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