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]

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

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Rubi [A]  time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 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 {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*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \[ \frac {1}{6} \log \left (3 e^x+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/(4 + 6*E^x),x]

[Out]

Log[2 + 3*E^x]/6

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fricas [A]  time = 0.39, size = 9, normalized size = 0.75 \[ \frac {1}{6} \, \log \left (3 \, e^{x} + 2\right ) \]

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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giac [A]  time = 0.24, size = 9, normalized size = 0.75 \[ \frac {1}{6} \, \log \left (3 \, e^{x} + 2\right ) \]

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)

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maple [A]  time = 0.00, size = 10, normalized size = 0.83 \[ \frac {\ln \left (3 \,{\mathrm e}^{x}+2\right )}{6} \]

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 = 0.43, size = 9, normalized size = 0.75 \[ \frac {1}{6} \, \log \left (3 \, e^{x} + 2\right ) \]

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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mupad [B]  time = 0.05, size = 9, normalized size = 0.75 \[ \frac {\ln \left (3\,{\mathrm {e}}^x+2\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(6*exp(x) + 4),x)

[Out]

log(3*exp(x) + 2)/6

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sympy [A]  time = 0.09, size = 8, normalized size = 0.67 \[ \frac {\log {\left (e^{x} + \frac {2}{3} \right )}}{6} \]

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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