3.320 \(\int \sin (x) \sin (2 x) \sin (3 x) \, dx\)

Optimal. Leaf size=25 \[ -\frac{1}{8} \cos (2 x)-\frac{1}{16} \cos (4 x)+\frac{1}{24} \cos (6 x) \]

[Out]

-Cos[2*x]/8 - Cos[4*x]/16 + Cos[6*x]/24

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Rubi [A]  time = 0.0296515, 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 = {4355, 2638} \[ -\frac{1}{8} \cos (2 x)-\frac{1}{16} \cos (4 x)+\frac{1}{24} \cos (6 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[2*x]/8 - Cos[4*x]/16 + Cos[6*x]/24

Rule 4355

Int[(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.)*(H_)[(e_.) + (f_.)*(x_)]^(r_.), x_Symbol] :>
Int[ExpandTrigReduce[ActivateTrig[F[a + b*x]^p*G[c + d*x]^q*H[e + f*x]^r], x], x] /; FreeQ[{a, b, c, d, e, f}
, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, sin] || EqQ[G, cos]) && (EqQ[H, sin] || EqQ[H, cos]) && IGtQ[p
, 0] && IGtQ[q, 0] && IGtQ[r, 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 \sin (x) \sin (2 x) \sin (3 x) \, dx &=\int \left (\frac{1}{4} \sin (2 x)+\frac{1}{4} \sin (4 x)-\frac{1}{4} \sin (6 x)\right ) \, dx\\ &=\frac{1}{4} \int \sin (2 x) \, dx+\frac{1}{4} \int \sin (4 x) \, dx-\frac{1}{4} \int \sin (6 x) \, dx\\ &=-\frac{1}{8} \cos (2 x)-\frac{1}{16} \cos (4 x)+\frac{1}{24} \cos (6 x)\\ \end{align*}

Mathematica [A]  time = 0.0096997, size = 25, normalized size = 1. \[ -\frac{1}{8} \cos (2 x)-\frac{1}{16} \cos (4 x)+\frac{1}{24} \cos (6 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[2*x]/8 - Cos[4*x]/16 + Cos[6*x]/24

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Maple [A]  time = 0.046, size = 20, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( 2\,x \right ) }{8}}-{\frac{\cos \left ( 4\,x \right ) }{16}}+{\frac{\cos \left ( 6\,x \right ) }{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*cos(2*x)-1/16*cos(4*x)+1/24*cos(6*x)

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Maxima [A]  time = 0.928093, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{24} \, \cos \left (6 \, x\right ) - \frac{1}{16} \, \cos \left (4 \, x\right ) - \frac{1}{8} \, \cos \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*cos(6*x) - 1/16*cos(4*x) - 1/8*cos(2*x)

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Fricas [A]  time = 2.309, size = 54, normalized size = 2.16 \begin{align*} \frac{4}{3} \, \cos \left (x\right )^{6} - \frac{5}{2} \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*cos(x)^6 - 5/2*cos(x)^4 + cos(x)^2

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Sympy [B]  time = 15.0418, size = 114, normalized size = 4.56 \begin{align*} \frac{x \sin{\left (x \right )} \sin{\left (2 x \right )} \sin{\left (3 x \right )}}{4} + \frac{x \sin{\left (x \right )} \cos{\left (2 x \right )} \cos{\left (3 x \right )}}{4} + \frac{x \sin{\left (2 x \right )} \cos{\left (x \right )} \cos{\left (3 x \right )}}{4} - \frac{x \sin{\left (3 x \right )} \cos{\left (x \right )} \cos{\left (2 x \right )}}{4} - \frac{5 \sin{\left (x \right )} \sin{\left (2 x \right )} \cos{\left (3 x \right )}}{24} - \frac{\sin{\left (2 x \right )} \sin{\left (3 x \right )} \cos{\left (x \right )}}{8} - \frac{\cos{\left (x \right )} \cos{\left (2 x \right )} \cos{\left (3 x \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sin(x)*sin(2*x)*sin(3*x)/4 + x*sin(x)*cos(2*x)*cos(3*x)/4 + x*sin(2*x)*cos(x)*cos(3*x)/4 - x*sin(3*x)*cos(x)
*cos(2*x)/4 - 5*sin(x)*sin(2*x)*cos(3*x)/24 - sin(2*x)*sin(3*x)*cos(x)/8 - cos(x)*cos(2*x)*cos(3*x)/6

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Giac [A]  time = 1.05124, size = 18, normalized size = 0.72 \begin{align*} -\frac{4}{3} \, \sin \left (x\right )^{6} + \frac{3}{2} \, \sin \left (x\right )^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*sin(x)^6 + 3/2*sin(x)^4