3.1536 \(\int \frac{1}{x^6 \sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=239 \[ -\frac{\sqrt{x^8+1}}{5 x^5}+\frac{\sqrt{\frac{\left (x^2+1\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2} x^4-2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (x^2+1\right ) \sqrt{x^8+1}}+\frac{\sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2} x^4+2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{x^8+1}} \]

[Out]

-Sqrt[1 + x^8]/(5*x^5) + (x^3*Sqrt[(1 + x^2)^2/x^2]*Sqrt[-((1 + x^8)/x^4)]*Ellip
ticF[ArcSin[Sqrt[-((Sqrt[2] - 2*x^2 + Sqrt[2]*x^4)/x^2)]/2], -2*(1 - Sqrt[2])])/
(10*Sqrt[2 + Sqrt[2]]*(1 + x^2)*Sqrt[1 + x^8]) + (x^3*Sqrt[-((1 - x^2)^2/x^2)]*S
qrt[-((1 + x^8)/x^4)]*EllipticF[ArcSin[Sqrt[(Sqrt[2] + 2*x^2 + Sqrt[2]*x^4)/x^2]
/2], -2*(1 - Sqrt[2])])/(10*Sqrt[2 + Sqrt[2]]*(1 - x^2)*Sqrt[1 + x^8])

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Rubi [A]  time = 0.16191, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{x^8+1}}{5 x^5}+\frac{\sqrt{\frac{\left (x^2+1\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2} x^4-2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (x^2+1\right ) \sqrt{x^8+1}}+\frac{\sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2} x^4+2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{x^8+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(5*x^5) + (x^3*Sqrt[(1 + x^2)^2/x^2]*Sqrt[-((1 + x^8)/x^4)]*Ellip
ticF[ArcSin[Sqrt[-((Sqrt[2] - 2*x^2 + Sqrt[2]*x^4)/x^2)]/2], -2*(1 - Sqrt[2])])/
(10*Sqrt[2 + Sqrt[2]]*(1 + x^2)*Sqrt[1 + x^8]) + (x^3*Sqrt[-((1 - x^2)^2/x^2)]*S
qrt[-((1 + x^8)/x^4)]*EllipticF[ArcSin[Sqrt[(Sqrt[2] + 2*x^2 + Sqrt[2]*x^4)/x^2]
/2], -2*(1 - Sqrt[2])])/(10*Sqrt[2 + Sqrt[2]]*(1 - x^2)*Sqrt[1 + x^8])

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Rubi in Sympy [A]  time = 13.9414, size = 202, normalized size = 0.85 \[ \frac{x^{3} \sqrt{\frac{- x^{8} - 1}{x^{4}}} \sqrt{\frac{\left (x^{2} + 1\right )^{2}}{x^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{- \frac{\sqrt{2} x^{4} - 2 x^{2} + \sqrt{2}}{x^{2}}}}{2} \right )}\middle | -2 + 2 \sqrt{2}\right )}{10 \sqrt{\sqrt{2} + 2} \left (x^{2} + 1\right ) \sqrt{x^{8} + 1}} + \frac{x^{3} \sqrt{\frac{- x^{8} - 1}{x^{4}}} \sqrt{- \frac{\left (- x^{2} + 1\right )^{2}}{x^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{\frac{\sqrt{2} x^{4} + 2 x^{2} + \sqrt{2}}{x^{2}}}}{2} \right )}\middle | -2 + 2 \sqrt{2}\right )}{10 \sqrt{\sqrt{2} + 2} \left (- x^{2} + 1\right ) \sqrt{x^{8} + 1}} - \frac{\sqrt{x^{8} + 1}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(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))/(10*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)
)/(10*sqrt(sqrt(2) + 2)*(-x**2 + 1)*sqrt(x**8 + 1)) - sqrt(x**8 + 1)/(5*x**5)

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Mathematica [A]  time = 1.41224, size = 221, normalized size = 0.92 \[ \frac{-\frac{\sqrt{-\frac{\left (x^2-1\right )^2}{x^2}} \left (x^2+1\right ) \sqrt{-\frac{x^8+1}{x^4}} x^8 F\left (\sin ^{-1}\left (\frac{\sqrt{x^2+\sqrt{2}+\frac{1}{x^2}}}{2^{3/4}}\right )|2 \left (-1+\sqrt{2}\right )\right )}{\sqrt{2+\sqrt{2}}}-\left (x^2-1\right ) \left (2 \left (x^{10}+x^8+x^2+1\right )+\sqrt{x^2+\frac{1}{x^2}+\sqrt{2}} \sqrt{\frac{\left (x^2+1\right )^2 \left (x^4-\sqrt{2} x^2+1\right )}{\left (\sqrt{2}-2\right ) x^4}} x^8 F\left (\sin ^{-1}\left (\frac{\sqrt{x^2+\sqrt{2}+\frac{1}{x^2}}}{2^{3/4}}\right )|-2 \left (1+\sqrt{2}\right )\right )\right )}{10 x^5 \left (x^4-1\right ) \sqrt{x^8+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^6*Sqrt[1 + x^8]),x]

[Out]

(-((x^8*Sqrt[-((-1 + x^2)^2/x^2)]*(1 + x^2)*Sqrt[-((1 + x^8)/x^4)]*EllipticF[Arc
Sin[Sqrt[Sqrt[2] + x^(-2) + x^2]/2^(3/4)], 2*(-1 + Sqrt[2])])/Sqrt[2 + Sqrt[2]])
- (-1 + x^2)*(2*(1 + x^2 + x^8 + x^10) + x^8*Sqrt[Sqrt[2] + x^(-2) + x^2]*Sqrt[
((1 + x^2)^2*(1 - Sqrt[2]*x^2 + x^4))/((-2 + Sqrt[2])*x^4)]*EllipticF[ArcSin[Sqr
t[Sqrt[2] + x^(-2) + x^2]/2^(3/4)], -2*(1 + Sqrt[2])]))/(10*x^5*(-1 + x^4)*Sqrt[
1 + x^8])

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Maple [C]  time = 0.042, size = 30, normalized size = 0.1 \[ -{\frac{1}{5\,{x}^{5}}\sqrt{{x}^{8}+1}}-{\frac{{x}^{3}}{15}{\mbox{$_2$F$_1$}({\frac{3}{8}},{\frac{1}{2}};\,{\frac{11}{8}};\,-{x}^{8})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^8+1)^(1/2),x)

[Out]

-1/5*(x^8+1)^(1/2)/x^5-1/15*x^3*hypergeom([3/8,1/2],[11/8],-x^8)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{8} + 1} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^6),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^8 + 1)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{8} + 1} x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^6),x, algorithm="fricas")

[Out]

integral(1/(sqrt(x^8 + 1)*x^6), x)

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Sympy [A]  time = 2.80788, size = 32, normalized size = 0.13 \[ \frac{\Gamma \left (- \frac{5}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{8}, \frac{1}{2} \\ \frac{3}{8} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 x^{5} \Gamma \left (\frac{3}{8}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(x**8+1)**(1/2),x)

[Out]

gamma(-5/8)*hyper((-5/8, 1/2), (3/8,), x**8*exp_polar(I*pi))/(8*x**5*gamma(3/8))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{8} + 1} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^6),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^8 + 1)*x^6), x)