Optimal. Leaf size=192 \[ -\frac{1}{3 x^3}-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )+\frac{1}{5} \log (x+1)+\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} x+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )+\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x\right ) \]
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Rubi [A] time = 0.535079, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ -\frac{1}{3 x^3}-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )+\frac{1}{5} \log (x+1)+\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} x+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )+\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(1 + x^5)),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(x**5+1),x)
[Out]
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Mathematica [A] time = 0.402624, size = 150, normalized size = 0.78 \[ \frac{1}{60} \left (-\frac{20}{x^3}-3 \left (1+\sqrt{5}\right ) \log \left (x^2+\frac{1}{2} \left (\sqrt{5}-1\right ) x+1\right )+3 \left (\sqrt{5}-1\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )+12 \log (x+1)+6 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{-4 x+\sqrt{5}+1}{\sqrt{10-2 \sqrt{5}}}\right )+6 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{4 x+\sqrt{5}-1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )\right ) \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^4*(1 + x^5)),x]
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Maple [A] time = 0.022, size = 161, normalized size = 0.8 \[ -{\frac{1}{3\,{x}^{3}}}+{\frac{\ln \left ( 1+x \right ) }{5}}-{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) \sqrt{5}}{20}}-{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{20}}+{\frac{2\,\sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) }+{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) \sqrt{5}}{20}}-{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{20}}-{\frac{2\,\sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(x^5+1),x)
[Out]
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Maxima [A] time = 1.60261, size = 173, normalized size = 0.9 \[ \frac{2 \, \sqrt{5} \arctan \left (\frac{4 \, x + \sqrt{5} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right )}{5 \, \sqrt{2 \, \sqrt{5} + 10}} - \frac{2 \, \sqrt{5} \arctan \left (\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right )}{5 \, \sqrt{-2 \, \sqrt{5} + 10}} + \frac{\log \left (2 \, x^{2} - x{\left (\sqrt{5} + 1\right )} + 2\right )}{5 \, \sqrt{5} + 5} - \frac{\log \left (2 \, x^{2} + x{\left (\sqrt{5} - 1\right )} + 2\right )}{5 \,{\left (\sqrt{5} - 1\right )}} - \frac{1}{3 \, x^{3}} + \frac{1}{5} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^5 + 1)*x^4),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^5 + 1)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.61751, size = 42, normalized size = 0.22 \[ \frac{\log{\left (x + 1 \right )}}{5} + \operatorname{RootSum}{\left (625 t^{4} + 125 t^{3} + 25 t^{2} + 5 t + 1, \left ( t \mapsto t \log{\left (125 t^{3} + x \right )} \right )\right )} - \frac{1}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(x**5+1),x)
[Out]
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GIAC/XCAS [A] time = 0.236997, size = 178, normalized size = 0.93 \[ \frac{1}{10} \, \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{4 \, x + \sqrt{5} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) - \frac{1}{10} \, \sqrt{2 \, \sqrt{5} + 10} \arctan \left (\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) + \frac{1}{20} \, \sqrt{5}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) - \frac{1}{20} \, \sqrt{5}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} - 1\right )} + 1\right ) - \frac{1}{3 \, x^{3}} - \frac{1}{20} \,{\rm ln}\left (x^{4} - x^{3} + x^{2} - x + 1\right ) + \frac{1}{5} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^5 + 1)*x^4),x, algorithm="giac")
[Out]