3.486 \(\int x^2 \cos (x) \sin ^2(x) \, dx\)

Optimal. Leaf size=44 \[ \frac{1}{3} x^2 \sin ^3(x)-\frac{2 \sin ^3(x)}{27}-\frac{4 \sin (x)}{9}+\frac{4}{9} x \cos (x)+\frac{2}{9} x \sin ^2(x) \cos (x) \]

[Out]

(4*x*Cos[x])/9 - (4*Sin[x])/9 + (2*x*Cos[x]*Sin[x]^2)/9 - (2*Sin[x]^3)/27 + (x^2*Sin[x]^3)/3

________________________________________________________________________________________

Rubi [A]  time = 0.0420206, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3443, 3310, 3296, 2637} \[ \frac{1}{3} x^2 \sin ^3(x)-\frac{2 \sin ^3(x)}{27}-\frac{4 \sin (x)}{9}+\frac{4}{9} x \cos (x)+\frac{2}{9} x \sin ^2(x) \cos (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x*Cos[x])/9 - (4*Sin[x])/9 + (2*x*Cos[x]*Sin[x]^2)/9 - (2*Sin[x]^3)/27 + (x^2*Sin[x]^3)/3

Rule 3443

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[(x^(m - n
+ 1)*Sin[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sin[a + b*x^n]^
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

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 (x) \sin ^2(x) \, dx &=\frac{1}{3} x^2 \sin ^3(x)-\frac{2}{3} \int x \sin ^3(x) \, dx\\ &=\frac{2}{9} x \cos (x) \sin ^2(x)-\frac{2 \sin ^3(x)}{27}+\frac{1}{3} x^2 \sin ^3(x)-\frac{4}{9} \int x \sin (x) \, dx\\ &=\frac{4}{9} x \cos (x)+\frac{2}{9} x \cos (x) \sin ^2(x)-\frac{2 \sin ^3(x)}{27}+\frac{1}{3} x^2 \sin ^3(x)-\frac{4}{9} \int \cos (x) \, dx\\ &=\frac{4}{9} x \cos (x)-\frac{4 \sin (x)}{9}+\frac{2}{9} x \cos (x) \sin ^2(x)-\frac{2 \sin ^3(x)}{27}+\frac{1}{3} x^2 \sin ^3(x)\\ \end{align*}

Mathematica [A]  time = 0.121809, size = 39, normalized size = 0.89 \[ \frac{1}{54} \left (\sin (x) \left (9 x^2+\left (2-9 x^2\right ) \cos (2 x)-26\right )+27 x \cos (x)-3 x \cos (3 x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(27*x*Cos[x] - 3*x*Cos[3*x] + (-26 + 9*x^2 + (2 - 9*x^2)*Cos[2*x])*Sin[x])/54

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 32, normalized size = 0.7 \begin{align*}{\frac{{x}^{2} \left ( \sin \left ( x \right ) \right ) ^{3}}{3}}+{\frac{2\,x \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{9}}-{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{3}}{27}}-{\frac{4\,\sin \left ( x \right ) }{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^2*sin(x)^3+2/9*x*(2+sin(x)^2)*cos(x)-2/27*sin(x)^3-4/9*sin(x)

________________________________________________________________________________________

Maxima [A]  time = 0.95696, size = 47, normalized size = 1.07 \begin{align*} -\frac{1}{18} \, x \cos \left (3 \, x\right ) + \frac{1}{2} \, x \cos \left (x\right ) - \frac{1}{108} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac{1}{4} \,{\left (x^{2} - 2\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*x*cos(3*x) + 1/2*x*cos(x) - 1/108*(9*x^2 - 2)*sin(3*x) + 1/4*(x^2 - 2)*sin(x)

________________________________________________________________________________________

Fricas [A]  time = 2.25054, size = 111, normalized size = 2.52 \begin{align*} -\frac{2}{9} \, x \cos \left (x\right )^{3} + \frac{2}{3} \, x \cos \left (x\right ) - \frac{1}{27} \,{\left ({\left (9 \, x^{2} - 2\right )} \cos \left (x\right )^{2} - 9 \, x^{2} + 14\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*x*cos(x)^3 + 2/3*x*cos(x) - 1/27*((9*x^2 - 2)*cos(x)^2 - 9*x^2 + 14)*sin(x)

________________________________________________________________________________________

Sympy [A]  time = 1.20993, size = 53, normalized size = 1.2 \begin{align*} \frac{x^{2} \sin ^{3}{\left (x \right )}}{3} + \frac{2 x \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{3} + \frac{4 x \cos ^{3}{\left (x \right )}}{9} - \frac{14 \sin ^{3}{\left (x \right )}}{27} - \frac{4 \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*sin(x)**3/3 + 2*x*sin(x)**2*cos(x)/3 + 4*x*cos(x)**3/9 - 14*sin(x)**3/27 - 4*sin(x)*cos(x)**2/9

________________________________________________________________________________________

Giac [A]  time = 1.08224, size = 47, normalized size = 1.07 \begin{align*} -\frac{1}{18} \, x \cos \left (3 \, x\right ) + \frac{1}{2} \, x \cos \left (x\right ) - \frac{1}{108} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac{1}{4} \,{\left (x^{2} - 2\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*x*cos(3*x) + 1/2*x*cos(x) - 1/108*(9*x^2 - 2)*sin(3*x) + 1/4*(x^2 - 2)*sin(x)