3.107 \(\int \cos (x) \sin (3 x) \, dx\)

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

[Out]

-Cos[2*x]/4 - Cos[4*x]/8

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Rubi [A]  time = 0.0153035, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{1}{4} \cos (2 x)-\frac{1}{8} \cos (4 x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]*Sin[3*x],x]

[Out]

-Cos[2*x]/4 - Cos[4*x]/8

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Rubi in Sympy [A]  time = 0.996206, size = 14, normalized size = 0.82 \[ - \frac{\cos{\left (2 x \right )}}{4} - \frac{\cos{\left (4 x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-cos(2*x)/4 - cos(4*x)/8

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Mathematica [A]  time = 0.00944942, size = 17, normalized size = 1. \[ -\frac{1}{2} \cos ^2(x)-\frac{1}{8} \cos (4 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]*Sin[3*x],x]

[Out]

-Cos[x]^2/2 - Cos[4*x]/8

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Maple [A]  time = 0.046, size = 14, normalized size = 0.8 \[ - \left ( \cos \left ( x \right ) \right ) ^{4}+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)*sin(3*x),x)

[Out]

-cos(x)^4+1/2*cos(x)^2

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Maxima [A]  time = 1.36308, size = 18, normalized size = 1.06 \[ -\frac{1}{8} \, \cos \left (4 \, x\right ) - \frac{1}{4} \, \cos \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*cos(4*x) - 1/4*cos(2*x)

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Fricas [A]  time = 0.228532, size = 18, normalized size = 1.06 \[ -\cos \left (x\right )^{4} + \frac{1}{2} \, \cos \left (x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-cos(x)^4 + 1/2*cos(x)^2

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Sympy [A]  time = 0.727638, size = 22, normalized size = 1.29 \[ - \frac{\sin{\left (x \right )} \sin{\left (3 x \right )}}{8} - \frac{3 \cos{\left (x \right )} \cos{\left (3 x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)*sin(3*x),x)

[Out]

-sin(x)*sin(3*x)/8 - 3*cos(x)*cos(3*x)/8

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GIAC/XCAS [A]  time = 0.199836, size = 18, normalized size = 1.06 \[ -\frac{1}{8} \, \cos \left (4 \, x\right ) - \frac{1}{4} \, \cos \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*cos(4*x) - 1/4*cos(2*x)