3.914 \(\int \frac{1}{\left (1-x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{5 x}{12 \sqrt{1-x^4}}+\frac{x}{6 \left (1-x^4\right )^{3/2}}+\frac{5}{12} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

x/(6*(1 - x^4)^(3/2)) + (5*x)/(12*Sqrt[1 - x^4]) + (5*EllipticF[ArcSin[x], -1])/
12

_______________________________________________________________________________________

Rubi [A]  time = 0.0185232, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{5 x}{12 \sqrt{1-x^4}}+\frac{x}{6 \left (1-x^4\right )^{3/2}}+\frac{5}{12} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^4)^(-5/2),x]

[Out]

x/(6*(1 - x^4)^(3/2)) + (5*x)/(12*Sqrt[1 - x^4]) + (5*EllipticF[ArcSin[x], -1])/
12

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.42995, size = 34, normalized size = 0.83 \[ \frac{5 x}{12 \sqrt{- x^{4} + 1}} + \frac{x}{6 \left (- x^{4} + 1\right )^{\frac{3}{2}}} + \frac{5 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**4+1)**(5/2),x)

[Out]

5*x/(12*sqrt(-x**4 + 1)) + x/(6*(-x**4 + 1)**(3/2)) + 5*elliptic_f(asin(x), -1)/
12

_______________________________________________________________________________________

Mathematica [A]  time = 0.0666771, size = 41, normalized size = 1. \[ \frac{-5 x^5+5 \left (1-x^4\right )^{3/2} F\left (\left .\sin ^{-1}(x)\right |-1\right )+7 x}{12 \left (1-x^4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^4)^(-5/2),x]

[Out]

(7*x - 5*x^5 + 5*(1 - x^4)^(3/2)*EllipticF[ArcSin[x], -1])/(12*(1 - x^4)^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.013, size = 64, normalized size = 1.6 \[{\frac{x}{6\, \left ({x}^{4}-1 \right ) ^{2}}\sqrt{-{x}^{4}+1}}+{\frac{5\,x}{12}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{5\,{\it EllipticF} \left ( x,i \right ) }{12}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^4+1)^(5/2),x)

[Out]

1/6*x*(-x^4+1)^(1/2)/(x^4-1)^2+5/12*x/(-x^4+1)^(1/2)+5/12*(-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. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^4 + 1)^(-5/2),x, algorithm="maxima")

[Out]

integrate((-x^4 + 1)^(-5/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{8} - 2 \, x^{4} + 1\right )} \sqrt{-x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^4 + 1)^(-5/2),x, algorithm="fricas")

[Out]

integral(1/((x^8 - 2*x^4 + 1)*sqrt(-x^4 + 1)), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.38035, size = 29, normalized size = 0.71 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**4+1)**(5/2),x)

[Out]

x*gamma(1/4)*hyper((1/4, 5/2), (5/4,), x**4*exp_polar(2*I*pi))/(4*gamma(5/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^4 + 1)^(-5/2),x, algorithm="giac")

[Out]

integrate((-x^4 + 1)^(-5/2), x)