\(\int (-1+x^3)^{2/3} \, dx\) [1294]

Optimal result

Integrand size = 9, antiderivative size = 94 \[ \int \left (-1+x^3\right )^{2/3} \, dx=\frac {1}{3} x \left (-1+x^3\right )^{2/3}-\frac {2 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{9} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]

[Out]

1/3*x*(x^3-1)^(2/3)-2/9*arctan(3^(1/2)*x/(x+2*(x^3-1)^(1/3)))*3^(1/2)+2/9*ln(-x+(x^3-1)^(1/3))-1/9*ln(x^2+x*(x
^3-1)^(1/3)+(x^3-1)^(2/3))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {201, 245} \[ \int \left (-1+x^3\right )^{2/3} \, dx=-\frac {2 \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \left (x^3-1\right )^{2/3} x+\frac {1}{3} \log \left (\sqrt [3]{x^3-1}-x\right ) \]

[In]

Int[(-1 + x^3)^(2/3),x]

[Out]

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

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 245

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x \left (-1+x^3\right )^{2/3}-\frac {2}{3} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx \\ & = \frac {1}{3} x \left (-1+x^3\right )^{2/3}-\frac {2 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05 \[ \int \left (-1+x^3\right )^{2/3} \, dx=\frac {3 (-1+x) \left (-1+x^3\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{3},-\frac {2}{3},-\frac {2}{3},\frac {8}{3},-\frac {-1+x}{1-(-1)^{2/3}},-\frac {-1+x}{1+\sqrt [3]{-1}}\right )}{5 \left (1+\frac {-1+x}{1+\sqrt [3]{-1}}\right )^{2/3} \left (1+\frac {-1+x}{1-(-1)^{2/3}}\right )^{2/3}} \]

[In]

Integrate[(-1 + x^3)^(2/3),x]

[Out]

(3*(-1 + x)*(-1 + x^3)^(2/3)*AppellF1[5/3, -2/3, -2/3, 8/3, -((-1 + x)/(1 - (-1)^(2/3))), -((-1 + x)/(1 + (-1)
^(1/3)))])/(5*(1 + (-1 + x)/(1 + (-1)^(1/3)))^(2/3)*(1 + (-1 + x)/(1 - (-1)^(2/3)))^(2/3))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.68 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.32

method	result	size
meijerg	\(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} x \operatorname {hypergeom}\left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}}}\)	\(30\)
risch	\(\frac {x \left (x^{3}-1\right )^{\frac {2}{3}}}{3}-\frac {2 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{3 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\)	\(42\)
pseudoelliptic	\(\frac {-3 x \left (x^{3}-1\right )^{\frac {2}{3}}-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )-2 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )}{9 \left (-x +\left (x^{3}-1\right )^{\frac {1}{3}}\right ) \left (x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )}\)	\(117\)
trager	\(\frac {x \left (x^{3}-1\right )^{\frac {2}{3}}}{3}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+4 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right )}{9}-\frac {2 \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{9}-\frac {2 \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{9}\)	\(280\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \left (-1+x^3\right )^{2/3} \, dx=\frac {1}{3} \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {2}{9} \, \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{9} \, \log \left (\frac {x^{2} + {\left (x^{3} - 1\right )}^{\frac {1}{3}} x + {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]

[In]

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

[Out]

1/3*(x^3 - 1)^(2/3)*x + 2/9*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - 1)^(1/3))/x) + 2/9*log(-(x - (x^3
- 1)^(1/3))/x) - 1/9*log((x^2 + (x^3 - 1)^(1/3)*x + (x^3 - 1)^(2/3))/x^2)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.34 \[ \int \left (-1+x^3\right )^{2/3} \, dx=- \frac {x e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \]

[In]

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

[Out]

-x*exp(-I*pi/3)*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), x**3)/(3*gamma(4/3))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \left (-1+x^3\right )^{2/3} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} - 1}{x^{3}} - 1\right )}} - \frac {1}{9} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {2}{9} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]

[In]

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

[Out]

2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3)/x + 1)) - 1/3*(x^3 - 1)^(2/3)/(x^2*((x^3 - 1)/x^3 - 1)) - 1/
9*log((x^3 - 1)^(1/3)/x + (x^3 - 1)^(2/3)/x^2 + 1) + 2/9*log((x^3 - 1)^(1/3)/x - 1)

Giac [F]

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

[In]

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

[Out]

integrate((x^3 - 1)^(2/3), x)

Mupad [B] (verification not implemented)

Time = 5.86 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.28 \[ \int \left (-1+x^3\right )^{2/3} \, dx=\frac {x\,{\left (x^3-1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {1}{3};\ \frac {4}{3};\ x^3\right )}{{\left (1-x^3\right )}^{2/3}} \]

[In]

int((x^3 - 1)^(2/3),x)

[Out]

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