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

Optimal. Leaf size=33 \[ \frac{3}{40} \cos (5 x)-\frac{3}{56} \cos (7 x)+\frac{1}{136} \cos (17 x)-\frac{1}{152} \cos (19 x) \]

[Out]

(3*Cos[5*x])/40 - (3*Cos[7*x])/56 + Cos[17*x]/136 - Cos[19*x]/152

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Rubi [A]  time = 0.0314779, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4354, 2638} \[ \frac{3}{40} \cos (5 x)-\frac{3}{56} \cos (7 x)+\frac{1}{136} \cos (17 x)-\frac{1}{152} \cos (19 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Cos[5*x])/40 - (3*Cos[7*x])/56 + Cos[17*x]/136 - Cos[19*x]/152

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 (x) \, dx &=\int \left (-\frac{3}{8} \sin (5 x)+\frac{3}{8} \sin (7 x)-\frac{1}{8} \sin (17 x)+\frac{1}{8} \sin (19 x)\right ) \, dx\\ &=-\left (\frac{1}{8} \int \sin (17 x) \, dx\right )+\frac{1}{8} \int \sin (19 x) \, dx-\frac{3}{8} \int \sin (5 x) \, dx+\frac{3}{8} \int \sin (7 x) \, dx\\ &=\frac{3}{40} \cos (5 x)-\frac{3}{56} \cos (7 x)+\frac{1}{136} \cos (17 x)-\frac{1}{152} \cos (19 x)\\ \end{align*}

Mathematica [A]  time = 0.0172737, size = 33, normalized size = 1. \[ \frac{3}{40} \cos (5 x)-\frac{3}{56} \cos (7 x)+\frac{1}{136} \cos (17 x)-\frac{1}{152} \cos (19 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Cos[5*x])/40 - (3*Cos[7*x])/56 + Cos[17*x]/136 - Cos[19*x]/152

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Maple [A]  time = 0.092, size = 26, normalized size = 0.8 \begin{align*}{\frac{3\,\cos \left ( 5\,x \right ) }{40}}-{\frac{3\,\cos \left ( 7\,x \right ) }{56}}+{\frac{\cos \left ( 17\,x \right ) }{136}}-{\frac{\cos \left ( 19\,x \right ) }{152}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/40*cos(5*x)-3/56*cos(7*x)+1/136*cos(17*x)-1/152*cos(19*x)

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Maxima [A]  time = 0.99366, size = 34, normalized size = 1.03 \begin{align*} -\frac{1}{152} \, \cos \left (19 \, x\right ) + \frac{1}{136} \, \cos \left (17 \, x\right ) - \frac{3}{56} \, \cos \left (7 \, x\right ) + \frac{3}{40} \, \cos \left (5 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/152*cos(19*x) + 1/136*cos(17*x) - 3/56*cos(7*x) + 3/40*cos(5*x)

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Fricas [B]  time = 2.796, size = 234, normalized size = 7.09 \begin{align*} -\frac{32768}{19} \, \cos \left (x\right )^{19} + \frac{147456}{17} \, \cos \left (x\right )^{17} - 18432 \, \cos \left (x\right )^{15} + 21504 \, \cos \left (x\right )^{13} - 14976 \, \cos \left (x\right )^{11} + 6336 \, \cos \left (x\right )^{9} - \frac{11112}{7} \, \cos \left (x\right )^{7} + \frac{1116}{5} \, \cos \left (x\right )^{5} - 18 \, \cos \left (x\right )^{3} + \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32768/19*cos(x)^19 + 147456/17*cos(x)^17 - 18432*cos(x)^15 + 21504*cos(x)^13 - 14976*cos(x)^11 + 6336*cos(x)^
9 - 11112/7*cos(x)^7 + 1116/5*cos(x)^5 - 18*cos(x)^3 + cos(x)

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Sympy [B]  time = 11.4344, size = 63, normalized size = 1.91 \begin{align*} \frac{1296 \sin{\left (x \right )} \sin ^{3}{\left (6 x \right )}}{11305} + \frac{1926 \sin{\left (x \right )} \sin{\left (6 x \right )} \cos ^{2}{\left (6 x \right )}}{11305} + \frac{216 \sin ^{2}{\left (6 x \right )} \cos{\left (x \right )} \cos{\left (6 x \right )}}{11305} + \frac{251 \cos{\left (x \right )} \cos ^{3}{\left (6 x \right )}}{11305} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1296*sin(x)*sin(6*x)**3/11305 + 1926*sin(x)*sin(6*x)*cos(6*x)**2/11305 + 216*sin(6*x)**2*cos(x)*cos(6*x)/11305
+ 251*cos(x)*cos(6*x)**3/11305

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Giac [A]  time = 1.11997, size = 34, normalized size = 1.03 \begin{align*} -\frac{1}{152} \, \cos \left (19 \, x\right ) + \frac{1}{136} \, \cos \left (17 \, x\right ) - \frac{3}{56} \, \cos \left (7 \, x\right ) + \frac{3}{40} \, \cos \left (5 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/152*cos(19*x) + 1/136*cos(17*x) - 3/56*cos(7*x) + 3/40*cos(5*x)