Integrand size = 132, antiderivative size = 25 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=-20+\frac {\log \left (-6+\frac {\log (x)}{x}\right )}{-4-e^x \log (3 x)} \] Output:
ln(ln(x)/x-6)/(-4-ln(3*x)*exp(x))-20
Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=-\frac {\log \left (-6+\frac {\log (x)}{x}\right )}{4+e^x \log (3 x)} \] Input:
Integrate[(-4 + 4*Log[x] + (-E^x + E^x*Log[x])*Log[3*x] + (-6*E^x*x + E^x* Log[x] + (-6*E^x*x^2 + E^x*x*Log[x])*Log[3*x])*Log[(-6*x + Log[x])/x])/(-9 6*x^2 + 16*x*Log[x] + (-48*E^x*x^2 + 8*E^x*x*Log[x])*Log[3*x] + (-6*E^(2*x )*x^2 + E^(2*x)*x*Log[x])*Log[3*x]^2),x]
Output:
-(Log[-6 + Log[x]/x]/(4 + E^x*Log[3*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.
| \(\displaystyle \int \frac {\left (\left (e^x x \log (x)-6 e^x x^2\right ) \log (3 x)-6 e^x x+e^x \log (x)\right ) \log \left (\frac {\log (x)-6 x}{x}\right )+4 \log (x)+\left (e^x \log (x)-e^x\right ) \log (3 x)-4}{-96 x^2+\left (e^{2 x} x \log (x)-6 e^{2 x} x^2\right ) \log ^2(3 x)+\left (8 e^x x \log (x)-48 e^x x^2\right ) \log (3 x)+16 x \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (\left (e^x x \log (x)-6 e^x x^2\right ) \log (3 x)-6 e^x x+e^x \log (x)\right ) \log \left (\frac {\log (x)-6 x}{x}\right )-4 \log (x)-\left (e^x \log (x)-e^x\right ) \log (3 x)+4}{x (6 x-\log (x)) \left (e^x \log (3 x)+4\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {6 x^2 \log (3 x) \log \left (\frac {\log (x)}{x}-6\right )-x \log (x) \log (3 x) \log \left (\frac {\log (x)}{x}-6\right )+6 x \log \left (\frac {\log (x)}{x}-6\right )-\log (x) \log (3 x)+\log (3 x)-\log (x) \log \left (\frac {\log (x)}{x}-6\right )}{x (6 x-\log (x)) \log (3 x) \left (e^x \log (3 x)+4\right )}-\frac {4 (x \log (3 x)+1) \log \left (\frac {\log (x)}{x}-6\right )}{x \log (3 x) \left (e^x \log (3 x)+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {1}{x (6 x-\log (x)) \left (e^x \log (3 x)+4\right )}dx-\int \frac {\log (x)}{x (6 x-\log (x)) \left (e^x \log (3 x)+4\right )}dx-4 \int \frac {\log \left (\frac {\log (x)}{x}-6\right )}{\left (e^x \log (3 x)+4\right )^2}dx-4 \int \frac {\log \left (\frac {\log (x)}{x}-6\right )}{x \log (3 x) \left (e^x \log (3 x)+4\right )^2}dx+6 \int \frac {x \log \left (\frac {\log (x)}{x}-6\right )}{(6 x-\log (x)) \left (e^x \log (3 x)+4\right )}dx-\int \frac {\log (x) \log \left (\frac {\log (x)}{x}-6\right )}{(6 x-\log (x)) \left (e^x \log (3 x)+4\right )}dx+6 \int \frac {\log \left (\frac {\log (x)}{x}-6\right )}{(6 x-\log (x)) \log (3 x) \left (e^x \log (3 x)+4\right )}dx-\int \frac {\log (x) \log \left (\frac {\log (x)}{x}-6\right )}{x (6 x-\log (x)) \log (3 x) \left (e^x \log (3 x)+4\right )}dx\) |
Input:
Int[(-4 + 4*Log[x] + (-E^x + E^x*Log[x])*Log[3*x] + (-6*E^x*x + E^x*Log[x] + (-6*E^x*x^2 + E^x*x*Log[x])*Log[3*x])*Log[(-6*x + Log[x])/x])/(-96*x^2 + 16*x*Log[x] + (-48*E^x*x^2 + 8*E^x*x*Log[x])*Log[3*x] + (-6*E^(2*x)*x^2 + E^(2*x)*x*Log[x])*Log[3*x]^2),x]
Output:
$Aborted
Time = 30.99 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
| method | result | size |
| parallelrisch | \(-\frac {\ln \left (\frac {\ln \left (x \right )-6 x}{x}\right )}{\ln \left (3 x \right ) {\mathrm e}^{x}+4}\) | \(25\) |
| risch | \(\frac {2 i \ln \left (-\frac {\ln \left (x \right )}{6}+x \right )}{-2 i \ln \left (3\right ) {\mathrm e}^{x}-2 i {\mathrm e}^{x} \ln \left (x \right )-8 i}-\frac {-i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (\frac {\ln \left (x \right )}{6}-x \right )\right ) \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )^{2}-2 i \pi \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )^{2}-i \pi \,\operatorname {csgn}\left (i \left (\frac {\ln \left (x \right )}{6}-x \right )\right ) \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )}{6}-x \right )}{x}\right )^{3}+2 i \pi +2 \ln \left (3\right )+2 \ln \left (2\right )-2 \ln \left (x \right )}{8+2 \ln \left (3\right ) {\mathrm e}^{x}+2 \,{\mathrm e}^{x} \ln \left (x \right )}\) | \(207\) |
Input:
int((((x*exp(x)*ln(x)-6*exp(x)*x^2)*ln(3*x)+exp(x)*ln(x)-6*exp(x)*x)*ln((l n(x)-6*x)/x)+(exp(x)*ln(x)-exp(x))*ln(3*x)+4*ln(x)-4)/((x*exp(x)^2*ln(x)-6 *exp(x)^2*x^2)*ln(3*x)^2+(8*x*exp(x)*ln(x)-48*exp(x)*x^2)*ln(3*x)+16*x*ln( x)-96*x^2),x,method=_RETURNVERBOSE)
Output:
-ln((ln(x)-6*x)/x)/(ln(3*x)*exp(x)+4)
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=-\frac {\log \left (-\frac {6 \, x - \log \left (x\right )}{x}\right )}{e^{x} \log \left (3\right ) + e^{x} \log \left (x\right ) + 4} \] Input:
integrate((((x*exp(x)*log(x)-6*exp(x)*x^2)*log(3*x)+exp(x)*log(x)-6*exp(x) *x)*log((log(x)-6*x)/x)+(exp(x)*log(x)-exp(x))*log(3*x)+4*log(x)-4)/((x*ex p(x)^2*log(x)-6*exp(x)^2*x^2)*log(3*x)^2+(8*x*exp(x)*log(x)-48*exp(x)*x^2) *log(3*x)+16*x*log(x)-96*x^2),x, algorithm="fricas")
Output:
-log(-(6*x - log(x))/x)/(e^x*log(3) + e^x*log(x) + 4)
Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=- \frac {\log {\left (\frac {- 6 x + \log {\left (x \right )}}{x} \right )}}{\left (\log {\left (x \right )} + \log {\left (3 \right )}\right ) e^{x} + 4} \] Input:
integrate((((x*exp(x)*ln(x)-6*exp(x)*x**2)*ln(3*x)+exp(x)*ln(x)-6*exp(x)*x )*ln((ln(x)-6*x)/x)+(exp(x)*ln(x)-exp(x))*ln(3*x)+4*ln(x)-4)/((x*exp(x)**2 *ln(x)-6*exp(x)**2*x**2)*ln(3*x)**2+(8*x*exp(x)*ln(x)-48*exp(x)*x**2)*ln(3 *x)+16*x*ln(x)-96*x**2),x)
Output:
-log((-6*x + log(x))/x)/((log(x) + log(3))*exp(x) + 4)
Time = 0.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=\frac {\log \left (x\right ) - \log \left (-6 \, x + \log \left (x\right )\right )}{{\left (\log \left (3\right ) + \log \left (x\right )\right )} e^{x} + 4} \] Input:
integrate((((x*exp(x)*log(x)-6*exp(x)*x^2)*log(3*x)+exp(x)*log(x)-6*exp(x) *x)*log((log(x)-6*x)/x)+(exp(x)*log(x)-exp(x))*log(3*x)+4*log(x)-4)/((x*ex p(x)^2*log(x)-6*exp(x)^2*x^2)*log(3*x)^2+(8*x*exp(x)*log(x)-48*exp(x)*x^2) *log(3*x)+16*x*log(x)-96*x^2),x, algorithm="maxima")
Output:
(log(x) - log(-6*x + log(x)))/((log(3) + log(x))*e^x + 4)
Time = 0.46 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=\frac {\log \left (x\right ) - \log \left (-6 \, x + \log \left (x\right )\right )}{e^{x} \log \left (3\right ) + e^{x} \log \left (x\right ) + 4} \] Input:
integrate((((x*exp(x)*log(x)-6*exp(x)*x^2)*log(3*x)+exp(x)*log(x)-6*exp(x) *x)*log((log(x)-6*x)/x)+(exp(x)*log(x)-exp(x))*log(3*x)+4*log(x)-4)/((x*ex p(x)^2*log(x)-6*exp(x)^2*x^2)*log(3*x)^2+(8*x*exp(x)*log(x)-48*exp(x)*x^2) *log(3*x)+16*x*log(x)-96*x^2),x, algorithm="giac")
Output:
(log(x) - log(-6*x + log(x)))/(e^x*log(3) + e^x*log(x) + 4)
Time = 3.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=-\frac {\ln \left (-\frac {6\,x-\ln \left (x\right )}{x}\right )}{\ln \left (3\,x\right )\,{\mathrm {e}}^x+4} \] Input:
int((log(3*x)*(exp(x) - exp(x)*log(x)) - 4*log(x) + log(-(6*x - log(x))/x) *(log(3*x)*(6*x^2*exp(x) - x*exp(x)*log(x)) - exp(x)*log(x) + 6*x*exp(x)) + 4)/(log(3*x)*(48*x^2*exp(x) - 8*x*exp(x)*log(x)) - 16*x*log(x) + log(3*x )^2*(6*x^2*exp(2*x) - x*exp(2*x)*log(x)) + 96*x^2),x)
Output:
-log(-(6*x - log(x))/x)/(log(3*x)*exp(x) + 4)
Time = 0.22 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.92 \[ \int \frac {-4+4 \log (x)+\left (-e^x+e^x \log (x)\right ) \log (3 x)+\left (-6 e^x x+e^x \log (x)+\left (-6 e^x x^2+e^x x \log (x)\right ) \log (3 x)\right ) \log \left (\frac {-6 x+\log (x)}{x}\right )}{-96 x^2+16 x \log (x)+\left (-48 e^x x^2+8 e^x x \log (x)\right ) \log (3 x)+\left (-6 e^{2 x} x^2+e^{2 x} x \log (x)\right ) \log ^2(3 x)} \, dx=\frac {-e^{x} \mathrm {log}\left (\mathrm {log}\left (x \right )-6 x \right ) \mathrm {log}\left (3 x \right )+e^{x} \mathrm {log}\left (\frac {\mathrm {log}\left (x \right )-6 x}{x}\right ) \mathrm {log}\left (3 x \right )+e^{x} \mathrm {log}\left (3 x \right ) \mathrm {log}\left (x \right )-4 \,\mathrm {log}\left (\mathrm {log}\left (x \right )-6 x \right )+4 \,\mathrm {log}\left (x \right )}{4 e^{x} \mathrm {log}\left (3 x \right )+16} \] Input:
int((((x*exp(x)*log(x)-6*exp(x)*x^2)*log(3*x)+exp(x)*log(x)-6*exp(x)*x)*lo g((log(x)-6*x)/x)+(exp(x)*log(x)-exp(x))*log(3*x)+4*log(x)-4)/((x*exp(x)^2 *log(x)-6*exp(x)^2*x^2)*log(3*x)^2+(8*x*exp(x)*log(x)-48*exp(x)*x^2)*log(3 *x)+16*x*log(x)-96*x^2),x)
Output:
( - e**x*log(log(x) - 6*x)*log(3*x) + e**x*log((log(x) - 6*x)/x)*log(3*x) + e**x*log(3*x)*log(x) - 4*log(log(x) - 6*x) + 4*log(x))/(4*(e**x*log(3*x) + 4))