3.8.9 \(\int e^x \sqrt {9-e^{2 x}} \, dx\) [709]

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)

________________________________________________________________________________________

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 {9}{2} \text {ArcSin}\left (\frac {e^x}{3}\right )+\frac {1}{2} e^x \sqrt {9-e^{2 x}} \end {gather*}

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 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^x \sqrt {9-e^{2 x}} \, dx &=\text {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} \text {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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 45, normalized size = 1.36 \begin {gather*} \frac {1}{2} e^x \sqrt {9-e^{2 x}}-9 \tan ^{-1}\left (\frac {\sqrt {9-e^{2 x}}}{3+e^x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*Sqrt[9 - E^(2*x)],x]

[Out]

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

________________________________________________________________________________________

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 22, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, \sqrt {-e^{\left (2 \, x\right )} + 9} e^{x} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, e^{x}\right ) \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 35, normalized size = 1.06 \begin {gather*} \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 ) \end {gather*}

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))

________________________________________________________________________________________

Sympy [A]
time = 0.63, size = 32, normalized size = 0.97 \begin {gather*} \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} > -3 \wedge e^{x} < 3 \end {cases} \end {gather*}

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) > -3) & (exp(x) < 3)))

________________________________________________________________________________________

Giac [A]
time = 4.60, size = 22, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, \sqrt {-e^{\left (2 \, x\right )} + 9} e^{x} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, e^{x}\right ) \end {gather*}

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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________