3.906 \(\int \frac{1}{x^4 \left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{5 \sqrt{1-x^4}}{6 x^3}+\frac{1}{2 x^3 \sqrt{1-x^4}}+\frac{5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

1/(2*x^3*Sqrt[1 - x^4]) - (5*Sqrt[1 - x^4])/(6*x^3) + (5*EllipticF[ArcSin[x], -1
])/6

_______________________________________________________________________________________

Rubi [A]  time = 0.034282, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 \sqrt{1-x^4}}{6 x^3}+\frac{1}{2 x^3 \sqrt{1-x^4}}+\frac{5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(1 - x^4)^(3/2)),x]

[Out]

1/(2*x^3*Sqrt[1 - x^4]) - (5*Sqrt[1 - x^4])/(6*x^3) + (5*EllipticF[ArcSin[x], -1
])/6

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.41573, size = 39, normalized size = 0.87 \[ \frac{5 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{6} - \frac{5 \sqrt{- x^{4} + 1}}{6 x^{3}} + \frac{1}{2 x^{3} \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(-x**4+1)**(3/2),x)

[Out]

5*elliptic_f(asin(x), -1)/6 - 5*sqrt(-x**4 + 1)/(6*x**3) + 1/(2*x**3*sqrt(-x**4
+ 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0607152, size = 33, normalized size = 0.73 \[ \frac{1}{6} \left (\frac{5 x^4-2}{x^3 \sqrt{1-x^4}}+5 F\left (\left .\sin ^{-1}(x)\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(1 - x^4)^(3/2)),x]

[Out]

((-2 + 5*x^4)/(x^3*Sqrt[1 - x^4]) + 5*EllipticF[ArcSin[x], -1])/6

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 59, normalized size = 1.3 \[{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{1}{3\,{x}^{3}}\sqrt{-{x}^{4}+1}}+{\frac{5\,{\it EllipticF} \left ( x,i \right ) }{6}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-x^4+1)^(3/2),x)

[Out]

1/2*x/(-x^4+1)^(1/2)-1/3*(-x^4+1)^(1/2)/x^3+5/6*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x
^4+1)^(1/2)*EllipticF(x,I)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((-x^4 + 1)^(3/2)*x^4), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (x^{8} - x^{4}\right )} \sqrt{-x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^4),x, algorithm="fricas")

[Out]

integral(-1/((x^8 - x^4)*sqrt(-x^4 + 1)), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.93578, size = 34, normalized size = 0.76 \[ \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{1}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(-x**4+1)**(3/2),x)

[Out]

gamma(-3/4)*hyper((-3/4, 3/2), (1/4,), x**4*exp_polar(2*I*pi))/(4*x**3*gamma(1/4
))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^4),x, algorithm="giac")

[Out]

integrate(1/((-x^4 + 1)^(3/2)*x^4), x)