3.339 \(\int x^3 \sin (x) \, dx\)

Optimal. Leaf size=24 \[ 3 x^2 \sin (x)+x^3 (-\cos (x))-6 \sin (x)+6 x \cos (x) \]

[Out]

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

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Rubi [A]  time = 0.0371059, antiderivative size = 24, 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, Rules used = {3296, 2637} \[ 3 x^2 \sin (x)+x^3 (-\cos (x))-6 \sin (x)+6 x \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sin[x],x]

[Out]

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

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^3 \sin (x) \, dx &=-x^3 \cos (x)+3 \int x^2 \cos (x) \, dx\\ &=-x^3 \cos (x)+3 x^2 \sin (x)-6 \int x \sin (x) \, dx\\ &=6 x \cos (x)-x^3 \cos (x)+3 x^2 \sin (x)-6 \int \cos (x) \, dx\\ &=6 x \cos (x)-x^3 \cos (x)-6 \sin (x)+3 x^2 \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0149542, size = 20, normalized size = 0.83 \[ 3 \left (x^2-2\right ) \sin (x)-x \left (x^2-6\right ) \cos (x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sin[x],x]

[Out]

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

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Maple [A]  time = 0.004, size = 25, normalized size = 1. \begin{align*} 6\,x\cos \left ( x \right ) -{x}^{3}\cos \left ( x \right ) -6\,\sin \left ( x \right ) +3\,{x}^{2}\sin \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*sin(x),x)

[Out]

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

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Maxima [A]  time = 0.952481, size = 28, normalized size = 1.17 \begin{align*} -{\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) + 3 \,{\left (x^{2} - 2\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sin(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.96472, size = 57, normalized size = 2.38 \begin{align*} -{\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) + 3 \,{\left (x^{2} - 2\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sin(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.554758, size = 26, normalized size = 1.08 \begin{align*} - x^{3} \cos{\left (x \right )} + 3 x^{2} \sin{\left (x \right )} + 6 x \cos{\left (x \right )} - 6 \sin{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*sin(x),x)

[Out]

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

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Giac [A]  time = 1.08333, size = 28, normalized size = 1.17 \begin{align*} -{\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) + 3 \,{\left (x^{2} - 2\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sin(x),x, algorithm="giac")

[Out]

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