3.622 \(\int \frac{e^x}{3-4 e^{2 x}} \, dx\)

Optimal. Leaf size=20 \[ \frac{\tanh ^{-1}\left (\frac{2 e^x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

ArcTanh[(2*E^x)/Sqrt[3]]/(2*Sqrt[3])

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Rubi [A]  time = 0.0388341, antiderivative size = 20, 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 \[ \frac{\tanh ^{-1}\left (\frac{2 e^x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[E^x/(3 - 4*E^(2*x)),x]

[Out]

ArcTanh[(2*E^x)/Sqrt[3]]/(2*Sqrt[3])

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Rubi in Sympy [A]  time = 7.19289, size = 19, normalized size = 0.95 \[ \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{2 \sqrt{3} e^{x}}{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)/(3-4*exp(2*x)),x)

[Out]

sqrt(3)*atanh(2*sqrt(3)*exp(x)/3)/6

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Mathematica [A]  time = 0.0229991, size = 36, normalized size = 1.8 \[ \frac{\log \left (2 e^x+\sqrt{3}\right )-\log \left (\sqrt{3}-2 e^x\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/(3 - 4*E^(2*x)),x]

[Out]

(-Log[Sqrt[3] - 2*E^x] + Log[Sqrt[3] + 2*E^x])/(4*Sqrt[3])

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Maple [A]  time = 0.006, size = 14, normalized size = 0.7 \[{\frac{\sqrt{3}}{6}{\it Artanh} \left ({\frac{2\,{{\rm e}^{x}}\sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(3-4*exp(2*x)),x)

[Out]

1/6*arctanh(2/3*exp(x)*3^(1/2))*3^(1/2)

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Maxima [A]  time = 0.912627, size = 35, normalized size = 1.75 \[ -\frac{1}{12} \, \sqrt{3} \log \left (-\frac{\sqrt{3} - 2 \, e^{x}}{\sqrt{3} + 2 \, e^{x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-e^x/(4*e^(2*x) - 3),x, algorithm="maxima")

[Out]

-1/12*sqrt(3)*log(-(sqrt(3) - 2*e^x)/(sqrt(3) + 2*e^x))

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Fricas [A]  time = 0.244315, size = 49, normalized size = 2.45 \[ \frac{1}{12} \, \sqrt{3} \log \left (\frac{4 \, \sqrt{3} e^{\left (2 \, x\right )} + 3 \, \sqrt{3} + 12 \, e^{x}}{4 \, e^{\left (2 \, x\right )} - 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-e^x/(4*e^(2*x) - 3),x, algorithm="fricas")

[Out]

1/12*sqrt(3)*log((4*sqrt(3)*e^(2*x) + 3*sqrt(3) + 12*e^x)/(4*e^(2*x) - 3))

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Sympy [A]  time = 0.110476, size = 15, normalized size = 0.75 \[ \operatorname{RootSum}{\left (48 z^{2} - 1, \left ( i \mapsto i \log{\left (6 i + e^{x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)/(3-4*exp(2*x)),x)

[Out]

RootSum(48*_z**2 - 1, Lambda(_i, _i*log(6*_i + exp(x))))

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GIAC/XCAS [A]  time = 0.224921, size = 41, normalized size = 2.05 \[ \frac{1}{12} \, \sqrt{3}{\rm ln}\left (\frac{1}{2} \, \sqrt{3} + e^{x}\right ) - \frac{1}{12} \, \sqrt{3}{\rm ln}\left ({\left | -\frac{1}{2} \, \sqrt{3} + e^{x} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-e^x/(4*e^(2*x) - 3),x, algorithm="giac")

[Out]

1/12*sqrt(3)*ln(1/2*sqrt(3) + e^x) - 1/12*sqrt(3)*ln(abs(-1/2*sqrt(3) + e^x))