3.38 \(\int x \cos (2 x) \, dx\)

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

[Out]

Cos[2*x]/4 + (x*Sin[2*x])/2

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Rubi [A]  time = 0.0191007, antiderivative size = 18, 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 \[ \frac{1}{2} x \sin (2 x)+\frac{1}{4} \cos (2 x) \]

Antiderivative was successfully verified.

[In]  Int[x*Cos[2*x],x]

[Out]

Cos[2*x]/4 + (x*Sin[2*x])/2

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Rubi in Sympy [A]  time = 0.850406, size = 14, normalized size = 0.78 \[ \frac{x \sin{\left (2 x \right )}}{2} + \frac{\cos{\left (2 x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*cos(2*x),x)

[Out]

x*sin(2*x)/2 + cos(2*x)/4

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Mathematica [A]  time = 0.00491014, size = 18, normalized size = 1. \[ \frac{1}{2} x \sin (2 x)+\frac{1}{4} \cos (2 x) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Cos[2*x],x]

[Out]

Cos[2*x]/4 + (x*Sin[2*x])/2

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Maple [A]  time = 0.007, size = 15, normalized size = 0.8 \[{\frac{\cos \left ( 2\,x \right ) }{4}}+{\frac{x\sin \left ( 2\,x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*cos(2*x),x)

[Out]

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

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Maxima [A]  time = 1.34922, size = 19, normalized size = 1.06 \[ \frac{1}{2} \, x \sin \left (2 \, x\right ) + \frac{1}{4} \, \cos \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*cos(2*x),x, algorithm="maxima")

[Out]

1/2*x*sin(2*x) + 1/4*cos(2*x)

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Fricas [A]  time = 0.22197, size = 19, normalized size = 1.06 \[ \frac{1}{2} \, x \sin \left (2 \, x\right ) + \frac{1}{4} \, \cos \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*cos(2*x),x, algorithm="fricas")

[Out]

1/2*x*sin(2*x) + 1/4*cos(2*x)

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Sympy [A]  time = 0.188105, size = 14, normalized size = 0.78 \[ \frac{x \sin{\left (2 x \right )}}{2} + \frac{\cos{\left (2 x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*cos(2*x),x)

[Out]

x*sin(2*x)/2 + cos(2*x)/4

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GIAC/XCAS [A]  time = 0.232345, size = 19, normalized size = 1.06 \[ \frac{1}{2} \, x \sin \left (2 \, x\right ) + \frac{1}{4} \, \cos \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*cos(2*x),x, algorithm="giac")

[Out]

1/2*x*sin(2*x) + 1/4*cos(2*x)