\(\int \sqrt [3]{x (1-x^2)} \, dx\) [306]

Optimal result

Integrand size = 13, antiderivative size = 93 \[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\frac {1}{2} x \sqrt [3]{x \left (1-x^2\right )}+\frac {\arctan \left (\frac {2 x-\sqrt [3]{x \left (1-x^2\right )}}{\sqrt {3} \sqrt [3]{x \left (1-x^2\right )}}\right )}{2 \sqrt {3}}+\frac {\log (x)}{12}-\frac {1}{4} \log \left (x+\sqrt [3]{x \left (1-x^2\right )}\right ) \] Output:

1/2*x*(x*(-x^2+1))^(1/3)+1/12*ln(x)-1/4*ln(x+(x*(-x^2+1))^(1/3))+1/6*arcta 
n(1/3*(2*x-(x*(-x^2+1))^(1/3))/(x*(-x^2+1))^(1/3)*3^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.47 \[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\frac {\sqrt [3]{x-x^3} \left (6 x^{4/3} \sqrt [3]{-1+x^2}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )+2 \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-\log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )\right )}{12 \sqrt [3]{x} \sqrt [3]{-1+x^2}} \] Input:

Integrate[(x*(1 - x^2))^(1/3),x]
 

Output:

((x - x^3)^(1/3)*(6*x^(4/3)*(-1 + x^2)^(1/3) + 2*Sqrt[3]*ArcTan[(Sqrt[3]*x 
^(2/3))/(x^(2/3) + 2*(-1 + x^2)^(1/3))] + 2*Log[-x^(2/3) + (-1 + x^2)^(1/3 
)] - Log[x^(4/3) + x^(2/3)*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)]))/(12*x^(1 
/3)*(-1 + x^2)^(1/3))
 

Rubi [A] (warning: unable to verify)

Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2078, 1910, 1938, 266, 807, 853}

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.

\(\displaystyle \int \sqrt [3]{x \left (1-x^2\right )} \, dx\)

\(\Big \downarrow \) 2078

\(\displaystyle \int \sqrt [3]{x-x^3}dx\)

\(\Big \downarrow \) 1910

\(\displaystyle \frac {1}{3} \int \frac {x}{\left (x-x^3\right )^{2/3}}dx+\frac {1}{2} \sqrt [3]{x-x^3} x\)

\(\Big \downarrow \) 1938

\(\displaystyle \frac {\left (1-x^2\right )^{2/3} x^{2/3} \int \frac {\sqrt [3]{x}}{\left (1-x^2\right )^{2/3}}dx}{3 \left (x-x^3\right )^{2/3}}+\frac {1}{2} \sqrt [3]{x-x^3} x\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\left (1-x^2\right )^{2/3} x^{2/3} \int \frac {x}{\left (1-x^2\right )^{2/3}}d\sqrt [3]{x}}{\left (x-x^3\right )^{2/3}}+\frac {1}{2} \sqrt [3]{x-x^3} x\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {\left (1-x^2\right )^{2/3} x^{2/3} \int \frac {x^{2/3}}{(1-x)^{2/3}}dx^{2/3}}{2 \left (x-x^3\right )^{2/3}}+\frac {1}{2} \sqrt [3]{x-x^3} x\)

\(\Big \downarrow \) 853

\(\displaystyle \frac {\left (1-x^2\right )^{2/3} x^{2/3} \left (-\frac {\arctan \left (\frac {1-\frac {2 x^{2/3}}{\sqrt [3]{1-x}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x^{2/3}-\sqrt [3]{1-x}\right )\right )}{2 \left (x-x^3\right )^{2/3}}+\frac {1}{2} \sqrt [3]{x-x^3} x\)

Input:

Int[(x*(1 - x^2))^(1/3),x]
 

Output:

(x*(x - x^3)^(1/3))/2 + (x^(2/3)*(1 - x^2)^(2/3)*(-(ArcTan[(1 - (2*x^(2/3) 
)/(1 - x)^(1/3))/Sqrt[3]]/Sqrt[3]) - Log[-(1 - x)^(1/3) - x^(2/3)]/2))/(2* 
(x - x^3)^(2/3))
 

Defintions of rubi rules used

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, 
x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
 

rule 853

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim 
p[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp 
[Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]
 

rule 1910

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[x*((a*x^j 
 + b*x^n)^p/(n*p + 1)), x] + Simp[a*(n - j)*(p/(n*p + 1))   Int[x^j*(a*x^j 
+ b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] &&  !IntegerQ[p] && LtQ[0, j, 
n] && GtQ[p, 0] && NeQ[n*p + 1, 0]
 

rule 1938

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F 
racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]))   Int[x^(m + j* 
p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !Inte 
gerQ[p] && NeQ[n, j] && PosQ[n - j]
 

rule 2078

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && G 
eneralizedBinomialQ[u, x] &&  !GeneralizedBinomialMatchQ[u, x]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3.

Time = 3.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.16

method	result	size
meijerg	\(\frac {3 x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{2}\right )}{4}\)	\(15\)
pseudoelliptic	\(\frac {x \left (2 \sqrt {3}\, \arctan \left (\frac {\left (-2 \left (-x^{3}+x \right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+6 \left (-x^{3}+x \right )^{\frac {1}{3}} x +\ln \left (\frac {\left (-x^{3}+x \right )^{\frac {2}{3}}-\left (-x^{3}+x \right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )-2 \ln \left (\frac {\left (-x^{3}+x \right )^{\frac {1}{3}}+x}{x}\right )\right )}{12 \left (\left (-x^{3}+x \right )^{\frac {2}{3}}-\left (-x^{3}+x \right )^{\frac {1}{3}} x +x^{2}\right ) \left (\left (-x^{3}+x \right )^{\frac {1}{3}}+x \right )}\)	\(132\)
trager	\(\frac {\left (-x^{3}+x \right )^{\frac {1}{3}} x}{2}-\frac {\ln \left (4959 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+x \right )^{\frac {2}{3}}-22833 \left (-x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -17718 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+7611 \left (-x^{3}+x \right )^{\frac {2}{3}}+5355 \left (-x^{3}+x \right )^{\frac {1}{3}} x -19836 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-1705 x^{2}+9711 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+1085\right )}{6}+\frac {\operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-6354 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+x \right )^{\frac {2}{3}}-16065 \left (-x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -20715 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+5355 \left (-x^{3}+x \right )^{\frac {2}{3}}+7611 \left (-x^{3}+x \right )^{\frac {1}{3}} x +25416 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+1550 x^{2}+4494 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-465\right )}{2}\)	\(305\)
risch	\(\frac {x {\left (-\left (x^{2}-1\right ) x \right )}^{\frac {1}{3}}}{2}+\frac {\left (\frac {\ln \left (-\frac {-35 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-1956 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-4104 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+175 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+23364 x^{4}+5850 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+2010 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+10476 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+4104 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-140 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-38232 x^{2}+35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-54 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+14868}{\left (-1+x \right ) \left (1+x \right )}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \ln \left (\frac {59 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-3750 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-1746 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}-295 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+12600 x^{4}+5850 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+5652 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+24624 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+1746 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+236 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-16380 x^{2}+35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-1902 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+3780}{\left (-1+x \right ) \left (1+x \right )}\right )}{36}\right ) {\left (-\left (x^{2}-1\right ) x \right )}^{\frac {1}{3}} {\left (\left (x^{2}-1\right )^{2} x^{2}\right )}^{\frac {1}{3}}}{\left (x^{2}-1\right ) x}\)	\(538\)

Input:

int((x*(-x^2+1))^(1/3),x,method=_RETURNVERBOSE)
 

Output:

3/4*x^(4/3)*hypergeom([-1/3,2/3],[5/3],x^2)
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.06 \[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {44032959556 \, \sqrt {3} {\left (-x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} + 10524305234 \, \sqrt {3} {\left (-x^{3} + x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + \frac {1}{2} \, {\left (-x^{3} + x\right )}^{\frac {1}{3}} x - \frac {1}{12} \, \log \left (3 \, {\left (-x^{3} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (-x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) \] Input:

integrate((x*(-x^2+1))^(1/3),x, algorithm="fricas")
 

Output:

-1/6*sqrt(3)*arctan((44032959556*sqrt(3)*(-x^3 + x)^(1/3)*x - sqrt(3)*(167 
54327161*x^2 - 2707204793) + 10524305234*sqrt(3)*(-x^3 + x)^(2/3))/(818358 
97185*x^2 - 1102302937)) + 1/2*(-x^3 + x)^(1/3)*x - 1/12*log(3*(-x^3 + x)^ 
(1/3)*x + 3*(-x^3 + x)^(2/3) + 1)
 

Sympy [F]

\[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\int \sqrt [3]{x \left (1 - x^{2}\right )}\, dx \] Input:

integrate((x*(-x**2+1))**(1/3),x)
 

Output:

Integral((x*(1 - x**2))**(1/3), x)
 

Maxima [F]

\[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\int { \left (-{\left (x^{2} - 1\right )} x\right )^{\frac {1}{3}} \,d x } \] Input:

integrate((x*(-x^2+1))^(1/3),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((-(x^2 - 1)*x)^(1/3), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.74 \[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\frac {1}{2} \, x^{2} {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {1}{3}} - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{12} \, \log \left ({\left (\frac {1}{x^{2}} - 1\right )}^{\frac {2}{3}} - {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left | {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \] Input:

integrate((x*(-x^2+1))^(1/3),x, algorithm="giac")
 

Output:

1/2*x^2*(1/x^2 - 1)^(1/3) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(1/x^2 - 1)^ 
(1/3) - 1)) + 1/12*log((1/x^2 - 1)^(2/3) - (1/x^2 - 1)^(1/3) + 1) - 1/6*lo 
g(abs((1/x^2 - 1)^(1/3) + 1))
 

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.31 \[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\frac {3\,x\,{\left (x-x^3\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {2}{3};\ \frac {5}{3};\ x^2\right )}{4\,{\left (1-x^2\right )}^{1/3}} \] Input:

int((-x*(x^2 - 1))^(1/3),x)
 

Output:

(3*x*(x - x^3)^(1/3)*hypergeom([-1/3, 2/3], 5/3, x^2))/(4*(1 - x^2)^(1/3))
 

Reduce [F]

\[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\frac {x^{\frac {4}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}}{2}-\frac {\left (\int \frac {x^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}}{x^{2}-1}d x \right )}{3} \] Input:

int((x*(-x^2+1))^(1/3),x)
 

Output:

(3*x**(1/3)*( - x**2 + 1)**(1/3)*x - 2*int((x**(1/3)*( - x**2 + 1)**(1/3)) 
/(x**2 - 1),x))/6