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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 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 (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 = 0.01, size = 23, normalized size = 1.00 \[ \frac {\sin (x)}{2}-\frac {1}{44} \sin (11 x)-\frac {1}{52} \sin (13 x) \]

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

________________________________________________________________________________________

fricas [B]  time = 0.60, size = 42, normalized size = 1.83 \[ -\frac {4}{143} \, {\left (2816 \, \cos \relax (x)^{12} - 6912 \, \cos \relax (x)^{10} + 6048 \, \cos \relax (x)^{8} - 2240 \, \cos \relax (x)^{6} + 315 \, \cos \relax (x)^{4} - 9 \, \cos \relax (x)^{2} - 18\right )} \sin \relax (x) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.13, size = 17, normalized size = 0.74 \[ -\frac {1}{52} \, \sin \left (13 \, x\right ) - \frac {1}{44} \, \sin \left (11 \, x\right ) + \frac {1}{2} \, \sin \relax (x) \]

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)

________________________________________________________________________________________

maple [A]  time = 0.13, size = 18, normalized size = 0.78 \[ \frac {\sin \relax (x )}{2}-\frac {\sin \left (11 x \right )}{44}-\frac {\sin \left (13 x \right )}{52} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 17, normalized size = 0.74 \[ -\frac {1}{52} \, \sin \left (13 \, x\right ) - \frac {1}{44} \, \sin \left (11 \, x\right ) + \frac {1}{2} \, \sin \relax (x) \]

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)

________________________________________________________________________________________

mupad [B]  time = 2.49, size = 17, normalized size = 0.74 \[ \frac {\sin \relax (x)}{2}-\frac {\sin \left (13\,x\right )}{52}-\frac {\sin \left (11\,x\right )}{44} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 1.64, size = 42, normalized size = 1.83 \[ \frac {71 \sin {\relax (x )} \sin ^{2}{\left (6 x \right )}}{143} + \frac {72 \sin {\relax (x )} \cos ^{2}{\left (6 x \right )}}{143} - \frac {12 \sin {\left (6 x \right )} \cos {\relax (x )} \cos {\left (6 x \right )}}{143} \]

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

________________________________________________________________________________________