Optimal. Leaf size=53 \[ -\frac{3 \sqrt{1-x^4}}{4 x^4}+\frac{1}{2 x^4 \sqrt{1-x^4}}-\frac{3}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0607043, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{3 \sqrt{1-x^4}}{4 x^4}+\frac{1}{2 x^4 \sqrt{1-x^4}}-\frac{3}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(x^5*(1 - x^4)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.87212, size = 42, normalized size = 0.79 \[ - \frac{3 \operatorname{atanh}{\left (\sqrt{- x^{4} + 1} \right )}}{4} - \frac{3 \sqrt{- x^{4} + 1}}{4 x^{4}} + \frac{1}{2 x^{4} \sqrt{- x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**5/(-x**4+1)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0909456, size = 41, normalized size = 0.77 \[ \frac{1}{4} \left (\frac{3 x^4-1}{x^4 \sqrt{1-x^4}}-3 \tanh ^{-1}\left (\sqrt{1-x^4}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^5*(1 - x^4)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.024, size = 82, normalized size = 1.6 \[ -{\frac{1}{4\,{x}^{4}}\sqrt{-{x}^{4}+1}}-{\frac{3}{4}{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{4}+1}}} \right ) }+{\frac{1}{4\,{x}^{2}+4}\sqrt{- \left ({x}^{2}+1 \right ) ^{2}+2+2\,{x}^{2}}}-{\frac{1}{4\,{x}^{2}-4}\sqrt{- \left ({x}^{2}-1 \right ) ^{2}-2\,{x}^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^5/(-x^4+1)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.4393, size = 82, normalized size = 1.55 \[ -\frac{3 \, x^{4} - 1}{4 \,{\left ({\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{-x^{4} + 1}\right )}} - \frac{3}{8} \, \log \left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{3}{8} \, \log \left (\sqrt{-x^{4} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x^4 + 1)^(3/2)*x^5),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.283338, size = 99, normalized size = 1.87 \[ -\frac{3 \, \sqrt{-x^{4} + 1} x^{4} \log \left (\sqrt{-x^{4} + 1} + 1\right ) - 3 \, \sqrt{-x^{4} + 1} x^{4} \log \left (\sqrt{-x^{4} + 1} - 1\right ) - 6 \, x^{4} + 2}{8 \, \sqrt{-x^{4} + 1} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x^4 + 1)^(3/2)*x^5),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 8.95723, size = 95, normalized size = 1.79 \[ \begin{cases} - \frac{3 \operatorname{acosh}{\left (\frac{1}{x^{2}} \right )}}{4} + \frac{3}{4 x^{2} \sqrt{-1 + \frac{1}{x^{4}}}} - \frac{1}{4 x^{6} \sqrt{-1 + \frac{1}{x^{4}}}} & \text{for}\: \left |{\frac{1}{x^{4}}}\right | > 1 \\\frac{3 i \operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{4} - \frac{3 i}{4 x^{2} \sqrt{1 - \frac{1}{x^{4}}}} + \frac{i}{4 x^{6} \sqrt{1 - \frac{1}{x^{4}}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**5/(-x**4+1)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.216231, size = 85, normalized size = 1.6 \[ -\frac{3 \, x^{4} - 1}{4 \,{\left ({\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{-x^{4} + 1}\right )}} - \frac{3}{8} \,{\rm ln}\left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{3}{8} \,{\rm ln}\left (-\sqrt{-x^{4} + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x^4 + 1)^(3/2)*x^5),x, algorithm="giac")
[Out]