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]

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

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Rubi [A]  time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 2638

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

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]

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*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 1.00 \[ \frac {3}{40} \cos (5 x)-\frac {3}{56} \cos (7 x)+\frac {1}{136} \cos (17 x)-\frac {1}{152} \cos (19 x) \]

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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fricas [B]  time = 3.97, size = 57, normalized size = 1.73 \[ -\frac {32768}{19} \, \cos \relax (x)^{19} + \frac {147456}{17} \, \cos \relax (x)^{17} - 18432 \, \cos \relax (x)^{15} + 21504 \, \cos \relax (x)^{13} - 14976 \, \cos \relax (x)^{11} + 6336 \, \cos \relax (x)^{9} - \frac {11112}{7} \, \cos \relax (x)^{7} + \frac {1116}{5} \, \cos \relax (x)^{5} - 18 \, \cos \relax (x)^{3} + \cos \relax (x) \]

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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giac [A]  time = 0.13, size = 25, normalized size = 0.76 \[ -\frac {1}{152} \, \cos \left (19 \, x\right ) + \frac {1}{136} \, \cos \left (17 \, x\right ) - \frac {3}{56} \, \cos \left (7 \, x\right ) + \frac {3}{40} \, \cos \left (5 \, x\right ) \]

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)

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maple [A]  time = 0.25, size = 26, normalized size = 0.79 \[ \frac {3 \cos \left (5 x \right )}{40}-\frac {3 \cos \left (7 x \right )}{56}+\frac {\cos \left (17 x \right )}{136}-\frac {\cos \left (19 x \right )}{152} \]

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 = 0.45, size = 25, normalized size = 0.76 \[ -\frac {1}{152} \, \cos \left (19 \, x\right ) + \frac {1}{136} \, \cos \left (17 \, x\right ) - \frac {3}{56} \, \cos \left (7 \, x\right ) + \frac {3}{40} \, \cos \left (5 \, x\right ) \]

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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mupad [B]  time = 0.08, size = 57, normalized size = 1.73 \[ -\frac {32768\,{\cos \relax (x)}^{19}}{19}+\frac {147456\,{\cos \relax (x)}^{17}}{17}-18432\,{\cos \relax (x)}^{15}+21504\,{\cos \relax (x)}^{13}-14976\,{\cos \relax (x)}^{11}+6336\,{\cos \relax (x)}^9-\frac {11112\,{\cos \relax (x)}^7}{7}+\frac {1116\,{\cos \relax (x)}^5}{5}-18\,{\cos \relax (x)}^3+\cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 5.21, size = 63, normalized size = 1.91 \[ \frac {1296 \sin {\relax (x )} \sin ^{3}{\left (6 x \right )}}{11305} + \frac {1926 \sin {\relax (x )} \sin {\left (6 x \right )} \cos ^{2}{\left (6 x \right )}}{11305} + \frac {216 \sin ^{2}{\left (6 x \right )} \cos {\relax (x )} \cos {\left (6 x \right )}}{11305} + \frac {251 \cos {\relax (x )} \cos ^{3}{\left (6 x \right )}}{11305} \]

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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