3.131 \(\int \cos (3 x) \sin (2 x) \, dx\)

Optimal. Leaf size=15 \[ \frac{\cos (x)}{2}-\frac{1}{10} \cos (5 x) \]

[Out]

Cos[x]/2 - Cos[5*x]/10

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Rubi [A]  time = 0.0076803, antiderivative size = 15, 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, Rules used = {4284} \[ \frac{\cos (x)}{2}-\frac{1}{10} \cos (5 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[3*x]*Sin[2*x],x]

[Out]

Cos[x]/2 - Cos[5*x]/10

Rule 4284

Int[cos[(c_.) + (d_.)*(x_)]*sin[(a_.) + (b_.)*(x_)], x_Symbol] :> -Simp[Cos[a - c + (b - d)*x]/(2*(b - d)), x]
- Simp[Cos[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps

\begin{align*} \int \cos (3 x) \sin (2 x) \, dx &=\frac{\cos (x)}{2}-\frac{1}{10} \cos (5 x)\\ \end{align*}

Mathematica [A]  time = 0.0061278, size = 15, normalized size = 1. \[ \frac{\cos (x)}{2}-\frac{1}{10} \cos (5 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[3*x]*Sin[2*x],x]

[Out]

Cos[x]/2 - Cos[5*x]/10

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Maple [A]  time = 0.036, size = 12, normalized size = 0.8 \begin{align*}{\frac{\cos \left ( x \right ) }{2}}-{\frac{\cos \left ( 5\,x \right ) }{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3*x)*sin(2*x),x)

[Out]

1/2*cos(x)-1/10*cos(5*x)

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Maxima [A]  time = 0.931286, size = 15, normalized size = 1. \begin{align*} -\frac{1}{10} \, \cos \left (5 \, x\right ) + \frac{1}{2} \, \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*sin(2*x),x, algorithm="maxima")

[Out]

-1/10*cos(5*x) + 1/2*cos(x)

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Fricas [A]  time = 1.90517, size = 38, normalized size = 2.53 \begin{align*} -\frac{8}{5} \, \cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*sin(2*x),x, algorithm="fricas")

[Out]

-8/5*cos(x)^5 + 2*cos(x)^3

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Sympy [B]  time = 0.525338, size = 26, normalized size = 1.73 \begin{align*} \frac{3 \sin{\left (2 x \right )} \sin{\left (3 x \right )}}{5} + \frac{2 \cos{\left (2 x \right )} \cos{\left (3 x \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*sin(2*x),x)

[Out]

3*sin(2*x)*sin(3*x)/5 + 2*cos(2*x)*cos(3*x)/5

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Giac [A]  time = 1.09231, size = 15, normalized size = 1. \begin{align*} -\frac{1}{10} \, \cos \left (5 \, x\right ) + \frac{1}{2} \, \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*sin(2*x),x, algorithm="giac")

[Out]

-1/10*cos(5*x) + 1/2*cos(x)