3.109 \(\int \sin (3 x) \sin (6 x) \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{6} \sin (3 x)-\frac{1}{18} \sin (9 x) \]

[Out]

Sin[3*x]/6 - Sin[9*x]/18

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Rubi [A]  time = 0.0160628, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{6} \sin (3 x)-\frac{1}{18} \sin (9 x) \]

Antiderivative was successfully verified.

[In]  Int[Sin[3*x]*Sin[6*x],x]

[Out]

Sin[3*x]/6 - Sin[9*x]/18

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Rubi in Sympy [A]  time = 1.02061, size = 12, normalized size = 0.71 \[ \frac{\sin{\left (3 x \right )}}{6} - \frac{\sin{\left (9 x \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(3*x)*sin(6*x),x)

[Out]

sin(3*x)/6 - sin(9*x)/18

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Mathematica [A]  time = 0.0112272, size = 17, normalized size = 1. \[ \frac{1}{6} \sin (3 x)-\frac{1}{18} \sin (9 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[3*x]*Sin[6*x],x]

[Out]

Sin[3*x]/6 - Sin[9*x]/18

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Maple [A]  time = 0.018, size = 9, normalized size = 0.5 \[{\frac{2\, \left ( \sin \left ( 3\,x \right ) \right ) ^{3}}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(3*x)*sin(6*x),x)

[Out]

2/9*sin(3*x)^3

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Maxima [A]  time = 1.34152, size = 18, normalized size = 1.06 \[ -\frac{1}{18} \, \sin \left (9 \, x\right ) + \frac{1}{6} \, \sin \left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(6*x)*sin(3*x),x, algorithm="maxima")

[Out]

-1/18*sin(9*x) + 1/6*sin(3*x)

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Fricas [A]  time = 0.215392, size = 19, normalized size = 1.12 \[ -\frac{2}{9} \,{\left (\cos \left (3 \, x\right )^{2} - 1\right )} \sin \left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(6*x)*sin(3*x),x, algorithm="fricas")

[Out]

-2/9*(cos(3*x)^2 - 1)*sin(3*x)

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Sympy [A]  time = 0.772494, size = 24, normalized size = 1.41 \[ - \frac{2 \sin{\left (3 x \right )} \cos{\left (6 x \right )}}{9} + \frac{\sin{\left (6 x \right )} \cos{\left (3 x \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(3*x)*sin(6*x),x)

[Out]

-2*sin(3*x)*cos(6*x)/9 + sin(6*x)*cos(3*x)/9

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GIAC/XCAS [A]  time = 0.202366, size = 18, normalized size = 1.06 \[ -\frac{1}{18} \, \sin \left (9 \, x\right ) + \frac{1}{6} \, \sin \left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(6*x)*sin(3*x),x, algorithm="giac")

[Out]

-1/18*sin(9*x) + 1/6*sin(3*x)