3.7.35 \(\int \frac {e^{-\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} (-6+8 x-2 x^2+e^{\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} (-4+4 x-x^2)+e^x (8-8 x+2 x^2))}{4-4 x+x^2} \, dx\)

Optimal. Leaf size=26 \[ 2 e^{e^x-\frac {5}{5+5 (-3+x)}-x}-x \]

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Rubi [F]  time = 8.09, 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 {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-6+8 x-2 x^2+e^{\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-4+4 x-x^2\right )+e^x \left (8-8 x+2 x^2\right )\right )}{4-4 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6 + 8*x - 2*x^2 + E^((1 + E^x*(2 - x) - 2*x + x^2)/(-2 + x))*(-4 + 4*x - x^2) + E^x*(8 - 8*x + 2*x^2))/(
E^((1 + E^x*(2 - x) - 2*x + x^2)/(-2 + x))*(4 - 4*x + x^2)),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-6+8 x-2 x^2+e^{\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-4+4 x-x^2\right )+e^x \left (8-8 x+2 x^2\right )\right )}{(-2+x)^2} \, dx\\ &=\int \left (-1+2 e^{e^x+\frac {1}{2-x}}-\frac {2 e^{e^x-\frac {(-1+x)^2}{-2+x}} \left (3-4 x+x^2\right )}{(-2+x)^2}\right ) \, dx\\ &=-x+2 \int e^{e^x+\frac {1}{2-x}} \, dx-2 \int \frac {e^{e^x-\frac {(-1+x)^2}{-2+x}} \left (3-4 x+x^2\right )}{(-2+x)^2} \, dx\\ &=-x+2 \int e^{e^x+\frac {1}{2-x}} \, dx-2 \int \left (e^{e^x-\frac {(-1+x)^2}{-2+x}}-\frac {e^{e^x-\frac {(-1+x)^2}{-2+x}}}{(-2+x)^2}\right ) \, dx\\ &=-x+2 \int e^{e^x+\frac {1}{2-x}} \, dx-2 \int e^{e^x-\frac {(-1+x)^2}{-2+x}} \, dx+2 \int \frac {e^{e^x-\frac {(-1+x)^2}{-2+x}}}{(-2+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 22, normalized size = 0.85 \begin {gather*} 2 e^{e^x-\frac {1}{-2+x}-x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 8*x - 2*x^2 + E^((1 + E^x*(2 - x) - 2*x + x^2)/(-2 + x))*(-4 + 4*x - x^2) + E^x*(8 - 8*x + 2*x
^2))/(E^((1 + E^x*(2 - x) - 2*x + x^2)/(-2 + x))*(4 - 4*x + x^2)),x]

[Out]

2*E^(E^x - (-2 + x)^(-1) - x) - x

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fricas [A]  time = 0.62, size = 29, normalized size = 1.12 \begin {gather*} -x + 2 \, e^{\left (-\frac {x^{2} - {\left (x - 2\right )} e^{x} - 2 \, x + 1}{x - 2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+4*x-4)*exp(((2-x)*exp(x)+x^2-2*x+1)/(x-2))+(2*x^2-8*x+8)*exp(x)-2*x^2+8*x-6)/(x^2-4*x+4)/exp(
((2-x)*exp(x)+x^2-2*x+1)/(x-2)),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, x^{2} - 2 \, {\left (x^{2} - 4 \, x + 4\right )} e^{x} + {\left (x^{2} - 4 \, x + 4\right )} e^{\left (\frac {x^{2} - {\left (x - 2\right )} e^{x} - 2 \, x + 1}{x - 2}\right )} - 8 \, x + 6\right )} e^{\left (-\frac {x^{2} - {\left (x - 2\right )} e^{x} - 2 \, x + 1}{x - 2}\right )}}{x^{2} - 4 \, x + 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+4*x-4)*exp(((2-x)*exp(x)+x^2-2*x+1)/(x-2))+(2*x^2-8*x+8)*exp(x)-2*x^2+8*x-6)/(x^2-4*x+4)/exp(
((2-x)*exp(x)+x^2-2*x+1)/(x-2)),x, algorithm="giac")

[Out]

integrate(-(2*x^2 - 2*(x^2 - 4*x + 4)*e^x + (x^2 - 4*x + 4)*e^((x^2 - (x - 2)*e^x - 2*x + 1)/(x - 2)) - 8*x +
6)*e^(-(x^2 - (x - 2)*e^x - 2*x + 1)/(x - 2))/(x^2 - 4*x + 4), x)

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maple [A]  time = 0.23, size = 32, normalized size = 1.23

method	result	size
risch	\(-x +2 \,{\mathrm e}^{\frac {{\mathrm e}^{x} x -x^{2}-2 \,{\mathrm e}^{x}+2 x -1}{x -2}}\)	\(32\)
norman	\(\frac {\left (-4+4 \,{\mathrm e}^{\frac {\left (2-x \right ) {\mathrm e}^{x}+x^{2}-2 x +1}{x -2}}+2 x -x^{2} {\mathrm e}^{\frac {\left (2-x \right ) {\mathrm e}^{x}+x^{2}-2 x +1}{x -2}}\right ) {\mathrm e}^{-\frac {\left (2-x \right ) {\mathrm e}^{x}+x^{2}-2 x +1}{x -2}}}{x -2}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2+4*x-4)*exp(((2-x)*exp(x)+x^2-2*x+1)/(x-2))+(2*x^2-8*x+8)*exp(x)-2*x^2+8*x-6)/(x^2-4*x+4)/exp(((2-x)
*exp(x)+x^2-2*x+1)/(x-2)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.63, size = 20, normalized size = 0.77 \begin {gather*} -x + 2 \, e^{\left (-x - \frac {1}{x - 2} + e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+4*x-4)*exp(((2-x)*exp(x)+x^2-2*x+1)/(x-2))+(2*x^2-8*x+8)*exp(x)-2*x^2+8*x-6)/(x^2-4*x+4)/exp(
((2-x)*exp(x)+x^2-2*x+1)/(x-2)),x, algorithm="maxima")

[Out]

-x + 2*e^(-x - 1/(x - 2) + e^x)

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mupad [B]  time = 0.82, size = 54, normalized size = 2.08 \begin {gather*} 2\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^x}{x-2}}\,{\mathrm {e}}^{\frac {2\,x}{x-2}}\,{\mathrm {e}}^{-\frac {x^2}{x-2}}\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^x}{x-2}}\,{\mathrm {e}}^{-\frac {1}{x-2}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((2*x + exp(x)*(x - 2) - x^2 - 1)/(x - 2))*(2*x^2 - exp(x)*(2*x^2 - 8*x + 8) - 8*x + exp(-(2*x + exp(
x)*(x - 2) - x^2 - 1)/(x - 2))*(x^2 - 4*x + 4) + 6))/(x^2 - 4*x + 4),x)

[Out]

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

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sympy [A]  time = 0.35, size = 22, normalized size = 0.85 \begin {gather*} - x + 2 e^{- \frac {x^{2} - 2 x + \left (2 - x\right ) e^{x} + 1}{x - 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2+4*x-4)*exp(((2-x)*exp(x)+x**2-2*x+1)/(x-2))+(2*x**2-8*x+8)*exp(x)-2*x**2+8*x-6)/(x**2-4*x+4)
/exp(((2-x)*exp(x)+x**2-2*x+1)/(x-2)),x)

[Out]

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

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