3.129 \(\int \cos ^3(6 x) \sin (9 x) \, dx\)

Optimal. Leaf size=33 \[ -\frac{1}{8} \cos (3 x)+\frac{1}{72} \cos (9 x)-\frac{1}{40} \cos (15 x)-\frac{1}{216} \cos (27 x) \]

[Out]

-Cos[3*x]/8 + Cos[9*x]/72 - Cos[15*x]/40 - Cos[27*x]/216

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Rubi [A]  time = 0.0328019, antiderivative size = 33, normalized size of antiderivative = 1., 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 = {4354, 2638} \[ -\frac{1}{8} \cos (3 x)+\frac{1}{72} \cos (9 x)-\frac{1}{40} \cos (15 x)-\frac{1}{216} \cos (27 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[6*x]^3*Sin[9*x],x]

[Out]

-Cos[3*x]/8 + Cos[9*x]/72 - Cos[15*x]/40 - Cos[27*x]/216

Rule 4354

Int[(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[ActivateT
rig[F[a + b*x]^p*G[c + d*x]^q], x], x] /; FreeQ[{a, b, c, d}, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, si
n] || EqQ[G, cos]) && IGtQ[p, 0] && IGtQ[q, 0]

Rule 2638

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

Rubi steps

\begin{align*} \int \cos ^3(6 x) \sin (9 x) \, dx &=\int \left (\frac{3}{8} \sin (3 x)-\frac{1}{8} \sin (9 x)+\frac{3}{8} \sin (15 x)+\frac{1}{8} \sin (27 x)\right ) \, dx\\ &=-\left (\frac{1}{8} \int \sin (9 x) \, dx\right )+\frac{1}{8} \int \sin (27 x) \, dx+\frac{3}{8} \int \sin (3 x) \, dx+\frac{3}{8} \int \sin (15 x) \, dx\\ &=-\frac{1}{8} \cos (3 x)+\frac{1}{72} \cos (9 x)-\frac{1}{40} \cos (15 x)-\frac{1}{216} \cos (27 x)\\ \end{align*}

Mathematica [A]  time = 0.0173943, size = 33, normalized size = 1. \[ -\frac{1}{8} \cos (3 x)+\frac{1}{72} \cos (9 x)-\frac{1}{40} \cos (15 x)-\frac{1}{216} \cos (27 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[6*x]^3*Sin[9*x],x]

[Out]

-Cos[3*x]/8 + Cos[9*x]/72 - Cos[15*x]/40 - Cos[27*x]/216

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Maple [A]  time = 0.038, size = 26, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( 3\,x \right ) }{8}}+{\frac{\cos \left ( 9\,x \right ) }{72}}-{\frac{\cos \left ( 15\,x \right ) }{40}}-{\frac{\cos \left ( 27\,x \right ) }{216}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(6*x)^3*sin(9*x),x)

[Out]

-1/8*cos(3*x)+1/72*cos(9*x)-1/40*cos(15*x)-1/216*cos(27*x)

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Maxima [A]  time = 1.01214, size = 34, normalized size = 1.03 \begin{align*} -\frac{1}{216} \, \cos \left (27 \, x\right ) - \frac{1}{40} \, \cos \left (15 \, x\right ) + \frac{1}{72} \, \cos \left (9 \, x\right ) - \frac{1}{8} \, \cos \left (3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(6*x)^3*sin(9*x),x, algorithm="maxima")

[Out]

-1/216*cos(27*x) - 1/40*cos(15*x) + 1/72*cos(9*x) - 1/8*cos(3*x)

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Fricas [A]  time = 2.44613, size = 117, normalized size = 3.55 \begin{align*} -\frac{32}{27} \, \cos \left (3 \, x\right )^{9} + \frac{8}{3} \, \cos \left (3 \, x\right )^{7} - \frac{12}{5} \, \cos \left (3 \, x\right )^{5} + \frac{10}{9} \, \cos \left (3 \, x\right )^{3} - \frac{1}{3} \, \cos \left (3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(6*x)^3*sin(9*x),x, algorithm="fricas")

[Out]

-32/27*cos(3*x)^9 + 8/3*cos(3*x)^7 - 12/5*cos(3*x)^5 + 10/9*cos(3*x)^3 - 1/3*cos(3*x)

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Sympy [B]  time = 23.3911, size = 71, normalized size = 2.15 \begin{align*} - \frac{16 \sin ^{3}{\left (6 x \right )} \sin{\left (9 x \right )}}{135} - \frac{8 \sin ^{2}{\left (6 x \right )} \cos{\left (6 x \right )} \cos{\left (9 x \right )}}{45} - \frac{2 \sin{\left (6 x \right )} \sin{\left (9 x \right )} \cos ^{2}{\left (6 x \right )}}{45} - \frac{19 \cos ^{3}{\left (6 x \right )} \cos{\left (9 x \right )}}{135} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(6*x)**3*sin(9*x),x)

[Out]

-16*sin(6*x)**3*sin(9*x)/135 - 8*sin(6*x)**2*cos(6*x)*cos(9*x)/45 - 2*sin(6*x)*sin(9*x)*cos(6*x)**2/45 - 19*co
s(6*x)**3*cos(9*x)/135

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Giac [A]  time = 1.12231, size = 34, normalized size = 1.03 \begin{align*} -\frac{1}{216} \, \cos \left (27 \, x\right ) - \frac{1}{40} \, \cos \left (15 \, x\right ) + \frac{1}{72} \, \cos \left (9 \, x\right ) - \frac{1}{8} \, \cos \left (3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(6*x)^3*sin(9*x),x, algorithm="giac")

[Out]

-1/216*cos(27*x) - 1/40*cos(15*x) + 1/72*cos(9*x) - 1/8*cos(3*x)