3.21.35 \(\int \frac {1+e^{4+x}}{e^4} \, dx\)

Optimal. Leaf size=10 \[ -1+e^x+\frac {x}{e^4} \]

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2194} \begin {gather*} \frac {x}{e^4}+e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + E^(4 + x))/E^4,x]

[Out]

E^x + x/E^4

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

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Mathematica [A]  time = 0.00, size = 9, normalized size = 0.90 \begin {gather*} e^x+\frac {x}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^(4 + x))/E^4,x]

[Out]

E^x + x/E^4

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fricas [A]  time = 0.73, size = 9, normalized size = 0.90 \begin {gather*} {\left (x + e^{\left (x + 4\right )}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4)*exp(x)+1)/exp(4),x, algorithm="fricas")

[Out]

(x + e^(x + 4))*e^(-4)

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giac [A]  time = 0.18, size = 9, normalized size = 0.90 \begin {gather*} {\left (x + e^{\left (x + 4\right )}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4)*exp(x)+1)/exp(4),x, algorithm="giac")

[Out]

(x + e^(x + 4))*e^(-4)

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maple [A]  time = 0.03, size = 8, normalized size = 0.80

method	result	size
risch	\(x \,{\mathrm e}^{-4}+{\mathrm e}^{x}\)	\(8\)
norman	\(x \,{\mathrm e}^{-4}+{\mathrm e}^{x}\)	\(10\)
default	\({\mathrm e}^{-4} \left (x +{\mathrm e}^{4} {\mathrm e}^{x}\right )\)	\(13\)
derivativedivides	\({\mathrm e}^{-4} \left ({\mathrm e}^{4} {\mathrm e}^{x}+\ln \left ({\mathrm e}^{x}\right )\right )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4)*exp(x)+1)/exp(4),x,method=_RETURNVERBOSE)

[Out]

x*exp(-4)+exp(x)

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maxima [A]  time = 0.54, size = 9, normalized size = 0.90 \begin {gather*} {\left (x + e^{\left (x + 4\right )}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4)*exp(x)+1)/exp(4),x, algorithm="maxima")

[Out]

(x + e^(x + 4))*e^(-4)

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mupad [B]  time = 1.15, size = 7, normalized size = 0.70 \begin {gather*} {\mathrm {e}}^x+x\,{\mathrm {e}}^{-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-4)*(exp(4)*exp(x) + 1),x)

[Out]

exp(x) + x*exp(-4)

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sympy [A]  time = 0.08, size = 7, normalized size = 0.70 \begin {gather*} \frac {x}{e^{4}} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4)*exp(x)+1)/exp(4),x)

[Out]

x*exp(-4) + exp(x)

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