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 = 0.0363133, 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 \[ -\frac{1}{3} (4-\sin (2 x))^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[Cos[2*x]*Sqrt[4 - Sin[2*x]],x]

[Out]

-(4 - Sin[2*x])^(3/2)/3

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Rubi in Sympy [A]  time = 1.88455, size = 12, normalized size = 0.75 \[ - \frac{\left (- \sin{\left (2 x \right )} + 4\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(2*x)*(4-sin(2*x))**(1/2),x)

[Out]

-(-sin(2*x) + 4)**(3/2)/3

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Mathematica [A]  time = 0.0184198, 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 = 0.021, size = 13, normalized size = 0.8 \[ -{\frac{1}{3} \left ( 4-\sin \left ( 2\,x \right ) \right ) ^{{\frac{3}{2}}}} \]

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 = 1.35489, size = 16, normalized size = 1. \[ -\frac{1}{3} \,{\left (-\sin \left (2 \, x\right ) + 4\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-sin(2*x) + 4)*cos(2*x),x, algorithm="maxima")

[Out]

-1/3*(-sin(2*x) + 4)^(3/2)

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Fricas [A]  time = 0.233611, size = 24, normalized size = 1.5 \[ \frac{1}{3} \,{\left (\sin \left (2 \, x\right ) - 4\right )} \sqrt{-\sin \left (2 \, x\right ) + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-sin(2*x) + 4)*cos(2*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.399012, size = 29, normalized size = 1.81 \[ \frac{\sqrt{- \sin{\left (2 x \right )} + 4} \sin{\left (2 x \right )}}{3} - \frac{4 \sqrt{- \sin{\left (2 x \right )} + 4}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219949, size = 16, normalized size = 1. \[ -\frac{1}{3} \,{\left (-\sin \left (2 \, x\right ) + 4\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-sin(2*x) + 4)*cos(2*x),x, algorithm="giac")

[Out]

-1/3*(-sin(2*x) + 4)^(3/2)