3.709 \(\int e^x \sqrt {9-e^{2 x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.03, 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 = {2249, 195, 216} \[ \frac {1}{2} e^x \sqrt {9-e^{2 x}}+\frac {9}{2} \sin ^{-1}\left (\frac {e^x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Sqrt[9 - E^(2*x)],x]

[Out]

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

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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 32, normalized size = 0.97 \[ \frac {1}{2} \left (e^x \sqrt {9-e^{2 x}}+9 \sin ^{-1}\left (\frac {e^x}{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Sqrt[9 - E^(2*x)],x]

[Out]

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

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fricas [A]  time = 0.41, size = 35, normalized size = 1.06 \[ \frac {1}{2} \, \sqrt {-e^{\left (2 \, x\right )} + 9} e^{x} - 9 \, \arctan \left ({\left (\sqrt {-e^{\left (2 \, x\right )} + 9} - 3\right )} e^{\left (-x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 22, normalized size = 0.67 \[ \frac {1}{2} \, \sqrt {-e^{\left (2 \, x\right )} + 9} e^{x} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 23, normalized size = 0.70 \[ \frac {9 \arcsin \left (\frac {{\mathrm e}^{x}}{3}\right )}{2}+\frac {\sqrt {-{\mathrm e}^{2 x}+9}\, {\mathrm e}^{x}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(9-exp(2*x))^(1/2),x)

[Out]

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

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maxima [A]  time = 2.31, size = 22, normalized size = 0.67 \[ \frac {1}{2} \, \sqrt {-e^{\left (2 \, x\right )} + 9} e^{x} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.09, size = 22, normalized size = 0.67 \[ \frac {9\,\mathrm {asin}\left (\frac {{\mathrm {e}}^x}{3}\right )}{2}+\frac {{\mathrm {e}}^x\,\sqrt {9-{\mathrm {e}}^{2\,x}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(9 - exp(2*x))^(1/2),x)

[Out]

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

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sympy [A]  time = 1.41, size = 29, normalized size = 0.88 \[ \begin {cases} \frac {\sqrt {9 - e^{2 x}} e^{x}}{2} + \frac {9 \operatorname {asin}{\left (\frac {e^{x}}{3} \right )}}{2} & \text {for}\: e^{x} < \log {\relax (3 )} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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