Optimal. Leaf size=32 \[ -\frac{3}{4 x^2}+\frac{3}{4} \tanh ^{-1}\left (x^2\right )+\frac{1}{4 x^2 \left (1-x^4\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0345444, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{3}{4 x^2}+\frac{3}{4} \tanh ^{-1}\left (x^2\right )+\frac{1}{4 x^2 \left (1-x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(1 - 2*x^4 + x^8)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.86682, size = 26, normalized size = 0.81 \[ \frac{3 \operatorname{atanh}{\left (x^{2} \right )}}{4} - \frac{3}{4 x^{2}} + \frac{1}{4 x^{2} \left (- x^{4} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(x**8-2*x**4+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0327269, size = 41, normalized size = 1.28 \[ \frac{1}{8} \left (-3 \log \left (1-x^2\right )+3 \log \left (x^2+1\right )+\frac{4-6 x^4}{x^2 \left (x^4-1\right )}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(1 - 2*x^4 + x^8)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.024, size = 50, normalized size = 1.6 \[ -{\frac{1}{-16+16\,x}}-{\frac{3\,\ln \left ( -1+x \right ) }{8}}+{\frac{1}{16+16\,x}}-{\frac{3\,\ln \left ( 1+x \right ) }{8}}-{\frac{1}{2\,{x}^{2}}}-{\frac{1}{8\,{x}^{2}+8}}+{\frac{3\,\ln \left ({x}^{2}+1 \right ) }{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(x^8-2*x^4+1),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.770033, size = 50, normalized size = 1.56 \[ -\frac{3 \, x^{4} - 2}{4 \,{\left (x^{6} - x^{2}\right )}} + \frac{3}{8} \, \log \left (x^{2} + 1\right ) - \frac{3}{8} \, \log \left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - 2*x^4 + 1)*x^3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.284432, size = 73, normalized size = 2.28 \[ -\frac{6 \, x^{4} - 3 \,{\left (x^{6} - x^{2}\right )} \log \left (x^{2} + 1\right ) + 3 \,{\left (x^{6} - x^{2}\right )} \log \left (x^{2} - 1\right ) - 4}{8 \,{\left (x^{6} - x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - 2*x^4 + 1)*x^3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.399905, size = 36, normalized size = 1.12 \[ - \frac{3 x^{4} - 2}{4 x^{6} - 4 x^{2}} - \frac{3 \log{\left (x^{2} - 1 \right )}}{8} + \frac{3 \log{\left (x^{2} + 1 \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(x**8-2*x**4+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.287366, size = 51, normalized size = 1.59 \[ -\frac{3 \, x^{4} - 2}{4 \,{\left (x^{6} - x^{2}\right )}} + \frac{3}{8} \,{\rm ln}\left (x^{2} + 1\right ) - \frac{3}{8} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - 2*x^4 + 1)*x^3),x, algorithm="giac")
[Out]