\(\int \frac {18055103407718400-268738560 x \log ^4(x)+x^2 \log ^8(x)+e^{-\frac {e^{5+x} \log ^4(x)}{-134369280+x \log ^4(x)}} (537477120 e^{5+x} \log ^3(x)[phone] e^{5+x} x \log ^4(x)+e^{5+x} (x-x^2) \log ^8(x))}{18055103407718400 x-268738560 x^2 \log ^4(x)+x^3 \log ^8(x)} \, dx\) [2623]

Optimal result

Integrand size = 106, antiderivative size = 23 \[ \int \frac {18055103407718400-268738560 x \log ^4(x)+x^2 \log ^8(x)+e^{-\frac {e^{5+x} \log ^4(x)}{-134369280+x \log ^4(x)}} \left (537477120 e^{5+x} \log ^3(x)[phone] e^{5+x} x \log ^4(x)+e^{5+x} \left (x-x^2\right ) \log ^8(x)\right )}{18055103407718400 x-268738560 x^2 \log ^4(x)+x^3 \log ^8(x)} \, dx=e^{\frac {e^{5+x}}{-x+\frac {134369280}{\log ^4(x)}}}+\log (x) \] Output:

ln(x)+exp(exp(5+x)/(134369280/ln(x)^4-x))
 

Mathematica [A] (verified)

Time = 2.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {18055103407718400-268738560 x \log ^4(x)+x^2 \log ^8(x)+e^{-\frac {e^{5+x} \log ^4(x)}{-134369280+x \log ^4(x)}} \left (537477120 e^{5+x} \log ^3(x)[phone] e^{5+x} x \log ^4(x)+e^{5+x} \left (x-x^2\right ) \log ^8(x)\right )}{18055103407718400 x-268738560 x^2 \log ^4(x)+x^3 \log ^8(x)} \, dx=e^{\frac {e^{5+x} \log ^4(x)}{134369280-x \log ^4(x)}}+\log (x) \] Input:

Integrate[(18055103407718400 - 268738560*x*Log[x]^4 + x^2*Log[x]^8 + (5374 
77120*E^(5 + x)*Log[x]^3 + 134369280*E^(5 + x)*x*Log[x]^4 + E^(5 + x)*(x - 
 x^2)*Log[x]^8)/E^((E^(5 + x)*Log[x]^4)/(-134369280 + x*Log[x]^4)))/(18055 
103407718400*x - 268738560*x^2*Log[x]^4 + x^3*Log[x]^8),x]
 

Output:

E^((E^(5 + x)*Log[x]^4)/(134369280 - x*Log[x]^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.

\(\displaystyle \int \frac {x^2 \log ^8(x)+e^{-\frac {e^{x+5} \log ^4(x)}{x \log ^4(x)-134369280}} \left (e^{x+5} \left (x-x^2\right ) \log ^8(x)[phone] e^{x+5} x \log ^4(x)[phone] e^{x+5} \log ^3(x)\right )-268738560 x \log ^4(x)+18055103407718400}{x^3 \log ^8(x)-268738560 x^2 \log ^4(x)+18055103407718400 x} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {x^2 \log ^8(x)+e^{-\frac {e^{x+5} \log ^4(x)}{x \log ^4(x)-134369280}} \left (e^{x+5} \left (x-x^2\right ) \log ^8(x)[phone] e^{x+5} x \log ^4(x)[phone] e^{x+5} \log ^3(x)\right )-268738560 x \log ^4(x)+18055103407718400}{x \left (134369280-x \log ^4(x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {1}{x}-\frac {e^{x+\frac {e^{x+5} \log ^4(x)}{134369280-x \log ^4(x)}+5} \log ^3(x) \left (x^2 \log ^5(x)-x \log ^5(x)-134369280 x \log (x)-537477120\right )}{x \left (x \log ^4(x)-134369280\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {e^{\frac {e^{x+5} \log ^4(x)}{134369280-x \log ^4(x)}+x+5}}{x^2}dx+18055103407718400 \int \frac {e^{\frac {e^{x+5} \log ^4(x)}{134369280-x \log ^4(x)}+x+5}}{x^2 \left (x \log ^4(x)-134369280\right )^2}dx+268738560 \int \frac {e^{\frac {e^{x+5} \log ^4(x)}{134369280-x \log ^4(x)}+x+5}}{x^2 \left (x \log ^4(x)-134369280\right )}dx-\int \frac {e^{\frac {e^{x+5} \log ^4(x)}{134369280-x \log ^4(x)}+x+5}}{x}dx-134369280 \int \frac {e^{\frac {e^{x+5} \log ^4(x)}{134369280-x \log ^4(x)}+x+5}}{x \left (x \log ^4(x)-134369280\right )}dx+537477120 \int \frac {e^{\frac {e^{x+5} \log ^4(x)}{134369280-x \log ^4(x)}+x+5} \log ^3(x)}{x \left (x \log ^4(x)-134369280\right )^2}dx+\log (x)\)

Input:

Int[(18055103407718400 - 268738560*x*Log[x]^4 + x^2*Log[x]^8 + (537477120* 
E^(5 + x)*Log[x]^3 + 134369280*E^(5 + x)*x*Log[x]^4 + E^(5 + x)*(x - x^2)* 
Log[x]^8)/E^((E^(5 + x)*Log[x]^4)/(-134369280 + x*Log[x]^4)))/(18055103407 
718400*x - 268738560*x^2*Log[x]^4 + x^3*Log[x]^8),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09

\[\ln \left (x \right )+{\mathrm e}^{-\frac {{\mathrm e}^{5+x} \ln \left (x \right )^{4}}{x \ln \left (x \right )^{4}-134369280}}\]

Input:

int((((-x^2+x)*exp(5+x)*ln(x)^8+134369280*x*exp(5+x)*ln(x)^4+537477120*exp 
(5+x)*ln(x)^3)*exp(-exp(5+x)*ln(x)^4/(x*ln(x)^4-134369280))+x^2*ln(x)^8-26 
8738560*x*ln(x)^4+18055103407718400)/(x^3*ln(x)^8-268738560*x^2*ln(x)^4+18 
055103407718400*x),x)
 

Output:

ln(x)+exp(-exp(5+x)*ln(x)^4/(x*ln(x)^4-134369280))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {18055103407718400-268738560 x \log ^4(x)+x^2 \log ^8(x)+e^{-\frac {e^{5+x} \log ^4(x)}{-134369280+x \log ^4(x)}} \left (537477120 e^{5+x} \log ^3(x)[phone] e^{5+x} x \log ^4(x)+e^{5+x} \left (x-x^2\right ) \log ^8(x)\right )}{18055103407718400 x-268738560 x^2 \log ^4(x)+x^3 \log ^8(x)} \, dx=e^{\left (-\frac {e^{\left (x + 5\right )} \log \left (x\right )^{4}}{x \log \left (x\right )^{4} - 134369280}\right )} + \log \left (x\right ) \] Input:

integrate((((-x^2+x)*exp(5+x)*log(x)^8+134369280*x*exp(5+x)*log(x)^4+53747 
7120*exp(5+x)*log(x)^3)*exp(-exp(5+x)*log(x)^4/(x*log(x)^4-134369280))+x^2 
*log(x)^8-268738560*x*log(x)^4+18055103407718400)/(x^3*log(x)^8-268738560* 
x^2*log(x)^4+18055103407718400*x),x, algorithm="fricas")
 

Output:

e^(-e^(x + 5)*log(x)^4/(x*log(x)^4 - 134369280)) + log(x)
 

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {18055103407718400-268738560 x \log ^4(x)+x^2 \log ^8(x)+e^{-\frac {e^{5+x} \log ^4(x)}{-134369280+x \log ^4(x)}} \left (537477120 e^{5+x} \log ^3(x)[phone] e^{5+x} x \log ^4(x)+e^{5+x} \left (x-x^2\right ) \log ^8(x)\right )}{18055103407718400 x-268738560 x^2 \log ^4(x)+x^3 \log ^8(x)} \, dx=\log {\left (x \right )} + e^{- \frac {e^{x + 5} \log {\left (x \right )}^{4}}{x \log {\left (x \right )}^{4} - 134369280}} \] Input:

integrate((((-x**2+x)*exp(5+x)*ln(x)**8+134369280*x*exp(5+x)*ln(x)**4+5374 
77120*exp(5+x)*ln(x)**3)*exp(-exp(5+x)*ln(x)**4/(x*ln(x)**4-134369280))+x* 
*2*ln(x)**8-268738560*x*ln(x)**4+18055103407718400)/(x**3*ln(x)**8-2687385 
60*x**2*ln(x)**4+18055103407718400*x),x)
 

Output:

log(x) + exp(-exp(x + 5)*log(x)**4/(x*log(x)**4 - 134369280))
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {18055103407718400-268738560 x \log ^4(x)+x^2 \log ^8(x)+e^{-\frac {e^{5+x} \log ^4(x)}{-134369280+x \log ^4(x)}} \left (537477120 e^{5+x} \log ^3(x)[phone] e^{5+x} x \log ^4(x)+e^{5+x} \left (x-x^2\right ) \log ^8(x)\right )}{18055103407718400 x-268738560 x^2 \log ^4(x)+x^3 \log ^8(x)} \, dx=e^{\left (-\frac {e^{\left (x + 5\right )} \log \left (x\right )^{4}}{x \log \left (x\right )^{4} - 134369280}\right )} + \log \left (x\right ) \] Input:

integrate((((-x^2+x)*exp(5+x)*log(x)^8+134369280*x*exp(5+x)*log(x)^4+53747 
7120*exp(5+x)*log(x)^3)*exp(-exp(5+x)*log(x)^4/(x*log(x)^4-134369280))+x^2 
*log(x)^8-268738560*x*log(x)^4+18055103407718400)/(x^3*log(x)^8-268738560* 
x^2*log(x)^4+18055103407718400*x),x, algorithm="maxima")
 

Output:

e^(-e^(x + 5)*log(x)^4/(x*log(x)^4 - 134369280)) + log(x)
 

Giac [F]

\[ \int \frac {18055103407718400-268738560 x \log ^4(x)+x^2 \log ^8(x)+e^{-\frac {e^{5+x} \log ^4(x)}{-134369280+x \log ^4(x)}} \left (537477120 e^{5+x} \log ^3(x)[phone] e^{5+x} x \log ^4(x)+e^{5+x} \left (x-x^2\right ) \log ^8(x)\right )}{18055103407718400 x-268738560 x^2 \log ^4(x)+x^3 \log ^8(x)} \, dx=\int { \frac {x^{2} \log \left (x\right )^{8} - 268738560 \, x \log \left (x\right )^{4} - {\left ({\left (x^{2} - x\right )} e^{\left (x + 5\right )} \log \left (x\right )^{8} - 134369280 \, x e^{\left (x + 5\right )} \log \left (x\right )^{4} - 537477120 \, e^{\left (x + 5\right )} \log \left (x\right )^{3}\right )} e^{\left (-\frac {e^{\left (x + 5\right )} \log \left (x\right )^{4}}{x \log \left (x\right )^{4} - 134369280}\right )} + 18055103407718400}{x^{3} \log \left (x\right )^{8} - 268738560 \, x^{2} \log \left (x\right )^{4} + 18055103407718400 \, x} \,d x } \] Input:

integrate((((-x^2+x)*exp(5+x)*log(x)^8+134369280*x*exp(5+x)*log(x)^4+53747 
7120*exp(5+x)*log(x)^3)*exp(-exp(5+x)*log(x)^4/(x*log(x)^4-134369280))+x^2 
*log(x)^8-268738560*x*log(x)^4+18055103407718400)/(x^3*log(x)^8-268738560* 
x^2*log(x)^4+18055103407718400*x),x, algorithm="giac")
 

Output:

integrate((x^2*log(x)^8 - 268738560*x*log(x)^4 - ((x^2 - x)*e^(x + 5)*log( 
x)^8 - 134369280*x*e^(x + 5)*log(x)^4 - 537477120*e^(x + 5)*log(x)^3)*e^(- 
e^(x + 5)*log(x)^4/(x*log(x)^4 - 134369280)) + 18055103407718400)/(x^3*log 
(x)^8 - 268738560*x^2*log(x)^4 + 18055103407718400*x), x)
 

Mupad [B] (verification not implemented)

Time = 2.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {18055103407718400-268738560 x \log ^4(x)+x^2 \log ^8(x)+e^{-\frac {e^{5+x} \log ^4(x)}{-134369280+x \log ^4(x)}} \left (537477120 e^{5+x} \log ^3(x)[phone] e^{5+x} x \log ^4(x)+e^{5+x} \left (x-x^2\right ) \log ^8(x)\right )}{18055103407718400 x-268738560 x^2 \log ^4(x)+x^3 \log ^8(x)} \, dx={\mathrm {e}}^{-\frac {{\mathrm {e}}^5\,{\mathrm {e}}^x\,{\ln \left (x\right )}^4}{x\,{\ln \left (x\right )}^4-134369280}}+\ln \left (x\right ) \] Input:

int((exp(-(exp(x + 5)*log(x)^4)/(x*log(x)^4 - 134369280))*(537477120*exp(x 
 + 5)*log(x)^3 + 134369280*x*exp(x + 5)*log(x)^4 + exp(x + 5)*log(x)^8*(x 
- x^2)) - 268738560*x*log(x)^4 + x^2*log(x)^8 + 18055103407718400)/(180551 
03407718400*x - 268738560*x^2*log(x)^4 + x^3*log(x)^8),x)
 

Output:

exp(-(exp(5)*exp(x)*log(x)^4)/(x*log(x)^4 - 134369280)) + log(x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.35 \[ \int \frac {18055103407718400-268738560 x \log ^4(x)+x^2 \log ^8(x)+e^{-\frac {e^{5+x} \log ^4(x)}{-134369280+x \log ^4(x)}} \left (537477120 e^{5+x} \log ^3(x)[phone] e^{5+x} x \log ^4(x)+e^{5+x} \left (x-x^2\right ) \log ^8(x)\right )}{18055103407718400 x-268738560 x^2 \log ^4(x)+x^3 \log ^8(x)} \, dx=\frac {e^{\frac {e^{x} \mathrm {log}\left (x \right )^{4} e^{5}}{\mathrm {log}\left (x \right )^{4} x -134369280}} \mathrm {log}\left (x \right )+1}{e^{\frac {e^{x} \mathrm {log}\left (x \right )^{4} e^{5}}{\mathrm {log}\left (x \right )^{4} x -134369280}}} \] Input:

int((((-x^2+x)*exp(5+x)*log(x)^8+134369280*x*exp(5+x)*log(x)^4+537477120*e 
xp(5+x)*log(x)^3)*exp(-exp(5+x)*log(x)^4/(x*log(x)^4-134369280))+x^2*log(x 
)^8-268738560*x*log(x)^4+18055103407718400)/(x^3*log(x)^8-268738560*x^2*lo 
g(x)^4+18055103407718400*x),x)
 

Output:

(e**((e**x*log(x)**4*e**5)/(log(x)**4*x - 134369280))*log(x) + 1)/e**((e** 
x*log(x)**4*e**5)/(log(x)**4*x - 134369280))