\(\int \frac {x}{(1-x^3)^{2/3}} \, dx\) [581]

Optimal result

Integrand size = 13, antiderivative size = 53 \[ \int \frac {x}{\left (1-x^3\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x-\sqrt [3]{1-x^3}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {337} \[ \int \frac {x}{\left (1-x^3\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-\sqrt [3]{1-x^3}-x\right ) \]

[In]

Int[x/(1 - x^3)^(2/3),x]

[Out]

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

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 {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x-\sqrt [3]{1-x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.57 \[ \int \frac {x}{\left (1-x^3\right )^{2/3}} \, dx=\frac {1}{6} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3}}\right )-2 \log \left (x+\sqrt [3]{1-x^3}\right )+\log \left (x^2-x \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )\right ) \]

[In]

Integrate[x/(1 - x^3)^(2/3),x]

[Out]

(-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)])/6

Maple [C] (verified)

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

Time = 4.54 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.28

method	result	size
meijerg	\(\frac {x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};x^{3}\right )}{2}\)	\(15\)
pseudoelliptic	\(-\frac {\ln \left (\frac {x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{3}+\frac {\ln \left (\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}-\left (-x^{3}+1\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (-2 \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )}{3}\)	\(79\)
trager	\(-\frac {\ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \left (-x^{3}+1\right )^{\frac {2}{3}} x -2 x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right )}{3}\)	\(199\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.55 \[ \int \frac {x}{\left (1-x^3\right )^{2/3}} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {1}{3} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]

[In]

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

[Out]

-1/3*sqrt(3)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(-x^3 + 1)^(1/3))/x) - 1/3*log((x + (-x^3 + 1)^(1/3))/x) + 1/6
*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 = 0.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.58 \[ \int \frac {x}{\left (1-x^3\right )^{2/3}} \, dx=\frac {x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} \]

[In]

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

[Out]

x**2*gamma(2/3)*hyper((2/3, 2/3), (5/3,), x**3*exp_polar(2*I*pi))/(3*gamma(5/3))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.47 \[ \int \frac {x}{\left (1-x^3\right )^{2/3}} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right )}\right ) - \frac {1}{3} \, \log \left (\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right ) + \frac {1}{6} \, \log \left (-\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(x/(-x^3 + 1)^(2/3), x)

Mupad [F(-1)]

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

[In]

int(x/(1 - x^3)^(2/3),x)

[Out]

int(x/(1 - x^3)^(2/3), x)