3.162 \(\int \frac{1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx\)

Optimal. Leaf size=74 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt{3}}-\frac{1}{12} \tanh ^{-1}\left (\frac{\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right )+\frac{1}{12} \tanh ^{-1}\left (\frac{x}{3}\right ) \]

[Out]

ArcTan[(Sqrt[3]*(1 - (1 - x^2)^(1/3)))/x]/(4*Sqrt[3]) + ArcTanh[x/3]/12 - ArcTan
h[(1 - (1 - x^2)^(1/3))^2/(3*x)]/12

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Rubi [A]  time = 0.0370406, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt{3}}-\frac{1}{12} \tanh ^{-1}\left (\frac{\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right )+\frac{1}{12} \tanh ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - x^2)^(1/3)*(9 - x^2)),x]

[Out]

ArcTan[(Sqrt[3]*(1 - (1 - x^2)^(1/3)))/x]/(4*Sqrt[3]) + ArcTanh[x/3]/12 - ArcTan
h[(1 - (1 - x^2)^(1/3))^2/(3*x)]/12

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Rubi in Sympy [A]  time = 8.49567, size = 15, normalized size = 0.2 \[ \frac{x \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{3},1,\frac{3}{2},x^{2},\frac{x^{2}}{9} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**2+1)**(1/3)/(-x**2+9),x)

[Out]

x*appellf1(1/2, 1/3, 1, 3/2, x**2, x**2/9)/9

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Mathematica [C]  time = 0.0704206, size = 125, normalized size = 1.69 \[ \frac{\sqrt [3]{\frac{x-1}{x-3}} \sqrt [3]{\frac{x+1}{x-3}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{4}{x-3},-\frac{2}{x-3}\right )-\sqrt [3]{\frac{x-1}{x+3}} \sqrt [3]{\frac{x+1}{x+3}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2}{x+3},\frac{4}{x+3}\right )}{4 \sqrt [3]{1-x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((1 - x^2)^(1/3)*(9 - x^2)),x]

[Out]

(((-1 + x)/(-3 + x))^(1/3)*((1 + x)/(-3 + x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3,
-4/(-3 + x), -2/(-3 + x)] - ((-1 + x)/(3 + x))^(1/3)*((1 + x)/(3 + x))^(1/3)*Ap
pellF1[2/3, 1/3, 1/3, 5/3, 2/(3 + x), 4/(3 + x)])/(4*(1 - x^2)^(1/3))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{1}{-{x}^{2}+9}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^2+1)^(1/3)/(-x^2+9),x)

[Out]

int(1/(-x^2+1)^(1/3)/(-x^2+9),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (x^{2} - 9\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^2 - 9)*(-x^2 + 1)^(1/3)),x, algorithm="maxima")

[Out]

-integrate(1/((x^2 - 9)*(-x^2 + 1)^(1/3)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^2 - 9)*(-x^2 + 1)^(1/3)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{x^{2} \sqrt [3]{- x^{2} + 1} - 9 \sqrt [3]{- x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**2+1)**(1/3)/(-x**2+9),x)

[Out]

-Integral(1/(x**2*(-x**2 + 1)**(1/3) - 9*(-x**2 + 1)**(1/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{1}{{\left (x^{2} - 9\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^2 - 9)*(-x^2 + 1)^(1/3)),x, algorithm="giac")

[Out]

integrate(-1/((x^2 - 9)*(-x^2 + 1)^(1/3)), x)