3.367 \(\int \sqrt{-1+e^{2 x}} \, dx\)

Optimal. Leaf size=26 \[ \sqrt{e^{2 x}-1}-\tan ^{-1}\left (\sqrt{e^{2 x}-1}\right ) \]

[Out]

Sqrt[-1 + E^(2*x)] - ArcTan[Sqrt[-1 + E^(2*x)]]

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Rubi [A]  time = 0.0298099, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \sqrt{e^{2 x}-1}-\tan ^{-1}\left (\sqrt{e^{2 x}-1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-1 + E^(2*x)],x]

[Out]

Sqrt[-1 + E^(2*x)] - ArcTan[Sqrt[-1 + E^(2*x)]]

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Rubi in Sympy [A]  time = 2.03819, size = 20, normalized size = 0.77 \[ \sqrt{e^{2 x} - 1} - \operatorname{atan}{\left (\sqrt{e^{2 x} - 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(exp(2*x) - 1) - atan(sqrt(exp(2*x) - 1))

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Mathematica [A]  time = 0.0127561, size = 26, normalized size = 1. \[ \sqrt{e^{2 x}-1}-\tan ^{-1}\left (\sqrt{e^{2 x}-1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-1 + E^(2*x)],x]

[Out]

Sqrt[-1 + E^(2*x)] - ArcTan[Sqrt[-1 + E^(2*x)]]

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Maple [A]  time = 0.009, size = 21, normalized size = 0.8 \[ -\arctan \left ( \sqrt{-1+{{\rm e}^{2\,x}}} \right ) +\sqrt{-1+{{\rm e}^{2\,x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+exp(2*x))^(1/2),x)

[Out]

-arctan((-1+exp(2*x))^(1/2))+(-1+exp(2*x))^(1/2)

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Maxima [A]  time = 1.6402, size = 27, normalized size = 1.04 \[ \sqrt{e^{\left (2 \, x\right )} - 1} - \arctan \left (\sqrt{e^{\left (2 \, x\right )} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23769, size = 27, normalized size = 1.04 \[ \sqrt{e^{\left (2 \, x\right )} - 1} - \arctan \left (\sqrt{e^{\left (2 \, x\right )} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.33753, size = 19, normalized size = 0.73 \[ \begin{cases} \sqrt{e^{2 x} - 1} - \operatorname{acos}{\left (e^{- x} \right )} & \text{for}\: e^{x} < 0 \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((sqrt(exp(2*x) - 1) - acos(exp(-x)), exp(x) < 0))

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GIAC/XCAS [A]  time = 0.204701, size = 27, normalized size = 1.04 \[ \sqrt{e^{\left (2 \, x\right )} - 1} - \arctan \left (\sqrt{e^{\left (2 \, x\right )} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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