\(\int \frac {e^{e^{\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}}+\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}} (-6-\log (2))+(3+\log (2)) \log (6)}{(3+\log (2)) \log (6)} \, dx\) [908]

Optimal result

Integrand size = 67, antiderivative size = 28 \[ \int \frac {e^{e^{\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}}+\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}} (-6-\log (2))+(3+\log (2)) \log (6)}{(3+\log (2)) \log (6)} \, dx=-1+x-\frac {e^{e^{-2+x+\frac {1+3 x}{3+\log (2)}}}}{\log (6)} \] Output:

x-exp(exp((1+3*x)/(3+ln(2))+x-2))/ln(6)-1
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {e^{e^{\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}}+\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}} (-6-\log (2))+(3+\log (2)) \log (6)}{(3+\log (2)) \log (6)} \, dx=x+\frac {e^{e^{\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}}} (-6-\log (2))}{(6+\log (2)) \log (6)} \] Input:

Integrate[(E^(E^((-5 + 6*x + (-2 + x)*Log[2])/(3 + Log[2])) + (-5 + 6*x + 
(-2 + x)*Log[2])/(3 + Log[2]))*(-6 - Log[2]) + (3 + Log[2])*Log[6])/((3 + 
Log[2])*Log[6]),x]
 

Output:

x + (E^E^((-5 + 6*x + (-2 + x)*Log[2])/(3 + Log[2]))*(-6 - Log[2]))/((6 + 
Log[2])*Log[6])
 

Rubi [F]

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.

\(\displaystyle \int \frac {(-6-\log (2)) \exp \left (\frac {6 x+(x-2) \log (2)-5}{3+\log (2)}+e^{\frac {6 x+(x-2) \log (2)-5}{3+\log (2)}}\right )+(3+\log (2)) \log (6)}{(3+\log (2)) \log (6)} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \left (-\exp \left (2^{-\frac {2-x}{3+\log (2)}} e^{-\frac {5-6 x}{3+\log (2)}}-\frac {\log (2) (2-x)-6 x+5}{3+\log (2)}\right ) (6+\log (2))+(3+\log (2)) \log (6)\right )dx}{(3+\log (2)) \log (6)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x (3+\log (2)) \log (6)-(6+\log (2)) \int \exp \left (2^{-\frac {2-x}{3+\log (2)}} e^{-\frac {5-6 x}{3+\log (2)}}-\frac {\log (2) (2-x)-6 x+5}{3+\log (2)}\right )dx}{(3+\log (2)) \log (6)}\)

Input:

Int[(E^(E^((-5 + 6*x + (-2 + x)*Log[2])/(3 + Log[2])) + (-5 + 6*x + (-2 + 
x)*Log[2])/(3 + Log[2]))*(-6 - Log[2]) + (3 + Log[2])*Log[6])/((3 + Log[2] 
)*Log[6]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04

method	result	size
norman	\(x -\frac {{\mathrm e}^{{\mathrm e}^{\frac {\left (-2+x \right ) \ln \left (2\right )+6 x -5}{3+\ln \left (2\right )}}}}{\ln \left (6\right )}\)	\(29\)
parts	\(x -\frac {{\mathrm e}^{{\mathrm e}^{\frac {\left (-2+x \right ) \ln \left (2\right )+6 x -5}{3+\ln \left (2\right )}}}}{\ln \left (6\right )}\)	\(29\)
default	\(\frac {\left (3+\ln \left (2\right )\right ) \ln \left (6\right ) x +\frac {\left (-\ln \left (2\right )-6\right ) {\mathrm e}^{{\mathrm e}^{\frac {\left (-2+x \right ) \ln \left (2\right )+6 x -5}{3+\ln \left (2\right )}}} \left (3+\ln \left (2\right )\right )}{\ln \left (2\right )+6}}{\left (3+\ln \left (2\right )\right ) \ln \left (6\right )}\)	\(58\)
parallelrisch	\(\frac {\frac {\left (-\ln \left (2\right )-6\right ) \left ({\mathrm e}^{{\mathrm e}^{\frac {\left (-2+x \right ) \ln \left (2\right )+6 x -5}{3+\ln \left (2\right )}}} \ln \left (2\right )+3 \,{\mathrm e}^{{\mathrm e}^{\frac {\left (-2+x \right ) \ln \left (2\right )+6 x -5}{3+\ln \left (2\right )}}}\right )}{\ln \left (2\right )+6}+\left (3+\ln \left (2\right )\right ) \ln \left (6\right ) x}{\left (3+\ln \left (2\right )\right ) \ln \left (6\right )}\)	\(80\)
derivativedivides	\(\frac {\ln \left (6\right ) \ln \left (2\right ) \ln \left ({\mathrm e}^{\frac {\left (-2+x \right ) \ln \left (2\right )+6 x -5}{3+\ln \left (2\right )}}\right )+3 \ln \left (6\right ) \ln \left ({\mathrm e}^{\frac {\left (-2+x \right ) \ln \left (2\right )+6 x -5}{3+\ln \left (2\right )}}\right )-{\mathrm e}^{{\mathrm e}^{\frac {\left (-2+x \right ) \ln \left (2\right )+6 x -5}{3+\ln \left (2\right )}}} \ln \left (2\right )-6 \,{\mathrm e}^{{\mathrm e}^{\frac {\left (-2+x \right ) \ln \left (2\right )+6 x -5}{3+\ln \left (2\right )}}}}{\ln \left (6\right ) \left (\ln \left (2\right )+6\right )}\)	\(108\)
risch	\(\frac {x \ln \left (2\right ) \ln \left (3\right )}{\left (3+\ln \left (2\right )\right ) \left (\ln \left (2\right )+\ln \left (3\right )\right )}+\frac {x \ln \left (2\right )^{2}}{\left (3+\ln \left (2\right )\right ) \left (\ln \left (2\right )+\ln \left (3\right )\right )}+\frac {3 x \ln \left (3\right )}{\left (3+\ln \left (2\right )\right ) \left (\ln \left (2\right )+\ln \left (3\right )\right )}+\frac {3 x \ln \left (2\right )}{\left (3+\ln \left (2\right )\right ) \left (\ln \left (2\right )+\ln \left (3\right )\right )}-\frac {{\mathrm e}^{{\mathrm e}^{\frac {x \ln \left (2\right )-2 \ln \left (2\right )+6 x -5}{3+\ln \left (2\right )}}} \ln \left (2\right )}{\left (3+\ln \left (2\right )\right ) \left (\ln \left (2\right )+\ln \left (3\right )\right )}-\frac {3 \,{\mathrm e}^{{\mathrm e}^{\frac {x \ln \left (2\right )-2 \ln \left (2\right )+6 x -5}{3+\ln \left (2\right )}}}}{\left (3+\ln \left (2\right )\right ) \left (\ln \left (2\right )+\ln \left (3\right )\right )}\)	\(152\)

Input:

int(((-ln(2)-6)*exp(((-2+x)*ln(2)+6*x-5)/(3+ln(2)))*exp(exp(((-2+x)*ln(2)+ 
6*x-5)/(3+ln(2))))+(3+ln(2))*ln(6))/(3+ln(2))/ln(6),x,method=_RETURNVERBOS 
E)
 

Output:

x-1/ln(6)*exp(exp(((-2+x)*ln(2)+6*x-5)/(3+ln(2))))
                                                                                    
                                                                                    
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (26) = 52\).

Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.36 \[ \int \frac {e^{e^{\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}}+\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}} (-6-\log (2))+(3+\log (2)) \log (6)}{(3+\log (2)) \log (6)} \, dx=\frac {{\left (x e^{\left (\frac {{\left (x - 2\right )} \log \left (2\right ) + 6 \, x - 5}{\log \left (2\right ) + 3}\right )} \log \left (6\right ) - e^{\left (\frac {{\left (\log \left (2\right ) + 3\right )} e^{\left (\frac {{\left (x - 2\right )} \log \left (2\right ) + 6 \, x - 5}{\log \left (2\right ) + 3}\right )} + {\left (x - 2\right )} \log \left (2\right ) + 6 \, x - 5}{\log \left (2\right ) + 3}\right )}\right )} e^{\left (-\frac {{\left (x - 2\right )} \log \left (2\right ) + 6 \, x - 5}{\log \left (2\right ) + 3}\right )}}{\log \left (6\right )} \] Input:

integrate(((-log(2)-6)*exp(((-2+x)*log(2)+6*x-5)/(3+log(2)))*exp(exp(((-2+ 
x)*log(2)+6*x-5)/(3+log(2))))+(3+log(2))*log(6))/(3+log(2))/log(6),x, algo 
rithm="fricas")
 

Output:

(x*e^(((x - 2)*log(2) + 6*x - 5)/(log(2) + 3))*log(6) - e^(((log(2) + 3)*e 
^(((x - 2)*log(2) + 6*x - 5)/(log(2) + 3)) + (x - 2)*log(2) + 6*x - 5)/(lo 
g(2) + 3)))*e^(-((x - 2)*log(2) + 6*x - 5)/(log(2) + 3))/log(6)
 

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^{e^{\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}}+\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}} (-6-\log (2))+(3+\log (2)) \log (6)}{(3+\log (2)) \log (6)} \, dx=x - \frac {e^{e^{\frac {6 x + \left (x - 2\right ) \log {\left (2 \right )} - 5}{\log {\left (2 \right )} + 3}}}}{\log {\left (6 \right )}} \] Input:

integrate(((-ln(2)-6)*exp(((-2+x)*ln(2)+6*x-5)/(3+ln(2)))*exp(exp(((-2+x)* 
ln(2)+6*x-5)/(3+ln(2))))+(3+ln(2))*ln(6))/(3+ln(2))/ln(6),x)
 

Output:

x - exp(exp((6*x + (x - 2)*log(2) - 5)/(log(2) + 3)))/log(6)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (26) = 52\).

Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.46 \[ \int \frac {e^{e^{\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}}+\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}} (-6-\log (2))+(3+\log (2)) \log (6)}{(3+\log (2)) \log (6)} \, dx=\frac {x {\left (\log \left (2\right ) + 3\right )} \log \left (6\right ) - {\left (\log \left (2\right ) + 3\right )} e^{\left (\frac {e^{\left (\frac {x \log \left (2\right )}{\log \left (2\right ) + 3} + \frac {6 \, x}{\log \left (2\right ) + 3} - \frac {5}{\log \left (2\right ) + 3}\right )}}{2^{\frac {2}{\log \left (2\right ) + 3}}}\right )}}{{\left (\log \left (2\right ) + 3\right )} \log \left (6\right )} \] Input:

integrate(((-log(2)-6)*exp(((-2+x)*log(2)+6*x-5)/(3+log(2)))*exp(exp(((-2+ 
x)*log(2)+6*x-5)/(3+log(2))))+(3+log(2))*log(6))/(3+log(2))/log(6),x, algo 
rithm="maxima")
 

Output:

(x*(log(2) + 3)*log(6) - (log(2) + 3)*e^(e^(x*log(2)/(log(2) + 3) + 6*x/(l 
og(2) + 3) - 5/(log(2) + 3))/2^(2/(log(2) + 3))))/((log(2) + 3)*log(6))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (26) = 52\).

Time = 0.20 (sec) , antiderivative size = 254, normalized size of antiderivative = 9.07 \[ \int \frac {e^{e^{\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}}+\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}} (-6-\log (2))+(3+\log (2)) \log (6)}{(3+\log (2)) \log (6)} \, dx=\frac {2 \, x {\left (\log \left (2\right ) + 3\right )} \log \left (6\right ) - \frac {{\left (2^{\frac {8}{9}} e^{\left (\frac {x \log \left (2\right ) + e^{\left (\frac {x \log \left (2\right ) + 6 \, x - 2 \, \log \left (2\right ) - 5}{\log \left (2\right ) + 3}\right )} \log \left (2\right ) + 6 \, x + 3 \, e^{\left (\frac {x \log \left (2\right ) + 6 \, x - 2 \, \log \left (2\right ) - 5}{\log \left (2\right ) + 3}\right )}}{\log \left (2\right ) + 3} + \frac {\log \left (2\right )^{2} - 15 \, \log \left (2\right ) - 45}{9 \, {\left (\log \left (2\right ) + 3\right )}}\right )} \log \left (2\right ) + 3 \cdot 2^{\frac {8}{9}} e^{\left (\frac {x \log \left (2\right ) + e^{\left (\frac {x \log \left (2\right ) + 6 \, x - 2 \, \log \left (2\right ) - 5}{\log \left (2\right ) + 3}\right )} \log \left (2\right ) + 6 \, x + 3 \, e^{\left (\frac {x \log \left (2\right ) + 6 \, x - 2 \, \log \left (2\right ) - 5}{\log \left (2\right ) + 3}\right )}}{\log \left (2\right ) + 3} + \frac {\log \left (2\right )^{2} - 15 \, \log \left (2\right ) - 45}{9 \, {\left (\log \left (2\right ) + 3\right )}}\right )}\right )} {\left (\log \left (2\right ) + 6\right )}}{e^{\left (\frac {x \log \left (2\right ) + 6 \, x - 2 \, \log \left (2\right ) - 5}{\log \left (2\right ) + 3}\right )} \log \left (2\right ) + 6 \, e^{\left (\frac {x \log \left (2\right ) + 6 \, x - 2 \, \log \left (2\right ) - 5}{\log \left (2\right ) + 3}\right )}}}{2 \, {\left (\log \left (2\right ) + 3\right )} \log \left (6\right )} \] Input:

integrate(((-log(2)-6)*exp(((-2+x)*log(2)+6*x-5)/(3+log(2)))*exp(exp(((-2+ 
x)*log(2)+6*x-5)/(3+log(2))))+(3+log(2))*log(6))/(3+log(2))/log(6),x, algo 
rithm="giac")
 

Output:

1/2*(2*x*(log(2) + 3)*log(6) - (2^(8/9)*e^((x*log(2) + e^((x*log(2) + 6*x 
- 2*log(2) - 5)/(log(2) + 3))*log(2) + 6*x + 3*e^((x*log(2) + 6*x - 2*log( 
2) - 5)/(log(2) + 3)))/(log(2) + 3) + 1/9*(log(2)^2 - 15*log(2) - 45)/(log 
(2) + 3))*log(2) + 3*2^(8/9)*e^((x*log(2) + e^((x*log(2) + 6*x - 2*log(2) 
- 5)/(log(2) + 3))*log(2) + 6*x + 3*e^((x*log(2) + 6*x - 2*log(2) - 5)/(lo 
g(2) + 3)))/(log(2) + 3) + 1/9*(log(2)^2 - 15*log(2) - 45)/(log(2) + 3)))* 
(log(2) + 6)/(e^((x*log(2) + 6*x - 2*log(2) - 5)/(log(2) + 3))*log(2) + 6* 
e^((x*log(2) + 6*x - 2*log(2) - 5)/(log(2) + 3))))/((log(2) + 3)*log(6))
 

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.18 \[ \int \frac {e^{e^{\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}}+\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}} (-6-\log (2))+(3+\log (2)) \log (6)}{(3+\log (2)) \log (6)} \, dx=x-\frac {{\mathrm {e}}^{\frac {2^{\frac {x}{\ln \left (2\right )+3}}\,{\mathrm {e}}^{\frac {6\,x}{\ln \left (2\right )+3}}\,{\mathrm {e}}^{-\frac {5}{\ln \left (2\right )+3}}}{2^{\frac {2}{\ln \left (2\right )+3}}}}\,\left (\ln \left (2\right )+3\right )}{\ln \left (216\right )+\ln \left (2\right )\,\ln \left (6\right )} \] Input:

int((log(6)*(log(2) + 3) - exp(exp((6*x + log(2)*(x - 2) - 5)/(log(2) + 3) 
))*exp((6*x + log(2)*(x - 2) - 5)/(log(2) + 3))*(log(2) + 6))/(log(6)*(log 
(2) + 3)),x)
 

Output:

x - (exp((2^(x/(log(2) + 3))*exp((6*x)/(log(2) + 3))*exp(-5/(log(2) + 3))) 
/2^(2/(log(2) + 3)))*(log(2) + 3))/(log(216) + log(2)*log(6))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{e^{\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}}+\frac {-5+6 x+(-2+x) \log (2)}{3+\log (2)}} (-6-\log (2))+(3+\log (2)) \log (6)}{(3+\log (2)) \log (6)} \, dx=\frac {-e^{\frac {e^{\frac {\mathrm {log}\left (2\right ) x +6 x +1}{\mathrm {log}\left (2\right )+3}}}{e^{2}}}+\mathrm {log}\left (6\right ) x}{\mathrm {log}\left (6\right )} \] Input:

int(((-log(2)-6)*exp(((-2+x)*log(2)+6*x-5)/(3+log(2)))*exp(exp(((-2+x)*log 
(2)+6*x-5)/(3+log(2))))+(3+log(2))*log(6))/(3+log(2))/log(6),x)
 

Output:

( - e**(e**((log(2)*x + 6*x + 1)/(log(2) + 3))/e**2) + log(6)*x)/log(6)