Integrand size = 76, antiderivative size = 18 \begin {dmath*} \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^2}{\log (8 (x-\log (x \log (x))))} \end {dmath*}
Time = 1.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^2}{\log (8 (x-\log (x \log (x))))} \end {dmath*}
Integrate[(-x + (-x + x^2)*Log[x] + (-2*x^2*Log[x] + 2*x*Log[x]*Log[x*Log[ x]])*Log[8*x - 8*Log[x*Log[x]]])/((-(x*Log[x]) + Log[x]*Log[x*Log[x]])*Log [8*x - 8*Log[x*Log[x]]]^2),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (x^2-x\right ) \log (x)+\left (2 x \log (x) \log (x \log (x))-2 x^2 \log (x)\right ) \log (8 x-8 \log (x \log (x)))-x}{(\log (x) \log (x \log (x))-x \log (x)) \log ^2(8 x-8 \log (x \log (x)))} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-\left (x^2-x\right ) \log (x)-\left (2 x \log (x) \log (x \log (x))-2 x^2 \log (x)\right ) \log (8 x-8 \log (x \log (x)))+x}{\log (x) (x-\log (x \log (x))) \log ^2(8 (x-\log (x \log (x))))}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {x^2}{(x-\log (x \log (x))) \log ^2(8 (x-\log (x \log (x))))}+\frac {2 x^2}{(x-\log (x \log (x))) \log (8 (x-\log (x \log (x))))}+\frac {x}{\log (x) (x-\log (x \log (x))) \log ^2(8 (x-\log (x \log (x))))}+\frac {x}{(x-\log (x \log (x))) \log ^2(8 (x-\log (x \log (x))))}-\frac {2 x \log (x \log (x))}{(x-\log (x \log (x))) \log (8 (x-\log (x \log (x))))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {x^2}{(x-\log (x \log (x))) \log ^2(8 (x-\log (x \log (x))))}dx+2 \int \frac {x^2}{(x-\log (x \log (x))) \log (8 (x-\log (x \log (x))))}dx+\int \frac {x}{(x-\log (x \log (x))) \log ^2(8 (x-\log (x \log (x))))}dx+\int \frac {x}{\log (x) (x-\log (x \log (x))) \log ^2(8 (x-\log (x \log (x))))}dx-2 \int \frac {x \log (x \log (x))}{(x-\log (x \log (x))) \log (8 (x-\log (x \log (x))))}dx\) |
Int[(-x + (-x + x^2)*Log[x] + (-2*x^2*Log[x] + 2*x*Log[x]*Log[x*Log[x]])*L og[8*x - 8*Log[x*Log[x]]])/((-(x*Log[x]) + Log[x]*Log[x*Log[x]])*Log[8*x - 8*Log[x*Log[x]]]^2),x]
3.13.9.3.1 Defintions of rubi rules used
Timed out.
\[\int \frac {\left (2 x \ln \left (x \right ) \ln \left (x \ln \left (x \right )\right )-2 x^{2} \ln \left (x \right )\right ) \ln \left (-8 \ln \left (x \ln \left (x \right )\right )+8 x \right )+\ln \left (x \right ) \left (x^{2}-x \right )-x}{\left (\ln \left (x \right ) \ln \left (x \ln \left (x \right )\right )-x \ln \left (x \right )\right ) \ln \left (-8 \ln \left (x \ln \left (x \right )\right )+8 x \right )^{2}}d x\]
int(((2*x*ln(x)*ln(x*ln(x))-2*x^2*ln(x))*ln(-8*ln(x*ln(x))+8*x)+ln(x)*(x^2 -x)-x)/(ln(x)*ln(x*ln(x))-x*ln(x))/ln(-8*ln(x*ln(x))+8*x)^2,x)
int(((2*x*ln(x)*ln(x*ln(x))-2*x^2*ln(x))*ln(-8*ln(x*ln(x))+8*x)+ln(x)*(x^2 -x)-x)/(ln(x)*ln(x*ln(x))-x*ln(x))/ln(-8*ln(x*ln(x))+8*x)^2,x)
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^{2}}{\log \left (8 \, x - 8 \, \log \left (x \log \left (x\right )\right )\right )} \end {dmath*}
integrate(((2*x*log(x)*log(x*log(x))-2*x^2*log(x))*log(-8*log(x*log(x))+8* x)+log(x)*(x^2-x)-x)/(log(x)*log(x*log(x))-x*log(x))/log(-8*log(x*log(x))+ 8*x)^2,x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \begin {dmath*} \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^{2}}{\log {\left (8 x - 8 \log {\left (x \log {\left (x \right )} \right )} \right )}} \end {dmath*}
integrate(((2*x*ln(x)*ln(x*ln(x))-2*x**2*ln(x))*ln(-8*ln(x*ln(x))+8*x)+ln( x)*(x**2-x)-x)/(ln(x)*ln(x*ln(x))-x*ln(x))/ln(-8*ln(x*ln(x))+8*x)**2,x)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \begin {dmath*} \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^{2}}{i \, \pi + 3 \, \log \left (2\right ) + \log \left (-x + \log \left (x\right ) + \log \left (\log \left (x\right )\right )\right )} \end {dmath*}
integrate(((2*x*log(x)*log(x*log(x))-2*x^2*log(x))*log(-8*log(x*log(x))+8* x)+log(x)*(x^2-x)-x)/(log(x)*log(x*log(x))-x*log(x))/log(-8*log(x*log(x))+ 8*x)^2,x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \begin {dmath*} \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^{2}}{\log \left (8 \, x - 8 \, \log \left (x\right ) - 8 \, \log \left (\log \left (x\right )\right )\right )} \end {dmath*}
integrate(((2*x*log(x)*log(x*log(x))-2*x^2*log(x))*log(-8*log(x*log(x))+8* x)+log(x)*(x^2-x)-x)/(log(x)*log(x*log(x))-x*log(x))/log(-8*log(x*log(x))+ 8*x)^2,x, algorithm=\
Timed out. \begin {dmath*} \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\int -\frac {x+\ln \left (x\right )\,\left (x-x^2\right )+\ln \left (8\,x-8\,\ln \left (x\,\ln \left (x\right )\right )\right )\,\left (2\,x^2\,\ln \left (x\right )-2\,x\,\ln \left (x\,\ln \left (x\right )\right )\,\ln \left (x\right )\right )}{{\ln \left (8\,x-8\,\ln \left (x\,\ln \left (x\right )\right )\right )}^2\,\left (\ln \left (x\,\ln \left (x\right )\right )\,\ln \left (x\right )-x\,\ln \left (x\right )\right )} \,d x \end {dmath*}
int(-(x + log(x)*(x - x^2) + log(8*x - 8*log(x*log(x)))*(2*x^2*log(x) - 2* x*log(x*log(x))*log(x)))/(log(8*x - 8*log(x*log(x)))^2*(log(x*log(x))*log( x) - x*log(x))),x)