\(\int \cos (x) \cos (2 x) \, dx\) [101]

Optimal result

Integrand size = 7, antiderivative size = 15 \[ \int \cos (x) \cos (2 x) \, dx=\frac {\sin (x)}{2}+\frac {1}{6} \sin (3 x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4368} \[ \int \cos (x) \cos (2 x) \, dx=\frac {\sin (x)}{2}+\frac {1}{6} \sin (3 x) \]

[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*} \text {integral}& = \frac {\sin (x)}{2}+\frac {1}{6} \sin (3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (2 x) \, dx=\frac {\sin (x)}{2}+\frac {1}{6} \sin (3 x) \]

[In]

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

[Out]

Sin[x]/2 + Sin[3*x]/6

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \cos (x) \cos (2 x) \, dx=\frac {1}{3} \, {\left (2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) \]

[In]

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

[Out]

1/3*(2*cos(x)^2 + 1)*sin(x)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int \cos (x) \cos (2 x) \, dx=- \frac {\sin {\left (x \right )} \cos {\left (2 x \right )}}{3} + \frac {2 \sin {\left (2 x \right )} \cos {\left (x \right )}}{3} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \cos (x) \cos (2 x) \, dx=\frac {1}{6} \, \sin \left (3 \, x\right ) + \frac {1}{2} \, \sin \left (x\right ) \]

[In]

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

[Out]

1/6*sin(3*x) + 1/2*sin(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \cos (x) \cos (2 x) \, dx=\frac {1}{6} \, \sin \left (3 \, x\right ) + \frac {1}{2} \, \sin \left (x\right ) \]

[In]

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

[Out]

1/6*sin(3*x) + 1/2*sin(x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \cos (x) \cos (2 x) \, dx=\sin \left (x\right )-\frac {2\,{\sin \left (x\right )}^3}{3} \]

[In]

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

[Out]

sin(x) - (2*sin(x)^3)/3