3.229 \(\int \cos ^2(x) \sin (3+2 x) \, dx\)

Optimal. Leaf size=28 \[ \frac{1}{4} x \sin (3)-\frac{1}{4} \cos (2 x+3)-\frac{1}{16} \cos (4 x+3) \]

[Out]

-Cos[3 + 2*x]/4 - Cos[3 + 4*x]/16 + (x*Sin[3])/4

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Rubi [A]  time = 0.0381084, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{4} x \sin (3)-\frac{1}{4} \cos (2 x+3)-\frac{1}{16} \cos (4 x+3) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^2*Sin[3 + 2*x],x]

[Out]

-Cos[3 + 2*x]/4 - Cos[3 + 4*x]/16 + (x*Sin[3])/4

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\cos{\left (2 x + 3 \right )}}{4} - \frac{\cos{\left (4 x + 3 \right )}}{16} + \sin{\left (3 \right )} \int \frac{1}{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**2*sin(3+2*x),x)

[Out]

-cos(2*x + 3)/4 - cos(4*x + 3)/16 + sin(3)*Integral(1/4, x)

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Mathematica [A]  time = 0.0142453, size = 28, normalized size = 1. \[ \frac{1}{4} x \sin (3)-\frac{1}{4} \cos (2 x+3)-\frac{1}{16} \cos (4 x+3) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^2*Sin[3 + 2*x],x]

[Out]

-Cos[3 + 2*x]/4 - Cos[3 + 4*x]/16 + (x*Sin[3])/4

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Maple [A]  time = 0.008, size = 23, normalized size = 0.8 \[ -{\frac{\cos \left ( 3+2\,x \right ) }{4}}-{\frac{\cos \left ( 3+4\,x \right ) }{16}}+{\frac{x\sin \left ( 3 \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^2*sin(3+2*x),x)

[Out]

-1/4*cos(3+2*x)-1/16*cos(3+4*x)+1/4*x*sin(3)

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Maxima [A]  time = 1.33387, size = 30, normalized size = 1.07 \[ \frac{1}{4} \, x \sin \left (3\right ) - \frac{1}{16} \, \cos \left (4 \, x + 3\right ) - \frac{1}{4} \, \cos \left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*sin(2*x + 3),x, algorithm="maxima")

[Out]

1/4*x*sin(3) - 1/16*cos(4*x + 3) - 1/4*cos(2*x + 3)

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Fricas [A]  time = 0.227144, size = 43, normalized size = 1.54 \[ -\frac{1}{2} \, \cos \left (3\right ) \cos \left (x\right )^{4} + \frac{1}{4} \, x \sin \left (3\right ) + \frac{1}{4} \,{\left (2 \, \cos \left (x\right )^{3} \sin \left (3\right ) + \cos \left (x\right ) \sin \left (3\right )\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*sin(2*x + 3),x, algorithm="fricas")

[Out]

-1/2*cos(3)*cos(x)^4 + 1/4*x*sin(3) + 1/4*(2*cos(x)^3*sin(3) + cos(x)*sin(3))*si
n(x)

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Sympy [A]  time = 3.05223, size = 76, normalized size = 2.71 \[ - \frac{x \sin ^{2}{\left (x \right )} \sin{\left (2 x + 3 \right )}}{4} - \frac{x \sin{\left (x \right )} \cos{\left (x \right )} \cos{\left (2 x + 3 \right )}}{2} + \frac{x \sin{\left (2 x + 3 \right )} \cos ^{2}{\left (x \right )}}{4} - \frac{\sin ^{2}{\left (x \right )} \cos{\left (2 x + 3 \right )}}{2} + \frac{3 \sin{\left (x \right )} \sin{\left (2 x + 3 \right )} \cos{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**2*sin(3+2*x),x)

[Out]

-x*sin(x)**2*sin(2*x + 3)/4 - x*sin(x)*cos(x)*cos(2*x + 3)/2 + x*sin(2*x + 3)*co
s(x)**2/4 - sin(x)**2*cos(2*x + 3)/2 + 3*sin(x)*sin(2*x + 3)*cos(x)/4

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GIAC/XCAS [A]  time = 0.211553, size = 30, normalized size = 1.07 \[ \frac{1}{4} \, x \sin \left (3\right ) - \frac{1}{16} \, \cos \left (4 \, x + 3\right ) - \frac{1}{4} \, \cos \left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*sin(2*x + 3),x, algorithm="giac")

[Out]

1/4*x*sin(3) - 1/16*cos(4*x + 3) - 1/4*cos(2*x + 3)