Integrand size = 8, antiderivative size = 86 \[ \int \cos ^3\left (\sqrt [3]{x}\right ) \, dx=4 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )+\frac {2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )-\frac {14}{3} \sin \left (\sqrt [3]{x}\right )+2 x^{2/3} \sin \left (\sqrt [3]{x}\right )+x^{2/3} \cos ^2\left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )+\frac {2}{9} \sin ^3\left (\sqrt [3]{x}\right ) \]
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
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77 \[ \int \cos ^3\left (\sqrt [3]{x}\right ) \, dx=\frac {1}{36} \left (162 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \cos \left (3 \sqrt [3]{x}\right )+81 \left (-2+x^{2/3}\right ) \sin \left (\sqrt [3]{x}\right )+\left (-2+9 x^{2/3}\right ) \sin \left (3 \sqrt [3]{x}\right )\right ) \]
(162*x^(1/3)*Cos[x^(1/3)] + 6*x^(1/3)*Cos[3*x^(1/3)] + 81*(-2 + x^(2/3))*S in[x^(1/3)] + (-2 + 9*x^(2/3))*Sin[3*x^(1/3)])/36
Time = 0.49 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {3843, 3042, 3792, 3042, 3113, 2009, 3777, 25, 3042, 3777, 3042, 3117}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cos ^3\left (\sqrt [3]{x}\right ) \, dx\) |
| \(\Big \downarrow \) 3843 |
| \(\displaystyle 3 \int x^{2/3} \cos ^3\left (\sqrt [3]{x}\right )d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \int x^{2/3} \sin \left (\sqrt [3]{x}+\frac {\pi }{2}\right )^3d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle 3 \left (\frac {2}{3} \int x^{2/3} \cos \left (\sqrt [3]{x}\right )d\sqrt [3]{x}-\frac {2}{9} \int \cos ^3\left (\sqrt [3]{x}\right )d\sqrt [3]{x}+\frac {1}{3} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac {2}{9} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (\frac {2}{3} \int x^{2/3} \sin \left (\sqrt [3]{x}+\frac {\pi }{2}\right )d\sqrt [3]{x}-\frac {2}{9} \int \sin \left (\sqrt [3]{x}+\frac {\pi }{2}\right )^3d\sqrt [3]{x}+\frac {1}{3} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac {2}{9} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )\right )\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle 3 \left (\frac {2}{9} \int \left (1-x^{2/3}\right )d\left (-\sin \left (\sqrt [3]{x}\right )\right )+\frac {2}{3} \int x^{2/3} \sin \left (\sqrt [3]{x}+\frac {\pi }{2}\right )d\sqrt [3]{x}+\frac {1}{3} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac {2}{9} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {2}{3} \int x^{2/3} \sin \left (\sqrt [3]{x}+\frac {\pi }{2}\right )d\sqrt [3]{x}+\frac {1}{3} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (-\frac {x}{3}-\sin \left (\sqrt [3]{x}\right )\right )+\frac {2}{9} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle 3 \left (\frac {2}{3} \left (2 \int -\sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )d\sqrt [3]{x}+x^{2/3} \sin \left (\sqrt [3]{x}\right )\right )+\frac {1}{3} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (-\frac {x}{3}-\sin \left (\sqrt [3]{x}\right )\right )+\frac {2}{9} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (\frac {2}{3} \left (x^{2/3} \sin \left (\sqrt [3]{x}\right )-2 \int \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )d\sqrt [3]{x}\right )+\frac {1}{3} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (-\frac {x}{3}-\sin \left (\sqrt [3]{x}\right )\right )+\frac {2}{9} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (\frac {2}{3} \left (x^{2/3} \sin \left (\sqrt [3]{x}\right )-2 \int \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )d\sqrt [3]{x}\right )+\frac {1}{3} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (-\frac {x}{3}-\sin \left (\sqrt [3]{x}\right )\right )+\frac {2}{9} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle 3 \left (\frac {2}{3} \left (x^{2/3} \sin \left (\sqrt [3]{x}\right )-2 \left (\int \cos \left (\sqrt [3]{x}\right )d\sqrt [3]{x}-\sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )\right )\right )+\frac {1}{3} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (-\frac {x}{3}-\sin \left (\sqrt [3]{x}\right )\right )+\frac {2}{9} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (\frac {2}{3} \left (x^{2/3} \sin \left (\sqrt [3]{x}\right )-2 \left (\int \sin \left (\sqrt [3]{x}+\frac {\pi }{2}\right )d\sqrt [3]{x}-\sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )\right )\right )+\frac {1}{3} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (-\frac {x}{3}-\sin \left (\sqrt [3]{x}\right )\right )+\frac {2}{9} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )\right )\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle 3 \left (\frac {1}{3} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac {2}{3} \left (x^{2/3} \sin \left (\sqrt [3]{x}\right )-2 \left (\sin \left (\sqrt [3]{x}\right )-\sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )\right )\right )+\frac {2}{9} \left (-\frac {x}{3}-\sin \left (\sqrt [3]{x}\right )\right )+\frac {2}{9} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )\right )\) |
3*((2*x^(1/3)*Cos[x^(1/3)]^3)/9 + (2*(-1/3*x - Sin[x^(1/3)]))/9 + (x^(2/3) *Cos[x^(1/3)]^2*Sin[x^(1/3)])/3 + (2*(x^(2/3)*Sin[x^(1/3)] - 2*(-(x^(1/3)* Cos[x^(1/3)]) + Sin[x^(1/3)])))/3)
3.1.61.3.1 Defintions of rubi rules used
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.), x_S ymbol] :> Simp[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] && Intege rQ[1/n]
Time = 1.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 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 | \(x^{\frac {2}{3}} \left (2+\cos ^{2}\left (x^{\frac {1}{3}}\right )\right ) \sin \left (x^{\frac {1}{3}}\right )-4 \sin \left (x^{\frac {1}{3}}\right )+4 x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right )+\frac {2 x^{\frac {1}{3}} \left (\cos ^{3}\left (x^{\frac {1}{3}}\right )\right )}{3}-\frac {2 \left (2+\cos ^{2}\left (x^{\frac {1}{3}}\right )\right ) \sin \left (x^{\frac {1}{3}}\right )}{9}\) | \(58\) |
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))
Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.56 \[ \int \cos ^3\left (\sqrt [3]{x}\right ) \, dx=\frac {2}{3} \, x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right )^{3} + \frac {1}{9} \, {\left ({\left (9 \, x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right )^{2} + 18 \, x^{\frac {2}{3}} - 40\right )} \sin \left (x^{\frac {1}{3}}\right ) + 4 \, x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right ) \]
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)*cos(x^(1/3))
Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (85) = 170\).
Time = 0.99 (sec) , antiderivative size = 513, normalized size of antiderivative = 5.97 \[ \int \cos ^3\left (\sqrt [3]{x}\right ) \, dx=\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} \]
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*t an(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)*tan(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)
Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.55 \[ \int \cos ^3\left (\sqrt [3]{x}\right ) \, dx=\frac {1}{36} \, {\left (9 \, x^{\frac {2}{3}} - 2\right )} \sin \left (3 \, x^{\frac {1}{3}}\right ) + \frac {9}{4} \, {\left (x^{\frac {2}{3}} - 2\right )} \sin \left (x^{\frac {1}{3}}\right ) + \frac {1}{6} \, x^{\frac {1}{3}} \cos \left (3 \, x^{\frac {1}{3}}\right ) + \frac {9}{2} \, x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right ) \]
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))
Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.55 \[ \int \cos ^3\left (\sqrt [3]{x}\right ) \, dx=\frac {1}{36} \, {\left (9 \, x^{\frac {2}{3}} - 2\right )} \sin \left (3 \, x^{\frac {1}{3}}\right ) + \frac {9}{4} \, {\left (x^{\frac {2}{3}} - 2\right )} \sin \left (x^{\frac {1}{3}}\right ) + \frac {1}{6} \, x^{\frac {1}{3}} \cos \left (3 \, x^{\frac {1}{3}}\right ) + \frac {9}{2} \, x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right ) \]
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))
Time = 13.56 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.72 \[ \int \cos ^3\left (\sqrt [3]{x}\right ) \, dx=4\,x^{1/3}\,\cos \left (x^{1/3}\right )-\frac {2\,{\cos \left (x^{1/3}\right )}^2\,\sin \left (x^{1/3}\right )}{9}-\frac {40\,\sin \left (x^{1/3}\right )}{9}+2\,x^{2/3}\,\sin \left (x^{1/3}\right )+\frac {2\,x^{1/3}\,{\cos \left (x^{1/3}\right )}^3}{3}+x^{2/3}\,{\cos \left (x^{1/3}\right )}^2\,\sin \left (x^{1/3}\right ) \]