3.27 \(\int x \cos (x) \sin (x) \, dx\)

Optimal. Leaf size=23 \[ -\frac{x}{4}+\frac{1}{2} x \sin ^2(x)+\frac{1}{4} \sin (x) \cos (x) \]

[Out]

-x/4 + (Cos[x]*Sin[x])/4 + (x*Sin[x]^2)/2

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Rubi [A]  time = 0.0204367, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{x}{4}+\frac{1}{2} x \sin ^2(x)+\frac{1}{4} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]  Int[x*Cos[x]*Sin[x],x]

[Out]

-x/4 + (Cos[x]*Sin[x])/4 + (x*Sin[x]^2)/2

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Rubi in Sympy [A]  time = 1.03985, size = 19, normalized size = 0.83 \[ \frac{x \sin ^{2}{\left (x \right )}}{2} - \frac{x}{4} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sin(x)**2/2 - x/4 + sin(x)*cos(x)/4

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Mathematica [A]  time = 0.00437193, size = 18, normalized size = 0.78 \[ \frac{1}{8} \sin (2 x)-\frac{1}{4} x \cos (2 x) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Cos[x]*Sin[x],x]

[Out]

-(x*Cos[2*x])/4 + Sin[2*x]/8

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Maple [A]  time = 0.004, size = 18, normalized size = 0.8 \[ -{\frac{x \left ( \cos \left ( x \right ) \right ) ^{2}}{2}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{4}}+{\frac{x}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*cos(x)*sin(x),x)

[Out]

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

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Maxima [A]  time = 1.39561, size = 19, normalized size = 0.83 \[ -\frac{1}{4} \, x \cos \left (2 \, x\right ) + \frac{1}{8} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*x*cos(2*x) + 1/8*sin(2*x)

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Fricas [A]  time = 0.225672, size = 23, normalized size = 1. \[ -\frac{1}{2} \, x \cos \left (x\right )^{2} + \frac{1}{4} \, \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{4} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.412793, size = 24, normalized size = 1.04 \[ \frac{x \sin ^{2}{\left (x \right )}}{4} - \frac{x \cos ^{2}{\left (x \right )}}{4} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*cos(x)*sin(x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.213684, size = 19, normalized size = 0.83 \[ -\frac{1}{4} \, x \cos \left (2 \, x\right ) + \frac{1}{8} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*x*cos(2*x) + 1/8*sin(2*x)