3.1.56 \(\int x \sqrt [3]{1-x^3} \, dx\) [56]

Optimal. Leaf size=73 \[ \frac {1}{3} x^2 \sqrt [3]{1-x^3}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6} \log \left (-x-\sqrt [3]{1-x^3}\right ) \]

[Out]

1/3*x^2*(-x^3+1)^(1/3)-1/6*ln(-x-(-x^3+1)^(1/3))-1/9*arctan(1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {285, 337} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6} \log \left (-\sqrt [3]{1-x^3}-x\right )+\frac {1}{3} \sqrt [3]{1-x^3} x^2 \end {gather*}

Antiderivative was successfully verified.

[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*} \int x \sqrt [3]{1-x^3} \, dx &=\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 {1}{3} \text {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1-x^3}-\frac {1}{9} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{9} \text {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1-x^3}-\frac {1}{9} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{18} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{6} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1-x^3}+\frac {1}{18} \log \left (1+\frac {x^2}{\left (1-x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{9} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1-x^3}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{18} \log \left (1+\frac {x^2}{\left (1-x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{9} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 99, normalized size = 1.36 \begin {gather*} \frac {1}{18} \left (6 x^2 \sqrt [3]{1-x^3}-2 \sqrt {3} \tan ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[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

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 3.
time = 1.00, size = 15, normalized size = 0.21

method	result	size
meijerg	\(\frac {x^{2} \hypergeom \left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{2}\)	\(15\)
risch	\(-\frac {x^{2} \left (x^{3}-1\right )}{3 \left (-x^{3}+1\right )^{\frac {2}{3}}}+\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{6 \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {2}{3}}}\)	\(69\)
trager	\(\frac {x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}}{3}-\frac {\ln \left (-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}+x^{3}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{9}+\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 x \left (-x^{3}+1\right )^{\frac {2}{3}}-2 x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right )}{9}\)	\(213\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 105, normalized size = 1.44 \begin {gather*} -\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}{9} \, \log \left (\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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
/9*log((-x^3 + 1)^(1/3)/x + 1) + 1/18*log(-(-x^3 + 1)^(1/3)/x + (-x^3 + 1)^(2/3)/x^2 + 1)

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Fricas [A]
time = 2.04, size = 96, normalized size = 1.32 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [C] Result contains complex when optimal does not.
time = 0.49, size = 32, normalized size = 0.44 \begin {gather*} \frac {x^{2} \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} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x^3 + 1)^(1/3)*x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\left (1-x^3\right )}^{1/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(1 - x^3)^(1/3),x)

[Out]

int(x*(1 - x^3)^(1/3), x)

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