### 3.24 $$\int x^2 \cos (3 x) \, dx$$

Optimal. Leaf size=29 $\frac{1}{3} x^2 \sin (3 x)-\frac{2}{27} \sin (3 x)+\frac{2}{9} x \cos (3 x)$

[Out]

(2*x*Cos[3*x])/9 - (2*Sin[3*x])/27 + (x^2*Sin[3*x])/3

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Rubi [A]  time = 0.025313, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {3296, 2637} $\frac{1}{3} x^2 \sin (3 x)-\frac{2}{27} \sin (3 x)+\frac{2}{9} x \cos (3 x)$

Antiderivative was successfully veriﬁed.

[In]

Int[x^2*Cos[3*x],x]

[Out]

(2*x*Cos[3*x])/9 - (2*Sin[3*x])/27 + (x^2*Sin[3*x])/3

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^2 \cos (3 x) \, dx &=\frac{1}{3} x^2 \sin (3 x)-\frac{2}{3} \int x \sin (3 x) \, dx\\ &=\frac{2}{9} x \cos (3 x)+\frac{1}{3} x^2 \sin (3 x)-\frac{2}{9} \int \cos (3 x) \, dx\\ &=\frac{2}{9} x \cos (3 x)-\frac{2}{27} \sin (3 x)+\frac{1}{3} x^2 \sin (3 x)\\ \end{align*}

Mathematica [A]  time = 0.0267297, size = 25, normalized size = 0.86 $\frac{1}{27} \left (9 x^2-2\right ) \sin (3 x)+\frac{2}{9} x \cos (3 x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^2*Cos[3*x],x]

[Out]

(2*x*Cos[3*x])/9 + ((-2 + 9*x^2)*Sin[3*x])/27

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Maple [A]  time = 0.008, size = 24, normalized size = 0.8 \begin{align*}{\frac{2\,x\cos \left ( 3\,x \right ) }{9}}-{\frac{2\,\sin \left ( 3\,x \right ) }{27}}+{\frac{{x}^{2}\sin \left ( 3\,x \right ) }{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cos(3*x),x)

[Out]

2/9*x*cos(3*x)-2/27*sin(3*x)+1/3*x^2*sin(3*x)

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Maxima [A]  time = 0.941103, size = 28, normalized size = 0.97 \begin{align*} \frac{2}{9} \, x \cos \left (3 \, x\right ) + \frac{1}{27} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(3*x),x, algorithm="maxima")

[Out]

2/9*x*cos(3*x) + 1/27*(9*x^2 - 2)*sin(3*x)

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Fricas [A]  time = 2.02833, size = 59, normalized size = 2.03 \begin{align*} \frac{2}{9} \, x \cos \left (3 \, x\right ) + \frac{1}{27} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(3*x),x, algorithm="fricas")

[Out]

2/9*x*cos(3*x) + 1/27*(9*x^2 - 2)*sin(3*x)

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Sympy [A]  time = 0.303979, size = 27, normalized size = 0.93 \begin{align*} \frac{x^{2} \sin{\left (3 x \right )}}{3} + \frac{2 x \cos{\left (3 x \right )}}{9} - \frac{2 \sin{\left (3 x \right )}}{27} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cos(3*x),x)

[Out]

x**2*sin(3*x)/3 + 2*x*cos(3*x)/9 - 2*sin(3*x)/27

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Giac [A]  time = 1.04041, size = 28, normalized size = 0.97 \begin{align*} \frac{2}{9} \, x \cos \left (3 \, x\right ) + \frac{1}{27} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(3*x),x, algorithm="giac")

[Out]

2/9*x*cos(3*x) + 1/27*(9*x^2 - 2)*sin(3*x)