3.2.7 \(\int \frac {(1-x^3)^{2/3}}{x} \, dx\) [107]

Optimal. Leaf size=70 \[ \frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 52, 57, 632, 210, 31} \begin {gather*} \frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x^3)^(2/3)/x,x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 90, normalized size = 1.29 \begin {gather*} \frac {1}{6} \left (3 \left (1-x^3\right )^{2/3}+2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1-x^3}\right )-\log \left (1+\sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^3)^(2/3)/x,x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.14, size = 23, normalized size = 0.33 \begin {gather*} \text {latex error} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(1 - x^3)^(2/3)/x,x]')

[Out]

-1 ^ (2 / 3) x ^ 2 Gamma[-2 / 3] hyper[{-2 / 3, -2 / 3}, {1 / 3}, 1 / x ^ 3] / (3 Gamma[1 / 3])

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 3.
time = 1.71, size = 66, normalized size = 0.94

method	result	size
meijerg	\(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], x^{3}\right )}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{9 \pi }\)	\(66\)
trager	\(\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{2}+\frac {\ln \left (\frac {-211 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3126 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \left (-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-11543 x^{3}-14247 \left (-x^{3}+1\right )^{\frac {2}{3}}-19749 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}+1688 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-5502 \left (-x^{3}+1\right )^{\frac {1}{3}}+15935 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+21437}{x^{3}}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {13 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-70 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+39 \left (-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-48 x^{3}-105 \left (-x^{3}+1\right )^{\frac {2}{3}}-144 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-104 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-39 \left (-x^{3}+1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+40}{x^{3}}\right )}{3}\)	\(254\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9/Pi*3^(1/2)*GAMMA(2/3)*(-(3/2-1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*Pi)*Pi*3^(1/2)/GAMMA(2/3)+2/3*Pi*3^(1/2)/
GAMMA(2/3)*x^3*hypergeom([1/3,1,1],[2,2],x^3))

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Maxima [A]
time = 0.33, size = 73, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.32, size = 75, normalized size = 1.07 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.51, size = 41, normalized size = 0.59 \begin {gather*} - \frac {x^{2} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {1}{x^{3}}} \right )}}{3 \Gamma \left (\frac {1}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 91, normalized size = 1.30 \begin {gather*} \frac {\ln \left |\left (-x^{3}+1\right )^{\frac {1}{3}}-1\right |}{3}-\frac {\ln \left (\left (\left (-x^{3}+1\right )^{\frac {1}{3}}\right )^{2}+\left (-x^{3}+1\right )^{\frac {1}{3}}+1\right )}{6}+\frac {\arctan \left (\frac {2 \left (-x^{3}+1\right )^{\frac {1}{3}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (\left (-x^{3}+1\right )^{\frac {1}{3}}\right )^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.40, size = 91, normalized size = 1.30 \begin {gather*} \frac {\ln \left ({\left (1-x^3\right )}^{1/3}-1\right )}{3}+\ln \left ({\left (1-x^3\right )}^{1/3}-9\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left ({\left (1-x^3\right )}^{1/3}-9\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\frac {{\left (1-x^3\right )}^{2/3}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^3)^(2/3)/x,x)

[Out]

log((1 - x^3)^(1/3) - 1)/3 + log((1 - x^3)^(1/3) - 9*((3^(1/2)*1i)/6 - 1/6)^2)*((3^(1/2)*1i)/6 - 1/6) - log((1
- x^3)^(1/3) - 9*((3^(1/2)*1i)/6 + 1/6)^2)*((3^(1/2)*1i)/6 + 1/6) + (1 - x^3)^(2/3)/2

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