\(\int e^x \sqrt {3-4 e^{2 x}} \, dx\) [640]

Optimal result

Integrand size = 17, antiderivative size = 36 \[ \int e^x \sqrt {3-4 e^{2 x}} \, dx=\frac {1}{2} e^x \sqrt {3-4 e^{2 x}}+\frac {3}{4} \arcsin \left (\frac {2 e^x}{\sqrt {3}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2281, 201, 222} \[ \int e^x \sqrt {3-4 e^{2 x}} \, dx=\frac {3}{4} \arcsin \left (\frac {2 e^x}{\sqrt {3}}\right )+\frac {1}{2} e^x \sqrt {3-4 e^{2 x}} \]

[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 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*} \text {integral}& = \text {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} \text {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] (verified)

Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.47 \[ \int e^x \sqrt {3-4 e^{2 x}} \, dx=\frac {1}{2} e^x \sqrt {3-4 e^{2 x}}+\frac {3}{2} \arctan \left (\frac {-\sqrt {3}+2 e^x}{\sqrt {3-4 e^{2 x}}}\right ) \]

[In]

Integrate[E^x*Sqrt[3 - 4*E^(2*x)],x]

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72

method	result	size
default	\(\frac {3 \arcsin \left (\frac {2 \,{\mathrm e}^{x} \sqrt {3}}{3}\right )}{4}+\frac {{\mathrm e}^{x} \sqrt {3-4 \,{\mathrm e}^{2 x}}}{2}\)	\(26\)
risch	\(-\frac {{\mathrm e}^{x} \left (-3+4 \,{\mathrm e}^{2 x}\right )}{2 \sqrt {3-4 \,{\mathrm e}^{2 x}}}+\frac {3 \arcsin \left (\frac {2 \,{\mathrm e}^{x} \sqrt {3}}{3}\right )}{4}\)	\(34\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int e^x \sqrt {3-4 e^{2 x}} \, dx=\frac {1}{2} \, \sqrt {-4 \, e^{\left (2 \, x\right )} + 3} e^{x} - \frac {3}{4} \, \arctan \left (\frac {1}{2} \, \sqrt {-4 \, e^{\left (2 \, x\right )} + 3} e^{\left (-x\right )}\right ) \]

[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] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int e^x \sqrt {3-4 e^{2 x}} \, dx=\frac {\sqrt {3 - 4 e^{2 x}} e^{x}}{2} + \frac {3 \operatorname {asin}{\left (\frac {2 \sqrt {3} e^{x}}{3} \right )}}{4} \]

[In]

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

[Out]

sqrt(3 - 4*exp(2*x))*exp(x)/2 + 3*asin(2*sqrt(3)*exp(x)/3)/4

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69 \[ \int e^x \sqrt {3-4 e^{2 x}} \, dx=\frac {1}{2} \, \sqrt {-4 \, e^{\left (2 \, x\right )} + 3} e^{x} + \frac {3}{4} \, \arcsin \left (\frac {2}{3} \, \sqrt {3} e^{x}\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69 \[ \int e^x \sqrt {3-4 e^{2 x}} \, dx=\frac {1}{2} \, \sqrt {-4 \, e^{\left (2 \, x\right )} + 3} e^{x} + \frac {3}{4} \, \arcsin \left (\frac {2}{3} \, \sqrt {3} e^{x}\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int e^x \sqrt {3-4 e^{2 x}} \, dx=\frac {3\,\mathrm {asin}\left (\frac {2\,\sqrt {3}\,{\mathrm {e}}^x}{3}\right )}{4}+{\mathrm {e}}^x\,\sqrt {\frac {3}{4}-{\mathrm {e}}^{2\,x}} \]

[In]

int(exp(x)*(3 - 4*exp(2*x))^(1/2),x)

[Out]

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