3.217 \(\int x \sin (a+x) \, dx\)

Optimal. Leaf size=12 \[ \sin (a+x)-x \cos (a+x) \]

[Out]

-(x*Cos[a + x]) + Sin[a + x]

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Rubi [A]  time = 0.0155563, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \sin (a+x)-x \cos (a+x) \]

Antiderivative was successfully verified.

[In]  Int[x*Sin[a + x],x]

[Out]

-(x*Cos[a + x]) + Sin[a + x]

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Rubi in Sympy [A]  time = 0.875497, size = 10, normalized size = 0.83 \[ - x \cos{\left (a + x \right )} + \sin{\left (a + x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*sin(a+x),x)

[Out]

-x*cos(a + x) + sin(a + x)

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Mathematica [A]  time = 0.0182112, size = 12, normalized size = 1. \[ \sin (a+x)-x \cos (a+x) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sin[a + x],x]

[Out]

-(x*Cos[a + x]) + Sin[a + x]

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Maple [A]  time = 0.005, size = 21, normalized size = 1.8 \[ a\cos \left ( a+x \right ) +\sin \left ( a+x \right ) - \left ( a+x \right ) \cos \left ( a+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*sin(a+x),x)

[Out]

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

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Maxima [A]  time = 1.38148, size = 27, normalized size = 2.25 \[ -{\left (a + x\right )} \cos \left (a + x\right ) + a \cos \left (a + x\right ) + \sin \left (a + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(a + x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.228348, size = 16, normalized size = 1.33 \[ -x \cos \left (a + x\right ) + \sin \left (a + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(a + x),x, algorithm="fricas")

[Out]

-x*cos(a + x) + sin(a + x)

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Sympy [A]  time = 0.186598, size = 10, normalized size = 0.83 \[ - x \cos{\left (a + x \right )} + \sin{\left (a + x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(a+x),x)

[Out]

-x*cos(a + x) + sin(a + x)

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GIAC/XCAS [A]  time = 0.198172, size = 16, normalized size = 1.33 \[ -x \cos \left (a + x\right ) + \sin \left (a + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(a + x),x, algorithm="giac")

[Out]

-x*cos(a + x) + sin(a + x)