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

Optimal. Leaf size=6 \[ -\tanh ^{-1}\left (e^x\right ) \]

[Out]

-ArcTanh[E^x]

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Rubi [A]  time = 0.0309683, antiderivative size = 6, 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 \[ -\tanh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x/(-1 + E^(2*x)),x]

[Out]

-ArcTanh[E^x]

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Rubi in Sympy [A]  time = 3.017, size = 5, normalized size = 0.83 \[ - \operatorname{atanh}{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(exp(x))

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Mathematica [B]  time = 0.00554211, size = 23, normalized size = 3.83 \[ \frac{1}{2} \log \left (1-e^x\right )-\frac{1}{2} \log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/(-1 + E^(2*x)),x]

[Out]

Log[1 - E^x]/2 - Log[1 + E^x]/2

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Maple [A]  time = 0., size = 6, normalized size = 1. \[ -{\it Artanh} \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(-1+exp(2*x)),x)

[Out]

-arctanh(exp(x))

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Maxima [A]  time = 1.33882, size = 20, normalized size = 3.33 \[ -\frac{1}{2} \, \log \left (e^{x} + 1\right ) + \frac{1}{2} \, \log \left (e^{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*log(e^x + 1) + 1/2*log(e^x - 1)

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Fricas [A]  time = 0.212988, size = 20, normalized size = 3.33 \[ -\frac{1}{2} \, \log \left (e^{x} + 1\right ) + \frac{1}{2} \, \log \left (e^{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*log(e^x + 1) + 1/2*log(e^x - 1)

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Sympy [A]  time = 0.095184, size = 15, normalized size = 2.5 \[ \frac{\log{\left (e^{x} - 1 \right )}}{2} - \frac{\log{\left (e^{x} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(exp(x) - 1)/2 - log(exp(x) + 1)/2

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GIAC/XCAS [A]  time = 0.198903, size = 22, normalized size = 3.67 \[ -\frac{1}{2} \,{\rm ln}\left (e^{x} + 1\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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