\(\int \frac {(-3+6 x-3 x^2+e^{\frac {2 e^e x}{-1+x}} (1-2 x+x^2)) \log (9-6 e^{\frac {2 e^e x}{-1+x}}+e^{\frac {4 e^e x}{-1+x}})+e^{\frac {2 e^e x}{-1+x}} (e^e (-16-8 x)+e^e (-8-4 x) \log (\frac {2+x}{5}))}{-6+9 x-3 x^3+e^{\frac {2 e^e x}{-1+x}} (2-3 x+x^3)} \, dx\) [51]

Optimal result

Integrand size = 141, antiderivative size = 31 \[ \int \frac {\left (-3+6 x-3 x^2+e^{\frac {2 e^e x}{-1+x}} \left (1-2 x+x^2\right )\right ) \log \left (9-6 e^{\frac {2 e^e x}{-1+x}}+e^{\frac {4 e^e x}{-1+x}}\right )+e^{\frac {2 e^e x}{-1+x}} \left (e^e (-16-8 x)+e^e (-8-4 x) \log \left (\frac {2+x}{5}\right )\right )}{-6+9 x-3 x^3+e^{\frac {2 e^e x}{-1+x}} \left (2-3 x+x^3\right )} \, dx=\log \left (\left (3-e^{\frac {2 e^e x}{-1+x}}\right )^2\right ) \left (2+\log \left (\frac {2+x}{5}\right )\right ) \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.58 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.87 \[ \int \frac {\left (-3+6 x-3 x^2+e^{\frac {2 e^e x}{-1+x}} \left (1-2 x+x^2\right )\right ) \log \left (9-6 e^{\frac {2 e^e x}{-1+x}}+e^{\frac {4 e^e x}{-1+x}}\right )+e^{\frac {2 e^e x}{-1+x}} \left (e^e (-16-8 x)+e^e (-8-4 x) \log \left (\frac {2+x}{5}\right )\right )}{-6+9 x-3 x^3+e^{\frac {2 e^e x}{-1+x}} \left (2-3 x+x^3\right )} \, dx=4 \log \left (3-e^{2 e^e+\frac {2 e^e}{-1+x}}\right )+\log \left (\left (-3+e^{2 e^e+\frac {2 e^e}{-1+x}}\right )^2\right ) \log \left (\frac {2+x}{5}\right ) \] Input:

Integrate[((-3 + 6*x - 3*x^2 + E^((2*E^E*x)/(-1 + x))*(1 - 2*x + x^2))*Log 
[9 - 6*E^((2*E^E*x)/(-1 + x)) + E^((4*E^E*x)/(-1 + x))] + E^((2*E^E*x)/(-1 
 + x))*(E^E*(-16 - 8*x) + E^E*(-8 - 4*x)*Log[(2 + x)/5]))/(-6 + 9*x - 3*x^ 
3 + E^((2*E^E*x)/(-1 + x))*(2 - 3*x + x^3)),x]
 

Output:

4*Log[3 - E^(2*E^E + (2*E^E)/(-1 + x))] + Log[(-3 + E^(2*E^E + (2*E^E)/(-1 
 + x)))^2]*Log[(2 + 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.

\(\displaystyle \int \frac {\left (-3 x^2+e^{\frac {2 e^e x}{x-1}} \left (x^2-2 x+1\right )+6 x-3\right ) \log \left (-6 e^{\frac {2 e^e x}{x-1}}+e^{\frac {4 e^e x}{x-1}}+9\right )+e^{\frac {2 e^e x}{x-1}} \left (e^e (-8 x-16)+e^e (-4 x-8) \log \left (\frac {x+2}{5}\right )\right )}{-3 x^3+e^{\frac {2 e^e x}{x-1}} \left (x^3-3 x+2\right )+9 x-6} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2009

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

Input:

Int[((-3 + 6*x - 3*x^2 + E^((2*E^E*x)/(-1 + x))*(1 - 2*x + x^2))*Log[9 - 6 
*E^((2*E^E*x)/(-1 + x)) + E^((4*E^E*x)/(-1 + x))] + E^((2*E^E*x)/(-1 + x)) 
*(E^E*(-16 - 8*x) + E^E*(-8 - 4*x)*Log[(2 + x)/5]))/(-6 + 9*x - 3*x^3 + E^ 
((2*E^E*x)/(-1 + x))*(2 - 3*x + x^3)),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 20.50 (sec) , antiderivative size = 179, normalized size of antiderivative = 5.77

method	result	size
risch	\(2 \ln \left (\frac {2}{5}+\frac {x}{5}\right ) \ln \left ({\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}}}{-1+x}}-3\right )-\frac {i \pi \ln \left (-2-x \right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}}}{-1+x}}-3\right )\right )}^{2} \operatorname {csgn}\left (i \left ({\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}}}{-1+x}}-3\right )^{2}\right )}{2}+i \pi \ln \left (-2-x \right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}}}{-1+x}}-3\right )\right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}}}{-1+x}}-3\right )^{2}\right )}^{2}-\frac {i \pi \ln \left (-2-x \right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}}}{-1+x}}-3\right )^{2}\right )}^{3}}{2}-8 \,{\mathrm e}^{{\mathrm e}}+4 \ln \left ({\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}}}{-1+x}}-3\right )\)	\(179\)

Input:

int((((x^2-2*x+1)*exp(x*exp(exp(1))/(-1+x))^2-3*x^2+6*x-3)*ln(exp(x*exp(ex 
p(1))/(-1+x))^4-6*exp(x*exp(exp(1))/(-1+x))^2+9)+((-4*x-8)*exp(exp(1))*ln( 
2/5+1/5*x)+(-8*x-16)*exp(exp(1)))*exp(x*exp(exp(1))/(-1+x))^2)/((x^3-3*x+2 
)*exp(x*exp(exp(1))/(-1+x))^2-3*x^3+9*x-6),x,method=_RETURNVERBOSE)
 

Output:

2*ln(2/5+1/5*x)*ln(exp(2*x*exp(exp(1))/(-1+x))-3)-1/2*I*Pi*ln(-2-x)*csgn(I 
*(exp(2*x*exp(exp(1))/(-1+x))-3))^2*csgn(I*(exp(2*x*exp(exp(1))/(-1+x))-3) 
^2)+I*Pi*ln(-2-x)*csgn(I*(exp(2*x*exp(exp(1))/(-1+x))-3))*csgn(I*(exp(2*x* 
exp(exp(1))/(-1+x))-3)^2)^2-1/2*I*Pi*ln(-2-x)*csgn(I*(exp(2*x*exp(exp(1))/ 
(-1+x))-3)^2)^3-8*exp(exp(1))+4*ln(exp(2*x*exp(exp(1))/(-1+x))-3)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-3+6 x-3 x^2+e^{\frac {2 e^e x}{-1+x}} \left (1-2 x+x^2\right )\right ) \log \left (9-6 e^{\frac {2 e^e x}{-1+x}}+e^{\frac {4 e^e x}{-1+x}}\right )+e^{\frac {2 e^e x}{-1+x}} \left (e^e (-16-8 x)+e^e (-8-4 x) \log \left (\frac {2+x}{5}\right )\right )}{-6+9 x-3 x^3+e^{\frac {2 e^e x}{-1+x}} \left (2-3 x+x^3\right )} \, dx={\left (\log \left (\frac {1}{5} \, x + \frac {2}{5}\right ) + 2\right )} \log \left (e^{\left (\frac {4 \, x e^{e}}{x - 1}\right )} - 6 \, e^{\left (\frac {2 \, x e^{e}}{x - 1}\right )} + 9\right ) \] Input:

integrate((((x^2-2*x+1)*exp(x*exp(exp(1))/(-1+x))^2-3*x^2+6*x-3)*log(exp(x 
*exp(exp(1))/(-1+x))^4-6*exp(x*exp(exp(1))/(-1+x))^2+9)+((-4*x-8)*exp(exp( 
1))*log(2/5+1/5*x)+(-8*x-16)*exp(exp(1)))*exp(x*exp(exp(1))/(-1+x))^2)/((x 
^3-3*x+2)*exp(x*exp(exp(1))/(-1+x))^2-3*x^3+9*x-6),x, algorithm="fricas")
 

Output:

(log(1/5*x + 2/5) + 2)*log(e^(4*x*e^e/(x - 1)) - 6*e^(2*x*e^e/(x - 1)) + 9 
)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).

Time = 0.56 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.87 \[ \int \frac {\left (-3+6 x-3 x^2+e^{\frac {2 e^e x}{-1+x}} \left (1-2 x+x^2\right )\right ) \log \left (9-6 e^{\frac {2 e^e x}{-1+x}}+e^{\frac {4 e^e x}{-1+x}}\right )+e^{\frac {2 e^e x}{-1+x}} \left (e^e (-16-8 x)+e^e (-8-4 x) \log \left (\frac {2+x}{5}\right )\right )}{-6+9 x-3 x^3+e^{\frac {2 e^e x}{-1+x}} \left (2-3 x+x^3\right )} \, dx=\log {\left (\frac {x}{5} + \frac {2}{5} \right )} \log {\left (e^{\frac {4 x e^{e}}{x - 1}} - 6 e^{\frac {2 x e^{e}}{x - 1}} + 9 \right )} + 4 \log {\left (e^{\frac {2 x e^{e}}{x - 1}} - 3 \right )} \] Input:

integrate((((x**2-2*x+1)*exp(x*exp(exp(1))/(-1+x))**2-3*x**2+6*x-3)*ln(exp 
(x*exp(exp(1))/(-1+x))**4-6*exp(x*exp(exp(1))/(-1+x))**2+9)+((-4*x-8)*exp( 
exp(1))*ln(2/5+1/5*x)+(-8*x-16)*exp(exp(1)))*exp(x*exp(exp(1))/(-1+x))**2) 
/((x**3-3*x+2)*exp(x*exp(exp(1))/(-1+x))**2-3*x**3+9*x-6),x)
 

Output:

log(x/5 + 2/5)*log(exp(4*x*exp(E)/(x - 1)) - 6*exp(2*x*exp(E)/(x - 1)) + 9 
) + 4*log(exp(2*x*exp(E)/(x - 1)) - 3)
 

Maxima [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {\left (-3+6 x-3 x^2+e^{\frac {2 e^e x}{-1+x}} \left (1-2 x+x^2\right )\right ) \log \left (9-6 e^{\frac {2 e^e x}{-1+x}}+e^{\frac {4 e^e x}{-1+x}}\right )+e^{\frac {2 e^e x}{-1+x}} \left (e^e (-16-8 x)+e^e (-8-4 x) \log \left (\frac {2+x}{5}\right )\right )}{-6+9 x-3 x^3+e^{\frac {2 e^e x}{-1+x}} \left (2-3 x+x^3\right )} \, dx=-2 \, {\left (\log \left (5\right ) - \log \left (x + 2\right ) - 2\right )} \log \left (e^{\left (\frac {2 \, e^{e}}{x - 1} + 2 \, e^{e}\right )} - 3\right ) \] Input:

integrate((((x^2-2*x+1)*exp(x*exp(exp(1))/(-1+x))^2-3*x^2+6*x-3)*log(exp(x 
*exp(exp(1))/(-1+x))^4-6*exp(x*exp(exp(1))/(-1+x))^2+9)+((-4*x-8)*exp(exp( 
1))*log(2/5+1/5*x)+(-8*x-16)*exp(exp(1)))*exp(x*exp(exp(1))/(-1+x))^2)/((x 
^3-3*x+2)*exp(x*exp(exp(1))/(-1+x))^2-3*x^3+9*x-6),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (26) = 52\).

Time = 2.18 (sec) , antiderivative size = 178, normalized size of antiderivative = 5.74 \[ \int \frac {\left (-3+6 x-3 x^2+e^{\frac {2 e^e x}{-1+x}} \left (1-2 x+x^2\right )\right ) \log \left (9-6 e^{\frac {2 e^e x}{-1+x}}+e^{\frac {4 e^e x}{-1+x}}\right )+e^{\frac {2 e^e x}{-1+x}} \left (e^e (-16-8 x)+e^e (-8-4 x) \log \left (\frac {2+x}{5}\right )\right )}{-6+9 x-3 x^3+e^{\frac {2 e^e x}{-1+x}} \left (2-3 x+x^3\right )} \, dx=\frac {x \log \left (x + 2\right ) \log \left (e^{\left (\frac {4 \, x e^{e}}{x - 1}\right )} - 6 \, e^{\left (\frac {2 \, x e^{e}}{x - 1}\right )} + 9\right ) - 2 \, x \log \left (5\right ) \log \left (e^{\left (\frac {2 \, x e^{e}}{x - 1}\right )} - 3\right ) + 4 \, e^{e} \log \left (5\right ) - 4 \, e^{e} \log \left (x + 2\right ) + 4 \, e^{e} \log \left (\frac {1}{5} \, x + \frac {2}{5}\right ) - \log \left (x + 2\right ) \log \left (e^{\left (\frac {4 \, x e^{e}}{x - 1}\right )} - 6 \, e^{\left (\frac {2 \, x e^{e}}{x - 1}\right )} + 9\right ) + 4 \, x \log \left (e^{\left (\frac {2 \, x e^{e}}{x - 1}\right )} - 3\right ) + 2 \, \log \left (5\right ) \log \left (e^{\left (\frac {2 \, x e^{e}}{x - 1}\right )} - 3\right ) - 4 \, \log \left (e^{\left (\frac {2 \, x e^{e}}{x - 1}\right )} - 3\right )}{x - 1} \] Input:

integrate((((x^2-2*x+1)*exp(x*exp(exp(1))/(-1+x))^2-3*x^2+6*x-3)*log(exp(x 
*exp(exp(1))/(-1+x))^4-6*exp(x*exp(exp(1))/(-1+x))^2+9)+((-4*x-8)*exp(exp( 
1))*log(2/5+1/5*x)+(-8*x-16)*exp(exp(1)))*exp(x*exp(exp(1))/(-1+x))^2)/((x 
^3-3*x+2)*exp(x*exp(exp(1))/(-1+x))^2-3*x^3+9*x-6),x, algorithm="giac")
 

Output:

(x*log(x + 2)*log(e^(4*x*e^e/(x - 1)) - 6*e^(2*x*e^e/(x - 1)) + 9) - 2*x*l 
og(5)*log(e^(2*x*e^e/(x - 1)) - 3) + 4*e^e*log(5) - 4*e^e*log(x + 2) + 4*e 
^e*log(1/5*x + 2/5) - log(x + 2)*log(e^(4*x*e^e/(x - 1)) - 6*e^(2*x*e^e/(x 
 - 1)) + 9) + 4*x*log(e^(2*x*e^e/(x - 1)) - 3) + 2*log(5)*log(e^(2*x*e^e/( 
x - 1)) - 3) - 4*log(e^(2*x*e^e/(x - 1)) - 3))/(x - 1)
 

Mupad [B] (verification not implemented)

Time = 3.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {\left (-3+6 x-3 x^2+e^{\frac {2 e^e x}{-1+x}} \left (1-2 x+x^2\right )\right ) \log \left (9-6 e^{\frac {2 e^e x}{-1+x}}+e^{\frac {4 e^e x}{-1+x}}\right )+e^{\frac {2 e^e x}{-1+x}} \left (e^e (-16-8 x)+e^e (-8-4 x) \log \left (\frac {2+x}{5}\right )\right )}{-6+9 x-3 x^3+e^{\frac {2 e^e x}{-1+x}} \left (2-3 x+x^3\right )} \, dx=4\,\ln \left ({\mathrm {e}}^{\frac {2\,x\,{\mathrm {e}}^{\mathrm {e}}}{x-1}}-3\right )+\ln \left (\frac {x}{5}+\frac {2}{5}\right )\,\ln \left ({\mathrm {e}}^{\frac {4\,x\,{\mathrm {e}}^{\mathrm {e}}}{x-1}}-6\,{\mathrm {e}}^{\frac {2\,x\,{\mathrm {e}}^{\mathrm {e}}}{x-1}}+9\right ) \] Input:

int((log(exp((4*x*exp(exp(1)))/(x - 1)) - 6*exp((2*x*exp(exp(1)))/(x - 1)) 
 + 9)*(6*x + exp((2*x*exp(exp(1)))/(x - 1))*(x^2 - 2*x + 1) - 3*x^2 - 3) - 
 exp((2*x*exp(exp(1)))/(x - 1))*(exp(exp(1))*(8*x + 16) + exp(exp(1))*log( 
x/5 + 2/5)*(4*x + 8)))/(9*x + exp((2*x*exp(exp(1)))/(x - 1))*(x^3 - 3*x + 
2) - 3*x^3 - 6),x)
 

Output:

4*log(exp((2*x*exp(exp(1)))/(x - 1)) - 3) + log(x/5 + 2/5)*log(exp((4*x*ex 
p(exp(1)))/(x - 1)) - 6*exp((2*x*exp(exp(1)))/(x - 1)) + 9)
 

Reduce [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.35 \[ \int \frac {\left (-3+6 x-3 x^2+e^{\frac {2 e^e x}{-1+x}} \left (1-2 x+x^2\right )\right ) \log \left (9-6 e^{\frac {2 e^e x}{-1+x}}+e^{\frac {4 e^e x}{-1+x}}\right )+e^{\frac {2 e^e x}{-1+x}} \left (e^e (-16-8 x)+e^e (-8-4 x) \log \left (\frac {2+x}{5}\right )\right )}{-6+9 x-3 x^3+e^{\frac {2 e^e x}{-1+x}} \left (2-3 x+x^3\right )} \, dx=4 \,\mathrm {log}\left (e^{\frac {2 e^{e} x -e^{e}}{x -1}}-e^{e^{e}} \sqrt {3}\right )+4 \,\mathrm {log}\left (e^{\frac {2 e^{e} x -e^{e}}{x -1}}+e^{e^{e}} \sqrt {3}\right )+\mathrm {log}\left (e^{\frac {4 e^{e} x}{x -1}}-6 e^{\frac {2 e^{e} x}{x -1}}+9\right ) \mathrm {log}\left (\frac {x}{5}+\frac {2}{5}\right ) \] Input:

int((((x^2-2*x+1)*exp(x*exp(exp(1))/(-1+x))^2-3*x^2+6*x-3)*log(exp(x*exp(e 
xp(1))/(-1+x))^4-6*exp(x*exp(exp(1))/(-1+x))^2+9)+((-4*x-8)*exp(exp(1))*lo 
g(2/5+1/5*x)+(-8*x-16)*exp(exp(1)))*exp(x*exp(exp(1))/(-1+x))^2)/((x^3-3*x 
+2)*exp(x*exp(exp(1))/(-1+x))^2-3*x^3+9*x-6),x)
 

Output:

4*log(e**((2*e**e*x - e**e)/(x - 1)) - e**(e**e)*sqrt(3)) + 4*log(e**((2*e 
**e*x - e**e)/(x - 1)) + e**(e**e)*sqrt(3)) + log(e**((4*e**e*x)/(x - 1)) 
- 6*e**((2*e**e*x)/(x - 1)) + 9)*log((x + 2)/5)