3.2.50 \(\int e^t \sqrt {9-e^{2 t}} \, dt\) [150]

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]

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

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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 = {2281, 201, 222} \begin {gather*} \frac {1}{2} e^t \sqrt {9-e^{2 t}}+\frac {9}{2} \sin ^{-1}\left (\frac {e^t}{3}\right ) \end {gather*}

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 201

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 222

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]

Rule 2281

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]

Rubi steps

\begin {align*} \int e^t \sqrt {9-e^{2 t}} \, dt &=\text {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} \text {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*}

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Mathematica [A]
time = 0.05, size = 45, normalized size = 1.36 \begin {gather*} \frac {1}{2} e^t \sqrt {9-e^{2 t}}-9 \tan ^{-1}\left (\frac {\sqrt {9-e^{2 t}}}{3+e^t}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^t*Sqrt[9 - E^(2*t)],t]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.65, size = 37, normalized size = 1.12 \begin {gather*} \text {ConditionalExpression}\left [\frac {E^t \sqrt {9-E^{2 t}}}{2}+\frac {9 \text {ArcSin}\left [\frac {E^t}{3}\right ]}{2},E^t>-3\text {\&\&}E^t<3\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[E^t*Sqrt[9 - E^(2*t)],t]')

[Out]

ConditionalExpression[E ^ t Sqrt[9 - E ^ (2 t)] / 2 + 9 ArcSin[E ^ t / 3] / 2, E ^ t > -3 && E ^ t < 3]

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Maple [A]
time = 0.02, size = 23, normalized size = 0.70

method	result	size
default	\(\frac {9 \arcsin \left (\frac {{\mathrm e}^{t}}{3}\right )}{2}+\frac {{\mathrm e}^{t} \sqrt {9-{\mathrm e}^{2 t}}}{2}\)	\(23\)
risch	\(-\frac {{\mathrm e}^{t} \left (-9+{\mathrm e}^{2 t}\right )}{2 \sqrt {9-{\mathrm e}^{2 t}}}+\frac {9 \arcsin \left (\frac {{\mathrm e}^{t}}{3}\right )}{2}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(t)*(9-exp(2*t))^(1/2),t,method=_RETURNVERBOSE)

[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 = 0.35, size = 22, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, \sqrt {-e^{\left (2 \, t\right )} + 9} e^{t} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, e^{t}\right ) \end {gather*}

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 = 0.35, size = 35, normalized size = 1.06 \begin {gather*} \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 {gather*}

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 = 0.66, size = 32, normalized size = 0.97 \begin {gather*} \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} > -3 \wedge e^{t} < 3 \end {cases} \end {gather*}

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) > -3) & (exp(t) < 3)))

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Giac [A]
time = 0.00, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{2} \mathrm {e}^{t} \sqrt {-\left (\mathrm {e}^{t}\right )^{2}+9}+\frac {9}{2} \arcsin \left (\frac {\mathrm {e}^{t}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.21, size = 22, normalized size = 0.67 \begin {gather*} \frac {9\,\mathrm {asin}\left (\frac {{\mathrm {e}}^t}{3}\right )}{2}+\frac {{\mathrm {e}}^t\,\sqrt {9-{\mathrm {e}}^{2\,t}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(t)*(9 - exp(2*t))^(1/2),t)

[Out]

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

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