Integrand size = 119, antiderivative size = 23 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=x-\log \left (\frac {4 \log (x)}{\log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}\right ) \]
Time = 0.50 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=x-\log (\log (x))+\log \left (\log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )\right ) \]
Integrate[((-8 + 2*x)*Log[x] + x*Log[x]*Log[x^2]*Log[Log[x^2]] + ((-2 + 2* x*Log[x])*Log[x^2] + (4 - x + (-4*x + x^2)*Log[x])*Log[x^2]*Log[Log[x^2]]) *Log[2 + (-4 + x)*Log[Log[x^2]]])/((2*x*Log[x]*Log[x^2] + (-4*x + x^2)*Log [x]*Log[x^2]*Log[Log[x^2]])*Log[2 + (-4 + x)*Log[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.
\(\displaystyle \int \frac {x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log (x)+\left ((2 x \log (x)-2) \log \left (x^2\right )+\left (\left (x^2-4 x\right ) \log (x)-x+4\right ) \log \left (\log \left (x^2\right )\right ) \log \left (x^2\right )\right ) \log \left ((x-4) \log \left (\log \left (x^2\right )\right )+2\right )+(2 x-8) \log (x)}{\left (2 x \log (x) \log \left (x^2\right )+\left (x^2-4 x\right ) \log (x) \log \left (\log \left (x^2\right )\right ) \log \left (x^2\right )\right ) \log \left ((x-4) \log \left (\log \left (x^2\right )\right )+2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log (x)+\left ((2 x \log (x)-2) \log \left (x^2\right )+\left (\left (x^2-4 x\right ) \log (x)-x+4\right ) \log \left (\log \left (x^2\right )\right ) \log \left (x^2\right )\right ) \log \left ((x-4) \log \left (\log \left (x^2\right )\right )+2\right )+(2 x-8) \log (x)}{x \log (x) \log \left (x^2\right ) \left (x \log \left (\log \left (x^2\right )\right )-4 \log \left (\log \left (x^2\right )\right )+2\right ) \log \left ((x-4) \log \left (\log \left (x^2\right )\right )+2\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+2 x-8}{x \log \left (x^2\right ) \left (x \log \left (\log \left (x^2\right )\right )-4 \log \left (\log \left (x^2\right )\right )+2\right ) \log \left ((x-4) \log \left (\log \left (x^2\right )\right )+2\right )}+\frac {x \log (x)-1}{x \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \frac {1}{\log \left (x^2\right ) \left (x \log \left (\log \left (x^2\right )\right )-4 \log \left (\log \left (x^2\right )\right )+2\right ) \log \left ((x-4) \log \left (\log \left (x^2\right )\right )+2\right )}dx-8 \int \frac {1}{x \log \left (x^2\right ) \left (x \log \left (\log \left (x^2\right )\right )-4 \log \left (\log \left (x^2\right )\right )+2\right ) \log \left ((x-4) \log \left (\log \left (x^2\right )\right )+2\right )}dx+\int \frac {\log \left (\log \left (x^2\right )\right )}{\left (x \log \left (\log \left (x^2\right )\right )-4 \log \left (\log \left (x^2\right )\right )+2\right ) \log \left ((x-4) \log \left (\log \left (x^2\right )\right )+2\right )}dx+x-\log (\log (x))\) |
Int[((-8 + 2*x)*Log[x] + x*Log[x]*Log[x^2]*Log[Log[x^2]] + ((-2 + 2*x*Log[ x])*Log[x^2] + (4 - x + (-4*x + x^2)*Log[x])*Log[x^2]*Log[Log[x^2]])*Log[2 + (-4 + x)*Log[Log[x^2]]])/((2*x*Log[x]*Log[x^2] + (-4*x + x^2)*Log[x]*Lo g[x^2]*Log[Log[x^2]])*Log[2 + (-4 + x)*Log[Log[x^2]]]),x]
3.24.72.3.1 Defintions of rubi rules used
Time = 104.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(2-\ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (\left (x -4\right ) \ln \left (\ln \left (x^{2}\right )\right )+2\right )\right )+x\) | \(22\) |
risch | \(x -\ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (\left (x -4\right ) \ln \left (2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )+2\right )\right )\) | \(50\) |
int(((((x^2-4*x)*ln(x)-x+4)*ln(x^2)*ln(ln(x^2))+(2*x*ln(x)-2)*ln(x^2))*ln( (x-4)*ln(ln(x^2))+2)+x*ln(x)*ln(x^2)*ln(ln(x^2))+(2*x-8)*ln(x))/((x^2-4*x) *ln(x)*ln(x^2)*ln(ln(x^2))+2*x*ln(x)*ln(x^2))/ln((x-4)*ln(ln(x^2))+2),x,me thod=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=x - \log \left (2 \, \log \left (x\right )\right ) + \log \left (\log \left ({\left (x - 4\right )} \log \left (2 \, \log \left (x\right )\right ) + 2\right )\right ) \]
integrate(((((x^2-4*x)*log(x)-x+4)*log(x^2)*log(log(x^2))+(2*x*log(x)-2)*l og(x^2))*log((x-4)*log(log(x^2))+2)+x*log(x)*log(x^2)*log(log(x^2))+(2*x-8 )*log(x))/((x^2-4*x)*log(x)*log(x^2)*log(log(x^2))+2*x*log(x)*log(x^2))/lo g((x-4)*log(log(x^2))+2),x, algorithm=\
Time = 0.45 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=x - \log {\left (\log {\left (x \right )} \right )} + \log {\left (\log {\left (\left (x - 4\right ) \log {\left (2 \log {\left (x \right )} \right )} + 2 \right )} \right )} \]
integrate(((((x**2-4*x)*ln(x)-x+4)*ln(x**2)*ln(ln(x**2))+(2*x*ln(x)-2)*ln( x**2))*ln((x-4)*ln(ln(x**2))+2)+x*ln(x)*ln(x**2)*ln(ln(x**2))+(2*x-8)*ln(x ))/((x**2-4*x)*ln(x)*ln(x**2)*ln(ln(x**2))+2*x*ln(x)*ln(x**2))/ln((x-4)*ln (ln(x**2))+2),x)
Time = 0.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=x + \log \left (\log \left (x \log \left (2\right ) + {\left (x - 4\right )} \log \left (\log \left (x\right )\right ) - 4 \, \log \left (2\right ) + 2\right )\right ) - \log \left (\log \left (x\right )\right ) \]
integrate(((((x^2-4*x)*log(x)-x+4)*log(x^2)*log(log(x^2))+(2*x*log(x)-2)*l og(x^2))*log((x-4)*log(log(x^2))+2)+x*log(x)*log(x^2)*log(log(x^2))+(2*x-8 )*log(x))/((x^2-4*x)*log(x)*log(x^2)*log(log(x^2))+2*x*log(x)*log(x^2))/lo g((x-4)*log(log(x^2))+2),x, algorithm=\
\[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=\int { \frac {x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\log \left (x^{2}\right )\right ) + {\left ({\left ({\left (x^{2} - 4 \, x\right )} \log \left (x\right ) - x + 4\right )} \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right ) + 2 \, {\left (x \log \left (x\right ) - 1\right )} \log \left (x^{2}\right )\right )} \log \left ({\left (x - 4\right )} \log \left (\log \left (x^{2}\right )\right ) + 2\right ) + 2 \, {\left (x - 4\right )} \log \left (x\right )}{{\left ({\left (x^{2} - 4 \, x\right )} \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\log \left (x^{2}\right )\right ) + 2 \, x \log \left (x^{2}\right ) \log \left (x\right )\right )} \log \left ({\left (x - 4\right )} \log \left (\log \left (x^{2}\right )\right ) + 2\right )} \,d x } \]
integrate(((((x^2-4*x)*log(x)-x+4)*log(x^2)*log(log(x^2))+(2*x*log(x)-2)*l og(x^2))*log((x-4)*log(log(x^2))+2)+x*log(x)*log(x^2)*log(log(x^2))+(2*x-8 )*log(x))/((x^2-4*x)*log(x)*log(x^2)*log(log(x^2))+2*x*log(x)*log(x^2))/lo g((x-4)*log(log(x^2))+2),x, algorithm=\
integrate((x*log(x^2)*log(x)*log(log(x^2)) + (((x^2 - 4*x)*log(x) - x + 4) *log(x^2)*log(log(x^2)) + 2*(x*log(x) - 1)*log(x^2))*log((x - 4)*log(log(x ^2)) + 2) + 2*(x - 4)*log(x))/(((x^2 - 4*x)*log(x^2)*log(x)*log(log(x^2)) + 2*x*log(x^2)*log(x))*log((x - 4)*log(log(x^2)) + 2)), x)
Time = 14.41 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=x+\ln \left (\ln \left (\ln \left (\ln \left (x^2\right )\right )\,\left (x-4\right )+2\right )\right )-\ln \left (\ln \left (x\right )\right ) \]