3.151 \(\int \sqrt{-9+e^{2 t}} \, dt\)

Optimal. Leaf size=30 \[ \sqrt{e^{2 t}-9}-3 \tan ^{-1}\left (\frac{1}{3} \sqrt{e^{2 t}-9}\right ) \]

[Out]

Sqrt[-9 + E^(2*t)] - 3*ArcTan[Sqrt[-9 + E^(2*t)]/3]

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Rubi [A]  time = 0.0333224, antiderivative size = 30, 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 \[ \sqrt{e^{2 t}-9}-3 \tan ^{-1}\left (\frac{1}{3} \sqrt{e^{2 t}-9}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-9 + E^(2*t)],t]

[Out]

Sqrt[-9 + E^(2*t)] - 3*ArcTan[Sqrt[-9 + E^(2*t)]/3]

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Rubi in Sympy [A]  time = 2.09252, size = 24, normalized size = 0.8 \[ \sqrt{e^{2 t} - 9} - 3 \operatorname{atan}{\left (\frac{\sqrt{e^{2 t} - 9}}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-9+exp(2*t))**(1/2),t)

[Out]

sqrt(exp(2*t) - 9) - 3*atan(sqrt(exp(2*t) - 9)/3)

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Mathematica [A]  time = 0.0134914, size = 30, normalized size = 1. \[ \sqrt{e^{2 t}-9}-3 \tan ^{-1}\left (\frac{1}{3} \sqrt{e^{2 t}-9}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-9 + E^(2*t)],t]

[Out]

Sqrt[-9 + E^(2*t)] - 3*ArcTan[Sqrt[-9 + E^(2*t)]/3]

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Maple [A]  time = 0.009, size = 23, normalized size = 0.8 \[ -3\,\arctan \left ( 1/3\,\sqrt{-9+{{\rm e}^{2\,t}}} \right ) +\sqrt{-9+{{\rm e}^{2\,t}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-9+exp(2*t))^(1/2),t)

[Out]

-3*arctan(1/3*(-9+exp(2*t))^(1/2))+(-9+exp(2*t))^(1/2)

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Maxima [A]  time = 1.53706, size = 30, normalized size = 1. \[ \sqrt{e^{\left (2 \, t\right )} - 9} - 3 \, \arctan \left (\frac{1}{3} \, \sqrt{e^{\left (2 \, t\right )} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e^(2*t) - 9),t, algorithm="maxima")

[Out]

sqrt(e^(2*t) - 9) - 3*arctan(1/3*sqrt(e^(2*t) - 9))

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Fricas [A]  time = 0.212037, size = 30, normalized size = 1. \[ \sqrt{e^{\left (2 \, t\right )} - 9} - 3 \, \arctan \left (\frac{1}{3} \, \sqrt{e^{\left (2 \, t\right )} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e^(2*t) - 9),t, algorithm="fricas")

[Out]

sqrt(e^(2*t) - 9) - 3*arctan(1/3*sqrt(e^(2*t) - 9))

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Sympy [A]  time = 1.361, size = 22, normalized size = 0.73 \[ \begin{cases} \sqrt{e^{2 t} - 9} - 3 \operatorname{acos}{\left (3 e^{- t} \right )} & \text{for}\: e^{t} < \log{\left (3 \right )} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-9+exp(2*t))**(1/2),t)

[Out]

Piecewise((sqrt(exp(2*t) - 9) - 3*acos(3*exp(-t)), exp(t) < log(3)))

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GIAC/XCAS [A]  time = 0.208623, size = 30, normalized size = 1. \[ \sqrt{e^{\left (2 \, t\right )} - 9} - 3 \, \arctan \left (\frac{1}{3} \, \sqrt{e^{\left (2 \, t\right )} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e^(2*t) - 9),t, algorithm="giac")

[Out]

sqrt(e^(2*t) - 9) - 3*arctan(1/3*sqrt(e^(2*t) - 9))