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