3.106 \(\int \sin (2 x) \sin (5 x) \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{6} \sin (3 x)-\frac{1}{14} \sin (7 x) \]

[Out]

Sin[3*x]/6 - Sin[7*x]/14

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Rubi [A]  time = 0.0149131, 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 \[ \frac{1}{6} \sin (3 x)-\frac{1}{14} \sin (7 x) \]

Antiderivative was successfully verified.

[In]  Int[Sin[2*x]*Sin[5*x],x]

[Out]

Sin[3*x]/6 - Sin[7*x]/14

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Rubi in Sympy [A]  time = 1.01965, size = 12, normalized size = 0.71 \[ \frac{\sin{\left (3 x \right )}}{6} - \frac{\sin{\left (7 x \right )}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(2*x)*sin(5*x),x)

[Out]

sin(3*x)/6 - sin(7*x)/14

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Mathematica [A]  time = 0.0112471, size = 17, normalized size = 1. \[ \frac{1}{6} \sin (3 x)-\frac{1}{14} \sin (7 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[2*x]*Sin[5*x],x]

[Out]

Sin[3*x]/6 - Sin[7*x]/14

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Maple [A]  time = 0.059, size = 14, normalized size = 0.8 \[{\frac{\sin \left ( 3\,x \right ) }{6}}-{\frac{\sin \left ( 7\,x \right ) }{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(2*x)*sin(5*x),x)

[Out]

1/6*sin(3*x)-1/14*sin(7*x)

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Maxima [A]  time = 1.39357, size = 18, normalized size = 1.06 \[ -\frac{1}{14} \, \sin \left (7 \, x\right ) + \frac{1}{6} \, \sin \left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(5*x)*sin(2*x),x, algorithm="maxima")

[Out]

-1/14*sin(7*x) + 1/6*sin(3*x)

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Fricas [A]  time = 0.226308, size = 32, normalized size = 1.88 \[ -\frac{2}{21} \,{\left (48 \, \cos \left (x\right )^{6} - 60 \, \cos \left (x\right )^{4} + 11 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(5*x)*sin(2*x),x, algorithm="fricas")

[Out]

-2/21*(48*cos(x)^6 - 60*cos(x)^4 + 11*cos(x)^2 + 1)*sin(x)

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Sympy [A]  time = 0.717103, size = 26, normalized size = 1.53 \[ - \frac{5 \sin{\left (2 x \right )} \cos{\left (5 x \right )}}{21} + \frac{2 \sin{\left (5 x \right )} \cos{\left (2 x \right )}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(2*x)*sin(5*x),x)

[Out]

-5*sin(2*x)*cos(5*x)/21 + 2*sin(5*x)*cos(2*x)/21

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GIAC/XCAS [A]  time = 0.19943, size = 18, normalized size = 1.06 \[ -\frac{1}{14} \, \sin \left (7 \, x\right ) + \frac{1}{6} \, \sin \left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(5*x)*sin(2*x),x, algorithm="giac")

[Out]

-1/14*sin(7*x) + 1/6*sin(3*x)