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

Optimal. Leaf size=73 \[ \frac{1}{3} x^2 \sin ^3(x)+2 x^2 \sin (x)-\frac{2}{3} x^3 \cos (x)-\frac{1}{3} x^3 \sin ^2(x) \cos (x)-\frac{2 \sin ^3(x)}{27}-\frac{40 \sin (x)}{9}+\frac{40}{9} x \cos (x)+\frac{2}{9} x \sin ^2(x) \cos (x) \]

[Out]

(40*x*Cos[x])/9 - (2*x^3*Cos[x])/3 - (40*Sin[x])/9 + 2*x^2*Sin[x] + (2*x*Cos[x]*Sin[x]^2)/9 - (x^3*Cos[x]*Sin[
x]^2)/3 - (2*Sin[x]^3)/27 + (x^2*Sin[x]^3)/3

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Rubi [A]  time = 0.0842397, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3311, 3296, 2637, 3310} \[ \frac{1}{3} x^2 \sin ^3(x)+2 x^2 \sin (x)-\frac{2}{3} x^3 \cos (x)-\frac{1}{3} x^3 \sin ^2(x) \cos (x)-\frac{2 \sin ^3(x)}{27}-\frac{40 \sin (x)}{9}+\frac{40}{9} x \cos (x)+\frac{2}{9} x \sin ^2(x) \cos (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(40*x*Cos[x])/9 - (2*x^3*Cos[x])/3 - (40*Sin[x])/9 + 2*x^2*Sin[x] + (2*x*Cos[x]*Sin[x]^2)/9 - (x^3*Cos[x]*Sin[
x]^2)/3 - (2*Sin[x]^3)/27 + (x^2*Sin[x]^3)/3

Rule 3311

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

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]

Rubi steps

\begin{align*} \int x^3 \sin ^3(x) \, dx &=-\frac{1}{3} x^3 \cos (x) \sin ^2(x)+\frac{1}{3} x^2 \sin ^3(x)+\frac{2}{3} \int x^3 \sin (x) \, dx-\frac{2}{3} \int x \sin ^3(x) \, dx\\ &=-\frac{2}{3} x^3 \cos (x)+\frac{2}{9} x \cos (x) \sin ^2(x)-\frac{1}{3} x^3 \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+2 \int x^2 \cos (x) \, dx\\ &=\frac{4}{9} x \cos (x)-\frac{2}{3} x^3 \cos (x)+2 x^2 \sin (x)+\frac{2}{9} x \cos (x) \sin ^2(x)-\frac{1}{3} x^3 \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-4 \int x \sin (x) \, dx\\ &=\frac{40}{9} x \cos (x)-\frac{2}{3} x^3 \cos (x)-\frac{4 \sin (x)}{9}+2 x^2 \sin (x)+\frac{2}{9} x \cos (x) \sin ^2(x)-\frac{1}{3} x^3 \cos (x) \sin ^2(x)-\frac{2 \sin ^3(x)}{27}+\frac{1}{3} x^2 \sin ^3(x)-4 \int \cos (x) \, dx\\ &=\frac{40}{9} x \cos (x)-\frac{2}{3} x^3 \cos (x)-\frac{40 \sin (x)}{9}+2 x^2 \sin (x)+\frac{2}{9} x \cos (x) \sin ^2(x)-\frac{1}{3} x^3 \cos (x) \sin ^2(x)-\frac{2 \sin ^3(x)}{27}+\frac{1}{3} x^2 \sin ^3(x)\\ \end{align*}

Mathematica [A]  time = 0.0958343, size = 51, normalized size = 0.7 \[ \frac{1}{108} \left (243 \left (x^2-2\right ) \sin (x)-\left (9 x^2-2\right ) \sin (3 x)-81 x \left (x^2-6\right ) \cos (x)+3 x \left (3 x^2-2\right ) \cos (3 x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-81*x*(-6 + x^2)*Cos[x] + 3*x*(-2 + 3*x^2)*Cos[3*x] + 243*(-2 + x^2)*Sin[x] - (-2 + 9*x^2)*Sin[3*x])/108

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Maple [A]  time = 0.023, size = 57, normalized size = 0.8 \begin{align*} -{\frac{{x}^{3} \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{3}}+2\,{x}^{2}\sin \left ( x \right ) -{\frac{40\,\sin \left ( x \right ) }{9}}+4\,x\cos \left ( x \right ) +{\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*x^3*(2+sin(x)^2)*cos(x)+2*x^2*sin(x)-40/9*sin(x)+4*x*cos(x)+1/3*x^2*sin(x)^3+2/9*x*(2+sin(x)^2)*cos(x)-2/
27*sin(x)^3

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Maxima [A]  time = 0.976939, size = 66, normalized size = 0.9 \begin{align*} \frac{1}{36} \,{\left (3 \, x^{3} - 2 \, x\right )} \cos \left (3 \, x\right ) - \frac{3}{4} \,{\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) - \frac{1}{108} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac{9}{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^3*sin(x)^3,x, algorithm="maxima")

[Out]

1/36*(3*x^3 - 2*x)*cos(3*x) - 3/4*(x^3 - 6*x)*cos(x) - 1/108*(9*x^2 - 2)*sin(3*x) + 9/4*(x^2 - 2)*sin(x)

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Fricas [A]  time = 2.24924, size = 146, normalized size = 2. \begin{align*} \frac{1}{9} \,{\left (3 \, x^{3} - 2 \, x\right )} \cos \left (x\right )^{3} - \frac{1}{3} \,{\left (3 \, x^{3} - 14 \, x\right )} \cos \left (x\right ) - \frac{1}{27} \,{\left ({\left (9 \, x^{2} - 2\right )} \cos \left (x\right )^{2} - 63 \, x^{2} + 122\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(3*x^3 - 2*x)*cos(x)^3 - 1/3*(3*x^3 - 14*x)*cos(x) - 1/27*((9*x^2 - 2)*cos(x)^2 - 63*x^2 + 122)*sin(x)

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Sympy [A]  time = 1.95342, size = 92, normalized size = 1.26 \begin{align*} - x^{3} \sin ^{2}{\left (x \right )} \cos{\left (x \right )} - \frac{2 x^{3} \cos ^{3}{\left (x \right )}}{3} + \frac{7 x^{2} \sin ^{3}{\left (x \right )}}{3} + 2 x^{2} \sin{\left (x \right )} \cos ^{2}{\left (x \right )} + \frac{14 x \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{3} + \frac{40 x \cos ^{3}{\left (x \right )}}{9} - \frac{122 \sin ^{3}{\left (x \right )}}{27} - \frac{40 \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**3*sin(x)**3,x)

[Out]

-x**3*sin(x)**2*cos(x) - 2*x**3*cos(x)**3/3 + 7*x**2*sin(x)**3/3 + 2*x**2*sin(x)*cos(x)**2 + 14*x*sin(x)**2*co
s(x)/3 + 40*x*cos(x)**3/9 - 122*sin(x)**3/27 - 40*sin(x)*cos(x)**2/9

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Giac [A]  time = 1.07071, size = 66, normalized size = 0.9 \begin{align*} \frac{1}{36} \,{\left (3 \, x^{3} - 2 \, x\right )} \cos \left (3 \, x\right ) - \frac{3}{4} \,{\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) - \frac{1}{108} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac{9}{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^3*sin(x)^3,x, algorithm="giac")

[Out]

1/36*(3*x^3 - 2*x)*cos(3*x) - 3/4*(x^3 - 6*x)*cos(x) - 1/108*(9*x^2 - 2)*sin(3*x) + 9/4*(x^2 - 2)*sin(x)