3.10.7 \(\int \frac {1}{(1-x^4)^{3/2}} \, dx\) [907]

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

[Out]

1/2*EllipticF(x,I)+1/2*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 = {205, 227} \begin {gather*} \frac {1}{2} F(\text {ArcSin}(x)|-1)+\frac {x}{2 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x^4)^(-3/2),x]

[Out]

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

Rule 205

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 3.21, size = 30, normalized size = 1.20 \begin {gather*} \frac {1}{2} x \left (\frac {1}{\sqrt {1-x^4}}+\, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^4)^(-3/2),x]

[Out]

(x*(1/Sqrt[1 - x^4] + Hypergeometric2F1[1/4, 1/2, 5/4, x^4]))/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. 44 vs. \(2 (19 ) = 38\).
time = 0.15, size = 45, normalized size = 1.80

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*x/(-x^4+1)^(1/2)+1/2*(-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(1/(-x^4+1)^(3/2),x, algorithm="maxima")

[Out]

integrate((-x^4 + 1)^(-3/2), x)

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Fricas [A]
time = 0.07, size = 32, normalized size = 1.28 \begin {gather*} \frac {{\left (x^{4} - 1\right )} F(\arcsin \left (x\right )\,|\,-1) - \sqrt {-x^{4} + 1} x}{2 \, {\left (x^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

1/2*((x^4 - 1)*elliptic_f(arcsin(x), -1) - sqrt(-x^4 + 1)*x)/(x^4 - 1)

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Sympy [A]
time = 0.33, size = 29, normalized size = 1.16 \begin {gather*} \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{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)**(3/2),x)

[Out]

x*gamma(1/4)*hyper((1/4, 3/2), (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(1/(-x^4+1)^(3/2),x, algorithm="giac")

[Out]

integrate((-x^4 + 1)^(-3/2), x)

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Mupad [B]
time = 1.07, size = 10, normalized size = 0.40 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{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)^(3/2),x)

[Out]

x*hypergeom([1/4, 3/2], 5/4, x^4)

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