3.1.34 \(\int \frac {e^{e^{-\frac {2 x^2}{-2+6 x}}-\frac {2 x^2}{-2+6 x}} (-2 x+3 x^2)}{1-6 x+9 x^2} \, dx\)

Optimal. Leaf size=20 \[ 3-e^{e^{\frac {2 x}{-6+\frac {2}{x}}}} \]

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Rubi [F]  time = 1.06, 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 {e^{e^{-\frac {2 x^2}{-2+6 x}}-\frac {2 x^2}{-2+6 x}} \left (-2 x+3 x^2\right )}{1-6 x+9 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^((-2*x^2)/(-2 + 6*x)) - (2*x^2)/(-2 + 6*x))*(-2*x + 3*x^2))/(1 - 6*x + 9*x^2),x]

[Out]

Defer[Int][E^(E^((-2*x^2)/(-2 + 6*x)) - (2*x^2)/(-2 + 6*x)), x]/3 - Defer[Int][E^(E^((-2*x^2)/(-2 + 6*x)) - (2
*x^2)/(-2 + 6*x))/(-1 + 3*x)^2, x]/3

Rubi steps

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

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Mathematica [A]  time = 0.19, size = 17, normalized size = 0.85 \begin {gather*} -e^{e^{\frac {x^2}{1-3 x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^((-2*x^2)/(-2 + 6*x)) - (2*x^2)/(-2 + 6*x))*(-2*x + 3*x^2))/(1 - 6*x + 9*x^2),x]

[Out]

-E^E^(x^2/(1 - 3*x))

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fricas [B]  time = 0.74, size = 48, normalized size = 2.40 \begin {gather*} -e^{\left (\frac {x^{2}}{3 \, x - 1} - \frac {x^{2} - {\left (3 \, x - 1\right )} e^{\left (-\frac {x^{2}}{3 \, x - 1}\right )}}{3 \, x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-2*x)*exp(-x^2/(6*x-2))^2*exp(exp(-x^2/(6*x-2))^2)/(9*x^2-6*x+1),x, algorithm="fricas")

[Out]

-e^(x^2/(3*x - 1) - (x^2 - (3*x - 1)*e^(-x^2/(3*x - 1)))/(3*x - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{2} - 2 \, x\right )} e^{\left (-\frac {x^{2}}{3 \, x - 1} + e^{\left (-\frac {x^{2}}{3 \, x - 1}\right )}\right )}}{9 \, x^{2} - 6 \, x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-2*x)*exp(-x^2/(6*x-2))^2*exp(exp(-x^2/(6*x-2))^2)/(9*x^2-6*x+1),x, algorithm="giac")

[Out]

integrate((3*x^2 - 2*x)*e^(-x^2/(3*x - 1) + e^(-x^2/(3*x - 1)))/(9*x^2 - 6*x + 1), x)

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maple [A]  time = 0.11, size = 17, normalized size = 0.85

method	result	size
risch	\(-{\mathrm e}^{{\mathrm e}^{-\frac {x^{2}}{3 x -1}}}\)	\(17\)
norman	\(\frac {-3 x \,{\mathrm e}^{{\mathrm e}^{-\frac {2 x^{2}}{6 x -2}}}+{\mathrm e}^{{\mathrm e}^{-\frac {2 x^{2}}{6 x -2}}}}{3 x -1}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2-2*x)*exp(-x^2/(6*x-2))^2*exp(exp(-x^2/(6*x-2))^2)/(9*x^2-6*x+1),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.65, size = 18, normalized size = 0.90 \begin {gather*} -e^{\left (e^{\left (-\frac {1}{3} \, x - \frac {1}{9 \, {\left (3 \, x - 1\right )}} - \frac {1}{9}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-2*x)*exp(-x^2/(6*x-2))^2*exp(exp(-x^2/(6*x-2))^2)/(9*x^2-6*x+1),x, algorithm="maxima")

[Out]

-e^(e^(-1/3*x - 1/9/(3*x - 1) - 1/9))

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mupad [B]  time = 0.44, size = 16, normalized size = 0.80 \begin {gather*} -{\mathrm {e}}^{{\mathrm {e}}^{-\frac {2\,x^2}{6\,x-2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(2*x^2)/(6*x - 2))*exp(exp(-(2*x^2)/(6*x - 2)))*(2*x - 3*x^2))/(9*x^2 - 6*x + 1),x)

[Out]

-exp(exp(-(2*x^2)/(6*x - 2)))

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sympy [A]  time = 0.35, size = 15, normalized size = 0.75 \begin {gather*} - e^{e^{- \frac {2 x^{2}}{6 x - 2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2-2*x)*exp(-x**2/(6*x-2))**2*exp(exp(-x**2/(6*x-2))**2)/(9*x**2-6*x+1),x)

[Out]

-exp(exp(-2*x**2/(6*x - 2)))

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