3.1424 \(\int \frac{1}{\left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=179 \[ \frac{x}{6 \sqrt{x^6+2}}+\frac{\left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{6 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

[Out]

x/(6*Sqrt[2 + x^6]) + (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])/(6*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 = 0.0817227, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{x}{6 \sqrt{x^6+2}}+\frac{\left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{6 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + x^6)^(-3/2),x]

[Out]

x/(6*Sqrt[2 + x^6]) + (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])/(6*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 = 2.69903, size = 153, normalized size = 0.85 \[ \frac{3^{\frac{3}{4}} x \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \left (x^{2} + \sqrt [3]{2}\right ) F\left (\operatorname{acos}{\left (\frac{x^{2} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{36 \sqrt{\frac{x^{2} \left (x^{2} + \sqrt [3]{2}\right )}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \sqrt{x^{6} + 2}} + \frac{x}{6 \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**6+2)**(3/2),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)/(36*sqrt(x**2*(x**2 + 2**(1/3)
)/(x**2*(1 + sqrt(3)) + 2**(1/3))**2)*sqrt(x**6 + 2)) + x/(6*sqrt(x**6 + 2))

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Mathematica [A]  time = 0.71569, size = 166, normalized size = 0.93 \[ \frac{x \left (\frac{2^{2/3} 3^{3/4} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (-1+\sqrt{3}\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}}}+6\right )}{36 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + x^6)^(-3/2),x]

[Out]

(x*(6 + (2^(2/3)*3^(3/4)*(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]))/(36*Sqrt[2 + x^6])

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Maple [C]  time = 0.023, size = 18, normalized size = 0.1 \[{\frac{x\sqrt{2}}{4}{\mbox{$_2$F$_1$}({\frac{1}{6}},{\frac{3}{2}};\,{\frac{7}{6}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^6+2)^(3/2),x)

[Out]

1/4*2^(1/2)*x*hypergeom([1/6,3/2],[7/6],-1/2*x^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^6 + 2)^(-3/2),x, algorithm="maxima")

[Out]

integrate((x^6 + 2)^(-3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^6 + 2)^(-3/2),x, algorithm="fricas")

[Out]

integral((x^6 + 2)^(-3/2), x)

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Sympy [A]  time = 1.94845, size = 34, normalized size = 0.19 \[ \frac{\sqrt{2} x \Gamma \left (\frac{1}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{6}, \frac{3}{2} \\ \frac{7}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac{7}{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*x*gamma(1/6)*hyper((1/6, 3/2), (7/6,), x**6*exp_polar(I*pi)/2)/(24*gamma
(7/6))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^6 + 2)^(-3/2),x, algorithm="giac")

[Out]

integrate((x^6 + 2)^(-3/2), x)