3.108 \(\int \cos (3 x) \cos (4 x) \, dx\)

Optimal. Leaf size=15 \[ \frac{\sin (x)}{2}+\frac{1}{14} \sin (7 x) \]

[Out]

Sin[x]/2 + Sin[7*x]/14

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Rubi [A]  time = 0.0149115, antiderivative size = 15, 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 \[ \frac{\sin (x)}{2}+\frac{1}{14} \sin (7 x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[3*x]*Cos[4*x],x]

[Out]

Sin[x]/2 + Sin[7*x]/14

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Rubi in Sympy [A]  time = 1.04208, size = 10, normalized size = 0.67 \[ \frac{\sin{\left (x \right )}}{2} + \frac{\sin{\left (7 x \right )}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(3*x)*cos(4*x),x)

[Out]

sin(x)/2 + sin(7*x)/14

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Mathematica [A]  time = 0.00982412, size = 15, normalized size = 1. \[ \frac{\sin (x)}{2}+\frac{1}{14} \sin (7 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[3*x]*Cos[4*x],x]

[Out]

Sin[x]/2 + Sin[7*x]/14

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Maple [A]  time = 0.078, size = 12, normalized size = 0.8 \[{\frac{\sin \left ( x \right ) }{2}}+{\frac{\sin \left ( 7\,x \right ) }{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(3*x)*cos(4*x),x)

[Out]

1/2*sin(x)+1/14*sin(7*x)

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Maxima [A]  time = 1.40762, size = 15, normalized size = 1. \[ \frac{1}{14} \, \sin \left (7 \, x\right ) + \frac{1}{2} \, \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)*cos(3*x),x, algorithm="maxima")

[Out]

1/14*sin(7*x) + 1/2*sin(x)

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Fricas [A]  time = 0.22198, size = 32, normalized size = 2.13 \[ \frac{1}{7} \,{\left (32 \, \cos \left (x\right )^{6} - 40 \, \cos \left (x\right )^{4} + 12 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)*cos(3*x),x, algorithm="fricas")

[Out]

1/7*(32*cos(x)^6 - 40*cos(x)^4 + 12*cos(x)^2 + 3)*sin(x)

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Sympy [A]  time = 0.716822, size = 26, normalized size = 1.73 \[ - \frac{3 \sin{\left (3 x \right )} \cos{\left (4 x \right )}}{7} + \frac{4 \sin{\left (4 x \right )} \cos{\left (3 x \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*x)*cos(4*x),x)

[Out]

-3*sin(3*x)*cos(4*x)/7 + 4*sin(4*x)*cos(3*x)/7

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GIAC/XCAS [A]  time = 0.204454, size = 15, normalized size = 1. \[ \frac{1}{14} \, \sin \left (7 \, x\right ) + \frac{1}{2} \, \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)*cos(3*x),x, algorithm="giac")

[Out]

1/14*sin(7*x) + 1/2*sin(x)