3.234 \(\int \cos (a+x) \sin (x) \, dx\)

Optimal. Leaf size=18 \[ -\frac{1}{2} x \sin (a)-\frac{1}{4} \cos (a+2 x) \]

[Out]

-Cos[a + 2*x]/4 - (x*Sin[a])/2

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Rubi [A]  time = 0.0128961, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4574, 2638} \[ -\frac{1}{2} x \sin (a)-\frac{1}{4} \cos (a+2 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + x]*Sin[x],x]

[Out]

-Cos[a + 2*x]/4 - (x*Sin[a])/2

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]))

Rule 2638

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

Rubi steps

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

Mathematica [A]  time = 0.0123611, size = 18, normalized size = 1. \[ \frac{1}{4} (-2 x \sin (a)-\cos (a+2 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + x]*Sin[x],x]

[Out]

(-Cos[a + 2*x] - 2*x*Sin[a])/4

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Maple [A]  time = 0.015, size = 15, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( a+2\,x \right ) }{4}}-{\frac{x\sin \left ( a \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a+x)*sin(x),x)

[Out]

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

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Maxima [A]  time = 0.947506, size = 19, normalized size = 1.06 \begin{align*} -\frac{1}{2} \, x \sin \left (a\right ) - \frac{1}{4} \, \cos \left (a + 2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+x)*sin(x),x, algorithm="maxima")

[Out]

-1/2*x*sin(a) - 1/4*cos(a + 2*x)

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Fricas [A]  time = 1.78751, size = 103, normalized size = 5.72 \begin{align*} -\frac{1}{2} \, \cos \left (a + x\right )^{2} \cos \left (a\right ) - \frac{1}{2} \, \cos \left (a + x\right ) \sin \left (a + x\right ) \sin \left (a\right ) - \frac{1}{2} \, x \sin \left (a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+x)*sin(x),x, algorithm="fricas")

[Out]

-1/2*cos(a + x)^2*cos(a) - 1/2*cos(a + x)*sin(a + x)*sin(a) - 1/2*x*sin(a)

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Sympy [B]  time = 0.544237, size = 32, normalized size = 1.78 \begin{align*} \frac{x \sin{\left (x \right )} \cos{\left (a + x \right )}}{2} - \frac{x \sin{\left (a + x \right )} \cos{\left (x \right )}}{2} + \frac{\sin{\left (x \right )} \sin{\left (a + x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+x)*sin(x),x)

[Out]

x*sin(x)*cos(a + x)/2 - x*sin(a + x)*cos(x)/2 + sin(x)*sin(a + x)/2

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Giac [A]  time = 1.06884, size = 19, normalized size = 1.06 \begin{align*} -\frac{1}{2} \, x \sin \left (a\right ) - \frac{1}{4} \, \cos \left (a + 2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+x)*sin(x),x, algorithm="giac")

[Out]

-1/2*x*sin(a) - 1/4*cos(a + 2*x)