Optimal. Leaf size=360 \[ \frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.694282, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(1 - x^4 + x^8)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 99.6204, size = 484, normalized size = 1.34 \[ \frac{\sqrt{3} \left (- \frac{\sqrt{3}}{2} + 1\right ) \log{\left (x^{2} - x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{12 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \left (- \frac{\sqrt{3}}{2} + 1\right ) \log{\left (x^{2} + x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{12 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \left (\frac{\sqrt{3}}{2} + 1\right ) \log{\left (x^{2} - x \sqrt{\sqrt{3} + 2} + 1 \right )}}{12 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (\frac{\sqrt{3}}{2} + 1\right ) \log{\left (x^{2} + x \sqrt{\sqrt{3} + 2} + 1 \right )}}{12 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (- \frac{\left (\sqrt{3} + 2\right )^{\frac{3}{2}}}{2} + 2 \sqrt{\sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (- \frac{\left (\sqrt{3} + 2\right )^{\frac{3}{2}}}{2} + 2 \sqrt{\sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} - \frac{\sqrt{3} \left (- \frac{\left (- \sqrt{3} + 2\right )^{\frac{3}{2}}}{2} + 2 \sqrt{- \sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} - \frac{\sqrt{3} \left (- \frac{\left (- \sqrt{3} + 2\right )^{\frac{3}{2}}}{2} + 2 \sqrt{- \sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} - \frac{1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(x**8-x**4+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0218228, size = 61, normalized size = 0.17 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}}\&\right ]-\frac{1}{x} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(1 - x^4 + x^8)),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.015, size = 52, normalized size = 0.1 \[ -{x}^{-1}-{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ({{\it \_R}}^{6}-{{\it \_R}}^{2} \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(x^8-x^4+1),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{1}{x} - \int \frac{x^{6} - x^{2}}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - x^4 + 1)*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.282247, size = 1311, normalized size = 3.64 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - x^4 + 1)*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.91863, size = 29, normalized size = 0.08 \[ \operatorname{RootSum}{\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log{\left (- 442368 t^{7} + 384 t^{3} + x \right )} \right )\right )} - \frac{1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(x**8-x**4+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.292224, size = 348, normalized size = 0.97 \[ -\frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - x^4 + 1)*x^2),x, algorithm="giac")
[Out]