Integrand size = 105, antiderivative size = 29 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=-x+x \log \left (-3+\log \left (\frac {e^x x}{5-\frac {2-x}{x}}\right )\right ) \]
Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=-x+x \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \]
Integrate[(-5 + 11*x + 3*x^2 + (1 - 3*x)*Log[(E^x*x^2)/(-2 + 6*x)] + (3 - 9*x + (-1 + 3*x)*Log[(E^x*x^2)/(-2 + 6*x)])*Log[-3 + Log[(E^x*x^2)/(-2 + 6 *x)]])/(3 - 9*x + (-1 + 3*x)*Log[(E^x*x^2)/(-2 + 6*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.
\(\displaystyle \int \frac {3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{6 x-2}\right )+\left ((3 x-1) \log \left (\frac {e^x x^2}{6 x-2}\right )-9 x+3\right ) \log \left (\log \left (\frac {e^x x^2}{6 x-2}\right )-3\right )+11 x-5}{(3 x-1) \log \left (\frac {e^x x^2}{6 x-2}\right )-9 x+3} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{6 x-2}\right )+\left ((3 x-1) \log \left (\frac {e^x x^2}{6 x-2}\right )-9 x+3\right ) \log \left (\log \left (\frac {e^x x^2}{6 x-2}\right )-3\right )+11 x-5}{(1-3 x) \left (3-\log \left (\frac {e^x x^2}{6 x-2}\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 x^2-3 x \log \left (\frac {e^x x^2}{6 x-2}\right )+\log \left (\frac {e^x x^2}{6 x-2}\right )+11 x-5}{(3 x-1) \left (\log \left (\frac {e^x x^2}{6 x-2}\right )-3\right )}+\log \left (\log \left (\frac {e^x x^2}{6 x-2}\right )-3\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {1}{\log \left (\frac {e^x x^2}{6 x-2}\right )-3}dx+\int \frac {x}{\log \left (\frac {e^x x^2}{6 x-2}\right )-3}dx-\int \frac {1}{(3 x-1) \left (\log \left (\frac {e^x x^2}{6 x-2}\right )-3\right )}dx+\int \log \left (\log \left (\frac {e^x x^2}{6 x-2}\right )-3\right )dx-x\) |
Int[(-5 + 11*x + 3*x^2 + (1 - 3*x)*Log[(E^x*x^2)/(-2 + 6*x)] + (3 - 9*x + (-1 + 3*x)*Log[(E^x*x^2)/(-2 + 6*x)])*Log[-3 + Log[(E^x*x^2)/(-2 + 6*x)]]) /(3 - 9*x + (-1 + 3*x)*Log[(E^x*x^2)/(-2 + 6*x)]),x]
3.20.99.3.1 Defintions of rubi rules used
Time = 3.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(-\frac {1}{6}+\ln \left (\ln \left (\frac {x^{2} {\mathrm e}^{x}}{6 x -2}\right )-3\right ) x -x\) | \(26\) |
risch | \(\ln \left (-\ln \left (2\right )-\ln \left (3\right )+2 \ln \left (x \right )+\ln \left ({\mathrm e}^{x}\right )-\ln \left (x -\frac {1}{3}\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right ) \left (-\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\operatorname {csgn}\left (i {\mathrm e}^{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\operatorname {csgn}\left (\frac {i}{x -\frac {1}{3}}\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x^{2} {\mathrm e}^{x}}{x -\frac {1}{3}}\right ) \left (-\operatorname {csgn}\left (\frac {i x^{2} {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\operatorname {csgn}\left (i x^{2}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i x^{2} {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right )\right )}{2}-3\right ) x -x\) | \(187\) |
int((((-1+3*x)*ln(x^2*exp(x)/(6*x-2))-9*x+3)*ln(ln(x^2*exp(x)/(6*x-2))-3)+ (1-3*x)*ln(x^2*exp(x)/(6*x-2))+3*x^2+11*x-5)/((-1+3*x)*ln(x^2*exp(x)/(6*x- 2))-9*x+3),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=x \log \left (\log \left (\frac {x^{2} e^{x}}{2 \, {\left (3 \, x - 1\right )}}\right ) - 3\right ) - x \]
integrate((((-1+3*x)*log(x^2*exp(x)/(6*x-2))-9*x+3)*log(log(x^2*exp(x)/(6* x-2))-3)+(1-3*x)*log(x^2*exp(x)/(6*x-2))+3*x^2+11*x-5)/((-1+3*x)*log(x^2*e xp(x)/(6*x-2))-9*x+3),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
Time = 0.73 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=- x + \left (x - \frac {1}{18}\right ) \log {\left (\log {\left (\frac {x^{2} e^{x}}{6 x - 2} \right )} - 3 \right )} + \frac {\log {\left (\log {\left (\frac {x^{2} e^{x}}{6 x - 2} \right )} - 3 \right )}}{18} \]
integrate((((-1+3*x)*ln(x**2*exp(x)/(6*x-2))-9*x+3)*ln(ln(x**2*exp(x)/(6*x -2))-3)+(1-3*x)*ln(x**2*exp(x)/(6*x-2))+3*x**2+11*x-5)/((-1+3*x)*ln(x**2*e xp(x)/(6*x-2))-9*x+3),x)
-x + (x - 1/18)*log(log(x**2*exp(x)/(6*x - 2)) - 3) + log(log(x**2*exp(x)/ (6*x - 2)) - 3)/18
Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=x \log \left (x - \log \left (2\right ) - \log \left (3 \, x - 1\right ) + 2 \, \log \left (x\right ) - 3\right ) - x \]
integrate((((-1+3*x)*log(x^2*exp(x)/(6*x-2))-9*x+3)*log(log(x^2*exp(x)/(6* x-2))-3)+(1-3*x)*log(x^2*exp(x)/(6*x-2))+3*x^2+11*x-5)/((-1+3*x)*log(x^2*e xp(x)/(6*x-2))-9*x+3),x, algorithm=\
Time = 0.48 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=x \log \left (x + \log \left (\frac {x^{2}}{2 \, {\left (3 \, x - 1\right )}}\right ) - 3\right ) - x \]
integrate((((-1+3*x)*log(x^2*exp(x)/(6*x-2))-9*x+3)*log(log(x^2*exp(x)/(6* x-2))-3)+(1-3*x)*log(x^2*exp(x)/(6*x-2))+3*x^2+11*x-5)/((-1+3*x)*log(x^2*e xp(x)/(6*x-2))-9*x+3),x, algorithm=\
Time = 13.65 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=x\,\left (\ln \left (\ln \left (\frac {x^2\,{\mathrm {e}}^x}{6\,x-2}\right )-3\right )-1\right ) \]