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

Optimal. Leaf size=74 \[ \frac{\sqrt{x^4-1}}{3 x^3}+\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{x^4-1}} \]

[Out]

Sqrt[-1 + x^4]/(3*x^3) + (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]
*x)/Sqrt[-1 + x^2]], 1/2])/(3*Sqrt[2]*Sqrt[-1 + x^4])

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Rubi [A]  time = 0.032677, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{x^4-1}}{3 x^3}+\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[-1 + x^4]),x]

[Out]

Sqrt[-1 + x^4]/(3*x^3) + (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]
*x)/Sqrt[-1 + x^2]], 1/2])/(3*Sqrt[2]*Sqrt[-1 + x^4])

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Rubi in Sympy [A]  time = 2.84377, size = 37, normalized size = 0.5 \[ \frac{\sqrt{- x^{4} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{3 \sqrt{x^{4} - 1}} + \frac{\sqrt{x^{4} - 1}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-x**4 + 1)*elliptic_f(asin(x), -1)/(3*sqrt(x**4 - 1)) + sqrt(x**4 - 1)/(3*x
**3)

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Mathematica [A]  time = 0.0311596, size = 40, normalized size = 0.54 \[ \frac{x^4+\sqrt{1-x^4} x^3 F\left (\left .\sin ^{-1}(x)\right |-1\right )-1}{3 x^3 \sqrt{x^4-1}} \]

Antiderivative was successfully verified.

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

[Out]

(-1 + x^4 + x^3*Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(3*x^3*Sqrt[-1 + x^4])

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Maple [C]  time = 0.014, size = 47, normalized size = 0.6 \[{\frac{1}{3\,{x}^{3}}\sqrt{{x}^{4}-1}}-{{\frac{i}{3}}{\it EllipticF} \left ( ix,i \right ) \sqrt{{x}^{2}+1}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^4-1)^(1/2),x)

[Out]

1/3*(x^4-1)^(1/2)/x^3-1/3*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., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} - 1} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^4 - 1)*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(x^4 - 1)*x^4), x)

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Sympy [A]  time = 2.28507, size = 31, normalized size = 0.42 \[ - \frac{i \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{x^{4}} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^4 - 1)*x^4), x)