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

Optimal. Leaf size=25 \[ \frac{1}{2} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )+\frac{x}{2 \sqrt{1-x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0029822, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {199, 221} \[ \frac{x}{2 \sqrt{1-x^4}}+\frac{1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^4)^(-3/2),x]

[Out]

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (1-x^4\right )^{3/2}} \, dx &=\frac{x}{2 \sqrt{1-x^4}}+\frac{1}{2} \int \frac{1}{\sqrt{1-x^4}} \, dx\\ &=\frac{x}{2 \sqrt{1-x^4}}+\frac{1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.007066, size = 30, normalized size = 1.2 \[ \frac{1}{2} x \left (\, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^4\right )+\frac{1}{\sqrt{1-x^4}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^4)^(-3/2),x]

[Out]

(x*(1/Sqrt[1 - x^4] + Hypergeometric2F1[1/4, 1/2, 5/4, x^4]))/2

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Maple [B]  time = 0.006, size = 45, normalized size = 1.8 \begin{align*}{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{{\it EllipticF} \left ( x,i \right ) }{2}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x^4 + 1)^(-3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.775779, size = 29, normalized size = 1.16 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), x**4*exp_polar(2*I*pi))/(4*gamma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x^4 + 1)^(-3/2), x)