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]

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

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Rubi [A]  time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 (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*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 1.00 \[ \frac {1}{4} \sin (2 x)-\frac {1}{40} \sin (10 x)-\frac {1}{56} \sin (14 x) \]

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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fricas [A]  time = 2.01, size = 32, normalized size = 1.28 \[ -\frac {1}{70} \, {\left (80 \, \cos \left (2 \, x\right )^{6} - 72 \, \cos \left (2 \, x\right )^{4} + 9 \, \cos \left (2 \, x\right )^{2} - 17\right )} \sin \left (2 \, x\right ) \]

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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giac [A]  time = 0.12, size = 19, normalized size = 0.76 \[ -\frac {1}{56} \, \sin \left (14 \, x\right ) - \frac {1}{40} \, \sin \left (10 \, x\right ) + \frac {1}{4} \, \sin \left (2 \, x\right ) \]

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)

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maple [A]  time = 0.12, size = 20, normalized size = 0.80 \[ \frac {\sin \left (2 x \right )}{4}-\frac {\sin \left (10 x \right )}{40}-\frac {\sin \left (14 x \right )}{56} \]

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 = 0.32, size = 19, normalized size = 0.76 \[ -\frac {1}{56} \, \sin \left (14 \, x\right ) - \frac {1}{40} \, \sin \left (10 \, x\right ) + \frac {1}{4} \, \sin \left (2 \, x\right ) \]

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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mupad [B]  time = 2.29, size = 25, normalized size = 1.00 \[ \frac {8\,{\sin \left (2\,x\right )}^7}{7}-\frac {12\,{\sin \left (2\,x\right )}^5}{5}+\frac {3\,{\sin \left (2\,x\right )}^3}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*sin(2*x)^3)/2 - (12*sin(2*x)^5)/5 + (8*sin(2*x)^7)/7

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sympy [B]  time = 1.73, size = 48, normalized size = 1.92 \[ \frac {17 \sin {\left (2 x \right )} \sin ^{2}{\left (6 x \right )}}{70} + \frac {9 \sin {\left (2 x \right )} \cos ^{2}{\left (6 x \right )}}{35} - \frac {3 \sin {\left (6 x \right )} \cos {\left (2 x \right )} \cos {\left (6 x \right )}}{35} \]

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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