\(\int \frac {4+4 x+x^2+e^{\frac {2 (-4 x-2 x^2+2 x^3+(-2 x-x^2+x^3) \log (4)+e^x (-2 x^2-x^2 \log (4)))}{2+x}} (-16-16 x+20 x^2+8 x^3+(-8-8 x+10 x^2+4 x^3) \log (4)+e^x (-16 x-12 x^2-4 x^3+(-8 x-6 x^2-2 x^3) \log (4)))}{4+4 x+x^2} \, dx\) [7421]

Optimal result

Integrand size = 143, antiderivative size = 31 \[ \int \frac {4+4 x+x^2+e^{\frac {2 \left (-4 x-2 x^2+2 x^3+\left (-2 x-x^2+x^3\right ) \log (4)+e^x \left (-2 x^2-x^2 \log (4)\right )\right )}{2+x}} \left (-16-16 x+20 x^2+8 x^3+\left (-8-8 x+10 x^2+4 x^3\right ) \log (4)+e^x \left (-16 x-12 x^2-4 x^3+\left (-8 x-6 x^2-2 x^3\right ) \log (4)\right )\right )}{4+4 x+x^2} \, dx=-2+e^{2 \left (-x+\frac {x^2 \left (-e^x+x\right )}{2+x}\right ) (2+\log (4))}+x \]

[Out]

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

Rubi [F]

\[ \int \frac {4+4 x+x^2+e^{\frac {2 \left (-4 x-2 x^2+2 x^3+\left (-2 x-x^2+x^3\right ) \log (4)+e^x \left (-2 x^2-x^2 \log (4)\right )\right )}{2+x}} \left (-16-16 x+20 x^2+8 x^3+\left (-8-8 x+10 x^2+4 x^3\right ) \log (4)+e^x \left (-16 x-12 x^2-4 x^3+\left (-8 x-6 x^2-2 x^3\right ) \log (4)\right )\right )}{4+4 x+x^2} \, dx=\int \frac {4+4 x+x^2+\exp \left (\frac {2 \left (-4 x-2 x^2+2 x^3+\left (-2 x-x^2+x^3\right ) \log (4)+e^x \left (-2 x^2-x^2 \log (4)\right )\right )}{2+x}\right ) \left (-16-16 x+20 x^2+8 x^3+\left (-8-8 x+10 x^2+4 x^3\right ) \log (4)+e^x \left (-16 x-12 x^2-4 x^3+\left (-8 x-6 x^2-2 x^3\right ) \log (4)\right )\right )}{4+4 x+x^2} \, dx \]

[In]

Int[(4 + 4*x + x^2 + E^((2*(-4*x - 2*x^2 + 2*x^3 + (-2*x - x^2 + x^3)*Log[4] + E^x*(-2*x^2 - x^2*Log[4])))/(2
+ x))*(-16 - 16*x + 20*x^2 + 8*x^3 + (-8 - 8*x + 10*x^2 + 4*x^3)*Log[4] + E^x*(-16*x - 12*x^2 - 4*x^3 + (-8*x
- 6*x^2 - 2*x^3)*Log[4])))/(4 + 4*x + x^2),x]

[Out]

x + (2 + Log[4])*Defer[Int][2^((2 - 7*x - 4*x^2 + 4*x^3)/(2 + x))*E^((x*(-6 - 3*x + 4*x^2 - 4*E^x*x*(1 + Log[2
])))/(2 + x)), x] - (2 + Log[4])*Defer[Int][2^((4*(2 - x - x^2 + x^3))/(2 + x))/E^((2*x*(4 - 2*x^2 + x*(2 + E^
x*(2 + Log[4]))))/(2 + x)), x] + 5*(2 + Log[4])*Defer[Int][2^((2 - 7*x - 4*x^2 + 4*x^3)/(2 + x))/E^((2*x*(4 -
2*x^2 + x*(2 + E^x*(2 + Log[4]))))/(2 + x)), x] - (2 + Log[4])*Defer[Int][2^((2 - 7*x - 4*x^2 + 4*x^3)/(2 + x)
)*E^((x*(-6 - 3*x + 4*x^2 - 4*E^x*x*(1 + Log[2])))/(2 + x))*x, x] + (2 + Log[4])*Defer[Int][(2^((2*(2 - 3*x -
2*x^2 + 2*x^3))/(2 + x))*x)/E^((2*x*(4 - 2*x^2 + x*(2 + E^x*(2 + Log[4]))))/(2 + x)), x] + (2 + Log[4])*Defer[
Int][(2^((6 - 5*x - 4*x^2 + 4*x^3)/(2 + x))*E^((x*(-6 - 3*x + 4*x^2 - 4*E^x*x*(1 + Log[2])))/(2 + x)))/(2 + x)
^2, x] + (2 + Log[4])*Defer[Int][2^((4*(2 - x - x^2 + x^3))/(2 + x))/(E^((2*x*(4 - 2*x^2 + x*(2 + E^x*(2 + Log
[4]))))/(2 + x))*(2 + x)^2), x] + 4*(2 + Log[4])*Defer[Int][2^((6 - 5*x - 4*x^2 + 4*x^3)/(2 + x))/(E^((2*x*(4
- 2*x^2 + x*(2 + E^x*(2 + Log[4]))))/(2 + x))*(2 + x)^2), x] - (2 + Log[4])*Defer[Int][2^((10 - 3*x - 4*x^2 +
4*x^3)/(2 + x))/(E^((2*x*(4 - 2*x^2 + x*(2 + E^x*(2 + Log[4]))))/(2 + x))*(2 + x)^2), x] - (2 + Log[4])*Defer[
Int][(2^((6 - 5*x - 4*x^2 + 4*x^3)/(2 + x))*E^((x*(-6 - 3*x + 4*x^2 - 4*E^x*x*(1 + Log[2])))/(2 + x)))/(2 + x)
, x] + 3*(2 + Log[4])*Defer[Int][2^((4*(2 - x - x^2 + x^3))/(2 + x))/(E^((2*x*(4 - 2*x^2 + x*(2 + E^x*(2 + Log
[4]))))/(2 + x))*(2 + x)), x] - 6*(2 + Log[4])*Defer[Int][2^((6 - 5*x - 4*x^2 + 4*x^3)/(2 + x))/(E^((2*x*(4 -
2*x^2 + x*(2 + E^x*(2 + Log[4]))))/(2 + x))*(2 + x)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {4+4 x+x^2+\exp \left (\frac {2 \left (-4 x-2 x^2+2 x^3+\left (-2 x-x^2+x^3\right ) \log (4)+e^x \left (-2 x^2-x^2 \log (4)\right )\right )}{2+x}\right ) \left (-16-16 x+20 x^2+8 x^3+\left (-8-8 x+10 x^2+4 x^3\right ) \log (4)+e^x \left (-16 x-12 x^2-4 x^3+\left (-8 x-6 x^2-2 x^3\right ) \log (4)\right )\right )}{(2+x)^2} \, dx \\ & = \int \frac {4+4 x+x^2-2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) \left (4+4 \left (1+e^x\right ) x+\left (-5+3 e^x\right ) x^2+\left (-2+e^x\right ) x^3\right ) (2+\log (4))}{(2+x)^2} \, dx \\ & = \int \left (1+\frac {2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) \left (4+4 x+4 e^x x-5 x^2+3 e^x x^2-2 x^3+e^x x^3\right ) (-2-\log (4))}{(2+x)^2}\right ) \, dx \\ & = x+(-2-\log (4)) \int \frac {2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) \left (4+4 x+4 e^x x-5 x^2+3 e^x x^2-2 x^3+e^x x^3\right )}{(2+x)^2} \, dx \\ & = x+(-2-\log (4)) \int \left (\frac {2^{2+\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right )}{(2+x)^2}+\frac {2^{2+\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) x}{(2+x)^2}-\frac {5\ 2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) x^2}{(2+x)^2}-\frac {2^{1+\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) x^3}{(2+x)^2}+\frac {2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (x-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) x \left (4+3 x+x^2\right )}{(2+x)^2}\right ) \, dx \\ & = x+(-2-\log (4)) \int \frac {2^{2+\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right )}{(2+x)^2} \, dx+(-2-\log (4)) \int \frac {2^{2+\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) x}{(2+x)^2} \, dx+(-2-\log (4)) \int \frac {2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (x-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) x \left (4+3 x+x^2\right )}{(2+x)^2} \, dx+(2+\log (4)) \int \frac {2^{1+\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) x^3}{(2+x)^2} \, dx+(5 (2+\log (4))) \int \frac {2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) x^2}{(2+x)^2} \, dx \\ & = x+(-2-\log (4)) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right )}{(2+x)^2} \, dx+(-2-\log (4)) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) x}{(2+x)^2} \, dx+(-2-\log (4)) \int \frac {2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}\right ) x \left (4+3 x+x^2\right )}{(2+x)^2} \, dx+(2+\log (4)) \int \frac {2^{\frac {2 \left (2-3 x-2 x^2+2 x^3\right )}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right ) x^3}{(2+x)^2} \, dx+(5 (2+\log (4))) \int \left (2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right )+\frac {2^{2+\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right )}{(2+x)^2}-\frac {2^{2+\frac {2-7 x-4 x^2+4 x^3}{2+x}} \exp \left (-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}\right )}{2+x}\right ) \, dx \\ & = x+(-2-\log (4)) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{(2+x)^2} \, dx+(-2-\log (4)) \int \left (-2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}}+2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}} x-\frac {2^{2+\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}}}{(2+x)^2}+\frac {2^{2+\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}}}{2+x}\right ) \, dx+(-2-\log (4)) \int \left (-\frac {2^{1+\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{(2+x)^2}+\frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{2+x}\right ) \, dx+(2+\log (4)) \int \left (-2^{2+\frac {2 \left (2-3 x-2 x^2+2 x^3\right )}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}+2^{\frac {2 \left (2-3 x-2 x^2+2 x^3\right )}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}} x-\frac {2^{3+\frac {2 \left (2-3 x-2 x^2+2 x^3\right )}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{(2+x)^2}+\frac {3\ 2^{2+\frac {2 \left (2-3 x-2 x^2+2 x^3\right )}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{2+x}\right ) \, dx+(5 (2+\log (4))) \int 2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}} \, dx+(5 (2+\log (4))) \int \frac {2^{2+\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{(2+x)^2} \, dx-(5 (2+\log (4))) \int \frac {2^{2+\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{2+x} \, dx \\ & = x+(-2-\log (4)) \int 2^{2+\frac {2 \left (2-3 x-2 x^2+2 x^3\right )}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}} \, dx+(-2-\log (4)) \int 2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}} x \, dx+(-2-\log (4)) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{(2+x)^2} \, dx+(-2-\log (4)) \int \frac {2^{3+\frac {2 \left (2-3 x-2 x^2+2 x^3\right )}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{(2+x)^2} \, dx+(-2-\log (4)) \int \frac {2^{2+\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}}}{2+x} \, dx+(-2-\log (4)) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{2+x} \, dx+(2+\log (4)) \int 2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}} \, dx+(2+\log (4)) \int 2^{\frac {2 \left (2-3 x-2 x^2+2 x^3\right )}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}} x \, dx+(2+\log (4)) \int \frac {2^{2+\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}}}{(2+x)^2} \, dx+(2+\log (4)) \int \frac {2^{1+\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{(2+x)^2} \, dx+(3 (2+\log (4))) \int \frac {2^{2+\frac {2 \left (2-3 x-2 x^2+2 x^3\right )}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{2+x} \, dx+(5 (2+\log (4))) \int 2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}} \, dx+(5 (2+\log (4))) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{(2+x)^2} \, dx-(5 (2+\log (4))) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{2+x} \, dx \\ & = x+(-2-\log (4)) \int 2^{\frac {4 \left (2-x-x^2+x^3\right )}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}} \, dx+(-2-\log (4)) \int 2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}} x \, dx+(-2-\log (4)) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{(2+x)^2} \, dx+(-2-\log (4)) \int \frac {2^{\frac {10-3 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{(2+x)^2} \, dx+(-2-\log (4)) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}}}{2+x} \, dx+(-2-\log (4)) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{2+x} \, dx+(2+\log (4)) \int 2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}} \, dx+(2+\log (4)) \int 2^{\frac {2 \left (2-3 x-2 x^2+2 x^3\right )}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}} x \, dx+(2+\log (4)) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{\frac {x \left (-6-3 x+4 x^2-4 e^x x (1+\log (2))\right )}{2+x}}}{(2+x)^2} \, dx+(2+\log (4)) \int \frac {2^{\frac {4 \left (2-x-x^2+x^3\right )}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{(2+x)^2} \, dx+(3 (2+\log (4))) \int \frac {2^{\frac {4 \left (2-x-x^2+x^3\right )}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{2+x} \, dx+(5 (2+\log (4))) \int 2^{\frac {2-7 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}} \, dx+(5 (2+\log (4))) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{(2+x)^2} \, dx-(5 (2+\log (4))) \int \frac {2^{\frac {6-5 x-4 x^2+4 x^3}{2+x}} e^{-\frac {2 x \left (4-2 x^2+x \left (2+e^x (2+\log (4))\right )\right )}{2+x}}}{2+x} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {4+4 x+x^2+e^{\frac {2 \left (-4 x-2 x^2+2 x^3+\left (-2 x-x^2+x^3\right ) \log (4)+e^x \left (-2 x^2-x^2 \log (4)\right )\right )}{2+x}} \left (-16-16 x+20 x^2+8 x^3+\left (-8-8 x+10 x^2+4 x^3\right ) \log (4)+e^x \left (-16 x-12 x^2-4 x^3+\left (-8 x-6 x^2-2 x^3\right ) \log (4)\right )\right )}{4+4 x+x^2} \, dx=\int \frac {4+4 x+x^2+e^{\frac {2 \left (-4 x-2 x^2+2 x^3+\left (-2 x-x^2+x^3\right ) \log (4)+e^x \left (-2 x^2-x^2 \log (4)\right )\right )}{2+x}} \left (-16-16 x+20 x^2+8 x^3+\left (-8-8 x+10 x^2+4 x^3\right ) \log (4)+e^x \left (-16 x-12 x^2-4 x^3+\left (-8 x-6 x^2-2 x^3\right ) \log (4)\right )\right )}{4+4 x+x^2} \, dx \]

[In]

Integrate[(4 + 4*x + x^2 + E^((2*(-4*x - 2*x^2 + 2*x^3 + (-2*x - x^2 + x^3)*Log[4] + E^x*(-2*x^2 - x^2*Log[4])
))/(2 + x))*(-16 - 16*x + 20*x^2 + 8*x^3 + (-8 - 8*x + 10*x^2 + 4*x^3)*Log[4] + E^x*(-16*x - 12*x^2 - 4*x^3 +
(-8*x - 6*x^2 - 2*x^3)*Log[4])))/(4 + 4*x + x^2),x]

[Out]

Integrate[(4 + 4*x + x^2 + E^((2*(-4*x - 2*x^2 + 2*x^3 + (-2*x - x^2 + x^3)*Log[4] + E^x*(-2*x^2 - x^2*Log[4])
))/(2 + x))*(-16 - 16*x + 20*x^2 + 8*x^3 + (-8 - 8*x + 10*x^2 + 4*x^3)*Log[4] + E^x*(-16*x - 12*x^2 - 4*x^3 +
(-8*x - 6*x^2 - 2*x^3)*Log[4])))/(4 + 4*x + x^2), x]

Maple [A] (verified)

Time = 1.88 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90

method	result	size
risch	\({\mathrm e}^{-\frac {4 x \left (1+\ln \left (2\right )\right ) \left ({\mathrm e}^{x} x -x^{2}+x +2\right )}{2+x}}+x\)	\(28\)
parallelrisch	\({\mathrm e}^{\frac {2 \left (-2 x^{2} \ln \left (2\right )-2 x^{2}\right ) {\mathrm e}^{x}+2 \left (2 x^{3}-2 x^{2}-4 x \right ) \ln \left (2\right )+4 x^{3}-4 x^{2}-8 x}{2+x}}+x -8\)	\(59\)
norman	\(\frac {x^{2}+x \,{\mathrm e}^{\frac {2 \left (-2 x^{2} \ln \left (2\right )-2 x^{2}\right ) {\mathrm e}^{x}+2 \left (2 x^{3}-2 x^{2}-4 x \right ) \ln \left (2\right )+4 x^{3}-4 x^{2}-8 x}{2+x}}+2 \,{\mathrm e}^{\frac {2 \left (-2 x^{2} \ln \left (2\right )-2 x^{2}\right ) {\mathrm e}^{x}+2 \left (2 x^{3}-2 x^{2}-4 x \right ) \ln \left (2\right )+4 x^{3}-4 x^{2}-8 x}{2+x}}-4}{2+x}\)	\(126\)

[In]

int((((2*(-2*x^3-6*x^2-8*x)*ln(2)-4*x^3-12*x^2-16*x)*exp(x)+2*(4*x^3+10*x^2-8*x-8)*ln(2)+8*x^3+20*x^2-16*x-16)
*exp(((-2*x^2*ln(2)-2*x^2)*exp(x)+2*(x^3-x^2-2*x)*ln(2)+2*x^3-2*x^2-4*x)/(2+x))^2+x^2+4*x+4)/(x^2+4*x+4),x,met
hod=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {4+4 x+x^2+e^{\frac {2 \left (-4 x-2 x^2+2 x^3+\left (-2 x-x^2+x^3\right ) \log (4)+e^x \left (-2 x^2-x^2 \log (4)\right )\right )}{2+x}} \left (-16-16 x+20 x^2+8 x^3+\left (-8-8 x+10 x^2+4 x^3\right ) \log (4)+e^x \left (-16 x-12 x^2-4 x^3+\left (-8 x-6 x^2-2 x^3\right ) \log (4)\right )\right )}{4+4 x+x^2} \, dx=x + e^{\left (\frac {4 \, {\left (x^{3} - x^{2} - {\left (x^{2} \log \left (2\right ) + x^{2}\right )} e^{x} + {\left (x^{3} - x^{2} - 2 \, x\right )} \log \left (2\right ) - 2 \, x\right )}}{x + 2}\right )} \]

[In]

integrate((((2*(-2*x^3-6*x^2-8*x)*log(2)-4*x^3-12*x^2-16*x)*exp(x)+2*(4*x^3+10*x^2-8*x-8)*log(2)+8*x^3+20*x^2-
16*x-16)*exp(((-2*x^2*log(2)-2*x^2)*exp(x)+2*(x^3-x^2-2*x)*log(2)+2*x^3-2*x^2-4*x)/(2+x))^2+x^2+4*x+4)/(x^2+4*
x+4),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).

Time = 0.34 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.87 \[ \int \frac {4+4 x+x^2+e^{\frac {2 \left (-4 x-2 x^2+2 x^3+\left (-2 x-x^2+x^3\right ) \log (4)+e^x \left (-2 x^2-x^2 \log (4)\right )\right )}{2+x}} \left (-16-16 x+20 x^2+8 x^3+\left (-8-8 x+10 x^2+4 x^3\right ) \log (4)+e^x \left (-16 x-12 x^2-4 x^3+\left (-8 x-6 x^2-2 x^3\right ) \log (4)\right )\right )}{4+4 x+x^2} \, dx=x + e^{\frac {2 \cdot \left (2 x^{3} - 2 x^{2} - 4 x + \left (- 2 x^{2} - 2 x^{2} \log {\left (2 \right )}\right ) e^{x} + \left (2 x^{3} - 2 x^{2} - 4 x\right ) \log {\left (2 \right )}\right )}{x + 2}} \]

[In]

integrate((((2*(-2*x**3-6*x**2-8*x)*ln(2)-4*x**3-12*x**2-16*x)*exp(x)+2*(4*x**3+10*x**2-8*x-8)*ln(2)+8*x**3+20
*x**2-16*x-16)*exp(((-2*x**2*ln(2)-2*x**2)*exp(x)+2*(x**3-x**2-2*x)*ln(2)+2*x**3-2*x**2-4*x)/(2+x))**2+x**2+4*
x+4)/(x**2+4*x+4),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (29) = 58\).

Time = 0.55 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.74 \[ \int \frac {4+4 x+x^2+e^{\frac {2 \left (-4 x-2 x^2+2 x^3+\left (-2 x-x^2+x^3\right ) \log (4)+e^x \left (-2 x^2-x^2 \log (4)\right )\right )}{2+x}} \left (-16-16 x+20 x^2+8 x^3+\left (-8-8 x+10 x^2+4 x^3\right ) \log (4)+e^x \left (-16 x-12 x^2-4 x^3+\left (-8 x-6 x^2-2 x^3\right ) \log (4)\right )\right )}{4+4 x+x^2} \, dx=x + 65536 \, e^{\left (4 \, x^{2} \log \left (2\right ) - 4 \, x e^{x} \log \left (2\right ) + 4 \, x^{2} - 4 \, x e^{x} - 12 \, x \log \left (2\right ) + 8 \, e^{x} \log \left (2\right ) - 12 \, x - \frac {16 \, e^{x} \log \left (2\right )}{x + 2} - \frac {16 \, e^{x}}{x + 2} - \frac {32 \, \log \left (2\right )}{x + 2} - \frac {32}{x + 2} + 8 \, e^{x} + 16\right )} \]

[In]

integrate((((2*(-2*x^3-6*x^2-8*x)*log(2)-4*x^3-12*x^2-16*x)*exp(x)+2*(4*x^3+10*x^2-8*x-8)*log(2)+8*x^3+20*x^2-
16*x-16)*exp(((-2*x^2*log(2)-2*x^2)*exp(x)+2*(x^3-x^2-2*x)*log(2)+2*x^3-2*x^2-4*x)/(2+x))^2+x^2+4*x+4)/(x^2+4*
x+4),x, algorithm="maxima")

[Out]

x + 65536*e^(4*x^2*log(2) - 4*x*e^x*log(2) + 4*x^2 - 4*x*e^x - 12*x*log(2) + 8*e^x*log(2) - 12*x - 16*e^x*log(
2)/(x + 2) - 16*e^x/(x + 2) - 32*log(2)/(x + 2) - 32/(x + 2) + 8*e^x + 16)

Giac [F]

\[ \int \frac {4+4 x+x^2+e^{\frac {2 \left (-4 x-2 x^2+2 x^3+\left (-2 x-x^2+x^3\right ) \log (4)+e^x \left (-2 x^2-x^2 \log (4)\right )\right )}{2+x}} \left (-16-16 x+20 x^2+8 x^3+\left (-8-8 x+10 x^2+4 x^3\right ) \log (4)+e^x \left (-16 x-12 x^2-4 x^3+\left (-8 x-6 x^2-2 x^3\right ) \log (4)\right )\right )}{4+4 x+x^2} \, dx=\int { \frac {x^{2} + 4 \, {\left (2 \, x^{3} + 5 \, x^{2} - {\left (x^{3} + 3 \, x^{2} + {\left (x^{3} + 3 \, x^{2} + 4 \, x\right )} \log \left (2\right ) + 4 \, x\right )} e^{x} + {\left (2 \, x^{3} + 5 \, x^{2} - 4 \, x - 4\right )} \log \left (2\right ) - 4 \, x - 4\right )} e^{\left (\frac {4 \, {\left (x^{3} - x^{2} - {\left (x^{2} \log \left (2\right ) + x^{2}\right )} e^{x} + {\left (x^{3} - x^{2} - 2 \, x\right )} \log \left (2\right ) - 2 \, x\right )}}{x + 2}\right )} + 4 \, x + 4}{x^{2} + 4 \, x + 4} \,d x } \]

[In]

integrate((((2*(-2*x^3-6*x^2-8*x)*log(2)-4*x^3-12*x^2-16*x)*exp(x)+2*(4*x^3+10*x^2-8*x-8)*log(2)+8*x^3+20*x^2-
16*x-16)*exp(((-2*x^2*log(2)-2*x^2)*exp(x)+2*(x^3-x^2-2*x)*log(2)+2*x^3-2*x^2-4*x)/(2+x))^2+x^2+4*x+4)/(x^2+4*
x+4),x, algorithm="giac")

[Out]

integrate((x^2 + 4*(2*x^3 + 5*x^2 - (x^3 + 3*x^2 + (x^3 + 3*x^2 + 4*x)*log(2) + 4*x)*e^x + (2*x^3 + 5*x^2 - 4*
x - 4)*log(2) - 4*x - 4)*e^(4*(x^3 - x^2 - (x^2*log(2) + x^2)*e^x + (x^3 - x^2 - 2*x)*log(2) - 2*x)/(x + 2)) +
4*x + 4)/(x^2 + 4*x + 4), x)

Mupad [B] (verification not implemented)

Time = 12.42 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.26 \[ \int \frac {4+4 x+x^2+e^{\frac {2 \left (-4 x-2 x^2+2 x^3+\left (-2 x-x^2+x^3\right ) \log (4)+e^x \left (-2 x^2-x^2 \log (4)\right )\right )}{2+x}} \left (-16-16 x+20 x^2+8 x^3+\left (-8-8 x+10 x^2+4 x^3\right ) \log (4)+e^x \left (-16 x-12 x^2-4 x^3+\left (-8 x-6 x^2-2 x^3\right ) \log (4)\right )\right )}{4+4 x+x^2} \, dx=x+\frac {2^{\frac {4\,x^3}{x+2}}\,{\mathrm {e}}^{-\frac {8\,x}{x+2}}\,{\mathrm {e}}^{-\frac {4\,x^2\,{\mathrm {e}}^x}{x+2}}\,{\mathrm {e}}^{-\frac {4\,x^2}{x+2}}\,{\mathrm {e}}^{\frac {4\,x^3}{x+2}}}{2^{\frac {8\,x}{x+2}}\,2^{\frac {4\,x^2\,{\mathrm {e}}^x}{x+2}}\,2^{\frac {4\,x^2}{x+2}}} \]

[In]

int((4*x + x^2 - exp(-(2*(4*x + 2*log(2)*(2*x + x^2 - x^3) + exp(x)*(2*x^2*log(2) + 2*x^2) + 2*x^2 - 2*x^3))/(
x + 2))*(16*x + exp(x)*(16*x + 2*log(2)*(8*x + 6*x^2 + 2*x^3) + 12*x^2 + 4*x^3) + 2*log(2)*(8*x - 10*x^2 - 4*x
^3 + 8) - 20*x^2 - 8*x^3 + 16) + 4)/(4*x + x^2 + 4),x)

[Out]

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