\(\int \frac {e^{e^{\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}}+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}} (48+361 x+152 x \log (x)+16 x \log ^2(x))}{361 x+152 x \log (x)+16 x \log ^2(x)} \, dx\) [431]

Optimal result

Integrand size = 85, antiderivative size = 21 \[ \int \frac {e^{e^{\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}}+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}} \left (48+361 x+152 x \log (x)+16 x \log ^2(x)\right )}{361 x+152 x \log (x)+16 x \log ^2(x)} \, dx=e^{e^{-5+x+\frac {12}{6+\log (x)-5 (5+\log (x))}}} \] Output:

exp(exp(4/(-19/3-4/3*ln(x))-5+x))
 

Mathematica [A] (verified)

Time = 5.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {e^{e^{\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}}+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}} \left (48+361 x+152 x \log (x)+16 x \log ^2(x)\right )}{361 x+152 x \log (x)+16 x \log ^2(x)} \, dx=e^{e^{-5+x-\frac {12}{19+4 \log (x)}}} \] Input:

Integrate[(E^(E^((-107 + 19*x + (-20 + 4*x)*Log[x])/(19 + 4*Log[x])) + (-1 
07 + 19*x + (-20 + 4*x)*Log[x])/(19 + 4*Log[x]))*(48 + 361*x + 152*x*Log[x 
] + 16*x*Log[x]^2))/(361*x + 152*x*Log[x] + 16*x*Log[x]^2),x]
 

Output:

E^E^(-5 + x - 12/(19 + 4*Log[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.

\(\displaystyle \int \frac {\left (361 x+16 x \log ^2(x)+152 x \log (x)+48\right ) \exp \left (\frac {19 x+(4 x-20) \log (x)-107}{4 \log (x)+19}+e^{\frac {19 x+(4 x-20) \log (x)-107}{4 \log (x)+19}}\right )}{361 x+16 x \log ^2(x)+152 x \log (x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (361 x+16 x \log ^2(x)+152 x \log (x)+48\right ) \exp \left (\frac {19 x+(4 x-20) \log (x)-107}{4 \log (x)+19}+e^{\frac {19 x+(4 x-20) \log (x)-107}{4 \log (x)+19}}\right )}{x (4 \log (x)+19)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\exp \left (\frac {19 x+(4 x-20) \log (x)-107}{4 \log (x)+19}+e^{\frac {19 x+(4 x-20) \log (x)-107}{4 \log (x)+19}}\right )+\frac {48 \exp \left (\frac {19 x+(4 x-20) \log (x)-107}{4 \log (x)+19}+e^{\frac {19 x+(4 x-20) \log (x)-107}{4 \log (x)+19}}\right )}{x (4 \log (x)+19)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \exp \left (\frac {19 x+(4 x-20) \log (x)-107}{4 \log (x)+19}+e^{\frac {19 x+(4 x-20) \log (x)-107}{4 \log (x)+19}}\right )dx+48 \int \frac {\exp \left (\frac {19 x+(4 x-20) \log (x)-107}{4 \log (x)+19}+e^{\frac {19 x+(4 x-20) \log (x)-107}{4 \log (x)+19}}\right )}{x (4 \log (x)+19)^2}dx\)

Input:

Int[(E^(E^((-107 + 19*x + (-20 + 4*x)*Log[x])/(19 + 4*Log[x])) + (-107 + 1 
9*x + (-20 + 4*x)*Log[x])/(19 + 4*Log[x]))*(48 + 361*x + 152*x*Log[x] + 16 
*x*Log[x]^2))/(361*x + 152*x*Log[x] + 16*x*Log[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.75 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19

method	result	size
parallelrisch	\({\mathrm e}^{{\mathrm e}^{\frac {\left (4 x -20\right ) \ln \left (x \right )+19 x -107}{4 \ln \left (x \right )+19}}}\)	\(25\)
risch	\({\mathrm e}^{{\mathrm e}^{\frac {4 x \ln \left (x \right )-20 \ln \left (x \right )+19 x -107}{4 \ln \left (x \right )+19}}}\)	\(26\)

Input:

int((16*x*ln(x)^2+152*x*ln(x)+361*x+48)*exp(((4*x-20)*ln(x)+19*x-107)/(4*l 
n(x)+19))*exp(exp(((4*x-20)*ln(x)+19*x-107)/(4*ln(x)+19)))/(16*x*ln(x)^2+1 
52*x*ln(x)+361*x),x,method=_RETURNVERBOSE)
 

Output:

exp(exp(((4*x-20)*ln(x)+19*x-107)/(4*ln(x)+19)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (15) = 30\).

Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.52 \[ \int \frac {e^{e^{\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}}+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}} \left (48+361 x+152 x \log (x)+16 x \log ^2(x)\right )}{361 x+152 x \log (x)+16 x \log ^2(x)} \, dx=e^{\left (\frac {{\left (4 \, \log \left (x\right ) + 19\right )} e^{\left (\frac {4 \, {\left (x - 5\right )} \log \left (x\right ) + 19 \, x - 107}{4 \, \log \left (x\right ) + 19}\right )} + 4 \, {\left (x - 5\right )} \log \left (x\right ) + 19 \, x - 107}{4 \, \log \left (x\right ) + 19} - \frac {4 \, {\left (x - 5\right )} \log \left (x\right ) + 19 \, x - 107}{4 \, \log \left (x\right ) + 19}\right )} \] Input:

integrate((16*x*log(x)^2+152*x*log(x)+361*x+48)*exp(((4*x-20)*log(x)+19*x- 
107)/(4*log(x)+19))*exp(exp(((4*x-20)*log(x)+19*x-107)/(4*log(x)+19)))/(16 
*x*log(x)^2+152*x*log(x)+361*x),x, algorithm="fricas")
 

Output:

e^(((4*log(x) + 19)*e^((4*(x - 5)*log(x) + 19*x - 107)/(4*log(x) + 19)) + 
4*(x - 5)*log(x) + 19*x - 107)/(4*log(x) + 19) - (4*(x - 5)*log(x) + 19*x 
- 107)/(4*log(x) + 19))
 

Sympy [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {e^{e^{\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}}+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}} \left (48+361 x+152 x \log (x)+16 x \log ^2(x)\right )}{361 x+152 x \log (x)+16 x \log ^2(x)} \, dx=e^{e^{\frac {19 x + \left (4 x - 20\right ) \log {\left (x \right )} - 107}{4 \log {\left (x \right )} + 19}}} \] Input:

integrate((16*x*ln(x)**2+152*x*ln(x)+361*x+48)*exp(((4*x-20)*ln(x)+19*x-10 
7)/(4*ln(x)+19))*exp(exp(((4*x-20)*ln(x)+19*x-107)/(4*ln(x)+19)))/(16*x*ln 
(x)**2+152*x*ln(x)+361*x),x)
 

Output:

exp(exp((19*x + (4*x - 20)*log(x) - 107)/(4*log(x) + 19)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (15) = 30\).

Time = 0.69 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33 \[ \int \frac {e^{e^{\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}}+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}} \left (48+361 x+152 x \log (x)+16 x \log ^2(x)\right )}{361 x+152 x \log (x)+16 x \log ^2(x)} \, dx=e^{\left (e^{\left (\frac {4 \, x \log \left (x\right )}{4 \, \log \left (x\right ) + 19} + \frac {19 \, x}{4 \, \log \left (x\right ) + 19} - \frac {20 \, \log \left (x\right )}{4 \, \log \left (x\right ) + 19} - \frac {107}{4 \, \log \left (x\right ) + 19}\right )}\right )} \] Input:

integrate((16*x*log(x)^2+152*x*log(x)+361*x+48)*exp(((4*x-20)*log(x)+19*x- 
107)/(4*log(x)+19))*exp(exp(((4*x-20)*log(x)+19*x-107)/(4*log(x)+19)))/(16 
*x*log(x)^2+152*x*log(x)+361*x),x, algorithm="maxima")
 

Output:

e^(e^(4*x*log(x)/(4*log(x) + 19) + 19*x/(4*log(x) + 19) - 20*log(x)/(4*log 
(x) + 19) - 107/(4*log(x) + 19)))
 

Giac [F]

\[ \int \frac {e^{e^{\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}}+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}} \left (48+361 x+152 x \log (x)+16 x \log ^2(x)\right )}{361 x+152 x \log (x)+16 x \log ^2(x)} \, dx=\int { \frac {{\left (16 \, x \log \left (x\right )^{2} + 152 \, x \log \left (x\right ) + 361 \, x + 48\right )} e^{\left (\frac {4 \, {\left (x - 5\right )} \log \left (x\right ) + 19 \, x - 107}{4 \, \log \left (x\right ) + 19} + e^{\left (\frac {4 \, {\left (x - 5\right )} \log \left (x\right ) + 19 \, x - 107}{4 \, \log \left (x\right ) + 19}\right )}\right )}}{16 \, x \log \left (x\right )^{2} + 152 \, x \log \left (x\right ) + 361 \, x} \,d x } \] Input:

integrate((16*x*log(x)^2+152*x*log(x)+361*x+48)*exp(((4*x-20)*log(x)+19*x- 
107)/(4*log(x)+19))*exp(exp(((4*x-20)*log(x)+19*x-107)/(4*log(x)+19)))/(16 
*x*log(x)^2+152*x*log(x)+361*x),x, algorithm="giac")
 

Output:

integrate((16*x*log(x)^2 + 152*x*log(x) + 361*x + 48)*e^((4*(x - 5)*log(x) 
 + 19*x - 107)/(4*log(x) + 19) + e^((4*(x - 5)*log(x) + 19*x - 107)/(4*log 
(x) + 19)))/(16*x*log(x)^2 + 152*x*log(x) + 361*x), x)
 

Mupad [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90 \[ \int \frac {e^{e^{\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}}+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}} \left (48+361 x+152 x \log (x)+16 x \log ^2(x)\right )}{361 x+152 x \log (x)+16 x \log ^2(x)} \, dx={\mathrm {e}}^{x^{\frac {4\,\left (x-5\right )}{4\,\ln \left (x\right )+19}}\,{\mathrm {e}}^{\frac {19\,x}{4\,\ln \left (x\right )+19}}\,{\mathrm {e}}^{-\frac {107}{4\,\ln \left (x\right )+19}}} \] Input:

int((exp((19*x + log(x)*(4*x - 20) - 107)/(4*log(x) + 19))*exp(exp((19*x + 
 log(x)*(4*x - 20) - 107)/(4*log(x) + 19)))*(361*x + 16*x*log(x)^2 + 152*x 
*log(x) + 48))/(361*x + 16*x*log(x)^2 + 152*x*log(x)),x)
 

Output:

exp(x^((4*(x - 5))/(4*log(x) + 19))*exp((19*x)/(4*log(x) + 19))*exp(-107/( 
4*log(x) + 19)))
 

Reduce [F]

\[ \int \frac {e^{e^{\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}}+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}} \left (48+361 x+152 x \log (x)+16 x \log ^2(x)\right )}{361 x+152 x \log (x)+16 x \log ^2(x)} \, dx=\frac {48 \left (\int \frac {e^{\frac {e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} e^{5} x +e^{x}}{e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} e^{5}}}}{16 e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} \mathrm {log}\left (x \right )^{2} x +152 e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} \mathrm {log}\left (x \right ) x +361 e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} x}d x \right )+361 \left (\int \frac {e^{\frac {e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} e^{5} x +e^{x}}{e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} e^{5}}}}{16 e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} \mathrm {log}\left (x \right )^{2}+152 e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} \mathrm {log}\left (x \right )+361 e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}}}d x \right )+16 \left (\int \frac {e^{\frac {e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} e^{5} x +e^{x}}{e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} e^{5}}} \mathrm {log}\left (x \right )^{2}}{16 e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} \mathrm {log}\left (x \right )^{2}+152 e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} \mathrm {log}\left (x \right )+361 e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}}}d x \right )+152 \left (\int \frac {e^{\frac {e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} e^{5} x +e^{x}}{e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} e^{5}}} \mathrm {log}\left (x \right )}{16 e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} \mathrm {log}\left (x \right )^{2}+152 e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}} \mathrm {log}\left (x \right )+361 e^{\frac {12}{4 \,\mathrm {log}\left (x \right )+19}}}d x \right )}{e^{5}} \] Input:

int((16*x*log(x)^2+152*x*log(x)+361*x+48)*exp(((4*x-20)*log(x)+19*x-107)/( 
4*log(x)+19))*exp(exp(((4*x-20)*log(x)+19*x-107)/(4*log(x)+19)))/(16*x*log 
(x)^2+152*x*log(x)+361*x),x)
 

Output:

(48*int(e**((e**(12/(4*log(x) + 19))*e**5*x + e**x)/(e**(12/(4*log(x) + 19 
))*e**5))/(16*e**(12/(4*log(x) + 19))*log(x)**2*x + 152*e**(12/(4*log(x) + 
 19))*log(x)*x + 361*e**(12/(4*log(x) + 19))*x),x) + 361*int(e**((e**(12/( 
4*log(x) + 19))*e**5*x + e**x)/(e**(12/(4*log(x) + 19))*e**5))/(16*e**(12/ 
(4*log(x) + 19))*log(x)**2 + 152*e**(12/(4*log(x) + 19))*log(x) + 361*e**( 
12/(4*log(x) + 19))),x) + 16*int((e**((e**(12/(4*log(x) + 19))*e**5*x + e* 
*x)/(e**(12/(4*log(x) + 19))*e**5))*log(x)**2)/(16*e**(12/(4*log(x) + 19)) 
*log(x)**2 + 152*e**(12/(4*log(x) + 19))*log(x) + 361*e**(12/(4*log(x) + 1 
9))),x) + 152*int((e**((e**(12/(4*log(x) + 19))*e**5*x + e**x)/(e**(12/(4* 
log(x) + 19))*e**5))*log(x))/(16*e**(12/(4*log(x) + 19))*log(x)**2 + 152*e 
**(12/(4*log(x) + 19))*log(x) + 361*e**(12/(4*log(x) + 19))),x))/e**5