\(\int \frac {(\frac {e^x}{2+e^x+x})^{\frac {2 e^{-3+x^2}}{x}} (e^{-3+x^2} (2 x+2 x^2)+e^{-3+x^2} (-4-2 x+8 x^2+4 x^3+e^x (-2+4 x^2)) \log (\frac {e^x}{2+e^x+x}))}{2 x^2+e^x x^2+x^3} \, dx\) [4754]

Optimal result

Integrand size = 109, antiderivative size = 25 \[ \int \frac {\left (\frac {e^x}{2+e^x+x}\right )^{\frac {2 e^{-3+x^2}}{x}} \left (e^{-3+x^2} \left (2 x+2 x^2\right )+e^{-3+x^2} \left (-4-2 x+8 x^2+4 x^3+e^x \left (-2+4 x^2\right )\right ) \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{2 x^2+e^x x^2+x^3} \, dx=\left (\frac {e^x}{2+e^x+x}\right )^{\frac {2 e^{-3+x^2}}{x}} \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (\frac {e^x}{2+e^x+x}\right )^{\frac {2 e^{-3+x^2}}{x}} \left (e^{-3+x^2} \left (2 x+2 x^2\right )+e^{-3+x^2} \left (-4-2 x+8 x^2+4 x^3+e^x \left (-2+4 x^2\right )\right ) \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{2 x^2+e^x x^2+x^3} \, dx=\int \frac {\left (\frac {e^x}{2+e^x+x}\right )^{\frac {2 e^{-3+x^2}}{x}} \left (e^{-3+x^2} \left (2 x+2 x^2\right )+e^{-3+x^2} \left (-4-2 x+8 x^2+4 x^3+e^x \left (-2+4 x^2\right )\right ) \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{2 x^2+e^x x^2+x^3} \, dx \]

[In]

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

[Out]

4*Log[E^x/(2 + E^x + x)]*Defer[Int][E^(-3 + x^2)*(E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x^2))/x), x] + 2*Defer[In
t][E^(-3 - x + x^2)*(E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x^2))/x), x] + 8*Log[E^x/(2 + E^x + x)]*Defer[Int][E^(
-3 - x + x^2)*(E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x^2))/x), x] - 2*Log[E^x/(2 + E^x + x)]*Defer[Int][(E^(-3 +
x^2)*(E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x^2))/x))/x^2, x] - 4*Log[E^x/(2 + E^x + x)]*Defer[Int][(E^(-3 - x +
x^2)*(E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x^2))/x))/x^2, x] + 2*Defer[Int][(E^(-3 - x + x^2)*(E^x/(2 + E^x + x)
)^(1 + (2*E^(-3 + x^2))/x))/x, x] - 2*Log[E^x/(2 + E^x + x)]*Defer[Int][(E^(-3 - x + x^2)*(E^x/(2 + E^x + x))^
(1 + (2*E^(-3 + x^2))/x))/x, x] + 4*Log[E^x/(2 + E^x + x)]*Defer[Int][E^(-3 - x + x^2)*x*(E^x/(2 + E^x + x))^(
1 + (2*E^(-3 + x^2))/x), x] - 4*Defer[Int][Defer[Int][E^(-3 + x^2)*(E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x^2))/x
), x]/(2 + E^x + x), x] - 4*Defer[Int][(x*Defer[Int][E^(-3 + x^2)*(E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x^2))/x)
, x])/(2 + E^x + x), x] - 8*Defer[Int][Defer[Int][E^(-3 - x + x^2)*(E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x^2))/x
), x]/(2 + E^x + x), x] - 8*Defer[Int][(x*Defer[Int][E^(-3 - x + x^2)*(E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x^2)
)/x), x])/(2 + E^x + x), x] + 2*Defer[Int][Defer[Int][(E^(-3 + x^2)*(E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x^2))/
x))/x^2, x]/(2 + E^x + x), x] + 2*Defer[Int][(x*Defer[Int][(E^(-3 + x^2)*(E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x
^2))/x))/x^2, x])/(2 + E^x + x), x] + 4*Defer[Int][Defer[Int][(E^(-3 - x + x^2)*(E^x/(2 + E^x + x))^(1 + (2*E^
(-3 + x^2))/x))/x^2, x]/(2 + E^x + x), x] + 4*Defer[Int][(x*Defer[Int][(E^(-3 - x + x^2)*(E^x/(2 + E^x + x))^(
1 + (2*E^(-3 + x^2))/x))/x^2, x])/(2 + E^x + x), x] + 2*Defer[Int][Defer[Int][(E^(-3 - x + x^2)*(E^x/(2 + E^x
+ x))^(1 + (2*E^(-3 + x^2))/x))/x, x]/(2 + E^x + x), x] + 2*Defer[Int][(x*Defer[Int][(E^(-3 - x + x^2)*(E^x/(2
+ E^x + x))^(1 + (2*E^(-3 + x^2))/x))/x, x])/(2 + E^x + x), x] - 4*Defer[Int][Defer[Int][E^(-3 - x + x^2)*x*(
E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x^2))/x), x]/(2 + E^x + x), x] - 4*Defer[Int][(x*Defer[Int][E^(-3 - x + x^2
)*x*(E^x/(2 + E^x + x))^(1 + (2*E^(-3 + x^2))/x), x])/(2 + E^x + x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \left (x (1+x)+\left (2+e^x+x\right ) \left (-1+2 x^2\right ) \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{x^2} \, dx \\ & = 2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \left (x (1+x)+\left (2+e^x+x\right ) \left (-1+2 x^2\right ) \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{x^2} \, dx \\ & = 2 \int \left (\frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \left (-1+2 x^2\right ) \log \left (\frac {e^x}{2+e^x+x}\right )}{x^2}+\frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \left (x+x^2-2 \log \left (\frac {e^x}{2+e^x+x}\right )-x \log \left (\frac {e^x}{2+e^x+x}\right )+4 x^2 \log \left (\frac {e^x}{2+e^x+x}\right )+2 x^3 \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{x^2}\right ) \, dx \\ & = 2 \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \left (-1+2 x^2\right ) \log \left (\frac {e^x}{2+e^x+x}\right )}{x^2} \, dx+2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \left (x+x^2-2 \log \left (\frac {e^x}{2+e^x+x}\right )-x \log \left (\frac {e^x}{2+e^x+x}\right )+4 x^2 \log \left (\frac {e^x}{2+e^x+x}\right )+2 x^3 \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{x^2} \, dx \\ & = 2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \left (x (1+x)+\left (-2-x+4 x^2+2 x^3\right ) \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{x^2} \, dx-2 \int \frac {(1+x) \left (2 \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-\int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx\right )}{2+e^x+x} \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx \\ & = 2 \int \left (\frac {e^{-3-x+x^2} (1+x) \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x}+\frac {e^{-3-x+x^2} (2+x) \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \left (-1+2 x^2\right ) \log \left (\frac {e^x}{2+e^x+x}\right )}{x^2}\right ) \, dx-2 \int \left (\frac {2 \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-\int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x}+\frac {x \left (2 \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-\int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx\right )}{2+e^x+x}\right ) \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx \\ & = 2 \int \frac {e^{-3-x+x^2} (1+x) \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+2 \int \frac {e^{-3-x+x^2} (2+x) \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \left (-1+2 x^2\right ) \log \left (\frac {e^x}{2+e^x+x}\right )}{x^2} \, dx-2 \int \frac {2 \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-\int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x} \, dx-2 \int \frac {x \left (2 \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-\int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx\right )}{2+e^x+x} \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx \\ & = 2 \int \left (e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}+\frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x}\right ) \, dx-2 \int \left (\frac {2 \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x}-\frac {\int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x}\right ) \, dx-2 \int \left (\frac {2 x \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x}-\frac {x \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x}\right ) \, dx-2 \int \frac {(1+x) \left (4 \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx-\int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+2 \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx\right )}{2+e^x+x} \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx+\left (8 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx \\ & = 2 \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx+2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+2 \int \frac {\int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x} \, dx+2 \int \frac {x \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x} \, dx-2 \int \left (\frac {4 \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx-\int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+2 \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x}+\frac {x \left (4 \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx-\int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+2 \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx\right )}{2+e^x+x}\right ) \, dx-4 \int \frac {\int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx-4 \int \frac {x \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx+\left (8 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx \\ & = 2 \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx+2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+2 \int \frac {\int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x} \, dx+2 \int \frac {x \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x} \, dx-2 \int \frac {4 \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx-\int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+2 \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx-2 \int \frac {x \left (4 \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx-\int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+2 \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx\right )}{2+e^x+x} \, dx-4 \int \frac {\int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx-4 \int \frac {x \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx+\left (8 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx \\ & = 2 \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx+2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+2 \int \frac {\int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x} \, dx+2 \int \frac {x \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x} \, dx-2 \int \left (\frac {4 \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x}-\frac {2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x}-\frac {\int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx}{2+e^x+x}+\frac {2 \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x}\right ) \, dx-2 \int \left (\frac {4 x \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x}-\frac {2 x \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x}-\frac {x \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx}{2+e^x+x}+\frac {2 x \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x}\right ) \, dx-4 \int \frac {\int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx-4 \int \frac {x \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx+\left (8 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx \\ & = 2 \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx+2 \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+2 \int \frac {\int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x} \, dx+2 \int \frac {x \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x} \, dx+2 \int \frac {\int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx}{2+e^x+x} \, dx+2 \int \frac {x \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx}{2+e^x+x} \, dx-4 \int \frac {\int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx-4 \int \frac {x \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx+4 \int \frac {\int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x} \, dx+4 \int \frac {x \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx}{2+e^x+x} \, dx-4 \int \frac {\int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx-4 \int \frac {x \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx-8 \int \frac {\int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx-8 \int \frac {x \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx}{2+e^x+x} \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx-\left (2 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx-\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int \frac {e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}}}{x^2} \, dx+\left (4 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3-x+x^2} x \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx+\left (8 \log \left (\frac {e^x}{2+e^x+x}\right )\right ) \int e^{-3-x+x^2} \left (\frac {e^x}{2+e^x+x}\right )^{1+\frac {2 e^{-3+x^2}}{x}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\frac {e^x}{2+e^x+x}\right )^{\frac {2 e^{-3+x^2}}{x}} \left (e^{-3+x^2} \left (2 x+2 x^2\right )+e^{-3+x^2} \left (-4-2 x+8 x^2+4 x^3+e^x \left (-2+4 x^2\right )\right ) \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{2 x^2+e^x x^2+x^3} \, dx=\left (\frac {e^x}{2+e^x+x}\right )^{\frac {2 e^{-3+x^2}}{x}} \]

[In]

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

[Out]

(E^x/(2 + E^x + x))^((2*E^(-3 + x^2))/x)

Maple [A] (verified)

Time = 34.56 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\({\mathrm e}^{\frac {2 \,{\mathrm e}^{x^{2}-3} \ln \left (\frac {{\mathrm e}^{x}}{{\mathrm e}^{x}+2+x}\right )}{x}}\)	\(25\)
risch	\(\left ({\mathrm e}^{x}+2+x \right )^{-\frac {2 \,{\mathrm e}^{x^{2}-3}}{x}} \left ({\mathrm e}^{x}\right )^{\frac {2 \,{\mathrm e}^{x^{2}-3}}{x}} {\mathrm e}^{-\frac {i {\mathrm e}^{x^{2}-3} \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{x}+2+x}\right ) \left (-\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{x}+2+x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{x}+2+x}\right )+\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{x}+2+x}\right )\right )}{x}}\)	\(112\)

[In]

int((((4*x^2-2)*exp(x)+4*x^3+8*x^2-2*x-4)*exp(x^2-3)*ln(exp(x)/(exp(x)+2+x))+(2*x^2+2*x)*exp(x^2-3))*exp(exp(x
^2-3)*ln(exp(x)/(exp(x)+2+x))/x)^2/(exp(x)*x^2+x^3+2*x^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\left (\frac {e^x}{2+e^x+x}\right )^{\frac {2 e^{-3+x^2}}{x}} \left (e^{-3+x^2} \left (2 x+2 x^2\right )+e^{-3+x^2} \left (-4-2 x+8 x^2+4 x^3+e^x \left (-2+4 x^2\right )\right ) \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{2 x^2+e^x x^2+x^3} \, dx=\left (\frac {e^{x}}{x + e^{x} + 2}\right )^{\frac {2 \, e^{\left (x^{2} - 3\right )}}{x}} \]

[In]

integrate((((4*x^2-2)*exp(x)+4*x^3+8*x^2-2*x-4)*exp(x^2-3)*log(exp(x)/(exp(x)+2+x))+(2*x^2+2*x)*exp(x^2-3))*ex
p(exp(x^2-3)*log(exp(x)/(exp(x)+2+x))/x)^2/(exp(x)*x^2+x^3+2*x^2),x, algorithm="fricas")

[Out]

(e^x/(x + e^x + 2))^(2*e^(x^2 - 3)/x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (\frac {e^x}{2+e^x+x}\right )^{\frac {2 e^{-3+x^2}}{x}} \left (e^{-3+x^2} \left (2 x+2 x^2\right )+e^{-3+x^2} \left (-4-2 x+8 x^2+4 x^3+e^x \left (-2+4 x^2\right )\right ) \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{2 x^2+e^x x^2+x^3} \, dx=\text {Timed out} \]

[In]

integrate((((4*x**2-2)*exp(x)+4*x**3+8*x**2-2*x-4)*exp(x**2-3)*ln(exp(x)/(exp(x)+2+x))+(2*x**2+2*x)*exp(x**2-3
))*exp(exp(x**2-3)*ln(exp(x)/(exp(x)+2+x))/x)**2/(exp(x)*x**2+x**3+2*x**2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\left (\frac {e^x}{2+e^x+x}\right )^{\frac {2 e^{-3+x^2}}{x}} \left (e^{-3+x^2} \left (2 x+2 x^2\right )+e^{-3+x^2} \left (-4-2 x+8 x^2+4 x^3+e^x \left (-2+4 x^2\right )\right ) \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{2 x^2+e^x x^2+x^3} \, dx=e^{\left (-\frac {2 \, e^{\left (x^{2} - 3\right )} \log \left (x + e^{x} + 2\right )}{x} + 2 \, e^{\left (x^{2} - 3\right )}\right )} \]

[In]

integrate((((4*x^2-2)*exp(x)+4*x^3+8*x^2-2*x-4)*exp(x^2-3)*log(exp(x)/(exp(x)+2+x))+(2*x^2+2*x)*exp(x^2-3))*ex
p(exp(x^2-3)*log(exp(x)/(exp(x)+2+x))/x)^2/(exp(x)*x^2+x^3+2*x^2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((((4*x^2-2)*exp(x)+4*x^3+8*x^2-2*x-4)*exp(x^2-3)*log(exp(x)/(exp(x)+2+x))+(2*x^2+2*x)*exp(x^2-3))*ex
p(exp(x^2-3)*log(exp(x)/(exp(x)+2+x))/x)^2/(exp(x)*x^2+x^3+2*x^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {\left (\frac {e^x}{2+e^x+x}\right )^{\frac {2 e^{-3+x^2}}{x}} \left (e^{-3+x^2} \left (2 x+2 x^2\right )+e^{-3+x^2} \left (-4-2 x+8 x^2+4 x^3+e^x \left (-2+4 x^2\right )\right ) \log \left (\frac {e^x}{2+e^x+x}\right )\right )}{2 x^2+e^x x^2+x^3} \, dx={\mathrm {e}}^{2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-3}}\,{\left (\frac {1}{x+{\mathrm {e}}^x+2}\right )}^{\frac {2\,{\mathrm {e}}^{x^2-3}}{x}} \]

[In]

int((exp((2*exp(x^2 - 3)*log(exp(x)/(x + exp(x) + 2)))/x)*(exp(x^2 - 3)*(2*x + 2*x^2) + exp(x^2 - 3)*log(exp(x
)/(x + exp(x) + 2))*(exp(x)*(4*x^2 - 2) - 2*x + 8*x^2 + 4*x^3 - 4)))/(x^2*exp(x) + 2*x^2 + x^3),x)

[Out]

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