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

Optimal. Leaf size=25 \[ \frac{1}{4} \sin (2 x)-\frac{1}{40} \sin (10 x)-\frac{1}{56} \sin (14 x) \]

[Out]

Sin[2*x]/4 - Sin[10*x]/40 - Sin[14*x]/56

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Rubi [A]  time = 0.027917, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4354, 2637} \[ \frac{1}{4} \sin (2 x)-\frac{1}{40} \sin (10 x)-\frac{1}{56} \sin (14 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[2*x]/4 - Sin[10*x]/40 - Sin[14*x]/56

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 (2 x) \sin ^2(6 x) \, dx &=\int \left (\frac{1}{2} \cos (2 x)-\frac{1}{4} \cos (10 x)-\frac{1}{4} \cos (14 x)\right ) \, dx\\ &=-\left (\frac{1}{4} \int \cos (10 x) \, dx\right )-\frac{1}{4} \int \cos (14 x) \, dx+\frac{1}{2} \int \cos (2 x) \, dx\\ &=\frac{1}{4} \sin (2 x)-\frac{1}{40} \sin (10 x)-\frac{1}{56} \sin (14 x)\\ \end{align*}

Mathematica [A]  time = 0.0163452, size = 25, normalized size = 1. \[ \frac{1}{4} \sin (2 x)-\frac{1}{40} \sin (10 x)-\frac{1}{56} \sin (14 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[2*x]/4 - Sin[10*x]/40 - Sin[14*x]/56

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Maple [A]  time = 0.048, size = 20, normalized size = 0.8 \begin{align*}{\frac{\sin \left ( 2\,x \right ) }{4}}-{\frac{\sin \left ( 10\,x \right ) }{40}}-{\frac{\sin \left ( 14\,x \right ) }{56}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sin(2*x)-1/40*sin(10*x)-1/56*sin(14*x)

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Maxima [A]  time = 0.982074, size = 26, normalized size = 1.04 \begin{align*} -\frac{1}{56} \, \sin \left (14 \, x\right ) - \frac{1}{40} \, \sin \left (10 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/56*sin(14*x) - 1/40*sin(10*x) + 1/4*sin(2*x)

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Fricas [A]  time = 2.36754, size = 92, normalized size = 3.68 \begin{align*} -\frac{1}{70} \,{\left (80 \, \cos \left (2 \, x\right )^{6} - 72 \, \cos \left (2 \, x\right )^{4} + 9 \, \cos \left (2 \, x\right )^{2} - 17\right )} \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/70*(80*cos(2*x)^6 - 72*cos(2*x)^4 + 9*cos(2*x)^2 - 17)*sin(2*x)

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Sympy [B]  time = 12.7746, size = 48, normalized size = 1.92 \begin{align*} \frac{17 \sin{\left (2 x \right )} \sin ^{2}{\left (6 x \right )}}{70} + \frac{9 \sin{\left (2 x \right )} \cos ^{2}{\left (6 x \right )}}{35} - \frac{3 \sin{\left (6 x \right )} \cos{\left (2 x \right )} \cos{\left (6 x \right )}}{35} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

17*sin(2*x)*sin(6*x)**2/70 + 9*sin(2*x)*cos(6*x)**2/35 - 3*sin(6*x)*cos(2*x)*cos(6*x)/35

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Giac [A]  time = 1.12103, size = 26, normalized size = 1.04 \begin{align*} -\frac{1}{56} \, \sin \left (14 \, x\right ) - \frac{1}{40} \, \sin \left (10 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/56*sin(14*x) - 1/40*sin(10*x) + 1/4*sin(2*x)