Optimal. Leaf size=347 \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}} \]
[Out]
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Rubi [A] time = 0.549372, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}} \]
Antiderivative was successfully verified.
[In] Int[x^4/(1 + x^8),x]
[Out]
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Rubi in Sympy [A] time = 56.0535, size = 311, normalized size = 0.9 \[ - \frac{\sqrt{2} \log{\left (x^{2} - x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{16 \sqrt{- \sqrt{2} + 2}} + \frac{\sqrt{2} \log{\left (x^{2} + x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{16 \sqrt{- \sqrt{2} + 2}} + \frac{\sqrt{2} \log{\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1 \right )}}{16 \sqrt{\sqrt{2} + 2}} - \frac{\sqrt{2} \log{\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1 \right )}}{16 \sqrt{\sqrt{2} + 2}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{8 \sqrt{- \sqrt{2} + 2}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{8 \sqrt{- \sqrt{2} + 2}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{8 \sqrt{\sqrt{2} + 2}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{8 \sqrt{\sqrt{2} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(x**8+1),x)
[Out]
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Mathematica [A] time = 0.00982412, size = 209, normalized size = 0.6 \[ -\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x-\cos \left (\frac{\pi }{8}\right )\right )\right )+\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x+\cos \left (\frac{\pi }{8}\right )\right )\right )-\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x-\sin \left (\frac{\pi }{8}\right )\right )\right )-\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x+\sin \left (\frac{\pi }{8}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(1 + x^8),x]
[Out]
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Maple [C] time = 0.006, size = 22, normalized size = 0.1 \[{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(x^8+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{x^{8} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(x^8 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240611, size = 1345, normalized size = 3.88 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(x^8 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.33024, size = 15, normalized size = 0.04 \[ \operatorname{RootSum}{\left (16777216 t^{8} + 1, \left ( t \mapsto t \log{\left (- 32768 t^{5} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(x**8+1),x)
[Out]
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GIAC/XCAS [A] time = 0.261436, size = 323, normalized size = 0.93 \[ -\frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) - \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) + \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) + \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) - \frac{1}{16} \, \sqrt{-\sqrt{2} + 2}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1\right ) + \frac{1}{16} \, \sqrt{-\sqrt{2} + 2}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1\right ) + \frac{1}{16} \, \sqrt{\sqrt{2} + 2}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{\sqrt{2} + 2}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(x^8 + 1),x, algorithm="giac")
[Out]