3.61 \(\int \cos ^3(\sqrt [3]{x}) \, dx\)

Optimal. Leaf size=86 \[ 2 x^{2/3} \sin \left (\sqrt [3]{x}\right )+x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac{2}{9} \sin ^3\left (\sqrt [3]{x}\right )-\frac{14}{3} \sin \left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0682848, antiderivative size = 86, 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 = {3362, 3311, 3296, 2637, 2633} \[ 2 x^{2/3} \sin \left (\sqrt [3]{x}\right )+x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac{2}{9} \sin ^3\left (\sqrt [3]{x}\right )-\frac{14}{3} \sin \left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3362

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.), x_Symbol] :> Dist[1/(n*f), Subst[Int[x
^(1/n - 1)*(a + b*Cos[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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 \cos ^3\left (\sqrt [3]{x}\right ) \, dx &=3 \operatorname{Subst}\left (\int x^2 \cos ^3(x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )+x^{2/3} \cos ^2\left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )-\frac{2}{3} \operatorname{Subst}\left (\int \cos ^3(x) \, dx,x,\sqrt [3]{x}\right )+2 \operatorname{Subst}\left (\int x^2 \cos (x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )+2 x^{2/3} \sin \left (\sqrt [3]{x}\right )+x^{2/3} \cos ^2\left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )+\frac{2}{3} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin \left (\sqrt [3]{x}\right )\right )-4 \operatorname{Subst}\left (\int x \sin (x) \, dx,x,\sqrt [3]{x}\right )\\ &=4 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )-\frac{2}{3} \sin \left (\sqrt [3]{x}\right )+2 x^{2/3} \sin \left (\sqrt [3]{x}\right )+x^{2/3} \cos ^2\left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )+\frac{2}{9} \sin ^3\left (\sqrt [3]{x}\right )-4 \operatorname{Subst}\left (\int \cos (x) \, dx,x,\sqrt [3]{x}\right )\\ &=4 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )-\frac{14}{3} \sin \left (\sqrt [3]{x}\right )+2 x^{2/3} \sin \left (\sqrt [3]{x}\right )+x^{2/3} \cos ^2\left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )+\frac{2}{9} \sin ^3\left (\sqrt [3]{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0992082, size = 66, normalized size = 0.77 \[ \frac{1}{36} \left (81 \left (x^{2/3}-2\right ) \sin \left (\sqrt [3]{x}\right )+\left (9 x^{2/3}-2\right ) \sin \left (3 \sqrt [3]{x}\right )+162 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \cos \left (3 \sqrt [3]{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.029, size = 58, normalized size = 0.7 \begin{align*}{x}^{{\frac{2}{3}}} \left ( 2+ \left ( \cos \left ( \sqrt [3]{x} \right ) \right ) ^{2} \right ) \sin \left ( \sqrt [3]{x} \right ) -4\,\sin \left ( \sqrt [3]{x} \right ) +4\,\sqrt [3]{x}\cos \left ( \sqrt [3]{x} \right ) +{\frac{2}{3}\sqrt [3]{x} \left ( \cos \left ( \sqrt [3]{x} \right ) \right ) ^{3}}-{\frac{2}{9} \left ( 2+ \left ( \cos \left ( \sqrt [3]{x} \right ) \right ) ^{2} \right ) \sin \left ( \sqrt [3]{x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.78985, size = 63, normalized size = 0.73 \begin{align*} \frac{1}{36} \,{\left (9 \, x^{\frac{2}{3}} - 2\right )} \sin \left (3 \, x^{\frac{1}{3}}\right ) + \frac{9}{4} \,{\left (x^{\frac{2}{3}} - 2\right )} \sin \left (x^{\frac{1}{3}}\right ) + \frac{1}{6} \, x^{\frac{1}{3}} \cos \left (3 \, x^{\frac{1}{3}}\right ) + \frac{9}{2} \, x^{\frac{1}{3}} \cos \left (x^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.90552, size = 166, normalized size = 1.93 \begin{align*} \frac{2}{3} \, x^{\frac{1}{3}} \cos \left (x^{\frac{1}{3}}\right )^{3} + \frac{1}{9} \,{\left ({\left (9 \, x^{\frac{2}{3}} - 2\right )} \cos \left (x^{\frac{1}{3}}\right )^{2} + 18 \, x^{\frac{2}{3}} - 40\right )} \sin \left (x^{\frac{1}{3}}\right ) + 4 \, x^{\frac{1}{3}} \cos \left (x^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(1/3)*cos(x^(1/3))^3 + 1/9*((9*x^(2/3) - 2)*cos(x^(1/3))^2 + 18*x^(2/3) - 40)*sin(x^(1/3)) + 4*x^(1/3)*c
os(x^(1/3))

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Sympy [B]  time = 5.29064, size = 513, normalized size = 5.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

54*x**(2/3)*tan(x**(1/3)/2)**5/(9*tan(x**(1/3)/2)**6 + 27*tan(x**(1/3)/2)**4 + 27*tan(x**(1/3)/2)**2 + 9) + 36
*x**(2/3)*tan(x**(1/3)/2)**3/(9*tan(x**(1/3)/2)**6 + 27*tan(x**(1/3)/2)**4 + 27*tan(x**(1/3)/2)**2 + 9) + 54*x
**(2/3)*tan(x**(1/3)/2)/(9*tan(x**(1/3)/2)**6 + 27*tan(x**(1/3)/2)**4 + 27*tan(x**(1/3)/2)**2 + 9) - 42*x**(1/
3)*tan(x**(1/3)/2)**6/(9*tan(x**(1/3)/2)**6 + 27*tan(x**(1/3)/2)**4 + 27*tan(x**(1/3)/2)**2 + 9) - 18*x**(1/3)
*tan(x**(1/3)/2)**4/(9*tan(x**(1/3)/2)**6 + 27*tan(x**(1/3)/2)**4 + 27*tan(x**(1/3)/2)**2 + 9) + 18*x**(1/3)*t
an(x**(1/3)/2)**2/(9*tan(x**(1/3)/2)**6 + 27*tan(x**(1/3)/2)**4 + 27*tan(x**(1/3)/2)**2 + 9) + 42*x**(1/3)/(9*
tan(x**(1/3)/2)**6 + 27*tan(x**(1/3)/2)**4 + 27*tan(x**(1/3)/2)**2 + 9) - 84*tan(x**(1/3)/2)**5/(9*tan(x**(1/3
)/2)**6 + 27*tan(x**(1/3)/2)**4 + 27*tan(x**(1/3)/2)**2 + 9) - 152*tan(x**(1/3)/2)**3/(9*tan(x**(1/3)/2)**6 +
27*tan(x**(1/3)/2)**4 + 27*tan(x**(1/3)/2)**2 + 9) - 84*tan(x**(1/3)/2)/(9*tan(x**(1/3)/2)**6 + 27*tan(x**(1/3
)/2)**4 + 27*tan(x**(1/3)/2)**2 + 9)

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Giac [A]  time = 1.15009, size = 63, normalized size = 0.73 \begin{align*} \frac{1}{36} \,{\left (9 \, x^{\frac{2}{3}} - 2\right )} \sin \left (3 \, x^{\frac{1}{3}}\right ) + \frac{9}{4} \,{\left (x^{\frac{2}{3}} - 2\right )} \sin \left (x^{\frac{1}{3}}\right ) + \frac{1}{6} \, x^{\frac{1}{3}} \cos \left (3 \, x^{\frac{1}{3}}\right ) + \frac{9}{2} \, x^{\frac{1}{3}} \cos \left (x^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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