3.971 \(\int \frac{1}{\sqrt{16-x^4}} \, dx\)

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

[Out]

EllipticF[ArcSin[x/2], -1]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0016934, antiderivative size = 12, normalized size of antiderivative = 1., 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} \[ \frac{1}{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[16 - x^4],x]

[Out]

EllipticF[ArcSin[x/2], -1]/2

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

Mathematica [A]  time = 0.0120847, size = 12, normalized size = 1. \[ \frac{1}{2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[16 - x^4],x]

[Out]

EllipticF[ArcSin[x/2], -1]/2

________________________________________________________________________________________

Maple [B]  time = 0.004, size = 34, normalized size = 2.8 \begin{align*}{\frac{1}{2}\sqrt{-{x}^{2}+4}\sqrt{{x}^{2}+4}{\it EllipticF} \left ({\frac{x}{2}},i \right ){\frac{1}{\sqrt{-{x}^{4}+16}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{4} + 16}}\,{d x} \end{align*}

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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{4} + 16}}{x^{4} - 16}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-x^4 + 16)/(x^4 - 16), x)

________________________________________________________________________________________

Sympy [B]  time = 0.535752, size = 31, normalized size = 2.58 \begin{align*} \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{align*}

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))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{4} + 16}}\,{d x} \end{align*}

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)