3.131 \(\int \cos (x) \sin ^2(6 x) \, dx\)

Optimal. Leaf size=23 \[ \frac{\sin (x)}{2}-\frac{1}{44} \sin (11 x)-\frac{1}{52} \sin (13 x) \]

[Out]

Sin[x]/2 - Sin[11*x]/44 - Sin[13*x]/52

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Rubi [A]  time = 0.0263351, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4354, 2637} \[ \frac{\sin (x)}{2}-\frac{1}{44} \sin (11 x)-\frac{1}{52} \sin (13 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Sin[6*x]^2,x]

[Out]

Sin[x]/2 - Sin[11*x]/44 - Sin[13*x]/52

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cos (x) \sin ^2(6 x) \, dx &=\int \left (\frac{\cos (x)}{2}-\frac{1}{4} \cos (11 x)-\frac{1}{4} \cos (13 x)\right ) \, dx\\ &=-\left (\frac{1}{4} \int \cos (11 x) \, dx\right )-\frac{1}{4} \int \cos (13 x) \, dx+\frac{1}{2} \int \cos (x) \, dx\\ &=\frac{\sin (x)}{2}-\frac{1}{44} \sin (11 x)-\frac{1}{52} \sin (13 x)\\ \end{align*}

Mathematica [A]  time = 0.0119901, size = 23, normalized size = 1. \[ \frac{\sin (x)}{2}-\frac{1}{44} \sin (11 x)-\frac{1}{52} \sin (13 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Sin[6*x]^2,x]

[Out]

Sin[x]/2 - Sin[11*x]/44 - Sin[13*x]/52

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Maple [A]  time = 0.051, size = 18, normalized size = 0.8 \begin{align*}{\frac{\sin \left ( x \right ) }{2}}-{\frac{\sin \left ( 11\,x \right ) }{44}}-{\frac{\sin \left ( 13\,x \right ) }{52}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*sin(6*x)^2,x)

[Out]

1/2*sin(x)-1/44*sin(11*x)-1/52*sin(13*x)

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Maxima [A]  time = 1.00829, size = 23, normalized size = 1. \begin{align*} -\frac{1}{52} \, \sin \left (13 \, x\right ) - \frac{1}{44} \, \sin \left (11 \, x\right ) + \frac{1}{2} \, \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(6*x)^2,x, algorithm="maxima")

[Out]

-1/52*sin(13*x) - 1/44*sin(11*x) + 1/2*sin(x)

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Fricas [B]  time = 2.48903, size = 154, normalized size = 6.7 \begin{align*} -\frac{4}{143} \,{\left (2816 \, \cos \left (x\right )^{12} - 6912 \, \cos \left (x\right )^{10} + 6048 \, \cos \left (x\right )^{8} - 2240 \, \cos \left (x\right )^{6} + 315 \, \cos \left (x\right )^{4} - 9 \, \cos \left (x\right )^{2} - 18\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(6*x)^2,x, algorithm="fricas")

[Out]

-4/143*(2816*cos(x)^12 - 6912*cos(x)^10 + 6048*cos(x)^8 - 2240*cos(x)^6 + 315*cos(x)^4 - 9*cos(x)^2 - 18)*sin(
x)

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Sympy [B]  time = 12.8496, size = 42, normalized size = 1.83 \begin{align*} \frac{71 \sin{\left (x \right )} \sin ^{2}{\left (6 x \right )}}{143} + \frac{72 \sin{\left (x \right )} \cos ^{2}{\left (6 x \right )}}{143} - \frac{12 \sin{\left (6 x \right )} \cos{\left (x \right )} \cos{\left (6 x \right )}}{143} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(6*x)**2,x)

[Out]

71*sin(x)*sin(6*x)**2/143 + 72*sin(x)*cos(6*x)**2/143 - 12*sin(6*x)*cos(x)*cos(6*x)/143

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Giac [A]  time = 1.11927, size = 23, normalized size = 1. \begin{align*} -\frac{1}{52} \, \sin \left (13 \, x\right ) - \frac{1}{44} \, \sin \left (11 \, x\right ) + \frac{1}{2} \, \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(6*x)^2,x, algorithm="giac")

[Out]

-1/52*sin(13*x) - 1/44*sin(11*x) + 1/2*sin(x)