Optimal. Leaf size=24 \[ \frac{1}{4 \left (x^4+1\right )}-\frac{1}{4} \log \left (x^4+1\right )+\log (x) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.028266, antiderivative size = 24, 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 \[ \frac{1}{4 \left (x^4+1\right )}-\frac{1}{4} \log \left (x^4+1\right )+\log (x) \]
Antiderivative was successfully verified.
[In] Int[1/(x*(1 + 2*x^4 + x^8)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.37042, size = 22, normalized size = 0.92 \[ \frac{\log{\left (x^{4} \right )}}{4} - \frac{\log{\left (x^{4} + 1 \right )}}{4} + \frac{1}{4 \left (x^{4} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(x**8+2*x**4+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0133068, size = 24, normalized size = 1. \[ \frac{1}{4 \left (x^4+1\right )}-\frac{1}{4} \log \left (x^4+1\right )+\log (x) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(1 + 2*x^4 + x^8)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.019, size = 21, normalized size = 0.9 \[{\frac{1}{4\,{x}^{4}+4}}+\ln \left ( x \right ) -{\frac{\ln \left ({x}^{4}+1 \right ) }{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(x^8+2*x^4+1),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.764091, size = 32, normalized size = 1.33 \[ \frac{1}{4 \,{\left (x^{4} + 1\right )}} - \frac{1}{4} \, \log \left (x^{4} + 1\right ) + \frac{1}{4} \, \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + 2*x^4 + 1)*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.253871, size = 43, normalized size = 1.79 \[ -\frac{{\left (x^{4} + 1\right )} \log \left (x^{4} + 1\right ) - 4 \,{\left (x^{4} + 1\right )} \log \left (x\right ) - 1}{4 \,{\left (x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + 2*x^4 + 1)*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.301816, size = 19, normalized size = 0.79 \[ \log{\left (x \right )} - \frac{\log{\left (x^{4} + 1 \right )}}{4} + \frac{1}{4 x^{4} + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(x**8+2*x**4+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.303119, size = 39, normalized size = 1.62 \[ \frac{x^{4} + 2}{4 \,{\left (x^{4} + 1\right )}} - \frac{1}{4} \,{\rm ln}\left (x^{4} + 1\right ) + \frac{1}{4} \,{\rm ln}\left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + 2*x^4 + 1)*x),x, algorithm="giac")
[Out]