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

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

[Out]

x/4 - Cos[Pi/6 + 2*x]/4

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Rubi [A]  time = 0.026581, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x}{4}-\frac{1}{4} \cos \left (2 x+\frac{\pi }{6}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]*Sin[Pi/6 + x],x]

[Out]

x/4 - Cos[Pi/6 + 2*x]/4

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\cos{\left (2 x + \frac{\pi }{6} \right )}}{4} + \int \frac{1}{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)*sin(1/6*pi+x),x)

[Out]

-cos(2*x + pi/6)/4 + Integral(1/4, x)

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Mathematica [A]  time = 0.0118138, size = 20, normalized size = 1. \[ \frac{x}{4}-\frac{1}{4} \cos \left (2 x+\frac{\pi }{6}\right ) \]

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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Maple [A]  time = 0.014, size = 15, normalized size = 0.8 \[{\frac{x}{4}}-{\frac{1}{4}\cos \left ({\frac{\pi }{6}}+2\,x \right ) } \]

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 = 1.37012, size = 19, normalized size = 0.95 \[ \frac{1}{4} \, x - \frac{1}{4} \, \cos \left (\frac{1}{6} \, \pi + 2 \, x\right ) \]

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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Fricas [A]  time = 0.245362, size = 77, normalized size = 3.85 \[ -\frac{{\left (362 \, \sqrt{3} + 627\right )} \cos \left (\frac{1}{6} \, \pi + x\right )^{2} +{\left (209 \, \sqrt{3} + 362\right )} \cos \left (\frac{1}{6} \, \pi + x\right ) \sin \left (\frac{1}{6} \, \pi + x\right ) - 209 \, \sqrt{3} x - 362 \, x}{4 \,{\left (209 \, \sqrt{3} + 362\right )}} \]

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*((362*sqrt(3) + 627)*cos(1/6*pi + x)^2 + (209*sqrt(3) + 362)*cos(1/6*pi + x
)*sin(1/6*pi + x) - 209*sqrt(3)*x - 362*x)/(209*sqrt(3) + 362)

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Sympy [A]  time = 0.760257, size = 37, normalized size = 1.85 \[ - \frac{x \sin{\left (x \right )} \cos{\left (x + \frac{\pi }{6} \right )}}{2} + \frac{x \sin{\left (x + \frac{\pi }{6} \right )} \cos{\left (x \right )}}{2} - \frac{\cos{\left (x \right )} \cos{\left (x + \frac{\pi }{6} \right )}}{2} \]

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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GIAC/XCAS [A]  time = 0.20282, size = 19, normalized size = 0.95 \[ \frac{1}{4} \, x - \frac{1}{4} \, \cos \left (\frac{1}{6} \, \pi + 2 \, x\right ) \]

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)