3.33 \(\int \frac{1}{-1+2 x^3} \, dx\)

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]

-(ArcTan[(1 + 2*2^(1/3)*x)/Sqrt[3]]/(2^(1/3)*Sqrt[3])) + Log[1 - 2^(1/3)*x]/(3*2
^(1/3)) - Log[1 + 2^(1/3)*x + 2^(2/3)*x^2]/(6*2^(1/3))

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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]

-(ArcTan[(1 + 2*2^(1/3)*x)/Sqrt[3]]/(2^(1/3)*Sqrt[3])) + Log[1 - 2^(1/3)*x]/(3*2
^(1/3)) - Log[1 + 2^(1/3)*x + 2^(2/3)*x^2]/(6*2^(1/3))

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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]

2**(2/3)*log(-2**(1/3)*x + 1)/6 - 2**(2/3)*log(2**(2/3)*x**2 + 2**(1/3)*x + 1)/1
2 - 2**(2/3)*sqrt(3)*atan(sqrt(3)*(2*2**(1/3)*x/3 + 1/3))/6

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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]

-(2*Sqrt[3]*ArcTan[(1 + 2*2^(1/3)*x)/Sqrt[3]] - 2*Log[1 - 2^(1/3)*x] + Log[1 + 2
^(1/3)*x + 2^(2/3)*x^2])/(6*2^(1/3))

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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]

1/6*2^(2/3)*ln(x-1/2*2^(2/3))-1/12*2^(2/3)*ln(x^2+1/2*2^(2/3)*x+1/2*2^(1/3))-1/6
*arctan(1/3*(1+2*2^(1/3)*x)*3^(1/2))*2^(2/3)*3^(1/2)

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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]

-1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2*2^(2/3)*x + 2^(1/3))) - 1/12*
2^(2/3)*log(2^(2/3)*x^2 + 2^(1/3)*x + 1) + 1/6*2^(2/3)*log(1/2*2^(2/3)*(2^(1/3)*
x - 1))

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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]

-1/36*sqrt(3)*2^(2/3)*(sqrt(3)*log(2^(2/3)*x^2 + 2^(1/3)*x + 1) - 2*sqrt(3)*log(
2^(1/3)*x - 1) + 6*arctan(1/3*sqrt(3)*(2*2^(1/3)*x + 1)))

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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]

2**(2/3)*log(x - 2**(2/3)/2)/6 - 2**(2/3)*log(x**2 + 2**(2/3)*x/2 + 2**(1/3)/2)/
6 - 2**(2/3)*sqrt(3)*atan(2*2**(1/3)*sqrt(3)*x/3 + sqrt(3)/3)/3

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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]

Exception raised: NotImplementedError