3.1 \(\int \frac{e^x}{4+6 e^x} \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{6} \log \left (3 e^x+2\right ) \]

[Out]

Log[2 + 3*E^x]/6

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Rubi [A]  time = 0.0168879, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2246, 31} \[ \frac{1}{6} \log \left (3 e^x+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x/(4 + 6*E^x),x]

[Out]

Log[2 + 3*E^x]/6

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]

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{align*} \int \frac{e^x}{4+6 e^x} \, dx &=\operatorname{Subst}\left (\int \frac{1}{4+6 x} \, dx,x,e^x\right )\\ &=\frac{1}{6} \log \left (2+3 e^x\right )\\ \end{align*}

Mathematica [A]  time = 0.0056498, size = 12, normalized size = 1. \[ \frac{1}{6} \log \left (3 e^x+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/(4 + 6*E^x),x]

[Out]

Log[2 + 3*E^x]/6

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Maple [A]  time = 0.001, size = 10, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 2+3\,{{\rm e}^{x}} \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(4+6*exp(x)),x)

[Out]

1/6*ln(2+3*exp(x))

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Maxima [A]  time = 1.13738, size = 12, normalized size = 1. \begin{align*} \frac{1}{6} \, \log \left (3 \, e^{x} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(4+6*exp(x)),x, algorithm="maxima")

[Out]

1/6*log(3*e^x + 2)

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Fricas [A]  time = 1.44771, size = 27, normalized size = 2.25 \begin{align*} \frac{1}{6} \, \log \left (3 \, e^{x} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(4+6*exp(x)),x, algorithm="fricas")

[Out]

1/6*log(3*e^x + 2)

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Sympy [A]  time = 0.12272, size = 8, normalized size = 0.67 \begin{align*} \frac{\log{\left (e^{x} + \frac{2}{3} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(4+6*exp(x)),x)

[Out]

log(exp(x) + 2/3)/6

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Giac [A]  time = 1.28391, size = 12, normalized size = 1. \begin{align*} \frac{1}{6} \, \log \left (3 \, e^{x} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(4+6*exp(x)),x, algorithm="giac")

[Out]

1/6*log(3*e^x + 2)