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

Optimal. Leaf size=8 \[ \frac {\log (4+x)}{e^3} \]

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {12, 31} \begin {gather*} \frac {\log (x+4)}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^3*(4 + x)),x]

[Out]

Log[4 + x]/E^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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {\log (4+x)}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^3*(4 + x)),x]

[Out]

Log[4 + x]/E^3

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fricas [A]  time = 0.54, size = 7, normalized size = 0.88 \begin {gather*} e^{\left (-3\right )} \log \left (x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+x)/exp(3),x, algorithm="fricas")

[Out]

e^(-3)*log(x + 4)

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giac [A]  time = 0.12, size = 8, normalized size = 1.00 \begin {gather*} e^{\left (-3\right )} \log \left ({\left | x + 4 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+x)/exp(3),x, algorithm="giac")

[Out]

e^(-3)*log(abs(x + 4))

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maple [A]  time = 0.49, size = 8, normalized size = 1.00

method	result	size
risch	\(\ln \left (4+x \right ) {\mathrm e}^{-3}\)	\(8\)
default	\(\ln \left (4+x \right ) {\mathrm e}^{-3}\)	\(10\)
norman	\(\ln \left (4+x \right ) {\mathrm e}^{-3}\)	\(10\)
meijerg	\({\mathrm e}^{-3} \ln \left (1+\frac {x}{4}\right )\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4+x)/exp(3),x,method=_RETURNVERBOSE)

[Out]

ln(4+x)*exp(-3)

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maxima [A]  time = 0.62, size = 7, normalized size = 0.88 \begin {gather*} e^{\left (-3\right )} \log \left (x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+x)/exp(3),x, algorithm="maxima")

[Out]

e^(-3)*log(x + 4)

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mupad [B]  time = 0.13, size = 7, normalized size = 0.88 \begin {gather*} \ln \left (x+4\right )\,{\mathrm {e}}^{-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-3)/(x + 4),x)

[Out]

log(x + 4)*exp(-3)

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sympy [A]  time = 0.06, size = 14, normalized size = 1.75 \begin {gather*} \frac {\log {\left (x e^{3} + 4 e^{3} \right )}}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+x)/exp(3),x)

[Out]

exp(-3)*log(x*exp(3) + 4*exp(3))

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