\(\int \frac {800+50 e^{-4 x+2 x^2}+400 x+50 x^2+e^{-2 x+x^2} (400+100 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} (-75 x^2-50 e^{-2 x+x^2} x^2-5 e^{-4 x+2 x^2} x^2-40 x^3-5 x^4)}{1600+100 e^{-4 x+2 x^2}+800 x+100 x^2+e^{-2 x+x^2} (800+200 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} (320 x+20 e^{-4 x+2 x^2} x+160 x^2+20 x^3+e^{-2 x+x^2} (160 x+40 x^2))+e^{\frac {2 (1+4 x+e^{-2 x+x^2} x+x^2)}{4+e^{-2 x+x^2}+x}} (16 x^2+e^{-4 x+2 x^2} x^2+8 x^3+x^4+e^{-2 x+x^2} (8 x^2+2 x^3))} \, dx\) [928]

Optimal result

Integrand size = 334, antiderivative size = 27 \[ \int \frac {800+50 e^{-4 x+2 x^2}+400 x+50 x^2+e^{-2 x+x^2} (400+100 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (-75 x^2-50 e^{-2 x+x^2} x^2-5 e^{-4 x+2 x^2} x^2-40 x^3-5 x^4\right )}{1600+100 e^{-4 x+2 x^2}+800 x+100 x^2+e^{-2 x+x^2} (800+200 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (320 x+20 e^{-4 x+2 x^2} x+160 x^2+20 x^3+e^{-2 x+x^2} \left (160 x+40 x^2\right )\right )+e^{\frac {2 \left (1+4 x+e^{-2 x+x^2} x+x^2\right )}{4+e^{-2 x+x^2}+x}} \left (16 x^2+e^{-4 x+2 x^2} x^2+8 x^3+x^4+e^{-2 x+x^2} \left (8 x^2+2 x^3\right )\right )} \, dx=\frac {x}{2+\frac {1}{5} e^{x+\frac {1}{4+e^{(-2+x) x}+x}} x} \]

[Out]

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

Rubi [F]

\[ \int \frac {800+50 e^{-4 x+2 x^2}+400 x+50 x^2+e^{-2 x+x^2} (400+100 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (-75 x^2-50 e^{-2 x+x^2} x^2-5 e^{-4 x+2 x^2} x^2-40 x^3-5 x^4\right )}{1600+100 e^{-4 x+2 x^2}+800 x+100 x^2+e^{-2 x+x^2} (800+200 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (320 x+20 e^{-4 x+2 x^2} x+160 x^2+20 x^3+e^{-2 x+x^2} \left (160 x+40 x^2\right )\right )+e^{\frac {2 \left (1+4 x+e^{-2 x+x^2} x+x^2\right )}{4+e^{-2 x+x^2}+x}} \left (16 x^2+e^{-4 x+2 x^2} x^2+8 x^3+x^4+e^{-2 x+x^2} \left (8 x^2+2 x^3\right )\right )} \, dx=\int \frac {800+50 e^{-4 x+2 x^2}+400 x+50 x^2+e^{-2 x+x^2} (400+100 x)+\exp \left (\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}\right ) \left (-75 x^2-50 e^{-2 x+x^2} x^2-5 e^{-4 x+2 x^2} x^2-40 x^3-5 x^4\right )}{1600+100 e^{-4 x+2 x^2}+800 x+100 x^2+e^{-2 x+x^2} (800+200 x)+\exp \left (\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}\right ) \left (320 x+20 e^{-4 x+2 x^2} x+160 x^2+20 x^3+e^{-2 x+x^2} \left (160 x+40 x^2\right )\right )+\exp \left (\frac {2 \left (1+4 x+e^{-2 x+x^2} x+x^2\right )}{4+e^{-2 x+x^2}+x}\right ) \left (16 x^2+e^{-4 x+2 x^2} x^2+8 x^3+x^4+e^{-2 x+x^2} \left (8 x^2+2 x^3\right )\right )} \, dx \]

[In]

Int[(800 + 50*E^(-4*x + 2*x^2) + 400*x + 50*x^2 + E^(-2*x + x^2)*(400 + 100*x) + E^((1 + 4*x + E^(-2*x + x^2)*
x + x^2)/(4 + E^(-2*x + x^2) + x))*(-75*x^2 - 50*E^(-2*x + x^2)*x^2 - 5*E^(-4*x + 2*x^2)*x^2 - 40*x^3 - 5*x^4)
)/(1600 + 100*E^(-4*x + 2*x^2) + 800*x + 100*x^2 + E^(-2*x + x^2)*(800 + 200*x) + E^((1 + 4*x + E^(-2*x + x^2)
*x + x^2)/(4 + E^(-2*x + x^2) + x))*(320*x + 20*E^(-4*x + 2*x^2)*x + 160*x^2 + 20*x^3 + E^(-2*x + x^2)*(160*x
+ 40*x^2)) + E^((2*(1 + 4*x + E^(-2*x + x^2)*x + x^2))/(4 + E^(-2*x + x^2) + x))*(16*x^2 + E^(-4*x + 2*x^2)*x^
2 + 8*x^3 + x^4 + E^(-2*x + x^2)*(8*x^2 + 2*x^3))),x]

[Out]

800*Defer[Int][E^(4*x)/((4*E^(2*x) + E^x^2 + E^(2*x)*x)^2*(10 + E^((E^x^2*x)/(E^x^2 + E^(2*x)*(4 + x)) + (E^(2
*x)*(1 + 4*x + x^2))/(E^x^2 + E^(2*x)*(4 + x)))*x)^2), x] + 400*Defer[Int][E^(2*x + x^2)/((4*E^(2*x) + E^x^2 +
E^(2*x)*x)^2*(10 + E^((E^x^2*x)/(E^x^2 + E^(2*x)*(4 + x)) + (E^(2*x)*(1 + 4*x + x^2))/(E^x^2 + E^(2*x)*(4 + x
)))*x)^2), x] + 400*Defer[Int][(E^(4*x)*x)/((4*E^(2*x) + E^x^2 + E^(2*x)*x)^2*(10 + E^((E^x^2*x)/(E^x^2 + E^(2
*x)*(4 + x)) + (E^(2*x)*(1 + 4*x + x^2))/(E^x^2 + E^(2*x)*(4 + x)))*x)^2), x] + 100*Defer[Int][(E^(2*x + x^2)*
x)/((4*E^(2*x) + E^x^2 + E^(2*x)*x)^2*(10 + E^((E^x^2*x)/(E^x^2 + E^(2*x)*(4 + x)) + (E^(2*x)*(1 + 4*x + x^2))
/(E^x^2 + E^(2*x)*(4 + x)))*x)^2), x] + 50*Defer[Int][(E^(4*x)*x^2)/((4*E^(2*x) + E^x^2 + E^(2*x)*x)^2*(10 + E
^((E^x^2*x)/(E^x^2 + E^(2*x)*(4 + x)) + (E^(2*x)*(1 + 4*x + x^2))/(E^x^2 + E^(2*x)*(4 + x)))*x)^2), x] - 75*De
fer[Int][(E^((E^(2*x) + 20*E^(2*x)*x + 5*E^x^2*x + 5*E^(2*x)*x^2)/(4*E^(2*x) + E^x^2 + E^(2*x)*x))*x^2)/((4*E^
(2*x) + E^x^2 + E^(2*x)*x)^2*(10 + E^((E^x^2*x)/(E^x^2 + E^(2*x)*(4 + x)) + (E^(2*x)*(1 + 4*x + x^2))/(E^x^2 +
E^(2*x)*(4 + x)))*x)^2), x] - 5*Defer[Int][(E^(4*x + (E^x^2*x*(-3 + 2*x))/(E^x^2 + E^(2*x)*(4 + x)) + (E^(2*x
)*(1 - 12*x + 5*x^2 + 2*x^3))/(E^x^2 + E^(2*x)*(4 + x)))*x^2)/((4*E^(2*x) + E^x^2 + E^(2*x)*x)^2*(10 + E^((E^x
^2*x)/(E^x^2 + E^(2*x)*(4 + x)) + (E^(2*x)*(1 + 4*x + x^2))/(E^x^2 + E^(2*x)*(4 + x)))*x)^2), x] - 40*Defer[In
t][(E^((E^(2*x) + 20*E^(2*x)*x + 5*E^x^2*x + 5*E^(2*x)*x^2)/(4*E^(2*x) + E^x^2 + E^(2*x)*x))*x^3)/((4*E^(2*x)
+ E^x^2 + E^(2*x)*x)^2*(10 + E^((E^x^2*x)/(E^x^2 + E^(2*x)*(4 + x)) + (E^(2*x)*(1 + 4*x + x^2))/(E^x^2 + E^(2*
x)*(4 + x)))*x)^2), x] - 5*Defer[Int][(E^((E^(2*x) + 20*E^(2*x)*x + 5*E^x^2*x + 5*E^(2*x)*x^2)/(4*E^(2*x) + E^
x^2 + E^(2*x)*x))*x^4)/((4*E^(2*x) + E^x^2 + E^(2*x)*x)^2*(10 + E^((E^x^2*x)/(E^x^2 + E^(2*x)*(4 + x)) + (E^(2
*x)*(1 + 4*x + x^2))/(E^x^2 + E^(2*x)*(4 + x)))*x)^2), x] + 50*Defer[Int][E^(2*x^2)/((10 + E^((E^x^2*x + E^(2*
x)*(1 + 4*x + x^2))/(E^x^2 + E^(2*x)*(4 + x)))*x)^2*(E^x^2 + E^(2*x)*(4 + x))^2), x] - 50*Defer[Int][(E^((E^x^
2*x*(3 + x) + E^(2*x)*(1 + 12*x + 7*x^2 + x^3))/(E^x^2 + E^(2*x)*(4 + x)))*x^2)/((10 + E^((E^x^2*x + E^(2*x)*(
1 + 4*x + x^2))/(E^x^2 + E^(2*x)*(4 + x)))*x)^2*(E^x^2 + E^(2*x)*(4 + x))^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{4 x} \left (800+50 e^{-4 x+2 x^2}+400 x+50 x^2+e^{-2 x+x^2} (400+100 x)+\exp \left (\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}\right ) \left (-75 x^2-50 e^{-2 x+x^2} x^2-5 e^{-4 x+2 x^2} x^2-40 x^3-5 x^4\right )\right )}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+\exp \left (\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}\right ) x\right )^2} \, dx \\ & = \int \left (\frac {800 e^{4 x}}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2}+\frac {50 e^{4 x+2 (-2+x) x}}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2}+\frac {400 e^{2 x+x^2}}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2}+\frac {400 e^{4 x} x}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2}+\frac {100 e^{2 x+x^2} x}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2}+\frac {50 e^{4 x} x^2}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2}-\frac {75 e^{4 x+\frac {e^{2 x}+4 e^{2 x} x+e^{x^2} x+e^{2 x} x^2}{4 e^{2 x}+e^{x^2}+e^{2 x} x}} x^2}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2}-\frac {50 e^{4 x+\frac {e^{x^2} (-1+x) x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1-4 x+3 x^2+x^3\right )}{e^{x^2}+e^{2 x} (4+x)}} x^2}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2}-\frac {5 e^{4 x+\frac {e^{x^2} x (-3+2 x)}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1-12 x+5 x^2+2 x^3\right )}{e^{x^2}+e^{2 x} (4+x)}} x^2}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2}-\frac {40 e^{4 x+\frac {e^{2 x}+4 e^{2 x} x+e^{x^2} x+e^{2 x} x^2}{4 e^{2 x}+e^{x^2}+e^{2 x} x}} x^3}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2}-\frac {5 e^{4 x+\frac {e^{2 x}+4 e^{2 x} x+e^{x^2} x+e^{2 x} x^2}{4 e^{2 x}+e^{x^2}+e^{2 x} x}} x^4}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2}\right ) \, dx \\ & = -\left (5 \int \frac {e^{4 x+\frac {e^{x^2} x (-3+2 x)}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1-12 x+5 x^2+2 x^3\right )}{e^{x^2}+e^{2 x} (4+x)}} x^2}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx\right )-5 \int \frac {e^{4 x+\frac {e^{2 x}+4 e^{2 x} x+e^{x^2} x+e^{2 x} x^2}{4 e^{2 x}+e^{x^2}+e^{2 x} x}} x^4}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx-40 \int \frac {e^{4 x+\frac {e^{2 x}+4 e^{2 x} x+e^{x^2} x+e^{2 x} x^2}{4 e^{2 x}+e^{x^2}+e^{2 x} x}} x^3}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx+50 \int \frac {e^{4 x+2 (-2+x) x}}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx+50 \int \frac {e^{4 x} x^2}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx-50 \int \frac {e^{4 x+\frac {e^{x^2} (-1+x) x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1-4 x+3 x^2+x^3\right )}{e^{x^2}+e^{2 x} (4+x)}} x^2}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx-75 \int \frac {e^{4 x+\frac {e^{2 x}+4 e^{2 x} x+e^{x^2} x+e^{2 x} x^2}{4 e^{2 x}+e^{x^2}+e^{2 x} x}} x^2}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx+100 \int \frac {e^{2 x+x^2} x}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx+400 \int \frac {e^{2 x+x^2}}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx+400 \int \frac {e^{4 x} x}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx+800 \int \frac {e^{4 x}}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx \\ & = -\left (5 \int \frac {e^{4 x+\frac {e^{x^2} x (-3+2 x)}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1-12 x+5 x^2+2 x^3\right )}{e^{x^2}+e^{2 x} (4+x)}} x^2}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx\right )-5 \int \frac {e^{\frac {e^{2 x}+20 e^{2 x} x+5 e^{x^2} x+5 e^{2 x} x^2}{4 e^{2 x}+e^{x^2}+e^{2 x} x}} x^4}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx-40 \int \frac {e^{\frac {e^{2 x}+20 e^{2 x} x+5 e^{x^2} x+5 e^{2 x} x^2}{4 e^{2 x}+e^{x^2}+e^{2 x} x}} x^3}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx+50 \int \frac {e^{4 x} x^2}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx+50 \int \frac {e^{2 x^2}}{\left (10+e^{\frac {e^{x^2} x+e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2 \left (e^{x^2}+e^{2 x} (4+x)\right )^2} \, dx-50 \int \frac {e^{\frac {e^{x^2} x (3+x)+e^{2 x} \left (1+12 x+7 x^2+x^3\right )}{e^{x^2}+e^{2 x} (4+x)}} x^2}{\left (10+e^{\frac {e^{x^2} x+e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2 \left (e^{x^2}+e^{2 x} (4+x)\right )^2} \, dx-75 \int \frac {e^{\frac {e^{2 x}+20 e^{2 x} x+5 e^{x^2} x+5 e^{2 x} x^2}{4 e^{2 x}+e^{x^2}+e^{2 x} x}} x^2}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx+100 \int \frac {e^{2 x+x^2} x}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx+400 \int \frac {e^{2 x+x^2}}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx+400 \int \frac {e^{4 x} x}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx+800 \int \frac {e^{4 x}}{\left (4 e^{2 x}+e^{x^2}+e^{2 x} x\right )^2 \left (10+e^{\frac {e^{x^2} x}{e^{x^2}+e^{2 x} (4+x)}+\frac {e^{2 x} \left (1+4 x+x^2\right )}{e^{x^2}+e^{2 x} (4+x)}} x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.71 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {800+50 e^{-4 x+2 x^2}+400 x+50 x^2+e^{-2 x+x^2} (400+100 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (-75 x^2-50 e^{-2 x+x^2} x^2-5 e^{-4 x+2 x^2} x^2-40 x^3-5 x^4\right )}{1600+100 e^{-4 x+2 x^2}+800 x+100 x^2+e^{-2 x+x^2} (800+200 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (320 x+20 e^{-4 x+2 x^2} x+160 x^2+20 x^3+e^{-2 x+x^2} \left (160 x+40 x^2\right )\right )+e^{\frac {2 \left (1+4 x+e^{-2 x+x^2} x+x^2\right )}{4+e^{-2 x+x^2}+x}} \left (16 x^2+e^{-4 x+2 x^2} x^2+8 x^3+x^4+e^{-2 x+x^2} \left (8 x^2+2 x^3\right )\right )} \, dx=\frac {5 x}{10+e^{x+\frac {e^{2 x}}{e^{x^2}+e^{2 x} (4+x)}} x} \]

[In]

Integrate[(800 + 50*E^(-4*x + 2*x^2) + 400*x + 50*x^2 + E^(-2*x + x^2)*(400 + 100*x) + E^((1 + 4*x + E^(-2*x +
x^2)*x + x^2)/(4 + E^(-2*x + x^2) + x))*(-75*x^2 - 50*E^(-2*x + x^2)*x^2 - 5*E^(-4*x + 2*x^2)*x^2 - 40*x^3 -
5*x^4))/(1600 + 100*E^(-4*x + 2*x^2) + 800*x + 100*x^2 + E^(-2*x + x^2)*(800 + 200*x) + E^((1 + 4*x + E^(-2*x
+ x^2)*x + x^2)/(4 + E^(-2*x + x^2) + x))*(320*x + 20*E^(-4*x + 2*x^2)*x + 160*x^2 + 20*x^3 + E^(-2*x + x^2)*(
160*x + 40*x^2)) + E^((2*(1 + 4*x + E^(-2*x + x^2)*x + x^2))/(4 + E^(-2*x + x^2) + x))*(16*x^2 + E^(-4*x + 2*x
^2)*x^2 + 8*x^3 + x^4 + E^(-2*x + x^2)*(8*x^2 + 2*x^3))),x]

[Out]

(5*x)/(10 + E^(x + E^(2*x)/(E^x^2 + E^(2*x)*(4 + x)))*x)

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44

method	result	size
risch	\(\frac {5 x}{x \,{\mathrm e}^{\frac {x \,{\mathrm e}^{\left (-2+x \right ) x}+x^{2}+4 x +1}{4+{\mathrm e}^{\left (-2+x \right ) x}+x}}+10}\)	\(39\)
parallelrisch	\(\frac {5 x}{x \,{\mathrm e}^{\frac {x \,{\mathrm e}^{x^{2}-2 x}+x^{2}+4 x +1}{{\mathrm e}^{x^{2}-2 x}+4+x}}+10}\)	\(43\)

[In]

int(((-5*x^2*exp(x^2-2*x)^2-50*x^2*exp(x^2-2*x)-5*x^4-40*x^3-75*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2
*x)+4+x))+50*exp(x^2-2*x)^2+(100*x+400)*exp(x^2-2*x)+50*x^2+400*x+800)/((x^2*exp(x^2-2*x)^2+(2*x^3+8*x^2)*exp(
x^2-2*x)+x^4+8*x^3+16*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))^2+(20*x*exp(x^2-2*x)^2+(40*x^2+1
60*x)*exp(x^2-2*x)+20*x^3+160*x^2+320*x)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))+100*exp(x^2-2*x)^2
+(200*x+800)*exp(x^2-2*x)+100*x^2+800*x+1600),x,method=_RETURNVERBOSE)

[Out]

5*x/(x*exp((x*exp((-2+x)*x)+x^2+4*x+1)/(4+exp((-2+x)*x)+x))+10)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {800+50 e^{-4 x+2 x^2}+400 x+50 x^2+e^{-2 x+x^2} (400+100 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (-75 x^2-50 e^{-2 x+x^2} x^2-5 e^{-4 x+2 x^2} x^2-40 x^3-5 x^4\right )}{1600+100 e^{-4 x+2 x^2}+800 x+100 x^2+e^{-2 x+x^2} (800+200 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (320 x+20 e^{-4 x+2 x^2} x+160 x^2+20 x^3+e^{-2 x+x^2} \left (160 x+40 x^2\right )\right )+e^{\frac {2 \left (1+4 x+e^{-2 x+x^2} x+x^2\right )}{4+e^{-2 x+x^2}+x}} \left (16 x^2+e^{-4 x+2 x^2} x^2+8 x^3+x^4+e^{-2 x+x^2} \left (8 x^2+2 x^3\right )\right )} \, dx=\frac {5 \, x}{x e^{\left (\frac {x^{2} + x e^{\left (x^{2} - 2 \, x\right )} + 4 \, x + 1}{x + e^{\left (x^{2} - 2 \, x\right )} + 4}\right )} + 10} \]

[In]

integrate(((-5*x^2*exp(x^2-2*x)^2-50*x^2*exp(x^2-2*x)-5*x^4-40*x^3-75*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp
(x^2-2*x)+4+x))+50*exp(x^2-2*x)^2+(100*x+400)*exp(x^2-2*x)+50*x^2+400*x+800)/((x^2*exp(x^2-2*x)^2+(2*x^3+8*x^2
)*exp(x^2-2*x)+x^4+8*x^3+16*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))^2+(20*x*exp(x^2-2*x)^2+(40
*x^2+160*x)*exp(x^2-2*x)+20*x^3+160*x^2+320*x)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))+100*exp(x^2-
2*x)^2+(200*x+800)*exp(x^2-2*x)+100*x^2+800*x+1600),x, algorithm="fricas")

[Out]

5*x/(x*e^((x^2 + x*e^(x^2 - 2*x) + 4*x + 1)/(x + e^(x^2 - 2*x) + 4)) + 10)

Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {800+50 e^{-4 x+2 x^2}+400 x+50 x^2+e^{-2 x+x^2} (400+100 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (-75 x^2-50 e^{-2 x+x^2} x^2-5 e^{-4 x+2 x^2} x^2-40 x^3-5 x^4\right )}{1600+100 e^{-4 x+2 x^2}+800 x+100 x^2+e^{-2 x+x^2} (800+200 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (320 x+20 e^{-4 x+2 x^2} x+160 x^2+20 x^3+e^{-2 x+x^2} \left (160 x+40 x^2\right )\right )+e^{\frac {2 \left (1+4 x+e^{-2 x+x^2} x+x^2\right )}{4+e^{-2 x+x^2}+x}} \left (16 x^2+e^{-4 x+2 x^2} x^2+8 x^3+x^4+e^{-2 x+x^2} \left (8 x^2+2 x^3\right )\right )} \, dx=\frac {5 x}{x e^{\frac {x^{2} + x e^{x^{2} - 2 x} + 4 x + 1}{x + e^{x^{2} - 2 x} + 4}} + 10} \]

[In]

integrate(((-5*x**2*exp(x**2-2*x)**2-50*x**2*exp(x**2-2*x)-5*x**4-40*x**3-75*x**2)*exp((x*exp(x**2-2*x)+x**2+4
*x+1)/(exp(x**2-2*x)+4+x))+50*exp(x**2-2*x)**2+(100*x+400)*exp(x**2-2*x)+50*x**2+400*x+800)/((x**2*exp(x**2-2*
x)**2+(2*x**3+8*x**2)*exp(x**2-2*x)+x**4+8*x**3+16*x**2)*exp((x*exp(x**2-2*x)+x**2+4*x+1)/(exp(x**2-2*x)+4+x))
**2+(20*x*exp(x**2-2*x)**2+(40*x**2+160*x)*exp(x**2-2*x)+20*x**3+160*x**2+320*x)*exp((x*exp(x**2-2*x)+x**2+4*x
+1)/(exp(x**2-2*x)+4+x))+100*exp(x**2-2*x)**2+(200*x+800)*exp(x**2-2*x)+100*x**2+800*x+1600),x)

[Out]

5*x/(x*exp((x**2 + x*exp(x**2 - 2*x) + 4*x + 1)/(x + exp(x**2 - 2*x) + 4)) + 10)

Maxima [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {800+50 e^{-4 x+2 x^2}+400 x+50 x^2+e^{-2 x+x^2} (400+100 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (-75 x^2-50 e^{-2 x+x^2} x^2-5 e^{-4 x+2 x^2} x^2-40 x^3-5 x^4\right )}{1600+100 e^{-4 x+2 x^2}+800 x+100 x^2+e^{-2 x+x^2} (800+200 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (320 x+20 e^{-4 x+2 x^2} x+160 x^2+20 x^3+e^{-2 x+x^2} \left (160 x+40 x^2\right )\right )+e^{\frac {2 \left (1+4 x+e^{-2 x+x^2} x+x^2\right )}{4+e^{-2 x+x^2}+x}} \left (16 x^2+e^{-4 x+2 x^2} x^2+8 x^3+x^4+e^{-2 x+x^2} \left (8 x^2+2 x^3\right )\right )} \, dx=\frac {5 \, x}{x e^{\left (x + \frac {e^{\left (2 \, x\right )}}{{\left (x + 4\right )} e^{\left (2 \, x\right )} + e^{\left (x^{2}\right )}}\right )} + 10} \]

[In]

integrate(((-5*x^2*exp(x^2-2*x)^2-50*x^2*exp(x^2-2*x)-5*x^4-40*x^3-75*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp
(x^2-2*x)+4+x))+50*exp(x^2-2*x)^2+(100*x+400)*exp(x^2-2*x)+50*x^2+400*x+800)/((x^2*exp(x^2-2*x)^2+(2*x^3+8*x^2
)*exp(x^2-2*x)+x^4+8*x^3+16*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))^2+(20*x*exp(x^2-2*x)^2+(40
*x^2+160*x)*exp(x^2-2*x)+20*x^3+160*x^2+320*x)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))+100*exp(x^2-
2*x)^2+(200*x+800)*exp(x^2-2*x)+100*x^2+800*x+1600),x, algorithm="maxima")

[Out]

5*x/(x*e^(x + e^(2*x)/((x + 4)*e^(2*x) + e^(x^2))) + 10)

Giac [F(-1)]

Timed out. \[ \int \frac {800+50 e^{-4 x+2 x^2}+400 x+50 x^2+e^{-2 x+x^2} (400+100 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (-75 x^2-50 e^{-2 x+x^2} x^2-5 e^{-4 x+2 x^2} x^2-40 x^3-5 x^4\right )}{1600+100 e^{-4 x+2 x^2}+800 x+100 x^2+e^{-2 x+x^2} (800+200 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (320 x+20 e^{-4 x+2 x^2} x+160 x^2+20 x^3+e^{-2 x+x^2} \left (160 x+40 x^2\right )\right )+e^{\frac {2 \left (1+4 x+e^{-2 x+x^2} x+x^2\right )}{4+e^{-2 x+x^2}+x}} \left (16 x^2+e^{-4 x+2 x^2} x^2+8 x^3+x^4+e^{-2 x+x^2} \left (8 x^2+2 x^3\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(((-5*x^2*exp(x^2-2*x)^2-50*x^2*exp(x^2-2*x)-5*x^4-40*x^3-75*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp
(x^2-2*x)+4+x))+50*exp(x^2-2*x)^2+(100*x+400)*exp(x^2-2*x)+50*x^2+400*x+800)/((x^2*exp(x^2-2*x)^2+(2*x^3+8*x^2
)*exp(x^2-2*x)+x^4+8*x^3+16*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))^2+(20*x*exp(x^2-2*x)^2+(40
*x^2+160*x)*exp(x^2-2*x)+20*x^3+160*x^2+320*x)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))+100*exp(x^2-
2*x)^2+(200*x+800)*exp(x^2-2*x)+100*x^2+800*x+1600),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 9.78 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.93 \[ \int \frac {800+50 e^{-4 x+2 x^2}+400 x+50 x^2+e^{-2 x+x^2} (400+100 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (-75 x^2-50 e^{-2 x+x^2} x^2-5 e^{-4 x+2 x^2} x^2-40 x^3-5 x^4\right )}{1600+100 e^{-4 x+2 x^2}+800 x+100 x^2+e^{-2 x+x^2} (800+200 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} \left (320 x+20 e^{-4 x+2 x^2} x+160 x^2+20 x^3+e^{-2 x+x^2} \left (160 x+40 x^2\right )\right )+e^{\frac {2 \left (1+4 x+e^{-2 x+x^2} x+x^2\right )}{4+e^{-2 x+x^2}+x}} \left (16 x^2+e^{-4 x+2 x^2} x^2+8 x^3+x^4+e^{-2 x+x^2} \left (8 x^2+2 x^3\right )\right )} \, dx=\frac {5\,x\,\left (8\,x+{\mathrm {e}}^{2\,x^2-4\,x}+x^2+16\right )+5\,x\,{\mathrm {e}}^{x^2-2\,x}\,\left (2\,x+8\right )}{\left (x\,{\mathrm {e}}^{\frac {1}{x+{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}+4}+\frac {x^2}{x+{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}+4}+\frac {4\,x}{x+{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}+4}+\frac {x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}}{x+{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}+4}}+10\right )\,{\left (x+{\mathrm {e}}^{x^2-2\,x}+4\right )}^2} \]

[In]

int((400*x + 50*exp(2*x^2 - 4*x) + exp(x^2 - 2*x)*(100*x + 400) + 50*x^2 - exp((4*x + x*exp(x^2 - 2*x) + x^2 +
1)/(x + exp(x^2 - 2*x) + 4))*(5*x^2*exp(2*x^2 - 4*x) + 50*x^2*exp(x^2 - 2*x) + 75*x^2 + 40*x^3 + 5*x^4) + 800
)/(800*x + 100*exp(2*x^2 - 4*x) + exp(x^2 - 2*x)*(200*x + 800) + exp((2*(4*x + x*exp(x^2 - 2*x) + x^2 + 1))/(x
+ exp(x^2 - 2*x) + 4))*(x^2*exp(2*x^2 - 4*x) + exp(x^2 - 2*x)*(8*x^2 + 2*x^3) + 16*x^2 + 8*x^3 + x^4) + exp((
4*x + x*exp(x^2 - 2*x) + x^2 + 1)/(x + exp(x^2 - 2*x) + 4))*(320*x + exp(x^2 - 2*x)*(160*x + 40*x^2) + 20*x*ex
p(2*x^2 - 4*x) + 160*x^2 + 20*x^3) + 100*x^2 + 1600),x)

[Out]

(5*x*(8*x + exp(2*x^2 - 4*x) + x^2 + 16) + 5*x*exp(x^2 - 2*x)*(2*x + 8))/((x*exp(1/(x + exp(-2*x)*exp(x^2) + 4
) + x^2/(x + exp(-2*x)*exp(x^2) + 4) + (4*x)/(x + exp(-2*x)*exp(x^2) + 4) + (x*exp(-2*x)*exp(x^2))/(x + exp(-2
*x)*exp(x^2) + 4)) + 10)*(x + exp(x^2 - 2*x) + 4)^2)