Integrand size = 107, antiderivative size = 20 \[ \int \frac {2 x \log (3)+6 e^x x \log (3)+\left (-6 e^x \log (3)-2 x \log (3)\right ) \log \left (3 e^x+x\right )+\left (3 e^x \log (3)+x \log (3)\right ) \log \left (3 e^x+x\right ) \log \left (-\frac {x}{\log \left (3 e^x+x\right )}\right )}{\left (3 e^x+x\right ) \log \left (3 e^x+x\right ) \log ^3\left (-\frac {x}{\log \left (3 e^x+x\right )}\right )} \, dx=\frac {x \log (3)}{\log ^2\left (-\frac {x}{\log \left (3 e^x+x\right )}\right )} \]
Time = 0.55 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {2 x \log (3)+6 e^x x \log (3)+\left (-6 e^x \log (3)-2 x \log (3)\right ) \log \left (3 e^x+x\right )+\left (3 e^x \log (3)+x \log (3)\right ) \log \left (3 e^x+x\right ) \log \left (-\frac {x}{\log \left (3 e^x+x\right )}\right )}{\left (3 e^x+x\right ) \log \left (3 e^x+x\right ) \log ^3\left (-\frac {x}{\log \left (3 e^x+x\right )}\right )} \, dx=\frac {x \log (3)}{\log ^2\left (-\frac {x}{\log \left (3 e^x+x\right )}\right )} \]
Integrate[(2*x*Log[3] + 6*E^x*x*Log[3] + (-6*E^x*Log[3] - 2*x*Log[3])*Log[ 3*E^x + x] + (3*E^x*Log[3] + x*Log[3])*Log[3*E^x + x]*Log[-(x/Log[3*E^x + x])])/((3*E^x + x)*Log[3*E^x + x]*Log[-(x/Log[3*E^x + x])]^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.
| \(\displaystyle \int \frac {6 e^x x \log (3)+2 x \log (3)+\left (-2 x \log (3)-6 e^x \log (3)\right ) \log \left (x+3 e^x\right )+\left (x \log (3)+3 e^x \log (3)\right ) \log \left (x+3 e^x\right ) \log \left (-\frac {x}{\log \left (x+3 e^x\right )}\right )}{\left (x+3 e^x\right ) \log \left (x+3 e^x\right ) \log ^3\left (-\frac {x}{\log \left (x+3 e^x\right )}\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\log (3) \left (2 x-2 \log \left (x+3 e^x\right )+\log \left (x+3 e^x\right ) \log \left (-\frac {x}{\log \left (x+3 e^x\right )}\right )\right )}{\log \left (x+3 e^x\right ) \log ^3\left (-\frac {x}{\log \left (x+3 e^x\right )}\right )}-\frac {2 (x-1) x \log (3)}{\left (x+3 e^x\right ) \log \left (x+3 e^x\right ) \log ^3\left (-\frac {x}{\log \left (x+3 e^x\right )}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \log (3) \int \frac {x^2}{\left (x+3 e^x\right ) \log \left (x+3 e^x\right ) \log ^3\left (-\frac {x}{\log \left (x+3 e^x\right )}\right )}dx-2 \log (3) \int \frac {1}{\log ^3\left (-\frac {x}{\log \left (x+3 e^x\right )}\right )}dx+2 \log (3) \int \frac {x}{\log \left (x+3 e^x\right ) \log ^3\left (-\frac {x}{\log \left (x+3 e^x\right )}\right )}dx+2 \log (3) \int \frac {x}{\left (x+3 e^x\right ) \log \left (x+3 e^x\right ) \log ^3\left (-\frac {x}{\log \left (x+3 e^x\right )}\right )}dx+\log (3) \int \frac {1}{\log ^2\left (-\frac {x}{\log \left (x+3 e^x\right )}\right )}dx\) |
Int[(2*x*Log[3] + 6*E^x*x*Log[3] + (-6*E^x*Log[3] - 2*x*Log[3])*Log[3*E^x + x] + (3*E^x*Log[3] + x*Log[3])*Log[3*E^x + x]*Log[-(x/Log[3*E^x + x])])/ ((3*E^x + x)*Log[3*E^x + x]*Log[-(x/Log[3*E^x + x])]^3),x]
3.1.76.3.1 Defintions of rubi rules used
Time = 9.79 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
| method | result | size |
| parallelrisch | \(\frac {\ln \left (3\right ) x}{{\ln \left (-\frac {x}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right )}^{2}}\) | \(20\) |
| risch | \(-\frac {4 \ln \left (3\right ) x}{{\left (\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right )-\pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right )}^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right )}^{2}-\pi {\operatorname {csgn}\left (\frac {i x}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right )}^{3}+2 \pi {\operatorname {csgn}\left (\frac {i x}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right )}^{2}-2 \pi +2 i \ln \left (x \right )-2 i \ln \left (\ln \left (3 \,{\mathrm e}^{x}+x \right )\right )\right )}^{2}}\) | \(156\) |
int(((3*ln(3)*exp(x)+x*ln(3))*ln(3*exp(x)+x)*ln(-x/ln(3*exp(x)+x))+(-6*ln( 3)*exp(x)-2*x*ln(3))*ln(3*exp(x)+x)+6*x*ln(3)*exp(x)+2*x*ln(3))/(3*exp(x)+ x)/ln(3*exp(x)+x)/ln(-x/ln(3*exp(x)+x))^3,x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {2 x \log (3)+6 e^x x \log (3)+\left (-6 e^x \log (3)-2 x \log (3)\right ) \log \left (3 e^x+x\right )+\left (3 e^x \log (3)+x \log (3)\right ) \log \left (3 e^x+x\right ) \log \left (-\frac {x}{\log \left (3 e^x+x\right )}\right )}{\left (3 e^x+x\right ) \log \left (3 e^x+x\right ) \log ^3\left (-\frac {x}{\log \left (3 e^x+x\right )}\right )} \, dx=\frac {x \log \left (3\right )}{\log \left (-\frac {x}{\log \left (x + 3 \, e^{x}\right )}\right )^{2}} \]
integrate(((3*log(3)*exp(x)+x*log(3))*log(3*exp(x)+x)*log(-x/log(3*exp(x)+ x))+(-6*log(3)*exp(x)-2*x*log(3))*log(3*exp(x)+x)+6*x*log(3)*exp(x)+2*x*lo g(3))/(3*exp(x)+x)/log(3*exp(x)+x)/log(-x/log(3*exp(x)+x))^3,x, algorithm= \
Time = 0.46 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {2 x \log (3)+6 e^x x \log (3)+\left (-6 e^x \log (3)-2 x \log (3)\right ) \log \left (3 e^x+x\right )+\left (3 e^x \log (3)+x \log (3)\right ) \log \left (3 e^x+x\right ) \log \left (-\frac {x}{\log \left (3 e^x+x\right )}\right )}{\left (3 e^x+x\right ) \log \left (3 e^x+x\right ) \log ^3\left (-\frac {x}{\log \left (3 e^x+x\right )}\right )} \, dx=\frac {x \log {\left (3 \right )}}{\log {\left (- \frac {x}{\log {\left (x + 3 e^{x} \right )}} \right )}^{2}} \]
integrate(((3*ln(3)*exp(x)+x*ln(3))*ln(3*exp(x)+x)*ln(-x/ln(3*exp(x)+x))+( -6*ln(3)*exp(x)-2*x*ln(3))*ln(3*exp(x)+x)+6*x*ln(3)*exp(x)+2*x*ln(3))/(3*e xp(x)+x)/ln(3*exp(x)+x)/ln(-x/ln(3*exp(x)+x))**3,x)
Time = 0.41 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {2 x \log (3)+6 e^x x \log (3)+\left (-6 e^x \log (3)-2 x \log (3)\right ) \log \left (3 e^x+x\right )+\left (3 e^x \log (3)+x \log (3)\right ) \log \left (3 e^x+x\right ) \log \left (-\frac {x}{\log \left (3 e^x+x\right )}\right )}{\left (3 e^x+x\right ) \log \left (3 e^x+x\right ) \log ^3\left (-\frac {x}{\log \left (3 e^x+x\right )}\right )} \, dx=\frac {x \log \left (3\right )}{\log \left (-x\right )^{2} - 2 \, \log \left (-x\right ) \log \left (\log \left (x + 3 \, e^{x}\right )\right ) + \log \left (\log \left (x + 3 \, e^{x}\right )\right )^{2}} \]
integrate(((3*log(3)*exp(x)+x*log(3))*log(3*exp(x)+x)*log(-x/log(3*exp(x)+ x))+(-6*log(3)*exp(x)-2*x*log(3))*log(3*exp(x)+x)+6*x*log(3)*exp(x)+2*x*lo g(3))/(3*exp(x)+x)/log(3*exp(x)+x)/log(-x/log(3*exp(x)+x))^3,x, algorithm= \
Time = 0.50 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {2 x \log (3)+6 e^x x \log (3)+\left (-6 e^x \log (3)-2 x \log (3)\right ) \log \left (3 e^x+x\right )+\left (3 e^x \log (3)+x \log (3)\right ) \log \left (3 e^x+x\right ) \log \left (-\frac {x}{\log \left (3 e^x+x\right )}\right )}{\left (3 e^x+x\right ) \log \left (3 e^x+x\right ) \log ^3\left (-\frac {x}{\log \left (3 e^x+x\right )}\right )} \, dx=\frac {x \log \left (3\right )}{\log \left (-x\right )^{2} - 2 \, \log \left (-x\right ) \log \left (\log \left (x + 3 \, e^{x}\right )\right ) + \log \left (\log \left (x + 3 \, e^{x}\right )\right )^{2}} \]
integrate(((3*log(3)*exp(x)+x*log(3))*log(3*exp(x)+x)*log(-x/log(3*exp(x)+ x))+(-6*log(3)*exp(x)-2*x*log(3))*log(3*exp(x)+x)+6*x*log(3)*exp(x)+2*x*lo g(3))/(3*exp(x)+x)/log(3*exp(x)+x)/log(-x/log(3*exp(x)+x))^3,x, algorithm= \
Timed out. \[ \int \frac {2 x \log (3)+6 e^x x \log (3)+\left (-6 e^x \log (3)-2 x \log (3)\right ) \log \left (3 e^x+x\right )+\left (3 e^x \log (3)+x \log (3)\right ) \log \left (3 e^x+x\right ) \log \left (-\frac {x}{\log \left (3 e^x+x\right )}\right )}{\left (3 e^x+x\right ) \log \left (3 e^x+x\right ) \log ^3\left (-\frac {x}{\log \left (3 e^x+x\right )}\right )} \, dx=\int \frac {2\,x\,\ln \left (3\right )-\ln \left (x+3\,{\mathrm {e}}^x\right )\,\left (2\,x\,\ln \left (3\right )+6\,{\mathrm {e}}^x\,\ln \left (3\right )\right )+6\,x\,{\mathrm {e}}^x\,\ln \left (3\right )+\ln \left (-\frac {x}{\ln \left (x+3\,{\mathrm {e}}^x\right )}\right )\,\ln \left (x+3\,{\mathrm {e}}^x\right )\,\left (x\,\ln \left (3\right )+3\,{\mathrm {e}}^x\,\ln \left (3\right )\right )}{{\ln \left (-\frac {x}{\ln \left (x+3\,{\mathrm {e}}^x\right )}\right )}^3\,\ln \left (x+3\,{\mathrm {e}}^x\right )\,\left (x+3\,{\mathrm {e}}^x\right )} \,d x \]
int((2*x*log(3) - log(x + 3*exp(x))*(2*x*log(3) + 6*exp(x)*log(3)) + 6*x*e xp(x)*log(3) + log(-x/log(x + 3*exp(x)))*log(x + 3*exp(x))*(x*log(3) + 3*e xp(x)*log(3)))/(log(-x/log(x + 3*exp(x)))^3*log(x + 3*exp(x))*(x + 3*exp(x ))),x)