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

Optimal. Leaf size=72 \[ \frac{1}{3} \sqrt{x^4-1} x+\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{x^4-1}} \]

[Out]

(x*Sqrt[-1 + x^4])/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 = 0.0328226, antiderivative size = 72, 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 \[ \frac{1}{3} \sqrt{x^4-1} x+\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[-1 + x^4],x]

[Out]

(x*Sqrt[-1 + x^4])/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 = 2.94839, size = 36, normalized size = 0.5 \[ \frac{x \sqrt{x^{4} - 1}}{3} + \frac{\sqrt{- x^{4} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{3 \sqrt{x^{4} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.033852, size = 36, normalized size = 0.5 \[ \frac{x^5+\sqrt{1-x^4} F\left (\left .\sin ^{-1}(x)\right |-1\right )-x}{3 \sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[-1 + x^4],x]

[Out]

(-x + x^5 + Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(3*Sqrt[-1 + x^4])

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Maple [C]  time = 0.012, size = 45, normalized size = 0.6 \[{\frac{x}{3}\sqrt{{x}^{4}-1}}-{{\frac{i}{3}}{\it EllipticF} \left ( ix,i \right ) \sqrt{{x}^{2}+1}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^4-1)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/sqrt(x^4 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^4/sqrt(x^4 - 1), x)

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Sympy [A]  time = 1.91683, size = 27, normalized size = 0.38 \[ - \frac{i x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{x^{4}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), x**4)/(4*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/sqrt(x^4 - 1), x)