3.25 \(\int x^2 \sin (2 x) \, dx\)

Optimal. Leaf size=29 \[ -\frac{1}{2} x^2 \cos (2 x)+\frac{1}{2} x \sin (2 x)+\frac{1}{4} \cos (2 x) \]

[Out]

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

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Rubi [A]  time = 0.0352794, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{1}{2} x^2 \cos (2 x)+\frac{1}{2} x \sin (2 x)+\frac{1}{4} \cos (2 x) \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sin[2*x],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.46451, size = 24, normalized size = 0.83 \[ - \frac{x^{2} \cos{\left (2 x \right )}}{2} + \frac{x \sin{\left (2 x \right )}}{2} + \frac{\cos{\left (2 x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*sin(2*x),x)

[Out]

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

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Mathematica [A]  time = 0.0253343, size = 25, normalized size = 0.86 \[ \frac{1}{2} x \sin (2 x)-\frac{1}{4} \left (2 x^2-1\right ) \cos (2 x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sin[2*x],x]

[Out]

-((-1 + 2*x^2)*Cos[2*x])/4 + (x*Sin[2*x])/2

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Maple [A]  time = 0.007, size = 24, normalized size = 0.8 \[{\frac{\cos \left ( 2\,x \right ) }{4}}-{\frac{{x}^{2}\cos \left ( 2\,x \right ) }{2}}+{\frac{x\sin \left ( 2\,x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*sin(2*x),x)

[Out]

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

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Maxima [A]  time = 1.32902, size = 28, normalized size = 0.97 \[ -\frac{1}{4} \,{\left (2 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) + \frac{1}{2} \, x \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.223277, size = 28, normalized size = 0.97 \[ -\frac{1}{4} \,{\left (2 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) + \frac{1}{2} \, x \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.405627, size = 24, normalized size = 0.83 \[ - \frac{x^{2} \cos{\left (2 x \right )}}{2} + \frac{x \sin{\left (2 x \right )}}{2} + \frac{\cos{\left (2 x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*sin(2*x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.201599, size = 28, normalized size = 0.97 \[ -\frac{1}{4} \,{\left (2 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) + \frac{1}{2} \, x \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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