\(\int \frac {x+2 x^3 \log (\frac {2}{x})+((-x^3+x \log (5)) \log (\frac {2}{x})+x \log (\frac {2}{x}) \log (\log (\frac {2}{x}))) \log (-x^2+\log (5)+\log (\log (\frac {2}{x})))+((-x^2+x^3+(1-x) \log (5)) \log (\frac {2}{x})+(1-x) \log (\frac {2}{x}) \log (\log (\frac {2}{x}))) \log ^2(-x^2+\log (5)+\log (\log (\frac {2}{x})))}{((-x^4+x^2 \log (5)) \log (\frac {2}{x})+x^2 \log (\frac {2}{x}) \log (\log (\frac {2}{x}))) \log (-x^2+\log (5)+\log (\log (\frac {2}{x})))+((-5 x^3+x^4+(5 x-x^2) \log (5)) \log (\frac {2}{x})+(-x^3+x \log (5)) \log (\frac {2}{x}) \log (x)+((5 x-x^2) \log (\frac {2}{x})+x \log (\frac {2}{x}) \log (x)) \log (\log (\frac {2}{x}))) \log ^2(-x^2+\log (5)+\log (\log (\frac {2}{x})))} \, dx\) [213]

Optimal result

Integrand size = 286, antiderivative size = 28 \[ \int \frac {x+2 x^3 \log \left (\frac {2}{x}\right )+\left (\left (-x^3+x \log (5)\right ) \log \left (\frac {2}{x}\right )+x \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log \left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )+\left (\left (-x^2+x^3+(1-x) \log (5)\right ) \log \left (\frac {2}{x}\right )+(1-x) \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log ^2\left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )}{\left (\left (-x^4+x^2 \log (5)\right ) \log \left (\frac {2}{x}\right )+x^2 \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log \left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )+\left (\left (-5 x^3+x^4+\left (5 x-x^2\right ) \log (5)\right ) \log \left (\frac {2}{x}\right )+\left (-x^3+x \log (5)\right ) \log \left (\frac {2}{x}\right ) \log (x)+\left (\left (5 x-x^2\right ) \log \left (\frac {2}{x}\right )+x \log \left (\frac {2}{x}\right ) \log (x)\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log ^2\left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )} \, dx=\log \left (5-x+\log (x)+\frac {x}{\log \left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )}\right ) \] Output:

ln(5+x/ln(ln(ln(2/x))+ln(5)-x^2)+ln(x)-x)
 

Mathematica [F]

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

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

Output:

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

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.

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

Input:

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

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(28)=56\).

Time = 0.71 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14

\[\ln \left (x -\ln \left (x \right )-5\right )+\ln \left (\ln \left (\ln \left (\ln \left (2\right )-\ln \left (x \right )\right )+\ln \left (5\right )-x^{2}\right )+\frac {x}{\ln \left (x \right )-x +5}\right )-\ln \left (\ln \left (\ln \left (\ln \left (2\right )-\ln \left (x \right )\right )+\ln \left (5\right )-x^{2}\right )\right )\]

Input:

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

Output:

ln(x-ln(x)-5)+ln(ln(ln(ln(2)-ln(x))+ln(5)-x^2)+x/(ln(x)-x+5))-ln(ln(ln(ln( 
2)-ln(x))+ln(5)-x^2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (28) = 56\).

Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.04 \[ \int \frac {x+2 x^3 \log \left (\frac {2}{x}\right )+\left (\left (-x^3+x \log (5)\right ) \log \left (\frac {2}{x}\right )+x \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log \left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )+\left (\left (-x^2+x^3+(1-x) \log (5)\right ) \log \left (\frac {2}{x}\right )+(1-x) \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log ^2\left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )}{\left (\left (-x^4+x^2 \log (5)\right ) \log \left (\frac {2}{x}\right )+x^2 \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log \left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )+\left (\left (-5 x^3+x^4+\left (5 x-x^2\right ) \log (5)\right ) \log \left (\frac {2}{x}\right )+\left (-x^3+x \log (5)\right ) \log \left (\frac {2}{x}\right ) \log (x)+\left (\left (5 x-x^2\right ) \log \left (\frac {2}{x}\right )+x \log \left (\frac {2}{x}\right ) \log (x)\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log ^2\left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )} \, dx=\log \left (x - \log \left (2\right ) + \log \left (\frac {2}{x}\right ) - 5\right ) + \log \left (\frac {{\left (x - \log \left (2\right ) + \log \left (\frac {2}{x}\right ) - 5\right )} \log \left (-x^{2} + \log \left (5\right ) + \log \left (\log \left (\frac {2}{x}\right )\right )\right ) - x}{x - \log \left (2\right ) + \log \left (\frac {2}{x}\right ) - 5}\right ) - \log \left (\log \left (-x^{2} + \log \left (5\right ) + \log \left (\log \left (\frac {2}{x}\right )\right )\right )\right ) \] Input:

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

Output:

log(x - log(2) + log(2/x) - 5) + log(((x - log(2) + log(2/x) - 5)*log(-x^2 
 + log(5) + log(log(2/x))) - x)/(x - log(2) + log(2/x) - 5)) - log(log(-x^ 
2 + log(5) + log(log(2/x))))
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (28) = 56\).

Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.46 \[ \int \frac {x+2 x^3 \log \left (\frac {2}{x}\right )+\left (\left (-x^3+x \log (5)\right ) \log \left (\frac {2}{x}\right )+x \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log \left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )+\left (\left (-x^2+x^3+(1-x) \log (5)\right ) \log \left (\frac {2}{x}\right )+(1-x) \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log ^2\left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )}{\left (\left (-x^4+x^2 \log (5)\right ) \log \left (\frac {2}{x}\right )+x^2 \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log \left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )+\left (\left (-5 x^3+x^4+\left (5 x-x^2\right ) \log (5)\right ) \log \left (\frac {2}{x}\right )+\left (-x^3+x \log (5)\right ) \log \left (\frac {2}{x}\right ) \log (x)+\left (\left (5 x-x^2\right ) \log \left (\frac {2}{x}\right )+x \log \left (\frac {2}{x}\right ) \log (x)\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log ^2\left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )} \, dx=\log \left (-x + \log \left (x\right ) + 5\right ) + \log \left (\frac {{\left (x - \log \left (x\right ) - 5\right )} \log \left (-x^{2} + \log \left (5\right ) + \log \left (\log \left (2\right ) - \log \left (x\right )\right )\right ) - x}{x - \log \left (x\right ) - 5}\right ) - \log \left (\log \left (-x^{2} + \log \left (5\right ) + \log \left (\log \left (2\right ) - \log \left (x\right )\right )\right )\right ) \] Input:

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

Output:

log(-x + log(x) + 5) + log(((x - log(x) - 5)*log(-x^2 + log(5) + log(log(2 
) - log(x))) - x)/(x - log(x) - 5)) - log(log(-x^2 + log(5) + log(log(2) - 
 log(x))))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (28) = 56\).

Time = 0.41 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.04 \[ \int \frac {x+2 x^3 \log \left (\frac {2}{x}\right )+\left (\left (-x^3+x \log (5)\right ) \log \left (\frac {2}{x}\right )+x \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log \left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )+\left (\left (-x^2+x^3+(1-x) \log (5)\right ) \log \left (\frac {2}{x}\right )+(1-x) \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log ^2\left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )}{\left (\left (-x^4+x^2 \log (5)\right ) \log \left (\frac {2}{x}\right )+x^2 \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log \left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )+\left (\left (-5 x^3+x^4+\left (5 x-x^2\right ) \log (5)\right ) \log \left (\frac {2}{x}\right )+\left (-x^3+x \log (5)\right ) \log \left (\frac {2}{x}\right ) \log (x)+\left (\left (5 x-x^2\right ) \log \left (\frac {2}{x}\right )+x \log \left (\frac {2}{x}\right ) \log (x)\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log ^2\left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )} \, dx=\log \left (x \log \left (-x^{2} + \log \left (5\right ) + \log \left (\log \left (2\right ) - \log \left (x\right )\right )\right ) - \log \left (-x^{2} + \log \left (5\right ) + \log \left (\log \left (2\right ) - \log \left (x\right )\right )\right ) \log \left (x\right ) - x - 5 \, \log \left (-x^{2} + \log \left (5\right ) + \log \left (\log \left (2\right ) - \log \left (x\right )\right )\right )\right ) - \log \left (\log \left (-x^{2} + \log \left (5\right ) + \log \left (\log \left (2\right ) - \log \left (x\right )\right )\right )\right ) \] Input:

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

Output:

log(x*log(-x^2 + log(5) + log(log(2) - log(x))) - log(-x^2 + log(5) + log( 
log(2) - log(x)))*log(x) - x - 5*log(-x^2 + log(5) + log(log(2) - log(x))) 
) - log(log(-x^2 + log(5) + log(log(2) - log(x))))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.82 \[ \int \frac {x+2 x^3 \log \left (\frac {2}{x}\right )+\left (\left (-x^3+x \log (5)\right ) \log \left (\frac {2}{x}\right )+x \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log \left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )+\left (\left (-x^2+x^3+(1-x) \log (5)\right ) \log \left (\frac {2}{x}\right )+(1-x) \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log ^2\left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )}{\left (\left (-x^4+x^2 \log (5)\right ) \log \left (\frac {2}{x}\right )+x^2 \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log \left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )+\left (\left (-5 x^3+x^4+\left (5 x-x^2\right ) \log (5)\right ) \log \left (\frac {2}{x}\right )+\left (-x^3+x \log (5)\right ) \log \left (\frac {2}{x}\right ) \log (x)+\left (\left (5 x-x^2\right ) \log \left (\frac {2}{x}\right )+x \log \left (\frac {2}{x}\right ) \log (x)\right ) \log \left (\log \left (\frac {2}{x}\right )\right )\right ) \log ^2\left (-x^2+\log (5)+\log \left (\log \left (\frac {2}{x}\right )\right )\right )} \, dx=-\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\frac {2}{x}\right )\right )+\mathrm {log}\left (5\right )-x^{2}\right )\right )+\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\frac {2}{x}\right )\right )+\mathrm {log}\left (5\right )-x^{2}\right ) \mathrm {log}\left (x \right )-\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\frac {2}{x}\right )\right )+\mathrm {log}\left (5\right )-x^{2}\right ) x +5 \,\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\frac {2}{x}\right )\right )+\mathrm {log}\left (5\right )-x^{2}\right )+x \right ) \] Input:

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

Output:

 - log(log(log(log(2/x)) + log(5) - x**2)) + log(log(log(log(2/x)) + log(5 
) - x**2)*log(x) - log(log(log(2/x)) + log(5) - x**2)*x + 5*log(log(log(2/ 
x)) + log(5) - x**2) + x)