Optimal. Leaf size=74 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{2} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{2} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0896653, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{2} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{2} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 - x^2 + x^4)^(-1),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 16.7216, size = 63, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{2} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**4-x**2+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.121835, size = 77, normalized size = 1.04 \[ \frac{i \left (\sqrt{-1-i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x\right )-\sqrt{-1+i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x\right )\right )}{\sqrt{6}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 - x^2 + x^4)^(-1),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.035, size = 57, normalized size = 0.8 \[{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{2}}+{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{2}}-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^4-x^2+1),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} - x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^4 - x^2 + 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.267376, size = 143, normalized size = 1.93 \[ -\frac{1}{12} \, \sqrt{3}{\left (4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x + 2 \, \sqrt{3} \sqrt{x^{2} + \sqrt{3} x + 1} + 3}\right ) + 4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x + 2 \, \sqrt{3} \sqrt{x^{2} - \sqrt{3} x + 1} - 3}\right ) - \log \left (x^{2} + \sqrt{3} x + 1\right ) + \log \left (x^{2} - \sqrt{3} x + 1\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^4 - x^2 + 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.526391, size = 63, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{2} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**4-x**2+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} - x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^4 - x^2 + 1),x, algorithm="giac")
[Out]