Integrand size = 135, antiderivative size = 27 \[ \int \frac {12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)+\left (12 e^{e^{\frac {1}{3} (12-7 x)}}+112 e^{\frac {1}{3} (12-7 x)} x\right ) \log \left (e^{-2 e^{\frac {1}{3} (12-7 x)}} \left (320+160 e^{e^{\frac {1}{3} (12-7 x)}} \log (x)+20 e^{2 e^{\frac {1}{3} (12-7 x)}} \log ^2(x)\right )\right )}{12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)} \, dx=x+\log ^2\left (20 \left (4 e^{-e^{4-\frac {7 x}{3}}}+\log (x)\right )^2\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(166\) vs. \(2(27)=54\).
Time = 0.32 (sec) , antiderivative size = 166, normalized size of antiderivative = 6.15 \[ \int \frac {12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)+\left (12 e^{e^{\frac {1}{3} (12-7 x)}}+112 e^{\frac {1}{3} (12-7 x)} x\right ) \log \left (e^{-2 e^{\frac {1}{3} (12-7 x)}} \left (320+160 e^{e^{\frac {1}{3} (12-7 x)}} \log (x)+20 e^{2 e^{\frac {1}{3} (12-7 x)}} \log ^2(x)\right )\right )}{12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)} \, dx=-4 e^{8-\frac {14 x}{3}}+x+\log ^2\left (\left (4+e^{e^{4-\frac {7 x}{3}}} \log (x)\right )^2\right )-4 e^{4-\frac {7 x}{3}} \log \left (20 e^{-2 e^{4-\frac {7 x}{3}}} \left (4+e^{e^{4-\frac {7 x}{3}}} \log (x)\right )^2\right )+\log \left (4+e^{e^{4-\frac {7 x}{3}}} \log (x)\right ) \left (8 e^{4-\frac {7 x}{3}}-4 \log \left (\left (4+e^{e^{4-\frac {7 x}{3}}} \log (x)\right )^2\right )+4 \log \left (20 e^{-2 e^{4-\frac {7 x}{3}}} \left (4+e^{e^{4-\frac {7 x}{3}}} \log (x)\right )^2\right )\right ) \]
Integrate[(12*x + 3*E^E^((12 - 7*x)/3)*x*Log[x] + (12*E^E^((12 - 7*x)/3) + 112*E^((12 - 7*x)/3)*x)*Log[(320 + 160*E^E^((12 - 7*x)/3)*Log[x] + 20*E^( 2*E^((12 - 7*x)/3))*Log[x]^2)/E^(2*E^((12 - 7*x)/3))])/(12*x + 3*E^E^((12 - 7*x)/3)*x*Log[x]),x]
-4*E^(8 - (14*x)/3) + x + Log[(4 + E^E^(4 - (7*x)/3)*Log[x])^2]^2 - 4*E^(4 - (7*x)/3)*Log[(20*(4 + E^E^(4 - (7*x)/3)*Log[x])^2)/E^(2*E^(4 - (7*x)/3) )] + Log[4 + E^E^(4 - (7*x)/3)*Log[x]]*(8*E^(4 - (7*x)/3) - 4*Log[(4 + E^E ^(4 - (7*x)/3)*Log[x])^2] + 4*Log[(20*(4 + E^E^(4 - (7*x)/3)*Log[x])^2)/E^ (2*E^(4 - (7*x)/3))])
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.
| \(\displaystyle \int \frac {12 x+\left (112 e^{\frac {1}{3} (12-7 x)} x+12 e^{e^{\frac {1}{3} (12-7 x)}}\right ) \log \left (e^{-2 e^{\frac {1}{3} (12-7 x)}} \left (20 e^{2 e^{\frac {1}{3} (12-7 x)}} \log ^2(x)+160 e^{e^{\frac {1}{3} (12-7 x)}} \log (x)+320\right )\right )+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)}{12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {12 x+\left (112 e^{\frac {1}{3} (12-7 x)} x+12 e^{e^{\frac {1}{3} (12-7 x)}}\right ) \log \left (e^{-2 e^{\frac {1}{3} (12-7 x)}} \left (20 e^{2 e^{\frac {1}{3} (12-7 x)}} \log ^2(x)+160 e^{e^{\frac {1}{3} (12-7 x)}} \log (x)+320\right )\right )+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)}{3 x \left (e^{e^{4-\frac {7 x}{3}}} \log (x)+4\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {3 e^{e^{\frac {1}{3} (12-7 x)}} \log (x) x+12 x+4 \left (28 e^{\frac {1}{3} (12-7 x)} x+3 e^{e^{\frac {1}{3} (12-7 x)}}\right ) \log \left (20 e^{-2 e^{\frac {1}{3} (12-7 x)}} \left (e^{2 e^{\frac {1}{3} (12-7 x)}} \log ^2(x)+8 e^{e^{\frac {1}{3} (12-7 x)}} \log (x)+16\right )\right )}{x \left (e^{e^{4-\frac {7 x}{3}}} \log (x)+4\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \int \left (\frac {112 e^{4-\frac {7 x}{3}} \log \left (20 e^{-2 e^{4-\frac {7 x}{3}}} \left (e^{e^{4-\frac {7 x}{3}}} \log (x)+4\right )^2\right )}{e^{e^{4-\frac {7 x}{3}}} \log (x)+4}+\frac {3 \left (e^{e^{4-\frac {7 x}{3}}} \log (x) x+4 x+4 e^{e^{4-\frac {7 x}{3}}} \log \left (20 e^{-2 e^{4-\frac {7 x}{3}}} \left (e^{e^{4-\frac {7 x}{3}}} \log (x)+4\right )^2\right )\right )}{x \left (e^{e^{4-\frac {7 x}{3}}} \log (x)+4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (12 \int \frac {\log \left (20 e^{-2 e^{4-\frac {7 x}{3}}} \left (e^{e^{4-\frac {7 x}{3}}} \log (x)+4\right )^2\right )}{x \log (x)}dx+112 \int \frac {e^{4-\frac {7 x}{3}} \log \left (20 e^{-2 e^{4-\frac {7 x}{3}}} \left (e^{e^{4-\frac {7 x}{3}}} \log (x)+4\right )^2\right )}{e^{e^{4-\frac {7 x}{3}}} \log (x)+4}dx-48 \int \frac {\log \left (20 e^{-2 e^{4-\frac {7 x}{3}}} \left (e^{e^{4-\frac {7 x}{3}}} \log (x)+4\right )^2\right )}{x \log (x) \left (e^{e^{4-\frac {7 x}{3}}} \log (x)+4\right )}dx+3 x\right )\) |
Int[(12*x + 3*E^E^((12 - 7*x)/3)*x*Log[x] + (12*E^E^((12 - 7*x)/3) + 112*E ^((12 - 7*x)/3)*x)*Log[(320 + 160*E^E^((12 - 7*x)/3)*Log[x] + 20*E^(2*E^(( 12 - 7*x)/3))*Log[x]^2)/E^(2*E^((12 - 7*x)/3))])/(12*x + 3*E^E^((12 - 7*x) /3)*x*Log[x]),x]
3.9.55.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
\[\int \frac {\left (12 \,{\mathrm e}^{{\mathrm e}^{-\frac {7 x}{3}+4}}+112 x \,{\mathrm e}^{-\frac {7 x}{3}+4}\right ) \ln \left (\left (20 \ln \left (x \right )^{2} {\mathrm e}^{2 \,{\mathrm e}^{-\frac {7 x}{3}+4}}+160 \ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{-\frac {7 x}{3}+4}}+320\right ) {\mathrm e}^{-2 \,{\mathrm e}^{-\frac {7 x}{3}+4}}\right )+3 x \ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{-\frac {7 x}{3}+4}}+12 x}{3 x \ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{-\frac {7 x}{3}+4}}+12 x}d x\]
int(((12*exp(exp(-7/3*x+4))+112*x*exp(-7/3*x+4))*ln((20*ln(x)^2*exp(exp(-7 /3*x+4))^2+160*ln(x)*exp(exp(-7/3*x+4))+320)/exp(exp(-7/3*x+4))^2)+3*x*ln( x)*exp(exp(-7/3*x+4))+12*x)/(3*x*ln(x)*exp(exp(-7/3*x+4))+12*x),x)
int(((12*exp(exp(-7/3*x+4))+112*x*exp(-7/3*x+4))*ln((20*ln(x)^2*exp(exp(-7 /3*x+4))^2+160*ln(x)*exp(exp(-7/3*x+4))+320)/exp(exp(-7/3*x+4))^2)+3*x*ln( x)*exp(exp(-7/3*x+4))+12*x)/(3*x*ln(x)*exp(exp(-7/3*x+4))+12*x),x)
Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)+\left (12 e^{e^{\frac {1}{3} (12-7 x)}}+112 e^{\frac {1}{3} (12-7 x)} x\right ) \log \left (e^{-2 e^{\frac {1}{3} (12-7 x)}} \left (320+160 e^{e^{\frac {1}{3} (12-7 x)}} \log (x)+20 e^{2 e^{\frac {1}{3} (12-7 x)}} \log ^2(x)\right )\right )}{12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)} \, dx=\log \left (20 \, {\left (e^{\left (2 \, e^{\left (-\frac {7}{3} \, x + 4\right )}\right )} \log \left (x\right )^{2} + 8 \, e^{\left (e^{\left (-\frac {7}{3} \, x + 4\right )}\right )} \log \left (x\right ) + 16\right )} e^{\left (-2 \, e^{\left (-\frac {7}{3} \, x + 4\right )}\right )}\right )^{2} + x \]
integrate(((12*exp(exp(-7/3*x+4))+112*x*exp(-7/3*x+4))*log((20*log(x)^2*ex p(exp(-7/3*x+4))^2+160*log(x)*exp(exp(-7/3*x+4))+320)/exp(exp(-7/3*x+4))^2 )+3*x*log(x)*exp(exp(-7/3*x+4))+12*x)/(3*x*log(x)*exp(exp(-7/3*x+4))+12*x) ,x, algorithm=\
log(20*(e^(2*e^(-7/3*x + 4))*log(x)^2 + 8*e^(e^(-7/3*x + 4))*log(x) + 16)* e^(-2*e^(-7/3*x + 4)))^2 + x
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (22) = 44\).
Time = 1.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)+\left (12 e^{e^{\frac {1}{3} (12-7 x)}}+112 e^{\frac {1}{3} (12-7 x)} x\right ) \log \left (e^{-2 e^{\frac {1}{3} (12-7 x)}} \left (320+160 e^{e^{\frac {1}{3} (12-7 x)}} \log (x)+20 e^{2 e^{\frac {1}{3} (12-7 x)}} \log ^2(x)\right )\right )}{12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)} \, dx=x + \log {\left (\left (20 e^{2 e^{4 - \frac {7 x}{3}}} \log {\left (x \right )}^{2} + 160 e^{e^{4 - \frac {7 x}{3}}} \log {\left (x \right )} + 320\right ) e^{- 2 e^{4 - \frac {7 x}{3}}} \right )}^{2} \]
integrate(((12*exp(exp(-7/3*x+4))+112*x*exp(-7/3*x+4))*ln((20*ln(x)**2*exp (exp(-7/3*x+4))**2+160*ln(x)*exp(exp(-7/3*x+4))+320)/exp(exp(-7/3*x+4))**2 )+3*x*ln(x)*exp(exp(-7/3*x+4))+12*x)/(3*x*ln(x)*exp(exp(-7/3*x+4))+12*x),x )
x + log((20*exp(2*exp(4 - 7*x/3))*log(x)**2 + 160*exp(exp(4 - 7*x/3))*log( x) + 320)*exp(-2*exp(4 - 7*x/3)))**2
\[ \int \frac {12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)+\left (12 e^{e^{\frac {1}{3} (12-7 x)}}+112 e^{\frac {1}{3} (12-7 x)} x\right ) \log \left (e^{-2 e^{\frac {1}{3} (12-7 x)}} \left (320+160 e^{e^{\frac {1}{3} (12-7 x)}} \log (x)+20 e^{2 e^{\frac {1}{3} (12-7 x)}} \log ^2(x)\right )\right )}{12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)} \, dx=\int { \frac {3 \, x e^{\left (e^{\left (-\frac {7}{3} \, x + 4\right )}\right )} \log \left (x\right ) + 4 \, {\left (28 \, x e^{\left (-\frac {7}{3} \, x + 4\right )} + 3 \, e^{\left (e^{\left (-\frac {7}{3} \, x + 4\right )}\right )}\right )} \log \left (20 \, {\left (e^{\left (2 \, e^{\left (-\frac {7}{3} \, x + 4\right )}\right )} \log \left (x\right )^{2} + 8 \, e^{\left (e^{\left (-\frac {7}{3} \, x + 4\right )}\right )} \log \left (x\right ) + 16\right )} e^{\left (-2 \, e^{\left (-\frac {7}{3} \, x + 4\right )}\right )}\right ) + 12 \, x}{3 \, {\left (x e^{\left (e^{\left (-\frac {7}{3} \, x + 4\right )}\right )} \log \left (x\right ) + 4 \, x\right )}} \,d x } \]
integrate(((12*exp(exp(-7/3*x+4))+112*x*exp(-7/3*x+4))*log((20*log(x)^2*ex p(exp(-7/3*x+4))^2+160*log(x)*exp(exp(-7/3*x+4))+320)/exp(exp(-7/3*x+4))^2 )+3*x*log(x)*exp(exp(-7/3*x+4))+12*x)/(3*x*log(x)*exp(exp(-7/3*x+4))+12*x) ,x, algorithm=\
1/3*integrate((3*x*e^(e^(-7/3*x + 4))*log(x) + 4*(28*x*e^(-7/3*x + 4) + 3* e^(e^(-7/3*x + 4)))*log(20*(e^(2*e^(-7/3*x + 4))*log(x)^2 + 8*e^(e^(-7/3*x + 4))*log(x) + 16)*e^(-2*e^(-7/3*x + 4))) + 12*x)/(x*e^(e^(-7/3*x + 4))*l og(x) + 4*x), x)
\[ \int \frac {12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)+\left (12 e^{e^{\frac {1}{3} (12-7 x)}}+112 e^{\frac {1}{3} (12-7 x)} x\right ) \log \left (e^{-2 e^{\frac {1}{3} (12-7 x)}} \left (320+160 e^{e^{\frac {1}{3} (12-7 x)}} \log (x)+20 e^{2 e^{\frac {1}{3} (12-7 x)}} \log ^2(x)\right )\right )}{12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)} \, dx=\int { \frac {3 \, x e^{\left (e^{\left (-\frac {7}{3} \, x + 4\right )}\right )} \log \left (x\right ) + 4 \, {\left (28 \, x e^{\left (-\frac {7}{3} \, x + 4\right )} + 3 \, e^{\left (e^{\left (-\frac {7}{3} \, x + 4\right )}\right )}\right )} \log \left (20 \, {\left (e^{\left (2 \, e^{\left (-\frac {7}{3} \, x + 4\right )}\right )} \log \left (x\right )^{2} + 8 \, e^{\left (e^{\left (-\frac {7}{3} \, x + 4\right )}\right )} \log \left (x\right ) + 16\right )} e^{\left (-2 \, e^{\left (-\frac {7}{3} \, x + 4\right )}\right )}\right ) + 12 \, x}{3 \, {\left (x e^{\left (e^{\left (-\frac {7}{3} \, x + 4\right )}\right )} \log \left (x\right ) + 4 \, x\right )}} \,d x } \]
integrate(((12*exp(exp(-7/3*x+4))+112*x*exp(-7/3*x+4))*log((20*log(x)^2*ex p(exp(-7/3*x+4))^2+160*log(x)*exp(exp(-7/3*x+4))+320)/exp(exp(-7/3*x+4))^2 )+3*x*log(x)*exp(exp(-7/3*x+4))+12*x)/(3*x*log(x)*exp(exp(-7/3*x+4))+12*x) ,x, algorithm=\
integrate(1/3*(3*x*e^(e^(-7/3*x + 4))*log(x) + 4*(28*x*e^(-7/3*x + 4) + 3* e^(e^(-7/3*x + 4)))*log(20*(e^(2*e^(-7/3*x + 4))*log(x)^2 + 8*e^(e^(-7/3*x + 4))*log(x) + 16)*e^(-2*e^(-7/3*x + 4))) + 12*x)/(x*e^(e^(-7/3*x + 4))*l og(x) + 4*x), x)
Time = 10.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)+\left (12 e^{e^{\frac {1}{3} (12-7 x)}}+112 e^{\frac {1}{3} (12-7 x)} x\right ) \log \left (e^{-2 e^{\frac {1}{3} (12-7 x)}} \left (320+160 e^{e^{\frac {1}{3} (12-7 x)}} \log (x)+20 e^{2 e^{\frac {1}{3} (12-7 x)}} \log ^2(x)\right )\right )}{12 x+3 e^{e^{\frac {1}{3} (12-7 x)}} x \log (x)} \, dx={\ln \left (20\,{\ln \left (x\right )}^2+160\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^4}{{\left ({\mathrm {e}}^x\right )}^{7/3}}}\,\ln \left (x\right )+320\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^4}{{\left ({\mathrm {e}}^x\right )}^{7/3}}}\right )}^2+x \]
int((12*x + log(exp(-2*exp(4 - (7*x)/3))*(20*exp(2*exp(4 - (7*x)/3))*log(x )^2 + 160*exp(exp(4 - (7*x)/3))*log(x) + 320))*(12*exp(exp(4 - (7*x)/3)) + 112*x*exp(4 - (7*x)/3)) + 3*x*exp(exp(4 - (7*x)/3))*log(x))/(12*x + 3*x*e xp(exp(4 - (7*x)/3))*log(x)),x)