Optimal. Leaf size=154 \[ -\frac{1}{7 x^7}+\frac{1}{3 x^3}-\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.284647, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714 \[ -\frac{1}{7 x^7}+\frac{1}{3 x^3}-\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(x^8*(1 + x^4 + x^8)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 62.7571, size = 141, normalized size = 0.92 \[ - \frac{\log{\left (x^{2} - x + 1 \right )}}{8} + \frac{\log{\left (x^{2} + x + 1 \right )}}{8} + \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{24} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{24} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{12} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{4} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{4} + \frac{1}{3 x^{3}} - \frac{1}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**8/(x**8+x**4+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.693319, size = 171, normalized size = 1.11 \[ -\frac{1}{7 x^7}+\frac{1}{3 x^3}-\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )+\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x\right )}{2 \sqrt{-6+6 i \sqrt{3}}}+\frac{\left (\sqrt{3}-i\right ) \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x\right )}{2 \sqrt{-6-6 i \sqrt{3}}}-\frac{\tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^8*(1 + x^4 + x^8)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.012, size = 119, normalized size = 0.8 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{4}}-{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{4}}-{\frac{1}{7\,{x}^{7}}}+{\frac{1}{3\,{x}^{3}}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^8/(x^8+x^4+1),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{7 \, x^{4} - 3}{21 \, x^{7}} + \frac{1}{2} \, \int \frac{x^{2}}{x^{4} - x^{2} + 1}\,{d x} + \frac{1}{8} \, \log \left (x^{2} + x + 1\right ) - \frac{1}{8} \, \log \left (x^{2} - x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + x^4 + 1)*x^8),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.274457, size = 250, normalized size = 1.62 \[ -\frac{\sqrt{3}{\left (84 \, \sqrt{3} x^{7} \arctan \left (\frac{1}{2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}}\right ) + 84 \, \sqrt{3} x^{7} \arctan \left (\frac{1}{2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}}\right ) - 21 \, \sqrt{3} x^{7} \log \left (x^{2} + x + 1\right ) + 21 \, \sqrt{3} x^{7} \log \left (x^{2} - x + 1\right ) + 42 \, x^{7} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 42 \, x^{7} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + 21 \, x^{7} \log \left (x^{2} + \sqrt{3} x + 1\right ) - 21 \, x^{7} \log \left (x^{2} - \sqrt{3} x + 1\right ) - 8 \, \sqrt{3}{\left (7 \, x^{4} - 3\right )}\right )}}{504 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + x^4 + 1)*x^8),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.12639, size = 209, normalized size = 1.36 \[ \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x - \frac{1}{2} + \frac{\sqrt{3} i}{6} - 18432 \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{5} \right )} + \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x - \frac{1}{2} - 18432 \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{5} - \frac{\sqrt{3} i}{6} \right )} + \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x + \frac{1}{2} + \frac{\sqrt{3} i}{6} - 18432 \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{5} \right )} + \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x + \frac{1}{2} - 18432 \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{5} - \frac{\sqrt{3} i}{6} \right )} + \operatorname{RootSum}{\left (2304 t^{4} + 48 t^{2} + 1, \left ( t \mapsto t \log{\left (- 18432 t^{5} - 4 t + x \right )} \right )\right )} + \frac{7 x^{4} - 3}{21 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**8/(x**8+x**4+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{8} + x^{4} + 1\right )} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + x^4 + 1)*x^8),x, algorithm="giac")
[Out]