3.98 \(\int x^2 \cos (x) \, dx\)

Optimal. Leaf size=16 \[ x^2 \sin (x)-2 \sin (x)+2 x \cos (x) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3296, 2637} \[ x^2 \sin (x)-2 \sin (x)+2 x \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[x^2*Cos[x],x]

[Out]

2*x*Cos[x] - 2*Sin[x] + x^2*Sin[x]

Rule 2637

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.88 \[ \left (x^2-2\right ) \sin (x)+2 x \cos (x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cos[x],x]

[Out]

2*x*Cos[x] + (-2 + x^2)*Sin[x]

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fricas [A]  time = 0.43, size = 14, normalized size = 0.88 \[ 2 \, x \cos \relax (x) + {\left (x^{2} - 2\right )} \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(x),x, algorithm="fricas")

[Out]

2*x*cos(x) + (x^2 - 2)*sin(x)

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giac [A]  time = 1.21, size = 14, normalized size = 0.88 \[ 2 \, x \cos \relax (x) + {\left (x^{2} - 2\right )} \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(x),x, algorithm="giac")

[Out]

2*x*cos(x) + (x^2 - 2)*sin(x)

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maple [A]  time = 0.01, size = 17, normalized size = 1.06 \[ x^{2} \sin \relax (x )+2 x \cos \relax (x )-2 \sin \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cos(x),x)

[Out]

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

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maxima [A]  time = 0.43, size = 14, normalized size = 0.88 \[ 2 \, x \cos \relax (x) + {\left (x^{2} - 2\right )} \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(x),x, algorithm="maxima")

[Out]

2*x*cos(x) + (x^2 - 2)*sin(x)

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mupad [B]  time = 0.03, size = 14, normalized size = 0.88 \[ \sin \relax (x)\,\left (x^2-2\right )+2\,x\,\cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cos(x),x)

[Out]

sin(x)*(x^2 - 2) + 2*x*cos(x)

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sympy [A]  time = 0.32, size = 17, normalized size = 1.06 \[ x^{2} \sin {\relax (x )} + 2 x \cos {\relax (x )} - 2 \sin {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cos(x),x)

[Out]

x**2*sin(x) + 2*x*cos(x) - 2*sin(x)

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