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

Optimal. Leaf size=33 \[ \frac{1}{2} e^t \sqrt{9-e^{2 t}}+\frac{9}{2} \sin ^{-1}\left (\frac{e^t}{3}\right ) \]

[Out]

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

_______________________________________________________________________________________

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]

[Out]

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

_______________________________________________________________________________________

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)

[Out]

sqrt(-exp(2*t) + 9)*exp(t)/2 + 9*asin(exp(t)/3)/2

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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")

[Out]

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

_______________________________________________________________________________________

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")

[Out]

-1/2*(18*(6*sqrt(-e^(2*t) + 9) + e^(2*t) - 18)*arctan((sqrt(-e^(2*t) + 9) - 3)*e
^(-t)) - (e^(3*t) - 18*e^t)*sqrt(-e^(2*t) + 9) + 6*e^(3*t) - 54*e^t)/(6*sqrt(-e^
(2*t) + 9) + e^(2*t) - 18)

_______________________________________________________________________________________

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)

[Out]

Piecewise((sqrt(-exp(2*t) + 9)*exp(t)/2 + 9*asin(exp(t)/3)/2, exp(t) < log(3)))

_______________________________________________________________________________________

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]

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