3.980 \(\int \frac{1}{x^2 \sqrt{-1+x^4}} \, dx\)

Optimal. Leaf size=140 \[ \frac{\sqrt{x^4-1}}{x}-\frac{x \left (x^2+1\right )}{\sqrt{x^4-1}}-\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4-1}}+\frac{\sqrt{2} \sqrt{x^2-1} \sqrt{x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{x^4-1}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0613276, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\sqrt{x^4-1}}{x}-\frac{x \left (x^2+1\right )}{\sqrt{x^4-1}}-\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4-1}}+\frac{\sqrt{2} \sqrt{x^2-1} \sqrt{x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[-1 + x^4]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.84739, size = 99, normalized size = 0.71 \[ - \frac{x \left (x^{2} + 1\right )}{\sqrt{x^{4} - 1}} + \frac{\sqrt{2} \sqrt{x^{2} - 1} \sqrt{x^{2} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{\sqrt{x^{2} - 1}} \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{4} - 1}} - \frac{\sqrt{- x^{4} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{\sqrt{x^{4} - 1}} + \frac{\sqrt{x^{4} - 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(x**4-1)**(1/2),x)

[Out]

-x*(x**2 + 1)/sqrt(x**4 - 1) + sqrt(2)*sqrt(x**2 - 1)*sqrt(x**2 + 1)*elliptic_e(
asin(sqrt(2)*x/sqrt(x**2 - 1)), 1/2)/sqrt(x**4 - 1) - sqrt(-x**4 + 1)*elliptic_f
(asin(x), -1)/sqrt(x**4 - 1) + sqrt(x**4 - 1)/x

_______________________________________________________________________________________

Mathematica [A]  time = 0.0460273, size = 55, normalized size = 0.39 \[ \frac{\sqrt{x^4-1}}{x}+\frac{\sqrt{1-x^2} \sqrt{x^2+1} \left (F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right )\right )}{\sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*Sqrt[-1 + x^4]),x]

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.013, size = 56, normalized size = 0.4 \[{\frac{1}{x}\sqrt{{x}^{4}-1}}+{i \left ({\it EllipticF} \left ( ix,i \right ) -{\it EllipticE} \left ( ix,i \right ) \right ) \sqrt{{x}^{2}+1}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(x^4-1)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} - 1} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 - 1)*x^2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^4 - 1)*x^2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{4} - 1} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 - 1)*x^2),x, algorithm="fricas")

[Out]

integral(1/(sqrt(x^4 - 1)*x^2), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.00875, size = 29, normalized size = 0.21 \[ - \frac{i \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{x^{4}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(x**4-1)**(1/2),x)

[Out]

-I*gamma(-1/4)*hyper((-1/4, 1/2), (3/4,), x**4)/(4*x*gamma(3/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} - 1} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 - 1)*x^2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^4 - 1)*x^2), x)