Integrand size = 8, antiderivative size = 87 \[ \int \sin ^3\left (\sqrt [3]{x}\right ) \, dx=\frac {14}{3} \cos \left (\sqrt [3]{x}\right )-2 x^{2/3} \cos \left (\sqrt [3]{x}\right )-\frac {2}{9} \cos ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin ^2\left (\sqrt [3]{x}\right )+\frac {2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right ) \]
14/3*cos(x^(1/3))-2*x^(2/3)*cos(x^(1/3))-2/9*cos(x^(1/3))^3+4*x^(1/3)*sin( x^(1/3))-x^(2/3)*cos(x^(1/3))*sin(x^(1/3))^2+2/3*x^(1/3)*sin(x^(1/3))^3
Time = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \sin ^3\left (\sqrt [3]{x}\right ) \, dx=\frac {1}{36} \left (-81 \left (-2+x^{2/3}\right ) \cos \left (\sqrt [3]{x}\right )+\left (-2+9 x^{2/3}\right ) \cos \left (3 \sqrt [3]{x}\right )-6 \sqrt [3]{x} \left (-27 \sin \left (\sqrt [3]{x}\right )+\sin \left (3 \sqrt [3]{x}\right )\right )\right ) \]
(-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
Time = 0.48 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {3842, 3042, 3792, 3042, 3113, 2009, 3777, 3042, 3777, 25, 3042, 3118}
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.
\(\displaystyle \int \sin ^3\left (\sqrt [3]{x}\right ) \, dx\) |
\(\Big \downarrow \) 3842 |
\(\displaystyle 3 \int x^{2/3} \sin ^3\left (\sqrt [3]{x}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 \int x^{2/3} \sin \left (\sqrt [3]{x}\right )^3d\sqrt [3]{x}\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle 3 \left (\frac {2}{3} \int x^{2/3} \sin \left (\sqrt [3]{x}\right )d\sqrt [3]{x}-\frac {2}{9} \int \sin ^3\left (\sqrt [3]{x}\right )d\sqrt [3]{x}-\frac {1}{3} x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {2}{9} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 \left (\frac {2}{3} \int x^{2/3} \sin \left (\sqrt [3]{x}\right )d\sqrt [3]{x}-\frac {2}{9} \int \sin \left (\sqrt [3]{x}\right )^3d\sqrt [3]{x}-\frac {1}{3} x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {2}{9} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )\right )\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle 3 \left (\frac {2}{3} \int x^{2/3} \sin \left (\sqrt [3]{x}\right )d\sqrt [3]{x}+\frac {2}{9} \int \left (1-x^{2/3}\right )d\cos \left (\sqrt [3]{x}\right )-\frac {1}{3} x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {2}{9} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {2}{3} \int x^{2/3} \sin \left (\sqrt [3]{x}\right )d\sqrt [3]{x}-\frac {1}{3} x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {2}{9} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (\cos \left (\sqrt [3]{x}\right )-\frac {x}{3}\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle 3 \left (\frac {2}{3} \left (2 \int \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )d\sqrt [3]{x}-x^{2/3} \cos \left (\sqrt [3]{x}\right )\right )-\frac {1}{3} x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {2}{9} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (\cos \left (\sqrt [3]{x}\right )-\frac {x}{3}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 \left (\frac {2}{3} \left (2 \int \sqrt [3]{x} \sin \left (\sqrt [3]{x}+\frac {\pi }{2}\right )d\sqrt [3]{x}-x^{2/3} \cos \left (\sqrt [3]{x}\right )\right )-\frac {1}{3} x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {2}{9} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (\cos \left (\sqrt [3]{x}\right )-\frac {x}{3}\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle 3 \left (\frac {2}{3} \left (2 \left (\int -\sin \left (\sqrt [3]{x}\right )d\sqrt [3]{x}+\sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )\right )-x^{2/3} \cos \left (\sqrt [3]{x}\right )\right )-\frac {1}{3} x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {2}{9} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (\cos \left (\sqrt [3]{x}\right )-\frac {x}{3}\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 3 \left (\frac {2}{3} \left (2 \left (\sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-\int \sin \left (\sqrt [3]{x}\right )d\sqrt [3]{x}\right )-x^{2/3} \cos \left (\sqrt [3]{x}\right )\right )-\frac {1}{3} x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {2}{9} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (\cos \left (\sqrt [3]{x}\right )-\frac {x}{3}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 \left (\frac {2}{3} \left (2 \left (\sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-\int \sin \left (\sqrt [3]{x}\right )d\sqrt [3]{x}\right )-x^{2/3} \cos \left (\sqrt [3]{x}\right )\right )-\frac {1}{3} x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {2}{9} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (\cos \left (\sqrt [3]{x}\right )-\frac {x}{3}\right )\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle 3 \left (-\frac {1}{3} x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {2}{3} \left (2 \left (\sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )+\cos \left (\sqrt [3]{x}\right )\right )-x^{2/3} \cos \left (\sqrt [3]{x}\right )\right )+\frac {2}{9} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+\frac {2}{9} \left (\cos \left (\sqrt [3]{x}\right )-\frac {x}{3}\right )\right )\) |
3*((2*(-1/3*x + Cos[x^(1/3)]))/9 - (x^(2/3)*Cos[x^(1/3)]*Sin[x^(1/3)]^2)/3 + (2*x^(1/3)*Sin[x^(1/3)]^3)/9 + (2*(-(x^(2/3)*Cos[x^(1/3)]) + 2*(Cos[x^( 1/3)] + x^(1/3)*Sin[x^(1/3)])))/3)
3.2.28.3.1 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[((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_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_S ymbol] :> Simp[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] && Intege rQ[1/n]
Time = 0.36 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.68
method | result | size |
derivativedivides | \(-x^{\frac {2}{3}} \left (2+\sin ^{2}\left (x^{\frac {1}{3}}\right )\right ) \cos \left (x^{\frac {1}{3}}\right )+4 \cos \left (x^{\frac {1}{3}}\right )+4 x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right )+\frac {2 x^{\frac {1}{3}} \left (\sin ^{3}\left (x^{\frac {1}{3}}\right )\right )}{3}+\frac {2 \left (2+\sin ^{2}\left (x^{\frac {1}{3}}\right )\right ) \cos \left (x^{\frac {1}{3}}\right )}{9}\) | \(59\) |
default | \(-x^{\frac {2}{3}} \left (2+\sin ^{2}\left (x^{\frac {1}{3}}\right )\right ) \cos \left (x^{\frac {1}{3}}\right )+4 \cos \left (x^{\frac {1}{3}}\right )+4 x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right )+\frac {2 x^{\frac {1}{3}} \left (\sin ^{3}\left (x^{\frac {1}{3}}\right )\right )}{3}+\frac {2 \left (2+\sin ^{2}\left (x^{\frac {1}{3}}\right )\right ) \cos \left (x^{\frac {1}{3}}\right )}{9}\) | \(59\) |
-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))
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.59 \[ \int \sin ^3\left (\sqrt [3]{x}\right ) \, dx=\frac {1}{9} \, {\left (9 \, x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right )^{3} - \frac {1}{3} \, {\left (9 \, x^{\frac {2}{3}} - 14\right )} \cos \left (x^{\frac {1}{3}}\right ) - \frac {2}{3} \, {\left (x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right )^{2} - 7 \, x^{\frac {1}{3}}\right )} \sin \left (x^{\frac {1}{3}}\right ) \]
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))
Time = 3.36 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.82 \[ \int \sin ^3\left (\sqrt [3]{x}\right ) \, dx=3 x^{\frac {2}{3}} \left (\frac {\cos ^{3}{\left (\sqrt [3]{x} \right )}}{3} - \cos {\left (\sqrt [3]{x} \right )}\right ) - 2 \sqrt [3]{x} \left (- \frac {\sin ^{3}{\left (\sqrt [3]{x} \right )}}{3} - 2 \sin {\left (\sqrt [3]{x} \right )}\right ) - \frac {2 \cos ^{3}{\left (\sqrt [3]{x} \right )}}{9} + \frac {14 \cos {\left (\sqrt [3]{x} \right )}}{3} \]
3*x**(2/3)*(cos(x**(1/3))**3/3 - cos(x**(1/3))) - 2*x**(1/3)*(-sin(x**(1/3 ))**3/3 - 2*sin(x**(1/3))) - 2*cos(x**(1/3))**3/9 + 14*cos(x**(1/3))/3
Time = 0.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.54 \[ \int \sin ^3\left (\sqrt [3]{x}\right ) \, dx=\frac {1}{36} \, {\left (9 \, x^{\frac {2}{3}} - 2\right )} \cos \left (3 \, x^{\frac {1}{3}}\right ) - \frac {9}{4} \, {\left (x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right ) - \frac {1}{6} \, x^{\frac {1}{3}} \sin \left (3 \, x^{\frac {1}{3}}\right ) + \frac {9}{2} \, x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right ) \]
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))
Time = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.54 \[ \int \sin ^3\left (\sqrt [3]{x}\right ) \, dx=\frac {1}{36} \, {\left (9 \, x^{\frac {2}{3}} - 2\right )} \cos \left (3 \, x^{\frac {1}{3}}\right ) - \frac {9}{4} \, {\left (x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right ) - \frac {1}{6} \, x^{\frac {1}{3}} \sin \left (3 \, x^{\frac {1}{3}}\right ) + \frac {9}{2} \, x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right ) \]
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))
Time = 6.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \sin ^3\left (\sqrt [3]{x}\right ) \, dx=\frac {14\,\cos \left (x^{1/3}\right )}{3}-3\,x^{2/3}\,\cos \left (x^{1/3}\right )+\frac {14\,x^{1/3}\,\sin \left (x^{1/3}\right )}{3}-\frac {2\,{\cos \left (x^{1/3}\right )}^3}{9}+x^{2/3}\,{\cos \left (x^{1/3}\right )}^3-\frac {2\,x^{1/3}\,{\cos \left (x^{1/3}\right )}^2\,\sin \left (x^{1/3}\right )}{3} \]