3.331 \(\int \frac{1}{1-e^{-x}+2 e^x} \, dx\)

Optimal. Leaf size=23 \[ \frac{1}{3} \log \left (1-2 e^x\right )-\frac{1}{3} \log \left (e^x+1\right ) \]

[Out]

Log[1 - 2*E^x]/3 - Log[1 + E^x]/3

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Rubi [A]  time = 0.0304393, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{3} \log \left (1-2 e^x\right )-\frac{1}{3} \log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - E^(-x) + 2*E^x)^(-1),x]

[Out]

Log[1 - 2*E^x]/3 - Log[1 + E^x]/3

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Rubi in Sympy [A]  time = 2.50401, size = 17, normalized size = 0.74 \[ \frac{\log{\left (- 2 e^{x} + 1 \right )}}{3} - \frac{\log{\left (e^{x} + 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-1/exp(x)+2*exp(x)),x)

[Out]

log(-2*exp(x) + 1)/3 - log(exp(x) + 1)/3

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Mathematica [A]  time = 0.0151064, size = 21, normalized size = 0.91 \[ \frac{1}{3} \left (\log \left (1-2 e^x\right )-\log \left (e^x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - E^(-x) + 2*E^x)^(-1),x]

[Out]

(Log[1 - 2*E^x] - Log[1 + E^x])/3

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Maple [A]  time = 0.01, size = 18, normalized size = 0.8 \[ -{\frac{\ln \left ( 1+{{\rm e}^{x}} \right ) }{3}}+{\frac{\ln \left ( 2\,{{\rm e}^{x}}-1 \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-1/exp(x)+2*exp(x)),x)

[Out]

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

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Maxima [A]  time = 1.34847, size = 26, normalized size = 1.13 \[ -\frac{1}{3} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{3} \, \log \left (e^{\left (-x\right )} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*log(e^(-x) + 1) + 1/3*log(e^(-x) - 2)

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Fricas [A]  time = 0.226399, size = 23, normalized size = 1. \[ \frac{1}{3} \, \log \left (2 \, e^{x} - 1\right ) - \frac{1}{3} \, \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*log(2*e^x - 1) - 1/3*log(e^x + 1)

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Sympy [A]  time = 0.133984, size = 17, normalized size = 0.74 \[ \frac{\log{\left (e^{x} - \frac{1}{2} \right )}}{3} - \frac{\log{\left (e^{x} + 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-1/exp(x)+2*exp(x)),x)

[Out]

log(exp(x) - 1/2)/3 - log(exp(x) + 1)/3

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GIAC/XCAS [A]  time = 0.212068, size = 24, normalized size = 1.04 \[ -\frac{1}{3} \,{\rm ln}\left (e^{x} + 1\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | 2 \, e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*ln(e^x + 1) + 1/3*ln(abs(2*e^x - 1))