\(\int \frac {1+3 x \log (4)+e^{3+e^{3-x \log (\frac {1}{27} (1+3 x \log (4)))}-x \log (\frac {1}{27} (1+3 x \log (4)))} (-3 x \log (4)+(-1-3 x \log (4)) \log (\frac {1}{27} (1+3 x \log (4))))}{1+3 x \log (4)} \, dx\) [1916]

Optimal result

Integrand size = 82, antiderivative size = 32 \[ \int \frac {1+3 x \log (4)+e^{3+e^{3-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )}-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )} \left (-3 x \log (4)+(-1-3 x \log (4)) \log \left (\frac {1}{27} (1+3 x \log (4))\right )\right )}{1+3 x \log (4)} \, dx=e^{e^{x \left (\frac {3}{x}-\log \left (\frac {1}{9} x \left (\frac {1}{3 x}+\log (4)\right )\right )\right )}}+x \] Output:

x+exp(exp((3/x-ln(x*(1/27/x+2/9*ln(2))))*x))
 

Mathematica [F]

\[ \int \frac {1+3 x \log (4)+e^{3+e^{3-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )}-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )} \left (-3 x \log (4)+(-1-3 x \log (4)) \log \left (\frac {1}{27} (1+3 x \log (4))\right )\right )}{1+3 x \log (4)} \, dx=\int \frac {1+3 x \log (4)+e^{3+e^{3-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )}-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )} \left (-3 x \log (4)+(-1-3 x \log (4)) \log \left (\frac {1}{27} (1+3 x \log (4))\right )\right )}{1+3 x \log (4)} \, dx \] Input:

Integrate[(1 + 3*x*Log[4] + E^(3 + E^(3 - x*Log[(1 + 3*x*Log[4])/27]) - x* 
Log[(1 + 3*x*Log[4])/27])*(-3*x*Log[4] + (-1 - 3*x*Log[4])*Log[(1 + 3*x*Lo 
g[4])/27]))/(1 + 3*x*Log[4]),x]
 

Output:

Integrate[(1 + 3*x*Log[4] + E^(3 + E^(3 - x*Log[(1 + 3*x*Log[4])/27]) - x* 
Log[(1 + 3*x*Log[4])/27])*(-3*x*Log[4] + (-1 - 3*x*Log[4])*Log[(1 + 3*x*Lo 
g[4])/27]))/(1 + 3*x*Log[4]), x]
 

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.

\(\displaystyle \int \frac {\left ((-3 x \log (4)-1) \log \left (\frac {1}{27} (3 x \log (4)+1)\right )-3 x \log (4)\right ) \exp \left (-x \log \left (\frac {1}{27} (3 x \log (4)+1)\right )+e^{3-x \log \left (\frac {1}{27} (3 x \log (4)+1)\right )}+3\right )+3 x \log (4)+1}{3 x \log (4)+1} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (1-27^x e^{e^3 27^x (3 x \log (4)+1)^{-x}+3} (x \log (64)+1)^{-x-1} \left (x \log (64) \log \left (\frac {2}{9} x \log (2)+\frac {1}{27}\right )+x \log (64)+\log \left (\frac {2}{9} x \log (2)+\frac {1}{27}\right )\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\log \left (\frac {2}{9} x \log (2)+\frac {1}{27}\right ) \int 27^x e^{27^x e^3 (3 \log (4) x+1)^{-x}+3} (\log (64) x+1)^{-x-1}dx-\log (64) \log \left (\frac {2}{9} x \log (2)+\frac {1}{27}\right ) \int 27^x e^{27^x e^3 (3 \log (4) x+1)^{-x}+3} x (\log (64) x+1)^{-x-1}dx-\log (64) \int 27^x e^{27^x e^3 (3 \log (4) x+1)^{-x}+3} x (\log (64) x+1)^{-x-1}dx+6 \log (2) \int \frac {\int 27^x e^{27^x e^3 (3 \log (4) x+1)^{-x}+3} (\log (64) x+1)^{-x-1}dx}{\log (64) x+1}dx+6 \log (2) \log (64) \int \frac {\int 27^x e^{27^x e^3 (3 \log (4) x+1)^{-x}+3} x (\log (64) x+1)^{-x-1}dx}{\log (64) x+1}dx+x\)

Input:

Int[(1 + 3*x*Log[4] + E^(3 + E^(3 - x*Log[(1 + 3*x*Log[4])/27]) - x*Log[(1 
 + 3*x*Log[4])/27])*(-3*x*Log[4] + (-1 - 3*x*Log[4])*Log[(1 + 3*x*Log[4])/ 
27]))/(1 + 3*x*Log[4]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.56

method	result	size
default	\(x +{\mathrm e}^{{\mathrm e}^{-x \ln \left (\frac {2 x \ln \left (2\right )}{9}+\frac {1}{27}\right )+3}}\)	\(18\)
norman	\(x +{\mathrm e}^{{\mathrm e}^{-x \ln \left (\frac {2 x \ln \left (2\right )}{9}+\frac {1}{27}\right )+3}}\)	\(18\)
risch	\(x +{\mathrm e}^{\left (\frac {2 x \ln \left (2\right )}{9}+\frac {1}{27}\right )^{-x} {\mathrm e}^{3}}\)	\(18\)
parallelrisch	\(x +{\mathrm e}^{{\mathrm e}^{-x \ln \left (\frac {2 x \ln \left (2\right )}{9}+\frac {1}{27}\right )+3}}\)	\(18\)
parts	\(x +{\mathrm e}^{{\mathrm e}^{-x \ln \left (\frac {2 x \ln \left (2\right )}{9}+\frac {1}{27}\right )+3}}\)	\(18\)

Input:

int((((-6*x*ln(2)-1)*ln(2/9*x*ln(2)+1/27)-6*x*ln(2))*exp(-x*ln(2/9*x*ln(2) 
+1/27)+3)*exp(exp(-x*ln(2/9*x*ln(2)+1/27)+3))+6*x*ln(2)+1)/(6*x*ln(2)+1),x 
,method=_RETURNVERBOSE)
 

Output:

x+exp(exp(-x*ln(2/9*x*ln(2)+1/27)+3))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).

Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.84 \[ \int \frac {1+3 x \log (4)+e^{3+e^{3-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )}-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )} \left (-3 x \log (4)+(-1-3 x \log (4)) \log \left (\frac {1}{27} (1+3 x \log (4))\right )\right )}{1+3 x \log (4)} \, dx={\left (x e^{\left (-x \log \left (\frac {2}{9} \, x \log \left (2\right ) + \frac {1}{27}\right ) + 3\right )} + e^{\left (-x \log \left (\frac {2}{9} \, x \log \left (2\right ) + \frac {1}{27}\right ) + e^{\left (-x \log \left (\frac {2}{9} \, x \log \left (2\right ) + \frac {1}{27}\right ) + 3\right )} + 3\right )}\right )} e^{\left (x \log \left (\frac {2}{9} \, x \log \left (2\right ) + \frac {1}{27}\right ) - 3\right )} \] Input:

integrate((((-6*x*log(2)-1)*log(2/9*x*log(2)+1/27)-6*x*log(2))*exp(-x*log( 
2/9*x*log(2)+1/27)+3)*exp(exp(-x*log(2/9*x*log(2)+1/27)+3))+6*x*log(2)+1)/ 
(6*x*log(2)+1),x, algorithm="fricas")
 

Output:

(x*e^(-x*log(2/9*x*log(2) + 1/27) + 3) + e^(-x*log(2/9*x*log(2) + 1/27) + 
e^(-x*log(2/9*x*log(2) + 1/27) + 3) + 3))*e^(x*log(2/9*x*log(2) + 1/27) - 
3)
 

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {1+3 x \log (4)+e^{3+e^{3-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )}-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )} \left (-3 x \log (4)+(-1-3 x \log (4)) \log \left (\frac {1}{27} (1+3 x \log (4))\right )\right )}{1+3 x \log (4)} \, dx=x + e^{e^{- x \log {\left (\frac {2 x \log {\left (2 \right )}}{9} + \frac {1}{27} \right )} + 3}} \] Input:

integrate((((-6*x*ln(2)-1)*ln(2/9*x*ln(2)+1/27)-6*x*ln(2))*exp(-x*ln(2/9*x 
*ln(2)+1/27)+3)*exp(exp(-x*ln(2/9*x*ln(2)+1/27)+3))+6*x*ln(2)+1)/(6*x*ln(2 
)+1),x)
 

Output:

x + exp(exp(-x*log(2*x*log(2)/9 + 1/27) + 3))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (26) = 52\).

Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {1+3 x \log (4)+e^{3+e^{3-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )}-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )} \left (-3 x \log (4)+(-1-3 x \log (4)) \log \left (\frac {1}{27} (1+3 x \log (4))\right )\right )}{1+3 x \log (4)} \, dx=\frac {1}{6} \, {\left (\frac {6 \, x}{\log \left (2\right )} - \frac {\log \left (6 \, x \log \left (2\right ) + 1\right )}{\log \left (2\right )^{2}}\right )} \log \left (2\right ) + \frac {\log \left (6 \, x \log \left (2\right ) + 1\right )}{6 \, \log \left (2\right )} + e^{\left (e^{\left (3 \, x \log \left (3\right ) - x \log \left (6 \, x \log \left (2\right ) + 1\right ) + 3\right )}\right )} \] Input:

integrate((((-6*x*log(2)-1)*log(2/9*x*log(2)+1/27)-6*x*log(2))*exp(-x*log( 
2/9*x*log(2)+1/27)+3)*exp(exp(-x*log(2/9*x*log(2)+1/27)+3))+6*x*log(2)+1)/ 
(6*x*log(2)+1),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1+3 x \log (4)+e^{3+e^{3-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )}-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )} \left (-3 x \log (4)+(-1-3 x \log (4)) \log \left (\frac {1}{27} (1+3 x \log (4))\right )\right )}{1+3 x \log (4)} \, dx=\int { -\frac {{\left (6 \, x \log \left (2\right ) + {\left (6 \, x \log \left (2\right ) + 1\right )} \log \left (\frac {2}{9} \, x \log \left (2\right ) + \frac {1}{27}\right )\right )} e^{\left (-x \log \left (\frac {2}{9} \, x \log \left (2\right ) + \frac {1}{27}\right ) + e^{\left (-x \log \left (\frac {2}{9} \, x \log \left (2\right ) + \frac {1}{27}\right ) + 3\right )} + 3\right )} - 6 \, x \log \left (2\right ) - 1}{6 \, x \log \left (2\right ) + 1} \,d x } \] Input:

integrate((((-6*x*log(2)-1)*log(2/9*x*log(2)+1/27)-6*x*log(2))*exp(-x*log( 
2/9*x*log(2)+1/27)+3)*exp(exp(-x*log(2/9*x*log(2)+1/27)+3))+6*x*log(2)+1)/ 
(6*x*log(2)+1),x, algorithm="giac")
 

Output:

integrate(-((6*x*log(2) + (6*x*log(2) + 1)*log(2/9*x*log(2) + 1/27))*e^(-x 
*log(2/9*x*log(2) + 1/27) + e^(-x*log(2/9*x*log(2) + 1/27) + 3) + 3) - 6*x 
*log(2) - 1)/(6*x*log(2) + 1), x)
 

Mupad [B] (verification not implemented)

Time = 4.31 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.53 \[ \int \frac {1+3 x \log (4)+e^{3+e^{3-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )}-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )} \left (-3 x \log (4)+(-1-3 x \log (4)) \log \left (\frac {1}{27} (1+3 x \log (4))\right )\right )}{1+3 x \log (4)} \, dx=x+{\mathrm {e}}^{\frac {{\mathrm {e}}^3}{{\left (\frac {2\,x\,\ln \left (2\right )}{9}+\frac {1}{27}\right )}^x}} \] Input:

int((6*x*log(2) - exp(exp(3 - x*log((2*x*log(2))/9 + 1/27)))*exp(3 - x*log 
((2*x*log(2))/9 + 1/27))*(log((2*x*log(2))/9 + 1/27)*(6*x*log(2) + 1) + 6* 
x*log(2)) + 1)/(6*x*log(2) + 1),x)
 

Output:

x + exp(exp(3)/((2*x*log(2))/9 + 1/27)^x)
 

Reduce [F]

\[ \int \frac {1+3 x \log (4)+e^{3+e^{3-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )}-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )} \left (-3 x \log (4)+(-1-3 x \log (4)) \log \left (\frac {1}{27} (1+3 x \log (4))\right )\right )}{1+3 x \log (4)} \, dx=-6 \left (\int \frac {e^{\frac {27^{x} e^{3}}{\left (6 \,\mathrm {log}\left (2\right ) x +1\right )^{x}}} 27^{x} \mathrm {log}\left (\frac {2 \,\mathrm {log}\left (2\right ) x}{9}+\frac {1}{27}\right ) x}{6 \left (6 \,\mathrm {log}\left (2\right ) x +1\right )^{x} \mathrm {log}\left (2\right ) x +\left (6 \,\mathrm {log}\left (2\right ) x +1\right )^{x}}d x \right ) \mathrm {log}\left (2\right ) e^{3}-\left (\int \frac {e^{\frac {27^{x} e^{3}}{\left (6 \,\mathrm {log}\left (2\right ) x +1\right )^{x}}} 27^{x} \mathrm {log}\left (\frac {2 \,\mathrm {log}\left (2\right ) x}{9}+\frac {1}{27}\right )}{6 \left (6 \,\mathrm {log}\left (2\right ) x +1\right )^{x} \mathrm {log}\left (2\right ) x +\left (6 \,\mathrm {log}\left (2\right ) x +1\right )^{x}}d x \right ) e^{3}-6 \left (\int \frac {e^{\frac {27^{x} e^{3}}{\left (6 \,\mathrm {log}\left (2\right ) x +1\right )^{x}}} 27^{x} x}{6 \left (6 \,\mathrm {log}\left (2\right ) x +1\right )^{x} \mathrm {log}\left (2\right ) x +\left (6 \,\mathrm {log}\left (2\right ) x +1\right )^{x}}d x \right ) \mathrm {log}\left (2\right ) e^{3}+x \] Input:

int((((-6*x*log(2)-1)*log(2/9*x*log(2)+1/27)-6*x*log(2))*exp(-x*log(2/9*x* 
log(2)+1/27)+3)*exp(exp(-x*log(2/9*x*log(2)+1/27)+3))+6*x*log(2)+1)/(6*x*l 
og(2)+1),x)
 

Output:

 - 6*int((e**((27**x*e**3)/(6*log(2)*x + 1)**x)*27**x*log((6*log(2)*x + 1) 
/27)*x)/(6*(6*log(2)*x + 1)**x*log(2)*x + (6*log(2)*x + 1)**x),x)*log(2)*e 
**3 - int((e**((27**x*e**3)/(6*log(2)*x + 1)**x)*27**x*log((6*log(2)*x + 1 
)/27))/(6*(6*log(2)*x + 1)**x*log(2)*x + (6*log(2)*x + 1)**x),x)*e**3 - 6* 
int((e**((27**x*e**3)/(6*log(2)*x + 1)**x)*27**x*x)/(6*(6*log(2)*x + 1)**x 
*log(2)*x + (6*log(2)*x + 1)**x),x)*log(2)*e**3 + x