3.1399 \(\int \frac{1}{x^6 \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=181 \[ -\frac{\sqrt{x^6+2}}{10 x^5}-\frac{x \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{10 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

[Out]

-Sqrt[2 + x^6]/(10*x^5) - (x*(2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/3)*x^2 + x^4)/
(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*EllipticF[ArcCos[(2^(1/3) + (1 - Sqrt[3])*x^2)/
(2^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(10*2^(1/3)*3^(1/4)*Sqrt[(x^2*
(2^(1/3) + x^2))/(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*Sqrt[2 + x^6])

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Rubi [A]  time = 0.0891908, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\sqrt{x^6+2}}{10 x^5}-\frac{x \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{10 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*Sqrt[2 + x^6]),x]

[Out]

-Sqrt[2 + x^6]/(10*x^5) - (x*(2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/3)*x^2 + x^4)/
(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*EllipticF[ArcCos[(2^(1/3) + (1 - Sqrt[3])*x^2)/
(2^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(10*2^(1/3)*3^(1/4)*Sqrt[(x^2*
(2^(1/3) + x^2))/(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*Sqrt[2 + x^6])

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Rubi in Sympy [A]  time = 4.15691, size = 156, normalized size = 0.86 \[ - \frac{3^{\frac{3}{4}} x \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \left (x^{2} + \sqrt [3]{2}\right ) F\left (\operatorname{acos}{\left (\frac{x^{2} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{60 \sqrt{\frac{x^{2} \left (x^{2} + \sqrt [3]{2}\right )}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \sqrt{x^{6} + 2}} - \frac{\sqrt{x^{6} + 2}}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3**(3/4)*x*sqrt((2*2**(1/3)*x**4 - 2*2**(2/3)*x**2 + 4)/(x**2*(1 + sqrt(3)) + 2
**(1/3))**2)*(x**2 + 2**(1/3))*elliptic_f(acos((x**2*(-sqrt(3) + 1) + 2**(1/3))/
(x**2*(1 + sqrt(3)) + 2**(1/3))), sqrt(3)/4 + 1/2)/(60*sqrt(x**2*(x**2 + 2**(1/3
))/(x**2*(1 + sqrt(3)) + 2**(1/3))**2)*sqrt(x**6 + 2)) - sqrt(x**6 + 2)/(10*x**5
)

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Mathematica [A]  time = 0.833676, size = 178, normalized size = 0.98 \[ \frac{-6 \left (x^6+2\right )-\frac{2^{2/3} 3^{3/4} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x^6 F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (-1+\sqrt{3}\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}}}}{60 x^5 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*Sqrt[2 + x^6]),x]

[Out]

(-6*(2 + x^6) - (2^(2/3)*3^(3/4)*x^6*(2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/3)*x^2
+ x^4)/(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*EllipticF[ArcCos[(2^(1/3) - (-1 + Sqrt[
3])*x^2)/(2^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/Sqrt[(x^2*(2^(1/3) +
x^2))/(2^(1/3) + (1 + Sqrt[3])*x^2)^2])/(60*x^5*Sqrt[2 + x^6])

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Maple [C]  time = 0.041, size = 31, normalized size = 0.2 \[ -{\frac{1}{10\,{x}^{5}}\sqrt{{x}^{6}+2}}-{\frac{x\sqrt{2}}{10}{\mbox{$_2$F$_1$}({\frac{1}{6}},{\frac{1}{2}};\,{\frac{7}{6}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^6+2)^(1/2),x)

[Out]

-1/10*(x^6+2)^(1/2)/x^5-1/10*2^(1/2)*x*hypergeom([1/6,1/2],[7/6],-1/2*x^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{6} + 2} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^6),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^6 + 2)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{6} + 2} x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^6),x, algorithm="fricas")

[Out]

integral(1/(sqrt(x^6 + 2)*x^6), x)

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Sympy [A]  time = 2.82298, size = 39, normalized size = 0.22 \[ \frac{\sqrt{2} \Gamma \left (- \frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{6}, \frac{1}{2} \\ \frac{1}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{12 x^{5} \Gamma \left (\frac{1}{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*gamma(-5/6)*hyper((-5/6, 1/2), (1/6,), x**6*exp_polar(I*pi)/2)/(12*x**5*
gamma(1/6))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{6} + 2} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^6),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^6 + 2)*x^6), x)