Optimal. Leaf size=33 \[ \frac{1}{2} e^t \sqrt{9-e^{2 t}}+\frac{9}{2} \sin ^{-1}\left (\frac{e^t}{3}\right ) \]
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Rubi [A] time = 0.0459188, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{2} e^t \sqrt{9-e^{2 t}}+\frac{9}{2} \sin ^{-1}\left (\frac{e^t}{3}\right ) \]
Antiderivative was successfully verified.
[In] Int[E^t*Sqrt[9 - E^(2*t)],t]
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Rubi in Sympy [A] time = 3.10135, size = 24, normalized size = 0.73 \[ \frac{\sqrt{- e^{2 t} + 9} e^{t}}{2} + \frac{9 \operatorname{asin}{\left (\frac{e^{t}}{3} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(exp(t)*(9-exp(2*t))**(1/2),t)
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Mathematica [A] time = 0.0242512, size = 33, normalized size = 1. \[ \frac{1}{2} e^t \sqrt{9-e^{2 t}}+\frac{9}{2} \sin ^{-1}\left (\frac{e^t}{3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[E^t*Sqrt[9 - E^(2*t)],t]
[Out]
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Maple [A] time = 0.014, size = 23, normalized size = 0.7 \[{\frac{{{\rm e}^{t}}}{2}\sqrt{9- \left ({{\rm e}^{t}} \right ) ^{2}}}+{\frac{9}{2}\arcsin \left ({\frac{{{\rm e}^{t}}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(exp(t)*(9-exp(2*t))^(1/2),t)
[Out]
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Maxima [A] time = 1.59341, size = 30, normalized size = 0.91 \[ \frac{1}{2} \, \sqrt{-e^{\left (2 \, t\right )} + 9} e^{t} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, e^{t}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^(2*t) + 9)*e^t,t, algorithm="maxima")
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Fricas [A] time = 0.211551, size = 124, normalized size = 3.76 \[ -\frac{18 \,{\left (6 \, \sqrt{-e^{\left (2 \, t\right )} + 9} + e^{\left (2 \, t\right )} - 18\right )} \arctan \left ({\left (\sqrt{-e^{\left (2 \, t\right )} + 9} - 3\right )} e^{\left (-t\right )}\right ) -{\left (e^{\left (3 \, t\right )} - 18 \, e^{t}\right )} \sqrt{-e^{\left (2 \, t\right )} + 9} + 6 \, e^{\left (3 \, t\right )} - 54 \, e^{t}}{2 \,{\left (6 \, \sqrt{-e^{\left (2 \, t\right )} + 9} + e^{\left (2 \, t\right )} - 18\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^(2*t) + 9)*e^t,t, algorithm="fricas")
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Sympy [A] time = 1.63177, size = 29, normalized size = 0.88 \[ \begin{cases} \frac{\sqrt{- e^{2 t} + 9} e^{t}}{2} + \frac{9 \operatorname{asin}{\left (\frac{e^{t}}{3} \right )}}{2} & \text{for}\: e^{t} < \log{\left (3 \right )} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(exp(t)*(9-exp(2*t))**(1/2),t)
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GIAC/XCAS [A] time = 0.211744, size = 30, normalized size = 0.91 \[ \frac{1}{2} \, \sqrt{-e^{\left (2 \, t\right )} + 9} e^{t} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, e^{t}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^(2*t) + 9)*e^t,t, algorithm="giac")
[Out]