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 = 0.0228637, 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 \[ -\frac{1}{2} x \sin (a)-\frac{1}{4} \cos (a+2 x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[a + x]*Sin[x],x]

[Out]

-Cos[a + 2*x]/4 - (x*Sin[a])/2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0145576, 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 = 0.006, size = 15, normalized size = 0.8 \[ -{\frac{\cos \left ( a+2\,x \right ) }{4}}-{\frac{x\sin \left ( a \right ) }{2}} \]

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 = 1.39566, size = 19, normalized size = 1.06 \[ -\frac{1}{2} \, x \sin \left (a\right ) - \frac{1}{4} \, \cos \left (a + 2 \, x\right ) \]

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 = 0.231597, size = 38, normalized size = 2.11 \[ -\frac{1}{2} \, \cos \left (a + x\right )^{2} \cos \left (a\right ) - \frac{1}{2} \, \cos \left (a + x\right ) \sin \left (a + x\right ) \sin \left (a\right ) - \frac{1}{2} \, x \sin \left (a\right ) \]

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 [A]  time = 0.740763, size = 32, normalized size = 1.78 \[ \frac{x \sin{\left (x \right )} \cos{\left (a + x \right )}}{2} - \frac{x \sin{\left (a + x \right )} \cos{\left (x \right )}}{2} - \frac{\cos{\left (x \right )} \cos{\left (a + x \right )}}{2} \]

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/XCAS [A]  time = 0.199018, size = 19, normalized size = 1.06 \[ -\frac{1}{2} \, x \sin \left (a\right ) - \frac{1}{4} \, \cos \left (a + 2 \, x\right ) \]

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)