3.3.34 \(\int \cos (a+x) \sin (x) \, dx\) [234]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 = {4670, 2718} \begin {gather*} -\frac {1}{2} x \sin (a)-\frac {1}{4} \cos (a+2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + x]*Sin[x],x]

[Out]

-1/4*Cos[a + 2*x] - (x*Sin[a])/2

Rule 2718

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

Rule 4670

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 (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*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{4} (-\cos (a+2 x)-2 x \sin (a)) \end {gather*}

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 = 0.05, size = 15, normalized size = 0.83

method	result	size
default	\(-\frac {\cos \left (a +2 x \right )}{4}-\frac {x \sin \left (a \right )}{2}\)	\(15\)
risch	\(-\frac {\cos \left (a +2 x \right )}{4}-\frac {x \sin \left (a \right )}{2}\)	\(15\)
meijerg	\(\frac {\cos \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (2 x \right )}{\sqrt {\pi }}\right )}{4}-\frac {\sin \left (a \right ) \sqrt {\pi }\, \left (\frac {2 x}{\sqrt {\pi }}-\frac {\sin \left (2 x \right )}{\sqrt {\pi }}\right )}{4}\)	\(45\)
norman	\(\frac {x \tan \left (\frac {x}{2}\right )+x \left (\tan ^{2}\left (\frac {x}{2}\right )\right ) \tan \left (\frac {a}{2}+\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right ) \tan \left (\frac {a}{2}+\frac {x}{2}\right )-x \tan \left (\frac {a}{2}+\frac {x}{2}\right )-x \tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {a}{2}+\frac {x}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {a}{2}+\frac {x}{2}\right )\right ) \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a+x)*sin(x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 2.22, size = 14, normalized size = 0.78 \begin {gather*} -\frac {1}{2} \, x \sin \left (a\right ) - \frac {1}{4} \, \cos \left (a + 2 \, x\right ) \end {gather*}

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 = 0.99, size = 28, normalized size = 1.56 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).
time = 0.13, size = 32, normalized size = 1.78 \begin {gather*} \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 {\cos {\left (x \right )} \cos {\left (a + x \right )}}{2} \end {gather*}

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 - cos(x)*cos(a + x)/2

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Giac [A]
time = 0.46, size = 14, normalized size = 0.78 \begin {gather*} -\frac {1}{2} \, x \sin \left (a\right ) - \frac {1}{4} \, \cos \left (a + 2 \, x\right ) \end {gather*}

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)

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Mupad [B]
time = 0.02, size = 14, normalized size = 0.78 \begin {gather*} -\frac {\cos \left (a+2\,x\right )}{4}-\frac {x\,\sin \left (a\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + x)*sin(x),x)

[Out]

- cos(a + 2*x)/4 - (x*sin(a))/2

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