3.24 \(\int x^3 \cos (x) \, dx\)

Optimal. Leaf size=23 \[ x^3 \sin (x)+3 x^2 \cos (x)-6 x \sin (x)-6 \cos (x) \]

[Out]

-6*Cos[x] + 3*x^2*Cos[x] - 6*x*Sin[x] + x^3*Sin[x]

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Rubi [A]  time = 0.0532141, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ x^3 \sin (x)+3 x^2 \cos (x)-6 x \sin (x)-6 \cos (x) \]

Antiderivative was successfully verified.

[In]  Int[x^3*Cos[x],x]

[Out]

-6*Cos[x] + 3*x^2*Cos[x] - 6*x*Sin[x] + x^3*Sin[x]

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Rubi in Sympy [A]  time = 1.92231, size = 26, normalized size = 1.13 \[ x^{3} \sin{\left (x \right )} + 3 x^{2} \cos{\left (x \right )} - 6 x \sin{\left (x \right )} - 6 \cos{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*cos(x),x)

[Out]

x**3*sin(x) + 3*x**2*cos(x) - 6*x*sin(x) - 6*cos(x)

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Mathematica [A]  time = 0.0118285, size = 19, normalized size = 0.83 \[ x \left (x^2-6\right ) \sin (x)+3 \left (x^2-2\right ) \cos (x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Cos[x],x]

[Out]

3*(-2 + x^2)*Cos[x] + x*(-6 + x^2)*Sin[x]

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Maple [A]  time = 0.01, size = 24, normalized size = 1. \[ -6\,\cos \left ( x \right ) +3\,{x}^{2}\cos \left ( x \right ) -6\,x\sin \left ( x \right ) +{x}^{3}\sin \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*cos(x),x)

[Out]

-6*cos(x)+3*x^2*cos(x)-6*x*sin(x)+x^3*sin(x)

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Maxima [A]  time = 1.33521, size = 27, normalized size = 1.17 \[ 3 \,{\left (x^{2} - 2\right )} \cos \left (x\right ) +{\left (x^{3} - 6 \, x\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*cos(x),x, algorithm="maxima")

[Out]

3*(x^2 - 2)*cos(x) + (x^3 - 6*x)*sin(x)

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Fricas [A]  time = 0.239757, size = 27, normalized size = 1.17 \[ 3 \,{\left (x^{2} - 2\right )} \cos \left (x\right ) +{\left (x^{3} - 6 \, x\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*cos(x),x, algorithm="fricas")

[Out]

3*(x^2 - 2)*cos(x) + (x^3 - 6*x)*sin(x)

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Sympy [A]  time = 0.870053, size = 26, normalized size = 1.13 \[ x^{3} \sin{\left (x \right )} + 3 x^{2} \cos{\left (x \right )} - 6 x \sin{\left (x \right )} - 6 \cos{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*cos(x),x)

[Out]

x**3*sin(x) + 3*x**2*cos(x) - 6*x*sin(x) - 6*cos(x)

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GIAC/XCAS [A]  time = 0.215839, size = 27, normalized size = 1.17 \[ 3 \,{\left (x^{2} - 2\right )} \cos \left (x\right ) +{\left (x^{3} - 6 \, x\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*cos(x),x, algorithm="giac")

[Out]

3*(x^2 - 2)*cos(x) + (x^3 - 6*x)*sin(x)