\(\int \cos (x) \cos (2 x) \cos (3 x) \, dx\) [143]

Optimal result

Integrand size = 11, antiderivative size = 30 \[ \int \cos (x) \cos (2 x) \cos (3 x) \, dx=\frac {x}{4}+\frac {1}{8} \sin (2 x)+\frac {1}{16} \sin (4 x)+\frac {1}{24} \sin (6 x) \]

[Out]

1/4*x+1/8*sin(2*x)+1/16*sin(4*x)+1/24*sin(6*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4440, 2717} \[ \int \cos (x) \cos (2 x) \cos (3 x) \, dx=\frac {x}{4}+\frac {1}{8} \sin (2 x)+\frac {1}{16} \sin (4 x)+\frac {1}{24} \sin (6 x) \]

[In]

Int[Cos[x]*Cos[2*x]*Cos[3*x],x]

[Out]

x/4 + Sin[2*x]/8 + Sin[4*x]/16 + Sin[6*x]/24

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4440

Int[(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.)*(H_)[(e_.) + (f_.)*(x_)]^(r_.), x_Symbol] :>
Int[ExpandTrigReduce[ActivateTrig[F[a + b*x]^p*G[c + d*x]^q*H[e + f*x]^r], x], x] /; FreeQ[{a, b, c, d, e, f}
, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, sin] || EqQ[G, cos]) && (EqQ[H, sin] || EqQ[H, cos]) && IGtQ[p
, 0] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{4}+\frac {1}{4} \cos (2 x)+\frac {1}{4} \cos (4 x)+\frac {1}{4} \cos (6 x)\right ) \, dx \\ & = \frac {x}{4}+\frac {1}{4} \int \cos (2 x) \, dx+\frac {1}{4} \int \cos (4 x) \, dx+\frac {1}{4} \int \cos (6 x) \, dx \\ & = \frac {x}{4}+\frac {1}{8} \sin (2 x)+\frac {1}{16} \sin (4 x)+\frac {1}{24} \sin (6 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (2 x) \cos (3 x) \, dx=\frac {x}{4}+\frac {1}{8} \sin (2 x)+\frac {1}{16} \sin (4 x)+\frac {1}{24} \sin (6 x) \]

[In]

Integrate[Cos[x]*Cos[2*x]*Cos[3*x],x]

[Out]

x/4 + Sin[2*x]/8 + Sin[4*x]/16 + Sin[6*x]/24

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77

method	result	size
default	\(\frac {x}{4}+\frac {\sin \left (2 x \right )}{8}+\frac {\sin \left (4 x \right )}{16}+\frac {\sin \left (6 x \right )}{24}\)	\(23\)
risch	\(\frac {x}{4}+\frac {\sin \left (2 x \right )}{8}+\frac {\sin \left (4 x \right )}{16}+\frac {\sin \left (6 x \right )}{24}\)	\(23\)
parallelrisch	\(\frac {x}{4}+\frac {\sin \left (2 x \right )}{8}+\frac {\sin \left (4 x \right )}{16}+\frac {\sin \left (6 x \right )}{24}\)	\(23\)

[In]

int(cos(x)*cos(2*x)*cos(3*x),x,method=_RETURNVERBOSE)

[Out]

1/4*x+1/8*sin(2*x)+1/16*sin(4*x)+1/24*sin(6*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \cos (x) \cos (2 x) \cos (3 x) \, dx=\frac {1}{12} \, {\left (16 \, \cos \left (x\right )^{5} - 10 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {1}{4} \, x \]

[In]

integrate(cos(x)*cos(2*x)*cos(3*x),x, algorithm="fricas")

[Out]

1/12*(16*cos(x)^5 - 10*cos(x)^3 + 3*cos(x))*sin(x) + 1/4*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (22) = 44\).

Time = 1.14 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.80 \[ \int \cos (x) \cos (2 x) \cos (3 x) \, dx=- \frac {x \sin {\left (x \right )} \sin {\left (2 x \right )} \cos {\left (3 x \right )}}{4} + \frac {x \sin {\left (x \right )} \sin {\left (3 x \right )} \cos {\left (2 x \right )}}{4} + \frac {x \sin {\left (2 x \right )} \sin {\left (3 x \right )} \cos {\left (x \right )}}{4} + \frac {x \cos {\left (x \right )} \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{4} - \frac {\sin {\left (x \right )} \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{24} - \frac {\sin {\left (2 x \right )} \cos {\left (x \right )} \cos {\left (3 x \right )}}{6} + \frac {3 \sin {\left (3 x \right )} \cos {\left (x \right )} \cos {\left (2 x \right )}}{8} \]

[In]

integrate(cos(x)*cos(2*x)*cos(3*x),x)

[Out]

-x*sin(x)*sin(2*x)*cos(3*x)/4 + x*sin(x)*sin(3*x)*cos(2*x)/4 + x*sin(2*x)*sin(3*x)*cos(x)/4 + x*cos(x)*cos(2*x
)*cos(3*x)/4 - sin(x)*cos(2*x)*cos(3*x)/24 - sin(2*x)*cos(x)*cos(3*x)/6 + 3*sin(3*x)*cos(x)*cos(2*x)/8

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \cos (x) \cos (2 x) \cos (3 x) \, dx=\frac {1}{4} \, x + \frac {1}{24} \, \sin \left (6 \, x\right ) + \frac {1}{16} \, \sin \left (4 \, x\right ) + \frac {1}{8} \, \sin \left (2 \, x\right ) \]

[In]

integrate(cos(x)*cos(2*x)*cos(3*x),x, algorithm="maxima")

[Out]

1/4*x + 1/24*sin(6*x) + 1/16*sin(4*x) + 1/8*sin(2*x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \cos (x) \cos (2 x) \cos (3 x) \, dx=\frac {1}{4} \, x + \frac {1}{24} \, \sin \left (6 \, x\right ) + \frac {1}{16} \, \sin \left (4 \, x\right ) + \frac {1}{8} \, \sin \left (2 \, x\right ) \]

[In]

integrate(cos(x)*cos(2*x)*cos(3*x),x, algorithm="giac")

[Out]

1/4*x + 1/24*sin(6*x) + 1/16*sin(4*x) + 1/8*sin(2*x)

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \cos (x) \cos (2 x) \cos (3 x) \, dx=\frac {x}{4}+\frac {\sin \left (2\,x\right )}{8}+\frac {\sin \left (4\,x\right )}{16}+\frac {\sin \left (6\,x\right )}{24} \]

[In]

int(cos(2*x)*cos(3*x)*cos(x),x)

[Out]

x/4 + sin(2*x)/8 + sin(4*x)/16 + sin(6*x)/24