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

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Rubi [A]  time = 0.0246832, 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, Rules used = {2249, 195, 216} \[ \frac{1}{2} e^t \sqrt{9-e^{2 t}}+\frac{9}{2} \sin ^{-1}\left (\frac{e^t}{3}\right ) \]

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

Rule 2249

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(d*e*Log[F])/(g*h*Log[G])]}, Dist[Denominator[m]/(g*h*Log[G]), Subst[Int[x^(Denominator[m]
- 1)*(a + b*F^(c*e - (d*e*f)/g)*x^Numerator[m])^p, x], x, G^((h*(f + g*x))/Denominator[m])], x] /; LtQ[m, -1]
|| GtQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int e^t \sqrt{9-e^{2 t}} \, dt &=\operatorname{Subst}\left (\int \sqrt{9-t^2} \, dt,t,e^t\right )\\ &=\frac{1}{2} e^t \sqrt{9-e^{2 t}}+\frac{9}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{9-t^2}} \, dt,t,e^t\right )\\ &=\frac{1}{2} e^t \sqrt{9-e^{2 t}}+\frac{9}{2} \sin ^{-1}\left (\frac{e^t}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0127705, size = 32, normalized size = 0.97 \[ \frac{1}{2} \left (e^t \sqrt{9-e^{2 t}}+9 \sin ^{-1}\left (\frac{e^t}{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^t*Sqrt[9 - E^(2*t)],t]

[Out]

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

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Maple [A]  time = 0.008, size = 23, normalized size = 0.7 \begin{align*}{\frac{{{\rm e}^{t}}}{2}\sqrt{9- \left ({{\rm e}^{t}} \right ) ^{2}}}+{\frac{9}{2}\arcsin \left ({\frac{{{\rm e}^{t}}}{3}} \right ) } \end{align*}

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))

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Maxima [A]  time = 1.40656, size = 30, normalized size = 0.91 \begin{align*} \frac{1}{2} \, \sqrt{-e^{\left (2 \, t\right )} + 9} e^{t} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, e^{t}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)*(9-exp(2*t))^(1/2),t, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.28943, size = 97, normalized size = 2.94 \begin{align*} \frac{1}{2} \, \sqrt{-e^{\left (2 \, t\right )} + 9} e^{t} - 9 \, \arctan \left ({\left (\sqrt{-e^{\left (2 \, t\right )} + 9} - 3\right )} e^{\left (-t\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)*(9-exp(2*t))^(1/2),t, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.23372, size = 29, normalized size = 0.88 \begin{align*} \begin{cases} \frac{\sqrt{9 - e^{2 t}} e^{t}}{2} + \frac{9 \operatorname{asin}{\left (\frac{e^{t}}{3} \right )}}{2} & \text{for}\: e^{t} < \log{\left (3 \right )} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)*(9-exp(2*t))**(1/2),t)

[Out]

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

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Giac [A]  time = 1.05804, size = 30, normalized size = 0.91 \begin{align*} \frac{1}{2} \, \sqrt{-e^{\left (2 \, t\right )} + 9} e^{t} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, e^{t}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)*(9-exp(2*t))^(1/2),t, algorithm="giac")

[Out]

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