\(\int \frac {e^{\frac {1}{\log ^4(2 x^5-x^6+(2 x-x^2) \log (36))}} (40 x^4-24 x^5+(8-8 x) \log (36))}{(-2 x^5+x^6+(-2 x+x^2) \log (36)) \log ^5(2 x^5-x^6+(2 x-x^2) \log (36))} \, dx\) [1084]

Optimal result

Integrand size = 95, antiderivative size = 18 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=e^{\frac {1}{\log ^4\left ((2-x) x \left (x^4+\log (36)\right )\right )}} \] Output:

exp(1/ln((x^4+2*ln(6))*(2-x)*x)^4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}} \] Input:

Integrate[(E^Log[2*x^5 - x^6 + (2*x - x^2)*Log[36]]^(-4)*(40*x^4 - 24*x^5 
+ (8 - 8*x)*Log[36]))/((-2*x^5 + x^6 + (-2*x + x^2)*Log[36])*Log[2*x^5 - x 
^6 + (2*x - x^2)*Log[36]]^5),x]
 

Output:

E^Log[-((-2 + x)*x*(x^4 + Log[36]))]^(-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.

\(\displaystyle \int \frac {\left (-24 x^5+40 x^4+(8-8 x) \log (36)\right ) \exp \left (\frac {1}{\log ^4\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}\right )}{\left (x^6-2 x^5+\left (x^2-2 x\right ) \log (36)\right ) \log ^5\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (-24 x^5+40 x^4+(8-8 x) \log (36)\right ) \exp \left (\frac {1}{\log ^4\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}\right )}{x \left (x^5-2 x^4+x \log (36)-2 \log (36)\right ) \log ^5\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {\left (-24 x^5+40 x^4+(8-8 x) \log (36)\right ) \exp \left (\frac {1}{\log ^4\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}\right )}{(x-2) x (16+\log (36)) \log ^5\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}+\frac {\left (-x^3-2 x^2-4 x-8\right ) \left (-24 x^5+40 x^4+(8-8 x) \log (36)\right ) \exp \left (\frac {1}{\log ^4\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}\right )}{x (16+\log (36)) \left (x^4+\log (36)\right ) \log ^5\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{(x-2) \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx-\frac {4 \log (36) \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{x \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx}{16+\log (36)}-\frac {64 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{x \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx}{16+\log (36)}+4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{\left (\sqrt [4]{-\log (36)}-x\right ) \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx-4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{\left (x+\sqrt [4]{-\log (36)}\right ) \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx+4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{\left (-x-(-1)^{3/4} \sqrt [4]{\log (36)}\right ) \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx-4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{\left (x-(-1)^{3/4} \sqrt [4]{\log (36)}\right ) \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx\)

Input:

Int[(E^Log[2*x^5 - x^6 + (2*x - x^2)*Log[36]]^(-4)*(40*x^4 - 24*x^5 + (8 - 
 8*x)*Log[36]))/((-2*x^5 + x^6 + (-2*x + x^2)*Log[36])*Log[2*x^5 - x^6 + ( 
2*x - x^2)*Log[36]]^5),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 245.65 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61

method	result	size
parallelrisch	\({\mathrm e}^{\frac {1}{{\ln \left (2 \left (-x^{2}+2 x \right ) \ln \left (6\right )-x^{6}+2 x^{5}\right )}^{4}}}\)	\(29\)
risch	\({\mathrm e}^{\frac {1}{{\ln \left (2 \left (-x^{2}+2 x \right ) \left (\ln \left (2\right )+\ln \left (3\right )\right )-x^{6}+2 x^{5}\right )}^{4}}}\)	\(32\)

Input:

int((2*(-8*x+8)*ln(6)-24*x^5+40*x^4)*exp(1/ln(2*(-x^2+2*x)*ln(6)-x^6+2*x^5 
)^4)/(2*(x^2-2*x)*ln(6)+x^6-2*x^5)/ln(2*(-x^2+2*x)*ln(6)-x^6+2*x^5)^5,x,me 
thod=_RETURNVERBOSE)
 

Output:

exp(1/ln(2*(-x^2+2*x)*ln(6)-x^6+2*x^5)^4)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=e^{\left (\frac {1}{\log \left (-x^{6} + 2 \, x^{5} - 2 \, {\left (x^{2} - 2 \, x\right )} \log \left (6\right )\right )^{4}}\right )} \] Input:

integrate((2*(-8*x+8)*log(6)-24*x^5+40*x^4)*exp(1/log(2*(-x^2+2*x)*log(6)- 
x^6+2*x^5)^4)/(2*(x^2-2*x)*log(6)+x^6-2*x^5)/log(2*(-x^2+2*x)*log(6)-x^6+2 
*x^5)^5,x, algorithm="fricas")
 

Output:

e^(log(-x^6 + 2*x^5 - 2*(x^2 - 2*x)*log(6))^(-4))
 

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=e^{\frac {1}{\log {\left (- x^{6} + 2 x^{5} + \left (- 2 x^{2} + 4 x\right ) \log {\left (6 \right )} \right )}^{4}}} \] Input:

integrate((2*(-8*x+8)*ln(6)-24*x**5+40*x**4)*exp(1/ln(2*(-x**2+2*x)*ln(6)- 
x**6+2*x**5)**4)/(2*(x**2-2*x)*ln(6)+x**6-2*x**5)/ln(2*(-x**2+2*x)*ln(6)-x 
**6+2*x**5)**5,x)
 

Output:

exp(log(-x**6 + 2*x**5 + (-2*x**2 + 4*x)*log(6))**(-4))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1148 vs. \(2 (18) = 36\).

Time = 0.38 (sec) , antiderivative size = 1148, normalized size of antiderivative = 63.78 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=\text {Too large to display} \] Input:

integrate((2*(-8*x+8)*log(6)-24*x^5+40*x^4)*exp(1/log(2*(-x^2+2*x)*log(6)- 
x^6+2*x^5)^4)/(2*(x^2-2*x)*log(6)+x^6-2*x^5)/log(2*(-x^2+2*x)*log(6)-x^6+2 
*x^5)^5,x, algorithm="maxima")
 

Output:

3*x^5*e^(1/(log(x^4 + 2*log(3) + 2*log(2))^4 + 4*(log(x^4 + 2*log(3) + 2*l 
og(2)) + log(-x + 2))*log(x)^3 + log(x)^4 + 4*log(x^4 + 2*log(3) + 2*log(2 
))^3*log(-x + 2) + 6*log(x^4 + 2*log(3) + 2*log(2))^2*log(-x + 2)^2 + 4*lo 
g(x^4 + 2*log(3) + 2*log(2))*log(-x + 2)^3 + log(-x + 2)^4 + 6*(log(x^4 + 
2*log(3) + 2*log(2))^2 + 2*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2) + lo 
g(-x + 2)^2)*log(x)^2 + 4*(log(x^4 + 2*log(3) + 2*log(2))^3 + 3*log(x^4 + 
2*log(3) + 2*log(2))^2*log(-x + 2) + 3*log(x^4 + 2*log(3) + 2*log(2))*log( 
-x + 2)^2 + log(-x + 2)^3)*log(x)))/(3*x^5 - 5*x^4 + 2*x*(log(3) + log(2)) 
 - 2*log(3) - 2*log(2)) - 5*x^4*e^(1/(log(x^4 + 2*log(3) + 2*log(2))^4 + 4 
*(log(x^4 + 2*log(3) + 2*log(2)) + log(-x + 2))*log(x)^3 + log(x)^4 + 4*lo 
g(x^4 + 2*log(3) + 2*log(2))^3*log(-x + 2) + 6*log(x^4 + 2*log(3) + 2*log( 
2))^2*log(-x + 2)^2 + 4*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2)^3 + log 
(-x + 2)^4 + 6*(log(x^4 + 2*log(3) + 2*log(2))^2 + 2*log(x^4 + 2*log(3) + 
2*log(2))*log(-x + 2) + log(-x + 2)^2)*log(x)^2 + 4*(log(x^4 + 2*log(3) + 
2*log(2))^3 + 3*log(x^4 + 2*log(3) + 2*log(2))^2*log(-x + 2) + 3*log(x^4 + 
 2*log(3) + 2*log(2))*log(-x + 2)^2 + log(-x + 2)^3)*log(x)))/(3*x^5 - 5*x 
^4 + 2*x*(log(3) + log(2)) - 2*log(3) - 2*log(2)) + 2*x*e^(1/(log(x^4 + 2* 
log(3) + 2*log(2))^4 + 4*(log(x^4 + 2*log(3) + 2*log(2)) + log(-x + 2))*lo 
g(x)^3 + log(x)^4 + 4*log(x^4 + 2*log(3) + 2*log(2))^3*log(-x + 2) + 6*log 
(x^4 + 2*log(3) + 2*log(2))^2*log(-x + 2)^2 + 4*log(x^4 + 2*log(3) + 2*...
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=e^{\left (\frac {1}{\log \left (-x^{6} + 2 \, x^{5} - 2 \, x^{2} \log \left (6\right ) + 4 \, x \log \left (6\right )\right )^{4}}\right )} \] Input:

integrate((2*(-8*x+8)*log(6)-24*x^5+40*x^4)*exp(1/log(2*(-x^2+2*x)*log(6)- 
x^6+2*x^5)^4)/(2*(x^2-2*x)*log(6)+x^6-2*x^5)/log(2*(-x^2+2*x)*log(6)-x^6+2 
*x^5)^5,x, algorithm="giac")
 

Output:

e^(log(-x^6 + 2*x^5 - 2*x^2*log(6) + 4*x*log(6))^(-4))
 

Mupad [B] (verification not implemented)

Time = 4.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx={\mathrm {e}}^{\frac {1}{{\ln \left (-x^6+2\,x^5-2\,\ln \left (6\right )\,x^2+4\,\ln \left (6\right )\,x\right )}^4}} \] Input:

int((exp(1/log(2*log(6)*(2*x - x^2) + 2*x^5 - x^6)^4)*(2*log(6)*(8*x - 8) 
- 40*x^4 + 24*x^5))/(log(2*log(6)*(2*x - x^2) + 2*x^5 - x^6)^5*(2*log(6)*( 
2*x - x^2) + 2*x^5 - x^6)),x)
 

Output:

exp(1/log(4*x*log(6) - 2*x^2*log(6) + 2*x^5 - x^6)^4)
 

Reduce [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=e^{\frac {1}{\mathrm {log}\left (-2 \,\mathrm {log}\left (6\right ) x^{2}+4 \,\mathrm {log}\left (6\right ) x -x^{6}+2 x^{5}\right )^{4}}} \] Input:

int((2*(-8*x+8)*log(6)-24*x^5+40*x^4)*exp(1/log(2*(-x^2+2*x)*log(6)-x^6+2* 
x^5)^4)/(2*(x^2-2*x)*log(6)+x^6-2*x^5)/log(2*(-x^2+2*x)*log(6)-x^6+2*x^5)^ 
5,x)
 

Output:

e**(1/log( - 2*log(6)*x**2 + 4*log(6)*x - x**6 + 2*x**5)**4)