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

Optimal. Leaf size=19 \[ \frac {1}{3} e^{4+4 e^4+x-x^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {12, 2236} \begin {gather*} \frac {1}{3} e^{-x^2+x+4 \left (1+e^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(4*(1 + E^4) + x - x^2)/3

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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{3} e^{4+4 e^4+x-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(4 + 4*E^4 + x - x^2)/3

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fricas [A]  time = 0.63, size = 17, normalized size = 0.89 \begin {gather*} e^{\left (-x^{2} + x + 4 \, e^{4} - \log \relax (3) + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x^2 + x + 4*e^4 - log(3) + 4)

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giac [A]  time = 0.12, size = 17, normalized size = 0.89 \begin {gather*} e^{\left (-x^{2} + x + 4 \, e^{4} - \log \relax (3) + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x^2 + x + 4*e^4 - log(3) + 4)

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maple [A]  time = 0.04, size = 16, normalized size = 0.84

method	result	size
risch	\(\frac {{\mathrm e}^{4+4 \,{\mathrm e}^{4}-x^{2}+x}}{3}\)	\(16\)
gosper	\({\mathrm e}^{-\ln \relax (3)+4 \,{\mathrm e}^{4}-x^{2}+x +4}\)	\(18\)
derivativedivides	\({\mathrm e}^{-\ln \relax (3)+4 \,{\mathrm e}^{4}-x^{2}+x +2} {\mathrm e}^{2}\)	\(21\)
default	\({\mathrm e}^{-\ln \relax (3)+4 \,{\mathrm e}^{4}-x^{2}+x +2} {\mathrm e}^{2}\)	\(21\)
norman	\({\mathrm e}^{-\ln \relax (3)+4 \,{\mathrm e}^{4}-x^{2}+x +2} {\mathrm e}^{2}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.35, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{3} \, e^{\left (-x^{2} + x + 4 \, e^{4} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*e^(-x^2 + x + 4*e^4 + 4)

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mupad [B]  time = 0.10, size = 17, normalized size = 0.89 \begin {gather*} \frac {{\mathrm {e}}^{4\,{\mathrm {e}}^4}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(4*exp(4))*exp(4)*exp(-x^2)*exp(x))/3

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sympy [A]  time = 0.10, size = 17, normalized size = 0.89 \begin {gather*} \frac {e^{2} e^{- x^{2} + x + 2 + 4 e^{4}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(2)*exp(-x**2 + x + 2 + 4*exp(4))/3

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