3.3.4 \(\int \cos (x) \cos (2 x) \cos (3 x) \, dx\) [204]

Optimal. Leaf size=30 \[ \frac {x}{4}+\frac {1}{8} \sin (2 x)+\frac {1}{16} \sin (4 x)+\frac {1}{24} \sin (6 x) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4440, 2717} \begin {gather*} \frac {x}{4}+\frac {1}{8} \sin (2 x)+\frac {1}{16} \sin (4 x)+\frac {1}{24} \sin (6 x) \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 30, normalized size = 1.00 \begin {gather*} \frac {x}{4}+\frac {1}{8} \sin (2 x)+\frac {1}{16} \sin (4 x)+\frac {1}{24} \sin (6 x) \end {gather*}

Antiderivative was successfully verified.

[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

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Maple [A]
time = 0.66, size = 23, normalized size = 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	\(\frac {x}{4}+\frac {\sin \left (2 x \right )}{8}+\frac {\sin \left (4 x \right )}{16}+\frac {\sin \left (6 x \right )}{24}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]
time = 0.34, size = 22, normalized size = 0.73 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 0.63, size = 25, normalized size = 0.83 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (22) = 44\).
time = 1.05, size = 114, normalized size = 3.80 \begin {gather*} - \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 )} \sin {\left (2 x \right )} \sin {\left (3 x \right )}}{6} + \frac {\sin {\left (x \right )} \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{8} + \frac {5 \sin {\left (3 x \right )} \cos {\left (x \right )} \cos {\left (2 x \right )}}{24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)*sin(2*x)*sin(3*x)/6 + sin(x)*cos(2*x)*cos(3*x)/8 + 5*sin(3*x)*cos(x)*cos(2*x)/24

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Giac [A]
time = 0.47, size = 22, normalized size = 0.73 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [B]
time = 0.25, size = 22, normalized size = 0.73 \begin {gather*} \frac {x}{4}+\frac {\sin \left (2\,x\right )}{8}+\frac {\sin \left (4\,x\right )}{16}+\frac {\sin \left (6\,x\right )}{24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Chatgpt [F] Failed to verify
time = 1.00, size = 38, normalized size = 1.27 \begin {gather*} -\frac {2 \sin \left (x \right ) \sin \left (2 x \right ) \sin \left (3 x \right )}{23}-\frac {\cos \left (x \right ) \sin \left (3 x \right ) \left (3 \cos \left (2 x \right ) \sin \left (x \right )-2 \sin \left (2 x \right ) \cos \left (x \right )\right )}{23} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-2/23*sin(x)*sin(2*x)*sin(3*x)-1/23*cos(x)*sin(3*x)*(3*cos(2*x)*sin(x)-2*sin(2*x)*cos(x))

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