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

Optimal. Leaf size=26 \[ -\frac{1}{2} \sin (\cos (x))+\frac{1}{44} \sin (11 \cos (x))+\frac{1}{52} \sin (13 \cos (x)) \]

[Out]

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

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Rubi [A]  time = 0.0464201, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4335, 4354, 2637} \[ -\frac{1}{2} \sin (\cos (x))+\frac{1}{44} \sin (11 \cos (x))+\frac{1}{52} \sin (13 \cos (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4335

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

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

Mathematica [A]  time = 4.68363, size = 26, normalized size = 1. \[ -\frac{1}{2} \sin (\cos (x))+\frac{1}{44} \sin (11 \cos (x))+\frac{1}{52} \sin (13 \cos (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.067, size = 21, normalized size = 0.8 \begin{align*} -{\frac{\sin \left ( \cos \left ( x \right ) \right ) }{2}}+{\frac{\sin \left ( 11\,\cos \left ( x \right ) \right ) }{44}}+{\frac{\sin \left ( 13\,\cos \left ( x \right ) \right ) }{52}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.961604, size = 27, normalized size = 1.04 \begin{align*} \frac{1}{52} \, \sin \left (13 \, \cos \left (x\right )\right ) + \frac{1}{44} \, \sin \left (11 \, \cos \left (x\right )\right ) - \frac{1}{2} \, \sin \left (\cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.37467, size = 494, normalized size = 19. \begin{align*} -\frac{4}{143} \,{\left (2816 \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{12} - 6912 \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{10} + 6048 \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{8} - 2240 \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{6} + 315 \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{4} - 9 \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{2} - 18\right )} \sin \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/143*(2816*cos((tan(1/2*x)^2 - 1)/(tan(1/2*x)^2 + 1))^12 - 6912*cos((tan(1/2*x)^2 - 1)/(tan(1/2*x)^2 + 1))^1
0 + 6048*cos((tan(1/2*x)^2 - 1)/(tan(1/2*x)^2 + 1))^8 - 2240*cos((tan(1/2*x)^2 - 1)/(tan(1/2*x)^2 + 1))^6 + 31
5*cos((tan(1/2*x)^2 - 1)/(tan(1/2*x)^2 + 1))^4 - 9*cos((tan(1/2*x)^2 - 1)/(tan(1/2*x)^2 + 1))^2 - 18)*sin((tan
(1/2*x)^2 - 1)/(tan(1/2*x)^2 + 1))

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Sympy [B]  time = 70.251, size = 54, normalized size = 2.08 \begin{align*} - \frac{71 \sin{\left (\cos{\left (x \right )} \right )} \sin ^{2}{\left (6 \cos{\left (x \right )} \right )}}{143} - \frac{72 \sin{\left (\cos{\left (x \right )} \right )} \cos ^{2}{\left (6 \cos{\left (x \right )} \right )}}{143} + \frac{12 \sin{\left (6 \cos{\left (x \right )} \right )} \cos{\left (\cos{\left (x \right )} \right )} \cos{\left (6 \cos{\left (x \right )} \right )}}{143} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09441, size = 27, normalized size = 1.04 \begin{align*} \frac{1}{52} \, \sin \left (13 \, \cos \left (x\right )\right ) + \frac{1}{44} \, \sin \left (11 \, \cos \left (x\right )\right ) - \frac{1}{2} \, \sin \left (\cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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