3.1401 \(\int \frac{1}{x^6 \sqrt{2+x^6}} \, dx\)

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

[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 = 0.0302554, 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, Rules used = {325, 225} \[ -\frac{\sqrt{x^6+2}}{10 x^5}-\frac{x \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{10 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

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])

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 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(x*(s
+ r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2
)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[(r*x^2*(s + r*x^2))/(s + (1
+ Sqrt[3])*r*x^2)^2]), x]] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [C]  time = 0.0044754, size = 29, normalized size = 0.16 \[ -\frac{\, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{1}{6};-\frac{x^6}{2}\right )}{5 \sqrt{2} x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*Sqrt[2 + x^6]),x]

[Out]

-Hypergeometric2F1[-5/6, 1/2, 1/6, -x^6/2]/(5*Sqrt[2]*x^5)

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Maple [C]  time = 0.024, size = 31, normalized size = 0.2 \begin{align*} -{\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}})}} \end{align*}

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. \begin{align*} \int \frac{1}{\sqrt{x^{6} + 2} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^6+2)^(1/2),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. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{6} + 2}}{x^{12} + 2 \, x^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x^6 + 2)/(x^12 + 2*x^6), x)

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Sympy [C]  time = 0.829708, size = 39, normalized size = 0.22 \begin{align*} \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 )} \end{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x^6 + 2)*x^6), x)