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

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

[Out]

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

________________________________________________________________________________________

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 = {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 2638

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

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]

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 = 0.01, size = 25, normalized size = 1.00 \[ -\frac {1}{8} \cos (2 x)-\frac {1}{16} \cos (4 x)+\frac {1}{24} \cos (6 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*Cos[2*x] - Cos[4*x]/16 + Cos[6*x]/24

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 17, normalized size = 0.68 \[ \frac {4}{3} \, \cos \relax (x)^{6} - \frac {5}{2} \, \cos \relax (x)^{4} + \cos \relax (x)^{2} \]

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

________________________________________________________________________________________

giac [A]  time = 1.11, size = 13, normalized size = 0.52 \[ -\frac {4}{3} \, \sin \relax (x)^{6} + \frac {3}{2} \, \sin \relax (x)^{4} \]

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

________________________________________________________________________________________

maple [A]  time = 0.12, size = 20, normalized size = 0.80 \[ -\frac {\cos \left (2 x \right )}{8}-\frac {\cos \left (4 x \right )}{16}+\frac {\cos \left (6 x \right )}{24} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 19, normalized size = 0.76 \[ \frac {1}{24} \, \cos \left (6 \, x\right ) - \frac {1}{16} \, \cos \left (4 \, x\right ) - \frac {1}{8} \, \cos \left (2 \, x\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.17, size = 14, normalized size = 0.56 \[ -\frac {{\sin \relax (x)}^4\,\left (8\,{\sin \relax (x)}^2-9\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sin(x)^4*(8*sin(x)^2 - 9))/6

________________________________________________________________________________________

sympy [B]  time = 12.67, size = 114, normalized size = 4.56 \[ \frac {x \sin {\relax (x )} \sin {\left (2 x \right )} \sin {\left (3 x \right )}}{4} + \frac {x \sin {\relax (x )} \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{4} + \frac {x \sin {\left (2 x \right )} \cos {\relax (x )} \cos {\left (3 x \right )}}{4} - \frac {x \sin {\left (3 x \right )} \cos {\relax (x )} \cos {\left (2 x \right )}}{4} - \frac {5 \sin {\relax (x )} \sin {\left (2 x \right )} \cos {\left (3 x \right )}}{24} - \frac {\sin {\left (2 x \right )} \sin {\left (3 x \right )} \cos {\relax (x )}}{8} - \frac {\cos {\relax (x )} \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{6} \]

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

________________________________________________________________________________________