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

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 = 0.0081178, 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, Rules used = {4284} \[ \frac{1}{4} \cos (2 x)-\frac{1}{8} \cos (4 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[2*x]/4 - Cos[4*x]/8

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 (x) \, dx &=\frac{1}{4} \cos (2 x)-\frac{1}{8} \cos (4 x)\\ \end{align*}

Mathematica [A]  time = 0.0055286, size = 17, normalized size = 1. \[ \frac{\cos ^2(x)}{2}-\frac{1}{8} \cos (4 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[x]^2/2 - Cos[4*x]/8

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Maple [A]  time = 0.035, size = 14, normalized size = 0.8 \begin{align*}{\frac{\cos \left ( 2\,x \right ) }{4}}-{\frac{\cos \left ( 4\,x \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.985178, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{8} \, \cos \left (4 \, x\right ) + \frac{1}{4} \, \cos \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*cos(4*x) + 1/4*cos(2*x)

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Fricas [A]  time = 2.2507, size = 35, normalized size = 2.06 \begin{align*} -\cos \left (x\right )^{4} + \frac{3}{2} \, \cos \left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)^4 + 3/2*cos(x)^2

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Sympy [A]  time = 0.896868, size = 20, normalized size = 1.18 \begin{align*} \frac{3 \sin{\left (x \right )} \sin{\left (3 x \right )}}{8} + \frac{\cos{\left (x \right )} \cos{\left (3 x \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14821, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{8} \, \cos \left (4 \, x\right ) + \frac{1}{4} \, \cos \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*cos(4*x) + 1/4*cos(2*x)