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

Optimal. Leaf size=58 \[ \frac{3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )-\frac{\log (x)}{2} \]

[Out]

-(Sqrt[3]*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]])/2 - Log[x]/2 + (3*Log[1 - (1 - x^2)^(1/3)])/4

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Rubi [A]  time = 0.0346961, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 57, 618, 204, 31} \[ \frac{3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )-\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 - x^2)^(2/3)),x]

[Out]

-(Sqrt[3]*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]])/2 - Log[x]/2 + (3*Log[1 - (1 - x^2)^(1/3)])/4

Rule 266

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 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), 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 618

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]

Rule 204

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

Rule 31

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

Rubi steps

\begin{align*} \int \frac{1}{x \left (1-x^2\right )^{2/3}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{2/3} x} \, dx,x,x^2\right )\\ &=-\frac{\log (x)}{2}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\sqrt [3]{1-x^2}\right )-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )\\ &=-\frac{\log (x)}{2}+\frac{3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^2}\right )\\ &=-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{1+2 \sqrt [3]{1-x^2}}{\sqrt{3}}\right )-\frac{\log (x)}{2}+\frac{3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0124386, size = 81, normalized size = 1.4 \[ \frac{1}{2} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{1}{4} \log \left (\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1\right )-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 - x^2)^(2/3)),x]

[Out]

-(Sqrt[3]*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]])/2 + Log[1 - (1 - x^2)^(1/3)]/2 - Log[1 + (1 - x^2)^(1/3) +
(1 - x^2)^(2/3)]/4

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Maple [C]  time = 0.041, size = 48, normalized size = 0.8 \begin{align*}{\frac{1}{2\,\Gamma \left ( 2/3 \right ) } \left ( \left ({\frac{\pi \,\sqrt{3}}{6}}-{\frac{3\,\ln \left ( 3 \right ) }{2}}+2\,\ln \left ( x \right ) +i\pi \right ) \Gamma \left ({\frac{2}{3}} \right ) +{\frac{2\,\Gamma \left ( 2/3 \right ){x}^{2}}{3}{\mbox{$_3$F$_2$}(1,1,{\frac{5}{3}};\,2,2;\,{x}^{2})}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/GAMMA(2/3)*((1/6*Pi*3^(1/2)-3/2*ln(3)+2*ln(x)+I*Pi)*GAMMA(2/3)+2/3*GAMMA(2/3)*x^2*hypergeom([1,1,5/3],[2,2
],x^2))

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Maxima [A]  time = 1.41876, size = 84, normalized size = 1.45 \begin{align*} -\frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{2} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-x^2 + 1)^(1/3) + 1)) - 1/4*log((-x^2 + 1)^(2/3) + (-x^2 + 1)^(1/3) + 1) +
1/2*log((-x^2 + 1)^(1/3) - 1)

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Fricas [A]  time = 2.15924, size = 200, normalized size = 3.45 \begin{align*} -\frac{1}{2} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - \frac{1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{2} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(3)*arctan(2/3*sqrt(3)*(-x^2 + 1)^(1/3) + 1/3*sqrt(3)) - 1/4*log((-x^2 + 1)^(2/3) + (-x^2 + 1)^(1/3)
+ 1) + 1/2*log((-x^2 + 1)^(1/3) - 1)

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Sympy [C]  time = 0.949689, size = 37, normalized size = 0.64 \begin{align*} - \frac{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{5}{3} \end{matrix}\middle |{\frac{1}{x^{2}}} \right )}}{2 x^{\frac{4}{3}} \Gamma \left (\frac{5}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(-2*I*pi/3)*gamma(2/3)*hyper((2/3, 2/3), (5/3,), x**(-2))/(2*x**(4/3)*gamma(5/3))

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Giac [A]  time = 1.06623, size = 86, normalized size = 1.48 \begin{align*} -\frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{2} \, \log \left (-{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-x^2 + 1)^(1/3) + 1)) - 1/4*log((-x^2 + 1)^(2/3) + (-x^2 + 1)^(1/3) + 1) +
1/2*log(-(-x^2 + 1)^(1/3) + 1)