3.128 \(\int \sin ^3(\sqrt [3]{x}) \, dx\)

Optimal. Leaf size=87 \[ -2 x^{2/3} \cos \left (\sqrt [3]{x}\right )-x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-\frac{2}{9} \cos ^3\left (\sqrt [3]{x}\right )+\frac{14}{3} \cos \left (\sqrt [3]{x}\right ) \]

[Out]

(14*Cos[x^(1/3)])/3 - 2*x^(2/3)*Cos[x^(1/3)] - (2*Cos[x^(1/3)]^3)/9 + 4*x^(1/3)*Sin[x^(1/3)] - x^(2/3)*Cos[x^(
1/3)]*Sin[x^(1/3)]^2 + (2*x^(1/3)*Sin[x^(1/3)]^3)/3

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Rubi [A]  time = 0.0636996, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {3361, 3311, 3296, 2638, 2633} \[ -2 x^{2/3} \cos \left (\sqrt [3]{x}\right )-x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-\frac{2}{9} \cos ^3\left (\sqrt [3]{x}\right )+\frac{14}{3} \cos \left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x^(1/3)]^3,x]

[Out]

(14*Cos[x^(1/3)])/3 - 2*x^(2/3)*Cos[x^(1/3)] - (2*Cos[x^(1/3)]^3)/9 + 4*x^(1/3)*Sin[x^(1/3)] - x^(2/3)*Cos[x^(
1/3)]*Sin[x^(1/3)]^2 + (2*x^(1/3)*Sin[x^(1/3)]^3)/3

Rule 3361

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :> Dist[1/(n*f), Subst[Int[x
^(1/n - 1)*(a + b*Sin[c + d*x])^p, x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && In
tegerQ[1/n]

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 2638

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

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \sin ^3\left (\sqrt [3]{x}\right ) \, dx &=3 \operatorname{Subst}\left (\int x^2 \sin ^3(x) \, dx,x,\sqrt [3]{x}\right )\\ &=-x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin ^2\left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )-\frac{2}{3} \operatorname{Subst}\left (\int \sin ^3(x) \, dx,x,\sqrt [3]{x}\right )+2 \operatorname{Subst}\left (\int x^2 \sin (x) \, dx,x,\sqrt [3]{x}\right )\\ &=-2 x^{2/3} \cos \left (\sqrt [3]{x}\right )-x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin ^2\left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+\frac{2}{3} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (\sqrt [3]{x}\right )\right )+4 \operatorname{Subst}\left (\int x \cos (x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2}{3} \cos \left (\sqrt [3]{x}\right )-2 x^{2/3} \cos \left (\sqrt [3]{x}\right )-\frac{2}{9} \cos ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin ^2\left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )-4 \operatorname{Subst}\left (\int \sin (x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{14}{3} \cos \left (\sqrt [3]{x}\right )-2 x^{2/3} \cos \left (\sqrt [3]{x}\right )-\frac{2}{9} \cos ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin ^2\left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0550805, size = 62, normalized size = 0.71 \[ \frac{1}{36} \left (-81 \left (x^{2/3}-2\right ) \cos \left (\sqrt [3]{x}\right )+\left (9 x^{2/3}-2\right ) \cos \left (3 \sqrt [3]{x}\right )-6 \sqrt [3]{x} \left (\sin \left (3 \sqrt [3]{x}\right )-27 \sin \left (\sqrt [3]{x}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x^(1/3)]^3,x]

[Out]

(-81*(-2 + x^(2/3))*Cos[x^(1/3)] + (-2 + 9*x^(2/3))*Cos[3*x^(1/3)] - 6*x^(1/3)*(-27*Sin[x^(1/3)] + Sin[3*x^(1/
3)]))/36

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Maple [A]  time = 0.01, size = 59, normalized size = 0.7 \begin{align*} -{x}^{{\frac{2}{3}}} \left ( 2+ \left ( \sin \left ( \sqrt [3]{x} \right ) \right ) ^{2} \right ) \cos \left ( \sqrt [3]{x} \right ) +4\,\cos \left ( \sqrt [3]{x} \right ) +4\,\sqrt [3]{x}\sin \left ( \sqrt [3]{x} \right ) +{\frac{2}{3}\sqrt [3]{x} \left ( \sin \left ( \sqrt [3]{x} \right ) \right ) ^{3}}+{\frac{2}{9} \left ( 2+ \left ( \sin \left ( \sqrt [3]{x} \right ) \right ) ^{2} \right ) \cos \left ( \sqrt [3]{x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x^(1/3))^3,x)

[Out]

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

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Maxima [A]  time = 0.972562, size = 63, normalized size = 0.72 \begin{align*} \frac{1}{36} \,{\left (9 \, x^{\frac{2}{3}} - 2\right )} \cos \left (3 \, x^{\frac{1}{3}}\right ) - \frac{9}{4} \,{\left (x^{\frac{2}{3}} - 2\right )} \cos \left (x^{\frac{1}{3}}\right ) - \frac{1}{6} \, x^{\frac{1}{3}} \sin \left (3 \, x^{\frac{1}{3}}\right ) + \frac{9}{2} \, x^{\frac{1}{3}} \sin \left (x^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x^(1/3))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.67498, size = 173, normalized size = 1.99 \begin{align*} \frac{1}{9} \,{\left (9 \, x^{\frac{2}{3}} - 2\right )} \cos \left (x^{\frac{1}{3}}\right )^{3} - \frac{1}{3} \,{\left (9 \, x^{\frac{2}{3}} - 14\right )} \cos \left (x^{\frac{1}{3}}\right ) - \frac{2}{3} \,{\left (x^{\frac{1}{3}} \cos \left (x^{\frac{1}{3}}\right )^{2} - 7 \, x^{\frac{1}{3}}\right )} \sin \left (x^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x^(1/3))^3,x, algorithm="fricas")

[Out]

1/9*(9*x^(2/3) - 2)*cos(x^(1/3))^3 - 1/3*(9*x^(2/3) - 14)*cos(x^(1/3)) - 2/3*(x^(1/3)*cos(x^(1/3))^2 - 7*x^(1/
3))*sin(x^(1/3))

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Sympy [A]  time = 8.2981, size = 80, normalized size = 0.92 \begin{align*} - \frac{9 x^{\frac{2}{3}} \cos{\left (\sqrt [3]{x} \right )}}{4} + \frac{x^{\frac{2}{3}} \cos{\left (3 \sqrt [3]{x} \right )}}{4} + \frac{9 \sqrt [3]{x} \sin{\left (\sqrt [3]{x} \right )}}{2} - \frac{\sqrt [3]{x} \sin{\left (3 \sqrt [3]{x} \right )}}{6} + \frac{9 \cos{\left (\sqrt [3]{x} \right )}}{2} - \frac{\cos{\left (3 \sqrt [3]{x} \right )}}{18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x**(1/3))**3,x)

[Out]

-9*x**(2/3)*cos(x**(1/3))/4 + x**(2/3)*cos(3*x**(1/3))/4 + 9*x**(1/3)*sin(x**(1/3))/2 - x**(1/3)*sin(3*x**(1/3
))/6 + 9*cos(x**(1/3))/2 - cos(3*x**(1/3))/18

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Giac [A]  time = 1.09136, size = 63, normalized size = 0.72 \begin{align*} \frac{1}{36} \,{\left (9 \, x^{\frac{2}{3}} - 2\right )} \cos \left (3 \, x^{\frac{1}{3}}\right ) - \frac{9}{4} \,{\left (x^{\frac{2}{3}} - 2\right )} \cos \left (x^{\frac{1}{3}}\right ) - \frac{1}{6} \, x^{\frac{1}{3}} \sin \left (3 \, x^{\frac{1}{3}}\right ) + \frac{9}{2} \, x^{\frac{1}{3}} \sin \left (x^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x^(1/3))^3,x, algorithm="giac")

[Out]

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