3.1405 \(\int \frac{1}{x^2 \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=392 \[ -\frac{\sqrt{x^6+2}}{2 x}+\frac{\left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{2 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}-\frac{\left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}-\frac{\sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.209219, antiderivative size = 392, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{\sqrt{x^6+2}}{2 x}+\frac{\left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{2 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}-\frac{\left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}-\frac{\sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[2 + x^6]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.1309, size = 350, normalized size = 0.89 \[ - \frac{2^{\frac{2}{3}} \sqrt [4]{3} x \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \left (x^{2} + \sqrt [3]{2}\right ) E\left (\operatorname{acos}{\left (\frac{x^{2} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{4 \sqrt{\frac{x^{2} \left (x^{2} + \sqrt [3]{2}\right )}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \sqrt{x^{6} + 2}} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{3}{4}} x \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \left (- 4 \sqrt{3} + 4\right ) \left (x^{2} + \sqrt [3]{2}\right ) F\left (\operatorname{acos}{\left (\frac{x^{2} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{96 \sqrt{\frac{x^{2} \left (x^{2} + \sqrt [3]{2}\right )}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \sqrt{x^{6} + 2}} + \frac{x \left (\frac{1}{2} + \frac{\sqrt{3}}{2}\right ) \sqrt{x^{6} + 2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} - \frac{\sqrt{x^{6} + 2}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(2/3)*3**(1/4)*x*sqrt((2*2**(1/3)*x**4 - 2*2**(2/3)*x**2 + 4)/(x**2*(1 + sqr
t(3)) + 2**(1/3))**2)*(x**2 + 2**(1/3))*elliptic_e(acos((x**2*(-sqrt(3) + 1) + 2
**(1/3))/(x**2*(1 + sqrt(3)) + 2**(1/3))), sqrt(3)/4 + 1/2)/(4*sqrt(x**2*(x**2 +
2**(1/3))/(x**2*(1 + sqrt(3)) + 2**(1/3))**2)*sqrt(x**6 + 2)) - 2**(2/3)*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)*(-4*sqrt(3) + 4)*(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)/(96*sqrt(x**2*(x**2
+ 2**(1/3))/(x**2*(1 + sqrt(3)) + 2**(1/3))**2)*sqrt(x**6 + 2)) + x*(1/2 + sqrt
(3)/2)*sqrt(x**6 + 2)/(x**2*(1 + sqrt(3)) + 2**(1/3)) - sqrt(x**6 + 2)/(2*x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.712329, size = 276, normalized size = 0.7 \[ \frac{-6 \left (x^6+2\right )+\frac{6 \left (1+\sqrt{3}\right ) \left (x^6+2\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}-\frac{\sqrt [3]{2} \sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x^2 \left (\left (\sqrt{3}-3\right ) F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (-1+\sqrt{3}\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )+6 E\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (-1+\sqrt{3}\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )\right )}{\sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}}}}{12 x \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*(2 + x^6) + (6*(1 + Sqrt[3])*x^2*(2 + x^6))/(2^(1/3) + (1 + Sqrt[3])*x^2) -
(2^(1/3)*3^(1/4)*x^2*(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]*(6*EllipticE[ArcCos[(2^(1/3) - (-1 + Sqrt[3])*x^2)/(2^(
1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4] + (-3 + Sqrt[3])*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])/(12*x*Sqrt[2 + x^6
])

_______________________________________________________________________________________

Maple [C]  time = 0.021, size = 33, normalized size = 0.1 \[ -{\frac{1}{2\,x}\sqrt{{x}^{6}+2}}+{\frac{\sqrt{2}{x}^{5}}{10}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{5}{6}};\,{\frac{11}{6}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(x^6+2)^(1/2),x)

[Out]

-1/2*(x^6+2)^(1/2)/x+1/10*2^(1/2)*x^5*hypergeom([1/2,5/6],[11/6],-1/2*x^6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^6 + 2)*x^2), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(x^6 + 2)*x^2), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.02498, size = 37, normalized size = 0.09 \[ \frac{\sqrt{2} \Gamma \left (- \frac{1}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{6}, \frac{1}{2} \\ \frac{5}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{12 x \Gamma \left (\frac{5}{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^6 + 2)*x^2), x)