3.1.24 \(\int x^3 \cos (x) \, dx\) [24]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {3377, 2718} \begin {gather*} x^3 \sin (x)+3 x^2 \cos (x)-6 x \sin (x)-6 \cos (x) \end {gather*}

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]

Rule 2718

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

Rule 3377

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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 19, normalized size = 0.83 \begin {gather*} 3 \left (-2+x^2\right ) \cos (x)+x \left (-6+x^2\right ) \sin (x) \end {gather*}

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 = 0.02, size = 24, normalized size = 1.04

method	result	size
risch	\(3 \left (x^{2}-2\right ) \cos \left (x \right )+x \left (x^{2}-6\right ) \sin \left (x \right )\)	\(20\)
default	\(-6 \cos \left (x \right )+3 x^{2} \cos \left (x \right )-6 x \sin \left (x \right )+x^{3} \sin \left (x \right )\)	\(24\)
meijerg	\(8 \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (-\frac {3 x^{2}}{2}+3\right ) \cos \left (x \right )}{4 \sqrt {\pi }}-\frac {x \left (-\frac {x^{2}}{2}+3\right ) \sin \left (x \right )}{4 \sqrt {\pi }}\right )\)	\(41\)
norman	\(\frac {3 x^{2}-12 x \tan \left (\frac {x}{2}\right )-3 x^{2} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 x^{3} \tan \left (\frac {x}{2}\right )-12}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cos(x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 4.80, size = 20, normalized size = 0.87 \begin {gather*} 3 \, {\left (x^{2} - 2\right )} \cos \left (x\right ) + {\left (x^{3} - 6 \, x\right )} \sin \left (x\right ) \end {gather*}

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 = 0.41, size = 20, normalized size = 0.87 \begin {gather*} 3 \, {\left (x^{2} - 2\right )} \cos \left (x\right ) + {\left (x^{3} - 6 \, x\right )} \sin \left (x\right ) \end {gather*}

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 = 0.13, size = 26, normalized size = 1.13 \begin {gather*} x^{3} \sin {\left (x \right )} + 3 x^{2} \cos {\left (x \right )} - 6 x \sin {\left (x \right )} - 6 \cos {\left (x \right )} \end {gather*}

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 [A]
time = 0.55, size = 20, normalized size = 0.87 \begin {gather*} 3 \, {\left (x^{2} - 2\right )} \cos \left (x\right ) + {\left (x^{3} - 6 \, x\right )} \sin \left (x\right ) \end {gather*}

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)

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Mupad [B]
time = 0.03, size = 24, normalized size = 1.04 \begin {gather*} \cos \left (x\right )\,\left (3\,x^2-6\right )-\sin \left (x\right )\,\left (6\,x-x^3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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