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]

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

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Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4574, 2638} \[ \frac {1}{4} x \sin (3)-\frac {1}{4} \cos (2 x+3)-\frac {1}{16} \cos (4 x+3) \]

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

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 ^2(x) \sin (3+2 x) \, dx &=\int \left (\frac {\sin (3)}{4}+\frac {1}{2} \sin (3+2 x)+\frac {1}{4} \sin (3+4 x)\right ) \, dx\\ &=\frac {1}{4} x \sin (3)+\frac {1}{4} \int \sin (3+4 x) \, dx+\frac {1}{2} \int \sin (3+2 x) \, dx\\ &=-\frac {1}{4} \cos (3+2 x)-\frac {1}{16} \cos (3+4 x)+\frac {1}{4} x \sin (3)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 1.00 \[ \frac {1}{4} x \sin (3)-\frac {1}{4} \cos (2 x+3)-\frac {1}{16} \cos (4 x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^2*Sin[3 + 2*x],x]

[Out]

-1/4*Cos[3 + 2*x] - Cos[3 + 4*x]/16 + (x*Sin[3])/4

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fricas [A]  time = 0.45, size = 32, normalized size = 1.14 \[ -\frac {1}{2} \, \cos \relax (3) \cos \relax (x)^{4} + \frac {1}{4} \, x \sin \relax (3) + \frac {1}{4} \, {\left (2 \, \cos \relax (x)^{3} \sin \relax (3) + \cos \relax (x) \sin \relax (3)\right )} \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*sin(3+2*x),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))*sin(x)

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giac [A]  time = 1.44, size = 22, normalized size = 0.79 \[ \frac {1}{4} \, x \sin \relax (3) - \frac {1}{16} \, \cos \left (4 \, x + 3\right ) - \frac {1}{4} \, \cos \left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 23, normalized size = 0.82 \[ \frac {\sin \relax (3) x}{4}-\frac {\cos \left (2 x +3\right )}{4}-\frac {\cos \left (4 x +3\right )}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2*sin(2*x+3),x)

[Out]

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

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maxima [A]  time = 0.47, size = 22, normalized size = 0.79 \[ \frac {1}{4} \, x \sin \relax (3) - \frac {1}{16} \, \cos \left (4 \, x + 3\right ) - \frac {1}{4} \, \cos \left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*sin(3+2*x),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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mupad [B]  time = 0.33, size = 22, normalized size = 0.79 \[ \frac {x\,\sin \relax (3)}{4}-\frac {\cos \left (4\,x+3\right )}{16}-\frac {\cos \left (2\,x+3\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x + 3)*cos(x)^2,x)

[Out]

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

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sympy [B]  time = 2.20, size = 75, normalized size = 2.68 \[ - \frac {x \sin ^{2}{\relax (x )} \sin {\left (2 x + 3 \right )}}{4} - \frac {x \sin {\relax (x )} \cos {\relax (x )} \cos {\left (2 x + 3 \right )}}{2} + \frac {x \sin {\left (2 x + 3 \right )} \cos ^{2}{\relax (x )}}{4} - \frac {\sin {\relax (x )} \sin {\left (2 x + 3 \right )} \cos {\relax (x )}}{4} - \frac {\cos ^{2}{\relax (x )} \cos {\left (2 x + 3 \right )}}{2} \]

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

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