3.429 \(\int \cos (x) \sqrt{\cos (2 x)} \, dx\)

Optimal. Leaf size=33 \[ \frac{\sin ^{-1}\left (\sqrt{2} \sin (x)\right )}{2 \sqrt{2}}+\frac{1}{2} \sin (x) \sqrt{\cos (2 x)} \]

[Out]

ArcSin[Sqrt[2]*Sin[x]]/(2*Sqrt[2]) + (Sqrt[Cos[2*x]]*Sin[x])/2

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Rubi [A]  time = 0.0384552, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\sin ^{-1}\left (\sqrt{2} \sin (x)\right )}{2 \sqrt{2}}+\frac{1}{2} \sin (x) \sqrt{\cos (2 x)} \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]*Sqrt[Cos[2*x]],x]

[Out]

ArcSin[Sqrt[2]*Sin[x]]/(2*Sqrt[2]) + (Sqrt[Cos[2*x]]*Sin[x])/2

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.0705124, size = 33, normalized size = 1. \[ \frac{\sin ^{-1}\left (\sqrt{2} \sin (x)\right )}{2 \sqrt{2}}+\frac{1}{2} \sin (x) \sqrt{\cos (2 x)} \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]*Sqrt[Cos[2*x]],x]

[Out]

ArcSin[Sqrt[2]*Sin[x]]/(2*Sqrt[2]) + (Sqrt[Cos[2*x]]*Sin[x])/2

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Maple [B]  time = 0.086, size = 62, normalized size = 1.9 \[ -{\frac{1}{8\,\sin \left ( x \right ) }\sqrt{ \left ( 2\, \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( -\sqrt{2}\arcsin \left ( 4\, \left ( \sin \left ( x \right ) \right ) ^{2}-1 \right ) -4\,\sqrt{-2\, \left ( \sin \left ( x \right ) \right ) ^{4}+ \left ( \sin \left ( x \right ) \right ) ^{2}} \right ){\frac{1}{\sqrt{2\, \left ( \cos \left ( x \right ) \right ) ^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)*cos(2*x)^(1/2),x)

[Out]

-1/8*((2*cos(x)^2-1)*sin(x)^2)^(1/2)*(-2^(1/2)*arcsin(4*sin(x)^2-1)-4*(-2*sin(x)
^4+sin(x)^2)^(1/2))/sin(x)/(2*cos(x)^2-1)^(1/2)

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Maxima [A]  time = 1.95078, size = 659, normalized size = 19.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(2)*(2*(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*(cos(1/2*arctan
2(sin(4*x), cos(4*x) + 1))*sin(2*x) - (cos(2*x) - 1)*sin(1/2*arctan2(sin(4*x), c
os(4*x) + 1))) + arctan2(-(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*(cos(
1/2*arctan2(sin(4*x), cos(4*x) + 1))*sin(2*x) - cos(2*x)*sin(1/2*arctan2(sin(4*x
), cos(4*x) + 1))), (cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*(cos(2*x)*c
os(1/2*arctan2(sin(4*x), cos(4*x) + 1)) + sin(2*x)*sin(1/2*arctan2(sin(4*x), cos
(4*x) + 1))) + 1) - arctan2(-(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*(c
os(1/2*arctan2(sin(4*x), cos(4*x) + 1))*sin(2*x) - cos(2*x)*sin(1/2*arctan2(sin(
4*x), cos(4*x) + 1))), (cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*(cos(2*x
)*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1)) + sin(2*x)*sin(1/2*arctan2(sin(4*x),
cos(4*x) + 1))) - 1) - arctan2((cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*
sin(1/2*arctan2(sin(4*x), cos(4*x) + 1)), (cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x)
+ 1)^(1/4)*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1)) + 1) + arctan2((cos(4*x)^2 +
sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*sin(1/2*arctan2(sin(4*x), cos(4*x) + 1)), (c
os(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*cos(1/2*arctan2(sin(4*x), cos(4*x
) + 1)) - 1))

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Fricas [A]  time = 0.238081, size = 96, normalized size = 2.91 \[ -\frac{1}{16} \, \sqrt{2} \arctan \left (\frac{32 \, \sqrt{2} \cos \left (x\right )^{4} - 48 \, \sqrt{2} \cos \left (x\right )^{2} + 17 \, \sqrt{2}}{8 \,{\left (4 \, \cos \left (x\right )^{2} - 3\right )} \sqrt{2 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right )}\right ) + \frac{1}{2} \, \sqrt{2 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*sqrt(2)*arctan(1/8*(32*sqrt(2)*cos(x)^4 - 48*sqrt(2)*cos(x)^2 + 17*sqrt(2)
)/((4*cos(x)^2 - 3)*sqrt(2*cos(x)^2 - 1)*sin(x))) + 1/2*sqrt(2*cos(x)^2 - 1)*sin
(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \cos{\left (x \right )} \sqrt{\cos{\left (2 x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(cos(x)*sqrt(cos(2*x)), x)

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GIAC/XCAS [A]  time = 0.207651, size = 36, normalized size = 1.09 \[ \frac{1}{4} \, \sqrt{2} \arcsin \left (\sqrt{2} \sin \left (x\right )\right ) + \frac{1}{2} \, \sqrt{-2 \, \sin \left (x\right )^{2} + 1} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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