3.9 \(\int \cos (2 x) \sqrt{4-\sin (2 x)} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{3} (4-\sin (2 x))^{3/2} \]

[Out]

-(4 - Sin[2*x])^(3/2)/3

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Rubi [A]  time = 0.0230421, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2668, 32} \[ -\frac{1}{3} (4-\sin (2 x))^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[2*x]*Sqrt[4 - Sin[2*x]],x]

[Out]

-(4 - Sin[2*x])^(3/2)/3

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0119864, size = 16, normalized size = 1. \[ -\frac{1}{3} (4-\sin (2 x))^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(4 - Sin[2*x])^(3/2)/3

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Maple [A]  time = 0.013, size = 13, normalized size = 0.8 \begin{align*} -{\frac{1}{3} \left ( 4-\sin \left ( 2\,x \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.946658, size = 16, normalized size = 1. \begin{align*} -\frac{1}{3} \,{\left (-\sin \left (2 \, x\right ) + 4\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-sin(2*x) + 4)^(3/2)

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Fricas [A]  time = 0.46015, size = 54, normalized size = 3.38 \begin{align*} \frac{1}{3} \,{\left (\sin \left (2 \, x\right ) - 4\right )} \sqrt{-\sin \left (2 \, x\right ) + 4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(sin(2*x) - 4)*sqrt(-sin(2*x) + 4)

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Sympy [B]  time = 0.29622, size = 29, normalized size = 1.81 \begin{align*} \frac{\sqrt{4 - \sin{\left (2 x \right )}} \sin{\left (2 x \right )}}{3} - \frac{4 \sqrt{4 - \sin{\left (2 x \right )}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(4 - sin(2*x))*sin(2*x)/3 - 4*sqrt(4 - sin(2*x))/3

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Giac [A]  time = 1.08493, size = 16, normalized size = 1. \begin{align*} -\frac{1}{3} \,{\left (-\sin \left (2 \, x\right ) + 4\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-sin(2*x) + 4)^(3/2)