Optimal. Leaf size=63 \[ -\frac{9 \sqrt{1-x^4}}{14 x^7}+\frac{1}{2 x^7 \sqrt{1-x^4}}-\frac{15 \sqrt{1-x^4}}{14 x^3}+\frac{15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0517454, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{9 \sqrt{1-x^4}}{14 x^7}+\frac{1}{2 x^7 \sqrt{1-x^4}}-\frac{15 \sqrt{1-x^4}}{14 x^3}+\frac{15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(x^8*(1 - x^4)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.65612, size = 54, normalized size = 0.86 \[ \frac{15 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{14} - \frac{15 \sqrt{- x^{4} + 1}}{14 x^{3}} - \frac{9 \sqrt{- x^{4} + 1}}{14 x^{7}} + \frac{1}{2 x^{7} \sqrt{- x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**8/(-x**4+1)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.045149, size = 50, normalized size = 0.79 \[ \frac{15 x^8-6 x^4+15 \sqrt{1-x^4} x^7 F\left (\left .\sin ^{-1}(x)\right |-1\right )-2}{14 x^7 \sqrt{1-x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^8*(1 - x^4)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.023, size = 73, normalized size = 1.2 \[{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{1}{7\,{x}^{7}}\sqrt{-{x}^{4}+1}}-{\frac{4}{7\,{x}^{3}}\sqrt{-{x}^{4}+1}}+{\frac{15\,{\it EllipticF} \left ( x,i \right ) }{14}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^8/(-x^4+1)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x^4 + 1)^(3/2)*x^8),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (x^{12} - x^{8}\right )} \sqrt{-x^{4} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x^4 + 1)^(3/2)*x^8),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 5.94527, size = 37, normalized size = 0.59 \[ \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{3}{2} \\ - \frac{3}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**8/(-x**4+1)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x^4 + 1)^(3/2)*x^8),x, algorithm="giac")
[Out]