\(\int \frac {-2 e^{x+2 e^{-x} (-2 x+2 x^2)} x-50 e^x x^3+e^{e^{-x} (-2 x+2 x^2)} (-4 x+12 x^2-4 x^3+e^x (-2+20 x^2))}{e^{x+2 e^{-x} (-2 x+2 x^2)} x^4+e^{x+e^{-x} (-2 x+2 x^2)} (4 x^3-10 x^5)+e^x (4 x^2-20 x^4+25 x^6)} \, dx\) [1112]

Optimal result

Integrand size = 154, antiderivative size = 30 \[ \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=\frac {1}{\frac {2}{-5+\frac {e^{e^{-x} x (-2+2 x)}}{x}}+x^2} \] Output:

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

Mathematica [F]

\[ \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=\int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx \] Input:

Integrate[(-2*E^(x + (2*(-2*x + 2*x^2))/E^x)*x - 50*E^x*x^3 + E^((-2*x + 2 
*x^2)/E^x)*(-4*x + 12*x^2 - 4*x^3 + E^x*(-2 + 20*x^2)))/(E^(x + (2*(-2*x + 
 2*x^2))/E^x)*x^4 + E^(x + (-2*x + 2*x^2)/E^x)*(4*x^3 - 10*x^5) + E^x*(4*x 
^2 - 20*x^4 + 25*x^6)),x]
 

Output:

Integrate[(-2*E^(x + (2*(-2*x + 2*x^2))/E^x)*x - 50*E^x*x^3 + E^((-2*x + 2 
*x^2)/E^x)*(-4*x + 12*x^2 - 4*x^3 + E^x*(-2 + 20*x^2)))/(E^(x + (2*(-2*x + 
 2*x^2))/E^x)*x^4 + E^(x + (-2*x + 2*x^2)/E^x)*(4*x^3 - 10*x^5) + E^x*(4*x 
^2 - 20*x^4 + 25*x^6)), 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.

\(\displaystyle \int \frac {-50 e^x x^3-2 e^{2 e^{-x} \left (2 x^2-2 x\right )+x} x+e^{e^{-x} \left (2 x^2-2 x\right )} \left (-4 x^3+12 x^2+e^x \left (20 x^2-2\right )-4 x\right )}{e^{2 e^{-x} \left (2 x^2-2 x\right )+x} x^4+e^x \left (25 x^6-20 x^4+4 x^2\right )+e^{e^{-x} \left (2 x^2-2 x\right )+x} \left (4 x^3-10 x^5\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{-e^{-x} \left (e^x-4\right ) x} \left (-50 e^x x^3-2 e^{2 e^{-x} \left (2 x^2-2 x\right )+x} x+e^{e^{-x} \left (2 x^2-2 x\right )} \left (-4 x^3+12 x^2+e^x \left (20 x^2-2\right )-4 x\right )\right )}{x^2 \left (-5 e^{2 e^{-x} x} x^2+e^{2 e^{-x} x^2} x+2 e^{2 e^{-x} x}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {2}{x^3}+\frac {2 e^{-2 e^{-x} x-e^{-x} \left (e^x-4\right ) x} \left (2 x^3-6 x^2+2 x-3 e^x\right )}{x^3 \left (5 e^{2 e^{-x} x} x^2-e^{2 e^{-x} x^2} x-2 e^{2 e^{-x} x}\right )}-\frac {2 e^{-e^{-x} \left (e^x-4\right ) x} \left (10 x^5-30 x^4+6 x^3+5 e^x x^2+12 x^2-4 x+2 e^x\right )}{x^3 \left (5 e^{2 e^{-x} x} x^2-e^{2 e^{-x} x^2} x-2 e^{2 e^{-x} x}\right )^2}\right )dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \left (-\frac {2}{x^3}+\frac {2 e^{-e^{-x} \left (e^x-2\right ) x} \left (2 x^3-6 x^2+2 x-3 e^x\right )}{x^3 \left (5 e^{2 e^{-x} x} x^2-e^{2 e^{-x} x^2} x-2 e^{2 e^{-x} x}\right )}-\frac {2 e^{-e^{-x} \left (e^x-4\right ) x} \left (10 x^5-30 x^4+6 x^3+5 e^x x^2+12 x^2-4 x+2 e^x\right )}{x^3 \left (5 e^{2 e^{-x} x} x^2-e^{2 e^{-x} x^2} x-2 e^{2 e^{-x} x}\right )^2}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {2}{x^3}-\frac {2 e^{-x} \left (-2 x^3+6 x^2-2 x+3 e^x\right )}{x^3 \left (5 x^2-e^{2 e^{-x} (x-1) x} x-2\right )}+\frac {2 e^{-e^{-x} \left (e^x-4\right ) x} \left (-10 x^5+30 x^4-6 x^3-5 e^x x^2-12 x^2+4 x-2 e^x\right )}{x^3 \left (-5 e^{2 e^{-x} x} x^2+e^{2 e^{-x} x^2} x+2 e^{2 e^{-x} x}\right )^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (-\frac {2}{x^3}-\frac {2 e^{-x} \left (-2 x^3+6 x^2-2 x+3 e^x\right )}{x^3 \left (5 x^2-e^{2 e^{-x} (x-1) x} x-2\right )}+\frac {2 e^{-e^{-x} \left (e^x-4\right ) x} \left (-10 x^5+30 x^4-6 x^3-5 e^x x^2-12 x^2+4 x-2 e^x\right )}{x^3 \left (-5 e^{2 e^{-x} x} x^2+e^{2 e^{-x} x^2} x+2 e^{2 e^{-x} x}\right )^2}\right )dx\)

Input:

Int[(-2*E^(x + (2*(-2*x + 2*x^2))/E^x)*x - 50*E^x*x^3 + E^((-2*x + 2*x^2)/ 
E^x)*(-4*x + 12*x^2 - 4*x^3 + E^x*(-2 + 20*x^2)))/(E^(x + (2*(-2*x + 2*x^2 
))/E^x)*x^4 + E^(x + (-2*x + 2*x^2)/E^x)*(4*x^3 - 10*x^5) + E^x*(4*x^2 - 2 
0*x^4 + 25*x^6)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10

method	result	size
risch	\(\frac {1}{x^{2}}+\frac {2}{x^{2} \left (5 x^{2}-{\mathrm e}^{2 x \left (-1+x \right ) {\mathrm e}^{-x}} x -2\right )}\)	\(33\)
parallelrisch	\(\frac {5 x -{\mathrm e}^{\left (2 x^{2}-2 x \right ) {\mathrm e}^{-x}}}{x \left (5 x^{2}-x \,{\mathrm e}^{\left (2 x^{2}-2 x \right ) {\mathrm e}^{-x}}-2\right )}\)	\(53\)

Input:

int((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+12*x^2 
-4*x)*exp((2*x^2-2*x)/exp(x))-50*exp(x)*x^3)/(x^4*exp(x)*exp((2*x^2-2*x)/e 
xp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20*x^4+4*x 
^2)*exp(x)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (27) = 54\).

Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27 \[ \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=-\frac {5 \, x e^{x} - e^{\left ({\left (2 \, x^{2} + x e^{x} - 2 \, x\right )} e^{\left (-x\right )}\right )}}{x^{2} e^{\left ({\left (2 \, x^{2} + x e^{x} - 2 \, x\right )} e^{\left (-x\right )}\right )} - {\left (5 \, x^{3} - 2 \, x\right )} e^{x}} \] Input:

integrate((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+ 
12*x^2-4*x)*exp((2*x^2-2*x)/exp(x))-50*exp(x)*x^3)/(x^4*exp(x)*exp((2*x^2- 
2*x)/exp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20*x 
^4+4*x^2)*exp(x)),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=- \frac {2}{- 5 x^{4} + x^{3} e^{\left (2 x^{2} - 2 x\right ) e^{- x}} + 2 x^{2}} + \frac {1}{x^{2}} \] Input:

integrate((-2*x*exp(x)*exp((2*x**2-2*x)/exp(x))**2+((20*x**2-2)*exp(x)-4*x 
**3+12*x**2-4*x)*exp((2*x**2-2*x)/exp(x))-50*exp(x)*x**3)/(x**4*exp(x)*exp 
((2*x**2-2*x)/exp(x))**2+(-10*x**5+4*x**3)*exp(x)*exp((2*x**2-2*x)/exp(x)) 
+(25*x**6-20*x**4+4*x**2)*exp(x)),x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).

Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \[ \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=-\frac {5 \, x e^{\left (2 \, x e^{\left (-x\right )}\right )} - e^{\left (2 \, x^{2} e^{\left (-x\right )}\right )}}{x^{2} e^{\left (2 \, x^{2} e^{\left (-x\right )}\right )} - {\left (5 \, x^{3} - 2 \, x\right )} e^{\left (2 \, x e^{\left (-x\right )}\right )}} \] Input:

integrate((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+ 
12*x^2-4*x)*exp((2*x^2-2*x)/exp(x))-50*exp(x)*x^3)/(x^4*exp(x)*exp((2*x^2- 
2*x)/exp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20*x 
^4+4*x^2)*exp(x)),x, algorithm="maxima")
 

Output:

-(5*x*e^(2*x*e^(-x)) - e^(2*x^2*e^(-x)))/(x^2*e^(2*x^2*e^(-x)) - (5*x^3 - 
2*x)*e^(2*x*e^(-x)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4292 vs. \(2 (27) = 54\).

Time = 0.29 (sec) , antiderivative size = 4292, normalized size of antiderivative = 143.07 \[ \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=\text {Too large to display} \] Input:

integrate((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+ 
12*x^2-4*x)*exp((2*x^2-2*x)/exp(x))-50*exp(x)*x^3)/(x^4*exp(x)*exp((2*x^2- 
2*x)/exp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20*x 
^4+4*x^2)*exp(x)),x, algorithm="giac")
 

Output:

(12500*x^15*e^(4*x*e^(-x) + 5/2*x) - 7500*x^14*e^(2*x^2*e^(-x) + 2*x*e^(-x 
) + 5/2*x) - 75000*x^14*e^(4*x*e^(-x) + 5/2*x) + 1500*x^13*e^(4*x^2*e^(-x) 
 + 5/2*x) + 45000*x^13*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) + 117500*x^13 
*e^(4*x*e^(-x) + 5/2*x) - 100*x^12*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 5/2*x) - 
 9000*x^12*e^(4*x^2*e^(-x) + 5/2*x) - 72500*x^12*e^(2*x^2*e^(-x) + 2*x*e^( 
-x) + 5/2*x) + 12500*x^12*e^(4*x*e^(-x) + 7/2*x) + 45000*x^12*e^(4*x*e^(-x 
) + 5/2*x) + 600*x^11*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 5/2*x) + 14900*x^11*e 
^(4*x^2*e^(-x) + 5/2*x) - 7500*x^11*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 7/2*x) 
- 15000*x^11*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) - 37500*x^11*e^(4*x*e^( 
-x) + 7/2*x) - 195500*x^11*e^(4*x*e^(-x) + 5/2*x) - 1020*x^10*e^(6*x^2*e^( 
-x) - 2*x*e^(-x) + 5/2*x) + 1500*x^10*e^(4*x^2*e^(-x) + 7/2*x) + 600*x^10* 
e^(4*x^2*e^(-x) + 5/2*x) + 22500*x^10*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 7/2*x 
) + 97700*x^10*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) + 2500*x^10*e^(4*x*e^ 
(-x) + 7/2*x) + 48000*x^10*e^(4*x*e^(-x) + 5/2*x) - 100*x^9*e^(6*x^2*e^(-x 
) - 2*x*e^(-x) + 7/2*x) + 120*x^9*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 5/2*x) - 
4500*x^9*e^(4*x^2*e^(-x) + 7/2*x) - 15540*x^9*e^(4*x^2*e^(-x) + 5/2*x) - 3 
500*x^9*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 7/2*x) - 31200*x^9*e^(2*x^2*e^(-x) 
+ 2*x*e^(-x) + 5/2*x) + 3125*x^9*e^(4*x*e^(-x) + 9/2*x) + 30000*x^9*e^(4*x 
*e^(-x) + 7/2*x) + 108800*x^9*e^(4*x*e^(-x) + 5/2*x) + 300*x^8*e^(6*x^2*e^ 
(-x) - 2*x*e^(-x) + 7/2*x) + 764*x^8*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 5/2...
 

Mupad [B] (verification not implemented)

Time = 7.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=\frac {1}{x^2}-\frac {2}{2\,x^2-5\,x^4+x^3\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^{-x}}} \] Input:

int(-(50*x^3*exp(x) + exp(-exp(-x)*(2*x - 2*x^2))*(4*x - exp(x)*(20*x^2 - 
2) - 12*x^2 + 4*x^3) + 2*x*exp(-2*exp(-x)*(2*x - 2*x^2))*exp(x))/(exp(x)*( 
4*x^2 - 20*x^4 + 25*x^6) + exp(-exp(-x)*(2*x - 2*x^2))*exp(x)*(4*x^3 - 10* 
x^5) + x^4*exp(-2*exp(-x)*(2*x - 2*x^2))*exp(x)),x)
 

Output:

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

Reduce [F]

\[ \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=-4 \left (\int \frac {e^{\frac {2 x^{2}+2 x}{e^{x}}}}{e^{\frac {e^{x} x +4 x^{2}}{e^{x}}} x^{3}-10 e^{\frac {e^{x} x +2 x^{2}+2 x}{e^{x}}} x^{4}+4 e^{\frac {e^{x} x +2 x^{2}+2 x}{e^{x}}} x^{2}+25 e^{\frac {e^{x} x +4 x}{e^{x}}} x^{5}-20 e^{\frac {e^{x} x +4 x}{e^{x}}} x^{3}+4 e^{\frac {e^{x} x +4 x}{e^{x}}} x}d x \right )+12 \left (\int \frac {e^{\frac {2 x^{2}+2 x}{e^{x}}}}{e^{\frac {e^{x} x +4 x^{2}}{e^{x}}} x^{2}-10 e^{\frac {e^{x} x +2 x^{2}+2 x}{e^{x}}} x^{3}+4 e^{\frac {e^{x} x +2 x^{2}+2 x}{e^{x}}} x +25 e^{\frac {e^{x} x +4 x}{e^{x}}} x^{4}-20 e^{\frac {e^{x} x +4 x}{e^{x}}} x^{2}+4 e^{\frac {e^{x} x +4 x}{e^{x}}}}d x \right )-2 \left (\int \frac {e^{\frac {2 x^{2}+2 x}{e^{x}}}}{e^{\frac {4 x^{2}}{e^{x}}} x^{4}-10 e^{\frac {2 x^{2}+2 x}{e^{x}}} x^{5}+4 e^{\frac {2 x^{2}+2 x}{e^{x}}} x^{3}+25 e^{\frac {4 x}{e^{x}}} x^{6}-20 e^{\frac {4 x}{e^{x}}} x^{4}+4 e^{\frac {4 x}{e^{x}}} x^{2}}d x \right )+20 \left (\int \frac {e^{\frac {2 x^{2}+2 x}{e^{x}}}}{e^{\frac {4 x^{2}}{e^{x}}} x^{2}-10 e^{\frac {2 x^{2}+2 x}{e^{x}}} x^{3}+4 e^{\frac {2 x^{2}+2 x}{e^{x}}} x +25 e^{\frac {4 x}{e^{x}}} x^{4}-20 e^{\frac {4 x}{e^{x}}} x^{2}+4 e^{\frac {4 x}{e^{x}}}}d x \right )-2 \left (\int \frac {e^{\frac {4 x^{2}}{e^{x}}}}{e^{\frac {4 x^{2}}{e^{x}}} x^{3}-10 e^{\frac {2 x^{2}+2 x}{e^{x}}} x^{4}+4 e^{\frac {2 x^{2}+2 x}{e^{x}}} x^{2}+25 e^{\frac {4 x}{e^{x}}} x^{5}-20 e^{\frac {4 x}{e^{x}}} x^{3}+4 e^{\frac {4 x}{e^{x}}} x}d x \right )-4 \left (\int \frac {e^{\frac {2 x^{2}+2 x}{e^{x}}} x}{e^{\frac {e^{x} x +4 x^{2}}{e^{x}}} x^{2}-10 e^{\frac {e^{x} x +2 x^{2}+2 x}{e^{x}}} x^{3}+4 e^{\frac {e^{x} x +2 x^{2}+2 x}{e^{x}}} x +25 e^{\frac {e^{x} x +4 x}{e^{x}}} x^{4}-20 e^{\frac {e^{x} x +4 x}{e^{x}}} x^{2}+4 e^{\frac {e^{x} x +4 x}{e^{x}}}}d x \right )-50 \left (\int \frac {e^{\frac {4 x}{e^{x}}} x}{e^{\frac {4 x^{2}}{e^{x}}} x^{2}-10 e^{\frac {2 x^{2}+2 x}{e^{x}}} x^{3}+4 e^{\frac {2 x^{2}+2 x}{e^{x}}} x +25 e^{\frac {4 x}{e^{x}}} x^{4}-20 e^{\frac {4 x}{e^{x}}} x^{2}+4 e^{\frac {4 x}{e^{x}}}}d x \right ) \] Input:

int((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+12*x^2 
-4*x)*exp((2*x^2-2*x)/exp(x))-50*exp(x)*x^3)/(x^4*exp(x)*exp((2*x^2-2*x)/e 
xp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20*x^4+4*x 
^2)*exp(x)),x)
 

Output:

2*( - 2*int(e**((2*x**2 + 2*x)/e**x)/(e**((e**x*x + 4*x**2)/e**x)*x**3 - 1 
0*e**((e**x*x + 2*x**2 + 2*x)/e**x)*x**4 + 4*e**((e**x*x + 2*x**2 + 2*x)/e 
**x)*x**2 + 25*e**((e**x*x + 4*x)/e**x)*x**5 - 20*e**((e**x*x + 4*x)/e**x) 
*x**3 + 4*e**((e**x*x + 4*x)/e**x)*x),x) + 6*int(e**((2*x**2 + 2*x)/e**x)/ 
(e**((e**x*x + 4*x**2)/e**x)*x**2 - 10*e**((e**x*x + 2*x**2 + 2*x)/e**x)*x 
**3 + 4*e**((e**x*x + 2*x**2 + 2*x)/e**x)*x + 25*e**((e**x*x + 4*x)/e**x)* 
x**4 - 20*e**((e**x*x + 4*x)/e**x)*x**2 + 4*e**((e**x*x + 4*x)/e**x)),x) - 
 int(e**((2*x**2 + 2*x)/e**x)/(e**((4*x**2)/e**x)*x**4 - 10*e**((2*x**2 + 
2*x)/e**x)*x**5 + 4*e**((2*x**2 + 2*x)/e**x)*x**3 + 25*e**((4*x)/e**x)*x** 
6 - 20*e**((4*x)/e**x)*x**4 + 4*e**((4*x)/e**x)*x**2),x) + 10*int(e**((2*x 
**2 + 2*x)/e**x)/(e**((4*x**2)/e**x)*x**2 - 10*e**((2*x**2 + 2*x)/e**x)*x* 
*3 + 4*e**((2*x**2 + 2*x)/e**x)*x + 25*e**((4*x)/e**x)*x**4 - 20*e**((4*x) 
/e**x)*x**2 + 4*e**((4*x)/e**x)),x) - int(e**((4*x**2)/e**x)/(e**((4*x**2) 
/e**x)*x**3 - 10*e**((2*x**2 + 2*x)/e**x)*x**4 + 4*e**((2*x**2 + 2*x)/e**x 
)*x**2 + 25*e**((4*x)/e**x)*x**5 - 20*e**((4*x)/e**x)*x**3 + 4*e**((4*x)/e 
**x)*x),x) - 2*int((e**((2*x**2 + 2*x)/e**x)*x)/(e**((e**x*x + 4*x**2)/e** 
x)*x**2 - 10*e**((e**x*x + 2*x**2 + 2*x)/e**x)*x**3 + 4*e**((e**x*x + 2*x* 
*2 + 2*x)/e**x)*x + 25*e**((e**x*x + 4*x)/e**x)*x**4 - 20*e**((e**x*x + 4* 
x)/e**x)*x**2 + 4*e**((e**x*x + 4*x)/e**x)),x) - 25*int((e**((4*x)/e**x)*x 
)/(e**((4*x**2)/e**x)*x**2 - 10*e**((2*x**2 + 2*x)/e**x)*x**3 + 4*e**((...