\(\int \frac {-x^2-4 x^3-4 x^4+e^2 (-1-4 x-4 x^2)+e^{\frac {2-x}{-x-2 x^2+e (1+2 x)}} (-4+10 e-16 x+4 x^2)+e^{\frac {2 (2-x)}{-x-2 x^2+e (1+2 x)}} (-4+10 e-16 x+4 x^2)+e (2 x+8 x^2+8 x^3)}{x^2+4 x^3+4 x^4+e^2 (1+4 x+4 x^2)+e (-2 x-8 x^2-8 x^3)} \, dx\) [82]

Optimal result

Integrand size = 174, antiderivative size = 31 \[ \int \frac {-x^2-4 x^3-4 x^4+e^2 \left (-1-4 x-4 x^2\right )+e^{\frac {2-x}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e^{\frac {2 (2-x)}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e \left (2 x+8 x^2+8 x^3\right )}{x^2+4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (-2 x-8 x^2-8 x^3\right )} \, dx=3-\left (1+e^{\frac {-2+x}{(-e+x) (1+2 x)}}\right )^2-x \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61 \[ \int \frac {-x^2-4 x^3-4 x^4+e^2 \left (-1-4 x-4 x^2\right )+e^{\frac {2-x}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e^{\frac {2 (2-x)}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e \left (2 x+8 x^2+8 x^3\right )}{x^2+4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (-2 x-8 x^2-8 x^3\right )} \, dx=-e^{-\frac {2 (-2+x)}{(e-x) (1+2 x)}}-2 e^{-\frac {-2+x}{(e-x) (1+2 x)}}-x \] Input:

Integrate[(-x^2 - 4*x^3 - 4*x^4 + E^2*(-1 - 4*x - 4*x^2) + E^((2 - x)/(-x 
- 2*x^2 + E*(1 + 2*x)))*(-4 + 10*E - 16*x + 4*x^2) + E^((2*(2 - x))/(-x - 
2*x^2 + E*(1 + 2*x)))*(-4 + 10*E - 16*x + 4*x^2) + E*(2*x + 8*x^2 + 8*x^3) 
)/(x^2 + 4*x^3 + 4*x^4 + E^2*(1 + 4*x + 4*x^2) + E*(-2*x - 8*x^2 - 8*x^3)) 
,x]
 

Output:

-E^((-2*(-2 + x))/((E - x)*(1 + 2*x))) - 2/E^((-2 + x)/((E - x)*(1 + 2*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.

\(\displaystyle \int \frac {-4 x^4-4 x^3-x^2+e^2 \left (-4 x^2-4 x-1\right )+e^{\frac {2-x}{-2 x^2-x+e (2 x+1)}} \left (4 x^2-16 x+10 e-4\right )+e^{\frac {2 (2-x)}{-2 x^2-x+e (2 x+1)}} \left (4 x^2-16 x+10 e-4\right )+e \left (8 x^3+8 x^2+2 x\right )}{4 x^4+4 x^3+x^2+e^2 \left (4 x^2+4 x+1\right )+e \left (-8 x^3-8 x^2-2 x\right )} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {4 \left (-4 x^4-4 x^3-x^2+e^2 \left (-4 x^2-4 x-1\right )+e^{\frac {2-x}{-2 x^2-x+e (2 x+1)}} \left (4 x^2-16 x+10 e-4\right )+e^{\frac {2 (2-x)}{-2 x^2-x+e (2 x+1)}} \left (4 x^2-16 x+10 e-4\right )+e \left (8 x^3+8 x^2+2 x\right )\right )}{(1+2 e)^3 (e-x)}+\frac {8 \left (-4 x^4-4 x^3-x^2+e^2 \left (-4 x^2-4 x-1\right )+e^{\frac {2-x}{-2 x^2-x+e (2 x+1)}} \left (4 x^2-16 x+10 e-4\right )+e^{\frac {2 (2-x)}{-2 x^2-x+e (2 x+1)}} \left (4 x^2-16 x+10 e-4\right )+e \left (8 x^3+8 x^2+2 x\right )\right )}{(1+2 e)^3 (2 x+1)}+\frac {-4 x^4-4 x^3-x^2+e^2 \left (-4 x^2-4 x-1\right )+e^{\frac {2-x}{-2 x^2-x+e (2 x+1)}} \left (4 x^2-16 x+10 e-4\right )+e^{\frac {2 (2-x)}{-2 x^2-x+e (2 x+1)}} \left (4 x^2-16 x+10 e-4\right )+e \left (8 x^3+8 x^2+2 x\right )}{(1+2 e)^2 (e-x)^2}+\frac {4 \left (-4 x^4-4 x^3-x^2+e^2 \left (-4 x^2-4 x-1\right )+e^{\frac {2-x}{-2 x^2-x+e (2 x+1)}} \left (4 x^2-16 x+10 e-4\right )+e^{\frac {2 (2-x)}{-2 x^2-x+e (2 x+1)}} \left (4 x^2-16 x+10 e-4\right )+e \left (8 x^3+8 x^2+2 x\right )\right )}{(1+2 e)^2 (2 x+1)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {8 \int e^{-\frac {2 (x-2)}{(e-x) (2 x+1)}}dx}{(1+2 e)^2}+\frac {16 (4-e) \int e^{-\frac {2 (x-2)}{(e-x) (2 x+1)}}dx}{(1+2 e)^3}-\frac {72 \int e^{-\frac {2 (x-2)}{(e-x) (2 x+1)}}dx}{(1+2 e)^3}+\frac {8 \int e^{-\frac {x-2}{(e-x) (2 x+1)}}dx}{(1+2 e)^2}+\frac {16 (4-e) \int e^{-\frac {x-2}{(e-x) (2 x+1)}}dx}{(1+2 e)^3}-\frac {72 \int e^{-\frac {x-2}{(e-x) (2 x+1)}}dx}{(1+2 e)^3}-\frac {2 (2-e) \int \frac {e^{-\frac {2 (x-2)}{(e-x) (2 x+1)}}}{(e-x)^2}dx}{1+2 e}-\frac {2 (2-e) \int \frac {e^{-\frac {x-2}{(e-x) (2 x+1)}}}{(e-x)^2}dx}{1+2 e}+\frac {20 \int \frac {e^{-\frac {2 (x-2)}{(e-x) (2 x+1)}}}{(2 x+1)^2}dx}{1+2 e}+\frac {20 \int \frac {e^{-\frac {x-2}{(e-x) (2 x+1)}}}{(2 x+1)^2}dx}{1+2 e}-\frac {4 x^3}{3 (1+2 e)^2}-\frac {2 x^2}{(1+2 e)^2}-\frac {4 (e-x)^4}{(1+2 e)^3}+\frac {4 (e-x)^3}{(1+2 e)^2}+\frac {(2 x+1)^4}{4 (1+2 e)^3}-\frac {(2 x+1)^3}{3 (1+2 e)^2}+\frac {8 e x}{1+2 e}-\frac {16 e^2 x}{(1+2 e)^2}-\frac {8 e x}{(1+2 e)^2}-\frac {x}{(1+2 e)^2}-\frac {4 e^4}{(1+2 e)^2 (e-x)}-\frac {4 e^3}{(1+2 e)^2 (e-x)}-\frac {e^2}{(1+2 e)^2 (e-x)}+\frac {e^2}{e-x}+\frac {2 e (1+6 e) \log (e-x)}{1+2 e}-\frac {4 e^2 \log (e-x)}{1+2 e}-\frac {16 e^3 \log (e-x)}{(1+2 e)^2}-\frac {12 e^2 \log (e-x)}{(1+2 e)^2}-\frac {2 e \log (e-x)}{(1+2 e)^2}\)

Input:

Int[(-x^2 - 4*x^3 - 4*x^4 + E^2*(-1 - 4*x - 4*x^2) + E^((2 - x)/(-x - 2*x^ 
2 + E*(1 + 2*x)))*(-4 + 10*E - 16*x + 4*x^2) + E^((2*(2 - x))/(-x - 2*x^2 
+ E*(1 + 2*x)))*(-4 + 10*E - 16*x + 4*x^2) + E*(2*x + 8*x^2 + 8*x^3))/(x^2 
 + 4*x^3 + 4*x^4 + E^2*(1 + 4*x + 4*x^2) + E*(-2*x - 8*x^2 - 8*x^3)),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(240\) vs. \(2(31)=62\).

Time = 0.51 (sec) , antiderivative size = 241, normalized size of antiderivative = 7.77

\[\frac {\left (-\frac {1}{2}-2 \,{\mathrm e}^{2}+{\mathrm e}\right ) x +\left (2-4 \,{\mathrm e}\right ) x \,{\mathrm e}^{\frac {2-x}{\left (1+2 x \right ) {\mathrm e}-2 x^{2}-x}}+\left (-2 \,{\mathrm e}+1\right ) x \,{\mathrm e}^{\frac {4-2 x}{\left (1+2 x \right ) {\mathrm e}-2 x^{2}-x}}+2 x^{3}+4 x^{2} {\mathrm e}^{\frac {2-x}{\left (1+2 x \right ) {\mathrm e}-2 x^{2}-x}}+2 x^{2} {\mathrm e}^{\frac {4-2 x}{\left (1+2 x \right ) {\mathrm e}-2 x^{2}-x}}-2 \,{\mathrm e} \,{\mathrm e}^{\frac {2-x}{\left (1+2 x \right ) {\mathrm e}-2 x^{2}-x}}-{\mathrm e} \,{\mathrm e}^{\frac {4-2 x}{\left (1+2 x \right ) {\mathrm e}-2 x^{2}-x}}-{\mathrm e}^{2}+\frac {{\mathrm e}}{2}}{\left (1+2 x \right ) \left ({\mathrm e}-x \right )}\]

Input:

int(((10*exp(1)+4*x^2-16*x-4)*exp((2-x)/((1+2*x)*exp(1)-2*x^2-x))^2+(10*ex 
p(1)+4*x^2-16*x-4)*exp((2-x)/((1+2*x)*exp(1)-2*x^2-x))+(-4*x^2-4*x-1)*exp( 
1)^2+(8*x^3+8*x^2+2*x)*exp(1)-4*x^4-4*x^3-x^2)/((4*x^2+4*x+1)*exp(1)^2+(-8 
*x^3-8*x^2-2*x)*exp(1)+4*x^4+4*x^3+x^2),x)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \frac {-x^2-4 x^3-4 x^4+e^2 \left (-1-4 x-4 x^2\right )+e^{\frac {2-x}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e^{\frac {2 (2-x)}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e \left (2 x+8 x^2+8 x^3\right )}{x^2+4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (-2 x-8 x^2-8 x^3\right )} \, dx=-x - e^{\left (\frac {2 \, {\left (x - 2\right )}}{2 \, x^{2} - {\left (2 \, x + 1\right )} e + x}\right )} - 2 \, e^{\left (\frac {x - 2}{2 \, x^{2} - {\left (2 \, x + 1\right )} e + x}\right )} \] Input:

integrate(((10*exp(1)+4*x^2-16*x-4)*exp((2-x)/((1+2*x)*exp(1)-2*x^2-x))^2+ 
(10*exp(1)+4*x^2-16*x-4)*exp((2-x)/((1+2*x)*exp(1)-2*x^2-x))+(-4*x^2-4*x-1 
)*exp(1)^2+(8*x^3+8*x^2+2*x)*exp(1)-4*x^4-4*x^3-x^2)/((4*x^2+4*x+1)*exp(1) 
^2+(-8*x^3-8*x^2-2*x)*exp(1)+4*x^4+4*x^3+x^2),x, algorithm="fricas")
 

Output:

-x - e^(2*(x - 2)/(2*x^2 - (2*x + 1)*e + x)) - 2*e^((x - 2)/(2*x^2 - (2*x 
+ 1)*e + x))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (20) = 40\).

Time = 0.49 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {-x^2-4 x^3-4 x^4+e^2 \left (-1-4 x-4 x^2\right )+e^{\frac {2-x}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e^{\frac {2 (2-x)}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e \left (2 x+8 x^2+8 x^3\right )}{x^2+4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (-2 x-8 x^2-8 x^3\right )} \, dx=- x - e^{\frac {2 \cdot \left (2 - x\right )}{- 2 x^{2} - x + e \left (2 x + 1\right )}} - 2 e^{\frac {2 - x}{- 2 x^{2} - x + e \left (2 x + 1\right )}} \] Input:

integrate(((10*exp(1)+4*x**2-16*x-4)*exp((2-x)/((1+2*x)*exp(1)-2*x**2-x))* 
*2+(10*exp(1)+4*x**2-16*x-4)*exp((2-x)/((1+2*x)*exp(1)-2*x**2-x))+(-4*x**2 
-4*x-1)*exp(1)**2+(8*x**3+8*x**2+2*x)*exp(1)-4*x**4-4*x**3-x**2)/((4*x**2+ 
4*x+1)*exp(1)**2+(-8*x**3-8*x**2-2*x)*exp(1)+4*x**4+4*x**3+x**2),x)
 

Output:

-x - exp(2*(2 - x)/(-2*x**2 - x + E*(2*x + 1))) - 2*exp((2 - x)/(-2*x**2 - 
 x + E*(2*x + 1)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1252 vs. \(2 (31) = 62\).

Time = 0.17 (sec) , antiderivative size = 1252, normalized size of antiderivative = 40.39 \[ \int \frac {-x^2-4 x^3-4 x^4+e^2 \left (-1-4 x-4 x^2\right )+e^{\frac {2-x}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e^{\frac {2 (2-x)}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e \left (2 x+8 x^2+8 x^3\right )}{x^2+4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (-2 x-8 x^2-8 x^3\right )} \, dx=\text {Too large to display} \] Input:

integrate(((10*exp(1)+4*x^2-16*x-4)*exp((2-x)/((1+2*x)*exp(1)-2*x^2-x))^2+ 
(10*exp(1)+4*x^2-16*x-4)*exp((2-x)/((1+2*x)*exp(1)-2*x^2-x))+(-4*x^2-4*x-1 
)*exp(1)^2+(8*x^3+8*x^2+2*x)*exp(1)-4*x^4-4*x^3-x^2)/((4*x^2+4*x+1)*exp(1) 
^2+(-8*x^3-8*x^2-2*x)*exp(1)+4*x^4+4*x^3+x^2),x, algorithm="maxima")
 

Output:

-4*((2*e - 1)*log(2*x + 1)/(8*e^3 + 12*e^2 + 6*e + 1) - (2*e - 1)*log(x - 
e)/(8*e^3 + 12*e^2 + 6*e + 1) - (x*(2*e - 1) + 2*e)/(2*x^2*(4*e^2 + 4*e + 
1) - x*(8*e^3 + 4*e^2 - 2*e - 1) - 4*e^3 - 4*e^2 - e))*e^2 + 2*(4*e*log(2* 
x + 1)/(8*e^3 + 12*e^2 + 6*e + 1) - 4*e*log(x - e)/(8*e^3 + 12*e^2 + 6*e + 
 1) + (x*(4*e^2 + 1) + 2*e^2 - e)/(2*x^2*(4*e^2 + 4*e + 1) - x*(8*e^3 + 4* 
e^2 - 2*e - 1) - 4*e^3 - 4*e^2 - e))*e^2 + ((4*x - 2*e + 1)/(2*x^2*(4*e^2 
+ 4*e + 1) - x*(8*e^3 + 4*e^2 - 2*e - 1) - 4*e^3 - 4*e^2 - e) - 4*log(2*x 
+ 1)/(8*e^3 + 12*e^2 + 6*e + 1) + 4*log(x - e)/(8*e^3 + 12*e^2 + 6*e + 1)) 
*e^2 + 2*((6*e + 1)*log(2*x + 1)/(8*e^3 + 12*e^2 + 6*e + 1) + 4*(2*e^3 + 3 
*e^2)*log(x - e)/(8*e^3 + 12*e^2 + 6*e + 1) - (x*(8*e^3 - 1) + 4*e^3 + e)/ 
(2*x^2*(4*e^2 + 4*e + 1) - x*(8*e^3 + 4*e^2 - 2*e - 1) - 4*e^3 - 4*e^2 - e 
))*e + 2*((2*e - 1)*log(2*x + 1)/(8*e^3 + 12*e^2 + 6*e + 1) - (2*e - 1)*lo 
g(x - e)/(8*e^3 + 12*e^2 + 6*e + 1) - (x*(2*e - 1) + 2*e)/(2*x^2*(4*e^2 + 
4*e + 1) - x*(8*e^3 + 4*e^2 - 2*e - 1) - 4*e^3 - 4*e^2 - e))*e - 4*(4*e*lo 
g(2*x + 1)/(8*e^3 + 12*e^2 + 6*e + 1) - 4*e*log(x - e)/(8*e^3 + 12*e^2 + 6 
*e + 1) + (x*(4*e^2 + 1) + 2*e^2 - e)/(2*x^2*(4*e^2 + 4*e + 1) - x*(8*e^3 
+ 4*e^2 - 2*e - 1) - 4*e^3 - 4*e^2 - e))*e - (e^(2*e/(x*(2*e + 1) - 2*e^2 
- e) + 10/(2*x*(2*e + 1) + 2*e + 1)) + 2*e^(e/(x*(2*e + 1) - 2*e^2 - e) + 
5/(2*x*(2*e + 1) + 2*e + 1) + 2/(x*(2*e + 1) - 2*e^2 - e)))*e^(-4/(x*(2*e 
+ 1) - 2*e^2 - e)) - x - (6*e + 1)*log(2*x + 1)/(8*e^3 + 12*e^2 + 6*e +...
 

Giac [F]

\[ \int \frac {-x^2-4 x^3-4 x^4+e^2 \left (-1-4 x-4 x^2\right )+e^{\frac {2-x}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e^{\frac {2 (2-x)}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e \left (2 x+8 x^2+8 x^3\right )}{x^2+4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (-2 x-8 x^2-8 x^3\right )} \, dx=\int { -\frac {4 \, x^{4} + 4 \, x^{3} + x^{2} + {\left (4 \, x^{2} + 4 \, x + 1\right )} e^{2} - 2 \, {\left (4 \, x^{3} + 4 \, x^{2} + x\right )} e - 2 \, {\left (2 \, x^{2} - 8 \, x + 5 \, e - 2\right )} e^{\left (\frac {2 \, {\left (x - 2\right )}}{2 \, x^{2} - {\left (2 \, x + 1\right )} e + x}\right )} - 2 \, {\left (2 \, x^{2} - 8 \, x + 5 \, e - 2\right )} e^{\left (\frac {x - 2}{2 \, x^{2} - {\left (2 \, x + 1\right )} e + x}\right )}}{4 \, x^{4} + 4 \, x^{3} + x^{2} + {\left (4 \, x^{2} + 4 \, x + 1\right )} e^{2} - 2 \, {\left (4 \, x^{3} + 4 \, x^{2} + x\right )} e} \,d x } \] Input:

integrate(((10*exp(1)+4*x^2-16*x-4)*exp((2-x)/((1+2*x)*exp(1)-2*x^2-x))^2+ 
(10*exp(1)+4*x^2-16*x-4)*exp((2-x)/((1+2*x)*exp(1)-2*x^2-x))+(-4*x^2-4*x-1 
)*exp(1)^2+(8*x^3+8*x^2+2*x)*exp(1)-4*x^4-4*x^3-x^2)/((4*x^2+4*x+1)*exp(1) 
^2+(-8*x^3-8*x^2-2*x)*exp(1)+4*x^4+4*x^3+x^2),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.00 \[ \int \frac {-x^2-4 x^3-4 x^4+e^2 \left (-1-4 x-4 x^2\right )+e^{\frac {2-x}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e^{\frac {2 (2-x)}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e \left (2 x+8 x^2+8 x^3\right )}{x^2+4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (-2 x-8 x^2-8 x^3\right )} \, dx=-x-2\,{\mathrm {e}}^{-\frac {2}{x-\mathrm {e}-2\,x\,\mathrm {e}+2\,x^2}}\,{\mathrm {e}}^{\frac {x}{x-\mathrm {e}-2\,x\,\mathrm {e}+2\,x^2}}-{\mathrm {e}}^{-\frac {4}{x-\mathrm {e}-2\,x\,\mathrm {e}+2\,x^2}}\,{\mathrm {e}}^{\frac {2\,x}{x-\mathrm {e}-2\,x\,\mathrm {e}+2\,x^2}} \] Input:

int(-(exp(2)*(4*x + 4*x^2 + 1) - exp(1)*(2*x + 8*x^2 + 8*x^3) + exp((x - 2 
)/(x + 2*x^2 - exp(1)*(2*x + 1)))*(16*x - 10*exp(1) - 4*x^2 + 4) + exp((2* 
(x - 2))/(x + 2*x^2 - exp(1)*(2*x + 1)))*(16*x - 10*exp(1) - 4*x^2 + 4) + 
x^2 + 4*x^3 + 4*x^4)/(exp(2)*(4*x + 4*x^2 + 1) - exp(1)*(2*x + 8*x^2 + 8*x 
^3) + x^2 + 4*x^3 + 4*x^4),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.06 \[ \int \frac {-x^2-4 x^3-4 x^4+e^2 \left (-1-4 x-4 x^2\right )+e^{\frac {2-x}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e^{\frac {2 (2-x)}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e \left (2 x+8 x^2+8 x^3\right )}{x^2+4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (-2 x-8 x^2-8 x^3\right )} \, dx=\frac {-e^{\frac {2 x}{2 e x -2 x^{2}+e -x}} x -2 e^{\frac {x +2}{2 e x -2 x^{2}+e -x}}-e^{\frac {4}{2 e x -2 x^{2}+e -x}}}{e^{\frac {2 x}{2 e x -2 x^{2}+e -x}}} \] Input:

int(((10*exp(1)+4*x^2-16*x-4)*exp((2-x)/((1+2*x)*exp(1)-2*x^2-x))^2+(10*ex 
p(1)+4*x^2-16*x-4)*exp((2-x)/((1+2*x)*exp(1)-2*x^2-x))+(-4*x^2-4*x-1)*exp( 
1)^2+(8*x^3+8*x^2+2*x)*exp(1)-4*x^4-4*x^3-x^2)/((4*x^2+4*x+1)*exp(1)^2+(-8 
*x^3-8*x^2-2*x)*exp(1)+4*x^4+4*x^3+x^2),x)
 

Output:

( - e**((2*x)/(2*e*x + e - 2*x**2 - x))*x - 2*e**((x + 2)/(2*e*x + e - 2*x 
**2 - x)) - e**(4/(2*e*x + e - 2*x**2 - x)))/e**((2*x)/(2*e*x + e - 2*x**2 
 - x))