Optimal. Leaf size=370 \[ -\frac{1}{3 x^3}+\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]
[Out]
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Rubi [A] time = 0.54314, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ -\frac{1}{3 x^3}+\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(1 - x^4 + x^8)),x]
[Out]
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Rubi in Sympy [A] time = 81.1481, size = 549, normalized size = 1.48 \[ \frac{\sqrt{3} \left (- \frac{3 \sqrt{3}}{2} + 3\right ) \log{\left (x^{2} - x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{36 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \left (- \frac{3 \sqrt{3}}{2} + 3\right ) \log{\left (x^{2} + x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{36 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \left (\frac{3 \sqrt{3}}{2} + 3\right ) \log{\left (x^{2} - x \sqrt{\sqrt{3} + 2} + 1 \right )}}{36 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (\frac{3 \sqrt{3}}{2} + 3\right ) \log{\left (x^{2} + x \sqrt{\sqrt{3} + 2} + 1 \right )}}{36 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (- \frac{\sqrt{\sqrt{3} + 2} \left (3 \sqrt{3} + 6\right )}{2} + 3 \sqrt{3} \sqrt{\sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{18 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (- \frac{\sqrt{\sqrt{3} + 2} \left (3 \sqrt{3} + 6\right )}{2} + 3 \sqrt{3} \sqrt{\sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{18 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (\frac{\left (- 3 \sqrt{3} + 6\right ) \sqrt{- \sqrt{3} + 2}}{2} + 3 \sqrt{3} \sqrt{- \sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{18 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (\frac{\left (- 3 \sqrt{3} + 6\right ) \sqrt{- \sqrt{3} + 2}}{2} + 3 \sqrt{3} \sqrt{- \sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{18 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} - \frac{1}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(x**8-x**4+1),x)
[Out]
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Mathematica [C] time = 0.0193475, size = 65, normalized size = 0.18 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right ]-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(1 - x^4 + x^8)),x]
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Maple [C] time = 0.014, size = 50, normalized size = 0.1 \[ -{\frac{1}{3\,{x}^{3}}}+{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(x^8-x^4+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{1}{3 \, x^{3}} - \int \frac{x^{4} - 1}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - x^4 + 1)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285668, size = 1354, normalized size = 3.66 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - x^4 + 1)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.13596, size = 31, normalized size = 0.08 \[ \operatorname{RootSum}{\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log{\left (- 9216 t^{5} + 8 t + x \right )} \right )\right )} - \frac{1}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(x**8-x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.29846, size = 348, normalized size = 0.94 \[ \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - x^4 + 1)*x^4),x, algorithm="giac")
[Out]