Optimal. Leaf size=85 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{x}+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
[Out]
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Rubi [A] time = 0.507893, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{x}+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(1 + x^6)),x]
[Out]
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Rubi in Sympy [A] time = 106.733, size = 71, normalized size = 0.84 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} - \frac{\operatorname{atan}{\left (x \right )}}{3} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} - \frac{1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(x**6+1),x)
[Out]
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Mathematica [A] time = 0.0415165, size = 82, normalized size = 0.96 \[ -\frac{\sqrt{3} x \log \left (x^2-\sqrt{3} x+1\right )-\sqrt{3} x \log \left (x^2+\sqrt{3} x+1\right )-2 x \tan ^{-1}\left (\sqrt{3}-2 x\right )+4 x \tan ^{-1}(x)+2 x \tan ^{-1}\left (2 x+\sqrt{3}\right )+12}{12 x} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(1 + x^6)),x]
[Out]
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Maple [A] time = 0.02, size = 66, normalized size = 0.8 \[ -{x}^{-1}-{\frac{\arctan \left ( x \right ) }{3}}-{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{6}}-{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{6}}-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(x^6+1),x)
[Out]
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Maxima [A] time = 1.59405, size = 88, normalized size = 1.04 \[ \frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) - \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) - \frac{1}{x} - \frac{1}{6} \, \arctan \left (2 \, x + \sqrt{3}\right ) - \frac{1}{6} \, \arctan \left (2 \, x - \sqrt{3}\right ) - \frac{1}{3} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^6 + 1)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234086, size = 139, normalized size = 1.64 \[ \frac{\sqrt{3} x \log \left (x^{2} + \sqrt{3} x + 1\right ) - \sqrt{3} x \log \left (x^{2} - \sqrt{3} x + 1\right ) - 4 \, x \arctan \left (x\right ) + 4 \, x \arctan \left (\frac{1}{2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}}\right ) + 4 \, x \arctan \left (\frac{1}{2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}}\right ) - 12}{12 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^6 + 1)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.7333, size = 71, normalized size = 0.84 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} - \frac{\operatorname{atan}{\left (x \right )}}{3} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} - \frac{1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(x**6+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{6} + 1\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^6 + 1)*x^2),x, algorithm="giac")
[Out]