3.978 \(\int \frac{x^6}{\sqrt{-1+x^4}} \, dx\)

Optimal. Leaf size=150 \[ \frac{1}{5} \sqrt{x^4-1} x^3+\frac{3 \left (x^2+1\right ) x}{5 \sqrt{x^4-1}}+\frac{3 \sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{5 \sqrt{2} \sqrt{x^4-1}}-\frac{3 \sqrt{2} \sqrt{x^2-1} \sqrt{x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{5 \sqrt{x^4-1}} \]

[Out]

(3*x*(1 + x^2))/(5*Sqrt[-1 + x^4]) + (x^3*Sqrt[-1 + x^4])/5 - (3*Sqrt[2]*Sqrt[-1
+ x^2]*Sqrt[1 + x^2]*EllipticE[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(5*Sqr
t[-1 + x^4]) + (3*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt
[-1 + x^2]], 1/2])/(5*Sqrt[2]*Sqrt[-1 + x^4])

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Rubi [A]  time = 0.0739881, antiderivative size = 150, 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 \[ \frac{1}{5} \sqrt{x^4-1} x^3+\frac{3 \left (x^2+1\right ) x}{5 \sqrt{x^4-1}}+\frac{3 \sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{5 \sqrt{2} \sqrt{x^4-1}}-\frac{3 \sqrt{2} \sqrt{x^2-1} \sqrt{x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{5 \sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/Sqrt[-1 + x^4],x]

[Out]

(3*x*(1 + x^2))/(5*Sqrt[-1 + x^4]) + (x^3*Sqrt[-1 + x^4])/5 - (3*Sqrt[2]*Sqrt[-1
+ x^2]*Sqrt[1 + x^2]*EllipticE[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(5*Sqr
t[-1 + x^4]) + (3*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt
[-1 + x^2]], 1/2])/(5*Sqrt[2]*Sqrt[-1 + x^4])

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Rubi in Sympy [A]  time = 7.12724, size = 112, normalized size = 0.75 \[ \frac{x^{3} \sqrt{x^{4} - 1}}{5} + \frac{3 x \left (x^{2} + 1\right )}{5 \sqrt{x^{4} - 1}} - \frac{3 \sqrt{2} \sqrt{x^{2} - 1} \sqrt{x^{2} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{\sqrt{x^{2} - 1}} \right )}\middle | \frac{1}{2}\right )}{5 \sqrt{x^{4} - 1}} + \frac{3 \sqrt{- x^{4} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{5 \sqrt{x^{4} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(x**4-1)**(1/2),x)

[Out]

x**3*sqrt(x**4 - 1)/5 + 3*x*(x**2 + 1)/(5*sqrt(x**4 - 1)) - 3*sqrt(2)*sqrt(x**2
- 1)*sqrt(x**2 + 1)*elliptic_e(asin(sqrt(2)*x/sqrt(x**2 - 1)), 1/2)/(5*sqrt(x**4
- 1)) + 3*sqrt(-x**4 + 1)*elliptic_f(asin(x), -1)/(5*sqrt(x**4 - 1))

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Mathematica [A]  time = 0.0455134, size = 56, normalized size = 0.37 \[ \frac{x^7-3 \sqrt{1-x^4} F\left (\left .\sin ^{-1}(x)\right |-1\right )+3 \sqrt{1-x^4} E\left (\left .\sin ^{-1}(x)\right |-1\right )-x^3}{5 \sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/Sqrt[-1 + x^4],x]

[Out]

(-x^3 + x^7 + 3*Sqrt[1 - x^4]*EllipticE[ArcSin[x], -1] - 3*Sqrt[1 - x^4]*Ellipti
cF[ArcSin[x], -1])/(5*Sqrt[-1 + x^4])

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Maple [C]  time = 0.011, size = 57, normalized size = 0.4 \[{\frac{{x}^{3}}{5}\sqrt{{x}^{4}-1}}-{{\frac{3\,i}{5}} \left ({\it EllipticF} \left ( ix,i \right ) -{\it EllipticE} \left ( ix,i \right ) \right ) \sqrt{{x}^{2}+1}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(x^4-1)^(1/2),x)

[Out]

1/5*x^3*(x^4-1)^(1/2)-3/5*I*(x^2+1)^(1/2)*(-x^2+1)^(1/2)/(x^4-1)^(1/2)*(Elliptic
F(I*x,I)-EllipticE(I*x,I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{x^{4} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/sqrt(x^4 - 1),x, algorithm="maxima")

[Out]

integrate(x^6/sqrt(x^4 - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{\sqrt{x^{4} - 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/sqrt(x^4 - 1),x, algorithm="fricas")

[Out]

integral(x^6/sqrt(x^4 - 1), x)

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Sympy [A]  time = 2.22213, size = 27, normalized size = 0.18 \[ - \frac{i x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{x^{4}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(x**4-1)**(1/2),x)

[Out]

-I*x**7*gamma(7/4)*hyper((1/2, 7/4), (11/4,), x**4)/(4*gamma(11/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{x^{4} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/sqrt(x^4 - 1),x, algorithm="giac")

[Out]

integrate(x^6/sqrt(x^4 - 1), x)