3.47.68 \(\int e^{3-x} (1-e-x) \, dx\)

Optimal. Leaf size=11 \[ e^{3-x} (e+x) \]

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Rubi [B]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 2.27, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2176, 2194} \begin {gather*} e^{3-x}-e^{3-x} (-x-e+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(3 - x)*(1 - E - x),x]

[Out]

E^(3 - x) - E^(3 - x)*(1 - E - x)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} e^{3-x} (e+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(3 - x)*(1 - E - x),x]

[Out]

E^(3 - x)*(E + x)

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fricas [A]  time = 0.85, size = 11, normalized size = 1.00 \begin {gather*} {\left (x + e\right )} e^{\left (-x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + e)*e^(-x + 3)

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giac [A]  time = 0.15, size = 15, normalized size = 1.36 \begin {gather*} x e^{\left (-x + 3\right )} + e^{\left (-x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(-x + 3) + e^(-x + 4)

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maple [A]  time = 0.06, size = 12, normalized size = 1.09

method	result	size
gosper	\({\mathrm e}^{3-x} \left (x +{\mathrm e}\right )\)	\(12\)
risch	\({\mathrm e}^{3-x} \left (x +{\mathrm e}\right )\)	\(12\)
norman	\(x \,{\mathrm e}^{3-x}+{\mathrm e}^{3-x} {\mathrm e}\)	\(19\)
derivativedivides	\(-{\mathrm e}^{3-x} \left (3-x \right )+3 \,{\mathrm e}^{3-x}+{\mathrm e}^{3-x} {\mathrm e}\)	\(32\)
default	\(-{\mathrm e}^{3-x} \left (3-x \right )+3 \,{\mathrm e}^{3-x}+{\mathrm e}^{3-x} {\mathrm e}\)	\(32\)
meijerg	\(-{\mathrm e}^{4} \left (1-{\mathrm e}^{-x}\right )-{\mathrm e}^{3} \left (1-\frac {\left (2 x +2\right ) {\mathrm e}^{-x}}{2}\right )+{\mathrm e}^{3} \left (1-{\mathrm e}^{-x}\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(3-x)*(x+exp(1))

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maxima [B]  time = 0.37, size = 27, normalized size = 2.45 \begin {gather*} {\left (x e^{3} + e^{3}\right )} e^{\left (-x\right )} + e^{\left (-x + 4\right )} - e^{\left (-x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e^3 + e^3)*e^(-x) + e^(-x + 4) - e^(-x + 3)

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mupad [B]  time = 3.37, size = 11, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^{3-x}\,\left (x+\mathrm {e}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(3 - x)*(x + exp(1) - 1),x)

[Out]

exp(3 - x)*(x + exp(1))

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sympy [A]  time = 0.09, size = 8, normalized size = 0.73 \begin {gather*} \left (x + e\right ) e^{3 - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + E)*exp(3 - x)

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