Integrand size = 97, antiderivative size = 32 \[ \int \frac {9-6 x^2-3 \log (x)+\left (-6+8 x-8 x^2+6 x^3+(3-3 x) \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )}{\left (-6 x+2 x^2-6 x^3+3 x \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )} \, dx=\log \left (5 e^{-x} x \log \left (1+3 \left (x+\frac {2-x+x^2-\log (x)}{x}\right )\right )\right ) \]
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {9-6 x^2-3 \log (x)+\left (-6+8 x-8 x^2+6 x^3+(3-3 x) \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )}{\left (-6 x+2 x^2-6 x^3+3 x \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )} \, dx=-x+\log (x)+\log \left (\log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )\right ) \]
Integrate[(9 - 6*x^2 - 3*Log[x] + (-6 + 8*x - 8*x^2 + 6*x^3 + (3 - 3*x)*Lo g[x])*Log[(6 - 2*x + 6*x^2 - 3*Log[x])/x])/((-6*x + 2*x^2 - 6*x^3 + 3*x*Lo g[x])*Log[(6 - 2*x + 6*x^2 - 3*Log[x])/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.
\(\displaystyle \int \frac {-6 x^2+\left (6 x^3-8 x^2+8 x+(3-3 x) \log (x)-6\right ) \log \left (\frac {6 x^2-2 x-3 \log (x)+6}{x}\right )-3 \log (x)+9}{\left (-6 x^3+2 x^2-6 x+3 x \log (x)\right ) \log \left (\frac {6 x^2-2 x-3 \log (x)+6}{x}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {6 x^2-\left (6 x^3-8 x^2+8 x+(3-3 x) \log (x)-6\right ) \log \left (\frac {6 x^2-2 x-3 \log (x)+6}{x}\right )+3 \log (x)-9}{\left (6 x^3-2 x^2+6 x-3 x \log (x)\right ) \log \left (6 x+\frac {6}{x}-\frac {3 \log (x)}{x}-2\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {6 x^2}{6 x^2-2 x-3 \log (x)+6}+\frac {8 x}{6 x^2-2 x-3 \log (x)+6}+\frac {6 x}{\left (6 x^2-2 x-3 \log (x)+6\right ) \log \left (6 x+\frac {6}{x}-\frac {3 \log (x)}{x}-2\right )}+\frac {3 \log (x)}{6 x^2-2 x-3 \log (x)+6}-\frac {8}{6 x^2-2 x-3 \log (x)+6}-\frac {3 \log (x)}{x \left (6 x^2-2 x-3 \log (x)+6\right )}+\frac {6}{x \left (6 x^2-2 x-3 \log (x)+6\right )}+\frac {3 \log (x)}{x \left (6 x^2-2 x-3 \log (x)+6\right ) \log \left (6 x+\frac {6}{x}-\frac {3 \log (x)}{x}-2\right )}+\frac {9}{x \left (-6 x^2+2 x+3 \log (x)-6\right ) \log \left (6 x+\frac {6}{x}-\frac {3 \log (x)}{x}-2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \int \frac {1}{6 x^2-2 x-3 \log (x)+6}dx-2 \int \frac {1}{-6 x^2+2 x+3 \log (x)-6}dx+6 \int \frac {x}{\left (6 x^2-2 x-3 \log (x)+6\right ) \log \left (6 x-2-\frac {3 \log (x)}{x}+\frac {6}{x}\right )}dx+3 \int \frac {\log (x)}{x \left (6 x^2-2 x-3 \log (x)+6\right ) \log \left (6 x-2-\frac {3 \log (x)}{x}+\frac {6}{x}\right )}dx+9 \int \frac {1}{x \left (-6 x^2+2 x+3 \log (x)-6\right ) \log \left (6 x-2-\frac {3 \log (x)}{x}+\frac {6}{x}\right )}dx-x+\log (x)\) |
Int[(9 - 6*x^2 - 3*Log[x] + (-6 + 8*x - 8*x^2 + 6*x^3 + (3 - 3*x)*Log[x])* Log[(6 - 2*x + 6*x^2 - 3*Log[x])/x])/((-6*x + 2*x^2 - 6*x^3 + 3*x*Log[x])* Log[(6 - 2*x + 6*x^2 - 3*Log[x])/x]),x]
3.30.87.3.1 Defintions of rubi rules used
Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
method | result | size |
default | \(\ln \left (x \right )-x +\ln \left (\ln \left (\frac {-3 \ln \left (x \right )+6 x^{2}-2 x +6}{x}\right )\right )\) | \(27\) |
norman | \(\ln \left (x \right )-x +\ln \left (\ln \left (\frac {-3 \ln \left (x \right )+6 x^{2}-2 x +6}{x}\right )\right )\) | \(27\) |
parallelrisch | \(4+\ln \left (\ln \left (-\frac {-6 x^{2}+3 \ln \left (x \right )+2 x -6}{x}\right )\right )-x +\ln \left (x \right )\) | \(29\) |
risch | \(\ln \left (x \right )-x +\ln \left (\ln \left (x^{2}-\frac {x}{3}-\frac {\ln \left (x \right )}{2}+1\right )+\frac {i \left (-\pi \,\operatorname {csgn}\left (i \left (-x^{2}+\frac {x}{3}+\frac {\ln \left (x \right )}{2}-1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+\frac {x}{3}+\frac {\ln \left (x \right )}{2}-1\right )}{x}\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (-x^{2}+\frac {x}{3}+\frac {\ln \left (x \right )}{2}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-x^{2}+\frac {x}{3}+\frac {\ln \left (x \right )}{2}-1\right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )+\pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+\frac {x}{3}+\frac {\ln \left (x \right )}{2}-1\right )}{x}\right )}^{3}+\pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+\frac {x}{3}+\frac {\ln \left (x \right )}{2}-1\right )}{x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-2 i \ln \left (2\right )-2 i \ln \left (3\right )+2 i \ln \left (x \right )\right )}{2}\right )\) | \(191\) |
int((((-3*x+3)*ln(x)+6*x^3-8*x^2+8*x-6)*ln((-3*ln(x)+6*x^2-2*x+6)/x)-3*ln( x)-6*x^2+9)/(3*x*ln(x)-6*x^3+2*x^2-6*x)/ln((-3*ln(x)+6*x^2-2*x+6)/x),x,met hod=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {9-6 x^2-3 \log (x)+\left (-6+8 x-8 x^2+6 x^3+(3-3 x) \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )}{\left (-6 x+2 x^2-6 x^3+3 x \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )} \, dx=-x + \log \left (x\right ) + \log \left (\log \left (\frac {6 \, x^{2} - 2 \, x - 3 \, \log \left (x\right ) + 6}{x}\right )\right ) \]
integrate((((-3*x+3)*log(x)+6*x^3-8*x^2+8*x-6)*log((-3*log(x)+6*x^2-2*x+6) /x)-3*log(x)-6*x^2+9)/(3*x*log(x)-6*x^3+2*x^2-6*x)/log((-3*log(x)+6*x^2-2* x+6)/x),x, algorithm=\
Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {9-6 x^2-3 \log (x)+\left (-6+8 x-8 x^2+6 x^3+(3-3 x) \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )}{\left (-6 x+2 x^2-6 x^3+3 x \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )} \, dx=- x + \log {\left (x \right )} + \log {\left (\log {\left (\frac {6 x^{2} - 2 x - 3 \log {\left (x \right )} + 6}{x} \right )} \right )} \]
integrate((((-3*x+3)*ln(x)+6*x**3-8*x**2+8*x-6)*ln((-3*ln(x)+6*x**2-2*x+6) /x)-3*ln(x)-6*x**2+9)/(3*x*ln(x)-6*x**3+2*x**2-6*x)/ln((-3*ln(x)+6*x**2-2* x+6)/x),x)
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {9-6 x^2-3 \log (x)+\left (-6+8 x-8 x^2+6 x^3+(3-3 x) \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )}{\left (-6 x+2 x^2-6 x^3+3 x \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )} \, dx=-x + \log \left (x\right ) + \log \left (\log \left (6 \, x^{2} - 2 \, x - 3 \, \log \left (x\right ) + 6\right ) - \log \left (x\right )\right ) \]
integrate((((-3*x+3)*log(x)+6*x^3-8*x^2+8*x-6)*log((-3*log(x)+6*x^2-2*x+6) /x)-3*log(x)-6*x^2+9)/(3*x*log(x)-6*x^3+2*x^2-6*x)/log((-3*log(x)+6*x^2-2* x+6)/x),x, algorithm=\
Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {9-6 x^2-3 \log (x)+\left (-6+8 x-8 x^2+6 x^3+(3-3 x) \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )}{\left (-6 x+2 x^2-6 x^3+3 x \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )} \, dx=-x + \log \left (x\right ) + \log \left (\log \left (6 \, x^{2} - 2 \, x - 3 \, \log \left (x\right ) + 6\right ) - \log \left (x\right )\right ) \]
integrate((((-3*x+3)*log(x)+6*x^3-8*x^2+8*x-6)*log((-3*log(x)+6*x^2-2*x+6) /x)-3*log(x)-6*x^2+9)/(3*x*log(x)-6*x^3+2*x^2-6*x)/log((-3*log(x)+6*x^2-2* x+6)/x),x, algorithm=\
Time = 14.91 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {9-6 x^2-3 \log (x)+\left (-6+8 x-8 x^2+6 x^3+(3-3 x) \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )}{\left (-6 x+2 x^2-6 x^3+3 x \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )} \, dx=\ln \left (\ln \left (-\frac {2\,x+3\,\ln \left (x\right )-6\,x^2-6}{x}\right )\right )-x+\ln \left (x\right ) \]