3.127 \(\int \sin ^2(\sqrt [3]{x}) \, dx\)

Optimal. Leaf size=69 \[ -\frac{3}{2} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac{x}{2}-\frac{3 \sqrt [3]{x}}{4}+\frac{3}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right )+\frac{3}{4} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.043028, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {3361, 3311, 30, 2635, 8} \[ -\frac{3}{2} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac{x}{2}-\frac{3 \sqrt [3]{x}}{4}+\frac{3}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right )+\frac{3}{4} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x^(1/3)]^2,x]

[Out]

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

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2635

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.049048, size = 41, normalized size = 0.59 \[ \frac{1}{8} \left (\left (3-6 x^{2/3}\right ) \sin \left (2 \sqrt [3]{x}\right )+4 x-6 \sqrt [3]{x} \cos \left (2 \sqrt [3]{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x^(1/3)]^2,x]

[Out]

(4*x - 6*x^(1/3)*Cos[2*x^(1/3)] + (3 - 6*x^(2/3))*Sin[2*x^(1/3)])/8

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Maple [A]  time = 0.01, size = 52, normalized size = 0.8 \begin{align*} 3\,{x}^{2/3} \left ( -1/2\,\cos \left ( \sqrt [3]{x} \right ) \sin \left ( \sqrt [3]{x} \right ) +1/2\,\sqrt [3]{x} \right ) -{\frac{3}{2}\sqrt [3]{x} \left ( \cos \left ( \sqrt [3]{x} \right ) \right ) ^{2}}+{\frac{3}{4}\cos \left ( \sqrt [3]{x} \right ) \sin \left ( \sqrt [3]{x} \right ) }+{\frac{3}{4}\sqrt [3]{x}}-x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x^(1/3))^2,x)

[Out]

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

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Maxima [A]  time = 0.950069, size = 41, normalized size = 0.59 \begin{align*} -\frac{3}{8} \,{\left (2 \, x^{\frac{2}{3}} - 1\right )} \sin \left (2 \, x^{\frac{1}{3}}\right ) - \frac{3}{4} \, x^{\frac{1}{3}} \cos \left (2 \, x^{\frac{1}{3}}\right ) + \frac{1}{2} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x^(1/3))^2,x, algorithm="maxima")

[Out]

-3/8*(2*x^(2/3) - 1)*sin(2*x^(1/3)) - 3/4*x^(1/3)*cos(2*x^(1/3)) + 1/2*x

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Fricas [A]  time = 1.62251, size = 134, normalized size = 1.94 \begin{align*} -\frac{3}{4} \,{\left (2 \, x^{\frac{2}{3}} - 1\right )} \cos \left (x^{\frac{1}{3}}\right ) \sin \left (x^{\frac{1}{3}}\right ) - \frac{3}{2} \, x^{\frac{1}{3}} \cos \left (x^{\frac{1}{3}}\right )^{2} + \frac{1}{2} \, x + \frac{3}{4} \, x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x^(1/3))^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.51308, size = 379, normalized size = 5.49 \begin{align*} \frac{12 x^{\frac{2}{3}} \tan ^{3}{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} - \frac{12 x^{\frac{2}{3}} \tan{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} - \frac{3 \sqrt [3]{x} \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} + \frac{18 \sqrt [3]{x} \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} - \frac{3 \sqrt [3]{x}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} + \frac{2 x \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} + \frac{4 x \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} + \frac{2 x}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} - \frac{6 \tan ^{3}{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} + \frac{6 \tan{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x**(1/3))**2,x)

[Out]

12*x**(2/3)*tan(x**(1/3)/2)**3/(4*tan(x**(1/3)/2)**4 + 8*tan(x**(1/3)/2)**2 + 4) - 12*x**(2/3)*tan(x**(1/3)/2)
/(4*tan(x**(1/3)/2)**4 + 8*tan(x**(1/3)/2)**2 + 4) - 3*x**(1/3)*tan(x**(1/3)/2)**4/(4*tan(x**(1/3)/2)**4 + 8*t
an(x**(1/3)/2)**2 + 4) + 18*x**(1/3)*tan(x**(1/3)/2)**2/(4*tan(x**(1/3)/2)**4 + 8*tan(x**(1/3)/2)**2 + 4) - 3*
x**(1/3)/(4*tan(x**(1/3)/2)**4 + 8*tan(x**(1/3)/2)**2 + 4) + 2*x*tan(x**(1/3)/2)**4/(4*tan(x**(1/3)/2)**4 + 8*
tan(x**(1/3)/2)**2 + 4) + 4*x*tan(x**(1/3)/2)**2/(4*tan(x**(1/3)/2)**4 + 8*tan(x**(1/3)/2)**2 + 4) + 2*x/(4*ta
n(x**(1/3)/2)**4 + 8*tan(x**(1/3)/2)**2 + 4) - 6*tan(x**(1/3)/2)**3/(4*tan(x**(1/3)/2)**4 + 8*tan(x**(1/3)/2)*
*2 + 4) + 6*tan(x**(1/3)/2)/(4*tan(x**(1/3)/2)**4 + 8*tan(x**(1/3)/2)**2 + 4)

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Giac [A]  time = 1.09981, size = 41, normalized size = 0.59 \begin{align*} -\frac{3}{8} \,{\left (2 \, x^{\frac{2}{3}} - 1\right )} \sin \left (2 \, x^{\frac{1}{3}}\right ) - \frac{3}{4} \, x^{\frac{1}{3}} \cos \left (2 \, x^{\frac{1}{3}}\right ) + \frac{1}{2} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x^(1/3))^2,x, algorithm="giac")

[Out]

-3/8*(2*x^(2/3) - 1)*sin(2*x^(1/3)) - 3/4*x^(1/3)*cos(2*x^(1/3)) + 1/2*x