3.647 \(\int \frac{e^x}{\sqrt{-3+e^{2 x}}} \, dx\)

Optimal. Leaf size=16 \[ \tanh ^{-1}\left (\frac{e^x}{\sqrt{e^{2 x}-3}}\right ) \]

[Out]

ArcTanh[E^x/Sqrt[-3 + E^(2*x)]]

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Rubi [A]  time = 0.0385788, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \tanh ^{-1}\left (\frac{e^x}{\sqrt{e^{2 x}-3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x/Sqrt[-3 + E^(2*x)],x]

[Out]

ArcTanh[E^x/Sqrt[-3 + E^(2*x)]]

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Rubi in Sympy [A]  time = 5.49344, size = 14, normalized size = 0.88 \[ \operatorname{atanh}{\left (\frac{e^{x}}{\sqrt{e^{2 x} - 3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)/(-3+exp(2*x))**(1/2),x)

[Out]

atanh(exp(x)/sqrt(exp(2*x) - 3))

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Mathematica [A]  time = 0.0126313, size = 16, normalized size = 1. \[ \log \left (\sqrt{e^{2 x}-3}+e^x\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[E^x + Sqrt[-3 + E^(2*x)]]

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Maple [A]  time = 0.013, size = 13, normalized size = 0.8 \[ \ln \left ({{\rm e}^{x}}+\sqrt{-3+ \left ({{\rm e}^{x}} \right ) ^{2}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(-3+exp(2*x))^(1/2),x)

[Out]

ln(exp(x)+(-3+exp(x)^2)^(1/2))

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Maxima [A]  time = 0.807099, size = 22, normalized size = 1.38 \[ \log \left (2 \, \sqrt{e^{\left (2 \, x\right )} - 3} + 2 \, e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(2*sqrt(e^(2*x) - 3) + 2*e^x)

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Fricas [A]  time = 0.234694, size = 22, normalized size = 1.38 \[ -\log \left (\sqrt{e^{\left (2 \, x\right )} - 3} - e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(sqrt(e^(2*x) - 3) - e^x)

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Sympy [A]  time = 0.513242, size = 10, normalized size = 0.62 \[ \operatorname{acosh}{\left (\frac{\sqrt{3} e^{x}}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)/(-3+exp(2*x))**(1/2),x)

[Out]

acosh(sqrt(3)*exp(x)/3)

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GIAC/XCAS [A]  time = 0.227905, size = 22, normalized size = 1.38 \[ -{\rm ln}\left (-\sqrt{e^{\left (2 \, x\right )} - 3} + e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(-sqrt(e^(2*x) - 3) + e^x)