Optimal. Leaf size=78 \[ -\frac{\log \left (2^{2/3} x^2+\sqrt [3]{2} x+1\right )}{6 \sqrt [3]{2}}+\frac{\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{2} x+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0952237, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{\log \left (2^{2/3} x^2+\sqrt [3]{2} x+1\right )}{6 \sqrt [3]{2}}+\frac{\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{2} x+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(-1 + 2*x^3)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 4.7508, size = 73, normalized size = 0.94 \[ \frac{2^{\frac{2}{3}} \log{\left (- \sqrt [3]{2} x + 1 \right )}}{6} - \frac{2^{\frac{2}{3}} \log{\left (2^{\frac{2}{3}} x^{2} + \sqrt [3]{2} x + 1 \right )}}{12} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3} + \frac{1}{3}\right ) \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*x**3-1),x)
[Out]
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Mathematica [A] time = 0.0745608, size = 66, normalized size = 0.85 \[ -\frac{\log \left (2^{2/3} x^2+\sqrt [3]{2} x+1\right )-2 \log \left (1-\sqrt [3]{2} x\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{2} x+1}{\sqrt{3}}\right )}{6 \sqrt [3]{2}} \]
Antiderivative was successfully verified.
[In] Integrate[(-1 + 2*x^3)^(-1),x]
[Out]
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Maple [A] time = 0.004, size = 58, normalized size = 0.7 \[{\frac{{2}^{{\frac{2}{3}}}}{6}\ln \left ( x-{\frac{{2}^{{\frac{2}{3}}}}{2}} \right ) }-{\frac{{2}^{{\frac{2}{3}}}}{12}\ln \left ({x}^{2}+{\frac{{2}^{{\frac{2}{3}}}x}{2}}+{\frac{\sqrt [3]{2}}{2}} \right ) }-{\frac{{2}^{{\frac{2}{3}}}\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,\sqrt [3]{2}x \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*x^3-1),x)
[Out]
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Maxima [A] time = 1.57509, size = 89, normalized size = 1.14 \[ -\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2 \cdot 2^{\frac{2}{3}} x + 2^{\frac{1}{3}}\right )}\right ) - \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (2^{\frac{2}{3}} x^{2} + 2^{\frac{1}{3}} x + 1\right ) + \frac{1}{6} \cdot 2^{\frac{2}{3}} \log \left (\frac{1}{2} \cdot 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} x - 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*x^3 - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213589, size = 77, normalized size = 0.99 \[ -\frac{1}{36} \, \sqrt{3} 2^{\frac{2}{3}}{\left (\sqrt{3} \log \left (2^{\frac{2}{3}} x^{2} + 2^{\frac{1}{3}} x + 1\right ) - 2 \, \sqrt{3} \log \left (2^{\frac{1}{3}} x - 1\right ) + 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \cdot 2^{\frac{1}{3}} x + 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*x^3 - 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.652428, size = 78, normalized size = 1. \[ \frac{2^{\frac{2}{3}} \log{\left (x - \frac{2^{\frac{2}{3}}}{2} \right )}}{6} - \frac{2^{\frac{2}{3}} \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x}{2} + \frac{\sqrt [3]{2}}{2} \right )}}{6} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt [3]{2} \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*x**3-1),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(1/(2*x^3 - 1),x, algorithm="giac")
[Out]