Optimal. Leaf size=169 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{14 \left (\sqrt{7}-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{14 \left (\sqrt{7}-\sqrt{3}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}} \]
[Out]
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Rubi [A] time = 0.292635, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{14 \left (\sqrt{7}-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{14 \left (\sqrt{7}-\sqrt{3}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}} \]
Antiderivative was successfully verified.
[In] Int[(1 - x^4)/(1 - 5*x^4 + x^8),x]
[Out]
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Rubi in Sympy [A] time = 18.8649, size = 168, normalized size = 0.99 \[ \frac{\sqrt{14} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{3} + \sqrt{7}}} \right )}}{14 \sqrt{- \sqrt{3} + \sqrt{7}}} + \frac{\sqrt{14} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{3} + \sqrt{7}}} \right )}}{14 \sqrt{\sqrt{3} + \sqrt{7}}} + \frac{\sqrt{14} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{3} + \sqrt{7}}} \right )}}{14 \sqrt{- \sqrt{3} + \sqrt{7}}} + \frac{\sqrt{14} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{3} + \sqrt{7}}} \right )}}{14 \sqrt{\sqrt{3} + \sqrt{7}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**4+1)/(x**8-5*x**4+1),x)
[Out]
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Mathematica [C] time = 0.0207839, size = 57, normalized size = 0.34 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-5 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-5 \text{$\#$1}^3}\&\right ] \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x^4)/(1 - 5*x^4 + x^8),x]
[Out]
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Maple [C] time = 0.01, size = 44, normalized size = 0.3 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-5\,{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-5\,{{\it \_R}}^{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^4+1)/(x^8-5*x^4+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{4} - 1}{x^{8} - 5 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^4 - 1)/(x^8 - 5*x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.310394, size = 693, normalized size = 4.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^4 - 1)/(x^8 - 5*x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.57599, size = 26, normalized size = 0.15 \[ - \operatorname{RootSum}{\left (157351936 t^{8} - 62720 t^{4} + 1, \left ( t \mapsto t \log{\left (50176 t^{5} - 24 t + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**4+1)/(x**8-5*x**4+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{4} - 1}{x^{8} - 5 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^4 - 1)/(x^8 - 5*x^4 + 1),x, algorithm="giac")
[Out]