3.67 \(\int \cos (x) \sin (\frac {\pi }{6}+x) \, dx\)

Optimal. Leaf size=20 \[ \frac {x}{4}-\frac {1}{4} \cos \left (2 x+\frac {\pi }{6}\right ) \]

[Out]

1/4*x-1/4*cos(1/6*Pi+2*x)

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4574, 2638} \[ \frac {x}{4}-\frac {1}{4} \cos \left (2 x+\frac {\pi }{6}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Sin[Pi/6 + x],x]

[Out]

x/4 - Cos[Pi/6 + 2*x]/4

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4574

Int[Cos[w_]^(q_.)*Sin[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sin[v]^p*Cos[w]^q, x], x] /; IGtQ[p, 0] &&
IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w],
x]))

Rubi steps

\begin {align*} \int \cos (x) \sin \left (\frac {\pi }{6}+x\right ) \, dx &=\int \left (\frac {1}{4}+\frac {1}{2} \sin \left (\frac {\pi }{6}+2 x\right )\right ) \, dx\\ &=\frac {x}{4}+\frac {1}{2} \int \sin \left (\frac {\pi }{6}+2 x\right ) \, dx\\ &=\frac {x}{4}-\frac {1}{4} \cos \left (\frac {\pi }{6}+2 x\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \[ \frac {x}{4}-\frac {1}{4} \cos \left (2 x+\frac {\pi }{6}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Sin[Pi/6 + x],x]

[Out]

x/4 - Cos[Pi/6 + 2*x]/4

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fricas [B]  time = 0.43, size = 31, normalized size = 1.55 \[ -\frac {1}{4} \, \sqrt {3} \cos \left (\frac {1}{6} \, \pi + x\right )^{2} - \frac {1}{4} \, \cos \left (\frac {1}{6} \, \pi + x\right ) \sin \left (\frac {1}{6} \, \pi + x\right ) + \frac {1}{4} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(1/6*pi+x),x, algorithm="fricas")

[Out]

-1/4*sqrt(3)*cos(1/6*pi + x)^2 - 1/4*cos(1/6*pi + x)*sin(1/6*pi + x) + 1/4*x

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giac [A]  time = 0.89, size = 14, normalized size = 0.70 \[ \frac {1}{4} \, x - \frac {1}{4} \, \cos \left (\frac {1}{6} \, \pi + 2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(1/6*pi+x),x, algorithm="giac")

[Out]

1/4*x - 1/4*cos(1/6*pi + 2*x)

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maple [A]  time = 0.08, size = 15, normalized size = 0.75 \[ \frac {x}{4}-\frac {\cos \left (2 x +\frac {\pi }{6}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*sin(1/6*Pi+x),x)

[Out]

1/4*x-1/4*cos(1/6*Pi+2*x)

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maxima [A]  time = 0.44, size = 14, normalized size = 0.70 \[ \frac {1}{4} \, x - \frac {1}{4} \, \cos \left (\frac {1}{6} \, \pi + 2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(1/6*pi+x),x, algorithm="maxima")

[Out]

1/4*x - 1/4*cos(1/6*pi + 2*x)

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mupad [B]  time = 0.15, size = 18, normalized size = 0.90 \[ \frac {x\,\sin \left (\frac {\Pi }{6}\right )}{2}-\frac {\cos \left (\frac {\Pi }{6}+2\,x\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*sin(Pi/6 + x),x)

[Out]

(x*sin(Pi/6))/2 - cos(Pi/6 + 2*x)/4

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sympy [B]  time = 0.56, size = 37, normalized size = 1.85 \[ - \frac {x \sin {\relax (x )} \cos {\left (x + \frac {\pi }{6} \right )}}{2} + \frac {x \sin {\left (x + \frac {\pi }{6} \right )} \cos {\relax (x )}}{2} - \frac {\cos {\relax (x )} \cos {\left (x + \frac {\pi }{6} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(1/6*pi+x),x)

[Out]

-x*sin(x)*cos(x + pi/6)/2 + x*sin(x + pi/6)*cos(x)/2 - cos(x)*cos(x + pi/6)/2

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