3.63.66 \(\int \frac {-8 x+10 x^2-2 x^3+e^{2+2 x} (-20-6 x+2 x^2)+e^{1+x} (16-2 x-8 x^2+2 x^3)+e^{\frac {x^2}{4}} (-2 x+2 x^2+e^{1+x} (4-3 x^2)+e^{2+2 x} (-4+x^2))+(e^{2+2 x} (10-2 x)+e^{1+x} (-8+10 x-2 x^2)+e^{\frac {x^2}{4}} (e^{2+2 x} (2-x)+e^{1+x} (-2+2 x))) \log (-4-e^{\frac {x^2}{4}}+x)}{e^{2+2 x+\frac {x^2}{4}}+e^{2+2 x} (4-x)} \, dx\)

Optimal. Leaf size=35 \[ 8-\left (-2-x+e^{-1-x} x+\log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )^2 \]

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Rubi [F]  time = 22.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-8 x+10 x^2-2 x^3+e^{2+2 x} \left (-20-6 x+2 x^2\right )+e^{1+x} \left (16-2 x-8 x^2+2 x^3\right )+e^{\frac {x^2}{4}} \left (-2 x+2 x^2+e^{1+x} \left (4-3 x^2\right )+e^{2+2 x} \left (-4+x^2\right )\right )+\left (e^{2+2 x} (10-2 x)+e^{1+x} \left (-8+10 x-2 x^2\right )+e^{\frac {x^2}{4}} \left (e^{2+2 x} (2-x)+e^{1+x} (-2+2 x)\right )\right ) \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{e^{2+2 x+\frac {x^2}{4}}+e^{2+2 x} (4-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*x + 10*x^2 - 2*x^3 + E^(2 + 2*x)*(-20 - 6*x + 2*x^2) + E^(1 + x)*(16 - 2*x - 8*x^2 + 2*x^3) + E^(x^2/4
)*(-2*x + 2*x^2 + E^(1 + x)*(4 - 3*x^2) + E^(2 + 2*x)*(-4 + x^2)) + (E^(2 + 2*x)*(10 - 2*x) + E^(1 + x)*(-8 +
10*x - 2*x^2) + E^(x^2/4)*(E^(2 + 2*x)*(2 - x) + E^(1 + x)*(-2 + 2*x)))*Log[-4 - E^(x^2/4) + x])/(E^(2 + 2*x +
x^2/4) + E^(2 + 2*x)*(4 - x)),x]

[Out]

4*E^(-1 - x)*x - E^(-2 - 2*x)*x^2 + E^(-1 - x)*x^2 - 2*E^(-1 - x)*Log[-4 - E^(x^2/4) + x] + 2*E^(-1 - x)*(1 -
x)*Log[-4 - E^(x^2/4) + x] - 20*Defer[Int][(4 + E^(x^2/4) - x)^(-1), x] + 10*Log[-4 - E^(x^2/4) + x]*Defer[Int
][(4 + E^(x^2/4) - x)^(-1), x] - 4*Defer[Int][E^(x^2/4)/(4 + E^(x^2/4) - x), x] + 2*Log[-4 - E^(x^2/4) + x]*De
fer[Int][E^(x^2/4)/(4 + E^(x^2/4) - x), x] - 6*Defer[Int][x/(4 + E^(x^2/4) - x), x] - 2*Log[-4 - E^(x^2/4) + x
]*Defer[Int][x/(4 + E^(x^2/4) - x), x] + 8*Defer[Int][(E^(-1 - x)*x)/(4 + E^(x^2/4) - x), x] - Log[-4 - E^(x^2
/4) + x]*Defer[Int][(E^(x^2/4)*x)/(4 + E^(x^2/4) - x), x] + 2*Defer[Int][(E^(-1 - x + x^2/4)*x)/(4 + E^(x^2/4)
- x), x] + 2*Defer[Int][x^2/(4 + E^(x^2/4) - x), x] - 6*Defer[Int][(E^(-1 - x)*x^2)/(4 + E^(x^2/4) - x), x] +
Defer[Int][(E^(x^2/4)*x^2)/(4 + E^(x^2/4) - x), x] - Defer[Int][(E^(-1 - x + x^2/4)*x^2)/(4 + E^(x^2/4) - x),
x] + Defer[Int][(E^(-1 - x)*x^3)/(4 + E^(x^2/4) - x), x] + 10*Defer[Int][Defer[Int][(4 + E^(x^2/4) - x)^(-1),
x]/(4 + E^(x^2/4) - x), x] - 5*Defer[Int][x*Defer[Int][(4 + E^(x^2/4) - x)^(-1), x], x] + 20*Defer[Int][(x*De
fer[Int][(4 + E^(x^2/4) - x)^(-1), x])/(4 + E^(x^2/4) - x), x] - 5*Defer[Int][(x^2*Defer[Int][(4 + E^(x^2/4) -
x)^(-1), x])/(4 + E^(x^2/4) - x), x] + 2*Defer[Int][Defer[Int][E^(x^2/4)/(4 + E^(x^2/4) - x), x]/(4 + E^(x^2/
4) - x), x] - Defer[Int][x*Defer[Int][E^(x^2/4)/(4 + E^(x^2/4) - x), x], x] + 4*Defer[Int][(x*Defer[Int][E^(x^
2/4)/(4 + E^(x^2/4) - x), x])/(4 + E^(x^2/4) - x), x] - Defer[Int][(x^2*Defer[Int][E^(x^2/4)/(4 + E^(x^2/4) -
x), x])/(4 + E^(x^2/4) - x), x] - 2*Defer[Int][Defer[Int][x/(4 + E^(x^2/4) - x), x]/(4 + E^(x^2/4) - x), x] +
Defer[Int][x*Defer[Int][x/(4 + E^(x^2/4) - x), x], x] - 4*Defer[Int][(x*Defer[Int][x/(4 + E^(x^2/4) - x), x])/
(4 + E^(x^2/4) - x), x] + Defer[Int][(x^2*Defer[Int][x/(4 + E^(x^2/4) - x), x])/(4 + E^(x^2/4) - x), x] - Defe
r[Int][Defer[Int][(E^(x^2/4)*x)/(4 + E^(x^2/4) - x), x]/(4 + E^(x^2/4) - x), x] + Defer[Int][x*Defer[Int][(E^(
x^2/4)*x)/(4 + E^(x^2/4) - x), x], x]/2 - 2*Defer[Int][(x*Defer[Int][(E^(x^2/4)*x)/(4 + E^(x^2/4) - x), x])/(4
+ E^(x^2/4) - x), x] + Defer[Int][(x^2*Defer[Int][(E^(x^2/4)*x)/(4 + E^(x^2/4) - x), x])/(4 + E^(x^2/4) - x),
x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-2-2 x} \left (2 e^{1+x} (-5+x)+e^{\frac {1}{4} (2+x)^2} (-2+x)-2 e^{\frac {x^2}{4}} (-1+x)+2 \left (4-5 x+x^2\right )\right ) \left (-x+e^{1+x} (2+x)-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x} \, dx\\ &=\int \left (\frac {e^{-2-2 x+\frac {1}{4} (2+x)^2} (-2+x) \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x}-\frac {2 e^{-2-2 x} \left (-4-e^{\frac {x^2}{4}}+5 e^{1+x}+5 x+e^{\frac {x^2}{4}} x-e^{1+x} x-x^2\right ) \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x}\right ) \, dx\\ &=-\left (2 \int \frac {e^{-2-2 x} \left (-4-e^{\frac {x^2}{4}}+5 e^{1+x}+5 x+e^{\frac {x^2}{4}} x-e^{1+x} x-x^2\right ) \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x} \, dx\right )+\int \frac {e^{-2-2 x+\frac {1}{4} (2+x)^2} (-2+x) \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x} \, dx\\ &=-\left (2 \int \frac {e^{-2-2 x} \left (4+e^{1+x} (-5+x)-e^{\frac {x^2}{4}} (-1+x)-5 x+x^2\right ) \left (x-e^{1+x} (2+x)+e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x} \, dx\right )+\int \frac {e^{-1-x+\frac {x^2}{4}} (2-x) \left (-2 e^{1+x}+x-e^{1+x} x+e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x} \, dx\\ &=-\left (2 \int \left (-\frac {e^{-1-x} (-5+x) \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x}+e^{-2-2 x} (-1+x) \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )\right ) \, dx\right )+\int \left (-\frac {2 e^{-1-x+\frac {x^2}{4}} \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x}+\frac {e^{-1-x+\frac {x^2}{4}} x \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x}\right ) \, dx\\ &=-\left (2 \int \frac {e^{-1-x+\frac {x^2}{4}} \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x} \, dx\right )+2 \int \frac {e^{-1-x} (-5+x) \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x} \, dx-2 \int e^{-2-2 x} (-1+x) \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right ) \, dx+\int \frac {e^{-1-x+\frac {x^2}{4}} x \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x} \, dx\\ &=-\left (2 \int \left (-e^{-2-2 x} (-1+x) x+e^{-1-x} (-1+x) \left (2+x-\log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )\right ) \, dx\right )-2 \int \left (\frac {2 e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x}+\frac {e^{\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x}-\frac {e^{-1-x+\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x}-\frac {e^{\frac {x^2}{4}} \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{4+e^{\frac {x^2}{4}}-x}\right ) \, dx+2 \int \left (-\frac {5 e^{-1-x} \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x}+\frac {e^{-1-x} x \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x}\right ) \, dx+\int \left (\frac {2 e^{\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x}+\frac {e^{\frac {x^2}{4}} x^2}{4+e^{\frac {x^2}{4}}-x}-\frac {e^{-1-x+\frac {x^2}{4}} x^2}{4+e^{\frac {x^2}{4}}-x}-\frac {e^{\frac {x^2}{4}} x \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{4+e^{\frac {x^2}{4}}-x}\right ) \, dx\\ &=2 \int \frac {e^{-1-x+\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx+2 \int e^{-2-2 x} (-1+x) x \, dx-2 \int e^{-1-x} (-1+x) \left (2+x-\log \left (-4-e^{\frac {x^2}{4}}+x\right )\right ) \, dx+2 \int \frac {e^{\frac {x^2}{4}} \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{4+e^{\frac {x^2}{4}}-x} \, dx+2 \int \frac {e^{-1-x} x \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x} \, dx-4 \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx-10 \int \frac {e^{-1-x} \left (2 e^{1+x}-x+e^{1+x} x-e^{1+x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x} \, dx+\int \frac {e^{\frac {x^2}{4}} x^2}{4+e^{\frac {x^2}{4}}-x} \, dx-\int \frac {e^{-1-x+\frac {x^2}{4}} x^2}{4+e^{\frac {x^2}{4}}-x} \, dx-\int \frac {e^{\frac {x^2}{4}} x \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{4+e^{\frac {x^2}{4}}-x} \, dx\\ &=2 \int \frac {e^{-1-x+\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx+2 \int \left (-e^{-2-2 x} x+e^{-2-2 x} x^2\right ) \, dx+2 \int \frac {x \left (2+x-e^{-1-x} x-\log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )}{4+e^{\frac {x^2}{4}}-x} \, dx-2 \int \left (-2 e^{-1-x}+e^{-1-x} x+e^{-1-x} x^2-e^{-1-x} (-1+x) \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right ) \, dx-2 \int \frac {\left (-2+e^{\frac {x^2}{4}} x\right ) \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx}{2 \left (4+e^{\frac {x^2}{4}}-x\right )} \, dx-4 \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx-10 \int \frac {2+x-e^{-1-x} x-\log \left (-4-e^{\frac {x^2}{4}}+x\right )}{4+e^{\frac {x^2}{4}}-x} \, dx-\log \left (-4-e^{\frac {x^2}{4}}+x\right ) \int \frac {e^{\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx+\left (2 \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right ) \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx+\int \frac {e^{\frac {x^2}{4}} x^2}{4+e^{\frac {x^2}{4}}-x} \, dx-\int \frac {e^{-1-x+\frac {x^2}{4}} x^2}{4+e^{\frac {x^2}{4}}-x} \, dx+\int \frac {\left (-2+e^{\frac {x^2}{4}} x\right ) \int \frac {e^{\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx}{2 \left (4+e^{\frac {x^2}{4}}-x\right )} \, dx\\ &=\frac {1}{2} \int \frac {\left (-2+e^{\frac {x^2}{4}} x\right ) \int \frac {e^{\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx}{4+e^{\frac {x^2}{4}}-x} \, dx-2 \int e^{-2-2 x} x \, dx-2 \int e^{-1-x} x \, dx+2 \int \frac {e^{-1-x+\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx+2 \int e^{-2-2 x} x^2 \, dx-2 \int e^{-1-x} x^2 \, dx+2 \int e^{-1-x} (-1+x) \log \left (-4-e^{\frac {x^2}{4}}+x\right ) \, dx+2 \int \left (\frac {2 x}{4+e^{\frac {x^2}{4}}-x}+\frac {x^2}{4+e^{\frac {x^2}{4}}-x}-\frac {e^{-1-x} x^2}{4+e^{\frac {x^2}{4}}-x}-\frac {x \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{4+e^{\frac {x^2}{4}}-x}\right ) \, dx+4 \int e^{-1-x} \, dx-4 \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx-10 \int \left (\frac {2}{4+e^{\frac {x^2}{4}}-x}+\frac {x}{4+e^{\frac {x^2}{4}}-x}-\frac {e^{-1-x} x}{4+e^{\frac {x^2}{4}}-x}-\frac {\log \left (-4-e^{\frac {x^2}{4}}+x\right )}{4+e^{\frac {x^2}{4}}-x}\right ) \, dx-\log \left (-4-e^{\frac {x^2}{4}}+x\right ) \int \frac {e^{\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx+\left (2 \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right ) \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx+\int \frac {e^{\frac {x^2}{4}} x^2}{4+e^{\frac {x^2}{4}}-x} \, dx-\int \frac {e^{-1-x+\frac {x^2}{4}} x^2}{4+e^{\frac {x^2}{4}}-x} \, dx-\int \frac {\left (-2+e^{\frac {x^2}{4}} x\right ) \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx}{4+e^{\frac {x^2}{4}}-x} \, dx\\ &=-4 e^{-1-x}+e^{-2-2 x} x+2 e^{-1-x} x-e^{-2-2 x} x^2+2 e^{-1-x} x^2-2 e^{-1-x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )+2 e^{-1-x} (1-x) \log \left (-4-e^{\frac {x^2}{4}}+x\right )+\frac {1}{2} \int \left (x \int \frac {e^{\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx+\frac {\left (-2-4 x+x^2\right ) \int \frac {e^{\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx}{4+e^{\frac {x^2}{4}}-x}\right ) \, dx-2 \int e^{-1-x} \, dx+2 \int e^{-2-2 x} x \, dx+2 \int \frac {e^{-1-x+\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx+2 \int \frac {x^2}{4+e^{\frac {x^2}{4}}-x} \, dx-2 \int \frac {e^{-1-x} x^2}{4+e^{\frac {x^2}{4}}-x} \, dx-2 \int \frac {e^{-1-x} x \left (2-e^{\frac {x^2}{4}} x\right )}{2 \left (4+e^{\frac {x^2}{4}}-x\right )} \, dx-2 \int \frac {x \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{4+e^{\frac {x^2}{4}}-x} \, dx-4 \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx-4 \int e^{-1-x} x \, dx+4 \int \frac {x}{4+e^{\frac {x^2}{4}}-x} \, dx-10 \int \frac {x}{4+e^{\frac {x^2}{4}}-x} \, dx+10 \int \frac {e^{-1-x} x}{4+e^{\frac {x^2}{4}}-x} \, dx+10 \int \frac {\log \left (-4-e^{\frac {x^2}{4}}+x\right )}{4+e^{\frac {x^2}{4}}-x} \, dx-20 \int \frac {1}{4+e^{\frac {x^2}{4}}-x} \, dx-\log \left (-4-e^{\frac {x^2}{4}}+x\right ) \int \frac {e^{\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx+\left (2 \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right ) \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx-\int e^{-2-2 x} \, dx+\int \frac {e^{\frac {x^2}{4}} x^2}{4+e^{\frac {x^2}{4}}-x} \, dx-\int \frac {e^{-1-x+\frac {x^2}{4}} x^2}{4+e^{\frac {x^2}{4}}-x} \, dx-\int \left (x \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx+\frac {\left (-2-4 x+x^2\right ) \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx}{4+e^{\frac {x^2}{4}}-x}\right ) \, dx\\ &=\frac {1}{2} e^{-2-2 x}-2 e^{-1-x}+6 e^{-1-x} x-e^{-2-2 x} x^2+2 e^{-1-x} x^2-2 e^{-1-x} \log \left (-4-e^{\frac {x^2}{4}}+x\right )+2 e^{-1-x} (1-x) \log \left (-4-e^{\frac {x^2}{4}}+x\right )+\frac {1}{2} \int x \int \frac {e^{\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx \, dx+\frac {1}{2} \int \frac {\left (-2-4 x+x^2\right ) \int \frac {e^{\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx}{4+e^{\frac {x^2}{4}}-x} \, dx+2 \int \frac {e^{-1-x+\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx+2 \int \frac {x^2}{4+e^{\frac {x^2}{4}}-x} \, dx-2 \int \frac {e^{-1-x} x^2}{4+e^{\frac {x^2}{4}}-x} \, dx+2 \int \frac {\left (-2+e^{\frac {x^2}{4}} x\right ) \int \frac {x}{4+e^{\frac {x^2}{4}}-x} \, dx}{2 \left (4+e^{\frac {x^2}{4}}-x\right )} \, dx-4 \int e^{-1-x} \, dx-4 \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx+4 \int \frac {x}{4+e^{\frac {x^2}{4}}-x} \, dx-10 \int \frac {x}{4+e^{\frac {x^2}{4}}-x} \, dx+10 \int \frac {e^{-1-x} x}{4+e^{\frac {x^2}{4}}-x} \, dx-10 \int \frac {\left (-2+e^{\frac {x^2}{4}} x\right ) \int \frac {1}{4+e^{\frac {x^2}{4}}-x} \, dx}{2 \left (4+e^{\frac {x^2}{4}}-x\right )} \, dx-20 \int \frac {1}{4+e^{\frac {x^2}{4}}-x} \, dx-\log \left (-4-e^{\frac {x^2}{4}}+x\right ) \int \frac {e^{\frac {x^2}{4}} x}{4+e^{\frac {x^2}{4}}-x} \, dx+\left (2 \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right ) \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx-\left (2 \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right ) \int \frac {x}{4+e^{\frac {x^2}{4}}-x} \, dx+\left (10 \log \left (-4-e^{\frac {x^2}{4}}+x\right )\right ) \int \frac {1}{4+e^{\frac {x^2}{4}}-x} \, dx+\int e^{-2-2 x} \, dx+\int \frac {e^{\frac {x^2}{4}} x^2}{4+e^{\frac {x^2}{4}}-x} \, dx-\int \frac {e^{-1-x+\frac {x^2}{4}} x^2}{4+e^{\frac {x^2}{4}}-x} \, dx-\int \frac {e^{-1-x} x \left (2-e^{\frac {x^2}{4}} x\right )}{4+e^{\frac {x^2}{4}}-x} \, dx-\int x \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx \, dx-\int \frac {\left (-2-4 x+x^2\right ) \int \frac {e^{\frac {x^2}{4}}}{4+e^{\frac {x^2}{4}}-x} \, dx}{4+e^{\frac {x^2}{4}}-x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 2.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-8 x+10 x^2-2 x^3+e^{2+2 x} \left (-20-6 x+2 x^2\right )+e^{1+x} \left (16-2 x-8 x^2+2 x^3\right )+e^{\frac {x^2}{4}} \left (-2 x+2 x^2+e^{1+x} \left (4-3 x^2\right )+e^{2+2 x} \left (-4+x^2\right )\right )+\left (e^{2+2 x} (10-2 x)+e^{1+x} \left (-8+10 x-2 x^2\right )+e^{\frac {x^2}{4}} \left (e^{2+2 x} (2-x)+e^{1+x} (-2+2 x)\right )\right ) \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{e^{2+2 x+\frac {x^2}{4}}+e^{2+2 x} (4-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-8*x + 10*x^2 - 2*x^3 + E^(2 + 2*x)*(-20 - 6*x + 2*x^2) + E^(1 + x)*(16 - 2*x - 8*x^2 + 2*x^3) + E^
(x^2/4)*(-2*x + 2*x^2 + E^(1 + x)*(4 - 3*x^2) + E^(2 + 2*x)*(-4 + x^2)) + (E^(2 + 2*x)*(10 - 2*x) + E^(1 + x)*
(-8 + 10*x - 2*x^2) + E^(x^2/4)*(E^(2 + 2*x)*(2 - x) + E^(1 + x)*(-2 + 2*x)))*Log[-4 - E^(x^2/4) + x])/(E^(2 +
2*x + x^2/4) + E^(2 + 2*x)*(4 - x)),x]

[Out]

Integrate[(-8*x + 10*x^2 - 2*x^3 + E^(2 + 2*x)*(-20 - 6*x + 2*x^2) + E^(1 + x)*(16 - 2*x - 8*x^2 + 2*x^3) + E^
(x^2/4)*(-2*x + 2*x^2 + E^(1 + x)*(4 - 3*x^2) + E^(2 + 2*x)*(-4 + x^2)) + (E^(2 + 2*x)*(10 - 2*x) + E^(1 + x)*
(-8 + 10*x - 2*x^2) + E^(x^2/4)*(E^(2 + 2*x)*(2 - x) + E^(1 + x)*(-2 + 2*x)))*Log[-4 - E^(x^2/4) + x])/(E^(2 +
2*x + x^2/4) + E^(2 + 2*x)*(4 - x)), x]

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fricas [B]  time = 1.62, size = 132, normalized size = 3.77 \begin {gather*} -{\left (e^{\left (2 \, x + 2\right )} \log \left ({\left ({\left (x - 4\right )} e^{\left (2 \, x + 2\right )} - e^{\left (\frac {1}{4} \, x^{2} + 2 \, x + 2\right )}\right )} e^{\left (-2 \, x - 2\right )}\right )^{2} + x^{2} + {\left (x^{2} + 4 \, x\right )} e^{\left (2 \, x + 2\right )} - 2 \, {\left (x^{2} + 2 \, x\right )} e^{\left (x + 1\right )} - 2 \, {\left ({\left (x + 2\right )} e^{\left (2 \, x + 2\right )} - x e^{\left (x + 1\right )}\right )} \log \left ({\left ({\left (x - 4\right )} e^{\left (2 \, x + 2\right )} - e^{\left (\frac {1}{4} \, x^{2} + 2 \, x + 2\right )}\right )} e^{\left (-2 \, x - 2\right )}\right )\right )} e^{\left (-2 \, x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2-x)*exp(x+1)^2+(2*x-2)*exp(x+1))*exp(1/4*x^2)+(-2*x+10)*exp(x+1)^2+(-2*x^2+10*x-8)*exp(x+1))*lo
g(-exp(1/4*x^2)+x-4)+((x^2-4)*exp(x+1)^2+(-3*x^2+4)*exp(x+1)+2*x^2-2*x)*exp(1/4*x^2)+(2*x^2-6*x-20)*exp(x+1)^2
+(2*x^3-8*x^2-2*x+16)*exp(x+1)-2*x^3+10*x^2-8*x)/(exp(x+1)^2*exp(1/4*x^2)+(-x+4)*exp(x+1)^2),x, algorithm="fri
cas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{3} - 10 \, x^{2} - {\left (2 \, x^{2} + {\left (x^{2} - 4\right )} e^{\left (2 \, x + 2\right )} - {\left (3 \, x^{2} - 4\right )} e^{\left (x + 1\right )} - 2 \, x\right )} e^{\left (\frac {1}{4} \, x^{2}\right )} - 2 \, {\left (x^{2} - 3 \, x - 10\right )} e^{\left (2 \, x + 2\right )} - 2 \, {\left (x^{3} - 4 \, x^{2} - x + 8\right )} e^{\left (x + 1\right )} + {\left ({\left ({\left (x - 2\right )} e^{\left (2 \, x + 2\right )} - 2 \, {\left (x - 1\right )} e^{\left (x + 1\right )}\right )} e^{\left (\frac {1}{4} \, x^{2}\right )} + 2 \, {\left (x - 5\right )} e^{\left (2 \, x + 2\right )} + 2 \, {\left (x^{2} - 5 \, x + 4\right )} e^{\left (x + 1\right )}\right )} \log \left (x - e^{\left (\frac {1}{4} \, x^{2}\right )} - 4\right ) + 8 \, x}{{\left (x - 4\right )} e^{\left (2 \, x + 2\right )} - e^{\left (\frac {1}{4} \, x^{2} + 2 \, x + 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2-x)*exp(x+1)^2+(2*x-2)*exp(x+1))*exp(1/4*x^2)+(-2*x+10)*exp(x+1)^2+(-2*x^2+10*x-8)*exp(x+1))*lo
g(-exp(1/4*x^2)+x-4)+((x^2-4)*exp(x+1)^2+(-3*x^2+4)*exp(x+1)+2*x^2-2*x)*exp(1/4*x^2)+(2*x^2-6*x-20)*exp(x+1)^2
+(2*x^3-8*x^2-2*x+16)*exp(x+1)-2*x^3+10*x^2-8*x)/(exp(x+1)^2*exp(1/4*x^2)+(-x+4)*exp(x+1)^2),x, algorithm="gia
c")

[Out]

integrate((2*x^3 - 10*x^2 - (2*x^2 + (x^2 - 4)*e^(2*x + 2) - (3*x^2 - 4)*e^(x + 1) - 2*x)*e^(1/4*x^2) - 2*(x^2
- 3*x - 10)*e^(2*x + 2) - 2*(x^3 - 4*x^2 - x + 8)*e^(x + 1) + (((x - 2)*e^(2*x + 2) - 2*(x - 1)*e^(x + 1))*e^
(1/4*x^2) + 2*(x - 5)*e^(2*x + 2) + 2*(x^2 - 5*x + 4)*e^(x + 1))*log(x - e^(1/4*x^2) - 4) + 8*x)/((x - 4)*e^(2
*x + 2) - e^(1/4*x^2 + 2*x + 2)), x)

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maple [B]  time = 0.12, size = 114, normalized size = 3.26

method	result	size
risch	\(-\ln \left (-{\mathrm e}^{\frac {x^{2}}{4}}+x -4\right )^{2}+2 x \left ({\mathrm e}^{x +1}-1\right ) {\mathrm e}^{-x -1} \ln \left (-{\mathrm e}^{\frac {x^{2}}{4}}+x -4\right )+\left (-x^{2} {\mathrm e}^{2 x +2}+4 \ln \left ({\mathrm e}^{\frac {x^{2}}{4}}-x +4\right ) {\mathrm e}^{2 x +2}+2 x^{2} {\mathrm e}^{x +1}-4 x \,{\mathrm e}^{2 x +2}-x^{2}+4 x \,{\mathrm e}^{x +1}\right ) {\mathrm e}^{-2 x -2}\)	\(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((2-x)*exp(x+1)^2+(2*x-2)*exp(x+1))*exp(1/4*x^2)+(-2*x+10)*exp(x+1)^2+(-2*x^2+10*x-8)*exp(x+1))*ln(-exp(
1/4*x^2)+x-4)+((x^2-4)*exp(x+1)^2+(-3*x^2+4)*exp(x+1)+2*x^2-2*x)*exp(1/4*x^2)+(2*x^2-6*x-20)*exp(x+1)^2+(2*x^3
-8*x^2-2*x+16)*exp(x+1)-2*x^3+10*x^2-8*x)/(exp(x+1)^2*exp(1/4*x^2)+(-x+4)*exp(x+1)^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2-x)*exp(x+1)^2+(2*x-2)*exp(x+1))*exp(1/4*x^2)+(-2*x+10)*exp(x+1)^2+(-2*x^2+10*x-8)*exp(x+1))*lo
g(-exp(1/4*x^2)+x-4)+((x^2-4)*exp(x+1)^2+(-3*x^2+4)*exp(x+1)+2*x^2-2*x)*exp(1/4*x^2)+(2*x^2-6*x-20)*exp(x+1)^2
+(2*x^3-8*x^2-2*x+16)*exp(x+1)-2*x^3+10*x^2-8*x)/(exp(x+1)^2*exp(1/4*x^2)+(-x+4)*exp(x+1)^2),x, algorithm="max
ima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {8\,x+\ln \left (x-{\mathrm {e}}^{\frac {x^2}{4}}-4\right )\,\left ({\mathrm {e}}^{x+1}\,\left (2\,x^2-10\,x+8\right )-{\mathrm {e}}^{\frac {x^2}{4}}\,\left ({\mathrm {e}}^{x+1}\,\left (2\,x-2\right )-{\mathrm {e}}^{2\,x+2}\,\left (x-2\right )\right )+{\mathrm {e}}^{2\,x+2}\,\left (2\,x-10\right )\right )+{\mathrm {e}}^{x+1}\,\left (-2\,x^3+8\,x^2+2\,x-16\right )+{\mathrm {e}}^{2\,x+2}\,\left (-2\,x^2+6\,x+20\right )+{\mathrm {e}}^{\frac {x^2}{4}}\,\left (2\,x+{\mathrm {e}}^{x+1}\,\left (3\,x^2-4\right )-{\mathrm {e}}^{2\,x+2}\,\left (x^2-4\right )-2\,x^2\right )-10\,x^2+2\,x^3}{{\mathrm {e}}^{2\,x+2}\,{\mathrm {e}}^{\frac {x^2}{4}}-{\mathrm {e}}^{2\,x+2}\,\left (x-4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x + log(x - exp(x^2/4) - 4)*(exp(x + 1)*(2*x^2 - 10*x + 8) - exp(x^2/4)*(exp(x + 1)*(2*x - 2) - exp(2*
x + 2)*(x - 2)) + exp(2*x + 2)*(2*x - 10)) + exp(x + 1)*(2*x + 8*x^2 - 2*x^3 - 16) + exp(2*x + 2)*(6*x - 2*x^2
+ 20) + exp(x^2/4)*(2*x + exp(x + 1)*(3*x^2 - 4) - exp(2*x + 2)*(x^2 - 4) - 2*x^2) - 10*x^2 + 2*x^3)/(exp(2*x
+ 2)*exp(x^2/4) - exp(2*x + 2)*(x - 4)),x)

[Out]

int(-(8*x + log(x - exp(x^2/4) - 4)*(exp(x + 1)*(2*x^2 - 10*x + 8) - exp(x^2/4)*(exp(x + 1)*(2*x - 2) - exp(2*
x + 2)*(x - 2)) + exp(2*x + 2)*(2*x - 10)) + exp(x + 1)*(2*x + 8*x^2 - 2*x^3 - 16) + exp(2*x + 2)*(6*x - 2*x^2
+ 20) + exp(x^2/4)*(2*x + exp(x + 1)*(3*x^2 - 4) - exp(2*x + 2)*(x^2 - 4) - 2*x^2) - 10*x^2 + 2*x^3)/(exp(2*x
+ 2)*exp(x^2/4) - exp(2*x + 2)*(x - 4)), x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ShapeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2-x)*exp(x+1)**2+(2*x-2)*exp(x+1))*exp(1/4*x**2)+(-2*x+10)*exp(x+1)**2+(-2*x**2+10*x-8)*exp(x+1)
)*ln(-exp(1/4*x**2)+x-4)+((x**2-4)*exp(x+1)**2+(-3*x**2+4)*exp(x+1)+2*x**2-2*x)*exp(1/4*x**2)+(2*x**2-6*x-20)*
exp(x+1)**2+(2*x**3-8*x**2-2*x+16)*exp(x+1)-2*x**3+10*x**2-8*x)/(exp(x+1)**2*exp(1/4*x**2)+(-x+4)*exp(x+1)**2)
,x)

[Out]

Exception raised: ShapeError

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