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

Optimal. Leaf size=81 \[ \frac{1}{4} \tan ^{-1}\left (\frac{1-\sqrt [3]{1-3 x^2}}{x}\right )-\frac{\tanh ^{-1}\left (\frac{\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt{3} x}\right )}{4 \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0404513, antiderivative size = 81, 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 \[ \frac{1}{4} \tan ^{-1}\left (\frac{1-\sqrt [3]{1-3 x^2}}{x}\right )-\frac{\tanh ^{-1}\left (\frac{\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt{3} x}\right )}{4 \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.60271, size = 17, normalized size = 0.21 \[ \frac{x \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{3},1,\frac{3}{2},3 x^{2},\frac{x^{2}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.151867, size = 126, normalized size = 1.56 \[ -\frac{9 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};3 x^2,\frac{x^2}{3}\right )}{\sqrt [3]{1-3 x^2} \left (x^2-3\right ) \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};3 x^2,\frac{x^2}{3}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};3 x^2,\frac{x^2}{3}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};3 x^2,\frac{x^2}{3}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-9*x*AppellF1[1/2, 1/3, 1, 3/2, 3*x^2, x^2/3])/((1 - 3*x^2)^(1/3)*(-3 + x^2)*(9
*AppellF1[1/2, 1/3, 1, 3/2, 3*x^2, x^2/3] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, 3*
x^2, x^2/3] + 3*AppellF1[3/2, 4/3, 1, 5/2, 3*x^2, x^2/3])))

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{\frac{1}{-{x}^{2}+3}{\frac{1}{\sqrt [3]{-3\,{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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