Optimal. Leaf size=56 \[ -\frac{x^2}{2}-\frac{1}{6} \log \left (1-x^2\right )+\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{12} \log \left (x^4+x^2+1\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.084001, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{x^2}{2}-\frac{1}{6} \log \left (1-x^2\right )+\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{12} \log \left (x^4+x^2+1\right ) \]
Antiderivative was successfully verified.
[In] Int[x^7/(1 - x^6),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 9.38667, size = 48, normalized size = 0.86 \[ - \frac{x^{2}}{2} - \frac{\log{\left (- x^{2} + 1 \right )}}{6} + \frac{\log{\left (x^{4} + x^{2} + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**7/(-x**6+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0339387, size = 78, normalized size = 1.39 \[ \frac{1}{12} \left (-6 x^2+\log \left (x^2-x+1\right )+\log \left (x^2+x+1\right )-2 \log (1-x)-2 \log (x+1)+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^7/(1 - x^6),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.017, size = 71, normalized size = 1.3 \[ -{\frac{{x}^{2}}{2}}+{\frac{\ln \left ({x}^{2}+x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( -1+x \right ) }{6}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 1+x \right ) }{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^7/(-x^6+1),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.59159, size = 58, normalized size = 1.04 \[ -\frac{1}{2} \, x^{2} + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) + \frac{1}{12} \, \log \left (x^{4} + x^{2} + 1\right ) - \frac{1}{6} \, \log \left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^7/(x^6 - 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.225738, size = 73, normalized size = 1.3 \[ -\frac{1}{36} \, \sqrt{3}{\left (6 \, \sqrt{3} x^{2} - \sqrt{3} \log \left (x^{4} + x^{2} + 1\right ) + 2 \, \sqrt{3} \log \left (x^{2} - 1\right ) - 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^7/(x^6 - 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.427112, size = 51, normalized size = 0.91 \[ - \frac{x^{2}}{2} - \frac{\log{\left (x^{2} - 1 \right )}}{6} + \frac{\log{\left (x^{4} + x^{2} + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} + \frac{\sqrt{3}}{3} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**7/(-x**6+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.227228, size = 59, normalized size = 1.05 \[ -\frac{1}{2} \, x^{2} + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) + \frac{1}{12} \,{\rm ln}\left (x^{4} + x^{2} + 1\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^7/(x^6 - 1),x, algorithm="giac")
[Out]