Integrand size = 152, antiderivative size = 30 \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\frac {\log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x} \]
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\frac {\log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x} \]
Integrate[(-18*x + 9*x*Log[3*x] - E^(3 - E^((E^3*x)/3)/3 + (E^3*x)/3)*x*Lo g[3*x]^3 + (-9*x*Log[3*x] - (9*Log[3*x]^3)/E^(E^((E^3*x)/3)/3))*Log[(x + L og[3*x]^2/E^(E^((E^3*x)/3)/3))/Log[3*x]^2])/(9*x^3*Log[3*x] + (9*x^2*Log[3 *x]^3)/E^(E^((E^3*x)/3)/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.
| \(\displaystyle \int \frac {-18 x-e^{\frac {e^3 x}{3}-\frac {1}{3} e^{\frac {e^3 x}{3}}+3} x \log ^3(3 x)+\left (-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)-9 x \log (3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )+9 x \log (3 x)}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x \log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right ) \log (3 x)+\log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right ) \log ^3(3 x)-e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x \log (3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right ) \log (3 x)}-\frac {e^{\frac {e^3 x}{3}+3} \log ^2(3 x)}{9 x \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {\log (3 x)}{x^2 \left (\log ^2(3 x)+e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x\right )}dx-\int \frac {\log ^2(3 x)}{x^2 \left (\log ^2(3 x)+e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x\right )}dx-\int \frac {\log \left (\frac {x}{\log ^2(3 x)}+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}\right )}{x^2}dx-\frac {1}{9} \int \frac {e^{\frac {e^3 x}{3}+3} \log ^2(3 x)}{x \left (\log ^2(3 x)+e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x\right )}dx-3 (2-\log (3 x)) \operatorname {ExpIntegralEi}(-\log (3 x))-3 \log (3 x) \operatorname {ExpIntegralEi}(-\log (3 x))-\frac {1}{x}\) |
Int[(-18*x + 9*x*Log[3*x] - E^(3 - E^((E^3*x)/3)/3 + (E^3*x)/3)*x*Log[3*x] ^3 + (-9*x*Log[3*x] - (9*Log[3*x]^3)/E^(E^((E^3*x)/3)/3))*Log[(x + Log[3*x ]^2/E^(E^((E^3*x)/3)/3))/Log[3*x]^2])/(9*x^3*Log[3*x] + (9*x^2*Log[3*x]^3) /E^(E^((E^3*x)/3)/3)),x]
3.14.95.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 300, normalized size of antiderivative = 10.00
\[\frac {\ln \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )}{x}-\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (3 x \right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )}{\ln \left (3 x \right )^{2}}\right )-i \pi \,\operatorname {csgn}\left (i \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )\right ) {\operatorname {csgn}\left (\frac {i \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )}{\ln \left (3 x \right )^{2}}\right )}^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (3 x \right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )}{\ln \left (3 x \right )^{2}}\right )}^{2}-i \pi \operatorname {csgn}\left (i \ln \left (3 x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (3 x \right )^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i \ln \left (3 x \right )\right ) \operatorname {csgn}\left (i \ln \left (3 x \right )^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i \ln \left (3 x \right )^{2}\right )^{3}+i \pi {\operatorname {csgn}\left (\frac {i \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )}{\ln \left (3 x \right )^{2}}\right )}^{3}+4 \ln \left (\ln \left (3 x \right )\right )}{2 x}\]
int(((-9*ln(3*x)^3*exp(-1/3*exp(1/3*x*exp(3)))-9*x*ln(3*x))*ln((ln(3*x)^2* exp(-1/3*exp(1/3*x*exp(3)))+x)/ln(3*x)^2)-x*exp(3)*exp(1/3*x*exp(3))*ln(3* x)^3*exp(-1/3*exp(1/3*x*exp(3)))+9*x*ln(3*x)-18*x)/(9*x^2*ln(3*x)^3*exp(-1 /3*exp(1/3*x*exp(3)))+9*x^3*ln(3*x)),x)
1/x*ln(ln(3*x)^2*exp(-1/3*exp(1/3*x*exp(3)))+x)-1/2*(I*Pi*csgn(I*(ln(3*x)^ 2*exp(-1/3*exp(1/3*x*exp(3)))+x))*csgn(I/ln(3*x)^2)*csgn(I/ln(3*x)^2*(ln(3 *x)^2*exp(-1/3*exp(1/3*x*exp(3)))+x))-I*Pi*csgn(I*(ln(3*x)^2*exp(-1/3*exp( 1/3*x*exp(3)))+x))*csgn(I/ln(3*x)^2*(ln(3*x)^2*exp(-1/3*exp(1/3*x*exp(3))) +x))^2-I*Pi*csgn(I/ln(3*x)^2)*csgn(I/ln(3*x)^2*(ln(3*x)^2*exp(-1/3*exp(1/3 *x*exp(3)))+x))^2-I*Pi*csgn(I*ln(3*x))^2*csgn(I*ln(3*x)^2)+2*I*Pi*csgn(I*l n(3*x))*csgn(I*ln(3*x)^2)^2-I*Pi*csgn(I*ln(3*x)^2)^3+I*Pi*csgn(I/ln(3*x)^2 *(ln(3*x)^2*exp(-1/3*exp(1/3*x*exp(3)))+x))^3+4*ln(ln(3*x)))/x
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\frac {\log \left (\frac {{\left (e^{\left (\frac {1}{3} \, x e^{3} - \frac {1}{3} \, e^{\left (\frac {1}{3} \, x e^{3}\right )} + 3\right )} \log \left (3 \, x\right )^{2} + x e^{\left (\frac {1}{3} \, x e^{3} + 3\right )}\right )} e^{\left (-\frac {1}{3} \, x e^{3} - 3\right )}}{\log \left (3 \, x\right )^{2}}\right )}{x} \]
integrate(((-9*log(3*x)^3*exp(-1/3*exp(1/3*x*exp(3)))-9*x*log(3*x))*log((l og(3*x)^2*exp(-1/3*exp(1/3*x*exp(3)))+x)/log(3*x)^2)-x*exp(3)*exp(1/3*x*ex p(3))*log(3*x)^3*exp(-1/3*exp(1/3*x*exp(3)))+9*x*log(3*x)-18*x)/(9*x^2*log (3*x)^3*exp(-1/3*exp(1/3*x*exp(3)))+9*x^3*log(3*x)),x, algorithm=\
log((e^(1/3*x*e^3 - 1/3*e^(1/3*x*e^3) + 3)*log(3*x)^2 + x*e^(1/3*x*e^3 + 3 ))*e^(-1/3*x*e^3 - 3)/log(3*x)^2)/x
Time = 4.62 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\frac {\log {\left (\frac {x + e^{- \frac {e^{\frac {x e^{3}}{3}}}{3}} \log {\left (3 x \right )}^{2}}{\log {\left (3 x \right )}^{2}} \right )}}{x} \]
integrate(((-9*ln(3*x)**3*exp(-1/3*exp(1/3*x*exp(3)))-9*x*ln(3*x))*ln((ln( 3*x)**2*exp(-1/3*exp(1/3*x*exp(3)))+x)/ln(3*x)**2)-x*exp(3)*exp(1/3*x*exp( 3))*ln(3*x)**3*exp(-1/3*exp(1/3*x*exp(3)))+9*x*ln(3*x)-18*x)/(9*x**2*ln(3* x)**3*exp(-1/3*exp(1/3*x*exp(3)))+9*x**3*ln(3*x)),x)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).
Time = 0.34 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=-\frac {e^{\left (\frac {1}{3} \, x e^{3}\right )} - 3 \, \log \left (x e^{\left (\frac {1}{3} \, e^{\left (\frac {1}{3} \, x e^{3}\right )}\right )} + \log \left (3\right )^{2} + 2 \, \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right ) + 6 \, \log \left (\log \left (3\right ) + \log \left (x\right )\right )}{3 \, x} \]
integrate(((-9*log(3*x)^3*exp(-1/3*exp(1/3*x*exp(3)))-9*x*log(3*x))*log((l og(3*x)^2*exp(-1/3*exp(1/3*x*exp(3)))+x)/log(3*x)^2)-x*exp(3)*exp(1/3*x*ex p(3))*log(3*x)^3*exp(-1/3*exp(1/3*x*exp(3)))+9*x*log(3*x)-18*x)/(9*x^2*log (3*x)^3*exp(-1/3*exp(1/3*x*exp(3)))+9*x^3*log(3*x)),x, algorithm=\
-1/3*(e^(1/3*x*e^3) - 3*log(x*e^(1/3*e^(1/3*x*e^3)) + log(3)^2 + 2*log(3)* log(x) + log(x)^2) + 6*log(log(3) + log(x)))/x
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).
Time = 13.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\frac {\log \left ({\left (e^{\left (\frac {1}{3} \, x e^{3} - \frac {1}{3} \, e^{\left (\frac {1}{3} \, x e^{3}\right )}\right )} \log \left (3 \, x\right )^{2} + x e^{\left (\frac {1}{3} \, x e^{3}\right )}\right )} e^{\left (-\frac {1}{3} \, x e^{3}\right )}\right ) - \log \left (\log \left (3 \, x\right )^{2}\right )}{x} \]
integrate(((-9*log(3*x)^3*exp(-1/3*exp(1/3*x*exp(3)))-9*x*log(3*x))*log((l og(3*x)^2*exp(-1/3*exp(1/3*x*exp(3)))+x)/log(3*x)^2)-x*exp(3)*exp(1/3*x*ex p(3))*log(3*x)^3*exp(-1/3*exp(1/3*x*exp(3)))+9*x*log(3*x)-18*x)/(9*x^2*log (3*x)^3*exp(-1/3*exp(1/3*x*exp(3)))+9*x^3*log(3*x)),x, algorithm=\
(log((e^(1/3*x*e^3 - 1/3*e^(1/3*x*e^3))*log(3*x)^2 + x*e^(1/3*x*e^3))*e^(- 1/3*x*e^3)) - log(log(3*x)^2))/x
Timed out. \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\int -\frac {18\,x-9\,x\,\ln \left (3\,x\right )+\ln \left (\frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^3}{3}}}{3}}\,{\ln \left (3\,x\right )}^2+x}{{\ln \left (3\,x\right )}^2}\right )\,\left (9\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^3}{3}}}{3}}\,{\ln \left (3\,x\right )}^3+9\,x\,\ln \left (3\,x\right )\right )+x\,{\ln \left (3\,x\right )}^3\,{\mathrm {e}}^3\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^3}{3}}}{3}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^3}{3}}}{9\,x^3\,\ln \left (3\,x\right )+9\,x^2\,{\ln \left (3\,x\right )}^3\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^3}{3}}}{3}}} \,d x \]
int(-(18*x - 9*x*log(3*x) + log((x + log(3*x)^2*exp(-exp((x*exp(3))/3)/3)) /log(3*x)^2)*(9*x*log(3*x) + 9*log(3*x)^3*exp(-exp((x*exp(3))/3)/3)) + x*l og(3*x)^3*exp(3)*exp(-exp((x*exp(3))/3)/3)*exp((x*exp(3))/3))/(9*x^3*log(3 *x) + 9*x^2*log(3*x)^3*exp(-exp((x*exp(3))/3)/3)),x)