3.660 \(\int e^x \sqrt{9+e^{2 x}} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{2} e^x \sqrt{e^{2 x}+9}+\frac{9}{2} \sinh ^{-1}\left (\frac{e^x}{3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0250647, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2249, 195, 215} \[ \frac{1}{2} e^x \sqrt{e^{2 x}+9}+\frac{9}{2} \sinh ^{-1}\left (\frac{e^x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Sqrt[9 + E^(2*x)],x]

[Out]

(E^x*Sqrt[9 + E^(2*x)])/2 + (9*ArcSinh[E^x/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 215

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

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} \sinh ^{-1}\left (\frac{e^x}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0104095, size = 30, normalized size = 0.97 \[ \frac{1}{2} \left (e^x \sqrt{e^{2 x}+9}+9 \sinh ^{-1}\left (\frac{e^x}{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Sqrt[9 + E^(2*x)],x]

[Out]

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

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Maple [A]  time = 0.059, size = 21, normalized size = 0.7 \begin{align*}{\frac{{{\rm e}^{x}}}{2}\sqrt{9+ \left ({{\rm e}^{x}} \right ) ^{2}}}+{\frac{9}{2}{\it Arcsinh} \left ({\frac{{{\rm e}^{x}}}{3}} \right ) } \end{align*}

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*arcsinh(1/3*exp(x))

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Maxima [A]  time = 1.46023, size = 27, normalized size = 0.87 \begin{align*} \frac{1}{2} \, \sqrt{e^{\left (2 \, x\right )} + 9} e^{x} + \frac{9}{2} \, \operatorname{arsinh}\left (\frac{1}{3} \, e^{x}\right ) \end{align*}

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*arcsinh(1/3*e^x)

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Fricas [A]  time = 0.777622, size = 84, normalized size = 2.71 \begin{align*} \frac{1}{2} \, \sqrt{e^{\left (2 \, x\right )} + 9} e^{x} - \frac{9}{2} \, \log \left (\sqrt{e^{\left (2 \, x\right )} + 9} - e^{x}\right ) \end{align*}

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 - 9/2*log(sqrt(e^(2*x) + 9) - e^x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e^{2 x} + 9} e^{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(exp(2*x) + 9)*exp(x), x)

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Giac [A]  time = 1.24677, size = 39, normalized size = 1.26 \begin{align*} \frac{1}{2} \, \sqrt{e^{\left (2 \, x\right )} + 9} e^{x} - \frac{9}{2} \, \log \left (\sqrt{e^{\left (2 \, x\right )} + 9} - e^{x}\right ) \end{align*}

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*log(sqrt(e^(2*x) + 9) - e^x)