\(\int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} (-20+80 x-120 x^2+80 x^3-20 x^4)+e^{\frac {e^4 x}{4-8 x+4 x^2}} (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 (x^2+x^3))}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} (40-120 x+120 x^2-40 x^3)+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} (-4+12 x-12 x^2+4 x^3)+e^x (-100+300 x-300 x^2+100 x^3)} \, dx\) [7637]

Optimal result

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

[Out]

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

Rubi [F]

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

[In]

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 - 12
0*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]

[Out]

5/E^x - (5*(1 - x))/E^x + (5*Defer[Int][E^(4 - x)/(-5 + E^((E^4*x)/(4*(-1 + x)^2)))^2, x])/4 + Defer[Int][E^(4
- x)/(-5 + E^((E^4*x)/(4*(-1 + x)^2))), x]/4 + (5*Defer[Int][E^(4 - x)/((-5 + E^((E^4*x)/(4*(-1 + x)^2)))^2*(
-1 + x)^3), x])/2 + Defer[Int][E^(4 - x)/((-5 + E^((E^4*x)/(4*(-1 + x)^2)))*(-1 + x)^3), x]/2 + (25*Defer[Int]
[E^(4 - x)/((-5 + E^((E^4*x)/(4*(-1 + x)^2)))^2*(-1 + x)^2), x])/4 + (5*Defer[Int][E^(4 - x)/((-5 + E^((E^4*x)
/(4*(-1 + x)^2)))*(-1 + x)^2), x])/4 + 5*Defer[Int][E^(4 - x)/((-5 + E^((E^4*x)/(4*(-1 + x)^2)))^2*(-1 + x)),
x] + Defer[Int][E^(4 - x)/((-5 + E^((E^4*x)/(4*(-1 + x)^2)))*(-1 + x)), x] + 2*Defer[Int][x/(E^x*(-5 + E^((E^4
*x)/(4*(-1 + x)^2)))), x] - Defer[Int][x^2/(E^x*(-5 + E^((E^4*x)/(4*(-1 + x)^2)))), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (20 e^{\frac {e^4 x}{2 (-1+x)^2}} (-1+x)^4-e^{4+\frac {e^4 x}{4 (-1+x)^2}} x^2 (1+x)+4 e^{\frac {e^4 x}{4 (-1+x)^2}} (-1+x)^3 \left (50-52 x+x^2\right )-20 (-1+x)^3 \left (25-27 x+x^2\right )\right )}{4 \left (5-e^{\frac {e^4 x}{4 (-1+x)^2}}\right )^2 (1-x)^3} \, dx \\ & = \frac {1}{4} \int \frac {e^{-x} \left (20 e^{\frac {e^4 x}{2 (-1+x)^2}} (-1+x)^4-e^{4+\frac {e^4 x}{4 (-1+x)^2}} x^2 (1+x)+4 e^{\frac {e^4 x}{4 (-1+x)^2}} (-1+x)^3 \left (50-52 x+x^2\right )-20 (-1+x)^3 \left (25-27 x+x^2\right )\right )}{\left (5-e^{\frac {e^4 x}{4 (-1+x)^2}}\right )^2 (1-x)^3} \, dx \\ & = \frac {1}{4} \int \left (-20 e^{-x} (-1+x)+\frac {5 e^{4-x} x^2 (1+x)}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right )^2 (-1+x)^3}+\frac {e^{-x} x \left (-8+\left (28+e^4\right ) x-\left (36-e^4\right ) x^2+20 x^3-4 x^4\right )}{\left (5-e^{\frac {e^4 x}{4 (-1+x)^2}}\right ) (1-x)^3}\right ) \, dx \\ & = \frac {1}{4} \int \frac {e^{-x} x \left (-8+\left (28+e^4\right ) x-\left (36-e^4\right ) x^2+20 x^3-4 x^4\right )}{\left (5-e^{\frac {e^4 x}{4 (-1+x)^2}}\right ) (1-x)^3} \, dx+\frac {5}{4} \int \frac {e^{4-x} x^2 (1+x)}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right )^2 (-1+x)^3} \, dx-5 \int e^{-x} (-1+x) \, dx \\ & = -5 e^{-x} (1-x)+\frac {1}{4} \int \left (\frac {e^{4-x}}{-5+e^{\frac {e^4 x}{4 (-1+x)^2}}}+\frac {2 e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right ) (-1+x)^3}+\frac {5 e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right ) (-1+x)^2}+\frac {4 e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right ) (-1+x)}+\frac {8 e^{-x} x}{-5+e^{\frac {e^4 x}{4 (-1+x)^2}}}-\frac {4 e^{-x} x^2}{-5+e^{\frac {e^4 x}{4 (-1+x)^2}}}\right ) \, dx+\frac {5}{4} \int \left (\frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right )^2}+\frac {2 e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right )^2 (-1+x)^3}+\frac {5 e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right )^2 (-1+x)^2}+\frac {4 e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right )^2 (-1+x)}\right ) \, dx-5 \int e^{-x} \, dx \\ & = 5 e^{-x}-5 e^{-x} (1-x)+\frac {1}{4} \int \frac {e^{4-x}}{-5+e^{\frac {e^4 x}{4 (-1+x)^2}}} \, dx+\frac {1}{2} \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right ) (-1+x)^3} \, dx+\frac {5}{4} \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right )^2} \, dx+\frac {5}{4} \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right ) (-1+x)^2} \, dx+2 \int \frac {e^{-x} x}{-5+e^{\frac {e^4 x}{4 (-1+x)^2}}} \, dx+\frac {5}{2} \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right )^2 (-1+x)^3} \, dx+5 \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right )^2 (-1+x)} \, dx+\frac {25}{4} \int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right )^2 (-1+x)^2} \, dx+\int \frac {e^{4-x}}{\left (-5+e^{\frac {e^4 x}{4 (-1+x)^2}}\right ) (-1+x)} \, dx-\int \frac {e^{-x} x^2}{-5+e^{\frac {e^4 x}{4 (-1+x)^2}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=\frac {e^{-x} x \left (-25+5 e^{\frac {e^4 x}{4 (-1+x)^2}}+x\right )}{-5+e^{\frac {e^4 x}{4 (-1+x)^2}}} \]

[In]

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 + 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]

[Out]

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

Maple [A] (verified)

Time = 9.88 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94

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

[In]

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+12
28*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,method=_RETURNVERBOSE)

[Out]

5*x*exp(-x)+x^2*exp(-x)/(exp(1/4*x*exp(4)/(-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 = 0.30 (sec) , antiderivative size = 203, normalized size of antiderivative = 5.97 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=\frac {{\left (x^{2} - 25 \, x\right )} e^{\left (\frac {4 \, x^{3} - 8 \, x^{2} + x e^{4} + 4 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, x^{3} - 4 \, x^{2} + x e^{4} + 2 \, x}{2 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )} + 5 \, x e^{\left (\frac {2 \, x^{3} - 4 \, x^{2} + x e^{4} + 2 \, x}{x^{2} - 2 \, x + 1}\right )}}{e^{\left (\frac {4 \, x^{3} - 8 \, x^{2} + x e^{4} + 4 \, x}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, x^{3} - 4 \, x^{2} + x e^{4} + 2 \, x}{2 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )} - 5 \, e^{\left (\frac {3 \, {\left (4 \, x^{3} - 8 \, x^{2} + x e^{4} + 4 \, x\right )}}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )}} \]

[In]

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, algorithm="fricas")

[Out]

((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 = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=\frac {x^{2}}{e^{x} e^{\frac {x e^{4}}{4 x^{2} - 8 x + 4}} - 5 e^{x}} + 5 x e^{- x} \]

[In]

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-50
0)/((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*e
xp(4)/(4*x**2-8*x+4))+(100*x**3-300*x**2+300*x-100)*exp(x)),x)

[Out]

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 = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.03 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=\frac {x^{2} + 5 \, x e^{\left (\frac {e^{4}}{4 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {e^{4}}{4 \, {\left (x - 1\right )}}\right )} - 25 \, x}{e^{\left (x + \frac {e^{4}}{4 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {e^{4}}{4 \, {\left (x - 1\right )}}\right )} - 5 \, e^{x}} \]

[In]

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, algorithm="maxima")

[Out]

(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 = 0.36 (sec) , antiderivative size = 1417, normalized size of antiderivative = 41.68 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=\text {Too large to display} \]

[In]

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, algorithm="giac")

[Out]

-(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 - 2*x + 1) + 4) + 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) + 4) - 5*x*e^((2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^
2 - 2*x + 1) + 4) - 2500*e^(3/2*x + 1/4*(2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 1)) + 2500*e^(x + 1/4*(4*x^
3 - 8*x^2 + x*e^4 + 4*x)/(x^2 - 2*x + 1)))/(25*x*e^(3/2*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) - 5*x*e^(3/2*x + 3/4*(2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x
^2 - 2*x + 1) + 4) - 5*x*e^(x + 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) + x*e^(x + (2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 1) + 4) + 25*e^(3/2*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) - 5*e^
(3/2*x + 3/4*(2*x^3 - 4*x^2 + x*e^4 + 2*x)/(x^2 - 2*x + 1) + 4) - 5*e^(x + 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) + e^(x + (2*x^3 - 4*x^2 + x*e^4 + 2*x)
/(x^2 - 2*x + 1) + 4))

Mupad [B] (verification not implemented)

Time = 14.59 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {-500+2040 x-3140 x^2+2180 x^3-600 x^4+20 x^5+e^{\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-20+80 x-120 x^2+80 x^3-20 x^4\right )+e^{\frac {e^4 x}{4-8 x+4 x^2}} \left (200-808 x+1228 x^2-836 x^3+220 x^4-4 x^5+e^4 \left (x^2+x^3\right )\right )}{e^{x+\frac {e^4 x}{4-8 x+4 x^2}} \left (40-120 x+120 x^2-40 x^3\right )+e^{x+\frac {2 e^4 x}{4-8 x+4 x^2}} \left (-4+12 x-12 x^2+4 x^3\right )+e^x \left (-100+300 x-300 x^2+100 x^3\right )} \, dx=5\,x\,{\mathrm {e}}^{-x}+\frac {x^2\,{\mathrm {e}}^{-x}}{{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^4}{4\,x^2-8\,x+4}}-5} \]

[In]

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))*ex
p(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
)

[Out]

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