Optimal. Leaf size=100 \[ -\frac{\log \left (x^2-\sqrt{2} x+1\right )}{8 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2} x+1\right )}{8 \sqrt{2}}-x+\frac{1}{4} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{1}{4} \tanh ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.1225, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.846 \[ -\frac{\log \left (x^2-\sqrt{2} x+1\right )}{8 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2} x+1\right )}{8 \sqrt{2}}-x+\frac{1}{4} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{1}{4} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[x^8/(1 - x^8),x]
[Out]
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Rubi in Sympy [A] time = 17.7037, size = 85, normalized size = 0.85 \[ - x - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{16} + \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{16} + \frac{\operatorname{atan}{\left (x \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{8} + \frac{\operatorname{atanh}{\left (x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8/(-x**8+1),x)
[Out]
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Mathematica [A] time = 0.0516753, size = 101, normalized size = 1.01 \[ \frac{1}{16} \left (-\sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )+\sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )-16 x-2 \log (1-x)+2 \log (x+1)+4 \tan ^{-1}(x)-2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )+2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^8/(1 - x^8),x]
[Out]
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Maple [A] time = 0.007, size = 77, normalized size = 0.8 \[ -x-{\frac{\ln \left ( -1+x \right ) }{8}}+{\frac{\ln \left ( 1+x \right ) }{8}}+{\frac{\arctan \left ( x \right ) }{4}}+{\frac{\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{8}}+{\frac{\arctan \left ( x\sqrt{2}-1 \right ) \sqrt{2}}{8}}+{\frac{\sqrt{2}}{16}\ln \left ({\frac{1+{x}^{2}+x\sqrt{2}}{1+{x}^{2}-x\sqrt{2}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8/(-x^8+1),x)
[Out]
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Maxima [A] time = 1.59145, size = 123, normalized size = 1.23 \[ \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{16} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{16} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) - x + \frac{1}{4} \, \arctan \left (x\right ) + \frac{1}{8} \, \log \left (x + 1\right ) - \frac{1}{8} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^8/(x^8 - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237907, size = 159, normalized size = 1.59 \[ -\frac{1}{16} \, \sqrt{2}{\left (8 \, \sqrt{2} x - 2 \, \sqrt{2} \arctan \left (x\right ) - \sqrt{2} \log \left (x + 1\right ) + \sqrt{2} \log \left (x - 1\right ) + 4 \, \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} + 1}\right ) + 4 \, \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} - 1}\right ) - \log \left (x^{2} + \sqrt{2} x + 1\right ) + \log \left (x^{2} - \sqrt{2} x + 1\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^8/(x^8 - 1),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8/(-x**8+1),x)
[Out]
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GIAC/XCAS [A] time = 0.219697, size = 126, normalized size = 1.26 \[ \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{16} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{16} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) - x + \frac{1}{4} \, \arctan \left (x\right ) + \frac{1}{8} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^8/(x^8 - 1),x, algorithm="giac")
[Out]