Integrand size = 109, antiderivative size = 24 \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=x \log \left (\frac {4}{3 x+\frac {\log (x)}{2}+\log (4+\log (2 x))}\right ) \]
Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=x \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right ) \]
Integrate[(-6 - 24*x + (-1 - 6*x)*Log[2*x] + (24*x + 4*Log[x] + (6*x + Log [x])*Log[2*x] + (8 + 2*Log[2*x])*Log[4 + Log[2*x]])*Log[8/(6*x + Log[x] + 2*Log[4 + Log[2*x]])])/(24*x + 4*Log[x] + (6*x + Log[x])*Log[2*x] + (8 + 2 *Log[2*x])*Log[4 + Log[2*x]]),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.
\(\displaystyle \int \frac {-24 x+(-6 x-1) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(2 \log (2 x)+8) \log (\log (2 x)+4)) \log \left (\frac {8}{6 x+\log (x)+2 \log (\log (2 x)+4)}\right )-6}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(2 \log (2 x)+8) \log (\log (2 x)+4)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-24 x+(-6 x-1) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(2 \log (2 x)+8) \log (\log (2 x)+4)) \log \left (\frac {8}{6 x+\log (x)+2 \log (\log (2 x)+4)}\right )-6}{(\log (2 x)+4) (6 x+\log (x)+2 \log (\log (2 x)+4))}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-24 x-6 x \log (2 x)-\log (2 x)-6}{(\log (2 x)+4) (6 x+\log (x)+2 \log (\log (2 x)+4))}+\log \left (\frac {8}{6 x+\log (x)+2 \log (\log (2 x)+4)}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {1}{6 x+\log (x)+2 \log (\log (2 x)+4)}dx+6 \int \frac {x}{6 x+\log (x)+2 \log (\log (2 x)+4)}dx-4 \int \frac {1}{(\log (2 x)+4) (6 x+\log (x)+2 \log (\log (2 x)+4))}dx-24 \int \frac {x}{(\log (2 x)+4) (6 x+\log (x)+2 \log (\log (2 x)+4))}dx-\int \frac {\log (2 x)}{(\log (2 x)+4) (6 x+\log (x)+2 \log (\log (2 x)+4))}dx-6 \int \frac {x \log (2 x)}{(\log (2 x)+4) (6 x+\log (x)+2 \log (\log (2 x)+4))}dx+x \log \left (\frac {8}{6 x+\log (x)+2 \log (\log (2 x)+4)}\right )\) |
Int[(-6 - 24*x + (-1 - 6*x)*Log[2*x] + (24*x + 4*Log[x] + (6*x + Log[x])*L og[2*x] + (8 + 2*Log[2*x])*Log[4 + Log[2*x]])*Log[8/(6*x + Log[x] + 2*Log[ 4 + Log[2*x]])])/(24*x + 4*Log[x] + (6*x + Log[x])*Log[2*x] + (8 + 2*Log[2 *x])*Log[4 + Log[2*x]]),x]
3.29.49.3.1 Defintions of rubi rules used
Time = 14.41 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(\ln \left (\frac {8}{2 \ln \left (\ln \left (2 x \right )+4\right )+\ln \left (x \right )+6 x}\right ) x\) | \(23\) |
risch | \(-x \ln \left (\frac {\ln \left (\ln \left (2\right )+\ln \left (x \right )+4\right )}{3}+\frac {\ln \left (x \right )}{6}+x \right )-x \ln \left (3\right )+2 x \ln \left (2\right )\) | \(31\) |
int((((2*ln(2*x)+8)*ln(ln(2*x)+4)+(ln(x)+6*x)*ln(2*x)+4*ln(x)+24*x)*ln(8/( 2*ln(ln(2*x)+4)+ln(x)+6*x))+(-6*x-1)*ln(2*x)-24*x-6)/((2*ln(2*x)+8)*ln(ln( 2*x)+4)+(ln(x)+6*x)*ln(2*x)+4*ln(x)+24*x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=x \log \left (\frac {8}{6 \, x + \log \left (x\right ) + 2 \, \log \left (\log \left (2\right ) + \log \left (x\right ) + 4\right )}\right ) \]
integrate((((2*log(2*x)+8)*log(log(2*x)+4)+(log(x)+6*x)*log(2*x)+4*log(x)+ 24*x)*log(8/(2*log(log(2*x)+4)+log(x)+6*x))+(-6*x-1)*log(2*x)-24*x-6)/((2* log(2*x)+8)*log(log(2*x)+4)+(log(x)+6*x)*log(2*x)+4*log(x)+24*x),x, algori thm=\
Exception generated. \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=\text {Exception raised: TypeError} \]
integrate((((2*ln(2*x)+8)*ln(ln(2*x)+4)+(ln(x)+6*x)*ln(2*x)+4*ln(x)+24*x)* ln(8/(2*ln(ln(2*x)+4)+ln(x)+6*x))+(-6*x-1)*ln(2*x)-24*x-6)/((2*ln(2*x)+8)* ln(ln(2*x)+4)+(ln(x)+6*x)*ln(2*x)+4*ln(x)+24*x),x)
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=3 \, x \log \left (2\right ) - x \log \left (6 \, x + \log \left (x\right ) + 2 \, \log \left (\log \left (2\right ) + \log \left (x\right ) + 4\right )\right ) \]
integrate((((2*log(2*x)+8)*log(log(2*x)+4)+(log(x)+6*x)*log(2*x)+4*log(x)+ 24*x)*log(8/(2*log(log(2*x)+4)+log(x)+6*x))+(-6*x-1)*log(2*x)-24*x-6)/((2* log(2*x)+8)*log(log(2*x)+4)+(log(x)+6*x)*log(2*x)+4*log(x)+24*x),x, algori thm=\
Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=3 \, x \log \left (2\right ) - x \log \left (6 \, x + \log \left (x\right ) + 2 \, \log \left (\log \left (2\right ) + \log \left (x\right ) + 4\right )\right ) \]
integrate((((2*log(2*x)+8)*log(log(2*x)+4)+(log(x)+6*x)*log(2*x)+4*log(x)+ 24*x)*log(8/(2*log(log(2*x)+4)+log(x)+6*x))+(-6*x-1)*log(2*x)-24*x-6)/((2* log(2*x)+8)*log(log(2*x)+4)+(log(x)+6*x)*log(2*x)+4*log(x)+24*x),x, algori thm=\
Time = 9.61 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=x\,\left (\ln \left (\frac {1}{6\,x+2\,\ln \left (\ln \left (2\,x\right )+4\right )+\ln \left (x\right )}\right )+3\,\ln \left (2\right )\right ) \]