3.91.67 \(\int \frac {e^{-\frac {x^2}{-1+x}} (2+16 x-4 x^2-2 x^3+e^4 (4 x-2 x^2))}{1-2 x+x^2} \, dx\)

Optimal. Leaf size=23 \[ 2 e^{-x+\frac {x}{1-x}} \left (5+e^4+x\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.54, 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 {e^{-\frac {x^2}{-1+x}} \left (2+16 x-4 x^2-2 x^3+e^4 \left (4 x-2 x^2\right )\right )}{1-2 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 + 16*x - 4*x^2 - 2*x^3 + E^4*(4*x - 2*x^2))/(E^(x^2/(-1 + x))*(1 - 2*x + x^2)),x]

[Out]

-2*(4 + E^4)*Defer[Int][E^(-(x^2/(-1 + x))), x] + 2*(6 + E^4)*Defer[Int][1/(E^(x^2/(-1 + x))*(-1 + x)^2), x] +
2*Defer[Int][1/(E^(x^2/(-1 + x))*(-1 + x)), x] - 2*Defer[Int][x/E^(x^2/(-1 + x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {x^2}{-1+x}} \left (2+16 x-4 x^2-2 x^3+e^4 \left (4 x-2 x^2\right )\right )}{(-1+x)^2} \, dx\\ &=\int \frac {e^{-\frac {x^2}{-1+x}} \left (2+4 \left (4+e^4\right ) x-2 \left (2+e^4\right ) x^2-2 x^3\right )}{(1-x)^2} \, dx\\ &=\int \left (-2 e^{-\frac {x^2}{-1+x}} \left (4+e^4\right )+\frac {2 e^{-\frac {x^2}{-1+x}} \left (6+e^4\right )}{(-1+x)^2}+\frac {2 e^{-\frac {x^2}{-1+x}}}{-1+x}-2 e^{-\frac {x^2}{-1+x}} x\right ) \, dx\\ &=2 \int \frac {e^{-\frac {x^2}{-1+x}}}{-1+x} \, dx-2 \int e^{-\frac {x^2}{-1+x}} x \, dx-\left (2 \left (4+e^4\right )\right ) \int e^{-\frac {x^2}{-1+x}} \, dx+\left (2 \left (6+e^4\right )\right ) \int \frac {e^{-\frac {x^2}{-1+x}}}{(-1+x)^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.26, size = 21, normalized size = 0.91 \begin {gather*} 2 e^{\frac {x^2}{1-x}} \left (5+e^4+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 16*x - 4*x^2 - 2*x^3 + E^4*(4*x - 2*x^2))/(E^(x^2/(-1 + x))*(1 - 2*x + x^2)),x]

[Out]

2*E^(x^2/(1 - x))*(5 + E^4 + x)

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 18, normalized size = 0.78 \begin {gather*} 2 \, {\left (x + e^{4} + 5\right )} e^{\left (-\frac {x^{2}}{x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+4*x)*exp(4)-2*x^3-4*x^2+16*x+2)/(x^2-2*x+1)/exp(x^2/(x-1)),x, algorithm="fricas")

[Out]

2*(x + e^4 + 5)*e^(-x^2/(x - 1))

________________________________________________________________________________________

giac [B]  time = 0.12, size = 43, normalized size = 1.87 \begin {gather*} 2 \, x e^{\left (-\frac {x^{2}}{x - 1}\right )} + 2 \, e^{\left (-\frac {x^{2}}{x - 1} + 4\right )} + 10 \, e^{\left (-\frac {x^{2}}{x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+4*x)*exp(4)-2*x^3-4*x^2+16*x+2)/(x^2-2*x+1)/exp(x^2/(x-1)),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.57, size = 20, normalized size = 0.87

method	result	size
gosper	\(2 \left ({\mathrm e}^{4}+x +5\right ) {\mathrm e}^{-\frac {x^{2}}{x -1}}\)	\(20\)
risch	\(\left (2 x +2 \,{\mathrm e}^{4}+10\right ) {\mathrm e}^{-\frac {x^{2}}{x -1}}\)	\(22\)
norman	\(\frac {\left (\left (2 \,{\mathrm e}^{4}+8\right ) x +2 x^{2}-10-2 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-\frac {x^{2}}{x -1}}}{x -1}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2+4*x)*exp(4)-2*x^3-4*x^2+16*x+2)/(x^2-2*x+1)/exp(x^2/(x-1)),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 20, normalized size = 0.87 \begin {gather*} 2 \, {\left (x + e^{4} + 5\right )} e^{\left (-x - \frac {1}{x - 1} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+4*x)*exp(4)-2*x^3-4*x^2+16*x+2)/(x^2-2*x+1)/exp(x^2/(x-1)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 5.94, size = 18, normalized size = 0.78 \begin {gather*} 2\,{\mathrm {e}}^{-\frac {x^2}{x-1}}\,\left (x+{\mathrm {e}}^4+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x^2/(x - 1))*(16*x + exp(4)*(4*x - 2*x^2) - 4*x^2 - 2*x^3 + 2))/(x^2 - 2*x + 1),x)

[Out]

2*exp(-x^2/(x - 1))*(x + exp(4) + 5)

________________________________________________________________________________________

sympy [A]  time = 0.29, size = 17, normalized size = 0.74 \begin {gather*} \left (2 x + 10 + 2 e^{4}\right ) e^{- \frac {x^{2}}{x - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2+4*x)*exp(4)-2*x**3-4*x**2+16*x+2)/(x**2-2*x+1)/exp(x**2/(x-1)),x)

[Out]

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

________________________________________________________________________________________