3.648 \(\int \frac{e^x}{16-e^{2 x}} \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{4} \tanh ^{-1}\left (\frac{e^x}{4}\right ) \]

[Out]

ArcTanh[E^x/4]/4

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Rubi [A]  time = 0.0336353, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{4} \tanh ^{-1}\left (\frac{e^x}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x/(16 - E^(2*x)),x]

[Out]

ArcTanh[E^x/4]/4

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Rubi in Sympy [A]  time = 7.29467, size = 7, normalized size = 0.58 \[ \frac{\operatorname{atanh}{\left (\frac{e^{x}}{4} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)/(16-exp(2*x)),x)

[Out]

atanh(exp(x)/4)/4

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Mathematica [A]  time = 0.00605024, size = 23, normalized size = 1.92 \[ \frac{1}{8} \log \left (e^x+4\right )-\frac{1}{8} \log \left (4-e^x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/(16 - E^(2*x)),x]

[Out]

-Log[4 - E^x]/8 + Log[4 + E^x]/8

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Maple [B]  time = 0.01, size = 16, normalized size = 1.3 \[ -{\frac{\ln \left ({{\rm e}^{x}}-4 \right ) }{8}}+{\frac{\ln \left ( 4+{{\rm e}^{x}} \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(16-exp(2*x)),x)

[Out]

-1/8*ln(exp(x)-4)+1/8*ln(4+exp(x))

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Maxima [A]  time = 0.817722, size = 20, normalized size = 1.67 \[ \frac{1}{8} \, \log \left (e^{x} + 4\right ) - \frac{1}{8} \, \log \left (e^{x} - 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-e^x/(e^(2*x) - 16),x, algorithm="maxima")

[Out]

1/8*log(e^x + 4) - 1/8*log(e^x - 4)

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Fricas [A]  time = 0.231038, size = 20, normalized size = 1.67 \[ \frac{1}{8} \, \log \left (e^{x} + 4\right ) - \frac{1}{8} \, \log \left (e^{x} - 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-e^x/(e^(2*x) - 16),x, algorithm="fricas")

[Out]

1/8*log(e^x + 4) - 1/8*log(e^x - 4)

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Sympy [A]  time = 0.095026, size = 15, normalized size = 1.25 \[ - \frac{\log{\left (e^{x} - 4 \right )}}{8} + \frac{\log{\left (e^{x} + 4 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)/(16-exp(2*x)),x)

[Out]

-log(exp(x) - 4)/8 + log(exp(x) + 4)/8

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GIAC/XCAS [A]  time = 0.23847, size = 22, normalized size = 1.83 \[ \frac{1}{8} \,{\rm ln}\left (e^{x} + 4\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | e^{x} - 4 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-e^x/(e^(2*x) - 16),x, algorithm="giac")

[Out]

1/8*ln(e^x + 4) - 1/8*ln(abs(e^x - 4))