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

Optimal. Leaf size=126 \[ \frac{x \left (x^2+1\right )}{\sqrt{x^4-1}}+\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4-1}}-\frac{\sqrt{2} \sqrt{x^2-1} \sqrt{x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{x^4-1}} \]

[Out]

(x*(1 + x^2))/Sqrt[-1 + x^4] - (Sqrt[2]*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticE[A
rcSin[(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])

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Rubi [A]  time = 0.0501161, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{x \left (x^2+1\right )}{\sqrt{x^4-1}}+\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4-1}}-\frac{\sqrt{2} \sqrt{x^2-1} \sqrt{x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[-1 + x^4],x]

[Out]

(x*(1 + x^2))/Sqrt[-1 + x^4] - (Sqrt[2]*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticE[A
rcSin[(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])

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Rubi in Sympy [A]  time = 5.77124, size = 88, normalized size = 0.7 \[ \frac{x \left (x^{2} + 1\right )}{\sqrt{x^{4} - 1}} - \frac{\sqrt{2} \sqrt{x^{2} - 1} \sqrt{x^{2} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{\sqrt{x^{2} - 1}} \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{4} - 1}} + \frac{\sqrt{- x^{4} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{\sqrt{x^{4} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(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(a
sin(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)

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Mathematica [A]  time = 0.0307353, size = 32, normalized size = 0.25 \[ \frac{\sqrt{1-x^4} \left (E\left (\left .\sin ^{-1}(x)\right |-1\right )-F\left (\left .\sin ^{-1}(x)\right |-1\right )\right )}{\sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[-1 + x^4],x]

[Out]

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

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Maple [C]  time = 0.009, size = 44, normalized size = 0.4 \[{-i \left ({\it EllipticF} \left ( ix,i \right ) -{\it EllipticE} \left ( ix,i \right ) \right ) \sqrt{{x}^{2}+1}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^4-1)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/sqrt(x^4 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2/sqrt(x^4 - 1), x)

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Sympy [A]  time = 1.8077, size = 27, normalized size = 0.21 \[ - \frac{i x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{x^{4}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), x**4)/(4*gamma(7/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/sqrt(x^4 - 1), x)