3.10.78 \(\int \frac {1}{\sqrt {-1+x^4}} \, dx\) [978]

Optimal. Leaf size=54 \[ \frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1+x^4}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {228} \begin {gather*} \frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 + x^4],x]

[Out]

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

Rule 228

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*b, 2]}, Simp[Sqrt[-a + q*x^2]*(Sqrt[(a + q*x^2
)/q]/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4]))*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2], x] /; IntegerQ[q]]
/; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

Rubi steps

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

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Mathematica [A]
time = 10.03, size = 25, normalized size = 0.46 \begin {gather*} \frac {\sqrt {1-x^4} F\left (\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {-1+x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 + x^4],x]

[Out]

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

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Maple [C] Result contains complex when optimal does not.
time = 0.17, size = 34, normalized size = 0.63

method	result	size
meijerg	\(\frac {\sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], x^{4}\right )}{\sqrt {\mathrm {signum}\left (x^{4}-1\right )}}\)	\(30\)
default	\(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{\sqrt {x^{4}-1}}\)	\(34\)
elliptic	\(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{\sqrt {x^{4}-1}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/sqrt(x^4 - 1), x)

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Fricas [F]
time = 0.07, size = 1, normalized size = 0.02 \begin {gather*} 0 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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Sympy [C] Result contains complex when optimal does not.
time = 0.29, size = 26, normalized size = 0.48 \begin {gather*} - \frac {i 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 | {x^{4}} \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)**(1/2),x)

[Out]

-I*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), x**4)/(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)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(x^4 - 1), x)

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Mupad [B]
time = 1.11, size = 26, normalized size = 0.48 \begin {gather*} \frac {x\,\sqrt {1-x^4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ x^4\right )}{\sqrt {x^4-1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4 - 1)^(1/2),x)

[Out]

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

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