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

Optimal. Leaf size=25 \[ \frac{2}{3} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )+\frac{1}{3} \sqrt{1-x^4} x \]

[Out]

(x*Sqrt[1 - x^4])/3 + (2*EllipticF[ArcSin[x], -1])/3

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Rubi [A]  time = 0.0030338, 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 = {195, 221} \[ \frac{1}{3} \sqrt{1-x^4} x+\frac{2}{3} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^4],x]

[Out]

(x*Sqrt[1 - x^4])/3 + (2*EllipticF[ArcSin[x], -1])/3

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[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 \sqrt{1-x^4} \, dx &=\frac{1}{3} x \sqrt{1-x^4}+\frac{2}{3} \int \frac{1}{\sqrt{1-x^4}} \, dx\\ &=\frac{1}{3} x \sqrt{1-x^4}+\frac{2}{3} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}

Mathematica [A]  time = 0.0177769, size = 39, normalized size = 1.56 \[ \frac{2 \sqrt{1-x^4} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )-x^5+x}{3 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^4],x]

[Out]

(x - x^5 + 2*Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(3*Sqrt[1 - x^4])

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Maple [B]  time = 0.004, size = 45, normalized size = 1.8 \begin{align*}{\frac{x}{3}\sqrt{-{x}^{4}+1}}+{\frac{2\,{\it EllipticF} \left ( x,i \right ) }{3}\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((-x^4+1)^(1/2),x)

[Out]

1/3*x*(-x^4+1)^(1/2)+2/3*(-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 \sqrt{-x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-x^4 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-x^4 + 1), x)

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Sympy [A]  time = 0.745521, size = 31, normalized size = 1.24 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \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((-x**4+1)**(1/2),x)

[Out]

x*gamma(1/4)*hyper((-1/2, 1/4), (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 \sqrt{-x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-x^4 + 1), x)