3.631 \(\int \frac{e^x}{\sqrt{1-e^{2 x}}} \, dx\)

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

[Out]

ArcSin[E^x]

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Rubi [A]  time = 0.0362115, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \sin ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x/Sqrt[1 - E^(2*x)],x]

[Out]

ArcSin[E^x]

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Rubi in Sympy [A]  time = 5.80158, size = 3, normalized size = 0.75 \[ \operatorname{asin}{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asin(exp(x))

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Mathematica [A]  time = 0.0116867, size = 4, normalized size = 1. \[ \sin ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/Sqrt[1 - E^(2*x)],x]

[Out]

ArcSin[E^x]

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Maple [A]  time = 0.01, size = 4, normalized size = 1. \[ \arcsin \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsin(exp(x))

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Maxima [A]  time = 0.886182, size = 4, normalized size = 1. \[ \arcsin \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/sqrt(-e^(2*x) + 1),x, algorithm="maxima")

[Out]

arcsin(e^x)

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Fricas [A]  time = 0.388679, size = 27, normalized size = 6.75 \[ -2 \, \arctan \left ({\left (\sqrt{-e^{\left (2 \, x\right )} + 1} - 1\right )} e^{\left (-x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/sqrt(-e^(2*x) + 1),x, algorithm="fricas")

[Out]

-2*arctan((sqrt(-e^(2*x) + 1) - 1)*e^(-x))

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Sympy [A]  time = 0.532435, size = 3, normalized size = 0.75 \[ \operatorname{asin}{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asin(exp(x))

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GIAC/XCAS [A]  time = 0.238143, size = 4, normalized size = 1. \[ \arcsin \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/sqrt(-e^(2*x) + 1),x, algorithm="giac")

[Out]

arcsin(e^x)