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

Optimal. Leaf size=17 \[ \frac {-e^{4-x}+3 x}{e^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {12, 2194} \begin {gather*} \frac {3 x}{e^2}-e^{2-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + E^(4 - x))/E^2,x]

[Out]

-E^(2 - x) + (3*x)/E^2

Rule 12

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

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

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {-e^{4-x}+3 x}{e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + E^(4 - x))/E^2,x]

[Out]

(-E^(4 - x) + 3*x)/E^2

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fricas [A]  time = 0.66, size = 15, normalized size = 0.88 \begin {gather*} {\left (3 \, x - e^{\left (-x + 4\right )}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-x+4)+3)/exp(2),x, algorithm="fricas")

[Out]

(3*x - e^(-x + 4))*e^(-2)

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giac [A]  time = 0.14, size = 15, normalized size = 0.88 \begin {gather*} {\left (3 \, x - e^{\left (-x + 4\right )}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-x+4)+3)/exp(2),x, algorithm="giac")

[Out]

(3*x - e^(-x + 4))*e^(-2)

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maple [A]  time = 0.05, size = 15, normalized size = 0.88

method	result	size
risch	\(3 \,{\mathrm e}^{-2} x -{\mathrm e}^{2-x}\)	\(15\)
default	\(\left (3 x -{\mathrm e}^{-x +4}\right ) {\mathrm e}^{-2}\)	\(18\)
norman	\(3 \,{\mathrm e}^{-2} x -{\mathrm e}^{-x +4} {\mathrm e}^{-2}\)	\(21\)
derivativedivides	\(-{\mathrm e}^{-2} \left ({\mathrm e}^{-x +4}+3 \ln \left ({\mathrm e}^{-x +4}\right )\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x+4)+3)/exp(2),x,method=_RETURNVERBOSE)

[Out]

3*exp(-2)*x-exp(2-x)

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maxima [A]  time = 0.36, size = 15, normalized size = 0.88 \begin {gather*} {\left (3 \, x - e^{\left (-x + 4\right )}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-x+4)+3)/exp(2),x, algorithm="maxima")

[Out]

(3*x - e^(-x + 4))*e^(-2)

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mupad [B]  time = 0.09, size = 14, normalized size = 0.82 \begin {gather*} 3\,x\,{\mathrm {e}}^{-2}-{\mathrm {e}}^{2-x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-2)*(exp(4 - x) + 3),x)

[Out]

3*x*exp(-2) - exp(2 - x)

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sympy [A]  time = 0.08, size = 14, normalized size = 0.82 \begin {gather*} \frac {3 x}{e^{2}} - \frac {e^{4 - x}}{e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-x+4)+3)/exp(2),x)

[Out]

3*x*exp(-2) - exp(-2)*exp(4 - x)

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