3.9.10 \(\int \sqrt {1-x^4} \, dx\) [810]

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

[Out]

2/3*EllipticF(x,I)+1/3*x*(-x^4+1)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 = {201, 227} \begin {gather*} \frac {2}{3} F(\text {ArcSin}(x)|-1)+\frac {1}{3} \sqrt {1-x^4} x \end {gather*}

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 201

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 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 \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*}

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Mathematica [A]
time = 2.89, size = 39, normalized size = 1.56 \begin {gather*} \frac {x-x^5+2 \sqrt {1-x^4} F\left (\left .\sin ^{-1}(x)\right |-1\right )}{3 \sqrt {1-x^4}} \end {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19 ) = 38\).
time = 0.15, size = 45, normalized size = 1.80

method	result	size
meijerg	\(x \hypergeom \left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], x^{4}\right )\)	\(12\)
default	\(\frac {x \sqrt {-x^{4}+1}}{3}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{3 \sqrt {-x^{4}+1}}\)	\(45\)
elliptic	\(\frac {x \sqrt {-x^{4}+1}}{3}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{3 \sqrt {-x^{4}+1}}\)	\(45\)
risch	\(-\frac {x \left (x^{4}-1\right )}{3 \sqrt {-x^{4}+1}}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{3 \sqrt {-x^{4}+1}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, 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((-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^4 + 1), x)

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Fricas [A]
time = 0.07, size = 12, normalized size = 0.48 \begin {gather*} \frac {1}{3} \, \sqrt {-x^{4} + 1} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(-x^4 + 1)*x

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Sympy [A]
time = 0.31, size = 31, normalized size = 1.24 \begin {gather*} \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 {gather*}

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 = 0.00, 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((-x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-x^4 + 1), x)

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Mupad [B]
time = 1.01, size = 10, normalized size = 0.40 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {1}{4};\ \frac {5}{4};\ x^4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^4)^(1/2),x)

[Out]

x*hypergeom([-1/2, 1/4], 5/4, x^4)

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