3.111 \(\int \cos (x) \cos (2 x) \cos (3 x) \, dx\)

Optimal. Leaf size=30 \[ \frac{x}{4}+\frac{1}{8} \sin (2 x)+\frac{1}{16} \sin (4 x)+\frac{1}{24} \sin (6 x) \]

[Out]

x/4 + Sin[2*x]/8 + Sin[4*x]/16 + Sin[6*x]/24

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Rubi [A]  time = 0.0482573, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x}{4}+\frac{1}{8} \sin (2 x)+\frac{1}{16} \sin (4 x)+\frac{1}{24} \sin (6 x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]*Cos[2*x]*Cos[3*x],x]

[Out]

x/4 + Sin[2*x]/8 + Sin[4*x]/16 + Sin[6*x]/24

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Rubi in Sympy [A]  time = 2.2711, size = 27, normalized size = 0.9 \[ \frac{x}{4} + \frac{\sin{\left (2 x \right )}}{4} + \frac{\sin{\left (3 x \right )} \cos{\left (3 x \right )}}{12} + \frac{\sin{\left (4 x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)*cos(2*x)*cos(3*x),x)

[Out]

x/4 + sin(2*x)/4 + sin(3*x)*cos(3*x)/12 + sin(4*x)/8

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Mathematica [A]  time = 0.0137906, size = 30, normalized size = 1. \[ \frac{x}{4}+\frac{1}{8} \sin (2 x)+\frac{1}{16} \sin (4 x)+\frac{1}{24} \sin (6 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]*Cos[2*x]*Cos[3*x],x]

[Out]

x/4 + Sin[2*x]/8 + Sin[4*x]/16 + Sin[6*x]/24

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Maple [A]  time = 0.003, size = 23, normalized size = 0.8 \[{\frac{x}{4}}+{\frac{\sin \left ( 2\,x \right ) }{8}}+{\frac{\sin \left ( 4\,x \right ) }{16}}+{\frac{\sin \left ( 6\,x \right ) }{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)*cos(2*x)*cos(3*x),x)

[Out]

1/4*x+1/8*sin(2*x)+1/16*sin(4*x)+1/24*sin(6*x)

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Maxima [A]  time = 1.38854, size = 30, normalized size = 1. \[ \frac{1}{4} \, x + \frac{1}{24} \, \sin \left (6 \, x\right ) + \frac{1}{16} \, \sin \left (4 \, x\right ) + \frac{1}{8} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*x)*cos(2*x)*cos(x),x, algorithm="maxima")

[Out]

1/4*x + 1/24*sin(6*x) + 1/16*sin(4*x) + 1/8*sin(2*x)

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Fricas [A]  time = 0.25978, size = 34, normalized size = 1.13 \[ \frac{1}{12} \,{\left (16 \, \cos \left (x\right )^{5} - 10 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{1}{4} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*x)*cos(2*x)*cos(x),x, algorithm="fricas")

[Out]

1/12*(16*cos(x)^5 - 10*cos(x)^3 + 3*cos(x))*sin(x) + 1/4*x

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Sympy [A]  time = 22.0666, size = 114, normalized size = 3.8 \[ - \frac{x \sin{\left (x \right )} \sin{\left (2 x \right )} \cos{\left (3 x \right )}}{4} + \frac{x \sin{\left (x \right )} \sin{\left (3 x \right )} \cos{\left (2 x \right )}}{4} + \frac{x \sin{\left (2 x \right )} \sin{\left (3 x \right )} \cos{\left (x \right )}}{4} + \frac{x \cos{\left (x \right )} \cos{\left (2 x \right )} \cos{\left (3 x \right )}}{4} - \frac{\sin{\left (x \right )} \cos{\left (2 x \right )} \cos{\left (3 x \right )}}{24} - \frac{\sin{\left (2 x \right )} \cos{\left (x \right )} \cos{\left (3 x \right )}}{6} + \frac{3 \sin{\left (3 x \right )} \cos{\left (x \right )} \cos{\left (2 x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)*cos(2*x)*cos(3*x),x)

[Out]

-x*sin(x)*sin(2*x)*cos(3*x)/4 + x*sin(x)*sin(3*x)*cos(2*x)/4 + x*sin(2*x)*sin(3*
x)*cos(x)/4 + x*cos(x)*cos(2*x)*cos(3*x)/4 - sin(x)*cos(2*x)*cos(3*x)/24 - sin(2
*x)*cos(x)*cos(3*x)/6 + 3*sin(3*x)*cos(x)*cos(2*x)/8

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GIAC/XCAS [A]  time = 0.204311, size = 30, normalized size = 1. \[ \frac{1}{4} \, x + \frac{1}{24} \, \sin \left (6 \, x\right ) + \frac{1}{16} \, \sin \left (4 \, x\right ) + \frac{1}{8} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*x)*cos(2*x)*cos(x),x, algorithm="giac")

[Out]

1/4*x + 1/24*sin(6*x) + 1/16*sin(4*x) + 1/8*sin(2*x)