Integrand size = 8, antiderivative size = 69 \[ \int \sin ^2\left (\sqrt [3]{x}\right ) \, dx=-\frac {3 \sqrt [3]{x}}{4}+\frac {x}{2}+\frac {3}{4} \cos \left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )-\frac {3}{2} x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )+\frac {3}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right ) \] Output:
-3/4*x^(1/3)+1/2*x+3/4*cos(x^(1/3))*sin(x^(1/3))-3/2*x^(2/3)*cos(x^(1/3))* sin(x^(1/3))+3/2*x^(1/3)*sin(x^(1/3))^2
Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.59 \[ \int \sin ^2\left (\sqrt [3]{x}\right ) \, dx=\frac {1}{8} \left (4 x-6 \sqrt [3]{x} \cos \left (2 \sqrt [3]{x}\right )+\left (3-6 x^{2/3}\right ) \sin \left (2 \sqrt [3]{x}\right )\right ) \] Input:
Integrate[Sin[x^(1/3)]^2,x]
Output:
(4*x - 6*x^(1/3)*Cos[2*x^(1/3)] + (3 - 6*x^(2/3))*Sin[2*x^(1/3)])/8
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {3842, 3042, 3792, 15, 3042, 3115, 24}
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.
\(\displaystyle \int \sin ^2\left (\sqrt [3]{x}\right ) \, dx\) |
\(\Big \downarrow \) 3842 |
\(\displaystyle 3 \int x^{2/3} \sin ^2\left (\sqrt [3]{x}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 \int x^{2/3} \sin \left (\sqrt [3]{x}\right )^2d\sqrt [3]{x}\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle 3 \left (\frac {1}{2} \int x^{2/3}d\sqrt [3]{x}-\frac {1}{2} \int \sin ^2\left (\sqrt [3]{x}\right )d\sqrt [3]{x}-\frac {1}{2} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {1}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right )\right )\) |
\(\Big \downarrow \) 15 |
\(\displaystyle 3 \left (-\frac {1}{2} \int \sin ^2\left (\sqrt [3]{x}\right )d\sqrt [3]{x}-\frac {1}{2} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {x}{6}+\frac {1}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 \left (-\frac {1}{2} \int \sin \left (\sqrt [3]{x}\right )^2d\sqrt [3]{x}-\frac {1}{2} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {x}{6}+\frac {1}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right )\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle 3 \left (\frac {1}{2} \left (\frac {1}{2} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )-\frac {\int 1d\sqrt [3]{x}}{2}\right )-\frac {1}{2} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {x}{6}+\frac {1}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right )\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle 3 \left (-\frac {1}{2} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac {x}{6}+\frac {1}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right )+\frac {1}{2} \left (\frac {1}{2} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )-\frac {\sqrt [3]{x}}{2}\right )\right )\) |
Input:
Int[Sin[x^(1/3)]^2,x]
Output:
3*(x/6 - (x^(2/3)*Cos[x^(1/3)]*Sin[x^(1/3)])/2 + (x^(1/3)*Sin[x^(1/3)]^2)/ 2 + (-1/2*x^(1/3) + (Cos[x^(1/3)]*Sin[x^(1/3)])/2)/2)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 0.39 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.28
method | result | size |
meijerg | \(\frac {3 x^{\frac {5}{3}} \operatorname {hypergeom}\left (\left [1, \frac {5}{2}\right ], \left [\frac {3}{2}, 2, \frac {7}{2}\right ], -x^{\frac {2}{3}}\right )}{5}\) | \(19\) |
derivativedivides | \(3 x^{\frac {2}{3}} \left (-\frac {\cos \left (x^{\frac {1}{3}}\right ) \sin \left (x^{\frac {1}{3}}\right )}{2}+\frac {x^{\frac {1}{3}}}{2}\right )-\frac {3 x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right )^{2}}{2}+\frac {3 \cos \left (x^{\frac {1}{3}}\right ) \sin \left (x^{\frac {1}{3}}\right )}{4}+\frac {3 x^{\frac {1}{3}}}{4}-x\) | \(52\) |
default | \(3 x^{\frac {2}{3}} \left (-\frac {\cos \left (x^{\frac {1}{3}}\right ) \sin \left (x^{\frac {1}{3}}\right )}{2}+\frac {x^{\frac {1}{3}}}{2}\right )-\frac {3 x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right )^{2}}{2}+\frac {3 \cos \left (x^{\frac {1}{3}}\right ) \sin \left (x^{\frac {1}{3}}\right )}{4}+\frac {3 x^{\frac {1}{3}}}{4}-x\) | \(52\) |
Input:
int(sin(x^(1/3))^2,x,method=_RETURNVERBOSE)
Output:
3/5*x^(5/3)*hypergeom([1,5/2],[3/2,2,7/2],-x^(2/3))
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.54 \[ \int \sin ^2\left (\sqrt [3]{x}\right ) \, dx=-\frac {3}{4} \, {\left (2 \, x^{\frac {2}{3}} - 1\right )} \cos \left (x^{\frac {1}{3}}\right ) \sin \left (x^{\frac {1}{3}}\right ) - \frac {3}{2} \, x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right )^{2} + \frac {1}{2} \, x + \frac {3}{4} \, x^{\frac {1}{3}} \] Input:
integrate(sin(x^(1/3))^2,x, algorithm="fricas")
Output:
-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)
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (66) = 132\).
Time = 0.41 (sec) , antiderivative size = 379, normalized size of antiderivative = 5.49 \[ \int \sin ^2\left (\sqrt [3]{x}\right ) \, dx=\frac {12 x^{\frac {2}{3}} \tan ^{3}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 4} - \frac {12 x^{\frac {2}{3}} \tan {\left (\frac {\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 4} - \frac {3 \sqrt [3]{x} \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 4} + \frac {18 \sqrt [3]{x} \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 4} - \frac {3 \sqrt [3]{x}}{4 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 4} + \frac {2 x \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 4} + \frac {4 x \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 4} + \frac {2 x}{4 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 4} - \frac {6 \tan ^{3}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 4} + \frac {6 \tan {\left (\frac {\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 4} \] Input:
integrate(sin(x**(1/3))**2,x)
Output:
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*tan (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*tan(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)
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.43 \[ \int \sin ^2\left (\sqrt [3]{x}\right ) \, dx=-\frac {3}{8} \, {\left (2 \, x^{\frac {2}{3}} - 1\right )} \sin \left (2 \, x^{\frac {1}{3}}\right ) - \frac {3}{4} \, x^{\frac {1}{3}} \cos \left (2 \, x^{\frac {1}{3}}\right ) + \frac {1}{2} \, x \] Input:
integrate(sin(x^(1/3))^2,x, algorithm="maxima")
Output:
-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
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.43 \[ \int \sin ^2\left (\sqrt [3]{x}\right ) \, dx=-\frac {3}{8} \, {\left (2 \, x^{\frac {2}{3}} - 1\right )} \sin \left (2 \, x^{\frac {1}{3}}\right ) - \frac {3}{4} \, x^{\frac {1}{3}} \cos \left (2 \, x^{\frac {1}{3}}\right ) + \frac {1}{2} \, x \] Input:
integrate(sin(x^(1/3))^2,x, algorithm="giac")
Output:
-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
Time = 41.81 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.49 \[ \int \sin ^2\left (\sqrt [3]{x}\right ) \, dx=\frac {x}{2}+\frac {3\,\sin \left (2\,x^{1/3}\right )}{8}-\frac {3\,x^{1/3}\,\cos \left (2\,x^{1/3}\right )}{4}-\frac {3\,x^{2/3}\,\sin \left (2\,x^{1/3}\right )}{4} \] Input:
int(sin(x^(1/3))^2,x)
Output:
x/2 + (3*sin(2*x^(1/3)))/8 - (3*x^(1/3)*cos(2*x^(1/3)))/4 - (3*x^(2/3)*sin (2*x^(1/3)))/4
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.62 \[ \int \sin ^2\left (\sqrt [3]{x}\right ) \, dx=-\frac {3 x^{\frac {2}{3}} \cos \left (x^{\frac {1}{3}}\right ) \sin \left (x^{\frac {1}{3}}\right )}{2}+\frac {3 \cos \left (x^{\frac {1}{3}}\right ) \sin \left (x^{\frac {1}{3}}\right )}{4}+\frac {3 x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right )^{2}}{2}-\frac {3 x^{\frac {1}{3}}}{4}+\frac {x}{2} \] Input:
int(sin(x^(1/3))^2,x)
Output:
( - 6*x**(2/3)*cos(x**(1/3))*sin(x**(1/3)) + 3*cos(x**(1/3))*sin(x**(1/3)) + 6*x**(1/3)*sin(x**(1/3))**2 - 3*x**(1/3) + 2*x)/4