\(\int \frac {3^{-\frac {2 (x^3-x^4)}{1+4 x^3 \log (\frac {x}{3})}} x^{\frac {2 (x^3-x^4)}{1+4 x^3 \log (\frac {x}{3})}} (2 x^2-2 x^3+(6 x^2-8 x^3) \log (\frac {x}{3})-8 x^6 \log ^2(\frac {x}{3}))}{1+8 x^3 \log (\frac {x}{3})+16 x^6 \log ^2(\frac {x}{3})} \, dx\) [1762]

Optimal result

Integrand size = 127, antiderivative size = 27 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=1+e^{\frac {2 (1-x)}{4+\frac {1}{x^3 \log \left (\frac {x}{3}\right )}}} \] Output:

exp((1-x)/(1/x^3/ln(1/3*x)+4))^2+1
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=\int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx \] Input:

Integrate[(x^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(2*x^2 - 2*x^3 + (6*x^ 
2 - 8*x^3)*Log[x/3] - 8*x^6*Log[x/3]^2))/(3^((2*(x^3 - x^4))/(1 + 4*x^3*Lo 
g[x/3]))*(1 + 8*x^3*Log[x/3] + 16*x^6*Log[x/3]^2)),x]
 

Output:

Integrate[(x^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(2*x^2 - 2*x^3 + (6*x^ 
2 - 8*x^3)*Log[x/3] - 8*x^6*Log[x/3]^2))/(3^((2*(x^3 - x^4))/(1 + 4*x^3*Lo 
g[x/3]))*(1 + 8*x^3*Log[x/3] + 16*x^6*Log[x/3]^2)), 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.

\(\displaystyle \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{4 x^3 \log \left (\frac {x}{3}\right )+1}} x^{\frac {2 \left (x^3-x^4\right )}{4 x^3 \log \left (\frac {x}{3}\right )+1}} \left (-8 x^6 \log ^2\left (\frac {x}{3}\right )-2 x^3+2 x^2+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )\right )}{16 x^6 \log ^2\left (\frac {x}{3}\right )+8 x^3 \log \left (\frac {x}{3}\right )+1} \, dx\)

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(2*x^2 - 2*x^3 + (6*x^2 - 8* 
x^3)*Log[x/3] - 8*x^6*Log[x/3]^2))/(3^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3] 
))*(1 + 8*x^3*Log[x/3] + 16*x^6*Log[x/3]^2)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 7.87 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\({\mathrm e}^{-\frac {2 x^{3} \left (-1+x \right ) \ln \left (\frac {x}{3}\right )}{4 x^{3} \ln \left (\frac {x}{3}\right )+1}}\)	\(29\)
risch	\(\left (\frac {x}{3}\right )^{-\frac {2 x^{3} \left (-1+x \right )}{4 x^{3} \ln \left (\frac {x}{3}\right )+1}}\)	\(33\)

Input:

int((-8*x^6*ln(1/3*x)^2+(-8*x^3+6*x^2)*ln(1/3*x)-2*x^3+2*x^2)*exp((-x^4+x^ 
3)*ln(1/3*x)/(4*x^3*ln(1/3*x)+1))^2/(16*x^6*ln(1/3*x)^2+8*x^3*ln(1/3*x)+1) 
,x,method=_RETURNVERBOSE)
 

Output:

exp(-x^3*(-1+x)*ln(1/3*x)/(4*x^3*ln(1/3*x)+1))^2
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=\frac {1}{\left (\frac {1}{3} \, x\right )^{\frac {2 \, {\left (x^{4} - x^{3}\right )}}{4 \, x^{3} \log \left (\frac {1}{3} \, x\right ) + 1}}} \] Input:

integrate((-8*x^6*log(1/3*x)^2+(-8*x^3+6*x^2)*log(1/3*x)-2*x^3+2*x^2)*exp( 
(-x^4+x^3)*log(1/3*x)/(4*x^3*log(1/3*x)+1))^2/(16*x^6*log(1/3*x)^2+8*x^3*l 
og(1/3*x)+1),x, algorithm="fricas")
 

Output:

1/((1/3*x)^(2*(x^4 - x^3)/(4*x^3*log(1/3*x) + 1)))
 

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=e^{\frac {2 \left (- x^{4} + x^{3}\right ) \log {\left (\frac {x}{3} \right )}}{4 x^{3} \log {\left (\frac {x}{3} \right )} + 1}} \] Input:

integrate((-8*x**6*ln(1/3*x)**2+(-8*x**3+6*x**2)*ln(1/3*x)-2*x**3+2*x**2)* 
exp((-x**4+x**3)*ln(1/3*x)/(4*x**3*ln(1/3*x)+1))**2/(16*x**6*ln(1/3*x)**2+ 
8*x**3*ln(1/3*x)+1),x)
 

Output:

exp(2*(-x**4 + x**3)*log(x/3)/(4*x**3*log(x/3) + 1))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).

Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=e^{\left (-\frac {1}{2} \, x - \frac {x}{2 \, {\left (4 \, x^{3} \log \left (3\right ) - 4 \, x^{3} \log \left (x\right ) - 1\right )}} + \frac {1}{2 \, {\left (4 \, x^{3} \log \left (3\right ) - 4 \, x^{3} \log \left (x\right ) - 1\right )}} + \frac {1}{2}\right )} \] Input:

integrate((-8*x^6*log(1/3*x)^2+(-8*x^3+6*x^2)*log(1/3*x)-2*x^3+2*x^2)*exp( 
(-x^4+x^3)*log(1/3*x)/(4*x^3*log(1/3*x)+1))^2/(16*x^6*log(1/3*x)^2+8*x^3*l 
og(1/3*x)+1),x, algorithm="maxima")
 

Output:

e^(-1/2*x - 1/2*x/(4*x^3*log(3) - 4*x^3*log(x) - 1) + 1/2/(4*x^3*log(3) - 
4*x^3*log(x) - 1) + 1/2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).

Time = 0.84 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=\frac {\left (\frac {1}{3} \, x\right )^{\frac {2 \, x^{3}}{4 \, x^{3} \log \left (\frac {1}{3} \, x\right ) + 1}}}{\left (\frac {1}{3} \, x\right )^{\frac {2 \, x^{4}}{4 \, x^{3} \log \left (\frac {1}{3} \, x\right ) + 1}}} \] Input:

integrate((-8*x^6*log(1/3*x)^2+(-8*x^3+6*x^2)*log(1/3*x)-2*x^3+2*x^2)*exp( 
(-x^4+x^3)*log(1/3*x)/(4*x^3*log(1/3*x)+1))^2/(16*x^6*log(1/3*x)^2+8*x^3*l 
og(1/3*x)+1),x, algorithm="giac")
 

Output:

(1/3*x)^(2*x^3/(4*x^3*log(1/3*x) + 1))/(1/3*x)^(2*x^4/(4*x^3*log(1/3*x) + 
1))
 

Mupad [B] (verification not implemented)

Time = 3.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.30 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx={\left (\frac {1}{9}\right )}^{\frac {x^3-x^4}{4\,x^3\,\ln \left (x\right )-4\,x^3\,\ln \left (3\right )+1}}\,x^{\frac {2\,\left (x^3-x^4\right )}{4\,x^3\,\ln \left (x\right )-4\,x^3\,\ln \left (3\right )+1}} \] Input:

int((exp((2*log(x/3)*(x^3 - x^4))/(4*x^3*log(x/3) + 1))*(log(x/3)*(6*x^2 - 
 8*x^3) + 2*x^2 - 2*x^3 - 8*x^6*log(x/3)^2))/(8*x^3*log(x/3) + 16*x^6*log( 
x/3)^2 + 1),x)
 

Output:

(1/9)^((x^3 - x^4)/(4*x^3*log(x) - 4*x^3*log(3) + 1))*x^((2*(x^3 - x^4))/( 
4*x^3*log(x) - 4*x^3*log(3) + 1))
 

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=\frac {e^{\frac {2 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{3}}{4 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{3}+1}}}{e^{\frac {2 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{4}}{4 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{3}+1}}} \] Input:

int((-8*x^6*log(1/3*x)^2+(-8*x^3+6*x^2)*log(1/3*x)-2*x^3+2*x^2)*exp((-x^4+ 
x^3)*log(1/3*x)/(4*x^3*log(1/3*x)+1))^2/(16*x^6*log(1/3*x)^2+8*x^3*log(1/3 
*x)+1),x)
 

Output:

e**((2*log(x/3)*x**3)/(4*log(x/3)*x**3 + 1))/e**((2*log(x/3)*x**4)/(4*log( 
x/3)*x**3 + 1))