3.886 \(\int \frac {1}{\sqrt {1-x^4}} \, dx\)

Optimal. Leaf size=4 \[ F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

EllipticF(x,I)

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Rubi [A]  time = 0.00, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {221} \[ F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - x^4],x]

[Out]

EllipticF[ArcSin[x], -1]

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}{\sqrt {1-x^4}} \, dx &=F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 4, normalized size = 1.00 \[ F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - x^4],x]

[Out]

EllipticF[ArcSin[x], -1]

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fricas [F]  time = 1.23, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 1}}{x^{4} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-x^4 + 1), x)

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maple [B]  time = 0.00, size = 31, normalized size = 7.75 \[ \frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{\sqrt {-x^{4}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-x^4 + 1), x)

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mupad [B]  time = 1.07, size = 10, normalized size = 2.50 \[ x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ x^4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1 - x^4)^(1/2),x)

[Out]

x*hypergeom([1/4, 1/2], 5/4, x^4)

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sympy [B]  time = 0.83, size = 29, normalized size = 7.25 \[ \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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