3.134 \(\int \cos ^2(3 x) \sin ^3(2 x) \, dx\)

Optimal. Leaf size=41 \[ -\frac{3}{16} \cos (2 x)+\frac{3}{64} \cos (4 x)+\frac{1}{48} \cos (6 x)-\frac{3}{128} \cos (8 x)+\frac{1}{192} \cos (12 x) \]

[Out]

(-3*Cos[2*x])/16 + (3*Cos[4*x])/64 + Cos[6*x]/48 - (3*Cos[8*x])/128 + Cos[12*x]/192

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Rubi [A]  time = 0.0433548, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4354, 2638} \[ -\frac{3}{16} \cos (2 x)+\frac{3}{64} \cos (4 x)+\frac{1}{48} \cos (6 x)-\frac{3}{128} \cos (8 x)+\frac{1}{192} \cos (12 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[3*x]^2*Sin[2*x]^3,x]

[Out]

(-3*Cos[2*x])/16 + (3*Cos[4*x])/64 + Cos[6*x]/48 - (3*Cos[8*x])/128 + Cos[12*x]/192

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 2638

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

Rubi steps

\begin{align*} \int \cos ^2(3 x) \sin ^3(2 x) \, dx &=\int \left (\frac{3}{8} \sin (2 x)-\frac{3}{16} \sin (4 x)-\frac{1}{8} \sin (6 x)+\frac{3}{16} \sin (8 x)-\frac{1}{16} \sin (12 x)\right ) \, dx\\ &=-\left (\frac{1}{16} \int \sin (12 x) \, dx\right )-\frac{1}{8} \int \sin (6 x) \, dx-\frac{3}{16} \int \sin (4 x) \, dx+\frac{3}{16} \int \sin (8 x) \, dx+\frac{3}{8} \int \sin (2 x) \, dx\\ &=-\frac{3}{16} \cos (2 x)+\frac{3}{64} \cos (4 x)+\frac{1}{48} \cos (6 x)-\frac{3}{128} \cos (8 x)+\frac{1}{192} \cos (12 x)\\ \end{align*}

Mathematica [A]  time = 0.0185465, size = 41, normalized size = 1. \[ -\frac{3}{16} \cos (2 x)+\frac{3}{64} \cos (4 x)+\frac{1}{48} \cos (6 x)-\frac{3}{128} \cos (8 x)+\frac{1}{192} \cos (12 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[3*x]^2*Sin[2*x]^3,x]

[Out]

(-3*Cos[2*x])/16 + (3*Cos[4*x])/64 + Cos[6*x]/48 - (3*Cos[8*x])/128 + Cos[12*x]/192

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Maple [A]  time = 0.066, size = 32, normalized size = 0.8 \begin{align*} -{\frac{3\,\cos \left ( 2\,x \right ) }{16}}+{\frac{3\,\cos \left ( 4\,x \right ) }{64}}+{\frac{\cos \left ( 6\,x \right ) }{48}}-{\frac{3\,\cos \left ( 8\,x \right ) }{128}}+{\frac{\cos \left ( 12\,x \right ) }{192}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3*x)^2*sin(2*x)^3,x)

[Out]

-3/16*cos(2*x)+3/64*cos(4*x)+1/48*cos(6*x)-3/128*cos(8*x)+1/192*cos(12*x)

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Maxima [A]  time = 1.0121, size = 42, normalized size = 1.02 \begin{align*} \frac{1}{192} \, \cos \left (12 \, x\right ) - \frac{3}{128} \, \cos \left (8 \, x\right ) + \frac{1}{48} \, \cos \left (6 \, x\right ) + \frac{3}{64} \, \cos \left (4 \, x\right ) - \frac{3}{16} \, \cos \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)^2*sin(2*x)^3,x, algorithm="maxima")

[Out]

1/192*cos(12*x) - 3/128*cos(8*x) + 1/48*cos(6*x) + 3/64*cos(4*x) - 3/16*cos(2*x)

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Fricas [A]  time = 2.39628, size = 80, normalized size = 1.95 \begin{align*} \frac{32}{3} \, \cos \left (x\right )^{12} - 32 \, \cos \left (x\right )^{10} + 33 \, \cos \left (x\right )^{8} - 12 \, \cos \left (x\right )^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)^2*sin(2*x)^3,x, algorithm="fricas")

[Out]

32/3*cos(x)^12 - 32*cos(x)^10 + 33*cos(x)^8 - 12*cos(x)^6

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Sympy [B]  time = 52.9172, size = 228, normalized size = 5.56 \begin{align*} - \frac{x \sin ^{3}{\left (2 x \right )} \sin ^{2}{\left (3 x \right )}}{16} + \frac{x \sin ^{3}{\left (2 x \right )} \cos ^{2}{\left (3 x \right )}}{16} - \frac{3 x \sin ^{2}{\left (2 x \right )} \sin{\left (3 x \right )} \cos{\left (2 x \right )} \cos{\left (3 x \right )}}{8} + \frac{3 x \sin{\left (2 x \right )} \sin ^{2}{\left (3 x \right )} \cos ^{2}{\left (2 x \right )}}{16} - \frac{3 x \sin{\left (2 x \right )} \cos ^{2}{\left (2 x \right )} \cos ^{2}{\left (3 x \right )}}{16} + \frac{x \sin{\left (3 x \right )} \cos ^{3}{\left (2 x \right )} \cos{\left (3 x \right )}}{8} - \frac{\sin ^{2}{\left (2 x \right )} \sin ^{2}{\left (3 x \right )} \cos{\left (2 x \right )}}{32} - \frac{15 \sin ^{2}{\left (2 x \right )} \cos{\left (2 x \right )} \cos ^{2}{\left (3 x \right )}}{32} + \frac{9 \sin{\left (2 x \right )} \sin{\left (3 x \right )} \cos ^{2}{\left (2 x \right )} \cos{\left (3 x \right )}}{16} - \frac{13 \sin ^{2}{\left (3 x \right )} \cos ^{3}{\left (2 x \right )}}{48} - \frac{\cos ^{3}{\left (2 x \right )} \cos ^{2}{\left (3 x \right )}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)**2*sin(2*x)**3,x)

[Out]

-x*sin(2*x)**3*sin(3*x)**2/16 + x*sin(2*x)**3*cos(3*x)**2/16 - 3*x*sin(2*x)**2*sin(3*x)*cos(2*x)*cos(3*x)/8 +
3*x*sin(2*x)*sin(3*x)**2*cos(2*x)**2/16 - 3*x*sin(2*x)*cos(2*x)**2*cos(3*x)**2/16 + x*sin(3*x)*cos(2*x)**3*cos
(3*x)/8 - sin(2*x)**2*sin(3*x)**2*cos(2*x)/32 - 15*sin(2*x)**2*cos(2*x)*cos(3*x)**2/32 + 9*sin(2*x)*sin(3*x)*c
os(2*x)**2*cos(3*x)/16 - 13*sin(3*x)**2*cos(2*x)**3/48 - cos(2*x)**3*cos(3*x)**2/16

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Giac [A]  time = 1.16051, size = 42, normalized size = 1.02 \begin{align*} \frac{1}{192} \, \cos \left (12 \, x\right ) - \frac{3}{128} \, \cos \left (8 \, x\right ) + \frac{1}{48} \, \cos \left (6 \, x\right ) + \frac{3}{64} \, \cos \left (4 \, x\right ) - \frac{3}{16} \, \cos \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)^2*sin(2*x)^3,x, algorithm="giac")

[Out]

1/192*cos(12*x) - 3/128*cos(8*x) + 1/48*cos(6*x) + 3/64*cos(4*x) - 3/16*cos(2*x)