3.9.86 \(\int \frac {1}{\sqrt {1-x^4}} \, dx\) [886]

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

[Out]

EllipticF(x,I)

________________________________________________________________________________________

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 = {227} \begin {gather*} F(\text {ArcSin}(x)|-1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - x^4],x]

[Out]

EllipticF[ArcSin[x], -1]

Rule 227

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*}

________________________________________________________________________________________

Mathematica [A]
time = 10.03, size = 4, normalized size = 1.00 \begin {gather*} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - x^4],x]

[Out]

EllipticF[ArcSin[x], -1]

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (4 ) = 8\).
time = 0.16, size = 31, normalized size = 7.75

method	result	size
meijerg	\(x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], x^{4}\right )\)	\(12\)
default	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{\sqrt {-x^{4}+1}}\)	\(31\)
elliptic	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{\sqrt {-x^{4}+1}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.07, size = 4, normalized size = 1.00 \begin {gather*} F(\arcsin \left (x\right )\,|\,-1) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

elliptic_f(arcsin(x), -1)

________________________________________________________________________________________

Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (2) = 4\).
time = 0.30, size = 29, normalized size = 7.25 \begin {gather*} \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 )} \end {gather*}

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))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________