\(\int x \sqrt [3]{-1+x^3} \, dx\) [1326]

Optimal result

Integrand size = 11, antiderivative size = 96 \[ \int x \sqrt [3]{-1+x^3} \, dx=\frac {1}{3} x^2 \sqrt [3]{-1+x^3}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{18} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]

[Out]

1/3*x^2*(x^3-1)^(1/3)+1/9*arctan(3^(1/2)*x/(x+2*(x^3-1)^(1/3)))*3^(1/2)+1/9*ln(-x+(x^3-1)^(1/3))-1/18*ln(x^2+x
*(x^3-1)^(1/3)+(x^3-1)^(2/3))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.68, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {285, 337} \[ \int x \sqrt [3]{-1+x^3} \, dx=\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{6} \log \left (x-\sqrt [3]{x^3-1}\right )+\frac {1}{3} \sqrt [3]{x^3-1} x^2 \]

[In]

Int[x*(-1 + x^3)^(1/3),x]

[Out]

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

Rule 285

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^2 \sqrt [3]{-1+x^3}-\frac {1}{3} \int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = \frac {1}{3} x^2 \sqrt [3]{-1+x^3}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{6} \log \left (x-\sqrt [3]{-1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int x \sqrt [3]{-1+x^3} \, dx=\frac {1}{18} \left (6 x^2 \sqrt [3]{-1+x^3}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )+2 \log \left (-x+\sqrt [3]{-1+x^3}\right )-\log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]

[In]

Integrate[x*(-1 + x^3)^(1/3),x]

[Out]

(6*x^2*(-1 + x^3)^(1/3) + 2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))] + 2*Log[-x + (-1 + x^3)^(1/3)
] - Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/18

Maple [C] (warning: unable to verify)

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

Time = 2.79 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.34

method	result	size
meijerg	\(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{2 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\)	\(33\)
risch	\(\frac {x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}}{3}-\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{6 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\)	\(46\)
pseudoelliptic	\(\frac {-6 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )-2 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )}{18 \left (-x +\left (x^{3}-1\right )^{\frac {1}{3}}\right ) \left (x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )}\)	\(119\)
trager	\(\frac {x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}}{3}+\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{9}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+4 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right )}{9}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+4 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right )}{9}\)	\(308\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.92 \[ \int x \sqrt [3]{-1+x^3} \, dx=\frac {1}{3} \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{9} \, \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{18} \, \log \left (\frac {x^{2} + {\left (x^{3} - 1\right )}^{\frac {1}{3}} x + {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]

[In]

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

[Out]

1/3*(x^3 - 1)^(1/3)*x^2 - 1/9*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - 1)^(1/3))/x) + 1/9*log(-(x - (x
^3 - 1)^(1/3))/x) - 1/18*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 = 0.75 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.38 \[ \int x \sqrt [3]{-1+x^3} \, dx=- \frac {x^{2} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} \]

[In]

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

[Out]

-x**2*exp(-2*I*pi/3)*gamma(2/3)*hyper((-1/3, 2/3), (5/3,), x**3)/(3*gamma(5/3))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.98 \[ \int x \sqrt [3]{-1+x^3} \, dx=-\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x {\left (\frac {x^{3} - 1}{x^{3}} - 1\right )}} - \frac {1}{18} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {1}{9} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate((x^3 - 1)^(1/3)*x, x)

Mupad [F(-1)]

Timed out. \[ \int x \sqrt [3]{-1+x^3} \, dx=\int x\,{\left (x^3-1\right )}^{1/3} \,d x \]

[In]

int(x*(x^3 - 1)^(1/3),x)

[Out]

int(x*(x^3 - 1)^(1/3), x)