3.25.24 \(\int e^{\frac {2 (-1+e^5 (3 x-x^2)-e^5 \log (3))}{e^5}} (6-4 x) \, dx\)

Optimal. Leaf size=20 \[ \frac {1}{9} e^{-\frac {2}{e^5}+6 x-2 x^2} \]

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Rubi [A]  time = 0.06, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2244, 12, 2236} \begin {gather*} \frac {1}{9} e^{-2 x^2+6 x-\frac {2}{e^5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-2/E^5 + 6*x - 2*x^2)/9

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2244

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1}{9} e^{-\frac {2}{e^5}+6 x-2 x^2} (6-4 x) \, dx\\ &=\frac {1}{9} \int e^{-\frac {2}{e^5}+6 x-2 x^2} (6-4 x) \, dx\\ &=\frac {1}{9} e^{-\frac {2}{e^5}+6 x-2 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 20, normalized size = 1.00 \begin {gather*} \frac {1}{9} e^{-\frac {2}{e^5}+6 x-2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((2*(-1 + E^5*(3*x - x^2) - E^5*Log[3]))/E^5)*(6 - 4*x),x]

[Out]

E^(-2/E^5 + 6*x - 2*x^2)/9

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fricas [A]  time = 0.66, size = 22, normalized size = 1.10 \begin {gather*} e^{\left (-2 \, {\left ({\left (x^{2} - 3 \, x\right )} e^{5} + e^{5} \log \relax (3) + 1\right )} e^{\left (-5\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6-4*x)*exp((-exp(5)*log(3)+(-x^2+3*x)*exp(5)-1)/exp(5))^2,x, algorithm="fricas")

[Out]

e^(-2*((x^2 - 3*x)*e^5 + e^5*log(3) + 1)*e^(-5))

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giac [A]  time = 1.30, size = 18, normalized size = 0.90 \begin {gather*} e^{\left (-2 \, x^{2} + 6 \, x - 2 \, e^{\left (-5\right )} - 2 \, \log \relax (3)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6-4*x)*exp((-exp(5)*log(3)+(-x^2+3*x)*exp(5)-1)/exp(5))^2,x, algorithm="giac")

[Out]

e^(-2*x^2 + 6*x - 2*e^(-5) - 2*log(3))

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maple [A]  time = 0.03, size = 17, normalized size = 0.85

method	result	size
risch	\(\frac {{\mathrm e}^{-2 x^{2}+6 x -2 \,{\mathrm e}^{-5}}}{9}\)	\(17\)
gosper	\({\mathrm e}^{-2 \left (x^{2} {\mathrm e}^{5}+{\mathrm e}^{5} \ln \relax (3)-3 x \,{\mathrm e}^{5}+1\right ) {\mathrm e}^{-5}}\)	\(28\)
default	\({\mathrm e}^{2 \left (-{\mathrm e}^{5} \ln \relax (3)+\left (-x^{2}+3 x \right ) {\mathrm e}^{5}-1\right ) {\mathrm e}^{-5}}\)	\(29\)
norman	\({\mathrm e}^{2 \left (-{\mathrm e}^{5} \ln \relax (3)+\left (-x^{2}+3 x \right ) {\mathrm e}^{5}-1\right ) {\mathrm e}^{-5}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6-4*x)*exp((-exp(5)*ln(3)+(-x^2+3*x)*exp(5)-1)/exp(5))^2,x,method=_RETURNVERBOSE)

[Out]

1/9*exp(-2*x^2+6*x-2*exp(-5))

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maxima [A]  time = 0.48, size = 22, normalized size = 1.10 \begin {gather*} e^{\left (-2 \, {\left ({\left (x^{2} - 3 \, x\right )} e^{5} + e^{5} \log \relax (3) + 1\right )} e^{\left (-5\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6-4*x)*exp((-exp(5)*log(3)+(-x^2+3*x)*exp(5)-1)/exp(5))^2,x, algorithm="maxima")

[Out]

e^(-2*((x^2 - 3*x)*e^5 + e^5*log(3) + 1)*e^(-5))

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mupad [B]  time = 0.14, size = 17, normalized size = 0.85 \begin {gather*} \frac {{\mathrm {e}}^{-2\,{\mathrm {e}}^{-5}}\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{-2\,x^2}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-2*exp(-5)*(exp(5)*log(3) - exp(5)*(3*x - x^2) + 1))*(4*x - 6),x)

[Out]

(exp(-2*exp(-5))*exp(6*x)*exp(-2*x^2))/9

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sympy [A]  time = 0.12, size = 26, normalized size = 1.30 \begin {gather*} e^{\frac {2 \left (\left (- x^{2} + 3 x\right ) e^{5} - e^{5} \log {\relax (3 )} - 1\right )}{e^{5}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6-4*x)*exp((-exp(5)*ln(3)+(-x**2+3*x)*exp(5)-1)/exp(5))**2,x)

[Out]

exp(2*((-x**2 + 3*x)*exp(5) - exp(5)*log(3) - 1)*exp(-5))

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