3.10.71 \(\int \frac {1}{\sqrt {16-x^4}} \, dx\) [971]

Optimal. Leaf size=12 \[ \frac {1}{2} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right ) \]

[Out]

1/2*EllipticF(1/2*x,I)

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Rubi [A]
time = 0.00, antiderivative size = 12, 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*} \frac {1}{2} F\left (\left .\text {ArcSin}\left (\frac {x}{2}\right )\right |-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[16 - x^4],x]

[Out]

EllipticF[ArcSin[x/2], -1]/2

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 {16-x^4}} \, dx &=\frac {1}{2} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )\\ \end {align*}

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Mathematica [A]
time = 10.02, size = 12, normalized size = 1.00 \begin {gather*} \frac {1}{2} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[16 - x^4],x]

[Out]

EllipticF[ArcSin[x/2], -1]/2

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (8 ) = 16\).
time = 0.15, size = 34, normalized size = 2.83

method	result	size
meijerg	\(\frac {x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], \frac {x^{4}}{16}\right )}{4}\)	\(15\)
default	\(\frac {\sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \EllipticF \left (\frac {x}{2}, i\right )}{2 \sqrt {-x^{4}+16}}\)	\(34\)
elliptic	\(\frac {\sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \EllipticF \left (\frac {x}{2}, i\right )}{2 \sqrt {-x^{4}+16}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^4+16)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(-x^2+4)^(1/2)*(x^2+4)^(1/2)/(-x^4+16)^(1/2)*EllipticF(1/2*x,I)

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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+16)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-x^4 + 16), x)

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Fricas [A]
time = 0.07, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, F(\arcsin \left (\frac {1}{2} \, x\right )\,|\,-1) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+16)^(1/2),x, algorithm="fricas")

[Out]

1/2*elliptic_f(arcsin(1/2*x), -1)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (5) = 10\).
time = 0.31, size = 31, normalized size = 2.58 \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 | {\frac {x^{4} e^{2 i \pi }}{16}} \right )}}{16 \Gamma \left (\frac {5}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**4+16)**(1/2),x)

[Out]

x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), x**4*exp_polar(2*I*pi)/16)/(16*gamma(5/4))

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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+16)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-x^4 + 16), x)

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Mupad [B]
time = 0.07, size = 13, normalized size = 1.08 \begin {gather*} \frac {x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ \frac {x^4}{16}\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(16 - x^4)^(1/2),x)

[Out]

(x*hypergeom([1/4, 1/2], 5/4, x^4/16))/4

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