3.627 \(\int \frac{e^x}{-4+e^{2 x}} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{2} \tanh ^{-1}\left (\frac{e^x}{2}\right ) \]

[Out]

-ArcTanh[E^x/2]/2

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Rubi [A]  time = 0.031038, antiderivative size = 12, 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 \[ -\frac{1}{2} \tanh ^{-1}\left (\frac{e^x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x/(-4 + E^(2*x)),x]

[Out]

-ArcTanh[E^x/2]/2

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Rubi in Sympy [A]  time = 5.61878, size = 8, normalized size = 0.67 \[ - \frac{\operatorname{atanh}{\left (\frac{e^{x}}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)/(-4+exp(2*x)),x)

[Out]

-atanh(exp(x)/2)/2

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Mathematica [A]  time = 0.00522724, size = 23, normalized size = 1.92 \[ \frac{1}{4} \log \left (2-e^x\right )-\frac{1}{4} \log \left (e^x+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/(-4 + E^(2*x)),x]

[Out]

Log[2 - E^x]/4 - Log[2 + E^x]/4

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Maple [B]  time = 0.009, size = 16, normalized size = 1.3 \[ -{\frac{\ln \left ( 2+{{\rm e}^{x}} \right ) }{4}}+{\frac{\ln \left ( -2+{{\rm e}^{x}} \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(-4+exp(2*x)),x)

[Out]

-1/4*ln(2+exp(x))+1/4*ln(-2+exp(x))

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Maxima [A]  time = 0.777038, size = 20, normalized size = 1.67 \[ -\frac{1}{4} \, \log \left (e^{x} + 2\right ) + \frac{1}{4} \, \log \left (e^{x} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(e^(2*x) - 4),x, algorithm="maxima")

[Out]

-1/4*log(e^x + 2) + 1/4*log(e^x - 2)

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Fricas [A]  time = 0.250054, size = 20, normalized size = 1.67 \[ -\frac{1}{4} \, \log \left (e^{x} + 2\right ) + \frac{1}{4} \, \log \left (e^{x} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(e^(2*x) - 4),x, algorithm="fricas")

[Out]

-1/4*log(e^x + 2) + 1/4*log(e^x - 2)

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Sympy [A]  time = 0.090813, size = 15, normalized size = 1.25 \[ \frac{\log{\left (e^{x} - 2 \right )}}{4} - \frac{\log{\left (e^{x} + 2 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)/(-4+exp(2*x)),x)

[Out]

log(exp(x) - 2)/4 - log(exp(x) + 2)/4

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GIAC/XCAS [A]  time = 0.234528, size = 22, normalized size = 1.83 \[ -\frac{1}{4} \,{\rm ln}\left (e^{x} + 2\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | e^{x} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(e^(2*x) - 4),x, algorithm="giac")

[Out]

-1/4*ln(e^x + 2) + 1/4*ln(abs(e^x - 2))