3.314 \(\int \frac{1}{\left (-1+x^3\right ) \sqrt [3]{2+x^3}} \, dx\)

Optimal. Leaf size=107 \[ \frac{\log \left (1-\frac{\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}\right )}{3 \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt [6]{3} \sqrt [3]{x^3+2}}+\frac{1}{\sqrt{3}}\right )}{3^{5/6}}-\frac{\log \left (\frac{\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+\frac{3^{2/3} x^2}{\left (x^3+2\right )^{2/3}}+1\right )}{6 \sqrt [3]{3}} \]

[Out]

-(ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*(2 + x^3)^(1/3))]/3^(5/6)) + Log[1 - (3^(1/3
)*x)/(2 + x^3)^(1/3)]/(3*3^(1/3)) - Log[1 + (3^(2/3)*x^2)/(2 + x^3)^(2/3) + (3^(
1/3)*x)/(2 + x^3)^(1/3)]/(6*3^(1/3))

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Rubi [A]  time = 0.149767, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ \frac{\log \left (1-\frac{\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}\right )}{3 \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt [6]{3} \sqrt [3]{x^3+2}}+\frac{1}{\sqrt{3}}\right )}{3^{5/6}}-\frac{\log \left (\frac{\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+\frac{3^{2/3} x^2}{\left (x^3+2\right )^{2/3}}+1\right )}{6 \sqrt [3]{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((-1 + x^3)*(2 + x^3)^(1/3)),x]

[Out]

-(ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*(2 + x^3)^(1/3))]/3^(5/6)) + Log[1 - (3^(1/3
)*x)/(2 + x^3)^(1/3)]/(3*3^(1/3)) - Log[1 + (3^(2/3)*x^2)/(2 + x^3)^(2/3) + (3^(
1/3)*x)/(2 + x^3)^(1/3)]/(6*3^(1/3))

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Rubi in Sympy [A]  time = 6.84566, size = 102, normalized size = 0.95 \[ \frac{3^{\frac{2}{3}} \log{\left (- \frac{\sqrt [3]{3} x}{\sqrt [3]{x^{3} + 2}} + 1 \right )}}{9} - \frac{3^{\frac{2}{3}} \log{\left (\frac{3^{\frac{2}{3}} x^{2}}{\left (x^{3} + 2\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{3} x}{\sqrt [3]{x^{3} + 2}} + 1 \right )}}{18} - \frac{\sqrt [6]{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{3} x}{3 \sqrt [3]{x^{3} + 2}} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**3-1)/(x**3+2)**(1/3),x)

[Out]

3**(2/3)*log(-3**(1/3)*x/(x**3 + 2)**(1/3) + 1)/9 - 3**(2/3)*log(3**(2/3)*x**2/(
x**3 + 2)**(2/3) + 3**(1/3)*x/(x**3 + 2)**(1/3) + 1)/18 - 3**(1/6)*atan(sqrt(3)*
(2*3**(1/3)*x/(3*(x**3 + 2)**(1/3)) + 1/3))/3

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Mathematica [A]  time = 0.186317, size = 112, normalized size = 1.05 \[ \frac{\sqrt{3} \left (2 \log \left (1-\frac{\sqrt [3]{3} x}{\sqrt [3]{2 x^3+1}}\right )-\log \left (\frac{\sqrt [3]{3} x}{\sqrt [3]{2 x^3+1}}+\frac{3^{2/3} x^2}{\left (2 x^3+1\right )^{2/3}}+1\right )\right )-6 \tan ^{-1}\left (\frac{2 x}{\sqrt [6]{3} \sqrt [3]{2 x^3+1}}+\frac{1}{\sqrt{3}}\right )}{6\ 3^{5/6}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((-1 + x^3)*(2 + x^3)^(1/3)),x]

[Out]

(-6*ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*(1 + 2*x^3)^(1/3))] + Sqrt[3]*(2*Log[1 - (
3^(1/3)*x)/(1 + 2*x^3)^(1/3)] - Log[1 + (3^(2/3)*x^2)/(1 + 2*x^3)^(2/3) + (3^(1/
3)*x)/(1 + 2*x^3)^(1/3)]))/(6*3^(5/6))

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}-1}{\frac{1}{\sqrt [3]{{x}^{3}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^3-1)/(x^3+2)^(1/3),x)

[Out]

int(1/(x^3-1)/(x^3+2)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 2\right )}^{\frac{1}{3}}{\left (x^{3} - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 2)^(1/3)*(x^3 - 1)),x, algorithm="maxima")

[Out]

integrate(1/((x^3 + 2)^(1/3)*(x^3 - 1)), x)

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Fricas [A]  time = 1.92608, size = 274, normalized size = 2.56 \[ \frac{1}{54} \cdot 3^{\frac{1}{6}}{\left (2 \, \sqrt{3} \log \left (-\frac{3 \cdot 3^{\frac{2}{3}}{\left (x^{3} + 2\right )}^{\frac{2}{3}} x - 9 \,{\left (x^{3} + 2\right )}^{\frac{1}{3}} x^{2} + 2 \cdot 3^{\frac{1}{3}}{\left (x^{3} - 1\right )}}{x^{3} - 1}\right ) - \sqrt{3} \log \left (\frac{3^{\frac{2}{3}}{\left (31 \, x^{6} + 46 \, x^{3} + 4\right )} + 9 \cdot 3^{\frac{1}{3}}{\left (5 \, x^{5} + 4 \, x^{2}\right )}{\left (x^{3} + 2\right )}^{\frac{1}{3}} + 9 \,{\left (7 \, x^{4} + 2 \, x\right )}{\left (x^{3} + 2\right )}^{\frac{2}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) + 6 \, \arctan \left (\frac{9 \, \sqrt{3}{\left (x^{3} + 2\right )}^{\frac{1}{3}} x^{2} - 2 \cdot 3^{\frac{5}{6}}{\left (x^{3} - 1\right )} + 18 \cdot 3^{\frac{1}{6}}{\left (x^{3} + 2\right )}^{\frac{2}{3}} x}{3 \,{\left (9 \,{\left (x^{3} + 2\right )}^{\frac{1}{3}} x^{2} + 2 \cdot 3^{\frac{1}{3}}{\left (x^{3} - 1\right )}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 2)^(1/3)*(x^3 - 1)),x, algorithm="fricas")

[Out]

1/54*3^(1/6)*(2*sqrt(3)*log(-(3*3^(2/3)*(x^3 + 2)^(2/3)*x - 9*(x^3 + 2)^(1/3)*x^
2 + 2*3^(1/3)*(x^3 - 1))/(x^3 - 1)) - sqrt(3)*log((3^(2/3)*(31*x^6 + 46*x^3 + 4)
+ 9*3^(1/3)*(5*x^5 + 4*x^2)*(x^3 + 2)^(1/3) + 9*(7*x^4 + 2*x)*(x^3 + 2)^(2/3))/
(x^6 - 2*x^3 + 1)) + 6*arctan(1/3*(9*sqrt(3)*(x^3 + 2)^(1/3)*x^2 - 2*3^(5/6)*(x^
3 - 1) + 18*3^(1/6)*(x^3 + 2)^(2/3)*x)/(9*(x^3 + 2)^(1/3)*x^2 + 2*3^(1/3)*(x^3 -
1))))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x - 1\right ) \sqrt [3]{x^{3} + 2} \left (x^{2} + x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**3-1)/(x**3+2)**(1/3),x)

[Out]

Integral(1/((x - 1)*(x**3 + 2)**(1/3)*(x**2 + x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 2\right )}^{\frac{1}{3}}{\left (x^{3} - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 2)^(1/3)*(x^3 - 1)),x, algorithm="giac")

[Out]

integrate(1/((x^3 + 2)^(1/3)*(x^3 - 1)), x)