3.1533 \(\int \frac{1}{\sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=223 \[ \frac{x^3 \sqrt{\frac{\left (x^2+1\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2} x^4-2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{2 \sqrt{2+\sqrt{2}} \left (x^2+1\right ) \sqrt{x^8+1}}-\frac{x^3 \sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2} x^4+2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{2 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{x^8+1}} \]

[Out]

(x^3*Sqrt[(1 + x^2)^2/x^2]*Sqrt[-((1 + x^8)/x^4)]*EllipticF[ArcSin[Sqrt[-((Sqrt[
2] - 2*x^2 + Sqrt[2]*x^4)/x^2)]/2], -2*(1 - Sqrt[2])])/(2*Sqrt[2 + Sqrt[2]]*(1 +
x^2)*Sqrt[1 + x^8]) - (x^3*Sqrt[-((1 - x^2)^2/x^2)]*Sqrt[-((1 + x^8)/x^4)]*Elli
pticF[ArcSin[Sqrt[(Sqrt[2] + 2*x^2 + Sqrt[2]*x^4)/x^2]/2], -2*(1 - Sqrt[2])])/(2
*Sqrt[2 + Sqrt[2]]*(1 - x^2)*Sqrt[1 + x^8])

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Rubi [A]  time = 0.124886, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{x^3 \sqrt{\frac{\left (x^2+1\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2} x^4-2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{2 \sqrt{2+\sqrt{2}} \left (x^2+1\right ) \sqrt{x^8+1}}-\frac{x^3 \sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2} x^4+2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{2 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{x^8+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 + x^8],x]

[Out]

(x^3*Sqrt[(1 + x^2)^2/x^2]*Sqrt[-((1 + x^8)/x^4)]*EllipticF[ArcSin[Sqrt[-((Sqrt[
2] - 2*x^2 + Sqrt[2]*x^4)/x^2)]/2], -2*(1 - Sqrt[2])])/(2*Sqrt[2 + Sqrt[2]]*(1 +
x^2)*Sqrt[1 + x^8]) - (x^3*Sqrt[-((1 - x^2)^2/x^2)]*Sqrt[-((1 + x^8)/x^4)]*Elli
pticF[ArcSin[Sqrt[(Sqrt[2] + 2*x^2 + Sqrt[2]*x^4)/x^2]/2], -2*(1 - Sqrt[2])])/(2
*Sqrt[2 + Sqrt[2]]*(1 - x^2)*Sqrt[1 + x^8])

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Rubi in Sympy [A]  time = 11.542, size = 189, normalized size = 0.85 \[ \frac{x^{3} \sqrt{\frac{- x^{8} - 1}{x^{4}}} \sqrt{\frac{\left (x^{2} + 1\right )^{2}}{x^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{- \frac{\sqrt{2} x^{4} - 2 x^{2} + \sqrt{2}}{x^{2}}}}{2} \right )}\middle | -2 + 2 \sqrt{2}\right )}{2 \sqrt{\sqrt{2} + 2} \left (x^{2} + 1\right ) \sqrt{x^{8} + 1}} - \frac{x^{3} \sqrt{\frac{- x^{8} - 1}{x^{4}}} \sqrt{- \frac{\left (- x^{2} + 1\right )^{2}}{x^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{\frac{\sqrt{2} x^{4} + 2 x^{2} + \sqrt{2}}{x^{2}}}}{2} \right )}\middle | -2 + 2 \sqrt{2}\right )}{2 \sqrt{\sqrt{2} + 2} \left (- x^{2} + 1\right ) \sqrt{x^{8} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**8+1)**(1/2),x)

[Out]

x**3*sqrt((-x**8 - 1)/x**4)*sqrt((x**2 + 1)**2/x**2)*elliptic_f(asin(sqrt(-(sqrt
(2)*x**4 - 2*x**2 + sqrt(2))/x**2)/2), -2 + 2*sqrt(2))/(2*sqrt(sqrt(2) + 2)*(x**
2 + 1)*sqrt(x**8 + 1)) - x**3*sqrt((-x**8 - 1)/x**4)*sqrt(-(-x**2 + 1)**2/x**2)*
elliptic_f(asin(sqrt((sqrt(2)*x**4 + 2*x**2 + sqrt(2))/x**2)/2), -2 + 2*sqrt(2))
/(2*sqrt(sqrt(2) + 2)*(-x**2 + 1)*sqrt(x**8 + 1))

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Mathematica [A]  time = 0.433544, size = 167, normalized size = 0.75 \[ \frac{x^3 \sqrt{-\frac{x^8+1}{x^4}} \left (\sqrt{\frac{\left (\sqrt{2}-2\right ) \left (x^2-1\right )^2}{x^2}} \left (x^2+1\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{x^2+\sqrt{2}+\frac{1}{x^2}}}{2^{3/4}}\right )|2 \left (-1+\sqrt{2}\right )\right )-\sqrt{2+\sqrt{2}} \left (x^2-1\right ) \sqrt{x^2+\frac{1}{x^2}+2} F\left (\sin ^{-1}\left (\frac{\sqrt{x^2+\sqrt{2}+\frac{1}{x^2}}}{2^{3/4}}\right )|-2 \left (1+\sqrt{2}\right )\right )\right )}{2 \sqrt{2} \left (x^4-1\right ) \sqrt{x^8+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/Sqrt[1 + x^8],x]

[Out]

(x^3*Sqrt[-((1 + x^8)/x^4)]*(Sqrt[((-2 + Sqrt[2])*(-1 + x^2)^2)/x^2]*(1 + x^2)*E
llipticF[ArcSin[Sqrt[Sqrt[2] + x^(-2) + x^2]/2^(3/4)], 2*(-1 + Sqrt[2])] - Sqrt[
2 + Sqrt[2]]*(-1 + x^2)*Sqrt[2 + x^(-2) + x^2]*EllipticF[ArcSin[Sqrt[Sqrt[2] + x
^(-2) + x^2]/2^(3/4)], -2*(1 + Sqrt[2])]))/(2*Sqrt[2]*(-1 + x^4)*Sqrt[1 + x^8])

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Maple [C]  time = 0.013, size = 14, normalized size = 0.1 \[ x{\mbox{$_2$F$_1$}({\frac{1}{8}},{\frac{1}{2}};\,{\frac{9}{8}};\,-{x}^{8})} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^8+1)^(1/2),x)

[Out]

x*hypergeom([1/8,1/2],[9/8],-x^8)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{8} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^8 + 1),x, algorithm="maxima")

[Out]

integrate(1/sqrt(x^8 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{8} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^8 + 1),x, algorithm="fricas")

[Out]

integral(1/sqrt(x^8 + 1), x)

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Sympy [A]  time = 1.67372, size = 27, normalized size = 0.12 \[ \frac{x \Gamma \left (\frac{1}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{8}, \frac{1}{2} \\ \frac{9}{8} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac{9}{8}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**8+1)**(1/2),x)

[Out]

x*gamma(1/8)*hyper((1/8, 1/2), (9/8,), x**8*exp_polar(I*pi))/(8*gamma(9/8))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{8} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^8 + 1),x, algorithm="giac")

[Out]

integrate(1/sqrt(x^8 + 1), x)