\(\int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} (-20+80 x-120 x^2+80 x^3-20 x^4)+e^{\frac {e^4 x}{4-8 x+4 x^2}} (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 (x^2+x^3))}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} (40-120 x+120 x^2-40 x^3)+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} (-4+12 x-12 x^2+4 x^3)+e^x (-100+300 x-300 x^2+100 x^3)} \, dx\) [1637]

Optimal result

Integrand size = 220, antiderivative size = 34 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=4-e^{-x} x \left (-5+\frac {x}{5-e^{\frac {e^4 x}{(-2+2 x)^2}}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=\frac {e^{-x} x \left (-25+5 e^{\frac {e^4 x}{4 (-1+x)^2}}+x\right )}{-5+e^{\frac {e^4 x}{4 (-1+x)^2}}} \] Input:

Integrate[(-500 + 2040*x - 3140*x^2 + 2180*x^3 - 600*x^4 + 20*x^5 + E^((2* 
E^4*x)/(4 - 8*x + 4*x^2))*(-20 + 80*x - 120*x^2 + 80*x^3 - 20*x^4) + E^((E 
^4*x)/(4 - 8*x + 4*x^2))*(200 - 808*x + 1228*x^2 - 836*x^3 + 220*x^4 - 4*x 
^5 + E^4*(x^2 + x^3)))/(E^(x + (E^4*x)/(4 - 8*x + 4*x^2))*(40 - 120*x + 12 
0*x^2 - 40*x^3) + E^(x + (2*E^4*x)/(4 - 8*x + 4*x^2))*(-4 + 12*x - 12*x^2 
+ 4*x^3) + E^x*(-100 + 300*x - 300*x^2 + 100*x^3)),x]
 

Output:

(x*(-25 + 5*E^((E^4*x)/(4*(-1 + x)^2)) + x))/(E^x*(-5 + E^((E^4*x)/(4*(-1 
+ 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.

\(\displaystyle \int \frac {20 x^5-600 x^4+2180 x^3-3140 x^2+e^{\frac {2 e^4 x}{4 x^2-8 x+4}} \left (-20 x^4+80 x^3-120 x^2+80 x-20\right )+e^{\frac {e^4 x}{4 x^2-8 x+4}} \left (-4 x^5+220 x^4-836 x^3+1228 x^2+e^4 \left (x^3+x^2\right )-808 x+200\right )+2040 x-500}{e^{\frac {e^4 x}{4 x^2-8 x+4}+x} \left (-40 x^3+120 x^2-120 x+40\right )+e^{\frac {2 e^4 x}{4 x^2-8 x+4}+x} \left (4 x^3-12 x^2+12 x-4\right )+e^x \left (100 x^3-300 x^2+300 x-100\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{-x} \left (4 e^{\frac {e^4 x}{4 (x-1)^2}} \left (x^2-52 x+50\right ) (x-1)^3-20 \left (x^2-27 x+25\right ) (x-1)^3-e^{\frac {e^4 x}{4 (x-1)^2}+4} x^2 (x+1)+20 e^{\frac {e^4 x}{2 (x-1)^2}} (x-1)^4\right )}{4 \left (5-e^{\frac {e^4 x}{4 (x-1)^2}}\right )^2 (1-x)^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int \frac {e^{-x} \left (20 e^{\frac {e^4 x}{2 (1-x)^2}} (1-x)^4-4 e^{\frac {e^4 x}{4 (1-x)^2}} \left (x^2-52 x+50\right ) (1-x)^3+20 \left (x^2-27 x+25\right ) (1-x)^3-e^{\frac {e^4 x}{4 (1-x)^2}+4} x^2 (x+1)\right )}{\left (5-e^{\frac {e^4 x}{4 (1-x)^2}}\right )^2 (1-x)^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{4} \int \left (\frac {5 e^{4-x} (x+1) x^2}{\left (-5+e^{\frac {e^4 x}{4 (x-1)^2}}\right )^2 (x-1)^3}+\frac {e^{-x} \left (-4 x^4+20 x^3-\left (36-e^4\right ) x^2+\left (28+e^4\right ) x-8\right ) x}{\left (5-e^{\frac {e^4 x}{4 (x-1)^2}}\right ) (1-x)^3}-20 e^{-x} (x-1)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} \left (-4 \int \frac {e^{-x} x^2}{-5+e^{\frac {e^4 x}{4 (x-1)^2}}}dx+5 \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (x-1)^2}}\right )^2}dx+\int \frac {e^{4-x}}{-5+e^{\frac {e^4 x}{4 (x-1)^2}}}dx+10 \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (x-1)^2}}\right )^2 (x-1)^3}dx+2 \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (x-1)^2}}\right ) (x-1)^3}dx+25 \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (x-1)^2}}\right )^2 (x-1)^2}dx+5 \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (x-1)^2}}\right ) (x-1)^2}dx+20 \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (x-1)^2}}\right )^2 (x-1)}dx+4 \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (x-1)^2}}\right ) (x-1)}dx+8 \int \frac {e^{-x} x}{-5+e^{\frac {e^4 x}{4 (x-1)^2}}}dx-20 e^{-x} (1-x)+20 e^{-x}\right )\)

Input:

Int[(-500 + 2040*x - 3140*x^2 + 2180*x^3 - 600*x^4 + 20*x^5 + E^((2*E^4*x) 
/(4 - 8*x + 4*x^2))*(-20 + 80*x - 120*x^2 + 80*x^3 - 20*x^4) + E^((E^4*x)/ 
(4 - 8*x + 4*x^2))*(200 - 808*x + 1228*x^2 - 836*x^3 + 220*x^4 - 4*x^5 + E 
^4*(x^2 + x^3)))/(E^(x + (E^4*x)/(4 - 8*x + 4*x^2))*(40 - 120*x + 120*x^2 
- 40*x^3) + E^(x + (2*E^4*x)/(4 - 8*x + 4*x^2))*(-4 + 12*x - 12*x^2 + 4*x^ 
3) + E^x*(-100 + 300*x - 300*x^2 + 100*x^3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 92.95 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94

method	result	size
risch	\(5 \,{\mathrm e}^{-x} x +\frac {x^{2} {\mathrm e}^{-x}}{{\mathrm e}^{\frac {{\mathrm e}^{4} x}{4 \left (-1+x \right )^{2}}}-5}\)	\(32\)
parallelrisch	\(\frac {\left (4 x^{2}+20 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{4}}{4 x^{2}-8 x +4}} x -100 x \right ) {\mathrm e}^{-x}}{4 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{4}}{4 x^{2}-8 x +4}}-20}\)	\(55\)

Input:

int(((-20*x^4+80*x^3-120*x^2+80*x-20)*exp(x*exp(4)/(4*x^2-8*x+4))^2+((x^3+ 
x^2)*exp(4)-4*x^5+220*x^4-836*x^3+1228*x^2-808*x+200)*exp(x*exp(4)/(4*x^2- 
8*x+4))+20*x^5-600*x^4+2180*x^3-3140*x^2+2040*x-500)/((4*x^3-12*x^2+12*x-4 
)*exp(x)*exp(x*exp(4)/(4*x^2-8*x+4))^2+(-40*x^3+120*x^2-120*x+40)*exp(x)*e 
xp(x*exp(4)/(4*x^2-8*x+4))+(100*x^3-300*x^2+300*x-100)*exp(x)),x,method=_R 
ETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (27) = 54\).

Time = 0.09 (sec) , antiderivative size = 203, normalized size of antiderivative = 5.97 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=\frac {{\left (x^{2} - 25 \, x\right )} e^{\left (\frac {4 \, x^{3} - 8 \, x^{2} + x e^{4} + 4 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, x^{3} - 4 \, x^{2} + x e^{4} + 2 \, x}{2 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )} + 5 \, x e^{\left (\frac {2 \, x^{3} - 4 \, x^{2} + x e^{4} + 2 \, x}{x^{2} - 2 \, x + 1}\right )}}{e^{\left (\frac {4 \, x^{3} - 8 \, x^{2} + x e^{4} + 4 \, x}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, x^{3} - 4 \, x^{2} + x e^{4} + 2 \, x}{2 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )} - 5 \, e^{\left (\frac {3 \, {\left (4 \, x^{3} - 8 \, x^{2} + x e^{4} + 4 \, x\right )}}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )}} \] Input:

integrate(((-20*x^4+80*x^3-120*x^2+80*x-20)*exp(x*exp(4)/(4*x^2-8*x+4))^2+ 
((x^3+x^2)*exp(4)-4*x^5+220*x^4-836*x^3+1228*x^2-808*x+200)*exp(x*exp(4)/( 
4*x^2-8*x+4))+20*x^5-600*x^4+2180*x^3-3140*x^2+2040*x-500)/((4*x^3-12*x^2+ 
12*x-4)*exp(x)*exp(x*exp(4)/(4*x^2-8*x+4))^2+(-40*x^3+120*x^2-120*x+40)*ex 
p(x)*exp(x*exp(4)/(4*x^2-8*x+4))+(100*x^3-300*x^2+300*x-100)*exp(x)),x, al 
gorithm="fricas")
 

Output:

((x^2 - 25*x)*e^(1/4*(4*x^3 - 8*x^2 + x*e^4 + 4*x)/(x^2 - 2*x + 1) + 1/2*( 
2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 1)) + 5*x*e^((2*x^3 - 4*x^2 + x* 
e^4 + 2*x)/(x^2 - 2*x + 1)))/(e^(1/2*(4*x^3 - 8*x^2 + x*e^4 + 4*x)/(x^2 - 
2*x + 1) + 1/2*(2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 1)) - 5*e^(3/4*( 
4*x^3 - 8*x^2 + x*e^4 + 4*x)/(x^2 - 2*x + 1)))
 

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=\frac {x^{2}}{e^{x} e^{\frac {x e^{4}}{4 x^{2} - 8 x + 4}} - 5 e^{x}} + 5 x e^{- x} \] Input:

integrate(((-20*x**4+80*x**3-120*x**2+80*x-20)*exp(x*exp(4)/(4*x**2-8*x+4) 
)**2+((x**3+x**2)*exp(4)-4*x**5+220*x**4-836*x**3+1228*x**2-808*x+200)*exp 
(x*exp(4)/(4*x**2-8*x+4))+20*x**5-600*x**4+2180*x**3-3140*x**2+2040*x-500) 
/((4*x**3-12*x**2+12*x-4)*exp(x)*exp(x*exp(4)/(4*x**2-8*x+4))**2+(-40*x**3 
+120*x**2-120*x+40)*exp(x)*exp(x*exp(4)/(4*x**2-8*x+4))+(100*x**3-300*x**2 
+300*x-100)*exp(x)),x)
 

Output:

x**2/(exp(x)*exp(x*exp(4)/(4*x**2 - 8*x + 4)) - 5*exp(x)) + 5*x*exp(-x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (27) = 54\).

Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.03 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=\frac {x^{2} + 5 \, x e^{\left (\frac {e^{4}}{4 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {e^{4}}{4 \, {\left (x - 1\right )}}\right )} - 25 \, x}{e^{\left (x + \frac {e^{4}}{4 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {e^{4}}{4 \, {\left (x - 1\right )}}\right )} - 5 \, e^{x}} \] Input:

integrate(((-20*x^4+80*x^3-120*x^2+80*x-20)*exp(x*exp(4)/(4*x^2-8*x+4))^2+ 
((x^3+x^2)*exp(4)-4*x^5+220*x^4-836*x^3+1228*x^2-808*x+200)*exp(x*exp(4)/( 
4*x^2-8*x+4))+20*x^5-600*x^4+2180*x^3-3140*x^2+2040*x-500)/((4*x^3-12*x^2+ 
12*x-4)*exp(x)*exp(x*exp(4)/(4*x^2-8*x+4))^2+(-40*x^3+120*x^2-120*x+40)*ex 
p(x)*exp(x*exp(4)/(4*x^2-8*x+4))+(100*x^3-300*x^2+300*x-100)*exp(x)),x, al 
gorithm="maxima")
 

Output:

(x^2 + 5*x*e^(1/4*e^4/(x^2 - 2*x + 1) + 1/4*e^4/(x - 1)) - 25*x)/(e^(x + 1 
/4*e^4/(x^2 - 2*x + 1) + 1/4*e^4/(x - 1)) - 5*e^x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1417 vs. \(2 (27) = 54\).

Time = 0.20 (sec) , antiderivative size = 1417, normalized size of antiderivative = 41.68 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=\text {Too large to display} \] Input:

integrate(((-20*x^4+80*x^3-120*x^2+80*x-20)*exp(x*exp(4)/(4*x^2-8*x+4))^2+ 
((x^3+x^2)*exp(4)-4*x^5+220*x^4-836*x^3+1228*x^2-808*x+200)*exp(x*exp(4)/( 
4*x^2-8*x+4))+20*x^5-600*x^4+2180*x^3-3140*x^2+2040*x-500)/((4*x^3-12*x^2+ 
12*x-4)*exp(x)*exp(x*exp(4)/(4*x^2-8*x+4))^2+(-40*x^3+120*x^2-120*x+40)*ex 
p(x)*exp(x*exp(4)/(4*x^2-8*x+4))+(100*x^3-300*x^2+300*x-100)*exp(x)),x, al 
gorithm="giac")
 

Output:

-(100*x^5*e^(3/2*x + 1/4*(2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 1)) - 
100*x^5*e^(x + 1/4*(4*x^3 - 8*x^2 + x*e^4 + 4*x)/(x^2 - 2*x + 1)) - 3000*x 
^4*e^(3/2*x + 1/4*(2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 1)) + 3000*x^ 
4*e^(x + 1/4*(4*x^3 - 8*x^2 + x*e^4 + 4*x)/(x^2 - 2*x + 1)) + 10900*x^3*e^ 
(3/2*x + 1/4*(2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 1)) - 10900*x^3*e^ 
(x + 1/4*(4*x^3 - 8*x^2 + x*e^4 + 4*x)/(x^2 - 2*x + 1)) + 5*x^3*e^(1/2*x + 
 1/4*(4*x^3 - 8*x^2 + x*e^4 + 4*x)/(x^2 - 2*x + 1) + 1/4*(2*x^3 - 4*x^2 + 
x*e^4 + 2*x)/(x^2 - 2*x + 1) + 4) - x^3*e^(1/2*x + 3/4*(2*x^3 - 4*x^2 + x* 
e^4 + 2*x)/(x^2 - 2*x + 1) + 4) - 15700*x^2*e^(3/2*x + 1/4*(2*x^3 - 4*x^2 
+ x*e^4 + 2*x)/(x^2 - 2*x + 1)) + 15700*x^2*e^(x + 1/4*(4*x^3 - 8*x^2 + x* 
e^4 + 4*x)/(x^2 - 2*x + 1)) - 120*x^2*e^(1/2*x + 1/4*(4*x^3 - 8*x^2 + x*e^ 
4 + 4*x)/(x^2 - 2*x + 1) + 1/4*(2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 
1) + 4) + 24*x^2*e^(1/2*x + 3/4*(2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 
 1) + 4) + 25*x^2*e^(1/4*(4*x^3 - 8*x^2 + x*e^4 + 4*x)/(x^2 - 2*x + 1) + 1 
/2*(2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 1) + 4) - 5*x^2*e^((2*x^3 - 
4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 1) + 4) + 10200*x*e^(3/2*x + 1/4*(2*x^3 
- 4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 1)) - 10200*x*e^(x + 1/4*(4*x^3 - 8*x^ 
2 + x*e^4 + 4*x)/(x^2 - 2*x + 1)) - 125*x*e^(1/2*x + 1/4*(4*x^3 - 8*x^2 + 
x*e^4 + 4*x)/(x^2 - 2*x + 1) + 1/4*(2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - 2* 
x + 1) + 4) + 25*x*e^(1/2*x + 3/4*(2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - ...
 

Mupad [B] (verification not implemented)

Time = 3.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=5\,x\,{\mathrm {e}}^{-x}+\frac {x^2\,{\mathrm {e}}^{-x}}{{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^4}{4\,x^2-8\,x+4}}-5} \] Input:

int((2040*x - exp((2*x*exp(4))/(4*x^2 - 8*x + 4))*(120*x^2 - 80*x - 80*x^3 
 + 20*x^4 + 20) + exp((x*exp(4))/(4*x^2 - 8*x + 4))*(exp(4)*(x^2 + x^3) - 
808*x + 1228*x^2 - 836*x^3 + 220*x^4 - 4*x^5 + 200) - 3140*x^2 + 2180*x^3 
- 600*x^4 + 20*x^5 - 500)/(exp(x)*(300*x - 300*x^2 + 100*x^3 - 100) + exp( 
(2*x*exp(4))/(4*x^2 - 8*x + 4))*exp(x)*(12*x - 12*x^2 + 4*x^3 - 4) - exp(( 
x*exp(4))/(4*x^2 - 8*x + 4))*exp(x)*(120*x - 120*x^2 + 40*x^3 - 40)),x)
 

Output:

5*x*exp(-x) + (x^2*exp(-x))/(exp((x*exp(4))/(4*x^2 - 8*x + 4)) - 5)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.59 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=\frac {x \left (5 e^{\frac {e^{4} x}{4 x^{2}-8 x +4}}+x -25\right )}{e^{x} \left (e^{\frac {e^{4} x}{4 x^{2}-8 x +4}}-5\right )} \] Input:

int(((-20*x^4+80*x^3-120*x^2+80*x-20)*exp(x*exp(4)/(4*x^2-8*x+4))^2+((x^3+ 
x^2)*exp(4)-4*x^5+220*x^4-836*x^3+1228*x^2-808*x+200)*exp(x*exp(4)/(4*x^2- 
8*x+4))+20*x^5-600*x^4+2180*x^3-3140*x^2+2040*x-500)/((4*x^3-12*x^2+12*x-4 
)*exp(x)*exp(x*exp(4)/(4*x^2-8*x+4))^2+(-40*x^3+120*x^2-120*x+40)*exp(x)*e 
xp(x*exp(4)/(4*x^2-8*x+4))+(100*x^3-300*x^2+300*x-100)*exp(x)),x)
 

Output:

(x*(5*e**((e**4*x)/(4*x**2 - 8*x + 4)) + x - 25))/(e**x*(e**((e**4*x)/(4*x 
**2 - 8*x + 4)) - 5))