Optimal. Leaf size=96 \[ -\frac{1}{6 x^6}-\frac{1}{2 x^2}+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x^2\right )-\frac{1}{4} \tan ^{-1}\left (2 x^2+\sqrt{3}\right )-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{8 \sqrt{3}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.185119, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625 \[ -\frac{1}{6 x^6}-\frac{1}{2 x^2}+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x^2\right )-\frac{1}{4} \tan ^{-1}\left (2 x^2+\sqrt{3}\right )-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{8 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*(1 - x^4 + x^8)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 40.9123, size = 83, normalized size = 0.86 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{24} - \frac{\operatorname{atan}{\left (2 x^{2} - \sqrt{3} \right )}}{4} - \frac{\operatorname{atan}{\left (2 x^{2} + \sqrt{3} \right )}}{4} - \frac{1}{2 x^{2}} - \frac{1}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**7/(x**8-x**4+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0271525, size = 56, normalized size = 0.58 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right ]-\frac{1}{6 x^6}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^7*(1 - x^4 + x^8)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 75, normalized size = 0.8 \[ -{\frac{1}{6\,{x}^{6}}}-{\frac{1}{2\,{x}^{2}}}-{\frac{\arctan \left ( 2\,{x}^{2}-\sqrt{3} \right ) }{4}}-{\frac{\arctan \left ( 2\,{x}^{2}+\sqrt{3} \right ) }{4}}-{\frac{\ln \left ( 1+{x}^{4}-{x}^{2}\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\ln \left ( 1+{x}^{4}+{x}^{2}\sqrt{3} \right ) \sqrt{3}}{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^7/(x^8-x^4+1),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{3 \, x^{4} + 1}{6 \, x^{6}} - \int \frac{x^{5}}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - x^4 + 1)*x^7),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.269092, size = 198, normalized size = 2.06 \[ \frac{\sqrt{3}{\left (12 \, \sqrt{3} x^{6} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x^{2} + 2 \, \sqrt{3} \sqrt{x^{4} + \sqrt{3} x^{2} + 1} + 3}\right ) + 12 \, \sqrt{3} x^{6} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x^{2} + 2 \, \sqrt{3} \sqrt{x^{4} - \sqrt{3} x^{2} + 1} - 3}\right ) + 3 \, x^{6} \log \left (x^{4} + \sqrt{3} x^{2} + 1\right ) - 3 \, x^{6} \log \left (x^{4} - \sqrt{3} x^{2} + 1\right ) - 4 \, \sqrt{3}{\left (3 \, x^{4} + 1\right )}\right )}}{72 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - x^4 + 1)*x^7),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.861041, size = 82, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{24} - \frac{\operatorname{atan}{\left (2 x^{2} - \sqrt{3} \right )}}{4} - \frac{\operatorname{atan}{\left (2 x^{2} + \sqrt{3} \right )}}{4} - \frac{3 x^{4} + 1}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**7/(x**8-x**4+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.344287, size = 358, normalized size = 3.73 \[ -\frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{96} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{96} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{96} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) + \frac{1}{96} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{3 \, x^{4} + 1}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - x^4 + 1)*x^7),x, algorithm="giac")
[Out]