3.1.35 \(\int \frac {1}{\sqrt [3]{1-x^3}} \, dx\) [35]

Optimal. Leaf size=49 \[ -\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right ) \]

[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)

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Rubi [A]
time = 0.00, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {245} \begin {gather*} \frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x^3)^(-1/3),x]

[Out]

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

Rule 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*} \int \frac {1}{\sqrt [3]{1-x^3}} \, dx &=-\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*}

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Mathematica [A]
time = 0.03, size = 86, normalized size = 1.76 \begin {gather*} \frac {\tan ^{-1}\left (\frac {-1+\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (1+\frac {x^2}{\left (1-x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{3} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^3)^(-1/3),x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.03, size = 16, normalized size = 0.33 \begin {gather*} x \text {hyper}\left [\left \{\frac {1}{3},\frac {1}{3}\right \},\left \{\frac {4}{3}\right \},x^3 \text {exp\_polar}\left [2 I \text {Pi}\right ]\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(1 - x^3)^(1/3),x]')

[Out]

x hyper[{1 / 3, 1 / 3}, {4 / 3}, x ^ 3 exp_polar[2 I Pi]]

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 3.
time = 1.20, size = 12, normalized size = 0.24

method	result	size
meijerg	\(x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )\)	\(12\)
trager	\(\frac {\ln \left (-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (-x^{3}+1\right )^{\frac {2}{3}}+3 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}-2 x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}-2 x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2\right )}{3}\)	\(194\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.34, size = 78, normalized size = 1.59 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (40) = 80\).
time = 0.48, size = 82, normalized size = 1.67 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.41, size = 29, normalized size = 0.59 \begin {gather*} \frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F] N/A
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(1/(-x^3+1)^(1/3),x)

[Out]

Could not integrate

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Mupad [B]
time = 0.33, size = 10, normalized size = 0.20 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ x^3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1 - x^3)^(1/3),x)

[Out]

x*hypergeom([1/3, 1/3], 4/3, x^3)

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