3.1538 \(\int \frac {1}{x^6 \sqrt {1+x^8}} \, dx\)

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]

-1/5*(x^8+1)^(1/2)/x^5+1/10*x^3*EllipticF(1/2*((2*x^2+2^(1/2)+x^4*2^(1/2))/x^2)^(1/2),(-2+2*2^(1/2))^(1/2))*(-
(-x^2+1)^2/x^2)^(1/2)*((-x^8-1)/x^4)^(1/2)/(-x^2+1)/(x^8+1)^(1/2)/(2+2^(1/2))^(1/2)+1/10*x^3*EllipticF(1/2*((2
*x^2-2^(1/2)-x^4*2^(1/2))/x^2)^(1/2),(-2+2*2^(1/2))^(1/2))*((x^2+1)^2/x^2)^(1/2)*((-x^8-1)/x^4)^(1/2)/(x^2+1)/
(x^8+1)^(1/2)/(2+2^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {325, 309, 1883} \[ -\frac {\sqrt {x^8+1}}{5 x^5}+\frac {\sqrt {\frac {\left (x^2+1\right )^2}{x^2}} \sqrt {-\frac {x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {-\frac {\sqrt {2} x^4-2 x^2+\sqrt {2}}{x^2}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{10 \sqrt {2+\sqrt {2}} \left (x^2+1\right ) \sqrt {x^8+1}}+\frac {\sqrt {-\frac {\left (1-x^2\right )^2}{x^2}} \sqrt {-\frac {x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {\frac {\sqrt {2} x^4+2 x^2+\sqrt {2}}{x^2}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{10 \sqrt {2+\sqrt {2}} \left (1-x^2\right ) \sqrt {x^8+1}} \]

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)]*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]) + (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])

Rule 309

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^8], x_Symbol] :> Dist[1/(2*Rt[b/a, 4]), Int[(1 + Rt[b/a, 4]*x^2)/Sqrt[a + b*
x^8], x], x] - Dist[1/(2*Rt[b/a, 4]), Int[(1 - Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x], x] /; FreeQ[{a, b}, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 1883

Int[((c_) + (d_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^8], x_Symbol] :> -Simp[(c*d*x^3*Sqrt[-((c - d*x^2)^2/(c*d*x^2
))]*Sqrt[-((d^2*(a + b*x^8))/(b*c^2*x^4))]*EllipticF[ArcSin[(1*Sqrt[(Sqrt[2]*c^2 + 2*c*d*x^2 + Sqrt[2]*d^2*x^4
)/(c*d*x^2)])/2], -2*(1 - Sqrt[2])])/(Sqrt[2 + Sqrt[2]]*(c - d*x^2)*Sqrt[a + b*x^8]), x] /; FreeQ[{a, b, c, d}
, x] && EqQ[b*c^4 - a*d^4, 0]

Rubi steps

\begin {align*} \int \frac {1}{x^6 \sqrt {1+x^8}} \, dx &=-\frac {\sqrt {1+x^8}}{5 x^5}-\frac {1}{5} \int \frac {x^2}{\sqrt {1+x^8}} \, dx\\ &=-\frac {\sqrt {1+x^8}}{5 x^5}+\frac {1}{10} \int \frac {1-x^2}{\sqrt {1+x^8}} \, dx-\frac {1}{10} \int \frac {1+x^2}{\sqrt {1+x^8}} \, dx\\ &=-\frac {\sqrt {1+x^8}}{5 x^5}+\frac {x^3 \sqrt {\frac {\left (1+x^2\right )^2}{x^2}} \sqrt {-\frac {1+x^8}{x^4}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {-\frac {\sqrt {2}-2 x^2+\sqrt {2} x^4}{x^2}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{10 \sqrt {2+\sqrt {2}} \left (1+x^2\right ) \sqrt {1+x^8}}+\frac {x^3 \sqrt {-\frac {\left (1-x^2\right )^2}{x^2}} \sqrt {-\frac {1+x^8}{x^4}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {\frac {\sqrt {2}+2 x^2+\sqrt {2} x^4}{x^2}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{10 \sqrt {2+\sqrt {2}} \left (1-x^2\right ) \sqrt {1+x^8}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.00, size = 22, normalized size = 0.09 \[ -\frac {\, _2F_1\left (-\frac {5}{8},\frac {1}{2};\frac {3}{8};-x^8\right )}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*Sqrt[1 + x^8]),x]

[Out]

-1/5*Hypergeometric2F1[-5/8, 1/2, 3/8, -x^8]/x^5

________________________________________________________________________________________

fricas [F]  time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{8} + 1}}{x^{14} + x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(x^8 + 1)/(x^14 + x^6), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{8} + 1} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^8+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^8 + 1)*x^6), x)

________________________________________________________________________________________

maple [C]  time = 0.14, size = 30, normalized size = 0.13 \[ -\frac {x^{3} \hypergeom \left (\left [\frac {3}{8}, \frac {1}{2}\right ], \left [\frac {11}{8}\right ], -x^{8}\right )}{15}-\frac {\sqrt {x^{8}+1}}{5 x^{5}} \]

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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{8} + 1} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^8+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^8 + 1)*x^6), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^6\,\sqrt {x^8+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(x^8 + 1)^(1/2)),x)

[Out]

int(1/(x^6*(x^8 + 1)^(1/2)), x)

________________________________________________________________________________________

sympy [C]  time = 0.95, 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))

________________________________________________________________________________________