Optimal. Leaf size=81 \[ \frac{1}{2} \sqrt{\frac{1}{10} \left (3-\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )-\frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right ) \]
[Out]
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Rubi [A] time = 0.147157, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{2} \sqrt{\frac{1}{10} \left (3-\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )-\frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right ) \]
Antiderivative was successfully verified.
[In] Int[x^5/(1 - 3*x^4 + x^8),x]
[Out]
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Rubi in Sympy [A] time = 15.4152, size = 99, normalized size = 1.22 \[ - \frac{\sqrt{2} \left (- \frac{3 \sqrt{5}}{10} + \frac{1}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{- \sqrt{5} + 3}} \right )}}{2 \sqrt{- \sqrt{5} + 3}} - \frac{\sqrt{2} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{\sqrt{5} + 3}} \right )}}{2 \sqrt{\sqrt{5} + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(x**8-3*x**4+1),x)
[Out]
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Mathematica [A] time = 0.0504037, size = 91, normalized size = 1.12 \[ \frac{1}{40} \left (\left (\sqrt{5}-5\right ) \log \left (-2 x^2+\sqrt{5}-1\right )+\left (5+\sqrt{5}\right ) \log \left (-2 x^2+\sqrt{5}+1\right )-\left (\sqrt{5}-5\right ) \log \left (2 x^2+\sqrt{5}-1\right )-\left (5+\sqrt{5}\right ) \log \left (2 x^2+\sqrt{5}+1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^5/(1 - 3*x^4 + x^8),x]
[Out]
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Maple [A] time = 0.005, size = 62, normalized size = 0.8 \[{\frac{\ln \left ({x}^{4}-{x}^{2}-1 \right ) }{8}}-{\frac{\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{5}}{5}} \right ) }-{\frac{\ln \left ({x}^{4}+{x}^{2}-1 \right ) }{8}}-{\frac{\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{5}}{5}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(x^8-3*x^4+1),x)
[Out]
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Maxima [A] time = 0.826103, size = 117, normalized size = 1.44 \[ \frac{1}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} + 1}{2 \, x^{2} + \sqrt{5} + 1}\right ) + \frac{1}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} - 1}{2 \, x^{2} + \sqrt{5} - 1}\right ) - \frac{1}{8} \, \log \left (x^{4} + x^{2} - 1\right ) + \frac{1}{8} \, \log \left (x^{4} - x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(x^8 - 3*x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.31288, size = 155, normalized size = 1.91 \[ -\frac{1}{40} \, \sqrt{5}{\left (\sqrt{5} \log \left (x^{4} + x^{2} - 1\right ) - \sqrt{5} \log \left (x^{4} - x^{2} - 1\right ) - \log \left (-\frac{10 \, x^{2} - \sqrt{5}{\left (2 \, x^{4} + 2 \, x^{2} + 3\right )} + 5}{x^{4} + x^{2} - 1}\right ) - \log \left (-\frac{10 \, x^{2} - \sqrt{5}{\left (2 \, x^{4} - 2 \, x^{2} + 3\right )} - 5}{x^{4} - x^{2} - 1}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(x^8 - 3*x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.73989, size = 165, normalized size = 2.04 \[ \left (- \frac{1}{8} - \frac{\sqrt{5}}{40}\right ) \log{\left (x^{2} - \frac{3}{2} - \frac{3 \sqrt{5}}{10} - 640 \left (- \frac{1}{8} - \frac{\sqrt{5}}{40}\right )^{3} \right )} + \left (- \frac{1}{8} + \frac{\sqrt{5}}{40}\right ) \log{\left (x^{2} - \frac{3}{2} - 640 \left (- \frac{1}{8} + \frac{\sqrt{5}}{40}\right )^{3} + \frac{3 \sqrt{5}}{10} \right )} + \left (- \frac{\sqrt{5}}{40} + \frac{1}{8}\right ) \log{\left (x^{2} - \frac{3 \sqrt{5}}{10} - 640 \left (- \frac{\sqrt{5}}{40} + \frac{1}{8}\right )^{3} + \frac{3}{2} \right )} + \left (\frac{\sqrt{5}}{40} + \frac{1}{8}\right ) \log{\left (x^{2} - 640 \left (\frac{\sqrt{5}}{40} + \frac{1}{8}\right )^{3} + \frac{3 \sqrt{5}}{10} + \frac{3}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(x**8-3*x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.294872, size = 124, normalized size = 1.53 \[ \frac{1}{40} \, \sqrt{5}{\rm ln}\left (\frac{{\left | 2 \, x^{2} - \sqrt{5} + 1 \right |}}{2 \, x^{2} + \sqrt{5} + 1}\right ) + \frac{1}{40} \, \sqrt{5}{\rm ln}\left (\frac{{\left | 2 \, x^{2} - \sqrt{5} - 1 \right |}}{{\left | 2 \, x^{2} + \sqrt{5} - 1 \right |}}\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x^{4} + x^{2} - 1 \right |}\right ) + \frac{1}{8} \,{\rm ln}\left ({\left | x^{4} - x^{2} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(x^8 - 3*x^4 + 1),x, algorithm="giac")
[Out]