3.52.46 \(\int \frac {4 e^{-16+4 x}}{3+e^{-16+4 x}} \, dx\)

Optimal. Leaf size=16 \[ \log \left (\left (3+e^{4 (-4+x)}\right ) (2+\log (\log (4)))\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 12, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {12, 2246, 31} \begin {gather*} \log \left (e^{4 x}+3 e^{16}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^(-16 + 4*x))/(3 + E^(-16 + 4*x)),x]

[Out]

Log[3*E^16 + E^(4*x)]

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]

Rule 2246

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4 \int \frac {e^{-16+4 x}}{3+e^{-16+4 x}} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{3+x} \, dx,x,e^{-16+4 x}\right )\\ &=\log \left (3 e^{16}+e^{4 x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 0.75 \begin {gather*} \log \left (3 e^{16}+e^{4 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^(-16 + 4*x))/(3 + E^(-16 + 4*x)),x]

[Out]

Log[3*E^16 + E^(4*x)]

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fricas [A]  time = 0.50, size = 9, normalized size = 0.56 \begin {gather*} \log \left (e^{\left (4 \, x - 16\right )} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(4*x-16)/(exp(4*x-16)+3),x, algorithm="fricas")

[Out]

log(e^(4*x - 16) + 3)

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giac [A]  time = 0.95, size = 9, normalized size = 0.56 \begin {gather*} \log \left (e^{\left (4 \, x - 16\right )} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(4*x-16)/(exp(4*x-16)+3),x, algorithm="giac")

[Out]

log(e^(4*x - 16) + 3)

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maple [A]  time = 0.02, size = 10, normalized size = 0.62

method	result	size
derivativedivides	\(\ln \left ({\mathrm e}^{4 x -16}+3\right )\)	\(10\)
default	\(\ln \left ({\mathrm e}^{4 x -16}+3\right )\)	\(10\)
norman	\(\ln \left ({\mathrm e}^{4 x -16}+3\right )\)	\(10\)
risch	\(16+\ln \left ({\mathrm e}^{4 x -16}+3\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*exp(4*x-16)/(exp(4*x-16)+3),x,method=_RETURNVERBOSE)

[Out]

ln(exp(4*x-16)+3)

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maxima [A]  time = 0.35, size = 9, normalized size = 0.56 \begin {gather*} \log \left (e^{\left (4 \, x - 16\right )} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(4*x-16)/(exp(4*x-16)+3),x, algorithm="maxima")

[Out]

log(e^(4*x - 16) + 3)

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mupad [B]  time = 0.10, size = 9, normalized size = 0.56 \begin {gather*} \ln \left ({\mathrm {e}}^{4\,x-16}+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp(4*x - 16))/(exp(4*x - 16) + 3),x)

[Out]

log(exp(4*x - 16) + 3)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.50 \begin {gather*} \log {\left (e^{4 x - 16} + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(4*x-16)/(exp(4*x-16)+3),x)

[Out]

log(exp(4*x - 16) + 3)

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