3.42 \(\int \cos (5 x) \sin (3 x) \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{4} \cos (2 x)-\frac{1}{16} \cos (8 x) \]

[Out]

Cos[2*x]/4 - Cos[8*x]/16

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Rubi [A]  time = 0.0150587, 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 \[ \frac{1}{4} \cos (2 x)-\frac{1}{16} \cos (8 x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[5*x]*Sin[3*x],x]

[Out]

Cos[2*x]/4 - Cos[8*x]/16

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Rubi in Sympy [A]  time = 1.04645, size = 12, normalized size = 0.71 \[ \frac{\cos{\left (2 x \right )}}{4} - \frac{\cos{\left (8 x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(5*x)*sin(3*x),x)

[Out]

cos(2*x)/4 - cos(8*x)/16

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Mathematica [A]  time = 0.0122521, size = 17, normalized size = 1. \[ \frac{\cos ^2(x)}{2}-\frac{1}{16} \cos (8 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[5*x]*Sin[3*x],x]

[Out]

Cos[x]^2/2 - Cos[8*x]/16

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Maple [A]  time = 0.104, size = 14, normalized size = 0.8 \[{\frac{\cos \left ( 2\,x \right ) }{4}}-{\frac{\cos \left ( 8\,x \right ) }{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(5*x)*sin(3*x),x)

[Out]

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

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Maxima [A]  time = 1.35434, size = 18, normalized size = 1.06 \[ -\frac{1}{16} \, \cos \left (8 \, x\right ) + \frac{1}{4} \, \cos \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(5*x)*sin(3*x),x, algorithm="maxima")

[Out]

-1/16*cos(8*x) + 1/4*cos(2*x)

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Fricas [A]  time = 0.291506, size = 34, normalized size = 2. \[ -8 \, \cos \left (x\right )^{8} + 16 \, \cos \left (x\right )^{6} - 10 \, \cos \left (x\right )^{4} + \frac{5}{2} \, \cos \left (x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(5*x)*sin(3*x),x, algorithm="fricas")

[Out]

-8*cos(x)^8 + 16*cos(x)^6 - 10*cos(x)^4 + 5/2*cos(x)^2

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Sympy [A]  time = 0.716914, size = 26, normalized size = 1.53 \[ \frac{5 \sin{\left (3 x \right )} \sin{\left (5 x \right )}}{16} + \frac{3 \cos{\left (3 x \right )} \cos{\left (5 x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(5*x)*sin(3*x),x)

[Out]

5*sin(3*x)*sin(5*x)/16 + 3*cos(3*x)*cos(5*x)/16

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GIAC/XCAS [A]  time = 0.198772, size = 18, normalized size = 1.06 \[ -\frac{1}{16} \, \cos \left (8 \, x\right ) + \frac{1}{4} \, \cos \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(5*x)*sin(3*x),x, algorithm="giac")

[Out]

-1/16*cos(8*x) + 1/4*cos(2*x)