3.22 \(\int x \sin (4 x) \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{16} \sin (4 x)-\frac{1}{4} x \cos (4 x) \]

[Out]

-(x*Cos[4*x])/4 + Sin[4*x]/16

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Rubi [A]  time = 0.0175443, 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 \[ \frac{1}{16} \sin (4 x)-\frac{1}{4} x \cos (4 x) \]

Antiderivative was successfully verified.

[In]  Int[x*Sin[4*x],x]

[Out]

-(x*Cos[4*x])/4 + Sin[4*x]/16

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Rubi in Sympy [A]  time = 0.82888, size = 14, normalized size = 0.78 \[ - \frac{x \cos{\left (4 x \right )}}{4} + \frac{\sin{\left (4 x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*sin(4*x),x)

[Out]

-x*cos(4*x)/4 + sin(4*x)/16

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Mathematica [A]  time = 0.00524932, size = 18, normalized size = 1. \[ \frac{1}{16} \sin (4 x)-\frac{1}{4} x \cos (4 x) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sin[4*x],x]

[Out]

-(x*Cos[4*x])/4 + Sin[4*x]/16

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Maple [A]  time = 0.007, size = 15, normalized size = 0.8 \[ -{\frac{x\cos \left ( 4\,x \right ) }{4}}+{\frac{\sin \left ( 4\,x \right ) }{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*sin(4*x),x)

[Out]

-1/4*x*cos(4*x)+1/16*sin(4*x)

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Maxima [A]  time = 1.37156, size = 19, normalized size = 1.06 \[ -\frac{1}{4} \, x \cos \left (4 \, x\right ) + \frac{1}{16} \, \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(4*x),x, algorithm="maxima")

[Out]

-1/4*x*cos(4*x) + 1/16*sin(4*x)

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Fricas [A]  time = 0.227059, size = 19, normalized size = 1.06 \[ -\frac{1}{4} \, x \cos \left (4 \, x\right ) + \frac{1}{16} \, \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(4*x),x, algorithm="fricas")

[Out]

-1/4*x*cos(4*x) + 1/16*sin(4*x)

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Sympy [A]  time = 0.19319, size = 14, normalized size = 0.78 \[ - \frac{x \cos{\left (4 x \right )}}{4} + \frac{\sin{\left (4 x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(4*x),x)

[Out]

-x*cos(4*x)/4 + sin(4*x)/16

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GIAC/XCAS [A]  time = 0.199063, size = 19, normalized size = 1.06 \[ -\frac{1}{4} \, x \cos \left (4 \, x\right ) + \frac{1}{16} \, \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(4*x),x, algorithm="giac")

[Out]

-1/4*x*cos(4*x) + 1/16*sin(4*x)