\(\int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+(16 x+32 x^2-8 x^3) \log (x)-16 \log ^2(x)}{4 x}} (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+(32-32 x^2+16 x^3) \log (x)-16 \log ^2(x))}{20 x} \, dx\) [25]

Optimal result

Integrand size = 110, antiderivative size = 36 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {1}{5} e^{-x+\frac {\left (x \left (-1-2 x+\frac {x^2}{2}\right )+2 \log (x)\right )^2}{x}} x \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {1}{5} e^{4 x^2+3 x^3-2 x^4+\frac {x^5}{4}+\frac {4 \log ^2(x)}{x}} x^{-3+2 (-4+x) x} \] Input:

Integrate[(-12*x - 32*x^2 + 40*x^3 + 36*x^4 - 32*x^5 + 5*x^6 + (32 - 32*x^ 
2 + 16*x^3)*Log[x] - 16*Log[x]^2)/(20*E^((-16*x^3 - 12*x^4 + 8*x^5 - x^6 + 
 (16*x + 32*x^2 - 8*x^3)*Log[x] - 16*Log[x]^2)/(4*x))*x),x]
 

Output:

(E^(4*x^2 + 3*x^3 - 2*x^4 + x^5/4 + (4*Log[x]^2)/x)*x^(-3 + 2*(-4 + x)*x)) 
/5
 

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.

\(\displaystyle \int \frac {\left (5 x^6-32 x^5+36 x^4+40 x^3-32 x^2+\left (16 x^3-32 x^2+32\right ) \log (x)-12 x-16 \log ^2(x)\right ) \exp \left (-\frac {-x^6+8 x^5-12 x^4-16 x^3+\left (-8 x^3+32 x^2+16 x\right ) \log (x)-16 \log ^2(x)}{4 x}\right )}{20 x} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{20} \int -e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{-\frac {-8 x^3+32 x^2+16 x}{4 x}-1} \left (-5 x^6+32 x^5-36 x^4-40 x^3+32 x^2+12 x+16 \log ^2(x)-16 \left (x^3-2 x^2+2\right ) \log (x)\right )dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{20} \int e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{-\frac {2 \left (-x^3+4 x^2+2 x\right )}{x}-1} \left (-5 x^6+32 x^5-36 x^4-40 x^3+32 x^2+12 x+16 \log ^2(x)-16 \left (x^3-2 x^2+2\right ) \log (x)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{20} \int \left (12 e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{-\frac {2 \left (-x^3+4 x^2+2 x\right )}{x}}+16 e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} \log ^2(x) x^{-\frac {2 \left (-x^3+4 x^2+2 x\right )}{x}-1}-16 e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} \left (x^3-2 x^2+2\right ) \log (x) x^{-\frac {2 \left (-x^3+4 x^2+2 x\right )}{x}-1}+32 e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{1-\frac {2 \left (-x^3+4 x^2+2 x\right )}{x}}-40 e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{2-\frac {2 \left (-x^3+4 x^2+2 x\right )}{x}}-36 e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{3-\frac {2 \left (-x^3+4 x^2+2 x\right )}{x}}+32 e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{4-\frac {2 \left (-x^3+4 x^2+2 x\right )}{x}}-5 e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{5-\frac {2 \left (-x^3+4 x^2+2 x\right )}{x}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{20} \left (-32 \int e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{2 (x-4) x}dx-12 \int e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{2 x^2-8 x-4}dx-32 \int e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{2 x^2-8 x-3}dx+40 \int e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{2 x^2-8 x-2}dx+36 \int e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{2 x^2-8 x-1}dx+5 \int e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{2 x^2-8 x+1}dx+32 \int e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{2 x^2-8 x-5} \log (x)dx-32 \int e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{2 x^2-8 x-3} \log (x)dx+16 \int e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{2 x^2-8 x-2} \log (x)dx-16 \int e^{\frac {x^6-8 x^5+12 x^4+16 x^3+16 \log ^2(x)}{4 x}} x^{2 x^2-8 x-5} \log ^2(x)dx\right )\)

Input:

Int[(-12*x - 32*x^2 + 40*x^3 + 36*x^4 - 32*x^5 + 5*x^6 + (32 - 32*x^2 + 16 
*x^3)*Log[x] - 16*Log[x]^2)/(20*E^((-16*x^3 - 12*x^4 + 8*x^5 - x^6 + (16*x 
 + 32*x^2 - 8*x^3)*Log[x] - 16*Log[x]^2)/(4*x))*x),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {\left (x^{3}-4 x^{2}+4 \ln \left (x \right )\right ) \left (x^{3}-4 x^{2}+4 \ln \left (x \right )-4 x \right )}{4 x}} x}{5}\)	\(39\)
parallelrisch	\(\frac {{\mathrm e}^{-\frac {-16 \ln \left (x \right )^{2}+\left (-8 x^{3}+32 x^{2}+16 x \right ) \ln \left (x \right )-x^{6}+8 x^{5}-12 x^{4}-16 x^{3}}{4 x}} x}{5}\)	\(56\)

Input:

int(1/20*(-16*ln(x)^2+(16*x^3-32*x^2+32)*ln(x)+5*x^6-32*x^5+36*x^4+40*x^3- 
32*x^2-12*x)/x/exp(1/4*(-16*ln(x)^2+(-8*x^3+32*x^2+16*x)*ln(x)-x^6+8*x^5-1 
2*x^4-16*x^3)/x),x,method=_RETURNVERBOSE)
 

Output:

1/5*exp(1/4*(x^3-4*x^2+4*ln(x))*(x^3-4*x^2+4*ln(x)-4*x)/x)*x
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {1}{5} \, x e^{\left (\frac {x^{6} - 8 \, x^{5} + 12 \, x^{4} + 16 \, x^{3} + 8 \, {\left (x^{3} - 4 \, x^{2} - 2 \, x\right )} \log \left (x\right ) + 16 \, \log \left (x\right )^{2}}{4 \, x}\right )} \] Input:

integrate(1/20*(-16*log(x)^2+(16*x^3-32*x^2+32)*log(x)+5*x^6-32*x^5+36*x^4 
+40*x^3-32*x^2-12*x)/x/exp(1/4*(-16*log(x)^2+(-8*x^3+32*x^2+16*x)*log(x)-x 
^6+8*x^5-12*x^4-16*x^3)/x),x, algorithm="fricas")
 

Output:

1/5*x*e^(1/4*(x^6 - 8*x^5 + 12*x^4 + 16*x^3 + 8*(x^3 - 4*x^2 - 2*x)*log(x) 
 + 16*log(x)^2)/x)
 

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.42 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {x e^{- \frac {- \frac {x^{6}}{4} + 2 x^{5} - 3 x^{4} - 4 x^{3} + \frac {\left (- 8 x^{3} + 32 x^{2} + 16 x\right ) \log {\left (x \right )}}{4} - 4 \log {\left (x \right )}^{2}}{x}}}{5} \] Input:

integrate(1/20*(-16*ln(x)**2+(16*x**3-32*x**2+32)*ln(x)+5*x**6-32*x**5+36* 
x**4+40*x**3-32*x**2-12*x)/x/exp(1/4*(-16*ln(x)**2+(-8*x**3+32*x**2+16*x)* 
ln(x)-x**6+8*x**5-12*x**4-16*x**3)/x),x)
 

Output:

x*exp(-(-x**6/4 + 2*x**5 - 3*x**4 - 4*x**3 + (-8*x**3 + 32*x**2 + 16*x)*lo 
g(x)/4 - 4*log(x)**2)/x)/5
 

Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {e^{\left (\frac {1}{4} \, x^{5} - 2 \, x^{4} + 3 \, x^{3} + 2 \, x^{2} \log \left (x\right ) + 4 \, x^{2} - 8 \, x \log \left (x\right ) + \frac {4 \, \log \left (x\right )^{2}}{x}\right )}}{5 \, x^{3}} \] Input:

integrate(1/20*(-16*log(x)^2+(16*x^3-32*x^2+32)*log(x)+5*x^6-32*x^5+36*x^4 
+40*x^3-32*x^2-12*x)/x/exp(1/4*(-16*log(x)^2+(-8*x^3+32*x^2+16*x)*log(x)-x 
^6+8*x^5-12*x^4-16*x^3)/x),x, algorithm="maxima")
 

Output:

1/5*e^(1/4*x^5 - 2*x^4 + 3*x^3 + 2*x^2*log(x) + 4*x^2 - 8*x*log(x) + 4*log 
(x)^2/x)/x^3
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.47 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {1}{5} \, x e^{\left (\frac {x^{6} - 8 \, x^{5} + 12 \, x^{4} + 8 \, x^{3} \log \left (x\right ) + 16 \, x^{3} - 32 \, x^{2} \log \left (x\right ) - 16 \, x \log \left (x\right ) + 16 \, \log \left (x\right )^{2}}{4 \, x}\right )} \] Input:

integrate(1/20*(-16*log(x)^2+(16*x^3-32*x^2+32)*log(x)+5*x^6-32*x^5+36*x^4 
+40*x^3-32*x^2-12*x)/x/exp(1/4*(-16*log(x)^2+(-8*x^3+32*x^2+16*x)*log(x)-x 
^6+8*x^5-12*x^4-16*x^3)/x),x, algorithm="giac")
 

Output:

1/5*x*e^(1/4*(x^6 - 8*x^5 + 12*x^4 + 8*x^3*log(x) + 16*x^3 - 32*x^2*log(x) 
 - 16*x*log(x) + 16*log(x)^2)/x)
 

Mupad [B] (verification not implemented)

Time = 2.37 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {x^{2\,x^2-8\,x-3}\,{\mathrm {e}}^{4\,x^2}\,{\mathrm {e}}^{3\,x^3}\,{\mathrm {e}}^{-2\,x^4}\,{\mathrm {e}}^{\frac {x^5}{4}}\,{\mathrm {e}}^{\frac {4\,{\ln \left (x\right )}^2}{x}}}{5} \] Input:

int(-(exp((4*log(x)^2 + 4*x^3 + 3*x^4 - 2*x^5 + x^6/4 - (log(x)*(16*x + 32 
*x^2 - 8*x^3))/4)/x)*((3*x)/5 - (log(x)*(16*x^3 - 32*x^2 + 32))/20 + (4*lo 
g(x)^2)/5 + (8*x^2)/5 - 2*x^3 - (9*x^4)/5 + (8*x^5)/5 - x^6/4))/x,x)
 

Output:

(x^(2*x^2 - 8*x - 3)*exp(4*x^2)*exp(3*x^3)*exp(-2*x^4)*exp(x^5/4)*exp((4*l 
og(x)^2)/x))/5
 

Reduce [B] (verification not implemented)

Time = 0.94 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.53 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {x^{2 x^{2}} e^{\frac {16 \mathrm {log}\left (x \right )^{2}+x^{6}+12 x^{4}+16 x^{3}}{4 x}}}{5 x^{8 x} e^{2 x^{4}} x^{3}} \] Input:

int(1/20*(-16*log(x)^2+(16*x^3-32*x^2+32)*log(x)+5*x^6-32*x^5+36*x^4+40*x^ 
3-32*x^2-12*x)/x/exp(1/4*(-16*log(x)^2+(-8*x^3+32*x^2+16*x)*log(x)-x^6+8*x 
^5-12*x^4-16*x^3)/x),x)
 

Output:

(x**(2*x**2)*e**((16*log(x)**2 + x**6 + 12*x**4 + 16*x**3)/(4*x)))/(5*x**( 
8*x)*e**(2*x**4)*x**3)