3.1.61 \(\int \cos ^3(\sqrt [3]{x}) \, dx\) [61]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, 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 = {3443, 3392, 3377, 2717, 2713} \begin {gather*} 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 ) \end {gather*}

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 2713

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]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

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 3392

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 3443

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]

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 66, normalized size = 0.77 \begin {gather*} \frac {1}{36} \left (162 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \cos \left (3 \sqrt [3]{x}\right )+81 \left (-2+x^{2/3}\right ) \sin \left (\sqrt [3]{x}\right )+\left (-2+9 x^{2/3}\right ) \sin \left (3 \sqrt [3]{x}\right )\right ) \end {gather*}

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 = 0.04, size = 58, normalized size = 0.67

method	result	size
derivativedivides	\(x^{\frac {2}{3}} \left (2+\cos ^{2}\left (x^{\frac {1}{3}}\right )\right ) \sin \left (x^{\frac {1}{3}}\right )-4 \sin \left (x^{\frac {1}{3}}\right )+4 x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right )+\frac {2 x^{\frac {1}{3}} \left (\cos ^{3}\left (x^{\frac {1}{3}}\right )\right )}{3}-\frac {2 \left (2+\cos ^{2}\left (x^{\frac {1}{3}}\right )\right ) \sin \left (x^{\frac {1}{3}}\right )}{9}\)	\(58\)
default	\(x^{\frac {2}{3}} \left (2+\cos ^{2}\left (x^{\frac {1}{3}}\right )\right ) \sin \left (x^{\frac {1}{3}}\right )-4 \sin \left (x^{\frac {1}{3}}\right )+4 x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right )+\frac {2 x^{\frac {1}{3}} \left (\cos ^{3}\left (x^{\frac {1}{3}}\right )\right )}{3}-\frac {2 \left (2+\cos ^{2}\left (x^{\frac {1}{3}}\right )\right ) \sin \left (x^{\frac {1}{3}}\right )}{9}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x^(1/3))^3,x,method=_RETURNVERBOSE)

[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 = 0.30, size = 47, normalized size = 0.55 \begin {gather*} \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 {gather*}

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 = 0.35, size = 48, normalized size = 0.56 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (85) = 170\).
time = 0.97, size = 513, normalized size = 5.97 \begin {gather*} \frac {54 x^{\frac {2}{3}} \tan ^{5}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} + \frac {36 x^{\frac {2}{3}} \tan ^{3}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} + \frac {54 x^{\frac {2}{3}} \tan {\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} - \frac {42 \sqrt [3]{x} \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} - \frac {18 \sqrt [3]{x} \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} + \frac {18 \sqrt [3]{x} \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} + \frac {42 \sqrt [3]{x}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} - \frac {84 \tan ^{5}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} - \frac {152 \tan ^{3}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} - \frac {84 \tan {\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} \end {gather*}

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 = 0.42, size = 47, normalized size = 0.55 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 0.50, size = 62, normalized size = 0.72 \begin {gather*} 4\,x^{1/3}\,\cos \left (x^{1/3}\right )-\frac {2\,{\cos \left (x^{1/3}\right )}^2\,\sin \left (x^{1/3}\right )}{9}-\frac {40\,\sin \left (x^{1/3}\right )}{9}+2\,x^{2/3}\,\sin \left (x^{1/3}\right )+\frac {2\,x^{1/3}\,{\cos \left (x^{1/3}\right )}^3}{3}+x^{2/3}\,{\cos \left (x^{1/3}\right )}^2\,\sin \left (x^{1/3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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