\(\int \frac {(-2 x+x^2+(2-2 x) \log (x)) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+((2 x-x^2) \log (x)+(-2 x+x^2) \log (x) \log (3 x)+((2 x^2-x^3) \log (x)+(-2 x^2+x^3) \log (x) \log (3 x)) \log (\log (x))) \log (\frac {-1-x \log (\log (x))}{-2 x+x^2}) \log (\log (\frac {-1-x \log (\log (x))}{-2 x+x^2}))+((2-x) \log (x)+(2 x-x^2) \log (x) \log (\log (x))) \log (\frac {-1-x \log (\log (x))}{-2 x+x^2}) \log (\log (\frac {-1-x \log (\log (x))}{-2 x+x^2})) \log (\log (\log (\frac {-1-x \log (\log (x))}{-2 x+x^2})))}{((-2 x+x^2) \log (x) \log ^2(3 x)+(-2 x^2+x^3) \log (x) \log ^2(3 x) \log (\log (x))) \log (\frac {-1-x \log (\log (x))}{-2 x+x^2}) \log (\log (\frac {-1-x \log (\log (x))}{-2 x+x^2}))} \, dx\) [9922]

Optimal result

Integrand size = 306, antiderivative size = 27 \[ \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\frac {x+\log \left (\log \left (\log \left (\frac {\frac {1}{x}+\log (\log (x))}{2-x}\right )\right )\right )}{\log (3 x)} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+\log (3 x)+\frac {((-2+x) x-2 (-1+x) \log (x)) \log (3 x)}{(-2+x) x \log (x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}-\frac {x \log (3 x) \log (\log (x))}{(-2+x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}-\frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x}}{\log ^2(3 x)} \, dx \\ & = \int \left (\frac {-2 x \log (3 x)+x^2 \log (3 x)+2 \log (x) \log (3 x)-2 x \log (x) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+2 x \log (x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-x^2 \log (x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-2 x \log (x) \log (3 x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+x^2 \log (x) \log (3 x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+2 x^2 \log (x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-x^3 \log (x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-2 x^2 \log (x) \log (3 x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+x^3 \log (x) \log (3 x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}{(-2+x) x \log (x) \log ^2(3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}-\frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)}\right ) \, dx \\ & = \int \frac {-2 x \log (3 x)+x^2 \log (3 x)+2 \log (x) \log (3 x)-2 x \log (x) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+2 x \log (x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-x^2 \log (x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-2 x \log (x) \log (3 x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+x^2 \log (x) \log (3 x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+2 x^2 \log (x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-x^3 \log (x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-2 x^2 \log (x) \log (3 x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+x^3 \log (x) \log (3 x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}{(-2+x) x \log (x) \log ^2(3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx \\ & = \int \frac {-1+\frac {\log (3 x) \left ((-2+x) x+\log (x) \left (2-2 x+(-2+x) x \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+x^2 \log (\log (x)) \left (-1+(-2+x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )\right )\right )}{(-2+x) x \log (x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}}{\log ^2(3 x)} \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx \\ & = \int \left (\frac {-1+\log (3 x)}{\log ^2(3 x)}+\frac {-2 x+x^2+2 \log (x)-2 x \log (x)-x^2 \log (x) \log (\log (x))}{(-2+x) x \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}\right ) \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx \\ & = \int \frac {-1+\log (3 x)}{\log ^2(3 x)} \, dx+\int \frac {-2 x+x^2+2 \log (x)-2 x \log (x)-x^2 \log (x) \log (\log (x))}{(-2+x) x \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx \\ & = \int \left (-\frac {1}{\log ^2(3 x)}+\frac {1}{\log (3 x)}\right ) \, dx+\int \left (\frac {-2 x+x^2+2 \log (x)-2 x \log (x)-x^2 \log (x) \log (\log (x))}{2 (-2+x) \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}+\frac {2 x-x^2-2 \log (x)+2 x \log (x)+x^2 \log (x) \log (\log (x))}{2 x \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}\right ) \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx \\ & = \frac {1}{2} \int \frac {-2 x+x^2+2 \log (x)-2 x \log (x)-x^2 \log (x) \log (\log (x))}{(-2+x) \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx+\frac {1}{2} \int \frac {2 x-x^2-2 \log (x)+2 x \log (x)+x^2 \log (x) \log (\log (x))}{x \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx-\int \frac {1}{\log ^2(3 x)} \, dx+\int \frac {1}{\log (3 x)} \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx \\ & = \frac {x}{\log (3 x)}+\frac {\operatorname {LogIntegral}(3 x)}{3}+\frac {1}{2} \int \left (\frac {2}{\log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}-\frac {2}{x \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}+\frac {2}{\log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}-\frac {x}{\log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}+\frac {x \log (\log (x))}{\log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}\right ) \, dx+\frac {1}{2} \int \left (\frac {2}{(-2+x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}-\frac {2 x}{(-2+x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}-\frac {2 x}{(-2+x) \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}+\frac {x^2}{(-2+x) \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}-\frac {x^2 \log (\log (x))}{(-2+x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}\right ) \, dx-\int \frac {1}{\log (3 x)} \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx \\ & = \frac {x}{\log (3 x)}-\frac {1}{2} \int \frac {x}{\log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx+\frac {1}{2} \int \frac {x^2}{(-2+x) \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx+\frac {1}{2} \int \frac {x \log (\log (x))}{\log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx-\frac {1}{2} \int \frac {x^2 \log (\log (x))}{(-2+x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx+\int \frac {1}{\log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx+\int \frac {1}{(-2+x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx-\int \frac {1}{x \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx-\int \frac {x}{(-2+x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx+\int \frac {1}{\log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx-\int \frac {x}{(-2+x) \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\frac {x}{\log (3 x)}+\frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{\log (3 x)} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 11.71 (sec) , antiderivative size = 274, normalized size of antiderivative = 10.15

\[\frac {2 i \ln \left (\ln \left (i \pi -\ln \left (x \right )-\ln \left (-2+x \right )+\ln \left (x \ln \left (\ln \left (x \right )\right )+1\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{-2+x}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{-2+x}\right )+\operatorname {csgn}\left (\frac {i}{-2+x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{-2+x}\right )+\operatorname {csgn}\left (i \left (x \ln \left (\ln \left (x \right )\right )+1\right )\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x \left (-2+x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x \left (-2+x \right )}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x \left (-2+x \right )}\right )+\operatorname {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{-2+x}\right )\right )}{2}+i \pi \operatorname {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x \left (-2+x \right )}\right )^{2} \left (\operatorname {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x \left (-2+x \right )}\right )-1\right )\right )\right )}{2 i \ln \left (3\right )+2 i \ln \left (x \right )}+\frac {2 i x}{2 i \ln \left (3\right )+2 i \ln \left (x \right )}\]

[In]

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

[Out]

2*I/(2*I*ln(3)+2*I*ln(x))*ln(ln(I*Pi-ln(x)-ln(-2+x)+ln(x*ln(ln(x))+1)-1/2*I*Pi*csgn(I/(-2+x)*(x*ln(ln(x))+1))*
(-csgn(I/(-2+x)*(x*ln(ln(x))+1))+csgn(I/(-2+x)))*(-csgn(I/(-2+x)*(x*ln(ln(x))+1))+csgn(I*(x*ln(ln(x))+1)))-1/2
*I*Pi*csgn(I/x/(-2+x)*(x*ln(ln(x))+1))*(-csgn(I/x/(-2+x)*(x*ln(ln(x))+1))+csgn(I/x))*(-csgn(I/x/(-2+x)*(x*ln(l
n(x))+1))+csgn(I/(-2+x)*(x*ln(ln(x))+1)))+I*Pi*csgn(I/x/(-2+x)*(x*ln(ln(x))+1))^2*(csgn(I/x/(-2+x)*(x*ln(ln(x)
)+1))-1)))+2*I*x/(2*I*ln(3)+2*I*ln(x))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\frac {x + \log \left (\log \left (\log \left (-\frac {x \log \left (\log \left (x\right )\right ) + 1}{x^{2} - 2 \, x}\right )\right )\right )}{\log \left (3\right ) + \log \left (x\right )} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\frac {x + \log \left (\log \left (\log \left (x \log \left (\log \left (x\right )\right ) + 1\right ) - \log \left (x\right ) - \log \left (-x + 2\right )\right )\right )}{\log \left (3\right ) + \log \left (x\right )} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 1.84 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\frac {x}{\log \left (3\right ) + \log \left (x\right )} + \frac {\log \left (\log \left (\log \left (-x \log \left (\log \left (x\right )\right ) - 1\right ) - \log \left (x - 2\right ) - \log \left (x\right )\right )\right )}{\log \left (3\right ) + \log \left (x\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 20.56 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\frac {x+\ln \left (\ln \left (\ln \left (\frac {x\,\ln \left (\ln \left (x\right )\right )+1}{2\,x-x^2}\right )\right )\right )}{\ln \left (3\,x\right )} \]

[In]

int((log(3*x)*(2*x + log(x)*(2*x - 2) - x^2) - log((x*log(log(x)) + 1)/(2*x - x^2))*log(log((x*log(log(x)) + 1
)/(2*x - x^2)))*(log(log(x))*(log(x)*(2*x^2 - x^3) - log(3*x)*log(x)*(2*x^2 - x^3)) + log(x)*(2*x - x^2) - log
(3*x)*log(x)*(2*x - x^2)) + log((x*log(log(x)) + 1)/(2*x - x^2))*log(log((x*log(log(x)) + 1)/(2*x - x^2)))*log
(log(log((x*log(log(x)) + 1)/(2*x - x^2))))*(log(x)*(x - 2) - log(log(x))*log(x)*(2*x - x^2)) + x^2*log(3*x)*l
og(log(x))*log(x))/(log((x*log(log(x)) + 1)/(2*x - x^2))*log(log((x*log(log(x)) + 1)/(2*x - x^2)))*(log(3*x)^2
*log(x)*(2*x - x^2) + log(3*x)^2*log(log(x))*log(x)*(2*x^2 - x^3))),x)

[Out]

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