log(x)+2*x^2)*log(3*x))*exp((x*log(log(3*x)/log(x))+log(x^ 
4/log(x)^2)-x^3)/(log(log(3*x)/log(x))-x^2))/(x*log(x)*log(3*x)*log(log(3* 
x)/log(x))^2-2*x^3*log(x)*log(3*x)*log(log(3*x)/log(x))+x^5*log(x)*log(3*x 
)),x)
 

Output:

e**((log(x**4/log(x)**2) + log(log(3*x)/log(x))*x - x**3)/(log(log(3*x)/lo 
g(x)) - x**2))