Integrand size = 49, antiderivative size = 23 \[ \int \frac {-16 \log (3)+e^{4+e^{\frac {x+(e+x) \log (3)}{\log (3)}}+\frac {x+(e+x) \log (3)}{\log (3)}} (16+16 \log (3))}{\log (3)} \, dx=2+16 \left (e^{4+e^{e+x+\frac {x}{\log (3)}}}-x\right ) \]
Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-16 \log (3)+e^{4+e^{\frac {x+(e+x) \log (3)}{\log (3)}}+\frac {x+(e+x) \log (3)}{\log (3)}} (16+16 \log (3))}{\log (3)} \, dx=16 \left (e^{4+e^{e+x+\frac {x}{\log (3)}}}-x\right ) \]
Integrate[(-16*Log[3] + E^(4 + E^((x + (E + x)*Log[3])/Log[3]) + (x + (E + x)*Log[3])/Log[3])*(16 + 16*Log[3]))/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.
\(\displaystyle \int \frac {(16+16 \log (3)) \exp \left (\frac {x+(x+e) \log (3)}{\log (3)}+e^{\frac {x+(x+e) \log (3)}{\log (3)}}+4\right )-16 \log (3)}{\log (3)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \left (-16 \log (3)+16 \exp \left (\frac {x+(x+e) \log (3)}{\log (3)}+3^{\frac {x+e}{\log (3)}} e^{\frac {x}{\log (3)}}+4\right ) (1+\log (3))\right )dx}{\log (3)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {16 e^{4+e} (1+\log (3)) \int \exp \left (\frac {(1+\log (3)) x+e^{\frac {x}{\log (3)}+x+e} \log (3)}{\log (3)}\right )dx-16 x \log (3)}{\log (3)}\) |
Int[(-16*Log[3] + E^(4 + E^((x + (E + x)*Log[3])/Log[3]) + (x + (E + x)*Lo g[3])/Log[3])*(16 + 16*Log[3]))/Log[3],x]
3.1.24.3.1 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]]
Time = 0.56 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09
method | result | size |
norman | \(-16 x +16 \,{\mathrm e}^{{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \left (3\right )+x}{\ln \left (3\right )}}+4}\) | \(25\) |
parts | \(-16 x +16 \,{\mathrm e}^{{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \left (3\right )+x}{\ln \left (3\right )}}+4}\) | \(25\) |
risch | \(-16 x +16 \,{\mathrm e}^{{\mathrm e}^{\frac {\ln \left (3\right ) {\mathrm e}+x \ln \left (3\right )+x}{\ln \left (3\right )}}+4}\) | \(27\) |
default | \(\frac {\frac {\left (16 \ln \left (3\right )+16\right ) {\mathrm e}^{{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \left (3\right )+x}{\ln \left (3\right )}}+4} \ln \left (3\right )}{\ln \left (3\right )+1}-16 x \ln \left (3\right )}{\ln \left (3\right )}\) | \(45\) |
parallelrisch | \(\frac {\frac {\left (16 \ln \left (3\right )+16\right ) {\mathrm e}^{{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \left (3\right )+x}{\ln \left (3\right )}}+4} \ln \left (3\right )}{\ln \left (3\right )+1}-16 x \ln \left (3\right )}{\ln \left (3\right )}\) | \(45\) |
derivativedivides | \(\frac {-16 \ln \left (3\right ) \ln \left ({\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \left (3\right )+x}{\ln \left (3\right )}}\right )+16 \,{\mathrm e}^{{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \left (3\right )+x}{\ln \left (3\right )}}+4}+64 \ln \left (3\right ) {\mathrm e}^{4} \operatorname {Ei}_{1}\left (-{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \left (3\right )+x}{\ln \left (3\right )}}\right )+16 \ln \left (3\right ) \left ({\mathrm e}^{{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \left (3\right )+x}{\ln \left (3\right )}}+4}-4 \,{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \left (3\right )+x}{\ln \left (3\right )}}\right )\right )}{\ln \left (3\right )+1}\) | \(120\) |
int(((16*ln(3)+16)*exp(((x+exp(1))*ln(3)+x)/ln(3))*exp(exp(((x+exp(1))*ln( 3)+x)/ln(3))+4)-16*ln(3))/ln(3),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.13 \[ \int \frac {-16 \log (3)+e^{4+e^{\frac {x+(e+x) \log (3)}{\log (3)}}+\frac {x+(e+x) \log (3)}{\log (3)}} (16+16 \log (3))}{\log (3)} \, dx=-16 \, {\left (x e^{\left (\frac {{\left (x + e\right )} \log \left (3\right ) + x}{\log \left (3\right )}\right )} - e^{\left (\frac {{\left (x + e + 4\right )} \log \left (3\right ) + e^{\left (\frac {{\left (x + e\right )} \log \left (3\right ) + x}{\log \left (3\right )}\right )} \log \left (3\right ) + x}{\log \left (3\right )}\right )}\right )} e^{\left (-\frac {{\left (x + e\right )} \log \left (3\right ) + x}{\log \left (3\right )}\right )} \]
integrate(((16*log(3)+16)*exp(((x+exp(1))*log(3)+x)/log(3))*exp(exp(((x+ex p(1))*log(3)+x)/log(3))+4)-16*log(3))/log(3),x, algorithm=\
-16*(x*e^(((x + e)*log(3) + x)/log(3)) - e^(((x + e + 4)*log(3) + e^(((x + e)*log(3) + x)/log(3))*log(3) + x)/log(3)))*e^(-((x + e)*log(3) + x)/log( 3))
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-16 \log (3)+e^{4+e^{\frac {x+(e+x) \log (3)}{\log (3)}}+\frac {x+(e+x) \log (3)}{\log (3)}} (16+16 \log (3))}{\log (3)} \, dx=- 16 x + 16 e^{e^{\frac {x + \left (x + e\right ) \log {\left (3 \right )}}{\log {\left (3 \right )}}} + 4} \]
integrate(((16*ln(3)+16)*exp(((x+exp(1))*ln(3)+x)/ln(3))*exp(exp(((x+exp(1 ))*ln(3)+x)/ln(3))+4)-16*ln(3))/ln(3),x)
Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {-16 \log (3)+e^{4+e^{\frac {x+(e+x) \log (3)}{\log (3)}}+\frac {x+(e+x) \log (3)}{\log (3)}} (16+16 \log (3))}{\log (3)} \, dx=-\frac {16 \, {\left (x \log \left (3\right ) - e^{\left (e^{\left (x + \frac {x}{\log \left (3\right )} + e\right )} + 4\right )} \log \left (3\right )\right )}}{\log \left (3\right )} \]
integrate(((16*log(3)+16)*exp(((x+exp(1))*log(3)+x)/log(3))*exp(exp(((x+ex p(1))*log(3)+x)/log(3))+4)-16*log(3))/log(3),x, algorithm=\
\[ \int \frac {-16 \log (3)+e^{4+e^{\frac {x+(e+x) \log (3)}{\log (3)}}+\frac {x+(e+x) \log (3)}{\log (3)}} (16+16 \log (3))}{\log (3)} \, dx=\int { \frac {16 \, {\left ({\left (\log \left (3\right ) + 1\right )} e^{\left (\frac {{\left (x + e\right )} \log \left (3\right ) + x}{\log \left (3\right )} + e^{\left (\frac {{\left (x + e\right )} \log \left (3\right ) + x}{\log \left (3\right )}\right )} + 4\right )} - \log \left (3\right )\right )}}{\log \left (3\right )} \,d x } \]
integrate(((16*log(3)+16)*exp(((x+exp(1))*log(3)+x)/log(3))*exp(exp(((x+ex p(1))*log(3)+x)/log(3))+4)-16*log(3))/log(3),x, algorithm=\
integrate(16*((log(3) + 1)*e^(((x + e)*log(3) + x)/log(3) + e^(((x + e)*lo g(3) + x)/log(3)) + 4) - log(3))/log(3), x)
Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-16 \log (3)+e^{4+e^{\frac {x+(e+x) \log (3)}{\log (3)}}+\frac {x+(e+x) \log (3)}{\log (3)}} (16+16 \log (3))}{\log (3)} \, dx=16\,{\mathrm {e}}^4\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{\ln \left (3\right )}}\,{\mathrm {e}}^{\mathrm {e}}\,{\mathrm {e}}^x}-16\,x \]