3.47 \(\int \frac{1}{-2+x^6} \, dx\)

Optimal. Leaf size=138 \[ \frac{\log \left (x^2-\sqrt [6]{2} x+\sqrt [3]{2}\right )}{12\ 2^{5/6}}-\frac{\log \left (x^2+\sqrt [6]{2} x+\sqrt [3]{2}\right )}{12\ 2^{5/6}}+\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{5/6} x}{\sqrt{3}}\right )}{2\ 2^{5/6} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2^{5/6} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right )}{2\ 2^{5/6} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}} \]

[Out]

ArcTan[1/Sqrt[3] - (2^(5/6)*x)/Sqrt[3]]/(2*2^(5/6)*Sqrt[3]) - ArcTan[1/Sqrt[3] +
(2^(5/6)*x)/Sqrt[3]]/(2*2^(5/6)*Sqrt[3]) - ArcTanh[x/2^(1/6)]/(3*2^(5/6)) + Log
[2^(1/3) - 2^(1/6)*x + x^2]/(12*2^(5/6)) - Log[2^(1/3) + 2^(1/6)*x + x^2]/(12*2^
(5/6))

_______________________________________________________________________________________

Rubi [A]  time = 0.405045, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857 \[ \frac{\log \left (x^2-\sqrt [6]{2} x+\sqrt [3]{2}\right )}{12\ 2^{5/6}}-\frac{\log \left (x^2+\sqrt [6]{2} x+\sqrt [3]{2}\right )}{12\ 2^{5/6}}+\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{5/6} x}{\sqrt{3}}\right )}{2\ 2^{5/6} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2^{5/6} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right )}{2\ 2^{5/6} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}} \]

Antiderivative was successfully verified.

[In]  Int[(-2 + x^6)^(-1),x]

[Out]

ArcTan[1/Sqrt[3] - (2^(5/6)*x)/Sqrt[3]]/(2*2^(5/6)*Sqrt[3]) - ArcTan[1/Sqrt[3] +
(2^(5/6)*x)/Sqrt[3]]/(2*2^(5/6)*Sqrt[3]) - ArcTanh[x/2^(1/6)]/(3*2^(5/6)) + Log
[2^(1/3) - 2^(1/6)*x + x^2]/(12*2^(5/6)) - Log[2^(1/3) + 2^(1/6)*x + x^2]/(12*2^
(5/6))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.2348, size = 131, normalized size = 0.95 \[ \frac{\sqrt [6]{2} \log{\left (x^{2} - \sqrt [6]{2} x + \sqrt [3]{2} \right )}}{24} - \frac{\sqrt [6]{2} \log{\left (x^{2} + \sqrt [6]{2} x + \sqrt [3]{2} \right )}}{24} - \frac{\sqrt [6]{2} \sqrt{3} \operatorname{atan}{\left (2^{\frac{5}{6}} \sqrt{3} \left (\frac{x}{3} - \frac{\sqrt [6]{2}}{6}\right ) \right )}}{12} - \frac{\sqrt [6]{2} \sqrt{3} \operatorname{atan}{\left (2^{\frac{5}{6}} \sqrt{3} \left (\frac{x}{3} + \frac{\sqrt [6]{2}}{6}\right ) \right )}}{12} - \frac{\sqrt [6]{2} \operatorname{atanh}{\left (\frac{2^{\frac{5}{6}} x}{2} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**6-2),x)

[Out]

2**(1/6)*log(x**2 - 2**(1/6)*x + 2**(1/3))/24 - 2**(1/6)*log(x**2 + 2**(1/6)*x +
2**(1/3))/24 - 2**(1/6)*sqrt(3)*atan(2**(5/6)*sqrt(3)*(x/3 - 2**(1/6)/6))/12 -
2**(1/6)*sqrt(3)*atan(2**(5/6)*sqrt(3)*(x/3 + 2**(1/6)/6))/12 - 2**(1/6)*atanh(2
**(5/6)*x/2)/6

_______________________________________________________________________________________

Mathematica [A]  time = 0.0541312, size = 122, normalized size = 0.88 \[ -\frac{-\log \left (2^{2/3} x^2-2^{5/6} x+2\right )+\log \left (2^{2/3} x^2+2^{5/6} x+2\right )-2 \log \left (2-2^{5/6} x\right )+2 \log \left (2^{5/6} x+2\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2^{5/6} x-1}{\sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2^{5/6} x+1}{\sqrt{3}}\right )}{12\ 2^{5/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-2 + x^6)^(-1),x]

[Out]

-(2*Sqrt[3]*ArcTan[(-1 + 2^(5/6)*x)/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 2^(5/6)*x)/
Sqrt[3]] - 2*Log[2 - 2^(5/6)*x] + 2*Log[2 + 2^(5/6)*x] - Log[2 - 2^(5/6)*x + 2^(
2/3)*x^2] + Log[2 + 2^(5/6)*x + 2^(2/3)*x^2])/(12*2^(5/6))

_______________________________________________________________________________________

Maple [A]  time = 0.056, size = 111, normalized size = 0.8 \[ -{\frac{\sqrt [6]{2}\ln \left ( x+\sqrt [6]{2} \right ) }{12}}+{\frac{\ln \left ( \sqrt [3]{2}-\sqrt [6]{2}x+{x}^{2} \right ) \sqrt [6]{2}}{24}}-{\frac{\sqrt{3}\sqrt [6]{2}}{12}\arctan \left ( -{\frac{\sqrt{3}}{3}}+{\frac{{2}^{{\frac{5}{6}}}x\sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( \sqrt [3]{2}+\sqrt [6]{2}x+{x}^{2} \right ) \sqrt [6]{2}}{24}}-{\frac{\sqrt{3}\sqrt [6]{2}}{12}\arctan \left ({\frac{\sqrt{3}}{3}}+{\frac{{2}^{{\frac{5}{6}}}x\sqrt{3}}{3}} \right ) }+{\frac{\sqrt [6]{2}\ln \left ( x-\sqrt [6]{2} \right ) }{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^6-2),x)

[Out]

-1/12*2^(1/6)*ln(x+2^(1/6))+1/24*ln(2^(1/3)-2^(1/6)*x+x^2)*2^(1/6)-1/12*arctan(-
1/3*3^(1/2)+1/3*2^(5/6)*x*3^(1/2))*2^(1/6)*3^(1/2)-1/24*ln(2^(1/3)+2^(1/6)*x+x^2
)*2^(1/6)-1/12*arctan(1/3*3^(1/2)+1/3*2^(5/6)*x*3^(1/2))*2^(1/6)*3^(1/2)+1/12*2^
(1/6)*ln(x-2^(1/6))

_______________________________________________________________________________________

Maxima [A]  time = 1.5752, size = 151, normalized size = 1.09 \[ -\frac{1}{12} \, \sqrt{3} 2^{\frac{1}{6}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{5}{6}}{\left (2 \, x + 2^{\frac{1}{6}}\right )}\right ) - \frac{1}{12} \, \sqrt{3} 2^{\frac{1}{6}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{5}{6}}{\left (2 \, x - 2^{\frac{1}{6}}\right )}\right ) - \frac{1}{24} \cdot 2^{\frac{1}{6}} \log \left (x^{2} + 2^{\frac{1}{6}} x + 2^{\frac{1}{3}}\right ) + \frac{1}{24} \cdot 2^{\frac{1}{6}} \log \left (x^{2} - 2^{\frac{1}{6}} x + 2^{\frac{1}{3}}\right ) - \frac{1}{12} \cdot 2^{\frac{1}{6}} \log \left (x + 2^{\frac{1}{6}}\right ) + \frac{1}{12} \cdot 2^{\frac{1}{6}} \log \left (x - 2^{\frac{1}{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 - 2),x, algorithm="maxima")

[Out]

-1/12*sqrt(3)*2^(1/6)*arctan(1/6*sqrt(3)*2^(5/6)*(2*x + 2^(1/6))) - 1/12*sqrt(3)
*2^(1/6)*arctan(1/6*sqrt(3)*2^(5/6)*(2*x - 2^(1/6))) - 1/24*2^(1/6)*log(x^2 + 2^
(1/6)*x + 2^(1/3)) + 1/24*2^(1/6)*log(x^2 - 2^(1/6)*x + 2^(1/3)) - 1/12*2^(1/6)*
log(x + 2^(1/6)) + 1/12*2^(1/6)*log(x - 2^(1/6))

_______________________________________________________________________________________

Fricas [A]  time = 0.224712, size = 207, normalized size = 1.5 \[ \frac{1}{384} \cdot 32^{\frac{5}{6}}{\left (4 \, \sqrt{3} \arctan \left (\frac{2 \, \sqrt{3}}{2 \cdot 32^{\frac{1}{6}} x + 32^{\frac{1}{6}} \sqrt{4^{\frac{2}{3}}{\left (4^{\frac{1}{3}} x^{2} + 32^{\frac{1}{6}} x + 2\right )}} + 2}\right ) + 4 \, \sqrt{3} \arctan \left (\frac{2 \, \sqrt{3}}{2 \cdot 32^{\frac{1}{6}} x + 32^{\frac{1}{6}} \sqrt{4^{\frac{2}{3}}{\left (4^{\frac{1}{3}} x^{2} - 32^{\frac{1}{6}} x + 2\right )}} - 2}\right ) - \log \left (2 \cdot 4^{\frac{1}{3}} x^{2} + 2 \cdot 32^{\frac{1}{6}} x + 4\right ) + \log \left (2 \cdot 4^{\frac{1}{3}} x^{2} - 2 \cdot 32^{\frac{1}{6}} x + 4\right ) - 2 \, \log \left (32^{\frac{1}{6}} x + 2\right ) + 2 \, \log \left (32^{\frac{1}{6}} x - 2\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 - 2),x, algorithm="fricas")

[Out]

1/384*32^(5/6)*(4*sqrt(3)*arctan(2*sqrt(3)/(2*32^(1/6)*x + 32^(1/6)*sqrt(4^(2/3)
*(4^(1/3)*x^2 + 32^(1/6)*x + 2)) + 2)) + 4*sqrt(3)*arctan(2*sqrt(3)/(2*32^(1/6)*
x + 32^(1/6)*sqrt(4^(2/3)*(4^(1/3)*x^2 - 32^(1/6)*x + 2)) - 2)) - log(2*4^(1/3)*
x^2 + 2*32^(1/6)*x + 4) + log(2*4^(1/3)*x^2 - 2*32^(1/6)*x + 4) - 2*log(32^(1/6)
*x + 2) + 2*log(32^(1/6)*x - 2))

_______________________________________________________________________________________

Sympy [A]  time = 0.989179, size = 14, normalized size = 0.1 \[ \operatorname{RootSum}{\left (1492992 t^{6} - 1, \left ( t \mapsto t \log{\left (- 12 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**6-2),x)

[Out]

RootSum(1492992*_t**6 - 1, Lambda(_t, _t*log(-12*_t + x)))

_______________________________________________________________________________________

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/(x^6 - 2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError