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

Optimal result

Integrand size = 9, antiderivative size = 54 \[ \int \frac {1}{\sqrt {-1+x^4}} \, dx=\frac {\sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1+x^4}} \]

[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)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {228} \[ \int \frac {1}{\sqrt {-1+x^4}} \, dx=\frac {\sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4-1}} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 10.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\sqrt {-1+x^4}} \, dx=\frac {\sqrt {1-x^4} \operatorname {EllipticF}(\arcsin (x),-1)}{\sqrt {-1+x^4}} \]

[In]

Integrate[1/Sqrt[-1 + x^4],x]

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 4.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.56

method	result	size
meijerg	\(\frac {\sqrt {-\operatorname {signum}\left (x^{4}-1\right )}\, x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^{4}\right )}{\sqrt {\operatorname {signum}\left (x^{4}-1\right )}}\)	\(30\)
default	\(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, F\left (i x , i\right )}{\sqrt {x^{4}-1}}\)	\(34\)
elliptic	\(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, F\left (i x , i\right )}{\sqrt {x^{4}-1}}\)	\(34\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.11 \[ \int \frac {1}{\sqrt {-1+x^4}} \, dx=-i \, F(\arcsin \left (x\right )\,|\,-1) \]

[In]

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

[Out]

-I*elliptic_f(arcsin(x), -1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.48 \[ \int \frac {1}{\sqrt {-1+x^4}} \, dx=- \frac {i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]

[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))

Maxima [F]

\[ \int \frac {1}{\sqrt {-1+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} - 1}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(x^4 - 1), x)

Giac [F]

\[ \int \frac {1}{\sqrt {-1+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} - 1}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(x^4 - 1), x)

Mupad [B] (verification not implemented)

Time = 5.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.48 \[ \int \frac {1}{\sqrt {-1+x^4}} \, dx=\frac {x\,\sqrt {1-x^4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ x^4\right )}{\sqrt {x^4-1}} \]

[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)