3.669 \(\int \frac{\cos (3 x)}{\sqrt{4-\sin ^2(3 x)}} \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \sin (3 x)\right ) \]

[Out]

ArcSin[Sin[3*x]/2]/3

________________________________________________________________________________________

Rubi [A]  time = 0.0263306, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3190, 216} \[ \frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \sin (3 x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[3*x]/Sqrt[4 - Sin[3*x]^2],x]

[Out]

ArcSin[Sin[3*x]/2]/3

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

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

Mathematica [A]  time = 0.029751, size = 13, normalized size = 1. \[ \frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \sin (3 x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[3*x]/Sqrt[4 - Sin[3*x]^2],x]

[Out]

ArcSin[Sin[3*x]/2]/3

________________________________________________________________________________________

Maple [A]  time = 0.031, size = 10, normalized size = 0.8 \begin{align*}{\frac{1}{3}\arcsin \left ({\frac{\sin \left ( 3\,x \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*arcsin(1/2*sin(3*x))

________________________________________________________________________________________

Maxima [A]  time = 1.43736, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{3} \, \arcsin \left (\frac{1}{2} \, \sin \left (3 \, x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*arcsin(1/2*sin(3*x))

________________________________________________________________________________________

Fricas [B]  time = 2.08545, size = 200, normalized size = 15.38 \begin{align*} \frac{1}{6} \, \arctan \left (\frac{\sqrt{\cos \left (3 \, x\right )^{2} + 3}{\left (\cos \left (3 \, x\right )^{2} + 1\right )} \sin \left (3 \, x\right ) - 4 \, \cos \left (3 \, x\right ) \sin \left (3 \, x\right )}{\cos \left (3 \, x\right )^{4} + 6 \, \cos \left (3 \, x\right )^{2} - 3}\right ) + \frac{1}{6} \, \arctan \left (\frac{\sin \left (3 \, x\right )}{\cos \left (3 \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*arctan((sqrt(cos(3*x)^2 + 3)*(cos(3*x)^2 + 1)*sin(3*x) - 4*cos(3*x)*sin(3*x))/(cos(3*x)^4 + 6*cos(3*x)^2 -
3)) + 1/6*arctan(sin(3*x)/cos(3*x))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (3 x \right )}}{\sqrt{- \left (\sin{\left (3 x \right )} - 2\right ) \left (\sin{\left (3 x \right )} + 2\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(3*x)/sqrt(-(sin(3*x) - 2)*(sin(3*x) + 2)), x)

________________________________________________________________________________________

Giac [A]  time = 1.16606, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{3} \, \arcsin \left (\frac{1}{2} \, \sin \left (3 \, x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*arcsin(1/2*sin(3*x))