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

Optimal. Leaf size=17 \[ 2 \log \left (e^x+2\right )-\log \left (e^x+1\right ) \]

[Out]

-Log[1 + E^x] + 2*Log[2 + E^x]

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Rubi [A]  time = 0.0546851, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ 2 \log \left (e^x+2\right )-\log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

-Log[1 + E^x] + 2*Log[2 + E^x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{2 x}}{e^{2 x} + 3 e^{x} + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(exp(2*x)/(exp(2*x) + 3*exp(x) + 2), x)

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Mathematica [A]  time = 0.00973932, size = 17, normalized size = 1. \[ 2 \log \left (e^x+2\right )-\log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

-Log[1 + E^x] + 2*Log[2 + E^x]

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Maple [A]  time = 0.011, size = 16, normalized size = 0.9 \[ -\ln \left ( 1+{{\rm e}^{x}} \right ) +2\,\ln \left ( 2+{{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.776713, size = 20, normalized size = 1.18 \[ 2 \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(e^x + 2) - log(e^x + 1)

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Fricas [A]  time = 0.241979, size = 20, normalized size = 1.18 \[ 2 \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(e^x + 2) - log(e^x + 1)

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Sympy [A]  time = 0.114886, size = 14, normalized size = 0.82 \[ - \log{\left (e^{x} + 1 \right )} + 2 \log{\left (e^{x} + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(exp(x) + 1) + 2*log(exp(x) + 2)

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GIAC/XCAS [A]  time = 0.264985, size = 20, normalized size = 1.18 \[ 2 \,{\rm ln}\left (e^{x} + 2\right ) -{\rm ln}\left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*ln(e^x + 2) - ln(e^x + 1)