\(\int \frac {e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}+e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}} x^2} (2 x-4 x^2+2 x^3+e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}} (3 x^4-6 x^5+3 x^6+e^6 (-8 x^3+4 x^4)))}{1-2 x+x^2} \, dx\) [339]

Optimal result

Integrand size = 141, antiderivative size = 30 \[ \int \frac {e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}+e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}} x^2} \left (2 x-4 x^2+2 x^3+e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}} \left (3 x^4-6 x^5+3 x^6+e^6 \left (-8 x^3+4 x^4\right )\right )\right )}{1-2 x+x^2} \, dx=2+e^{e^{e^{x^2 \left (-\frac {4 e^6}{1-x}+x\right )}} x^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}+e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}} x^2} \left (2 x-4 x^2+2 x^3+e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}} \left (3 x^4-6 x^5+3 x^6+e^6 \left (-8 x^3+4 x^4\right )\right )\right )}{1-2 x+x^2} \, dx=e^{e^{e^{x^2 \left (\frac {4 e^6}{-1+x}+x\right )}} x^2} \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 7277

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 107.97 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93

method	result	size
risch	\({\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{\frac {x^{2} \left (x^{2}+4 \,{\mathrm e}^{6}-x \right )}{-1+x}}}}\)	\(28\)
parallelrisch	\({\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{\frac {x^{2} \left (x^{2}+4 \,{\mathrm e}^{6}-x \right )}{-1+x}}}}\)	\(30\)

Input:

int((((4*x^4-8*x^3)*exp(3)^2+3*x^6-6*x^5+3*x^4)*exp((4*x^2*exp(3)^2+x^4-x^ 
3)/(-1+x))+2*x^3-4*x^2+2*x)*exp(exp((4*x^2*exp(3)^2+x^4-x^3)/(-1+x)))*exp( 
x^2*exp(exp((4*x^2*exp(3)^2+x^4-x^3)/(-1+x))))/(x^2-2*x+1),x,method=_RETUR 
NVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}+e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}} x^2} \left (2 x-4 x^2+2 x^3+e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}} \left (3 x^4-6 x^5+3 x^6+e^6 \left (-8 x^3+4 x^4\right )\right )\right )}{1-2 x+x^2} \, dx=e^{\left (x^{2} e^{\left (e^{\left (\frac {x^{4} - x^{3} + 4 \, x^{2} e^{6}}{x - 1}\right )}\right )}\right )} \] Input:

integrate((((4*x^4-8*x^3)*exp(3)^2+3*x^6-6*x^5+3*x^4)*exp((4*x^2*exp(3)^2+ 
x^4-x^3)/(-1+x))+2*x^3-4*x^2+2*x)*exp(exp((4*x^2*exp(3)^2+x^4-x^3)/(-1+x)) 
)*exp(x^2*exp(exp((4*x^2*exp(3)^2+x^4-x^3)/(-1+x))))/(x^2-2*x+1),x, algori 
thm="fricas")
 

Output:

e^(x^2*e^(e^((x^4 - x^3 + 4*x^2*e^6)/(x - 1))))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}+e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}} x^2} \left (2 x-4 x^2+2 x^3+e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}} \left (3 x^4-6 x^5+3 x^6+e^6 \left (-8 x^3+4 x^4\right )\right )\right )}{1-2 x+x^2} \, dx=\text {Timed out} \] Input:

integrate((((4*x**4-8*x**3)*exp(3)**2+3*x**6-6*x**5+3*x**4)*exp((4*x**2*ex 
p(3)**2+x**4-x**3)/(-1+x))+2*x**3-4*x**2+2*x)*exp(exp((4*x**2*exp(3)**2+x* 
*4-x**3)/(-1+x)))*exp(x**2*exp(exp((4*x**2*exp(3)**2+x**4-x**3)/(-1+x))))/ 
(x**2-2*x+1),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}+e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}} x^2} \left (2 x-4 x^2+2 x^3+e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}} \left (3 x^4-6 x^5+3 x^6+e^6 \left (-8 x^3+4 x^4\right )\right )\right )}{1-2 x+x^2} \, dx=e^{\left (x^{2} e^{\left (e^{\left (x^{3} + 4 \, x e^{6} + \frac {4 \, e^{6}}{x - 1} + 4 \, e^{6}\right )}\right )}\right )} \] Input:

integrate((((4*x^4-8*x^3)*exp(3)^2+3*x^6-6*x^5+3*x^4)*exp((4*x^2*exp(3)^2+ 
x^4-x^3)/(-1+x))+2*x^3-4*x^2+2*x)*exp(exp((4*x^2*exp(3)^2+x^4-x^3)/(-1+x)) 
)*exp(x^2*exp(exp((4*x^2*exp(3)^2+x^4-x^3)/(-1+x))))/(x^2-2*x+1),x, algori 
thm="maxima")
 

Output:

e^(x^2*e^(e^(x^3 + 4*x*e^6 + 4*e^6/(x - 1) + 4*e^6)))
 

Giac [F]

\[ \int \frac {e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}+e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}} x^2} \left (2 x-4 x^2+2 x^3+e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}} \left (3 x^4-6 x^5+3 x^6+e^6 \left (-8 x^3+4 x^4\right )\right )\right )}{1-2 x+x^2} \, dx=\int { \frac {{\left (2 \, x^{3} - 4 \, x^{2} + {\left (3 \, x^{6} - 6 \, x^{5} + 3 \, x^{4} + 4 \, {\left (x^{4} - 2 \, x^{3}\right )} e^{6}\right )} e^{\left (\frac {x^{4} - x^{3} + 4 \, x^{2} e^{6}}{x - 1}\right )} + 2 \, x\right )} e^{\left (x^{2} e^{\left (e^{\left (\frac {x^{4} - x^{3} + 4 \, x^{2} e^{6}}{x - 1}\right )}\right )} + e^{\left (\frac {x^{4} - x^{3} + 4 \, x^{2} e^{6}}{x - 1}\right )}\right )}}{x^{2} - 2 \, x + 1} \,d x } \] Input:

integrate((((4*x^4-8*x^3)*exp(3)^2+3*x^6-6*x^5+3*x^4)*exp((4*x^2*exp(3)^2+ 
x^4-x^3)/(-1+x))+2*x^3-4*x^2+2*x)*exp(exp((4*x^2*exp(3)^2+x^4-x^3)/(-1+x)) 
)*exp(x^2*exp(exp((4*x^2*exp(3)^2+x^4-x^3)/(-1+x))))/(x^2-2*x+1),x, algori 
thm="giac")
 

Output:

integrate((2*x^3 - 4*x^2 + (3*x^6 - 6*x^5 + 3*x^4 + 4*(x^4 - 2*x^3)*e^6)*e 
^((x^4 - x^3 + 4*x^2*e^6)/(x - 1)) + 2*x)*e^(x^2*e^(e^((x^4 - x^3 + 4*x^2* 
e^6)/(x - 1))) + e^((x^4 - x^3 + 4*x^2*e^6)/(x - 1)))/(x^2 - 2*x + 1), x)
 

Mupad [B] (verification not implemented)

Time = 2.82 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}+e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}} x^2} \left (2 x-4 x^2+2 x^3+e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}} \left (3 x^4-6 x^5+3 x^6+e^6 \left (-8 x^3+4 x^4\right )\right )\right )}{1-2 x+x^2} \, dx={\mathrm {e}}^{x^2\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {4\,x^2\,{\mathrm {e}}^6}{x-1}}\,{\mathrm {e}}^{\frac {x^4}{x-1}}\,{\mathrm {e}}^{-\frac {x^3}{x-1}}}} \] Input:

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

Output:

exp(x^2*exp(exp((4*x^2*exp(6))/(x - 1))*exp(x^4/(x - 1))*exp(-x^3/(x - 1)) 
))
 

Reduce [B] (verification not implemented)

Time = 1.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}+e^{e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}}} x^2} \left (2 x-4 x^2+2 x^3+e^{\frac {4 e^6 x^2-x^3+x^4}{-1+x}} \left (3 x^4-6 x^5+3 x^6+e^6 \left (-8 x^3+4 x^4\right )\right )\right )}{1-2 x+x^2} \, dx=e^{e^{e^{\frac {4 e^{6} x^{2}+x^{4}-x^{3}}{x -1}}} x^{2}} \] Input:

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

Output:

e**(e**(e**((4*e**6*x**2 + x**4 - x**3)/(x - 1)))*x**2)