\(\int \frac {(-1+x^3)^{2/3}}{x} \, dx\) [1157]

Optimal result

Integrand size = 13, antiderivative size = 86 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x} \, dx=\frac {1}{2} \left (-1+x^3\right )^{2/3}+\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {1}{6} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \] Output:

1/2*(x^3-1)^(2/3)-1/3*arctan(-1/3*3^(1/2)+2/3*(x^3-1)^(1/3)*3^(1/2))*3^(1/ 
2)+1/3*ln(1+(x^3-1)^(1/3))-1/6*ln(1-(x^3-1)^(1/3)+(x^3-1)^(2/3))
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x} \, dx=\frac {1}{6} \left (3 \left (-1+x^3\right )^{2/3}+2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+2 \log \left (1+\sqrt [3]{-1+x^3}\right )-\log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.80, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {798, 60, 68, 16, 1083, 217}

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.

\(\displaystyle \int \frac {\left (x^3-1\right )^{2/3}}{x} \, dx\)

\(\Big \downarrow \) 798

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 68

\(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {1}{\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}-\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}+\frac {3}{2} \left (x^3-1\right )^{2/3}-\frac {1}{2} \log \left (x^3\right )\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}+\frac {3}{2} \left (x^3-1\right )^{2/3}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{3} \left (3 \int \frac {1}{-x^6-3}d\left (2 \sqrt [3]{x^3-1}-1\right )+\frac {3}{2} \left (x^3-1\right )^{2/3}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )\)

\(\Big \downarrow \) 217

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 68

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[-(b*c - a*d)/b, 3]}, Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 798

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

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

Maple [A] (verified)

Time = 2.31 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(\frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{2}-\frac {\ln \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\ln \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3}\)	\(66\)
meijerg	\(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], x^{3}\right )}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}\right )}{9 \pi {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}}}\)	\(84\)
trager	\(\frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{2}+\frac {\ln \left (\frac {-211 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3126 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \left (x^{3}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-11543 x^{3}-14247 \left (x^{3}-1\right )^{\frac {2}{3}}+19749 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+1688 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+5502 \left (x^{3}-1\right )^{\frac {1}{3}}+15935 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+21437}{x^{3}}\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-1649 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+9683 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \left (x^{3}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1266 x^{3}+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-14247 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+13192 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+5502 \left (x^{3}-1\right )^{\frac {1}{3}}-6557 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1055}{x^{3}}\right )}{3}\)	\(239\)

Input:

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

Output:

1/2*(x^3-1)^(2/3)-1/6*ln(1-(x^3-1)^(1/3)+(x^3-1)^(2/3))-1/3*3^(1/2)*arctan 
(1/3*(2*(x^3-1)^(1/3)-1)*3^(1/2))+1/3*ln(1+(x^3-1)^(1/3))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.78 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{2} \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \] Input:

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

Output:

-1/3*sqrt(3)*arctan(2/3*sqrt(3)*(x^3 - 1)^(1/3) - 1/3*sqrt(3)) + 1/2*(x^3 
- 1)^(2/3) - 1/6*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 1/3*log((x^3 
 - 1)^(1/3) + 1)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.68 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.43 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x} \, dx=- \frac {x^{2} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 \Gamma \left (\frac {1}{3}\right )} \] Input:

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

Output:

-x**2*gamma(-2/3)*hyper((-2/3, -2/3), (1/3,), exp_polar(2*I*pi)/x**3)/(3*g 
amma(1/3))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{2} \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \] Input:

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

Output:

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/2*(x^3 - 1)^( 
2/3) - 1/6*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 1/3*log((x^3 - 1)^ 
(1/3) + 1)
 

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{2} \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x} \, dx=\frac {\ln \left ({\left (x^3-1\right )}^{1/3}+1\right )}{3}+\frac {{\left (x^3-1\right )}^{2/3}}{2}+\ln \left (9\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2+{\left (x^3-1\right )}^{1/3}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (9\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2+{\left (x^3-1\right )}^{1/3}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \] Input:

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

Output:

log((x^3 - 1)^(1/3) + 1)/3 + (x^3 - 1)^(2/3)/2 + log(9*((3^(1/2)*1i)/6 - 1 
/6)^2 + (x^3 - 1)^(1/3))*((3^(1/2)*1i)/6 - 1/6) - log(9*((3^(1/2)*1i)/6 + 
1/6)^2 + (x^3 - 1)^(1/3))*((3^(1/2)*1i)/6 + 1/6)
 

Reduce [F]

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

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

Output:

((x**3 - 1)**(2/3) - 2*int((x**3 - 1)**(2/3)/(x**4 - x),x))/2