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 = 0.032852, 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 \[ \frac{1}{6} \log \left (3 e^x+2\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x/(4 + 6*E^x),x]

[Out]

Log[2 + 3*E^x]/6

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Rubi in Sympy [A]  time = 6.9181, size = 8, normalized size = 0.67 \[ \frac{\log{\left (3 e^{x} + 2 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)/(4+6*exp(x)),x)

[Out]

log(3*exp(x) + 2)/6

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Mathematica [A]  time = 0.00338894, size = 12, normalized size = 1. \[ \frac{1}{6} \log \left (6 e^x+4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/(4 + 6*E^x),x]

[Out]

Log[4 + 6*E^x]/6

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Maple [A]  time = 0.003, size = 10, normalized size = 0.8 \[{\frac{\ln \left ( 2+3\,{{\rm e}^{x}} \right ) }{6}} \]

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 = 0.759142, size = 12, normalized size = 1. \[ \frac{1}{6} \, \log \left (3 \, e^{x} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2*e^x/(3*e^x + 2),x, algorithm="maxima")

[Out]

1/6*log(3*e^x + 2)

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Fricas [A]  time = 0.253753, size = 12, normalized size = 1. \[ \frac{1}{6} \, \log \left (3 \, e^{x} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2*e^x/(3*e^x + 2),x, algorithm="fricas")

[Out]

1/6*log(3*e^x + 2)

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Sympy [A]  time = 0.129865, 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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GIAC/XCAS [A]  time = 0.225902, size = 12, normalized size = 1. \[ \frac{1}{6} \,{\rm ln}\left (3 \, e^{x} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2*e^x/(3*e^x + 2),x, algorithm="giac")

[Out]

1/6*ln(3*e^x + 2)