3.669 \(\int \frac{e^x}{2+3 e^x+e^{2 x}} \, dx\)

Optimal. Leaf size=15 \[ \log \left (e^x+1\right )-\log \left (e^x+2\right ) \]

[Out]

Log[1 + E^x] - Log[2 + E^x]

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Rubi [A]  time = 0.0471533, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \log \left (e^x+1\right )-\log \left (e^x+2\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x/(2 + 3*E^x + E^(2*x)),x]

[Out]

Log[1 + E^x] - Log[2 + E^x]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)/(2+3*exp(x)+exp(2*x)),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0068166, size = 15, normalized size = 1. \[ \log \left (e^x+1\right )-\log \left (e^x+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/(2 + 3*E^x + E^(2*x)),x]

[Out]

Log[1 + E^x] - Log[2 + E^x]

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Maple [A]  time = 0.009, size = 14, normalized size = 0.9 \[ \ln \left ( 1+{{\rm e}^{x}} \right ) -\ln \left ( 2+{{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(2+3*exp(x)+exp(2*x)),x)

[Out]

ln(1+exp(x))-ln(2+exp(x))

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Maxima [A]  time = 0.771553, size = 18, normalized size = 1.2 \[ -\log \left (e^{x} + 2\right ) + \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(e^x + 2) + log(e^x + 1)

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Fricas [A]  time = 0.255223, size = 18, normalized size = 1.2 \[ -\log \left (e^{x} + 2\right ) + \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(e^x + 2) + log(e^x + 1)

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Sympy [A]  time = 0.102234, size = 12, normalized size = 0.8 \[ \log{\left (e^{x} + 1 \right )} - \log{\left (e^{x} + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)/(2+3*exp(x)+exp(2*x)),x)

[Out]

log(exp(x) + 1) - log(exp(x) + 2)

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GIAC/XCAS [A]  time = 0.296672, size = 18, normalized size = 1.2 \[ -{\rm ln}\left (e^{x} + 2\right ) +{\rm ln}\left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(e^x + 2) + ln(e^x + 1)