\(\int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} (-4 x^3+e^{-5 x^2+x^2 \log (x)} (-9 x^3+9 x^4+(2 x^3-2 x^4) \log (x)))+(-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} (-x+2 x^2)) \log (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x})}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} (x^3-2 x^4+x^5)} \, dx\) [1232]

Optimal result

Integrand size = 178, antiderivative size = 33 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {-4 x+\log \left (e^{e^{x^2 (-5+\log (x))}}+\frac {6}{x}\right )}{x-x^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4 x-\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(-1+x) x} \] Input:

Integrate[(-6 + 6*x - 24*x^2 + E^E^(-5*x^2 + x^2*Log[x])*(-4*x^3 + E^(-5*x 
^2 + x^2*Log[x])*(-9*x^3 + 9*x^4 + (2*x^3 - 2*x^4)*Log[x])) + (-6 + 12*x + 
 E^E^(-5*x^2 + x^2*Log[x])*(-x + 2*x^2))*Log[(6 + E^E^(-5*x^2 + x^2*Log[x] 
)*x)/x])/(6*x^2 - 12*x^3 + 6*x^4 + E^E^(-5*x^2 + x^2*Log[x])*(x^3 - 2*x^4 
+ x^5)),x]
 

Output:

(4*x - Log[E^(x^x^2/E^(5*x^2)) + 6/x])/((-1 + x)*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.

\(\displaystyle \int \frac {-24 x^2+\left (\left (2 x^2-x\right ) e^{e^{x^2 \log (x)-5 x^2}}+12 x-6\right ) \log \left (\frac {x e^{e^{x^2 \log (x)-5 x^2}}+6}{x}\right )+e^{e^{x^2 \log (x)-5 x^2}} \left (e^{x^2 \log (x)-5 x^2} \left (9 x^4-9 x^3+\left (2 x^3-2 x^4\right ) \log (x)\right )-4 x^3\right )+6 x-6}{6 x^4-12 x^3+6 x^2+\left (x^5-2 x^4+x^3\right ) e^{e^{x^2 \log (x)-5 x^2}}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-24 x^2+\left (\left (2 x^2-x\right ) e^{e^{x^2 \log (x)-5 x^2}}+12 x-6\right ) \log \left (\frac {x e^{e^{x^2 \log (x)-5 x^2}}+6}{x}\right )+e^{e^{x^2 \log (x)-5 x^2}} \left (e^{x^2 \log (x)-5 x^2} \left (9 x^4-9 x^3+\left (2 x^3-2 x^4\right ) \log (x)\right )-4 x^3\right )+6 x-6}{(1-x)^2 x^2 \left (e^{e^{-5 x^2} x^{x^2}} x+6\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {4 e^{e^{-5 x^2} x^{x^2}} x}{(x-1)^2 \left (e^{e^{-5 x^2} x^{x^2}} x+6\right )}-\frac {24}{(x-1)^2 \left (e^{e^{-5 x^2} x^{x^2}} x+6\right )}+\frac {6}{(x-1)^2 \left (e^{e^{-5 x^2} x^{x^2}} x+6\right ) x}-\frac {6}{(x-1)^2 \left (e^{e^{-5 x^2} x^{x^2}} x+6\right ) x^2}+\frac {(2 x-1) \log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(x-1)^2 x^2}-\frac {e^{e^{-5 x^2} x^{x^2}-5 x^2} x^{x^2+1} (2 \log (x)-9)}{(x-1) \left (e^{e^{-5 x^2} x^{x^2}} x+6\right )}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[(-6 + 6*x - 24*x^2 + E^E^(-5*x^2 + x^2*Log[x])*(-4*x^3 + E^(-5*x^2 + x 
^2*Log[x])*(-9*x^3 + 9*x^4 + (2*x^3 - 2*x^4)*Log[x])) + (-6 + 12*x + E^E^( 
-5*x^2 + x^2*Log[x])*(-x + 2*x^2))*Log[(6 + E^E^(-5*x^2 + x^2*Log[x])*x)/x 
])/(6*x^2 - 12*x^3 + 6*x^4 + E^E^(-5*x^2 + x^2*Log[x])*(x^3 - 2*x^4 + x^5) 
),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.05 (sec) , antiderivative size = 222, normalized size of antiderivative = 6.73

\[-\frac {\ln \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x \left (-1+x \right )}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x}\right )}^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x}\right )}^{3}+8 x +2 \ln \left (x \right )}{2 x \left (-1+x \right )}\]

Input:

int((((2*x^2-x)*exp(exp(x^2*ln(x)-5*x^2))+12*x-6)*ln((x*exp(exp(x^2*ln(x)- 
5*x^2))+6)/x)+(((-2*x^4+2*x^3)*ln(x)+9*x^4-9*x^3)*exp(x^2*ln(x)-5*x^2)-4*x 
^3)*exp(exp(x^2*ln(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)*exp(exp(x^2*l 
n(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x)
 

Output:

-1/x/(-1+x)*ln(exp(x^(x^2)*exp(-5*x^2))*x+6)+1/2*(I*Pi*csgn(I/x)*csgn(I*(e 
xp(x^(x^2)*exp(-5*x^2))*x+6))*csgn(I/x*(exp(x^(x^2)*exp(-5*x^2))*x+6))-I*P 
i*csgn(I/x)*csgn(I/x*(exp(x^(x^2)*exp(-5*x^2))*x+6))^2-I*Pi*csgn(I*(exp(x^ 
(x^2)*exp(-5*x^2))*x+6))*csgn(I/x*(exp(x^(x^2)*exp(-5*x^2))*x+6))^2+I*Pi*c 
sgn(I/x*(exp(x^(x^2)*exp(-5*x^2))*x+6))^3+8*x+2*ln(x))/x/(-1+x)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4 \, x - \log \left (\frac {x e^{\left (e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} + 6}{x}\right )}{x^{2} - x} \] Input:

integrate((((2*x^2-x)*exp(exp(x^2*log(x)-5*x^2))+12*x-6)*log((x*exp(exp(x^ 
2*log(x)-5*x^2))+6)/x)+(((-2*x^4+2*x^3)*log(x)+9*x^4-9*x^3)*exp(x^2*log(x) 
-5*x^2)-4*x^3)*exp(exp(x^2*log(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)*e 
xp(exp(x^2*log(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 13.87 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=- \frac {\log {\left (\frac {x e^{e^{x^{2} \log {\left (x \right )} - 5 x^{2}}} + 6}{x} \right )}}{x^{2} - x} + \frac {4}{x - 1} \] Input:

integrate((((2*x**2-x)*exp(exp(x**2*ln(x)-5*x**2))+12*x-6)*ln((x*exp(exp(x 
**2*ln(x)-5*x**2))+6)/x)+(((-2*x**4+2*x**3)*ln(x)+9*x**4-9*x**3)*exp(x**2* 
ln(x)-5*x**2)-4*x**3)*exp(exp(x**2*ln(x)-5*x**2))-24*x**2+6*x-6)/((x**5-2* 
x**4+x**3)*exp(exp(x**2*ln(x)-5*x**2))+6*x**4-12*x**3+6*x**2),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4 \, x - \log \left (x e^{\left (e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} + 6\right ) + \log \left (x\right )}{x^{2} - x} \] Input:

integrate((((2*x^2-x)*exp(exp(x^2*log(x)-5*x^2))+12*x-6)*log((x*exp(exp(x^ 
2*log(x)-5*x^2))+6)/x)+(((-2*x^4+2*x^3)*log(x)+9*x^4-9*x^3)*exp(x^2*log(x) 
-5*x^2)-4*x^3)*exp(exp(x^2*log(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)*e 
xp(exp(x^2*log(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (33) = 66\).

Time = 0.57 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.36 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4 \, x - \log \left ({\left (x e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2} + e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} + 6 \, e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} e^{\left (-x^{2} \log \left (x\right ) + 5 \, x^{2}\right )}\right ) + \log \left (x\right )}{x^{2} - x} \] Input:

integrate((((2*x^2-x)*exp(exp(x^2*log(x)-5*x^2))+12*x-6)*log((x*exp(exp(x^ 
2*log(x)-5*x^2))+6)/x)+(((-2*x^4+2*x^3)*log(x)+9*x^4-9*x^3)*exp(x^2*log(x) 
-5*x^2)-4*x^3)*exp(exp(x^2*log(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)*e 
xp(exp(x^2*log(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4\,x-\ln \left (\frac {x\,{\mathrm {e}}^{x^{x^2}\,{\mathrm {e}}^{-5\,x^2}}+6}{x}\right )}{x\,\left (x-1\right )} \] Input:

int(-(log((x*exp(exp(x^2*log(x) - 5*x^2)) + 6)/x)*(exp(exp(x^2*log(x) - 5* 
x^2))*(x - 2*x^2) - 12*x + 6) - exp(exp(x^2*log(x) - 5*x^2))*(exp(x^2*log( 
x) - 5*x^2)*(log(x)*(2*x^3 - 2*x^4) - 9*x^3 + 9*x^4) - 4*x^3) - 6*x + 24*x 
^2 + 6)/(exp(exp(x^2*log(x) - 5*x^2))*(x^3 - 2*x^4 + x^5) + 6*x^2 - 12*x^3 
 + 6*x^4),x)
 

Output:

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

Reduce [F]

\[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx =\text {Too large to display} \] Input:

int((((2*x^2-x)*exp(exp(x^2*log(x)-5*x^2))+12*x-6)*log((x*exp(exp(x^2*log( 
x)-5*x^2))+6)/x)+(((-2*x^4+2*x^3)*log(x)+9*x^4-9*x^3)*exp(x^2*log(x)-5*x^2 
)-4*x^3)*exp(exp(x^2*log(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)*exp(exp 
(x^2*log(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x)
 

Output:

 - 4*int(e**(x**(x**2)/e**(5*x**2))/(e**(x**(x**2)/e**(5*x**2))*x**3 - 2*e 
**(x**(x**2)/e**(5*x**2))*x**2 + e**(x**(x**2)/e**(5*x**2))*x + 6*x**2 - 1 
2*x + 6),x) - 6*int(log((e**(x**(x**2)/e**(5*x**2))*x + 6)/x)/(e**(x**(x** 
2)/e**(5*x**2))*x**5 - 2*e**(x**(x**2)/e**(5*x**2))*x**4 + e**(x**(x**2)/e 
**(5*x**2))*x**3 + 6*x**4 - 12*x**3 + 6*x**2),x) + 12*int(log((e**(x**(x** 
2)/e**(5*x**2))*x + 6)/x)/(e**(x**(x**2)/e**(5*x**2))*x**4 - 2*e**(x**(x** 
2)/e**(5*x**2))*x**3 + e**(x**(x**2)/e**(5*x**2))*x**2 + 6*x**3 - 12*x**2 
+ 6*x),x) - int((e**(x**(x**2)/e**(5*x**2))*log((e**(x**(x**2)/e**(5*x**2) 
)*x + 6)/x))/(e**(x**(x**2)/e**(5*x**2))*x**4 - 2*e**(x**(x**2)/e**(5*x**2 
))*x**3 + e**(x**(x**2)/e**(5*x**2))*x**2 + 6*x**3 - 12*x**2 + 6*x),x) + 2 
*int((e**(x**(x**2)/e**(5*x**2))*log((e**(x**(x**2)/e**(5*x**2))*x + 6)/x) 
)/(e**(x**(x**2)/e**(5*x**2))*x**3 - 2*e**(x**(x**2)/e**(5*x**2))*x**2 + e 
**(x**(x**2)/e**(5*x**2))*x + 6*x**2 - 12*x + 6),x) + 9*int((x**(x**2)*e** 
(x**(x**2)/e**(5*x**2))*x**2)/(e**((5*e**(5*x**2)*x**2 + x**(x**2))/e**(5* 
x**2))*x**3 - 2*e**((5*e**(5*x**2)*x**2 + x**(x**2))/e**(5*x**2))*x**2 + e 
**((5*e**(5*x**2)*x**2 + x**(x**2))/e**(5*x**2))*x + 6*e**(5*x**2)*x**2 - 
12*e**(5*x**2)*x + 6*e**(5*x**2)),x) - 2*int((x**(x**2)*e**(x**(x**2)/e**( 
5*x**2))*log(x)*x**2)/(e**((5*e**(5*x**2)*x**2 + x**(x**2))/e**(5*x**2))*x 
**3 - 2*e**((5*e**(5*x**2)*x**2 + x**(x**2))/e**(5*x**2))*x**2 + e**((5*e* 
*(5*x**2)*x**2 + x**(x**2))/e**(5*x**2))*x + 6*e**(5*x**2)*x**2 - 12*e*...