Optimal. Leaf size=47 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{3} x}{1-x^2}\right )}{2 \sqrt{3}}+\frac{1}{6} \tanh ^{-1}\left (\frac{x}{x^2+1}\right )+\frac{1}{3} \tanh ^{-1}(x) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.194203, antiderivative size = 73, normalized size of antiderivative = 1.55, number of steps used = 10, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{1}{12} \log \left (x^2-x+1\right )+\frac{1}{12} \log \left (x^2+x+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{3} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(1 - x^6)^(-1),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 35.3902, size = 68, normalized size = 1.45 \[ - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{\log{\left (x^{2} + x + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{6} + \frac{\operatorname{atanh}{\left (x \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-x**6+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0151938, size = 75, normalized size = 1.6 \[ \frac{1}{12} \left (-\log \left (x^2-x+1\right )+\log \left (x^2+x+1\right )-2 \log (1-x)+2 \log (x+1)+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x^6)^(-1),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 66, normalized size = 1.4 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( -1+x \right ) }{6}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+x \right ) }{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-x^6+1),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.58004, size = 88, normalized size = 1.87 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{12} \, \log \left (x^{2} + x + 1\right ) - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{6} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(x^6 - 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.22943, size = 101, normalized size = 2.15 \[ \frac{1}{36} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} + x + 1\right ) - \sqrt{3} \log \left (x^{2} - x + 1\right ) + 2 \, \sqrt{3} \log \left (x + 1\right ) - 2 \, \sqrt{3} \log \left (x - 1\right ) + 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(x^6 - 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.772314, size = 83, normalized size = 1.77 \[ - \frac{\log{\left (x - 1 \right )}}{6} + \frac{\log{\left (x + 1 \right )}}{6} - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{\log{\left (x^{2} + x + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-x**6+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.222549, size = 90, normalized size = 1.91 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{12} \,{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{12} \,{\rm ln}\left (x^{2} - x + 1\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(x^6 - 1),x, algorithm="giac")
[Out]