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

Optimal. Leaf size=12 \[ x-2 \log \left (1-e^x\right ) \]

[Out]

x - 2*Log[1 - E^x]

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Rubi [A]  time = 0.0432246, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ x-2 \log \left (1-e^x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + E^x)/(1 - E^x),x]

[Out]

x - 2*Log[1 - E^x]

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Rubi in Sympy [A]  time = 5.30623, size = 15, normalized size = 1.25 \[ - x - 2 \log{\left (- e^{x} + 1 \right )} + 2 \log{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00594752, size = 12, normalized size = 1. \[ x-2 \log \left (1-e^x\right ) \]

Antiderivative was successfully verified.

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

[Out]

x - 2*Log[1 - E^x]

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Maple [A]  time = 0.009, size = 12, normalized size = 1. \[ \ln \left ({{\rm e}^{x}} \right ) -2\,\ln \left ( -1+{{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+exp(x))/(1-exp(x)),x)

[Out]

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

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Maxima [A]  time = 1.34889, size = 12, normalized size = 1. \[ x - 2 \, \log \left (e^{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 2*log(e^x - 1)

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Fricas [A]  time = 0.222899, size = 12, normalized size = 1. \[ x - 2 \, \log \left (e^{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 2*log(e^x - 1)

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Sympy [A]  time = 0.072978, size = 8, normalized size = 0.67 \[ x - 2 \log{\left (e^{x} - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 2*log(exp(x) - 1)

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GIAC/XCAS [A]  time = 0.208635, size = 14, normalized size = 1.17 \[ x - 2 \,{\rm ln}\left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 2*ln(abs(e^x - 1))