3.2.25 \(\int \cos (x) \cos (2 x) \, dx\) [125]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = {4368} \begin {gather*} \frac {\sin (x)}{2}+\frac {1}{6} \sin (3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x]/2 + Sin[3*x]/6

Rule 4368

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*}

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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {\sin (x)}{2}+\frac {1}{6} \sin (3 x) \end {gather*}

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 = 0.04, size = 12, normalized size = 0.80

method	result	size
default	\(\frac {\sin \left (x \right )}{2}+\frac {\sin \left (3 x \right )}{6}\)	\(12\)
risch	\(\frac {\sin \left (x \right )}{2}+\frac {\sin \left (3 x \right )}{6}\)	\(12\)
norman	\(\frac {-\frac {4 \tan \left (x \right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3}+\frac {2 \left (\tan ^{2}\left (x \right )\right ) \tan \left (\frac {x}{2}\right )}{3}+\frac {4 \tan \left (x \right )}{3}-\frac {2 \tan \left (\frac {x}{2}\right )}{3}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan ^{2}\left (x \right )\right )}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*cos(2*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 7.03, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{6} \, \sin \left (3 \, x\right ) + \frac {1}{2} \, \sin \left (x\right ) \end {gather*}

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 = 0.62, size = 12, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, {\left (2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) \end {gather*}

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 = 0.12, size = 20, normalized size = 1.33 \begin {gather*} - \frac {\sin {\left (x \right )} \cos {\left (2 x \right )}}{3} + \frac {2 \sin {\left (2 x \right )} \cos {\left (x \right )}}{3} \end {gather*}

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 = 1.16, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{6} \, \sin \left (3 \, x\right ) + \frac {1}{2} \, \sin \left (x\right ) \end {gather*}

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)

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Mupad [B]
time = 0.03, size = 9, normalized size = 0.60 \begin {gather*} \sin \left (x\right )-\frac {2\,{\sin \left (x\right )}^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x) - (2*sin(x)^3)/3

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