3.43 \(\int \sin (2 x) \sin (4 x) \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{4} \sin (2 x)-\frac{1}{12} \sin (6 x) \]

[Out]

Sin[2*x]/4 - Sin[6*x]/12

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Rubi [A]  time = 0.0153777, 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 \[ \frac{1}{4} \sin (2 x)-\frac{1}{12} \sin (6 x) \]

Antiderivative was successfully verified.

[In]  Int[Sin[2*x]*Sin[4*x],x]

[Out]

Sin[2*x]/4 - Sin[6*x]/12

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Rubi in Sympy [A]  time = 1.01537, size = 12, normalized size = 0.71 \[ \frac{\sin{\left (2 x \right )}}{4} - \frac{\sin{\left (6 x \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(2*x)*sin(4*x),x)

[Out]

sin(2*x)/4 - sin(6*x)/12

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Mathematica [A]  time = 0.0104433, size = 17, normalized size = 1. \[ \frac{1}{4} \sin (2 x)-\frac{1}{12} \sin (6 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[2*x]*Sin[4*x],x]

[Out]

Sin[2*x]/4 - Sin[6*x]/12

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Maple [A]  time = 0.014, size = 9, normalized size = 0.5 \[{\frac{ \left ( \sin \left ( 2\,x \right ) \right ) ^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(2*x)*sin(4*x),x)

[Out]

1/3*sin(2*x)^3

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Maxima [A]  time = 1.33073, size = 18, normalized size = 1.06 \[ -\frac{1}{12} \, \sin \left (6 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(4*x)*sin(2*x),x, algorithm="maxima")

[Out]

-1/12*sin(6*x) + 1/4*sin(2*x)

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Fricas [A]  time = 0.237042, size = 19, normalized size = 1.12 \[ -\frac{1}{3} \,{\left (\cos \left (2 \, x\right )^{2} - 1\right )} \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(4*x)*sin(2*x),x, algorithm="fricas")

[Out]

-1/3*(cos(2*x)^2 - 1)*sin(2*x)

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Sympy [A]  time = 0.777018, size = 22, normalized size = 1.29 \[ - \frac{\sin{\left (2 x \right )} \cos{\left (4 x \right )}}{3} + \frac{\sin{\left (4 x \right )} \cos{\left (2 x \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(2*x)*sin(4*x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.201293, size = 18, normalized size = 1.06 \[ -\frac{1}{12} \, \sin \left (6 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(4*x)*sin(2*x),x, algorithm="giac")

[Out]

-1/12*sin(6*x) + 1/4*sin(2*x)