3.101 \(\int \cos (x) \cos (2 x) \, dx\)

Optimal. Leaf size=15 \[ \frac{\sin (x)}{2}+\frac{1}{6} \sin (3 x) \]

[Out]

Sin[x]/2 + Sin[3*x]/6

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Rubi [A]  time = 0.008827, antiderivative size = 15, 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, Rules used = {4283} \[ \frac{\sin (x)}{2}+\frac{1}{6} \sin (3 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Cos[2*x],x]

[Out]

Sin[x]/2 + Sin[3*x]/6

Rule 4283

Int[cos[(a_.) + (b_.)*(x_)]*cos[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[a - c + (b - d)*x]/(2*(b - d)), x]
+ Simp[Sin[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps

\begin{align*} \int \cos (x) \cos (2 x) \, dx &=\frac{\sin (x)}{2}+\frac{1}{6} \sin (3 x)\\ \end{align*}

Mathematica [A]  time = 0.0048753, size = 15, normalized size = 1. \[ \frac{\sin (x)}{2}+\frac{1}{6} \sin (3 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x]/2 + Sin[3*x]/6

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Maple [A]  time = 0.018, size = 12, normalized size = 0.8 \begin{align*}{\frac{\sin \left ( x \right ) }{2}}+{\frac{\sin \left ( 3\,x \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*cos(2*x),x)

[Out]

1/2*sin(x)+1/6*sin(3*x)

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Maxima [A]  time = 0.979609, size = 15, normalized size = 1. \begin{align*} \frac{1}{6} \, \sin \left (3 \, x\right ) + \frac{1}{2} \, \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(2*x),x, algorithm="maxima")

[Out]

1/6*sin(3*x) + 1/2*sin(x)

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Fricas [A]  time = 2.33741, size = 39, normalized size = 2.6 \begin{align*} \frac{1}{3} \,{\left (2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(2*x),x, algorithm="fricas")

[Out]

1/3*(2*cos(x)^2 + 1)*sin(x)

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Sympy [A]  time = 0.786458, size = 20, normalized size = 1.33 \begin{align*} - \frac{\sin{\left (x \right )} \cos{\left (2 x \right )}}{3} + \frac{2 \sin{\left (2 x \right )} \cos{\left (x \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(2*x),x)

[Out]

-sin(x)*cos(2*x)/3 + 2*sin(2*x)*cos(x)/3

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Giac [A]  time = 1.10561, size = 15, normalized size = 1. \begin{align*} \frac{1}{6} \, \sin \left (3 \, x\right ) + \frac{1}{2} \, \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(2*x),x, algorithm="giac")

[Out]

1/6*sin(3*x) + 1/2*sin(x)