3.368 \(\int \cos (x) \cos (4 x) \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{6} \sin (3 x)+\frac{1}{10} \sin (5 x) \]

[Out]

Sin[3*x]/6 + Sin[5*x]/10

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Rubi [A]  time = 0.0163079, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{6} \sin (3 x)+\frac{1}{10} \sin (5 x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]*Cos[4*x],x]

[Out]

Sin[3*x]/6 + Sin[5*x]/10

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Rubi in Sympy [A]  time = 0.965101, size = 12, normalized size = 0.71 \[ \frac{\sin{\left (3 x \right )}}{6} + \frac{\sin{\left (5 x \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)*cos(4*x),x)

[Out]

sin(3*x)/6 + sin(5*x)/10

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Mathematica [A]  time = 0.00964717, size = 17, normalized size = 1. \[ \frac{1}{6} \sin (3 x)+\frac{1}{10} \sin (5 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]*Cos[4*x],x]

[Out]

Sin[3*x]/6 + Sin[5*x]/10

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Maple [A]  time = 0.066, size = 14, normalized size = 0.8 \[{\frac{\sin \left ( 3\,x \right ) }{6}}+{\frac{\sin \left ( 5\,x \right ) }{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)*cos(4*x),x)

[Out]

1/6*sin(3*x)+1/10*sin(5*x)

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Maxima [A]  time = 1.37692, size = 18, normalized size = 1.06 \[ \frac{1}{10} \, \sin \left (5 \, x\right ) + \frac{1}{6} \, \sin \left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)*cos(x),x, algorithm="maxima")

[Out]

1/10*sin(5*x) + 1/6*sin(3*x)

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Fricas [A]  time = 0.231394, size = 24, normalized size = 1.41 \[ \frac{1}{15} \,{\left (24 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)*cos(x),x, algorithm="fricas")

[Out]

1/15*(24*cos(x)^4 - 8*cos(x)^2 - 1)*sin(x)

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Sympy [A]  time = 0.723057, size = 20, normalized size = 1.18 \[ - \frac{\sin{\left (x \right )} \cos{\left (4 x \right )}}{15} + \frac{4 \sin{\left (4 x \right )} \cos{\left (x \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)*cos(4*x),x)

[Out]

-sin(x)*cos(4*x)/15 + 4*sin(4*x)*cos(x)/15

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GIAC/XCAS [A]  time = 0.198041, size = 18, normalized size = 1.06 \[ \frac{1}{10} \, \sin \left (5 \, x\right ) + \frac{1}{6} \, \sin \left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)*cos(x),x, algorithm="giac")

[Out]

1/10*sin(5*x) + 1/6*sin(3*x)