3.283 \(\int \cos (3 x) \cos (5 x) \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{4} \sin (2 x)+\frac{1}{16} \sin (8 x) \]

[Out]

Sin[2*x]/4 + Sin[8*x]/16

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Rubi [A]  time = 0.0155857, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{4} \sin (2 x)+\frac{1}{16} \sin (8 x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[3*x]*Cos[5*x],x]

[Out]

Sin[2*x]/4 + Sin[8*x]/16

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Rubi in Sympy [A]  time = 1.0389, size = 12, normalized size = 0.71 \[ \frac{\sin{\left (2 x \right )}}{4} + \frac{\sin{\left (8 x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(3*x)*cos(5*x),x)

[Out]

sin(2*x)/4 + sin(8*x)/16

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Mathematica [A]  time = 0.0119763, size = 17, normalized size = 1. \[ \frac{1}{4} \sin (2 x)+\frac{1}{16} \sin (8 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[3*x]*Cos[5*x],x]

[Out]

Sin[2*x]/4 + Sin[8*x]/16

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Maple [A]  time = 0.049, size = 14, normalized size = 0.8 \[{\frac{\sin \left ( 2\,x \right ) }{4}}+{\frac{\sin \left ( 8\,x \right ) }{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(3*x)*cos(5*x),x)

[Out]

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

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Maxima [A]  time = 1.3483, size = 18, normalized size = 1.06 \[ \frac{1}{16} \, \sin \left (8 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(5*x)*cos(3*x),x, algorithm="maxima")

[Out]

1/16*sin(8*x) + 1/4*sin(2*x)

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Fricas [A]  time = 0.220826, size = 30, normalized size = 1.76 \[{\left (8 \, \cos \left (x\right )^{7} - 12 \, \cos \left (x\right )^{5} + 5 \, \cos \left (x\right )^{3}\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(5*x)*cos(3*x),x, algorithm="fricas")

[Out]

(8*cos(x)^7 - 12*cos(x)^5 + 5*cos(x)^3)*sin(x)

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Sympy [A]  time = 0.735587, size = 26, normalized size = 1.53 \[ - \frac{3 \sin{\left (3 x \right )} \cos{\left (5 x \right )}}{16} + \frac{5 \sin{\left (5 x \right )} \cos{\left (3 x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*x)*cos(5*x),x)

[Out]

-3*sin(3*x)*cos(5*x)/16 + 5*sin(5*x)*cos(3*x)/16

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GIAC/XCAS [A]  time = 0.225624, size = 18, normalized size = 1.06 \[ \frac{1}{16} \, \sin \left (8 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(5*x)*cos(3*x),x, algorithm="giac")

[Out]

1/16*sin(8*x) + 1/4*sin(2*x)