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

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

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Rubi [A]  time = 0.06, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6688, 2194} \begin {gather*} -x^2+x-e^{-x-e+3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-E^(3 - E - x) + x - x^2

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} -e^{3-e-x}+x-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-E^(3 - E - x) + x - x^2

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fricas [A]  time = 1.11, size = 28, normalized size = 1.47 \begin {gather*} -{\left ({\left (x^{2} - x\right )} e^{\left (x + e - 3\right )} + 1\right )} e^{\left (-x - e + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 19, normalized size = 1.00 \begin {gather*} -x^{2} + x - e^{\left (-x - e + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 + x - e^(-x - e + 3)

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maple [A]  time = 0.04, size = 20, normalized size = 1.05

method	result	size
risch	\(-x^{2}+x -{\mathrm e}^{3-{\mathrm e}-x}\)	\(20\)
norman	\(\left ({\mathrm e}^{x} x -{\mathrm e}^{x} x^{2}-{\mathrm e}^{-{\mathrm e}} {\mathrm e}^{3}\right ) {\mathrm e}^{-x}\)	\(29\)
default	\({\mathrm e}^{3-{\mathrm e}} \left (-{\mathrm e}^{-x}+{\mathrm e}^{{\mathrm e}} {\mathrm e}^{-3} x -{\mathrm e}^{{\mathrm e}} {\mathrm e}^{-3} x^{2}\right )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2+x-exp(3-exp(1)-x)

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maxima [A]  time = 0.35, size = 19, normalized size = 1.00 \begin {gather*} -x^{2} + x - e^{\left (-x - e + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 + x - e^(-x - e + 3)

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mupad [B]  time = 0.08, size = 20, normalized size = 1.05 \begin {gather*} x-x^2-{\mathrm {e}}^{-\mathrm {e}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x)*exp(3 - exp(1))*(exp(exp(1) - 3)*exp(x)*(2*x - 1) - 1),x)

[Out]

x - x^2 - exp(-exp(1))*exp(-x)*exp(3)

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sympy [A]  time = 0.13, size = 15, normalized size = 0.79 \begin {gather*} - x^{2} + x - \frac {e^{3} e^{- x}}{e^{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2 + x - exp(3)*exp(-E)*exp(-x)

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