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]

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

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Rubi [A]  time = 0.0472429, 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 \[ -\frac{1}{8} \cos (2 x)-\frac{1}{16} \cos (4 x)+\frac{1}{24} \cos (6 x) \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\cos{\left (2 x \right )}}{8} - \frac{\cos{\left (4 x \right )}}{16} - \frac{\int ^{\sin{\left (3 x \right )}} x\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(x)*sin(2*x)*sin(3*x),x)

[Out]

-cos(2*x)/8 - cos(4*x)/16 - Integral(x, (x, sin(3*x)))/3

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Mathematica [A]  time = 0.0144485, 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 = 0.037, size = 20, normalized size = 0.8 \[ -{\frac{\cos \left ( 2\,x \right ) }{8}}-{\frac{\cos \left ( 4\,x \right ) }{16}}+{\frac{\cos \left ( 6\,x \right ) }{24}} \]

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)

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Maxima [A]  time = 1.39326, size = 26, normalized size = 1.04 \[ \frac{1}{24} \, \cos \left (6 \, x\right ) - \frac{1}{16} \, \cos \left (4 \, x\right ) - \frac{1}{8} \, \cos \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(3*x)*sin(2*x)*sin(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 = 0.24567, size = 23, normalized size = 0.92 \[ \frac{4}{3} \, \cos \left (x\right )^{6} - \frac{5}{2} \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 21.3735, size = 114, normalized size = 4.56 \[ \frac{x \sin{\left (x \right )} \sin{\left (2 x \right )} \sin{\left (3 x \right )}}{4} + \frac{x \sin{\left (x \right )} \cos{\left (2 x \right )} \cos{\left (3 x \right )}}{4} + \frac{x \sin{\left (2 x \right )} \cos{\left (x \right )} \cos{\left (3 x \right )}}{4} - \frac{x \sin{\left (3 x \right )} \cos{\left (x \right )} \cos{\left (2 x \right )}}{4} - \frac{3 \sin{\left (x \right )} \sin{\left (2 x \right )} \cos{\left (3 x \right )}}{8} + \frac{\sin{\left (x \right )} \sin{\left (3 x \right )} \cos{\left (2 x \right )}}{6} + \frac{\sin{\left (2 x \right )} \sin{\left (3 x \right )} \cos{\left (x \right )}}{24} \]

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

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GIAC/XCAS [A]  time = 0.199975, size = 18, normalized size = 0.72 \[ -\frac{4}{3} \, \sin \left (x\right )^{6} + \frac{3}{2} \, \sin \left (x\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/3*sin(x)^6 + 3/2*sin(x)^4