3.1529 \(\int \frac{x^8}{\sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=237 \[ \frac{1}{5} \sqrt{x^8+1} x-\frac{\sqrt{\frac{\left (x^2+1\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2} x^4-2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (x^2+1\right ) \sqrt{x^8+1}}+\frac{\sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2} x^4+2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{x^8+1}} \]

[Out]

(x*Sqrt[1 + x^8])/5 - (x^3*Sqrt[(1 + x^2)^2/x^2]*Sqrt[-((1 + x^8)/x^4)]*Elliptic
F[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)]*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])])/(10*Sqrt[2 + Sqrt[2]]*(1 - x^2)*Sqrt[1 + x^8])

_______________________________________________________________________________________

Rubi [A]  time = 0.154955, antiderivative size = 237, 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 \[ \frac{1}{5} \sqrt{x^8+1} x-\frac{\sqrt{\frac{\left (x^2+1\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2} x^4-2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (x^2+1\right ) \sqrt{x^8+1}}+\frac{\sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2} x^4+2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{x^8+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^8/Sqrt[1 + x^8],x]

[Out]

(x*Sqrt[1 + x^8])/5 - (x^3*Sqrt[(1 + x^2)^2/x^2]*Sqrt[-((1 + x^8)/x^4)]*Elliptic
F[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)]*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])])/(10*Sqrt[2 + Sqrt[2]]*(1 - x^2)*Sqrt[1 + x^8])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.4595, size = 201, normalized size = 0.85 \[ - \frac{x^{3} \sqrt{\frac{- x^{8} - 1}{x^{4}}} \sqrt{\frac{\left (x^{2} + 1\right )^{2}}{x^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{- \frac{\sqrt{2} x^{4} - 2 x^{2} + \sqrt{2}}{x^{2}}}}{2} \right )}\middle | -2 + 2 \sqrt{2}\right )}{10 \sqrt{\sqrt{2} + 2} \left (x^{2} + 1\right ) \sqrt{x^{8} + 1}} + \frac{x^{3} \sqrt{\frac{- x^{8} - 1}{x^{4}}} \sqrt{- \frac{\left (- x^{2} + 1\right )^{2}}{x^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{\frac{\sqrt{2} x^{4} + 2 x^{2} + \sqrt{2}}{x^{2}}}}{2} \right )}\middle | -2 + 2 \sqrt{2}\right )}{10 \sqrt{\sqrt{2} + 2} \left (- x^{2} + 1\right ) \sqrt{x^{8} + 1}} + \frac{x \sqrt{x^{8} + 1}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(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(-(sqr
t(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)) + x*sqrt(x**8 + 1)/5

_______________________________________________________________________________________

Mathematica [A]  time = 1.94393, size = 218, normalized size = 0.92 \[ \frac{x \left (x^2-1\right ) \left (\sqrt{x^2+\frac{1}{x^2}+\sqrt{2}} \sqrt{\frac{\left (x^2+1\right )^2 \left (x^4-\sqrt{2} x^2+1\right )}{\left (\sqrt{2}-2\right ) x^4}} x^2 F\left (\sin ^{-1}\left (\frac{\sqrt{x^2+\sqrt{2}+\frac{1}{x^2}}}{2^{3/4}}\right )|-2 \left (1+\sqrt{2}\right )\right )+2 \left (x^{10}+x^8+x^2+1\right )\right )-\frac{x^3 \sqrt{-\frac{\left (x^2-1\right )^2}{x^2}} \left (x^2+1\right ) \sqrt{-\frac{x^8+1}{x^4}} F\left (\sin ^{-1}\left (\frac{\sqrt{x^2+\sqrt{2}+\frac{1}{x^2}}}{2^{3/4}}\right )|2 \left (-1+\sqrt{2}\right )\right )}{\sqrt{2+\sqrt{2}}}}{10 \left (x^4-1\right ) \sqrt{x^8+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^8/Sqrt[1 + x^8],x]

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.046, size = 26, normalized size = 0.1 \[{\frac{x}{5}\sqrt{{x}^{8}+1}}-{\frac{x}{5}{\mbox{$_2$F$_1$}({\frac{1}{8}},{\frac{1}{2}};\,{\frac{9}{8}};\,-{x}^{8})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(x^8+1)^(1/2),x)

[Out]

1/5*x*(x^8+1)^(1/2)-1/5*x*hypergeom([1/8,1/2],[9/8],-x^8)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^8/sqrt(x^8 + 1), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^8/sqrt(x^8 + 1), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.7386, size = 29, normalized size = 0.12 \[ \frac{x^{9} \Gamma \left (\frac{9}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{8} \\ \frac{17}{8} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac{17}{8}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**9*gamma(9/8)*hyper((1/2, 9/8), (17/8,), x**8*exp_polar(I*pi))/(8*gamma(17/8))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^8/sqrt(x^8 + 1), x)