Optimal. Leaf size=109 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}(x)}{6\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}} \]
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Rubi [A] time = 0.0456091, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}(x)}{6\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[1/((3 - x^2)*(1 + x^2)^(1/3)),x]
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Rubi in Sympy [A] time = 11.585, size = 192, normalized size = 1.76 \[ \frac{\sqrt [3]{2} \sqrt{3} \log{\left (- x + \sqrt{3} \right )}}{24} - \frac{\sqrt [3]{2} \sqrt{3} \log{\left (x + \sqrt{3} \right )}}{24} + \frac{\sqrt [3]{2} \sqrt{3} \log{\left (- x - \sqrt [3]{2} \sqrt{3} \sqrt [3]{x^{2} + 1} + \sqrt{3} \right )}}{24} - \frac{\sqrt [3]{2} \sqrt{3} \log{\left (x - \sqrt [3]{2} \sqrt{3} \sqrt [3]{x^{2} + 1} + \sqrt{3} \right )}}{24} - \frac{\sqrt [3]{2} \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \left (- x + \sqrt{3}\right )}{3 \sqrt [3]{x^{2} + 1}} + \frac{\sqrt{3}}{3} \right )}}{12} + \frac{\sqrt [3]{2} \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \left (x + \sqrt{3}\right )}{3 \sqrt [3]{x^{2} + 1}} + \frac{\sqrt{3}}{3} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-x**2+3)/(x**2+1)**(1/3),x)
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Mathematica [C] time = 0.0558719, size = 124, normalized size = 1.14 \[ -\frac{9 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right )}{\left (x^2-3\right ) \sqrt [3]{x^2+1} \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-x^2,\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-x^2,\frac{x^2}{3}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((3 - x^2)*(1 + x^2)^(1/3)),x]
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Maple [F] time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{1}{-{x}^{2}+3}{\frac{1}{\sqrt [3]{{x}^{2}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-x^2+3)/(x^2+1)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (x^{2} + 1\right )}^{\frac{1}{3}}{\left (x^{2} - 3\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((x^2 + 1)^(1/3)*(x^2 - 3)),x, algorithm="maxima")
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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 + 1)^(1/3)*(x^2 - 3)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{x^{2} \sqrt [3]{x^{2} + 1} - 3 \sqrt [3]{x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-x**2+3)/(x**2+1)**(1/3),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{1}{{\left (x^{2} + 1\right )}^{\frac{1}{3}}{\left (x^{2} - 3\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((x^2 + 1)^(1/3)*(x^2 - 3)),x, algorithm="giac")
[Out]