3.640 \(\int e^x \sqrt{3-4 e^{2 x}} \, dx\)

Optimal. Leaf size=36 \[ \frac{1}{2} e^x \sqrt{3-4 e^{2 x}}+\frac{3}{4} \sin ^{-1}\left (\frac{2 e^x}{\sqrt{3}}\right ) \]

[Out]

(E^x*Sqrt[3 - 4*E^(2*x)])/2 + (3*ArcSin[(2*E^x)/Sqrt[3]])/4

________________________________________________________________________________________

Rubi [A]  time = 0.0288832, antiderivative size = 36, 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} \[ \frac{1}{2} e^x \sqrt{3-4 e^{2 x}}+\frac{3}{4} \sin ^{-1}\left (\frac{2 e^x}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Sqrt[3 - 4*E^(2*x)],x]

[Out]

(E^x*Sqrt[3 - 4*E^(2*x)])/2 + (3*ArcSin[(2*E^x)/Sqrt[3]])/4

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^x \sqrt{3-4 e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \sqrt{3-4 x^2} \, dx,x,e^x\right )\\ &=\frac{1}{2} e^x \sqrt{3-4 e^{2 x}}+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3-4 x^2}} \, dx,x,e^x\right )\\ &=\frac{1}{2} e^x \sqrt{3-4 e^{2 x}}+\frac{3}{4} \sin ^{-1}\left (\frac{2 e^x}{\sqrt{3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0145496, size = 36, normalized size = 1. \[ \frac{1}{4} \left (2 e^x \sqrt{3-4 e^{2 x}}+3 \sin ^{-1}\left (\frac{2 e^x}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Sqrt[3 - 4*E^(2*x)],x]

[Out]

(2*E^x*Sqrt[3 - 4*E^(2*x)] + 3*ArcSin[(2*E^x)/Sqrt[3]])/4

________________________________________________________________________________________

Maple [A]  time = 0.059, size = 26, normalized size = 0.7 \begin{align*}{\frac{{{\rm e}^{x}}}{2}\sqrt{3-4\, \left ({{\rm e}^{x}} \right ) ^{2}}}+{\frac{3}{4}\arcsin \left ({\frac{2\,{{\rm e}^{x}}\sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(3-4*exp(2*x))^(1/2),x)

[Out]

1/2*exp(x)*(3-4*exp(x)^2)^(1/2)+3/4*arcsin(2/3*exp(x)*3^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.46051, size = 34, normalized size = 0.94 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, e^{\left (2 \, x\right )} + 3} e^{x} + \frac{3}{4} \, \arcsin \left (\frac{2}{3} \, \sqrt{3} e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(3-4*exp(2*x))^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(-4*e^(2*x) + 3)*e^x + 3/4*arcsin(2/3*sqrt(3)*e^x)

________________________________________________________________________________________

Fricas [A]  time = 0.824951, size = 103, normalized size = 2.86 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, e^{\left (2 \, x\right )} + 3} e^{x} - \frac{3}{4} \, \arctan \left (\frac{1}{2} \, \sqrt{-4 \, e^{\left (2 \, x\right )} + 3} e^{\left (-x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(3-4*exp(2*x))^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(-4*e^(2*x) + 3)*e^x - 3/4*arctan(1/2*sqrt(-4*e^(2*x) + 3)*e^(-x))

________________________________________________________________________________________

Sympy [A]  time = 1.48305, size = 42, normalized size = 1.17 \begin{align*} \begin{cases} \frac{\sqrt{3 - 4 e^{2 x}} e^{x}}{2} + \frac{3 \operatorname{asin}{\left (\frac{2 \sqrt{3} e^{x}}{3} \right )}}{4} & \text{for}\: e^{x} < \log{\left (\frac{\sqrt{3}}{2} \right )} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(3-4*exp(2*x))**(1/2),x)

[Out]

Piecewise((sqrt(3 - 4*exp(2*x))*exp(x)/2 + 3*asin(2*sqrt(3)*exp(x)/3)/4, exp(x) < log(sqrt(3)/2)))

________________________________________________________________________________________

Giac [A]  time = 1.21404, size = 34, normalized size = 0.94 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, e^{\left (2 \, x\right )} + 3} e^{x} + \frac{3}{4} \, \arcsin \left (\frac{2}{3} \, \sqrt{3} e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(3-4*exp(2*x))^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(-4*e^(2*x) + 3)*e^x + 3/4*arcsin(2/3*sqrt(3)*e^x)