3.70 \(\int \cos (2 x) \sin (x) \, dx\)

Optimal. Leaf size=15 \[ \frac{\cos (x)}{2}-\frac{1}{6} \cos (3 x) \]

[Out]

Cos[x]/2 - Cos[3*x]/6

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Rubi [A]  time = 0.0081673, antiderivative size = 15, 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, Rules used = {4284} \[ \frac{\cos (x)}{2}-\frac{1}{6} \cos (3 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[x]/2 - Cos[3*x]/6

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 (2 x) \sin (x) \, dx &=\frac{\cos (x)}{2}-\frac{1}{6} \cos (3 x)\\ \end{align*}

Mathematica [A]  time = 0.0051229, size = 15, normalized size = 1. \[ \frac{\cos (x)}{2}-\frac{1}{6} \cos (3 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[x]/2 - Cos[3*x]/6

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Maple [A]  time = 0.013, size = 12, normalized size = 0.8 \begin{align*}{\frac{\cos \left ( x \right ) }{2}}-{\frac{\cos \left ( 3\,x \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cos(x)-1/6*cos(3*x)

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Maxima [A]  time = 0.984009, size = 15, normalized size = 1. \begin{align*} -\frac{1}{6} \, \cos \left (3 \, x\right ) + \frac{1}{2} \, \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*cos(3*x) + 1/2*cos(x)

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Fricas [A]  time = 2.22676, size = 32, normalized size = 2.13 \begin{align*} -\frac{2}{3} \, \cos \left (x\right )^{3} + \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*cos(x)^3 + cos(x)

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Sympy [A]  time = 1.86378, size = 20, normalized size = 1.33 \begin{align*} \frac{2 \sin{\left (x \right )} \sin{\left (2 x \right )}}{3} + \frac{\cos{\left (x \right )} \cos{\left (2 x \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10839, size = 15, normalized size = 1. \begin{align*} -\frac{1}{6} \, \cos \left (3 \, x\right ) + \frac{1}{2} \, \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*cos(3*x) + 1/2*cos(x)