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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\cos (x)}{2}-\frac {1}{6} \cos (3 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[x]/2 - Cos[3*x]/6

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fricas [A]  time = 0.78, size = 9, normalized size = 0.60 \[ -\frac {2}{3} \, \cos \relax (x)^{3} + \cos \relax (x) \]

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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giac [A]  time = 0.13, size = 11, normalized size = 0.73 \[ -\frac {1}{6} \, \cos \left (3 \, x\right ) + \frac {1}{2} \, \cos \relax (x) \]

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)

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maple [A]  time = 0.07, size = 12, normalized size = 0.80 \[ \frac {\cos \relax (x )}{2}-\frac {\cos \left (3 x \right )}{6} \]

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 = 0.31, size = 11, normalized size = 0.73 \[ -\frac {1}{6} \, \cos \left (3 \, x\right ) + \frac {1}{2} \, \cos \relax (x) \]

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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mupad [B]  time = 0.02, size = 9, normalized size = 0.60 \[ \cos \relax (x)-\frac {2\,{\cos \relax (x)}^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(x) - (2*cos(x)^3)/3

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sympy [A]  time = 0.45, size = 20, normalized size = 1.33 \[ \frac {2 \sin {\relax (x )} \sin {\left (2 x \right )}}{3} + \frac {\cos {\relax (x )} \cos {\left (2 x \right )}}{3} \]

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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