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

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 = 0.0086686, 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[x]*Sin[2*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 (x) \sin (2 x) \, dx &=-\frac{\cos (x)}{2}-\frac{1}{6} \cos (3 x)\\ \end{align*}

Mathematica [A]  time = 0.0050567, size = 15, normalized size = 1. \[ -\frac{\cos (x)}{2}-\frac{1}{6} \cos (3 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[x]/2 - Cos[3*x]/6

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Maple [A]  time = 0.009, size = 7, normalized size = 0.5 \begin{align*} -{\frac{2\, \left ( \cos \left ( x \right ) \right ) ^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*cos(x)^3

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Maxima [A]  time = 0.994978, 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(x)*sin(2*x),x, algorithm="maxima")

[Out]

-1/6*cos(3*x) - 1/2*cos(x)

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Fricas [A]  time = 2.29816, size = 20, normalized size = 1.33 \begin{align*} -\frac{2}{3} \, \cos \left (x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*cos(x)^3

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Sympy [A]  time = 0.891557, size = 22, normalized size = 1.47 \begin{align*} - \frac{\sin{\left (x \right )} \sin{\left (2 x \right )}}{3} - \frac{2 \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(x)*sin(2*x),x)

[Out]

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

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Giac [A]  time = 1.12472, 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(x)*sin(2*x),x, algorithm="giac")

[Out]

-1/6*cos(3*x) - 1/2*cos(x)