\(\int \frac {4+e^4+x+(1+e^9 (-4-e^4-x)) \log (\frac {e^x}{4}) \log (\log (\frac {e^x}{4}))}{e^9 (-4 x-e^4 x-x^2) \log (\frac {e^x}{4}) \log (\log (\frac {e^x}{4}))+(4+e^4+x) \log (\frac {e^x}{4}) \log (\log (\frac {e^x}{4})) \log ((4+e^4+x) \log (\log (\frac {e^x}{4})))} \, dx\) [191]

Optimal result

Integrand size = 121, antiderivative size = 27 \[ \int \frac {4+e^4+x+\left (1+e^9 \left (-4-e^4-x\right )\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )}{e^9 \left (-4 x-e^4 x-x^2\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )+\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )} \, dx=5+\log \left (-e^9 x+\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right ) \] Output:

5+ln(ln((x+4+exp(4))*ln(ln(1/4*exp(x))))-x*exp(9))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {4+e^4+x+\left (1+e^9 \left (-4-e^4-x\right )\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )}{e^9 \left (-4 x-e^4 x-x^2\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )+\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )} \, dx=\log \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right ) \] Input:

Integrate[(4 + E^4 + x + (1 + E^9*(-4 - E^4 - x))*Log[E^x/4]*Log[Log[E^x/4 
]])/(E^9*(-4*x - E^4*x - x^2)*Log[E^x/4]*Log[Log[E^x/4]] + (4 + E^4 + x)*L 
og[E^x/4]*Log[Log[E^x/4]]*Log[(4 + E^4 + x)*Log[Log[E^x/4]]]),x]
 

Output:

Log[E^9*x - Log[(4 + E^4 + x)*Log[Log[E^x/4]]]]
 

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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle e^9 \int \frac {1}{e^9 x-\log \left (\left (x+e^4+4\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )}dx-\left (1-4 e^9-e^{13}\right ) \int \frac {1}{\left (x+e^4+4\right ) \left (e^9 x-\log \left (\left (x+e^4+4\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )}dx-e^9 \left (4+e^4\right ) \int \frac {1}{\left (x+e^4+4\right ) \left (e^9 x-\log \left (\left (x+e^4+4\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )}dx-\int \frac {1}{\log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \left (e^9 x-\log \left (\left (x+e^4+4\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )}dx\)

Input:

Int[(4 + E^4 + x + (1 + E^9*(-4 - E^4 - x))*Log[E^x/4]*Log[Log[E^x/4]])/(E 
^9*(-4*x - E^4*x - x^2)*Log[E^x/4]*Log[Log[E^x/4]] + (4 + E^4 + x)*Log[E^x 
/4]*Log[Log[E^x/4]]*Log[(4 + E^4 + x)*Log[Log[E^x/4]]]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 77.57 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(\ln \left (\left (x \,{\mathrm e}^{9}-\ln \left (\left (x +4+{\mathrm e}^{4}\right ) \ln \left (\ln \left (\frac {{\mathrm e}^{x}}{4}\right )\right )\right )\right ) {\mathrm e}^{-9}\right )\)	\(27\)

Input:

int((((-exp(4)-x-4)*exp(9)+1)*ln(1/4*exp(x))*ln(ln(1/4*exp(x)))+x+4+exp(4) 
)/((x+4+exp(4))*ln(1/4*exp(x))*ln(ln(1/4*exp(x)))*ln((x+4+exp(4))*ln(ln(1/ 
4*exp(x))))+(-x*exp(4)-x^2-4*x)*exp(9)*ln(1/4*exp(x))*ln(ln(1/4*exp(x)))), 
x,method=_RETURNVERBOSE)
 

Output:

ln((x*exp(9)-ln((x+4+exp(4))*ln(ln(1/4*exp(x)))))/exp(9))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {4+e^4+x+\left (1+e^9 \left (-4-e^4-x\right )\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )}{e^9 \left (-4 x-e^4 x-x^2\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )+\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )} \, dx=\log \left (-x e^{9} + \log \left ({\left (x + e^{4} + 4\right )} \log \left (x - 2 \, \log \left (2\right )\right )\right )\right ) \] Input:

integrate((((-exp(4)-x-4)*exp(9)+1)*log(1/4*exp(x))*log(log(1/4*exp(x)))+x 
+4+exp(4))/((x+4+exp(4))*log(1/4*exp(x))*log(log(1/4*exp(x)))*log((x+4+exp 
(4))*log(log(1/4*exp(x))))+(-x*exp(4)-x^2-4*x)*exp(9)*log(1/4*exp(x))*log( 
log(1/4*exp(x)))),x, algorithm="fricas")
 

Output:

log(-x*e^9 + log((x + e^4 + 4)*log(x - 2*log(2))))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {4+e^4+x+\left (1+e^9 \left (-4-e^4-x\right )\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )}{e^9 \left (-4 x-e^4 x-x^2\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )+\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )} \, dx=\text {Timed out} \] Input:

integrate((((-exp(4)-x-4)*exp(9)+1)*ln(1/4*exp(x))*ln(ln(1/4*exp(x)))+x+4+ 
exp(4))/((x+4+exp(4))*ln(1/4*exp(x))*ln(ln(1/4*exp(x)))*ln((x+4+exp(4))*ln 
(ln(1/4*exp(x))))+(-x*exp(4)-x**2-4*x)*exp(9)*ln(1/4*exp(x))*ln(ln(1/4*exp 
(x)))),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {4+e^4+x+\left (1+e^9 \left (-4-e^4-x\right )\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )}{e^9 \left (-4 x-e^4 x-x^2\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )+\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )} \, dx=\log \left (-x e^{9} + \log \left (x + e^{4} + 4\right ) + \log \left (\log \left (x - 2 \, \log \left (2\right )\right )\right )\right ) \] Input:

integrate((((-exp(4)-x-4)*exp(9)+1)*log(1/4*exp(x))*log(log(1/4*exp(x)))+x 
+4+exp(4))/((x+4+exp(4))*log(1/4*exp(x))*log(log(1/4*exp(x)))*log((x+4+exp 
(4))*log(log(1/4*exp(x))))+(-x*exp(4)-x^2-4*x)*exp(9)*log(1/4*exp(x))*log( 
log(1/4*exp(x)))),x, algorithm="maxima")
 

Output:

log(-x*e^9 + log(x + e^4 + 4) + log(log(x - 2*log(2))))
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {4+e^4+x+\left (1+e^9 \left (-4-e^4-x\right )\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )}{e^9 \left (-4 x-e^4 x-x^2\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )+\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )} \, dx=\log \left (-x e^{9} + \log \left (x + e^{4} + 4\right ) + \log \left (\log \left (x - 2 \, \log \left (2\right )\right )\right )\right ) \] Input:

integrate((((-exp(4)-x-4)*exp(9)+1)*log(1/4*exp(x))*log(log(1/4*exp(x)))+x 
+4+exp(4))/((x+4+exp(4))*log(1/4*exp(x))*log(log(1/4*exp(x)))*log((x+4+exp 
(4))*log(log(1/4*exp(x))))+(-x*exp(4)-x^2-4*x)*exp(9)*log(1/4*exp(x))*log( 
log(1/4*exp(x)))),x, algorithm="giac")
 

Output:

log(-x*e^9 + log(x + e^4 + 4) + log(log(x - 2*log(2))))
 

Mupad [B] (verification not implemented)

Time = 3.95 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {4+e^4+x+\left (1+e^9 \left (-4-e^4-x\right )\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )}{e^9 \left (-4 x-e^4 x-x^2\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )+\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )} \, dx=\ln \left (\ln \left (\ln \left (x-\ln \left (4\right )\right )\,\left (x+{\mathrm {e}}^4+4\right )\right )-x\,{\mathrm {e}}^9\right ) \] Input:

int((x + exp(4) - log(log(exp(x)/4))*log(exp(x)/4)*(exp(9)*(x + exp(4) + 4 
) - 1) + 4)/(log(log(exp(x)/4))*log(log(log(exp(x)/4))*(x + exp(4) + 4))*l 
og(exp(x)/4)*(x + exp(4) + 4) - log(log(exp(x)/4))*exp(9)*log(exp(x)/4)*(4 
*x + x*exp(4) + x^2)),x)
 

Output:

log(log(log(x - log(4))*(x + exp(4) + 4)) - x*exp(9))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {4+e^4+x+\left (1+e^9 \left (-4-e^4-x\right )\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )}{e^9 \left (-4 x-e^4 x-x^2\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )+\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\frac {e^{x}}{4}\right )\right ) e^{4}+\mathrm {log}\left (\mathrm {log}\left (\frac {e^{x}}{4}\right )\right ) x +4 \,\mathrm {log}\left (\mathrm {log}\left (\frac {e^{x}}{4}\right )\right )\right )-e^{9} x \right ) \] Input:

int((((-exp(4)-x-4)*exp(9)+1)*log(1/4*exp(x))*log(log(1/4*exp(x)))+x+4+exp 
(4))/((x+4+exp(4))*log(1/4*exp(x))*log(log(1/4*exp(x)))*log((x+4+exp(4))*l 
og(log(1/4*exp(x))))+(-x*exp(4)-x^2-4*x)*exp(9)*log(1/4*exp(x))*log(log(1/ 
4*exp(x)))),x)
 

Output:

log(log(log(log(e**x/4))*e**4 + log(log(e**x/4))*x + 4*log(log(e**x/4))) - 
 e**9*x)