Optimal. Leaf size=79 \[ -\frac{\log \left (x^2+3\right )}{4\ 2^{2/3}}+\frac{3 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]
[Out]
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Rubi [A] time = 0.120327, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\log \left (x^2+3\right )}{4\ 2^{2/3}}+\frac{3 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[x/((1 - x^2)^(1/3)*(3 + x^2)),x]
[Out]
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Rubi in Sympy [A] time = 8.78088, size = 73, normalized size = 0.92 \[ - \frac{\sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{8} + \frac{3 \sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{8} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(-x**2+1)**(1/3)/(x**2+3),x)
[Out]
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Mathematica [A] time = 0.0674892, size = 84, normalized size = 1.06 \[ \frac{2 \log \left (2-\sqrt [3]{2-2 x^2}\right )-\log \left (\left (2-2 x^2\right )^{2/3}+2 \sqrt [3]{2-2 x^2}+4\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{4\ 2^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x/((1 - x^2)^(1/3)*(3 + x^2)),x]
[Out]
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Maple [F] time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{x}{{x}^{2}+3}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(-x^2+1)^(1/3)/(x^2+3),x)
[Out]
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Maxima [A] time = 1.50847, size = 116, normalized size = 1.47 \[ \frac{1}{8} \cdot 4^{\frac{2}{3}} \sqrt{3} \arctan \left (\frac{1}{12} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (4^{\frac{1}{3}} + 2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right )}\right ) - \frac{1}{16} \cdot 4^{\frac{2}{3}} \log \left (4^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{8} \cdot 4^{\frac{2}{3}} \log \left (-4^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 + 3)*(-x^2 + 1)^(1/3)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236298, size = 109, normalized size = 1.38 \[ \frac{1}{16} \cdot 4^{\frac{2}{3}}{\left (2 \, \sqrt{3} \arctan \left (\frac{1}{6} \, \sqrt{3}{\left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 2\right )}\right ) - \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 4\right ) + 2 \, \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 + 3)*(-x^2 + 1)^(1/3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(-x**2+1)**(1/3)/(x**2+3),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 + 3)*(-x^2 + 1)^(1/3)),x, algorithm="giac")
[Out]