3.1.39 \(\int \cos (x) \cos (3 x) \cos (5 x) \, dx\) [39]

Optimal. Leaf size=31 \[ \frac {\sin (x)}{4}+\frac {1}{12} \sin (3 x)+\frac {1}{28} \sin (7 x)+\frac {1}{36} \sin (9 x) \]

[Out]

1/4*sin(x)+1/12*sin(3*x)+1/28*sin(7*x)+1/36*sin(9*x)

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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {\sin (x)}{4}+\frac {1}{12} \sin (3 x)+\frac {1}{28} \sin (7 x)+\frac {1}{36} \sin (9 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Cos[3*x]*Cos[5*x],x]

[Out]

Sin[x]/4 + Sin[3*x]/12 + Sin[7*x]/28 + Sin[9*x]/36

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 {\cos (x)}{4}+\frac {1}{4} \cos (3 x)+\frac {1}{4} \cos (7 x)+\frac {1}{4} \cos (9 x)\right ) \, dx\\ &=\frac {1}{4} \int \cos (x) \, dx+\frac {1}{4} \int \cos (3 x) \, dx+\frac {1}{4} \int \cos (7 x) \, dx+\frac {1}{4} \int \cos (9 x) \, dx\\ &=\frac {\sin (x)}{4}+\frac {1}{12} \sin (3 x)+\frac {1}{28} \sin (7 x)+\frac {1}{36} \sin (9 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 31, normalized size = 1.00 \begin {gather*} \frac {\sin (x)}{4}+\frac {1}{12} \sin (3 x)+\frac {1}{28} \sin (7 x)+\frac {1}{36} \sin (9 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Cos[3*x]*Cos[5*x],x]

[Out]

Sin[x]/4 + Sin[3*x]/12 + Sin[7*x]/28 + Sin[9*x]/36

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Maple [A]
time = 0.62, size = 24, normalized size = 0.77

method	result	size
default	\(\frac {\sin \left (x \right )}{4}+\frac {\sin \left (3 x \right )}{12}+\frac {\sin \left (7 x \right )}{28}+\frac {\sin \left (9 x \right )}{36}\)	\(24\)
risch	\(\frac {\sin \left (x \right )}{4}+\frac {\sin \left (3 x \right )}{12}+\frac {\sin \left (7 x \right )}{28}+\frac {\sin \left (9 x \right )}{36}\)	\(24\)
parallelrisch	\(\frac {\sin \left (x \right )}{4}+\frac {\sin \left (3 x \right )}{12}+\frac {\sin \left (7 x \right )}{28}+\frac {\sin \left (9 x \right )}{36}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*cos(3*x)*cos(5*x),x,method=_RETURNVERBOSE)

[Out]

1/4*sin(x)+1/12*sin(3*x)+1/28*sin(7*x)+1/36*sin(9*x)

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Maxima [A]
time = 0.39, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{36} \, \sin \left (9 \, x\right ) + \frac {1}{28} \, \sin \left (7 \, x\right ) + \frac {1}{12} \, \sin \left (3 \, x\right ) + \frac {1}{4} \, \sin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(3*x)*cos(5*x),x, algorithm="maxima")

[Out]

1/36*sin(9*x) + 1/28*sin(7*x) + 1/12*sin(3*x) + 1/4*sin(x)

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Fricas [A]
time = 0.62, size = 30, normalized size = 0.97 \begin {gather*} \frac {1}{63} \, {\left (448 \, \cos \left (x\right )^{8} - 640 \, \cos \left (x\right )^{6} + 240 \, \cos \left (x\right )^{4} + 5 \, \cos \left (x\right )^{2} + 10\right )} \sin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(3*x)*cos(5*x),x, algorithm="fricas")

[Out]

1/63*(448*cos(x)^8 - 640*cos(x)^6 + 240*cos(x)^4 + 5*cos(x)^2 + 10)*sin(x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
time = 1.34, size = 66, normalized size = 2.13 \begin {gather*} - \frac {10 \sin {\left (x \right )} \sin {\left (3 x \right )} \sin {\left (5 x \right )}}{63} - \frac {11 \sin {\left (x \right )} \cos {\left (3 x \right )} \cos {\left (5 x \right )}}{63} - \frac {17 \sin {\left (3 x \right )} \cos {\left (x \right )} \cos {\left (5 x \right )}}{63} + \frac {25 \sin {\left (5 x \right )} \cos {\left (x \right )} \cos {\left (3 x \right )}}{63} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(3*x)*cos(5*x),x)

[Out]

-10*sin(x)*sin(3*x)*sin(5*x)/63 - 11*sin(x)*cos(3*x)*cos(5*x)/63 - 17*sin(3*x)*cos(x)*cos(5*x)/63 + 25*sin(5*x
)*cos(x)*cos(3*x)/63

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Giac [A]
time = 0.48, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{36} \, \sin \left (9 \, x\right ) + \frac {1}{28} \, \sin \left (7 \, x\right ) + \frac {1}{12} \, \sin \left (3 \, x\right ) + \frac {1}{4} \, \sin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(3*x)*cos(5*x),x, algorithm="giac")

[Out]

1/36*sin(9*x) + 1/28*sin(7*x) + 1/12*sin(3*x) + 1/4*sin(x)

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Mupad [B]
time = 0.28, size = 27, normalized size = 0.87 \begin {gather*} \frac {64\,{\sin \left (x\right )}^9}{9}-\frac {128\,{\sin \left (x\right )}^7}{7}+16\,{\sin \left (x\right )}^5-\frac {17\,{\sin \left (x\right )}^3}{3}+\sin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3*x)*cos(5*x)*cos(x),x)

[Out]

sin(x) - (17*sin(x)^3)/3 + 16*sin(x)^5 - (128*sin(x)^7)/7 + (64*sin(x)^9)/9

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Chatgpt [F] Failed to verify
time = 1.00, size = 19, normalized size = 0.61 \begin {gather*} \frac {\sin \left (8 x \right )}{48}+\frac {\sin \left (6 x \right )}{12}+\frac {\sin \left (2 x \right )}{24} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(cos(x)*cos(3*x)*cos(5*x),x)

[Out]

1/48*sin(8*x)+1/12*sin(6*x)+1/24*sin(2*x)

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