Optimal. Leaf size=33 \[ -\frac{1}{4 \left (x^4+1\right )}-\frac{1}{4 x^4}+\frac{1}{2} \log \left (x^4+1\right )-2 \log (x) \]
[Out]
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Rubi [A] time = 0.0358058, antiderivative size = 33, 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 \[ -\frac{1}{4 \left (x^4+1\right )}-\frac{1}{4 x^4}+\frac{1}{2} \log \left (x^4+1\right )-2 \log (x) \]
Antiderivative was successfully verified.
[In] Int[1/(x^5*(1 + 2*x^4 + x^8)),x]
[Out]
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Rubi in Sympy [A] time = 5.74552, size = 29, normalized size = 0.88 \[ - \frac{\log{\left (x^{4} \right )}}{2} + \frac{\log{\left (x^{4} + 1 \right )}}{2} - \frac{1}{4 \left (x^{4} + 1\right )} - \frac{1}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**5/(x**8+2*x**4+1),x)
[Out]
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Mathematica [A] time = 0.0191513, size = 33, normalized size = 1. \[ -\frac{1}{4 \left (x^4+1\right )}-\frac{1}{4 x^4}+\frac{1}{2} \log \left (x^4+1\right )-2 \log (x) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^5*(1 + 2*x^4 + x^8)),x]
[Out]
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Maple [A] time = 0.021, size = 28, normalized size = 0.9 \[ -{\frac{1}{4\,{x}^{4}}}-{\frac{1}{4\,{x}^{4}+4}}-2\,\ln \left ( x \right ) +{\frac{\ln \left ({x}^{4}+1 \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^5/(x^8+2*x^4+1),x)
[Out]
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Maxima [A] time = 0.759771, size = 45, normalized size = 1.36 \[ -\frac{2 \, x^{4} + 1}{4 \,{\left (x^{8} + x^{4}\right )}} + \frac{1}{2} \, \log \left (x^{4} + 1\right ) - \frac{1}{2} \, \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + 2*x^4 + 1)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.252424, size = 59, normalized size = 1.79 \[ -\frac{2 \, x^{4} - 2 \,{\left (x^{8} + x^{4}\right )} \log \left (x^{4} + 1\right ) + 8 \,{\left (x^{8} + x^{4}\right )} \log \left (x\right ) + 1}{4 \,{\left (x^{8} + x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + 2*x^4 + 1)*x^5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.428095, size = 29, normalized size = 0.88 \[ - \frac{2 x^{4} + 1}{4 x^{8} + 4 x^{4}} - 2 \log{\left (x \right )} + \frac{\log{\left (x^{4} + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**5/(x**8+2*x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.29694, size = 45, normalized size = 1.36 \[ -\frac{2 \, x^{4} + 1}{4 \,{\left (x^{8} + x^{4}\right )}} + \frac{1}{2} \,{\rm ln}\left (x^{4} + 1\right ) - \frac{1}{2} \,{\rm ln}\left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + 2*x^4 + 1)*x^5),x, algorithm="giac")
[Out]